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tioned. There have since been large scale computer simula- tions which reveal that, while the predictions of the scaling argument may be correct, the phenomenon of percolation
14
niques to examine how the conductivity of a finite-sized sam- ple of the material varies as the sample size is systematically scaled up toward infinite size. A key assumption of the argu- ment is that as the size of the sample becomes infinite, inelas- tic scattering due to finite temperatures destroys the wave nature of the electrons so that the diffusion approximation becomes valid. However, the fundamental problem of waves in a static disordered system, where there is absolutely no inelastic scattering or attenuation of the waves, even at boundaries, remains a challenging problem. In any case, the scaling argument is important for systems which have some inelastic scattering. One of the most important results is that as the amount of disorder in a three-dimensional system is increased, there is a transition from a situation where at least some eigenfunctions are extended, to a situation where all the eigenfunctions are localized; this is called the Anderson tran- sition. Two dimensions is referred to as the critical dimen- sion, where the theory is particularly difficult; in a two- dimensional system with some disorder, it is believed that all eigenfunctions are localized (as for one dimension), so that there is no Anderson transition.
Coherent backscatter
When a plane wave in a uniform medium is incident upon the surface of a disordered system, then one might expect uniform diffuse reflection. But again the disordered system presents a surprising result. In this case it is found that the waves experience enhanced reflection in a direction opposite to the incident direction, and this is called “coherent backscatter.”16,17
Coherent backscatter was first noted in regard to the
brightness of the lunar surface at full moon, when the
observed reflection is nearly opposite the illumination. Lunar
as well as laboratory observations have been made since
1922, and the effect was theoretically explained as due to
reflection from a disordered medium by Hapke18 in 1963.
Experiments designed to specifically test Hapke’s theory were
made by Oetking19 in 1966. More extensive measurements
and a re-working of the theory were published by Ishimaru,
dered system in high dimensions.
For both coherent backscatter and Anderson localiza-
tion, quantitative theory is very difficult. However, coherent backscatter may be understood qualitatively with a simple picture, as shown in Fig. 6. Here a plane wave is incident from a uniform medium onto the surface of a disordered system at some angle from the normal. A ray picture is adopted, and two parts of the plane wave are represented by two rays, qin1 and qin2. Because of the obliqueness of the incidence, the two rays arrive at the surface of the disordered medium with a phase difference Δφ. The consequence of the scattering inside the disordered medium may be found by adding all possible internal paths for the rays. The important feature to note is that every path for one ray may be associated with a time reversed path for the other ray, as illustrated in one instance in Fig. 6. Thus outgoing rays qout1 and qout2 will have undergone the same phase change inside the disordered medium. If the outgoing rays exit at an angle other than the reverse of the incident direction, then the original phase shift Δφ will not be undone, and if results are averaged, the ran- dom phase shift will result in incoherent interference of waves. However, if the outgoing rays are in the backward direction, then the original phase shift Δφ will also be reversed and all waves will interfere constructively. Thus the averaged intensity in the backward direction will be enhanced relative to other scattering directions.
One very interesting feature of coherent backscatter is
may also play a role.
The scaling argument uses renormalization group15 tech-
20,21
Coherent backscatter was rediscovered by a number of
22–24
et al. in 1984.
This time the theoreticians involved were also engaged in studies of Anderson localiza- tion, and the effect was referred to as “weak localization” and was made very popular. The term “weak localization” is mis- leading because the effect occurs regardless of whether or not eigenfunctions are localized. As will be seen below, coherent backscatter is a direct consequence of time-reversal invari- ance, whereas, as Furstenberg’s theorem illustrates even for one-dimension, localization is a much more complicated phenomenon. “Weak localization” is a term for a perturba- tion approach which exists only because we, unlike Nature, are unable to exactly solve the problem of waves in a disor-
researchers around 1985.
Wave Propagation in Arrays 17