Page 18 - Winter 2010

Fig. 2. A band structure diagram providing eigenfrequencies as functions of Bloch wavenumber and band index (represented by the different symbols).
ever small) which may be transmitted past a scatterer will excite the next local oscillator into resonance, and that oscil- lator can grow to large amplitude. This is the case for all of the local oscillators, and hence the extended eigenfunction is possible.
4. Unlike the modulus, the eigenfunction itself (the real or imaginary part of a complex field) can exhibit interference phenomena, and have a non-trivial appearance, as illustrated in Fig. 1c.
5. Every eigenfunction has an associated eigenfrequency (or natural frequency). Each eigenfunction and eigenfre- quency can be labeled with two parameters: a “Bloch wavenumber” and a “band index.” The eigenfrequencies may be displayed graphically in a “band structure” diagram, as illustrated in Fig. 2. Discrete values along the horizontal axis are the Bloch wavenumbers, and the lines of connected sym- bols in Fig. 2 are the bands; counting the individual bands from the bottom upward gives the band index. The bare hor- izontal regions between the bands are called “gaps” or “stop bands.”
6. For a periodic system with N local oscillators, each band will have N discrete eigenfrequencies.
7. Waves may propagate at frequencies within a band, but not at frequencies which fall in a gap. If a wave at a gap fre- quency is incident on an interface between a uniform medi- um and a periodic array, then the wave cannot propagate into the array; however, the field does not stop abruptly, but can extend into the array as an evanescent wave8 decaying expo- nentially into the array region.
With the features of eigenfunctions in a periodic array reviewed in items 1 to 7 above, we can now consider what happens with a disordered array of scatterers. The model for a one-dimensional disordered system will be similar to the periodic system in Fig. 1a, but instead of the scatterers hav- ing a constant lattice spacing, a, the spacing between the scat- terers varies (i.e., the spacing is random), as illustrated in Fig. 3a. The system may still be thought of as consisting of cou-
pled local oscillators, but now each local oscillator is differ- ent. With reference to item 3 above, instead of having identi- cal local oscillators giving the same amplitude at each site, it would be expected that different local oscillators would give different amplitudes at each site. However, this is not at all what happens—what happens is that there is a maximum amplitude at one site, and moving away from that site, the amplitude of the eigenfunction exponentially decays, as illus- trated in Fig. 3b. The eigenfunctions for a disordered array of scatterers are “exponentially localized.”
The occurrence of exponential localization may seem quite unexpected—why does disorder in the local oscillators not simply duplicate Fig. 1b but vary the amplitude at differ- ent locations? The fact that the eigenfunctions become expo- nentially localized is the essence of Furstenberg’s theorem. It is a very significant result, and it was a crucial element in the 1977 Nobel Prize for Philip Anderson and Sir Neville Mott. The occurrence of exponentially localized eigenfunctions in one-, two- and three-dimensional disordered systems is now referred to as “Anderson localization.” This phenomenon and other advances in solving very difficult problems in long range phase coherence in disordered systems are very impor- tant results, and they have been the basis of landmark papers and numerous awards in condensed matter physics.
What is desired is a “balls-and-springs” explanation of how disorder leads to exponential localization of eigenfunc- tions. In violation of the experimentalist’s creed that “six months in the lab can save you a day in the library,” it is pos- sible to go to the library and search for papers on Anderson localization. There are very many such papers, but nearly all use Anderson localization as a way of explaining some phe- nomenon in a disordered system, but none provide the expla- nation for exponential localization. A look at Furstenberg’s paper shows that a researcher must be well versed in the pre- ceding developments in that field of mathematics to gain any insight. The consequence of Furstenberg’s theorem for waves in a one-dimensional disordered system can be stated sim- ply—proceeding to plus and minus infinity, eigenfunctions of a disordered system decay exponentially to zero with “probability one.” But understanding the manifestation of that statement and its probabilistic nature in a real experi- mental system of finite size is not easy—there is no “balls- and-springs” explanation.
While there is no simple physical explanation, it is possi-
Fig. 3. The model for a disordered array of scatterers and an exponentially localized eigenfunction. Unlike Fig. 1a, the scatterers in (a) do not have the same lattice spac- ing, so the local oscillators are different. The question is: how does the disorder lead to the exponential localization, as illustrated in (b)?
14 Acoustics Today, January 2010