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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 53 54 72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. 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Applied optics, 33(16):3604–3610, 1994. 118 BIBLIOGRAPHY 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. 155 156 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. 157 158 Acknowledgment

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