Analytical solution for the field in photonic structures containing cubic nonlinearity

In this work, we consider the exact solution of the stationary cubic nonlinear equation in a semi-infinite nonlinear medium in contact with a one-dimensional photonic crystal. Two kinds of analytical solutions are found for an arbitrary magnitude of the nonlinearity: a standing-wave-like one containing the inverse elliptic function Eli(?∣m), and a one-wave-type solution for transmitted TE-polarized waves. An approximate two-wave solution is proposed to describe the field propagation through the nonlinear film covering the photonic crystal. It is shown that the problem of a mixed linear-nonlinear structure may be reduced to a transcendental kernel equation determining the field inside the nonlinear part of the medium. The light reflection from a Si/SiO₂ layered structure in contact with an optically nonlinear medium is calculated. The angular-frequency photonic band diagram and power dependency are investigated. Local interface waveguide modes are considered.     

Technique for determining the cell parameters of the confined one-dimensional photonic crystal,based on the Floquet-Bloch formalism

The technique for determining the cell structural parameters of the confined one-dimensional photonic crystal, based on the strict analytical representation of Floquet-Bloch waves in the form of inhomogeneous waves is proposed.     

Spin Hall effect of light in one-dimensional photonic crystal

We established a general propagating model to investigate the spin Hall effect of light in one-dimensional photonic crystal. A polarized (spin dependent) Gaussian beam which was incident obliquely through one-dimensional photonic crystal was demonstrated. Having decomposed a polarized Gaussian beam into different plane wave components charactering individual wave vectors, we revealed the transmission coefficient and reflection coefficient of each plane wave which propagates through the one-dimensional photonic crystal. It enabled us to obtain exact solution to the electric field of transmitted and reflected beams, and the analytical formula of light intensity, accordingly. A method based upon the partial differentials with the intensity distribution of the transmitted and reflected Gaussian beams was presented to determine the transverse and longitude shifts explicitly. Spin dependent shifts in one-dimensional photonic crystal provide alternative evidence for the spin Hall effect of light.     

Physical optics of photonic crystals

A one-dimensional harmonic model of a photonic crystal is considered using the methods of physical optics. The theoretical formalism is based on the notion of the Fresnel volume reflection and the system of two first-order differential equations, which are equivalent to the wave equation. Using the Rayleigh layer as an example, it is shown that the volume reflection plays a role of the friction, similar to the friction in oscillations of a pendulum, and, in a strongly inhomogeneous medium, can suppress field oscillations and turn the group velocity to zero. In the approximation of small modulation factor, the models of two, four, and six waves are considered. In the two-wave model, the dispersion relation contains a zone of inhomogeneous waves, whose width is determined by the Fresnel reflection coefficient from one period. The refinement in terms of the four-wave or six-wave model yields only a small correction to the position of the zone, retaining its width unchanged. The wavenumber as a function of frequency is described by a circle inside the zone of inhomogeneous waves and by a hyperbola outside this zone. Mathematically, the method used is significantly simpler than those based on the application of the Floquet theorem to the wave equation. It is shown that the notion of forbidden zones is inconsistent with respect to photonic crystals, and the term zones of inhomogeneous waves is proposed instead.     

Modified phase matching conditions for second harmonic generation in a finite one-dimensional photonic crystal near the bragg condition: Weak and strong diffraction

Enhanced noncollinear second harmonic generation in a finite one-dimensional photonic crystal is analyzed theoretically under conditions of pump field localization near the Bragg reflection. It is shown numerically that phase-matched second-harmonic generation can be implemented in a finite one-dimensional photonic crystal that does not satisfy the conventional phase-matching conditions calculated for effective Bloch modes with narrow spectral lines. The intensity of the generated second-harmonic signal exceeds the second-harmonic intensity attained under the conventional phase-matching conditions by more than an order of magnitude. This phenomenon is explained by interference between Bloch modes having similar amplitudes, wavenumbers, and spectral widths. Since the spatial spectra of waves propagating in a bounded medium have finite widths, the broadened spectral lines of proximate effective Bloch modes resulting from Bragg diffraction of waves tuned to the first transmission resonances near the photonic bandgap edge overlap, merging into a spectral profile with center shifted relative to the original effective Bloch wavevectors. This effect leads to modified phase matching conditions for second harmonic generation in a finite photonic crystal, which are written for the centers of the spectral profiles resulting from modal overlap, rather than for individual effective wavevectors. Substantially different phase matching conditions are obtained for weakly and strongly diffracted beams, whereas conventional phase matching conditions hold only for transmitted signals in the case of weak diffraction.     

Leaky modes and directed modes in confined one-dimensional photonic crystal

Directed modes and leaky modes in a confined one-dimensional photonic crystal are analytically described. The dispersion relation of these modes is written as the transverse resonance condition of Floquet-Bloch waves, taking into account phase shifts at interfaces with adjacent media.     

Phase modulation of Bloch surface waves with the use of a diffraction microrelief at the boundary of a one-dimensional photonic crystal

Phase modulation methods for surface electromagnetic waves propagating at the interface between a homogeneous medium and a one-dimensional photonic crystal have been analyzed numerically and theoretically. Modulation is performed by changing the geometrical parameters of the microrelief on the surface of the photonic crystal. The phase modulation methods under consideration can be used to create optical elements for surface waves, in particular, lenses, prisms, and diffraction gratings. A Bragg grating has been calculated as an example. According to the simulation results, the average coefficient of reflection of a surface wave in the band gap is 0.95.     

Eigenwaves in one-dimensional photonic crystals with gain

The dispersion characteristics of a one-dimensional photonic crystal are analyzed in detail on the basis of the exactly solvable Kronig-Penney model for the particular case of a periodic system consisting of an infinite number of insulator-air layers arranged in the direction of propagation of radiation. An analytical solution is obtained, and the behavior of eigenwaves near the edges of the photonic band is thoroughly studied. The behavior of roots on a complex plane is qualitatively analyzed with the use of approximate dispersion characteristics obtained via expanding the solution of the wave equation in series in terms of the harmonics of the lattice spacing. It is shown that, despite the amplifying properties of the material, no gain in the photonic band gap is present, because of the interference quenching of waves.     

The dynamic Casimir effect within a vibrating metal photonic crystal

The lattice-vibrating metal photonic crystal is exactly a system of dynamical Casimir effect connected in series, and so we can expect that a dynamical Casimir effect is enhanced by the photonic band effect. In the present study, when an electromagnetic field between metal plates is in the ground state in a one-dimensional metal photonic crystal, the radiation of electromagnetic wave in excited states has been investigated by artificially introducing lattice vibration to the photonic crystal. In this case as well as a dynamical Casimir effect, it has been shown that the harmonics of a ground state are generated just by vibrating a photonic crystal even without an incident wave. The dependencies of the radiating power on the number of layers and on the wavenumber of the lattice vibration are remarkable. It has been found that the radiation amplitude on lower excited states is not necessarily large and radiation on specific excited levels is large.     

色散负折射特异介质的1维光子晶体缺陷态

在1维光子晶体中引入色散负折射特异介质,分析了其介电常数和磁导率在微波区（1～10 GHz）与频率的关系。分别对以色散负折射介质为缺陷的正折射率介质光子晶体及以正折射率介质为缺陷的正负折射率介质光子晶体进行了数值仿真,得出了均匀平面电磁波在正入射和斜入射时的透射谱。结果表明:在介电常数和磁导率均为负值的不同禁带中,分别出现了单、双缺陷模。利用透射谱的这一特点,可实现单、双通道滤波。     

Excitation of surface electromagnetic waves at an anisotropically conducting vacuum-metamaterial interface by the attenuated total internal reflection method

The excitation of well-localized oblique surface waves above the surface of a dielectric with a one-dimensional array of perfectly conducting wires is studied theoretically using the attenuated total internal reflection method. It is assumed that the distance between the wires and their diameter are much smaller than the surface wavelength. The frequencies of excited surface waves are much lower than the plasma frequency of the metal, and their electric field is orthogonal to the wires. It is shown that such surface waves can be excited with the help of a homogeneous TM wave as well as with the help of a homogeneous wave with an electric field polarized perpendicularly to the wires. It is found that in the course of excitation of oblique waves, the incident TM wave is partly polarized into a wave of the TE type.     

Narrowband spectral filters on the basis of threadlike metal Si photonic crystals

The scattering of transmitted and evanescent electromagnetic waves by threadlike photonic crystals, made of different materials, is studied theoretically. The reflection and transmission spectra of the waves, the electric vector of which is parallel to the threads, are calculated on the basis of a matrix Riccati equation. The physical reasons for the appearance of transparency peaks in the transmission spectra of threadlike photonic crystals are discussed. It is established that narrowband filters suggested previously and formed by a periodic system of Si threads about 1 μm in diameter and adjusted to a wavelength of 1.5 μm totally lose their filtering capabilities at small deviations (about 5%) from the thread diameter. However, the reflection and transmission spectra of filters containing periodically packed metallized Si threads are stable in the case of such variations in thread size. It is shown that threadlike crystals are an effective discrete mirror for radiation of the near- and mid-IR spectral range. The capabilities of threadlike crystals for filtering homogeneous and inhomogeneous waves are shown.     

Density of states of a one-dimensional disordered photonic crystal

An analytic theory of the density of states in one-dimensional disordered photonic crystals is proposed. It is shown that the problem of the density of optical modes can be reduced in the small dielectric contrast approximation to solving a generalized Fokker-Planck equation for the distribution function of the logarithmic derivative of the electric field (the wave phase). The exact analytic solution and density-of-states asymptotics deep in the band gap of the photonic crystal and close to the band gap edge are derived. The results obtained agree well with the empirical relations derived earlier from numerical experiments.     

Optical Tamm states at the interface between a photonic crystal and a nanocomposite with resonance dispersion

Optical Tamm states localized at the edges of a photonic crystal bounded from one or both sides by a nanocomposite have been studied. The nanocomposite consists of metallic nanoinclusions, which have a spherical or orientationally ordered spheroidal shape and are dispersed in a transparent matrix, and is characterized by the resonant effective permittivity. The transmission, reflection, and absorption spectra have been calculated for waves with longitudinal and transverse polarizations in such structures at the normal incidence of light. The spectral manifestation of Tamm states that is due to the existence of negative values of the real part of the effective permittivity has been analyzed for the visible spectral range. It has been established that the characteristics of Tamm states localized at the edge of the photonic crystal depend strongly both on the concentration of nanoballs in the nanocomposite film and on its thickness. Modes formed by two coupled Tamm plasmon polaritons localized at the edges of the photonic crystal adjacent to two nanocomposite layers have been examined. It has been shown that, in the case of the anisotropic nanocomposite layer adjacent to the photonic crystal, each of two orthogonal polarizations of the incident wave corresponds to a specific frequency of the Tamm state localized at the edge; owing to this property, the transmission spectra of such a structure are polarization sensitive.     

Quantum electron plasma in one-dimensional metallic–dielectric photonic crystal

The interaction of the electromagnetic radiation with one-dimensional photonic crystal consisting of metal and transparent dielectric medium is studied numerically. Dielectric permeabilities of the electron plasma in the metal are considered both in the quantum Mermin and in the classical Drude–Lorentz approaches. It is shown that the reflection, transmission and absorption-frequency zones of electromagnetic radiation appear in the photonic crystal. In addition, the reflectance, transmittance and absorptance optical coefficients for such photonic crystal in the quantum approach differ from those coefficients in the Drude–Lorentz approach.     

Zero reflection from layered anisotropic metamaterial structures

Mathematical formulation for analysis of electromagnetic wave scattering from multilayer anisotropic homogeneous metamaterial structures with general tensors is presented. The result is reflection and transmission matrices formulas. Then, closed form formulas are extracted for structures with diagonal tensors. We try to find structures with exact zero reflection for general incident waves. New cases with zero and low reflection are introduced.     

Polariton dynamics of a one-dimensional gyrotropic magnetic photonic crystal in a dc electric field: I. Peculiarities of refraction

It is shown that both TE and TM waves incident on the surface of a semi-infinite one-dimensional thin-layer gyrotropic magnetic photonic crystal of the magnetic-nonmagnetic dielectric type can experience a number of refraction anomalies in an external electric field due to the effect of hybridization of evanescent waves in magnetic and nonmagnetic layers. These anomalies are absent under the same conditions in spatially homogeneous media consisting of the same materials as the photonic crystal.     

可见光波段SiO2/CdSe一维光子晶体及缺陷模的研究究

采用SiO2/CdSe构建了可见光波段一维光子晶体结构,并在其中引入LiTaO3缺陷层。利用传输矩阵法,分析了电磁波在无缺陷与含LiTaO3缺陷层两种光子晶体中的带隙结构,系统地研究了缺陷层参数对光子晶体可见光波段带隙结构的影响规律。计算结果表明：LiTaO3的引入,有利于带隙宽度的增加,调整缺陷层结构参数,缺陷模的位置可在不同颜色区域出现,如红光、黄光等缺陷模。该结构有望用于制作可见光波段的滤波器。     

Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-microm wavelength

Highly efficient transmission of 1.5-microm light in a two-dimensional (2D) photonic crystal slab waveguide is experimentally demonstrated. Light waves are shown to be guided along a triple-line defect formed within a 2D crystal and vertically by a strong index-guiding mechanism. At certain wavelength ranges, complete transmission is observed, suggesting lossless guiding along this photonic one-dimensional conduction channel.     

The Sommerfeld precursor in photonic crystals

We calculate the Sommerfeld precursor that results after transmission of a generic electromagnetic plane wave pulse with transverse electric polarization, through a one-dimensional rectangular N-layer photonic crystal with two slabs per layer. The shape of this precursor equals the shape of the precursor that would result from transmission through a homogeneous medium. However, amplitude and period of the precursor are now influenced by the spatial average of the plasma frequency squared instead of the plasma frequency squared for the homogeneous case.