Page 24 - Winter 2010

Fig. 9. Illustration of a method for observing Anderson localization in two or three dimensions. Stacking plates machined in this pattern results in an array of coupled Helmholtz oscillators.
or three dimensions would be to begin with an array of high Q local oscillators with slightly different natural frequencies, and then weakly couple them together in a two- or three- dimensional pattern. With weak coupling, a periodic system would have narrow bands and mostly gaps; the introduction of sufficient disorder in the local oscillators would lead to Anderson localized eigenfunctions.
A method for accomplishing this with acoustical waves is illustrated in Fig. 9. A number of aluminum plates (one of which is shown in Fig. 9) are machined on one side with hemispherical depressions, together with interconnecting semi-cylindrical channels. Each plate is machined the same way on the opposite side, and holes are drilled through, con- necting the bottoms of the hemispheres. The plates are then stacked and sealed together, resulting in a three dimensional array of spherical cavities connected with cylindrical necks. Acoustically this would be an array of coupled Helmholtz resonators; narrow necks would correspond to weak cou- pling. The system as described would be periodic, having identical Helmholtz resonators, and could be tested for Bloch wave behavior. For a disordered system, “stuffing blocks” of various sizes could be placed inside the spherical cavities, shifting the local Helmholtz resonance frequencies and intro- ducing diagonal disorder.
There are large numbers of possible systems of two- or three-dimensional arrays of coupled mechanical or electro- magnetic oscillators that could be constructed. However, there is little point in actually using these classical systems for serious experimental research. The problem is that following the guidelines to readily observe Anderson localization leads to systems consisting of lumped elements connected with simple coupling, which can be treated exactly in linear theo- ry, and numerical computer calculations of eigenvalues and eigenfunctions could be carried out to high accuracy, much better than could be obtained experimentally. The funda- mental physics governing the wave nature of the classical experiments is well established, so that if any significant devi- ations from the calculated results were observed in an exper- iment, the only valid conclusion would be that the experi- ment must be cleaned up. Properly designed experiments
could be used for pedagogical purposes, student lab experi- ments or lecture demonstrations. Classical experiments could also be used to study systems where the theoretical cal- culations are not so tractable, as for media not consisting of local oscillators, for time-dependent potential fields or for the nonlinear wave equation, as discussed in Part 11 of this tutorial series.
In the literature the statement that it is difficult to observe classical Anderson localization is often encountered. In light of the discussion above it would seem that this state- ment is quite untrue, and in most cases it is incorrect. However, if this statement is made in regard to localization experiments which are motivated by the prospect of practical applications, such as optical waveguide multiplexers, etc., where constraints prevent the use of weakly coupled high Q local oscillators, then obtaining localization effects may be more of an experimental challenge. On the other hand, it is possible to follow the guidelines and be more clever in the design of practical devices utilizing the phenomena of disor- dered systems, as well as periodic systems.AT
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20 Acoustics Today, January 2010