A semi-definite optimization method for maximizing the shared band gap of topological photonic crystals

Photonic crystal is a kind of periodic optical nanostructure consisting of two or more materials with different dielectric constants, which has attracted great deal of attention because of its wide range of potential applications in the field of optics. Photonic crystal can be fabricated into one-, or two-, or three- dimensional one. Among them, the two-dimensional photonic crystal turns into a hot focus due to its fantastic optical and electrical properties and relatively simple fabrication technique. Since the tunable band gaps of two-dimensional photonic crystals are beneficial to designing the novel optical devices, to study their optical and electrical properties for controlling the electromagnetic wave is quite valuable in both theoretical and practical aspects. In this work, we propose a new type of two-dimensional function photonic crystal, which can tune the band gaps of photonic crystals. The two-dimensional function photonic crystal is different from the traditional photonic crystal composed of medium columns with spatially invariant dielectric constants, since the dielectric constants of medium column are the functions of space coordinates. Specifically, the photorefractive nonlinear optical effect or electro-optic effect is utilized to turn the dielectric constant of medium column into the function of space coordinates, which results in the formation of two-dimensional function photonic crystal. We use the plane-wave expansion method to derive the eigen-equations for the TE and TM mode. By the Fourier transform, we obtain the Fourier transform form (G) for the dielectric constant function (r) of two-dimensional function photonic crystal, which is more complicated than the Fourier transform in traditional two-dimensional photonic crystal. The calculation results indicate that when the dielectric constant of medium column is a constant, the Fourier transforms for both of them are the same, which implies that the traditional two-dimensional photonic crystal is a special case for the two-dimensional function photonic crystal. Based on the above theory, we calculate the band gap structure of two-dimensional function photonic crystal, especially investigate in detail the corresponding band gap structures of TE and TM modes. The function of dielectric constant can be described as (r) = kr + b, in which k and b are adjustable parameters. Through comparing the calculation results for both kinds of photonic crystals, we can find that the band structures of TE and TM modes in two-dimensional function photonic crystals are quite different from those in traditional two-dimensional photonic crystal. Adjusting parameter k, we can successfully change the number, locations and widths of band gaps, indicating that the band gap structure of two-dimensional function photonic crystal is tunable. These results provide an important design method and theoretical foundation for designing optical devices based on two-dimensional photonic crystal.

The photonic crystals of parylene and silicon dielectric media for infrared light localization are analyzed in this paper. The Bloch's theorem is adopted to calculate the infrared light transmission in two-dimensional photonic crystal. First, the band gap diagrams for photonic crystal of parylene and silicon are calculated and compared respectively. It is revealed that the photonic crystal of parylene rods in air has a bigger band gap for TM than that for TE mode. In the photonic crystal of air hole in dielectric slab, the stop band width for TE mode is bigger than that for TM wave, and the band gap of silicon photonic crystal is more obvious than that of parylene slab. The energy distribution and boundary condition of electrical field in the interface of dielectric media are considered to be responsible for the reason of the band gap differences for TE and TM wave. Second, the band gap vs. air hole radius of parylene and silicon photonic crystal is obtained, which shows the relationship of stop band width vs. air hole radius. Third, the infrared light localization in point defect is found, and the electrical field profiles for both parylene and silicon photonic crystals are shown. The central point defect in photonic crystal acts as a resonant cavity to confine infrared light and reach high photon density. Finally, the energy confinement efficiency vs. lattice arrangement of photonic crystal is calculated, which can be useful for photonic crystal design and fabrication.

3 - Design of bandgaps in photonic crystals

Modal dynamic residual-based model updating through regularized semidefinite programming with facial reduction

Important technological efforts have been made in the last five years for implementing the concept of photonic bandgap (PBG) crystals [1] in the optical frequency range. Air/semiconductor crystals are very attractive in view of a monolithic integration in optoelectronic integrated circuits (OEICs), because their large modulation of the refractive index potentially allows to obtain large PBGs for each or eventually both polarizations of light. In order to display a PBG in the near-infrared, the period P of such crystals must however be scaled down to submicron sizes. Photonic properties are very sensitive to the porosity of the crystal as well as to some details of its pattern, which makes the demands in terms of regularity and uniformity difficult to satisfy even for state of the art microfabrication techniques. For instance, the dry etching in a single step of 3D [2] or 2D [3–5] PBG crystals illustrates the current limits of etching techniques: the deviations from a perfect anisotropy limit the depth of good quality crystals to typically 1 µm. Concerning alternative approaches now, the electrochemical etching of deep 2D crystals, which is very successful in the mid-infrared (P≈8 µm) [6], might prove difficult to implement in the near-infrared due to the thinness of the semiconductor sidewalls. Finally, imperfect mask alignment will also plague planar period by period fabrication of 3D PBG crystals [7]. Hopefully, thin 2D PBG crystals are in principle sufficient for most potential applications of PBG crystals in OEICs. Hybrid 3D microcavities formed by a 2D PBG crystal sandwiched by two bragg mirrors have also been proposed as a route toward full spontaneous emission control [3]. The structural quality of thin 2D PBG crystals fabricated by electron-beam lithography and reactive ion etching [3,5] is presumably already good enough to test these proposals.

Topological photonic crystals inherit the unique properties of topological insulators, including topologically protected energy transfer and unidirectional propagation, which offer an excellent platform for exploring exotic physics and developing photonic devices. However, topological photonic crystals possessing mid-infrared edge modes that have potential applications in infrared imaging, biosensing, thermal radiation energy transfer, etc., are seldom brought into focus. In this work, we study the topological properties of a photonic crystal slab (PCS) consisting of silicon square veins in the mid-infrared, which is intended to mimic the two-dimensional Su–Schrieffer–Heeger model. By interfacing topologically trivial and nontrivial PCSs, mid-infrared edge modes can appear at domain wall, according to the principle of bulk-edge correspondence. It is also demonstrated high-efficiency mid-infrared light transport can be achieved by these edge modes. In addition, adjusting the vertical offset near the interface can manipulate the bandwidth for various applications and turns the connected PCS structure to a photonic realization of Rice–Mele model. We further fabricate the PCS and provide an experimental observation of transverse-electric-like edge modes in mid-infrared by using the scattering-type scanning near-field optical microscope. Additionally, we integrate it with phase change material of nanoscale thickness, Ge2Sb2Te5, to realize an ultrafast and switchable topological waveguide with zero static power. This work not only enriches the fundamental understanding of topological physics in mid-infrared optical settings, but also shows promising prospects in compact devices for energy transfer and information processing for light sources in these wavelengths, for instance, thermal radiation.

We develop a class of supercell photonic crystals supporting complete photonic bandgaps based on breaking spatial symmetries of the underlying primitive photonic crystal. One member of this class based on a two-dimensional honeycomb structure supports a complete bandgap for an index-contrast ratio as low as $n_{high}/n_{low} = 2.1$, making this the first such 2D photonic crystal to support a complete bandgap in lossless materials at visible frequencies. The complete bandgaps found in such supercell photonic crystals do not necessarily monotonically increase as the index-contrast in the system is increased, disproving a long-held conjecture of complete bandgaps in photonic crystals.

We propose a compact, high efficient full photonic crystal Mach-Zehnder (MZ) interferometer based on the self-collimation and photonic band gap in two-dimensional photonic crystals. Line defected photonic crystals are used as the beam splitter and the mirror. The interference theory is used to discuss the interferometer output mechanism, and compared with the finite-difference time-domain (FDTD) simulation results. The designed MZ interferometer can surve as micro-detectors of gas and liquid, which may play an important role in integrated optics.

The study of photonic crystals has emerged as an attractive area of research in nanoscience in the last years. In this work, we study the properties of a two-dimensional photonic crystal composed of dielectric rods. The unit cell of the system is composed of six rods organized on the sites of a C6 triangular lattice. We induce a topological phase by introducing an angular perturbation ϕ in the pristine system. The topology of the system is then determined by using the so-called k.p perturbed model. Our results show that the system presents a topological and a trivial phase, depending on the sign of the angular perturbation ϕ. The topological character of the system is probed by evaluating the electromagnetic energy density and analyzing its distribution in the real space, in particular on the maximal Wyckoff points. We also find two edge modes at the interface between the trivial and topological photonic crystals, which present a pseudospin topological behavior. By applying the bulk-edge correspondence, we study the pseudospin edge modes and conclude that they are robust against defects, disorder and reflection. Moreover, the localization of the edge modes leads to the confinement of light and the interface behaves as a waveguide for the propagation of electromagnetic waves. Finally, we show that the two edge modes present energy flux propagating in opposite directions, which is the photonic analogue of the quantum spin Hall effect.

A topological photonic crystal InGaAsP/InP core-shell nanowire array laser operating in the 1550 nm wavelength band is proposed and simulated. The structure is composed of an inner topological nontrivial photonic crystal and outer topological trivial photonic crystal. For a nanowire with height of 8 µm, high quality factor of 4.7 × 104 and side-mode suppression ratio of 11 dB are obtained, approximately 32.9 and 5.5 times that of the uniform photonic crystal nanowire array, respectively. Under optical pumping, the topological nanowire array laser exhibits a threshold 27.3% lower than that of the uniform nanowire array laser, due to the smaller nanowire slit width and stronger optical confinement. Moreover, the topological NW laser exhibits high tolerence to manufacturing errors. This work may pave the way for the development of low-threshold single-mode high-robustness nanolasers.

Topological photonics aims to utilize topological photonic bands and corresponding edge modes to implement robust light manipulation, which can be readily achieved in the linear regime of light-matter interaction. Importantly, unlike solid state physics, the common test bed for new ideas in topological physics, topological photonics provide an ideal platform to study wave mixing and other nonlinear interactions. These are well-known topics in classical nonlinear optics but largely unexplored in the context of topological photonics. Here, we investigate nonlinear interactions of one-way edge-modes in frequency mixing processes in topological photonic crystals. We present a detailed analysis of the band topology of two-dimensional photonic crystals with hexagonal symmetry and demonstrate that nonlinear optical processes, such as second- and third-harmonic generation can be conveniently implemented via one-way edge modes of this setup. Moreover, we demonstrate that more exotic phenomena, such as slow-light enhancement of nonlinear interactions and harmonic generation upon interaction of backward-propagating (left-handed) edge modes can also be realized. Our work opens up new avenues towards topology-protected frequency mixing processes in photonics.

A new kind of heterostructures containing 3D diamond and 2D holes structures, and diamond-structure photonic crystals and 2D holes-structure photonic crystals fabricated by stereolithography and gel-casting with alumina were studied at microwave range, respectively. The heterostructures were designed by 2D holes structure embedded in 3D diamond structure, in which the lattice of three kinds of structures was equivalent. It was found that the band gaps of photonic crystal heterostructure were broadened by 124.6% and 150% comparing to that of diamond-structure crystal and 2D aerial holes structure. Experimental results showed the band gap broadened was not connected with a linear superposition of the band gap of 2D and 3D photonic crystals, which was the superposition of partial overlap.

Two-dimensional (2-D) photonic crystals (PhCs) of square lattice have been investigated to find the complete photonic band gap. Band diagram calculations have been performed using the simulator "PWE band solver" of the software package optiFDTD, based on plane wave expansion method. Air holes in dielectric background with four different shapes have been considered. Parameters like filling fraction, rod/air hole orientation angle and material dielectric constant have been varied. It has been found that PhCs with square air holes have the largest gap; the gap-midgap ratio is 13.66% for GaAs (when filling fraction, f=0.680 and orientation angle ϑ = 30°). Finally, a proposal is made to improve the efficiency of thin film solar cell using this photonic crystal as back reflector. Full Text: PDF References J.D. Joannopoulos, S.G. Johnson, J. Winn, R. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, New Jersey 2008) L. Zeng, Y. Yi, C. Hong, J. Liu, N. Feng, X. Duan, L.C. Kimerlin, B.A. Alamariu, "Efficiency enhancement in Si solar cells by textured photonic crystal back reflector", Appl. Phys. Lett. 89, 111111 (2006) [CrossRef] P. Bermel, C. Luo, L. Zeng, L.C. Kimerling, J.D. Joannopoulos, "Improving thin-film crystalline silicon solar cell efficiencies with photonic crystals", Opt. Exp. 15 16986 (2007) [CrossRef] M. Qiu, S. He, "Large complete band gap in two-dimensional photonic crystals with elliptic air holes", Phys. Rev. B 60, 10610 (1999) [CrossRef] G. Qiua, F. Lina, Y.P. Licompelte, Complete two-dimensional bandgap of photonic crystals of a rectangular Bravais lattice (Elsevier Science, 2003) R. Wang, X.H. Wang, B.Y. Gu, G.Z. Yang, "Effects of shapes and orientations of scatterers and lattice symmetries on the photonic band gap in two-dimensional photonic crystals", J. Appl. Phys. 90, 4307 (2001) [CrossRef] J. Gee, "Optically enhanced absorption in thin silicon layers using photonic crystals", 29th IEEE Photovolt. Spec. Conf., pp. 150?153 (2002) J.I. Pankove, Optical Process in Semiconductors (Dover, New York 1971).

Manipulation of topological edge and corner states in photonic Kagome crystals through different combinations

Topological quantum matter can be realized by subjecting engineered systems to time-periodic modulations. In analogy with static systems, periodically driven quantum matter can be topologically classified by topological invariants, whose non-zero value guarantees the presence of robust edge modes. In the high-frequency limit of the drive, topology is described by standard topological invariants, such as Chern numbers. Away from this limit, these topological numbers become irrelevant, and novel topological invariants must be introduced to capture topological edge transport. The corresponding edge modes were coined anomalous topological edge modes, to highlight their intriguing origin. Here we demonstrate the experimental observation of these topological edge modes in a 2D photonic lattice, where these propagating edge states are shown to coexist with a quasi-localized bulk. Our work opens an exciting route for the exploration of topological physics in time-modulated systems operating away from the high-frequency regime.