Biaxial Ellipsometry

1
Biaxial Ellipsometry
Paula Smith, Russell Chipman University of
Arizona College of Optical Sciences

Introduction
Mueller Matrix Imaging Polarimeter
Results
The dielectric tensors of a biaxial substrate,
single-layer thin film, or multi-layer structure
can be characterized by measuring an
angle-of-incidence Mueller matrix with a Mueller
matrix imaging polarimeter (MMIP). The Mueller
matrix as a function of angle can be used to
determine the dielectric tensor (including its
orientation) similar to multi-angle ellipsometry.
Our MMIP uses a monochrometer source that cover
the range from 400 to 800nm. The polarization
state generator and analyzer have fixed
polarizers as well as retarders that rotate
through 64 positions. The images are captured at
each retarder position by a CCD camera. A pair
of microscope objectives (with low polarization
properties) are used to focus onto the sample and
collect reflected or transmitted light allowing
the system to measure as many as 10,000 incident
angles at once. Measurements taken without the
microscope objectives determine sample uniformity.
A Mueller matrix is calculated from the images
taken during the measurement and run through the
optimization program. The result is the
dielectric tensor that gives the best fit Mueller
Matrix to the Mueller Matrix of the sample. An
example of a measured angle-of-incidence Mueller
matrix is shown below.
Dielectric Tensor
The dielectric tensor consists of the real and
imaginary parts of the three principal indices
(nx kxi, ny kyi and nz kzi) as well as
three Euler angles (?????) to describe its
orientation.
Given a dielectric tensor and film thickness, a
Jones matrix can be calculated by solving
Maxwells equations at each surface. This gives
a Jones reflectivity matrix which can be
converted into Mueller matrix (dropping the
absolute phase).
Across a variety of materials the resulting best
fit Mueller matrix agree well with the measured
data.
The samples can be measured at various angles in
reflection or in transmission by swinging the
arm.
Conclusions
Biaxial ellipsometry determines the dielectric
tensor of samples. To accurately determine
dielectric tensors for multilayer samples, the
angle-of-incidence Mueller matrix images should
be measured for multiple wavelengths in
transmission or reflection incorporating
dispersion models.
An optimization algorithm finds the best fit
dielectric tensor to the measured
angle-of-incidence Mueller matrix image.