52
Mueller Matrix Ellipsometric imaging
Lars Martin Sandvik Aas
Department of Physics
NTNU
May 4, 2009
2
Preface
This report has been written to report on work done in the course TFY4510 Physics,Specialization Project. The work has been in collaboration with the Applied OpticsGroup, Department of physics at the Norwegian University of Science and Technology.
I would like to thank my supervisor Associate Professor Morten Kildemo and PhDstudent Frantz Stabo-Eeg for help and feedback.
Trondheim, May 4, 2009
Lars Martin Sandvik Aas
Summary
A near infrared Mueller matrix ellipsometric imaging system based on ferroelectric liq-uid crystals has been demonstrated. The ellipsometer has been demonstrated to be anextremely sensitive measurement technology for determination of stress in transparentcrystalline structures. Indications show that it might be a usefull tool to determine shearforces in transparent crystals.
The performance of the Mueller matrix ellipsometer(MME) has been analyzed bothregarding speed and stability. The MME can measure a stable Mueller matrix in 16ms,with some reduction in stability the measurement time can be reduced to 6.4ms. Thisis the state of the art within Mueller matrix ellipsometry. In order to make the MMEimaging system more user frendly a great deal of programming has been done to mini-mize the calibration time of the MME imaging system. This resulted in a reduction incomputation time from 7 hours to 25 minutes.
Contents
1 Introduction 1
2 Polarization 3
2.1 Polarization of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 The plane wave description of polarized light . . . . . . . . . . . . 3
2.2 Representations of polarized light . . . . . . . . . . . . . . . . . . . . . . . 52.2.1 The Jones representation . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Coherency matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.3 Stokes representation . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Mueller-Stokes calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Relation between the Jones and the Mueller matrix . . . . . . . . . 8
2.4 Polarization changing optical components . . . . . . . . . . . . . . . . . . 92.4.1 Linear polarizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.2 Optical retarder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.3 Liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Crystal optics 13
4 Tissue optics 15
4.1 Polarized light in tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Mueller matrix ellipsometry 19
5.1 Determination of system matrices . . . . . . . . . . . . . . . . . . . . . . . 195.1.1 Reference samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 Decomposition of Mueller Matrices . . . . . . . . . . . . . . . . . . . . . . 215.3 Mueller matrix imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6 Setup 25
6.1 Mueller Matrix ellipsometer . . . . . . . . . . . . . . . . . . . . . . . . . . 256.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.1.2 Light source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.1.3 Polarization state generator and analyzer . . . . . . . . . . . . . . 256.1.4 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.2 Instrumental improvements . . . . . . . . . . . . . . . . . . . . . . . . . . 28
vi Contents
6.2.1 New reference sample . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.4 Stressed isotropic crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7 Measurements on skin 33
A Mueller matrices for some optical components 41
List of Figures
2.1 The polarization ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Liquid crystal phases, all molecules are oriented in the same direction . . . 11
4.1 LAG ETT I 3D!! The scattered �eld Esca with an incoming planewaveEinc and scatter angle θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.1 Schematic drawing of an ellipsometer in re�ection mode. With light source(S), polarization state generator (PSG), sample, polarization state ana-lyzer (PSA) and detector or camera in case of imaging (D) . . . . . . . . . 20
6.1 The source of the collimated beam incidenting the PSG. One can turn themirror to change laser source, L1 collects the beam down to the coherencescrambler in order to remove speckle in the image. The lens L2 collimatesthe scattered beam from the scrambler. . . . . . . . . . . . . . . . . . . . . 26
6.2 The MME imaging system developed at NTNU. From left to right, laserlight source(Laser), convex lens (L1) collecting light down to the coher-ent scrambler (CS), convex lens (L2) collimating the laser beam before itenters the PSG which consist of a horizontal polarizer (P1), �xed QWPat 465nm(R1), FLC as HWP at 510nm (F1), �xed HWP at 1008nm(R2)and FLC as HWP at 1020nm(F2). The beam interfere with the samplebefore it enters the PSA which has the same con�guration as PSG but inoposit order. The telelens is imaging the sample in the object plane (OP)on to the CCD of camera(IP). . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.3 A Sellmaier model of a quarter waveplate at 1310nm . . . . . . . . . . . . 296.4 A CaF2 prism is put under pressure with a plug, the prism surface is rough
in the top front(a) and on the top(b). . . . . . . . . . . . . . . . . . . . . 306.5 A µm screw is mounted and bolted to the table in order to put a force on
the CaF2 prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7.1 Mueller matrix image of a non magliant mole. . . . . . . . . . . . . . . . 347.2 Mueller matrix image of a non magliant mole injected with dynospheres . 35
viii List of Figures
1. Introduction
The polarization is one of the fundamental properties of light, it describes the orientaionof the electromagnetic �eld. Polarization is invisible for the human eye, but we knowthat some insects and animals use polarization for navigation and signaling [11, 13].
An important application of polarized light is ellipsometry. The ellipsometer is aninstrument which can measure the optical properties of materials through measurementsof the change of the state of polarization upon re�ection or transmission through a sample.It is used for measuring e.g thickness of thin �lms, optical constant, surface roughness,composition and optical anisotropy.
The Mueller-Stokes formalism is a mathematical representation of polarized light andpolarizing properties of optical matter. The electric �eld is represented by a 4 elementStokes vector and the polarization changing properties are described by a 4 by 4 Muellermatrix. In comparsion with the more known Jones formalism, Mueller-Stokes includesdescription of intensity variation and depolarization. The Stokes vector describes thestate of polarization based on time averaged intensity measurements, which allow thesource of light to be partially incoherent.
Mueller matrix ellipsometry(MME) is a technique which can measure the completeMueller matrix of a sample. It is generally done by generating enough known polarizationstates and analyse the change in polarization after interfering with matter. MME hasshown great results in studies of nano surface structures [23] and remote sensing [15].
A Muller matrix ellipsometer based on liquid crystals(LC) has no moving parts andsupply a highly stable beam, suitable for direct imaging system. Ferroelectric LCs areextremely fast and allow switching speeds that are detector limited in low �ux applica-tions [17].
A Mueller matrix imaging(MMI) system gives the elements in the Mueller matrixas images. MMI has shown great results in biomedicine, such as for determination ofmalignant moles and lupus lesions [28], hepatic �brosis [19] etc. By measuring Muellermatrix images fast enough monitoring live and swift changing processes in production ofnanostructures, medicine and advanced high speed sensing, may be possible.
2 Introduction
2. Polarization
This chapter will outline the theory of polarized light and will give an introduction tothe Mueller-Stokes formalism used in this thesis.
2.1 Polarization of light
The famous Maxwell's equations from 1873 describes the full electromagnetic treatmentof light and introduces the transverse wave orientational nature of light known as polar-ization. Togheter with wavelength and speed, polarization is a fundamental property oflight.
2.1.1 The plane wave description of polarized light
A solution to Maxwell's equations are the monochromatic (single frequency) wave trav-elling in the z direction with no x and y dependence, called the plane wave. Using theconvention proposed by Hauge et al. [12] the electric �eld of a plane wave is
E(z, t) = xEx(z, t) + yEy(z, t). (2.1)
Here x and y are the unit vectors in x and y direction and the orthogonal �elds Ex(z, t)and Ey(z, t) are given by the real part of a complex �eld
Ex(z, t) = <{Ex0 exp i(ωt− kz + δx)}Ey(z, t) = <{Ey0 exp i(ωt− kz + δy)},
(2.2)
where Ex0 and Ey0 are complex amplitudes, k = 2π/λ is the wavenumber, λ the wave-length, ω the angular frequency and t time. δx and δy indicate phase factors for the twoorthogonal �elds. The polarization state is generally determined by the trajectory tracedout by the vector E(z, t) at each position z as a function of time t.
By manipulating and adding Ex and Ey from equation (2.2), the state of polarizationcan be described by Ey0, Ex0 and the phase di�erence δ = δy − δx:(
ExEx0
)2
+(EyEy0
)2
− 2ExEx0
EyEy0
cos δ = sin2 δ. (2.3)
Equation (2.3) is known as the ellipse equation, and shows that the trajectory of theelectric �eld in the x-y plane, as it propagates, is generally an ellipse. E.g. for δ = ±π/2,Ex and Ey out of phase, equation (2.3) becomes(
ExEx0
)2
+(EyEy0
)2
= 1,
4 Polarization
Figure 2.1: The polarization ellipse
Representations of polarized light 5
wich is, for Ex0 = Ey0 = E0, the equation of a circle. For δ = ±2πm,m = [0, 1, 2..],Ex and Ey are in phase, and the polarization state is linear with an angle given by
arctan(Ey0
Ex0
)relative to the x-axis. If δ > 0, Ey is delayed compared to Ex, and vise
versa.
2.2 Representations of polarized light
Due to the vector nature of polarized light it is useful to write the polarization state as avector. Whatever changing the state of polarizationcan then be represented as a matrixwhich can operate on an incoming polarization state. The two vectorized polarizationrepresentation that are most used are the Jones and Stokes vectors. In addition, a matrixrepresentation called coherency matrix, because of its posibility to describe partiallypolarized light, are used to represent polarized light.
2.2.1 The Jones representation
In 1941 R. C. Jones presented his calculus for polarized light [? ]. The Jones vector isde�ned as the vector of the plane wave (monochromatic) complex electric �eld of equation(2.1) with explicit time and space dependence removed
E =[ExEy
]=[Ex0e
iδx
Ey0eiδy
], (2.4)
where Ex0, Ey0, δx and δy still are the complex amplitudes and phases of the x- andy-direction. Optical interactions are de�ned by a 2 × 2 transfer matrix J, called theJones matrix, such that[? ]
E′ = JE[E′xE′y
]=[a bc d
] [ExEy
]=[aEx + bEycEx + dEy
] (2.5)
J is called the Jones matrix.
2.2.2 Coherency matrix
All light sources, natural and human made, have statistical properties caused by un-predictable �uctuations in the lightsource. Light is said to be totally incoherent if thephase dependence of the light is totally random. When the phase is totally random, thepolarization is totally random, and we have non-polarized light. For partially incoherentlight, the phase have a trend of coherency, but at some time the polarization might bechanged. Then the light is said to be partially polarized [3, 24]. Because of the stochasticproperties of paritally polarized light, the Jones vector presented above can only describe
6 Polarization
totally polarized light. To describe stochastic properties the most obvious thing to do isto introduce time averages over the cross correlated electric �eld components [4, 12]. Thiscan be synthesized to the Wiener coherency matrix JM , de�ned as the direct product ofthe Jones vector with its hermitian adjoint
JM ≡⟨E×E†
⟩=[〈ExE∗x〉
⟨ExE
∗y
⟩〈EyE∗x〉
⟨EyE
∗y
⟩ ] =[Jxx JxyJyx Jyy
]. (2.6)
The angular brackets denote averaging over several wave cycles. N. Wiener [32] was the�rst to introduce the coherency matrix. 1
2.2.3 Stokes representation
A third way to represent polarized light is by the four element Stokes vector S. The vectorelements were �rst de�ned by the famouse George G. Stokes 2 in 1952. The Stokes vectoris de�ned as
S =
s0
s1
s2
s3
=
I0◦ + I90◦
I0◦ − I90◦
I45◦ − I−45◦
IR − IL
=
⟨E2x0
⟩+⟨E2y0
⟩⟨E2x0
⟩−⟨E2y0
⟩2 〈Ex0Ey0〉 cos δ2 〈Ex0Ey0〉 sin δ
. (2.7)
The angular brackets again signify time averages. The subscripts of the intensity I referto polarization states relative to the x-axis(0◦), right(R) and left(L) circular polarization.The parameter s0 is the total intensity of the light, s1 the di�erence between the amountof linear horizontal and vertical polarized light, s2 the di�erence between the amount oflinear +45◦ and −45◦, and s3 the di�erence between the amount of right and left circularpolarized light. Since the Stokes parameteres are expressed in terms of intensities, theStokes representation are preferable in experimental work.
An important quantity when dealing with partially polarized light is the degree ofpolarization P [2]. It is given by the ratio of the intensity of the polarized light and thetotal intensity s0.
P =
√s2
1 + s22 + s2
3
s0. (2.8)
For fully polarized light s0 =√s2
1 + s22 + s2
3 and the degree of polarization is one.The Stokes representation is completely equivalent to the coherency matrix. By
writing the elements in the coherency matrix as a vector, it can be translated to theStokes vector by using a translation matrix
s0
s1
s2
s3
=
1 0 0 11 0 0 −10 1 1 00 i −i 0
JxxJxyJyxJyy
(2.9)
1E. Wolf [33] did independently reintroduce the concept on a more fundamental approach. For allpractial studies it is advantageous to use the Wiener coherency matrix[4].
2George G. Stokes did important contributions to �uid dynamics, mathematics as well as optics
Mueller-Stokes calculus 7
An example of a Stokes vector is the linear horizontally polarized light, when Ey0 = 0.Then from equation (2.7)
S =
⟨E2x0
⟩⟨E2x0
⟩00
= I0
1100
. (2.10)
where I0 =⟨E2x0
⟩.
2.3 Mueller-Stokes calculus
When describing the polarized light as a Stokes vector, an interaction of the light withsome matter will result in a linear transformation of the vector. The interaction can beput into a 4× 4 matrix known as the Mueller matrix (MM) [7]
M =
m11 m12 m13 m14
m21 m22 m23 m24
m31 m32 m33 m34
m41 m42 m43 m44
. (2.11)
All optical components, such as polarizers, retarders, re�ective surfaces, etc. can bedescribed by a Mueller matrix. For a known Mueller matrix and an incoming Stokesvector one can �nd the outgoing Stokes vector S′ by multiplying the matrix with thevector. The most general outgoing Stokes vector is found by acting the general Muellermatrix (2.11) on the general Stokes vector (2.7)
S′ = M ·S
=
m11s0 +m12s1 +m13s2 +m14s3
m21s0 +m22s1 +m23s2 +m24s3
m31s0 +m32s1 +m33s2 +m34s3
m41s0 +m42s1 +m43s2 +m44s3
. (2.12)
The Mueller matrix describes the optical component's polarizing properties. The totalMueller matrix (Msys) of an optical system is found by direct successive Mueller matrixmultiplication of the di�erent optical componets Mueller matrixes. With light interactionfrom component 1 to N, the system Mueller matrix is
Msys = MNMN−1 · · ·M1. (2.13)
8 Polarization
Rotation
A rotation of the coordinate system by an angle θ in the Mueller-Stokes calculus is carriedout by the rotation Mueller matrix R(θ) [9]
R(θ) =
1 0 0 00 cos(2θ) sin(2θ) 00 − sin(2θ) cos(2θ) 10 0 0 1
. (2.14)
A rotation of a Stokes vector by an angle θ is given by S(θ) = R(θ)S. R is also knownas the Mueller matrix of an optical rotator.
The Mueller matrix of a rotated optical element is found by �rst doing a rotation ofthe coordinate system, applying the original Mueller matrix and then rotate back to theoriginal coordinate system. Hence the rotated Mueller matrix is
M(θ) = R(−θ)MR(θ). (2.15)
2.3.1 Relation between the Jones and the Mueller matrix
By starting with the Jones matrix of equation (2.5), the transformation of the coherencyvector3 can be found by calculating J′ =
⟨E′ ×E′†
⟩J′M =FJJM
J ′xxJ ′xyJ ′yxJ ′yy
=
aa∗ ab∗ ba∗ bb∗
ac∗ ad∗ bc∗ bd∗
ca∗ cb∗ da∗ db∗
cc∗ cd∗ dc∗ dd∗
JxxJxyJyxJyy
, (2.16)
and then using equation (2.9) to transform the coherency transfer matrix to a Mueller-Jones matrix4
MJ = AFJA−1, (2.17)
such that
MJ =
12
aa∗ + cc∗ + bb∗ + dd∗ aa∗ + cc∗ − bb∗ − dd∗ ab∗ + cd∗ + ba∗ + dc∗ −i(ab∗ + cd∗ + ba∗ − dc∗)aa∗ + bb∗ − cc∗ − dd∗ aa∗ − bb∗ − cc∗ + dd∗ ab∗ − cd∗ + ba∗ − dc∗ −i(ab∗ − cd∗ − ba∗ + dc∗)ac∗ + ca∗ + bd∗ + db∗ ac∗ + ca∗ − bd∗ − db∗ ad∗ + bc∗ + cb∗ + da∗ −i(ad∗ − bc∗ + cb∗ − da∗)
i(ac∗ + bd∗ − ca∗ − db∗) i(ac∗ − bd∗ − ca∗ − db∗) i(ad∗ − bc∗ − cb∗ − da∗) ad∗ − bc∗ − cb∗ + da∗
.(2.18)
3The coherency vector is the elements of the coherency matrix in vector form,
JM =[
Jxx Jxy Jyx Jyy
]T4A Mueller matrix is called a Mueller-Jones matrix if, and only if, the Mueller matrix can be de�ned
by a Jones matrix.
Polarization changing optical components 9
MJ can be used for direct transformation of a Jones matrix to a Mueller matrix. AllJones matrices can be translated to a Mueller matrix but the oposite is not true5, e.g.depolarizing properties can not be described by a Jones matrix.
2.4 Polarization changing optical components
An optical component which has the property of changing the state of polarization ofincident light, through a change in the shape of the polarization ellipse in equation (2.3),is called a polarizing component. The polarizing elements used in the Mueller matrixellipsometer(MME) of this project will in the following be presented brie�y. For a morethorough discussion about why these elements were choosen, see [16, 29].
2.4.1 Linear polarizer
In a linear polarizer the absorption of light depends on the direction of the incidentelectric �eld. Polarizers used in the MME are made of a certain anisotropic media, calleddichroic. The most common dichroic material are the Polaroid H-sheet. It is producedby streching polyvinyl alcohol in a certain direction, such that the hydrocarbons lies ina grid. The transmittance through the polarizer is bandwith limited. Other solutions ofthe linear polarizer are the wired grid polarizer, used in the infrared region. The electric�eld orthogonal to the wires will when incidenting set up a current in the wires andpass through, and the parallel component will be absorbed. Other solutions for a linearpolarizer are re�ection at the Brewster angle and polarizing beamsplitters (e.g. Rochonor Wollaston) [3, ch. 6] but they are not suitable for imaging.
The Mueller matrix of a horizontal linear polarizer is
M =12
1 1 0 01 1 0 00 0 0 00 0 0 0
. (2.19)
The linear polarizer oriented at an angle θ are given using equation (2.15) as
M =12
1 cos 2θ sin 2θ 0
cos 2θ cos2 2θ cos 2θ sin 2θ 0sin 2θ cos 2θ sin 2θ sin2 2θ 0
0 0 0 0
. (2.20)
2.4.2 Optical retarder
An optical retarder changes the state of polarization by delaying Ex or Ey compared tothe other, i.e. change the phase di�erence δ such that the incident wave and the outgoingwave have di�erent phase [3].
5Anderson et al.[1] have stated the criteria for when a Mueller matrix can be transformed to a Jonesmatrix.
10 Polarization
Optical retarders are usually made from uniaxial crystals. A uniaxial material hasdi�erent refractive indexes for di�erent axes in the material, a property called birefrin-gence. Two axes have equal refractive indexes,no, where one of them are the optical axis,the other are called the ordinary axis. The third axis are called the extraordinary axis,and has refractive index ne.
Di�erent refractive indices cause di�erent propagation velocity, c0/no and c0/ne forthe electric �eld in the two directions. The axis with the lowest refractive index in amedium is called fast axis, the orthogonal, with the highest refractive index, is called theslow axis. In uniaxial crystals the fast axis is the same as the ordinary axis and the slowaxis is the same as the extraordinary axis. A measure for the birefringence is ∆n:
∆n = ne − no. (2.21)
When propagating through the crystal the ordinary and the extraordinary wave under-goes di�erent phase shifts. The di�erence in phase shift ∆ϕ is given by
∆ϕ =2πλ0d(|∆n|), (2.22)
where d is the material thickness and λ0 is the wavelength in vacuum. Birefringence andretardation in crystals will be further discussed in section 3.
Standard retarders are created with ∆ϕ = 2π, π or π/2, called respectively full-wave
plate(fwp), half-wave plate(hwp) and quarter-wave plate(qwp). A retarder is designedfor one wavelength λ0. If another retardance is desired one can use a model to determinewhich wavelength,λ0, your e.g. quarter-wave plate should hold [16].
The Mueller matrix of a linear retarder with retardance ∆ϕ is given by
M =12
1 0 0 00 1 0 00 0 cos ∆ϕ − sin ∆ϕ0 0 sin ∆ϕ cos ∆ϕ
. (2.23)
Rotated an angle θ it reads
M =12
1 0 0 00 cos2 2θ + sin2 2θ cos ∆ϕ (1− cos ∆ϕ) cos 2θ sin 2θ − sin 2θ sin ∆ϕ0 (1− cos ∆ϕ) cos 2θ sin 2θ sin2 2θ cos 2θ sin ∆ϕ0 sin 2θ sin ∆ϕ cos 2θ sin ∆ϕ cos ∆ϕ
.(2.24)
2.4.3 Liquid crystals
A liquid crystal(LC) is a collection of liquid molecules [30] with relative high orientationalorder and low positional order. Liquid crystals can have di�erent degrees of �uidity(moreor less isotropic). The two most used LC phases are called nematic and smectic-C.
In the smectic-C phase, the molecules are arranged in 2D layers and the moleculesare bound to a layer, but move freely within this, see Figure 2.2(b). A LC acts as an
Polarization changing optical components 11
(a) Nematic phase (b) Smectic-C phase
Figure 2.2: Liquid crystal phases, all molecules are oriented in the same direction
optical retarder, and has the Mueller matrix of equation (2.24). A smectic-C LC, hasferroelectric properties. If it is put between two glass plates, surface interaction permitonly two stable states of orientation of the molecules. This causes smectic LCs to onlyrotate a constant angle, i.e not change retardance. The switching between these twostates is done by switching an electric �eld across the LC.
In nematic phase, the molecules �oat freely in all directions but all have the sameorientation, see Figure 2.2(a). In a nematic LC one can change the retardance of thecrystall, i.e. change ∆n. This is done by rotating the molecules along with an appliedelectric �eld.
In ellipsometry LCs are used in low intensity setups. With too high intensity theLC will heat up and might be broken. Nematic LCs used for optical applications arecalled liquid crystal variable retarders (LCVR) and smectic-C ferroelectric liquid crystal(FLC)[16]. The advantage of the LCVR is the possibility to change the retardanceas function of applied voltage, and always obtain an optimal polarization state in theellipsometer [17]. The disadvantage is that it is quite slow and that the retardance hasa strong temperature dependence. The FLC has �xed retardance, and are created asa QWP or HWP for a desired wavelength. When a voltage is applied to the FLC theoptical axis of the retarder change to a �xed position. There are only two stable positionsi.e. the FLC has only two stable states. In a FLC one can not change the retardance,only the orientation. Since there are only two stable states, the FLC are much lesstemperature dependent than the LCVR. Changing state for the FLC is also much faster,typical < 20µs, than for the LCVR,10ms [17].
12 Polarization
3. Crystal optics
Crystals have molecules or atoms organized in spacein a lattice structure. If the latticehas di�erent lattice constants for di�erent directions in the medium it is called linearanisotropic. In such a medium, each component of the electric �ux density is a linearcombination of the three components of the electric �eld [3, ch 6].
Di =∑j
εijEj (3.1)
The dielectric properties of the medium are therfore described by a 3 by 3 electric per-mittivity tensor ε such that equation (3.1) can be written in vector form as D = εE.The elements of ε depend on how the coordinate system is choosen relative to the crystalstructure. The relationD = εE can be inverted and written in the form E = η
ε0D where
η = ε0ε−1 are the electric impermeability tensor and ε−1 is the invers of ε. Choosing a
cartesian coordinate system, and folowing Khan [14], the optical index ellipsoid is thequadratic representation of the impermeability tensor.∑
ij
ηijxixj = 1, i, j = 1, 2, 3. (3.2)
where ηij = ε0/εij = 1/n2ij is the i, j element in η and nij =
√εij/ε0 is the refractive
index. Equation (3.2) can be written as
η11x2 + η22y
2 + η33z2 + 2η23yz + 2η31zx+ 2η12xy = 1 (3.3)
in which summation over repeated indices is implicit and ηij = ηji.Assuming perfect elasticity, the generalized Hooke's law holds. According to Pockel's
phenomenological theory of photoelasticity [22], the di�erence between the optical param-eters in a deformed(ηij) and undeformed state(η0
ij) are linear functions of the componentsof the stress.
η11 − η011 = −(q11Pxx + q12Pyy + q13Pzz + q14Pyz + q15Pzx + q16Pxy)
η12 − η012 = −(q61Pxx + q62Pyy + q63Pzz + q64Pyz + q65Pzx + q66Pxy)...
(3.4)
where qij are the stress optical constants and Pxx, Pyy, ..., Pxy are the stress components.Depending on the geometry of the stress applied and the optical constants of the material,the birefringence(∆n) can be calculated from the relations above. For stress along thethe main axes in the crystal(x, y and z), one �nd that the change in the refractive indexand birefringence is proportional with the stress [14].
14 Crystal optics
4. Tissue optics
Biological tissues are optically inhomogeneous and absorbing media, with refractive indexhigher than that of air. This give rise to Fresnel re�ection in the interface tissue/air, whilethe remaining light penetrates the tissue and experience scattering. Cellular organellessuch as mitochondria are the main scatterers.
In the ultra violet(UV) and the infra red(IR)(λ > 2µm) spectral regions light isabsorbed very fast. In the wavelength range of 600nm to 1.5µm scattering is the majore�ect, and the light penetrates to a depth of 8-10mm.
A collimated beam is attenuated in a tissue layer of thickness d according to theBeer-Lambert law
I(d) = (1−RF )I0 exp(−µtd), (4.1)
where I(d) is the intensity of transmitted light, RF Fresnel re�ection coe�cient1. I0 is theincident light intensity and µt = µa +µs is the extinction coe�cient(or total attenuationcoe�cient), µa is the absorption coe�cient and µs is the scattering coe�cient. The meanfree path length between two interactions is denoted by lph = µ−1
t
4.1 Polarized light in tissue
The random inhomogeneous nature of tissue results in fast depolarization of polarizedlight travelling through it, therefore the polarization can usually be ignored when workingwith tissue optics. However, for transparent tissues(such as for the eye, membranes,super�cial skin layers and when working in the more transparent near infra red) thestate of polarization remains measureable even at mm thicknesses. Information abouttissue structure can then be extracted from the degree of depolarization, polarizationstate transformation and polarization signatures in the scattered light that all lie in theMueller matrix.
The theory of elastic scattering by small particles was originally derived by GustavMie. Bohren and Hu�man[5] derived the Mie theory using the Mueller-Stokes formalism,which states that the scattered �eld (Es) and the incoming plane wave �eld(Ei) relateslinearly [
E||sE⊥s
]=
exp {ik(r − z)}−ikr
[S2 S3
s4 S1
] [E||iE⊥i
](4.2)
where E|| and E⊥ are the electric �elds parallel and orthogonal to the scattering plane,S1−4 are the complex elements in the scattering/transfer Jones matrix and depend on the
1The Fresnel coe�cient for normal incident RF =[
n−1n+1
]2where n is the relative mean refractive
index of tissue and surrounding media.
16 Tissue optics
scattering angle θ and the azimuthal angle φ(�gure 4.1), k = 2π/λ is the wavenumber,λ =λ0/n the wavelength of the light in the scattering medium, λ0 is the wavelength in vacuumand n is the mean refractive index of the scattering medium.
In order to �nd the complex elements of the scattering Jones matrix for all θ andφ one need to measure both the amplitude and the phase of the scattered light in allspatial directions, this is complicated but can be done by measuring the interferencebetween light scattered from a scatterer with known properties and the particle with un-known properties. Because measurements of the Stokes vector just involves real intensitymeasurements, not phase measurements, the measurements of the scattering matrix canbe done with less experimental problems by translating the scattering matrix from theJones calculus to the Mueller-Stokes calculus by the use of equation (2.18). The scatteredStokes vector S is given by
Ss =MsSis0,s
s1,s
s2,s
s3,s
=1
k2r2
S11 S12 S13 S14
S21 S22 S23 S24
S31 S32 S33 S34
S41 S42 S43 S44
s0,i
s1,i
s2,i
s3,i
(4.3)
where the scattering coe�cients Sij are given by
S11 =12(|S1|2 + |S2|2 + |S3|2 + |S4|2
)S12 =
12(|S2|2 − |S1|2 + |S4|2 − |S3|2
)S13 =Re {S2S
∗3 + S1S
∗4}
S14 =Im {S2S∗3 − S1S
∗4}
S21 =12(|S2|2 − |S1|2 − |S4|2 + |S3|2
)S22 =
12(|S1|2 + |S2|2 − |S3|2 − |S4|2
)S23 =Re {S2S
∗3 − S1S
∗4}
S24 =Im {S2S∗3 + S1S
∗4}
S31 =Re {S2S∗4 + S1S
∗3}
S32 =Re {S2S∗4 − S1S
∗3}
S32 =Re {S2S∗4 − S1S
∗3}
S33 =Re {S1S∗2 + S3S
∗4}
S34 =Im {S2S∗1 + S4S
∗3}
S41 =Im {S∗2S4 + S∗3S1}S42 =Im {S∗2S4 − S∗3S1}S43 =Im {S1S
∗2 − S3S
∗4}
S44 =Re {S1S∗2 − S3S
∗4}
(4.4)
Polarized light in tissue 17
Organelle Refractive index
Extracellular �uid [21] 1.35-1.36Cytoplasm [6] 1.36-1.375Nucleus [6] 1.38-1.41Mitochondria and organelles[20] 1.38-1.41Melanin[31] 1.6-1.7
Table 4.1: Refrative indices for di�erent organelles
For optical inactive particles such as spheres and cylinders S3 and S4 are zero, andthe resulting elements for the Mueller scattering matrix are
S11 = S22 =12(|S2|2 + |S1|2
)S12 = S21 =
12(|S2|2 − |S1|2
)S33 = S44 =
12
(S∗2S1 + S2S∗1)
S34 = −S43 =12
(S1S∗2 − S2S
∗1) ,
(4.5)
with all other elements equal zero.
18 Tissue optics
Figure 4.1: LAG ETT I 3D!! The scattered �eld Esca with an incoming planewave Einc andscatter angle θ
5. Mueller matrix ellipsometry
Mueller matrix ellipsometry(MME) is a technique where the complete Mueller matrix(MM)of a sample can be measured. The MM holds all information about the polarizing proper-ties of the sample, compared to conventional ellipsometry which gives limited informationabout e.g. the depolarization and intensity variations, the latter is important in o� spec-ular scattering samples.
In general, you need a polarization state generator(PSG) to generate known Stokesvectors, and a polarization state analyzer(PSA) to analyze how the Stokes vectors changetheir polarization state after interfering with the sample. Since no light source withchangeable polarization state and no detector/camera with a polarization state analyzer isavailable, one use a sequence of optical components to create and analyze the polarization.A light source and a detector is used for creating and detecting the light. A principledrawing of an ellipsometer is given in �gure 5.1.
To determine all 16 elements of the samples MM a minimum of 16 measurements areneeded. 4 generated states which are analysed by 4 analyser states. The 16 measurementsis used to construct a measurement matrix B.
The PSG and the PSA is characterized by their own 4 by 4 matrix. These matri-ces, called W and A, can be determined by a calibration routine, for instance the onedeveloped by Compain et al. [10] wich will be discussed in section 5.1.
The system equation can be found by multiplying the MM of the PSG(W), thesample(M), and the analyzer(A) [10].
B = AMW (5.1)
Multiplying the inverse of A and W from each side gives
M = A−1BW−1 (5.2)
By measuring B the Mueller matrix, M, of a sample can be calculated for known Wand A.
5.1 Determination of system matrices
To determine the modulation matrix W and the analyzer matrix A, the Eigenvalue cal-ibration method(ECM) proposed by Compain et. al. [10] is widely used. The ECM is apowerful method with many attractive properties, Aand Wcan be determined indepen-dently, all systematic errors are accounted for and the choice of calibration samples doesnot depend on Aor W. The limitations of the ECM are that the forms of the Muellermatrices of the reference samples have to be known, the orientation of one of the reference
20 Mueller matrix ellipsometry
Figure 5.1: Schematic drawing of an ellipsometer in re�ection mode. With light source (S),polarization state generator (PSG), sample, polarization state analyzer (PSA) anddetector or camera in case of imaging (D)
samples must be precisely known. There have to be at least two reference samples thatare su�ciently di�erent in order to get unique Aand W.
Compain et. al. [10] sugests two di�erent approaches for the ECM, one for trans-mission mode calibration and one for re�ection mode calibration. Only the transmissionmode calibration method will be discussed here since it is the one implemented in oursetup.
Transmission mode ECM
In the case of transmission mode, the reference samples are choosen to be air, a linearpolarizer, a linear retarder, and rotations of these. Each reference sample gets its ownmeasured intensity matrix Bi which form a set {B} corresponding to the set of referencesamples {M},
Bi = AMiW, B0 = AW, i = 1, 2, 3 (5.3)
From {B} the two sets {C} and {C′} are constructed
Ci = B−10 Bi = W−1MiW, i = 1, 2, 3
C′i = BiB−10 = AMiA
−1, i = 1, 2, 3(5.4)
Ci is independent of A and C′i is independent of W. C′i, Ci and Mi are matrices withthe same eigenvalues. With some knowledge of the reference sample, one can reconstructits Mueller matric by associate the eigenvalues of Ci with the theoretical eigenvalues ofMi.
Once all Mi are reconstructed one can �nd �nd A and W by solving two sets oflinear equation given by
MiX−XCi = 0 (5.5)
X′Mi −C′iX′ = 0 (5.6)
Decomposition of Mueller Matrices 21
with solutionsX = W andX′ = A. It is su�cient to solve the equation for e.g. W. ThenAcan be obtained by using the measurement of air B0 = AW so that A = B0W
−1.The transmission mode ECM is implemented in MATLAB by F. Stabo-Eeg.
5.1.1 Reference samples
The choice of reference samples are important for the accuracy of the calibration. Ithas been shown [18] that for two polarizers and one retarder as reference samples theorientation and retardance are
θpol = 90/degree (5.7)
θret = 30.5◦ or 59◦ (5.8)
∆ = 109◦ (5.9)
5.2 Decomposition of Mueller Matrices
In order to analyze a measured Mueller matrix(MM), which may be very complex, dif-ferent tools have been developed. Lu and Chipman [8] suggests a polar decompositionprocedure where the measured MM is factorized into a diattenuation matrix MD, aretarding matrix MR and a depolarizing matrix M∆.
M = M∆MRMD (5.10)
Some polarizing properties can be deduced directly fromM, such as the diattenuation(D)and polarizance(P).
Diattenuation is the property of intensity transmittance of light depending on thepolarization state of the incident beam. The diattenuation, D, is de�ned as the length
of the diattenuation vector→D
→D=
DH
D45
DC
=1m11
m12
m13
m14
(5.11)
where m11..14 are the elements of M ( see equation (2.11)). The three elements of→D, describe the horizontal(DH), 45
◦linear(D45), and circular(DC) diattenuation of thecorresponding elements in the incoming Stokes vector.
The polarizance(P) describe how the optical component polarizes incident unpolar-
ized light. P is de�ned as the length of→P
→P=
PHP45
PC
=1m11
m21
m31
m41
(5.12)
The subscripts of the elements of→P is the same as of the diattenuation vector.
22 Mueller matrix ellipsometry
The retardance vector(→R) is de�ned in the same way as
→D and
→P .
→R= RR =
RHR45
RC
(5.13)
R is the length of→R, and R is the unit vector of
→R.
→R can not, as for
→D and
→P ,
be directly deduced from M, but can be found by following Lu and Chipman[8]. Thefactorized matrices in equation (5.10) are de�ned as.
M∆ =
[1
→0T
→P∆ m∆
],MR =
[1
→0T
→0 mR
],MD =
[1
→DT
→D mD
](5.14)
where→P∆=
→P −m
→D
1−→D
2 (5.15)
and m = mRmD. The matrices mR, mD and m∆ are 3 by 3 sub-matrices of theretardance, diattenuation and depolarization matrices.
The submatrice mD is given by the diattenuation vector and length of it as
mD =√
1−D2I3 + (1−√
1−D2)DDT (5.16)
where D =→D /|
→D | and I3 the 3 by 3 identity matrix.
Considering a Mueller matrix with no depolarization M∆ = I4, gives
M = MRMD = m11
[1
→0T
→P∆ mDmR
]= m11
[1
→0T
→P∆ m
](5.17)
then mR can be written as
mR =1√
1−D2
(m−
(1−
√1−D2
)P DT
)(5.18)
Considering a Mueller matrix with no diattenuation MD = I4, gives
M = M∆MR =
[1
→0T
→P∆ m∆mR
]= m11
[1
→0T
→P∆ m′
](5.19)
then m∆ can then be found as
m∆ =±{m′(m′)T +
(√λ1λ2 +
√λ2λ3 +
√λ3λ1
)I3
}−1
·
{(√λ1 +
√λ2 +
√λ3
)m′(m′)T +
√λ1λ2λ3I3
} (5.20)
Decomposition of Mueller Matrices 23
where λ1, λ2 and λ3 are the eigenvalues of m′(m′)T . The sign of m∆ is the same asthat of the determinant of m′.
The length of→R (equation (5.13)) is given as
R = arccos (trace(MR)
2− 1) (5.21)
The unit vector R is given by
R = R
a1
a2
a3
, ai =1
2 sinR
3∑j,k=1
εijk[mR]jk (5.22)
where εijk is the Levi-Civita symbol. To separate contributions of linear retardance andcircular retardance, the total retardance matrix MR can be written as the product of alinear retarder(equation (2.24), with retardance ∆ϕ and fast axis at an angle θ) and acircular retarder(equation (2.14), with optical rotation of ψ). The trace of MR can thenbe shown as
trace(MR) = 4 cos2 (∆ϕ2
) cos2 ψ (5.23)
Inserting Eq. (5.23) into (5.21) gives
R = arccos[2 cos2 (
∆ϕ2
) cos2 ψ − 1]
(5.24)
The third element of the retardance vector,→R, can now be expressed as
R2C =
sin2 ψ cos2 ∆ϕ2
1− cos2 ϕ cos2 ∆ϕ2
(5.25)
Both the total retardance R and the circular retardation RC are now expressed by thelinear retardance, ∆ϕ, and optical rotation,ψ. R and RC are independent of the orien-taion of the fast axis of the linear retarder. The linear retardance and optical rotationcan be expressed as
∆ϕ = 2 arccos
√a2
3
(1− cos2
R
2
)+ cos2
R
2(5.26)
ψ = arccoscos R2
cos ∆ϕ2
(5.27)
The orientation of the fast axis can be determined by R45 and RC
θ =12
arctan(R45
RC
), θ ∈ [0◦, 180◦] (5.28)
in reality θ ∈ [0◦, 360◦], so equation (5.28) leads to an ambiguity of the direction of thefast axis.
24 Mueller matrix ellipsometry
5.3 Mueller matrix imaging
6. Setup
6.1 Mueller Matrix ellipsometer
There are di�erent ways to realize a Mueller matrix ellipsometer(MME). When the MMEused in this project was developed [16], focus on speed and getting a relative wide nu-merical aperture was of importance, in order to make it possible to further develop thesystem into a Mueller matrix imaging system. The fact that the swiching time is ex-tremely fast for ferroelectric liquid crystal retarders compared to other solutions, such asliquid crystal variable retarders, rotating retarders etc., is the reason why this solutionwas choosen.
6.1.1
6.1.2 Light source
It is now possible to use both a 980nm diode laser and a 1.5µm diode laser in the MME.The source of light is changed by turning a mirror. The 1.5µm laser supply a morestable beam than the original 980nm laser, which was found to have some 100Hz noise.For imaging a Hamamatsu C4742-95 camera with an active Silicon CCD of 512x512pixels was used. The bandwith of Silicon limited the Mueller matrix imaging to the980nm laser. In order to go deeper into the near infrared and use the 1.5µm source, anInGaAs camera was put into the setup. The software for the use of this camera in theMME system was not �nished in time to present Mueller matrix images at 1.5µm in thisreport.
For the characterization of the switching time the 1.5µm source was used, with anextended InGaAs avalanche diode detector as sensor.
A visible, He-Ne, laser has been placed in the back of the setup, such that it can beused for lineup. In order to decorrelate speckle patterns, the laser beam is focused onto arotating di�user [26]. The beam is then collimated by a lens and enters the PSG. Figure6.1 shows a photo of the light source.
6.1.3 Polarization state generator and analyzer
The polarization state generator and analyser developed at NTNU, consist each of ahigh contrast dichroic polarizer(P), two 0th-order �xed waveplates(a quarter-waveplateat 465 nm(F1) and a half-waveplate at 1008 nm(F2)) and two 0th-order ferroelectricliquid crystals (FLC)(half-waveplates at 510 nm(LC1) and 1020 nm(LC2)) [16, 17]. Nochanges has been done to the PSG and PSA. The PSG and PSA are constructed asshown in �gure 6.2.
26 Setup
Figure 6.1: The source of the collimated beam incidenting the PSG. One can turn the mirrorto change laser source, L1 collects the beam down to the coherence scrambler inorder to remove speckle in the image. The lens L2 collimates the scattered beamfrom the scrambler.
Mueller Matrix ellipsometer 27
Figure6.2:TheMMEimagingsystem
developed
atNTNU.From
leftto
right,laserlightsource(Laser),convex
lens(L1)collecting
lightdow
nto
thecoherentscrambler(CS),convex
lens(L2)collimatingthelaserbeam
before
itenters
thePSG
which
consistofahorizontalpolarizer(P1),�xed
QWPat
465n
m(R
1),FLCasHWPat
510n
m(F1),�xed
HWPat
1008
nm(R
2)
andFLC
asHWPat
1020
nm(F2).
Thebeam
interferewiththesample
before
itenters
thePSA
whichhasthesame
con�gurationasPSG
butin
opositorder.Thetelelensisimagingthesamplein
theobject
plane(O
P)onto
theCCDof
camera(IP).
28 Setup
Waveplate AUV AIR λUV λIRQWP 110 50 134 11.16HWP 520 257 134 11.16
Table 6.1: The coe�cients of equation (6.1)
6.1.4 Imaging
The Mueller matrix ellipsometer has been set-up as an imaging system. It has a relativewide numerical aperture which make it possible to use a wide collimated beam. A telelens has been placed before the camera, after the PSG, in order to image the sample onto the active CCD of the camera. The current lens images the object in a 1:1 scale, suchthat the area imaged is 7mm · 7mm. This can easily be changed by replacing and/orchange the positions of the lenses.
6.2 Instrumental improvements
Some minor changes have been done to the Mueller matrix imageing system.
6.2.1 New reference sample
The fact that the waveplate used as reference sample only had a clear aperture of 8mmresulted in a relative small beam size, and a small area to be imaged. To increasethis area a new reference sample has been acquired.
Sellmeier model
A more accurate model for retardance in a waveplate than that of equation (2.22) arethe Sellmeier model[25]
δ ≈ AUV λ(λ2 − λ2
UV
)1/2 − AIRλ(λ2 − λ2
IR
)1/2 (6.1)
The constants AUV , AIR, λUV and λIR for a quarter wave plate and half waveplate were determined expermientaly by Ladstein et al. [17] and are presented in table6.1. A retardance of ∆ϕ ∈ [160 − 60]◦ for λ ∈ [800 − 1700]nm (ideally 109◦ for allewavelengths[18]). The waveplate retarder was modeled using equation (6.1) for di�erentstandard thicknesses1. The most suitable standard thickness waveplate with satisfactoryretardance for wavelengths in the near infra red(800nm-1700nm) was found to be the1310nm whose retardance spectrum is ploted in �gure 6.3
The clear aperture of the waveplate was choosen to be the same as that of the wave-plates in the PSA and PSG, 18mm, so that it is no longer the limiting part.
1Standard waveplates are being produced as qwp and hwp to standard laser wavelength, see www.
casix.com
Calibration 29
Figure 6.3: A Sellmaier model of a quarter waveplate at 1310nm
6.3 Calibration
The calibration theory developed by Compain et al. [10], described in section 5.1 wasimplemented in MatLab by Stabo-Eeg in 2006 and implemented for the transmissionimaging system by Skjerping [27]. The calibration routine was very time consuming.Calibration for all pixels on the camera was taking approximately 7 hours [27]. Tomake the calibration more e�cient one can now choose a region of interest directly ona live picture in the LabView program. This reduced the calibration time proportionalto the portion of the image. An investigation of the MatLab code resulted in �ndingsome slow parts. The program is written for a single detector. Running the routinemore than 266 thousand times(512 · 512) is time consuming. By writing out some of thematrix calculations done in for loops to direct evaluations(with help of Maple), time wasshortened with approximately 2 hours. MatLab's function fminunc is a relative slowfunction, it is a function for �nding minimum of an input function. In the calibrationroutine we use it for �nding the rotation of the reference sample. By �nding the rotationjust for every 100 pixel and assuming that the next 99 pixels are aprroximatly the sameas the one calculated, the total calibration time was reduced to less than 25 minutes(23min 49 s) for all pixels. For further optimalization of the calibration routine it shouldeither be vectorized in MatLab or rewritten in a more e�ective language.
6.4 Stressed isotropic crystal
Applying a mechanical force to a crystal will induce an anisotropic stress in the material.This will change the lattice constants and the dielectric tensor as discussed in section 3.Applying stress to an isotropic crystal will transform it to an anisotropic crystal.
30 Setup
(a) Front view (b) Top view, red dot indicates contactarea of the plug
Figure 6.4: A CaF2 prism is put under pressure with a plug, the prism surface is rough in thetop front(a) and on the top(b).
Figure 6.5: A µm screw is mounted and bolted to the table in order to put a force on the CaF2
prism
Stressed isotropic crystal 31
A calcium �uorid(CaF2) prism is put in the image plane of the MME. CaF2 is anisotropic crystal that is transparent from the UV(0.15µm) to the infrared (9µm). Inorder to apply a force onto the prism, a µm screw is mounted and bolted to the table asshown in �gure 6.5. Accidentally the CaF2 prism at hand has a rough strip on top of it.Therefor the data from right under the surface is useless. Front view and top view of theprism and plug, is sketched in �gure 6.4. Since the plug has a circular contact-surface,of which the diameter is much smaller than the width of the prism, it may be di�cult toanalyze the data any further.
32 Setup
7. Measurements on skin
34 Measurements on skin
Figure 7.1: Mueller matrix image of a non magliant mole.
35
Figure 7.2: Mueller matrix image of a non magliant mole injected with dynospheres
36 Measurements on skin
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40 Bibliography
A. Mueller matrices for some optical
components
Here are Mueller Matrices given for some optical components
Free space
M =
1 0 0 00 1 0 00 0 1 00 0 0 1
(A.1)
Isotropic absorbing material with transmittance k
M =
k 0 0 00 k 0 00 0 k 00 0 0 k
(A.2)
Linear polarizer at angle θ
M =12
1 cos 2θ sin 2θ 0
cos 2θ cos2 2θ cos 2θ sin 2θ 0sin 2θ cos 2θ sin 2θ sin2 2θ 0
0 0 0 0
(A.3)
Horizontal linear polarizer
M =12
1 1 0 01 1 0 00 0 0 00 0 0 0
(A.4)
Vertical linear polarizer
M =12
1 −1 0 0−1 1 0 00 0 0 00 0 0 0
(A.5)
Linear polarizer at angle 45◦
42 Mueller matrices for some optical components
M =12
1 0 1 00 0 0 01 0 1 00 0 0 0
(A.6)
Right circular polarizer
M =12
1 0 0 10 0 0 00 0 0 01 0 0 1
(A.7)
Left circular polarizer
M =12
1 0 0 −10 0 0 00 0 0 0−1 0 0 1
(A.8)
Linear retarder with fast axis at angle θ and retardation δ
M =12
1 0 0 00 cos2 2θ + sin2 2θ cos ∆ϕ (1− cos ∆ϕ) cos 2θ sin 2θ − sin 2θ sin ∆ϕ0 (1− cos ∆ϕ) cos 2θ sin 2θ sin2 2θ cos 2θ sin ∆ϕ0 sin 2θ sin ∆ϕ cos 2θ sin ∆ϕ cos ∆ϕ
(A.9)
Linear quarter wave retarder with fast axis at 0◦
M =12
1 0 0 00 1 0 00 0 0 10 0 −1 0
(A.10)
Linear half wave retarder with fast axis at 45◦
M =12
1 0 0 00 −1 0 00 0 1 00 0 0 −1
(A.11)
