ABSTRACT
The concept of quasicrystal as a nonperiodic structure with perfect long-ranged order was brought in solid-state physics by Levine and Steinhardt.1 At present, it has become clear that, in addition to crystalline and amorphous materials, there exists a third form of solids that unexpectedly lls the gap between the two well-dened condensed-matter states. Moreover, this intermediate class called aperiodic deterministic structures includes the famous Fibonacci sequence ABAAB . . . and other quasicrystals that can be described by a projection onto the n-dimensional (nD) space with n = 1, 2, or 3 of an mD periodic lattice with dimensionality m > n. Examples of aperiodic structures different from quasicrystals are Thue-Morse and period-doubling sequences. Discovery of quasicrystals and other deterministic aperiodic structures initiated new elds of research in photonics. The studies of aperiodic long-range-ordered systems were extended to optics in the work by Kohmoto et al.,2 where a 1D quasicrystal constructed of dielectric layers forming the Fibonacci sequence was proposed. Since then, photonic quasicrystals and other articial long-rangeordered aperiodic objects have aroused an increasing interest in optical spectroscopy of solids.3-5
In this chapter, we rst dene the quasicrystals and present their structure factors. Then we consider light propagation in aperiodic photonic structures and pay particular attention
CONTENTS
2.6.1 Denition and Structure Factor .................................................................................... 132 2.6.1.1 One-Dimensional Quasicrystals .................................................................... 132 2.6.1.2 Fibonacci Structures ......................................................................................... 133 2.6.1.3 Structure Factor................................................................................................. 134
2.6.2 Two-Wave Approximation ............................................................................................. 137 2.6.3 Reƒection and Transmission from Fibonacci Structures ........................................... 138
2.6.3.1 Nonresonant Fibonacci Binary Chains ......................................................... 139 2.6.3.2 Fibonacci QW Structures ................................................................................. 140
2.6.4 Scaling Features and Localization of Excitonic Polaritons........................................ 143 2.6.5 Variety of Aperiodic Long-Range-Order Photonic Structures ................................. 146
2.6.5.1 Non-Fibonacci Aperiodic Sequences ............................................................. 146 2.6.5.2 Two-Dimensional Structures .......................................................................... 147 2.6.5.3 Three-Dimensional Structures ....................................................................... 148
2.6.6 Summary .......................................................................................................................... 148 References ..................................................................................................................................... 149
to application of the two-wave approximation (TWA). The latter allows one to interpret the optical spectra of aperiodic structures in terms of the periodic objects and underline the specic features arising as a result of the nonperiodicity. To illustrate, we analyze not only the binary Fibonacci optical superlattices built of two constituent layers A and B but also the recently proposed articial objects, namely, Fibonacci multiple quantum-well (QW) structures. An important point is that the regimes where TWA is invalid demonstrate the properties of optical spectra that are forbidden for periodic structures: (i) the localization characteristic for disordered systems and (ii) scaling and self-similarity that are absent in both conventional crystals and disordered materials. In the nal part of this chapter, we brieƒy enumerate other proposed and studied photonic aperiodic long-range-ordered photonic structures-1D, 2D, and 3D.