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-dened 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 articial long-rangeordered aperiodic objects have aroused an increasing interest in optical spectroscopy of solids.3-5

In this chapter, we rst dene the quasicrystals and present their structure factors. Then we consider light propagation in aperiodic photonic structures and pay particular attention

CONTENTS

2.6.1 Denition 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 specic 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 articial 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.