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.