Far-field method for the characterisation of three

Far-field method for the characterisation of three
Far-field method for the
characterisation of
three-dimensional fields:
vectorial polarimetry
Oscar Gabriel Rodrı́guez Herrera
Supervisor:
Prof. Chris Dainty
A thesis submitted in partial fulfilment of the requirements
for the degree of Doctor of Philosophy,
School of Physics, College of Science,
National University of Ireland, Galway
August 2009
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . .
List of acronyms . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
1.1 High resolution microscopy . . . . . . . . .
1.1.1 STED fluorescence microscopy . . .
1.1.2 Two-photon absorption microscopy
1.2 Polarimetry . . . . . . . . . . . . . . . . .
1.2.1 Diattenuation and retardance . . .
1.2.2 Stokes vectors and Mueller matrices
1.2.3 Mueller matrix polarimetry . . . .
1.3 Vectorial polarimetry . . . . . . . . . . . .
2 Numerical analysis
2.1 Vectorial theory of diffraction . . . .
2.2 Evaluation of the diffraction integrals
2.3 Calculation of the scattered field . . .
2.3.1 FDTD method . . . . . . . .
2.4 Near- to far-field transformation . . .
2.5 McCutchen’s method . . . . . . . . .
2.6 Performance of the method . . . . . .
2.6.1 Point-scatterer . . . . . . . .
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3 Experimental setup
3.1 Vectorial polarimeter . . . . . . . . . . . . . . .
3.1.1 Polarisation state generator (PSG) . . .
3.1.2 Polarisation state analyser (PSA) . . . .
3.2 Modulation of the Pockels cells . . . . . . . . .
3.2.1 General remarks . . . . . . . . . . . . .
3.2.2 Theoretical modelling of the modulation
3.2.3 Azimuthal alignment of the Pockels cells
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CONTENTS
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4 System calibration
4.1 Eigenvalue calibration method . . . . . . . . . . . . . . .
4.2 Calibration of the pupil PSA . . . . . . . . . . . . . . . .
4.3 Characterisation of the objective . . . . . . . . . . . . .
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5 Experimental results
5.1 Scattering-angle-resolved Mueller matrix of a flat mirror
5.2 Scattering-angle-resolved Stokes parameters of a pointscatterer . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
81
3.3
3.4
3.2.4 Adjustment of the amplitude and bias . . .
3.2.5 Stability of the PSG . . . . . . . . . . . . .
Calculation of the Mueller matrix . . . . . . . . . .
Alignment and synchronization of the CCD cameras
6 Conclusions
6.1 Field distribution in the focal region
6.2 Calculation of the scattered field . . .
6.3 Gold nano-sphere as a point-scatterer
6.4 Future work . . . . . . . . . . . . . .
Bibliography
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Abstract
In the search for increasing the resolution of an optical system it is
usually necessary to increase its numerical aperture (NA). For high NA
the scalar theory of diffraction is no longer applicable and the vectorial
nature of the electromagnetic (EM) field cannot be ignored. In this case,
the EM field in the focal region of the system must be described using a
vectorial theory of diffraction.
An important consequence of focusing with high NA, as predicted by
the vectorial theory of diffraction, is the appearance of a longitudinal
component of the field, comparable to the transversal components in the
focal region. In conventional polarisation microscopy, even when high
NAs are used, only the transversal components are considered for the
analysis of the specimen. Although this approach has been proved to be
adequate in a number of applications, it ignores the potentially valuable
information contained in the longitudinal component.
The Z -polarized microscope, introduced by Huse et al. [J. Biomed.
Opt. 6 (2001) 480], showed the other side of the approach. That is, the Z polarized microscope only considered the interaction of the longitudinal
component with the sample, ignoring the information contained in the
transversal components.
In this work, we present an alternative far-field method for the analysis of the three-dimensional EM field resulting from the interaction of a
tightly focused field and a sub-resolution scatterer. The method proposed
here is based on the analysis of the scattering-angle-resolved polarisation
state distribution across the exit pupil of a high NA objective lens and
it is proved to yield high sensitivity in sub-resolution displacements of a
sub-resolution specimen.
iii
Acknowledgements
I am obliged to my supervisor, Prof. Chris Dainty, for his support,
guidance, encouragement and fruitful discussions. I am also indebted to
Dr. David Lara Saucedo for his support, guidance and comments as well
as for all the discussions that we had since he came up with the idea that
originated this work.
I thank Dr. Sumio Kumashiro for stimulating discussions on the
different aspects of the project and Mr. Masahiro Takebe for discussions
on FDTD modelling.
This project was funded by the Mexican National Council for Science
and Technology (CONACYT), PhD scholarship No. 177627, Science
Foundation Ireland (SFI), grants 01/PI.2/B039C and 07/IN.1/I906, and
Shimadzu Corporation, Japan.
iv
List of acronyms
ABC
Absorbing Boundary Condition
CCD
Charge Coupled Device
CMMP
Confocal Mueller Matrix Polarimeter
DDA
Discrete Dipole Approximation
DP-ECM Double-Pass Eigenvalue Calibration Method
ECM
Eigenvalue Calibration Method
EM
Electromagnetic
ENZ
Extended Nijboer-Zernike
FDTD
Finite-Difference Time-Domain
FFT
Fast Fourier Transform
FPS
Frames Per Second
NA
Numerical Aperture
NTFF
Near- to Far-Field Transformation
PEC
Perfect Electric Conductor
PSA
Polarisation State Analyser
PSF
Point-Spread-Function
PSG
Polarisation State Generator
SLM
Spatial Light Modulator
STED
Stimulated Emission Depletion
v
Chapter 1
Introduction
The objective of the vectorial polarimetry method is to obtain subresolution information about the specimen under observation. Because of
the large number of systems that have been built to image sub-resolution
objects, it is appropriate to start with a review of the high resolution microscopy techniques currently available. As we shall see, the range and
nature of the techniques is broad, as well as their applications.
In the second part of this chapter we shall discuss the concept of
polarimetry, which includes the polarisation formalism used in this research and the principles of Mueller matrix polarimetry. Finally, we shall
conclude with an introduction to the technique that we call vectorial polarimetry.
1.1
High resolution microscopy
Recent developments in areas such as nano-technology and biological
sciences have increased the need of high resolution imaging techniques
for the analysis of small structures. In this section we discuss some of
the high resolution microscopy techniques currently available although it
is not, and it does not intend to be, a comprehensive review.
Among the most popular high resolution imaging instruments are the
electron microscope, the tunneling microscope [1] and the atomic force
microscope [2]. These microscopes have resolutions as small as a fraction
of a nanometer but they present drawbacks that limit their application.
The electron microscope, for instance, is an excellent method to image
specimens made of, or coated with, a good electric conductor. If the specimen is made of a bad electric conductor, as it is the case for biological
samples, it has to be carefully prepared before the analysis. The prepara1
Chapter 1. Introduction
tion of the sample may introduce artefacts in the image of the specimen.
Thus, electron microscopy is not always suitable for the analysis of biological samples. The tunneling microscope, on the other hand, works
with specimens made with a good electric conductor or a semiconductor
limiting, as well, the range of applications where this instrument can be
used. Finally, the atomic force microscope, although does not have limitations on the material of which the specimen has to be made, relies on
the correct choice of the tip for the required resolution, complicating the
acquisition of good quality images of an unknown specimen.
The maximum resolution of far-field optical microscopes working in
the linear regime is limited by the diffraction limit. Therefore, the
maximum transversal resolution according to Rayleigh’s criterion is ≈
0.61λ/NA, where λ is the wavelength of the light and NA is the numerical aperture of the objective, whereas the longitudinal resolution is
≈ λ/(NA)2 [3]. Attempts to overcome these limits have produced a large
number of different microscopy techniques based on linear and non-linear
optical phenomena.
Since the NA is related to the maximum resolution of an optical system, it is possible to increase the resolution of an optical microscope by
increasing its NA. The use of oil-immersion and solid-immersion microscope objectives to increase the NA, and thus the resolution of optical
microscopes, has become a standard technique.
According to the vectorial theory of diffraction [4], the electromagnetic (EM) field distribution in the focal region of a lens depends, also,
on the polarisation state of the incident light. This property becomes
important for large values of the NA. Therefore, the reduction of the
point-spread-function (PSF) using radial polarisation [5], annular apertures [6] and the combination of both [7], in a high NA objective lens, has
been proposed as a way to increase the resolution. It has also been proposed the use of evanescent wave illumination combined with syntheticaperture microscopy [8] or optical diffraction tomography [9] as an alternative way to increase the resolution. To our knowledge, the former has
been proved experimentally whereas the latter is pending for experimental verification.
One of the most influential inventions of the last century, concerning
the field of optical microscopy, is the confocal microscope [10]. Its capability to image objects in three dimensions has made it one of the most
used microscopy techniques. The resolution of the confocal microscope
is strongly dependent of the detector size, ideally a point detector. However, as Hell pointed out [11], the confocal microscope does not break
2
Chapter 1. Introduction
the diffraction limit but rather pushes diffraction to its limits. Another
high resolution imaging technique that allows three-dimensional imaging
of the sample is optical coherence tomography (OCT) [12]. In this case,
the depth resolution depends on the coherence length of the illumination
and broadband sources are used for high resolution imaging.
Some of the most popular microscopy techniques nowadays, due to
the high resolutions that can be achieved, are based on non-linear phenomena. Stimulated-emission-depletion (STED) fluorescence microscopy
[13] and two-photon absorption microscopy are two of the most used. In
the following sections we shall discuss briefly the working principle of
these two techniques without going deep into the details.1
1.1.1
STED fluorescence microscopy
Fig. 1.1 shows the energy levels involved in the absorption and emission processes of a typical fluorophore. S0 is the ground state and S1
is the first excited electronic state. L0 is the low vibrational level of S0
and L3 its higher level. L1 and L2 are the directly excited and the relaxed vibrational levels, respectively, of S1 . Einstein’s equations govern
the absorption and emission of the fluorophore. In a typical fluorescence
scanning microscope, the fluorophores within the excited area spontaneously decay and emit light that is collected and focused onto a point
detector. A complete image is obtained by scanning the specimen. The
resolution of such systems is related to the area covered by the PSF of the
focusing lens. Thus, one way to increase the resolution of a fluorescence
microscope is by reducing the area of the specimen that participates in
the fluorescence.
The STED technique uses a secondary beam, known as the STED
beam, to stimulate the emission of the fluorophores in the outer region
of the excitation area before fluorescence occurs. Since only the fluorophores in the central region of the excitation area contribute to the
fluorescence signal, the STED beam increases the resolution of the microscope. In the original STED microscope, four mirrors were used to
split the STED beam, which was then focused in different positions of
the outer region of the excitation area. Currently, STED fluorescence
microscopes use doughnut modes for the STED beam [14], allowing a
more uniform illumination of the outer region.
Because of the nature of the technique, the excitation and STED
1
For a comprehensive discussion on non-linear high resolution microscopy techniques the reader is referred to Stefan Hell’s work [11].
3
Chapter 1. Introduction
L1
S1
L2
L3
S0
L0
Figure 1.1: Energy levels of a typical fluorophore.
beams are pulsed laser beams. In a typical fluorophore, the average
fluorescence lifetime is of the order of 2ns. Additionally, the vibrational
relaxations L1 → L2 and L3 → L0 occur in, approximately, 1-5ps. For a
maximum efficiency of the method, it is desirable for the STED beam to
stimulate the emission of the fluorophores in the outer region as soon as
they undergo the vibrational transition L1 → L2 . This can be achieved
by using short pulses for the excitation and STED beams and choosing
the appropriate time delay ∆t between them. The duration of the STED
pulses is, preferably, longer than 1-5ps since the rate at which L2 can be
depleted depends on the average lifetime of L3 . Resolutions as small as
35nm are achievable with STED fluorescence microscopy.
1.1.2
Two-photon absorption microscopy
Single-photon fluorescence microscopy may cause photo-bleaching and/or
photo-damage to the specimen if its absorption band is close to the absorption wavelength of the fluorophores. To avoid these effects, twophoton fluorescence microscopy may be used. In this technique, the
excitation of the fluorophores is done by the absorption of two photons
with half the energy necessary for one photon fluorescence. In this way,
the illumination wavelength is moved away from the absorption band of
the specimen. Fig. 1.2 is a diagram of the energy levels for two-photon
absorption of a typical fluorophore.
Two-photon fluorescence microscopy requires simultaneous absorption of two photons by the fluorophore. To satisfy this condition it is
necessary to place the fluorophore in a region with high density of photons. The focal region of a lens illuminated with a short pulses laser is
4
Chapter 1. Introduction
L1
S1
L2
L3
S0
L0
Figure 1.2: Energy levels of a typical fluorophore in single-photon and
two-photon absorption. The purple arrow represents single-photon absorption whereas the two red arrows represent two-photon absorption.
a good candidate. Because of the distribution of photons in the focal region, only a small volume around the focus has a sufficiently high density
for two-photon absorption to take place. Thus, the detected fluorescence
comes from that small volume increasing significantly the lateral and
longitudinal resolutions of the microscope.
Another two-photon absorption-based technique can be found in the
work by Ramsay et al. [15]. In this work, instead of fluorescence, the
detected signal is the current generated by the sample upon absorption
of two photons. The nature of this technique makes it suitable for the
analysis of integrated circuits. Resolutions below 1µm, at a wavelength
λ = 1530nm, have been reported with a relatively low NA objective
(NA=0.55) by combining it with a solid-immersion lens (SIL) designed
with the standard prescription given to obtain the maximum NA [3].
A further improvement of this technique is of interest to us since
it makes use of the properties of tightly focused polarised light. The
asymmetric irradiance distribution in the focal region of a high NA lens
for incident light linearly polarised [16], suggests an increment of the
resolution in the direction orthogonal to the incident polarisation, as
stated by Richards and Wolf. Using this property of tightly focused
linearly polarised light, Serrels et al. [17] increased the resolution of the
method presented by Ramsay et al.
5
Chapter 1. Introduction
1.2
Polarimetry
In the last decades, the study of light polarisation has gained importance
due to the high sensitivity of characterisation techniques based on the
measurement of the change in the polarisation state as result of the interaction with a sample. The range of techniques where the measurement
of the polarisation state of a light beam is used as the method to analyse
the sample is known as polarimetry. As a starting point to our discussion
on polarimetry, we shall present the basics of the formalism used in this
work.
1.2.1
Diattenuation and retardance
Two important concepts in polarimetry are those of diattenuation and
retardance. Since these concepts are extensively used in this work, it is
appropriate to present their definition.
The diattenuation of an optical element is the dependence of its transmittance on the incident polarisation state. The operational definition
of the diattenuation, as given by Chipman [18], is:
D≡
|τq − τr |
τq + τr
(1.1)
where τq and τr are the transmittances for the element’s eigenpolarisations. With this definition, the possible values for the diattenuation are
in the range 0 ≤ D ≤ 1, where D = 1 corresponds to a perfect polariser
and D = 0 to an optical element with no transmittance dependence on
the incident polarisation state.
Retardance is the relative phase difference introduced to the eigenpolarisations by an optical element. The corresponding operational definition, as given by Chipman [18], is:
R ≡ |δq − δr |
(1.2)
where δq and δr are the phase change for the eigenpolarisations. The
possible values for the retardance are in the range 0 ≤ R ≤ π. In
a wave-plate, the direction of the polarisation component that emerges
with the leading phase is said to be along the direction of the fast axis,
whereas the component polarised in the orthogonal direction is said to be
along the direction of the slow axis. Thus, the fast axis is the direction
for which the component of an incident wave polarised in this direction
travels fastest within the wave-plate.
6
Chapter 1. Introduction
1.2.2
Stokes vectors and Mueller matrices
The polarisation properties of an object, as well as the polarisation state
of a light beam, can be described in different ways. One possibility
is to represent the object with a matrix and the beam with a vector.
The interaction of the light beam with the object is then represented
by the product of the matrix and the vector. Jones formalism [3] is a
suitable tool to describe this interaction as long as the object is nondepolarising. Thus, Jones formalism is limited in the range of objects
and light beams that can be described. A more general formalism is
given by the combination of the Mueller matrix and the Stokes vectors.
The Stokes parameters, i.e. the elements of the Stokes vectors, were
introduced by Sir George Stokes as a set of measurable quantities that
describe the polarisation state of a light beam for completely polarised,
partially polarised and unpolarised light. The definition of the Stokes
vector, for quasi-monochromatic light, is [19]:


hEx Ex∗ + Ey Ey∗ i
 hEx Ex∗ − Ey Ey∗ i 

(1.3)
S=
 hEx Ey∗ + Ey Ex∗ i 
ihEx Ey∗ − Ey Ex∗ i
where Ex and Ey are the components of the electric field in the x- and
y-direction, respectively. The symbol hi indicates that the quantities are
ensemble averages but, assuming stationarity and ergodicity, they can
be replaced by time averages with the same result. The first element of
the Stokes vector is the total irradiance, the second one is the fraction
of light linearly polarised in the horizontal and/or vertical direction, the
third element is the fraction linearly polarised at ±45◦ and the the fourth
one is the fraction of light circularly polarised with right and/or left
handedness.
The Mueller matrix is a 4 × 4 real matrix of the form:


m11 m12 m13 m14
 m21 m22 m23 m24 

M=
(1.4)
 m31 m32 m33 m34 
m41 m42 m43 m44
that contains all the information concerning the polarisation linear properties of the object that represents. Depending on the polarisation properties of the object, there may be symmetries between elements of the
Mueller matrix. However, in the most general case, all the elements of
the matrix are different.
7
Chapter 1. Introduction
An important property of the Mueller matrix representation of polarisation is linearity.2 That is, the total effect over the polarisation of
a light beam due to a series of N optical elements, each one represented
by a Mueller matrix Mi , i = 1, . . . , N , is given by
M = MN · · · M2 M1
(1.5)
For instance, non-depolarising objects can be represented by the combination of a linear-diattenuator and a linear-retarder. Therefore, the
Mueller matrix of a linear-retarder, with its fast axis oriented in the
horizontal direction, followed by a linear-diattenuator is [20]


1
− cos 2Ψ
0
0
 − cos 2Ψ

1
0
0

R(τ, Ψ, ∆) = τ 

0
0
sin 2Ψ cos ∆ sin 2Ψ sin ∆ 
0
0
− sin 2Ψ sin ∆ sin 2Ψ cos ∆
(1.6)
where τ is the transmittance for unpolarised light, ∆ is the retardance
and Ψ, defined as
r
τp
(1.7)
tan Ψ =
τs
for τp and τs the transmittance for p and s polarisations, is an auxiliary
angle related to the diattenuation of the optical element.3 The eigenvalues of Mueller matrices of the same form as Eq. (1.6) are important
for the calibration of the system developed in this work (see section 4.1).
These eigenvalues are [21]
λr1 = 2τ sin2 (Ψ)
λr2 = 2τ cos2 (Ψ)
λi1 = τ sin(2Ψ) exp(i∆)
λi2 = τ sin(2Ψ) exp(−i∆)
(1.8)
from which we can determine τ , ∆ and Ψ.
To conclude this section we notice that not every 4 × 4 real matrix
represents a physically realizable object and thus, not every 4×4 real matrix is a Mueller matrix. Conditions for a particular matrix to represent a
Mueller matrix can be found in the literature [22–25]. Furthermore, the
physical interpretation of the Mueller matrix is not as straightforward
2
Linearity is not exclusive of the Mueller matrix formalism. It is, in fact, a property
of all matrix representations of polarisation.
3
∆ and Ψ are known as the ellipsometric angles.
8
Chapter 1. Introduction
Detector
θ
Sample
PSA
Light
source
PSG
Figure 1.3: Diagram of the basic configuration of a Mueller matrix polarimeter [27].
as the interpretation of the Stokes vector. Nevertheless, a Lu-Chipman
decomposition [26], for instance, reveals polarisation properties of the
object represented by the matrix, in this case its retardance and diattenuation.
1.2.3
Mueller matrix polarimetry
A typical polarimeter (a device built to do polarimetry) consists of a
polarisation state generator (PSG), a place to position the sample, and a
polarisation state analyser (PSA). The PSG determines the polarisation
state of the incident light whereas the PSA measures the change in this
state after interaction with the sample. A Mueller matrix polarimeter
is a polarimeter designed and built to measure the 16 elements of the
Mueller matrix.
The basic configuration of a Mueller matrix polarimeter requires 4 independent incident polarisation states and 4 independent analysers. Fig.
1.3 is a diagram of this configuration as presented in [27]. The combination of the 4 incident polarisation states with the 4 analysers yields
the 16 measurements required to obtain the complete Mueller matrix of
a general sample. Fig. 1.4 is a diagram of the combinations polariseranalyser used in the measurement of each element of the Mueller matrix
using the polarimeter in Fig. 1.3. Depending on the polarisation properties of a particular sample, and thus in the possible symmetries between
9
Chapter 1. Introduction
m11
m12
m13
m14
m21
m22
m23
m24
m31
m32
m33
m34
m41
m42
m43
m44
Note:
Unpolarised
Linear at 45 o
Linear horizontal
Right circular
Figure 1.4: Diagram of the combinations polariser-analyser, for the polarimeter in Fig. 1.3, used in the measurement of each element of the
Mueller matrix.
its elements, less measurements may be required.
The polarimeter presented above is helpful to understand the functioning of a Mueller matrix poarimeter. Nevertheless, in this configuration, as suggested in Fig. 1.3, the different polarisers and analysers are
interchanged between measurements making the system highly sensitive
to experimental errors.
Alternative configurations using rotating compensators [28, 29] or
variable retardance elements, such as liquid crystal modulators [30], photoelastic modulators [31] and Pockels cells [32, 33], for the PSG and/or the
PSA have been reported in the literature. The sensitivity to experimental errors of those polarimeters is lower and performance optimization
methods [34], as well as calibration methods [21], have been proposed.
The issue of calibration of a polarimeter shall be discussed further in
Chap. 4 within the context of the calibration of our system.
1.3
Vectorial polarimetry
Most polarimeters measure the interaction of beam-like fields with the
sample. Since light is a vectorial perturbation, in the most general case
it has three-dimensional structure; that is, it has a component in each of
the three spatial directions. However, this three-dimensional structure is
usually ignored even in polarisation microscopes with high NA objectives,
10
Chapter 1. Introduction
where the three-dimensional nature of the focused EM field is apparent.
To the best of our knowledge, the first attempt to completely characterise a three-dimensional EM field, and that is what we call vectorial
polarimetry, was done by Ellis and Dogariu [35]. Their method is based
on the use of a couple of near-field probes, in nine different configurations,
and produced excellent results. Nevertheless, as any near-field technique,
it is only suitable for applications where the distance between the sample
and the probes can be controlled with sufficiently high precision, making
apparent the need for an alternative far-field method.
Richards and Wolf showed [16], using the vectorial theory of diffraction developed by the latter [4], that in the focal region of a high NA
aplanatic lens (see Chap. 2), for incident light linearly polarised, there
is a longitudinal component comparable to the transversal components,
i.e., there is a three-dimensional field distribution. Other incident polarisation states produce a three-dimensional field distribution in the focal
region of a high NA lens as well [5, 36]. We propose to use this distribution as a non-mechanical probe to perform vectorial polarimetry in the
far-field.
In our method, a three-dimensional focused field interacts with the
specimen yielding, in the most general case, a three-dimensional scattered field. The resulting scattered field propagates to the detection system where a collector lens collimates it creating a beam-like field with no
component along the optical axis; the component along the optical axis
is projected over the transversal components by the collector lens (see
Fig. 1.5). This projection produces an angle-resolved field distribution.
Therefore, by analysing the polarisation state of the scattered light across
the exit pupil of the collector lens, using traditional polarimetry techniques, it is possible to retrieve information about the three-components
of the field scattered by the specimen and thus, about its interaction with
the focused field.
To assess the feasibility of our method, we modelled its performance
using a number of numerical tools.4 The modelling started with the
calculation of the field distribution in the focal region of a lens as a
function of its NA and the polarisation state of the incident light. This
was done using two different methods that shall be discussed in Chap. 2.
Then, we modelled the interaction of the focused field with the sample.
A number of different methods to describe this interaction are readily
available. In this work we chose to start with a point-scatterer and then
4
We modelled the performance of a vectorial polarimeter in the reflection configuration but this does not mean that only this configuration is possible.
11
Chapter 1. Introduction
E
Exit
pupil
(0)
E
(2)
(s)
E
(1)
Ez
(1)
Ex
E
(1)
Ez
(s)
(s)
Ex
Focal
region
x
Focusing
lens
Collector
lens
y
z
Figure 1.5: Bird’s-eye view diagram of the vectorial polarimetry method.
(s)
The longitudinal component of the scattered field, Ez , is projected over
the transversal components by the collector lens. The crosses indicate
sample points on the exit pupil where the polarisation state is analysed.
use the Finite-Difference Time-Domain (FDTD) method, which shall be
briefly discussed in §2.3.1, to model more complex samples. Due to the
limitations of the FDTD method, it is necessary to perform a near- to farfield transformation over the results obtained for the scattered field; this
transformation shall be discussed in §2.4. These three steps constitute
the basis of the system’s performance modelling.
A fundamental part of this work is the experimental verification of
the method. We built a vectorial polarimeter and performed a number
of tests to get a proof of concept. The experimental setup, its calibration, and the results obtained shall be presented in Chaps. 3, 4, and 5,
respectively. Finally, the conclusions of this work and comments on the
future of this research are presented in Chap. 6.
Conferences and publications
The vectorial polarimetry method has been presented for discussion with
the community in a number of international conferences and workshops.
It has also originated a European patent application and a paper describing the method is in preparation.
• David Lara, Oscar Rodrı́guez and Chris Dainty, “Far-field threedimensional optical polarimetry”, Workshop on Random Electromagnetic Fields, Orlando FL, USA, May 2007.
12
Chapter 1. Introduction
• Oscar Rodrı́guez, David Lara and Chris Dainty, “Far-field method
for the characterization of three-dimensional fields”, Progress in
Electromagnetics Research Symposium (PIERS), Prague, Czech Republic, August 2007.
• Oscar Rodrı́guez, David Lara and Chris Dainty, “Far-field method
for the characterization of three-dimensional fields”, 3rd European
Optical Society Topical Meeting on Advanced Imaging Techniques,
Lille, France, September 2007.
• David Lara, Oscar Rodrı́guez and Chris Dainty, “A vectorial polarimetry method and apparatus for analysing the three-dimensional
electromagnetic field resulting from an interaction between a focused illuminating field and a sample to be observed”, European
Patent Application No. 07107376.1.
• Oscar Rodrı́guez, David Lara and Chris Dainty, “Far-field method
for the characterization of three-dimensional fields”, Frontiers in
Optics 2008, Rochester NY, USA, October 2008.
• Oscar Rodrı́guez, David Lara and Chris Dainty, “Far-field vectorial polarimetry”, Novel Techniques in Microscopy, Vancouver BC,
Canada, April 2009.
• David Lara, Oscar Rodrı́guez and Chris Dainty, “Far-field threedimensional optical polarimetry”, Workshop on Partial Electromagnetic Coherence and 3D Polarization, Koli, Finland, May 2009.
• David Lara, Oscar Rodrı́guez and Chris Dainty, “Far-field threedimensional optical polarimetry”, paper in preparation.
13
Chapter 2
Numerical analysis
Before embarking on the rather difficult experimental verification of the
vectorial polarimetry method introduced in this work, we used a number
of numerical methods to model its performance and assess its feasibility.
The numerical tools used in this research were developed from scratch
based on known techniques and a number of tests were done to verify
the accuracy and reliability of our implementation of each technique, as
shall be discussed in this chapter.
2.1
Vectorial theory of diffraction
It is known that the scalar theory of diffraction is only applicable when
focusing with low NA. Therefore, to describe the EM field distribution
in the focal region of a high NA system it is necessary to use a vectorial
theory of diffraction.
Different approaches have been made to establish a vectorial theory
of diffraction although, currently, the most used is the theory developed
by Wolf [4]. In his theory, Wolf described the focused field as the coherent superposition of plane waves, each of which is related to a ray
directed from a point in the exit pupil of the high NA system towards its
geometrical focus. Wolf associated a weighted electric (or magnetic) field
with each of these rays and calculated the total field using the following
equations
ZZ
a(sx , sy )
ik
exp(ik[Φ(sx , sy ) + ŝ · rp ])dsx dsy
(2.1)
E(P ) = −
2π
sz
Ω
ZZ
b(sx , sy )
ik
H(P ) = −
exp(ik[Φ(sx , sy ) + ŝ · rp ])dsx dsy
(2.2)
2π
sz
Ω
15
Chapter 2. Numerical analysis
Ei
Er
φ
s
x
rp P
θ
y
O
z
Figure 2.1: Diagram of the light focused by a high NA system [37]. The
origin of the coordinate system is in the geometrical focus of the lens.
where k is the wave number, ŝ is a unitary vector in the direction of the
ray, Φ(sx , sy ) is the aberration function, i.e. the deviation of the actual
wavefront from a sphere centered in the geometrical focus, a(sx , sy ) and
b(sx , sy ) are the so-called strength vectors or strength factors, i.e. the
weighted fields, Ω is the solid angle subtended by the exit pupil as seen
from the geometrical focus and rp is the vector from the origin to the
observation point P (see Fig. 2.1).1 The integrals shown in Eqs. (2.1)
and (2.2) are usually referred to as the Debye-Wolf integrals due to the
resemblance that they bear with Debye’s integrals for scalar diffraction.
Using Debye-Wolf integrals, Richards and Wolf [16] calculated the field
distribution in the focal plane of an aplanatic system as a function of its
NA for incident light linearly polarised. An aplanatic system is a system
that obeys Abbe’s sine condition. Thus, in the context of Richards and
Wolf work, an aplanatic system is a system for which any ray entering
the system parallel to the optical axis at a height h, crosses a sphere with
radius f , where f is its focal length, at a height h in the image space.
Fig. 2.2 is a diagram of a system satisfying this condition. Richards and
Wolf found that, for an aplanatic system, the electric field components
can be written as follows
e2x = −iK[I0 + I2 cos 2φp ]
e2y = −iKI2 sin 2φp
e2z = −2KI1 cos φp
1
(2.3)
The geometrical focus is the gaussian focus obtained by tracing rays in an
aberration-free lens.
16
Chapter 2. Numerical analysis
Incident
ray
h
Focused
ray
Aplanatic
system
θ
f
Optical
axis
Figure 2.2: Diagram of the aplanatic system studied by Richards and
Wolf [16].
with the functions In given by
Z α
I0 =
(cos θ)1/2 sin θ(1 + cos θ)J0 (krp sin θ sin θp )
0
× exp[ikrp cos θ cos θp ]dθ
Z α
I1 =
(cos θ)1/2 (sin θ)2 J1 (krp sin θ sin θp )
0
× exp[ikrp cos θ cos θp ]dθ
Z α
(cos θ)1/2 sin θ(1 − cos θ)J2 (krp sin θ sin θp )
I2 =
0
× exp[ikrp cos θ cos θp ]dθ
(2.4)
where 0 ≤ θ, θp < π and 0 ≤ φp < 2π are the spherical and azimuthal
angles, respectively, α is the angular semi-aperture of the exit pupil,
and K is a constant factor that depends on the wavelength, the focal
length of the system, and the refractive index of the surrounding medium.
Equivalent relations can be readily found for the magnetic field.
An important result of Richards and Wolf work is that the irradiance distribution in the focal plane starts to differ from the Airy pattern
predicted by the scalar theory as the NA is increased, and is highly
asymmetric for high NA. In fact, their results showed that the principal
maximum elongates beyond the theoretical Airy disc radius in the direction of the incident polarisation and shortens below this radius in the
orthogonal direction. Fig. 2.3 shows the result presented by Richards
and Wolf for the limiting case NA→1 in air.2
2
If the optical system is immersed in a medium with higher refractive index, the
limiting NA is increased but the corresponding maximum angular semi-aperture is
17
Chapter 2. Numerical analysis
10
90
8
1 1
0.30.3
6
4 4
4
1 21 2
80
0.50.50.70.7
70
2 2
0.20.20.9
1 1
2
vy
0.40.4
7070
3030
0
5050
9090
60
50
3 3
40
1010
-2
30
-4
20
-6
10
-8
-10
-10
-8
-6
-4
-2
0
vx
2
4
6
8
10
Figure 2.3: Irradiance in the focal plane of a NA→1 (in air) lens for
incident light linearly polarised in the x-direction. The irradiance is
normalized to be 100 in the geometrical focus and the units in the axes
are optical units.
18
Chapter 2. Numerical analysis
An extension of Richards and Wolf work, to consider focusing through
a planar interface between two media of mismatched refractive indices,
was given by Török et al. in [37]. This case is useful, for instance, in
microscopy and optical data storage, where an intermediate medium has
an effect over the focused light. The generalization given by Török et
al. introduces the so-called generalized Jones matrices to compute the
strength factor of each ray. The generalized Jones matrices are 3 × 3
matrices that account for the geometrical transformations suffered by
the electric (or magnetic) field of a typical ray as it passes through the
optical system.
Following the formalism introduced in [37], and assuming that the
electric field remains on the same side and at constant angle with respect
to a meridional plane, the strength factor of a typical ray is found to be
(2)
(0)
E(x,y,z) = R−1 [P(2) ]−1 IA(θ1 )P(1) LRE(x,y,z)
(2.5)
where R, L, P(1) , P(2) and I are the generalized Jones matrices described
(0)
(2)
below, E(x,y,z) and E(x,y,z) are the incident and focused field in the second
medium, respectively, and A(θ1 ) is an apodization function that accounts
for energy conservation in an aplanatic system (see Fig. 2.4).3
The generalized Jones matrices describe the geometrical transformations suffered by the incident field during its pass through the lens. This
transformations can be explained as follows: R is a rotation around the
z-axis, by an angle φ1 , to make the incident field parallel to the meridional plane that contains the optical axis and the incident ray, L is a
rotation around the direction orthogonal to the meridional plane, by an
angle θ1 , to make the incident field orthogonal to the ray that goes from
a point in the pupil to the geometrical focus, P(1) is a coordinate system
transformation to obtain the field components in terms of components
parallel and orthogonal to the plane of incidence in the interface between
the two mismatched media, I is a matrix with the Fresnel transmission
coefficients for the aforementioned components, [P(2) ]−1 is the inverse
coordinate system transformation that yields the transmitted field components in the coordinate system defined by the meridional plane and
its orthogonal, and [R]−1 is the inverse rotation around the z-axis, by
an angle −φ1 , to obtain the focused field components in the global XYZ
coordinate system.
the same, namely, α = π/2. In this case, the value of K in Eqs. (2.3) is modified
since it depends on the refractive index of the immersion medium.
3
Richards and √
Wolf proved in [16] that the apodization function for an aplanatic
system is A(θ) = cos θ.
19
Chapter 2. Numerical analysis
E
(0)
E
(1)
u
s1
interface
z = -d
φ1
E
(2)
s2
x
rp P
θ2
y
n1
O
z
n2
Figure 2.4: Diagram of the light focused through a planar interface between two media of mismatched refractive indices [37].
With the strength factor given in Eq. (2.5), Török et al. calculated
the electric (and magnetic) field distribution in the focal region of the
system shown in Fig. 2.4, for incident light linearly polarised, by setting
the appropriate boundary conditions over the fields given by Debye-Wolf
integrals (Eqs. (2.1) and (2.2)) at the interface between the two media.
The result of their calculations can be written in the same way shown in
Eq. (2.3) but with the constant factor K depending also on the refractive
index of medium 1, n1 , and integrals In given by
Z α
I0 =
(cos θ1 )1/2 (sin θ1 ) exp[ik0 Ψ(θ1 , θ2 , −d)]
0
v sin θ1
× (τs + τp cos θ2 )J0
sin α
iu cos θ2
× exp
dθ1
sin2 α
Z α
(cos θ1 )1/2 (sin θ1 ) exp[ik0 Ψ(θ1 , θ2 , −d)]
I1 =
0
v sin θ1
iu cos θ2
× τp (sin θ2 )J1
exp
dθ1
sin α
sin2 α
Z α
I2 =
(cos θ1 )1/2 (sin θ1 ) exp[ik0 Ψ(θ1 , θ2 , −d)]
0
v sin θ1
× (τs − τp cos θ2 )J2
sin α
20
Chapter 2. Numerical analysis
× exp
iu cos θ2
sin2 α
dθ1
(2.6)
where v and u are the optical units defined as
v = k1 rp sin θp sin α
u = k2 rp cos θp sin2 α
(2.7)
for k1 and k2 the wave numbers in media 1 and 2, respectively, τp and
τs are the Fresnel transmission coefficients for the p and s polarisation
components, respectively, at the interface between media 1 and 2, θ1 and
θ2 are the spherical angles in media 1 and 2, respectively, α is the semiaperture of the exit pupil in medium 1 and Ψ(θ1 , θ2 , −d) is a spherical
aberration function due to the interface.
The introduction of spherical aberration in the definition of the In ’s is
the main difference between the results given by Richards and Wolf and
those by Török et al. It is worth noticing that, as pointed out by Török
et al., Eqs. (2.6) reduce to Eqs. (2.4) when the special case n1 = n2 is
considered, as should be expected. The numerical results obtained by
Török et al., based on their generalization of Debye-Wolf integrals, are
shown in [38].
Another interesting example of the application of the vectorial theory
of diffraction is the focusing of circularly polarised light. In that case,
the electric field components in the focal region are
K
e2x = √ [I0 + I2 exp(∓2iφp )]
2
∓iK
e2y = √ [I0 − I2 exp(∓2iφp )]
2
−2iK
e2z = √ I1 exp(∓iφp )
2
(2.8)
where the ∓ sign indicates the handedness: - for right and + for left
circular polarisation. The integrals In are the same as in the case of
linear polarisation.
Note that, because of the form of the electric field components in the
focal region given by Eqs. (2.3), for linear polarisation, and Eqs. (2.8),
for circular polarisation, the state of polarisation in the geometrical focus
is that of the incident light. This is to be expected since all the other
components of the rays reaching the focal region at an angle different from
zero interfere destructively in the focal point. From the mathematical
point of view, the Bessel functions of the first kind and order n, Jn (ξ),
21
Chapter 2. Numerical analysis
are identically zero at ξ = 0 for all n 6= 0 (J0 (0) = 1). Thus, the only In
different from zero at the focus is I0 , which corresponds to the incident
polarisation state.
2.2
Evaluation of the diffraction integrals
In the previous section, it was briefly discussed how to calculate the
electric field in the focal region of a high NA system. However, the
expressions for the field components are given in terms of the integrals In
for which no analytical solutions are known. To overcome this problem, it
is possible to perform the numerical integration of these three functions.
Nevertheless, it is necessary to be very careful with these integrations due
to the rapid variation of the exponential term in the integrals, which can
produce considerable numerical errors in the evaluation of the functions.
To simplify the evaluation of these integrals, Török et al. developed
in [39] an analytical way to transform the highly unstable integrals In
into more stable functions. The transformation proposed in [39] reduces
the integrals to a set of infinite series that can be accurately evaluated
with a relatively small number of terms due to its rapid convergence.
The transformation proposed by Török et al. changes the integrals to
the following functions
∞
n2 X (0) s (1/2)
a i Cs (cos ψ)js (ω)
I0 = 2
n1 s=0 s
∞
X
n2
s (3/2)
I1 = 2 (sin ψ)
a(1)
(cos ψ)js+1 (ω)
s i Cs
n1
s=0
∞
X
n2
2
s (5/2)
I2 = 2 (sin ψ)
a(2)
(cos ψ)js+2 (ω)
s i Cs
n1
s=0
(n+1/2)
(2.9)
where, for n = 0 . . . 2, Cs
is the Gegenbauer polynomial of the first
kind and order n + 1/2, js+n is the spherical Bessel function of the first
kind and order s + n, ω and ψ are generalized optical units, and the
expansion coefficients are given by
Z β
cos θ2
(0)
(0)
√
(sin θ1 )(τs + τp cos θ2 )
as = Ns
cos θ1
0
× exp[ik0 Ψ(cos θ2 )]Cs(1/2) (cos θ2 )dθ2
Z β
cos θ2
(1)
(1)
√
as = Ns
(sin θ1 )τp exp[ik0 Ψ(cos θ2 )]
cos θ1
0
22
Chapter 2. Numerical analysis
a(2)
s
× Cs(3/2) (cos θ2 )(sin2 θ2 )dθ2
Z β
cos θ2
(2)
√
= Ns
(sin θ1 )(τs − τp cos θ2 )
cos θ1
0
× exp[ik0 Ψ(cos θ2 )]Cs(5/2) (cos θ2 ) sin2 θ2 dθ2
(2.10)
(n)
with Ns a normalization factor. These expressions are, at least in
principle, more stable than those given by Eqs. (2.6) and thus, easier
to evaluate.4 Alternatively, the evaluation of the Debye-Wolf diffraction
integrals can be done using the eigenfunctions expansion introduced by
Sherif and Török [40] and by Sherif et al. [41].
Although the expressions given in Eqs. (2.9) and (2.10) can be implemented in a computer programme (in fact, during the development
of this research this approximation was used before any other) it is also
possible to evaluate the integrals In using the approximations shown in
[42]. In this work, Wilson et al. showed that the form and relative magnitude of the integrals do not depend strongly on the value of the aperture
and thus, they can be approximated by their values for small apertures.
Using this approximation, the integrals reduce to
J1 (v)
v
J
(v)
2
I1 (v, α) ≈ α3
v
4
α J3 (v)
I2 (v, α) ≈
2 v
I0 (v, α) ≈ 2α2
(2.11)
with v given by Eq. (2.7), which can be easily evaluated.
2.3
Calculation of the scattered field
There are many different approaches to perform scattering calculations.
Some of them are analytical (T-matrix, Mie scattering, etc.) and some
others numerical (Discrete Dipole Approximation, Finite Element, etc.)
but most of them are only applicable in particular cases. In this work, we
chose to use the Finite-Difference Time-Domain (FDTD) method [43, 44]
due to its relative ease of implementation in a computer programme and
its reliability proved in a number of applications [45–47].
4
The form of the spherical aberration function, Ψ(cos θ2 ), obtained by Török et
al. in [39], may introduce further instabilities in the calculations. However, a suitable
way to avoid these instabilities was presented by Török et al. in the same paper.
23
Chapter 2. Numerical analysis
z
Hx
Ez
Hy
(i∆, j∆, k∆)
Ex
Ey
y
Hz
x
Figure 2.5: Yee’s cubic unit-cell at location (i∆, j∆, k∆) for i, j, k positive integers and ∆ the cell’s side length. The electric and magnetic field
components are located in their actual position within the unit-cell.
2.3.1
FDTD method
The FDTD method is a numerical method for the solution of Maxwell’s
coupled curl equations introduced by Yee [48].5 A comprehensive discussion of the method is beyond the scope of this thesis and only the main
ideas shall be discussed here.
In the FDTD method, the region of space to be modelled is divided
in so-called unit-cells. Each unit-cell has associated three components
of the electric field and three components of the magnetic field. Fig.
2.5 is a schematic diagram of Yee’s cubic unit-cell with the electric and
magnetic fields positioned accordingly. To solve the Ampère-Maxwell
equation for the electric field in the whole modelled space at a time (n +
1/2)∆t, with n a positive integer and ∆t the duration of the time-step,
the space derivatives of the magnetic field components are approximated
by finite differences between their value in adjacent unit-cells at time
n∆t. Then, the time derivative of the electric field is approximated by a
finite difference between values of the field separated one time-step, and
the electric field at time (n+1/2)∆t is obtained as function of the electric
field at time (n−1/2)∆t and the magnetic field components at time n∆t.
For instance, in the notation of the FDTD method, the x component of
5
Although the FDTD method itself does not solve explicitly the divergence equations, it can be proved that the fields obtained with this method satisfy them. A
proof of this statement can be found in [43].
24
Chapter 2. Numerical analysis
the electric field at time (n + 1/2)∆t is written as
n+1/2
Ex |i+1/2,j,k
= Ca,Ex |i+1/2,j,k Ex |n−1/2
i+1/2,j,k
+ Cb,Ex |i+1/2,j,k [Hz |ni+1/2,j+1/2,k − Hz |ni+1/2,j−1/2,k − . . .
− Hy |ni+1/2,j,k+1/2 + Hy |ni+1/2,j,k−1/2 − Jsourcex |ni+1/2,j,k ∆]
(2.12)
where the subscripts indicate the unit-cell where the field is being evaluated, with its corresponding spatial offset, and the superscripts the
time-step at which the field is taken. In Eq. (2.12) Ca,Ex |i+1/2,j,k and
Cb,Ex |i+1/2,j,k are factors that depend on the cell’s side length, the duration of the time-step and the physical properties of the space at the
location of the unit-cell. Jsourcex |ni+1/2,j,k is a current density associated
to external sources. Similar relations can be written for the y and z
components of the electric field.
Once the electric field has been computed across the modelled space,
the magnetic field is updated, half a time-step later, using the values of
the electric field calculated as described above. Thus, the FDTD expression for the y component of the magnetic field, as given by Faraday’s
law, is written as
n
Hy |n+1
i+1/2,j,k+1/2 = Da,Hy |i+1/2,j,k+1/2 Hy |i+1/2,j,k+1/2
n+1/2
− Db,Hy |i+1/2,j,k+1/2 [Ez |n+1/2
i,j,k+1/2 − Ez |i+1,j,k+1/2 − . . .
n+1/2
n+1/2
− Ex |i+1/2,j,k
+ Ex |n+1/2
i+1/2,j,k+1 − Msourcey |i+1/2,j,k+1/2 ∆]
(2.13)
where Da,Hy |i+1/2,j,k+1/2 and Db,Hy |i+1/2,j,k+1/2 are factors that depend on
the parameters of the FDTD method (∆ and ∆t) and Msourcey |n+1/2
i+1/2,j,k+1/2
is a “magnetic current” introduced to represent external sources of magnetic field. Again, similar expressions can be written for the x and z
components.
The temporal staircasing nature of the method allows for the solution
of the coupled equations without solving a system of simultaneous equations. By repeating this process a sufficiently large number of time-steps,
we can compute the stationary solution for the scattered field.
A fundamental part of the FDTD method is the modelling of the
incident field. Several approaches to model this field have been proposed
in the past years; the point source condition, the total-field/scatteredfield formulation and the scattered-field formulation are three of the most
commonly used. In this work we used the scattered-field formulation
because it allows us to easily introduce fields with non-planar wavefronts,
25
Chapter 2. Numerical analysis
λ/2
/2
λ/6
/6
X
Y
Z
Figure 2.6: Sub-resolution rail-shaped scatterer made of a PEC.
as is the case in our tightly focused incident field. The scattered-field
formulation stems from the fact that the total field in the surface and
interior of the scatterer must satisfy the boundary conditions for the
EM field. For instance, for the scatterers modelled in this work —i.e.,
scatterers made of a perfect electric conductor (PEC)— the boundary
conditions can be written as
at the interface
−Eti
t
Es =
(2.14)
0
inside
where Ets and Eti are the components of the scattered and incident electric
field, respectively, tangential to the surface of the scatterer. Thus, with
prior knowledge of the incident field, we can set the initial conditions
within the FDTD space using Eq. (2.14) and start the time-stepping
process for the solution of Maxwell’s curl equations.
In its most simple formulation, as originally introduce by Yee and
used in this work, the space modelled is divided in cubic unit-cells. This
formulation simplifies the expressions for the calculation of the EM field
components but limits the shape of the scatterers that can be accurately
modelled.6 The cubic unit-cell is suitable for the scatterers modelled in
this work, namely: the sub-resolution rail-shaped scatterer in Fig. 2.6,
the square-shaped scatterer in Fig. 2.7, and the cross-shaped scatterer
in Fig. 2.8.
Another important aspect of the unit-cell is its size. To reduce numerical dispersion (i.e. the change in the propagation velocity as a function of the direction of propagation within the FDTD modelled space)
and achieve accurate results, the unit-cell must have a side length much
smaller than the smallest wavelength considered to interact with the
6
For instance, modelling scatterers with round edges may produce large numerical
errors.
26
Chapter 2. Numerical analysis
X
λ/6
/6
Y
λ/2
/2
Z
Figure 2.7: Sub-resolution square-shaped scatterer made of a PEC.
X
λ/6
/6
Y
λ/2
/2
Z
Figure 2.8: Sub-resolution cross-shaped scatterer made of a PEC.
scatterer.7 The actual side length of the unit-cell, compared to the wavelength, depends on the particular application and numerical experiments
have to be done to assess the effects of numerical dispersion. In our implementation of the FDTD method, we chose a side length ∆ = λ/30
which, according to our numerical experiments, reduces sufficiently the
numerical dispersion.
The length of the time-step is also important in the FDTD method.
Large time-steps generate numerical instability that yields noisy, inaccurate results. In fact, the shortest the time-step, the highest the stability
and accuracy of the results. However, if a very small time-step is chosen, a larger number of time-steps is required to reach the stationary
state, increasing the computation time. The upper limit for the duration
of the time-step, that guaranties the numerical stability of the FDTD
method, is given by the Courant factor. For a three-dimensional cubic
unit-cell space, the Courant factor limits the length of the time-step to
the following range
∆
∆t ≤ √
(2.15)
c 3
7
In our case, we modelled monochromatic incident light with λ = 532nm but the
FDTD method itself allows for the modelling of broadband sources.
27
Chapter 2. Numerical analysis
where c is the speed of light in vacuo.
In many applications it is necessary to model the propagation of the
scattered field in free space. However, if the simulations need to be
run for a long time, the computed fields reach the outer boundaries of
the modelled space and spurious, i.e. non-physical, reflections occur.
The spurious reflections propagate back into the computational space
corrupting the calculated fields. The effect of these reflections can be
reduced to a minimum using absorbing boundary conditions (ABCs).
The ABCs permit the absorption of the scattered field in the outer regions
of the modelled space and, in this way, reduce the spurious reflections.
The optimization of the ABCs is, perhaps, one of the most important
parts of its implementation. Nevertheless, no standard method for this
is available making it very tricky.
2.4
Near- to far-field transformation
Although the FDTD method is useful for the calculation of the scatteredfield, the size of the space region that can be modelled is limited by the
computer resources available, particularly by the memory. Therefore,
usually it is only possible to model small regions of space around the
scatterer.8 In many applications, the far-scattered field is required, as it
is in our case, and a near- to far-field (NTFF) transformation must be
performed.
Török et al. [49] introduced a method for the calculation of the
far-field from the near-field using a modified version of the StrattonChu integrals for diffraction [50]. In their method, as in any NTFF
transformation, the far-field at any point in the space outside a volume
containing all the sources and sinks of EM field can be calculated from
the value of the EM field over a surface enclosing the volume. The
advantage of Török et al. method over similar methods is that the farfield EM components are given in terms of the cartesian components of
the near-field, which simplifies its application to FDTD results.
Based on Török et al. method, we developed a programme to perform the NTFF transformation of our FDTD results. This programme
calculates the far-field over a surface at an arbitrary distance form the
scatterer. For instance, it can be used to calculate the far-field over a
plane parallel to the XY plane at a distance zp from the scatterer.9 To
8
By small we mean regions with dimensions comparable to the wavelength.
The size of the plane and the distance form the scatterer are parameters that can
be modified in the programme.
9
28
Chapter 2. Numerical analysis
6
x 1017
NTFF transformation
Analytical solution
5.5
Irradiance (a.u.)
5
4.5
4
3.5
3
2.5
−500
−400
−300
−200
−100
0
Y/λ
100
200
300
400
500
Figure 2.9: Comparison between the analytical solution and the NTFF
transformation for the electric field irradiance radiated by an electric
dipole with dipole moment oriented along the x-axis. The observation
plane is a square with side a = 1000λ at a distance zp = 500λ from the
dipole.
test our NTFF programme, we used the analytical solution of the field
radiated by a dipole to calculate the field components over the surface of
a small parallelogram (with sides comparable to the wavelength) enclosing the dipole. These results were used to feed the NTFF transformation
programme. Then, we calculated the far-field radiated by the dipole,
with the analytical solution, and compared it with the field obtained
with the NTFF transformation programme over a square plane of side
a = 1000λ at a distance zp = 500λ from the scatterer. The comparison
between the electric field irradiance distribution obtained using the analytical solution and the NTFF transformation along the y-axis is shown
in Fig. 2.9 for a dipole with dipole moment in the x-direction. The
relative error, defined by Török et al. as
N
X
=
|Esc (ri ) − EAn (ri )|2
i=1
N
X
,
|EAn (ri )|2
i=1
29
(2.16)
Chapter 2. Numerical analysis
was calculated and a maximum value = 1.3181 × 10−6 , which is small
enough for our application, was obtained. A close look at Fig. (2.9) shows
that the error between the analytical solution and the NTFF transformation is larger at the edges of the plot, even if it is hard to notice it
due to the scale, indicating that the numerical errors are smaller at the
center of the observation plane.
2.5
McCutchen’s method
The vectorial theory of diffraction introduced by Wolf has been proved
suitable for the calculation of the EM field distribution in the focal region of a high NA lens. However, the definition of the strength factor,
apart from a few well known cases such as linear, circular, azimuthal
and radial polarisation, can be difficult. This is particularly difficult for
non-homogeneous incident polarisation states.10 Thus, it is appropriate
to look for alternatives to simplify the calculation of the focused field for
non-homogeneous incident polarisations. McCutchen’s method [51, 52] is
a good alternative to Wolf’s vectorial theory of diffraction in these cases.
McCutchen’s method states that the field in the focal region of a lens
can be calculated as the three-dimensional Fourier transform of the socalled generalized aperture. The generalized aperture is the projection
of the exit pupil over the surface of a sphere limited by the solid angle
subtended by the exit pupil as seen from the geometrical focus, i.e.,
the angular semi-aperture of the generalized aperture is limited by the
maximum angle permitted by the NA.
The calculation of the generalized aperture in McCutchen’s method is
equivalent to the calculation of the strength factor in the vectorial theory
of diffraction. Thus, during the projection of the pupil over the spherical surface, the different components of the incident field are combined
giving the contribution of each ray to the different components of the
focused field. Each component is calculated independently of the others, which implies three Fourier transforms but, since the method can
be implemented in a computer programme using an FFT algorithm, the
calculations are considerably faster than direct integration of Debye-Wolf
integrals.
For the implementation of McCutchen’s method in a computer programme it is necessary to do a sampling of the generalized aperture. Note
10
A non-homogeneous incident polarisation state is that where the polarisation
varies across the aperture of the beam.
30
Chapter 2. Numerical analysis
FFT
x
y
z
Figure 2.10: Representation of the implementation of McCutchen’s
method in a computer programme. An FFT algorithm is applied to
the sampling of the generalized aperture to obtain the field distribution
in the focal region of the lens. The sampling is finer in the z-direction to
obtain the same number of sampling pixels over the generalized aperture
in the three spatial directions.
that the sampling is done in three-dimensions since the generalized aperture is a surface embedded in a three-dimensional space. In our case,
the sampling was done according to the Whittaker-Shannon sampling
theorem [53] (Fig. 2.10) for a pixel size of λ/30 in the focal region. This
pixel size is the same used for the evaluation of Debye-Wolf integrals and
is consistent with the unit-cell side length of our FDTD programme.
Since Debye-Wolf integrals have become a standard for the calculation of high NA focused fields, we tested our implementation of McCutchen’s method by comparing the results obtained with both methods
for incident light linearly polarised in the x-direction. Fig. 2.11 shows
this comparison as a profile along the x-direction in the focal plane of a
NA=0.95 aplanatic lens at λ = 532nm.
Although the profiles agree quite well, a close look at the results reveals slight differences between the two methods. The main reason for
the differences is the numerical errors due to the sampling of the generalized aperture. A finer sampling yields results more similar to Debye-Wolf
integrals but the demand for computer resources increases enormously.
The definition of non-homogeneous incident polarisation states in McCutchen’s method is relatively easy, compared to Debye-Wolf integrals,
since the polarisation state over the sampling of the generalized aperture
can be described pixel-wise and then the application of an FFT algorithm is straightforward. This characteristic can be used to tailor the
field distribution in the focal region by engineering the incident polari31
Chapter 2. Numerical analysis
1
Debye-Wolf
McCutchen
0.9
Normalized irradiance (a.u.)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-2
-1.5
-1
-0.5
0
X/λ
0.5
1
1.5
2
Figure 2.11: Comparison between McCutchen’s method and Debye-Wolf
integrals for incident light linearly polarised. The profile and incident
polarisation are taken along the x-direction.
sation [36]. The ability to tailor the EM field distribution in the focal
region of a high NA focusing system plays a key role in our method since
it can be used to improve its sensitivity by choosing the field distribution
that produces the largest effect over the polarisation pattern in the exit
pupil of the collector lens for a particular kind of specimens.
Another appealing characteristic of McCutchen’s method is the direct
connection that can be established between the sampling of the generalized aperture and the aperture of a pixel-wise spatial light modulator.
This connection has been investigated by Iglesias and Vohnsen [36] in
the context of using a tandem of pixel-wise spatial light modulators to
mimic continuous polarisation states. An example of this is the similar
EM field distributions obtained in the focal region of a high NA lens for
radial and pseudo-radial incident polarisations. Fig. 2.12 is a schematic
diagram of both polarisation states.
The largest component of the focused field for incident radial polarisation is the z component. The spot produced by this component
is smaller than the corresponding Airy disc and it has been proposed
as a way to increase the resolution of optical scanning microscopes (see
§1.1). Fig. 2.13 is a comparison between profiles of the z component’s
normalized modulus squared along the x-direction of the focal plane for
32
Chapter 2. Numerical analysis
Radial
polarisation
Pseudo-radial
polarisation
Figure 2.12: Radial and pseudo-radial polarisation.
incident radial and pseudo-radial polarisation. This figure shows that
the two profiles are different in the secondary maxima but have the same
central lobe. Since most of the energy is localized in the central lobe, and
is this lobe what gives the resolution of imaging systems with radial illumination, it is apparent that, to a large extent, the use of pseudo-radial
polarisation yields similar results as pure radial polarisation.
2.6
Performance of the method
The preceding sections introduced the basic ideas of the tools used to
model the performance of the vectorial polarimetry method. In this section we shall present examples of the polarisation patterns in the exit
pupil of the collector lens as obtained with the tools developed in this
research.
2.6.1
Point-scatterer
It is a common practice in the modelling of the performance of optical microscopes to analyse as a first specimen a point-scatterer. Such
specimen is usually modelled as a point-dipole with dipole moment, p,
proportional to the incident field. This assumption is valid as long as
the point-scatterer is much smaller than the wavelength of the illumination [54]. Under these circumstances, the scattered field can be obtained
using the analytical solution for the radiation of a dipole which, in the
far-field region, is given by [54]
(s)
E(x,y,z)
1
=−
4π0
k 2 eikr
r × (r × p)
r
33
(2.17)
Chapter 2. Numerical analysis
1
Pseudo-radial
Radial
0.9
0.8
Normalized |Ez |2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-2
-1.5
-1
-0.5
0
X/λ
0.5
1
1.5
2
Figure 2.13: Comparison between profiles of the z component’s normalized modulus squared in the focal plane along the x-direction for incident
radial and pseudo-radial polarisation.
where r is the unit vector in the direction of observation, r is the distance
from the dipole to the point of observation, k is the wavenumber and 0
is the free space permittivity.
Taking the dipole as placed in the geometrical focus, the scattered
field is calculated over the surface of a sphere, with radius equal to the
focal length, for each ray directed from the focus to a point in the exit
pupil of the collector lens.11 This field is then collected and collimated
by the collector lens. Thus, the field in the exit pupil of the collector lens
is obtained as
(2)
(s)
E(x,y,z) = B(θ)R−1 L−1 RE(x,y,z)
(2.18)
where R and L are the same as in Eq. (2.5) and B(θ) is the reciprocal of the apodization function A(θ), introduced to account for the
conservation of energy. Fig. 2.14 shows the Stokes parameters in the
exit pupil of the collector lens for an on axis point-scatterer in the focal
plane with incident light linearly polarised in the x-direction (p is given
by Eqs. (2.3)).12 Each point in the exit pupil corresponds to a particular
11
The calculation of the scattered field has to be done over a spherical surface to
satisfy the aplanatism assumed for the lens.
12
In this work the terms on axis and off axis refer to the position of the pointscatterer with respect to the optical axis, i.e. the z-axis in our coordinate system.
34
Chapter 2. Numerical analysis
S 0x
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
S 2x
Y/NA
0.9
0
0 0.2 0.4 0.6 0.8
X/NA
1
S 3x
1
1
0
−1
−1 −0.8−0.6−0.4−0.2
0
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
1
0.8
−1
−1 −0.8−0.6−0.4−0.2
Y/NA
S 1x
1
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2
0 0.2 0.4 0.6 0.8
X/NA
1
−1
−1 −0.8−0.6−0.4−0.2
−1
0 0.2 0.4 0.6 0.8
X/NA
1
−1
Figure 2.14: Stokes parameters in the exit pupil of the collector lens for
an on axis point-scatterer in the focal plane with incident light, i.e. before
focusing, linearly polarised in the x-direction. The Stokes parameters are
normalized with respect to the maximum irradiance.
35
Chapter 2. Numerical analysis
scattering angle, i.e. the four pupils in Fig. 2.14 are the scattering-angleresolved Stokes parameters of the scattered field. From the figure, it is
apparent that the polarisation state, described by the Stokes parameters, is not homogeneous across the exit pupil of the collector lens. This
inhomogeneity stems from the scattering-angle dependence of the scattered field components and the projection of the longitudinal component
over the transversal components; this is the kind of information that the
vectorial polarimetry method retrieves.
To understand the Stokes parameters shown in Fig. 2.14 recall that
the dipole field detected by an observer depends on the angle between
the dipole moment and the direction of observation, as well as on the
distance to the point of observation. That is, for a given direction of
observation, the observer sees a dipole moment equal to the projection
of the actual dipole moment over a plane orthogonal to the direction
of observation. Thus, its magnitude and orientation are different for
different directions of observation. This characteristic of the radiated
field gives the dipole its particular toroidal radiation distribution. On
the other hand, large values of θ correspond to large values of B(θ), the
reciprocal of the apodization function. Recall that B(θ) was introduced
for the conservation of energy. Therefore, the irradiance of the radiated
field is larger at the rim of the exit pupil. The combination of both effects
is observed in elements S0x , S1x and S2x of Fig. 2.14, whereas S3x equal
to zero indicates that there is no circularly polarised component of the
radiated field; this is to be expected, from the discussion above, for a
linearly polarised incident field. Fig. 2.15 shows the Stokes parameters
for the same case shown in Fig. 2.14 but normalized with respect to the
irradiance distribution, i.e. pixel by pixel. This kind of normalization
removes the irradiance variations in the Stokes parameters distribution
and is useful to analyse the polarisation state distribution. However, as
it shall be discussed in Chap. 5, we decided to analyse our results using
a normalization by the maximum irradiance in the pupil.
Another possibility which is of interest, as we shall see later in this
section, is to analyse the Stokes parameters in the exit pupil of the collector lens for incident light circularly polarised. Fig. 2.16 shows the
results obtained for an on axis point-scatterer in the focal plane with
right circular incident polarisation. From the figure, we see that S0r
is circularly symmetric whereas S1r and S2r have a two-fold symmetry.
Note that S2r is rotated 45◦ , around the optical axis, with respect to
S1r . These distributions can be understood using the same argument of
the direction of observation given above, but for a dipole whose dipole
36
Chapter 2. Numerical analysis
S 0x
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
Y/NA
0.9
0
−1
−1 −0.8−0.6−0.4−0.2
0
0
0 0.2 0.4 0.6 0.8
X/NA
S
S 2x
1
1
1
3x
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
1
0.8
−1
−1 −0.8−0.6−0.4−0.2
Y/NA
S 1x
1
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2
0 0.2 0.4 0.6 0.8
X/NA
1
−1
−1 −0.8−0.6−0.4−0.2
−1
0 0.2 0.4 0.6 0.8
X/NA
1
−1
Figure 2.15: Stokes parameters in the exit pupil of the collector lens for
an on axis point-scatterer in the focal plane with incident light, i.e. before
focusing, linearly polarised in the x-direction. The Stokes parameters,
except S0x , are normalized pixel by pixel with respect to the irradiance
distribution. S0x is normalized with respect to the maximum irradiance.
37
Chapter 2. Numerical analysis
S 0r
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
S 2r
Y/NA
0.9
0
0 0.2 0.4 0.6 0.8
X/NA
1
S 3r
1
1
0
−1
−1 −0.8−0.6−0.4−0.2
0
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
1
0.8
−1
−1 −0.8−0.6−0.4−0.2
Y/NA
S 1r
1
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2
0 0.2 0.4 0.6 0.8
X/NA
1
−1
−1 −0.8−0.6−0.4−0.2
−1
0 0.2 0.4 0.6 0.8
X/NA
1
−1
Figure 2.16: Stokes parameters in the exit pupil of the collector lens
for an on axis point-scatterer with incident light circularly polarised to
the right. The Stokes parameters are normalized with respect to the
maximum irradiance. The constant value in the distribution of S3r is the
same, in absolute value, as the normalized total irradiance in the centre
of the pupil, namely, 0.569.
38
Chapter 2. Numerical analysis
moment is describing a circle around the optical axis. Since the dipole is
describing a circle in the focal plane, an observer, at a distance equal to
the focal length from this plane, moving away from the optical axis sees
a dipole moment equal to the projection of the circular dipole moment
over a plane perpendicular to the direction of observation. Thus, when
the distance from the optical axis is increased, the ellipse described by
the projection of the circular dipole moment shortens in the direction
parallel to the direction of the displacement, remaining constant in the
orthogonal direction. As a consequence, as the distance from the optical
axis is increased, the component of the scattered field linearly polarised in
the direction orthogonal to the direction of the displacement is increased.
In Fig. 2.16, we can also see that S3r is different from zero with a
constant value across the pupil; this value corresponds to the total irradiance in the centre of the pupil (0.569 when normalized with respect
to the maximum irradiance in the pupil). The existence of a circularly
polarised component is not surprising since the incident light is circularly
polarised. The handedness, on the other hand, is changed because we
are modelling the performance of the vectorial polarimeter in a reflection configuration. Finally, we note that, although S3r remains constant
across the exit pupil, its relative irradiance, with respect to the total
irradiance given by S0r , is reduced as we go further from the centre of
the pupil.
So far, we have discussed two different illuminations for an on axis
point-scatterer. As we saw, the distribution of the Stokes parameters in
the exit pupil depends on the characteristics of the incident field. Since
the EM field distribution has a non-constant profile around the focal
region, it is worth analysing the polarisation distribution in the exit pupil
of the collector lens for a slightly off axis point-scatterer. Fig. 2.17 is
a diagram of the focal plane with the on axis and off axis positions of
the point-scatterer analysed in this work marked. For off axis positions,
the z components of the electric field for the different rays coming from
the pupil to the point of observation do not cancel out and an effective
z component appears. The change in the relative phase between the z
components, at positions different from the geometrical focus, is the main
reason for the existence of the z component.
Figs. 2.18 and 2.19 show the Stokes parameters in the exit pupil
for a point-scatterer in the focal plane at x = −λ/3 and x = +λ/3,
respectively, with incident light linearly polarised. From the figures, it
is clear that the Stokes vector for the off axis point-scatterer is different
from that for the on axis scatterer shown in Fig. 2.14. However, the first
39
Chapter 2. Numerical analysis
Y
−λ/3
+λ/3
X
Figure 2.17: Diagram of the focal plane marking the three positions of
the point-scatterer analysed. The origin of the XY coordinate system
corresponds to the on axis position.
three Stokes parameters are the same in both vectors indicating that
the information contained in them is not enough to differentiate between
the two positions of the scatterer. The fourth elements have a similar
distribution in both vectors but the handedness is reversed. Therefore,
S3x can be used to distinguish between the two off axis positions of the
scatterer.
A look at the components of the EM field in the focal region reveals
that the Ey component is zero along the x-axis and the Ex component
has the same value at both positions of the scatterer. The value of the
Ez component at x = −λ/3, on the other hand, is the negative of its
value at x = +λ/3. Thus, the key to distinguish between the two off
axis positions is in the Ez component of the focused field. The fact
that Ey is zero along the x-axis, for incident light linearly polarised in
the x-direction, whereas Ex and Ez are non-zero, shows that the electric
field along this axis oscillates in a plane orthogonal to the focal plane.13
Therefore, using the same arguments about the direction of observation,
we can see that, for an observer moving along the x-axis, the effective
dipole moment is always a linear dipole with magnitude that increases as
he moves away from the optical axis. If the observer now moves along the
y-axis, he sees an ellipse with increasing circularly polarised component
as the distance from the optical axis is increased. The handedness of the
circularly polarised component depends on the phase of the z component,
13
Richards and Wolf showed in [16] that for every point in the focal plane, the
polarisation ellipse is orthogonal to that plane.
40
Chapter 2. Numerical analysis
S 0x
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
S 2x
Y/NA
0.9
0
0 0.2 0.4 0.6 0.8
X/NA
1
S 3x
1
1
0
−1
−1 −0.8−0.6−0.4−0.2
0
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
1
0.8
−1
−1 −0.8−0.6−0.4−0.2
Y/NA
S 1x
1
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2
0 0.2 0.4 0.6 0.8
X/NA
1
−1
−1 −0.8−0.6−0.4−0.2
−1
0 0.2 0.4 0.6 0.8
X/NA
1
−1
Figure 2.18: Stokes parameters in the exit pupil of the collector lens
for a point-scatterer in the focal plane at x = −λ/3 with incident light
linearly polarised. The Stokes parameters are normalized with respect
to the maximum irradiance.
41
Chapter 2. Numerical analysis
S 0x
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
S 2x
Y/NA
0.9
0
0 0.2 0.4 0.6 0.8
X/NA
1
S 3x
1
1
0
−1
−1 −0.8−0.6−0.4−0.2
0
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
1
0.8
−1
−1 −0.8−0.6−0.4−0.2
Y/NA
S 1x
1
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2
0 0.2 0.4 0.6 0.8
X/NA
1
−1
−1 −0.8−0.6−0.4−0.2
−1
0 0.2 0.4 0.6 0.8
X/NA
1
−1
Figure 2.19: Stokes parameters in the exit pupil of the collector lens
for a point-scatterer in the focal plane at x = +λ/3 with incident light
linearly polarised. The Stokes parameters are normalized with respect
to the maximum irradiance.
42
Chapter 2. Numerical analysis
S 0r
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
S 2r
Y/NA
0.9
0
0 0.2 0.4 0.6 0.8
X/NA
1
S 3r
1
1
0
−1
−1 −0.8−0.6−0.4−0.2
0
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
1
0.8
−1
−1 −0.8−0.6−0.4−0.2
Y/NA
S 1r
1
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2
0 0.2 0.4 0.6 0.8
X/NA
1
−1
−1 −0.8−0.6−0.4−0.2
−1
0 0.2 0.4 0.6 0.8
X/NA
1
−1
Figure 2.20: Stokes parameters in the exit pupil of the collector lens
for a point-scatterer in the focal plane at x = −λ/3 with incident light
circularly polarised to the right. The Stokes parameters are normalized
with respect to the maximum irradiance.
which is consistent with the π phase difference between the two off axis
positions.
The Stokes vectors in the exit pupil of the collector lens for an off
axis point-scatterer with incident light circularly polarised to the right
are shown if Figs. 2.20, for x = −λ/3, and 2.21, for x = +λ/3.
In this case, it is also clear that the off axis Stokes parameters are
different from the on axis ones. Furthermore, the four Stokes parameters are different for the two positions of the off axis point-scatterer.
The analysis of the EM field distribution in the focal region reveals that
the Ex and Ey components are different from zero, and from each other,
and have the same value at both off axis positions. Furthermore, just
as for the case of incident light linearly polarised, the value of the Ez
component at x = −λ/3 is the negative of its value at x = +λ/3, indicating the importance of the Ez component to differentiate between
the two positions of the scatterer. Since the Ex and Ey components are
43
Chapter 2. Numerical analysis
S 0r
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
S 2r
Y/NA
0.9
0
0 0.2 0.4 0.6 0.8
X/NA
1
S 3r
1
1
0
−1
−1 −0.8−0.6−0.4−0.2
0
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
1
0.8
−1
−1 −0.8−0.6−0.4−0.2
Y/NA
S 1r
1
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2
0 0.2 0.4 0.6 0.8
X/NA
1
−1
−1 −0.8−0.6−0.4−0.2
−1
0 0.2 0.4 0.6 0.8
X/NA
1
−1
Figure 2.21: Stokes parameters in the exit pupil of the collector lens
for a point-scatterer in the focal plane at x = +λ/3 with incident light
circularly polarised to the right. The Stokes parameters are normalized
with respect to the maximum irradiance.
44
Chapter 2. Numerical analysis
different from zero, the polarisation ellipse will be tilted when compared
to the case of incident light linearly polarised. However, the scattered
field distribution in this case is not as straightforward to interpret as the
previous case and requires further analysis.
An important remark concerning the results presented in this section
is that, for all the cases analysed, the degree of polarisation remained
unity at every pixel. Thus, just as expected, the point-scatterer does not
affect the degree of polarisation.
It is important to notice that the displacement of the point-scatterer
in each direction (λ/3) is well below the diffraction limit (0.61λ according to Rayleigh’s criterion). Therefore, Figs. 2.18–2.21 show that the
vectorial polarimetry method can be used to retrieve sub-diffraction information of the scatterer’s position.
The maximum resolution that can be achieved with the vectorial polarimetry method depends on the variations of the focused field, which
can be engineered by tailoring the incident polarisation state (see §2.5),
and on the ability to differentiate between exit pupil distributions of the
Stokes parameters. In this section we have done a qualitative analysis
of the differences between pupil distributions. However, a good way to
determine the maximum resolution of the vectorial polarimetry method
would be to define a metric to measure, quantitatively, the difference
between two pupil distributions of a given Stokes parameter. This metric should consider the statistics of the distributions, the noise in the
measurements (not included in the analysis above) and an appropriate
threshold to determine if a change in the position has occurred.
With the FDTD method-based programme and the NTFF transformation we modelled the Stokes parameters in the exit pupil for the subresolution objects shown in Figs. 2.6–2.8. Nevertheless, we could not
reduce the spurious reflections (see §2.3.1) enough and thus, we obtained
unreliable results that we do not present here.
Rather than continuing with a theoretical analysis of the capabilities
and limitations of the vectorial polarimetry method, we decided to work
in getting an experimental proof of concept. The following chapters
present the work done in this direction, including the construction of
a vectorial polarimeter, its calibration and the first experimental results.
45
Chapter 3
Experimental setup
The numerical results presented in the previous chapter show that, using
the vectorial polarimetry technique, high sensitivity on sub-diffraction
displacements of a point-scatterer can be obtained from the analysis of
the polarisation distribution in the exit pupil of a high NA collector lens.
In this chapter, we shall describe the setup built to obtain an experimental proof of concept. The instrument developed, the vectorial
polarimeter, is a division of amplitude Mueller matrix polarimeter with
extended detectors to obtain, simultaneously, scattering-angle-resolved
information of the three-dimensional scattered field.
3.1
Vectorial polarimeter
Division of amplitude is one of the many possible configurations for a
polarimeter with the advantage of no moving parts and, thus, high speed
data acquisition. The main limitation of this type of polarimeters is that,
since the detected light is divided between all of the detection channels,
the light available for each channel is reduced and signal to noise ratio
might be an issue.
As mentioned in §1.2.3, the basic elements of a polarimeter are the
polarisation state generator (PSG) and the polarisation state analyser
(PSA). The following two subsections describe the configuration of the
PSG and the PSA in the vectorial polarimeter. The description of both
elements is given in reference to the diagram of the vectorial polarimeter
setup presented in Fig. 3.1.
47
Chapter 3. Experimental setup
D1
D3
D2
PBS1
P2
P3
BS3
D4
SMP
BS4
QWP1
OBJ2
L2
BS1
PC1
IRIS2
PH
M1
M2
IRIS3
L3
OBJ1
L1
P1
PC2
BS2
WNDF
L5
PBS2
BS5
IRIS1
NDF
M4
L4
CCD1
L6
LS
QWP2
BS6
M3
CCD4
CCD2
P5
P4
CCD3
Figure 3.1: Diagram of the vectorial polarimeter setup.
3.1.1
Polarisation state generator (PSG)
The configuration of the vectorial polarimeter’s PSG is the following: a
linearly polarised diode pumped solid state laser, LS, is the light source.1
Neutral density filters NDF and WNDF are used to control the amount
of light in the system; WNDF is a wheel with different neutral density
filters.
Because of the limited size of the optical table, the laser beam is
redirected towards the longest side of the table by mirror M1. Then,
a Glan-Taylor linear vertical polariser, P1, increases the purity of the
reference incident polarisation state. The advantage of using a crystal
linear polariser rather than, for instance, a dichroic polymer film polariser, is that the crystal polariser is less likely to produce interference
fringes; this point, which became and issue in the measurement of the
1
MellesGriot 85-GCA-005, λ=532nm. This laser was inherited from a previous
setup due to its high stability. However, as we shall discuss in Chap. 6, a more
suitable source might be used in future realizations of the vectorial polarimeter.
48
Chapter 3. Experimental setup
scattering-angle-resolved Mueller matrices, is further discussed in §4.2.
Two Pockels cells with their fast axis at −45◦ and 0◦ from the horizontal, PC1 and PC2, respectively, are modulated to convert the linearly polarised incident light into any polarisation state over the Poincaré
sphere. Details on the modulation of the Pockels cells are presented in
§3.2. Aperture stops IRIS1 and IRIS2 are used as aids in the alignment
of the system and to block undesired back reflections from the optical
elements. Mirror M2 is an auxiliary mirror used in the alignment of
the system. PC2 is the last element of the PSG, the rest of the optical
elements described in this subsection are only auxiliary optics.
The light transmitted by the second Pockels cell is spatially filtered
with the combination of an objective with focal length f=15.5mm, OBJ1,
and a 20µm pinhole, PH.2 Lens L1, at a distance equal to its focal length
from PH, collimates the filtered light and aperture stop IRIS3 chops off
the rim of the beam producing a more homogeneous irradiance beam
with an aperture diameter of ∼7.5mm. The light is then directed by
beam-splitter BS1 towards OBJ2, the high NA objective, which focuses
it onto the specimen, SMP. The high NA objective used in the first part of
this research is an MPLAPO 100x Olympus objective (of the MPlanApo
series) with focal length f=1.8mm and NA=0.95 in air.
The configuration of the vectorial polarimeter shown in Fig. 3.1 corresponds to a reflection type microscope. This is not a necessary condition
and the method works, in principle, on transmission as well. In the reflection configuration, OBJ2 is also the collector lens. Therefore, OBJ2
collects and collimates the backscattered light sending it to two different PSAs, namely: the confocal PSA and the pupil PSA, as shall be
described in the following subsection.
3.1.2
Polarisation state analyser (PSA)
The first PSA in the vectorial polarimeter measures the polarisation state
of the fraction of light collected by OBJ2 that is reflected by BS1 and
refocused by L1 onto the position of PH. Pinhole PH limits the light that
goes through to a small fraction corresponding to the conjugate of the
objective’s focus. This is the basic configuration of a confocal microscope.
Thus, the PSA associated to this part of the system is known as the
confocal PSA.
The light that passes through PH is collimated by OBJ1 and the
2
The objective is a Linos 038722 which, for a nominal laser beam diameter of
1.1mm, produces an Airy disc with a diameter of 18.8µm.
49
Chapter 3. Experimental setup
fraction reflected by BS2 is then refocused by lens L2 onto the position
of point detectors D1-D4.3 Each detector measures a different component
of the polarisation state of the light backscattered by the specimen. BS3
is a non-polarising beam-splitter that splits the light in two, sending 50%
to the polarising beam-splitter PBS1 and the rest to the non-polarising
beam-splitter BS4. PBS1 separates the linear horizontal component,
detected by D1, from the linear vertical component, detected by D2.
The fraction of light sent to BS4 is divided in two and sent to D3, which
has a linear polariser, P2, with its transmission axis at +45◦ in front of
it, and D4, with a combination of a quarter wave-plate with its fast axis
at +45◦ , QWP1, followed by a linear horizontal polariser, P3, in front of
it. The combination of QWP1 and P3 constitutes a left-circular analyser.
This part of the vectorial polarimeter’s PSA is a slight modification of
the PSA used in the CMMP developed by Lara [33], being different from
the latter only in the position of the confocal pinhole with respect to the
incident light. Note that the confocal PSA is formed by detectors D1-D4,
polarisers P2-P3 and quarter-wave-plate QWP1. The rest of the optical
elements are auxiliary optics.
For the pupil PSA (formed by cameras CCD1-CCD4, polarisers P4P5 and quarter-wave-plate QWP2), the exit pupil of the collector lens,
OBJ2, is imaged onto the position of the cameras by the combination of
two simple telescopes.4 The first telescope, formed by L3 and L4, has
a transverse magnification MT = 0.5 and the second one, formed by L5
and L6, has an MT = 1.5 Mirrors M3 and M4 are used to keep the light
within the limits of the optical table.6 Similarly to the case of the confocal
PSA, non-polarising beam-splitter BS5 splits the light sending 50% to
the Glan-Thompson polarising beam-splitter, PBS2, that separates the
linear vertical component, detected by CCD1, from the linear horizontal,
detected by CCD2. The rest of the light is sent to non-polarising beamsplitter BS6. The light impinging on BS6 is divided between CCD3, with
a linear polariser at +45◦ in front of it, P4, and CCD4, with a left-circular
analyser (just as the one described above for the confocal PSA) formed
by quarter-wave-plate QWP2 and linear polariser P5.
3
Model 2001 manufactured by New Focus, Inc.
The four CCD cameras are of the model Flea2-08S2M-C manufactured by Point
Grey Research, Inc.
5
L3 is a G063212000 achromat, L4 is a G063215000 achromat and L5 and L6 are
G063207000 achromats, all of them from the Linos catalogue.
6
Both mirrors are Newport Laser Line Dielectric Mirrors with Part No.
10D20DM.11.
4
50
Chapter 3. Experimental setup
3.2
3.2.1
Modulation of the Pockels cells
General remarks
The modulation of the Pockels cells used for the vectorial polarimeter is
similar to that presented in [33] with the difference that, in the present
work, only 6 of the 256 pairs of voltages, each pair defining an incident
polarisation state, were used. The 6 polarisation states chosen are: linear
horizontal (H), linear vertical (V ), linear at +45◦ (+), linear at −45◦ (−),
right circular (R), and left circular (L).
The main reason to limit the number of incident polarisation states
to only 6 is that a slower data acquisition rate is used in the pupil PSA
because of the limitations of the CCD cameras which have a maximum
frame rate of 30 frames per second (FPS). Thus, a larger number of incident polarisation states would increase the time spent in a set of measurements for a given sample. Another disadvantage of a larger number
of incident polarisation states is that it would increase the number of
image files created by the cameras, increasing the use of disc space for a
single measurement.
As it is known in polarimetry, the minimum number of measurements
required to determine the complete Mueller matrix of a general sample
is 16; the combination of 4 independent incident polarisation states and
4 independent analysers is required. Therefore, with the 6 incident polarisation states, and 4 detection channels in each PSA of the vectorial
polarimeter, the total number of measurements is 24 for each PSA, allowing us to obtain the complete Mueller matrix of the sample from
an over-determined set of measurements. The over-determination of the
measurements implies that some of them are equivalent. However, more
than a limitation this is an advantage since the extra information may be
used to reduce errors in the experimentally determined Mueller matrices.
A figure of merit commonly used as a measure of the sensitivity of
the PSG and PSA to random noise propagation is the condition number.
This quantity can be used as long as the set of polarisation states generated (for the PSG) or analysed (for the PSA) is linearly independent; if
the set is not linearly independent, the associated matrix is singular and
the condition number is not a good measure of the system’s sensitivity.
The modulation matrix associated to our 6 polarisation states PSG
51
Chapter 3. Experimental setup
is


1 1
0 0

(3.1)
0 0
1 −1
√
which has a condition number cond(PSG6 ) = 3. This value is exactly
the same as the value for the Tetrahedron that corresponds to the optimum configuration for a 4 reconstructed Stokes parameters polarimeter
[55, 56].
The analysis matrix associated to each PSA, recall that both PSAs
have the same configuration, is


1 1 0 0
1  1 −1 0 0 

(3.2)
PSA4 = 
2  21 0 12 0 
1
0 0 − 12
2
1 1
 1 −1
PSG6 = 
0 0
0 0
1 1
0 0
1 −1
0 0
with a condition number cond(PSA4 ) = 3.6123. This number is far from
the optimum value for a 4 detectors polarimeter but the 4 measurements
are completely independent and their experimental implementation is
straightforward.
3.2.2
Theoretical modelling of the modulation
Following the treatment presented in [33], each Pockels cell was modelled as a linear retarder with variable retardance. Therefore, its Mueller
matrix, when its fast axis is oriented at 0◦ from the horizontal, is given
by


1
0
0
0
0

1
0
0

P0◦ (τ, ∆(t)) = τ 
(3.3)
0
0
cos ∆(t)
sin ∆(t) 
0
0 − sin ∆(t) cos ∆(t)
where τ is the transmittance for unpolarised light and ∆(t) is the variable
retardance.
To calculate the Mueller matrix of a polariser (or a retarder) with
transmission (or fast) axis at an arbitrary angle θ, Mθ , the following
expression can be used
Mθ = R(θ)M0◦ R(−θ)
52
(3.4)
Chapter 3. Experimental setup
where M0◦ is the Mueller matrix of the optical element with its axis
(whether it is the transmission or the fast axis) parallel to the horizontal
direction and R(θ) is the rotation matrix given by


1
0
0
0
 0 cos 2θ − sin 2θ 0 

R(θ) = 
(3.5)
 0 sin 2θ
cos 2θ
0
0
0
0
1
From Eqs. (3.3) and (3.5), the Mueller matrix of the first Pockels cell,
PC1, with its fast axis at −45◦ from the horizontal, is given by


1
0
0
0
0
cos ∆1 (t)
0 sin ∆1 (t) 

P−45◦ (τ1 , ∆1 (t)) = τ1 
(3.6)
0

0
1
0
0 − sin ∆1 (t) 0 cos ∆1 (t)
where the variable retardance, ∆1 (t), and the transmittance, τ1 , have a
subscript to indicate that they correspond to the first Pockels cell.
Since the fast axis of the second Pockels cell, PC2, is oriented at 0◦ ,
its Mueller matrix is given by


1
0
0
0
0

1
0
0

P0◦ (τ2 , ∆2 (t)) = τ2 
(3.7)
0
0
cos ∆2 (t)
sin ∆2 (t) 
0
0 − sin ∆2 (t) cos ∆2 (t)
where the subscript in the variable retardance, ∆2 (t), and the transmittance, τ2 , indicates that they correspond to the second Pockels cell.
The light emerging from the Glan-Taylor polariser, P1, is linearly
polarised in the vertical direction. The Stokes vector corresponding to
this polarisation state is
 
1
 −1 

Si = 
(3.8)
 0
0
Therefore, the polarisation state of the light after PC2, as a function of
the two variable retardances, is given by
So (t) = P0◦ (τ2 , ∆2 (t)) · P−45◦ (τ1 , ∆1 (t)) · Si


1


− cos ∆1 (t)

= τ1 τ2 
 sin ∆1 (t) sin ∆2 (t) 
sin ∆1 (t) cos ∆2 (t)
53
(3.9)
Chapter 3. Experimental setup
Eq. (3.9) shows that, with the proper choice of the modulation for
the variable retardances, ∆1 (t) and ∆2 (t), it is possible to generate any
polarisation state over the surface of the Poincaré sphere. However, as
mentioned earlier in this chapter, only six linearly independent polarisation states were used in the measurements with the vectorial polarimeter. Their corresponding pairs of retardance, in wavelengths, are shown
in Table 3.1. The variable retardances, ∆1 and ∆2 , have been written
without time dependence to emphasize that their values were kept constant during the time taken for the measurements with a given incident
polarisation state.
Table 3.1: Retardance pairs, in wavelengths, used in the measurements
with the vectorial polarimeter.
Polarisation state
∆1
∆2
+
− 43
− 34
H
− 21
− 58
R
− 14
− 12
V
0
− 38
−
− 34
− 14
L
− 41
0
In the following two sections we present the azimuthal alignment of
the Pockels cells and the adjustment of the amplitude and bias of the
modulation functions. In these adjustments to the experimental setup,
we used the modulation programme developed in [33], where the following
54
Chapter 3. Experimental setup
0.25
Retardance (λ)
0
R
-
−0.25
L
V
−0.5
+
1st Pockels cell
2nd Pockels cell
H
−0.75
0
0.8
1.6
2.4
3.2
4
Time elapsed (milliseconds)
4.8
5.6
Figure 3.2: Retardance modulation of the the two Pockels cells as a function of time. The 6 pairs of retardances used during the measurements
are marked with green dots and labeled with their corresponding polarisation state. The blue and red solid lines are the retardance induced
in PC1 and PC2, respectively, during calibration and alignment of the
system.
sawtooth modulation parameters were used
2t
)−
T0
t
∆2 (t) = 2πfrac( ) −
T0
∆1 (t) = 4πfrac(
3π
2
3π
2
(3.10)
(3.11)
where frac(x) is the fractional part of x and T0 is the period of the
modulation for PC1. Fig. 3.2 is a plot of the retardance modulation,
as a function of time, with the 6 incident polarisation states used in the
measurements (+, H, R, V , − and L) marked and labeled.
3.2.3
Azimuthal alignment of the Pockels cells
To model the alignment of the two Pockels cells at the correct azimuthal
orientation of their fast axis, −45◦ and 0◦ , we start by noting that when
the reference linear vertical polarisation impinges on PC1 the polarisation
55
Chapter 3. Experimental setup
state of the output is given by

So1

1
 − cos ∆1 (t) 

= P−45◦ (τ1 , ∆1 (t)) · Si1 = τ1 


0
sin ∆1 (t)
(3.12)
where P−45◦ (τ1 , ∆1 (t)) is given by Eq. (3.6) and Si1 is the same as Si
given by Eq. (3.8). From Eq. (3.12) we can see that So1 does not have
a linear component at ±45◦ for any value of the modulation parameter
∆1 (t); the third element of the Stokes vector is identically zero. Thus,
by placing PC1 between the linear vertical Glan-Taylor polariser and a
linear polariser at, for instance, +45◦ , its fast axis can be aligned by
minimizing the transmitted irradiance modulation for all the values of
∆1 (t).
Similarly, if the light incident on PC2 is linearly polarised at +45◦ ,
the light transmitted by PC2 will be given by


1


0

So2 = P0◦ (τ2 , ∆2 (t)) · Si2 = τ2 
(3.13)
 cos ∆2 (t) 
− sin ∆2 (t)
where P0◦ (τ2 , ∆2 (t)) is given by Eq. (3.7) and Si2 is the Stokes vector of
light linearly polarised at +45◦ , i.e.
 
1
0

Si2 = 
(3.14)
1
0
We can see that the resulting polarisation state does not have a component linearly polarised in the horizontal and/or vertical direction for any
value of the modulation ∆2 (t); the second element of the Stokes vector is
identically zero. Thus, PC2 can be aligned by minimizing the transmitted irradiance modulation, for any value of ∆2 (t), when the cell is placed
between a linear polariser at +45◦ and, for instance, a linear horizontal
polariser.
3.2.4
Adjustment of the amplitude and bias
The modulation of the Pockels cells given by the ramp functions described
in §3.2.2 depends on the correct adjustment of the voltage functions sent
56
Chapter 3. Experimental setup
to the amplifiers connected to the cells. An error in the amplitude and/or
bias of these functions affects the polarisation states generated by the
Pockels cells.
To determine the correct amplitude and bias of the voltages sent to
the cells, each cell was placed between crossed polarisers, one at a time,
and the transmitted irradiance was measured for a set of consecutive
modulation cycles. The method can be explained as follows: placing
the first Pockels cell between the vertical Glan-Taylor polariser and a
horizontal polariser, the polarisation state of the transmitted light, as a
function of the retardance modulation, is given by


1 − cos ∆1 (t)
τ1  1 − cos ∆1 (t) 
0

(3.15)
So1 = 

0
2 
0
From the expression above, the irradiance transmitted through the horizontal polariser is, except for a constant factor, IH = 1 − cos ∆1 (t),
which corresponds to a cosine function displaced and mirror reflected in
a horizontal plane. Thus, the transmitted irradiance has a cosine profile
that can be measured with a power meter and an oscilloscope, and the
amplitude and bias (or initial phase) of the function sent to the amplifier
connected to PC1 can be adjusted to produce a continuous cosine profile, for the transmitted irradiance, with the appropriate initial phase. A
discontinuous profile is an indication that the modulation cycle has an
incorrect amplitude whereas the bias, or initial phase, can be adjusted
to make the voltage pass through zero at the beginning of each cycle.
Placing the second Pockels cell between polarisers at +45◦ and −45◦ ,
the Stokes vector after the second polariser is


1 − cos ∆2 (t)

τ2 
0
0

(3.16)
So2 = 

2 −(1 − cos ∆2 (t)) 
0
In this case, the transmitted irradiance is I− = 1 − cos ∆2 (t) which, just
as for PC1, corresponds to a cosine profile. Therefore, the amplitude
and bias of the function sent to the amplifier connected to PC2 can
be adjusted to obtain a continuous cosine variation of the transmitted
irradiance with the correct initial phase.
The experimental procedure used to adjust the amplitude and bias of
the functions sent to the amplifiers, using the method described above,
57
Chapter 3. Experimental setup
is discussed in detail in [33], where the Pockels cells were controlled by
sending analog voltages to their amplifiers via a digital-to-analog (D/A)
PCI board connected to a PC.7 The resolution of the D/A converter in
the board is 16-bit and the same board was also used to read the signals
from the point detectors in the confocal PSA.
3.2.5
Stability of the PSG
An important aspect of the vectorial polarimeter is the stability of the
PSG; the reliability of measurements taken with the system depends on
this property. Every time a set of measurements was done, the amplitude
and bias were adjusted and recorded in the logbook. The data recorded
showed that these quantities have to be readjusted after a few minutes
in order to get the correct modulation for the Pockels cells.8 This effect
was reported by Lara in [33] and it was suggested that it is related to the
increase in room temperature due to the heat produced by the equipment
and the operator.
Fig. 3.3 is a plot of the values for the adjusted amplitude and bias,
inequivalent wavelengths of retardance, for 19 measurements taken at
different times during two laboratory sessions. Measurements 1-9 correspond to the first laboratory session and the remaining measurements,
10-19, correspond to the second session. The results presented in the
figure show that the variations in the amplitude and bias of the function
sent to PC2 are larger than those of the function sent to PC1. If the only
source of these variations were the increment in room temperature, we
would expect that both Pockels cells would be affected in a similar way.
Since this is not the case, we conclude that the voltage function driving
PC2 is inherently more instable than the one driving PC1.
Among the possible sources for the instability are the DaqBoard/2000,
used to send the functions to the amplifiers, and the amplifier connected
to PC2 itself. However, with the currently available data is not possible
to determine which of the aforementioned sources, if any, is responsible
for the observed instability.
Finally, we note that there is a quite large jump between measurements number 9 and 10, except for the amplitude of PC2, which correspond to the final measurement of the first session and the first measurement of the second session, respectively, indicating that there is an actual
dependence of the adjusted amplitude and bias in room temperature.
7
Digital-to-analog PCI board DaqBoard/2000 manufactured by IOtech.
In some cases the variation in the amplitude and bias was noticeable after each
measurement cycle which takes, approximately, 4-5 minutes.
8
58
0.4706
0.4170
0.4681
0.4159
0.4657
0.4147
0.4632
0.4136
0.4608
0.4125
0
Amplitude PC1
Amplitude PC2
2
4
6
8
10
12
Measurement number
(a)
14
16
PC1 Bias (retardance wavelengths)
0.4727
18
20
0.4583
0.6568
Bias PC1
Bias PC2
0.4704
0.6519
0.4682
0.6470
0.4659
0.6421
0.4636
0.6372
0.4614
0.6323
0.4591
0
2
4
6
8
10
12
Measurement number
(b)
14
16
18
PC2 amplitude (retardance wavelengths)
0.4182
20
PC2 Bias (retardance wavelengths)
PC1 amplitude (retardance wavelengths)
Chapter 3. Experimental setup
0.6274
Figure 3.3: Variation of the adjusted (a) amplitude and (b) bias of the
functions sent to PC1 and PC2, in equivalent wavelengths of retardance,
for 19 different measurements taken in two laboratory sessions. Measurements 1-9 correspond to the first session and measurements 10-19
correspond to the second session.
59
Chapter 3. Experimental setup
3.3
Calculation of the Mueller matrix
The calculation of the Mueller matrix from experimental data depends
on the nature of the measurement technique. For instance, the Mueller
matrix can be obtained from the spectral analysis of the polarimetric
measurements [32] or from direct algebraic relations between the measurements [27].
The experimental data obtained with the vectorial polarimeter is suitable for the calculation of the Mueller matrix of the specimen using the
method presented in [27].9 The method presented by Bickel and Bailey
establishes four properties of the experimental data and their relation
with the elements of the Mueller matrix. However, only two of them are
relevant for our analysis, namely:
• The combination polariser-analyser used in a particular measurement determines uniquely the elements of the Mueller matrix mixed
in that measurement.
• Measurements taken with the complementary polariser-analyser
configurations produce combinations of the same elements of the
Mueller matrix that differ only in the sign of the elements.
The analytic relations describing the irradiance measured for a particular
combination polariser-analyser are used to determine what set of measurements is necessary to obtain each element of the Mueller matrix. For
instance, m12 can be obtained as:
1
m12 = (IH0 − IV 0 )
2
(3.17)
That is, m12 is obtained as the difference between the total scattered
irradiance for incident light polarised in the horizontal direction and the
total scattered irradiance for incident light polarised in the vertical direction. None of the PSAs in the vectorial polarimeter measures the total
irradiance and, thus, this quantity is obtained indirectly as the incoherent superposition of the horizontal and vertical components. Therefore,
m12 for the vectorial polarimeter is obtained as:
1
m12 = [(IHH + IHV ) − (IV H + IV V )]
2
9
(3.18)
The original CMMP obtained the Mueller matrix using spectral analysis. The
modified CMMP, included in the vectorial polarimeter as the confocal PSA, uses the
method described in this section.
60
Chapter 3. Experimental setup
Similar expressions can be found for the rest of the elements. Fig. 3.4
shows the measurements and operations necessary to calculate the complete Mueller matrix of a specimen. The expressions presented in the
figure are given in terms of measurements that are not available in the
vectorial polarimeter; none of the PSAs includes neither a −45◦ nor a
right-circular analyser. Nevertheless, since some of the measurements
are redundant, we can write most elements of the Mueller matrix, except
m11 , in terms of other elements previously calculated. As an example,
let us consider again the element m12 given by Eq. (3.18).
The first element of the Mueller matrix in the vectorial polarimeter
is given by
1
(3.19)
m11 = [(IHH + IHV ) + (IV H + IV V )]
2
Therefore,
m11 + m12 = IHH + IHV
(3.20)
from which
m12 = IHH + IHV − m11
(3.21)
Thus, m12 is given in terms of a couple of measurements and m11 . Again,
similar relations can be found for the rest of the elements of the Mueller
matrix.
Because of the configuration of the PSA in the vectorial polarimeter,
an extra clarification is required. Non-polarising beam-splitter BS6, in
the pupil PSA, halves the irradiance impinging on CCD3 and CCD4
compared to the irradiance impinging on CCD1 and CCD2.10 This extra
reduction in the relative irradiance has to be taken into account in the
calculation of the Mueller matrix.
3.4
Alignment and synchronization of the
CCD cameras
The operations involved in the calculation of the Mueller matrix from the
data obtained with the pupil PSA are done pixel by pixel. Therefore, the
correct alignment of the CCD cameras, to make sure that corresponding
pixels in the cameras coincide and that all of the images are in focus, is
paramount.
The cameras were focused using a custom-made alignment target and
an incoherent extended light source. The alignment target consisted of
10
BS4 has the same effect over the irradiance detected by D3 and D4, compared to
D1 and D2, in the confocal PSA.
61
Chapter 3. Experimental setup
m11
m12
I 00
m21
1
2
[I H0 - I V0]
m31
1
2
m41
1
2
[(I HH + I VV)-(I VH + IHV)]
1
2
1
2
[(I +H + I -V)-(I -H + I +V)]
m33
[(I H+ + I V-)-(I V+ + I H- )]
m42
I 0R - I 0L
[I +0 - I -0 ]
m23
m32
I 0+ - I 0-
m14
1
2
m22
I 0H - I 0V
Note:
m13
1
2
[(I ++ + I -- )-(I -+ + I +- )]
m43
[(I HR + I VL )-(I VR + IHL)]
1
2
[(I +R + I -L )-(I -R + I+L )]
Unpolarised
Linear at 45 o
Linear horizontal
Right circular
1
2
[I R0 - I L0]
m24
1
2
[(I RH + I LV)-(I LH + I RV)]
m34
1
2
[(I R+ + I L- )-(I L+ + I R- )]
m44
1
2
[(I RR + I LL)-(I LR + I RL)]
Figure 3.4: Measurements and operations necessary to compute each of
the 16 elements of the Mueller matrix [27]. From left to right, the first
symbol, and subscript of I, represents the polarisation state of the incident light whereas the second symbol, and subscript, represents the
analyser used in the corresponding measurement. The convention followed for the subscripts is the same as in §3.2.1 with the extra ‘0’ indicating unpolarised light, for the incident light, or total irradiance, for
the analyser.
62
Chapter 3. Experimental setup
a cross made with a piece of wire tied-up to a cage of the same type as
those used to built the vectorial polarimeter. The incoherent extended
light source was a torch with a piece of ground tape in front of it. The
custom-made alignment target was placed in the position were the exit
pupil of the microscope objective is, when the system is complete, and
the extended source was pointed from the position of the sample towards
the relay imaging optics of the pupil PSA.
Although this technique might seem rough at a first glance, it proved
to be appropriate to bring the CCD cameras to focus. In fact, the alignment of the cameras using the coherent light source of the vectorial polarimeter, LS, instead the incoherent extended light source, was more
difficult due to diffraction by the wire.
The right correspondence between pixels was achieved in two steps:
• Coincidence of the pupil images in the cameras, i.e. same spherical
angle, θ, for the corresponding set of pixels
• Correct orientation of the cameras, i.e. same azimuthal angle, φ,
for corresponding pixels.
The alignment of the spherical angle, θ, was done taking the crosscorrelation between three images (CCD2-CCD4) with the reference image
(CCD1) and adjusting their horizontal and vertical position, x and y, respectively, until the cross-correlation function had its maximum in the
center; this indicates that the two pupils are covering the same area of
the CCD detectors (Fig. 3.5a).
The azimuthal alignment of the cameras (Fig. 3.5b) was done using
a vertical wire passing through the center of the aperture and rotating the cameras until the image of the wire was vertical in all of them.
Then, a fine adjustment was done taking the cross-correlation of the
reference image (CCD1) with the other three images (CCD2-CCD4) and
adjusting the azimuthal orientation of the cameras to maximize the crosscorrelation.
The detection and analysis of sub-resolution features requires an active scanning of the specimen. Thus, the polarimetric measurements
have to be done synchronously. From the beginning, the Pockels cells
modulation was synchronized with the data acquisition of the confocal
PSA; the CMMP was built with this in mind [33]. The integration of the
CCD cameras with the rest of the system required the synchronization of
each camera with the others and with the modulation cycle. The camc
eras were synchronized using the programme MultiSync
, developed by
PointGrey Research, Inc., and the synchronization with the modulation
63
Chapter 3. Experimental setup
Y
θ θr
Y
φr
φ
X
X
(b)
(a)
Figure 3.5: Diagram of the (a) spherical angle, θ, and (b) azimuthal
angle, φ, alignment of the CCD cameras. θr and φr are the spherical and
azimuthal angles of pixels in the reference camera (CCD1).
cycle was done in software by keeping the modulation constant during
the acquisition of the images with the cameras. An extra line was included in the programme to beep whenever a picture, taken by any of the
cameras, is out of synchronization and to display the number of frames,
and camera number, out of sync.
The maximum frame rate achievable with the CCD cameras is 30FPS.
However, to avoid recording out of sync images, we reduced the frame
rate to the minimum, 1.875FPS. This does not represent a problem in
the measurements presented in this work since we are only considering
static samples. In this configuration, each measuring cycle, for a single
image recording, takes ∼36-40 seconds.
This concludes our discussion on the vectorial polarimeter setup. The
following chapter describes the calibration of the system.
64
Chapter 4
System calibration
Although the theoretical performance of a polarimeter can be optimized
to obtain accurate measurements of the Mueller matrix of a specimen,
the experimental realization of such a system is not always straightforward. Thus, the performance of a polarimeter is limited by experimental
constraints.
Due to the polarisation dependence of Fresnel coefficients, whenever
light passes through an optical element its polarisation state is affected.
The amount and exact nature of the artefacts introduced in the polarisation depend on the geometry and material properties of the optical element. To take into account these effects, the polarimeter has to
be properly calibrated. Compain et al. [21] introduced a calibration
method, to minimize the effects of the polarisation artefacts introduced
by the optical elements in a polarimeter, that does not require an accurate modelling of the optics in the system. Their method is known as
the eigenvalue calibration method and, with a slight modification with
respect to the original formulation, is the method used to calibrate the
vectorial polarimeter.
4.1
Eigenvalue calibration method
The modified eigenvalue calibration method (ECM) used in this work
is based on the idea that the polarisation artefacts introduced by the
optical elements between the PSG and the sample can be represented
by a system matrix W. In a similar way, the artefacts introduced by
the elements between the sample and the PSA can be represented by
another system matrix A. Therefore, if W and A are known, and they
are invertible (as it is expected for a complete polarimeter), the Mueller
65
Chapter 4. System calibration
matrix of the sample can be obtained from the experimental data by
multiplying the experimentally determined Mueller matrix, Mexp , by the
inverse of W and A.
The main difference between the original ECM and the modified version used in this work is that W and A are the modulation and analysis
matrices of the PSG and PSA, respectively, in the original formulation
whereas, in the variation used herein, they are 4 × 4 square matrices containing only the polarisation artefacts introduced by the system. That is,
in the absence of experimental errors, W and A reduce to Eqs. (3.1) and
(3.2), respectively, in the original ECM whereas, in the modified ECM,
they reduce to identity matrices.
The modified ECM can be further explained as follows: the Stokes
vector of the light impinging onto the sample is obtained as the product
of the Stokes vector generated by the PSG multiplied by W. After interaction with the sample, represented by the multiplication by its Mueller
matrix, the resulting Stokes vector is then multiplied by the second system matrix, A, producing the polarisation state that is measured with
the PSA. Mathematically, this can be expressed as follows:
SPSA = (A · M · W)SPSG
(4.1)
where SPSG and SPSA are the Stokes vectors that represent the polarisation state generated by the PSG and measured with the PSA, respectively, and M is the Mueller matrix of the sample. Thus, the Mueller
matrix obtained from the measurements with the system is
Mexp = A · M · W
(4.2)
The problem now is to determine W and A. Compain et al. [21]
proved that W can be obtained as the null space of the system
HM : X → (M) · X − X · (aw)−1 (a(m)w)
(4.3)
where the uppercase variables represent theoretical matrices and the lowercase italic variables represent their experimentally obtained counterparts. In Eq. (4.3), X is the unknown to be determined. When X is
equal to W, as Compain et al. proved it, the right hand side of Eq. (4.3)
is identically zero in the absence of experimental errors; that is the key
to find W.
To determine W unambiguously it is necessary to use a sufficiently
large number of linearly independent calibration samples with known
theoretical Mueller matrix. De Martino et al. [57] showed that W can
66
Chapter 4. System calibration
be obtained from the experimental Mueller matrices of four samples,
namely: free space, a linear horizontal polariser, a linear vertical polariser
and a linear retarder with fast axis at 30◦ . The details of Compain et al.
method, as well as De Martino et al. optimum selection of calibration
samples, are beyond the scope of this work and the interested reader is
referred to their original work.
Once the matrix W is known, A can be determined from the experimental Mueller matrix of free space as
A = aw · (W−1 )
(4.4)
where the Mueller matrix of free space is given by the identity matrix
and thus, it is not explicitly written in Eq. (4.4).
The vectorial polarimeter developed in this research was built to work
in reflection. This configuration was chosen due to the high reflectivity
of the specimens considered in the numerical modelling of the system.
Under these circumstances, it is not possible to calibrate the system
by direct application of the modified ECM since the light passes twice
through the calibration samples.
In [33], Lara introduced the double-pass eigenvalue calibration method
(DP-ECM) as an extension of the original ECM for systems working in
the double-pass reflection configuration. During the double-pass through
the calibration samples, an optical element with its axis at an angle θ for
the first-pass will have its axis at an angle −θ for the second-pass. This
introduces an extra constraint to the ECM, as Lara pointed out in his
original work, that has to be considered.
The analysis presented in [33] concluded that the double-pass Mueller
matrices of the calibration samples, Bdp
i , and the calibration matrix W
can be obtained in the same way as in the original ECM, whereas the
second calibration matrix, A, is given by
−1
A = Bdp
· MMirror
0 ·W
(4.5)
where the matrix Bdp
0 is the double-pass Mueller matrix measured for free
space and MMirror is the Mueller matrix of an auxiliary mirror used in
the calibration. By choosing the coordinate system for the representation
of the double-pass measurements as the system given by the first-pass,
Lara showed that the second calibration matrix is given by the following
expression
−1
Adp = A · MMirror = Bdp
(4.6)
0 ·W
Therefore, the calibrated matrices are obtained as
M = (Adp )−1 · Mexp · (W−1 )
67
(4.7)
Chapter 4. System calibration
where, as mentioned above, Mexp is the Mueller matrix of the specimen
as obtained from the measurements.
4.2
Calibration of the pupil PSA
As mentioned in the previous section, the ECM uses the experimentally determined Mueller matrix of four samples, with known theoretical
Mueller matrices, to calculate the system matrices W and A. Since the
system was built to work in reflection, an auxiliary mirror, normal to
the incident beam, was placed during the calibration, after the calibration samples, to reflect the light transmitted in the first-pass back to the
calibration samples and then to the relay optics.
High NA microscope objectives have a short focal length, and an even
shorter working distance, that makes impossible to place the calibration
samples, other than free space, between the objective and the auxiliary
mirror. Therefore, the calibration of the system was done without the
objective; the polarisation artefacts introduced by the objective were
measured separately (see §4.3).
The difference between the calibration of the confocal PSA, as discussed in [33], and the pupil PSA is that the latter has to be calibrated
pixel by pixel for every pixel within the aperture of the beam. That is,
the DP-ECM has to be applied to corresponding pixels in the 16 images
that form the scattering-angle-resolved Mueller matrix of the specimen.
Although the application of the DP-ECM might seem straightforward
once the method has been applied to measurements done with a point
detector, in practice it is important to set an appropriate threshold to
the values of the Mueller matrix elements to discriminate between pixels
within the aperture of the beam, where the matrix has to be calibrated,
and pixels outside, where no calibration is required.
To determine if a particular pixel was within the aperture of the
beam, a threshold was set on the minimum value of the determinant of
the raw Mueller matrix of free space, B0 , for that pixel.1 This matrix
was chosen because, as it is the identity matrix, it is easy to differentiate
between pixels within and outside the aperture of the beam. The actual
value of the threshold was established, according to the limitations of the
analysis programme, to take pixels with associated singular or near-tosingular matrices as pixels outside the aperture of the beam. That is,
pixels with a determinant of their corresponding Mueller matrix smaller
1
In what remains of this thesis, we shall omit the superscript in the double-pass
Mueller matrices since no single-pass matrices are considered in this work.
68
Chapter 4. System calibration
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
Y: millimeters
1.5
1
1
0.8
0.5
0.6
0
−0.5
0.4
−1
0.2
−1.5
Y: millimeters
0
1.5
−0.2
1
−0.4
0.5
−0.6
0
−0.8
−0.5
−1
−1
−1.5
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
Figure 4.1: Calibrated Mueller matrix of free space, B0 , within the aperture of the incident beam.
than the floating-point relative accuracy of the analysis programme are
considered as being outside of the aperture.
The raw images obtained from the experiment were reduced from
768 × 1024 pixels to 192 × 256 pixels by applying a 4 × 4 binning to
reduce noise and speed up the data analysis; the pixel by pixel calibration
routine takes ∼6 hours if the total number of pixels in the raw images is
used compared to ∼30 minutes for the reduced images. After the binning,
a total of 26248 pixels were obtained within the aperture of the pupil for
a 49152 pixels image.
The calibration of the system was done with the average of 30 measurements taken with each combination polariser-analyser for all the calibration samples. Then, an independent set of measurements with the
calibration samples was taken to use them as experimental data to be calibrated. Fig. 4.1 is the calibrated Mueller matrix of free space obtained as
the average of 30 measurements of each combination polariser-analyser.
This figure shows the stability of the vectorial polarimeter. Note that
most of the polarisation artefacts introduced by the optical elements in
the system have been reduced with the calibration. However, it is apparent that there are still some residual artefacts as can be seen in, for
instance, m42 . These artefacts have a very peculiar form that resembles
69
Chapter 4. System calibration
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
Y: millimeters
1.5
0.6
1
0.5
0.4
0
−0.5
0.2
−1
−1.5
0
Y: millimeters
1.5
−0.2
1
0.5
−0.4
0
−0.5
−0.6
−1
−1.5
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
Figure 4.2: Distribution of the calibration matrix W within the aperture
of the incident beam.
low frequency interference fringes and, in fact, that is their origin. The
presence of these fringes is due to a backreflection in the glass pane that
protects the CCD detectors and is clearer in elements of the raw Mueller
matrix whose calculation is related to the +45◦ polarised component of
the scattered light. Therefore, the low frequency fringes are experimental errors that propagate to the final results during the calibration of the
system. This effect can be observed in the distribution of the calibration matrices, W and A, shown in Figs. 4.2 and 4.3, respectively. As
discussed in §4.1, in the absence of errors, W and A are equal to the
identity matrix. From the figures, we can see that this is not the case for
the vectorial polarimeter. The distributions obtained for the calibration
matrices have the form of the Mueller matrix of a retarder.
The experimentally obtained Mueller matrices for the rest of the calibration samples, also obtained as the average of 30 measurements, are
shown in Figs. 4.4, 4.5, and 4.6, for the linear horizontal polariser, linear vertical polariser and linear retarder, respectively.2 The calibrated
Mueller matrices that we would expect for homogeneous specimens, in
2
The polarisers used in the calibration are Newport 10LP-VIS and the linear retarder is a 633nm third-order quarter-wave-plate that introduces an effective nominal
retardance, at 532nm, of ∼ 0.26λ in double-pass.
70
Chapter 4. System calibration
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
4
Y: millimeters
1.5
x 10
4
1
0.5
3
0
2
−0.5
−1
1
−1.5
0
Y: millimeters
1.5
−1
1
−2
0.5
0
−3
−0.5
−4
−1
−1.5
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
Figure 4.3: Distribution of the calibration matrix A within the aperture
of the incident beam.
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
Y: millimeters
1.5
1
1
0.8
0.5
0.6
0
−0.5
0.4
−1
0.2
−1.5
Y: millimeters
0
1.5
−0.2
1
−0.4
0.5
−0.6
0
−0.8
−0.5
−1
−1
−1.5
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
Figure 4.4: Calibrated Mueller matrix of the horizontal polariser, B1 ,
within the aperture of the incident beam.
71
Chapter 4. System calibration
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
Y: millimeters
1.5
1
1
0.8
0.5
0.6
0
−0.5
0.4
−1
0.2
−1.5
Y: millimeters
0
1.5
−0.2
1
−0.4
0.5
−0.6
0
−0.8
−0.5
−1
−1
−1.5
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
Figure 4.5: Calibrated Mueller matrix of the vertical polariser, B2 , within
the aperture of the incident beam.
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
Y: millimeters
1.5
1
1
0.8
0.5
0.6
0
−0.5
0.4
−1
0.2
−1.5
Y: millimeters
0
1.5
−0.2
1
−0.4
0.5
−0.6
0
−0.8
−0.5
−1
−1
−1.5
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
Figure 4.6: Calibrated Mueller matrix of the linear retarder at 30◦ , B3 ,
within the aperture of the incident beam.
72
Chapter 4. System calibration
the ideal case, are constant values across the aperture of the beam, as long
as diffraction during propagation is not important. However, variations
in the values, other than the fringes discussed above, can be observed
in the elements of the Mueller matrix for each calibration sample. Part
of these variations are concentric circles around the centre of the aperture, present in the irradiance distribution of the incident beam, that
appear as a consequence of chopping-off the rim of the incident beam
with IRIS3 (see Fig. 3.1). Other variations observed are high frequency
fringes at −45◦ . These fringes were traced back to the linear polariser at
+45◦ (a dichroic polymer film polariser) in front of CCD3; the internal
structure of the polymer film polariser is responsible for the high frequency fringes. Finally, dust particles in the optics are also a source of
variations in the measured irradiance distributions. Note that all these
sources of experimental errors are present in Fig. 4.3, the distribution of
calibration matrix A. Thus, most of the residual polarisation artefacts
are introduced by the optical elements between the sample and the PSA.
To get rid of the variations, it has been suggested to divide by m11 all
the elements of the Mueller matrix, except m11 itself [58]. This method
may help to reduce the effect of irradiance variations when a single detector is used in the measurements. However, division by m11 is not the
best option when different detectors are used since each detector, not
necessarily involved in the measurement of a particular element of the
matrix, measures different irradiance variations due to the optical elements in front of it. Thus, in this work we have chosen to present the
Mueller matrices just as obtained after calibration.
As part of the analysis of the experimental results, we calculated
the errors propagation in the vectorial polarimeter by associating to the
experimental data an error equal to the standard deviation of the 30 averaged measurements. Fig. 4.7 shows the absolute error in each element
of the Mueller matrix of free space. These errors are given in the range
[0, 0.1], i.e. the maximum variation is 10% of the maximum experimental
value for m11 , and have an almost radial distribution similar to the irradiance variations of the incident beam. From the figure, it is apparent
that the maxima errors occur at the rim of the beam. Note that the high
frequency fringes at −45◦ are also present. Thus, since the exit pupil of
the collector lens only covers a fraction of the centre, the effect of these
errors is negligible. It is interesting that the largest errors occur in the
elements of the Mueller matrix associated to the measurements with the
+45◦ polariser. This is an indication that the measurements with the
largest variations are taken with camera CCD3, which is consistent with
73
Chapter 4. System calibration
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
Y: millimeters
1.5
1
0.5
0
−0.5
−1
−1.5
Y: millimeters
1.5
0.1
1
0.09
0.5
0
0.08
−0.5
0.07
−1
0.06
−1.5
Y: millimeters
0.05
1.5
0.04
1
0.03
0.5
0.02
0
0.01
−0.5
0
−1
−1.5
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
−2 −1.5 −1 −0.5 0 0.5 1
X: millimeters
1.5 2
Figure 4.7: Experimental absolute errors in the aperture of the beam for
each element of the scattering-angle-resolved Mueller matrix of free space,
B0 . The maximum in the scale corresponds to 10% of the maximum value
measured for m11 . The area covered by the exit pupil of the collector
lens is in the central part of the aperture. A zoom in to this region does
not reveal further details of the distribution.
the variations discussed above. Similar errors where obtained for the
other calibration samples and, thus, are not presented here.
Fig. 4.8 is a matrix with a histogram of the pixel value distribution,
within the aperture of the beam, for B0 . The mean and standard deviation of the pixel value distribution, for the Mueller matrix of each
calibration sample, are shown in Table 4.1.
For the sake of comparison, we calculated the corresponding theoretical Mueller matrices fitted with the retardance, transmission coefficients,
and orientation angles obtained from the average of the values calculated
with the DP-ECM within the aperture of the beam. These results are
shown in Table 4.2.
The agreement between the theoretical and experimental values is
below the standard deviation for almost every element of the matrices;
the only exceptions are m24 and m32 in B0 . Thus, to a good extent,
the experimentally determined Mueller matrices correspond with their
theoretical counterparts.
74
10000
10000
8000
8000
8000
8000
6000
4000
0.9
10000
1
1.1
Pixel value
1.2
0
−0.1 −0.05
1.3
12000
0
0.05
Pixel value
0.1
0.15
4000
0
−0.2
0.1
6000
10000
5000
0
0.15
0.8
4
2 x10
1
1.1
Pixel value
1.2
0
−0.1
1.3
4000
1
0.5
0
−0.1 −0.05
0
0.05
Pixel value
0.1
0.15
0
−0.15 −0.1 −0.05 0
Pixel value
0.05
0
0.1
0.8
8000
8000
4000
2000
2000
−0.1
0
0.1
Pixel value
0.2
0
−0.08−0.06−0.04−0.02 0
Pixel value
Counts
10000
8000
Counts
10000
4000
2.5 x10
0.1
0.2
Pixel value
0.3
0.9
1
1.1
Pixel value
1.2
1
0
−0.3
1.3
−0.2
−0.1
0
Pixel value
0.1
0.9
1
1.1
Pixel value
1.2
10000
0.2
8000
4000
2000
0
−0.1
1.5
0.5
6000
0.02 0.04
4
2
4000
10000
6000
0
−0.1
0.3
6000
12000
6000
0.1
0.2
Pixel value
2000
2000
12000
0
4000
8000
Counts
Counts
6000
0
10000
1.5
8000
0.2
2000
0.9
12000
10000
0
0.1
Pixel value
6000
Counts
0.1
−0.1
10000
Counts
14000
0
−0.2
0.05
2000
0
0.05
Pixel value
4000
8000
8000
4000
0
−0.1 −0.05
6000
2000
0
−0.15 −0.1 −0.05 0
Pixel value
15000
Counts
Counts
Counts
6000
2000
Counts
4000
10000
8000
Counts
6000
2000
Counts
0.8
4000
2000
2000
0
6000
Counts
10000
10000
Counts
12000
Counts
Counts
Chapter 4. System calibration
6000
4000
2000
−0.05
0
0.05
Pixel value
0.1
0
0.8
1.3
Figure 4.8: Histogram of the pixel value distribution, within the aperture of the beam, for B0 . The mean and standard deviation of this
distribution are shown in Table 4.1.
75
Chapter 4. System calibration
Table 4.1: Mean and standard deviation of the pixel value distribution,
within the aperture of the beam, for the Mueller matrix of each calibration sample.
Sample
Mean M
Standard deviation
B0
B1
B2

1.0056
 −0.0005

 −0.0240
−0.0296
−0.0011
1.0031
0.0726
0.0389
0.0008
−0.0270
0.9838
−0.0375

0.0061
−0.0272 

−0.0003 
1.0229

0.0280
 0.0256

 0.0246
0.0355
0.0270
0.0286
0.0135
0.0269
0.0165
0.0098
0.0296
0.0240

0.0277
0.0114 

0.0222 
0.0343

0.4374
 0.4310

 −0.0012
0.0022
0.4431
0.4372
−0.0010
0.0029
−0.0006
−0.0004
−0.0007
0.0012

0.0009
0.0009 

−0.0009 
−0.0011

0.0316
 0.0325

 0.0387
0.0615
0.0317
0.0316
0.0392
0.0627
0.0275
0.0267
0.0071
0.0168

0.0360
0.0293 

0.0072 
0.0154

−0.4315
0.4249
0.0072
0.0168
0.0029
−0.0029
0.0046
0.0022

0.0062
−0.0063 

−0.0007 
−0.0007

0.0522
 0.0549

 0.0318
0.0529
0.0455
0.0479
0.0524
0.0528
0.0558
0.0545
0.0100
0.0157

0.0268
0.0263 

0.0069 
0.0117

0.0297
0.0776
0.0578
0.0666
0.0388
0.0523
0.0641
0.0880

0.0344
0.0701 

0.0807 
0.0639
0.4276
 −0.4213

 −0.0074
−0.0184

B3
1.0278
 0.0192

 0.0040
0.0255
−0.0128
0.2961
0.4603
−0.8482
−0.0056
0.4254
0.7579
0.5860

−0.0110
0.8939 

−0.4858 
0.0131
0.0889
 0.0532

 0.0807
0.0542
Since the specimens that the vectorial polarimeter is intended to analyse are well below the diffraction limit, it is convenient to have an extra
aid to localize the region of interest in the specimen; the confocal PSA
was included in the experimental setup to help in this task. However, the
calibration of the confocal PSA is tricky, as discussed by Lara in [33], and
the pinhole plays a key role in this process. In Lara’s work, the calibration of the system was finally done without the confocal pinhole to avoid
problems in the calibration. In the vectorial polarimeter the confocal
pinhole is part of the spatial filter for the incident light, making unpractical to remove it for calibration. As a consequence, the calibration of the
confocal PSA yield results in disagreement with the theoretical Mueller
matrices of the calibration samples. Nevertheless, getting correct polarimetric measurements with the confocal PSA is not paramount in the
vectorial polarimeter since these measurements are only used as an aid
to locate the feature of interest. Thus, excluding the calibration of the
confocal PSA from this work does not affect the results to be presented
in the next chapter.
76
Chapter 4. System calibration
Table 4.2: Fitted theoretical Mueller matrix of each calibration sample
with its corresponding parameters.
Sample
Theoretical MM
B0
τ =1
B1
τ = 0.8658
θ = 0.0002◦
B2
τ = 0.8530
θ = 90.8574◦
B3
τ = 1.0280
θ = 29.6550◦
∆ = −1.5236
Ψ = 0.7652
4.3

1.0000
 0.0000

 0.0000
0.0000
0.0000
1.0000
0.0000
0.0000
0.0000
0.0000
1.0000
0.0000

0.0000
0.0000 

0.0000 
1.0000

0.4329
0.4329
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

0.0000
0.0000 

0.0000 
0.0000
0.4329
 0.4329

 0.0000
0.0000

0.4265
 −0.4263

 −0.0128
0.0000

1.0280
 0.0000

 0.0000
0.0000
−0.4263
0.4261
0.0128
0.0000
0.0000
0.3037
0.4299
−0.8830
−0.0128
0.0128
0.0003
0.0000
0.0000
0.4299
0.7728
0.5241

0.0000
0.0000 

0.0000 
0.0000

0.0000
0.8830 

−0.5241 
0.0485
Characterisation of the objective
The characterisation of the objective is important to obtain reliable measurements of the Mueller matrix with the vectorial polarimeter. As mentioned in the previous section, it was not possible to include the objective
during the calibration with the DP-ECM due to practical limitations. Instead, we used a spherical reference surface to asses the polarisation artefacts introduced by the objective. The method used in this assessment
is described in this section.
When the focus of the objective coincides with the centre of curvature of the spherical reference surface, the polarisation state of the light
reflected by the surface will be the same, except for a constant phase resulting from the Fresnel reflection coefficients for normal incidence on the
air-glass interface, as that of the incident light. That is, the polarisation
77
Chapter 4. System calibration
Incident light
Focus
Reflected light
Reference sphere
Objective
(a)
(b)
Figure 4.9: (a) Diagram of the method used to characterise the polarisation properties of the microscope objective. The electric field of the
incident and reflected light is represented by the black and blue arrows,
respectively. (b) Interferogram (centre of the image) for the reference
sphere aligned. Note that the exit pupil of the objective is smaller than
the aperture of the incident beam.
state distribution of the reflected light across the exit pupil of the objective should be the same as the distribution of the incident light across
its entrance pupil. Any change in the polarisation distribution will be
directly related to artefacts introduced by the objective. Fig. 4.9a is a diagram showing the geometry of the method. The reference sphere used in
the characterisation of our objective is a custom-made BK7 sphere with
no coating and polished to high standards to avoid introducing extra
artefacts.3 To align the sphere we used the interference pattern generated by the superposition of the light reflected by the sphere, collimated
by the objective, and the light reflected by the reference mirror M2 (see
Fig. 3.1). Fig. 4.9b is a picture of the interference pattern when the
reference sphere is properly aligned.
Fig. 4.10 is the Mueller matrix measured for the reference sphere.
This matrix is similar to the matrix of free space with a quite homogeneous distribution. The value of the matrix coefficients is reduced, with
respect to the values for B0 obtained without the objective, because the
reflectivity of the uncoated reference sphere is lower than the reflectivity
of the calibration mirror. Note that the circularly symmetric irradiance
variations discussed in the previous section are also present in this matrix, confirming that they are part of the illumination.
Any non-depolarising optical element can be represented by a di3
Reference sphere manufactured by IC Optical Systems Ltd.
78
Y/NA
Y/NA
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Y/NA
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Y/NA
Chapter 4. System calibration
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1-0.8-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
X/NA
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1-0.8-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
X/NA
-1-0.8-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
X/NA
-1-0.8-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
X/NA
Figure 4.10: Mueller matrix of the reference sphere used to characterise
the polarisation artefacts introduced by the high NA microscope objective. The spot in the lower-left corner of the pupil is due to a scratch
in the surface of the sphere accidentally produced during the alignment.
Note that the images correspond to a zoom-in on the region of interest,
i.e., the pupil itself.
79
Chapter 4. System calibration
1
1
0.8
0.8
0.6
0.6
0.4
0.6
0.2
Y/NA
Y/NA
3π/4
0.4
0.2
0
−0.2
π/2
0
−0.2
0.4
−0.4
−0.4
−0.6
π/4
−0.6
0.2
−0.8
−1
−1 −0.8 −0.6 −0.4 −0.2
π
1
0.8
−0.8
0
X/NA
0.2
0.4
0.6
0.8
1
−1
−1 −0.8 −0.6 −0.4 −0.2
0
(a)
0
X/NA
0.2
0.4
0.6
0.8
1
0
(b)
Figure 4.11: (a) Diattenuation, Ψ, and (b) retardance, ∆, of the reference
sphere across the exit pupil of the objective.
attenuation, Ψ, and a retardance, ∆ [26]. Therefore, it is of interest
to determine Ψ and ∆ for the objective. Lu-Chipman decomposition
[26] is an analysis tool to obtain this kind of information from experimentally determined Mueller matrices that has become a standard in
the field of Mueller matrix polarimetry. Fig. 4.11 shows the diattenuation and retardance, across the pupil of the objective, obtained from
the Lu-Chipman decomposition of the experimental Mueller matrix for
the reference sphere. The diattenuation distribution shows variations
that do not have a well defined structure and follow the distribution of
the variations in m11 . Thus, the diattenuation distribution in Fig. 4.11
is a consequence of the residual polarisation artefacts, after calibration,
rather than a property of the objective. The retardance has a similar
distribution but with variations much smaller than the variations in the
diattenuation. Therefore, we conclude that the variations in the diattenuation and the retardance obtained for the reference sphere are due
to errors propagated during the calibration and that the objective introduces, at a first approximation, negligible polarisation artefacts in the
measurements.
80
Chapter 5
Experimental results
The experimental verification of the numerical results presented in Chap.
2 for a point-scatterer is presented in this chapter. However, before presenting the results for a point-scatterer —as a final test of the system—
we shall present the scattering-angle-resolved Mueller matrix of a flat
mirror. This kind of measurement, with flat surfaces, has been reported
by De Martino et al. [59] and Ben Hatit et al. [58] as a calibration
method for a high NA objective in a system built to measure critical
dimensions in diffraction gratings.
The measurements presented by De Martino et al. and Ben Hatit
et al. are based on the use of a high NA objective lens as a means to
measure, simultaneously, a large range of scattering-angles. Therefore,
their results show the kind of distributions for the elements of the Mueller
matrix that can be expected in the exit pupil of a high NA lens for a flat
surface specimen.
5.1
Scattering-angle-resolved Mueller matrix of a flat mirror
The flat mirror used in these measurements is a Newport Broadband
SuperMirror.1 The alignment of the mirror, with respect to the incident
beam, was done without the microscope objective. The method used is
the same as the one used for the alignment of the auxiliary mirror during
the calibration of the system (see Chap. 4). That is, the interference
pattern between the flat mirror and the reference beam reflected by M2
was used to adjust the tilt of the flat mirror and make its surface perpen1
Newport dielectric Broadband SuperMirror model 10CM00SB.1.
81
Chapter 5. Experimental results
dicular to the incident beam. Then, the microscope objective was placed
in its position and the mirror was brought into focus.
Two different methods were used to determine if the mirror was in
focus. In the first method, the image of the exit pupil in one of the CCD
cameras was used to assess the position of the mirror. As the mirror
approached the position of the focus, a light spot appeared at the center
of the pupil and increased its size until it filled completely the exit pupil
of the objective. This was an indication that the mirror was in focus
or, at least, very close to it. When the mirror continued moving closer
to the objective, the amount of light in the exit pupil reduced until it
disappeared. This, of course, indicated that the mirror had surpassed
the focal plane for a distance larger than half the depth of focus.
In the second method, the interference between the reference beam,
reflected from M2, and the light collimated by the objective was used
to determine the position of the mirror. This method is similar to the
method used for the alignment without the objective. The difference is
that, in this case, what varies is the distance between the flat mirror and
the objective rather than the tilt of the mirror.
The experimental results obtained for the flat mirror are presented
in Fig. 5.1. These results have symmetries similar to those reported in
[59] and [58] for flat semiconductor surfaces. The differences between the
results are due to the different physical properties of the samples. Note
that the results in Fig 5.1 are given as obtained directly from the experiment. Thus, they are presented in the global XY coordinate system.
Although this might seem to be a natural choice, at a first glance, this is
not the case as can be seen from the two-fold and four-fold symmetries
present in many of the Mueller matrix elements. This kind of symmetries
are not expected for the Mueller matrix of a homogeneous sample, as it
was pointed out by Ben Hatit et al. in [58]. Because of the nature of
the reflection on the flat mirror, it is more appropriate to present the
results in a local PS coordinate system, where P and S are the directions
parallel and perpendicular, respectively, to the plane of incidence.2
The definition of both coordinate systems is shown in Fig. 5.2, where
the exit pupil is represented by the green circles. The transformation
of the experimental results from the XY coordinate system to the PS
coordinate system is shown in Fig. 5.3. After the transformation, the
two-fold and four-fold symmetries present in many of the Mueller matrix
elements disappeared. The transformed matrix is more homogeneous,
2
The transformation of the coordinate system was also proposed by Ben Hatit et
al. in [58].
82
Y/NA
Y/NA
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Y/NA
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Y/NA
Chapter 5. Experimental results
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1-0.8-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
X/NA
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1-0.8-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
X/NA
-1-0.8-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
X/NA
-1-0.8-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
X/NA
Figure 5.1: Experimentally determined Mueller matrix in the exit pupil
of the collector lens for the flat mirror. The results are presented in the
global XY coordinate system as obtained from the experiment.
83
Chapter 5. Experimental results
Y
Y
Y
Y
Y
X
S
P
S
X
Y
S
Y
X
S
P
X
X
Y
Y
X
P
X
S
Y
S
X
X
P
P
X
S
P
P
(a)
S
P
(b)
Figure 5.2: Definition of the (a) global XY and (b) local PS coordinate
systems in the exit pupil of the collector lens.
despite the variations in irradiance discussed in Chap. 4, as it is expected
for a homogeneous sample.
The form of the Mueller matrix in Fig. 5.3 reveals a radial symmetry
that indicates that the mirror behaves like a linear retarder with fast axis
perpendicular to the plane of incidence and retardance dependent on the
angle of incidence. For instance, in the center of the images, which correspond to small angles of incidence, the Mueller matrix corresponds to
that of a perfect mirror.3 However, as the angle of incidence is increased,
the retardance, as can be seen in elements m34 and m43 , increases slowly
to π, then it drops quickly to zero and finally increases quickly until the
rim of the pupil. A Lu-Chipman decomposition of the matrix, which
is independent of the chosen coordinate system, can be used to better
appreciate the retardance distribution. Figs. 5.4a and 5.4b are the diattenuation and retardance distributions, respectively, in the exit pupil
of the collector lens for the flat mirror. While the retardance follows the
behaviour described above, the diattenuation exhibits a rather noisy distribution with no clear information about the properties of the sample.
The sudden increase in the diattenuation observed at the rim of the pupil
is an artefact due to a quick drop in irradiance within this region; this is
an indication that the mirror was not exactly in focus.
During the measurements with the flat mirror, the control on the
position of the mirror with respect to the objective was limited by the
3
Recall that the calibration matrices change the Mueller matrix of a mirror by the
matrix of free space.
84
Y/NA
Y/NA
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Y/NA
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Y/NA
Chapter 5. Experimental results
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1-0.8-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
X/NA
-0.2
-0.4
-0.6
-0.8
-1
-1-0.8-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
X/NA
-1-0.8-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
X/NA
-1-0.8-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
X/NA
Figure 5.3: Experimentally determined Mueller matrix in the exit pupil
of the collector lens for the flat mirror. The results are presented in the
local PS coordinate system as obtained from the transformation of the
experimental results in Fig. 5.1.
1
1
0.8
0.6
0.6
0.6
0.2
Y/NA
0.2
Y/NA
3π/4
0.4
0.4
0
−0.2
π/2
0
−0.2
0.4
−0.4
−0.4
−0.6
π/4
−0.6
0.2
−0.8
−0.8
−1
−1 −0.8 −0.6 −0.4 −0.2
π
1
0.8
0.8
0
X/NA
0.2
0.4
0.6
0.8
1
−1
−1 −0.8 −0.6 −0.4 −0.2
0
(a)
0
X/NA
0.2
0.4
0.6
0.8
1
0
(b)
Figure 5.4: (a) Diattenuation and (b) retardance of the flat mirror across
the exit pupil of the objective.
85
Chapter 5. Experimental results
minimum incremental motion and repeatability of the translation stages
used.4 The nominal depth of focus of the MPLAPO 100x Olympus objective used in these measurements is 0.3µm whereas the minimum incremental motion of the translation stages is 0.050µm with a bidirectional
repeatability of 2µm. Thus, since a scanning along the optical axis had
to be done to determine if the objective was in or out of focus, the bidirectional repeatability of the linear stages limited the performance of the
vectorial polarimeter.
5.2
Scattering-angle-resolved Stokes parameters of a point-scatterer
The limitation in the performance of the vectorial polarimeter described
in the previous section was acceptable for the measurements with the flat
mirror. However, it constitutes a real drawback for the measurements intended to verify the numerical results for a point-scatterer. Since this is
important to obtain an experimental proof of concept of the vectorial
polarimetry method, a piezo-positioning system —or piezo-stage—, with
a nominal resolution of 2nm in closed-loop, was included in the experimental setup to improve the control over the position of the sample.5
A microscope slide with 80nm diameter gold nano-spheres was used
as a sample; each gold nano-sphere was considered as a point-scatterer.6
The main assumption in the numerical analysis of the field scattered by a
point-scatterer is that the point-scatterer behaves like an electric dipole
with dipole moment proportional to the incident field. This condition is
fulfilled by a metallic sphere as long as the radius of the sphere is much
smaller than the wavelength [54]. The gold nano-spheres used in this
work satisfy this condition. Besides, this is consistent with a previous
work where the same kind of nano-spheres were used as point-scatterers
at visible wavelengths [60].
Because of practical limitations in the system, further modifications
to the experimental setup were necessary for the measurements with the
gold nano-spheres. Among these limitations is the correct positioning of
the piezo-stage.
4
A three-axis array formed by three Newport miniature linear stages model MFA-
CC.
5
The piezo-positioning system is a three-axis Tritor100SG manufactured by
PiezoSystem Jena.
6
The slides with gold nano-spheres were prepared and donated by Klas Lindfors
from the Helsinki University of Technology.
86
Chapter 5. Experimental results
Piezo-stages are made of ceramic materials that are very strong but
brittle when mechanical strain forces are applied. Thus, it is important
to position the piezo-stage in the right orientation and with a load within
the limits prescribed by the manufacturer. To place the piezo-stage in
the correct position it was necessary to modify the system to make the
microscope objective face down to the optical table. Therefore, an extra
mirror at 45◦ with respect to the incident beam, M5, was introduce before
the objective to reflect the light downwards (see Fig. 5.5).
The introduction of the extra mirror increased the distance between
the exit pupil of the objective and the first lens in the relay optical system
used to image the pupil in the CCD cameras. Thus, the two lenses on
the first relay telescope, L3 and L4 in Fig. 3.1, were replaced by a couple
of lenses, L3’ and L4’, respectively, with a larger focal length.7 The
new relay telescope has a transverse magnification slightly larger than
the original one (MT = 0.57 as compared to the original MT = 0.5).
Nevertheless, the final size of the pupil image remains within the limits
of the CCD detectors.
The modified system was re-calibrated to include the effects of the extra mirror and the new relay telescope in the system calibration matrices.
This calibration was done in the same way as described in Chap. 4 with
similar results. The actual results are not included herein for the sake of
brevity and because they do not contribute further to our discussion.
Finding a gold nano-sphere by scanning the sample with the light
focused by the high NA objective proved difficult. To facilitate the localization of the nano-spheres, the light from laser LS (see Fig. 3.1) was
blocked and an auxiliary laser beam, ALS, was included.8 The auxiliary
laser impinged on the microscope slide from one side rather than from
the top or the bottom. This arrangement is similar to darkfield illumination in the sense that the light collected by the objective is light purely
scattered by the sample, i.e. does not include neither transmitted nor
specularly reflected light. Fig. 5.5 is a diagram of the modified system.
A mirror in a flip-in mount, M6, was introduced between M3 and M4 to
deviate the light towards lens L7, and form an image of the microscope
slide in CCD5; this image is the same that would be obtained in a typical
optical microscope with darkfield illumination. When the flip-in mount is
in the upright position, the light is deviated towards L7; when it is lying
down, the light continues to form an image of the pupil in CCD1-CCD4.
7
L3 was replaced by a G063235000 achromat and L4 by a G063213000 achromat
from the Linos catalogue.
8
ALS is a Spectra-Physics He-Ne laser model 196-1 with λ = 633nm.
87
Chapter 5. Experimental results
D1
D3
D2
PBS1
P2
P3
ALS
BS3
D4
SMP
BS4
QWP1
OBJ2
L2
M5
BS1
PC1
IRIS2
PH
M1
M2
IRIS3
L3’
OBJ1
P1
PC2
BS2
L1
WNDF
L5
PBS2
BS5
IRIS1
NDF
M4
L4’
CCD1
L6
M6
LS
QWP2
BS6
M3
L7
CCD4
CCD2
P5
P4
CCD5
CCD3
Figure 5.5: Modified setup diagram. The part of the system not used
in the localization of the nano-spheres is shaded. The section within
the dashed lines box is orthogonal to the plane of the image due to
the reflection in M5 (the mirror at 45◦ included before the objective).
Auxiliary laser, ALS, impinges on the side of the microscope slide. Lenses
L3’ and L4’ are the replacements for L3 and L4, respectively, in the new
first relay telescope. Mirror M6 is in a flip-in mount to deviate the beam
towards L7 and form an image of the microscope slide in CCD5.
88
Chapter 5. Experimental results
Placing the nano-sphere in the focal region of the objective using the
image in CCD5 was straightforward. However, measuring the polarisation state distribution in the exit pupil for the nano-sphere was impossible
due to a strong backreflection in the microscope slide that screened the
light scattered by the nano-sphere. Therefore, an extra modification was
required.
To reduce the backreflection on the microscope slide, the MPLAPO
100x Olympus objective was replaced by an UPLSAPO 100x oil immersion Olympus objective —of the UPlanSApo series— with NA=1.40 in
n = 1.518 immersion oil. The refractive index matching between the
immersion oil and the microscope slide reduced the backreflection sufficiently to obtain usable information from the light scattered by the
nano-sphere.
It was impractical to fill the reference sphere used to calibrate the
MPLAPO objective with immersion oil to measure the Mueller matrix of
the UPLSAPO immersion objective. Thus, the assessment of the polarisation artefacts introduced by the UPLSAPO objective was not directly
done. The information provided by the manufacturer states that the polarisation properties of both objectives are within the same specifications.
Therefore, since the Mueller matrix of the MPLAPO objective does not
show any strong polarisation artefact, the effect of the UPLSAPO objective on the polarisation measurements was assumed to be negligible as
well.
With the dry objective, the microscope slide did not require any further preparation; the sample was just placed over the piezo-positioning
system and brought to focus. However, for the oil immersion objective
we put a drop of immersion oil over the slide and covered it with a cover
glass for protection. Fig. 5.6a is a diagram of the prepared sample illuminated by the auxiliary laser, ALS, in position to be analysed with the
oil immersion objective. Fig. 5.6b is the corresponding image obtained
with CCD5.
The image in CCD5 was useful to position the gold nano-sphere in
the focal region of the high NA objective. However, this image was not
sufficient to place the nano-sphere exactly at the focus. A fine adjustment
on the transversal position of the nano-sphere was done by maximizing
the irradiance in the CCD2 image of the pupil —i.e. the component
horizontally polarised of the scattered field— for incident light linearly
polarised in the horizontal direction.9
9
Note that other corresponding combinations polariser-analiser might have been
used for the fine adjustment as well; the linear horizontal case was chosen arbitrarily.
89
Chapter 5. Experimental results
Collected light
Objective
Cover glass
Immersion oil
Incident beam
Gold nano-sphere
Microscope slide
(b)
(a)
Figure 5.6: (a) Diagram showing the preparation of the sample for analysis with the oil immersion objective. The sample is being illuminated by
the auxiliary laser, ALS. (b) Corresponding image of the nano-spheres as
obtained by CCD5 with the flip-in mount mirror in the upright position.
The fine adjustment on the position of the nano-sphere along the
optical axis was done by verifying that the pupil was completely filled
with light, just as described in §5.1. Afterwards, also as described in
the aforementioned section, the interference pattern between the light
collected by the objective and the reference beam reflected by M2 was
checked.
Although the measurements performed with the vectorial polarimeter
may be used to reconstruct the complete Mueller matrix of the samples,
the results presented in §2.6 are limited to the pupil distribution of the
Stokes parameters. Thus, we shall omit the experimental Mueller matrix
of the gold nano-sphere and present the comparison between numerical
and experimental results for the Stokes parameters in the exit pupil.
Fig. 5.7a shows the experimental results obtained for the on axis
gold nano-sphere with incident light horizontally polarised. Fig. 5.7b
shows the corresponding numerical results for a point-scatterer under
the same oil-immersion conditions. These results are normalized with
respect to the maximum S0x and are similar to the numerical results
presented in Fig. 2.14.10 Despite the noise in the experimental results,
10
The experimental results for the gold nano-sphere presented in this chapter do not
correspond exactly to the numerical results presented in Chap. 2 due to the different
surrounding media. The experimental results were obtained with a NA=1.40 in a
medium with n = 1.518 whereas the numerical results in Chap. 2 were obtained for a
NA=0.95 in air, n = 1.0. Nevertheless, since the difference in angular semi-aperture
90
Chapter 5. Experimental results
S 1x
S 0x
1
0.9
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
1
Y/NA
0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
0
0
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
0
1
S 3x
S 2x
1
1
Y/NA
1
1
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
1
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
−1
1
−1
(a)
S 1x
S 0x
1
0.9
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
Y/NA
0.8
−1
−1 −0.8−0.6−0.4−0.2
0
−1
−1 −0.8−0.6−0.4−0.2
0
0
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
S 3x
S 2x
1
1
Y/NA
1
1
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2
0 0.2 0.4 0.6 0.8
X/NA
1
−1
−1 −0.8−0.6−0.4−0.2
−1
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
−1
(b)
Figure 5.7: (a) Experimentally obtained Stokes parameters distribution
in the exit pupil of the collector lens for an on axis gold nano-sphere with
incident light linearly polarised in the horizontal direction. (b) Corresponding numerical results for a point-scatterer. The Stokes parameters
are normalized with respect to the maximum S0x .
91
Chapter 5. Experimental results
and the asymmetry between the right and left hand sides of S0x and S1x
not present in the numerical results, the basic structure of the polarisation state distribution is apparent and in agreement with the results
presented in Fig. 5.7b. Furthermore, the noise in the experimental results is consistent with the noise in the Mueller matrix of the calibration
samples (see Chap. 4). For instance, even though the noise is relatively
high in S3x , which according to the numerical results should be zero, the
noise distribution follows the interference fringes discussed in §4.2. Thus,
this distribution does not contain information about the specimen under
observation but about the imperfections of our optical system.
Fig. 5.8a shows the experimental results obtained for the on axis
gold nano-sphere for incident light circularly polarised to the left. The
corresponding numerical results are shown in Fig. 5.8b. It is important to note that the numerical results are for incident light circularly
polarised to the right whereas the experimental results are for incident
light circularly polarised to the left. Even though these results have the
opposite handedness they correspond because in the numerical results
the reference coordinate system is different for incident and scattered
light whereas it is the same in the experimental results; the calibration
with the DP-ECM accounts for the difference. The noise in Fig. 5.8a
is, again, consistent with the residual polarisation artefacts that remain
after the calibration. However, once more, the basic structure of the
Stokes parameters distribution in the exit pupil of the collector lens is
apparent and in agreement with the numerical results.
So far, we have verified experimentally that it is possible, to a certain
extent, to measure the inhomogeneous polarisation state distribution in
the exit pupil of the collector lens for a point-scatterer. The next step is
to verify the high sensitivity on the position of the point-scatterer, within
the focal region of the objective, achieved with the analysis of the exit
pupil polarisation state distribution.
Figs. 5.9a and 5.10a show the Stokes parameters distribution in the
exit pupil for an off axis point-scatterer located in the focal plane at
x = −λ/3 and x = +λ/3, respectively, with incident light linearly polarised in the horizontal direction. The corresponding numerical results
are shown in Figs. 5.9b and 5.10b.
Despite the noise in the measurements, the general form of the measured distributions is the same
as in the numerical calculations. Furthermore, the mirror symmetry,
is only ≈ 3.81◦ , the Stokes parameters pupil distributions in both cases are similar,
as can be seen in Fig. 5.7b. These latter results were obtained assuming a perfect
matching between the refractive indices of the immersion oil and the cover glass.
92
Chapter 5. Experimental results
S 1r
S 0r
1
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
1
Y/NA
0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
0
0
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
0
1
S 3r
S 2r
1
1
Y/NA
1
1
0.9
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
1
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
−1
1
−1
(a)
S 1r
S 0r
1
0.9
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
Y/NA
0.8
−1
−1 −0.8−0.6−0.4−0.2
0
−1
−1 −0.8−0.6−0.4−0.2
0
0
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
S 3r
S 2r
1
1
Y/NA
1
1
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2
0 0.2 0.4 0.6 0.8
X/NA
1
−1
−1 −0.8−0.6−0.4−0.2
−1
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
−1
(b)
Figure 5.8: (a) Experimentally obtained Stokes parameters distribution
in the exit pupil of the collector lens for an on axis gold nano-sphere with
incident light circularly polarised to the left. (b) Corresponding numerical results for a point-scatterer. The Stokes parameters are normalized
with respect to the maximum S0r .
93
Chapter 5. Experimental results
S 1x
S 0x
1
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
1
S 2x
Y/NA
0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
0
0
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
0
1
S 3x
1
1
Y/NA
1
1
0.9
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
1
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
−1
1
−1
(a)
S 1x
S 0x
1
0.9
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
Y/NA
0.8
−1
−1 −0.8−0.6−0.4−0.2
0
−1
−1 −0.8−0.6−0.4−0.2
0
0
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
S 3x
S 2x
1
1
Y/NA
1
1
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2
0 0.2 0.4 0.6 0.8
X/NA
1
−1
−1 −0.8−0.6−0.4−0.2
−1
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
−1
(b)
Figure 5.9: (a) Experimentally obtained Stokes parameters in the exit
pupil of the collector lens for a gold nano-sphere in the focal plane at x =
−λ/3 with incident light linearly polarised in the horizontal direction.
(b) Corresponding numerical results for a point-scatterer. The Stokes
parameters are normalized with respect to the maximum S0x .
94
Chapter 5. Experimental results
S 1x
S 0x
1
0.9
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
1
S 2x
Y/NA
0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
0
0
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
0
1
S 3x
1
1
Y/NA
1
1
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
1
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
−1
1
−1
(a)
S 1x
S 0x
1
0.9
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
Y/NA
0.8
−1
−1 −0.8−0.6−0.4−0.2
0
−1
−1 −0.8−0.6−0.4−0.2
0
0
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
S 3x
S 2x
1
1
Y/NA
1
1
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2
0 0.2 0.4 0.6 0.8
X/NA
1
−1
−1 −0.8−0.6−0.4−0.2
−1
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
−1
(b)
Figure 5.10: (a) Experimentally obtained Stokes parameters in the exit
pupil of the collector lens for a gold nano-sphere in the focal plane at x =
+λ/3 with incident light linearly polarised in the horizontal direction.
(b) Corresponding numerical results for a point-scatterer. The Stokes
parameters are normalized with respect to the maximum S0x .
95
Chapter 5. Experimental results
along the horizontal direction, predicted for S3x between the x = −λ/3
and x = +λ/3 positions of the scatterer, is present in the experimental
results. Thus, the results show that it is possible to determine if the
sub-resolution scatterer has undergone a sub-resolution displacement, as
well as the direction of the sub-resolution displacement, by analysing the
polarisation state distribution in the exit pupil of the high NA collector
lens.
As discussed in Chap. 2, and shown in the previous results, the high
sensitivity on the position of the sub-resolution scatterer is limited to S3x
in the case of incident light linearly polarised; the other three elements
exhibit essentially the same distribution independently of the direction
of the displacement. It was shown in §2.6 that the use of incident light
circularly polarised allows us to obtain the high sensitivity in any of the
four Stokes parameters. Figs. 5.11a and 5.12a show the experimental
results obtained for this case. The corresponding numerical results are
shown in Figs. 5.11b and 5.12b. The agreement between numerical
and experimental results is apparent, in most of the Stokes parameters,
despite the noise in the measurements. The element with the worst
agreement with its numerical counterpart, especially in the results for the
gold nano-sphere at x = −λ/3, is S2r . Thus, due to the relatively high
noise level in the experimentally determined S2r , no conclusive remarks,
based on this element, can be done concerning the displacement of the
gold nano-sphere. Nevertheless, as can be seen in Figs. 5.11 and 5.12,
the direction of the sub-resolution displacement of the gold nano-sphere
is encoded in the other three Stokes parameters, as predicted by the
numerical calculations.
96
Chapter 5. Experimental results
S 1r
S 0r
1
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
1
S 2r
Y/NA
0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
0
0
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
0
1
S 3r
1
1
Y/NA
1
1
0.9
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
1
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
−1
1
−1
(a)
S 1r
S 0r
1
0.9
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
Y/NA
0.8
−1
−1 −0.8−0.6−0.4−0.2
0
−1
−1 −0.8−0.6−0.4−0.2
0
0
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
S 3r
S 2r
1
1
Y/NA
1
1
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2
0 0.2 0.4 0.6 0.8
X/NA
1
−1
−1 −0.8−0.6−0.4−0.2
−1
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
−1
(b)
Figure 5.11: (a) Experimentally obtained Stokes parameters in the exit
pupil of the collector lens for a gold nano-sphere in the focal plane at
x = −λ/3 with incident light circularly polarised to the left. (b) Corresponding numerical results for a point-scatterer. The Stokes parameters
are normalized with respect to the maximum S0r .
97
Chapter 5. Experimental results
S 1r
S 0r
1
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
1
Y/NA
0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
0
0
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
0
1
S 3r
S 2r
1
1
Y/NA
1
1
0.9
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
1
−0.8
−1
−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8
X/NA
−1
1
−1
(a)
S 1r
S 0r
0.8
0.8
0.6
0.8
0.6
0.6
0.4
0.7
0.4
0.4
0.2
0.6
0.2
0.2
0
0.5
−0.2
0.4
−0.2
−0.2
−0.4
0.3
−0.4
−0.4
−0.6
0.2
−0.6
−0.6
−0.8
0.1
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
Y/NA
0.9
0
−1
−1 −0.8−0.6−0.4−0.2
0
0
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
S 3r
S 2r
1
1
−1
1
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
Y/NA
Y/NA
0.8
−1
−1 −0.8−0.6−0.4−0.2
Y/NA
1
1
1
1
0
0
0
−0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
−1 −0.8−0.6−0.4−0.2
0 0.2 0.4 0.6 0.8
X/NA
1
−1
−1 −0.8−0.6−0.4−0.2
−1
−0.8
0 0.2 0.4 0.6 0.8
X/NA
1
−1
(b)
Figure 5.12: (a) Experimentally obtained Stokes parameters in the exit
pupil of the collector lens for a gold nano-sphere in the focal plane at
x = +λ/3 with incident light circularly polarised to the left. (b) Corresponding numerical results for a point-scatterer. The Stokes parameters
are normalized with respect to the maximum S0r .
98
Chapter 6
Conclusions
The results presented in the previous chapter constitute an experimental
proof of concept of the vectorial polarimetry method. Even with the
unavoidable experimental errors, the agreement between the numerical
and the experimental results is apparent.
In this chapter, we shall give some final remarks about the results
obtained and discuss possible directions for the future of this research.
6.1
Field distribution in the focal region
Although the most used method to calculate the field distribution in
the focal region of a high NA objective lens is the direct integration of
Debye-Wolf integrals, McCutchen’s method offers a suitable alternative.
The comparison between McCutchen’s method and Debye-Wolf integrals
presented in §2.5 shows that the field distribution in the focal region
is, in both cases, essentially the same. The slight differences observed
between the two methods are due to the different samplings used. An
alternative method for the calculation of the field distribution in the
focal region of a lens, based on the Fourier transform of a pupil function,
was introduced by Leutenegger et al. in [61]. This method, however, is
fundamentally the same as McCutchen’s method and does not represent
any real advantage.
One of the strengths of McCutchen’s method in comparison with the
direct integration of Debye-Wolf integrals, in the context of modern digital computers, is the possibility of applying an FFT algorithm to reduce
the computational time. Leutenegger et al. discussed in their work the
best way to implement their method in a digital computer to achieve
maximum speed in the calculations. They even included a comparison
99
Chapter 6. Conclusions
between their optimized implementation and a simple implementation of
an FFT algorithm. Their conclusion is that, whereas the latter takes
≈ 1min to complete the calculations, the former only takes ≈ 30sec.
Even though the optimized method halves the computational time, this
does not represent a real advantage in applications where the calculations
do not need to be done in real time.
Another appealing feature of McCutchen’s method is that it is relatively easy to calculate the focused field distribution for inhomogeneous
incident polarisation states; this is not the case in the direct integration
of Debye-Wolf integrals. As discussed in Chap. 2, the electric field distribution in the focal region is a function of the NA of the focusing lens as
well as the polarisation state of the incident light. In this work, only results for homogeneous incident polarisation states —i.e. states for which
the polarisation is the same at any position within the aperture of the
beam— have been presented. Using only this kind of incident distributions limits the amount of possible field distributions in the focal region;
recall that the sensitivity of the vectorial polarimetry method depends
on the electric field distribution in the focal region. Thus, by engineering
the polarisation state distribution of the incident light, it is possible to
tailor the field distribution in the focal region to increase the sensitivity
of the method. In the laboratory, incident polarisation state engineering
can be done using a tandem of spatial light modulators (SLMs), as proposed by Iglesias and Vohnsen [36]. In this case, the correct alignment
between corresponding pixels of the SLMs is an important practical issue
to be considered.
The engineering of the incident polarisation state, as a means to tailor
the field distribution in the focal region, has been explored by several
authors in the past [36, 61–63]. However, most work done in this field
has only addressed the problem of finding the field distribution in the
focal region for a given incident polarisation distribution. Increasing the
sensitivity of our method will require the solution of the inverse problem.
That is, finding the polarisation state distribution of the incident light
for a prescribed field distribution in the focal region. Albeit this problem
is not trivial, at least one paper tackling it can be found in the literature
[64].
To conclude our discussion on inhomogeneous incident polarisation
states we note that for homogeneous incident polarisation states, even
though the light is focused on the sample, the polarisation state distribution of the scattered light in the exit pupil of the collector lens can
be considered as a scattering-angle-resolved Mueller matrix; each pixel
100
Chapter 6. Conclusions
in the pupil corresponds to a different pair of azimuthal and spherical
scattering-angles. However, for incident light with an inhomogeneous
polarisation state, the interpretation of the pupil distribution of polarisation states for the scattered light is not straightforward and will require
further analysis.
6.2
Calculation of the scattered field
The calculation of the scattered field is an important part in the modelling of the performance of the vectorial polarimetry method. The results presented in this work, numerical and experimental, are limited to
the case of a point-scatterer. This was the first example examined because the analytical solution for the field radiated by a dipole is well
known [54]. Although interesting results were obtained in this simple
case, it is of interest to analyse the performance of the method for subresolution scatterers with a more complex structure. For spherical objects, for instance, the analytical solution for the scattered field is given
by Mie’s theory. However, Mie’s scattering solution is only given for incident monochromatic plane waves. Thus, the modelling of a focused field
interacting with a spherical sample should be obtained as the scattered
field resulting from the coherent superposition of Mie’s scattered fields
for a number of plane waves, with a given polarisation state, incident on
the sample at different angles. In this case, the calculation of the focused
field, using either of the methods discussed in the previous section, is
unnecessary.
As discussed in Chap. 2, in the most general case the interaction of
the focused field with the sample can be modelled using numerical methods. Even though several numerical methods for the solution of scattering problems are readily available, the method chosen in this work is the
FDTD method; the fundamental ideas of the method were introduced in
Chap. 2. In the same chapter, the NTFF transformation proposed by
Török et al., and used in this work, was discussed. Our implementation
of the NTFF transformation was tested with excellent results but, when
used in combination with the FDTD method, unreliable results were obtained. Part of the problem is in the poor performance of the ABCs
implemented in the programme. However, the main source of errors is
the incapability to model the boundary conditions accurately. That is,
within the context of the NTFF transformation, since the integration
surface has to enclose a volume containing all the sources and sinks of
EM field, it is impossible to model an infinite half-space made of a PEC.
101
Chapter 6. Conclusions
Thus, an integration surface enclosing a volume large enough, in comparison with the scatterer, can be used to approximate an infinite PEC
half-space. This solution is impractical since the size of the space region
modelled with the FDTD method is limited by the amount of computer
resources available. An alternative to avoid this problem is modelling
more realistic scatterers. This shall be done in the next stage of this
research to model, for instance, a metallic scatterer over a dielectric substrate.
The use of the FDTD method was an arbitrary choice. Thus, another
research direction worth exploring is the use of a different method for
the calculation of the scattered field. Discrete Dipole Approximation
(DDA) is a suitable alternative. An important characteristic of the DDA
method is that the solution of the scattered field is given only for incident
monochromatic plane waves. This is a characteristic that DDA shares
with Mie’s theory. Therefore, just as for Mie’s theory, the total scattered
field has to be obtained as the coherent superposition of the scattered
field obtained for a set of plane waves, with a given polarisation state,
incident at different angles on the sample. Again, as mentioned above
for Mie’s scattering, the use of any of the methods discussed for the
calculation of the focused field is unnecessary.
6.3
Gold nano-sphere as a point-scatterer
The numerical and experimental results obtained in this work show that
high sensitivity on sub-resolution displacements of a sub-resolution scatterer is achieved by analysing the polarisation state distribution in the
exit pupil of a high NA collector lens. The results also show that the high
sensitivity is linked to the longitudinal component of the focused field;
the symmetry break between the value of that component at x = −λ/3
and x = +λ/3 is the key to differentiate between the two positions (see
§2.6).
The main limitation of the high sensitivity achieved with the vectorial
polarimetry method is that there is not an obvious way to relate the pupil
polarisation state distribution with the actual value of the displacement
of the point-scatterer. This means that, at the moment, there is not a
quantitative way to differentiate between two polarisation distributions
corresponding to two slightly different positions of the point-scatterer.
As can be expected, this also limits the assessment of the minimum
sensitivity achievable for a given incident polarisation state.
The role of the NA in the vectorial polarimetry method is paramount.
102
Chapter 6. Conclusions
The information necessary to determine if the point-scatterer has undergone a displacement, as well as the direction of the displacement, is
mostly contained in the rim of the exit pupil, revealing the importance
of using a high NA collector lens. This can be further appreciated by
noting that in the center of the pupil, which corresponds to low NA, the
polarisation state distribution is mostly homogeneous. Thus, it is important to minimize the aberrations introduced by the imaging system at
the edge of the pupil. This can be done by keeping the aperture of the
pupil within the central region of the relay imaging system.
At this point it is appropriate to discuss the spot that can be observed
in the center of some of the experimental results (see Figs. 5.10 and
5.12). The spot is a backreflection produced in the interface between the
immersion oil and the microscope slide interface due to the finite size of
the gold nano-sphere. The experimental evidence available indicates that
the presence or absence of the central spot is related to a slight defocus
of the gold nano-sphere. Whether the sphere is better focused when the
central spot is present or not, is not clear from the experimental data.
Even though the calibration of the vectorial polarimeter with the
DP-ECM reduced the polarisation artefacts introduced by the elements
in the setup, residual artefacts can be observed in the experimental data.
These artefacts represent a drawback in the performance of the vectorial
polarimeter. For instance, the effect of the irradiance variations due to
the interference fringes is one of the most important limitations of the
system. The high frequency fringes, as discussed in Chap. 4, have their
origin in the transmission properties of the dichroic polariser at 45◦ in
front of CCD3. The low frequency fringes come from parasite reflections
in the glass cover attached to the CCD detectors. To get rid of the high
frequency fringes, the 45◦ dichroic polariser may be substituted with a
linear crystal polariser, like the Glan-Taylor polariser in the experimental
setup. The low frequency fringes may be reduced by using CCD cameras
with no glass cover attached to the CCD detector or by using a single
CCD camera with a sequential PSA. In the latter case, the experimental
results may be normalized, pixel by pixel, with respect to m11 to remove
the irradiance variations and keep only the polarisation distributions,
just as described in [59] and [58].
An alternative to reduce the effects of the interference fringes in the
measurements with the vectorial polarimeter is to change the highly temporally and spatially coherent laser source in the vectorial polarimeter
by a source with good spatial coherence but reduced temporal coherence.
Such a source might be, for instance, a superluminescent diode.
103
Chapter 6. Conclusions
Finally, it is possible to change the detection scheme to achieve coherent detection. In this case, a reference light beam can be use to interfere
with the image of the exit pupil of the collector lens in the plane of
the detectors. From the analysis of the interference pattern it would be
possible to retrieve information about a sub-resolution specimen.
6.4
Future work
A suitable quantitative method to relate the polarisation state distribution in the exit pupil with the actual displacement of the point-scatterer
shall be developed as part of the future of this project. This may be done
with an statistical analysis of the pupil distributions together with the
definition of an appropriate metric and a threshold.
Since the results presented in this work are limited to the case of a
point-scatterer, the range of applications in which they could be useful
is extremely limited. A possible application of the results is in material
inspection; this possibility is currently being explored in collaboration
with Shimadzu Corporation, Japan. However, to increase the range of
possible applications of the method, in the next stage of this research
we shall model and measure the pupil polarisation distribution for subresolution samples with a more complex structure.
Although having information about the three-dimensional field scattered by the sample is of interest per se, for most applications it is desirable to obtain an actual image of the specimen. The vectorial polarimeter
measures the scattering-angle-resolved polarisation state distribution of
the scattered field. Thus, this method can be used to retrieve the electric field components of the scattered filed and the relative phase between
them. The information about the electric field can then be used as the
input to solve the inverse problem and retrieve the shape of the specimen.
In the theory, the solution of the inverse problem is conditioned to
the uniqueness of the far-field distribution of the scattered field for subresolution specimens; this is part of the analysis that shall be done in
the future. In the experiment, the solution of the inverse problem is also
conditioned by the accuracy of the measurements and the signal-to-noise
ratio. Therefore, reconstructing an image of an unknown specimen from
the data obtained with the vectorial polarimeter, via the solution of an
inverse problem, is not trivial neither in the theory nor in the experiment.
The analysis presented in this work only considers aberration-free
systems. Even though this is an appropriate assumption for modern
high quality microscope objectives, the impact of aberrations in the per104
Chapter 6. Conclusions
formance of the method shall be studied in the future. An analytical
method for the calculation of the effect of aberrations in the focused field
distribution of a high NA system, based on the extended Nijboer-Zernike
(ENZ) representation, has been introduced by Braat et al. [65]. Alternatively, the effect of aberrations in the focusing system can be studied
by introducing appropriate phase variations in the generalized aperture
within the context of McCutchen’s method.
Finally, the further increase in the sensitivity of the method, by engineering the incident polarisation state, is part of the issues to be addressed in the future. The experimental control of the focused field distribution shall include a couple of SLMs to engineer the incident inhomogeneous polarisation state. The number of active pixels in the SLMs will
have an influence in the polarisation distributions that can be achieved.
However, the numerical results reported in [36] show that a sampling of
32 × 32 × 32 points in the object space is sufficient to control the field
distribution in the focal region. This does not represent a problem since
a sampling density of 32 × 32 pixels, across the aperture of the incident
beam, is well below the number of active pixels in currently available
SLMs.
105
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