This document presents an approach for modeling the propagation of light through stratified anisotropic media using Mueller matrices. It describes how to: 1) Set up the dielectric tensor in the laboratory frame using Euler rotations to relate the material and laboratory coordinate systems. 2) Calculate the propagation matrix Δ for each layer from the dielectric tensor and wavevector. 3) Determine the transition matrices at each interface by matching boundary conditions. 4) Obtain the Jones matrix for each layer and determine the overall Mueller matrix by matrix multiplication of the individual layer matrices.

1. Mueller matrix approach to the propagation of light in
stratified anisotropic media
F. Ferrieu*,1
, E. Garcia-Caurel**,2
, M. Stchakovsky3
1
STMicroelectronics, 850, rue Jean Monnet F-38926, Crolles Cedex France
2
LPICM, Ecole polytechnique, CNRS, 91128, Palaiseau, France
3
Division Couches Minces, Horiba Jobin Yvon SAS, 91380, Chilly-Mazarin,
France
* frederic.ferrieu@cea.fr, Phone +33 (0)4 38 78 40 56,Fax +33 (0)438785273
*enric.garcia-caurel@polytechnique.edu, Phone+33(0)169333134,Fax +33(0)19333006
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G. E. Jellison Thin Solid Films313-314,33-39(1998).and 450,42-50(2004).
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R. Barakat, Optics Com. 38, 3,159-161 (1981)
Current research in the propagation of light in anisotropic material is driven by the growing
interest of new synthesized materials for Nanotechnologies. For a large number of anisotropic
substrate, as well as in the case of thin nano films such as thin layers of ion beam deposited
materials, weak optical properties can be detected by new sensitive generalized ellipsometry and
polarimetry instruments The experimentalist has then to deal with a large number of acquired
data such as with these 4x4 Mueller matrix Mij The aim is here to revise the existing formalism
already detailed in several publications and give an overall view in order to build a general
algorithm in the Mij ( the Mueller matrices) through the Jones rss,rsp,rps,rpp matrix elements. The
explicit equations given hereafter can be applied to both transmitted and reflected amplitude
electromagnetic wave of light propagating in anisotropic multilayer media deposited on an
anisotropic substrate which can be implemented with the use of the most recent linear algebra
libraries commercially available.
Target
1.3kW, 3Torr H2C=CH-CH3/He
Following the curl operator in Maxwell equations for E and H the ∆ matrix depends on
the dielectric tensor and the component kx of the wave propagation vector k0,
The propagation of light through a general linear, and non-magnetic anisotropic media can
described using the Maxwell equations written in their wave form:
BJEcurl

ω=−
DjHcurl

ω=
)(
)(
zj
z
z
∆Ψ−=
∂
Ψ∂
ω
ck /0 ω=
T
yxyx HHEE ],,,[=Ψ










=
333231
232221
131211
}{
εεε
εεε
εεε
εij
),,(),,(}{ 1
θψϕεθψϕε −
= AAij
The exponential Matrice IS OBTAINED by Decomposition of the matrix
following Wöhler and Cayley Hamilton
Laboratory frame
Intrinsic media orientation
f
n
i
ipia LdTLT )(
1
1
∏=
−
−=
In general the coordinate system of the laboratory and the main coordinate system of
the anisotropic media are not the same they can be related by a rotation transformation.
A common way of writing a rotation transformation is to use the Euler matrices and the
associated Euler angles (θ,Ψ,Φ).
Multilayer case ( thickness di)
Transitions matrices respectively
between air and film and film with
substrate fa LL ,
•Build the ∆ Matrix :
For each layer
The value of the eigen-values βi can
be also found using commercially
available software solving the
corresponding linear equation
Tdkiq
Me i
},,,{ 3210
0
ββββ=
•Find the Transition matrices by matching the E,H components at each interface
Transition matrices air/film
. The continuity of the electric and magnetic components at the interface impose: the
relations between, the electric, E, and magnetic, H, field , . The projection of the fields on
both, s and p components yield:
.
)0()0( =Ψ+=Ψ=Ψ zzL refincaa
],cos,,cos[ paasasapinc AnAnAA Φ−Φ=Ψ
],cos,,cos[ paasasapref BnBnBB ΦΦ−=Ψ{
Transition matrices film/anisotropic substrate
Φa AOI (Angle of Incidence)
na=ambient index na=1
Ψf is a linear combination of the eigenvectors Ξ of ∆
Because in a semi-infinite substrate there are not light
that comes back, the eigenvectors associated with
back-travelling wave can not be used to build the
linear combinations Only two vectors have to be
selected with:
For a biaxial system eignevectors are easily obtained
Such as with axis aligned. To the laboratory frame one
gets thus for the ci:
Mueller matrices determination
Here the Tij elements correspond to the general transfer matrix= matrix product of each
layer thickness di : From P. Yeh , one obtain each of the four elements of the
corresponding Jones matrix and through he Kronecker product with matrices A and A
-1
Relation with the
Mueller Matrices
1−=j
aax nk Φ= sin
Solving the Problem in an anisotropic multilayer media
means :
:
•The dielectric tensor set up in the laboratory frame: Euler rotation
idik
e ∆0
fa LL ,
aL
fL
Müller Matrix
Calculate