Application of optical fibres in precision heterodyne laser

Application of optical fibres in precision heterodyne laser
Application of optical fibres in precision
heterodyne laser interferometry
CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN
Knarren, Bastiaan A.W.H.
Application of optical fibres in precision heterodyne laser interferometry
by Bastiaan Knarren – Eindhoven: Technische Universiteit Eindhoven, 2003.
Proefschrift. - ISBN 90-386-3044-1
NUR 987
Subject headings: laser interferometry ; optical fibres / heterodyne laser
interferometry / dimensional metrology ; nanometre uncertainty
This thesis was prepared with the LATEX 2ε documentation system.
Printed by Grafisch bedrijf Ponsen en Looijen, Wageningen, The Netherlands.
c
Copyright 2003
by B.A.W.H. Knarren
All rights reserved. No parts of this publication may be reproduced, utilised
or stored in any form or by any means, electronic or mechanical, including
photocopying, recording or by any information storage and retrieval system,
without written permission of the copyright holder.
Application of optical fibres in precision
heterodyne laser interferometry
P
ter verkrijging van de graad van doctor
aan de Technische Universiteit Eindhoven
op gezag van de Rector Magnificus, prof.dr. R.A. van Santen,
voor een commissie aangewezen door het College voor Promoties
in het openbaar te verdedigen op
dinsdag 24 juni 2003 om 16.00 uur
door
Bastiaan Andreas Wilhelmus Hubertus Knarren
geboren te Vlodrop
Dit proefschrift is goedgekeurd door de promotoren:
prof.dr.ir. P.H.J. Schellekens
en
prof.dr.ir. G.M.W. Kroesen
Copromotor:
dr. H. Haitjema
Samenvatting
In de halfgeleider industrie is de belichting met de waferscanner één van
de kritieke stappen in het productieproces van (computer)chips. Heterodyne laser interferometers worden in deze waferscanners gebruikt als primaire
positiemeetsystemen. De laserlichtbron van de interferometer is echter ook
een warmtebron die, door thermomechanische interactie met de machine, de
nauwkeurigheid van de machine beı̈nvloedt. Daarnaast is veel (dure) optiek
nodig die de laserbundel door de machine naar waferstage en reticlestage leidt.
Door toepassing van een optische glasvezel zou zowel de (warme) laserlichtbron buiten de machine geplaatst kunnen worden als ook een reductie van het
aantal optische componenten voor de ’beam-delivery’ mogelijk zijn.
In dit promotieonderzoek is onderzocht welke meetnauwkeurigheden met een
fiber gekoppelde heterodyne laserinterferometer mogelijk zijn. Daarvoor zijn
diverse soorten en lengtes polarisatie behoudende fibers onderzocht en is via
een Jones-model het karakteristieke gedrag van een dubbel brekende fiber beschreven.
Het is voor het splitsen van de laserbundel, voor het meten van meerdere assen,
van belang dat de lichtopbrengst aan het eind van de fiber zo hoog mogelijk
is. Omdat er geen fiber coupler beschikbaar was voor de koppeling van het
laser licht, in 6 graden van vrijheid t.o.v. de fiber te manipuleren, is ook een
fiber-coupler met hoge precisie ontwikkeld. Uit berekeningen, volgt dat met
het ontwikkelde inkoppel systeem een rendement van 47% mogelijk moet zijn.
Verbeteringen zijn met name te behalen in het gebruik van een bundel met een
kleinere numerieke apertuur, omdat de numerieke apertuur van de huidige
bundel erg afwijkt van dat van de fiber. Experimenteel zijn rendementen van
35 − 66% behaald bij het gebruik van 2 verschillende type lenzen.
Voor nauwkeurige verplaatsingsmetingen is een stabiele polarisatietoestand
essentieel. Dit wordt bereikt door het gebruik van de polarisatie behoudende
fibers. Door toepassing van deze zogenaamde PM fibers ontstaan echter ook
faseverschuivingen. Deze faseverschuivingen worden direct als een ’virtuele’
verplaatsing gemeten. Door toepassing van een externe referentiemeting is
deze belangrijkste foutenbron (grotendeels) geëlimineerd.
Door de ontwikkeling van een specifieke uitlijnprocedure kan aan de hand van
een parameter (de extinction ratio) de kwaliteit van de fiber worden bepaald.
De extinction ratio is de verhouding tussen de niet gewenste en gewenste polarisatie toestand. Deze parameter kan direct worden verkregen uit metingen
v
vi
Samenvatting
en/of door aanpassing van deze metingen aan het speciaal ontwikkelde analytisch model. Door toepassing van het analytisch model kunnen echter ook
zeer nauwkeurig de hoofdassen van de fiber worden bepaald, waardoor precisie uitlijning t.o.v. optiek en laser head mogelijk is. De kwaliteit van de
fiber is representatief voor de uiteindelijk te bereiken nauwkeurigheid met de
interferometer. Door selectie, op basis van de eerder genoemde parameter, kan
het effect van de andere foutenbron, het mengen van de twee orthogonale polarisaties (polarisatiemixing), tot de vereiste nauwkeurigheid verder worden
verkleind.
De kwaliteit van de fibers lijkt uit deze meting nauwelijks van het type, noch
van de lengte af te hangen. Wel zijn er grote verschillen tussen leveranciers
gevonden. Door toepassing van het ontwikkelde Jones model van een PM
fiber is onderzocht of de globale of locale verstoringen het gedrag van de fiber
verklaren. Ook de locatie van deze locale verstoringen kan worden gevarieerd.
Door het vergelijken van de resultaten van de simulaties en de metingen is
aannemelijk gemaakt dat de locale verstoringen aan begin en eind van de fiber
verantwoordelijk zijn voor de fiber kwaliteit. De connector lijkt de voornaamste
oorzaak van de verschillen in gemeten fiber-kwaliteit. Met dit model kunnen
ook goed het effecten van langere fibers worden beschreven.
Het zogenaamde ’fibre-fed’ laserinterferometer systeem is voor diverse fibers
vergeleken met een systeem zonder fibers. Dit is gedaan voor een afstand van
300 mm (maximale wafergrootte) en een afstand van enkele micrometers.
Resultaten laten zien dat op een afstand van 300 mm het maximale verschil
tussen een systeem met en zonder fiber, voor het gehele systeem inclusief optiek, brekingsindex verschillen, baan afwijkingen van de slede en fouten zoals
dode weglengte, slechts 7 nanometer bedraagt. Bij kalibraties over de korte
afstand komen de periodieke niet-lineariteiten, welk veroorzaakt worden door
mixingeffecten in de fiber, duidelijk naar voren. Het verschil tussen de fibers,
welke reeds door de extinction ratio metingen was vastgesteld, is ook hier
gevonden. Door optimale selectie van de fiber is een reductie van de nietlineariteit van 6 nm (top-top) naar minder dan 2 nm bereikt. Voor het meten
met een meetonzekerheid van kleiner dan 1 nm zijn fibers met een extinction
ratio van minimaal 1:850 nodig.
Samenvattend kan worden gesteld dat middels modellering en metingen veel
inzicht is verkregen in de (on)mogelijkheden die (commercieel verkrijgbare)
fibers bieden voor het transport van het heterodyne laserlicht van een commerciële laserinterferometer. Daarnaast is een krachtige meetmethode ontwikkeld voor de selectie van de fiber kwaliteit zodat men vooraf de geschiktheid
van een fiber kan bepalen. Daarbij is ook een model van de fiber gemaakt
welke het gedrag van een dubbel brekende fiber goed beschrijft, en de resultaten van deze (selectie)metingen goed voorspelt. Door korte en lange slag
kalibratiemetingen is aangetoond dat ook voor nauwkeurige metingen met een
meetonzekerheid van 1 nm een fiber gekoppelde heterodyne laserinterferometer kan worden gebruikt mits de rest van het systeem optimaal functioneert.
Bij gebruik van vlakke-spiegel optiek, worden alle hier gepresenteerde nietlineareiten en onzekerheden nog eens met een factor 2 gereduceerd.
Abstract
In the semi-conductor industry the photo-lithography, which is performed
with a wafer scanner, is one of the critical production steps within the production of (computer)chips. Heterodyne laser interferometers are used in these
wafer scanners as the primary displacement measurement system. The laser
light source of the interferometer however is also a heat source influencing
the achievable displacement measurement accuracy by the thermo-mechanical
interaction with the machine. In addition much (expensive) optics is needed
to deliver the laser beam, through the machine, to the wafer stage and reticle
stage. The use of an optical fibre would make it possible to position the laser
light source outside the machine, as well as reducing the number of optical
components needed for beam delivery.
In this research the achievable accuracies with a fibre fed heterodyne laser interferometer are investigated. Therefore several different types and lengths of
polarisation maintaining fibres have been investigated and a Jones-model has
been constructed which describes the characteristic behaviour of these birefringent fibres.
For the measurement of multiple interferometer axes it is important that the
beam intensity of the light emerging from the fibre is as high as possible. Because no fibre coupler was available to manipulate the laser light beam in 6
degrees of freedom in respect with the fibre, a high precision fibre coupler was
developed. Based on the measured resolutions of the coupler, a calculated efficiency of 47% should be possible. Improvements of the coupling efficiencies
are mainly achievable with special optics so that the numerical aperture of the
beam better match that of the fibre. Experimentally coupling efficiencies of
35 − 66% were achieved when using two different lenses.
A stable polarisation state is required for accurate displacement measurements.
This is achieved by using polarisation maintaining fibres. By applying these
PM fibres however also phase changes between the two polarisations occurs.
These phase changes are measured as a ’virtual’ displacement. By using an
external reference measurement, this main error source is mainly eliminated.
By the development of a specific alignment procedure with the use of only one
parameter (the extinction ratio), the quality of the fibre can be determined. The
extinction ratio is the ratio of the unwanted and wanted polarisation state. This
parameter can directly be obtained from measurement and/or fitting of these
measurements to the analytical model. By applying the analytical model also
vii
viii
Abstract
the orientation of the fibres main axes can be determined which allows the precise alignment of the fibre in respect with the optics and laser head. The quality
of the fibre measured, represents the achievable displacement uncertainty of
the fibre fed heterodyne laser interferometer. By selection, based on the before
mentioned parameter, the other main error source, the mixing of the two orthogonal polarisations (polarisation mixing), on the achievable displacement
measurement accuracy can be minimised.
Extinction ratio measurements have been performed at different fibre types and
length of PM fibres. From the measurements, the quality of the fibre turned out
to depend hardly on the type nor on the fibre length. Large difference between
suppliers were found however. By applying the developed Jones model of a
PM fibre the effect of global and local disturbances on the behaviour of the fibre
is investigated. Also the location of the local disturbances could be varied. By
comparing the results from the simulations and the measurements it could be
shown that the disturbance at the fibres ends are responsible for the fibre quality. The connector seems to be the main cause of the differences measured in
the fibre quality. With this model also the effects of long fibres can be described
adequately.
The fiber fed heterodyne laser interferometer system was compared to a comparable system without fibres. The validation measurements were done over
a range of 300 mm (wafer size) as well as over a range of several micrometres,
for detection of periodic errors due to polarisation mixing.
Results from measurements over a range of 300 mm show that the total difference between the system with and without fibre, for the complete system
including optics, refractive indices, stage form and dead path error, is only
7 nanometre. Over the short range of several micrometre, the periodic nonlinearities, which are caused by mixing effects within fibre cable, are measured.
The difference between fibre types, which already was measured with the
extinction ratio measurement was also found. By optimal selection of the fibre a reduction of the non-linearities from 6 nm (top-top) to below 2 nm was
achieved. For the development of a fibre fed heterodyne laser interferometer
with an measurement uncertainty less than 1 nm, fibres with an extinction ratio
of at least 1:850 are required.
By means of modelling and measurements the (im)possibilities which (commercial available) PM fibres offer for the development of a fibre fed heterodyne
laser interferometer are investigated. A powerful selection method has been
developed, where the suitability of a fibre can be determined. In addition a
model has been constructed which predicts the behaviour of the fibre very well.
By means of a short and long stroke validation measurement is was shown that
a fibre fed heterodyne laser interferometer, using dedicated polarisation maintaining fibres, also can be used for accurate displacement measurements, down
to the nanometre level.
With the use of plane mirror optics, all presented non-linearities and uncertainties are reduced by a factor 2.
Contents
Samenvatting
v
Abstract
1
2
3
4
Introduction
1.1 Laser interferometry
1.2 Project description .
1.3 Project goals . . . . .
1.4 Thesis outline . . . .
1.5 Summary . . . . . . .
vii
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1
3
8
9
9
9
Heterodyne laser interferometry and optical fibres
2.1 Axes definition . . . . . . . . . . . . . . . . . .
2.2 Heterodyne laser interferometry . . . . . . . .
2.3 Optical fibres . . . . . . . . . . . . . . . . . . . .
2.4 General introduction to optical fibres . . . . . .
2.5 Polarisation maintaining optical fibres . . . . .
2.6 PM fibres used for experiments . . . . . . . . .
2.7 Fibre model . . . . . . . . . . . . . . . . . . . .
2.8 Summary . . . . . . . . . . . . . . . . . . . . . .
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11
11
12
15
16
23
27
27
29
Optical fibre coupling
3.1 The coupling of laser light into the fibre . . . . .
3.2 Calculated in-coupling accuracies . . . . . . . . .
3.3 Design of the high precision fibre coupler . . . .
3.4 Validation of the coupler resolutions and strokes
3.5 Achievable coupling efficiency . . . . . . . . . .
3.6 Other effects . . . . . . . . . . . . . . . . . . . . .
3.7 Fibre beam splitter . . . . . . . . . . . . . . . . .
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . .
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31
32
35
39
44
48
51
52
52
Characterization of polarisation maintaining fibres
4.1 Basic characteristics of PM fibres . . . . . . . . . . . . . . . . . .
4.2 Measurement of the polarisation state after the fibre . . . . . . .
4.3 Discussion of measurement results . . . . . . . . . . . . . . . . .
53
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x
CONTENTS
4.4
4.5
5
6
Fibre modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Accuracy of a fibre fed heterodyne laser interferometer
5.1 Phase shifts . . . . . . . . . . . . . . . . . . . . . . .
5.2 Polarisation mixing . . . . . . . . . . . . . . . . . . .
5.3 System validation . . . . . . . . . . . . . . . . . . . .
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . .
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76
80
81
. 81
. 88
. 99
. 104
Conclusions and recommendations
107
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Bibliography
112
A List of used Nomenclature, Acronyms and Symbols
119
B Babinet Soleil compensator
123
C Component calibration
125
D Intensity calculations
133
E Laser head
139
F Mode coupling
143
G Computation of the AC-methods
145
H Fibre connector
153
Curriculum vitae
155
Acknowledgment
157
Chapter 1
Introduction
Due to the miniaturization, the accuracy demanded by industry for both product and production has rapidly increased the last decades. In production
systems accuracy positioning systems are becoming more and more important.
Examples are modern technologies such as making DVD-masters for digital
video or scanning electron microscopy (SEM) systems which require high accurate positioning systems. Micro system technology (MST) is the science
of producing miniaturised systems that contain sensor, signal-processing and
actuator functions all in one. This MST is a new but becoming increasingly
important tool, requiring also both high positioning and measuring systems.
The semiconductor industry is one of the most important industries requiring
high precision. The line width on the chip, which is the distance between the
two edges of a hardened photo resist, which is used standardly gradually reduced from several µm in the 80’s down to 130 nm today. Even smaller line
widths of only 90 nm are already in use to produce high precision chips for the
mobile phone industry. By 2007 the line width is expected to be decreased to
only 45 nanometres and in 2009 the line widths will be decreased to 32 nanometres [Sil02]. Although these decreasing line widths demands are in the future,
Time
1981
1987
1993
1997
2001
2003
2007
2009
Line width
2000 nm
1000 nm
500 nm
250 nm
130 nm
90 nm
45 nm
32 nm
Table 1.1: Time line of the minimum line width possible on a semiconductor [Sil02].
1
2
Introduction
they are todays technology demands. A semiconductor is built up of several
layers, which need to be positioned onto each other. The resist is exposed in a
wafer scanner, which projects an image of the reticle to the wafer processed. In
order to connect the lines of two layers they must overlap and this overlap is
called overlay. The overlay error should be approximately 30% of the linewidth
at maximum. The linewidth which will be decreased to 32 nm in 2009, means
that in 2009 the total position uncertainty must be within 10 nm.
In current precision displacement measurements, the laser interferometer is
one of the most accurate displacement measurement techniques. Therefore the
laser interferometer is also used as displacement measurement system in the
wafer scanner, as schematically shown in Figure 1.1. Because of the increasing
demand for accuracy, the laser sources themselves are becoming a limitation.
The laser head is, beside the measurement source also one of the heat sources
within the machine. This heat will affect both the measurement accuracy directly as well as by the thermo-mechanical interaction of the construction.
Illumination beam
Reticle stage
Receivers
Wafer stage
with wafer
Laser
Optics
Figure 1.1: Schematically the laser interferometric measuring system of
a wafer scanner, both for wafer stage and reticle stage.
The laser source can also be too large, or the machine too complex, to mount
the laser head directly to the measured path. As displacement can be measured
only along the laser beam axis, much (expensive) optics is needed to deliver
the laser beam to the right position. To overcome these problems the use of a
(flexible) coupling between laser head and interferometer optics will be investigated. This may reduce the amount of expensive optics, and on the other hand
the laser head can be positioned away from the scanner, without disturbing the
wafer scanner accuracy by thermal effects.
1.1 Laser interferometry
3
The Precision Engineering section of the Eindhoven University of Technology
is also interested in the measurement of the accuracy of a coordinate measuring machine (CMM) or a manufacturing machine (machine tool). Due to
high accuracies needed, this should be done with laser interferometers, especially for the high precision coordinate measuring machines under development [Rui99, Ver99]. Because realignment is not practical, or at least very time
consuming, a flexible coupling between the laser head and the interferometer
optics would have many advantages [Flo01].
Clearly there is a need for a flexible coupling between laser head and interferometer optics. However introducing this coupling should not reduce the
measurement accuracy of the system. An optical fibre could be used to couple
the light, but high demands such as optical polarisation stability are needed
for this fibre. In this thesis, the use of an optical fibre in precision heterodyne
interferometry will be examined in more detail.
For laser interferometric measurements, polarisation and phase stability are
crucial. Most attention is paid to those aspects. During the whole research the
achieved accuracy which should be at the nanometre level is the most important
factor, and this will be emphasised throughout this thesis.
1.1 Laser interferometry
The laser interferometer is used to measure displacements. Main advantages of
this technique are the long range and high accuracy, as well as the non contact
measurement method.
One of the most basic forms of (length) interferometry, is the (modified) Michelson interferometer as used in modern interferometers. This is shown in Figure 1.2. The light source used throughout this research is a wavelength staBeam splitter
A
Measuring arm
B
Laser
Reference Measurement
Detector Detector
System
System
C
L
Reference arm
Figure 1.2: A basic Michelson interferometer with optics, for measuring
the displacement L, as used in modern interferometers.
bilised laser [Sie86], for more information see Appendix E. The laser light, or
electromagnetic wave propagating along the z-axis is described described by
[PP93]:
~ =
E
En
=
Ex (z,t)~ei + E y (z,t)~e j
(1.1)
An cos(ωn t − kn z − φn ) , n = x,y
(1.2)
4
Introduction
With An the amplitude of the E-field, ωn the angular frequency, k the propagation constant, ωn the phase, z the coordinate along the propagation axis and t
the time.
In a beam splitter, the laser beam is split into a reference and a measuring beam.
The reference beam follows a constant or reference path. The measuring beam
propagates along the measuring path. After both beams have been reflected
on their mirror, the beams recombine at the beam splitter, where they interfere. This interference depends on the (relative) mirror position difference.
If the phases of the two beams are the same, constructive interference takes
place. When the beams are 180◦ out of phase, destructive interference takes
place. This phase difference is measured as an intensity change on the detector.
The intensity (I) of a wave is defined as the square of the amplitude of the E
field, thus the inner product of E with its Hermitian conjugated. Depending
on the interferometer used, the detector system varies and is therefore shown
schematically. It can be shown that the measuring signal Im is
(1.3)
Im = I0 1 + cos(∆φ)
Here ∆φ is the total phase difference between the measurement and reference
beam. The total phase difference is found by the initial phase difference and
the phase difference due to the mirror displacement or:
∆φ = φre f − φmeas − δφ
(1.4)
To calculate the mirror displacement L the initial phase difference δφ must be
measured. This initial phase can either be measured directly in front of the
interferometer or by measuring a second interference signal which has a 90◦
phase shift and these signals are then used to calculate this phase difference.
From these two measurements the displacement of the mirror can be calculated.
When the optical path difference ∆ is, the actual length difference L is:
∆ = 2k0 L with k0 = 1, 2 or 4
(1.5)
Where k0 is a factor depending on the optical configuration used, describing the
number of passes from the interferometer to the mirror. This factor describes the
extra number of passes of the beam from the interferometer to the measurement
mirror. For a standard interferometer with beam splitter and retro reflector k0 =
1, for plane mirror optics k0 = 2, while for high resolution optics k0 = 4. Results
in this thesis are always translated to standard interferometer equivalents, thus
for k0 = 1.
Changes in optical path length due to environmental changes are described by
the refractive index n. With λ the mean vacuum wavelength of the laser light
used, the phase difference ∆φ between the orthogonal polarisations is:
∆φ =
2πn∆ 4πnk0 L
=
λ
λ
(1.6)
So
L=
∆φλ
4πnk0
(1.7)
1.1 Laser interferometry
5
Distances are measured as a phase shift of two orthogonal waves. Any additional phase shift caused by a fibre is measured as an erroneous distance.
There are two different interferometric systems:
• The homodyne or one frequency laser interferometer
• The heterodyne or two frequency laser interferometer
In this thesis only the heterodyne laser interferometer will be described, however many aspects described are also applicable for the homodyne laser interferometer.
1.1.1 Polarisation
As most commercially available displacement interferometers (e.g Agilent, Heidenhain, Renishaw and Zygo) use polarising optics to split the beam, in this
section a brief overview of polarisation is given.
When the light is polarised, the electromagnetic field (Equation 1.2) has a determined orientation. This means, the plane in which the light waves are moving
is not random, but has a defined orientation. In the case of linear polarisation,
all the waves of the light are moving in one plane, e.g. the one shown in Figure 1.3a. This linear polarisation is described by an Ex and E y fields with a zero
phase difference. The projection of a linearly polarised light wave shows an
oscillating vector, which amplitude vary in time at a constant orientation.
x
x
y
Linear
x
y
Elliptical
y
Circular
Figure 1.3: Different polarisation states, the first being linearly polarised, the third circularly polarised. The most general
polarisation state is the second one and is called elliptically
polarised.
When the beam splitter in Figure 1.2 is a polarising beam splitter, the light from
the laser head is split into two orthogonal linearly polarised beams. The axes
of these beams are determined by the orientation of the beam splitter. After
recombination, the two polarisations do not interfere because the beams are
polarised orthogonally. By placing a polariser under 45 degrees in front of the
detector, the two polarisation direction are projected onto the transmission axis
of the polariser and interfere.
In general light is elliptically polarised, this means that the two E fields have,
6
Introduction
contrary to the linear polarisation, a phase difference. In elliptically polarised
light both the projected amplitude and orientation vary with time. This is illustrated in Figure 1.3b.
If there are two linearly polarised beams with the same amplitude and with
90 degree phase shifts, the resulting light is circularly polarised, see Figure 1.3c.
This beam has a constant projected amplitude but a varying orientation.
In fact both linearly polarised and circularly polarised are special cases of the
elliptic polarised light, with a phase difference of 0◦ respectively 90◦ between
the two orthogonal E-fields[Hua97].
1.1.2 Error sources
Until now a perfect interferometer was described without any errors. This
means no component errors, no alignment errors and fixed environment conditions. The most limiting factor for accurate displacement measurement is
the wavelength of light used. This wavelength of light is determined by the
wavelength of light in vacuum and the refractive index (change) of the medium
where the interferometer is used in. By using the Edlén [BD93, BP98] formula
and measuring environmental conditions like temperature en pressure, the refractive index of air can be calculated. Alternatively a wavelength tracker can
be used were the change in optical path length is measured proportional to refractive index changes. As the use of a tracker only changes of refractive index
are measured, an initial refractive index determined with the Edlén formula
is required to measure the absolute refractive index. Minimising environment
influences can be reached by minimising the lengths of section A, B and C of
Figure 1.2.
For high precision measurements the non-linearities caused by polarisation
mixing are of main interest. These non-linearities are the deviations from the
measured displacement, when a linear displacement or phase shift is made and
this intended displacement is subtracted from the measurement result. Resulting effects are position errors depending on the phase difference, as shown
in Figure 5.5. If e.g. the polarising beam splitters main axes are not aligned
with the lasers [JS98], polarisation mixing occur. Component imperfections
also can lead to non-linearities [PEC96]. Another main source for mixing are
laser beam imperfections [dF97, Lor02]. Other effects, such as data aging
[OKT93] and ghost reflections [WD98] also result in incorrect phase determination. In literature, influences causing non linearities are already studied
for a long time [Que83, Sut87, TYN89]. Predictions for achievable inaccuracies for several different separate error sources were given in various papers:
[RB90, HW92, dFP93], while others aimed on direct measurement and elimination of first and second order harmonics in the measurement signals as a result
of different separate influences [HZ94, PB95, BP00]. In the Precision Engineering section of the Eindhoven University of Technology extensive research is
done to further predict and measure these non-linearities caused by the combination of all (previously) reported effects [CHS02]. When using a fibre these
non linearities will also be important, both for the inherent fibre properties
1.1 Laser interferometry
7
(mode leaking) [PBH82] as well as for fibre alignment. Also the projection
of the non-orthogonal laser heads main polarisations to the orthogonal fibres
main axes will cause polarisation mixing [Lor02].
The last group of errors are errors like the Abbe [Abb90] and cosine error, which
are introduced by the measurement setup and/or measurement strategy.
Summarising, the following errors can be present in an Michelson-interferometer:
• Phase determining errors
– Unequal amplitude: E1 , E2
– Elliptical polarised beams of the light source, e.g. due to non ideal
quarter wave plate or not optimal aligned quarter wave plate in laser
head
~ 1 not perpendicular to E
~2
– Squareness error: E
– Alignment error optics (e.g. beam splitter)
– Polariser in receivers not at 45 degrees.
– Wave front errors
– Beam divergence
– Back reflection
– Ghost reflections
– Data aging
• Wavelength errors
– Vacuum wavelength error
– Refractive index compensation
• Measurement errors
– Cosine error
– Abbe error
As measurement errors are inherent to the measurement method, these are not
taken into account in this research. Also the wavelength errors are not taken
into account, because these also are not influenced by introducing a fibre. The
error source which is investigated here is the phase error, as this error can be
influenced by the behaviour of the fibre introduction.
1.1.3 Fibre introduction in interferometry
The flexible coupling will be between laser head and interferometer replacing
section A of Figure 1.2. Thereby having the advantages mentioned earlier, e.g.
less optics, removing heat source etc. The reference and measurement receiver
can also be equipped with fibre optics. These fibre optical receivers are already
commercially available, and do not influence the measurement accuracy, since
they only transport an interference signal. In Chapter 2, a detailed description
of a heterodyne laser interferometer is given, in a later chapter the difference
between the so-called optical receiver fibres and flexible coupling fibre will be
described in more detail.
8
Introduction
1.2 Project description
One of the possible future applications may be wafer and reticle stage displacement measurement. For this heterodyne laser interferometers are used because
of the high accuracy both at low and high speed. When using (heterodyne)
laser interferometry, the laser head itself also sets some limits on the system as
mentioned before: the laser head is a disturbing heat source and lots of extra
optics are required for beam delivery.
Illumination beam
Reticle stage
Fibre
Receivers
Wafer stage
Optics
Laser
Figure 1.4: Schematically the proposed possible laser interferometric
measuring system of a wafer scanner were the laser light is
transmitted by fibres, both for wafer stage and reticle stage.
In CMM calibration where the machine itself allows the laser head to be positioned directly to the target, the use of a flexible head could reduce calibration
(alignment) time drastically. For this the use of a medium which is flexible
and able to transport the laser beam should be used. Optical waveguides, also
called optical fibres, fulfil both demands for transmitting the beam.
As only heterodyne laser interferometers are considered in this research, both
frequencies could be transported in separate fibres, however in this project the
use of one single fibre to transport the two frequencies of a heterodyne laser
interferometer will be examined only. An example of the use of a fibre fed laser
interferometer is given in Figure 1.4.
When using optical fibres, the influence of these fibres on the measurement
accuracy of a heterodyne laser interferometer must be investigated; i.e. the
achievable accuracy and the limiting factors must be determined. Different
fibre types will be investigated, as well as independent methods for measurement of the fibre properties which determine the accuracy. After this, the
1.3 Project goals
9
complete fibre fed heterodyne interferometer must be calibrated.
Homodyne fibre fed laser interferometers are nowadays coming onto the market [Hei00, Ren00]. This thesis, however deals with fibre fed heterodyne laser
interferometer only, but will also give much knowledge for those who want to
use fibres in high precision interferometric measurements.
1.3 Project goals
The goal of the project is to develop and test a heterodyne laser interferometer,
where the light is transmitted through a single fibre from the laser head to the
interferometer. A boundary condition is that the fibre must be re-connectable
either at the laser head or at the interferometer or both. The fibre fed heterodyne
laser interferometer must be able to measure distances at least up to 300 mm.
The most important goal however is the achievable accuracy. In this project
therefore the limiting factors must be exploited. The goal is the development
and validation of a laser interferometric displacement measurement system
with an uncertainty of 1 nm, thus comparable to the interferometer without
fibre.
Finally a system validation is needed, where the developed system has to be
compared with the standard heterodyne laser interferometer. This is done over
a long range of at least 300 mm, and a short range of a few wavelengths (µm)
to validate the linearity.
1.4 Thesis outline
First an introduction into heterodyne laser interferometry and optical fibres is
given, and into the aspects of fibres which influence accuracy. After that, the
fibre coupling is discussed. Chapter 4, first describes the influence of phase
shifts. Then the polarisation mixing is described in more detail. Alignment,
simulations and fibre selection are described in this chapter. In Chapter 5
validation measurements for the short range as well as long range are presented.
Finally conclusions and recommendations are given in chapter 6.
As accuracy is the main aspect of this research throughout the chapters accuracy
on the distance measurement will be the main focus.
1.5 Summary
This project must be seen in close relation with other projects within the Precision Engineering section of the Eindhoven University of Technology, all to
improve measurement accuracy of the laser measuring systems.
Using fibres in laser interferometry could gain many advantages (flexible output, relocating heat source, decrease bending optic needed). Possible additional
influence of the fibre which could affect the accuracy need to be validated. The
goal is to develop a fibre fed heterodyne laser interferometer with nanometre
10
Introduction
uncertainty. The developed system should perform with the same accuracy
compared to the system without fibre, and sources which may cause differences will be identified. The developed system will be compared with system
without fibre at a long ranges of at least 300 mm to show the applicability of
the system and a short range to show the achievable uncertainties.
Chapter 2
Heterodyne laser
interferometry and optical
fibres
In Chapter 1 the need for the development of a fibre fed heterodyne laser
interferometer was given. Therefore in this chapter first the heterodyne laser
interferometer as displacement measurement system will be explained. After
giving the effects of polarisation mixing in heterodyne laser interferometry,
optical fibre theory is presented. Several aspects are discussed, which influence
choices made for the development of a fibre fed heterodyne laser interferometer.
2.1 Axes definition
In the following paragraphs both an ideal and a non-ideal heterodyne laser interferometer will be discussed. In this section axes and orientation are defined
which are used throughout the modeling in the next sections.
A heterodyne laser has two frequencies, one in each polarisation direction. In
the models, the E-field with frequency f1 is coincident with the vertical axis and
~ 1 , while the other field is coincident with the horizontal
is represented with E
~ 2 . As in a non-ideal interferaxis and has frequency f2 and is represented with E
ometer both frequencies are elliptically polarised, the out-of-phase component
~ 01 and E
~ 02 respectively. The non-orthogonality of the pois represented with E
larisations is represented by angle ε. The polariser (in the detector) is oriented
at an angle αP with respect to the reference coordinate system. All other optical
components are assumed to be perfect and aligned with the x, y coordinate
system. Graphically the axes and E-vector definitions are given in Figure 2.1.
In an ideal interferometer ε = 0 degrees and αP = 45 degrees. The two frequen~ 01 = E
~ 02 = 0.
cies are linearly polarised in an ideal interferometer, thus E
11
12
Heterodyne laser interferometry and optical fibres
x
~1
E
Polariser
αP
~ 02
E
~2
E
ε
y
~ 01
E
Figure 2.1: The definition of axes for the non ideal E-vectors in a heterodyne laser interferometer.
2.2 Heterodyne laser interferometry
In this section the working principle of an (ideal) heterodyne laser interferometer is described. In a heterodyne interferometer both polarisations have
different frequencies. One polarisation direction with frequency f1 , is deflected
to the reference arm, while the other polarisation direction, with frequency f2
is used in the measurement arm. This is achieved by using a polarising beam
splitter, in the interferometer as shown in Figure 2.2.
Polarising
Beam splitter
Measuring arm
Heterodyne Laser
Reference Measurement
Receiver Receiver
L
Reference arm
Figure 2.2: Schematically a heterodyne laser interferometer. In the polarising beam splitter one polarisation direction is deflected
to the reference arm, the other polarisation direction is transmitted to the measuring arm. After combining again in the
beam splitter the beam enters the measurement receiver.
After recombining, the two polarisations are fed through a polariser oriented
at 45◦ with respect to the reference coordinate system, onto the detector. The
interference signal on the detector has, due to the frequency difference, a beat
signal with a component f1 − f2 . As a result of the mirror displacement with
a speed v, the frequency of the measurement arm changes to f2 + ∆ f , with
∆ f = f2 · vc , thereby changing the beat signal on the detector to f1 − f2 − ∆ f . The
reference detector (normally located in the laser head), still measures the f1 − f2
2.2 Heterodyne laser interferometry
13
beat signal component. By combining these two measurements, the measurement electronics measures the frequency difference ∆ f as a phase difference
∆φ, by looking at the time difference between the zero crossings of the beat
signal. Due to band filters, the measurement electronics also eliminates dc and
high frequencies.
2.2.1 Homodyne versus heterodyne laser interferometry
There are four main advantages to use heterodyne laser interferometry. The first
advantage is that the interference signal is insensitive to intensity changes of the
light. The main advantage however is the high accuracy of the measurement at
both high speed (1 m/s) and very low speed, contrary to many other systems
where either high speed or very low speed is measured less accurate. This is
achieved by the high signal-to-noise ratio. Other advantages are the absence of
laser tube noise (up to 10 kHz) and heterodyne beams can be split more often,
thus more axes can be measured with one laser source.
Disadvantages of the heterodyne laser interferometer can be the maximal speed,
at which the measurement mirror can move. This depends on the frequency
difference used. Also compensation of non-linearities is more difficult than for
the homodyne systems[Hey81].
2.2.2 Ideal heterodyne interferometer
First an ideal heterodyne interferometer is described. In this interferometer, no
component errors nor alignment errors are assumed. In the next section, errors
will be introduced, in a non-ideal heterodyne interferometer.
In the HeNe laser tube, by the Zeeman-effect, two opposite circularly polarised
beams are generated. These two orthogonal polarisations have frequencies f1
respectively f2 and are left respectively right handed circularly polarised. After
the cavity a λ/4 plate (oriented at 45◦ ) transform these circular polarisations
into two orthogonal linearly polarised beams. The laser main axes E1 and E2
are assumed to coincide with the reference x,y coordinate system. These two
ideal beams are represented by:
E~x = E~1 = E0 sin(2π f1 t + φ01 )~ei
E~y = E~2 = E0 sin(2π f2 t + φ02 )~e j
(2.1)
(2.2)
With E0 the amplitude of the E-field, f the frequency, t the time, φ0i the initial
phase of the beam.
This beam is split into two identical beams by a non polarising beam splitter.
One of those serves as measuring beam one as reference beam. This reference
beam passes a polariser at 45 degrees and both polarisations interfere. This
interference signal strikes onto the detector resulting in an interference signal
of which only the alternating current is measured due to a band pass filter,
passing only the difference of the two optical frequencies. It can be shown (see
14
Heterodyne laser interferometry and optical fibres
Appendix D) that this reference interference AC-signal can be described by:
Sr = 1/2E20 cos(2π( f1 − f2 )t + (φ01 − φ02 ))
(2.3)
This reference AC-signal normally is extracted within the laser head.
The laser head’s output is only the measuring beam. This measuring beam
still contains the two polarisation directions. From this measuring beam, the
two waves are separated by a polarising beam splitter. While one is traveling
a constant or reference distance, here assumed the E1 -polarisation, the other
one is traveling the measuring path (E2 -polarisation). The reference beam, in
the reference arm of the interferometer gains a constant phase shift φre f . The
measurement beam, in the measuring arm of the interferometer, gains a phase
shift φmeas , proportional to the change in (optical) path length. After passing
the beam splitter again, the two beams are recombined.
The recombined beam then passes also through a polariser oriented at 45◦ and
the interference signal is detected. After the beams are recombined and have
passed the polariser, the electric field vector can be described by:
√
Em = 1/2 2E0 (sin(2π f1 t + φ01 + φre f ) + sin(2π f2 t + φ02 + φmeas ))
(2.4)
The AC-measuring signal Sm is (see Appendix D):
Sm = 1/2E20 cos(2π( f1 − f2 )t + (φ01 − φ02 ) + (φre f − φmeas ))
(2.5)
The measuring signal either can be measured using a separate receiver, or
(if available) received back into the receiver section of the laser head. By
combining Sr and Sm the phase difference φre f − φmeas can be calculated, where
with the use of Equation 1.6 the displacement of the mirror can be calculated.
2.2.3 Non-ideal heterodyne interferometer
In this section errors are introduced into the ideal heterodyne laser interferometer. For an overview of error sources refer to Section 1.1. Most important
are non-linearities due to polarisation mixing since this is one of the factors
influenced by introducing a fibre. For more information about the influences
of (other) error sources on the accuracy of laser interferometry refer to the work
of Cosijns [CHS02].
Polarisation mixing effects can be caused by several different sources, e.g.
slightly elliptical laser output polarisation, optical misalignment and, as will be
seen later, due to cross talk in a fibre. In this example only elliptically polarised
input beams (with ellipticity α) with their main axes parallel to the reference
coordinate system are modelled, as this is one of the effects later seen with the
introduction of fibres in an interferometer. No non-orthogonality nor optical
errors or misalignment are described but only equal ellipticity at both axis.
Visual this is represented by the axes and E-vectors definitions of Figure 2.1,
with ε = 0 degrees.
2.3 Optical fibres
15
The E-field of the mixed light is now described by:
~ x = E1 sin(2π f1 t + φ01 )~ei + E02 cos(2π f2 t + φ02 )~ei
E
~ y = E01 cos(2π f1 t + φ01 )~e j + E2 sin(2π f2 t + φ02 )~e j ,
E
(2.6)
(2.7)
with E1 = E2 = cos(α)E0 and E01 = E02 = sin(α)E0 . The measured AC-signals Sr
and Sm are (see Appendix D):
Sr
Sm
=
0.5E20
=




2
0.5E0 


with:
2 sin α cos α sin(2π∆ f t + ∆φ0 ) + cos(2π∆ f t + ∆φ0 )

ncos(2π∆ f t + ∆φ0 + ∆φ)·
o 
−2 sin α cos α sin(∆φ) + cos2 α + sin2 α cos(2∆φ) 


+

n sin((2π∆ f t + ∆φ0 + ∆φ)·
o

2
2 sin α cos α cos(∆φ) + sin α sin(2∆φ)
∆ f = f1 − f2
and
∆φ0 = φ01 − φ02
and
(2.8)
(2.9)
∆φ = φre f − φmeas
The introduction of the elliptically polarised beam introduces periodic errors.
The measured phase has periodic errors, also called non-linearities which have
periods of one or two cycles per optical fringe. This means that the error shows
per half wavelength (optical) displacement either 1 or 2 times.
From Equation 2.9 it can be seen that the polarisation mixing introduces both
first (∆φ) and second order (2∆φ) non-linearities. These non-linearities are
reducing the accuracy achievable with laser interferometers. In most applications thermal expansion and refractive index changes are larger error sources,
which are not influenced by introducing a fibre. However in high precision
positioning, e.g. in a wafer scanner, these non-linearities are becoming the
major limiting factor in the high-end laser interferometer systems operating in
vacuum.
2.3 Optical fibres
Optical fibres were developed at the end of the 19th century [Hec93], but only
after WW II research and applications with fibres were started. After the advent of the laser (1960) the use of optical fibres grew rapidly, especially for
the potential benefits of sending information from one place to an other as opposed to electric transmission. During the following decades, the transmission
capacity of the fibres increased quickly. Because of its low loss transmission,
high information capacity, small size and weight, immunity to electromagnetic
interference, signal security and the abundant availability of the raw material
(i.e. ordinary sand) ultra pure glass fibres have become the major communication medium end of the 20th century and are nowadays commonly used in
telecommunication and sensor applications.
Fibres can be made using the double crucible method, where core and cladding
(see Section 2.4.1) are melted simultaneously. Commonly used however is a
16
Heterodyne laser interferometry and optical fibres
three step process, preform forming, drawing and coating. Of the three, preforming is the most crucial.
A preform is an enlarged version of the fibre with the same geometrical shape,
core radius to cladding radius ratio and refractive index profile. This preform
can be made using 4 different methods:
•
•
•
•
Outside vapor deposition
Vapor phase axial deposition
Modified chemical vapor deposition
Plasma modified chemical vapor deposition
All these methods are used to dope the fibre controlled with oxide components. In the drawing step the metre long preform is extended to the final fibre
[Che96, Sen92].
In this chapter optical fibre theory is summarised. It is focused on application
to (heterodyne) laser interferometry, and the special demands (e.g. concerning
minimum polarisation mixing) on the emerging laser beam.
2.4 General introduction to optical fibres
In this section a short introduction to fibres is given. This basic knowledge is
needed to understand problems, choices made and give general background
for theoretical descriptions in the following sections.
2.4.1 Optical fibre structure
The optical fibres mechanical structure consists of an inner core surrounded by
a cladding needed to reflect the light. A coating protects this cladding. This is
schematically represented in Figure 2.3.
Coating
Cladding
Core
Figure 2.3: Schematically the cross section of a fibre with the fibre structure. The light is transmitted in the core. The coating only
serves as a protective layer.
The core and cladding normally both are made of very pure silica doped with
rare earth elements. The core refractive index is different from that of the
2.4 General introduction to optical fibres
17
cladding. Plastic optical fibres [Mar97] are used more and more, but are not
available in the same quality as the silica fibres yet. Especially the attenuation is
about 200 dB/km where as for all glass fibre this is 10−50 dB/km at λ = 633 nm.
Currently also hollow fibre structures are under research ([Har00]). Also fibre
bundles are used, these bundles of fibres are made of fibres without coating.
These systems are for transmitting images and are very costly, because the
fibres all need to be orientated precisely the same in the matrix at both ends.
The core diameter typically is several micro metres, whereas for standard glass
fibres the cladding diameter is 125 µm. Coating sizes are a few millimetres.
2.4.2 Light transmission by fibres
To introduce light transmission by fibres, in this section light transmission
based on ray optics is given. As the size of the fibre core reduces, the approximations given here may not always be valid, but the general ideas remain.
Describing the light in terms of rays is only a model; in fact the fibre is a waveguide for which the properties should be determined by rigorously solving the
Maxwell equations with the boundary conditions imposed by the fibre. Light
is transmitted in waveguides or fibres with total internal reflection at the corecladding boundary. To prevent the light from loosing its power and to avoid
losses through the scattering of light by impurities on the surface of the fibre,
the optical fibre (core) is covered with a layer of a (much) lower refractive index (cladding). Remember that reflection means bouncing off, while refraction
means bending. Confusion between the terms reflection and refraction sometimes occurs in this case because the total reflection that takes place in the fibre
is described by Snell’s law (Equation 2.10), which is generally taken as the law
of refraction.
n1 sin θ1 = n2 sin θ2
(2.10)
Because the refractive index of the core (ncore ) is greater than that of the cladding
(ncladding ), light at an angle larger than the critical angle (θc ) is no longer refracted,
but (totally) reflected into the core. See e.g. ray B in Figure 2.4. The total reflecA
n0
0
θm
θm
B
θc
ncladding
ncore
θ
B
A
Figure 2.4: Propagation of light through an optical fibre core.
tion allows the light to be transmitted over long distances by being reflected
18
Heterodyne laser interferometry and optical fibres
inward thousands of times.Light that has an angle of incidence θ that is smaller
than the critical angle θc , e.g. ray A in Figure 2.4, will only reflect partially and
refract partially. After many refractions the light will be lost in the cladding,
where this light is absorbed.
Total reflection only depends on the refractive indices and can be described by:
θ < θc = arcsin
ncladding
ncore
(2.11)
As light with an angle smaller than the critical angle is not transported, it is of
no use to launch light with that angle into the fibre. Note that the light which is
not transmitted has an angle greater than θm at the entrance surface, which may
seem to be contrary to what is stated earlier, but that depends on the critical
angle definition. The light striking the input face of the fibre only within in a
cone with an angle θm (see Figure 2.4) is transmitted. The angle θm also called
half acceptance angle is found by
0
n0 θm = ncore θm
(2.12)
The larger the half acceptance angle (θm ) the easier the coupling will be (and
thus the efficiency). With n0 = 1 (e.g. air) and some trigonometry the numerical
aperture, which is the sine of half acceptance angle by definition is:
q
NA ≡ n0 sin θm = n2core − n2cladding
(2.13)
Typically, the numerical aperture of a fibre is in the range of 0,1 to 0,4 [OZ 99c].
2.4.3 Attenuation
Not all light which is coupled into the fibre will emerge from the fibre. In this
section several attenuation mechanisms are discussed.
As described in the previous section, light striking the core cladding interface at
an angle larger than the critical angle (θc ) is no longer (completely) transmitted.
Imperfections, e.g. a bulb on the interface, cause diffraction of the light. This
diffracted light always has a component larger than the critical angle, and a part
of this component will be lost in the cladding because it will not completely
reflect.
Rayleigh scattering is the fundamental loss mechanism arising from density
fluctuations into the fibre during manufacturing [Hen89, Agr01]. This microscopic non-uniformity of the refractive index partially scatters the light into
many directions. The Rayleigh loss (1 − αR,dB ) is estimated to be [Agr01]:
αR,dB =
0,8
CR
=
= 4,98 ≈ 5 dB/km
4
λ
0,6334
(2.14)
Where λ is the wavelength used and CR is a constant depending on the fibre
core. For a 10 metre long fibre the Rayleigh loss is estimated to be αRl=10 = 0,9886.
For bending, two types are distinguished: micro and macro bending. In the
2.4 General introduction to optical fibres
19
case of micro bending, attenuation is due to imperfections in the geometry of the
fibre (core ellipticity, core diameter variations, rough core cladding boundary).
This is either caused by manufacturing or mechanical stress, such as pressure,
tension, twist.
When the fibre bend diameter reaches centimetres it is called macro bending.
The light traveling inside the fibre core loses optical power due to less than
total reflection at the core-cladding boundary. It strikes the outside surface
at an angle larger than the critical angle, so that not all the light is reflected
towards the inside of a fibre, but a small portion is refracted. In practice there
is no noticeable loss of energy by bending, when the diameter of the bend is
larger than 10 cm [Hen89].
Absorption, mainly due to OH-ions, also causes attenuation.
In the fibre fed laser interferometer attenuations are not a problem since the
fibres used are relatively short (only a few metres) and the fibre losses are
mainly determined by losses in the laser-coupling (see Chapter 3). Because the
wave length to be used is fixed, the attenuation only is important to understand
the mixing effects.
As only short length are used the output power (Pout ) due to attenuations,
which often are given in dB/km, can be calculated by:
αa,dB
)
(2.15)
Pout = Pin αLa ,
(2.16)
αa = 10(−
10
with Pout is the input power, αa,dB the attenuation in dB/km, α the attenuation
as fraction and L is the length in km. The output can also be calculated using
the attenuation as fraction over a length L, called αa,L . The output power is then
calculated by:
αa,dB L
)
(2.17)
Pout = Pin αa,L
(2.18)
αa,L = 10(−
10
Using Equations 2.15 and 2.16 (or 2.17 and 2.18), and an attenuation of 12 dB/km
[OZ 99c], losses in the fibre are calculated as small as 3% over lengths up to
10 m.
2.4.4 Number of modes in an optical fibre
A mode is a three dimensional electric field configuration, characterised by a
single propagation constant (velocity). A mode represents one of the possible
solutions of the Maxwell’s equations for a specific geometry and refractive index of the fibre. This can be explained with the help of the wavefront condition:
all electric field components in a wavefront of a specific mode must have the
same phase. This is visualised by the dotted lines in Figure 2.5. This requires
that the optical path length of the ray AB differs from that of CD by (a multiple
of) 2π. This condition can only be met for modes (rays) which form certain
20
Heterodyne laser interferometry and optical fibres
angles with the optical axis. Note that an internal reflection also results in
a phase change. This means that the length AB does not differ 2π from CD
but (much) less. Fibres are commonly mathematically characterised with the
Phase front
n1
Light ray
B
C
θ1
n2 d
A
θ1
θ1
n1
D
Figure 2.5: Rays in a (slab) waveguide to show that only certain angles
(or modes) are possible due to the wavefront condition.
V-number, also called normalised frequency. This V-number determines the
number of core modes, for V ≤ 2,405 only one mode is supported by the fibre.
The frequency V combines all essential fibre data and the wavelength in a single
number. The V-number is defined as [Hen89]:
V=
2πrcore q 2
ncore − n2cladding
λ
(2.19)
There can be a (small) confusion: a mono mode fibre has two orthogonal modes
(LP01 , also called HE01 or TEM11 , and LP00 ).
A mono mode fibre only has the fundamental mode as long as the wavelength
used is larger than the cut-off wavelength. For smaller wavelengths also higher
order modes can propagate. The cut-off wavelength is defined by [Hen89]:
s
λcuto f f = 3,7rcorencore
2
n2core − n2cladding
n2core
(2.20)
2.4.5 Optical waveguide subdivision
Fibres can be subdivided in several ways. Often a division based on the
propagation of the electromagnetic wave (light) through the medium is used.
Modeling the light in a fibre is usually based on wave optics and modes (see
Section 2.4.4). Based on the numbers of modes mono mode and multi mode
fibres are distinguished.
Multi-mode fibres
In multi mode fibres, there are many modes. When coupled into a multi mode
fibre each longitudinal mode of a laser separates into many fibre waveguide
modes. Due to their relatively large core radius these fibres are easier to
align. Because the coupling is not very critically there is little loss of energy.
For communication applications, there is a high bandwidth for transmitting
2.4 General introduction to optical fibres
21
signals. However, these fibres suffer from modal noise. Modal noise causes the
modes to interfere. This is called speckle. As the speckle pattern is based on
phase relations, it is sensitive to changes of phase. These changes are already
caused by small movement and temperature changes of the fibre. Therefore
the speckle pattern is always changing.
In addition, the multi mode fibre can further be divided, based on the refraction
index profile (over the fibres cross section), in:
• Step index fibre.
These fibres do have a step in refraction-index-profile. Nowadays however, these are only used rarely in multi mode optical fibres.
• Graded index fibre
A gradually refraction-index-profile reduction (often parabolic). In addition, the angle of acceptance depends on the distance from the core
center. The half acceptance angle is maximal at the center and zero at the
core-cladding boundary. Graded index fibres represent a compromise
in coupling efficiency and bandwidth. Graded index fibres suffer more
from speckle than step index fibres.
Due to speckle multi mode fibres are not suitable for the fibre fed heterodyne laser interferometer, despite the good coupling efficiencies which can be
achieved easily.
Mono mode fibres
Because of the small core radius, these fibres are difficult to align. Delivering
the light needs special attention to coupling. There can be (great) loss of
energy, especially with poor alignment. Mono mode fibres maintain a state of
polarisation if the following conditions are met:
•
•
•
•
•
perfectly circular core
no bend of fibre
no transverse pressure
constant temperature
no (randomly) varying intrinsic stress in fibre
Practically none of these will be met and the mono mode fibres are sensitive
to polarisation noise (also called birefringent1 noise). Due to this noise the
polarisation state in the fibre changes, due to the leaking of one mode to the
other, as a function of mechanical stress such as vibrations. Mono mode fibres
also show spectral noise, this is the wavelength dependent phase shift within
a mode. As the laser light used is monochromatic, this is of no interest for this
research. Mono mode fibres are always of the step index type.
It can be concluded that mono mode fibres are needed, because of speckle.
If mono mode fibres are used, polarisation noise has to be minimised in order
1 Birefringence
will be explained in Section 2.5.1
22
Heterodyne laser interferometry and optical fibres
to be used in the fibre fed interferometer.
For this, polarisation preserving fibres are developed. These fibres preserve
polarisation by a non axial-symmetrical refractive index profile. These fibres
are subdivided into:
• Polarisation maintaining fibres (PM)
• Polarising fibres (PZ)
Polarising fibres are only able to transmit one polarisation, while polarisation
maintaining fibres can deliver two orthogonal polarisations.
As for the development of the fibre fed heterodyne laser interferometer a stable
output of two polarisations is essential, polarisation maintaining fibres are
selected. These fibres are described in detail in Section 2.5
2.4.6 Back reflection
If light is coupled into the fibre always some light is reflected. If no coating is
applied, on the standard air-glass interface this reflections is about 4 %. If e.g.
the fibres end face, at the input side is parallel to the laserhead the light which
is reflected back can make the laser function instable. At the other side light
emerging from the fibre will also reflect partially and is then transported back to
the input side, where it agian is partially reflected and transmitted. The latter
would again result in laser instability due to disturbance of the laser effect.
The light reflected back into the fibre however would mix with the intented
light coupled in and, due to the mixing, reduce the displacement measurement
accuracy of the interferometer. Back reflection was prevented using special
APC (angle polished connector) type optical fibres. These fibres, as shown in
Figure 2.6, have a polish at an angles of 8 or 9 degree (90−θB ). Due to this polish,
the reflected light is not reflected back into the laser head. Reflected ligth at
the output side of the fibre is reflected into the cladding, where it is absorbed.
The fibre also has high anti-relection coatings, to minimize the reflection and
to optimize the transmition of light. All other optical components need proper
coating to minimize back reflection. Reflections from lenses, beamsplitters and
polarisers still can influence the system performance.
θB
Figure 2.6: The use of a fibre with an angled polish, to minimise the
effect of back reflection of a fibre.
2.5 Polarisation maintaining optical fibres
23
2.5 Polarisation maintaining optical fibres
Ordinary fibres do not preserve polarisation direction due to mode leak caused
by changes in internal stress distribution e.g. by change in environmental conditions. Polarisation maintaining fibres are fibres who preserve polarisation,
thus having a stable polarisation output. Polarisation is preserved in these
fibres by an asymmetry in the fibres cross section. These asymmetries cause
the fibre to raise birefringence. These polarisation maintaining fibres can be
subdivided into:
• shape birefringent fibres (asymmetrical [core] shape)
• stress-induced fibres (this means an asymmetrical stress distribution resulting in a circle asymmetrical refractive index profile due to materials
with different thermal expansion coefficient, as is seen by the parts next
to the core (dark dots) in the cladding as e.g. seen in Figure H.2.)
Speaking of shape birefringent fibres, often elliptical core fibres are meant.
Although other types are mentioned in literature [SB97, Gro89], no shape birefringent fibre was available commercially. Disadvantages of the elliptical core
fibres are their smaller diameter and a large difference in length-width resulting
in a strong orientation dependence. An advantage of elliptical core fibre would
be a smaller temperature dependence . Shape birefringent fibres do have stress
birefringence.
In stress-induced fibres the melting temperatures between the core and the
stress applying parts difference. This difference caused stresses because one is
solidified before the other. The birefringence is raised by the induced stresses.
High birefringent fibres preserve the polarisation direction because the variations of the stress are relatively small compared to the build-in stress. Thus
the changes of ∆n caused by the change of stress are small compared to the
birefringence, and so the polarisation direction is preserved.
2.5.1 Birefringence
To understand the principles of the polarisation maintaining fibres, first the
principle that make fibres maintain the polarisation state, called birefringence
is introduced. Many different materials are optically anisotropic which means
that their optical properties are not equal in all directions. This difference is
caused by not completely symmetrical binding forces of the electrons from
the atoms in the material. This can be represented by a simplified mechanical
model, where the charged electrons are bounded by springs of different stiffness (i.e. having different spring constants) [Hec93]. The speed of the wave
and therefore the refraction index is determined by this difference. This phenomenon causes different refraction indexes in the fibre at different main axes.
Only materials with different nx and n y , where z is the propagation direction, are
described here. Controlled by the atom arrangement in the fibre, two orthogonal axes with different refraction indexes are present. Those axes are called
fast and slow axis or main axes. A material whose indices of refraction differ
24
Heterodyne laser interferometry and optical fibres
in two different directions is called uni-axial birefringent1 . Birefringence can
be linear or circular, in this thesis only the linear birefringence are described,
as this may be found in an optical fibre.
The normalised linear birefringence or modal birefringence (B) is:
B = nx − n y = ∆n
(2.21)
The modal birefringence can be divided into [Kam81]:
B = BS + BG
(2.22)
BS is the strain contribution to the birefringence is and BG the shape (geometry)
contribution. These strain and shape contribution are used to make polarisation
maintaining fibres. Change in temperature or mechanical stresses, change the
original birefringence, thereby changing the polarisation maintainability of
the fibre. A large birefringence reduces the polarisation mixing, because the
fluctuations of the refractive index changes are then relatively smaller. The
following effects may cause birefringence:
• stress
• strain
• changes caused by:
– temperature dependence of the stress-optic coefficient
– difference in expansion coefficient of the core to the cladding
Mode Coupling
Due to imperfections energy is transferred from the x mode to the y mode and
this is called mode coupling. A perfect fibre has no mode coupling.
No structure is absolutely perfect, e.g. fibres with nominally circular cross
section are never perfectly round. There is always a slight ellipticity in the core
of the fibres. If the fibre core has an elliptical cross section, modes polarised in
the directions of the major and minor axes of the ellipse have slightly different
group velocities. A mode that is initially linearly polarised in an arbitrary direction can be decomposed into two modes, each of which is polarised in the
direction of one of the principal axes of the ellipse. At some distance into the fibre the two modes arrive with slightly different phases so that the superposed
field is no longer linearly polarised. This shows that fibres with slight core
ellipticity do not permit a linearly polarised guided wave to maintain its polarisation. Core ellipticity is not the only ”depolarising” effect in fibres, transverse
anisotropy (stress, strain and refraction index distribution) of the fibre-material
also contribute to this effect. Mode coupling will cause a transfer of one mode
to the other and as energy is transfered between the two modes, both carrying
their own polarisation and frequency, this results in polarisation or frequency
1 The word refringe is old English and is used in stead of our present-day term refraction. It
comes from Latin and means to break. So birefringence means double refractive
2.5 Polarisation maintaining optical fibres
25
mixing. The effects of this mixing will be measured in Chapter 4 and 5. Due to
the high birefringence in the fibre, the E-field of the mode leaking is, even after
a short section of the fibre out of phase with the E-field of the leaked mode of
a point somewhat earlier, an is eliminated due to the destructive interference
between the two.
A more detailed overview about sources causing mode coupling is given in Appendix F. The described effects are the most effective if applied at 45 degrees,
but because the angle at which the perturbation is applied is however not
known, in experiments this was of no use. If the perturbations have principle
axes that are coincident with the unperturbed system, no energy is coupled
between the modes, see [Kam81, PBH82].
Beat length
One of the characterising parameters of a birefringent fibre is the beat-length.
The beat-length is the length required to rotate the polarisation direction by 360◦
[Sen92]. The beat length is thus the length of the fibre after which the wave
in the slow axis is delayed by exactly one optical wavelength. The beat-length
Output
Fibre
1 beat length
Input
Fast axis
Slow axis
Figure 2.7: Schematically the definition of the beat length. The beat
length is the length of the fibre after which the wave in the
slow axis is in respect with the fast axis delayed by exactly
one optical wavelength.
(Lb ) is defined by:
Lb =
λ
B
(2.23)
Where B is given by Equation 2.21.
The shorter the beat length the sooner the leak components are out of phase
26
Heterodyne laser interferometry and optical fibres
and the ofter destructive interference takes place.
From [Kam81] it is found that the power spectrum of the disturbances has a lowpass shape with beat lengths larger than about 1 mm. High birefringent fibres
have small beat lengths. From various papers [Kam81], [Oko81], [PBH82] beat
lengths smaller than one, up to several mm are reported. A disturbance will
thus only result in a slight change of the initial output state. The waveguide then
essentially appears length invariant, since external perturbations are swamped
by the high level of internal birefringence.
Rotation of the direction of the resulting polarisation due to bending of the fibre
is an example of a disturbance. To see this, consider a fibre with the following
properties: ∆n = 10−3 , ncore = 1,46 and a core radius of rcore = 2,15 µm operating
at a vacuum wavelength λ = 0,633 µm. With these values, a beat length of
L = 0,63 mm is calculated by combining Equations 2.23 and 2.21. Assume a
bend diameter of 20 cm and using Equation F.5 the length needed to rotate the
resulting polarisation by this ellipticity is more than 5 km. So the effect on the
beat length and therefore on the output polarisation is negligible.
The beat length specified of the fibres used in this research is Lb = 2 mm
[OZ 99c].
Conclusions
Because polarisation maintaining fibres do have different refractive indices
(birefringence) their mutual (orthogonal) modes have different group velocities. At imaginary cross-section, the resulting E-field is changed from linear via
elliptic to circular and visa versa, if the input polarisation was linear at 45◦ and
the fibre is excited e.g. heated. This group velocity difference is caused by the
refractive index difference, which both modes encounter. Due to this refractive
index difference the phase of the mode leaking, is after a quarter beat length
out of phase of the mode leaking at that point and will destructively interfere
thereby preserving the polarisation.
Refractive index changes are caused by mechanical strain or temperature influences. If the initial differences in the refractive indices are relatively large (small
beat-length) compared to the disturbances, the change in refractive indices are
small compared to the birefringence and therefore the effects on the state of
output polarisation are small.
The initial refractive index difference is determined by an asymmetry in the
fibre. This can be a geometrical asymmetry (e.g. elliptical core fibre), or a stress
asymmetry (e.g. bow-tie fibre). Because the waveguide is made of one material (with oxide dopings), the refractive index differences are limited (in the
range of 10−5 to 10−3 ) in production. While giving a stable output polarisation
direction in high birefringent fibres, these fibres causes (large) phase changes
during stress change.
In Chapter 4, this phase change due to stress change will be used to determine
the fibres main axes, thus to align the fibre.
2.6 PM fibres used for experiments
27
2.6 PM fibres used for experiments
The tests described in Chapter 4 and 5 were subsequently carried out with:
1.
2.
3.
4.
5.
6.
7.
8.
Normal mono mode fibre, not polarisation maintaining 1
Bow-tie mono mode polarisation maintaining fibre 1
Oval inner clad mono mode polarisation maintaining fibre 1
Panda type mono mode polarisation maintaining fibre 2
Pure mode (Panda) type mono mode polarisation maintaining fibre 3
Pure mode (Panda) type mono mode PM fibre 3 , (length = 5 m)
Pure mode (Panda) type mono mode PM fibre 3 , (length = 15 m)
Pure mode (Panda) type mono mode PM fibre 3 , (length = 50 m)
The first 5 fibres are all nominally 3 metres long and were used to find differences between fibre types. The last 4 being only different in length are used to
find any length dependence influence. All fibres are commercially available.
Despite the above mentioned suppliers, the fibres are made by other manufactures 4 .
The different types of PM-fibres used are shown in Figure 2.8.
Bow-Tie
Panda
Elliptical inner clad
Figure 2.8: Schematic the different types of PM fibres used. Differences
are the means by which the stress is introduced.
2.7 Fibre model
In order to model the mode coupling of the fibre based on the influences
described in Appendix F, precise environmental and detailed fibre properties
are needed. However, it cannot be known what the specific core diameter at a
given point is, neither can the exact orientation of the main axes be given at that
point. As the fibre exhibits local (i.e. for a section of fibre) linear birefringence
induced by geometric anisotropy of the core, residual stress or bending, the
1 Wave
Optic[Wav99]
Source[Poi99]
3 OZ-Optics [OZ 99c]
4 e.g. Corning [Cor01]
2 Point
28
Heterodyne laser interferometry and optical fibres
fibre can be locally characterised by the simple Jones [Jon41] matrix of a linear
retarder. Because the fibre is not homogeneous, it has to be considered as a
succession of wave plates having arbitrary birefringence and orientation. In this
case the model is not supposed to give a local description of the polarisation
state of the fibre but must be seen as a global transfer function, linking the
Jones vector of the output state of polarisation to the input vector. The fibre
will be modeled as a series of optical elements, which can all represent for
effects described earlier, and still explain the output [VR99, CHO+ 01]. This is
graphical represented in Figure 2.9. The Jones matrix of the fibre, H f ibre, is the
Y
M1
Mn
Y
···
X
X
L
Figure 2.9: A model of the fibre. The fibre is represented by wave plates
having both an orientation and retardation. The wave plates
along the length of the fibre can vary both in orientation and
retardation.
product of linear retarders and rotation matrices as described by Equation 2.24.
H f ibre =
n
Y
R(−θi )Mi R(θi )
(2.24)
i=1
Were R(θ) is the rotation matrix of angle θ given by:
"
#
cos(θ) sin(θ)
R(θ) =
− sin(θ) cos(θ)
(2.25)
Mi is the matrix of a linear birefringent medium, with a phase difference of φi
 φi

 ei 2

0

Mi = 
(2.26)
φi 

0 e−i 2
Both θ and φi vary over the length L, thus also matrices R and Mi vary with the
position along the fibre.
If a plate has no rotation relative to its neighbouring plates, this plate will only
retard the beam. If the plate has however an orientation also mode coupling
will occur. This means energy is transfered from one main axis to the other,
and visa versa. In Section 4.4, with the use of the measurement data this model
will be expanded and explained in more detail.
2.8 Summary
29
2.8 Summary
The chapter was started by the description of an ideal heterodyne laser interferometer, then beam ellipticity was introduced in the interferometer. Beam
ellipticity was introduced because this is influenced by introducing a fibre in an
interferometer, due to the mixing within the fibre. It was shown that beam ellipticity causes the interferometer to measure errors. The errors measured due
to beam ellipticity are period with the laser wavelength used, and are called,
first and second order non-linearities, for more information see [CHS02].
Optical waveguides are divided into multi and mono mode fibres. Multi mode
fibres can not be used, due to the changing intensity profile over the cross
section of the fibre, caused by interference of different modes, called speckle.
Because the need for a stable output polarisation, polarisation maintaining
fibres are needed to transmit the laser beam from the laser head to the interferometer. These fibres show birefringence, this is the difference in refractive
index for the fibres main axes. By bending or heating the fibre, the internal
stress distribution changes and therefore the birefringence. Due to the high
birefringence present, small change introduced (e.g. due to stress or bending)
are of only minor influence. Using coated angle polished connector (APC)
fibres, back reflections is reduced, and the disturbing effect of it is eliminated.
A Jones model was used to represent birefringence, that accounts for both mode
coupling and phase changes.
30
Heterodyne laser interferometry and optical fibres
Chapter 3
Optical fibre coupling
The light which should be guided by the fibre has to be coupled into and out
of the fibre. In this chapter this process will be described. As the mono mode
optical fibre (core) diameter is only a few µm, high accuracies and stability of
the input section is demanded for coupling the light into the optical fibre.
Fibre
Fibre
output beam
(collimated)
Laser
Output section
Input section
Interferometer
Figure 3.1: Schematic diagram of the coupling of the light from laser
head to the interferometer via the optical fibre.
In many motion systems, e.g. in a wafer scanner, the laser interferometer is
used as a position measurement system. In these systems the laser beam is split
several times. Each split beam is used for measuring a rotation or a translation
axis. To enable the measurement of several axes, the output intensity of the
fibre must be maximal. This must be achieved by optimising the coupling
efficiency.
In addition it is difficult to increase the optical laser output from a physical
point of view. Increasing the output intensity of the ’Zeeman’ type laser will
decrease the split frequency (∆ f ). A lower split frequency causes a lower
maximum velocity to be measured. This means that the maximum achievable
optical power must be transmitted.
In the wafer lithography machines, every few years the laser head is replaced.
This replacement makes it necessary to disconnect the laser-fibre-optic chain at
least at one point. Coupling is thus also required for maintenance or component replacement.
The most important issue is an accurate rotation alignment. Rotational align31
32
Optical fibre coupling
ment is necessary for maximising the achievable displacement measurement
accuracy of the interferometer (by minimising optical mixing). As an accurate
rotational alignment for research purposes should be of high resolution and low
hysteresis and the relative motion between laser and fibre must be minimised
to achieve optimal performance. As this was not commercially available when
this research started, in Section 3.3 the development and realization of a fibre
coupler with six degrees of freedom for our experiments is described.
3.1 The coupling of laser light into the fibre
In this section achievable coupling efficiencies are discussed. In order to get
a good understanding of coupling efficiency related to a small focused beam,
first some general theory is presented.
3.1.1 Axes definition
In Figure 3.2, axes and dimensions are defined for the optical fibre coupling
system, these will be used in the following sections. Rl is the laser beam radius
β
Fibre
Light beam
Y
Rl
Z
γ
rcore
X
α
Figure 3.2: Definitions of axes and dimensions of the optical fibre coupling system.
while r is the fibre mode radius. The fibre mode radius is ideally rcore = 2,15 µm
and the laser mode radius from the laser head used is Rl = 3 mm. X, Y and
Z devote the reference coordinate system, where the Z-axis coincides with the
fibre propagation axis. The origin of this reference coordinate system is on the
center of the fibres entrance face. Rotations of the laser beam with respect to
the fibre are indicated with α, β and γ. All rotations and translations described
in the next sections are relative rotations and translations between laser beam
and fibre.
3.1 The coupling of laser light into the fibre
33
3.1.2 Coupling means of the laser light
To transmit the light from the laser through a fibre different optical components
are needed. At the input, the laser beam has to be reduced from a diameter of
about 6 mm (of the laser head used) to a fibre with a core diameter of 4,3 µm
(for λ = 0,633 µm, this is a mono mode fibre). At the output of the fibre this
4,3 µm beam has to be expanded to the original size of 6 mm. See for a graphical
representation Figure 3.1.
As the use of standard components is preferable, a connection at input and/or
output side should preferably be done with a so-called (A)FC-connector. These
connectors are universal and supported by a range of suppliers. An advantage
is also that it allows a relatively easy exchange of fibres.
Experimentally, intensity fluctuations caused by bending the fibre near the
connector are found. This can easily be suppressed by applying an extra fixation
after the connector. Most important when using connectors is the possible
influence of achievable displacement uncertainty of the laser interferometer
(see also Section 4.4 and Appendix H). A global orientation of the fibres main
axes is provided by the fibre keying, supplied by the manufacturer. This fibre
keying is within 3◦ aligned with the fibres main axes.
With the use of these connectors the exchange of fibres is relatively easy and the
connector (keying) also gives a good indication of the orientation of the fibres
main axes to start the rotational alignment with. A disadvantage of the use of
connectors is the possible influence on the achievable displacement uncertainty
of the laser interferometer.
3.1.3 The coupling of laser light into a fibre
For the coupling of the light into the fibre an optical system is needed. Most
easy this is done by the use of a collimator lens with connector assembly.
Standard commercially available collimator lenses focus only beam diameters
up to 2 mm into a single mode fibre. This can be solved either by cutting of
the rest of the light (this leads to 61% loss of energy), or to reduce the light
beam first to approximately 2 mm. Alternatively the beam expander within
the laser head could be removed, giving a 1 mm beam. At last also collimators
with larger input (and output) beams were found. Both a 4,5 mm and 6 mm
collimator were found, but in the developed coupler only the 4,5 mm can be
used because when the coupler was developed only the 4,5 mm collimator was
available.
The use of collimator lenses with diameters of 4,5 mm and 6 mm will give a
reduced coupling efficiency due to the relative large numerical aperture of the
lens (see Section 3.5).
3.1.4 Beam propagation
As the size of the fibre core is of the same magnitude as the wavelength of
the light used, the light propagation may no longer be described by ray or
34
Optical fibre coupling
geometric optics. The propagation of the light through the optical system,
e.g. the focussed beam at the fibres input, is described by Gaussian optics. In
Gaussian optics, the focal point is replaced by a focal plane, where the beam
has its smallest cross section (2w0 ) or beam waist, see Figure 3.3. The beam
1/e irradiance surface
θ
z
ne
c co
toti
p
m
asy
w0
w(z)
Figure 3.3: The focal plane of a small spot (2w0 ) described by Gaussian
optics. The beam radius at a distance z is w(z). The distance
√
at which the beam diameter has expanded to w(z) = 2w0 ,
is called the Rayleigh length. For large distances of z, the
beam propagation is along the asymptotic cone described
by the half acceptance angle (θ).
diameter 2w(z) at a distance z from the smallest cross section is found by [Sie86]:
v
t
w(z) = w0

2
 λz 

1 +  2 
πw
(3.1)
0
With λ the wavelength of the light used. The half acceptance angle (θ) is the
angle of the asymptote described by this function w(z) for large values of z, see
Figure 3.3. The larger the half acceptance angle the smaller the focal spot will
be, but the smaller the Rayleigh length is. The Rayleigh length is the length
√
by which the beam is expanded so that w(z) = 2w0 . The Rayleigh length is
an indication for the focal depth. In our system the half acceptance angle, the
diameter of the fibre as well as the wave length used are fixed.
At the entrance side the light beam has to be reduced from a diameter of about
6 mm (with the used heterodyne laser head) to a diameter of 4,3 µm (fibre
core). Using a lens, the focal distance is determined by the NA of the fibre
(Equation 2.13) and the laser beam radius:
f ≈
3
Rl
=
= 27 mm
NA 0,11
For more information about Gaussian beams refer to [PP93].
(3.2)
3.2 Calculated in-coupling accuracies
35
3.2 Calculated in-coupling accuracies
In this section the required manipulator adjustments and beam output intensities are calculated. The total coupling efficiency is described by:
ηt = Tηx η y
(3.3)
Where T means transmission losses and ηx and η y are efficiencies due to misalignment and mode mismatch for the orthogonal axes x and y. The most
accurate description of ηi is obtained by using the beam overlap theory as
presented in Equation 3.4 [CR98].
∞
2
R
EL E f di
−∞
ηi = ∞
R
R∞ 2
|EL |2 di · E f di
−∞
,i = x,y
(3.4)
−∞
Where EL the laser mode and E f the fibre mode are in direction i. As the
refracted laser beam mode is not known exactly, e.g. due to limited lens
diameter, the efficiency is estimated by:
Rr2
E2 (r) dr
P
r1
ηi =
=
P0
RR2
E2 (R) dR
(3.5)
R1
Where the boundaries of the laser beam mode are given by R1 and R2 while
the boundaries of the fibre mode are given by r1 and r2 . Because both laser
and fibre are nearly perfectly cylindrical no difference is made between x and
y direction. Efficiencies calculated in the following sections are assumed to be
equal for both directions.
ηx ≈ η y = η
(3.6)
To make an estimation of the required specifications for all errors a 90% intensity
efficiency criterium will be used. The efficiencies are calculated small deviations around the optimal specification and assuming independent influences,
using a two dimensional Gaussian approximation as described by Equation 3.5.
3.2.1 Transversal accuracy
The transversal adjustment is used to translate the focussed beam in x, y direction with respect to the fibre core. It is assumed that the collimator output
(beam waist) is a perfectly Gaussian beam with the size of the fibres core mode
but at the fibres entrance surface with a transversal offset (∆t ) or optical axes
mismatch, as In Figure 3.4 schematically is shown.
36
Optical fibre coupling
Collimated beam
∆t
Fibre
Figure 3.4: A transversal misalignment (∆t ) between laser beam and
fibre core as can be caused by a transversal offset or core
eccentricity.
In addition also core eccentricity, due to production tolerance needs to be
compensated for in order to achieve optimal alignment efficiency. By using
Equation 3.5 and the 0,9 efficiency criterion the required transverse adjustment
accuracy is ∆t = 1,2 µm.
3.2.2 Longitudinal accuracy
The longitudinal adjustment is used to translate the focussed beam in the zdirection onto the fibre entrance side. In Figure 3.5 schematically the beam is
defocussed resulting in a not optimal coupling efficiency. Also, if the focussed
beam has not the same diameter as the fibre core mode, no optimal coupling
efficiency is achieved. If again the 0,9 efficiency criterium is used and EquaCollimated beam
Fibre
∆L
Figure 3.5: Non optimal coupling efficiency due to a focus error ∆L . The
same effect can be caused by an error in the focussed beam
diameter.
tion 3.5 it can be calculated that the beam diameter may expand approximately
1 µm. The change of the beam waist at the entrance surface of the fibre is thus
0,5 µm. The focus beam diameter may differ maximal 1 µm at the entrance face
of the fibre, as resulting of both non ideal focussing and non ideal beam waist
due to the optics used.
Next the maximal defocussing distance of an ideally positioned beam is calculated. The beam may expand 0,5 µm or:
w(z) = w0 + 0,5 µm
(3.7)
Substituting into Equation 3.1 this results in:
2

 λz 
2 
2
2

(w0 + 0,5) = w0 + w0 ·  2 
πw
0
(3.8)
3.2 Calculated in-coupling accuracies
37
And then solving for z with the nominal fibre radius w0 = 2,15 µm, thus the
defocusing distance is:
s
(2w0 + 0,25)π2 w20
= 22,8 µm
(3.9)
z=
λ2
This means that in the longitudinal direction (z) the maximal focal offset is
22,8 µm.
3.2.3 Azimuth accuracy
When the beam is ideally focussed with the correct beam waist, but the beam
is tilted also no optimal coupling is achieved as depicted graphically in Figure 3.6. To calculate the efficiency, the far field overlap is estimated, therefore
ated be
Collim
∆a
am
Fibre
Figure 3.6: Schematic of the azimuth misalignment (∆a ) of the laser
beam with respect to the fibre to calculate the fibre coupler α
resolution. The azimuth error α is transferred to a transverse
offset α · z at distance z; then the efficiency is calculated as
with the transversal accuracy
.
the azimuth error α is transferred at this distance z to a transverse offset α · z.
Proceeded is as with the transversal accuracy, using Equation 3.5 and the 0,9 efficiency criterium, the required azimuth accuracy is then 0,7 degrees (0,01 rad).
For the azimuth β the same procedure is used, and due to the symmetry of the
problem, the required azimuth accuracy is also 0,7 degrees (0,01 rad).
3.2.4 Collimator alignment accuracy
In the developed fibre coupler at maximum a 4,5 mm collimator lens can be
used because at the time the coupler was developed only this collimator system
was available. Out of a 6 mm laser beam only 4,5 mm can be transmitted as
schematically shown in Figure 3.7. Also the non-ideal alignment of the lens with
respect to the center axes of the laser beam and fibre core is shown. Using the
same 0,9 efficiency criterium also for the collimator, the required positioning
accuracy is calculated. The assumption was made that for calculating the
efficiency the beam remains Gaussian. Aberration effects due to the limited
lens diameter are not taken into account. To increase the efficiency and to
minimise the aberration effects due to the cut off of the laser beam, a beam
reductor can be used between laser head and fibre collimator.
38
Optical fibre coupling
Collimated beam
∆c
Fibre
Figure 3.7: Schematically the misalignment of a 4,5 mm collimator lens
in front of the 6 mm laser beam as used in the fibre coupler.
To prevent the cut off of the laser beam a beam reductor can
be applied.
Using 4,5 mm nominal lens out of 6 mm nominal beam gives an alignment
accuracy of ∆c = 0,9 mm (radial). If the center of the lens and the laser beam are
within 0,9 mm, 90% coupling efficiency is possible despite of not transmitting
the complete beam due to the limited lens diameter.
3.2.5 Numerical Aperture mismatch
If the cone of the light entering the fibre is larger than the numerical aperture of
the fibre, light is lost in the cladding as shown in Figure 3.8. An estimation of
Lost power
0
2θm
Fibre
Figure 3.8: Schematically the effect of the numerical aperture mismatch
between laser beam and fibre. Light with an angle larger
0
than the numerical aperture (θm ) is lost in the cladding.
this loss can be made by transforming the perfect focal plane in z direction, and
then calculating at a distance z the mismatch of the transformed focal plane.
The fibre focal planes thereby increase less than the laser beam. Again using
the 0,9 coupling criterium requires a beam with an numerical aperture smaller
than 0,14 has to be used.
3.2.6 Rotational accuracy
Due to the symmetry of the fibre, any rotational alignment error in γ does not
influence coupling efficiency. Rotational alignment only influences frequency
mixing and this reduces the achievable displacement measurement uncertainty
of the interferometer.
For accurate measurements with the fibre fed heterodyne laser interferometer,
3.3 Design of the high precision fibre coupler
39
polarisation mixing is important. One of the sources of mixing is the rotational
alignment error of the fibres main axes to the lasers main axes. The mixing due
to a rotational misalignment is calculated by [SB97]:
ERalign,db = 10 10 log(tan2 γ)
ERalign = tan2 γ
(3.10)
Where ER the extinction ratio is (see Section 4.1) and γ is the rotational alignment of the fibres main axes with respect to the polarisation direction of the
laser beam.
To achieve measurement uncertainties at the nanometre level with the fibre
fed heterodyne laser interferometer the rotational alignment should be better
than 0,6◦ due to the polarisation mixing caused by this rotational alignment as
will be shown in chapter 5. In order to see what is possible, the adjustment is
designed to be better than 0,06◦ (1 mrad).
Instead of a rotation of the fibre or laser head, also an optical rotation with
a half wave plate can be used. A half wave plate was not used, to prevent
any (mixing) effects of this plate. In a commercial application the rotational
alignment could be replaced with a half wave plate which can be rotated.
3.2.7 Estimated coupling efficiency
For all the calculations, a 2D Gaussian instead of a 3D Gaussian beam was
used. The coupling efficiencies and the required adjustment resolutions are
estimated by Equation 3.5. The most accurate calculation (theoretically the
most correct) is provided by using the beam overlap integral theory with the
correct diffraction pattern. Using the approximations supplied by [Res95] to
calculate the efficiencies, nearly the same results as from the estimations were
found.
Other effects are the two air-glass interfaces and the attenuation calculated
in Section 2.4.3. Assumed is that these surfaces are coated to transmit 99%.
The attenuation factor of a 10 metre fibre is 0,97. If all accuracies of the nine
adjustments/effects as discussed before (∆tx , ∆ty , ∆L , ∆aα , ∆aβ , NAx , NA y , ∆Cx
and ∆Cy ) are as calculated each contribute to 90%. This means that in theory
that in a worst case only
ηT = 0,99 · 0,992 · 0,97 ≈ 0,37
(3.11)
of the input intensity is available at the output. In Section 3.5 an overall coupling
efficiency is calculated based on the measured accuracies of the coupler.
3.3 Design of the high precision fibre coupler
As the fibre must be adjusted in 6 degrees of freedom, the use of a commercial
coupler was not possible. First of all no 6 degrees of freedom laser to fibre
couplers were on the market when this research started. Secondly, most of the
40
Optical fibre coupling
couplers have coupled adjustments, making alignment a very time-consuming
task.
The coupler to be designed therefore must have uncoupled adjustments, with
at least the resolutions, as calculated in Section 3.2. Based on these calculations
Axis
X and Y fine
X and Y coarse
Z
α and β
γ fine
γ coarse
Stroke
> 5 µm
> 0,4 mm
> 0,4 mm
> 5 mrad
> 40 mrad
> 20◦
Resolution /20◦
< 0,1 µm
< 3 µm
< 4 µm
< 1 mrad
< 1 mrad
< 2◦
Table 3.1: The designed specifications of the fibre manipulator. For adjustments performed with rotational actuators, a resolution
per 20◦ rotational adjustment is used
the specifications of the strokes and resolutions of the fibre coupler to be designed for are shown in Table 3.1. These resolutions are at least 5 to 30 times
smaller than the calculated accuracies to ensure easy and precise adjustment.
For adjustments performed with rotational actuators, a resolution per 20◦ rotational adjustment is used. Strokes are based on experience but coarse resolution
must be smaller than the fine stroke.
Secondly, long term stability and predictable behaviour is preferable. Finally a
design was made [vdM00] and realised by the central technical facilities (GTD)
of Eindhoven University of Technology. A photograph of the realised coupler
is shown in Figure 3.9.
This statically determined design has 6 uncoupled degrees of freedom adjustment, which behave very repeatable by using elastic hinges. The coupler was
designed with a thermal centre at the focal point, to ensure long term stability.
The fibre coupler can be used with a separate lens as well as a fibre with integrated collimator lens.
By using a fibre with integrated collimator lens, x and y translations are only
of minor importance as they only center the lens with respect to the laser beam
and do not influence the focal point any more. The z translation is completely
redundant when using the fibre with integrated collimator lens.
In the following sections an overview of the functioning of the realised coupler is given. For more information about the used design principles refer to
[RR96].
In Section 3.4, the validations of the strokes and resolutions of the realised
coupler are described.
3.3.1 Definition of actuator and rotation axes on the coupler.
To describe the working of the apparatus that can adjust the laser beam in 6
degrees of freedom (DOF) with respect to the fibre, the axes definitions as given
3.3 Design of the high precision fibre coupler
41
Figure 3.9: Photo of the realised fibre coupler, note the holes in the topplane to cool down the laser which is mounted inside the
system.
in Figure 3.10 are used. The difference with Figure 3.2 is the location of the α
and β adjustments. These axes are rotated 45 degrees and performed by the
laser head. The rotation point of the α and β axes is around the focal point. All
other translations and rotations were done with the fibres.
3.3.2 X and Y-axes
The X- and Y-axes are used to translate the fibre core in the focal plane. The Xaxis is the horizontal translation perpendicular to optical axis, while the Y-axis
the vertical translation perpendicular to optical axis.
As only collimator lenses were used, this adjustment only centers beam axis
and lens axis, and is of minor importance. This changes if a separate lens and
fibre holder would be used.
The coarse adjustment is realised by applying differential threads, as shown
schematically in Figure 3.11a, where the inner and outer thread differ by
0,05 mm. The fine adjustment is done by elastic deformation of the thread.
In Figure 3.11b a photograph of the x-y manipulator is shown. The upper
wheel is for the coarse adjustment while the lower wheel is for the fine adjustment.
42
Optical fibre coupling
Y
Y’
β
45◦
γ
Laser
X’
Z
α
X
Figure 3.10: The translation and rotation axes of the fibre coupler. The
α and β adjustments are the only axes which are performed
with the laser head. The actuators of these adjustments are
rotated 45◦ with respect to the xy-coordinate system. The
x, y, z translations as well as the γ rotation is performed
with the fibre.
a
b
Figure 3.11: a) Schematic diagram of the principle of the coarse adjustment of the X and Y axes. The elastic constraints as shown
on the left are realised as elastic hinges to guide the coupler
in the X or Y axis direction only.
b) Photograph of the X and Y axes manipulator. Upper
screw is coarse adjustment, lower screw is fine adjustment.
3.3 Design of the high precision fibre coupler
43
3.3.3 Z-axis
The focal plane can be adjusted, using the z-manipulator. The z-axis is of no
importance if an integrated collimator lens is used, but is intended for the use
with a separate lens.
The focal adjustment (Figure 3.12) was done by rotating a lever with a micrometer screw gauge. The lever is connected with a sprout to the platform
supporting the fibre and the x, y and γ adjustments. The z-axis has a large play
of about 3,5 turns, but play was intentionally designed to prevent vibrations
from the base entering the stage over the lever and this play does not negatively
influence the adjustment accuracy .
base
Figure 3.12: On the left schematically the design principle is shown
with some constructional details, while a photograph of
the manipulator is found on the right. The manipulator
for z-axis, has some play to prevent vibrations entering the
xy-stage.
When using an integrated lens, the coupling efficiency can be optimised by
using fibres with adjustable focus connectors [OZ 99b]. This special connector
allows the spacing between the fibre and lens, to be precisely controlled without
rotating the fibre. This allows compensation for changes in beam waist location
due to manufacturing tolerances. These fibre connectors can be adjusted over a
range of 3,5 mm, with a resolution of 10 µm per 30◦ rotation of the adjustment.
3.3.4 α and β-axes
The α and β-axes are used for the rotational alignment. The actuators are
around the x0 and y0 axes respectively. With the use of an integrated collimator
lens these adjustments are of main importance.
The rotation is done by rotating the complete laser head. The point of rotation
is designed to be the focal point of the laser beam at which the fibre input is
44
Optical fibre coupling
located.
After alignment the α and β axes need to be fixed to make a stiff connection. The
Figure 3.13: The stiff box used to mount the laser head in. The manipulator for α and β-axes are located on the back (not shown).
The rotation axes are along the elastic hinges at the front.
The intersection of these axes is located at the focal point
(fibres input face).
manipulators are turned back to prevent vibrations to influence the stability.
The rotation axes of the α and β-axes are as shown in Figure 3.13. The intersection of these axes is at the focal point.
3.3.5 γ-axis
The γ adjustment, as shown in Figure 3.14, is used for the rotation of the
fibre round the optical propagation axis. This rotation is needed to align the
fibres main axes to the laser heads main axes. This rotational alignment is
not relevant for the coupling efficiency. As mentioned before the rotational
alignment is of main importance for the achievable accuracy of the fibre fed
laser interferometer.
The coarse rotational alignment is done by rotating the fibre within its clamp.
The fine adjustment is done by rotating the clamp elastically.
3.4 Validation of the coupler resolutions and strokes
After the realization of the fibre coupler, the coupler was tested to see whether
the designed specifications were met. In this section measurement setups are
3.4 Validation of the coupler resolutions and strokes
45
2
2
3
1
1
2
4
3
1
5
3
a
b
c
Figure 3.14: a) Schematically the fibre clamp (2) with fibre (4) and the
cantilever (1) for the γ adjustment (front view).
b) Schematically the elastic guiding (3) of the γ adjustment.
In the final design the leaf springs are fold back resulting
in a more compact design (side view).
c) Photographs of the γ actuator. The fibre clamp is rotated
with a cantilever. The wheel is connected over a Cardan
coupling to the rod (5) to drive the cantilever.
46
Optical fibre coupling
shown together with measurement results to determine the resolutions and
strokes of the realised coupler. For adjustments performed with rotational
actuators, a resolution per 20◦ rotational adjustment is used.
3.4.1 X and Y-axes
The X and Y-axes are used to translate the focussed laser beam onto the fibre
core within the focal plane. The X and Y displacements are measured using a
capacitive probe. The X and Y-axes displacements were measured with respect
to the laser head. Therefore the capacitive probe was mounted onto the box in
which the laser head is mounted. The fibre collimator system was replaced by
an iron dummy. In this way directly the displacement between laser head and
fibre can be measured. The measurement setup is shown in Figure 3.15a.
3
2
4
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
0
2
4
6
8
10
Displacement in µm
1
5
a
20
10
15
Measurement nr.
b
25
Figure 3.15: a) Measurement setup for the validation of the x and y-axes
(4). The capacitive probe (3) is mounted onto the box (2)
where the laser head (1) is mounted in. The probe measures the displacement of an iron dummy mounted in the
fibre clamp (not visible), with respect to the box.
b) Results setup x and y-axes (fine). Every turn 4 measurements were done. Upper right corner gives position of the
coarse adjustment (in turns).
The results are shown in Figure 3.15b. As from Figure 3.15b can be seen,
the total displacement of the fine adjustment for the X and Y axes is 9 µm.
The resolution of this fine adjustment was 0,08 µm/20◦ , which is well below
specification of 0,1 µm/20◦ .
For the coarse adjustment a stroke of 0,37 mm was measured. Because the
X and Y coarse adjustments are the only actuators without stroke restriction,
no further tests were done as design specifications were reached. The coarse
resolution was 2,1 µm/20◦ , which was well below the required resolution of
3 µm/20◦ .
3.4 Validation of the coupler resolutions and strokes
47
3.4.2 Z-axis
The z-axis can be used to translate the fibres entrance side onto the focal plane.
The z-translation of the complete chuck, containing also the X and Y translations as well as the γ rotation, was measured using an inductive probe (Mahr
Millitron nr. 1301). The measurement setup is shown in Figure 3.16a.
3 4
2
5
450
Measured displacement in µm
1
a
fit
data
400
350
300
250
200
150
100
50
0
1
1,5 2 2,5 3 3,5 4 4,5
Actuator displacement in mm
b
5
Figure 3.16: a) A photograph of the measurement setup used to measure the z-axis adjustment. The inductive probe (1)
mounted on a clamp (2) measures the displacement of the
stage (3) where the fibre is mounted on. On the stage also
the X and Y manipulator (4) and γ manipulator (5) are
mounted.
b) Measurement result for the z-axis adjustment.
Results from the measurements of the z-translation are found in Figure 3.16b.
The stroke of the z-translation was as designed 0,4 mm, while the resolution
was 2,8 µm/20◦ which was well within design specification of 4 µm/20◦ .
3.4.3 α and β-axes
The α and β-axes are used to rotate the laser beam with respect to the fibres
main axes. The α and β rotations are measured with two inductive probes as
shown in Figure 3.17a. The probes were mounted onto the frame at an angle
of 45 degrees.
The measured stroke was larger than 6 mrad. As the maximum measurement
range of the inductive probe was reached and because this stroke already was
larger than the design specification, no further measurements were done. The
total stroke of the coupler is about 24,3 mrad (based on number of turns possible
with the actuator). The measured resolution of 0,089 mrad/20◦ was also within
design specifications of 1 mrad/20◦ .
48
Optical fibre coupling
7
Rotation in mrad
6
5
4
3
2
1
0
1
2
3
a
4
0
2
3
1
Turns of the actuator
b
4
Figure 3.17: a) Photograph of the measurement setup to measure the
strokes and resolutions of the α and β-axes. The inductive
probes (1) and (4) measured the displacement of the box
(3) with the laser head in respect to the base (2).
b) Measurement result from the α and β-axes. One measurement per rotation of the actuator was done.
3.4.4 γ-axis
The γ-axis is of no use for the coupling efficiency, but is of major importance
for the achievable measurement accuracy of the interferometer due to mixing
effects. The γ rotation was measured with the setup shown in Figure 3.18a.
In the fibre clamp again a dummy was placed. Onto the dummy a cantilever
was mounted. The rotation was measured using 2 inductive probes, one at the
center, the other at the end of the cantilever.
The fine adjustment has a measured stroke, see Figure 3.18b, of 0,044 rad (2,52◦ )
and a resolution of 0,3 mrad/20◦ which is well below design specification.
In Table 3.2 an overview of all strokes and resolutions are given. Resolutions are given per 20 degrees rotation of the adjustment except for the γ coarse
adjustment were no rotational actuator is used.
3.5 Achievable coupling efficiency
With the resolutions measured in the previous section a coupling efficiency is
estimated. As an integrated collimator lens is used, a core alignment error is
not caused by the coupler but due to core eccentricity, which has a specification
of < 1 µm. Because also other effects in the connector may contribute, a 1 µm
per axis core alignment error is assumed. A major error is due to numerical
aperture mismatch between lens and fibre. The numerical aperture of the 6 mm
lens used is 0,2 while the numerical aperture of the fibre is specified at 0,11.
3.5 Achievable coupling efficiency
49
Rotation in mrad
25
20
15
10
5
0
-5
-10
-15
-20
-25
0
1
2
3
4
a
1
2
3
5
4
Turns
6
7
8
b
Figure 3.18: a) Photograph of the measurement setup of the γ-axis
(fine). The inductive probes (2) and (4) are used to measure
the displacement of a cantilever (3) mounted into the fibre
clamp. On the photo also the Y-actuator (1) is shown.
b) Measurement results of the γ-axis (fine).
Axis
X and Y fine
X and Y coarse
Z
α and β
γ fine
γ coarse
Stroke
Designed Measured
> 5 µm
9 µm
> 0,4 mm
0,37 mm
> 0,4 mm
0,4 mm
> 5 mrad
24 mrad
> 40 mrad
44 mrad
> 20◦
40◦
Resolution /20◦
Designed Measured
< 0,1 µm
0,08 µm
< 3 µm
2,1 µm
< 4 µm
2,8 µm
< 1 mrad
89 µrad
< 1 mrad
0,3 mrad
< 2◦
1◦
Table 3.2: Resolution and strokes of the realised fibre manipulator, both
measured and designed.
50
Optical fibre coupling
From the 4,5 mm collimator lens, which is used in the coupler, no data (e.g.
focus distance) are available therefore the data of the 6 mm collimator lens are
used instead. The numerical aperture of the fibre has a large uncertainty due to
the large fibre mode diameter range (4 ± 1 µm). Resulting coupling efficiencies
due to this numerical aperture mismatch were estimated to be between 0,8 and
0,9 for each axis. This is by far the most important influence on the couplings
efficiency. Another influence which can not be altered is the insertion loss of the
connector. The couplings efficiency due to this insertion loss is 0,93. All errors
and their contribution to the couplings efficiency are presented in Table 3.3.
error
NA error lens x
NA error lens y
insertion loss
dz
dx fibre
dy fibre
dx lens
dy lens
fibre attenuation
reflection lens
dα
dβ
reflection fibre
effect
NA mismatch: fibre 0,11; lens 0,2
NA mismatch: fibre 0,11; lens 0,2
connector spec (0,3 dB)
25 µm
1 µm
1 µm
0,2 mm offset, incl. 4,5 mm beam
0,2 mm offset, incl. 4,5 mm beam
12 dB/km, l = 10 m
coating spec
89 µrad
89 µrad
back reflection spec (60 dB)
efficiency
0,82
0,82
0,93
0,93
0,94
0,94
0,97
0,97
0,97
0,997
1,00
1,00
1,00
Table 3.3: Coupling efficiencies for various error effects. Efficiencies are
calculated based on measurements at the fibre coupler and
specifications from manufacturers.
Using the efficiencies from Table 3.3, the achievable coupling efficiency would
be:
ηT = 0,822 · 0,932 · 0,942 · 0,973 · 0,997 ≈ 0,47
(3.12)
From experiments, using several different setups with different collimator
lenses, coupling efficiencies of 35% up to 66% were obtained. The higher
coupling efficiency indicates that the lens used in that experiment has a better
numerical aperture match with the fibre and/or better alignment in x, y and
z. This however could not be validated as the fibre lens assembly (with also a
4,5 mm collimator lens) was glued together by the manufacturer. As no lens
data were obtained from that manufacturer the data from the 6 mm lens were
used instead to estimate the coupling efficiency.
3.6 Other effects
51
3.6 Other effects
In the previous sections the coupling efficiencies and coupler adjustment resolutions were calculated. In this section other effects which could influence
coupling efficiency are discussed.
3.6.1 Pointing stability
One of the effects know from lasers, which could effect coupling efficiency, is
the pointing stability of the laser. This is the tilting of the laser output beam.
Tilting the laser output is equal to the α and β adjustment, and if this tilting
is comparable to these adjustments this can influence the coupling efficiency.
Any tilt of the beam will result in a displacement of the beam in the focal plane.
If in addition the effect would then also be unrepeatable, this would require
realignment of the fibre. If the pointing stability effects are small compared to
the azimuth adjustment, the effect on the coupling efficiency is also negligible.
During warm up the specification on the beam rotation of the laser head used
were < 2 0 (581 µrad) [Com92]. For the long turn the specifications on the beam
rotation of the laser head are < 5 00 (24 µrad).
If the laser head is within specification, only the beam rotation caused by the
pointing stability during warm up affects the coupling efficiency. The long
turn (after warm up) specifications is below 30% of the coupler adjustment
resolution, and are therefore assumed not to affect the coupling efficiency.
Measurements were done to verify the laser head specifications. This was done
by measuring the displacement of the beam at a distance of 14,6 m. During
warm up a pointing stability of 205 µrad was measured. The warm up pointing
stability was well below the specification. The long turn pointing stability measured for more than 8 hours and was smaller than 68,5 µrad. No smaller angles
could be measured due to the measurement setup. From this measurement it
is not clear whether the long turn specification is met. It is however clear that
the measured pointing stability is below the coupler adjustment resolution.
The transverse beam displacement due to the pointing stability also could effect
the coupling efficiency. However because the collimator lens is located near
the laser head, this transverse beam displacements is only of minor influence.
During alignment and use, no effects of the long turn pointing stability on the
coupling efficiency were observed. The effects during warm up are described
in the next section.
3.6.2 Laser warm-up
During the warm up period of the laser, due to pointing stability and rest effects
of the coupler construction, the output intensity is not optimal. After warm
up, the laser is without realignment, aligned for optimal output intensity. After
cooling down and repowering, no realignment is needed. Both construction
and pointing stability behave repeatable during warmup. However during
warm up (±1 − 2 hours) the coupling efficiency and thus the intensity after the
52
Optical fibre coupling
fibre is fluctuating.
3.7 Fibre beam splitter
For many servo controlled applications, where the laser interferometer is
used as a displacement measurement system, more than one axis have to be
measured. In conventional setups the beam is somewhere split with a nonpolarising beam splitter in order to make one beam for each axis. When using
fibres, either the beam can be split first, and then transmitted to each axis, or
all light is coupled into a fibre, which is later split using fibre beam splitters
[OZ 99a]. The use of these fibre beam splitters is outside the scope of this research, but the output of each fibre could be analysed with the tools presented
in Chapter 4 and can be validated with the methods presented in Chapter 5.
3.8 Summary
Rotational alignment is of major importance for the achievable accuracy of the
fibre fed heterodyne laser interferometer due to the mixing of the polarisation.
As this rotational alignment is so important, in this chapter the development
of a laser to fibre coupler is described. The coupler has 6 degrees of freedom,
uncoupled adjustments and a thermal centre at the focal plane where the fibre
input is located.
The output intensity of the fibre is important for measuring more axes with one
laser source. Therefore in this chapter effects influencing coupling efficiency
were discussed.
Based on beam overlap integral and a 90% efficiency criterium, alignment resolutions were calculated. Using this efficiency criterium, reflection losses and
fibre attenuation, a calculated coupling efficiency of 37% should be achievable.
Finally, based on the measured coupler resolutions, fibre and lens specifications, an overall coupling efficiency of 47% is calculated. From experiments,
coupling efficiencies between 35% and 66% were achieved. Differences found
experimentally are probably also due to the use of different collimator lenses.
This could not be validated as only from one lens specifications are available.
From the measurements at the fibre coupler resolutions it can be shown that the
numerical aperture mismatch between lens and fibre is the major influence affecting the coupling efficiency. This however can be improved by using a beam
reductor and a smaller collimator lenses, with a smaller numerical aperture.
With an optimal match of the numerical aperture, theoretically efficiencies up
to 70% could be achieved with the setup.
For the development of a future coupler, the rotational alignment could be
done using a λ/2 plate. The effects of the half wave plate on the mixing then
also have to be investigated. When using an integrated collimator lens, both x,
y and z axes can be omitted also.
Chapter 4
Characterization of
polarisation maintaining
fibres
4.1 Basic characteristics of PM fibres
As explained in Chapter 2, polarisation maintaining (PM) fibres are needed
to optimally transmit polarised light from the laser to the interferometer. It
is important to know how well a given input polarisation is transmitted by
the fibre. The extinction ratio is used to describe the quality of polarisation
maintaining of an input polarisation by the fibre. The extinction ratio is defined
as the portion of the light which is emitted by the fibre in the unwanted mode
(Punwanted ) divided by the power in the wanted mode (Pwanted ). The extinction
ratio is always measured as intensity ratio.
The extinction ratio (ER), often expressed in dB, is given by Equation 4.1.
Punwanted
ERdB = −10 10 log
(4.1)
Pwanted
For the use with heterodyne lasers the extinction ratio for both polarisations
must be measured. The extinction ratio is directly proportional to the amount
of mixing of the polarisations of the fibre output. As shown in Section 2.2.3
this mixing causes non-linearities in the interferometer and thus reduces the
displacement measurement accuracy of the fibre fed interferometer. A low
extinction ratio is thus needed for high measurement accuracies with the fibre
fed laser heterodyne interferometer.
The use of the fibre keying always causes a large deviation of the optimal
achievable extinction ratio. This is due to the production tolerances of the fibre
keying (±1◦ ), and the limited fibre connector assembly production tolerances
(±3◦ ). Resulting rotational alignment is, even with a perfect fibre, insufficient
53
54
Characterization of polarisation maintaining fibres
for a fibre fed heterodyne laser interferometer with nanometre accuracy. In the
following sections therefore the exact extinction ratio which is achievable after
careful alignment is determined using different measurement techniques.
4.2 Measurement of the polarisation state after the
fibre
As described in the previous section, the extinction ratio for both axes and thus
the output polarisation state of the fibre need to be measured. In this section
different measurement methods and procedures to align a fibre, using these
measurement methods, are described.
4.2.1 Measurement plan
In this section an overview of the measurement methods as well as the measurement strategy are given.
To measure the state of polarisation and the change in state of polarisation,
three different approaches have been applied. From these methods the mode
coupling or mixing is then calculated. The first two methods are based on AC
measurements and the last method is based on DC measurements. In the DCmethod stable light intensities are measured. In the AC-methods the intensity
is measured with the frequency difference ∆ f = f2 − f1 of the heterodyne laser
beam or beat frequency components (∆ f1 = fr − f1 , ∆ f2 = fr − f2 ) with an other
laser source used with frequency fr .
In the DC-method, the change of state of polarisation caused by the fibre was
measured, for each polarisation direction independently, applying the “principle of superposition” [Hec93]. The change of the polarisation state for the
(heterodyne) laser beam is calculated by adding the results for both polarisation
directions together. The AC measurements were mainly developed to verify
the results of the DC method. The AC measurements are also needed for the
precise alignment from the fibre with respect to the laser head for the validation
measurements as presented in Chapter 5. In addition these methods are used
to measure the ellipticity and non orthogonality of the laser head used.
At first, three different types of PM fibres are measured, to see if there are differences in extinction ratio between different fibre types: Panda type, Bow-tie
type and elliptical inner clad PM fibre (see Section 2.6).
Secondly, different lengths of one fibre type are measured to find any length
dependency on the extinction ratio. The length measured ranged from 3 to
50 metre.
Based on Equation 4.1 the extinction ratio or cross talk (mode coupling) of a
fibre for both the main axis are calculated by:
ER1 =
P1
P01
(4.2)
4.2 Measurement of the polarisation state after the fibre
ER2 =
P2
,
P02
55
(4.3)
where Pi are the main axes with the wanted power, and P0i are the leaked
intensities. Assuming independent contributions from the uncertainties in P1
and P01 , denoted by u(P1 ) and u(P01 ) respectively, the uncertainty in ER1 , u(ER1 )
is given by:
s
!2
!2
u(ER1 )
u(P1 )
u(P01 )
+
=
ER1
P1
P01
s
!2
!2
dER1
dER1
u(ER1 ) =
∗ δP21 +
∗ δP201
dP1
dP01
v
t
2

 P1 
1 2
2

∗ δP1 +  2  ∗ δP201
u(ER1 ) =
P01
P01
(4.4)
(4.5)
(4.6)
Using a heterodyne source, due to the elliptical output the subscripts i are used
for the axis of the main component of the frequency fi and the minor components are described by the 0i indices. The used indices are thus equivalent to
the ones used in Chapter 2.
The orientations of the polariser or analyser with respect to the reference coordinate system at which the extinction ratio is measured is called αp respectively
αa .
4.2.2 AC-methods
Two AC- methods were developed [Lor02, LKC+ 03] to verify the results of
the DC method (Section 4.2.3). In addition these methods were used to see if
the assumption of measuring both polarisations independently is also valid if
both modes are used simultaneously. The developed methods are also used to
characterise the laser head used. In addition these techniques can be used to
measure the state of polarisation of any beam in the interferometer. The AC
method described secondly is, in a modified setup, also used to align the fibre
with respect to the laser head to do the validation measurements as presented
in Chapter 5.
In the first method the components of the laser beam under investigation are
beat with an external reference. In the second method the two laser frequencies
are beat against each other.
Method 1: Carrier frequency method
In the first method, a circularly polarised reference source is mixed with the
heterodyne laser beam emerging from the fibre. With a polariser the resulting
beat frequencies are analysed using an avalanche photo detector. By using a
spectrum analyser two beat signals of the lower and higher laser frequency are
56
Characterization of polarisation maintaining fibres
Heterodyne
Laser
recorded, for various polariser angles. The measurement setup is schematically
shown in Figure 4.1.
Fibre
Lens
Circular reference
source
Non-polarising
Beam reductor
beam splitter
avalanche
photo
detector
Polariser
Spectrum
analyser
Figure 4.1: Setup used in the 1st AC method. The beam to be measured
is mixed with a circular reference source, and with the use
of an avalanche photo detector, for various polariser angles
the beat signals is measured using a spectrum analyser.
The four different E-field components of the two frequencies are measured, with
respect to the reference E-field. From these four measurements, the amplitudes
per frequency at different angles were used to calculate both beam ellipticity
and beam non-orthogonality.
By using the Jones-formalism this method is mathematically described by:
~ out = PR(αp )E
~ in
E
(4.7)
Where the Jones-matrix of the polariser P is given by:
"
P=
1
0
0
0
#
(4.8)
R is the rotation matrix as given in Equation 2.25. The definition of the E-fields
is given in Figure 4.2. The Jones vector of the light emerging from the fibre
~ in , thus:
(E f ibre ) and the circular reference source (Ecrs ) are together E
~ in = E
~ crs + E
~ f ibre
E
(4.9)
where
"
~ crs = Ecrs
E
eiϕ
ei(ϕ+π/2)
#
(4.10)
4.2 Measurement of the polarisation state after the fibre
57
x
Polariser
~ crs
E
~1
E
αP
~ 02
E
~2
E
ε
~ 01
E
y
Figure 4.2: Definition of the E-fields used in method 1. The circular
~ crs , while the polariser is at an angle αp .
reference source is E
~1, E
~ 01 , E
~ 2 and E
~ 02 represent the light emerging from
The E
the fibre, with non-orthogonality .
with ϕ = fcrs t, and
~ f ibre = E
~ 1 + R(π/2 − )E
~2
E
(4.11)
while
"
~i =
E
Ei eiδi
E0i ei(δi +π/2)
#
,i = 1,2
(4.12)
with δi = fi t.
The resulting beat frequency components from the light emerging from the
fibre with the circular reference source are, as shown in Appendix G, given by
IAC = IAC f 1 + IAC f 2 . With:
IAC f 1
=
1/2 cos2 (αp )Ecrs E1 cos(ϕ − δ1 ) +
1/2 sin2 (αp )EcrsE01 cos(ϕ − δ1 ) +
1/2 sin(αp ) cos(αp )Ecrs E01 sin(ϕ − δ1 ) −
IAC f 2
=
1/2 sin(αp ) cos(αp )Ecrs E1 sin(ϕ − δ1 )
1/2 cos(αp )EcrsE02 sin(ϕ − δ2 ) cos(αp + ) +
(4.13)
1/2 sin(αp )Ecrs E02 cos(ϕ − δ2 ) cos(αp + )
(4.14)
1/2 cos(αp )EcrsE2 cos(ϕ − δ2 ) sin(αp + ) −
1/2 sin(αp )Ecrs E2 sin(ϕ − δ2 ) sin(αp + ) +
58
Characterization of polarisation maintaining fibres
The resulting minimum and maximum beat signals amplitudes are, when assuming << 1 thus sin() ≈ 0 and cos() ≈ 1:
IAC f 1,αp=0
IAC f 1,αp=π/2
IAC f 2,αp=−
IAC f 2,αp=π/2−
=
=
1/2EcrsE1
1/2EcrsE01
(4.15)
(4.16)
=
=
1/2EcrsE02
1/2EcrsE2
(4.17)
(4.18)
The extinction ratios are thus:
IAC f 1,αp=π/2
!2
Ecrs E01 2
E01 2
=
IAC f 1,αp=0
EcrsE1
E1
!2 2
IAC f 2,αp=−
E02 2
Ecrs E02
ER2 =
=
=
IAC f 2,αp =π/2−
EcrsE2
E2
ER1 =
=
(4.19)
(4.20)
Because the four different E-fields of the fibre output are measured with respect to the reference E-field, this requires a stable reference E-field. Therefore
the non-uniformity of the circular reference was measured, by measuring the
ellipticity of the intensity profile. In the processing of the results, for this nonuniformity was compensated.
Disadvantages of this method are the small detector surface, which requires the
addition of an extra lens, and the unknown detector and amplifier non-linearity,
but these are constant effects. Also the relative drifts of the frequencies of the
sources used caused problems. In addition only the combined effects of laser
head and fibre are measured. However the effects of the laser head showed
to be small. The main advantages of this method are the high extinction ratio
measurement possible and the ability to measure also after polarising components, e.g. a polarising beam splitter.
Because of the relatively low frequency stability of the laser sources used, compared to the frequency difference of the heterodyne laser head, the extinction
ratio was compared for several frequency intervals, after filtering erroneous
data caused by this large frequency shifting. Refer to [Lor02] for more information.
Due to the large number of measurements, the polarisation state of the beam
to be analysed needs to be stable in time. As measurements take time and the
fibres output polarisation which is varying due to phase changes within the
fibre caused by change in birefringence, the fibre output is by no means stable
for the time the measurements would last. Therefore the same warming up
and cooling down procedure of the fibre is applied as in detail will be described
in Section 4.2.4, in order to see the complete change of polarisation. During
the complete change of polarisation, the change in beat signal is observed. For
each polariser angle then the minimum and maximum E-fields are calculated
and with the use of these E-fields, the extinction ratios were calculated. Because both a minimum and maximum E-field were recorded, both an upper
and lower extinction ratio are found.
4.2 Measurement of the polarisation state after the fibre
59
The experiments done were carried out under ideal input alignment of the fibre. The fibre tested with this method is the Point-Source fibre, with integrated
lenses.
The main reason why this method was not used more often, is that the DCmethod is less complicated. With the DC-method the fibre can be measured
independently, without the influence of the laser head e.g. the mixing due to
the non-orthogonality of the laser head.
Method2: Direct beat measurement
In the second method, the beat signal was obtained from the two frequencies
of the heterodyne laser itself. In this method the beat signals are measured by
rotating a polariser in front of the avalanche photo detector. The setup as used
is shown in Figure 4.3. Using the same procedure with the Jones formalism as
Heterodyne
Laser
Fibre
Avalanche
photo
detector
Polariser
Spectrum
analyser
Figure 4.3: Setup used in the 2nd AC method. The beat signals are
measured with an avalanche photo detector for various polariser angles using a spectrum analyser.
for method 1, the E-field after the polariser is described by:
~ out = PR(αp )E
~ f ibre
E
(4.21)
Where P is the Jones matrix for the polariser which is given by Equation 4.8, R
~ f ibre the Jones vector of the
is the rotation matrix as given in Equation 2.25 and E
light emerging from the fibre as is described by Equation 4.11. In Appendix G
it is shown that the resulting beat signals are:
IAC
=
1/2E1 E02 sin(δ1 − δ2 ) cos(αp ) cos(αp + ) +
1/2E1 E2 cos(δ1 − δ2 ) cos(αp ) sin(αp + ) −
1/2E01 E2 sin(δ1 − δ2 ) sin(αp ) sin(αp + ) −
1/2E01 E02 cos(δ1 − δ2 ) sin(αp ) cos(αp + )
(4.22)
The minimum and maximum beat signal amplitudes are, when assuming
E01 E02 ≈ 0 and << 1 thus sin() ≈ 0 and cos() ≈ 1:
IAC,αp=0
IAC,αp=π/4−/2
=
=
1/2E1E02
1/4E1E2
(4.23)
(4.24)
IAC,αp=π/2−
=
1/2E01 E2
(4.25)
60
Characterization of polarisation maintaining fibres
The extinction ratios are then:
!2
E01 2
E01 E2 2
=
= 2
IAC,αp=π/4−/2
1/2E1E2
E1
!2 2
IAC,αp=0
E02 2
E1 E02
ER2 = 2
=
= 2
IAC,αp=π/4−/2
1/2E1E2
E2
ER1 = 2
IAC,αp=π/2−
(4.26)
(4.27)
The maximum error due to the simplifications is smaller than 1% [Lor02]. With
this method the E02 E1 and E2 E01 beat signals were measured for a polariser angle of nominal 0 degrees and 90 degrees. Together with these signals when the
polariser is oriented at 45 degrees the main beat E1 E2 was measured to compensate for unequal intensities of the two main polarisations. This requires
only 3 polariser angles whereas for the previous method 4 measurements were
needed. The second advantage is the use of an optimal designed detector
(the detector from the laser interferometer system), special for this frequency
difference, with integrated optics. The used detector was different from the
one used in method 1, which was a general purpose avalanche photo detector.
From th detector/amplifier used in this method also no linearity information
was available. With this second method both the extinction ratio of the optimal aligned Point-Source fibre and the laser head used for the interferometric
measurements were measured.
4.2.3 DC-method
A relatively ’simple’ setup compared to the AC-methods, would be measuring
only intensities. Therefore in this section a method based on intensity measurements to measure the change in state of polarisation caused by the fibre is
described. First the measurement setup used to measure the change in state of
polarisation will be explained in detail.
In this setup a polariser is used after the light source, to make a well defined
state of polarisation, with a known polarisation (direction) output. From this
linearly polarised light the changes, caused by the fibre are measured. Due
to misalignment and mode leaking both modes carry light. By applying temperature (stress) changes, due to the fibres birefringence, the phase difference
between the two orthogonal modes will change accordingly. The output polarisation state will therefore be gradually change from linear via elliptical
back to linear. By adding a second polariser, called analyser, a linear part is
blocked, and changes in intensity of the transmission axis are measured with a
power-meter. The complete measurement setup will now be described in more
detail.
DC-Measurement setup
For the measurement of the polarisation state change in the fibre, e.g. by bending or temperature change, the measurement setup as schematically given in
Figure 4.4) was used. As light source a stabilised HeNe laser was used. For
4.2 Measurement of the polarisation state after the fibre
61
Disturbance/environment
Circular source
Fibre
Polariser coupler
Fibre
coupler Analyser
0.00
Detector
0.00
Fibre
Figure 4.4: Schematical diagram of the measurement setup for measuring the change in polarisation state due to fibre effects.
unambiguous definition of the state of polarisation a Glan-Thompson [PP93]
polariser was used, which can be rotated around its optical axis, in order to
make a -highly- linearly polarised beam at any desired angle.
To prevent amplitude change during rotation, the output of the laser source
must be circular. This was achieved by using either a laser with circular output polarisation or by adding a quarter wave plate, oriented at an angle of
45 degrees to make the output circular. The light is circularly polarised now
and the intensity is independent of the polarisers orientation. Because only
small deviations around the optimal alignment are used, deviations due to not
perfectly circular polarisation may be neglected.
Using the Jones formalism the output E-field of the DC measurement setup can
be described by:
~ out
E
~ crs
= A · R(αa ) · H f ibre · R(−αp ) · P · R(αp ) · E
~ crs
= P · R(αa + π/2) · H f ibre · R(−αp ) · P · R(αp ) · E
(4.28)
(4.29)
Where P is the Jones matrix for the polariser and is given by Equation 4.8. A
is the Jones matrix for the analyser and R is the rotation matrix as given in
Equation 2.25 and Ecrs is the Jones vector of the circular light source described
by Equation 4.10. For now the Jones matrix of the fibre just is described as a
linear birefringent retarder:
#
" iφ
e x
0
(4.30)
H f ibre =
0 eiφy
In Section 4.4 the fibre model will be discussed in detail. In that section also
effects like polarisation mixing within the fibre are discussed.
After the fibre, a second, pivoted, Glan-Thompson polariser was placed, as an
analyser. By setting their mutual orientation perpendicular (αa + αp = π/2),
ideally all light should be blocked. To detect the light parallel polarised along
to the analyser transmission axis an intensity detector is used. A photograph
of the setup is shown in Figure 4.5. Before measurements were done first the
system itself was carefully examined. The complete system, without a fibre,
62
Characterization of polarisation maintaining fibres
had an extinction ratio of 1:200.000. This means that the measurement system
is able to measure changes in the intensity ratio down to 5 ppm.
1
2
3
4
5
6
7
8
Figure 4.5: Photograph of the DC- measurement setup. Linearly polarised light is obtained by placing a polariser (2) in front of
the circular source (1). The light is transmitted by a fibre (5)
through an other polarised called analyser (7) to the detector (8). After the collimator lens (3,6) an extra fixation (4) is
used to prevent intensity changes due to bending near the
connector.
When the fibre is added, the fibre is nominally (at sight) aligned with the
horizontal and vertical axes, by using the fibre keying. Now the main axes
of the laser, the polariser axes and fibres main axes are all in the same plane
within a few degrees. First the orientation of these axes to each other must be
determined. This is done with the use of the alignment procedure described in
the next paragraph.
4.2.4 Axes alignment procedure
To measure the extinction ratio of a fibre, first the main axes of the fibre must
be found precisely. In this section the procedure to find the main axes is described.
First the analyser axis is set to a(n arbitrary) position around the nominal
position (visual), while the polariser is rotated (about ±5 degrees) around its
nominal position. For each polariser orientation, the fibre is heated and cooled
down, resulting in phase changes between the two orthogonal principal axes of
the fibre. Thus at a given polariser and analyser angle, the state of polarisation
changes from linear to elliptical and back to linear (see also Figure 4.14) all the
time, due to the phase shifts introduced by the heating (and cooling) of the fibre.
When the output polarisation is linear, the intensity on the detector is minimal
(with the analyser’s transmission axis perpendicular to this linear axis). On
the other hand, the circular state of polarisation results in a maximal intensity.
If the intensity signal versus temperature (or time) is recorded a sinusoidal
relation is found, as plotted in Figure 4.6. For clarity, the difference between the
found maximum and minimum, this is the amplitude of the intensity change,
can be plotted. The minimum (marked with x in Figure 4.6) and the maximum
(marked with o in Figure 4.6) intensity are recorded for each polariser angle
4.2 Measurement of the polarisation state after the fibre
63
0,48
0,46
Intensity in µW
0,44
0,42
0,4
0,38
0,36
0,34
0,32
25
30
35
40
Time in sec
45
Figure 4.6: The intensity after the analyser, while the fibre is cooling down. The minimal (x) and maximal intensity (o) is
recorded, the polariser and analyser angle are fixed during
the experiment.
and plotted versus the polarisers orientation. An example of such a measurement result is found in Figure 4.7. The minimum of the maximal intensity
identifies the correct alignment between fibre input and polariser. Therefore
this orientation also defines the fibres main axis at the input side.
Now the polariser is altered to this orientation and fixed, and the orientation
of the analyser is changed, following the same procedure as for the polariser.
An example of a result for the rotation of the analyser is given Figure 4.8. The
minimum of the maximal intensity again is the optimal alignment between fibre output and analyser. Clearly the typical behaviour of the minimal intensity
is seen. In Section 4.3 this typical intensity profile is discussed in detail.
Both polariser and analyser are now aligned optimal. The intensity measured
at this optimal alignment gives the minimal polarisation mixing (or polarisation leaking) possible for this fibre. The fibres main axis, both for input and
output, are known with respect to the analyser and polarisers reference coordinate system. The same procedure can be repeated for the other main axis of
the fibre. In this way any non-orthogonality of both axes is measured.
Optimal axes orientation for alignment
In the previous section the fibres main axes were determined. In this section
an alignment strategy is discussed to start the alignment. The best (starting)
orientations for the polariser/analyser to identify their orientations respectively
to the fibres main axes are determined here.
First the orientation of the fibres main axis was determined where an optimal
64
Characterization of polarisation maintaining fibres
Intensities Imin and Imax in µW
4
3,5
3
2,5
2
1,5
1
0,5
0
-6
-4
-2
0
2
6
8
4
Polariser angle in degrees
10
12
Figure 4.7: The minimal (x) and maximal (o) intensity recorded after
the analyser while the fibre is cooling down to be used for
alignment of the polarisers main axes. The minimum of the
maximal intensity defines the fibres main axes at the input
side of the fibre. Results are from a different experiment as
shown in Figure 4.6.
Intensities Imin and Imax in µW
4,5
4
a
b
c
3,5
3
2,5
2
1,5
1
0,5
0
82
84
86
96
88
90
92
94
Analyser angle in degrees
98
100
Figure 4.8: The minimum (x) and maximum (o) intensity after analyser, while the fibre is cooling down, for alignment of the
analysers main axes. The typical minimum intensity profile
will be explained in detail in Section 4.3. The three lines
indicated by a, b, and c are the orientations used in the
experiments of Section 4.2.4.
4.2 Measurement of the polarisation state after the fibre
65
Intensity in µW
Intensity in µW Intensity in µW
alignment was made, using the method as was described in the previous paragraph. This orientation is indicated by the dashed line a in Figure 4.8.
Then deliberately the orientation of the analyser was altered by respectively 4
degrees (now the fibre is aligned along in the global minimum of the minimal
intensity, as indicated by the dashed line b in Figure 4.8) and 8 degrees (now the
fibre is presumed to be unaligned as shown by the dashed line c in Figure 4.8).
Note that the fibre keying is done within 3 degrees of accuracy. Therefore it is
always possible to align the fibre within 3 degrees of the optimal alignment.
Then for each of the three analyser orientations the optimal polariser alignment was measured using the method described in the previous section. The
result of these three measurements are found in Figure 4.9. For the latter (8
degrees rotation) the alignment turned out to be harder than with the nominal,
while the curves are flatter. For the one with the analyser 4 degrees rotation
1
Analyser aligned
0,8
0,6
0,4
0,2
-9
-8
-7
3
2
a
-5
-6
-3
-4
Polariser angle in degrees
-2
-1
0
Analyser: 4 degrees misaligned (global minimum)
b
1
0
-9
-8
-7
6
4
-6
-5
-3
-4
Polariser angle in degrees
-2
-1
0
Analyser unaligned
c
2
0
-9
-8
-7
-6
-5
-3
-4
Polariser angle in degrees
-2
-1
0
Figure 4.9: The minimum and maximum intensity after the analyser,
while the fibre is cooling down. The analyser is oriented as
indicated by the dashed lines in Figure 4.8 thus aligned (a),
misaligned (b), an aligned along 4 degrees(c).
misaligned, the alignment is found even more difficult while, the lower curve
was around zero, while the upper curve again was not as steep as it was for
the well aligned analyser. At optimal analyser alignment both minimum and
maximum are relatively most curved, making alignment more easy.
No measurements were done with different polarisers angles, while the nominal polariser angle can be identified from the measurements at a given analyser
angle and then can be used to align the analyser.
If alignment is difficult, the alignment should be done iteratively, so the analyser becomes more and more aligned, and thus makes it easier to align.
66
Characterization of polarisation maintaining fibres
4.2.5 Bending versus temperature change
In Section 2.5.1 already was stated that effects causing mode coupling (e.g.
bending) are most effective if applied at an angle of 45 degrees with respect to
the fibres main axis. Because the angle at which the perturbation is applied
is not known, it is also not determined what disturbance of the fibre is made.
The temperature change however can be made easily and repeatably. If the
mixing values found for temperature changes are a good approximation for
characterising the complete fibre, no other tests for disturbing the fibre (e.g. by
bending) are needed.
The following experiment is therefore conducted: first the fibre is placed at rest
(at standard laboratory conditions), then the fibre is moved and bent extremely.
Finally the fibre is heated, and cooled down again. Results of this experiment
are found in Figure 4.10. A total of around 45 sec is recorded, from the intensity
a
b
c
d
0,48
0,46
Intensity in µW
0,44
0,42
0,4
0,38
0,36
0,34
0,32
5
10
15
20
25
30
Time in sec
35
40
45
Figure 4.10: Intensity after analyser, while the fibre is held still (a), bend
extremely (b) and the fibre was heated (c) and is cooling
down (d).
detector output onto a scope. Clearly can be seen the fibre at rest, till 10 sec
(section a of Figure 4.10) and the time moving the fibre between 10 − 15 sec
(section b of Figure 4.10). The cooling down of the fibre is found from 25 sec
and upwards and is indicated by section d of Figure 4.10. The heating of the
fibre between 17 − 22 sec is not clearly visible (section c of Figure 4.10).
Because the minimum and maximum intensity for both bending and temperature change are the same, is it clear that the temperature change is a good
approximation for characterising the fibre.
A note is made to the first part of the figure where the initial mixing is seen.
This initial mixing can have any value between the maximum and the minimum intensity. The intensity need by no means, to be around the mean. This
4.2 Measurement of the polarisation state after the fibre
67
is due to the phase difference between the two main axes.
The radius of curvature of the fibre may not be small otherwise the output intensity reduces due to leaking into the cladding. The used radius of curvature
of 100 mm and more are adequately predicted by temperature changes.
4.2.6 Axes orthogonality
Intensity in nW
To prevent mixing in the interferometer and thus reducing displacement measurement accuracy of the laser interferometer, the output polarisations of the
fibre must be orthogonal. Otherwise the two frequencies (polarisations) can not
be split within the beam splitter and mix. Therefore the orthogonality of the polarisation output was measured. This was done by determining both the fibres
main axes (and their extinction ratios) with respect to the reference coordinate
system. Results from measurements done both for 0◦ and 90◦ are shown in
Figure 4.11. From our experiments no deviation of the orthogonality from 90◦
300
200
100
Intensity in nW
0
-2
-1
0
2
3
1
Orientation analyser in degrees
4
5
89
90
92
93
91
Orientation analyser in degrees
94
95
300
200
100
0
88
Figure 4.11: Results from the fibres axes orthogonality measurement.
The intensity after the analyser for both fibres main axes
is plotted respectively in upper and lower graph. The fibres main axes are perpendicular within our measurement
resolution.
could be measured and the orthogonality was within the measurement resolution of 0,1◦ . From results after fitting (see Section 4.3) even a non-orthogonality
() of less than 0,06◦ was obtained. Both are in good agreement of the expected
and reported [VR99] non-orthogonality. In [VR99] the reported orthogonality
of the output was within the measurement error of ±0,5◦ . The output state of
polarisation is thus within our measurement accuracy perpendicular. Theoretically the two main axes, both carrying their own frequency, are perpendicular.
Because the fibres principal axis are defined by the rotational symmetry of the
geometry of the fibre.
68
Characterization of polarisation maintaining fibres
4.2.7 AC and DC Measurement results compared
To verify the DC-method, the measurement results of the three methods are
compared. In addition it can be seen if the assumption of measuring both
polarisations independently is valid.
The results from the measurements for the Point-Source fibre of both ACmethods, as well as the result of the DC measuring method are given in Table 4.1. From these results it can be seen that all measurements are in the same
Method
DC (0◦ )
DC (90◦ )
AC-1, f1
AC-1, f2
AC-2, f1
AC-2, f2
Extinction ratio ± u(ER)
(worse)
1:(93 ± 7)
1:(120 ± 9)
1:(97 ± 18)
1:(121 ± 22)
1:(98 ± 13)
1:(110 ± 13)
orthogonality error
(in degrees)
0
1
0
Table 4.1: Results from measurements of the extinction ratio of the
Point-Source panda type for various measurement methods.
From the results it can be seen that these are all in good
agreement. The effects on the displacement uncertainty are
calculated in Chapter 5.
range, the only difference is the measured non-orthogonality with the use of
method 1, which was is probably due to a measurement error, as both other
methods measured the expected orthogonality. Also a rotational misalignment
of the non-polarising beam splitter, which influence only one of the two polarisation directions, could be responsible for this difference.
As shown, the measurements for all three methods are in good agreement.
Because the DC measurements are most easy and capable of measuring the
effects of the fibre without the influence of the laser head used, in the following
sections this method will be used. The effects of this mixing on the achievable
interferometric displacement uncertainty is calculated in Section 5.2.3.
4.2.8 Measurement results for different fibre types
As a high extinction ratio would result in high measurement accuracy of the
fibre fed laser interferometer, first different PM fibre types were tested to see if
there are differences between different types as stated in Section 4.2.1. For each
fibre the polarisation mixing (extinction ratio) is determined by dividing the
leak intensity (this is the maximum intensity while the variation is minimal,
e.g. ±0,7 µW at ±89,5◦ in Figure 4.8) by the maximum intensity. The maximal
intensity is found by rotating the analyser 90 degrees with respect to the orientation of the leak intensity. The extinction ratio is thus the ratio between the
intensities transmitted as plotted in Figure 4.14b and the maximum intensity.
4.2 Measurement of the polarisation state after the fibre
Fibre type
Panda (0◦ )
Panda (90◦ )
Bow-tie (0◦ )
Bow-tie (90◦ )
Elliptical inner clad (0◦ )
Elliptical inner clad (90◦ )
Extinction ratio ± u(ER)
(worse)
1:(94 ± 7)
1:(120 ± 9)
1:(129 ± 9)
1:(125 ± 9)
1:(103 ± 7)
1:(104 ± 7)
69
orthogonality error
(in degrees)
0
0
0
Table 4.2: Results of extinction measurements for various fibre types. The ratio
is given for the maximum intensity of the coupled mode. The effects
on the displacement uncertainty is calculated in chapter 5.
Results from several different tested fibres are found in Table 4.2.
From these results it can be seen that the polarisation mixing is less than 1 %,
which is standard for all manufactures, who guarantee at least an extinction
ratio of −20 dB. The relatively large difference between the two axes in the
Panda type fibre is probably due to effects within the (glued) fibre collimator
system, which was used with this fibre only. Using these results the maximal
achievable uncertainty will be calculated in Section 5.2.3.
4.2.9 Special selected fibre
The fibres in the previous section have extinction ratios around 1:100. From
measurements presented in Chapter 5, the maximal achievable displacement
uncertainty with heterodyne laser interferometry with such fibres is not at the
one nanometre level, but is up to five nanometers uncertainty. In order to
make a fibre fed laser interferometer with one nanometer uncertainty, a better
polarisation maintaining fibre is needed. After intensive search, a company
was found who could supply PM-fibres with better specifications. In the next
section this fibre is examined, to see if the fibre shows better performance.
If the measured fibre is more suitable, this would not only be measured in a
higher extinction ratio, but also the measured typical intensity profile as shown
in Figure 4.8 would change.
For a fibre with a higher extinction ratio the intensity variation at optimal
alignment would be smaller. In addition the angle between the global minimum
of the minimum intensity with respect to the optimal alignment would also be
smaller (this is the angle marked between the dashed lines a and b in this
figure). For more information refer to Section 4.3 and Figure 4.14.
70
Characterization of polarisation maintaining fibres
Measurement results
With the same setup as described in Section 4.2.3 the polariser in front of
the special selected fibre was aligned using the procedure described in that
section. In Figure 4.12 only the alignment of the analyser is shown. The
Intensity after analyser in nW
250
200
150
100
50
0
111
112
113 114 115 116 117
Analyser angle in degrees
118
119
Figure 4.12: The minimal and maximal intensity after the analyser, results from measurements for the special selected fibre. The
changes in the characteristic profile compared to the measurements of Figure 4.8 are as predicted are found. The
amplitude difference at optimal alignment is smaller and
the angle between the optimal alignment and the global
minimum is also smaller.
special selected fibres minimal intensity divided by the maximal (polariser 90
degrees rotated) is about 1:850. This will reduce measurement uncertainty of
the fibre fed heterodyne laser interferometer, compared to the fibres measured
in the previous section. As already predicted, the intensity change at optimal
alignment is smaller. Also the angle between the optimal alignment and the
absolute minimum intensity is smaller.
4.2.10 Length dependence
The second influence to be investigated according to Section 4.2.1 is the length
dependency on the extinction ratio. For several applications the goal of the
fibre fed heterodyne laser interferometer is to position the laser head outside
the machine. The laser head e.g. is a heat-source, and positioning the laser head
outside the machine would improve the thermo-mechanical stability of the
machine. In this section therefore length dependency of the fibre is investigated.
To see the influence of fibre length, several different length of fibre where
4.2 Measurement of the polarisation state after the fibre
71
measured. Length where varied between 3 and 50 metre. Again the same
experiments as described in Section 4.2.3 where done for all lengths. The
measurement result for the 50 metre fibre is given in Figure 4.13. Clearly the
0,25
Imin and Imax in µW
0,2
0,15
0,1
0,05
0
72
73
74
75
76
77
78
79
Analyser angle in degrees
80
81
Figure 4.13: The minimal and maximal intensity after the analyser, results from the 50 metre fibre.
lack of the two global minima is seen. This is one of the aspects of increasing
the fibre length, due to the unpredictable phase of the mode leaking process.
Results from these experiments are summarised in Table 4.3.
By increasing the fibre length, due to the relative large attenuation (12 dB/km)
of the PM fibres, the output intensity will reduce.
Fibre length
3m
5m
15 m
50 m
Extinction ratio ± u(ER)
(worse)
1:(900 ± 27)
1:(844 ± 17)
1:(563 ± 16)
1:(1642 ± 137)
Table 4.3: Extinction ration measured for different fibre length. From
the measurements no length dependence is found. As will be
shown in Section 4.2.11, the connector influence is probably
the dominating factor.
72
Characterization of polarisation maintaining fibres
4.2.11 Fibre connector replacement
To verify indications that, especially for short fibres, not the fibre but the fibre
connectors are responsible for the overall measured fibre quality in terms of
extinction ratio, the connectors of the 3 meter fibre of the previous paragraph
were replaced by the manufacturer. The fibres extinction ratio again was measured with the procedure as described in Section 4.2.4, and the extinction ratio
was changed from 1:900 to 1:300, thereby clearly showing the influence of the
fibres connectors.
It showed that the connectors are of major importance. If the measured extinction ratio is low, the extinction ratio could be improved by replacing the
connectors.
The influence of the connectors may be explained by carefully examining the
production. In the production the cleaved fibre is inserted in a ceramic tube
and both are glued together. The diameter tolerances between fibre and ceramics allow the glue to vary in thickness around the fibre. While hardening
the fibre may be deformed. This deformation can reduce the internal birefringence, especially if applied at (or near) 45 degrees. For more information refer
to Appendix H.
By using the measurement setup as described in Section 4.2.3, fibre selection
on polarisation maintaining quality is therefore possible.
4.3 Discussion of measurement results
In this section a simple model is presented, which describes the measurement
results found. This model is also well suited for determining the orientation of
the fibres main axes. In Section 4.4 this model is extended to a complete Jones
model of the fibre to give a better (physical) understanding of the fibre.
As can be seen from Figure 4.8, the minimum intensity at the optimal alignment is higher, compared to the intensity at a deviated angle (around 4◦ ). This
is explained by mode leaking. At the fibres entrance linearly polarised light
is coupled into the fibre at their main axes. Due to fibre imperfections, some
polarised light will leak to the opposite main fibre axis (polarisation direction).
Thus if light is coupled in only one main axis, at the output the other axis
always will guide some (unwanted) power.
Using standard (commercially available) fibres, typically the mode leaking is
smaller than 1 % (see previous sections). At the analyser the resulting E-field
will depend on the mutual phase(relation) of the two axes resulting in linearly
or elliptically polarised light. When both modes are in phase, and the output is
therefore linearly polarised, the intensity after the analyser can be zero. While
the analysers transmission axis is aligned perpendicular to the resulting E-fields
(these are the axes found by adding the two E-field together, with 0◦ or 180◦
phase shift), zero or minimum intensity is found, see Figure 4.14a respectively
4.14c. The axis found is not one of the fibres main axes but the axis depending
on the resulting E-field caused by the mixing. While the analyser is aligned
perpendicular to the fibres main axis, the leaking component is, independently
4.3 Discussion of measurement results
a
73
b
c
Analyser absorption axis
αa,a
αa,b
E1
αa,c
E1
E1
dα
dα
E01
E01
E01
Figure 4.14: Local maximum, and global minimum explained from Figure 4.8: a) and c) With 0◦ (or 180◦ ) phase shift the linearly
polarised light can be blocked by the analyser. b) the analyser is perpendicular to the fibres main axis: the leaking
(E01 ) is never blocked. αa is the absolute orientation of the
analyser, and dα the relative angle between position a) and
b).
of its phase, always recorded see Figure 4.14b. At this axis, ideally the resulting
minimum and maximum intensity would be equal.
When the analyser is aligned along the resulting linear E-field (around ±4◦
with respect to the optical alignment thus at point 2 of Figure 4.15), the contrast
of the interference signal can be 100 %. While the contrast is 100 %, the projected
E-fields (noted with E∗ ) are the same in magnitude or:
E∗1
=
E1 sin(dα) =
E∗01
(4.31)
E01 cos(dα)
(4.32)
Where E1 is the wanted and E01 is the unwanted mode and dα the angle of the
analyser in respect to optimal alignment. When αa is the absolute orientation (as
defined in Figure 4.14) of the analyser in respect with the reference coordinate
system, dα = αa,b − αa,a . At this point complete constructive and destructive
interference take place.
The maximal intensity measured at this analyser angle is thus:
Imax,contrast=1 = (E1 sin(dα) + E01 cos(dα))2 = (2E1 sin(dα))2 = (2E01 cos(dα))2(4.33)
The intensity change at optimal alignment can be explained by the leaking of
0
the leaked signal back to the original mode, but out of phase called Ii . The
74
Characterization of polarisation maintaining fibres
intensities calculated with the model are:
p
2
 Imax,contrast=1 

I1 = 
2 sin(dα) 
2
p
 Imax,contrast=1 
I01 = 

2 cos(dα) 
0
I1 =
I01
I01
I1
(4.34)
(4.35)
(4.36)
The minimum (Imin ) and maximum (Imax ) intensity are then given by:
2
0
Imin = E1 sin(φ) + E1 cos(φ) + E01 cos(φ)
2
0
Imax = E1 sin(φ) + E1 cos(φ) − E01 cos(φ)
(4.37)
(4.38)
Where φ the phase of the E-field as given by their corresponding intensities
in Equation 4.34-4.36. This model is implemented, by using the measurement
points that are automatically found by the simulation program. An example
of a measurement and simulation is given in Figure 4.15 and the measurement
points are high lighted by the circulated measurements dots:
For the measurements of maximum intensity:
1. the global minimum (of the maximum intensity): (89,5◦ ; 0,63 µW)
2. at global minimum (minimum intensity) left: (85◦ ; 1,76 µW)
3. at global minimum (minimum intensity) right: (94◦ ; 1,70 µW)
For the measurements of minimum intensity:
4. global minimum (left): (85◦ ; 0 µW)
5. global minimum (right): (94◦ ; 0 µW)
6. local maximum: (89,5◦ ; 0,26 µW)
From these six points a first estimation of 3 parameters, which are used to
describe the above mention model are made. The parameters which are used
to describe the model are Imax,contrast=1, αa,b and dα.
• Imax,contrast=1 = mean of intensities of the maximum intensities at global
minimum (left and right).
• dα = mean of angles global minimum intensity (left and right).
• αa,b = angle at global minimum of maximum intensity.
Imax,contrast=1 is the intensity at which the minimal intensity is minimal, dα is the
angle between that minimal intensity and optimal alignment, and at last αa,b ,
which is the angle of the analyser and the optimal alignment relative to the
reference coordinate system.
4.3 Discussion of measurement results
dα = 4,67◦ ; αa,b = 89,33◦ ; ER1 = 150
4,5
Intensity after analyser in µW
75
4
3,5
3
2,5
2
2
1,5
1
1
0,5
0
3
80
4
5
6
85
90
95
Analyser angle in degrees
100
Figure 4.15: The minimal and maximal intensity after the analyser, results from the simulation and measurements found in Figure 4.8. Solid line: simulation, dots: measured, numbered
circulated dots: for calculating starting parameters for estimation. From the fitting the orientation of the global
optimum is 89,33◦, while the angle between this optimum
and the global minimum is 4,67◦ . From the simulation
using Equation 4.34-4.38, an ER of 1:150 is calculated.
Using these 6 points, for this example, the following first estimation is made:
• Imax,contrast=1 = 1,73 µW
• dα = 4,5◦
• αa,b = 89,5◦
After fitting (the complete measurement data) the following intensities and
parameters were obtained Imax,contrast=1 = 1,66 µW, dα = 4,67◦, αa,b = 89,33◦ thus
0
I1 = 62,73 µW, I01 = 0,42 µW and I1 = 0,028 µW. This results in an extinction
ratio (ER1 ) of 1:150.
As from Figure 4.15 can be seen the model is in good agreement with the
measurements. The fit allows also precise determination of the orientation
of the main axes with only a limited number of measurements to be taken.
The extinction ratio estimated by this model however, is always higher due
to small differences of the location of the optimal alignment between model
and measurements. In addition some effects within the fibre may result in
a slightly different behaviour. The fibre then can not just be modelled with
leaking and leaking back. Also the assumption that the leaking back is always
directly proportional to the leaking, which was assumed in this model, need
not always to be valid. In the next section therefore all effects which (could)
influence the mixing in the fibre are described in more detail.
This model however quickly gives a parameter identifying the fibre adequately,
when bared in mind that this can be a slight over estimation. In addition this
76
Characterization of polarisation maintaining fibres
model can give the orientation of the fibres main axes.
4.4 Fibre modelling
In the previous sections of this chapter the fibre was modelled as a single
birefringent retarder plate. For the explanation of the methods used, this works
fine. In this section the fibre model is optimised to give a better understanding
of the effects seen and support the conclusions drawn.
In Section 2.7 was already explained that a fibre can be modelled using a series
of retarder plates. From experiments, as will be shown later, it turned out
that the fibre could be modeled adequately by assuming the first and last plate
to be half wave plates having an orientation depending on the fibre quality.
Contrary to the model presented in the previous section, the mixing caused by
these two half wave plates does not need to be the same. This is modeled by
the orientation difference between the half wave plates. The half wave plates
are rotated respectively τ and υ. The orientations of the wave plates, represents
a good piece of fiber (with a quality of e.g. −50 dB). The phase was varied
between 0 and 2π. Using the Jones formalism, the fibre is then modeled as:
n−1
Y
H f ibre = R(τ) · HWP · R(−τ) ·
i=2
(R(θi )Mi R(−θi )) · R(υ) · HWP · R(−υ) (4.39)
Where R and Mi are defined by Equation 2.25-2.26 and the Jones matrix of the
half wave plate(HWP) is given by:
"
#
1
0
−i π2
(4.40)
HWP = e
0 −1
This fibre model is then used to refine the results found by the DC measurement setup as described by Equation 4.29. First this model is compared to the
approach presented in Section 4.3. Therefore the results of this Jones model
were given as measurement data to the previously presented model. As from
Figure 4.16 can be seen, both models are in good agreement.
Because there were indications that the connector determines the overall fibre
quality (see also Section 4.2.11) the half wave plates are used to represent the
connector. Also the characteristic result (Figure 4.8) was expected to be caused
by the connectors and not by the fibre. Therefore this was verified by modeling
the fibre also without half wave plates and with the half wave plates in an
intermediate section.
When the half wave plates are omitted, the influence of the fibre is along its
entire length the same. Thus every section of the fibre causes equal mixing
represented by the wave plates not aligned perfectly. If the half wave plates are
omitted, the output intensity changes according to Figure 4.17 thereby clearly
not representing measurement performed.
The minimum and maximum intensity do not coincide because the wave plates
are not aligned perfectly, to simulate mixing. Assuming an uniform mixing
4.4 Fibre modelling
77
Intensity after analyser ×103
6
5
4
3
2
1
0
-6
-5
-3
-2
0
-4
-1
1
Orientation analyser in degrees
2
3
Figure 4.16: Intensity predicted for a Jones model of the fibre versus
analytical model after the polariser. The orientation of the
half wave plates was chosen to be: τ = αa . The results of
the Jones model were given as measurement data to the
analytical model.
Intensity after analyser ×103
8
7
6
5
4
3
2
1
0
-6
-2
0
2
6
4
-4
Orientation analyser in degrees
8
Figure 4.17: The minimum and maximum intensity predicted for a
model without HWP’s, after the polariser.
78
Characterization of polarisation maintaining fibres
along the entire fibre length does not represent the fibres characteristic behaviour and therefore also does not represent the fibre.
To see if local disturbance in the fibre could also represent the mixing found,
the model was changed. The half wave plates, representing the main mixing
are relocated not at the beginning and end, but somewhere in between. The
DC setup with this model is schematically shown in Figure 4.18. By relocating
the half wave plates, the influence of the connectors is minimised, and any production influence are highlighted hereby emphasising the possible dominant
presents of local disturbances.
wave plate
P
half wave plate
A
half wave plate
Figure 4.18: To see any dominant influence of local disturbances, e.g.
caused by the production of the fibre, a model with the half
wave plates somewhere in between of the fibre was made.
The complete DC measurement setup with both polariser
and analyser and fibre is schematically shown.
If both half wave plates are located somewhere in between (in the middle of
the fibre), the simulated output intensities are as given in Figure 4.19. This also
Intensity after analyser ×103
8
7
6
5
4
3
2
1
0
origin1
-6
-2
0
2
-4
4
Orientation analyser in degrees
6
Figure 4.19: Intensity predicted for a model with half wave plates in
between, after the polariser.
is not in accordance with the measurements done. Based on these simulations
4.4 Fibre modelling
79
and the measurements done in Section 4.2.11 it is clear that the mixing effects
are caused mainly by the connectors.
For the most simple model the intervening plates are modeled as one plate
having no orientation but only a continuous variable retardation. As described
other effects, e.g. imperfections in the fibre also cause mixing, nor is the phase
changing continuously along the fibre. To account for other effects these intermediate plates are modeled having a random orientation and phase retardation.
Due to the relative short fibre modeled, (few intermediate plates), results only
change negligible, for longer fibres however this is no longer true. For short
fibres a simple model of two half wave plates with one or more intermediate
plates with continuous phase change described the fibre adequately. For longer
fibres the model is extended with more intermediate plates having random orientation and phase changes. The orientation was uniformly distributed, and
chosen so that each piece represented between a perfect fibre or a very good
fibre (−50 dB). The phase was distributed between 0 and 2π. Good results
were obtained by 1-10 plates per metre fibre length.
With random phase and orientation the measurements done for the 50 metre
long fibre were also predicted adequately. In Figure 4.20 the minimal intensity
clearly show the same behaviour as measured. The minimum intensity tends
to flatten, as was seen also from the measurements. This is caused by the
mixing within the fibre. The Jones model showed to be a powerful tool to see
8
Intensity after analyser ×103
7
6
5
4
3
2
1
0
-8
origin1
-6
-2
0
2
6
-4
4
Orientation analyser in degrees
8
Figure 4.20: Intensity predicted for a model for 50 metre fibre, after the
polariser.
the effects of fibre imperfections and the influence of the connectors. The Jones
model is capable of showing the influence of global and local disturbances at
various locations along the fibre. Using the Jones model the effects seen from
the measurements done at longer fibres can be explained. The model in the
previous section however can be more useful to estimate the characteristics of
the fibre (orientation and extinction ratio).
80
Characterization of polarisation maintaining fibres
4.5 Summary
In this chapter a fibre characterization method is developed. This method was
verified with two other methods to measure the polarisation state of a polarised
beam. With this measurement method, both polarisation mixing (fibre quality)
and orthogonality were measured. This was done for several different PM
fibres types and also for one type of PM fibres for a range of length ranging
between 3 and 50 metre. From measurements no large difference was found
between different fibre types, nor was a real length dependent influence on the
extinction ratio measured. For longer length, beside lower output intensities,
minor changes were found, but these do not influence the fibre quality. This
measurement method can also be used as a selection method. As can be seen
from the different fibre qualities measured, the mixing ranged between 1:100
up to 1:1650.
Two models for this mixing were presented. One showing that the fibre quality,
e.g. used for fibre selection or identification can be done by only one parameter
(or two if the orientation is also needed). The second model, based on Jones
matrices gives more insight about the mixing. Based on measurement results
and this model, especially for short fibres (few metres) it could be shown that
the mixing was in the beginning and end of the fibre, while the intermediate
was nearly perfect. This indicates that the connectors determine the fibre
quality, especially for short fibre lengths. To prove if the fibre connector were
determining the overall fibre behaviour, from one fibre both connectors where
replaced. After careful examination, the fibre quality was changed more than
50% showing the influence of fibre connectors. This was confirmed using the
Jones model.
Chapter 5
Accuracy of a fibre fed
heterodyne laser
interferometer
In the previous chapter, among other things, fibres were characterised. To
characterise polarisation maintaining fibres three methods were used to identify extinction ratio (polarisation mixing) and beam orthogonality. From the
theory presented in chapter 2 and experiments two main aspects rise: phase
stability and polarisation mixing effects.
Due to stresses in the fibre, either due to temperature change or mechanical
interactions, the output phase is changing. Because the total phase measured
is used to calculate the displacement, this phase change would be measured as
a virtual displacement.
Due to the polarisation mixing the interferometer can no longer split the two
frequencies for both interferometer arms independently and this will give a
cyclic displacement measurement error. In this chapter the influences of these
and other effects, which can influence displacement accuracy will be examined.
5.1 Phase shifts
Due to the birefringence in the fibre the two orthogonal axis in the fibre will gain
different phase changes. These phase changes are caused by refractive index
changes due to changes internal stress distribution. These changes in internal
stress are caused by the temperature dependence of the refractive index and due
to the expansion of the material. The changes in the internal stress distribution
were used in the previous chapter to produce all output polarisation states of
the fibre caused by these phase changes. In the interferometer however this
change in output phase is measured as a virtual displacement.
81
82
Accuracy of a fibre fed heterodyne laser interferometer
5.1.1 Theory
To calculate the displacement of the measurement mirror of the interferometer,
the phase difference φ1 − φ2 was calculated (see Section 2.2.2). In the fibre the
phases of the main axes are changing due to the birefringence. Suppose that the
phase changes in the fibre along the main axes are φx respective φ y , the output
of a fibre without mixing is described by Equation 5.2 instead of Equation 2.2.
E~x = E~1 = E0 sin(2π f1 t + φ01 + φx )ei
E~y = E~2 = E0 sin(2π f2 t + φ02 + φ y )e j
(5.1)
(5.2)
The measured phase change after passing though a perfect interferometer is
then 2π( f1 − f2 )t+φ1 −φ2 +φx −φ y . The measured phase change φx −φ y is due to
the fibre birefringence and not due to a mirror displacement. This measurement
error could be compensated for if the reference signal also measures this phase
change. This is accomplished by measuring the reference signal Ir not in the
laser head but after the fibre as shown in Figure 5.1. This will be called external
reference hereafter. By doing so Equations 2.3 and 2.5 are transfered to:
Sr
Sm
= 1/2E20 cos(2π( f1 − f2 )t + (φ01 − φ02 ) + (φx − φ y ))
=
1/2E20 cos(2π( f1
(5.3)
− f2 )t + (φ01 − φ02 ) + (φx − φ y ) + (φ1 − φ2 )) (5.4)
From these AC-signals only the phase difference φ1 −φ2 , representing the mirror
displacement can be calculated. When introducing leaking the initial phases
and the phases introduced by the fibre are canceled out equivalently.
5.1.2 Internal reference
As standard a laser interferometer is used with its internal reference measurement system. In the previous section it however already was explained that
an external reference detector system would be needed. The use of an internal
reference gives however insight in the phase changes within the fibre. In addition such measurements illustrate the effect of omitting the external reference
and also to see differences in temperature sensitivity of several different fibre
types.
The tests with an internal reference were carried out by heating up and cooling
down the fibres.
Measurement setup to calibrate the internal reference
In this section the measurement setup to measure the influences of the phase
changes in the fibre on the measurement accuracy with internal reference are
examined. To measure the influence of omitting the external reference, the
measurement setup as schematically plotted in Figure 5.1 was built.
This setup is nearly equivalent to the interferometer system as used later in the
interferometric displacement system. In order to compare results, the internal
reference is measured also outside the fibre but in front of the fibre. To calibrate
5.1 Phase shifts
83
Non polarising
beam splitter Fibre
Non polarising Measurement
beam splitter
detector
Laser
Interferometer
Internal reference detector
External reference detector
Figure 5.1: Schematically the measurement setup used to measure
phase changes within the fibre. The setup is the same as for
a standard displacement measurement setup. The reference
signal is obtained by using either the internal or external
reference detector with integrated polariser. The internal
reference is also measured outside the laser head to make
the results comparable. The interferometer (cube corners
and polarising beam splitter) can be omitted to reduce errors.
the system in the interferometer section of Figure 5.1, no optics was installed.
Determining phase changes is done with a standard interferometer detector
systems with integrated polarisers as delivered by the manufacturer. Ideally
the measured displacement would be zero. Any other measured displacement
is then only caused by the phase change in the fibre.
First the fibre is aligned optimal at the input side to the laser main axes and at
output side to the polarisers axis.
The fibre between the reference detectors is then heated and cooled down
thereby changing the phase differences between the two orthogonal modes of
the laser beam in the fibre. In the mean time the phase difference is measured
using the interferometer systems detectors and electronics.
The fibre is placed in a tube of warm water which is cooling down to homogenise temperature and ensure that fibre (core) temperature can be measured approximately. To make sure that the water temperature and the fibres
core are the same, only fibres with a 0,9 mm thick coating were used. As the
fibre is cooling down a temperature sensor in the water was used to trigger a
measurement every 0,05 K temperature change.
This allows precise measurement of phase shift versus fibre temperature. As
proved in Section 4.2.5 this also represents bending of the fibre. With this setup
the ’virtual’ displacement of different types of PM fibres is measured in the
next sections.
Bow-tie polarisation maintaining fibre
Using the measuring setup as described in previous paragraph, first the phase
shift versus temperature change for the Bow-tie polarisation maintaining fibre is measured. This is done with an optimally aligned fibre with respect to
the laser and the receiver. A typical result of the measurement of the optical
84
Accuracy of a fibre fed heterodyne laser interferometer
path length change versus temperature change is found in Figure 5.2. Clearly
3,0
Displacement in µm
2,5
2,0
1,5
1,0
0,5
0
origin1
0
1
2
3
4
Temperature difference in K
5
Figure 5.2: Temperature change versus phase change, for a Bow-tie
(solid line) and elliptical inner clad (dots) PM fibre with
the use of an internal reference. Results are calculated back
to represent a heated fibre section of 1 metre.
visible is the good linear behaviour, the remaining deviations may rise from
non-linearities in the temperature sensor. Also it was assumed that the phase
change was recorded exactly when the temperature was changed 0,05 K and
temperature gradients are absent. In practice this is not completely correct.
In addition the length dependence of the heated section of the fibre on the phase
change was examined. For this the same experiment was repeated for several
lengths of fibre heated. The results of this experiment are found in Figure 5.3.
The deviation from the fit can be caused because the length of the heated section
of the fibre could not be determined very accurately. Using the results from this
experiment a temperature dependence of the Bow-tie fibre was determined to
be 540 nm/K/m.
Without correction these fibres are not suitable for measurements with nanometre accuracies. Even when the fibre temperature is kept constant within 0,1◦ C
over its full length, virtual displacements of 50 nm per metre fibre still can
take place. Correcting for temperature changes is however not very practical
because it requires not only the fibre temperature but also the temperature
gradients of the fibre. The solution for this problem is the use of the external
reference which was explained in Section 5.1.1. In addition would this also
compensate for phase changes caused by bending the fibre. Results from such
external reference experiments are found in Section 5.1.3.
5.1 Phase shifts
85
Optical path length change in µm/K
1,8
1,6
1,4
1,2
1,0
0,8
0,6
0,4
0,2
0
origin1
0
0,5
2
1
1,5
Fibre length in m
2,5
3
Figure 5.3: Phase shifts measured by changing the fibres temperature
versus fibre length for a Bow-tie fibre with the use of an
internal reference.
Elliptical inner clad polarisation maintaining fibre
To investigate temperature dependence for different types of polarisation maintaining fibres, the same experiment is carried out for an elliptical inner clad
polarisation maintaining fibre. From literature [Miz93, ZL94], a lower temperature dependence is expect to be found.
After repeating the experiment with the internal reference as done for the BowTie fibre, the temperature dependence turned out to be 230 nm/K/m.
To see if the birefringence is changed due to change in bending diameter and
bending axis in respect to the main axes this experiment is repeated for different winding diameters. In this experiment the influence of the unwinding
and rewinding of the fibre are thus examined. All winding diameters were
between 100 and 200 mm. For every experiment the temperature dependence
was again calculated.
The variations in temperature dependence measured after winding and unwinding was not large enough to explain the differences found from the deviations of the length dependence as shown in the previous paragraph. The main
influence in these variations is the uncertainty in the length of the heated fibre
section.
As predicted the temperature dependence found was lower than that of the
Bow-tie polarisation maintaining fibre, and was 230 nm/K/m.
Normal mono mode fibre
Although already theoretically was explained in Section 2.4.5, normal mono
mode fibres can not preserve polarisation, these fibres were tested to validate
86
Accuracy of a fibre fed heterodyne laser interferometer
this.
The same test as with the PM fibres is also performed with a normal mono mode
fibre. Because these fibres do not have main axes, no rotational alignment of
the fibre is needed at the input side. The output orientations were aligned with
respect to the polarisers.
While the fibre cools down after heating, no or very little (±20 nm) phase change
was observed. This however would be too much to be used directly for a fibre
fed heterodyne laser interferometer with nanometre uncertainty. If the fibre
however is moved, e.g. by bumping the table or by bending the fibre, even
signal drop occurs often because the output polarisation axes rotate. Also tests
have been done by applying a force to the fibre, but these results were also not
very reproducible. The only conclusion that can be drawn is that after applying
a force, the output state is rotated. It was not evident if there was a (linear) relation. Also reapplying the same force did not yield the same results. Therefore
the following explanation could be valid. While applying this force the fibre is
slightly bent or twisted. Because the influence of this bending/twisting is of the
same order or larger than that of the applied force, we can not find repeatable
measurements. In addition, it was noticed that the initial twist rate or twisting
the fibre considerable influences the change in output state. Because of the
signal loss due to the rotation of the polarisation, normal mono mode optical
fibres are as expected not suitable to be used in a fibre fed heterodyne laser
interferometer.
5.1.3 External reference
As explained in Section 5.1.1 the reference signal must be measured after the
fibre output to cancel the phase changes occurring in the fibre. Thus the external reference detector of Figure 5.1 must be used. To see if the external
reference detector does eliminate the phase changes, the same experiment as
described in the previous sections was done, but now with the use of the external reference detector instead of the internal reference detector. The results
from this experiment are found in Figure 5.4. The linear temperature dependence clearly is compensated within the resolution of the measurement system
of 1,2 nm. Residual effects can occur due to misalignment or imperfections of
the non-polarising beam splitter. From experiments it was seen that the result
is comparable to the results obtained with the same experiment without fibre.
5.1.4 Evaluation of the external reference
A polarisation maintaining fibre must be used to prevent the polarisation direction to rotate, as a normal mono mode fibre apparently does under bending.
Polarisation maintaining fibres exhibit large phase changes between the two
orthogonal modes due to temperature changes resulting in large measurement
errors with the use of the standard laser interferometer setup. If the reference
signal is measured in front of the input of the fibre, the resulting phase changes
5.1 Phase shifts
87
Displacement in nm
1,5
1
0,5
0
-0,5
-1
-1,5
0
1
2
3
Temperature change in K
4
5
Figure 5.4: Results from a PM-fibre with external reference, while the
fibre was cooling down. Every point represents a measurement point. The phase changes within the fibre as found in
Figure 5.2, are completely compensated for.
are not canceled out. To overcome measurement errors due to phase shifts
within the fibre, the reference phase needs to be measured at the fibres output,
just after the light has emerged from the fibre, but before it enters the interferometer.
By using this external reference receiver after the Bow-tie polarisation maintaining fibre the temperature dependence of typically 540 nm/K/m is successful
eliminated.
From measurements done in [Miz93] it was found that a polarisation maintaining fibre exhibits a ’virtual’ displacement of 34 nm by a 180 degrees bend,
and a phase change due to temperature change of 750 nm per degree. In
[Miz93] measurements were presented of a temperature dependence of a fibre
fed interferometer of 0,75 mm per ◦ C. However: ”Change in internal fibre
birefringence due to temperature does not cause mode coupling, since the axes
of fibre symmetry remain unchanged; nevertheless, if both modes carry power
their phase-relation, and hence the output polarisation state can vary considerably” [PBH82]. This property was used in Chapter 4. However no length
dependence nor a heated fibre length is reported, this seemed comparable with
the results found in this research. Our experiments show that the external
reference will compensate for both.
However the ’normal’ mono mode fibre (no PM) is very sensitive for bending,
resulting in signal drops, this fibre can not be used for polarisation interferometers. These fibres do have an excellent thermal behaviour by showing the
same refractive indices for both polarisations, contrary to PM fibres which are
very sensitive to temperature changes. Elliptical inner clad polarisation maintaining fibres are less temperature sensitive than Bow-tie fibres. The bending
sensitivity was not measured but these bending effects on the measurement
88
Accuracy of a fibre fed heterodyne laser interferometer
uncertainty are, as explained in Section 4.2.5, adequately estimated by changing the fibres temperature. Bending however can cause the beam intensity to
vary.
5.2 Polarisation mixing
In the previous chapter the influence of the fibres birefringence was measured.
The measurements in [Lor02] show that the beam emerging from the fibre is
more elliptically polarised than the beams of the laser head themselves. The
beams emerging from the laser laser head have an ellipticity of 1:10000, with a
non orthogonality of 0,2◦ . In the following section the influence of this mixing
on the displacement measurement accuracy will be investigated.
5.2.1 Theory
For the functioning of the interferometer, linearly polarised light was assumed.
The light emerging from the laser source is highly polarised and can be described by its state of polarisation as shown in Figure 2.1. Practically the light
will never be perfectly polarised, nor will the two principal planes be perfectly perpendicular. For a heterodyne laser interferometer both polarisation
directions ideally represent one frequency. In the output of the laser however
one polarisation direction may contain also a fraction of the other frequency.
Note that the deviations are small, in a commercially available laser, these are
smaller than 0,1%. The change in state of polarisation can be described by Jones
calculus as e.g. is shown in Section 4.4. In Section 2.2.3 the influence of this
mixing is explained in detail for the case of polarisation mixing. This theory is
used in the next section to predict the non-linearity in an interferometer.
5.2.2 Measurement of polarisation state
In Chapter 4 with three methods the polarisation state of several fibres was
measured. Globally there are two groups of fibres, the first one is slightly better
than 1:100 the second around 1:1000. For both categories of fibres the output
polarisations are measured to be perpendicular. In the following sections fibres
from each group are measured, as representative for fibres of these classes.
From the last group results from a 1:850 and a 1:1650 fibre are shown.
5.2.3 Simulations
With the measured results of the mixing of the PM fibres presented in the previous chapter, now the maximal achievable uncertainty resulting from these
mixing is calculated.
To predict the non-linearities the virtual laser interferometer developed by
Cosijns [CHS02] is used. In the simulations a measurement of 4λ optical path
5.2 Polarisation mixing
89
length (2λ mirror displacement) change is simulated with a standard interferometer. In Figure 5.5 results from a simulation are found if the laser beam has
a mixing of 1 % for both frequencies/polarisations. On the horizontal axis the
10
8
Non-linearity in nm
6
4
2
0
-2
-4
-6
-8
-10
origin1
0
200
600
800
400
Displacement in nm
1000
1200
Figure 5.5: Simulations for achievable uncertainty for fibres with an extinction ratio of 1:100 (solid line) and 1:850 (dotted line) for
both polarisation, from the virtual laser interferometer developed by Cosijns [CHS02]. The remaining non-linearity
is plotted versus the mirror displacement. From the simulations a non-linearity of 20 nm(t-t) for the 1:100 fibre and
7 nm(t-t) for the 1:850 fibre is predicted.
mirror displacement is plotted, while on the vertical axis the deviation from the
perfect interferometer is plotted. All other (optical) components are supposed
to be perfect and are also aligned perfect (e.g. causing no polarisation mixing,
nor (unpredictable) phase shifts). From the simulations presented in Figure 5.5
it can be seen that the maximal achievable accuracy of an interferometer with
a fibre that has a mixing of 1 % (intensities), is 20 nm(t-t).
Repeating this simulation for a fibre with a mixing ratio of 1:850 gives an maximal achievable accuracy of 7 nm(t-t). For achieving an uncertainty of better
than 2 nm(t-t), the required polarisation mixing is again determined with the
help of the before mentioned model. Assuming all optical components ideal,
the polarisation mixing may maximal be −40 dB or 0,01 %(1:10000). Allowing
some optical component imperfections as well as alignment errors the fibre
should preferably maintain the polarisation even somewhat better. This however is well above the maximal (standard) available fibres on the market. Note
that in this simulation all other components are perfect, and perfectly aligned.
To achieve this error, which is caused by the fibre itself, all other components
and environmental influences must be eliminated down to this level.
This means however that a perfect fibre is needed because the laser head al-
90
Accuracy of a fibre fed heterodyne laser interferometer
ready had an ellipticity of 1:10000. Due to the non-orthogonality of the laser
head this also means that the uncertainty possible with only the standard laser
head would already be more than 1 nm. Therefore some more simulations
done based on the laser interferometer system as presented in [CHS02] were
done. From these simulations it could be shown however that, if the beams
are both e.g. clock wise elliptically polarised, as is assumed in the simulations
presented here, the uncertainty is much larger than with beams both opposite
elliptically polarised. In the case of opposite elliptically polarised beams the
effects of both on the non-linearity compensate partly and the resulting uncertainty will be smaller. This is e.g. the case in the laser head, where ’linearly’
polarised light is made with the use of a quarter wave plate out of the opposite
circularly polarised beams within the laser tube. The resulting non-linearities
found with the simulations presented here are thus an upper limit.
5.2.4 Validation
In this section the predicted non-linearities of the fibre are verified. There
are several ways of generating an optical displacement of a few wavelengths.
Analysing only the beam properties is best done by using a Babinet-Soleil compensator (see Appendix B) because of the common path setup. In this way no
shielding is required, and a linear displacement is made easily without the need
of an reference measurement. Also the number of optical components to be
aligned is minimal. The change in refractive index also can be used to generate
(small) optical displacements. However this requires accurate measurement
of the actual refractive index change. At last the use of a displacement interferometer setup can be used. This requires also a good reference but most of
all, all errors caused by the optics such as misalignment and mixing are also
measured. Because in the validation measurements only the influence of the
fibre must be measured a Babinet-Soleil compensator setup is best suited for a
short range validation. This method is discussed in the next section.
Short range calibration
To measure the non linearities a modified standard interferometer setup is
used. To eliminate the phase shifts effects within the fibre the external reference detector is used. The Babinet-Soleil compensator is used to make the
optical phase changes between the two polarisations and is thus also located
where the interferometer is located normally. A schematic representation of
the setup is found in Figure 5.6.
The complete setup was optimally aligned with respect to the fibre’s main optical axis. In front of the reference receiver a polariser was placed, oriented at
precisely 45◦ with respect to the optical axis. In the measurement receiver also
the integrated polariser was removed, and replaced by an external polariser,
oriented at precisely 45◦ with respect to the optical axis. The Babinet-Soleil
compensator was aligned along one of the fibres main axes.
Mutual orientations were measured and aligned within 0,05◦ , in order to min-
5.2 Polarisation mixing
91
Fibre
Non polarising
beam splitter
Measurement
detector
Laser
External
reference
detector
Babinet Soleil
compensator
Figure 5.6: Setup for measuring the non linearity, using a Babinet Soleil
Compensator.
imise other contributions to the measured non linearity. As precaution the
fibre was neither excited mechanically nor by changing the fibres temperature
during this experiment.
First the system without fibre is measured, as a reference to see the system
performance of this measurement setup. Results are found in Figure 5.7, in this
figure the deviation of the linear fit (ideal interferometer) is plotted. From these
5
4
Nonlinearity in nm
3
2
1
0
-1
-2
-3
-4
-5
0
200
600
400
Displacement in nm
800
1000
Figure 5.7: Non linearity of the Agilent 5517C laser head, measured by
using a Babinet Soleil Compensator.
measurements it can be seen that the measurement setup, as well as the laser
head cause no measurable non-linearity. Due to the limited resolution of the
laser interferometer (1,2 nm) there is quite some deviation (±1,8 nm t-t) from
the ideal linear displacement.
Next the Panda type PM fibre as representative fibre with an extinction ratio
of 1:100 is installed between laser head and external reference and is aligned
optimal. In Figure 5.8 the results from this experiments are shown. Again the
deviation from the linear fit is plotted. The non-linearity found for the Panda
type PM fibre measured by using a Babinet Soleil compensator is ±5,5 nm(t-t).
92
Accuracy of a fibre fed heterodyne laser interferometer
5
4
Nonlinearity in nm
3
2
1
0
-1
-2
-3
-4
-5
0
200
600
400
Displacement in nm
800
1000
Figure 5.8: Non linearity of a Panda type fibre with an extinction ratio
of 1:100, measured by using a Babinet Soleil Compensator.
As can be seen from the measurement results the non linearity found is lower
than expected from the simulations. This is probably due to the optical compensation as described in that section.
Then a fibre with an extinction ratio of 1:850 was installed (and aligned optimal) The deviation from the linear fit is given in Figure 5.9. The remaining
non-linearity clearly has reduced, and is now about 2 nm(t-t). Also this fibre
has a measured non-linearity that is about a factor 2 lower than expected from
the simulations. With this fibre it would be possible to measure with an uncertainty of 1 nm.
Finally the fibre with an extinction ratio of 1:1650 was aligned optimal, and
again the deviation from an ideal interferometer was measured. Results from
this experiment are found in Figure 5.10. The non-linearity has disappeared
and the results are comparable with the one without fibre. Further testing
would require higher output resolution from the interferometer electronics.
But this was not available in our research.
All uncertainties are calculated by using cube corner interferometers. By using
plane mirror interferometers the resulting non linearities would reduce by a
factor 2 due to the optical configuration, where the beam travels not 1 but 2
times to the mirror.
5.2.5 Other effects due to beam behaviour
In this section the influence of other effects like beam divergence en intensity
distribution on the accuracy is described in detail. This is done to optimise and
guarantee the total system performance.
5.2 Polarisation mixing
93
5
4
Nonlinearity in nm
3
2
1
0
-1
-2
-3
-4
-5
0
200
600
400
Displacement in nm
800
1000
Figure 5.9: Non linearity of a fibre with an extinction ratio of 1:850,
measured by using a Babinet Soleil Compensator.
5
4
Nonlinearity in nm
3
2
1
0
-1
-2
-3
-4
-5
0
200
600
400
Displacement in nm
800
1000
Figure 5.10: Non linearity of a fibre with an extinction ratio of 1:1650.
The deviation from linearity shown was measured with a
Babinet-Soleil compensator.
94
Accuracy of a fibre fed heterodyne laser interferometer
Receiver using a fibre
After the beams have passed the interferometer and have recombined, the
phase differences need to be detected. This can be done either by using the
receiver section of a laser (the lasers internal receiver, if available), or by using
a separate receiver. From this external receiver, the detector and electronics
either can be mounted near the optics or can be positioned further away.
The use of fibres to deliver the light to the interferometer was also intended
to minimise the movable weight. Thus for minimising the weight of the fibre
fed interferometer, the mounted variant is not wanted. To position the detector
away from the detector standard fibre fed receivers are available as shown in
Figure 5.11: A photograph of a receiver. The beam is collimated and
then passes trough a polariser. The interfered beam is
guided by an optical fibre to the electronics (Photograph
courtesy of Agilent Technologies, Inc).
Figure 5.11. In additions the electronics, which also can be seen as a heat source
can be positioned outside of the machine.
The fibres used are plastic multi-mode optical fibres. To minimise the effects
of the optical fibres only one signal (the interference signal) is transmitted.
This is the interference signal that is gained by placing a polariser before the
fibre at an angle of 45 degrees. This device is expected not to influence the
displacement measurement accuracy because only intensity variations, due
to temperature variation and bending occur. The measurement accuracy is
unaffected by amplitude variations because both beams have already passed
a polariser under 45 degrees and have interfered. Amplitude changes of the
interference signals do not influence the measurement accuracy of heterodyne
laser interferometers. Tests have been done with a standard fibre fed receiver
to see any influence on the measurement accuracy.
From experiments, either by bending or heating the fibres, no influences on
the measurement signal were found. The fibre should not need a rotational
alignment within the collimator housing, this also has been proved. In addition,
connecting and reconnecting has been preformed without any problem.
From the observations made by using this receiver alignment of the collimator
housing, where the polariser is mounted in is critical but does not need to limit
5.2 Polarisation mixing
95
measurement accuracy. Because only the collimating lens has to be mounted
to the interferometer, this option is well suited for a lightweight interferometer.
For a flexible measurement system, this device for returning the measuring
signal(s) can be used without loss of accuracy.
Beam intensity profile
The interferometer accuracy is influenced by the intensity profile of the beams
emerging onto the detector due to electronic sensitivity specifications [Com99].
If the one of the beams would have a flat intensity profile during increase in
optical path length the intensity profile will gradually change to a Gaussian
intensity distribution as described later in this section. The system is designed
for beams with a Gaussian intensity distribution, as the laser beam has a Gaussian intensity distribution. It has thus to be validated that the beam emerging
from the fibre has also a Gaussian intensity distribution.
Theoretically the intensity profile of a mono mode optical fibre is nearly Gaussian. In the far field, which is defined as a length large compared to the surface
of the aperture divided by the wave length of the light (λ) used [PP93], the
intensity profile is Gaussian. The characteristic distance for the far field (L f ) is
thus:
L f >> r2 /λ
(5.5)
Where r is the radius of the aperture. The far field of the mono mode fibre is
thus after approximately 20 µm. The light beam then is collimated by a lens
and this of course also can influence the beam intensity distribution. Here also
it is assumed that the beam in the far field of the lens has a Gaussian intensity
distribution. The far field area of the lens is larger than 50 mm. In practice the
beam intensity profile from the beam emerging from the fibre fed heterodyne
laser interferometer can be assumed Gaussian.
To verify the assumption that the intensity profile can be assumed Gaussian, the
intensity profile was measured. The intensity profile is measured by imaging
the beam onto a screen. This image was recorded using a CCD camera. The
maximal intensity profile was determined for both x and y axes and compared
with a theoretical Gaussian intensity distribution. Results from this experiment
are shown in Figure 5.12.
The intensity profiles were determined at maximal beam intensity. Through
the acquired profiles a theoretical Gaussian intensity distribution was fitted.
As can be seen from Figure 5.12 the intensity profile can be assumed Gaussian.
Wavefront distortion
The phase front of the E-field has to be flat for proper working of the interferometer. In this section the influence of disaffects of this flat phase front on the
accuracy is described. Phase front disturbance can arise by optics (e.g. curved
surface) or surface defects (e.g. mirror imperfections). Because of the mono
mode character of the fibre wave front disturbances are not expected, but the
Accuracy of a fibre fed heterodyne laser interferometer
Normalised intensity
Normalised intensity
96
1
0,5
0
300
350
400
Pixel
450
1
0,5
0
150
200
250
Pixel
300
350
Figure 5.12: The intensity profile of the beam emerging from the fibre,
upper graph shows the x-profile while the lower the yprofile. The dots are measurement points while the solid
line is the fitted Gauss approximation.
collimating optics, beam splitter or mirrors could introduce any distortion.
The influence of wavefront distortion is modeled by two functions f1 (A) and
f2 (A), where A is the surface of the beam. These functions f1 (A) and f2 (A)
represent the present phase (distribution) of the E-field. E-fields of both polarisations of the perfect beam are therefore multiplied by these function f1 (A) and
f2 (A) respectively. The E-fields described by Equations 2.1 and 2.2 are then:
~x = E
~ 1 = E0 cos(2π f1 t + φ01 )~ei · f1 (A)
E
~y = E
~ 1 = E0 cos(2π f2 t + φ02 )~e j · f2 (A)
E
(5.6)
(5.7)
The functions f1 (A) and f2 (A) can be functions of the local x, y and z coordinates
which can due to beam or optics movement also be time dependent. These Efields are then interfering by passing through a polariser at 45 degrees. The
overall phase is determined by the integral over the total (interfered) beam
surface. Assuming ∆φ = φ01 − φ02 this would result in an AC-reference signal
of:
Z
Sr = E20
f1 (A) f2 (A) cos(∆φ) dA
(5.8)
A
If the functions f1 (A) and/or f2 (A) are not time dependent, and are also not
dependent of the position of the (measurement) mirror, this non-uniform wave
front does not influence the measurement accuracy. It however does reduce
the contrast of the interfered signal.
In addition, it is required that the beams do not shift relative to each other
and beam diameter variations are also not allowed. If the beams are moving
5.2 Polarisation mixing
97
relative to each other and/or the beam diameter are changing the integrated
phase change and there will be a measurement error. Also if the functions f1 (A)
and/or f2 (A) are time dependent, there will be a measurement error. Because
these functions are not known, the measurement error is also not known.
However by specifying, over time and position, a maximal deviation from the
ideal flat wave front, an estimation of the maximal error can be made.
The same procedure but with functions f3 (A) and f4 (A) can be applied to the
measurement beams (see Equations D.25 and D.26) after beams have passed the
measurement arm and reference arm and have recombined. For these signals
the same restrictions apply. As long as the interference areas are constant
and the functions do not vary over time or position, e.g. by a moving mirror
as with the interferometer blocks of a wafer scanner, this will not influence
measurement accuracy.
Fibre output beam divergence
As predicted before, the measurement accuracy also can be affected by a non
parallel beam. Especially for long range use and high accuracy displacement
measurement the fibre output beam must have a Rayleigh length (Section 3.1.4)
comparable with the non-fibre versions. For the non-fibre versions the Rayleigh
length is calculated (using Equation 3.1) to be 44 m. The fibre output should
have a comparable small output divergence.
The beam output divergence is determined by the collimator lens (system) after
the fibre and the distance between the fibre output and the lens. As the fibre
output can be regarded as a point source, in theory any desired collimated
beam can be made. Using a standard connectorised fibre and collimator, the
position of the fibre end face however is fixed. The fibre output can thus be
out of focus and the beam is not necessary parallel. This is determined by the
manufacturing of the fibre connector and the collimator. By exchanging one of
them the production tolerances are responsible for the beam to be divergent or
convergent.
To overcome this problem a pigtailed fibre collimator system with an integrated
lens was used. Here the manufacturer has optimised the distance between lens
and fibre. With this system a beam divergence of 50% was measured after
17 meters.
Another tested solution is the use of fibres with special connectors. These fibre
connectors [OZ 99b] have an adjustable focus, and allow fine tuning of the fibre
output. Using these connectors the location of the beam waist can be changed.
The Rayleigh length can thus be changed to meet a given specification. The
position of the lens with respect to the fibre can be adjusted over a range of
3,5 mm with a resolution of 10 µm. A convergent or divergent beam can be
corrected to locate the beam waist, where ever specified.
By using an adjustable connector [OZ 99b] the beam diameter was measured
at several positions along the beam for a distance up to 27 m. From the results
given in Figure 5.13 the Rayleigh length is calculated to be 15 m. However
this is smaller than from the laser head it was better than with the connectors
98
Accuracy of a fibre fed heterodyne laser interferometer
without adjustable focus. The use of a more complex lens system could be used
to decrease the output divergence even more. Due to the small divergence the
Beam diameter in mm
6
4
2
0
-2
-4
-6
0
5
10
20
15
Position in m
25
30
Figure 5.13: Measured beam diameter at several positions along the
beam, to measure the beam divergence.
effects on the displacement accuracy of a range of several hundred millimetres
(e.g. stroke of a wafer stage) will be small.
Back reflection
Back reflection would disturb the working of the laser tube. In Section 2.4.6
therefore it was already explained how Angle Polished Connectors (APC)fibres
reduce back reflection. Lenses used for coupling the light were also coated to
minimise reflection. By taking these precautions, during experiments no effects
of back reflection were identified.
Orthogonality of polarisation
The measurement displacement accuracy is directly influenced by polarisation
state. One of the aspects of the polarisation state is the orthogonality of the two
polarisations, because in the beam splitter the two polarisation, carrying each
one frequency must be split. If the polarisation directions are not orthogonal,
there will always be a mixing in the beam splitter. In Section 4.2.6 the orthogonality of the fibre output was measured. This non-orthogonality was smaller
than 0,1 degrees. A non-orthogonality of 0,1 degrees would result in a maximal
displacement error of 0,18 nm [CHS02].
The non orthogonality measured of the laser head is 0,2 degrees [LKC+ 03]. Because this non orthogonality is transfered in a mixing by coupling the light into
the fibre, the resulting non linearities are determined by the laser head. The
resulting non linearity as estimated for the non-orthogonality of the laser head
5.3 System validation
99
are 0,35 nm.
Both influences are to small too be measured with current displacement interferometers.
5.2.6 Short range evaluation
From the measurements shown in the previous section it is clear that the fibre
fed laser interferometer will meet its (accuracy) limits, due to this polarisation
mixing. Using a Babinet-Soleil compensator it was shown that measurement
uncertainty caused by the output beam of the fibre can be smaller than 1 nm
when using fibres with an extinction ratio of better than 1:850.
All non-linearities are given for cube corner interferometers, for plane mirror
interferometer uncertainties reduce by a factor 2.
The differences found compared to the simulations are probably due to effects
within the non-polarising beam splitter used to split the external reference and
effects as described in the Section 5.2.5.
To show the interferometer system performance, a complete interferometer
system is built. Note that not only fibre mixing, but also misalignment and
component imperfections, as well as refractive index changes, electronics and
noise will be measured. A setup build to show the overall performance consists
of the interferometer displacement system over 300 mm. The complete system
validation is presented in Section 5.3. First some other effects which could
(also) influence the displacement measurement accuracy are described.
5.3 System validation
To show the performance of the developed and realised fibre fed interferometer a complete interferometric displacement system was tested. The total
system performance was evaluated by the measurement of the displacement
of a moving mirror over 300 mm as is current practice in wafer scanners. This
measurement will show the suitability of the system for standard laser interferometer measurement tasks. Over a large range the refractive index and
the mixing caused by the optical components are dominant over non-linearity
effects. Therefore the accuracy (non-linearity) was evaluated by a small optical displacement with the Babinet-Soleil compensator in Section 5.2.4, where
refractive index variations are not a major influence.
5.3.1 Long range calibration setup
For the demonstration of laser interferometric measurements, with nanometre
resolution, a complete interferometer displacement measurement setup was
built. The position of a moving stage is simultaneously measured by the fibre
fed laser interferometer and by a ’classical’ laser interferometer. Both laser
interferometers are equipped with standard interferometers with cube corners.
The main purpose of this is the demonstration of the phase compensations as
100
Accuracy of a fibre fed heterodyne laser interferometer
well as studying the general behaviour of a fibre fed laser interferometer in a
real application.
The setup as schematically shown in Figure 5.14 consists of a classical Michelson
Fibre
Non polarising
beam splitter
Reference receiver
Laser
Mirror
Measurement receiver
Reference Laser
d
Figure 5.14: Schematically the measurement setup used to validate the
fibre fed laser interferometer. The lower laser will be the
reference. The upper laser with the fibre is used with the
external reference signal. The mirror is used to bend the
beam returning from the interferometer to the measurement receiver.
interferometer, using standard Agilent interferometer optics. The moving mirror is mounted on top of a motorised precision slide with an range of 300 mm.
The displacement of the moving mirror is measured with both the fibre fed heterodyne laser interferometer and a standard commercially available Agilent
displacement laser interferometer.
A photograph of the used setup is shown in Figure 5.15. On the left the Agilent
5528 laser head is found which will be the reference. In front, the fibre output
is found, with the external reference signal. Also the standard interferometer
optics, with polarising beam splitter and corner cubes is visible (mounted on
the movable stage). The reference signal from the fibre fed heterodyne laser
interferometer is measured after the fibre in order to cancel out phase differences between the two orthogonal polarisation directions. The reference and
measurement beams of the fibre fed laser interferometer are fed through standard high performance fibre optic receivers. From these fibres no errors (can)
occur because in front of this fibre a polariser already produces the interference
signal, which then is guided through a multi mode plastic optical fibre to the
detector.
To minimise refractive index influences, the beams of both interferometers are
positioned together as close as possible. To prevent mixing between the two
lasers, a beam configuration as plotted in Figure 5.16 was used.
The setup was thermally and mechanically isolated, to prevent vibrations (or
air noise) and thermal influence disturb the measurements. Due to the shield-
5.3 System validation
101
2
1
5
6
7
3
4
89
Figure 5.15: Photograph of the measurement setup used to validate the
fibre fed laser interferometer. On the left the Agilent 5528
(1) laser head is found which will be the reference. In
front, the fibre output (5) is found, with the non polarising beam splitter (6) and the external reference signal (9).
Also the standard interferometer optics, with polarising
beam splitter (2) and corner cubes (3, 4) is visible (one (4)
mounted on the movable precision stage). To measure the
measurement signal the beam is bent with mirror (7) to the
measurement receiver (8). Note that for measurements the
complete setup was mechanically and thermally isolated.
Figure 5.16: Beam configuration used in the interferometer setup as
presented in Figure 5.15. This was done to prevent mixing
between the to laser interferometers. For one laser, the
beams are in the horizontal plane, while for the other the
beams are in the vertical plane.
102
Accuracy of a fibre fed heterodyne laser interferometer
ing and the beam configuration used the refractive index differences between
the beams of both interferometers are minimised.
During this experiment the fibre was not mechanically (or thermally) excited.
Measurements where done every 0,1 mm. At every single position several
measurements where done.
Because hardware triggering can be used, the stage need not to stand still while
both interferometers measure the position of the mirror. For the experiments
presented in the next section however the stage was not moving while a position measurement was made. In other experiments with speeds up to 20 mm/s
no difference compared to these results were found.
5.3.2 Long range calibration results
In this section the measurement results are shown for the performance of the
complete fibre fed interferometer. Results are the difference between the Agilent 5528 and the fibre fed heterodyne laser interferometer. Due to the large
number of measurement points the measurements lasts for several hours.
First some results are shown with the reference laser in enhanced resolution
mode (straightness mode [Com92]). Due to range limitations of this mode, the
results found in Figure 5.17 are only up to 63 mm. From this result it can be
3
2
Difference in nm
1
0
-1
-2
-3
-4
-5
-6
-7
10
20
30
40
Position in mm
50
60
Figure 5.17: Results for a long range measurement with the setup
found in Figure 5.15. The vertical axis is the difference
between the HP5528 (reference system) in enhanced resolution mode and the fibre fed laser interferometer (with
external reference). The deviations may be caused by effect
of the straightness mode, refractive index variations and
dead path error.
5.3 System validation
103
seen that the fibre fed laser interferometer is within 7 nm(t-t) from the reference interferometer. This deviation can be caused by optic noise caused by the
moving mirror, dead path error, effects of the straightness mode and electronic
imperfections. Due to the long measurement time also changes in wavelength
and residual refractive index influence like temperature and pressure gradients
are responsible for these differences.
The same experiment was also done for the complete range of the stage. The
results from this experiment are found in Figure 5.18. Clearly the small devi30
25
Difference in nm
20
15
10
5
0
-5
-10
0
50
200
100
150
Displacement in mm
250
300
Figure 5.18: Results for the comparison between fibre fed heterodyne
laser interferometer and a standard heterodyne displacement interferometer for a long range measurement. The
setup used is found in Figure 5.15. On the vertical axis is
the difference between the reference system and the fibre
fed laser interferometer (with external reference) is shown.
The resolution of the laser interferometer systems used was
1,2 µm.
ation (few counts) over a small ranges is visible. Over the total range a global
behaviour was measured. The form of this global behaviour was also measured
while comparing two conventional laser interferometers. This global form is
due to the deviations in the zTy-straightness ([Ver93]) of the stage, resulting
in rotation of the cube corner in the measurement arm of the interferometer.
The rotation effects are different for the beams of the laser interferometer in the
horizontal plane as for the beams of the interferometer in the vertical beam.
The behaviour is explained by the rotation of the stage which was verified
with a straightness measurement. From these measurements, the stage form is
calculated. As from Figure 5.19) can be seen it showed the same global form
from as the deviation between the two interferometers.
For the other rotation the stage was about 10 times more flat, as was found by
104
Accuracy of a fibre fed heterodyne laser interferometer
80
70
Deviation in µm
60
50
40
30
20
10
0
-10
-20
0
50
200
100
150
Stage position in mm
250
300
Figure 5.19: The global form of the stage used, calculated from the
measured zTy-straightness.
the zTx-straightness measurements. The influences of the zTx-straightness on
the measured displacement uncertainty will thus also be much smaller.
From the measurements in this section the differences between the reference laser interferometer and the fibre fed heterodyne laser interferometer
are 7 nm(t-t) over a long range displacement. This well proves the use and
applicability of the fibre fed heterodyne laser interferometer with the external
reference signal.
5.3.3 Endurance test
For continuous reliable operation of the measurement system, an endurance
test was preformed. During this research a two weeks continuous operation
of the system was preformed without any problem. The laser output intensity
was stable and displacement measurements were done without the need of
any (re)alignment. After two weeks the experiment was shut down, as this
endurance test was successful.
5.4 Summary
In this chapter the performance of the developed fibre fed heterodyne laser
interferometer was validated.
In a short range displacement test, it was shown that cyclic errors due to polarisation mixing within the fibre, as measured in chapter 4 are found. Using
fibres with extinction ratios of 1:850 or better the amplitude of the non linearity
measured is below 1 nm. With the fibre with an extinction ratio of 1:1650 the
5.4 Summary
105
non linearity is so small that it can no longer be measured with the heterodyne
laser interferometer.
The complete system performance was shown in a comparison between a commercial heterodyne laser interferometer and the developed fibre fed heterodyne laser interferometer. For this comparison a displacement of a cube corner
mounted on a stage was measured simultaneously with both interferometers.
Over the range of 300 mm the good performance of the fibre fed heterodyne
laser interferometer was shown. From the measurements no effects of the fibres
on the displacement uncertainty over this range. Deviations found were due
to refractive index changes and (small) rotations of the stage used.
In this chapter also other effects, such as beam intensity profile, beam divergence and wavefront distortion, which could influence accuracy are investigated. It was shown that these effects are small, and thus are only of minor
influence on the displacement uncertainty.
106
Accuracy of a fibre fed heterodyne laser interferometer
Chapter 6
Conclusions and
recommendations
6.1 Conclusions
For decades the continuous reduction in product dimensions require positioning systems as used in manufacturing to be able to position the products with
increasing accuracy. In IC technology the continuous increase of processor
speed leads to a steady decrease of critical dimensions to be made. Numerous
measures have been applied to make the wafer steppers and scanners, used
in the lithography of wafers, more accurate. Heterodyne laser interferometers
are used in these wafer scanners as the primary displacement measurement
system. In this research the influence of the laser head on the machine performance is treated.
The laser head is a heat source and this heat source will influence the machine
performance due to its thermo-mechanical interaction with the machine. Secondly the laser beam delivery to different positions uses a considerable amount
of beam steering optics.
Both disadvantages could be eliminated by introducing optical fibres to deliver
the laser light from the laser head to the interferometer. In this research the
effects of the introduction of fibres on the measurement uncertainty is explored
in detail. The beam emerging from the laser head is delivered by this fibre to
the interferometer optics. Both bending and temperature of the fibre will not
be stable, so special attention is paid to these aspects.
In the research the achievable measurement accuracy of an optimised laser interferometer using fibres for the beam delivery is analysed. This was done by
carefully examine the physical properties of the laser beam.
Because of the small dimensions of the fibre core and the high requirements
to the fibre coupling, in this research a dedicated fibre coupler was developed.
The 6 degrees of freedom fibre coupler was designed using elastic elements, to
107
108
Conclusions and recommendations
prevent play and give a reproducible and predictable behaviour, as well as a
thermal center to minimise temperature effects on the coupling efficiency.
Based on a 90% coupling efficiency demand for each effect, all required resolutions of the fibre coupler were calculated. In worst case this would result
in an overall coupling efficiency of only 37%. After realisation the measured
actuator resolutions and the specifications of the optics were used to calculated
a couping efficiency of 47%. The numerical aperture mismatch between fibre
and beam is of major importance for this coupling efficiency. The numerical
aperture is large because the laser beam is coupled with one lens to the fibre.
If the laser beam is reduced first, the numerical aperture of the fibre and lens
better match. By doing so a coupling efficiency of 70% should be possible.
Experimentally, coupling efficiencies in the range of 35 to 66% were achieved,
mainly due to lens properties. Differences are caused by the use of different
lenses. However only from one lens specifications were available, for the other
lenses the same specifications were assumed.
Because the interferometer requires a stable polarisation state, the output polarisation of the fibre is of major importance. Using standard mono mode fibres
the output polarisation can change due to bending. To deliver a stable output
polarisation state, so called polarisation maintaining (PM) fibres must be used.
These fibres maintain the polarisation state by high birefringence. This birefringence is caused by a non-circular stress profile, that produces a different
refractive index of both main axes. The birefringence is large compared to the
influence of bending and temperature change. Therefore these effects on the
output polarisation can be neglected at ’normal’ use.
Because the refractive index of the two main axes in the fibre change differently,
the output phase depend on the internal stress distribution. As the interferometer measures the phase difference between the two polarisations, any change
caused by the fibre is measured as a ’virtual’ displacement. This measurement
error can successfully be eliminated by using an external reference measurement. It was shown that the linear temperature dependence of the fibre was
successfully eliminated.
Due to imperfections the output polarisation of a large linear input polarisation is not linear but slightly elliptically polarised. This polarisation mixing is
caused by mode leaking in the fibre-cable of which the effects are described in
the next sections.
6.1.1 Fibre selection (criteria)
The output polarisation of the fibre is slightly elliptical due to the mode leak in
the fibre. Because also the phase between the two modes is not constant, the
output polarisation state is not constant either. The output of the optical fibre
is characterised by the extinction ratio, this is the ratio of the intensity of the
unwanted polarisation and the wanted polarisation.
Three different measurement techniques were developed to measure the extinction ratio of an optical fibre. With all three methods it was possible to measure
6.1 Conclusions
109
a comparable extinction ratio of around 1:100 for a polarisation maintaining
fibre. Because the DC-method is most easy, most accurate and it allows to
measure the influence of the fibre without the influence of the laser head used,
this method then was used to measure different commercially available types
of PM-fibres. From measurements, as presented in Table 4.2, no significant
difference between different types was found. All three types of PM-fibres
showed an extinction ratio of about 1:100, which is also the specification generally guaranteed by the suppliers.
Finally, a supplier was found who could supply a fibre specially selected out
of a very good bulk fibre quality. The extinction ratio of that special selected
fibre was measured to be 1:900.
In addition, length dependence was investigated by measuring fibres (from
the same stock as the special selected one) with lengths ranging from 3 to 50
metre. Extinction ratios measured varied between 1:550 to 1:1650 and showed
no length dependency of the extinction ratio. Differences may be explained by
connector effects as discussed later. As shown by the validation measurements,
the extinction ratio is directly related to the achievable displacement measurement uncertainty.
The analytical model developed explained the characteristic measurement results found. Using this model, the fibre quality is characterised with only one
parameter and the orientation of the main axes with respect to the reference
coordinate system can be fitted. The model can thus be used to align the fibre
with respect to the laser head or optics.
Not only polarisation mixing but also axes non-orthogonality would reduce
measurement accuracy. From measurements no deviation from axis orthogonality for the fibres could be measured within our measurement uncertainty
of 0,03◦ , which is small compared to the used laser head which showed a nonorthogonality of 0,2◦ .
From analyses and measurements it became clear that the fibre connectors are
of major importance for the overall extinction ratio of the fibre. To check this the
connectors of a fibre where replaced by the manufacturer. After replacement,
the extinction ratio had dropped from 1:900 to 1:300 and clearly showing the
influence of the connectors.
To explain this better and to analyse other influences, a model of the fibre was
developed based on Jones matrices. From this model it could also be shown
that the characteristic results found while rotating the analyser could not be
explained by the fibre overall quality or local disturbances located within the
fibre. Comparing the experiments to the model showed also that the fibre
quality was determined by disturbance located at the fibre ends. It can be
concluded that, especially for relative short length of fibre, the connectors are
the dominant cause of mode leaking.
The developed measurement method can be used to predict the suitability of a
fibre. If the extinction ratio of the fibre is too low, by replacing the connectors,
the extinction ratio can be altered because we assume the connector assembly
process to be responsible to a certain extend.
110
Conclusions and recommendations
6.1.2 Achievable accuracies
To test the system performance the fibre fed heterodyne laser interferometer
was compared to a system without fibre, both at short and long range.
For the short range the refractive index changes of the air were minimised by
a common path interferometer setup, based on a Babinet-Soleil compensator.
With a 1:100 fibre and the external reference a periodic non-linearity error of
6 nm(t-t) was measured. The laser interferometer’s non-linearity error while
omitting the fibre was below the measurement resolution of 1,2 nm of the interferometer.
When a 1:850 fibre was used, the non-linearity error was reduced to 2 nm(t-t),
while the non-linearity of the 1:1650 fibre showed to be also below the measurement resolution of the interferometer.
For the long range validation the displacement of a cube corner was measured
simultaneously both with the developed fibre fed heterodyne laser interferometer and with a standard laser interferometer. On a range of 60 mm the
deviation between the two interferometers found was within a band of 7 nm.
This deviation is caused by refractive index changes, optical imperfections and
electrical noise.
On a range of 300 mm the deviation between the two was up to 20 nm. However most of this is due to the rotation of the stage used as was verified by
comparing two standard laser interferometers as well as straightness measurements of the stage. The deviation between the 2 interferometers over the total
range of 300 mm not explained by the stage and residual effects of the corner
cubes was also 7 nm which also is caused by refractive index variations, optic
and electronic noise. Other sources influencing this difference are the dead
path error and errors due to the long measurement time, like frequency change
of the laser.
When using a high quality PM fibre it was shown that the developed fibre
fed heterodyne laser interferometer can measure with nanometre uncertainty
over the intended measurement range. When using plane mirror optics, all
presented non-linearity errors and uncertainties are reduced by a factor of two.
6.1.3 Project goals achieved
The short and long range measurements show the suitability of the developed
fibre fed heterodyne laser interferometer. The uncertainty goal of 1 nm is
possible when using fibre with an extinction ratio of 1:850 or better. The overall
system performance was well illustrated with the long range validation.
The output intensity was not as good as intended. The main problem may be
the numerical aperture mismatch of the lens used. When using a lens with a
numerical aperture match, and using a beam expander, a coupling efficiency
of 70% should be possible.
6.2 Recommendations
111
6.2 Recommendations
As this research showed great possibilities of using optical fibres, several improvements could be tested when building prototypes.
The rotational alignment can be done using a half wave plate instead of a mechanical rotation of the fibre, as was done to prevent any influence of the half
wave plate on the polarisation state. Imperfections of the half wave plate will
also cause polarisation mixing.
The coupling efficiency could be increased by reducing the beam first and
thereby ensuring the numerical aperture of the beam to match the numerical
aperture of the fibre better.
The optimal output efficiency and the minimalisation of the cost (optical components) are the main challenges for future research.
It should be investigated if compensation can be used to reduce the nonlinearities. If the compensation works, also fibres with lower extinction ratios
may be used.
It could be investigated how well circularly polarised light is transmitted using
standard mono mode optical fibres. If the mixing of the circular polarisation is
low and the output polarisation remains circular under bending and temperature change, the fibre fed heterodyne interferometer could be realised without
any extra optics. The circularly polarised light from the gas tube is then first
transmitted by the fibre and after the fibre circular output polarisations are
transformed to linear polarisations with a quarter wave plate as done now
within the laser head.
At last the use of fibre beam splitters can be investigated to reduce the number
of optical components even more. The beams emerging of these beam splitters
can be investigated using the measurement methods described in this thesis.
It should be investigated how to measure small changes in the wavefront, in
order to quantify the effects on the displacement measurement accuracy.
112
Conclusions and recommendations
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Appendix A
List of used Nomenclature,
Acronyms and Symbols
Abbreviations, Acronyms
AFC
ER
PANDA
PM
PZ
TUE
HP
Angle polished Fibre Connector
Extinction ratio
Polarisation maintaining AND Absorbtion reducing
Polarisation Maintaining
Polarising
Technische Universiteit Eindhoven
Hewlett Packard nowadays Agilent Technologies
Symbols
roman
description
A
Ai
B
BG
BS
c
C
CR
Jones matrix analyser
amplitude
modal birefringence
shape induced birefringence
strain induced birefringence
speed of light : 299792458 m/s
stress optical coefficient silica at 633 nm: −3,5 · 10−11 m2 /kg
constant depending on fibre core,
for Rayleigh loss: 0,7 − 0,9 dB/km − µm4
diameter, distance
d
119
120
E
~
E
E0
ER1
ER2
~ei, j,k
f
F
f1
f2
g0
H
H f ibre
I
I1
I01
0
I1
Imin
Imax
k
k0
Mi
L
Lb
n
NA
ncore
ncladding
n⊥
n//
n0
P
Pwanted
rcore
R
Ret
Rl
Sm
Sr
t
T
u(x)
V
V0
w0
w(z)
List of used Nomenclature, Acronyms and Symbols
Young modulus
E-vector
amplitude E-field
extinction ratio of f1 polarisation
extinction ratio of f2 polarisation
unit vectors
frequency
Force
frequency of E1 -field
frequency of E2 -field
material constant silica for circular birefringence : 0,037
longitudinal magnetic field
Jones matrix fibre
intensity
intensity of wanted mode
intensity of unwanted mode
intensity leaked back (out phase)
minimum intensity
maximum intensity
wave number
constant depending on optical configuration
Jones matrix retarder
length
beat length
refractive index
numerical aperture
refractive index fibre core
refractive index fibre cladding
refractive index Babinet Soleil compensator
refractive index Babinet Soleil compensator
refractive index medium (e.g. air)
Jones matrix polariser
wanted output power
radius fibre core
rotation matrix
retardation
laser beam radius
AC-measurement signal
AC-reference signal
time
transmission losses
uncertainty of x
V-number
Verdet constant : 4,5 · 10−6 kg/m2
beam waist
beam radius at position z
121
x
x1
y
z
x-axis
position along x-axis
y-axis
z-axis or position
greek
description
α
α0
αA
αa
αp
αR
dα
β
βx
βy
γ
δβ
δφ
∆
∆aα,β
∆Cx,y
∆L
∆tx,y
∆f
∆φ
∆φ0
ηt
θi
θB
θc
λ
λcuto f f
φ
φre f
φmeas
φ01
φ02
ω
ξ
α-axis
optical rotation
attenuation
analyser orientation
polariser orientation
attenuation due to Rayleigh losses
fibre quality (comparable to ER)
β-axis
birefringence in x-axis
birefringence in y-axis
γ-axis
birefringence
initial phase difference
optical path length
azimuth error
collimator error
longitudinal error
transversal error
frequency difference
total phase difference
total initial phase difference
ellipticity
total coupling efficiency
angle
angle of fibre polish
critical angle
laser vacuum wavelength
cut-off wavelength
phase
phase reference arm interferometer
phase measurement arm interferometer
initial phase of f1 axis
initial phase of f2 axis
optical angular frequency
twist rate
122
List of used Nomenclature, Acronyms and Symbols
Appendix B
Babinet Soleil compensator
A Babinet Soleil compensator [AN97, AB99, Lon73, PP93] is made of birefringent crystal. If polarised light incidents on the crystal, both components
propagate through the crystal with different index of refraction and speed. On
emerging, the cumulative relative phase difference can be described in terms of
the difference between optical paths for the two components. If the thickness
of the crystal at the point of transmission is d1 and d2 as shown in Figure B.1,
the difference in optical path length (∆) is
∆ = ||n⊥ − n// ||(d1 − d2 ),
(B.1)
and the corresponding phase difference ∆ϕ is
∆ϕ = 2π
2π
∆
=
||n⊥ − n// ||d.
λ
λ
(B.2)
n⊥ and n// are the refractive indices of the two main axes of the crystal are and
λ the wave length of the light used.
d2
d1
Figure B.1: Schematically the representation of the two wedges, at two
positions, in the Babinet Soleil compensator. To make the
effective thickness, not to depend on the point of incident,
the compensator consists of two thin wedges with n⊥ and a
plane parallel plate with n// .
123
124
Babinet Soleil compensator
If the orientation of the optical axis is not parallel to the polarisation directions,
this results in a combined phase shift.
The fact that the effective thickness of the plate depends on the point of incident
is sometimes a disadvantage. To overcome this, the compensator consists of two
thin wedges and a plane parallel plate as shown in Figure B.1. The combined
effect of the wedges and the plate is than a plate of variable thickness but whose
thickness does not depend on the point of incident. Commonly the Babinet
Soleil compensator is used as a variable retarder. When oriented at 45 ◦ in
respect with the incoming linearly polarised beam this beam can be transformed
to a circularly polarised beam (retardation of λ/4) or the polarisation direction
can be rotated (retardation of λ/2).
When positioned at 0◦ in front of a laser interferometer with two orthogonal
beams, a relative phase difference between the beams can be made. With
the used Babinet Soleil compensator, an optical displacement of more than 2λ
could be made. Because the beams follow the same path, the only difference is
the slightly different refractive index in the two directions. This has the great
advantage that in a simple setup hardly any additional noise is measured under
normal laboratory conditions. When using the Babinet Soleil compensator as a
’virtual’ interferometer, the absence of additional noise allows the measurement
of small displacements without optical noise. By a rotational mis-alignment,
Laser
Reference receiver
Detector (receiver)
Babinet Soleil compensator
Figure B.2: Schematic representation of the use of a Babinet Soleil Compensator for simulating displacements. Due to the common
path setup, environmental fluctuations on the refractive index are minimised. This setup allows to simulate small optical displacements between the two polarisation directions
as normally is gained in the two arm of the interferometer.
the Babinet Soleil compensator can be used to simulate the effect of e.g. a
misaligned beam splitter. To do so a setup as shown schematically in Figure B.2
can then be used [CHS02]. Because alignment is very important the calibration
of the orientation of the main axes of the Babinet Soleil compensator is described
in Appendix C.
Appendix C
Component calibration
For the optical components used to measure the extinction ratio as well as the
linearity measurements, it is important to know the angular orientation relative
to the sample under investigation. In this appendix the relative orientation of
the polariser, analyser and Babinet Soleil compensator is determined. The goal
of calibration is to find the orientations (and behaviour) of all components.
Calibration component orientation
Before using the components, first the mutual orientations need to be examined.
This is done by using null-ellipsometry and four-zone averaging [AB99].
The setup used in null-ellipsometry is shown in Figure C.1. As light source
a circularly polarised homodyne laser source is used. The orientation of the
polariser is given by P, the orientation of the analyser is given by A. The
orientation of the Babinet-Soleil compensator is given by C, while the retardance
of the compensator is given by δc .
Babinet Soleil compensator
Laser
Detector
Polariser
Analyser
Figure C.1: Schematically the calibration setup as used to calibrate the
mutual orientation of the polariser, analyser and BabinetSoleil compensator. This setup is also used to determine
the retardance of the Babinet-Soleil compensator.
125
126
Component calibration
1. First, the orientation of the polariser and analyser is examined. This is
done by setting their mutual extinction axes perpendicular. The intensity behind the analyser gets zero. To do so the compensator has to be
removed. Noted is that only a relative orientation is determined, and not
an absolute. By using four-zone average this gives the following combinations:
P1= 0,00◦ A1= 90,21◦
P2= 90,00◦ A2=180,19◦
P3= 45,00◦ A3=135,22◦
P4=−45,00◦ A4= 45,22◦
If we define P0=0,00◦ then A0=0,21 ± 0,02◦ .
2. Next, the (relative) orientation of the compensator is examined. For this
the compensator is placed in between the polariser and analyser at an arbitrary retardance; these will be determined in the next step. After setting
the polariser and analyser perpendicular, the compensator is altered so
that it cause no polarisation mixing. By setting the axes of the polariser
parallel to the analyser, the compensator is used maximally in order to
obtain a π/2 polarisation rotation. Here also the goal is a minimum intensity behind the analyser. (δc was set to 0,00 at its readout scale)
P=P0
, A=A0+90◦ C1= −5,62◦
◦
P=P0+45 , A=A0−45◦ C2=−50,29◦
P=P0+90◦ , A=A0
C3= −5,10◦
◦
◦
P=P0−45 , A=A0+45
C2= 40,20◦
The main axis of the Babinet-Soleil compensator is found at C0= −5,2 ± 0,4◦ .
3. At last the retardance of the Babinet-Soleil compensator must be determined. This can be done at two ways:
(a) The retardance is changed so that the polarisation is not altered
(δc = n · λ) or rotated maximal (δc = n · λ + λ/2) while yielding
zero intensity. If the polariser and the analyser are perpendicular
and the compensator is set at an azimuth of 45 degrees, then the
increments where the compensator causes no polarisation rotation
are obtained. These are the increments where the intensity after
the analyser is minimal. If however the analyser is parallel to the
polariser, and the compensator is unaltered at 45 degrees, then a
δc = λ/2 is required for minimal intensity behind the analyser. The
distance between the subsequently found values is λ.
C=C0+45◦, P=P0, A=A0+90◦ C=C0+45◦, P=P0, A=A0
n · λ: δc = 17,15
n · λ + λ/2: δc = 8,85
δc = 0,52
δc =-7,82
δc =-16,08
The scale of δc is a displacement in mm, which is generated by a
screw-micrometer. For λ/4 is δc = 4,6. This means not that λ/4 is 4,6
but that the Babinet-Soleil compensator acts as a quarter wave plate
for δc = 4,6. A change of optical path length by λ/4 is 4,2.
127
(b) δc also is found by using a polariser, the compensator and a plane
mirror as shown in Figure C.2. If the compensator (oriented at 45
Polarising
beam splitter
Babinet Soleil
compensator
at 45 degrees
Laser
Detector 1
Plane mirror
(Zero intensity) detector 2
Figure C.2: Schematically the setup used for the calibration of the
Babinet-Soleil compensator (method 3b).
degrees) is set to δc = n · λ/2, then the wave is 90 degrees rotated.
The output at the detector 1 is then I=0. The output at detector 2 is
I=0, for the intermediate phases of δc = λ/4 + n · λ/2. Combining the
values found from method 3a and 3b we obtain the results as shown
in Figure C.3.
20
method 3a
method 3b
δc compensator
15
10
5
0
-5
-10
-15
-20
-1
-0,8 -0,6 -0,4 -0,2 0 0,2 0,4
Phase change in λ
0,6
0,8
1
Figure C.3: Retardances versus compensator displacement for method
3a and 3b.
128
Component calibration
Linearity of Babinet-Soleil compensator
Because the Babinet-Soleil compensator is also used as a variable retarder, its
linearity needs to be calibrated as well. This is necessary if the optical path
difference ||n⊥ − n// ||d (see Appendix B) is simulated. If non-linearities are need
to be measured we have to know the non linearity of the compensator itself.
Errors rise from material imperfections and the limitations of the wedges of
the Babinet-Soleil construction. In principle, there are 3 ways for obtaining the
linearity of the Babinet-Soleil compensator:
1. Measuring the total optical thickness as a function of the wedges displacements by a laser interferometer, where only one polarisation passes
the compensator.
2. Measuring the retardance as function of the wedges displacements by a
laser interferometer, where both polarisations pass the compensator.
3. Obtain discrete retardances, at integer, half integer and quart integer
wavelength, by ellipsometry.
Method 1:
In the followed measuring-setup the optical path length change n⊥ · d or n// · d is
measured. To calibrate the Babinet-Soleil compensator the measuring setup as
schematically is shown in Figure C.4 is used. The results of the displacement of
Babinet-Soleil
compensator
Laser
Receiver
Polarising
beam splitter
Cube corner
Figure C.4: Schematically the calibration setup for measuring the linearity of the Babinet-Soleil compensator.
the Babinet-Soleil compensator wedges versus the measured relative retardance
of the Babinet-Soleil compensator is given in Figure C.5.
Method 2:
By this method, the refractive index difference of the wedges is measured as
function of the wedge displacement. The used measurement setup is shown in
Figure C.6. Using this method, the measurement result as shown in Figure C.7
is obtained. Because in this method only the refractive index difference is measured while in method 1 the absolute optical path length change is measured,
129
Relative difference ×10−3
1,5
1
0,5
0
-0,5
-1
-1,5
-400 -300 -200 -100 0
100 200
Phase change in nm
300
400
Figure C.5: Residue after least square fit of the optical path length
change versus displacement Babinet-Soleil compensator.
Note the y-axis is in 10−3 .
Babinet Soleil compensator
Laser
Receiver
Figure C.6: Calibration of the Babinet-Soleil compensator.
130
Component calibration
0,8
0,6
Difference in nm
0,4
0,2
0
-0,2
-0,4
-0,6
-0,8
-1
-800
-700
-600
-500 -400 -300 -200 -100
Optical path length in nm
0
Figure C.7: Residue after least square fit by using method 2.
this method should give a better description of the Babinet-Soleil compensator
linearity. However the total optical path length difference is about 20 times
less, the residue is 100 times smaller. The results of method 1 are influenced by
environmental fluctuations, because both beams do not travel the same optical
paths. The optical path length of the beam, which did not pass the compensator,
can also change e.g. due to refractive index changes of the air.
Method 3:
For the third method, the ellipsometer arrangement as described in the previous section (also method 3) can be used. Using the results obtained in that
section, the difference of the linear fit of that data is plotted in Figure C.8 for
the retardance of every λ/4.
Combining the three methods, we see that method 1 suffers from environmental influences and that the measurements of method 3 show large deviations,
caused by component imperfections. The best results are obtained by method 2.
The non-linearity of the Babinet-Soleil compensator is ±10−3. This means that
with the refractive index difference a maximum deviation of the linearity of
1 nm is achievable by 1 µm optical path length.
In [CHS02] Figure 12, the measured non-linearities of the Babinet Soleil compensator is shown obtained by using two Babinet Soleil compensators. In this
method the phase change versus wedge displacement is measured, for the
compensator under calibration, at various retardations of the other compensator. By averaging over these measurements, the non-linearity effects within
the Babinet-Soleil compensator under investigation are remaining.
131
8
Difference in nm
6
4
2
0
-2
-4
-6
-8
-800 -600
residue method 3a
residue method 3b
-400
0
200
400
-200
Optical path length in nm
600
800
Figure C.8: Residuals of the measurements as shown in Figure C.3, for
the calibration of the Babinet-Soleil compensator. Deviations are mainly due to component imperfections
Component imperfections
In the preceding sections, ideal and imperfect components were presumed. In
addition, azimuth angle errors were neglected. Although perfect components
and no azimuth angle errors were presumed they are well influencing the
measurements as e.g. is the cause of the ’large’ deviations of the last method
described in the previous section. When making use of a four zone nulling
scheme most of the errors are disappear in first order. For an extensive analysis
of imperfections and errors, see [AB99].
132
Component calibration
Appendix D
Intensity calculations
In this appendix the receiver signals are calculated. In the first section the
reference receiver signal for an ideal interferometer is calculated. In the next
section the reference receiver signal for a non-ideal interferometer is given.
Then the measurement receiver signal for an ideal interferometer is given. In
the last section the measurement receiver signal for a non-ideal interferometer
are calculated.
Ideal reference receiver signal
As described in paragraph 2.2.2 the E-fields of an ideal laser beam, with equal
amplitude, are:
E~x = E~1 = E0 sin(2π f1 t + φ01 )~ei
E~y = E~2 = E0 sin(2π f2 t + φ02 )~e j
(D.1)
(D.2)
In this section the signal of the reference receiver is calculated. The total
intensity, after the polariser at 45◦ , on the detector is:
It
=
=
=
2
π
π
cos( )Ex + sin( )E y
4
4
2
1/2 Ex + E y
2
1/2E20 sin(2π f1 t + φ01 ) + sin(2π f2 t + φ02 )
(D.3)
(D.4)
(D.5)
Substituting 2π fi t + φ0i with xi :
It
=
=
=
1/2E20 (sin(x1 ) + sin(x2 ))2
1/2E20 sin2 (x1 ) + sin2 (x2 ) + 2 sin(x1 ) sin(x2 )
1/2E20 sin2 (x1 ) + sin2 (x2 ) − cos(x1 + x2 ) + cos(x1 − x2 )
133
(D.6)
(D.7)
(D.8)
134
Intensity calculations
The ideal AC signal on the reference receiver is:
Sr
= 1/2E20 (cos(x1 − x2 ))
=
=
1/2E20 cos(2π f1 t
1/2E20 cos(2π( f1
(D.9)
+ φ01 − 2π f2 t + φ02 )
(D.10)
− f2 )t + (φ01 − φ02 ))
(D.11)
Reference receiver signal of non ideal interferometer
In Section 2.2.3 an non-ideal interferometer is described. In this interferometer
the polarisation state of the heterodyne laser beam is not linear but elliptical,
and the two polarisations are not orthogonal. The ellipticity for both polarisations is assumed equal. All other components are assumed perfect and ideal
aligned with the reference coordinate system. A graphical representation of
all E-fields and orientations is given in Figure 2.1. The E-field in front of the
detector is given by:
~ x = E1 sin(2π f1 t + φ01 )~e j + E02 cos(2π f2 t + φ02 )~ei
E
(D.12)
~ y = E01 cos(2π f1 t + φ01 )~ei + E2 sin(2π f2 t + φ02 )~e j
E
(D.13)
The intensity on the reference receiver is, when Ei = cos αE0 and E0i = sin αE0 :
π
π
= cos( )Ex + sin( )E y
4
4
2
= 1/2 Ex + E y
It
= 1/2E20
2
(D.14)
(D.15)
cos α sin(2π f1 t+φ01 ) + sin α cos(2π f2 t + φ02 )+
sin α cos(2π f1 t+φ01 ) + cos α sin(2π f2 t + φ02 )
!2
(D.16)
Substituting 2π fi t + φ0i with xi :
It
=
1/2E20








= 1/2E20 






cos α sin(x1 ) + sin α cos(x2 )+
sin α cos(x1 ) + cos α sin(x2 )
!2
cos2 α sin2 (x1 ) + sin2 α cos2 (x2 )+
sin2 α cos2 (x1 ) + cos2 α sin2 (x2 )+
2 ∗ cos α sin(x1 ) ∗ sin α cos(x2 )+
2 ∗ cos α sin(x1 ) ∗ sin α cos(x1 )+
2 ∗ cos α sin(x1 ) ∗ cos α sin(x2 )+
2 ∗ sin α cos(x2 ) ∗ sin α cos(x1 )+
2 ∗ sin α cos(x2 ) ∗ cos α sin(x2 )+
2 ∗ sin α cos(x1 ) ∗ cos α sin(x2 )
(D.17)
















(D.18)
135














2
= 1/2E0 













cos2 α sin2 (x1 ) + sin2 α cos2 (x2 )+
sin2 α cos2 (x1 ) + cos2 α sin2 (x2 )+
cos α sin α sin(x1 + x2 )+
cos α sin α sin(x1 − x2 )+
cos α sin α sin(x1 + x1 )+
cos α sin α sin(x1 − x1 )−
cos2 α cos(x1 + x2 )+
cos2 α cos(x1 − x2 )+
sin2 α cos(x2 + x1 )+
sin2 α cos(x2 − x1 )+
sin α cos α sin(x2 + x2 )−
sin α cos α sin(x2 − x2 )+
sin α cos α sin(x1 + x2 )−
sin α cos α sin(x1 − x2 )





























(D.19)
The reference receivers AC signal of a non-ideal interferometer is:
Sac

 sin α cos α sin(x1 − x2 )+
 cos2 α cos(x − x )+

1
2
= 1/2E20 
 sin2 α cos(x2 − x1 )−

sin α cos α sin(x1 − x2 )

 sin α cos α sin(x1 − x2 )+
 cos2 α cos(x − x )+

1
2
= 1/2E20 
 sin2 α cos(x2 − x1 )−

sin α cos α sin(x1 − x2 )







(D.20)







(D.21)
Substituting x1 − x2 with ∆x:
Sr = 1/2E20 2 sin α cos α sin(∆x)+cos2 α cos(∆x)+sin2 α cos(−∆x) (D.22)
= 1/2E20 2 sin α cos α sin(∆x) + cos(∆x)
(D.23)
With: ∆x = 2π( f1 − f1 )t + (φ01 − φ02 ).
The reference measurement signal is thus:
Sr
=
1/2E20
2 sin α cos α sin(2π( f1 − f1 )t + (φ01 − φ02 ))+
cos(2π( f1 − f1 )t + (φ01 − φ02 ))
!
(D.24)
Ideal measurement receiver signal
In paragraph 2.2.2 an ideal laser beam are is described. The E-fields of the beam
in front of the measurement receiver is described by:
E~x = E~1 = E0 sin(2π f1 t + φ01 + φ1 )~ei
E~y = E~2 = E0 sin(2π f2 t + φ02 + φ2 )~e j
(D.25)
(D.26)
136
Intensity calculations
The total intensity on the detector, after the polariser at 45◦ is:
2
π
π
(D.27)
It = cos( )Ex + sin( )E y
4
4
2
(D.28)
= 1/2 Ex + E y
2
(D.29)
= 1/2E20 sin(2π f1 t + φ01 + φ1 ) + sin(2π f2 t + φ02 + φ2 )
Substituting 2π fi t + φ0i with xi :
2
It = 1/2E20 sin(x1 + φ1 ) + sin(x2 + φ2 )
= 1/2E20
sin2 (x1 + φ1 ) + sin2 (x2 + φ2 )+
2 ∗ sin(x1 + φ1 ) ∗ sin(x2 + φ2 )

2
2
 sin (x1 + φ1 ) + sin (x2 + φ2 )−

2
= 1/2E0  cos(x1 + φ1 + x2 + φ2 )+

cos(x1 + φ1 − x2 − φ2 )
(D.30)
!
(D.31)





The ideal AC signal on the measurement receiver:
Sm = 1/2E20 cos(x1 + φ1 − x2 − φ2 )
=
=
1/2E20 cos(2π f1 t
1/2E20 cos(2π( f1
+ φ01 + φ1 − 2π f2 t − φ02 − φ2 )
− f2 )t + (φ01 − φ02 ) + (φ1 − φ2 ))
(D.32)
(D.33)
(D.34)
(D.35)
Measurement receiver signal of non ideal interferometer
In Section 2.2.3 an non-ideal interferometer is described. In this interferometer
the polarisation state of the heterodyne laser beam is not linear but elliptical,
and the two polarisations are not orthogonal. The ellipticity for both polarisations is assumed equal. All other components are assumed perfect and
ideal aligned with the reference coordinate system. The E-field, after passing
through the interferometer, in front of the measurement detector is given by:
~ x = E1 sin(2π f1 t + φ01 )~e j + E02 cos(2π f2 t + φ02 )~ei
E
(D.36)
~ y = E01 cos(2π f1 t + φ01 )~ei + E2 sin(2π f2 t + φ02 )~e j
E
(D.37)
The intensity on the measurement receiver is, when Ei = cos(α)E0 and E0i =
sin(α)E0 :
2
π
π
(D.38)
It = cos( )Ex + sin( )E y
4
4
2
(D.39)
= 1/2 Ex + E y
!2
cos α sin(2π f1 t+φ01 +φ1 )+sin α cos(2π f2 t+φ02 +φ1 )+
= 1/2E20
(D.40)
sin α cos(2π f1 t+φ01 +φ2 )+cos α sin(2π f2 t+φ02 +φ2 )
137
Substituting 2π fi t + φ0i with xi :
It
=
1/2E20








2
= 1/2E0 





















= 1/2E20 













cos α sin(x1 + φ1 ) + sin α cos(x2 + φ1 )+
sin α cos(x1 + φ2 ) + cos α sin(x2 + φ2 )
!2
cos2 α sin2 (x1 + φ1 ) + sin2 α cos2 (x2 + φ1 )+
sin2 α cos2 (x1 + φ2 ) + cos2 α sin2 (x2 + φ2 )+
2 ∗ cos α sin(x1 + φ1 ) ∗ sin α cos(x2 + φ1 )+
2 ∗ cos α sin(x1 + φ1 ) ∗ sin α cos(x1 + φ2 )+
2 ∗ cos α sin(x1 + φ1 ) ∗ cos α sin(x2 + φ2 )+
2 ∗ sin α cos(x2 + φ1 ) ∗ sin α cos(x1 + φ2 )+
2 ∗ sin α cos(x2 + φ1 ) ∗ cos α sin(x2 + φ2 )+
2 ∗ sin α cos(x1 + φ2 ) ∗ cos α sin(x2 + φ2 )
cos2 α sin2 (x1 + φ1 ) + sin2 α cos2 (x2 + φ1 )+
sin2 α cos2 (x1 + φ2 ) + cos2 α sin2 (x2 + φ2 )+
cos α sin α sin(x1 + φ1 + x2 + φ1 )+
cos α sin α sin(x1 + φ1 − x2 − φ1 )+
cos α sin α sin(x1 + φ1 + x1 + φ2 )+
cos α sin α sin(x1 + φ1 − x1 − φ2 )−
cos2 α cos(x1 + φ1 + x2 + φ2 )+
cos2 α cos(x1 + φ1 − x2 − φ2 )+
sin2 α cos(x2 + φ1 + x1 + φ2 )+
sin2 α cos(x2 + φ1 − x1 − φ2 )+
sin α cos α sin(x2 + φ1 + x2 + φ2 )−
sin α cos α sin(x2 + φ1 − x2 − φ2 )+
sin α cos α sin(x1 + φ2 + x2 + φ2 )−
sin α cos α sin(x1 + φ2 − x2 − φ2 )
(D.41)
















(D.42)





























(D.43)
The measurement receivers AC signal of a non-ideal interferometer is:


 sin α cos α sin(x1 + φ1 − x2 − φ1 )+ 
 cos2 α cos(x + φ − x − φ )+



1
1
2
2
Sm = 1/2E20 
(D.44)

2
 sin α cos(x2 + φ1 − x1 − φ2 )−



sin α cos α sin(x1 + φ2 − x2 − φ2 )


 sin α cos α sin(x1 − x2 )+

 cos2 α cos(x − x + φ − φ )+ 

1
2
1
2
2

= 1/2E0 
(D.45)

 sin2 α cos(x2 − x1 + φ1 − φ2 )− 


sin α cos α sin(x1 − x2 )
Substituting x1 − x2 with ∆x and φ1 − φ2 with ∆φ:


 2 sin α cos α sin(∆x)+ 


Sm = 1/2E20  cos2 α cos(∆x + ∆φ)+ 


sin2 α cos(−∆x + ∆φ)


 2 sin α cos α sin(∆x)+ 


= 1/2E20  cos2 α cos(∆x + ∆φ)+ 


sin2 α cos(∆x − ∆φ)
(D.46)
(D.47)
138
Intensity calculations


 2 sin α cos α cos(∆x + ∆φ − ∆φ − π/2)+ 

2
cos α cos(∆x + ∆φ)+
=


2
sin α cos(∆x + ∆φ − 2∆φ)
n
o

 2 sin α cos α ncos(∆x + ∆φ) cos(∆φ + π/2)o +

 2 sin α cos α sin(∆x + ∆φ) sin(∆φ + π/2) +

n
o

2
= 1/2E0  cos2 α cos(∆x + ∆φ) +
n
o

 sin2 α cos(∆x + ∆φ) cos(2∆φ) +

n
o

sin2 α sin(∆x + ∆φ) sin(2∆φ)

1/2E20 

(D.48)












(D.49)
With: ∆x = 2π( f1 − f1 )t + (φ01 − φ02 ) and ∆φ = (φ1 − φ2 )
n
o

2
2


cos(∆x+∆φ)
−2
sinα
cosα
sin(∆φ)+cos
α+sin
α
cos(2∆φ)

 (D.50)
n
o
Sm = 1/2E20

2
sin(∆x + ∆φ) 2 sin α cos α cos(∆φ) + sin α sin(2∆φ)
Appendix E
Laser head
Specifications
The used heterodyne laser head is an Agilent 5517C laser head. In Table E.1 the
most relevant specifications [Com92, Com93] for this research are summoned.
Quantity
Nominal vacuum wavelength
Measured vacuum wavelength
Vacuum wavelength accuracy (life time)
Vacuum wavelength stability (1 hour)
Vacuum wavelength stability (life time)
Heat dissipation
Output power
Warm up time
Beam diameter
Frequency difference
Frequency f1
Resolution with 10897B-VME board
Maximum speed
Maximum range
Spec
632,991354 nm
632,9913652 nm
0,02 ppm
0,002 ppm
0,02 ppm
35 W (warm up)
18 W (operation)
180 µW - 1 mW
measured: 400 µW
10 minutes
6 mm
2,4 − 3 MHz
horizontal
1,23 nm (cube corner)
711 mm/s (cube corner)
±21,2 m (cube corner)
Table E.1: Specifications of the Agilent 5517C laser head.
Frequency calibration
In this section the frequency and the frequency difference between the two
orthogonal polarisations of the Agilent 5517C laser head are measured. This is
139
140
Laser head
done by measuring the frequency difference between the Agilent 5517C and a
primary standard (an iodine stabilised HeNe laser [Qui94]). A photograph of
this setup is found in Figure E.1. First the frequency is measured during laser
Figure E.1: Setup from verification measurement with the Agilent
5517C and an iodine stabilised HeNe laser (standard).
startup (warm up). This is done by measuring the frequency difference with
respect to the standard. In Figure E.2 the measured frequency difference over
a period of about 1 hour is found.
From this result it is seen that the average difference with the standard laser
102,3 MHz is. The iodine stabilised HeNe laser was used in the ’g-dip’ which
is 473 612 340,4 MHz. So the frequency of (horizontal polarisation direction
of) the Agilent 5517C is 473 612 238,1 MHz. The relative uncertainty in this
measurement, were the calculations are summed up in Table E.2, is 2 · 10−9. The
Quantity
Uncertainty in reference (∆ f = 150 kHz)
Range of frequency measurement (typical, see e.g. Figure E.2)
Deviation of the frequency counter (6 kHz)
Relative uncertainty (k=1)
u( f )in Hz
3 · 10−10 f
2 · 10−9 f
1,2 · 10−11 f
2 · 10−9 f
Table E.2: Uncertainty budget for the frequency calibration of the Agilent
5517C.
141
1.025
1.0245
Frequency in MHz ×100
1.024
1.0235
1.023
1.0225
1.022
1.0215
1.021
1.0205
1.02 0
20
10
30
40
Time in minutes
50
60
Figure E.2: Frequency difference between the Agilent 5517C and an
iodine stabilised HeNe laser (standard), during warm up of
the Agilent 5517C.
vacuum wavelength is found by:
λ=
c
299792458
=
f
f
(E.1)
This corresponds to:
λ = 632,991 367 0 ± 0,000 001 3 nm
(E.2)
This (λ = 632,991 367 nm, see Equation E.2) is the nominal wavelength of the
horizontal polarisation direction, hereafter noted with a subscript 1. The values
for the horizontal direction are summed up in Table E.3. The uncertainty based
on one standard deviation is u, while U is the extended uncertainty (k = 2).
With the same procedure the frequency difference and the absolute frequency
Quantity
f1
u f1
λ1
uλ1
U(k = 2) = 2 · uλ1 = Uλ1
Value
473 612 238,1 MHz
0,95 MHz
632,991 367 0 nm
0,000 001 3 nm
0,000 003 nm
Table E.3: Measured frequencies and uncertainties for the horizontal polarisation direction.
of the other polarisation direction were determined. For the vertical polarisation (in the future referred with subscript 2 or 1 + ∆) direction we can construct
142
Laser head
Quantity
f2
u f2
λ2
uλ2
U λ2
Value
473 612 240,6 Mhz
0,95 Mhz
632,991 363 6 nm
0,000 001 3 nm
0,000 003 nm
Table E.4: Measured frequencies and uncertainties for the vertical polarisation
direction.
Table E.4. Thus giving a frequency difference of 2,5 Mhz.
Using a polariser in front of only the Agilent 5517C the frequency difference
between the two orthogonal polarisations of this laser head was measured
directly, with a beat-measurement. The mean of the measured frequency difference (see Table E.5) is 2,67 MHz.
As it is not directly obvious which wavelength is used in displacement meadata point
1
2
3
4
5
∆ f in MHz
2,6758
2,6757
2,6701
2,6716
2,6722
Table E.5: Measured frequency difference.
surements (This depends on the (polarisation) optic used) the mean wavelength
is calculated. This results in a larger uncertainty but the absolute error can be
smaller. The mean wavelength and its uncertainties are given in Table E.6.
This approaches the specified nominal wavelength of 632,991 354 0 nm, up to
Quantity
f
uf
value
473 612 239,4 Mhz
1,6 Mhz
λ
uλ
Uλ
632,991 365 2 nm
0,000 002 1 nm
0,000 004 nm
Table E.6: Mean values for the wavelength.
17,8 · 10−9 f . For the Agilent 5517C with serial nr. 3744A04164 we use from now
on the measured wavelength of 632,991 365 2 nm.
Appendix F
Mode coupling
Linear birefringence
The retardance (Ret) of a linear birefringent fibre is [PBH82, Hen89]:
Ret(z) = δβ · z
(F.1)
Ex0 y0 = R · Exy
(F.2)
Where δβ the birefringence and z the coordinate along the fibre is. The birefringence δβ = βx − β y = Bλ
2π . Due to a wrong alignment, or by applying a torque
to the fibre, the main axes of the fibre (xy) do not coincide with those of the
EM-wave (x’y’). By a wrong alignment this results in a mode coupling of:
Where R is the rotation matrix (given by Equation 2.25). To a fibre under torque
this results in:
X
Ex0 y0 =
R · Exy (z)
(F.3)
The mode coupling can be described by a discrete number of wrongly orientated pieces of fibre with mutual rotation error R(z). Torque can be prevented
by using a fibre with high torsion stiffness (e.g. a steel armoured cable).
Circular birefringence
If the light is circularly polarised the use of circular birefringent fibres [PBH82,
US79] is preferable. These fibres do not need an alignment of the main axes,
but are not (yet) commercially available. As they need circular input, they also
give a circular output. The most common cause of circular birefringence is
fibre twist. The polarisation rotation increases linearly with the fibre length
and is described by Equation F.7. The two orthogonal modes are left and right
circularly polarised with the difference in propagation constant:
δβcirc = 2g0 ξ
0
Where g is a material property (for silica 0,073) and ξ uniform twist rate.
143
(F.4)
144
Mode coupling
Source of mode coupling
A good model of the fibre behaviour is not yet developed, however perturbation
theory [Vas91] and other approximations are found in the literature. One paper
[PBH82] gives a good survey of all sources of error approximations. Phase
changes between the orthogonal modes in a birefringent fibre is caused by
[PBH82]:
• Bending of fibres:
2
r
πEC r 2
= −1,35 · 106
(F.5)
λ R
R
Where E is the Youg’s modulus (silica: 7,75 · 109 kg/m2 ), C the stressoptical coefficient (silica at 633: −3,5 · 10−11 m2 /kg), r the fibre radius, R
the bend radius and λ the wavelength used.
• Pressure
8C F
F
δβPressure =
= −0,44
(F.6)
λ r
r
Where F is the transverse applied force per unit fibre length in kg/m.
• Twist
δβBend =
0
δβTwist = g ξz = α0 z
(F.7)
Where g0 a material property (silica 0,073), ξ twist rate and α0 optical
rotation per unit length.
• Magnetic fields
δβMagnet = V0 Hz
Where V0 is the Verdet constant (silica 4,5 · 10
tudinal magnetic field applied to the fibre.
• Temperature differences
(F.8)
−6
2
kg/m ) and H the longi-
– Thermal expansion between fibre and its support structure. This
transverse force is often the main cause of time variation in output
polarisation-state.
– The fibre internal birefringence is temperature sensitive owing to:
∗ temperature dependence op the stress-optic coefficient
∗ difference in thermal expansion between core and cladding
Minor effects that cause also mode coupling:
• Diameter variations. These variations are caused by core ellipticity, because the core can not be produced perfectly cylindrical and due to bending effects.
• Index profile variations
• Numerical aperture variations
Appendix G
Computation of the
AC-methods
In this appendix the AC methods as described in Section 4.2.2 are computed in
detail. To repeat the naming and orientation of the E-fields, refer to Figure G.1.
In method 1 the circular reference source is used, in method 2 this reference
source is not used.
x
Polariser
~ crs
E
~1
E
αP
~ 02
E
~2
E
ε
~ 01
E
y
Figure G.1: Definition of the E-fields used in method 1 and 2. The cir~ crs (only method 1), the polariser
cular reference source is E
~
~ 01 , E
~ 2 and E
~ 02 represent the light
is at an angle αp . The E1 , E
emerging from the fibre, with non-orthogonality .
145
146
Computation of the AC-methods
Method 1: Carrier frequency method
By using the Jones-formalism the input E field of this method is mathematically
described by:
"
~ in =
E
E1 eiδ1 + cos()E02 ei(δ2 +π/2) + sin()E2 eiδ2 + Ecrs eiϕ
i(δ1 +π/2)
E01 e
+ cos()E2 eiδ2 − sin()E02 ei(δ2 +π/2) + Ecrs ei(ϕ+π/2)
#
(G.1)
with ϕ = fcrs t and δi = fi t. The output E-field is described by Equation 4.7, thus
the output intensity is given by:
Iout = (PR(αp )Ein )2
(G.2)
or:


iδ
i(δ +π/2)
+ sin()E2 eiδ2 + Ecrseiϕ ) 
 cos(αp )(E1 e 1 + cos()E02 e 2


(G.3)
Iout =  sin(αp )(E01 ei(δ1 +π/2) + cos()E2eiδ2 − sin()E02 ei(δ2 +π/2) +  ∗


Ecrs ei(ϕ+π/2) )


−iδ
−i(δ +π/2)
+ sin()E2 e−iδ2 + Ecrs e−iϕ ) 
 cos(αp )(E1 e 1 + cos()E02 e 2


 (G.4)
 sin(αp )(E01 e−i(δ1 +π/2) + cos()E2 e−iδ2 − sin()E02 e−i(δ2 +π/2) +


Ecrs e−i(ϕ+π/2) )
The AC output intensity is given by:
IAC = IAC1 + IAC2 + IAC3 + IAC4
(G.5)
with
IAC1 =
IAC2 =
IAC3 =
IAC4 =


−i(δ2 +π/2)
−iδ2 
iϕ
−iδ1

+
cos()E
e
+
sin()E
e
E
e
∗
E
e
02
2
crs
1

 (G.6)
cos2 (αp ) 
i(δ2 +π/2)
iδ2 
−iϕ
iδ1
+ sin()E2 e
+Ecrse ∗ E1 e + cos()E02 e




Ecrs ei(ϕ+π/2) ∗ E01 e−i(δ1 +π/2) +



−i(δ
+π/2)
i(ϕ+π/2)
−iδ
 Ecrse
∗ cos()E2 e 2 − sin()E02 e 2


sin2 (αp ) 
(G.7)
−i(ϕ+π/2)
i(δ1 +π/2)


+
+E
e
∗
E
e
crs
01


 E e−i(ϕ+π/2) ∗ cos()E eiδ2 − sin()E ei(δ2 +π/2) 
crs
02
2




Ecrs eiϕ ∗ E01 e−i(δ1 +π/2) +


 E eiϕ ∗ cos()E e−iδ2 − sin()E e−i(δ2 +π/2) 
 crs

02
2
 (G.8)
sin(αp ) cos(αp ) 
−iϕ
i(δ1 +π/2)


+E
e
∗
E
e
+
crs
01


Ecrs e−iϕ ∗ cos()E2eiδ2 − sin()E02 ei(δ2 +π/2)




Ecrs ei(ϕ+π/2) ∗ E1 e−iδ1 +


E ei(ϕ+π/2) ∗ cos()E e−i(δ2 +π/2) + sin()E e−iδ2 
 crs

02
2
 (G.9)
sin(αp ) cos(αp ) 
−i(ϕ+π/2)
iδ1


+E
e
∗
E
e
+
crs
1


−i(ϕ+π/2)
i(δ2 +π/2)
iδ2 
Ecrs e
∗ cos()E02 e
+ sin()E2 e
147
Using eiφ + e−iφ = 1/2 cos(φ) this is:


E1 cos(ϕ − δ1 )+


 cos()E cos(ϕ − δ − π/2)+ 
IAC1 = 1/2 cos (αp )Ecrs 
(G.10)

02
2


sin()E2 cos(ϕ − δ2 )


E01 cos(ϕ + π/2 − δ1 − π/2)+ 



2
cos()E2 cos(ϕ + π/2 − δ2 )
(G.11)
IAC2 = 1/2 sin (αp )Ecrs 



− sin()E02 cos(ϕ + π/2 − δ2 − π/2)


E01 cos(ϕ − δ1 − π/2)+ 



cos()E2 cos(ϕ − δ2 )
IAC3 = 1/2 sin(αp ) cos(αp )Ecrs 
(G.12)



− sin()E02 cos(ϕ − δ2 − π/2)


E1 cos(ϕ + π/2 − δ1 )+




sin()E2 cos(ϕ + π/2 − δ2 )
IAC4 = 1/2 sin(αp ) cos(αp )Ecrs 
 (G.13)


+ cos()E02 cos(ϕ + π/2 − δ2 − π/2)
2
Simplify and using cos(φ ± π/2) = ∓ sin(φ):
E1 cos(ϕ − δ1 ) + cos()E02 sin(ϕ − δ2 )
+ sin()E2 cos(ϕ − δ2 )
2
IAC1 = 1/2 cos (αp )Ecrs
E01 cos(ϕ − δ1 ) − cos()E2 sin(ϕ − δ2 )
− sin()E02 cos(ϕ − δ2 )
IAC2 = 1/2 sin2 (αp )Ecrs
IAC3 = 1/2 sin(αp ) cos(αp )Ecrs
IAC4 = 1/2 sin(αp ) cos(αp )Ecrs
E01 sin(ϕ − δ1 ) + cos()E2 cos(ϕ − δ2 )
− sin()E02 sin(ϕ − δ2 )
−E1 sin(ϕ − δ1 ) + cos()E02 cos(ϕ − δ2 )
− sin()E2 sin(ϕ − δ2 )
!
(G.14)
!
(G.15)
!
(G.16)
!
(G.17)
Splitting based on the two beat frequencies: IAC = IAC f 1 + IAC f 2 . The first one
(IAC f 1 ) is given by:
IAC f 1 =
+
+
−
1/2 cos2 (αp )Ecrs E1 cos(ϕ − δ1 )
1/2 sin2 (αp )Ecrs E01 cos(ϕ − δ1 )
1/2 sin(αp ) cos(αp )Ecrs E01 sin(ϕ − δ1 )
1/2 sin(αp ) cos(αp )Ecrs E1 sin(ϕ − δ1 )
(G.18)
The maximal f1 fcrs beat signal:
IAC f 1,αp=0 =
1/2EcrsE1 cos(ϕ − δ1 )
(G.19)
1/2EcrsE01 cos(ϕ − δ1 )
(G.20)
The minimum f1 fcrs beat signal:
IAC f 1,αp=π/2 =
148
Computation of the AC-methods
The IAC f 2 beat signal is given by:
IAC f 2 =
1/2 cos2 (αp )Ecrs cos()E02 sin(ϕ − δ2 )
+ 1/2 cos2 (αp )Ecrs sin()E2 cos(ϕ − δ2 )
− 1/2 sin2 (αp )Ecrs cos()E2 sin(ϕ − δ2 )
− 1/2 sin2 (αp )Ecrs sin()E02 cos(ϕ − δ2 )
+ 1/2 sin(αp ) cos(αp )Ecrs cos()E2 cos(ϕ − δ2 )
− 1/2 sin(αp ) cos(αp )Ecrs sin()E02 sin(ϕ − δ2 )
+ 1/2 sin(αp ) cos(αp )Ecrs cos()E02 cos(ϕ − δ2 )
− 1/2 sin(αp ) cos(αp )Ecrs sin()E2 sin(ϕ − δ2 )
(G.21)
Rewriting as ∗ cos(α) cos(β) etc. fractions:
IAC f 2 =
1/2 cos(αp )EcrsE02 sin(ϕ − δ2 ) cos(αp ) cos()
+ 1/2 cos(αp )EcrsE2 cos(ϕ − δ2 ) cos(αp ) sin()
− 1/2 sin(αp )EcrsE2 sin(ϕ − δ2 ) sin(αp ) cos()
− 1/2 sin(αp )EcrsE02 cos(ϕ − δ2 ) sin(αp ) sin()
+ 1/2 cos(αp )EcrsE2 cos(ϕ − δ2 ) sin(αp ) cos()
− 1/2 cos(αp )EcrsE02 sin(ϕ − δ2 ) sin(αp ) sin()
+ 1/2 sin(αp )EcrsE02 cos(ϕ − δ2 ) cos(αp ) cos()
− 1/2 sin(αp )EcrsE2 sin(ϕ − δ2 ) cos(αp ) sin()
(G.22)
Rewriting with cos(α) cos(β) ± sin(α) sin(β) and sin(α) cos(β) ± cos(α) sin(β) fractions:
IAC f 2 = 1/2 cos(αp )Ecrs E02 sin(ϕ − δ2 )(cos(αp ) cos() − sin(αp ) sin())
+ 1/2 cos(αp )Ecrs E2 cos(ϕ − δ2 )(cos(αp ) sin() + sin(αp ) cos())
− 1/2 sin(αp )EcrsE2 sin(ϕ − δ2 ))(sin(αp ) cos() + cos(αp ) sin()
+ 1/2 sin(αp )EcrsE02 cos(ϕ − δ2 )(cos(αp ) cos() − sin(αp ) sin()) (G.23)
Simplifying using cos(α) cos(β) ± sin(α) sin(β) = cos(α ∓ β) and
sin(α) cos(β) ± cos(α) sin(β) = sin(α ± β):
IAC f 2 =
1/2 cos(αp )Ecrs E02 sin(ϕ − δ2 ) cos(α + )
+ 1/2 cos(αp )Ecrs E2 cos(ϕ − δ2 ) sin(α + )
− 1/2 sin(αp )Ecrs E2 sin(ϕ − δ2 ) sin(α + )
+ 1/2 sin(αp )Ecrs E02 cos(ϕ − δ2 ) cos(α + )
(G.24)
The minimum f2 fcrs beat signal:
IAC f 2,αp=− =
1/2 cos(−)EcrsE02 sin(ϕ − δ2 )
+ 1/2 sin(−)EcrsE02 cos(ϕ − δ2 )
(G.25)
149
Assume << 1, sin() ≈ 0, cos() ≈ 1
1/2EcrsE02 sin(ϕ − δ2 )
IAC f 2,αp=− =
(G.26)
The maximum f2 fcrs beat signal:
IAC f 2,αp=π/2− =
−
1/2 sin()EcrsE2 cos(ϕ − δ2 )
1/2 cos()EcrsE2 sin(ϕ − δ2 )
(G.27)
Assume << 1, sin() ≈ 0, cos() ≈ 1
IAC f 2,αp=π/2− =
−1/2EcrsE2 sin(ϕ − δ2 )
(G.28)
Method 2: Discrete beat measurement
By using the Jones-formalism the input E-field of this method is mathematically
described by:
"
#
iδ1
cos()E02 ei(δ2 +π/2) + sin()E2 eiδ2
~ in = E1 e i(δ++π/2)
E
(G.29)
E01 e 1
+ cos()E2 eiδ2 − sin()E02 ei(δ2 +π/2)
with δi = fi t. As with method 1, the output intensity is given by:
Iout = (PR(αp )Ein )2
(G.30)
or:
"
Iout =
"
cos(αp )(E1 eiδ1 + cos()E02 ei(δ2 +π/2) + sin()E2 eiδ2 )
sin(αp )(E01 ei(δ1 +π/2) + cos()E2 eiδ2 − sin()E02 ei(δ2 +π/2) )
#
cos(αp )(E1 e−iδ1 + cos()E02 e−i(δ2 +π/2) + sin()E2 e−iδ2 )
sin(αp )(E01 e−i(δ1 +π/2) + cos()E2 e−iδ2 − sin()E02 e−i(δ2 +π/2) )
∗
#
(G.31)
(G.32)
And the AC intensity components of the output intensity are given by:
IAC
=
IAC1 + IAC2 + IAC3 + IAC4
(G.33)
with:
IAC1
IAC2
IAC3
IAC4


iδ1
−i(δ2 +π/2)
−iδ2

+ 
cos()E
e
+
sin()E
e
E
e
02
2
1


= cos2 (αp ) 
(G.34)

E1 e−iδ1 cos()E02 ei(δ2 +π/2) + sin()E2 eiδ2


 E01 ei(δ1 +π/2) cos()E2 e−iδ2 − sin()E02 e−i(δ2 +π/2) + 
2
 (G.35)
= sin (αp ) 

E01 e−i(δ1 +π/2) cos()E2 eiδ2 − sin()E02 ei(δ2 +π/2)

 iδ E1 e 1 cos()E2 e−iδ2 − sin()E02e−i(δ2 +π/2) +
 (G.36)

= sin(αp ) cos(αp )  −iδ

E1 e 1 cos()E2 eiδ2 − sin()E02ei(δ2 +π/2)


E01 ei(δ1 +π/2) cos()E02 e−i(δ2 +π/2) +sin()E2e−iδ2 
 (G.37)
= sin(αp ) cos(αp ) 
+E01 e−i(δ1 +π/2) cos()E02 ei(δ2 +π/2) +sin()E2 eiδ2 
150
Computation of the AC-methods
Using eiφ + e−iφ = 1/2 cos(φ) this is:
!
cos()E02 cos(δ1 − δ2 − π/2)+
sin()E2 cos(δ1 − δ2 )
2
IAC1 = 1/2E1 cos (αp )
(G.38)
cos()E2 cos(δ1 + π/2 − δ2 )−
sin()E02 cos(δ1 + π/2 − δ2 − π/2)
IAC2 = 1/2E01 sin2 (αp )
IAC3 = 1/2E1 sin(αp ) cos(αp )
IAC4 = 1/2E01 sin(αp ) cos(αp )
cos()E2 cos(δ1 − δ2 )−
sin()E02 cos(δ1 − δ2 − π/2)
!
(G.39)
!
(G.40)
!
cos()E02 cos(δ1 + π/2 − δ2 − π/2)
(G.41)
+ sin()E2 cos(δ1 + π/2 − δ2 )
Simplify and using cos(φ ± π/2) = ∓ sin(φ):
IAC1
IAC2
=
=
1/2E1 cos2 (αp )
2
1/2E01 sin (αp )
cos()E02 sin(δ1 − δ2 )+
sin()E2 cos(δ1 − δ2 )
!
− cos()E2 sin(δ1 − δ2 )−
sin()E02 cos(δ1 − δ2 )
IAC3
=
1/2E1 sin(αp ) cos(αp )
IAC4
=
1/2E01 sin(αp ) cos(αp )
(G.42)
!
cos()E2 cos(δ1 − δ2 )−
sin()E02 sin(δ1 − δ2 )
(G.43)
!
cos()E02 cos(δ1 − δ2 )−
sin()E2 sin(δ1 − δ2 )
(G.44)
!
(G.45)
Expanding:
IAC =
+
−
−
+
−
+
−
1/2E1 cos2 (αp ) cos()E02 sin(δ1 − δ2 )
1/2E1 cos2 (αp ) sin()E2 cos(δ1 − δ2 )
1/2E01 sin2 (αp ) cos()E2 sin(δ1 − δ2 )
1/2E01 sin2 (αp ) sin()E02 cos(δ1 − δ2 )
1/2E1 sin(αp ) cos(αp ) cos()E2 cos(δ1 − δ2 )
1/2E1 sin(αp ) cos(αp ) sin()E02 sin(δ1 − δ2 )
1/2E01 sin(αp ) cos(αp ) cos()E02 cos(δ1 − δ2 )
1/2E01 sin(αp ) cos(αp ) sin()E2 sin(δ1 − δ2 )
(G.46)
The maximum f1 f02 beat signal:
IAC,αp=0 =
+
1/2E1 cos()E02 sin(δ1 − δ2 )
1/2E1 sin()E2 cos(δ1 − δ2 )
(G.47)
Assume << 1, sin() ≈ 0, cos() ≈ 1
IAC,αp =0 = 1/2E1 E02 sin(δ1 − δ2 )
(G.48)
151
Rewriting as ∗ cos(α) cos(β) etc. fractions:
1/2E1 E02 sin(δ1 − δ2 ) cos(αp )(cos(αp ) cos())
IAC =
+ 1/2E1 E2 cos(δ1 − δ2 ) cos(αp )(cos(αp ) sin())
− 1/2E01 E2 sin(δ1 − δ2 ) sin(αp )(sin(αp ) cos())
− 1/2E01 E02 cos(δ1 − δ2 ) sin(αp )(sin(αp ) sin())
+ 1/2E1 E2 cos(δ1 − δ2 ) cos(αp )(sin(αp ) cos())
− 1/2E1 E02 sin(δ1 − δ2 ) cos(αp )(sin(αp ) sin())
+ 1/2E01 E02 cos(δ1 − δ2 ) sin(αp )(cos(αp ) cos())
− 1/2E01 E2 sin(δ1 − δ2 ) sin(αp )(cos(αp ) sin())
(G.49)
Rewriting with cos(α) cos(β) ± sin(α) sin(β) and
sin(α) cos(β) ± cos(α) sin(β) fractions:
IAC =
1/2E1E02 sin(δ1 − δ2 ) cos(αp )(cos(αp ) cos() − sin(αp ) sin())
+ 1/2E1E2 cos(δ1 − δ2 ) cos(αp )(cos(αp ) sin() + sin(αp ) cos())
− 1/2E01E2 sin(δ1 − δ2 ) sin(αp )(sin(αp ) cos() + cos(αp ) sin())
− 1/2E01E02 cos(δ1 − δ2 ) sin(αp )(sin(αp ) sin() − cos(αp ) cos()) (G.50)
Simplifying using cos(α) cos(β) ± sin(α) sin(β) = cos(α ∓ β) and sin(α) cos(β) ±
cos(α) sin(β) = sin(α ± β):
IAC =
+
−
−
1/2E1E02 sin(δ1 − δ2 ) cos(αp ) cos(α + )
1/2E1E2 cos(δ1 − δ2 ) cos(αp ) sin(α + )
1/2E01E2 sin(δ1 − δ2 ) sin(αp ) sin(α + )
1/2E01E02 cos(δ1 − δ2 ) sin(αp ) cos(α + )
(G.51)
The maximum f01 f2 beat signal:
IAC,αp=π/2− =
1/2E1 E2 cos(δ1 − δ2 ) sin()
− 1/2E01 E2 sin(δ1 − δ2 ) cos()
(G.52)
Assume << 1, sin() ≈ 0, cos() ≈ 1
IAC,αp=π/2− = 1/2E01E2 sin(δ1 − δ2 )
(G.53)
The maximum f1 f2 beat signal:
IAC,αp=π/4−/2 = 1/2E1E02 sin(δ1 − δ2 ) cos(π/4 − /2) cos(π/4 − /2 + )
+ 1/2E1E2 cos(δ1 − δ2 ) cos(π/4 − /2) sin(π/4 − /2 + )
− 1/2E01E2 sin(δ1 − δ2 ) sin(π/4 − /2) sin(π/4 − /2 + )
− 1/2E01E02 cos(δ1 −δ2 ) sin(π/4−/2) cos(π/4−/2 + ) (G.54)
152
Computation of the AC-methods
Simplifying:
IAC,αp=π/4−/2 =
1/2E1E02 sin(δ1 − δ2 ) cos(π/4 − /2) cos(π/4 + /2)
+ 1/2E1E2 cos(δ1 − δ2 ) cos(π/4 − /2) sin(π/4 + /2)
− 1/2E01E2 sin(δ1 − δ2 ) sin(π/4 − /2) sin(π/4 + /2)
− 1/2E01E02 cos(δ1 − δ2 ) sin(π/4 − /2) cos(π/4 + /2) (G.55)
Simplifying using cos 1/2(α − β) sin 1/2((α − β) = 1/2 sin(α) − sin(β)),
sin 1/2(α − β) cos 1/2(α − β) = 1/2(sin(α) + sin(β)),
cos 1/2(α − β) cos 1/2(α − β) = 1/2(cos(α) + cos(β)) and
sin 1/2(α − β) sin 1/2(α − β) = −1/2(cos(α) − cos(β)):
IAC,αp=π/4−/2 =
+
−
−
1/4E1E02 sin(δ1 − δ2 ) cos()
1/4E1E2 cos(δ1 − δ2 )(1 + sin())
1/4E01E2 sin(δ1 − δ2 ) cos()
01/4E01E02 cos(δ1 − δ2 )(1 − sin())
(G.56)
Assume E01 E02 = 0,E1 E02 = 0, E01 E2 = 0, << 1, sin() ≈ 0
IAC,αp=π/4−/2 = 0,25E1E2 cos(δ1 − δ2 )
(G.57)
Appendix H
Fibre connector
As shown in Section 4.2.11, the connector is assumed to be of major influence
on the extinction ratio. The influence of the connectors may be explained by
carefully examining the production. In the connector assembly process, the
cleaved fibre is inserted in a ceramic tube and both are glued together. The
diameter tolerances (1 µm) between fibre and ceramics allow the glue to vary
in thickness around the cladding. While hardening the fibre may be deformed.
This deformation can reduce the internal birefringence, especially if applied
at (or near) 45 degrees. Schematically this is shown in Figure H.1. The fibre
Ceramic
Glue
Core
Stress
applying part
Figure H.1: The stripped fibre in the ceramic housing. Due to the production tolerances, the glue thickness between the cladding
and ceramic is not uniform. On the left figure the stress
caused by the glue coincident with the fibres main axes, on
the right the stress induced is applied at 45 degrees. In the
latter the internal birefringence may be reduced.
keying is shown in Figure H.2. The fibre keying by the supplier is done with an
accuracy of 3 degrees in respect to the fibres main axes. The tolerances of the
keying and key-way (0,05 mm) also allow a rotational misalignment of about 1
degree. The key-way is the contra form for the keying, which is present in the
connector insertion. The fibres main axes are thus only known within 4 degrees.
153
154
Fibre connector
This orientation is used as a starting point for the precise alignment of the fibre
in respect with the laser head and optics. This precise alignment is required for
optimal performance of the fibre fed interferometer. The alignment accuracy
of the keying only is inadequate for a fibre used in a fibre fed heterodyne laser
interferometer with nanometre uncertainty.
Connector
Ceramic
Keying
Core
Stress
applying part
Figure H.2: The fibre keying. The tolerances in the orientation of the
keying and the key way itself, requires the precise alignment of the fibre for a fibre fed heterodyne laser interferometer with nanometre uncertainty. The keying is used
as a starting point for the precise alignment of the fibre in
respect with the laser head and optics.
Curriculum vitae
Bastiaan Knarren was born on July 11th, 1975 in Vlodrop (L), the Netherlands.
In 1992 he obtained his HAVO certificate at the Serviam scholengemeenschap
in Sittard. At that school he also studied for the Atheneum degree, which
he received in 1994. He commenced his study of Mechanical Engineering at
Eindhoven University of Technology. On 25 February 1999 he finished his
masters project with a thesis titled ”Use of fibres in laser interferometry”. This
research project was continued in his PhD study which resulted in this thesis.
This doctoral study was performed at the section Precision Engineering of
Eindhoven University of Technology. This research was financially supported
mainly by Agilent Technologies and ASML.
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Curriculum vitae
Acknowledgment
The research described in this thesis has been performed at the section Precision
Engineering of Eindhoven University of Technology. This project was carried
out in close cooperation with, and funded by, ASML and Agilent Technologies.
First of all, I would like to thank my first promotor, prof.dr.ir. P.H.J. Schellekens
for giving me the opportunity to do my PhD work in his group. His personality
and confidence encouraged me greatly. I benefited from his experience and I
cherish good memories to these years.
Furthermore, a special word of thanks goes to Agilent Technologies, especially
to K. Bos. I would like to thank him for his support to the project and the
quick supply of equipment or knowledge. I would like to thank ir. M. Beems
of ASML for all his comments and ASML’s support to the project.
I would like to thank my copromotor dr. H. Haitjema for all his contributions.
I would like to express my thanks to all the members of the promotion committee for their support and for reviewing this thesis: prof.dr.ir. G.M.W. Kroesen
(second promotor), prof.dr. L.P.H. de Goey and prof.dr. K.A.H. van Leeuwen.
I also would like to thank the two students who contributed to the research
project as described in this thesis: ir. K. van de Meerakker and ir. D.J. Lorier.
Furthermore I would like to thank all my former and present colleagues, staff
members and student of the Precision Engineering section. The fine atmosphere in the group, especially between all PhD-students, made the work a lot
easier. Special thanks goes to my room mates Suzanne Cosijns, Maarten Jansen
and Guido Gubbels to whom I had many useful discussions.
Thanks also goes to the GTD, especially S. Plukker, who realised the fibre coupler (and some modifications). For their help to make several special tools, I
would like to thank ing. W. ter Elst and E. Reker.
Last but not least I would like to thank my family, especially my parents, my
wife Réanne and my friends for their unconditional support.
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Acknowledgment
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