KEYWORDS: Matrices, Binary data, 3D modeling, Algorithms, Monte Carlo methods, Systems modeling, Detection and tracking algorithms, Metamaterials, Current controlled current source, Refraction
This paper presents a comparison of one-dimensional optical localization effects for both a disordered quarter-wave
stack and disordered non-quarter-wave stack. Optical localization in these one-dimensional photonic bandgap structures
is studied using the transfer matrix formalism, where each matrix is a function of one or more random variables. As the
random matrix product model tends to infinity, Furstenberg's theorem on products of random matrices tells us that the
upper (and positive) Lyapunov exponent exists and is deterministic. This Lyapunov exponent is clearly identified as the
localization factor (inverse localization length) for the disordered photonic bandgap structure. The Lyapunov exponent
can be calculated via the Wolf algorithm which tracks the growth of a vector propagated by the long chain of random
matrices. Numerical results of the localization factor are provided using the Wolf algorithm. In the randomized models,
layer thicknesses are randomized, being drawn from both a uniform probability density function and a binary probability
mass function. Significant notches are noted for a number of the results. The Lyapunov exponent can also be found
from Furstenberg's integral formula, which involves integration with respect to the probability distribution of the
elements of the random matrices, and the so-called invariant probability measure of the direction of the vector
propagated by the long chain of random matrices. This invariant measure can be determined numerically from a bin
counting technique similar to the Wolf algorithm. Invariant measure plots based on the bin counting method are shown
at selected frequencies.
Access to the requested content is limited to institutions that have purchased or subscribe to SPIE eBooks.
You are receiving this notice because your organization may not have SPIE eBooks access.*
*Shibboleth/Open Athens users─please
sign in
to access your institution's subscriptions.
To obtain this item, you may purchase the complete book in print or electronic format on
SPIE.org.
INSTITUTIONAL Select your institution to access the SPIE Digital Library.
PERSONAL Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.