Berreman4x4

Berreman4x4

Olivier Castany(a)Department of Optics, Telecom Bretagne, 29238 Brest, France

June 16, 2013

Abstract

Electromagnetic plane wave propagation in stratified anisotropic media was described by Berreman with theuse of 4×4 matrices. Berreman4x4 is a numerical implementation of the method in Python. Examples ofapplications are ellipsometry analysis, design of Bragg mirrors or study of twisted liquid crystal structures.

(a)Electronic mail: [email protected]

Description of Berreman4x4Olivier Castany

Electromagnetic wave propagation in stratified media isimportant in several applications like ellipsometry anal-ysis, Bragg mirrors or twisted liquid crystal structures.Propagation of plane waves in isotropic media can besolved with 2×2 propagation matrix methods1. In thecase of anisotropic layers, a 4×4 propagation matrixmethod was developed2,3 and is now known as Berre-man’s method.The present work is an open-source implementation of

the method in Python. This programming language waschosen for ease of use, readability of the source code,portability and availability of scientific libraries (NumPyand SciPy). A drawback is slow speed, because the lan-guage is interpreted1.Calculations are based on articles from Berreman3 and

Schubert4. Application to ellipsometry is based on Fuji-wara’s book5. General references for optics in anisotropicmedia are found from Born and Wolf1, and Jackson6.

1 Presentation

As described on figure 1, we consider a stratified samplewith layers invariant in the (x, y) plane and stacked inthe z direction, starting from zf = 0. Because of thetranslation invariance, modes with bounded fields canbe classified according to wave numbers (kx, ky) ∈ R2.Without loss of generality, we consider plane waves inthe (x, z) direction, i.e. ky = 0.The front half-space is isotropic and a plane wave

can be decomposed into s and p polarizations. The spolarization is a wave with electric field perpendicular(senkrecht) to the plane of incidence, i.e. along y. Thep polarization is a wave with electric field parallel to theplane of incidence. A plane wave i is incident from thefront half-space with incidence angle φi and reflected intoa plane wave r with the same angle. Angles are orientedby the y direction. In the general case, the back half-space is anisotropic and two transmitted plane waves tare induced, with the same kx, but different angles φt.The electric vector is not necessarily perpendicular to ~kand s and p waves can not be considered.

2 Conventions and units

Formulas are mostly based on Schubert’s article4, withone difference being the orientation of the incident p po-larization. Schubert takes the convention used in ellip-sometry, where the base electric vectors for p polariza-

1High speed calculations are possible with NumPy and SciPyif array operations are vectorized. However the structure of Berre-man4x4 is difficult to vectorize with enough generality.

x

zy

i

r

t

φi

Ei

Er

Ex

p

s Back half-space,anisotropic

Fronthalf-space

Sample

Et

FIG. 1: Geometry of the sample, with input and outputplane waves.

tion point towards the sample in the front and exit halfspaces2. In our work, we used a different convention, inwhich the base electric vectors point towards the x di-rection in both half-spaces. With this convention, thebase electric vectors for the incident, reflected and trans-mitted waves are all in the same direction in the case ofreflection with normal incidence.

Gaussian units are used in the detail of the calcula-tion. This makes it easier to compare our results withpast literature, which uses Gaussian units for the mostpart. Of course, reflection and transmission coefficientshave no unit and do not depend on the choice of unit.Conversion between units is presented in the appendix onelectromagnetic units in reference 6. In Gaussian units,Maxwell’s equations read

∇× E = −1c

∂B

∂tand ∇×H = 1

c

∂D

∂t.

We consider linear materials with tensorial constitutiverelations D = εE and B = µH. In Gaussian units,ε0 = µ0 = 1 and the relation to S.I. units is

ES.I. = E√4πε0

and HS.I. = H√4πµ0

,

where ε0 and µ0 are the usual S.I. values. The conventionfor time-varying complex fields is taken as exp(−iωt),and Maxwell’s equations read

∇× E = ik0H and ∇×H = −ik0εE,

with k0 = ω/c. We consider non-magnetic materials, i.e.µ = 1. The permittivity ε may be complex. For example,isotropic materials with Im(ε) > 0 are lossy.

2In Fujiwara’s book, reference 5, p. 223, the base electric vec-tors for p polarization point towards the −z direction, which leadsto the same definition as Schubert’s.

1

2

3 Propagation inside the sample

In the stratified sample, Maxwell’s equations lead toa propagation equation for the transverse components(Ex, Ey, Hx, Hy). The demonstration can be found inreference 3 and leads to3

∂Ψ∂z

= ik0∆(z)Ψ(z), with Ψ =

ExEyHx

Hy

, (1)

where ∆(z) is a 4 × 4 matrix. For a general dielectrictensor, the matrix is4 ∆(z) =

−Kx

ε3,1

ε3,3−Kx

ε3,2

ε3,30 1− K2

x

ε3,3

0 0 −1 0

ε2,3ε3,1

ε3,3− ε2,1 K2

x − ε2,2 + ε2,3ε3,2

ε3,30 Kx

ε2,3

ε3,3

ε1,1 − ε1,3ε3,1

ε3,3ε1,2 − ε1,3

ε3,2

ε3,30 −Kx

ε1,3

ε3,3

The reduced wave number Kx = kx/k0 is a constantthroughout the sample and depends only on the angle ofthe incident wave.For a homogeneous slab z1 < z < z2, the matrix ∆(z)

is constant and equation 1 can be integrated into Ψ(z2) =Phs(z2, z1)×Ψ(z1), where the propagator is given by thematrix exponential

Phs(z2, z1) = exp(i (z2 − z1) k0 ∆

)The numerical computation of a matrix exponential isgenerally slow. Berreman suggested to diagonalize thematrix by searching the eigenvalues and eigenvectors3.However, the knowledge of the eigenvectors is not neces-sary and the result can be expressed based on the eigen-values only4. I. Abdulhalim et al. used the Lagrange-Sylvester interpolation polynomial7–9 and Wöhler et al.used Cayley-Hamilton’s theorem10,11. Both approacheslead to the same expression. The eigenvalues can becalculated numerically as the roots of the characteristicpolynomial. Literal expressions can be found for specificsituations, like for a uniaxial material10,11 or in the caseof normal incidence7. Also, in the case of a diagonaltensor, a specific solution for Phs is available8.If a part of the sample is inhomogeneous, it is subdi-

vided into slices over which the variation of ∆(z) is small.For such a slice, the propagator P (z2, z1) is approximatedby a homogeneous slab for which the ∆ matrix is evalu-ated in the middle of the interval,

P (z2, z1) ' exp(i (z2 − z1) k0 ∆

(z1 + z2

2

)).

3In Berreman’s article, there is a misprint in equation (3), itshould read C = MG.

4Knowledge of the eigenvectors is necessary only for the exittransition matrix if the back half-space is anisotropic.

The total propagator P (zN , z0) for N slices between z0and zN is approximated the product

Pa(zN , z0) = Phs(zN , zN−1)× · · · × Phs(z1, z0).

The order of the error is

P (zN , z0)− Pa(zN , z0) = O(1/N2)

and Z. Lu demonstrated12 that this does not depend onwhether the thin slab propagator Phs is the exact propa-gator or an approximation, possibly to first order. Con-sequently, the simplest solution to equation 1 is to takethe first order expansion

Ψ(z2) ' i(z2 − z1)k0 ∆(z1 + z2

2

)×Ψ(z1),

which corresponds to the first order expansion of the ex-ponential,

Phs(z2, z1) ' i(z2 − z1)k0 ∆(z1 + z2

2

)For improving convergence and efficiency, Z. Lu pre-sented an extrapolation method to eliminate the leadingterms of the error12. If the propagator used for the thinhomogeneous slabs is the exact propagator Phs, the erroris reduced to O(1/N4). Z. Lu presented another versionwith a symplectic integrator that showed improved con-vergence13. In this version, the propagator for a thin slabis simply the product of three homogeneous slab propa-gators evaluated for different thickness and position (seeequation (10) in the reference).

4 Transition to half-spaces

For isotropic half-spaces, the relation between the wavecoefficients and the vector Ψ at the boundary is given bythe transition matrices Lf and Lb with

Ψ(0) = Lf

EisErsEipErp

z=zf

and Ψ(zb) = Lb

Ets0Etp0

z=zb

The ordering of the components is (s+, s−, p+, p−), wherethe superscripts indicate the traveling direction along thez axis. The direction of each wave is characterized bya reduced wave vector ~K = (Kx, 0,Kz) satisfying theHelmholtz relation ~K2 = n2 = ε. An angle φ determines~K = n (cos(φ), 0, sin(φ)) and may be a complex numberwhen the material is lossy or when an evanescent waveoccurs. It is unique with φ′ ∈ ] − π, π] and φ′′ ∈ R.Considering the angle φ for the wave traveling in the zdirection, we have φ′ ∈ [−π/2, π/2] and

L =

0 0 cos(φ) cos(φ)1 1 0 0

−n cos(φ) n cos(φ) 0 00 0 n −n

3

When the back half-space is anisotropic, we can not de-compose the transmitted wave on s and p polarizations,but by analogy, we decompose Ψ(zb) over the eigenvec-tors Ψk of the ∆b matrix,

Ψ(zb) =4∑k=1

ckΨk

We sort the eigenvectors so that Ψ1 and Ψ3 correspondto waves propagating in the z direction, i.e. the eigen-values q1 and q3 have positive real part. This descrip-tion incorporates the isotropic case for which (c1, c3) =(Ets, Etp)(zb). We can write

Ψ(zb) = Lb

c10c30

with Lb =

Ψ11 0 Ψ31 0Ψ12 0 Ψ32 0Ψ13 0 Ψ33 0Ψ14 0 Ψ34 0

5 Matrix assembling and Jones matrices

The global propagation matrix and the two transitionmatrices are assembled in order to relate the coefficientsof the waves in the two half-spaces. We obtain the totaltransfer matrix T with

EisErsEipErp

zf

= L−1f P (zf , zb) Lb

c10c30

= T

c10c30

.

The two useful relations can be extracted,(EipEis

)zf

=(T33 T31T13 T11

)(c3c1

)= Tit

(c3c1

)(ErpErs

)zf

=(T43 T41T23 T21

)(c3c1

)= Trt

(c3c1

).

Reflection of the incident wave can be described by aJones matrix Tri, and if the back half-space is isotropic, aJones matrix Tti for transmission can also be defined5,14,(

ErpErs

)zf

=(rpp rpsrsp rss

)(EipEis

)zf

= Tri

(EipEis

)zf(

EtpEts

)zb

=(tpp tpstsp tss

)(EipEis

)zf

= Tti

(EipEis

)zf

These matrices contain all the information on reflectionand transmission of the sample. They are obtained bythe relations

Tri = TrtT−1it and Tti = T−1

it .

6 Circularly polarized light

When fields are decomposed over the s and p polariza-tions, the basis for the Jones vectors is (Ep,Es) with

Ep =(

10

)and Es =

(01

).

It is possible to consider other bases, for example theleft and right circular polarizations. For the incident andtransmitted waves the Jones vectors are

EiL,EtL = 1√2

(1i

)and EiR,EtR = 1√

2

(1−i

).

For the reflected wave, we have

ErL = 1√2

(1−i

)and ErR = 1√

2

(1i

).

The transformation matrix from the (s, p) basis to the(L,R) basis will be called C in the case of incident andreflected waves, and D for the reflected wave. We have

C = 1√2

(1 1i −i

)and D = 1√

2

(1 1−i i

).

The relations between the Jones vectors in the two basesare (

EipEis

)= C

(EiLEiR

),

(EtpEis

)= C

(EtLEiR

),

and(ErpEis

)= D

(ErLEiR

).

As a result, the Jones matrices for circularly polarizedlight are T cti = C−1 Tti C and T cri = D−1 Tri C.

7 Ellipsometry parameters

Ellipsometry parameters describe the reflection of thesample by the normalized reflection matrix Tri/rss, aspresented in Fujiwara’s book, reference 5, p. 220. How-ever, since we use an opposite orientation convention forErp, a change of sign is necessary for matching the con-vention of ellipsometry and we define(

tan(ψpp) e−i∆pp tan(ψps) e−i∆ps

tan(ψsp) e−i∆sp 1

)

=(−rpp/rss −rps/rssrsp/rss 1

).

The minus sign in front of ∆ is chosen due to theexp(−iωt) phase convention. The ellipsometry angles arechosen with ψ ∈ [0, π/2] and ∆ ∈ ]− π, π]. If the sampleis isotropic in the (x, y) direction, the off-diagonal coef-ficients sp and ps vanish and only two parameters areneeded to describe the reflection. We have

Tri =(rpp 00 rss

)and Tti =

(tpp 00 tss

)and we define the ellipsometry parameters ψ and ∆ with

tan(ψ) e−i∆ = −rpp/rss.

4

8 Installation and use

Berreman4x4 is offered as a Python module namedBerreman4x4.py, which can be imported and used inPython scripts with the command

import Berreman4x4

The module file should either be present in the workingdirectory or be accessible in the module path. A conve-nient organization is to store Python modules in a specialdirectory pointed by the PYTHONPATH environment vari-able. For example, my .bashrc script contains

export PYTHONPATH="/home/castany/.python"

and the .python directory contains symbolic links to thedifferent Python modules I use.Berreman4x4 depends on the standard Python mod-

ules NumPy15 and SciPy16. Application examples needmodule matplotlib17 for plotting.General workflow with Berreman4x4 consists of (1)

building the structure, (2) calculating Jones matrices, (3)extracting the desired coefficients and plotting.

9 Code documentation and examples

The source code contains detailed information on theclasses and functions, incorporated as docstrings. Theycan be conveniently displayed when working with a shelllike IPython18.Commented examples are bundled with the program

and can be run directly from the command line. Theyrange from simple situations to more complex structures:reflection on an interface, interferences in a thin layer,reflection on a Bragg mirror, 90◦ twisted nematic liquidcrystal. Whenever possible, result from Berreman4x4 iscompared with exact analytical result, and show excel-lent match. The code is also validated against resultspresented in reference 5.The UML class diagram of Berreman4x4 is pre-

sented on figures 2 and 3. A Structure is builtfrom a list of Layer objects, between one frontand one back HalfSpace objects. Material ob-jects are created for defining MaterialLayer objects.Several classes of non-dispersive materials are pro-vided, but dispersive materials may also be cre-ated, like the class IsotropicDispersive. Additionalclasses of materials can be created by deriving classesMaterial and DispersionLaw. Layers with a continu-ous spatial variation of the permittivity tensor are de-scribed with an InhomogeneousMaterial and form anInhomogeneousLayer object.

10 TODO

General:

• Check Berreman’s equations again. Which are theassumptions?

• Verify efficiency of Lu’s method. It seems to beworse than the midpoint method for the twisted ne-matic example. Strange.

• Check code if back half-space is lossy or total inter-nal reflection occurs. Complex φ?

• Provide function for displaying elementary plotsR, T = f(λ) for a structure.

Source code:

• Homogeneous layer: implement exact solution withCayley-Hamilton or Lagrange-Sylvester.

• Calculation of the fields inside the structure (not justt and r).

• Class MaterialLayer may be a bad idea, becauseclasses Material and InhomogeneousMaterial arenot brothers.

• Maintenant que j’ai créégetPermittivityProfile(), c’est peut-être possible de l’utiliser pour simplifiergetPropagationMatrix(). Se demander sic’est une bonne idée. Il faudra alors ajouter le choixde direction inv à getPermittivityProfile().

5

Material+getTensor(lbda)

IsotropicMaterial+getRefractiveIndex(lbda)

IsotropicDispersive+law

-__init__(self,law=None)+getTensor(lbda)+getRefractiveIndex(lbda)

IsotropicNonDispersiveMaterial+n

-__init__(self,n=1.5)+getRefractiveIndex(lbda=None)

NonDispersiveMaterial+epsilon

-__init__(self,epsilon=None)+getTensor(lbda=None)+rotated(R)

SellmeierLaw+A, B, lbda0

-__init__(self,coeff=(1.0, 1.0, 300e-9))+getValue(lbda)

UniaxialNonDispersiveMaterial+n

-__init__(self,no=1.5,ne=1.7)

DispersionLaw+getValue(lbda)Berreman4x4 module

+c, pi

+rotation_Euler(angles)+rotation_V(V)+rotation_v_theta(v,theta)+buildDeltaMatrix(Kx,eps)+hs_propagator(Delta,h,k0,method="linear",q=None)+hs_propagator_lin(Delta,h,k0,q=None)+hs_propagator_Pade(Delta,h,k0,q=7)+hs_propagator_Taylor(Delta,h,k0,q=5)+hs_propagator_eig(Delta,h,k0,q=None)+extractCoefficient(Jones,coeff_name)+circularJones(Jones)+extractEllipsoParam(Jr)

BiaxialNonDispersiveMaterial+n

-__init__(self,diag=(1.5,1.6,1.7))

Structure+frontHalfSpace+backHalfSpace+layers

-__init__(self,front=None,layers=None,back=None)+setFrontHalfSpace(halfSpace)+setBackHalfSpace(halfSpace)+setLayers(layers)+getPermittivityProfile(lbda)+getPropagationMatrix(Kx,k0=1e6,inv=False)+getStructureMatrix(Kx,k0=1e6)+getJones(Kx,k0=1e6)+getIndexProfile(lbda=1e-6,v=e_x)+drawStructure(lbda,margin=0.15)

FIG. 2: Class diagram of Berreman4x4: structure, materials, and module functions.

6

Berreman4x4.IsotropicHalfSpace+material

-__init__(self,material=None)+get_Kx_from_Phi(Phi,k0=1e6)+get_Phi_from_Kx(Kx,k0=1e6)+getTransitionMatrix(Kx,k0=1e6,inv=False)

InhomogeneousMaterial+getTensor(z,lbda)+getSlices()

HomogeneousIsotropicLayer+setThickness(h)+get_QWP_thickness(lbda=1e-6)

RepeatedLayers+n+before+after+layers

-__init__(self,layers=None,n=2,before=0,after=0)+setRepetition(n,before=0,after=0)+setLayers(layers)+getPermittivityProfile(lbda)+getPropagationMatrix(Kx,k0=1e6,inv=False)

HomogeneousLayer+h+material+hs_propagator: function+hs_order

-__init__(self,material=None,h=1e-6, hs_method="eig",hs_order=2)+setThickness(h)+setMethod(hs_method,hs_order=2)+getPermittivityProfile(lbda)+getPropagationMatrix(Kx,k0=1e6,inv=False)+getDeltaMatrix(Kx,k0=1e6)

Berreman4x4.HalfSpace+material

-__init__(self,material=None)+setMaterial(material)+getTransitionMatrix(Kx,k0=1e6)

MaterialLayer+material

+setMaterial(material)

InhomogeneousLayer+material+z+getSlicePropagator: function+hs_propagator: function+hs_order

+__init__(self,material=None,evaluation="midpoint", hs_method="Padé",q=2)+setMethod(evaluation,hs_method,q)+getPermittivityProfil(lbda)+getPropagationMatrix(Kx,k0=1e6,inv=False)+getSlicePropagator_mid(z2,z1,Kx,k0=1e6)+getSlicePropagator_sym(z2,z1,Kx,k0=1e6)

Layer+getPermittivityProfile(lbda)+getPropagationMatrix(Kx,k0,inv)

TwistedMaterial+material+d+angle+div

-__init__(self,material=None, d=4e-6,angle=pi/2,div=25)+setMaterial(material)+setThickness(d)+setAngle(angle)+setDivision(div)+getTensor(z,lbda=None)+getSlices(self)

FIG. 3: Class diagram of Berreman4x4: layers, inhomogeneous materials, and half-spaces.

Validation examples taken from Fujiwara’s book

Olivier Castany

We reproduce the situation presented by Fujiwara in hisbook Spectroscopic Ellipsometry 5, section 6.4.2, p. 241–243 and section 6.4.1, p. 237–239. He presents detailedcalculations and intermediate steps that are useful fortesting our code.

1 Example of section 6.4.2

The situation is depicted on figure 4. A uniaxial film isformed on a silicon substrate. The incident medium isair and the silicon substrate has refractive index nt =3.898 + 0.016i. The thickness of the film is d = 100 nmand the refractive indices are no = 2.0 and ne = 2.5. Theorientation of the film is given by Euler angles φE = π/4and θE = π/4. Angle φE is the first Euler angle, inducinga rotation of axis x around axis z, leading to axis x′.Angle θE is the second Euler angle, inducing a rotationof the axis of the material around x′.

In validation-Fujiwara-642.py, we reproduce Fuji-wara’s results with our code and obtain the same values,except for a few places where a sign is reversed. The rea-son is that Fujiwara uses the convention of ellipsometryfor orienting Erp, which is the opposite of our conven-tion. We also reproduce figure 6.19, p. 242, when theorientation of the anisotropic film is varied.

x

z

i

r

t

φi

Ei

Er

Ex

p

sSilicon substrateAir Film

Etφt

FIG. 4: Geometry of the situation treated by Fujiwarain his book, section 6.4.2.

2 Example of section 6.4.1

In validation-Fujiwara-641.py, light is reflected inthe air by an anisotropic substrate as depicted on fig-ure 5. We reproduce figure 6.16, p. 238, when the angleof incidence φi is varied and figure 6.17, p. 239, whenthe Euler angle φE of the anisotropic substrate is varied.The difference with section 6.4.2 is that the substrate isanisotropic and the code uses a general HalfSpace in-stead of an IsotropicHalfSpace.

x

z

i

r

φi

Ei

Er

p

s

t

Et

Air Anisotropic substrate

FIG. 5: Geometry of the situation treated by Fujiwarain his book, section 6.4.1.

7

Example of the frustrated total internal reflectionOlivier Castany, Céline Molinaro

Frustrated total internal reflection is used as a validationexample. Analytical and numerical results are compared.

1 Presentation

We consider the situation on figure 6, where two half-spaces with indices nf and nb are separated by a mediumof index ns and thickness d. The three media are as-sumed to be lossless. The incoming plane wave definesvector kx throughout the structure. The reduced wavevector is Kx = kx/k0 = nf sin(φi). The scalar Helmholtzequation (∆ + k2

0ε){E,H} = 0, holds separatley insidethe three media, and implies that there are at most twowaves in each medium, with wave vector ±kz, given byk2z = k2

0n2 − k2

x.

x

zφi

φt

t

i

s

p

s s

p p

r

Front half-space Back half-space

z = 0 z = d

Ei

Er

E+

E−

Et

nf nbns

d

FIG. 6: Frustrated total internal reflection with inputand output plane waves.

We consider the separation medium. Since kx is real,kz is either purely real or purely complex. The first casehappens for incidence angles φi smaller than the criticalangle φic, given by sin(φic) = ns/nf . In this case, twoplane waves are present, with ±kz wavenumber. The sec-ond case corresponds to the total internal reflexion in thefront half-space medium and these is an evanescent wavein the separation medium. If the third material is closeenough, a plane wave is transmitted from the evanescentwave. This phenomenon is the frustrated total internalreflexion. In this situation, two evanescent waves arepresent in the medium, with ±k′′z exponential evolution.Due to the (xz) mirror symmetry, s and p modes can

be considered separately. For s polarization, we have

~E = E(x, z) ~y and ~H = 1ik0

−∂zE0∂xE

.

For p polarization, we have

~H = H(x, z) ~y and ~E = i

k0ε

−∂zH0∂xH

.

This duality implies that expressions for p polarizationrelate naturally to H. However, transmission and reflec-tion coefficients are always defined with respect to E,

t = EtEi

and r = ErEi

, for both s and p polarizations.

In the next sections, we will study the anatomy of thewaves in detail. For that purpose, the Poynting vectorgives useful physical insight. In Gaussian units, the timeaverage of the Poynting vector is

〈~Π〉t = c

8π Re(~E × ~H∗

).

2 Anatomy of a single wave

If we consider an s-polarized wave defined by~E = E(x, y) ~y with E(x, y) = E0 e

−iωt+i(kxx+kzz),

we deduce ~H = E(x, y)

−Kz

0Kx

,

~H∗ = E∗0 eiωt−i(kxx+k∗zz)

−K∗z0Kx

,

and 〈~Π〉t = c

8π |E0|2e−2k′′z z

Kx

0K ′z

. (2)

If we consider a p-polarized wave defined by~H = H(x, y) ~y with H(x, y) = H0 e

−iωt+i(kxx+kzz),

we deduce ~E = H(x, y)ε

Kz

0−Kx

, (3)

~H∗ = H∗0 eiωt−i(kxx+k∗zz) ~y,

and 〈~Π〉t = c

8π|H0|2

εe−2k′′z z

Kx

0K ′z

. (4)

In these expressions, we defined kz = k′z + ik′′z . Weobserve that the Poynting vector is parallel to the realpart of the wave vector. If kz is real (k′′z = 0), the waveamplitude is constant along z. If kz is purely complex(k′z = 0), there is no energy flow in the z direction, andthe wave decays exponentially in the z direction. Figure 7represents the two cases.

8

9

x

z(a) Plane wave

x

z(b) Evanescent wave

kz real kz purely complex

FIG. 7: Anatomy of a homogeneous plane wave and anevanescent wave. The arrows show the real part of thewave vector. The thickness of the arrow indicates theintensity of the wave.

3 Reflection on an interface: fields andPoynting vector

We consider an incident wave partially reflected at z = 0by a structure that does not affect the parity of the wave(s or p polarization). The complex reflection coefficientsare called rs and rp. The waves may either be plane orevanescent waves. The incident and reflected waves arenamed with “+” and “−” subscripts. The total field is~E = ~E+ + ~E−. For the s polarisation, we consider{~E+ = E+ ~y~E− = E− ~y

with{E+ = E+

0 e−iωt+i(kxx+kzz)

E− = E−0 e−iωt+i(kxx−kzz).

The magnetic excitation is

~H = ~H+ + ~H− = E+

−Kz

0Kx

+ E−

Kz

0Kx

.

The amplitudes of the incident and reflected waves areconnected by rs = E−0 /E

+0 and the Poynting vector is

〈~Π〉t = c

8π |E+0 |2

e−2k′′z z

Kx

0K ′z

+ |rs|2e2k′′z z

Kx

0−K ′z

+

+2 |rs|

Kx cos(θs − 2k′zz)0

K ′′z sin(θs − 2k′zz)

where we defined rs = |rs| eiθs . The first and secondterms correspond to the incident and reflected waves.The third term arises from the interference of the twowaves. If Kz is real, the expression in curly braces be-comesKx

0Kz

+ |rs|2 Kx

0−Kz

+ 2 |rs|

Kx cos(θs − 2kzz)00

.

If Kz is purely complex, it becomes

e−2k′′z z

Kx

00

+ |rs|2e2k′′z z

Kx

00

+ 2

Kxr′s

0K ′′z r

′′s

,

which exhibits an energy flow in the z direction, propor-tionnal to K ′′z r′′s .For p polarisation, we consider{~H+ = H+ ~y~H− = H− ~y

with{H+ = H+

0 e−iωt+i(kxx+kzz)

H− = H−0 e−iωt+i(kxx−kzz).

The electric field is

~E = ~E+ + ~E− = H+

ε

Kz

0−Kx

+ H−

ε

−Kz

0−Kx

. (5)

The amplitudes of the incident and reflected waves areconnected by rp = E−p /E

+p = H−0 /H

+0 . The Poynting

vector has the same expression as for the s polarization,with rs replaced by rp = |rp| eiθp and |E+

0 |2 replaced by|H+

0 |2/ε,

〈~Π〉t = c

8π|H+

0 |2

ε

e−2k′′z z

Kx

0K ′z

+ |rp|2e2k′′z z

Kx

0−K ′z

+

+2 |rp|

Kx cos(θp − 2k′zz)0

K ′′z sin(θp − 2k′zz)

.

If Kz is purely complex, there is an energy flow in thez direction, proportionnal to K ′′z r′′p . Figure 8 representsthe variation of the Poynting vector along z, in the caseof evanescent waves.

z

x

〈Πx,int〉t〈Πz,int〉t

〈Π−x 〉t

〈Π+x 〉t

〈~Π〉tInterface

z = 0

FIG. 8: Refection of an evanescent wave on an interface(kz is purely complex and r′′ ≥ 0). The Poynting vectoris decomposed in different terms.

4 Expression of the reflexion coefficientsfor an interface between two media

We consider the reflexion on an interface between twomedia “1” and “2”, with light coming from medium “1”.The reflection and transmission coefficients for the s po-larization are19

rs = kz1 − kz2

kz1 + kz2and ts = 1 + rs = 2 kz1

kz1 + kz2.

10

The expressions are valid for complex wave vectors andwe deduce

r′s = |kz1|2 − |kz2|2

|kz1 + kz2|2and r′′s = 2(k′′z1k

′z2 − k′z1k

′′z2)

|kz1 + kz2|2.

In the case of an evanescent wave in region 1, the wavenumber is kz1 = ik′′z1 and we deduce

r′s = k′′2z1 − |kz2|2

|ik′′z1 + kz2|2and r′′s = 2k′′z1k

′z2

|ik′′z1 + kz2|2.

The last expression shows that r′′s ≥ 0, which impliesthat the energy flow is directed to the right, as expected.For p polarization, the coefficients for the magnetic

field are rHp = H−1p/H+1p and tHp = H+

2p/H+1p with

rHp = ε2kz1 − ε1kz2

ε2kz1 + ε1kz2and tHp = 1+rHp = 2ε2kz1

ε2kz1 + ε1kz2.

The direction and value of the electric field is deducedfrom equations 3 and 5, which leads to rp = −rHp andtp = n1/n2 × tHp and implies19

rp = ε1kz2 − ε2kz1

ε2kz1 + ε1kz2and tp = 2n1n2kz1

ε2kz1 + ε1kz2.

We deduce r′p = |ε1kz2|2 − |ε2kz1|2

|ε2kz1 + ε1kz2|2and

r′′p = 2ε1ε2(k′′z1k′z2 − k′z1k

′′z2)

|ε2kz1 + ε1kz2|2.

In the case of an evanescent wave in region 1, the wavenumber is kz1 = ik′′z1 and we deduce

r′p = |ε1kz2|2 − ε22k′′2z1|ε2ik′′z1 + ε1kz2|2

and r′′p = 2ε1ε2k′′z1k′z2

|ε2ik′′z1 + ε1kz2|2.

The last expression shows that r′′p ≥ 0, which impliesthat the energy flow is directed to the right, as expected.The power flow along the z direction is deduced from

equations 2 and 4, leading to the expressions for thepower coefficients for both polarizations,

R = 〈Π−z1〉t

〈Π+z1〉t

= |r|2 and T = 〈Π+z2〉t

〈Π+z1〉t

= k′z2k′z1× |t|2.

We verify that R + T = 1 for both polarizations whenthe materials are lossless5. Also, if we consider the coef-ficients for the reverse directions, we show that for bothpolarizations we have r12 = −r21, t12/k2 = t21/k1 andt12t21 + r2

12 = 1 .

5The relation only has a meaning when kz1 is real. The demon-stration uses the fact that kz2 is either real or purely complex. Thetwo cases are considered separately and both verify R + T = 1.

5 Application to the frustrated totalinternal reflection

We consider waves Ei and Er in the incident medium,E+ and E− in the separation medium, and Et in theback half-space. The relations at the interfaces can bewritten in the same fashion for both polarizations. Atthe z = 0 interface, we have{

E+(0) = rfs E−(0) + tsf Ei(0)

Er(0) = rsf Ei(0) + tfs E−(0)

and at the z = d interface, we have{Et(d) = tbs E

+(d)E−(d) = rbs E

+(d).

Propagation between planes z = 0 and z = d implies{E+(d) = E+(0) eikzd

E−(0) = E−(d) eikzd ,

where kz is the wavenumber in the separation medium.From these equations, we get6

E+(0)/Ei(0) = tsf1− rfsrbsei2kzd

E−(0)/Ei(0) = rbstsfei2kzd

1− rfsrbsei2kzd

r = Er(0)/Ei(0) = rsf + rbsei2kzd

1− rfsrbsei2kzd

t = Et(d)/Ei(0) = tbstsfeikzd

1− rfsrbsei2kzd

The power reflection and transmission coefficient are

R = |r|2 and T = 〈Πzt〉t〈Πzi〉t

= k′zbk′zf|t|2.

We verify that R + T = 1 when the materials arelossless. To demonstrate this, different cases are sep-arated, depending on the type of wave in the regions.In the case when there are plane waves in both half-spaces, we show that k′zb/k

′zf |tbstsf |2 exp(−2k′′z d) =

|tfstsf tbstsb| exp(−2k′′z d). When there is also a planewave in the separation region, the reflection coeffi-cients are real, and the term simplifies to (1 − r2

fs)(1 −r2bs). When there is an evanescent wave in the sep-aration medium, the reflection coefficients are com-plex with unitary modulus, and the term simplifies to4 r′′bsr′′fs exp(−2k′′z d). In both cases, putting this term inthe expansion of R+ T leads to the result R+ T = 1.

6We used the relation tfstsf − rfsrsf = 1 for simplifying theexpression of the coefficient r.

Various examplesCéline Molinaro, Olivier Castany

1 Glass layer

In Interferences.py, we consider the glass plate fromfigure 9, where light arrives with a 30° incidence angle(Kx = 0.5). The reflection and transmission coefficientsR and T are plotted as a function of the glass thickness,for both polarizations. Interferences are visible.

x

z

i

rt

h

30◦

Glassn = 1.5

Airn = 1.0

Air

FIG. 9: Glass plate with multiple reflections inside.

2 Reflection on an interface

A plane wave is reflected and refracted by an interface,as represented on figure 10.

n1 n2

z

x

i

t

r

φi

φt

FIG. 10: Reflection of a plane wave on an interface be-tween two materials.

In interface-Jones.py, we consider an interface withn1 = 1.0 and n2 = 1.5. The Jones matrix for reflection

and transmission are calculated, with linear or circularbases, for normal incidence.

In interface-reflection.py, the power reflectionand transmission coefficientsR and T are plotted for bothpolarizations as a function of the reduced wave vectorKx. We observe that R + T = 1. The coefficient |t2| isalso plotted and is clearly different from T . The resultfrom Berreman4x4 and the analytic solution are plottedtogether and are identical. The refractive indices can begiven from the command line and total internal reflectionis observed if n1 > n2.

3 Bragg Mirror

We consider TiO2/SiO2 Bragg mirrors as presented onfigure 11. In validation-Bragg.py, the reflection co-efficient R is calculated with Berreman4x4 and com-pared to the analytical result19, for different incidenceangles and polarizations. The results are identical. InBragg-example.py, the reflection and transmission coef-ficients R and T are calculated for a Bragg mirror with8.5 periods at normal incidence.

z

x

Air GlassSiO2TiO2

i

TiO2 SiO2

tr

period

FIG. 11: TiO2/SiO2 Bragg mirror with two periods.

4 Twisted nematic liquid crystal

We consider a twisted nematic liquid crystal betweentwo glass substrates as represented on figure 12. Intwisted-nematic.py, the transmission coefficient T cal-culated with Berreman4x4 is compared with the Gooch-Tarry law. Agreement is excellent when the twisted layeris divided into 18 divisions. When only 7 divisions areused, there is a slight discrepancy in the high wavenum-ber range.

11

12

d

x

z

Glass GlassLC

P A

FIG. 12: Twisted nematic liquid crystal between glasssubstrates with polarizer and analyzer.

5 Cholesteric liquid crystal

We consider the cholesteric liquid crystal from figure 13,with right handedness.

p2

x

z

GlassGlass

FIG. 13: Cholesteric liquid crystal aligned along the xdirection on the two glass substrates. The pitch p corre-sponds to a 360° turn of the helix.

In validation-cholesteric.py, the reflection spec-trum of a right-handed circular polarization is calculatedwith Berreman4x4 and compared to the analytic expres-sions from references 20 and 21. Agreement is good andthe little differences may be due to the fact that the ana-lytic expression does not take into account the reflectionat the glass boundaries. In Berreman4x4, the half-pitchof the twisted material was divided into 25 parts, whichis more than enough for an accurate result.In cholesteric-example.py, transmission and reflec-

tion of a cholesteric structure is simulated for righthanded circularly polarized light. Transmission of unpo-larized light is also simulated, and analyzed with differentpolarizers. Eigenvalues and eigenvectors of the transmis-sion matrix for different cases are calculated.

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