Page 22 - Winter 2010

Fig. 6. Qualitative picture of coherent backscatter. Time-reversal invariance results in each internal path having a mate with exactly the opposite phase shift. If the outgoing rays are in the opposite direction from the incident rays, then the phase shift Δφ is also reversed, and the backscattered rays always add constructively.
that the “incident” wave does not have to originate outside of the disordered system; Weaver25 has shown that the effect also occurs inside irregular reverberant rooms.
Experimental studies of Anderson localization
Before describing experiments which measure Anderson localization, it would be worthwhile to review some concepts which are important in the design of an experiment. The basic model involves coupled local oscillators—for the peri- odic system, the local oscillators and the coupling are all identical, and for a disordered system they are different. It is important that the local oscillators and the coupling mecha- nism have low damping (i.e., have a high quality factor, Q); this is necessary for the waves to have long-range phase coherence, so that interference effects are not degraded.
An important concept for the periodic system is that when beginning with uncoupled local oscillators that have discrete natural frequencies, and when the oscillators are coupled, the discrete frequencies broaden out into bands. The width of the bands (from the lowest frequency to the highest frequency in a band) scales with the strength of the coupling; weak coupling gives rise to narrow bands and wide gaps. There are modes at gap frequencies, but they are evanescent waves which decay exponentially. Suppose dis- order is introduced into a periodic system. Now eigenfre- quencies may appear in the regions which were formally gaps. Since the modes in the gaps already had exponential decay, expect that modes appearing in the gaps would have eigenfunctions which evolved from the evanescent waves and thus would be strongly localized. A more rigorous treatment by Thouless26 has shown that this is indeed the case. The conclusion is that it is easier to observe Anderson localization by beginning with narrow bands and wide gaps. Thus for an experiment on Anderson localization, start with weakly coupled local oscillators.
Another crucial undertaking when building an experi- ment to measure Anderson localization is to first build and test, using the same types of local oscillators and coupling, a periodic system. In this case Bloch’s theorem accurately pre- dicts what must be observed, and if the experimental meas- urements do not match the prediction, then the experiment must be fixed. Only when Bloch’s theorem is observed is it possible to confidently proceed to studies of Anderson local- ization by adding disorder to the system.
In adding disorder to a periodic system, there are two possibilities—put disorder in the local oscillators, or put dis- order in the coupling. The former method introduces what is called “diagonal disorder,” and the second is called “off-diag- onal disorder.” From the “balls-and springs” description of how extended eigenfunctions occur in a periodic system, it is expected that diagonal disorder would give stronger Anderson localization, and this is indeed the case. The final conclusion is that to experimentally observe Anderson local- ization, begin with high Q local oscillators with slightly dif- ferent natural frequencies, and couple them together weakly.
Acoustic Anderson localization in one dimension
An acoustic experiment which was used for studying waves in a periodic system, as well as a disordered system, was discussed in Part 1 of this series. The experimental sys- tem is just like the theoretical model system: a string (steel wire) stretched to some tension, with masses (lead shot) spaced along the string at intervals. Transverse waves are excited in the wire with an electromechanical shaker at one end of the wire. The wave field throughout the system is measured with an electromagnetic pickup which can travel the length of the steel wire, measuring the wave amplitude and phase as a function of position.
The actual system discussed in Part 1 was for lecture demonstrations—the one used for serious research was very similar but much longer, having about 50 masses on a wire about 15 m in length. If the masses were infinite in size, then the local oscillators (the sections of string between the mass- es) would be uncoupled; thus weak coupling corresponds to large masses. A measure of the size of the masses may be obtained by determining the reflection coefficient at a single mass; the experimentally measured coefficient of 0.9997 indicates that the masses very nearly, but not completely, iso- late the local oscillators.
At first the lead shot on the wire were arranged periodi- cally. Since the local oscillator is formed by the length of wire between the masses, calipers were used to insure that the wire lengths between the inside edges of the lead shot were as identical as possible. It should be noted that the size and mass of the individual lead shot were different, with variations of as much as 13%. However, this corresponds to off-diagonal disorder, which does not significantly disturb the otherwise periodic system of local oscillators.
The first step in the experiment was to measure the eigenfrequencies, at least in one band. This was done by plac- ing the pickup near the end of the wire away from the driver, then sweeping the frequency of the drive and monitoring the response of the system with the pickup. Sweeping the fre-
18 Acoustics Today, January 2010