This book is the result of our interest in understanding, mastering, and engineering randomness in photonic systems. It is a natural consequence of what we did in the past. In the late 1980s, while Lorenzo Pavesi was working on semiconductor superlattices he noticed that for some energies the vertical transport through the superlattice minibands was inhibited due to disorder (L. Pavesi et al. 1989. Phys. Rev. B, 39, 7788). Then, working on the recombination dynamics of excitons in porous silicon, he further noticed that the random arrangement of silicon quantum dots has a strong influence on the recombination dynamics of excitons(L. Pavesi et al. 1993. Phys. Rev. B, 48, 17625). After Mher Ghulinyan came to Trento in 2002, we developed the techniques to fabricate free-standing porous silicon dielectric multilayers of any stacking sequence (M. Ghulinyan et al. 2003. J. Appl. Phys., 93, 9724). This was the first time that we had the chance to designat will one-dimensional periodic, aperiodic, or random photonic systems. A fascinating new physics opened up for us: that of the analogy of photon propagationin complex dielectric systems with carrier transport in random electronic systems. Our latest results in the field are associated with sequences of ring resonators where randomness causes the formation of resonant coupling between different rings with the possibility of yielding the optical analog of the electromagnetic induced transparency (M. Mancinelli et al. 2011. Opt. Express, 19, 13664), or chaotic photon propagation.

Over all these years, we have had the chance to interact with many researchers active in the field of periodic, quasiperiodic, and random photonic systems. From these interactions the idea of this book was born. We have therefore collected together a series of self-contained chapters to cover the whole field with the specific aim of introducing the different aspects, showing the current status of the research, and envisaging future directions. All invited authors have responded to this challenge with great enthusiasm and professionalism.

The book opens with Chapter 1 by W. L. Vos, A. Lagendijk, and A. P. Mosk, which introduces the field and covers the timeline between the early studies on these systems and the very latest achievements in the field.

Introduction

Optical waveguides are used to transport both phase-insensitive and phase-sensitive information in today's communications networks and signal processing devices. Among applications needing high performance waveguides are delay lines and optical phase-locked loops, generating precise phase offsets for waveform sampling, analog-to-digital conversion, and beam-forming. In fundamental terms, high performance waveguides are based on the ballistic propagation of photons [4, 168], which may be capable of reaching the ultimate limits of computational speed and energy efficiency of interconnects [331, 332], since ballistic propagation of electrons [132] is difficult to achieve in practical chips. Therefore, quantifying the impact of disorder on light propagation, and distinguishing between ballistic and non-ballistic transport, is an important and ongoing research endeavor.

Ballistic transport requires that the phase-breaking length be greater than the device length; which is usually satisfied in optical fibers of short length, and in planar glass lightwave circuits, in which the refractive index difference between core and cladding materials is small [260, 261]. But light propagation in high-index contrast waveguides, e.g. in silicon photonics, can be highly susceptible to the effects of disorder. Here, we will focus on periodically structured nanophotonic waveguides, e.g. coupled microring resonators [188, 392, 548, 555] in which light can accumulate significantly more phase and delay per unit length than in conventional waveguides. Such devices can be valuable for many reasons, among which are the facts that on-chip footprint is a precious commodity in real-world devices, and that our ability to increase, via geometrical patterning of waveguiding structures, the optical phase accumulation per unit (structural) length will considerably benefit nonlinear multiwave interactions. However, such devices may also be more strongly susceptible to the effects of disorder [329], and the issue of whether ballistic propagation can be achieved in coupled-resonator structures is being investigated closely.

From a physics viewpoint, one-dimensional structures are interesting because multiple scattering in transverse dimensions can be restricted, e.g. by using single-transverse-mode waveguides as the constituent building blocks for linear chains of resonators, e.g. microring arrays.

General overview

In many areas in the physical sciences, the propagation of waves in complex mediaplays a central role. Acoustics, applied mathematics, elastics, environmental sciences, mechanics, marine sciences, medical sciences, microwaves, and seismology are just a few examples, and of course nanophotonics [6, 429, 448, 449]. Here, complex media are understood to exhibit a strongly inhomogeneous spatial structure, which determines to a large extent their linear and nonlinear properties. Examples of complex media are random, heterogeneous, porous, and fractal media. The challenges that researchers in these areas must surmount to describe, understand, and ultimately predict wave propagation are formidable. Right from the start of this field – in the early 1900s – it was clear that new concepts and major approximations had to be introduced. Famous examples of such concepts are the effective medium theory [55, 60, 321] and the radiative transport theory [81, 491]. These concepts are very much alive, even today. The fundamental challenge with these approximate concepts is that often the length scales of the inhomogeneities in the complex medium are comparable with the wavelength, whereas the range of validity of these approximations is restricted to situations where these length scales are much larger than the wavelength. Consequently, many relevant situations arise where either the effective medium theory, or radiative transport theory, or both, fail dramatically. Examples of such situations are given in this introduction and throughout this book.

The complexity of the medium that supports wave propagation can be classified in a number of ways. Many different types of spatial inhomogeneities in a host matrix can be envisioned, varying from completely random, via aperiodic and waveguide structures, to quasi-crystalline and even structures with long-range periodic order. The in homogeneities could be of a self-organized form, or the fruit of precise engineering. An additional classification is whether or not the complexity is confined to the surface (in two dimensions (2D)) or is present throughout the volume of the medium (in three dimensions (3D)), as in porous media.

Introduction

Suppressed transport and enhanced fluctuations of conductance and transmission are prominent features of random mesoscopic systems in which the wave is temporally coherent within the sample [6, 9, 501, 525]. The associated breakdowns of particle diffusion and of self-averaging of flux were first considered in the context of electronic conduction and for many years thought to be exclusively quantum phenomena [1, 2, 9, 11, 12, 113, 114, 476, 477, 525]. Independently, however, wave localization was demonstrated theoretically for a statistically inhomogeneous waveguide [156]. Over time, it became increasingly apparent that localization and mesoscopic fluctuations reflected general wave properties and might therefore be observed for classical waves as well [7, 22, 54, 78, 106, 130, 144, 145, 154, 156, 175, 221, 223, 225, 241, 263, 326, 353, 419, 462, 466, 487, 488, 501, 531, 535, 543]. In particular, the level and transmission eigenchannel descriptions proposed, respectively, by Thouless [476, 477] and Dorokhov [113, 114] to describe the scaling of conductance in electronic wires at zero temperature, are essentially wave descriptions involving the character of quasi-normal modes of excitation with in the sample and speckle patterns of the incident and transmitted field. Quasi-normal modes, which we will refer to as modes, are resonances of an open system. These modes decay at a constant rate due to the combined effects of leakage from the sample and dissipative processes. Eigenchannels of the transmission matrix, are obtained by finding the singular values of the field transmission matrix and they represent linked field speckle patterns at the input and output of the sample surfaces. Eigenchannels are linear combinations of phase coherent channels impinging upon and emerging from the sample. Examples of such channels may be propagating transverse modes of an empty waveguide, or transverse momentum states in the leads attached to a resistor. In measurements, input and output channels are often combinations of source and detectors at positions on the incident and output planes, respectively. Whereas modes are biorthogonal field speckle patterns over the volume of the sample [87, 293], eigenchannels are biorthogonal field speckle patterns at the input and output planes of the sample.

Introduction

Self-organization is one of the most extraordinary tools in Nature which assembles its elementary building blocks, atoms and molecules, in the form of solid substances. Crystals, in this sense, represent the class of a vast number of solid materials in which atoms (molecules) are brought together in a perfect, spatially periodic manner. A macroscopic crystal, therefore, is made by repeating its smallest microscopic period – the crystal unit cell – infinitely in space (Fig. 5.1(a)). Therefore, when translating the crystal by an integer number of unit cell size, the crystal coincides with itself. This property is in the basis of modern solid-state physics and crystallography and is known as the discrete translational symmetry. Crystals are thus ordered materials in which atoms show both short- (within a couple of nearest neighbors) and long-range order. Because of the periodic lattice potential, the energy states within a crystal are extended, and the electrons can diffuse freely through it.

At the opposite extreme of crystalline order there are the disordered or amorphous solid materials, in which the short-range order may still exist, while the long-range order is completely missing (Fig. 5.1(c)). In these kind of solids, e.g. in heavily doped semiconductors or glasses, periodicity is lacking and the atomic potential varies in a random manner. In certain cases, when the degree of randomness is large enough, electron wavefunctions localize exponentially near individual atomic sites and the diffusion of charge carriers vanishes (Anderson localization) [11].

With the advances in X-ray diffraction tools at the beginning of the twentieth century, scientists learned to image the signatures of long-range order of crystals by mapping the symmetries of their reciprocal space lattices. The crystal symmetries for all possible arrangements of perfect periodicity are classified in 230 space groups. Contrary to this, the diffraction form, a disordered solid, shows cloud-like concentric rings indicating diffraction from a totally random atomic lattice and, consequently, a lack of any symmetry or, equivalently, any preferential direction in the solid.

This chapter presents an overview of the state-of-the-art of photonic three-dimensional deterministic aperiodic structures with a special emphasis on photonic quasicrystals. Reaching out into the third dimension has its own challenges. As discussed in detail in the preceding chapters, generating a deterministic aperiodic structure in one and two dimensions requires correctly calculated layer thicknesses (one dimension) or positions on a plane, where material is removed (forming holes) or added (forming cylinders, dots or other objects). Because of solid substrates all of these structures are inherently stable and one can concentrate directly on their physical properties. Having a properly calculated point-set does not necessarily help in three dimensions. Without supporting structures holding the building blocks in position, all of these constructs will collapse. Hence, devising proper support structures which do not alter the overall properties under study is one challenge; the second being the actual fabrication. For photonic structures feature sizes have to be on a scale of 100 nm and separation between features should not exceed a few microns – hence, part of this chapter is dedicated to technological questions. Because of these challenges there are only a limited number of groups working on three-dimensional aperiodic structures, a field which is just starting to expand. However, the benefit of reaching into the third dimension is to create a real material showing perhaps surprising properties, like the three-dimensional quasicrystals Shechtman et al. [428] discovered in 1984.

In Section 7.1 we will briefly classify the different aperiodic sequences used later on with respect to their Fourier spectra, especially keeping in mind that three-dimensional structures can only be of finite dimensions. Section 7.2 is then dedicated to the generation of three-dimensional point-sets and proper support structures, presenting the cut-and-project method as a powerful tool for generating quasicrystalline structures with almost any desired symmetry properties. Fabrication technologies with respect to three-dimensional aperiodic structures are briefly summarized in Section 7.3, before Section 7.4 presents three-dimensional realizations of quasicrystals and deterministic aperiodic structures.

Introduction

The propagation of light in periodically ordered photonic crystals bears a strong analogy to the wave propagation of conduction electrons in atomic crystals. The development of Bloch modes and dispersion are determined by the interference of waves that are diffracted by lattice planes [19, 52]. The periodicity commensurate with the wavelength (a ≈ λ/2) gives rise to Bragg diffraction that is associated with frequency windows for which waves are forbidden to propagate in a particular direction. Such stop gaps have long been known to arise for electromagnetic waves, notably for X-rays in atomic crystals [214]. At optical frequencies, the stop gaps that occur in one-dimensional periodic structures, known as Bragg stacks, are widely used in optical laboratories as broadband highly reflecting dielectric mirrors [556].

The distinguishing feature of three-dimensional (3D) photonic crystals is that a common stop gap can be achieved for all directions and for all polarizations simultaneously: the widely pursued 3D photonic band gap. Historically a 3D band gap has never been considered for the propagation of X-rays in periodic media, since a gap requires a high refractive index contrast of order unity, whereas the refractive index varies by less then 10−4 in the X-ray range. At frequencies inside the 3D photonic band gap, the density of optical states (DOS) completely vanishes. As the density of states can also be interpreted as the density of vacuum fluctuations [334], a 3D band gap thus serves as an effective shield to these fluctuations. The total absence of optical modes in a photonic band gap has implications beyond classical optics that break the analogy between the behavior of light in photonic crystals and the behavior of electrons in atomic crystals.

3D photonic band gap crystals play an important role in cavity quantum electrodynamics (cQED) [64, 549], where they offer at least five prospects for new physics. Firstly, probably the most eagerly pursued phenomenon is the complete inhibition of spontaneous emission: an excited quantum emitter – such as an atom, molecule, or quantum dot – embedded in a crystal with its transition frequency tuned to within the 3D band gap remains forever excited since it cannot decay to the ground state by emitting a photon.

From lasers to random lasers

Lasers are among the most useful and popular of all optical devices, with countless applications ranging from biology to astronomy. First predicted by Letokhov [292] and measured experimentally by Lawandy [269] and others [174, 319], random lasers [495, 529] connected for the first time the physics of ordinary lasers with that of disordered systems, boosting spectacularly in the early 1990s the interest of the scientific community in complex photonics. The possibility of using intrinsically disordered structures to create novel optical systems is attractive, not only from the applications point of view, but also fundamentally, allowing scientists to connect the theoretical paradigms of complexity, nonlinearity, disorder, and even glass physics with photonics.

Laser stands for Light Amplification by Stimulated Emission of Radiation, thus the amplifier is its fundamental element. A coherent optical amplifier is capable of increasing the amplitude of an optical field while maintaining its phase. If a monochromatic beam is injected into such a device the output will have the same frequency, while the phase can be the same or shifted by a fixed amount. In contrast, an amplifier that increases the intensity of an optical wave without preserving the phase is an incoherent amplifier. An amplifier based on stimulated emission is coherent: the stimulation process allows a photon in a given mode to induce an atom lying in an excited state to undergo a transition to a lower energy level, emitting a photon that is identical to the exciting photon, thus preserving frequency, direction, polarization and phase. If stimulation happens in a material in which the population is inverted (i.e. the majority of the atoms lie at the excited level), then an avalanche process in which every photon creates a duplicate of itself, is ignited exponentially, increasing the amplitude in the mode.

Cleaning optical surfaces

You can clean a seriously stained or corroded surface by gentle polishing with a fine wet rag and a dab of rouge or cerirouge. This is a last resort of course, on a surface which would otherwise be discarded. Apart from this you should never, never touch a dirty optical surface with any solid other than a fine sable-hair artist's brush dipped in a suitable solvent such as analytical quality iso-propyl alcohol. The sweat on your fingers (which helps in fingerprinting) is acid enough to etch glass. There is never any reason to allow a cloth to touch an optical-quality glass surface, as it will almost certainly have abrasive silica particles embedded in it.

Camera lenses

Camera lenses are best kept with a permanent anti-ultra-violet filter in place. If it is necessary to clear dust, a blower-brush is the proper device. If the glass surface has stains or dried water-spots, then the solvent of choice is iso-propyl alcohol. Diethyl ether can also be used in a well-ventilated place and ethyl alcohol too, although it is not such a good solvent for lipids. Diethyl ether evaporates quickly and the consequent cooling of the surface may cause water-drop condensation in high humidity surroundings. Ethyl alcohol will dissolve water.

The most famous and illustrious mis-user of an optical instrument is Sherlock Holmes. There is an iconic figure of him with Inverness cape, deerstalker hat and calabash pipe, peering though a magnifying glass, the latter held at arm's length, to inspect a possible blood stain.

This is the wrong way to use a magnifying glass – which incidentally should always have a plano-convex lens rather than the biconvex lens with which many cheap versions are provided. The glass should be held close to the eye, plane side facing, and the object brought in until it is in clear focus at a comfortable distance for viewing. This gives the clearest image, the widest field and the minimum of optical aberrations. It is the attention to small detail like this which helps ensure success. Watchmakers do it properly, with a loupe, a lens held, like a monocle, in the eye socket.

In experimental science, especially in physics laboratories, it is sometimes found, when beginning a new piece of basic research, that no appropriate apparatus exists and that it is necessary to improvise. The traditional laboratory stand-and-clamp then comes into its own, followed, after some experimentation, by a properly designed system with an optical bench or table with lenses, mirrors and other basic optical elements for measuring and analysing radiation. Skills in the design and assembly of such provisional devices are part of the true experimenter's art.

Interferometry

Interferometers, as the name implies, rely on the wave nature of light and of the constructive and destructive interference between wave-trains. They are devices, often of great ingenuity, with many applications to fine measurement (~10−9 m) or ultra-fine measurement (~10−12 m) and as such they have led the way to many advances in science, not least in provoking the theory of relativity and in confirming the predictions of quantum mechanics. After overcoming formidable technical problems in construction and stability, interferometers are now in everyday use, particularly in spectroscopy, fine measurement and quality control.

Oscillation, phase and phase-difference

The word ‘oscillator’ applies as much to the electric vector of a beam of radiation as it does to a swinging pendulum. Two oscillators are ‘in phase’ when they reach a maximum displacement in the same direction at the same time and are ‘out of phase’ if their displacements are exactly opposite. We then say that their phase-difference is 180° or that they have π radians phase-difference. Their phase-difference is measured as an angle because it is convenient to use the mathematics of a circle to do so: there are no physical lines or angles involved.

Introduction: polarized light

James Clerk Maxwell (1831–1879) discovered the first unified field theory when in 1865 he showed, drawing on Faraday's experiments, the Biot–Savart law and other results, that interacting electric and magnetic fields and forces could be described by the same set of equations. Oliver Heaviside (1850–1925) reduced the twenty quaternion equations of Maxwell's theory to the four vector equations that ordinary mortals could understand. Maxwell showed that a wave equation could be deduced and that the consequent electromagnetic field propagates in a direction perpendicular to its electric and magnetic field vectors, and at a velocity which could be calculated from two quantities (permittivity and susceptibility) measurable in separate laboratory experiments. The resultant computed velocity was very close indeed to the experimentally measured speed of light. It was an easy conjecture then, that light was electromagnetic radiation. It was a further logical extension to predict radio waves, later confirmed by David Hughes' and Heinrich Hertz's spark-gap experiments.

The mutually perpendicular electric and magnetic field vectors oscillate in amplitude at about 5 × 1014 Hz for red light. If the direction of propagation is the positive z-direction, then the electric and magnetic vectors lie in the x−y plane. The light is said to be plane polarized if the angle the electric vector makes with the x-axis is constant in time.

Introduction: the photographic lens

A simple plano-convex lens has one surface plane and the other spherical. Its focal surface is also approximately spherical. The image of a point source at −∞ on the optic axis is never a point because of spherical aberration, and the image is more and more distorted by other aberrations as the field angle increases.

Eighteenth century telescope makers such as John Dollond had discovered these aberrations long before the first photo-sensitive chemical materials were discovered, and they knew that both chromatic aberration and spherical aberration could be corrected (i.e. reduced to a satisfactorily small level) by combining two lenses with spherical surfaces, a positive one made of crown glass and a negative of flint glass, into an achromatic doublet, usually but not always a cemented doublet, in which the second and third surfaces were complementary and the two were joined by a thin film of Canada balsam. This was the first step on a long road which has led to the wide-aperture zoom lenses of current cameras.

A significant milepost on this road was the Gauss achromat, a positive meniscus followed by a negative meniscus: an air-spaced doublet which by its ‘bendings’, i.e. changing the shapes of the elements without changing their powers, reduced the aberrations. It was originally designed by C. F. Gauss at the request of Repsold the Hamburg lens maker, who needed an achromatic telescope objective of improved performance.