1. (U) Uncertainty Quantification for an Isotropic Material in a Partially Filled
Waveguide
(U) Introduction
(U) During July, a new uncertainty estimation technique was written for the extraction of permittivity εr
and permeability μr from S parameter measurements of an isotropic material. This estimation is to be
performed from a partially filled rectangular waveguide with horizontal air gaps. The purpose of this
project is to transition Andrew Bogle’s Master of Science thesis to the LAB. As seen below in Figure 1, for
the case we are observing the sample must fully fill the waveguide in the vertical direction. Since the
material we are observing is isotropic only one εr and one μr are needed to extract S11 and S21.
(U) In the future we hope that this technique paves the way for future extractions with less strict
constraints. One day we hope to be able to extract S parameters from a material that does not span the
vertical or horizontal distance of the waveguide.
Figure 1: (U) From Andrew Bogle’s Master of Science thesis entitled “Electromagnetic Material
Characterization Using a Partially Filled Rectangular Waveguide”. The variables d, b, and L represent the
sample’s width, height, and length respectively. The size of the air gaps is irrelevant as long as the
sample is centered in the x direction.
(U) A user will input a frequency array, sample dimensions, waveguide aperture dimensions, number of
modes used in the mode matching technique, ε and μ along with biases and variances of S parameters.
(U) A MATLAB code (ProgramBogle based heavily off of Andrew Bogle’s thesis) was written which
determines the S parameters as a function of material parameters μ and ε and several auxiliary
parameters. This function is the basis for the uncertainty estimation. These S parameter estimates are
then used to calculate the S parameter mean values using the user input biases.
z=0 z=L
1 2S
x=a/2
x=-a/2
 ,d d 
Top Down view
x=d/2
x=-d/2
Cross-sectional view
x=-a/2 x=a/2
 ,d d 
x=d/2x=-d/2
y=b
y=0

2. (U) The extraction method is essentially an inversion of the function ProgramBogle using optimization. A
Newton’s Method is used to find the global minimum. After S parameters have been extracted the mean
S parameter values are calculated and used in the uncertainty analysis.
(U) The uncertainty analysis uses an iterated equal weight cubature rule to calculate the first and
seconds moments of the inverse process. These moments provide all that is needed to calculate the bias
and the variance of μ and ε. The user is able to specify how many subintervals the cubature uses to
approximate the uncertainty.
(U) Restrictions and current status
(U) The user must specify the dimensions of the sample, the dimensions of the waveguide aperture, bias
and variance values of S11 and S21 measurements, and the true values and guesses of μr and εr in order
to obtain an estimate of RMSE for μ and ε. This isotropic uncertainty analysis will then provide variance
and bias estimates of μ and ε separately.
(U) Currently we are working through several numerical bugs in the forward process (ProgramBogel) in
attempt to match the FORTRAN code provided by Andrew Bogel in his master’s thesis. In the future
after we fix these numerical errors the uncertainty analysis will be fully functional.