Monte Carlo simulations have long been used to study Anderson localization in models of one-dimensional random stacks. Because such simulations use substantial computational resources and because the randomness of random number generators for such simulations has been called into question, a non-Monte Carlo approach is of interest. This paper uses a non-Monte Carlo methodology, limited to discrete random variables, to determine the Lyapunov exponent, or its reciprocal, known as the localization length, for a one-dimensional random stack model, proposed by Asatryan, et al., consisting of various combinations of negative, imaginary and positive index materials that include the effects of dispersion and absorption, as well as off-axis incidence and polarization effects. Dielectric permittivity and magnetic permeability are the two variables randomized in the models. In the paper, Furstenberg’s integral formula is used to calculate the Lyapunov exponent of an infinite product of random matrices modeling the one-dimensional stack. The integral formula requires integration with respect to the probability distribution of the randomized layer parameters, as well as integration with respect to the so-called invariant probability measure of the direction of the vector propagated by the long chain of random matrices. The non-Monte Carlo approach uses a numerical procedure of Froyland and Aihara which calculates the invariant measure as the left eigenvector of a certain sparse row-stochastic matrix, thus avoiding the use of any random number generator. The results show excellent agreement with the Monte Carlo generated simulations which make use of continuous random variables, while frequently providing reductions in computation time.
KEYWORDS: Refraction, Matrices, Polarization, Monte Carlo methods, Metamaterials, Computer simulations, MATLAB, Systems modeling, Numerical simulations, Current controlled current source
We consider polarization and off-axis incidence effects for one-dimensional random stacks consisting of alternating and
non-alternating layers of positive and negative index materials, with index of refraction and thickness discretely
disordered. Such long randomly disordered systems exhibit Anderson localization, whose effects can be studied via the
Lyapunov exponent of the product of independent identically distributed random transfer matrices modeling the stack.
We use Furstenberg’s integral formula to calculate Lyapunov exponents for s and p polarizations, and for a range of
angles of incidence for these random matrix models. Furstenberg’s integral formula requires integration with respect to
the probability distribution of the randomized layer parameters, and integration with respect to the so-called invariant
probability measure of the direction of the vector propagated by the long chain of random matrices. This invariant
measure can rarely be calculated analytically, so some numerical technique must be used to produce the invariant
measure for a given random matrix product model. Here we use the algorithm of Froyland-Aihara, especially suited for
discretely disordered parameters, to calculate the invariant measure. This algorithm produces the invariant measure from
the left eigenvector of a certain sparse row-stochastic matrix. This sparse matrix represents the probabilities that a vector
in one of a number of discrete directions will be transferred to another discrete direction via the random transfer matrix.
The Froyland-Aihara algorithm thus provides a non-Monte Carlo method to calculate localization effects, with potential
reduction in computation time compared to traditional layer or vector iteration methods.
KEYWORDS: Refraction, Matrices, Monte Carlo methods, Numerical integration, Metamaterials, Computer simulations, MATLAB, Matrix multiplication, Time metrology, Systems modeling
We consider one-dimensional photonic bandgap structures with negative index of refraction materials modeled in every
layer, or in every other layer. When the index of refraction is randomized, and the number of layers becomes large, the
light waves undergo Anderson localization, resulting in confinement of the transmitted energy. Such a photonic
bandgap structure can be modeled by a long product of random transfer matrices, from which the (upper) Lyapunov
exponent can be calculated to characterize the localization effect. Furstenberg’s theorem gives a precise formula to
calculate the Lyapunov exponent when the random matrices, under general conditions, are independent and identically
distributed. Specifically, Furstenberg’s integral formula can be used to calculate the Lyapunov exponent via integration
with respect to the probability measure of the random matrices, and with respect to the so-called invariant probability
measure of the direction of the vector propagated by the long chain of random matrices. It is this latter invariant
probability measure, so fundamental to Furstenberg’s theorem, which is generally impossible to determine analytically.
Here we use a bin counting technique with Monte Carlo chosen random parameters from a continuous distribution to
numerically estimate the invariant measure and then calculate Lyapunov exponents from Furstenberg’s integral formula.
This result, one of the first times an invariant measure has been calculated for a continuously disordered structure made
of alternating layers of positive and negative index materials, is compared to results for all negative index or equivalently
all positive index structures.
KEYWORDS: Refraction, Matrices, Monte Carlo methods, Computer simulations, Systems modeling, Numerical integration, Binary data, Statistical analysis, Metamaterials, Current controlled current source
For one-dimensional photonic bandgap structures consisting of alternating layers of positive and negative index
materials, Anderson localization effects will appear when one or more parameters is disordered. Such long randomly
disordered systems can be modeled via a long chain of independent identically distributed random matrices. The
Lyapunov exponent of such a random matrix product characterizes the energy confinement due to Anderson localization.
Furstenberg’s integral formula gives, at least theoretically, the Lyapunov exponent precisely. Furstenberg’s integral
formula requires integration with respect to the probability distribution of the randomized layer parameters, and
integration with respect to the so-called invariant probability measure of the direction of the vector propagated by the
long chain of random matrices. This invariant measure can rarely be calculated analytically, so some numerical
technique must be used to produce the invariant measure for a given random matrix product model. Here we use the
Froyland-Aihara method to find the invariant measure. This method estimates the invariant measure from the left
eigenvector of a certain sparse row-stochastic matrix. This sparse matrix represents the probabilities that a vector in one
of a number of discrete directions will be transferred to another discrete direction via the random transfer matrix. This
paper, possibly for the first time, presents the numerically calculated invariant measure for a discretely disordered one-dimensional
photonic bandgap structure which includes negative index material in alternating layers. Results are
compared with the structure containing all positive index layers, as well as with the counterpart structure in which
random variables are drawn from a uniform probability density function.
Existing in the "gray area" between perfectly periodic and purely randomized photonic bandgap structures are the socalled
aperoidic structures whose layers are chosen according to some deterministic rule. We consider here a onedimensional
photonic bandgap structure, a quarter-wave stack, with the layer thickness of one of the bilayers subject to
being either thin or thick according to five deterministic sequence rules and binary random selection. To produce these
aperiodic structures we examine the following sequences: Fibonacci, Thue-Morse, Period doubling, Rudin-Shapiro, as
well as the triadic Cantor sequence. We model these structures numerically with a long chain (approximately 5,000,000)
of transfer matrices, and then use the reliable algorithm of Wolf to calculate the (upper) Lyapunov exponent for the long
product of matrices. The Lyapunov exponent is the statistically well-behaved variable used to characterize the Anderson
localization effect (exponential confinement) when the layers are randomized, so its calculation allows us to more
precisely compare the purely randomized structure with its aperiodic counterparts. It is found that the aperiodic photonic
systems show much fine structure in their Lyapunov exponents as a function of frequency, and, in a number of cases, the
exponents are quite obviously fractal.
KEYWORDS: Matrices, Monte Carlo methods, Binary data, Photon transport, Numerical simulations, Metamaterials, Current controlled current source, Refraction, Polarization, Numerical analysis
A long product of random transfer matrices is frequently used to model disordered one-dimensional photonic bandgap
structures in order to investigate optical Anderson localization. The Lyapunov exponent of this long matrix product,
known to exist from Furstenberg's theorem, is identified as the localization factor (inverse localization length). It is not
unusual to have 5,000,000 random matrices with Monte Carlo chosen elements in one product to calculate a single
Lyapunov exponent, and then have results averaged over as many as 10,000 ensembles. The entire process has to be
repeated for 100 or more frequencies to clearly show the frequency dependence of the optical localization effects. This
paper instead uses a non-Monte Carlo numerical technique to calculate the Lyapunov exponents. This technique, by
Froyland and Aihara, is especially suited to the case where the disorder in the photonic bandgap structure is discrete.
Namely, it is used to calculate the probability distribution of the direction of the vector propagated by the long chain of
random matrices by finding the left eigenvector of a certain sparse row-stochastic matrix. This distribution is used in
Furstenberg's integral formula to calculate the Lyapunov exponent. Now this technique is extended to the case where the
random elements of the photonic bandgap transfer matrices are intended to be chosen from a continuous distribution.
Specifically, discrete probability mass functions whose moments increasingly match those of a uniform probability
density function are used with the Froyland-Aihara method. A very significant savings in computation time is noted
compared to Monte Carlo approaches.
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.
In the one-dimensional optical analog to Anderson localization, a periodically layered medium has one or more
parameters randomly disordered. Such a medium can be modeled by an infinite product of 2x2 random transfer matrices
with the upper Lyapunov exponent of the matrix product identified as the localization factor (inverse localization
length). Furstenberg's integral formula for the Lyapunov exponent requires integration with respect to both the
probability measure of the random matrices and the invariant probability measure of the direction of the vector
propagated by the random matrix product. This invariant measure is difficult to find analytically, so one of several
numerical techniques must be used in its calculation. Here, we focus on one of those techniques, Ulam's method, which
sets up a sparse matrix of the probabilities that an entire interval of possible directions will be transferred to some other
interval of directions. The left eigenvector of this sparse matrix forms the estimated invariant measure. While Ulam's
method is shown to produce results as accurate as others, it suffers from long computation times. The Ulam method,
along with other approaches, is demonstrated on a random Fibonacci sequence having a known answer, and on a quarter-wave
stack model with discrete disorder in layer thickness.
KEYWORDS: Matrices, Monte Carlo methods, Binary data, Structural dynamics, Dynamical systems, Systems modeling, Signal attenuation, Analog electronics, Numerical integration, Phase measurement
In the one-dimensional classical analogs to Anderson localization, whether optical, acoustical or structural dynamic, the
periodic system has its periodicity disrupted by having one or more of its parameters randomly disordered. Such
randomized systems can be modeled via an infinite product of random transfer matrices. In the case where the transfer
matrices are 2x2, the upper (and positive) Lyapunov exponent of the random matrix product is identified as the
localization factor (inverse localization length) for the disordered one-dimensional model. It is this localization factor
which governs the confinement of energy transmission along the disordered system, and for which the localization
phenomenon has been of interest.
The theorem of Furstenberg for infinite products of random matrices allows us to calculate this upper Lyapunov
exponent. In Furstenberg's master formula we integrate with respect to the probability measure of the random matrices,
but also with respect to the invariant probability measure of the direction of the vector propagated by the long chain of
random matrices. This invariant measure is difficult to find analytically, and, as a result, either an approximating
assumption is frequently made, or, less frequently, the invariant measure is determined numerically.
Here we calculate the invariant measure numerically using a Monte Carlo bin counting technique and then numerically
integrate Furstenberg's formula to arrive at the localization factor for both continuous and discrete disorder of the mass.
This result is cross checked with the (modified) Wolf algorithm.
In the one-dimensional optical analog to Anderson localization, a periodically layered medium has one or more
parameters randomly disordered. Such a randomized system can be modeled by an infinite product of 2x2 random
transfer matrices with the upper Lyapunov exponent of the matrix product identified as the localization factor (inverse
localization length) for the model. The theorem of Furstenberg allows us, at least theoretically, to calculate this upper
Lyapunov exponent. In Furstenberg's formula we not only integrate with respect to the probability measure of the
random matrices, but also with respect to the invariant probability measure of the direction of the vector propagated by
the random matrices. This invariant measure is difficult to find analytically, and, as a result, the most successful
approach is to determine the invariant measure numerically. A Monte Carlo simulation which uses accumulated bin
counts to track the direction of the propagated vector through a long chain of random matrices does a good job of
estimating the invariant probability measure, but with a level of uncertainty. A potentially more accurate numerical
technique by Froyland and Aihara obtains the invariant measure as a left eigenvector of a large sparse matrix containing
probability values determined by the action of the random matrices on input vectors. We first apply these two
techniques to a random Fibonacci sequence whose Lyapunov exponent was determined by Viswanath. We then
demonstrate these techniques on a quarter-wave stack model with binary discrete disorder in layer thickness, and
compare results to the continuously disordered counterpart.
Optical localization in one-dimensional very long disordered photonic bandgap structures is characterized by the upper
Lyapunov exponent of the infinite random matrix product model. This upper (and positive) Lyapunov exponent, or
localization factor (inverse localization length), is, at least theoretically, calculable from Furstenberg's formula. This
formula not only requires the probability density function for the random variables of the random transfer matrix, but
also requires the invariant probability measure of the direction of the vector propagated by the long chain of random
matrices. This invariant measure is generally only calculable numerically, and when the transfer matrices are in the
usual real basis, or even a plane wave basis, the invariant measure takes no consistent form. However, by using a
similarity transformation which moves the fixed points of the bilinear transformation corresponding to the long random
matrix product, we can put the random matrix product in the so-called real hyperbolic-canonical basis. The aim of this
change of basis is to place the attracting fixed point at the origin and the repelling fixed point at infinity. The invariant
measure, as a result, is dramatically simplified, approaching a probability mass function with mass concentrated around
the angle zero.
The transfer matrix formalism has proven to be a powerful tool for analyzing one-dimensional photonic bandgap
structures, whether their multilayers are perfectly periodic or randomized in some fashion. In the randomized structure,
as the number of layers tends to infinity, Furstenberg's formula can be used, at least theoretically, to find the
deterministic Lyapunov exponent (localization factor, sometimes called the inverse localization length) governing the
confinement of energy transmission in the model. The challenge in using Furstenberg's formula is that it requires the
calculation of the invariant probability measure of the direction of the vector propagated by the chain of random
matrices. This invariant measure is usually impossible to find analytically, and so one must resort to numerical
simulation or some other approximating assumption. To aid in the numerical determination of this invariant probability
measure, we consider matrix similarity transformations based on the average plane wave transfer matrix at a given
frequency. These transformations, like the original transfer matrix, are elements of SU(1,1), the special pseudo-unitary
group, and are obtained by moving the fixed points of the bilinear (or Mobius) transformation of the original transfer
matrix to the corresponding fixed points of the canonical forms known from the Iwasawa decomposition. Amazingly, in
some situations, including a quarter-wave stack, such a transformation can cause the invariant probability measure to
become a nearly uniform probability density function, making the Furstenberg formula more readily useable.
Optical localization in a randomly disordered infinite length one-dimensional photonic band gap structure is studied
using the transfer matrix formalism. Asymptotically, the infinite product of random matrices acting on a nonrandom
input vector induces an invariant probability measure on the direction of the propagated vector. This invariant measure
is numerically calculated for use in Furstenberg's master formula giving the upper Lyapunov exponent (localization
factor) of the infinite random matrix product. A quarter-wave stack model with one of the bilayer thicknesses disordered
is used for simulation purposes. In this plane wave model the invariant measure is rarely a uniform probability density
function, as is sometimes assumed in the literature. Yet, the assumption of a uniform probability density function for the
invariant measure gives surprisingly good results for a highly disordered system in the UV region.
A conceptual attitude control subsystem design for the Pluto Fast Flyby spacecraft is described. Mass, cost, schedule and performance, approximately in that order, drove the mission, spacecraft, as well as the attitude control subsystem design. The paper discusses the key mission requirements impacting the attitude control subsystem design, as well as the important subsystem trades. The spacecraft is a three axis stabilized vehicle using cold gas jets for attitude control and hydrazine thrusters for trajectory correction maneuvers. Attitude determination relies heavily on a low mass star tracker capable of determining attitude by pointing anywhere in the celestial sphere. Tracking of planetary features with the star tracker may also be desirable. A small inertial reference unit and a sun sensor will accompany the tracker to complete the suite of components for attitude determination.
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