Nonlinear Photonic Bandgap Structure Research Articles

Application of transfer matrix method to second-harmonic generation in nonlinear photonic bandgap structures: oblique incidence

Generalization of the transfer matrix method is developed to analyze Type I second-harmonic generation in linear–nonlinear multilayer one-dimensional photonic bandgap structures for oblique incidence of a nondepleted fundamental. The advantage of the transfer matrix method is that it takes into account reflections and interferences between all forward and backward propagating fundamental and second-harmonic waves. The conversion efficiency is calculated as a function of the incident angle of the fundamental and the thicknesses of the linear and nonlinear layers. Specific incident angles and thicknesses may generate relatively high conversion efficiency inside nonlinear material. Our analytical and numerical analyses show that the conversion efficiency of second-harmonic generation depends on the fundamental pump power, second-order susceptibility, and field enhancement in the photonic bandgap structure. Upper bounds on pump intensity can be found for a given incidence angle and sample thickness where the nondepleted pump approximation can be used to model such a nonlinear structure.

An Exponential Time Differencing Algorithm for the FDTD-PML Analysis of Nonlinear Photonic Bandgap Structures

An exponential time differencing (ETD) algorithm is presented for the analysis of photonic crystal structures comprising the third-order nonlinear materials. Compared with the well known Z-transform method and auxiliary differential equation method for incorporating the nonlinearity, the proposed ETD algorithm has the least memory requirement and shows the same accuracy. The stretched coordinates perfectly matched layer (PML) with the complex frequency shifted variable implemented using the ETD algorithm is applied to terminate nonlinear photonic bandgap structures. Compared with the conventional PML and the convolutional PML, the proposed implementation can improve the absorbing performance significantly with similar memory requirement.

Continuation Finite Element Simulation of Second Harmonic Generation in Photonic Crystals

AbstractA computational study on the enhancement of the second harmonic generation (SHG) in one-dimensional (1D) photonic crystals is presented. The mathematical model is derived from a nonlinear system of Maxwell’s equations, which partly overcomes the shortcoming of some existing models based on the undepleted pump approximation. We designed an iterative scheme coupled with the finite element method which can be applied to simulate the SHG in one dimensional nonlinear photonic band gap structures in our previous work. For the case that the nonlinearity is strong which is desirable to enhance the conversion efficiency, a continuation method is introduced to ensure the convergence of the iterative procedure. The convergence of our method is fast. Numerical experiments also indicate the conversion efficiency of SHG can be significantly enhanced when the frequencies of the fundamental and the second harmonic wave are tuned at the photonic band edges. The maximum total conversion efficiency available reaches more than 50% in all the cases studied.

The transmission and dispersion characteristics in nonlinear photonic band gap structure with variable period

The variation of the transmission and the dispersion with respect to the frequency of one-dimensional nonlinear photonic band gap material with variable period is demonstrated by numerical simulation in this paper. It is shown that the position of the photonic band gap will shift to the red side with the increase of the thickness of the layers. This effect becomes obvious with the increasing of frequency. And for the same structure, it can be shown that the photonic band gaps shift to the red side and the transmission with respect to the frequency becomes larger with the increasing of the applied intensity.

Numerical investigation of a self-induced transparency soliton in a nonlinear photonic bandgap structure doped uniformly with two-level atoms

We numerically study the characteristics of a nonlinear pulse that propagates through a one-dimensional photonic bandgap (PBG) structure uniformly doped with two-level atoms. The numerical model adopted is the coupled system of nonlinear coupled-mode equations and atomic Bloch equations. The simulation results show that a self-induced transparency (SIT) soliton and a gap soliton can coexist in a nonlinear PBG structure uniformly doped with resonant atoms. Although this mixed state, known as a SIT-gap soliton, near the PBG edge has been theoretically predicted, we numerically show that such a solitary wave still exists even if its central frequency is located deep inside the PBG. The propagating characteristics of the SIT-gap soliton are also discussed.

Finite Element Analysis of Optical Bistability in One-Dimensional Nonlinear Photonic Band Gap Structures with a Defect

We present a new approach based on the recently reported finite element scheme16 to study the optical response of a finite one-dimensional nonlinear grating. Using the transmitted wave amplitude as a numerical input parameter, we are able to find all stable and unstable solutions related to a specific incident wave which build up the complete bistability curve. The method is applied to investigate the optical bistability in a nonlinear quarter-wave reflector with a defect. With a proper choice of the incident light frequency, a very low bistability threshold is predicted for an optimized defect structure.

Nonlinear photonic band-gap structure with extremely high efficiency for third-harmonic generation

A method of tailoring the electromagnetic states in photonic band-gap structures was used in designing a dielectric structure optimized for third-harmonic generation. The structure we considered was a stack of several kinds of χ(2) nonlinear bilayers approximately 10 μm in length. In the theoretical analysis and numerical experiment, a conversion efficiency enhancement of more than 14 orders of magnitude, as compared with that in a normal conventional setup of two crystals, was found. The possibility of efficient third-harmonic generation in a single structure of χ(2) materials and the phenomenal enhancement of the conversion efficiency show that the electromagnetic state control in photonic band-gap structures offers great potential in nonlinear optics.

Distortionless pulse-train propagation in a nonlinear photonic bandgap structure doped uniformly with inhomogeneously broadening two-level atoms

The pulse propagation in a one-dimensional nonlinear photonic bandgap (PBG) structure doped uniformly with inhomogeneously broadening two-level atoms is investigated. The Maxwell-Bloch equations describing pulse propagation in such a uniformly doped PBG structure are derived first and further reduced to effective nonlinear coupled-mode equations. An exact analytic pulse-train solution to these effective coupled-mode equations is obtained. Such a distortionless; pulse-train solution is given by sinusoidal functions with a DC background and a modulated phase. Numerical examples of the distortionless pulse train in a silica-based PBG structure doped uniformly with Lorentzian lineshape two-level atoms are shown.

Localized electromagnetic modes of one-dimensional modulated nonlinear photonic band-gap structures

The localized electromagnetic modes of a modulated one-dimensional photonic band-gap structure which consists of a sandwich of linear-nonlinear-linear multilayers were investigated. A number of solitonlike modes induced by the nonlinear effect have been found, and their exact analytical experssion has been derived. The intensity-dependent transmission has been investigated, and complete resonant transmission has been found.

Raman gap solitons

We show that an intense pump pulse, detuned far from the Bragg resonance of a nonlinear periodic structure, can excite a gap soliton at a wavelength within the band gap that corresponds to the Raman shift of the medium. This Raman gap soliton is a stable, long-lived, quasistationary excitation that exists within the grating even after the pump pulse has passed. We find both stationary solitons as well as slow Raman gap solitons with velocities as low as 1% of the speed of light. The predicted phenomena should be observable in fiber Bragg gratings and other nonlinear photonic band gap structures.

Dynamic cross-waveguide optical switching with a nonlinear photonic band-gap structure

We present a numerical study of a two dimensional all- optical switching device which consists of two crossed waveguides and a nonlinear photonic band{gap structure in the center. The switching mechanism is based on a dynamic shift of the photonic band edge by means of a strong pump pulse and is modeled on the basis of a two dimensional finite volume time domain method. With our arrangement we find a pronounced optical switching effect in which due to the cross-waveguide geometry the overlay of the probe beam by a pump pulse is significantly reduced.

Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures

We present a detailed analysis of finite energy solitary waves in two- and three-dimensional nonlinear periodic structures exhibiting a complete photonic band gap. Solitary waves in photonic crystals with a two-dimensional (2D) square and triangular symmetry group as well as a 3D fcc symmetry group are described in terms of an effective nonlinear Dirac equation derived using the slowly varying envelope approximation for the electromagnetic field. Unlike one-dimensional Bragg solitons, the multiple symmetry points of the 2D and 3D Brillouin zones give rise to two distinct classes of solitary wave solutions. Solutions associated with a higher order symmetry point of the crystal exist for both positive and negative Kerr nonlinearities, whereas solutions associated with a twofold symmetry point occur only for positive Kerr coefficient. Using a variational method we derive the important physical features such as the size, shape, peak intensity, and total energy of the solitary waves. This is then confirmed numerically using the finite element Ritz-Galerkin method. It is shown that the initial variational method and the finite element numerical method are in good agreement. We discuss the stability of these solitary waves with respect to small perturbations. It is suggested that an analytical stability criterion for spinor fields satisfying the nonlinear Dirac type of equation may exist, similar to the well known stability criterion for solitary waves in the nonlinear Schr\"odinger equation. Our stability criterion correctly reproduces the stability conditions of other nonlinear Dirac type of equations which have been studied numerically. Our study suggests that for an ideal Kerr medium, two-dimensional solitary waves in a band gap are stable, whereas three-dimensional ones are stable only in certain regions of the gap.

Optical limiting and switching of short pulses by use of a nonlinear photonic bandgap structure with a defect

An extension of the recently introduced nonlinear finite-difference time-domain technique [Opt. Lett.21, 1138 (1996)] for the study of electromagnetic wave propagation in a non-linear Kerr medium to include absorption is presented. The optical limiting and switching of short pulses by use of a nonlinear quarter-wave reflector (a one-dimensional photonic bandgap structure) with a defect is studied. Comparison with an optical limiter and with an optical switch with a perfect nonlinear quarter-wave reflector shows that introducing a defect can improve the performance of these devices.