Localization of optics: Quasiperiodic media

Suppression of alias and replica noises in phase holograms using fractal topologies

Two-dimensional fractal topologies featuring (scaling) self-similarity, dense set of Bragg (diffraction) peaks, and inherent rotation symmetry, which are not achievable with regular grid-matrix geometries, exhibit optical robustness against structural damage and noise immunity of optical transmission paths. In this work, we numerically and experimentally demonstrate phase holograms using fractal plane-divisions. By taking advantage of the symmetries of the fractal topology, we propose numerical algorithms to design the fractal holograms. This algorithm solves the inapplicability of the conventional iterative Fourier transform algorithm (IFTA) method and enables efficient optimizations of millions of adjustable parameters in the optical element. Experimental samples show that the alias and replica noises in the image plane of fractal holograms are clearly suppressed, facilitating applications for high-accuracy and compact requirements.

Tunable Multi-Band-Stop Filters Using Generalized Fibonacci Photonic Crystals for Optical Communication Applications

In this study, a numerical investigation of photonic quasi-periodic Generalized Fibonacci (GF) (m, n) sequences is carried out in the visible spectrum. The transfer matrix method is employed to study the behavior of wave propagation through the photonic structures. Firstly and to highlight the importance of the GF structure, its transmittance spectrum is compared to those of periodic and ordinary Fibonacci structures. It is shown that the GF structure permits one to obtain multi-photonic band gaps (PBGs) separated by several resonance modes. The variation in the parameter m of the GF (m, 1) structure allows for the tuning of the number, the position and the width of these bands. By changing the parameter m, the wavelengths (650, 850, 1300, and 1550 nm) of the plastic and glass optical fibers can be allowed or forbidden to transmit through the structure according to the value of this parameter. In contrast, the variation in the parameter n for GF (1, n) hides all PBGs and only permits the appearance of several Kiessig fringes. The proposed structures can find application as tunable multi-band-stop filters for optical fiber wavelengths.

Self-Similarity Properties of Complex Quasi-Periodic Fibonacci and Cantor Photonic Crystals

In this paper, the influence of structural modifications on basic quasi-periodic (QP) photonic crystals (PhC’s) on self-similarity feature in their spectral responses is examined. Investigated crystals are chosen based on a present knowledge on the QP crystals, and are classified according to their structure. One of the QP crystals considered for the calculations is a concatenation, Fibonacci structure. It characterizes with a self-similar spectra for its different orders, which means, that the spectral shape repeats itself and can be partially identical for a different orders of the Fibonacci QP crystal. The calculations were also performed for the fractal structure, based on a Cantor QP crystal. Just as for the case of the Fibonacci structure, it characterizes with a self-similar spectra for different orders of the structure. Considered photonic devices are next put through simple modification operations by multiplication, conjugation or mirror reflection. Resulting, modified structures are used for the calculations of their spectral response. Results show, that the self-similarity of the spectra is not affected by performed modifications, and thus spectral response of QP PhC can be designed without losing this feature. Moreover the regular expansion of the repeated central part of the spectrum that appears in higher-order Fibonacci QP PhC spectra (due to the self-similarity) with the increase Fibonacci crystal order is presented here for the first time.

Fundamental and multipole solitons in amplitude-modulated Fibonacci lattices

We investigated the existence and stability of fundamental and multipole solitons supported by amplitude-modulated Fibonacci lattices with self-focusing nonlinearity. Owing to the quasi-periodicity of Fibonacci lattices, families of solitons localized in different waveguides have different properties. We found that the existence domain of fundamental solitons localized in the central lattice is larger than that of solitons localized in the adjacent central waveguide. The former counterparts are completely stable in their existence region, while the latter have a narrow unstable region near the lower cut-off. Two families of dipole solitons were also comprehensively studied. We found the outer lattice distribution can significantly change the existence region of solitons. In addition, we specifically analyzed the properties of four complicated multipole solitons with pole numbers 3, 5, 7, and 9. In the Fibonacci lattice, their field moduli of multipole solitons are all asymmetrically distributed. The linear-stability analysis and direct simulations reveal that as the number of poles of the multipole soliton increases, its stable domain is compressed. Our results provide helpful insight for understanding the dynamics of nonlinear localized multipole modes in Fibonacci lattices with an optical nonlinearity.

Scaling Law, Confined and Surface Modes in Photonic Fibonacci Stub Structures: Theory and Experiment

We investigate both theoretically and experimentally the properties of electromagnetic waves propagation and localization in periodic and quasi-periodic stub structures of Fibonacci type. Each block constituting the Fibonacci sequence (FS) is composed of an horizontal segment and a vertical stub. The origin of the primary and secondary gaps shown in such systems is discussed. The behaviors and scattering properties of the electromagnetic modes are studied in two geometries, when the FS is inserted horizontally between two semi-infinite waveguides or grafted vertically along a guide. Typical properties of the Fibonacci systems such as the fragmentation of the frequency spectrum, the self-similarity following a scaling law are analyzed and discussed. It is found that certain modes inside these two geometries decrease according to a power law rather than an exponential law and the localization of these modes displays the property of self-similarity around the central gap frequency of the periodic structure where the quasi-periodicity is most effective. Also, the eigenmodes of the FS of different generation order are studied depending on the boundary conditions imposed on its extremities. It is shown that both geometries provide complementary information on the localization of the different modes inside the FS. In particular, in addition to bulk modes, some localized modes induced by both extremities of the system exhibit different behaviors depending on which surface they are localized. The theory is carried out using the Green’s function approach through an analysis of the dispersion relation, transmission coefficient and electric field distribution through such finite structures. The theoretical findings are in good agreement with the experimental results performed by measuring in the radio-frequency range the transmission along a waveguide in which the FS is inserted horizontally or grafted vertically.

Tuning of the polariton modes induced by longitudinal strong coupling in the graphene hybridized DBR cavity

In this Letter, we construct a graphene hybridized distributed Bragg reflector (DBR) cavity, where spatially longitudinal strong coupling occurs between the Tamm plasmon polaritons (TPPs) existing around the graphene layer and the cavity mode (CM) existing in the DBR cavity. As a result, two hybrid polariton modes emerge, which contain both the TPP and the CM components. In the simulation, we demonstrate that the resonant frequencies and the damping rates of the polariton modes can be actively tuned by the graphene Fermi level and the incident angle of light. Besides, the coupling strength and the damping rates are also passively tuned by the pair number of the layers in the DBR. Theoretically, we analyze the TPP-CM strong coupling by the coupled harmonic oscillator equations, which help to explain the regulation process. The controllable TPP-CM longitudinal strong coupling with two absorption bands may achieve potential applications in developing graphene-based active optoelectronic and polaritonic devices in terahertz waves.

Quantum Walks in Periodic and Quasiperiodic Fibonacci Fibers

Quantum walk is a key operation in quantum computing, simulation, communication and information. Here, we report for the first time the demonstration of quantum walks and localized quantum walks in a new type of optical fibers having a ring of cores constructed with both periodic and quasiperiodic Fibonacci sequences, respectively. Good agreement between theoretical and experimental results has been achieved. The new multicore ring fibers provide a new platform for experiments of quantum effects in low-loss optical fibers which is critical for scalability of real applications with large-size problems. Furthermore, our new quasiperiodic Fibonacci multicore ring fibers provide a new class of quasiperiodic photonics lattices possessing both on- and off-diagonal deterministic disorders for realizing localized quantum walks deterministically. The proposed Fibonacci fibers are simple and straightforward to fabricate and have a rich set of properties that are of potential use for quantum applications. Our simulation and experimental results show that, in contrast with randomly disordered structures, localized quantum walks in new proposed quasiperiodic photonics lattices are highly controllable due to the deterministic disordered nature of quasiperiodic systems.

Aperiodic-Order-Induced Multimode Effects and Their Applications in Optoelectronic Devices

Unlike periodic and random structures, many aperiodic structures exhibit unique hierarchical natures. Aperiodic photonic micro/nanostructures usually support optical multimodes due to either the rich variety of unit cells or their hierarchical structure. Mainly based on our recent studies on this topic, here we review some developments of aperiodic-order-induced multimode effects and their applications in optoelectronic devices. It is shown that self-similarity or mirror symmetry in aperiodic micro/nanostructures can lead to optical or plasmonic multimodes in a series of one-dimensional/two-dimensional (1D/2D) photonic or plasmonic systems. These multimode effects have been employed to achieve optical filters for the wavelength division multiplex, open cavities for light–matter strong coupling, multiband waveguides for trapping “rainbow”, high-efficiency plasmonic solar cells, and transmission-enhanced plasmonic arrays, etc. We expect that these investigations will be beneficial to the development of integrated photonic and plasmonic devices for optical communication, energy harvesting, nanoantennas, and photonic chips.

Evolution of orbital angular momentum in a soft quasi-periodic structure with topological defects

A continuously deformative space possesses trivial or nontrivial topological characteristics depending on the associated homotopy groups associated with spaces describing the physical processes. Moreover, the interaction of spatial warping and structural symmetry always presents fantastic phenomena, especially in the systems with unique symmetrical properties such as quasicrystals. Here, we propose a quasi-periodic structure (QPS) with topological defects. The analytical expression of the corresponding Fourier spectrum is derived, which reflects the combined effects of topological structure and quasi-translational symmetry. Light-matter interaction therein brings unusual diffraction characteristics with exotic evolution of orbital angular momentum (OAM). Long-range correlation of QPS resulted in multi-fractal and pairwise distribution of optical singularities. A general conservation law of OAM is revealed. A liquid crystal photopatterned QPS is fabricated to demonstrate the above characteristics. Dynamic reconfigurable manipulation of optical singularities is achieved. Our approach offers the opportunity to manipulate OAM with multiple degrees of freedom, which has promising applications in multi-channel quantum information processing and high-dimensional quantum state generation.

Localized quantum walks in quasi-periodic Fibonacci arrays of waveguides

New quasi-periodic arrays of waveguides (AWs) constructed with Fibonacci sequences are proposed to realize localized quantum walks (LQWs). The proposed Fibonacci arrays of waveguides (FAWs) are simple and straightforward to make, but have a rich set of properties that are of potential use for applications in quantum communication. Our simulations show that, in contrast with randomly disordered AWs, LQWs in FAWs are highly controllable due to the deterministic disorder nature of quasi-periodic systems. Furthermore, unique LQWs with symmetrical probability distribution can be conveniently realized in the FAWs.

Multiple adjustable optical Tamm states in one-dimensional photonic quasicrystals with predesigned bandgaps

We proposed an approach to get multiple and adjustable optical Tamm states (OTSs) by constructing a structure consisting of a metal layer and one-dimensional photonic quasicrystals with preassigned bandgaps. In the structure, multiple OTSs excited simultaneously in each bandgap were observed. We explored the physics mechanism of the multiple OTSs by analyzing the electric field intensity distribution in the structure. Besides, the results also show that the thickness of the top layer gives one more degree of freedom in designing multiple OTSs. Finally, we demonstrated that one additional OTS can be obtained independently by adding another bandgap to the proposed structure.

Multipurpose S-shaped solvable profiles of the refractive index: application to modeling of antireflection layers and quasi-crystals

A class of four-parameter solvable profiles of the electromagnetic admittance has recently been discovered by applying the newly developed Property & Field Darboux Transformation method (PROFIDT). These profiles are highly flexible. In addition, the related electromagnetic field solutions are exact, in closed form, and involve only elementary functions. In this paper, we focus on those that are S-shaped, and we provide all of the tools needed for easy implementation. These analytical bricks can be used for high-level modeling of lightwave propagation in photonic devices presenting a piecewise-sigmoidal refractive index profile, such as, for example, antireflection layers, rugate filters, chirped filters, and photonic crystals. For small amplitudes of the index modulation, these elementary profiles are very close to a cosine profile. They can therefore be considered as valuable surrogates for computing the scattering properties of components like Bragg filters and reflectors as well. In this paper we present an application for antireflection layers and another for 1D quasi-crystals (QC). The proposed S-shaped profiles can be easily manipulated for exploring the optical properties of smooth QC, a class of photonic devices that adds to the classical binary-level QC.

Investigation of one-way absorption properties in an asymmetric photonic crystal containing a semiconductor defect

We theoretically study the one-way absorption in two 1D defective asymmetric photonic crystals, air/(DB)^NA(BD)^M/air and air/(DB)^NA(BD)^MA(DB)^NA(BD)^M/air, where A and B are dielectrics, D is the semiconductor, n-InSb, and N, M are stack numbers with N≠M. It is revealed that their absorption spectra exhibit one-way properties. We also find that the number of one-way absorption peaks depends on the symmetry and number of defect layers, which are similar to the defect modes in the transmittance spectra of the usual symmetry photonic crystals. Additionally, effects of the incident angles for both TE and TM waves on the one-way feature are also presented. At a large incident angle, the TE wave is almost reflected, whereas the TM wave can have a partial absorption.

Wave properties in asymmetric single-negative-base photonic crystals

We theoretically study wave properties for one-dimensional defective asymmetric photonic crystals, air/(AB)^MG(BA)^N/air, air/(AQ)^MG(QA)^N/air, and air/(BQ)^MG(QB)^N/air, where A is a lossy epsilon-negative material, B is a lossy mu-negative material, G and Q are dielectrics with different refractive indexes, and M and N are stack numbers with M≠N. Special attention has been paid to their absorption spectra. It is found that at certain frequencies the absorption can exhibit unidirectional properties. Our calculated results show two kinds of unidirectional absorption peaks. One is a single absorption peak whose frequency depends on the thickness of defect layer G. For the other peaks, its frequency does not change when the defect layer's thickness changes. In addition, in the second kind of peaks, the peak numbers for forward and backward propagation are different, that is, there are (M-1) absorption peaks for forward propagation, while there are (N-1) absorption peaks for backward propagation. When the two kinds of unidirectional absorption peaks are merged, some new peaks appear, and both forward and backward propagation will have (M+N-1) absorption peaks.

Revealing the Topology of Quasicrystals with a Diffraction Experiment

Topological properties of crystals and quasicrystals is a subject of recent and growing interest. This Letter reports an experiment where, for certain quasicrystals, these properties can be directly retrieved from diffraction. We directly observe, using an interferometric approach, all of the topological invariants of finite-length Fibonacci chains in their diffraction pattern. We also quantitatively demonstrate the stability of these topological invariants with respect to structural disorder.

Terahertz bistability and multistability in graphene/dielectric Fibonacci multilayer

Here, we benefit from the strong nonlinear response of graphene and the rich variety of resonances provided by a graphene/dielectric Fibonacci multilayer to realize bistability and multistability in the terahertz (THz) frequency range. Toward this pursuit, we employ the nonlinear transfer matrix method. We examine the suitability of resonances in the Fibonacci multilayer for the bi/multistability purposes and determine the proper working point. We report various switching up/down manners via single or stepwise jumps between states of the same or different contrasts upon increasing followed by decreasing the intensity of the incident wave. We show that graphene samples of high quality are preferred for bi/multistable switching in terms of reducing the switch-up/-down thresholds and widening the multistable region. We also explore the possibility of tuning the bi/multistable behavior via the frequency and angle of the incident wave as well as the graphene Fermi level. We envision precious applications in THz switching, realizing logic gates, and so on for this system.

Multimode photon-exciton coupling in an organic-dye-attached photonic quasicrystal

In this Letter, we present hybrid strong coupling between multiple photonic modes and excitons in an organic-dye-attached photonic quasicrystal. The excitons effectively interact with the photonic modes offered by the photonic quasicrystal, and multiple hybrid polariton bands are demonstrated in both experiments and calculations. Furthermore, by retrieving the measured dispersion map, we get the mixing fractions of photonic modes and excitons, and show that the polariton bands inherit not only the energy dispersion features, but also the damping behaviors from both the photonic modes and the excitons. Our investigation may inspire related studies on multimode light-matter interactions and achieve some potential applications for multimode sensors.

Performance Characteristics of Bio-Inspired Metal Nanostructures as Surface-Enhanced Raman Scattered (SERS) Substrates

The fabrication of high-performance plasmonic nanomaterials for bio-sensing and trace chemical detection is a field of intense theoretical and experimental research. The use of metal-silicon nanopillar arrays as analytical sensors has been reported with reasonable results in recent years. The use of bio-inspired nanocomposite structures that follow the Fibonacci numerical architecture offers the opportunity to develop nanostructures with theoretically higher and more reproducible plasmonic fields over extended areas. The work presented here describes the nanofabrication process for a series of 40 µm × 40 µm bio-inspired arrays classified as asymmetric fractals (sunflower seeds and romanesco broccoli), bilaterally symmetric (acacia leaves and honeycombs), and radially symmetric (such as orchids and lily flowers) using electron beam lithography. In addition, analytical capabilities were evaluated using surface-enhanced Raman scattering (SERS). The substrate characterization and SERS performance of the developed substrates as the strategies to assess the design performance are presented and discussed.

Bulk and surface acoustic waves in solid-fluid Fibonacci layered materials

We study theoretically the propagation and localization of acoustic waves in quasi-periodic structures made of solid and fluid layers arranged according to a Fibonacci sequence. We consider two types of structures: either a given Fibonacci sequence or a periodic repetition of a given sequence called Fibonacci superlattice. Various properties of these systems such as: the scaling law and the self-similarity of the transmission spectra or the power law behavior of the measure of the energy spectrum have been highlighted for waves of sagittal polarization in normal and oblique incidence. In addition to the allowed modes which propagate along the system, we study surface modes induced by the surface of the Fibonacci superlattice. In comparison with solid-solid layered structures, the solid-fluid systems exhibit transmission zeros which can break the self-similarity behavior in the transmission spectra for a given sequence or induce additional gaps other than Bragg gaps in a periodic structure.

Photonic quasi-crystal terahertz lasers

Quasi-crystal structures do not present a full spatial periodicity but are nevertheless constructed starting from deterministic generation rules. When made of different dielectric materials, they often possess fascinating optical properties, which lie between those of periodic photonic crystals and those of a random arrangement of scatterers. Indeed, they can support extended band-like states with pseudogaps in the energy spectrum, but lacking translational invariance, they also intrinsically feature a pattern of 'defects', which can give rise to critically localized modes confined in space, similar to Anderson modes in random structures. If used as laser resonators, photonic quasi-crystals open up design possibilities that are simply not possible in a conventional periodic photonic crystal. In this letter, we exploit the concept of a 2D photonic quasi crystal in an electrically injected laser; specifically, we pattern the top surface of a terahertz quantum-cascade laser with a Penrose tiling of pentagonal rotational symmetry, reaching 0.1-0.2% wall-plug efficiencies and 65 mW peak output powers with characteristic surface-emitting conical beam profiles, result of the rich quasi-crystal Fourier spectrum.

Analysis of Fibonacci gratings and their diffraction patterns

Aperiodic and fractal optical elements are proving to be promising candidates in image-forming devices. In this paper, we analyze the diffraction patterns of Fibonacci gratings (FbGs), which are prototypical examples of aperiodicity. They exhibit novel characteristics such as redundancy and robustness that keep their imaging characteristics intact even when there is significant loss of information. FbGs also contain fractal signatures and are characterized by a fractal dimension. Our study suggests that aperiodic gratings may be better than their fractal counterparts in technologies based on such architectures. We also identify the demarcating features of aperiodic and fractal diffraction, which have been rather fuzzy in the literature so far.

Comparison of the transmission properties of self-similar, periodic, and random multilayers at normal incidence

The effect of self-similarity on acoustic and elastic wave propagation at normal incidence is investigated using Classical Cantor and Fibonacci multilayered structures. They are made of two sorts of orthotropic plies having differently oriented orthotropic axes with respect to the propagation direction. The properties of their transmission coefficient are presented using a unidirectional numerical model based on a transfer matrix formalism. It was found that stack self-similarity influences the acoustic transmission properties. Transmission coefficients of self-similar stacks present a self-similar shape and behavior. A self-similar process, applied to layer orientation allows multilayered stacks to be created. A thickness-equivalent model was developed to compare these structures with standard self-similar multilayers which are finally compared to periodic and random stacks. The transmission coefficient of a deterministic self-similar Fibonacci structure is similar to that of an averaged transmission coefficient of random stacks.

Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential

We report on the study of a polariton gas confined in a quasiperiodic one-dimensional cavity, described by a Fibonacci sequence. Imaging the polariton modes both in real and reciprocal space, we observe features characteristic of their fractal energy spectrum such as the opening of minigaps obeying the gap labeling theorem and log-periodic oscillations of the integrated density of states. These observations are accurately reproduced solving an effective 1D Schrödinger equation, illustrating the potential of cavity polaritons as a quantum simulator in complex topological geometries.

Resonant modes of 12-fold symmetric defect free photonic quasicrystal

This work investigates the resonant modes of a 12-fold symmetric defect free photonic quasicrystal (PQC) nanorod array using finite difference time domain (FDTD) simulation. Localized modes can exist in PQC without introducing defects due to the lack of translational symmetry. The resonant modes of the unit cell PQC and the one time expanded PQC from unit cell are systematically examined. The resonant spectrum is that of a single rod modified by the interaction among PQC nanorods. The mode confinement is contributed by guided resonance and destructive interference scattering. The self-scaling similarity of resonant spectrum and mode profile are also investigated.

Omnidirectional reflection from generalized Fibonacci quasicrystals

We determine the optimal thicknesses for which omnidirectional reflection from generalized Fibonacci quasicrystals occurs. By capitalizing on the idea of wavelength- and angle-averaged reflectance, we assess in a consistent way the performance of the different systems. Our results indicate that some of these aperiodic arrangements can largely over-perform the conventional photonic crystals as omnidirectional reflection is concerned.

High-Q filters with complete transmission by quasi-periodic dielectric multilayers

A multiple narrow bandpass filter with both high Q factor and complete transmission using quasi-periodic Thue-Morse dielectric multilayers is proposed. The Q factor of the system increases exponentially with the generation order. Even though the Q factor of resonances increases as the generation order of the multilayers increases, these resonances are still complete resonances. The number of resonance peaks for the bandpass filter of the system also increases as the generation order increases. These resonance peaks have a multifractal distribution throughout the frequency range, which is different from that in traditional periodical multilayers.

Band gaps and transmission spectra in generalized Fibonacci σ(p,q) one-dimensional magnonic quasicrystals

We employ a microscopic theory to investigate spin wave (magnon) propagation through their dispersion and transmission spectra in magnonic crystals arranged to display deterministic disorder. In this work the quasiperiodic arrangement investigated is the well-known generalized Fibonacci sequence, which is characterized by the σ(p,q) parameter, where p and q are non-zero integers. In order to determine the bulk modes and transmission spectra of the spin waves, the calculations are carried out for the exchange dominated regime within the framework of the Heisenberg model and taking into account the random phase approximation. We have considered magnetic materials that have a ferromagnetic order, and the transfer-matrix treatment is applied to simplify the algebra. The results reveal that spin wave spectra display a rich and interesting magnonic pass- and stop-bands structures, including an almost symmetric band gap distribution around of a mid-gap frequency, which depends on the Fibonacci sequence type.

Microfluidics integration of aperiodic plasmonic arrays for spatial-spectral optical detection

We demonstrate successful integration of aperiodic arrays of metal nanoparticles with microfluidics technology for optical sensing using the spectral-colorimetric responses of nanostructured arrays to refractive index variations. Different aperiodic arrays of gold (Au) nanoparticles with varying interparticle separations and Fourier spectral properties are fabricated using Electron Beam Lithography (EBL) and integrated with polydimethylsiloxane (PDMS) microfluidics structures by soft-lithographic micro-imprint techniques. The spectral shifts of scattering spectra and the distinctive modifications of structural color patterns induced by refractive index variations were simultaneously measured inside microfluidic flow cells by dark-field spectroscopy and image correlation analysis in the visible spectral range. The integration of engineered aperiodic arrays of Au nanoparticles with microfluidics devices provides a novel sensing platform with multiplexed spatial-spectral responses for opto-fluidics applications and lab-on-a-chip optical biosensing.

Measurement of surface plasmon correlation length differences using Fibonacci deterministic hole arrays

Using terahertz (THz) transmission measurements through two-dimensional Fibonacci deterministic subwavelength hole arrays fabricated in metal foils, we find that the surface plasmon-polariton (SPP) correlation lengths for aperiodic resonances are smaller than those associated with the underlying grid. The enhanced transmission spectra associated with these arrays contain two groups of Fano-type resonances: those related to the two-dimensional Fibonacci structure and those related to the underlying hole grid array upon which the aperiodic Fibonacci array is built. For both groups the destructive interference frequencies at which transmission minima occur closely match prominent reciprocal vectors in the hole array (HA) structure-factor in reciprocal space. However the Fibonacci-related transmission resonances are much weaker than both their calculated Fourier intensity in k space and the grid-related resonances. These differences may arise from the complex, multi-fractal dispersion relations and scattering from the underlying grid arrays. We also systematically studied and compared the transmission resonance strength of Fibonacci HA and periodic HA lattices as a function of the number of holes in the array structure. We found that the Fibonacci-related resonance strengths are an order of magnitude weaker than that of the periodic HA, consistent with the smaller SPP correlation length for the aperiodic structure.

Exploiting aperiodic designs in nanophotonic devices

In this work we consider the role of aperiodic order-order without periodicity-in the design of different optical devices in one, two and three dimensions. To this end, we will first study devices based on aperiodic multilayered structures. In many instances the recourse to Fibonacci, Thue-Morse or fractal arrangements of layers results in improved optical properties compared with their periodic counterparts. On this basis, the possibility of constructing optical devices based on a modular design of the multilayered structure, where periodic and quasiperiodic subunits are properly mixed, is analyzed, illustrating how this additional degree of freedom enhances the optical performance in some specific applications. This line of thought can be naturally extended to aperiodic arrangements of optical elements, such as nanospheres or dielectric rods in the plane, as well as to three-dimensional photonic quasicrystals based on polymer materials. In this way, plentiful possibilities for new tailored materials naturally appear, generally following suitable optimization algorithms. Then, we present a detailed discussion on the physical properties supporting the preferential use of aperiodic devices in a number of optical applications, opening new avenues for technological innovation. Finally we suggest some related emerging topics that deserve some attention in the years to come.

Multiple responses of TPP-assisted near-perfect absorption in metal/Fibonacci quasiperiodic photonic crystal

Absorption properties in one-dimensional quasiperiodic photonic crystal composed of a thin metallic layer and dielectric Fibonacci multilayers are investigated. It is found that a large number of photonic stopbands can occur at the dielectric Fibonacci multilayers. Tamm plasmon polaritons (TPPs) with the frequencies locating at each photonic stopband are excited at the interface between the metallic layer and the dielectric layer, leading to almost perfect absorption for the energy of incident wave. By adjusting the length of dielectric layer with higher refractive-index or the Fibonacci order, the number of absorption peaks can be tuned effectively and enlarged significantly.

Characterization of large array of plasmonic nanoparticles on layered substrate: dipole mode analysis integrated with complex image method

In this paper, an efficient analytical method for characterizing large array of plasmonic nanoparticles located over planarly layered substrate is introduced. The model is called dipole mode complex image (DMCI) method since the main idea lies in modeling a subwavelength spherical nanoparticle at its electric scattering resonance with an induced electric dipole and representing the electromagnetic (EM) fields of this electric dipole over the layered substrate in terms of finite complex images. The major advantages of the proposed method are its accuracy and rapid calculation in characterizing various kinds of large periodic and aperiodic arrays of nanoparticles on layered substrates. The computational time can be reduced significantly in compared to the traditional methods. The accuracy of the theoretical model is validated through comparison with numerical integration of Sommerfeld integrals. Moreover, the analytical results are compared well with those determined by full-wave finite difference time domain (FDTD) method. To demonstrate the capability of our technique, the performances of large arrays of nanoparticles on layered silicon substrates for efficient sunlight energy incoupling are studied.

Thermodynamics of photons on fractals

A thermodynamical treatment of a massless scalar field (a photon) confined to a fractal spatial manifold leads to an equation of state relating pressure to internal energy, PV(s) = U/d(s), where d(s) is the spectral dimension and V(s) defines the "spectral volume." For regular manifolds, V(s) coincides with the usual geometric spatial volume, but on a fractal this is not necessarily the case. This is further evidence that on a fractal, momentum space can have a different dimension than position space. Our analysis also provides a natural definition of the vacuum (Casimir) energy of a fractal. We suggest ways that these unusual properties might be probed experimentally.

Fractal spectrum of charge carriers in quasiperiodic graphene structures

In this work we investigate the interaction of charge carriers in graphene with a series of p-n-p junctions arranged according to a deterministic quasiperiodic substitutional Fibonacci sequence. The junctions create a potential landscape with quantum wells and barriers of different widths, allowing the existence of quasi-confined states. Spectra of quasi-confined states are calculated for several generations of the Fibonacci sequence as a function of the wavevector component parallel to the barrier interfaces. The results show that, as the Fibonacci generation is increased, the dispersion branches form energy bands distributed as a Cantor-like set. Besides, for a quasiperiodic set of potential barriers, we obtain the electronic tunneling probability as a function of energy, which shows a striking self-similar behavior for different generation numbers.

High-channel-count plasmonic filter with the metal-insulator-metal Fibonacci-sequence gratings

Fibonacci-sequence gratings based on metal-insulator-metal waveguides are proposed. The spectrum properties of this structure are numerically investigated by using the transfer matrix method. Numerical results demonstrate that the proposed structure can generate high-channel-count plasmonic stop bands and can find significant applications in highly integrated dense wavelength division multiplexing networks.

Optical transmission through generalized Thue-Morse superlattices

In this paper we study the transmission properties of light through generalized Thue-Morse (GTM (m, n)) aperiodic superlattices and obtain the formulae of the transmission coefficients (TCs) analytically at the central wavelength, which are confirmed by numerical simulations. It is found that: (1) all of the 1st generation systems are transparent to the substrates medium B; (2) GTM (m, 2 j) systems are transparent to the substrates medium B; (3) GTM (2i, 2j + 1) systems are translucent to the substrates medium B except the 1st generation; and (4) when i not equal j, transmission through GTM (2i + 1, 2j + 1) systems attenuates rapidly with the increase of generation number l and sequence parameters m, n. On the other hand, the positional correlations between the constituents of GTM (m, n) aperiodic superlattices responsible for the resonant states are also discussed. Based on the conclusions we study the properties of the amplitude of the electric field vector and find that they are different from those of periodic lattices and chaotic systems.

Broad-bandgap and low-sidelobe surface plasmon polariton reflector with Bragg-grating-based MIM waveguide

Surface plasmon polariton reflector (SPPR) based on metal-insulator-metal (MIM) Bragg grating waveguide is numerically studied. A quasi-chirped technique is applied to the engraved grooves in the surface of the MIM waveguide, and a new kind of broad-bandgap SPPR is achieved. Meanwhile, by optimizing the profile of gap width between the metal and dielectric, the spectral sidelobe of SPPR is effectively suppressed and thus the performance o f the SPPR is further improved.

Wave propagation through Cantor-set media: chaos, scaling, and fractal structures

Propagation of waves through Cantor-set media is investigated by renormalization-group analysis. For specific values of wave numbers, transmission coefficients are shown to be governed by the logistic map and, in the chaotic region, they show sensitive dependence on small changes in parameters of the system such as the index of refraction. For other values of wave numbers, our numerical results suggest that light transmits completely or reflects completely by the Cantor-set media Cinfinity. It is also shown that transmission coefficients exhibit a local scaling behavior near complete transmission if the complete transmission is achieved at a wave number kappa=kappa* with a rational kappa*/pi. The scaling function is obtained analytically by using the Euler totient function, and the local scaling behavior is confirmed numerically.

Perfect light transmission in Fibonacci arrays of dielectric multilayers

In this paper we study the propagation of light through an asymmetric array of dielectric multilayers built by joining two porous silicon substructures in a Fibonacci sequence. Each Fibonacci substructure follows the well-known recursive rule but in the second substructure dielectric layers A and B are exchanged. Even without mirror symmetry, this array gives rise to multiple transparent states, which follow the scaling properties and self-similar spectra of a single Fibonacci multilayer. We apply the transfer matrix formalism to calculate the transmittance. By setting the transfer matrix of the array equal to ± I, the identity matrix, frequencies of perfect light transmission are reproduced in our theoretical calculations. Although the light absorption of porous silicon in the optical range limits our experimental study to low Fibonacci generations, the positions of the transparent states are well predicted by the above-mentioned condition. We conclude that mirror symmetry in arrays of Fibonacci multilayers is sufficient but not necessary to generate multiple transparent states, opening broader applications of quasiperiodic systems as filters and microcavities of multiple frequencies.

One dimensional resonant Fibonacci quasicrystals: noncanonical linear and canonical nonlinear effects

A detailed experimental and theoretical study of the linear and nonlinear optical properties of different Fibonacci-spaced multiple-quantum-well structures is presented. Systematic numerical studies are performed for different average spacing and geometrical arrangement of the quantum wells. Measurements of the linear and nonlinear (carrier density dependent) reflectivity are shown to be in good agreement with the computational results. As the pump pulse energy increases, the excitation-induced dephasing broadens the exciton resonances resulting in a disappearance of sharp features and reduction in peak reflectivity.

Luminescence properties of a Fibonacci photonic quasicrystal

An active one-dimensional Fibonacci photonic quasi-crystal is realized via spin coating. Luminescence properties of an organic dye embedded in the quasi-crystal are studied experimentally and compared to theoretical simulations. The luminescence occurs via the pseudo-bandedge mode and follows the dispersion properties of the Fibonacci crystal. Time resolved luminescence measurement of the active structure shows faster spontaneous emission rate, indicating the effect of the large photon densities available at the bandedge due to the presence of critically localized states. The experimental results are in good agreement with the theoretical calculations for steady-state luminescence spectra.

Dense wavelength-division multiplexing dispersion compensators based on chirped and apodized Fibonacci structures: CA-FC(j,n)

Chromatic dispersion compensation is an essential feature of high speed dense wavelength-division multiplexing (DWDM) systems. We propose a dispersion compensator structure whose characteristics meet the optical DWDM system requirements. The proposed structure is based on Fibonacci quasi-periodic multilayer structures composed of layers with large index differences. Studying the dispersive properties of Fibonacci structures with generation numbers j=3 and 4, and calculating group delay (GD) and group velocity dispersion (GVD) of their reflection bands, we have demonstrated that to have a smooth GD and almost a constant GVD in each band of a DWDM system, one needs not only to suitably chirp the structure refractive index profile, but also must properly apodize it. We also demonstrate the possibility of achieving high slope GDs and large GVDs by means of high order Fibonacci structures with thicker layers. Finally, by varying the layer dimensions and refractive indices as well as Fibonacci's order, one can design DWDM dispersion compensators suitable for distances as long as 220 km.

Photonic-plasmonic scattering resonances in deterministic aperiodic structures

In this paper, we combine experimental dark-field scattering spectroscopy and accurate electrodynamics calculations to investigate the scattering properties of two-dimensional plasmonic lattices based on the concept of aperiodic order. In particular, by discussing visible light scattering from periodic, Fibonacci, Thue-Morse and Rudin-Shapiro lattices fabricated by electron-beam lithography on transparent quartz substrates, we demonstrate that deterministic aperiodic Au nanoparticle arrays give rise to broad plasmonic resonances spanning the entire visible spectrum. In addition, we show that far-field diffractive coupling is responsible for the formation of characteristic photonic-plasmonic scattering modes in aperiodic arrays of metal nanoparticles. Accurate scattering simulations based on the generalized Mie theory approach support our experimental results. The possibility of engineering complex metal nanoparticle arrays with distinctive plasmonic resonances extending across the entire visible spectrum can have a significant impact on the design and fabrication of novel nanodevices based on broadband plasmonic enhancement.

Constraints on transmission, dispersion, and density of states in dielectric multilayers and stepwise potential barriers with an arbitrary layer arrangement

Normal-incidence transmission and dispersion properties of optical multilayers and one-dimensional stepwise potential barriers in the nontunneling regime are analytically investigated. The optical paths of every constituent layer in a multilayer structure, as well as the parameters of every step of the stepwise potential barrier, are constrained by a generalized quarter-wave condition. No other restrictions on the structure geometry are imposed, i.e., the layers are arranged arbitrarily. We show that the density of states (DOS) spectra of the multilayer or barrier in question are subject to integral conservation rules similar to the Barnett-Loudon sum rule but occurring within a finite frequency or energy interval. In the optical case, these frequency intervals are regular. For the potential barriers, only nonperiodic energy intervals can be present in the spectrum of any given structure, and only if the parameters of constituent potential steps are properly chosen. The integral conservation relations derived analytically have also been verified numerically. The relations can be used in dispersion-engineered multilayer-based devices, e.g., ultrashort pulse compressors or ultracompact optical delay lines, as well as to design multiple-quantum-well electronic heterostructures with engineered DOS.

Electromagnetic wave propagation in quasi-periodic photonic circuits

We study theoretically and experimentally the properties of quasiperiodic one-dimensional serial loop structures made of segments and loops arranged according to a Fibonacci sequence (FS). Two systems are considered. (i) By inserting the FS horizontally between two waveguides, we give experimental evidence of the scaling behaviour of the amplitude and the phase of the transmission coefficient. (ii) By grafting the FS vertically along a guide, we obtain from the maxima of the transmission coefficient the eigenmodes of the finite structure (assuming the vanishing of the magnetic field at the boundaries of the FS). We show that these two systems (i) and (ii) exhibit the property of self-similarity of order three at certain frequencies where the quasiperiodicity is most effective. In addition, because of the different boundary conditions imposed on the ends of the FS, we show that horizontal and vertical structures give different information on the localization of the different modes inside the FS. Finally, we show that the eigenmodes of the finite FS coincide exactly with the surface modes of two semi-infinite superlattices obtained by the cleavage of an infinite superlattice formed by a periodic repetition of a given FS.

Fibonacci-like photonic structure for femtosecond pulse compression

The compression of femtosecond laser pulses by linear quasiperiodic and periodic photonic multilayer structures is studied both experimentally and theoretically. We compare the compression performance of a Fibonacci and a periodic structure with similar total thickness and the same number of layers, and find the performance to be higher in the Fibonacci case, as predicted by numerical simulation. This compression enhancement takes place due to the larger group velocity dispersion at a defect resonance of the transmission spectrum of the Fibonacci structure. We demonstrate that the Fibonacci structure with the thickness of only 2.8 microm can compress a phase-modulated laser pulse by up to 30%. The possibility for compression of laser pulses with different characteristics in a single multilayer is explored. The operation of the compressor in the reflection regime has been modeled, and we show numerically that the reflected laser pulse is subjected to real compression: not only does its duration decrease but also its amplitude rises.

Correlation between single-cylinder properties and bandgap formation in photonic structures

The origin of frequency gaps in the dispersion relation of periodic, quasi-periodic, and random photonic structures consisting of different arrangements of dielectric cylinders has been investigated. For TM polarization it was found that the formation and properties of gaps are strongly affected by Mie resonances of a single cylinder. Both the spectral position and size depend on the properties of this single scatterer. In contrast, for TE polarization no correlation between the scattering properties and bandgap formation was found, as Mie resonances are spectrally not well separated. For the inverted structure consisting of air cylinders in a dielectric material, the frequency gaps depend on the spatial arrangement of the cylinders because no pronounced Mie resonances exist in this case.

Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures

We simulate an evolutionary process in the lab for designing a novel high confinement photonic structure, starting with a set of completely random patterns, with no insight on the initial geometrical pattern. We show a spontaneous emergence of periodical patterns as well as previously unseen high confinement subwavelength bowtie regions. The evolved structure has a Q of 300 and an ultrasmall modal volume of 0.112 (lambda/2n)3. The emergence of the periodic patterns in the structure indicates that periodicity is a principal condition for effective control of the distribution of light.