Electromagnetic Waves Through Disordered Systems: Comparison of Intensity, Transmission and Conductance
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ELECTROMAGNETIC WAVES THROUGH DISORDERED SYSTEMS: COMPARISON OF INTENSITY, TRANSMISSION AND CONDUCTANCE FREDY R ZYPMAN AND GABRIEL CWILICH YESHIVA UNIVERSITY, DEPARTMENT OF PHYSICS, NEW YORK, NY 10033 ABSTRACT We obtain the statistics of the intensity, transmission and conductance for scalar electromagnetic waves propagating through a disordered collection of scatterers. Our results show that the probability distribution for these quantities x, follow a universal µ
form, YU ( x) = x n e − x . This family of functions includes the Rayleigh distribution (when α=0, µ=1) and the Dirac delta function (α + ∞), which are the expressions for intensity and transmission in the diffusive regime neglecting correlations. Finally, we find simple analytical expressions for the nth moment of the distributions and for to the ratio of the moments of the intensity and transmission, which generalizes the n! result valid in the previous case.
INTRODUCTION Real materials for electronic applications are intrinsically disordered. They are disordered in the sense that the positions of all atoms in the system are neither perfectly regular, nor in practice, known in detail. At most, we can have statistical information about the atomic positions. It is thus natural to try to find characterization probes that extract statistical information from the real system under study. In the last few years it has been envisaged the use of statistical parameters to characterize the behavior of waves in disordered systems. Most experimental and theoretical studies have concentrated in electron waves in semiconductors, microwaves in photonic materials, and sound waves in imperfect acoustic structures (much advance has been achieved in the understanding of electronic behavior by studying its model counterpart, the electromagnetic field). Regardless of the application and the actual nature of the wave, when it crosses a disordered system, its intensity fluctuates randomly (Figure 1) and, for not very large −
I I
values, follows a Rayleigh probability distribution e . However, for large values of I, the distribution departs slightly from Rayleigh. If has been found that this departure can be used as a signature of local disorder in the sample, which can cause localization of the wave. This was first discovered by Anderson in the context of electron transport in semiconductors and has lately been extended to microwaves in photonic structures. In addition, knowledge of the statistical distribution of light inside a cavity provides information on the efficiency of generating light in a quantum well and transmitting it through an embedding dielectric. The propagation of waves in disordered systems has interested scientists since Lord Rayleigh studied the diffusion of light in the atmosphere to explain the color of the sky1, and led to the development of the theory of Radiative Transfer2. K5.8.1
Figure 1. A small sample of 30 configurations (out of a run of 64,000) of the variation of intensity through a system of randomly distributed spheres taken from a g
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