Raytracing algorithms for modelling light propagation ...

PFL–Aachen/Hamburg/Paris Report 1632/2003 Date of issue: 08/2015

Raytracing algorithms for modelling propagation in birefringent media

Horst Greiner

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Koninklijke Philips Electronics N.V.2015

light

Authors’ address data:

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H. Greiner PFL; [email protected]

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Koninklijke Philips Electronics N.V. 2015 All rights are reserved. Reproduction in whole or in part is prohibited without the written consent of the copyright owner 2

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Koninklijke Philips Electronics N.V. 2015

PFL– Aachen/Hamburg/Paris Report Title:

Raytracing algorithms for modelling light propagation in birefringent media

Author(s):

Horst Greiner

Keywords:

Raytracing, birefringent, uniaxial, anisotropic materials, polarized light, polarized backlights, optical modelling

Abstract:

Raytracing algorithms for modelling the propagation of (partially) polarized light in systems with birefringent media are presented in detail. Their implementation into the simulation software CALPLAY which is used in Philips Research to model display illumination systems is discussed. With this enhancement CALPLAY allows a physically correct description of the optics of “polarised” front and backlights incorporating birefringent materials.

Conclusions:

The proposed implementation of raytracing algorithms for systems incorporating birefringent materials has been successfully applied to the modelling of backlights in which anisotropic outcoupling structures are used to produce polarized light, thus corroborating the validity, speed and robustness of the proposed algorithms. In the future developers of back- and frontlights have a unique modelling tool at their disposal. The application to polarized backlights should be the subject of another report..

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Contents 1 Introduction and overview

7

2 Light Propagation in isotropic media

11

3 Light propagation in linear anisotropic materials 3.1 Linear anisotropic materials 3.2 Uniaxial Materials

15 15 19

4 Reflection and Refraction at Material boundaries 4.1 Isotropic/isotropic Interfaces 4.2 Isotropic/Anisotropic Interfaces 4.3 Anisotropic/Isotropic Interfaces 4.4 Anisotropic/Anisotropic Interfaces 4.5 Light propagation through thin slabs of anisotropic material 4.6 Reflection and Refraction at anisotropic media with absorption

24 25 29 34 36 37 38

5 Implementation of Raytracing in systems with birefringent media 5.1 Material interfaces 5.2 Description of polarizers 5.3 Detector Screens

41 42 44 45

6 Appendix 6.1 Solving Fresnel’s equation 6.2 Proof of eigenvector properties 6.3 Propagation through absorbing birefringent crystals We then use the vector identities

46 46 48 50 52

7 References

54

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1 Introduction and overview The scientific observation of double refraction in calcite spars by Bertholinus in 1669 heralded a new era in optics. Accounting for the observed polarization effects required a whole new conception of light and provided the touchstone for many theories which are associated with the names of Huygens, Newton, Fresnel and Stokes, the final verdict being the electromagnetic theory of Maxwell published in 1864 nearly two centuries after the initial discovery. This theory provides a conceptually and experimentally valid description of light propagation in isotropic and anisotropic materials including the polarization dependent phenomena of reflection and refraction at material interfaces. For a concise history including modern developments and many interesting references we recommend the opening chapters of Brousseau10. The subject matter of light propagation in isotropic and anisotropic media is e.g. very well 22 explained in the now classic book on optics by Hecht The technological applications of polarized light are intimately connected with the development of optically anisotropic materials as these allow a precise manipulation of its properties. In the 19th century suitably shaped prisms of birefingent crystals were used for the production and manipulation of polarized light. At the end of this century the optical anisotropy of liquid crystals was studied. In the first half of the 20th century new sheet like polarizers with manifold technological applications were developed. In the jargon of modern technology managers the optical properties of birefringent materials are “enabling” many applications ranging from liquid crystal displays to optical storage. Light propagation in anisotropic materials can be described on the basis of geometrical or wave optics, as appropriate, using the well known laws of reflection and refraction. In devices like the polarisation microscope it is usually sufficient to analyse the propagation of a limited number of light rays through the system in the paraxial approximation. Effects arising from the “oblique” transversal of anisotropic structures are negligible. This is no longer the case in systems like liquid crystal displays52 which have to be optimised for their “off-axis” performance. Furthermore the evaluation of the performance of such appliances requires the analysis of millions of rays through the system which can naturally only be accomplished with a modern desk top computer. Unfortunately up to recently commercial simulation packages for light propagation did not allow for this: either they did not incorporate the possibility to model polarization effects properly or the raytracing was far too slow to cope with millions of rays in a reasonable time. Thus the modelling of novel outcoupling structures invented by Philips Research 9, 5, 6, 7, 25, 26 consisting of anisotropic microprisms which allow the realization of LCD backlights producing polarised light was not possible. For a survey on “polarized” backlights we recommend 6, 25. It was therefore decided by Philips Research to incorporate 


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raytracing in anisotropic media into my raytracing program CALPLAY which is unique in its speed to raytrace microfacetted geometries and widely used in display illumination research within Philips (lately its fast raytracer has also been integrated into the standard development software of Philips Luminaires). The physics behind such an implementation are straightforward, although many of the formulas of the trigonometric kind used in the physics literature are not suitable for an efficient numerical code. At first it was planned to “outsource” the code development, but after some initial contacts with the universities of Buenos Aires and Moscow which independently made offers for 60.000 US dollars (is this academic globalization?), it was decided to do the work within Philips Research, at last. With hindsight this resulted in much better concept as the one which was proposed to us by the academic vendors: firstly our code relies strictly on numerical linear algebra and trigonometric functions are not used at all. This greatly increases the speed of execution and avoids many unnecessary case distinctions. Secondly we describe the polarization properties of a light ray by its coherency matrix instead 8, 10, 13 . Mathematically of the Stokes parameters as explained in the literature these two descriptions are equivalent, but the former offers a number of advantages as the 2x2 coherency matrices can be transformed directly using 2x2 matrices whose elements are given by the field amplitudes at the material boundaries thus avoiding the somewhat cumbersome 4x4 Mueller matrices described e.g. in detail by Collett 13. As a result of our investigations we have also found that the raytracing approach generally fails for anisotropic media with absorption (with some exceptions), as the energy of a refracted ray is not split between the ordinary and extraordinary ray in such media 18. In this report we give a detailed description of the mathematics underlying the raytracer for anisotropic materials implemented into CALPLAY: explicit algorithms and formulas are given, so that a programmer should not have any difficulties to write his or her own code following the procedures indicated. The report is organized as follows: Chapter 2 recapitulates the basic concepts used for describing the propagation of light in isotropic media and presents the coherency matrix formalism for partially polarized light. Chapter 3 introduces linear anisotropic media: the Maxwell equations and their plane wave solutions are formulated and first discussed for general biaxial media and then for uniaxial media which allow for some simplification. Chapter 4 discusses the reflection and refraction of plane waves at the interfaces of isotropic and anisotropic materials. The following cases are discussed: isotropic/isotropic, isotropic/anisotropic, anisotropic/isotropic, anisotropic/anisotropic. The algorithms for obtaining the wave, electrical, magnetic field and Poynting vectors and the coherency matrices of the reflected and refracted waves for a given incident wave are presented in detail. By evaluating the normal components of the Poynting vectors of these waves the distribution of the incident energy into the reflected and refracted waves is evaluated. Finally light propagation through thin sheets of anisotropic material (wave plates) where the interference between the ordinary and extraordinary waves has to be taken into account is described. 8

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Chapter 5 gives a detailed account of the implementation of our raytracing algorithms for birefringent media into our simulation package CALPLAY. The algorithms habe been applied successfully to the modelling of polarized back7 lights . The appendix provides some detailed derivations used in the report. Finally there is an extensive set of references. The successful application of the code to the modelling of polarized backlights with anisotropic outcoupling structures will hopefully be the subject of a second report. Let me conclude with the following citation from the Feynman lectures on physics 15 (vol II, 33-3) which contains much of the subject in a nutshell: If you want to entertain yourself, you can try the following terrifying problem that was the ultimate test for graduate students back in 1890; solve Maxwell’s equation for plane waves in an anisotropic crystal, that is, when the polarization P is related to the electric field E by a tensor of polarizability. You should, of course, choose your axes along the principal axes of the tensor, so that the relations are simplest (then Px = αaEx, Py = αbEy, Pz = αcEz ), but let the waves have an arbitrary direction and polarization. You should be able to find the relations between E and B, and how k varies with direction and wave polarization. Then you will understand the optics of an anisotropic crystal. It would be best to start with the simpler case of a birefringent crystal-like calcite- for which two of the polarizabilities are equal (say, αb = αb ), and see if you can understand why you see double when you look through such a crystal. If you can do that, then try the hardest case, in which all three α’s are different. Then you will know whether you are up to the level of a graduate student of 1890…. Five comments are in order: 1) using the proper vectorial formulation as in this report it is not necessary to choose the principal axes of the tensor for your coordinate system, thus avoiding a lot of unnecessary coordinate transformations. 2) as recommended by Feynman we have restricted the implementation to uniaxial crystals as they occur in most applications. This simplifies life a lot as the surface of dispersion (Fresnel surface) decomposes into a sphere and an ellipsoid. In the general case of a biaxial medium (“the hardest”) we have to deal with an irreducible quartic algebraic surface with eight algebraic singularities which define the optical axes. To find the propagation vector of a refracted or reflected wave one then has to solve a fourth order equation instead of a quadratic equation in the uniaxial case. But the algorithms developed apply equally to the hardest case, once a solver for a fourth order polynomial is implemented (see section 4.2 for more details). 3) The following statement in the Feynman lectures 15 (vol I, 33-9) is in general not true, although most clever physicists think it is: 


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When an unpolarized beam strikes an anomalously refracting crystal, it is separated into an ordinary ray, which travels straight through in the normal manner, and an extraordinary ray which is displaced as it passes through the crystal. These two emergent rays rays are linearly polarized at right angles to each other. That this is true can be readily demonstrated with a sheet of polaroid to analyze the polarization of the emergent rays….. 4) I found it quite hard to reach the level of a graduate student of 1890…. 5) I am viewing the text I am writing on a 15 inch LCD display…..

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2 Light Propagation in isotropic media Formel-Kapitel 2 Abschnitt 1 In this section we present the formalism we employ for modelling light propagation and reflection/refraction in (an)isotropic materials in a summary fashion. For more detailed explanations and deductions we refer to standard text. 8, 10, 13, 22, 24, 28, 34, 50, 53 books . Our description is based on geometrical optics and therefore diffraction and interference effects arising from geometrical features of wave-length size are not accounted for. Geometrical optics and its relation to physical optics is discussed in some detail in chapter 3 in the classical treatise on optics by 8 Born and Wolf . In geometrical optics light energy propagates along straight lines in homogeneous media of constant refractive index. Locally the E and H field is given by plane waves of the form where E0 and H0 denote complex 3 vectors.

E = E0 exp ( i ( ω t − k ⋅ r ) )

H = H 0 exp ( i ( ω t − k ⋅ r ) )

.

(2.1.1)

The direction of energy propagation of a plane wave (2.1.1) is given by the time averaged Poynting vector

S = 0.5Re ( E × H ) .

(2.1.2)

In isotropic lossless media the ray direction, the E and H vector are orthogonal. If we assume that a quasi-monochromatic light wave propagates in the z direction, the degree of temporal correlation between the transversal x and y components of the electrical field vector E decribes the polarization state of 8 the wave (Born and Wolf , chapter 10.9.1). The elements of the coherency matrix

 J xx J=  J yx 

J xy   E x E x = J yy   E E  y x

ExEy   E yE y  

(2.1.3)

are amenable to direct measurement (the brackets denote time averages). They can be directly related to the so-called Stokes parameters S0, S1, S2, S3 by the transformations

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S0 = J xx + J yy , S1 = J xx − J yy ,

(2.1.4)

S2 = J xy + J yx , S3 = i(J yx − J xy ) 1 (S0 + S1 ) , 2 1 J yy = ( S0 − S1 ) , 2 1 J xy = ( S2 + iS3 ) , 2 1 J yx = ( S2 − iS1 ) 2 J xx =

(2.1.5)

For the convenience of the reader and later use we list a few of the basic properties of the coherncy matrix taken from reference 8: . i)

The Poynting vector can be expressed as ( k.E=0 !)

S = 0.5Re(k )E ⋅ E = 0.5Re(k ) trace(J ) ii)

(2.1.6)

completely unpolarized light is represented by the coherency matrix

 1 0  0 1  

(2.1.7)

or equivalently by Stokes parameters

S1 = S2 = S3 = 0 iii)

(2.1.8)

completely polarized light is characterized by the condition

det(J) = 0

(2.1.9)

S0 = S12 + S22 + S32

(2.1.10)

or equivalently by

The coherency matrix can then be expressed by 12

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 a12 a1a 2 exp ( iδ )    2  a a exp ( iδ )  a2  1 2 

(2.1.11)

with δ = integer multiple of π for linearly polarized light with vibration direction a2/a1 and δ = integer multiple of π/2 and a1=a2 for circularly polarized light. iv) Any coherency matrix J can be decomposed into an unpolarized part J (2) and a completely polarized part J of the form

 A 0   B D J = J (1) + J (2) =   + D C, 0 A    

(1)

(2.1.12)

where A,B,C and D are given by the expressions

A = 0.5 B = 0.5 C = 0.5

( J xx + J yy ) − W ( J xx − J yy ) + W ( J yy − J xx ) + W

(2.1.13)

D = J xy W = 0.5 tr ( J ) − 4det(J) 2

Hence the total intensity of the polarized part is given by

( )

Tr J (2) = B + C = W = 0.5

( J xx + J yy )

2

− 4det ( J )

(2.1.14)

and the degree of polarization P by

P = 1−

4det(J) tr(J)

2

.

(2.1.15)

Equivalently the four Stokes parameters s=(s0, s1, s2, s3) can be decomposed into an unpolarized and a completely polarized part as follows:

s = s(1) + s(2) =

 s − s 2 + s 2 + s 2 ,0,0,0  +  s 2 + s 2 + s 2 ,s ,s ,s   0   1 1 2 3 2 3 1 2 3     


(2.1.16)

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The degree of polarization is therefore given by

s12 + s 22 + s32 P= s0 iv)

(2.1.17)

transformation properties of the coherency matrix

If a basis change for the transversal electrical field vector

E = E1E1 + E 2 E2 → E′ = E1′ E′1 + E′2 E′2

= ( a11E1 + a12E 2 ) E′1 + ( a 21E1 + a 22E 2 ) E′2

(2.1.18)

is specified by a coefficient matrix

 E1′   E1   a11 a12   E1  = A  E′   =     2  E 2   a 21 a 22   E 2 

(2.1.19)

the coherency matrix transforms according to

J′ = AJA t

(2.1.20)

as can be verfied by direct substitution.

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3 Light propagation in linear anisotropic materials In this chapter electromagnetic wave propagation in linear anisotropic media is discussed. The first section deals with the general case of birefringent materials whereas the second sections deals with the simplifications of the formalism for uniaxial materials. Light propagation in linear anisotropic materials 8, 24, 28 . is presented in many textbooks, we particularly recommend

3.1 Linear anisotropic materials Formel-Kapitel 3 Abschnitt 1 In the following we use rationalized MKS units. Linear anisotropic materials are characterized by a linear relationship between the electric displacement vector D and the electrical field vector E: D = ε0 ε E (3.1.1) Here ε is a 3 by 3 hermitian tensor with complex entries εij satisfying

εij = ε ji for i, j = 1,...,3 .

(3.1.2)

For lossless media the permittivity tensor is therefore symmetrical and can always be expressed in diagonal form in an appropriate coordinate system (principal axes) with diagonal elements εx , εy and εz which leads to the following classification of linear anisotropic materials: Isotropic: εx = εy = εz Uniaxial: εx = εy ≠εz Biaxial: εx ≠ εy ≠εz We note in passing that for media with absorption the principal axes of the real and imaginary part of the permittivity tensor do not necessarily coincide 8 (Born and Wolf , chapter 15.6.1). The principal permittivities also define the principal refractive indices by the relations

n 2x = ε x , n 2y = ε y , n 2z = ε z .

(3.1.3)

We first discuss light propagation in general birefringent media. In the ensuing section the special relations which apply only to uniaxial materials are presented. The equations presented apply to lossless and absorptive media with 


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complex permittivities alike, although the physical interpretation of the solutions becomes quite involved in the case of absorption and some new phe1, 2, 3, 14, nomena arise. To get a taste the reader is referred to the publications 18, 37 and section 4.6. To identify monochromatic plane waves propagating in general biaxial media, we assume plane waves of the form

E exp i ( ω t − k ⋅ r )  and H exp i ( ω t − k ⋅ r ) 

(3.1.4)

Substitution of these expressions into Maxwell’s equations leads to the equations

k × E = ωµ H

(3.1.5)

k × H = −ωεE = −ωD

(3.1.6)

k ⋅ H = 0 and k ⋅ D = 0

(3.1.7)

Substitution of (3.1.5) into (3.1.6) the second equation then gives

ω2 µ D = −k × ( k × E ) = − ( k ⋅ E ) k + ( k ⋅ k ) E

(3.1.8)

which is equivalent to

k × ( k × E ) + ω2µεE = 0

(3.1.9)

From these relations the following can already be concluded: i) ii)

the three vectors k, D and H are mutually orthogonal the vector E is orthogonal to H and lies in the plane spanned by D and k iii) the Poynting vector S = ExH giving the direction of energy flow is also in the plane spanned by D and k (as H is perpendicular to k and D!) and its direction does not necessarily coincide with k. For a given wavevector k = (kx ,ky, kz) the electrical field E = (Ex , Ey , Ez) has to satisfy the linear equation

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ω2µε − k 2 − k 2 x y z  A(k , ω) E =  k yk x   k zk x 

k xk y

k xkz

ω2µε y − k 2x − k 2z

k yk z

k zk y

ω2µε z − k 2x

  E x   E  = 0  y   − k 2y   E z  

(3.1.10) The dispersion relation is therefore given by

det A ( k , ω) = 0

(3.1.11)

which is equivalent to.

) ( k2x + k 2y + k 2z ) (3.1.12) 2 2 2 2 2 2 2 2 2 2 2 2  2 2 2  −  n x k x ( n y + n z ) + n yk y ( n x + n z ) + n z k z ( n x + n y ) + n x n yn z = 0   (

) (

F k x , k y , k z = n 2x k 2x + n 2y k 2y + n 2z k 2z

The surface F (kx , ky , kz ) = 0 is called the normal surface and consists of an interior and an exterior shell which touch in four singular points. For a k vector on the surface the direction of the corresponding Poynting vector is given by the surface normal grad F in this point. This observation also proves that the E and H vectors are tangent to the normal surface. Introducing the expression

k = nω µ0ε0 k

(3.1.13)

where k is a unit vector into the determinant (3.1.11) we get upon expansion a biquadratic equation for n2 (Fresnel’s equation)

( n 2x k 2x + n 2y k 2y + n z2 k z2 ) n 4 −  n 2x k 2x ( n 2y + n 2z ) + n 2y k 2y ( n 2x + n z2 ) + n z2 k z2 ( n 2x + n 2y )   

n 2 (3.1.14)

+ n 2x n 2y n 2z = 0 which can readily be solved for n2. Hence for a given direction of propagation k there are two possible eigenmodes given by

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k i = n iω µ0ε0 k i = 1, 2 .

(3.1.15)

The tips of these vectors describe an algebraic surface which is symmetric with regard to the coordinate planes and consists of an interior and exterior shell of ellipsoidal shape. The interior and exterior surface touch in four singular points for which n1 = n2 . The corresponding directions are called optical axes (for some nice illustrations see 30, chapter 6.6.5) To obtain the eigenmodes, we have to solve equation (3.1.10). We write

 a1  A =  a 2  with row vectors ai . a   3

(3.1.16)

det( A ) = a1 ⋅ ( a 2 × a 3 ) = 0

(3.1.17)

From the relation

which holds for all permutations of the indices, it readily follows that any linear combination of the vectors

a1t × at2 , at2 × at3 , a1t × at3

(3.1.18)

represents a solution of the equation (3.1.10). For a lossless medium the matrix A and hence the components of the E vectors are real we are dealing with linearly polarized waves. From the eigenvalue equation

(

)

Kε −1D + ω2µD = k × k × ε −1D + ω2µD = 0

(3.1.19)

for the electric displacement vector D and the fact that the matrices

 −k 2 − k 2 z  y ε −1 and K =  k y k x    k zk x 

k xk y −k 2x − k 2z k zk y

  k yk z    − k 2x − k 2y   kxkz

(3.1.20)

are symmetric, it can be deduced that the vectors D1 and D2 are orthogonal and perpendicular to the wave propagation vector k. In geometric terms they can be constructed by intersecting the plane perpendicular to the wave propagation vector k with the index ellipsoid

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x12 n12

+

x 22 n 22

+

x 32 n 32

=1

(3.1.21)

The principal axes of the resulting ellipse then represent the directions of the vectors D1 and D2 The surface normals at the points where the principal axes intersect the index ellipsoid give the directions of the corresponding E fields. 24 34 For more details see , chapter 3.5.2, 3.5.2 and , chapitre 31.

3.2 Uniaxial Materials Formelabschnitt 2 For uniaxial materials we have by convention

no = n x = n y

ne = n z

(ordinary index ) and

(extraordin ary index ) 50, 51

with the z-axis representing the optical axis c. The books treat uniaxial materials in explicit detail. In general coordinates equation the dieelectric tensor can be written as

ε = εoI + ( εe − εo ) c ⊗ c ,

(3.2.1)

where by definition

( c ⊗ c )( x ) = ( c ⋅ x )

c

(3.2.2)

The uniaxial symmetry entails a number of simplifications of the expressions for the general case which we explicit in this section. For uniaxial materials Fresnel’s equation factors into two quadratic equations which can be readily solved:

(

n 2 k ⋅ k + n 2 − n 2 e o  o e e

) (c ⋅ k e )2 − ne2no2 

k ⋅ k − n 2  = 0 (3.2.3) o  o o

In the crystal frame of reference they become

( (

)

)

 n 2 n 2 k 2 + k 2 + n 2 k 2 − n 2n 2   n 2 − n 2  = 0 o x e z e o  o y  

(3.2.4)

and the indices of the ordinary and extraordinary wave are given by

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n = no n=

(

n on e

)

n o2 k 2x + k 2y + n e2 k 2z

=

none n o2 sin 2 ( θ ) + n e2 cos 2 ( θ )

(3.2.5)

where θ denotes the angle between the wave propagation vector and the zaxis (optical axis) of the uniaxial crystal. Furthermore the following relations can be obtained from these formulas for later reference:

( n 2 − ne2 ) ( no2 sin 2 ( θ) + ne2 cos2 ( θ)) = ne2 ( no2 − ne2 ) cos2 ( θ)

(3.2.6)

( n 2 − no2 ) ( no2 sin2 ( θ) + ne2 cos2 ( θ)) = no2 ( ne2 − no2 ) sin 2 ( θ)

(3.2.7)

and

For uniaxial crystals the electrical and magnetic field vectors of the ordinary and extraordinary waves can be explicitly formulated in the coordinate frame of the crystal as follows: The electric field vector of the ordinary wave expressed in crystal coordinates is given by

 k o,y   0   Eo =  − k o,x  = k o ×  0  1  0     

(3.2.8)

If c denotes the optical axis in the lab frame, E can be expressed in the lab frame as

Eo = k o × c

(3.2.9)

and is therefore perpendicular to the principal plane. The eigenvector property

k o × ( k o × Eo ) + ε Eo = 0

(3.2.10)

can be verified directly (see appendix 7.2). For the magnetic field vector we get in the crystal frame

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


 k k  k k  o,x o,z    o,x o,z  H o = k o × Eo =  k o,y k o,z  =  k o,y k o,z  = −n o2     2 2 2 2      −k o,x − k o,y   k o,z − n o 

 0  0 + k   o,z k o (3.2.11)   1

with

no2 = k o ⋅ k o In the lab frame this becomes

H o = k o × Eo = k o × ( k o × c ) = − n o2 c + ( c ⋅ k o ) k o

(3.2.12)

Hence the magnetic field vector of the ordinary wave lies in the principal plane. With a little algebra we obtain for the Poynting vector

So = Re ( Eo × Ho ) = 2 Re  n 2 − ( k ⋅ c )  k + {( c ⋅ k ) ( k ⋅ k ) − n 2 ( k ⋅ c )  

}

(3.2.13)

For n and k real the second term vanishes and we see that the Poynting vector of the ordinary wave points into the direction of the wave vector. The E vector of the extraordinary wave in crystal coordinates is given by

 n 2k x   0  e e Ee = n 2 k e −  n e2k ey  = n 2 − n e2 k e + n e2 − n o2 k ez  0  (3.2.14)   1 2 z  n ok e     

(

)

(

)

with

n2 = k e ⋅ k e In the lab frame this becomes

(

)

(

Ee = n 2 − n e2 k e + n e2 − n o2

) (c ⋅ k e ) c

(3.2.15)

and lies therefore in the principal plane defined by the optical axis and ke. 


21

Again the eigenvector property can be verified directly (see appendix 7.2) The magnetic field vector in the crystal frame is given by

 0 z 2 2 H e = k e × Ee = k e n e − n o k e ×  0  = k ez n e2 − n o2 1  

(

)

(

)

 ky   e   −k ex  (3.2.16)    0   

and in the lab frame by

(

H e = k e × Ee = n e2 − n o2

) (c ⋅ k e ) (ke × c)

(3.2.17)

and is perpendicular to the principal plane. With a little algebra the Poynting vector of the extraordinary wave in the crystal frame can now be deduced:

Se = Re ( Ee × H e ) =

( (

) )

  n 2 − n o2 k e,z k e,z k e,x     Re n e2 − n o2  n 2 − n o2 k e,z k e,z k e,y     − n 2 − n 2 k k + k k  e e,x e,x e,y e,y k e,z  

(

)

(

)(

(3.2.18)

)

If n and k are real we get with the help of (3.2.6) and (3.2.7) and the fact that

k z2 = n 2 cos 2 (θ ) and k x2 + k y2 = n 2 sin 2 (θ )  n2 k   o e,x  Se = Re ( Ee × H e ) ∝  n o2 k e,y     2  n e k e,z   

(3.2.19)

which demonstrates that Se points into the direction of the surface normal of the ellipsoid

{(

E = kx ,k y ,kz

22

) no2 k x2 + no2 k y2 + ne2 k y2 = no2 ne2 }. 


at k. And that the angle θs between the Poynting vector and the optical axis is given by

tan (θ s ) =

no2 ne2

tan (θ ) .

In the laboratory frame (3.2.19) becomes (assuming n and k to be real)

(

)

S e = no2 k + ne2 − no2 (c ⋅ k )c In the lab frame we obtain also the general expression

(

)

S e = Re  n 2 − no2  ne2 − no2 (c ⋅ k )(c ⋅ k ) k +      n 2 − n 2  n 2 − n 2 (c ⋅ k )(k ⋅ k ) + n 2 − n 2  n 2 − n 2 (c ⋅ k )(c ⋅ k )(c ⋅ k ) c   o  e o e o  e o      

(



)


(

)

23

4 Reflection and Refraction at Material boundaries Formel-Kapitel 4 Abschnitt 1 Consider two media separated by a plane boundary with normal η. To describe the reflection and refraction of a plane wave impinging from medium I onto medium II, the following cases have to be considered: isotropic/isotropic, isotropic/anisotropic, anisotropic/isotropic and anisotropic/anisotropic. In all cases the incident wave has to be matched with the reflected and refracted waves, where in anisotropic materials both ordinary and extraordinary waves have to be taken into account. The reflected and refracted waves have to fulfill the following conditions: i)

ii)

the tangential component of their wave propagation vector must equal the tangential component of the wave propagation vector of the incident wave. The tangential components of the total E and H fields must be continuous across the material boundary.

From condition i) and the dispersion relations the propagation vectors of the outgoing waves can be determined. Condition ii) furnishes the amplitude coefficients for the reflected and refracted waves. With the amplitude coefficients the coherency matrices of the outgoing waves can be easily obtained from the coherency matrix of the incident wave. The traces of these matrices also yield the relative probabilities of propagation of the outgoing waves. In the following we consider the various possible cases separately and establish explicit formulas for the outgoing waves. Light transmission through a thin slice of birefringent material is treated in section 4.5 as a special case where the interference between the ordinary and extraordinary ray must be taken into account. Explicit formulas for the reflection/refraction at isotropic/anisotropic boundaries have been discussed at length in the publications of C.Simon and col38, 39, 40, 41, 42, 43, 44, 45 . Here we adopt the approach of S.McClain 31, 32 laborators where the amplitude coefficients of the reflected and refracted waves are obtained from a system of linear equations expressing the boundary conditions for the electromagnetic fields. These linear equations can be numerically solved by any of the stadard methods as described for instance in Numerical Recipes 36. With these amplitudes the coherency matrices of the reflected and refracted rays can be deduced straightforwardly from the coherency matrix of the incident ray, a scheme first suggested by Wolff 49. Furthermore the Poynting vectors and their normal components and hence the distribution of the incident energy on the outgoing waves can be evaluated. This allows the random selection of an outgoing ray based on its energetic importance.

24




4.1 Isotropic/isotropic Interfaces This case corresponds to the well known Fresnel formulas for reflection and refraction given in most optics textbooks. In our formalism we start with the incident wave vector kinc and make the ansatz

k = k inc + Γη

(4.1.1)

for the outgoing waves. This ensures that the tangential component of the wavevector remains invariant. Equating

( k inc + Γ1η)2 = n12

and ( k inc + Γ 2 η) = n 22 2

(4.1.2)

one obtains quadratic equations

(

)

2 Γi2 + 2Γi k inc ⋅ η + k inc − n i2 = 0 i = 1, 2

(4.1.3)

For the reflected wave we get

Γ1 = −2k inc ⋅ η

(4.1.4)

and for the refracted wave

Γ 2 = −k inc ⋅ η +

2 ( k inc ⋅ η) + n 22 − k inc

(4.1.5)

with complex values corresponding to total internal reflection with

Im(k ) ⋅ η ≥ 0 . To obtain the amplitudes of reflection and transmission we note that in isotropic media any wave can be written as a superposition of s- and p-polarized waves. The following suffix notation will be used: t transmitted, r reflected, o ordinary, e extraordinary, p p-polarized and s spolarized wave. Thus Eto symbolizes the transmitted, ordinary wave etc. We therefore introduce the vectors




25

Einc =

Einc , Einc

Ers =

kr × η , kr × η

Erp =

k r × Ers , H rp = k r × Erp k r × Ers

E ts =

kt × η , kt × η

E tp =

k t × E ts , H tp = k t × E tp k t × E ts

Hinc = k inc × Einc H rs = k r × Ers (4.1.6)

H ts = k t × E ts

The reflection and transmission amplitudes are determined from the four linear equations

( ) t p ⋅ ( a ts E ts + a tp E tp − a rs Ers − a rp Erp ) = t p ⋅ Einc t s ⋅ ( a ts H ts + a tp H tp − a rs H rs − a rp H rp ) = ts ⋅ Hinc t p ⋅ ( a ts H ts + a tp H tp − a rs H rs − a rp H rp ) = t p ⋅ Hinc t s ⋅ a ts E ts + a tp E tp − a rs Ers − a rp Erp = ts ⋅ Einc

(4.1.7)

where ts and tp denote tangential vectors perpendicular and parallel to the plane of incidence:

ts =

k inc × η k inc × η

t p = η × ts

(4.1.8)

These equations can be recast in the linear equation

 ts ⋅ E ts   t p ⋅ E ts   ts ⋅ H ts t ⋅ H  p ts

t s ⋅ E tp

−t s ⋅ Ers

t p ⋅ E tp

−t p ⋅ Ers

ts ⋅ H tp

−t s ⋅ H rs

t p ⋅ H tp

−t p ⋅ H rs

−t s ⋅ Erp   −t p ⋅ Erp  −t s ⋅ H rp  −t p ⋅ H rp 

 a ts   ts ⋅ Einc  a   t ⋅ E   tp  =  p inc   a rs   ts ⋅ H  inc     a t ⋅ H  rp   p inc 

(4.1.9)

Due to the choice of the tangential vectors the coefficient matrix has a num26




ber of zero entries

 t ⋅E  s ts  0   0    t p ⋅ H ts

0

−ts ⋅ E rs

t p ⋅ E tp

0

t s ⋅ H tp

0

0

−t p ⋅ H rs

   −t p ⋅ E rp   −ts ⋅ H rp   0  0

 a ts   t s ⋅ Einc  a   t ⋅ E   tp  =  p inc   a rs   t s ⋅ H  inc     a rp  t p ⋅ Hinc 

(4.1.10)

so that the equations for the s and p coefficients decouple into two linear equations with two unknowns:

 ts ⋅ E ts t ⋅ H  p ts

−t s ⋅ Ers  −t p ⋅ H rs 

 a ts   t s ⋅ Einc  a  =  t ⋅ H   rs   p inc 

(4.1.11)

 −t p ⋅ E tp   t s ⋅ H tp

−t p ⋅ Erp   −t s ⋅ H rp 

 a tp   t p ⋅ Einc    = a  rp   t s ⋅ Hinc 

(4.1.12)

To obtain the amplitudes and coherency matrices of the reflected and refracted waves in terms of the incident wave, we proceed as follows: The incident ray is given by the incident wave vector kinc and the coherency matrix J which expressed with regard to the transversal unit vectors E1 and E2.(not necessarily aligned to the plane of incidence).We want to calculate the transmission and reflection coefficients for the refracted and reflected waves and their coherency matrices with respect to the sp frame given by the plane of incidence. Assume that the incident wave is given by

E = E1 E1 + E 2 E2 and H = k inc × E = E1 H1 + E 2 H 2 with

H1 = k inc × E1 and H 2 = k inc × E2 . and the coherency matrix by




27

 J ss J =   J ps

J sp   E1E1 = J pp   E2 E1

E1E2 E2 E2

 .  

From (4.1.11) and (4.1.12) we calculate the amplitudes

a1ts , a1tp , a1rs , a1rp for Einc = E1 and Hinc = H1 = k inc × E1 (4.1.13) 2 a 2ts , a 2tp , a 2rs , a rp for Einc = E2 and Hinc = H 2 = k inc × E2 (4.1.14) and obtain for the refracted wave

(

)

(

)

2 2 Et = E1 a1ts + E2 ats2 Ets + E1 a1tp + E2 atp Etp = E1 a1ts Ets + E2 atp Etp

(

)

(

)

2 H t = k t × Et = E1 a1ts + E2 ats2 H ts + E1 a1tp + E2 atp H tp

. We now define

 a1 ts A=  a1tp 

a 2ts  . 2  a tp 

(4.1.15)

In case that the vectors E1 and E2 are aligned to the plane of incidence (s+p) we have a1tp = ats2 = 0 and the matrix A becomes diagonal. The coherency matrix of the transmitted ray can be calculated as J t = AJA t and the Poynting vector as

1 S t = Re(E t × H t ) = 0.5 tr(J t ) k t 2

(4.1.16)

(4.1.17)

The corresponding expressions for the reflected wave are obtained by replacing the suffix t by the suffix r. The Poynting vector of the incident wave is given by

28




1 S inc = Re(Einc × H inc ) = 0.5 (E1E1 + E2 E2 ) k inc = 0.5 ( J11 + J 22 ) k inc 2 and energy conservation gives

η ⋅ S inc = η ⋅ S t − η ⋅ S r The coefficients for transmission and reflection are therefore

T=

η ⋅ St η ⋅ Sr and R = η ⋅ Sinc η ⋅ Sinc

(4.1.18)

4.2 Isotropic/Anisotropic Interfaces Formelabschnitt 2 Assume that the incident wave vector expressed in the crystal frame of reference is given by kinc and the interface normal (which points away from the incident medium) likewise by η. Introducing the ansatz

k = k inc + Γη

(4.2.1)

into Fresnel’s equation (3.1.12)

(

)

(

)(

F k x , k y , k z = n x2 k x2 + n 2y k y2 + n z2 k z2 k x2 + k y2 + k z2

[ (

)

(

)

)]

(

)

− n x2 k x2 n 2y + n z2 + n 2y k y2 n x2 + n z2 + n z2 k z2 n x2 + n 2y + n x2 n 2y n z2 = 0 yields

(

)

F kinc, x + Γη x , kinc, y + Γη y , kinc, z + Γη z = 0 , a quartic equation for Γ which has to be be solved numerically. The rather unwieldly coefficients of this equation are given in the appendix.In the case of uniaxial media this equation factors into two quadratic expressions

( (

)

)

(

)

 n 2 k 2 + k 2 + n 2k 2 − n 2n 2   k 2 + k 2 + k 2 − n 2  = 0 y e z e o   x y z o   o x 

(4.2.2)

and the ansatz (4.2.1) gives two quadratic equations for Γ which can be read


29

ily solved(we recommend the procedure in Numerical Recipes 5.6).

(

)

(

36

, Chapter

)

 n 2 + n 2 − n 2 η2  Γ 2 +  n 2 k ⋅ η + n 2 − n 2 k  e o z  e o inc,z ηz  Γ +  o  o inc

(

)

(4.2.3)

n o2 k inc ⋅ k inc + n e2 − n o2 k inc,z k inc,z − n e2n o2 = 0 Γ 2 + 2k inc ⋅ η Γ + k inc ⋅ k inc − n o2 = 0

(4.2.4)

In the lab frame these equations can be written as

(

) (c ⋅ η)2  Γ2 + no2 k inc ⋅ η + ( ne2 − no2 ) kinc ⋅ η 2 k inc ⋅ k inc + ( n e2 − n o2 ) ( k inc ⋅ c ) − n e2n o2 = 0

2 2  2  n o + n e − n o

n o2

Γ 2 + 2k inc ⋅ η Γ + k inc ⋅ k inc − n o2 = 0

Γ+ (4.2.5)

(4.2.6)

As for general biaxial media nx is close to ny, the roots corresponding to a uniaxial medium with

1  1 1  = + and no2 2  n x2 n 2y  1

ne = n z

can be used as starting values for a root polishing procedure for the quartic equation 33, 36. The roots ΓI (I=1,…,4) define the wave vectors

k i = k inc + Γi η i = 1,..., 4

(4.2.7)

and corresponding Poynting vectors

Si = Re ( Ei × Hi )

(4.2.8)

The Poynting vectors corresponding to refracted waves have to point into the second medium and therefore must satisfy the geometric inequalities

Si ⋅ η ≥ 0

(4.2.9)

with equality for totally reflected waves. Furthermore the imaginary part of the 30




wave vector has to satisfy the relation (the field amplitudes must not grow in the refracting medium)

Im ( k i ) ⋅ η ≥ 0

(4.2.10)

Simple geometrical considerations concerning the normal surface show that there are at most two refracted waves which propagate into the medium. For uniaxial media the analysis can be simplified: for the ordinary ray the analysis given in section 2.3.1 applies with

Γ r = −2k inc ⋅ η

(4.2.11)

for the reflected ray and

Γ to = −k inc ⋅ η +

2 ( k inc ⋅ η) + n o2 − k inc

(4.2.12)

for the refracted ordinary ray. For the extraordinary wave the the two possible propagation vectors have to be calculated by solving the quadratic equation (4.2.5) and the corresponding Poynting vectors have to be tested for conditions (4.2.9) and (4.2.10). One should keep in mind that at grazing incidence there may exist wavevectors k such that

k ⋅ η and S ⋅ η

(4.2.13)

carry different signs, i.e. point into opposite directions with regard to the material boundary. In case both refractive indices are real, the choice of the proper root becomes very simple: In case both roots Γ are real, the larger one has to be chosen. To see this consider the ellipsoid

{(

E = kx ,k y ,kz

) no2 k x2 + no2 k y2 + ne2 k y2 = no2 ne2 }

and let t denote the tangential component of the incident wavevector kinc. Now the following cases can arise: a) t lies inside E: then the line

L = {k = t + Γη Γ ∈ R} will intersect E in two points k1 and k2. As the direction of the Poynting vector is given by the exterior surface normal of the ellipsoid, clearly the root with the positive real part has to be chosen. b) t lies outside E: then L either intersects E in two points k1 and k2. Then the root with the larger real part has to be chosen. In case both roots are negative (this can happen if the index of the incident medium is bracketed 


31

by the ordinary and extraordinary index of the uniaxial medium), we are dealing with the case where the normal of the wave vector and the Poynting vector are pointing in opposite directions. This has been studied 42 in some detail by Simon . In case the roots are not real, they will be complex conjugates of each other and the root with the negative imaginary part has to be chosen (total internal reflection) The wavevectors of the ordinary and extraordinary waves are denoted by kto and kte .and the corresponding eigenvectors which can be calculated from (3.1.10) by Eto and Ete . In the uniaxial case they can be obtained directly from formulas (3.2.9) and (3.2.15) respectively. The corresponding H vectors are then calculated by forming the vector product kxE. To calculate the amplitudes for the transmitted o and e waves and the reflected s and p waves we proceed as in the isotropic/isotropic case by defining vectors

Einc =

Einc , Einc

Ers =

kr × η , kr × η

Erp =

k r × Ers , H rp = k r × Erp k r × Ers

E to =

E to , E to

H to = k t × E to

E te =

E te , E te

H te = k t × E te

Hinc = k inc × Einc H rs = k r × Ers (4.2.14)

The amplitudes for the various waves then have to satisfy the linear equations with ts and tp being defined by (4.1.8)

 ts ⋅ E to t ⋅E  p to t ⋅ H  s to t p ⋅ H to

32

t s ⋅ E te

−t s ⋅ Ers

t p ⋅ E te

0

ts ⋅ H te

0

t p ⋅ H te

−t p ⋅ H rs

 −t p ⋅ Erp   −ts ⋅ H rp    0 0



 a to   ts ⋅ Einc  a   t ⋅ E   te  =  p inc  (4.2.15)  a rs   t s ⋅ Hinc      a rp  t p ⋅ Hinc 


which can be readily solved for given vectors Einc and Hinc.by a standard 36 method as described in . An alternative method is to reduce the system of four linear equations to two linear equations for the unknowns ato and ate by eliminating ars and arp from the equations which can be readily accomplished due to the particular structure of the coefficient matrix. Assuming the incident wave to be defined by

E = E1 E1 + E2 E 2

J J =  11  J 21

and H = k inc × E = E1 H1 + E2 H 2

J12   E1E1 = J 22   E2 E1

 ,  

E1E2 E2 E2

and defining amplitudes by

a1to , a1te , a1rs , a1rp for Einc = E1 and Hinc = H1

(4.2.16)

a 2to , a 2te , a 2rs , a 2rp for Einc = E2 and Hinc = H 2 we obtain for the refracted and reflected waves the following expressions:

(

)

E to = E1 a1to + E 2 a 2to E to

(

)

H to = k to × E to = E1 a1to + E 2 a 2to H to

(

1 2 S to = Re(E to × H to ) = 0.5 E1E1a1to a1to + E 2E 2a 2to a to 2

) ( Eto × H to )

(4.2.17)

Identical formulas can be written for the extraordinary refracted wave by replacing the suffix o by e. Two observations are in order: As the o and e waves are linearly polarized the coherence matrix is trivial. We just retain the electrical field vector. The direction of the poynting vectors and the k vectors do not necessarily coincide (only for the ordinary wave in uniaxial media). For the reflected wave we get

(

)

(

)

2 2 Er = E1 a1rs + E 2 a rs Ers + E1 a1rp + E 2 a rp Erp

(

H r = k r × Er = E1 a1rs + E 2 a 2rs 


)

(

H rs + E1 a1rp + E 2 a 2rp

) Hrp

(4.2.18)

33

Writing

 a1 rs A=  a1rp 

2  a rs  2  a rp 

(4.2.19)

we obtain for the coherency matrix of the reflected ray

Jr = A J At

(4.2.20)

1 1 S r = Re(Er × H r ) = tr ( J r ) k r 2 2

(4.2.21)

The transfer amplitudes for the refracted and reflected rays follow from energy conservation

η ⋅ Sinc = η ⋅ S to + η ⋅ S te − η ⋅ S r

(4.2.22)

4.3 Anisotropic/Isotropic Interfaces Formelabschnitt 3 We assume a linearly polarized wave with wavevector kinc and field vector Einc propagating in an anisotropic medium which encounters an isotropic medium. The interface has a normal η pointing into the isotropic medium. In general there will exist a refracted, an ordinary and an extraordinary reflected wave. The propagation vectors kt , kro and kre of these waves can be calculated as described in the previous section 4.2. Define now the following vectors

34




Einc =

Einc , Einc

Hinc = k inc × Einc

E ts =

kt × η , kt × η

E tp =

k t × E ts , H tp = k t × E tp k t × E ts

Ero =

Ero , Ero

H ro = k ro × Ero

Ere =

Ere , Ere

H re = k re × Ere

H ts = k t × E ts (4.3.1)

and calculate amplitudes aro , are , ats , atp from

 ts ⋅ Ero t ⋅E  p ro t ⋅ H  s ro t p ⋅ H ro

t s ⋅ Ere

−t s ⋅ E ts

t p ⋅ Ere

0

t s ⋅ H re

0

t p ⋅ H re

−t p ⋅ H ts

 −t p ⋅ E tp   −t s ⋅ H tp    0 0

a ro   ts ⋅ Einc  a   t ⋅ E   re  =  p inc  (4.3.2)  a ts   t s ⋅ Hinc       a tp  t p ⋅ Hinc 

which can be readily solved for given vectors Einc and Hinc.(cf. The remarks following (4.2.15)). From the amplitudes the following expressions for the Poynting vectors of the reflected ordinary and extraordinary waves are obtained:

S ro = 0.5 a ro a ro Ero × H ro

(4.3.3)

S re = 0.5 a re a re Ere × H re

(4.3.4)

For the transmitted ray propagating in an isotropic medium we obtain the coherency matrix as

 a ts a ts Jt =   a ts a tp 

a tp a ts   a tp a tp 

(4.3.5)

and the Poynting vector




35

(

)

St = 0.5 trace ( J t ) k t = 0.5 a ts a ts + a tp a tp k t

(4.3.6)

4.4 Anisotropic/Anisotropic Interfaces Formelabschnitt 4 We assume a linearly polarized wave with wavevector kinc and field vector Einc propagating in an anisotropic medium which encounters a second anisotropic medium. The interface has a normal η pointing into the second medium. There will exist ordinary and extraordinary waves in refraction and reflection, whose propagation vectors are denoted by kto , kte ,kro and kre . The E vectors of these waves can be calculated as described in sections 3.1 and 3.2. Define now the following vectors

Einc =

Einc , Einc

Hinc = k inc × Einc

E to =

E to , E to

H to = k to × E to

E te =

E te , E te

H te = k te × E te

Ero =

Ero , Ero

H ro = k ro × Ero

Ere =

Ere , Ere

H re = k re × Ere

(4.4.1)

and calculate amplitudes aro , are , ato , ate by solving the linear equations

 ts ⋅ Ero t ⋅E  p ro  ts ⋅ H ro  t p ⋅ H ro

t s ⋅ Ere

−ts ⋅ E to

t p ⋅ Ere

−t p ⋅ E to

t s ⋅ H re

−t s ⋅ H to

t p ⋅ H re

−t p ⋅ H to

−t s ⋅ E te  −t p ⋅ E te   −t s ⋅ H te   −t p ⋅ H te 

a ro   ts ⋅ Einc  a   t ⋅ E   re  =  p inc  (4.4.2)  a to   ts ⋅ Hinc       a te  t p ⋅ Hinc 

With these amplitudes the Poynting vectors of the various waves can be written

36




S ro = 0.5 a ro a ro Ero × H ro S re = 0.5 a re are Ere × H re S to = 0.5 a to a to E to × H to

(4.4.3)

S te = 0.5 a te a te E te × H te

4.5 Light propagation through thin slabs of anisotropic material Formelabschnitt 5 When a light ray is transmitted through a thin plate of birefringent material, interference effects due to the phase difference between the ordinary and extraordinary ray can not be neglected. The formalism developed so far allows to describe this situation in a rather straightforward manner. Assuming an incident wave of the form

E = E1 E1 + E 2 E2

(4.5.1)

propagating in an isotropic medium, the wave transmitted into the birefringent medium (which we assume for simplicity’s sake to be uniaxial) can be written as

(

)

(

)

A to E to + A te E te = E1 a1to + E 2 a 2to E to + E1 a1te + E 2 a 2te E te

(4.5.2)

as explained in section 4.2 on isotropic/anisotropic interfaces. If we assume that the slab orientation is given by the normal η, its thickness by d and the wavevectors of the ordinary and extraordinary wave by ko and ke, the transmitted field upon transversal of the slab can be written as

Bto E to + Bte E te = exp ( −id η ⋅ k o ) A to E to + exp ( −id η ⋅ k e ) A te E te

(4.5.3)

Using the formalism of section 4.3 on anisotropic/isotropic interfaces, the wave emerging from the slab can be written

(

E′ = E′s E′s + E′p E′p = a ots Bto + a ets Bte

)

(

E′s + a ops Bto + a eps Bte

)

E′p

(4.5.4)

In matrix notation the transformation P: E→ →E’ can be written more compactly as 


37

o  E′s   a ts  E′  =  o  p   a tp

a ets   exp ( −id η ⋅ k o )   a sto 0   s e  0 exp − id η ⋅ k ( ) a tp   e   a te

a pto   Es     (4.5.5) p  Ep a te   

With the help of the product matrix P the accompanying transformation of the coherency matrix can be readily accomplished according to the formula

J′ = P J P t

(4.5.6)

As explained by Yeh 52, chapter 8 in more detail, this “extended” Jones method neglects multiple reflections at the slab interfaces which can be accounted for by employing a full 4x4 matrix method. Such approaches give the exact solution of the Maxwell equations for layered media with anisotropic layers. For more mathematical details the reader can consult the books 52, 23. The latter contains references to relevant MATLAB code.

4.6 Reflection and Refraction at anisotropic media with absorption Formelabschnitt 6 Consider for example a light ray impinging from an isotropic medium onto a lossless uniaxial medium. The energy of the incident ray will then be divided exactly between the ordinary and extraordinary refracted rays and the reflected ray. The amount of energy which goes into each ray is given the normal component of the Poynting vector of each individual wave. As we will show in this section this power splitting between the ordinary and extraordinary ray applies only for lossless media. For absorptive uniaxial media it is only valid if the optical axis and the surface normal point in the same direction of if the optical axis is perpendicular to the surface normal. In the latter case the ordinary refractive index must be real. Aspects of light propagation in anisotropic media with absorption are discussed in references 1, 2, 3, 8, 14, 19, 20, 35, 37, 46. Consider now the Poynting vector of the total energy flow across the boundary arising from the ordinary and extraordinary wave:

38




S = 0.5 Re ( Eo + Ee ) × ( H o + H e )  = 0.5 Re(Eo × H o ) So +0.5 Re(Ee × H e ) Se +0.5 Re(Eo × H e ) Soe

(4.6.1)

+0.5 Re(Ee × H o ) Seo So and Se represent the Poynting vectors of the ordinary and extraordinary wave whereas the „mixed“ terms Soe and Seo have no direct physical meaning. ηSo and ηSe represent the power flowing into the ordinary and extraordinary wave. Here we want to elucidate the conditions for splitting of the total power flow between the ordinary and extraordinary wave, i.e

η ⋅ S = η ⋅ So + η ⋅ Se

(4.6.2)

or equivalently for the vanishing of the normal component of the mixed terms

η ⋅ ( Soe + Seo )

(4.6.3)

As shown in appendix 7.3 the normal component of the mixed terms can be expressed as follows

η ⋅ ( Soe + Seo ) =

(

)

2 0.5Re  n o2 k e ⋅ k e + n e2 − n o2 ( c ⋅ k e ) − n o2n e2  c ⋅ ( t × η )  

(4.6.4)

From this expression the it can be easily deduced that the normal component of the mixed terms vanishes in the following cases: i)

no and ne are real:

Here we get

[ 0.5 Re[ n

( ) 2 ] 2 2 2 2 2 2 o k e ⋅ k e + (ne − no ) (c ⋅ k e ) − no ne ] = 0.5 Re[ no2 k e ⋅ k e + (ne2 − no2 ) (c ⋅ k e )2 − no2 ne2 ] = 0 0.5 Re no2 k e ⋅ k e + ne2 − no2 (c ⋅ k e ) − no2 ne2 =

where we have used the assumption




39

no = no

and

ne = ne

,

the trivial fact that for a complex number z we have

Re( z ) = Re( z ) , and finally (3.2.3). ii)

the surface normal and the optical axis point into the same direction:

In this case we clearly have

c ⋅ (t × η) = 0 , hence the assertion. iii)

the optical axis is perpendicular to the surface normal and no is real:

By assumption

c ⋅ k e = c ⋅ (t + γ e η) = c ⋅ t is real and the assertion follows similarly to case i).

40




5 Implementation of Raytracing in systems with birefringent media Formel-Kapitel 5 Abschnitt 1 We discuss the implementation of raytracing in systems incorporating birefringent materials. Raytracing with polarized light has been discussed by various authors 11, 12, 16, 49. Polarization raytracing through birefringent materials is 17, 21, 27, 29, 31, 32, 47, 48 addressed by references . The propagation of light in imaging and non-imaging systems can be accurately described by raytracing: light rays emanate from light sources and are reflected, refracted, scattered or absorbed by material objects in the light path. To analyze the distribution and properties of the light analysing screens can be positioned in the system space. In the last decade a large number of commercial software packages which allow the simulation of light propagation based on raytracing have become available and are successfully applied to optical design problems of all kinds. A raytrace algorithm comprises essentially a geometrical and an optical part: once a ray is launched its geometrical intersection with the surface of the nearest geometrical object has to be determined. In principle this is an easy task of applied geometry that can be complicated by the presence of many geometrical objects which have to be tested for intersection. This problem can be very much alleviated by the application of acceleration techniques like spatial subdivision. Once the intersection is found an optical model of the surface has to be applied to generate further rays emanating from this point, e.g. (Fresnel’s) laws of reflection and refraction at material boundaries and surface scattering. The energy of the incoming ray is split between a number of outgoing rays according to the physical model of the surface and an outgoing ray is chosen randomly according to the energy distribution. This so-called Monte-Carlo raytracing scheme is very appropriate for modelling illumination systems where stray light is not the primary concern. In the past years I have developed the raytracing software CALPLAY for modelling illumination systems. The program is particularly suited to simulate lightguide structures as they are employed for LCD back- and frontlights. Light outcoupling from a lightguide is usually accomplished by the application of optical microstructures on the lightguide which are difficult to raytrace because of their sheer number. Assuming that all structures can be approximated by planar microfacets I have rediscovered and implemented a particu4 larly efficient and simple ray trace acceleration scheme . With this scheme one obtains raytracing speeds one to two order of magnitudes faster than with commercial programs. The program also provides a number of standard models to describe the optical surfaces (Fresnel reflection and refraction, scattering, birefringence as described in this report) and allows to generate microstructured surfaces consisting of prismatic, pyramidal and cylindrical structures. CALPLAY has been applied successfully to the simulation of backand frontlights as described for instance in reference 7. 


41

Here we only want to describe in detail how rays interact with isotropic/isotropic, isotropic/anisotropic, anisotropic/isotropic and anisotropic/anisotropic interfaces. In our program a ray comprises the following data: a) b) c) d) e)

a three vector p specifying the starting point a unit vector d giving the ray direction the wave vector k two orthonormal vectors E1 and E2 perpendicular to d the coherency matrix J expressed with regard to E1 and E2

When a ray is emitted from an incoherent source, two orthonormal vectors E1 and E2 perpendicular to the ray direction are chosen arbitrarily and the coherency matrix is set to the identity matrix (cf.chapter 2).

5.1 Material interfaces When a ray impinges on an interface, an algorithm based on the physical description of the interface has to be applied to generate an outgoing ray. Here When a ray impinges on an interface, an algorithm based on the physical we want to explain these procedures in detail for the interfaces analyzed in this report: i)

isotropic/isotropic

We refer to subsection 4.1 for all explanations and formulas. For an incident ray kinc we proceed as follows: a) calculate the k vectors of the reflected and refracted wave using formulas (4.1.4) and (4.1.5). b) Calculate the field amplitudes (4.1.13) and (4.1.14) for

a1ts , a1tp , a1rs , a1rp for Einc = E1 and H inc = H1 = k inc × E1 2 2 2 ats2 , atp , ars , arp for Einc = E 2 and H inc = H 2 = k inc × E 2 With the field amplitudes calculate the coherency matrices of the trans mitted and reflected rays (4.1.16) c) calculate the Poynting vectors of the transmitted and reflected ray (4.1.17). calculate the transmission (T) and reflection (R) coefficient as in formula 42




(4.1.18). We remark that in the case of total internal reflection T=0 is automatically obtained from the algorithm. Use a random generator with weights T and R to decide whether the reflected or transmitted ray is propagated further

ii)

isotropic/anisotropic

We refer to subsection 4.2 for detailed explanations and formulae. For an incident ray kinc we proceed as follows: a)

b) c)

d) e)

iii)

calculate the k vectors of the reflected, refracted ordinary and extraordinary waves using formulas (4.2.1). To obtain the correct wave vector for the extraordinary wave, proceed as described in section 4.2: If Γ1 and Γ2 are real, choose Γ=max(Γ1, Γ2). If Γ1 and Γ2 are complex, calculate the Poynting vectors corresponding to k1 and k2 and choose the one with S η> 0. Calculate the field amplitudes (4.2.16) . With the field amplitudes calculate the coherency matrix of the reflected wave (4.2.20) and the electric field vector of the ordinary and extraordinary transmitted wave (4.2.17). Calculate the Poynting vector of the reflected wave (4.2.21) and of the ordinary and extraordinary transmitted wave (4.2.17). Calculate the normal components of the various Poynting vectors to determine the energy distribution between the reflected and transmitted waves and randomly choose a propagating ray with the corresponding probabilities. anisotropic/isotropic

We refer to subsection 4.3 for detailed explanations and formulae. For an incident ray kinc we proceed as follows: a)

b)

c)

iv)

Assuming a linearly polarized wave with electric field vector Einc , calculate the amplitudes aro, are, ats and atp for the reflected ordinary and extraordinary and transmitted waves from the linear equations (4.3.2). Use the field amplitudes to calculate the Poynting vectors of the ordinary and extraordinary reflected waves (4.3.3) and (4.3.4) and the coherency matrix (4.3.5) and the Poynting vector (4.3.6) of the transmitted wave. Calculate the normal components of the Poynting vectors and use them as probabilities to randomly determine the propagating ray. anisotropic/anisotropic

We refer to subsection 4.4 for detailed explanations and formulae. For an incident ray kinc we proceed as follows: 


43

a) Assuming a linearly polarized wave with electric field vector Einc , calculate the amplitudes aro, are, ato and ate for the reflected ordinary and extraordinary and transmitted ordinary and extraordinary waves from the linear equations (4.4.2). b) Use the field amplitudes to calculate the Poynting vectors of the ordinary and extraordinary reflected and transmitted waves (4.4.3). c) Calculate the normal components of the Poynting vectors and use them as probabilities to randomly determine the propagating ray.

5.2 Description of polarizers Formelabschnitt 2 To describe sheets of absorbing polarizers the pure raytracing approach in which the energy of a wave incident on a birefringent material is split between the ordinary and extraordinary wave is in general not applicable to materials with absorption as explained in section 4.6: raytracing only works if the optical axis is perpendicular or parallel to the sheet surface. In the latter case the ordinary index of refraction has to be real. To overcome this limitation one can describe the propagation of the entire incident wave through the sheet as described in section 4.5. In this case the energy of the incident wave is not split up and interference effects between the ordinary and extraordinary wave are accounted for. A third possibility are so-called “ideal polarizers” (Yeh 51 , p.252), where the E field of the incoming wave is projected on the transmission axis given by a unit vector t. Assuming the incident field E to have the the form

E = E1 E1 + E 2 E2

(5.2.1)

and writing

E t = t ⋅ E = E1 E1 ⋅ t + E 2 E2 ⋅ t

(5.2.2)

the “projected” power can be expressed as

Et ⋅ Et = E1E1 ( E1 ⋅ t ) + E 2E 2 ( E2 ⋅ t ) + 2Re(E1 ⋅ E 2 ) ( E1 ⋅ t )( E2 ⋅ t ) 2

2

(5.2.3) If we assume that the incident wave is normalized, i.e.

E1E1 + E 2E 2 = 1

(5.2.4)

this gives the probability for transmission through the ideal polarizer.

44




5.3 Detector Screens To analyse the flow of light through the system real or virtual rectangular detector screens can be placed into the light path. The impinging rays can be sorted according to location and direction by subdividing the phase space defined by the screen into spatial ∆x∆y and angular ∆ϕ∆θ bins. To evaluate the polarization properties of the rays belonging to a certain bin ∆x∆y∆ϕ∆θ their Stokes parameters which can be obtained from their coherency matrix via (2E4) are added. The normalized sum of the individual Stokes parameters then convey information on the state of polarization as explained in chapter 2.




45

6 Appendix Formel-Kapitel 6 Abschnitt 1

6.1 Solving Fresnel’s equation Introducing the ansatz

k = k inc + Γη

(6.1.1)

into Fresnel’s equation

) (

) ( k 2x + k 2y + k 2z ) −  n 2x k 2x ( n 2y + n z2 ) + n 2y k 2y ( n 2x + n z2 ) + n z2k z2 ( n 2x + n 2y )    (

F k x , k y , k z = n 2x k 2x + n 2y k 2y + n 2z k 2z

(6.1.2)

yields

(

)

F k inc,x + Γηx , k inc,y + Γηy , k inc,z + Γηz =

( aΓ2 + bΓ + c )( dΓ2 + eΓ + f ) − ( gΓ2 + hΓ + i ) + j = adΓ + ( ae + bd ) Γ 4

(6.1.3)

3

+ ( af + cd + be − g ) Γ 2 + ( bf + ce − h ) Γ + ( −i + j + cf ) = 0 with coefficients

a = n 2x η2x + n 2yη2y + n 2z η2z

(

b = 2 n 2x k x ηx + n 2y k yηy + n z2k z ηz

46

(6.1.4)

)

(6.1.5)

c = n 2x k 2x + n 2y k 2y + n z2k z2

(6.1.6)

d = η2x + η2y + ηz2 = 1

(6.1.7)




(

e = 2 k x ηx + k yηy + k z ηz

(

f = k 2x + k 2y + k 2z

)

)

(6.1.8)

2 (6.1.9)

( ) η2x + n 2y ( n 2x + n 2z ) η2y + n z2 ( n 2x + n 2y ) ηz2 (6.1.10) h = 2  n 2x ( n 2y + n z2 ) ηx + n 2y ( n 2x + n z2 ) ηy + n z2 ( n 2x + n 2y ) ηz  (6.1.11)   i = n 2x ( n 2y + n z2 ) k 2x + n 2y ( n 2x + n z2 ) k 2y + n z2 ( n 2x + n 2y ) k z2 (6.1.12) g = n 2x n 2y + n z2

j = n 2x n 2y n z2

(6.1.13)

To solve this quartic equation we recommend to work directly with equation (6.1.2) and to use a numerical solver. Starting values for Γ can be obtained from the quadratic equations

(

)

 2  2 2 2  2 2 2  n o  ( k x + Γηx ) + k y + Γηy  + n e ( k z + Γηz )  − n e n o      

(

)

= n o2 η2x + n o2 η2y + n e2 η2z Γ 2

(

)

(6.1.14)

− 2 n o2k x ηx + n o2k yηy + n e2k z ηz Γ +

( no2

k 2x + n o2 k 2y + n e2 k 2z − n e2 n o2

)=0

( ) = ( η2x + η2y + η2z ) Γ 2 − 2 ( k x ηx + k yηy + k z ηz ) Γ + ( k 2x + k 2y + k 2z − n o2 ) = 0  k2 + k2 + k2 − n2  y z o   x




(6.1.15)

47

6.2 Proof of eigenvector properties Formelabschnitt 2 To prove that

Eo = k o × c

(6.2.1)

is an eigenvector of the propagation equation (3.1.9) we have to show

k o × (k o × (k o × c)) + ε (k o × c) = 0

(6.2.2)

Introducing expression (3.2.1) for the dielectric tensor ε, we get

k o × ( k o × ( k o × c ) ) + ε o I ( k o × c ) + ( ε e − εo ) ( c ⋅ ( k o × c ) ) c = k o × ( k o × ( k o × c ) ) + εoI ( k o × c )

= ( k o ⋅ ( k o × c ) ) k o − ( k o ⋅ k o )( k o × c ) + εoI ( k o × c )

(6.2.3)

= − ( k o ⋅ k o )( k o × c ) + εoI ( k o × c ) = 0 because of (3.2.3).

To show the eigenvector property of the E field vector of the extraordinary wave (3.2.15)

(

)

(

Ee = n 2 − n e2 k e + n e2 − n o2

) (c ⋅ k e ) c

(6.2.4)

with

n2 = ke ⋅ ke

(6.2.5)

we proceed as follows: using standard vector identities we obtain

k e × ( k e × Ee )

( ) (c ⋅ k e ) (k e × (k e × c)) = ( n e2 − n o2 ) ( c ⋅ k e ) ( c ⋅ k e ) k e − ( k e ⋅ k e ) c  2 = ( n e2 − n o2 ) ( c ⋅ k e ) k e − ( n e2 − n o2 ) ( c ⋅ k e ) ( k e ⋅ k e ) c = n e2 − n o2

48



(6.2.6)


( ε o I + ( ε e − ε o ) c ⊗ c ) Ee = ( εoI + ( εe − εo ) c ⊗ c ) ( ( n 2 − n e2 )

( ) + n o2 ( n e2 − n o2 ) ( c ⋅ k e ) c + ( n 2 − n e2 ) ( n e2 − n o2 ) ( c ⋅ k e ) c + ( n e2 − n o2 ) ( n e2 − n o2 ) ( c ⋅ k e ) c

(

k e + n e2 − n o2

) (c ⋅ k e ) c)

= n o2 n 2 − n e2 k e

(6.2.7)

Adding terms from the previous two expressions we get the coefficient for ke

( ne2 − no2 ) (c ⋅ k e )2 + no2 ( n 2 − ne2 ) = 2 n o2 k e ⋅ k e + ( n e2 − n o2 ) ( c ⋅ k e ) − n o2 n e2 = 0

(6.2.8)

because of (3.2.3). The coefficient of c becomes

(

) (c ⋅ k e ) + ( ne2 − no2 ) ( n 2 − ne2 ) (c ⋅ ke ) + ( n e2 − n o2 ) ( n e2 − n o2 ) ( c ⋅ k e ) − ( n e2 − n o2 ) ( c ⋅ k e ) ( k e ⋅ k e ) (6.2.9) = ( n e2 − n o2 ) ( c ⋅ k e ) ( n o2 + ( n 2 − n e2 ) + ( n e2 − n o2 ) − ( k e ⋅ k e ) ) = 0 n o2 n e2 − n o2

As ke and c are linearly independent the proposition is proved.




49

6.3 Propagation through absorbing birefringent crystals Formelabschnitt 3 Consider a light ray which impinges from an isotropic incident medium on a uniaxial medium with absorption. Following the procedures given chapter 4.2 we can then calculate the ordinary and extraordinary refracted waves. If t denotes the tangential component of the wavevector of the incident wave and η the surface normal, we get for the ordinary wave

k o = t + γo η

(6.3.1)

Eo = k o × c

(6.3.2)

H o = − n o2 c + ( c ⋅ k o ) k o

(6.3.3)

and for the extraordinary wave

k e = t + γe η

(

)

(

Ee = n 2 − n e2 k e + n e2 − n o2

(

H e = k e × Ee = n e2 − n o2

(6.3.4)

) (c ⋅ k e ) c

) (c ⋅ k e ) (ke × c)

(6.3.5)

(6.3.6)

Here we want to prove the following expression for the normal components of the mixed Poynting vectors

η ⋅ ( Soe + Seo ) =

(

)

2 0.5Re  n o2 k e ⋅ k e + n e2 − n o2 ( c ⋅ k e ) − n o2n e2  c ⋅ ( t × η )  

(6.3.7)

To do so we begin by writing

η ⋅ Soe = 0.5 η ⋅ ( Eo × H e ) η ⋅ Seo = 0.5 η ⋅ ( Ee × H o )

(6.3.8) (6.3.9)

With expressions (6.3.5) and (6.3.3) we obtain with the help of the vector identities

a × a = 0 and

50

η × (η × a ) = 0 


Ee × H o =

(

)

(

) (c ⋅ k e ) c × −no2 c + (c ⋅ k o ) k o  = − n o2 ( n 2 − n e2 ) k e × c + ( c ⋅ k o ) ( n 2 − n e2 ) k e × k o + ( n e2 − n o2 ) ( c ⋅ k o ) ( c ⋅ k e ) c × k o (6.3.10) = − n o2 ( n 2 − n e2 ) t × c − γ e n o2 ( n 2 − n e2 ) η × c + ( c ⋅ k o ) ( n 2 − n e2 ) ( γo − γ e ) t × η +  γo ( n e2 − n o2 ) ( c ⋅ k o ) ( c ⋅ k e )  c × η   − ( n e2 − n o2 ) ( c ⋅ k o ) ( c ⋅ k e ) t × c  n2 − n2 k + n2 − n2 e e e o 

With the help of the vector identity

η ⋅ (t × c ) = c ⋅ (t × η) , we then obtain

η ⋅ ( Ee × H o )

) ( ) ) = ( n o2 ( n 2 − n e2 ) + ( n e2 − n o2 ) ( c ⋅ k o ) ( c ⋅ k e ) ) c ⋅ ( t × η )

( (

= − n o2 n 2 − n e2 + n e2 − n o2 ( c ⋅ k o ) ( c ⋅ k e ) η ⋅ ( t × c )

(6.3.11)

With expressions (6.3.2) and (6.3.6) we get

( ) = ( n e2 − n o2 ) ( c ⋅ k e ) ( ( t + γ o η ) × c ) × ( ( t + γe η ) × c ) = ( n e2 − n o2 ) ( c ⋅ k e ) ( t × c + γ o η × c ) × ( t × c + γe η × c ) = ( n e2 − n o2 ) ( c ⋅ k e ) ( γe − γ o ) ( t × c ) × ( η × c )

Eo × H e = n e2 − n o2 ( c ⋅ k e ) ( k o × c ) × ( k e × c )




(6.3.12)

51

We then use the vector identities

(t × c ) × (η × c) = ((η × c) ⋅ t ) c = (t × η) ⋅ c

,

which can be deduced from the general formulas

a × (b × c ) = (a ⋅ c) b − (a ⋅ b ) c setting

a = η × c,

b=t

c=c

and

and

((η × c) ⋅ t ) = −((η × t ) ⋅ c ) = ((t × η) ⋅ c) to obtain

(

)

Eo × H e = n e2 − n o2 ( c ⋅ k e ) ( γe − γ o ) ( ( t × η ) ⋅ c ) c

(6.3.13)

From this we get immediately

(

)

η ⋅ ( Eo × H e ) = n e2 − n o2 ( c ⋅ k e ) ( γe − γ o ) ( ( t × η ) ⋅ c ) ( c ⋅ η ) (6.3.14) Summing (6.3.11) and (6.3.14) we obtain

η ⋅ ( Ee × H o ) + η ⋅ ( Eo × H e ) =

(n (n 2 o

(

2

) (

)

)

− n e2 + n e2 − n o2 ( c ⋅ k o ) ( c ⋅ k e ) c ⋅ ( t × η )

)

+ n e2 − n o2 ( c ⋅ k e ) ( γe − γ o ) ( ( t × η) ⋅ c ) ( c ⋅ η) =

(

) (

)

(

)

 n 2 n 2 − n 2 + n 2 − n 2 ( c ⋅ k ) ( c ⋅ k ) + n 2 − n 2 ( c ⋅ k ) ( c ⋅ η) ( γ − γ ) e e o o e e o e e o   o

c ⋅ ( t × η) The term in square brackets can now be written as

52




(no2 (n 2 − ne2 )+ (ne2 − no2 )(c ⋅ k o )(c ⋅ k e ) + (ne2 − no2 )(c ⋅ k e )(c ⋅ η)(γ e − γ o )) = no2 k e ⋅ k e + (ne2 − no2 )(c ⋅ k o )(c ⋅ k e ) − no2 ne2 + (ne2 − no2 )(c ⋅ k e )(c ⋅ k e − c ⋅ k o ) 2 = no2 k e ⋅ k e + (ne2 − no2 )(c ⋅ k e ) − no2 ne2 + 2i Im[(ne2 − no2 )(c ⋅ k o )(c ⋅ k e )] which proves the proposition.

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7 References 1. C. Alberdi et al., Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dieelectric media, Journal of Modern Optics, 2002, vol.49, no 9, pp.1553-1566 2. V.I. Alshits, V.N. Lyubimow, W.Gierulski, A.Radowicz, ElectromagneticWave Surfaces in Absorptive Uniaxial Crystals. I. Cones of Zero Birefringence and Cones of Equal Absorption for Isonormal Waves, Crystallography Reports, Vol 45, No 6, 2000, pp.985-990 3. V.I. Alshits, V.N. Lyubimow, Electromagnetic-Wave Surfaces in Absorptive Uniaxial Crystals. II. Cones of Specific Ordinary Waves and Cones of Extreme Absorption, Crystallography Reports, Vol 45, No 6, 2000, pp.991994 4. J. Amanatides and A.Woo, A fast voxel traversal algorithm, Computer Animation: CG87. Proceedings of the Conference held at Computer Graphics 87, London, UK, Oct. 1987, p.149-55 5. S.M.P.Blom, H.P.M.Huck, H.J.Cornelissen, Towards Polarised Light Eimitting Back Lights, Micro-structured Anisotropic Layers, AsiaDisplay, IDW ’01, 525-528 (2001) 6. S.Blom and H.Cornelissen, Polarisation specific display illumination. A literature survey on methods to enhance the efficiency of edge-lit illumination systems for LCDs, Nat.Lab. Technical Note 2000/169 7. S.M.P.Blom, H.P.M.Huck, H.J.Cornelissen, H.Greiner, Towards a polarized lightemitting backligt: micro-structured anisotropic layers, Journal of the SID 10/3, 2002, pp.209-213 8. M.Born and E.Wolf, Principles of Optics, 7th (expanded) edition, Cambridge University Press, 1999 9. D. Broer et al., Illumination System for a Flat-Panel Picture Display Device, US Patent 5,729,311, March 17, 1998 10. C.Brousseau, Fundamentals of Polarized Light, a Statistical Approach, Wiley 1998 11. R.A. Chipman, Mechanics of polarization ray tracing, Optical Engineering/ June 1995/ vol.34 No.6/, pp.1636-1645 12. R.A. Chipman, Progress in polarization ray tracing, SPIE Proceedings, Vol.2265, pp.141-151

54




13. E.Collett, Polarized Light, Fundamentals and Applications, Marcel Dekker, 1993 14. R.Echarri and M.T. Garea, Behavior of the Poynting vector in uniaxial absorbent media, Pure Appl. Opt. 3 (1994), pp.931-941

15. R.Feynman, Feynman Lectures on physics, Addison-Wesley 1969 (International Student edition) 16. E.R.Freniere, G.G.Gregory, R.Hassler, Polarization models for Monte Carlo ray tracing, Optical Design and Analysis Software, Proceedings SPIE, volume 3780, 1999

17. F.X. Gentit, Litrani: a general purpose Monte-Carlo program simulating light-propagation in isotropic or anisotropic media, Nuclear-InstrumentsMethods-in Physics-Research-Section-A, vol.486. no 1-1, p.35-39, 2002, for more info see also http://gentit.home.cern.ch 18. H.Greiner, Power Splitting between Refracted Ordinary and Extraordinary Waves in Uniaxial Crystals with Absorption, submitted to Optik 19. P. Halevi, Plane electromagnetic waves in material media: Are they transverse waves?, Am. J. Phys. 48(10), Oct. 1980, pp.861-867 20. P. Halevi and A.Mendoza-Hernandez, Temporal and spatial behavior of the Poynting vector in dissipative media: refraction from vacuum into a medium, J.Opt.Soc.Am., Vol 71, No 10, pp.1238-1242

21. R.A. Hassler et al., Modelling birefringence in optomechanical design and analysis software, Proceedings of SPIE Vol.4769, for more info see also www.lambdares.com 22. E. Hecht, Optics, 4th edition, Addison Wesley, 2002 23. I.J. Hodgkinson and Q.H.Wu, Birefringent Thin Films and Polarizing Elements, World Scientific, 1997 24. S.Huard, Polarization of Light, Wiley, 1997 25. H.Jagt, Polymeric Polarisation Optics for Energy Efficient Liquid Crystal Display Illumination, Thesis, Eindhoven 2001 26. H.Jagt, H.J.Cornelissen, D.J.Broer, C.W.M.Bastiaansen, Micro-structured polymeric polarized light emitting lightguide for LCD illumination, SID 02 Digest 27. J.P. Lesso, A.J. Duncan, W. Sibbett, M.J. Padgett, Aberrations introduced by a lens from a birefringent material, Applied Optics, vol 39, No 4, pp.592-598, 2000

28. E.M. Lifschitz, L.P. Pitaevski, L.D. Landau, Electrodynamics of Continuous Media, Elsevier Science, 1985

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55

29. Q.Liang, Simple ray tracing formulas for uniaxial optical crystals, Applied Optics, Vol.29, No.7, pp.1008-1010 30. S.G. Lipson, H.Lipson, D.S. Tannhauser, Optical Physics, third edition, Cambridge University Press, 1995 31. S.McClain et al., Polarization ray tracing in anisotropic optically active media. I. Algorithms, J.Opt.Soc.Am.A., Vol. 10, No.11/November 1993, pp. 2371-2382 32. S.McClain et al., Polarization ray tracing in anisotropic optically acive media. II. Theory and physics, J. Opt. Soc. Am. A, Vol. 10, No 11/November 1993, pp.2383-2393 33. A.Morgan, Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems, Prentice Hall, 1987 34. J.-P. Perez, Optique, Fondements et Applications, 5 ieme edition, Masson,1996, chapitre 31 35. F. Pockels, Lehrbuch der Kristalloptik, B.G. Teubner, Leipzig, 1906 36. W.Press et al., Numerical Recipes in FORTRAN, The art of Scientific Computing, second edition, Cambridge University Press, 1992 37. G.M. Ramachandran, S.Ramaseshan, Crystal Optics, in Handbuch der Physik, Band XXV/1, Kristalloptik, Beugung, Springer 1961, editor S.Flügge, pp.1-217 38. M.C. Simon, Ray tracing formulas for monoaxial optical components, Applied Optics, Vol.22, No.2, 1983, pp. 354-360 39. M.C. Simon, Ray tracing formulas for monoaxial optical components: vectorial formulation, Applied Optics, Vol 25, No.12, 1986, pp.1935-1939 40. M.C. Simon, Refraction in biaxial crystals: a formula for the indices, J.Opt.Soc.Am.A, vol.4, No12 1987, pp. 2201-2204 41. M.C. Simon and R.M. Echarri, Internal total reflection in monoaxial crystals, Appl. Opt. 26 (1987), 3878 42. M.C. Simon, L.I. Perez, Total reflection in uniaxial crystals, Optik, 82, No.2 (1989), 37-42 43. M.C. Simon and Rodolfo M. Echarri, Inhibited reflection in uniaxial crystals, Optics Letters, vol.14, No.5, 1989, pp.257-259 44. M.C. Simon and Liliana I. Perez, Reflection and transmission coefficients in uniaxial crystals, Journal of Modern Optics, 1991, vol.38, No.3, pp.503-518 45. M.C. Simon et al., Isotropic medium-uniaxial crystal interfaces: light polarization in total external reflection, Optik 110, No.9 (1999), pp 439-445.

56




46. G. Szivessy, Kristalloptik, In Geiger und Scheel’s Handbuch der Physik, Vol.20, pp.635-904. Berlin, Springer 1928 47. J.D. Trolinger, R.A. Chipman, Polarization ray tracing in birefringent media, SPIE vol. 891 Polarization Considerations for Optical Systems (1988), pp.122-133 48. J.D. Trolinger, Russell A. Chipman, Donald K. Wilson, Polarization ray tracing in birefringent media, Optical Engineering/ April 1991/Vol.30 No.4, pp.461-466 49. L.B. Wolff and D.J. Kurlander, Ray Tracing with Polarization Parameters, IEEE Computer GraphicsApplications, 1990, pp.44-55 50. A.Yariv, P. Yeh, Optical Waves in Crystals, John Wiley, 1984 51. P. Yeh, Optical Waves in Layered Media, John Wiley, 1988 52. P. Yeh and C. Gu, Optics of Liquid Crystal Displays, John Wiley, 1999, chapter 3.7.2 53. Yu. P. Svirko, N.I. Zheludev, Polarization of Light in Nonlinear Optics, Wiley, 1998 54. W.Q. Zhang, General ray-tracing formulas for uniaxial crystal, Applied Optics, Vol.31, No.34, 1992, 7328-7331

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