Scaling properties of multilayer fractal structures

As regards light propagation in multilayer media in general, it should be noted that the majority of results here were obtained in the studies of the two opposite cases. On the one hand, the properties of perfectly periodic media (such as photonic crystals) have been generally described and understood, and those media are known to exhibit photonic band gaps and omnidirectional reflection, which offers a wide variety of applications [1-3]. On the other hand, there are numerous studies devoted to light localization and coherent backscattering in absolutely random multilayers (see the review [4] and references therein). However, there are structures that behave much like ‘disordered’ but are constructed according to a deterministic procedure. They are called non-periodic deterministic (NPD) media, and they constitute a separate field of research because they possess the properties of both periodic and random structures and also have some distinct features not found in traditional media. Not yet studied to a sufficient extent, NPD multilayer stacks lack a neat and generally accepted classification, but one can clearly distinguish several large groups in accordance with the algorithm used for the stack construction. There are, for instance, substitutional lattices, so called because they are ‘generated’ via a repeated substitution rule. These structures were first noticed and thus are best researched among NPD’s. In terms of the present paper it is vital that substitutional lattices possessing quasi-periodicity (Fibonacci multilayers) have self-similar optical transmission spectra, and the frequencies of resonance peaks form a fractal Cantor set [5]. At the same time, there are NPD multilayers that are fractal by themselves. They are called multilayer fractal structures because they are constructed in a similar way to well-known fractal sets. These structures are much less studied (see for example [6-9]), and are subject to investigation in this paper, which is organized as follows. Section 2 deals with the definition of fractal multilayers and certain mathematical conventions used later on. Section 3 describes spectral scalability, one of the two effects found that indicate a strong connection between geometrical and spectral features of multilayers. Discussion and analysis of the other such effect, sequential splitting, is what follows in Section 4. Finally, Section 5 summarizes the paper and outlines ways for future investigation.