Coupled-mode Approach Research Articles

Parametric Amplification in Four-Layered QPM Leaky Waveguides

Parametric amplification of signal wave in a four-layered leaky waveguide configuration in Z-cut LiNbO 3 , where the pump and signal waves are guided modes and the idler is radiated into the substrate has been studied using coupled mode approach. It is shown that in this configuration, the signal gain can be greatly enhanced as compared to the case of a three-layered QPM Cerenkov-idler configuration. The studies show that by appropriately choosing the waveguide parameters, one can have a trade-off between signal gain bandwidth and the peak signal gain achieved. These studies should find application in designing optical parametric amplifiers and oscillators.

STIMULATED BRILLOUIN SCATTERING IN n-TYPE III–V PIEZOELECTRIC SEMICONDUCTORS

Based upon the hydrodynamic model of an one-component semiconductor plasma, a detailed analytical study of stimulated Brillouin scattering (SBS) processes in a doped weakly piezoelectric semiconductor has been made following the coupled mode approach. The nonlinear effective polarization and the corresponding third-order optical susceptibility at Stokes frequency for the SBS processes are obtained. The analysis deals with the qualitative behavior of threshold pump electric field required for the onset of SBS and the Brillouin gain with respect to the doping concentration well above threshold. Numerical analysis has been made for n-type InSb crystal duly irradiated by a nanosecond pulsed 10.6 μm CO 2 laser. The influence of piezoelectric properties of the crystal medium on both threshold pump field and Brillouin gain has been explored and it is found that SBS with significant gain at lower threshold pump field could be achieved in the weakly piezoelectric moderately doped n-type semiconductor. The Brillouin gain is noted to be significantly large only over a very narrow doping range. The theoretical analysis shows that the doped III–V semiconductors have a good potentiality in the fabrication of SBS based phase conjugate mirrors with high reflectivity well below the optical damage threshold.

A two-way integral equation approach to a shelf break problem

Benchmark solutions play an important role in the development, testing, and application of numerical models for range-dependent problems in underwater acoustics. To produce benchmark accuracy propagation solutions requires careful attention to both the theoretical formalism and numerical implementation. The focus of this research is a ‘‘benchmark’’ solution for a two-dimensional (2-D) shelf-break problem in ocean acoustics. The method employed to construct a solution is a two-way energy-conserving coupled mode approach in the form of coupled inhomogeneous integral equations. Solutions of the coupled equations are obtained using a powerful approach originally introduced in nuclear theory and also used to solve simple nonseparable problems in underwater acoustics. The basic integral equations are slightly modified to permit a Lanczos expansion to form a solution. The solution of the original set of integral equations is then easily recovered from the solution of the modified equations. Benchmark quality demands that the errors in each step of the numerical calculation be well quantified, so that the resolution in the transmission loss can be determined. Consistency checks are made in the form of using different numerical schemes to compute the adiabatic or free propagator and mode coupling matrix operator.

Analysis and design of large leaky-mode array employing the coupled-mode approach

This paper presents the coupled-mode approach to the analysis and design of a large leaky-mode array, in which an N-element microstrip array above a common ground plane supports N coupled leaky modes (leaky EH/sub 1/ modes) at the first higher order. Following a brief description of a conventional full-wave nonstandard eigenvalue problem for solving the complex propagation constants, we present the detailed formulation of the coupled-mode approach, clearly showing the transformation of the nonstandard eigenvalue problem into a standard one. Thus, all the eigenvalues (complex propagation constants) and eigenvectors (modal current distributions) are solved simultaneously, regardless of the size of the array (N). Two key issues pertinent to the successful implementation of the proposed coupled-mode approach are addressed: the determination of the coupling coefficients and the uniqueness of the isolated uncoupled leaky mode, which represents the leaky modal solution of a single microstrip, but is derived from a system of coupled leaky-mode solutions provided the coupled microstrips have equal width and spacing. Closed-form expressions for obtaining the coupling coefficients of orders two, three, and four are presented. Theoretical studies of closely coupled microstrip arrays of two, three, and four elements show that the magnitude of the coupling coefficient /sub i,i+j/ between elements i and i+ decreases at the order of 10/sup -j/. These theoretical case studies also lead to the same isolated uncoupled leaky-mode solution as predicted. Furthermore, the dispersion characteristics of the microstrip array at the first higher order obtained by the coupled-mode approach and the full-wave approach agree excellently for all the case studies. Error analyses indicated that at least two coupling coefficients (C/sub i,i+1/ andC/sub i,i+2/) are required for obtaining accurate complex propagation constants with rms errors less than 1% for most of the leaky region of the particular array under investigation. An example of applying the proposed coupled-mode- approach for analyzing a corporate-fed leaky-mode array of eight elements is reported, revealing that only four out of the eight leaky modes are excited. The coupled-mode theory predicts the far-field radiation pattern in the main beam region in excellent agreement with the measured results.

INTEGRAL EQUATION COUPLED MODE APPROACH APPLIED TO INTERNAL WAVE PROBLEMS

A two-way coupled mode approach based on an integral equation formalism is applied to sound propagation through internal wave fields defined at the 1999 Shallow Water Acoustics Modeling Workshop. Solutions of the coupled equations are obtained using a powerful approach originally introduced in nuclear theory and also used to solve simple nonseparable problems in underwater acoustics. The basic integral equations are slightly modified to permit a Lanczos expansion to form a solution. The solution of the original set of integral equations is then easily recovered from the solution of the modified equations. Two important aspects of the integral equation method are revealed. First, the Lanczos expansion converges faster than a Born expansion of the original integral equations. Second, even when the Born expansion diverges due to strong mode coupling, the Lanczos expansion converges. It is shown that the internal wave problems examined are essentially one-way propagation problems because one observes good agreement between the coupled mode solutions and those provided by an energy-conserving parabolic equation algorithm. In the Workshop examples, at both 25 and 250 Hz, significantly greater coupling between modes occurs in the linear internal wave field case than the nonlinear soliton case.

Integral Equation Coupled Mode Approach Applied to Internal Wave Problems

Integral Equation Coupled Mode Approach Applied to Internal Wave Problems

Mode coupling in tilted planar waveguide gratings

Three different grating tilting formulas for predicting the optimum grating tilt angle for strong mode coupling in planar waveguide gratings are derived. The optimum tilt angles obtained by the ray-optics approach deviate ~1 degrees for transmissive mode coupling and ~1.5 degrees for reflective mode coupling from those computed by the coupled-mode approach. The coupled-mode analysis and ray-optics analysis of the tilted planar waveguide gratings show that the transmissive planar waveguide gratings should be tilted more than 84 degrees for strong TE(0)-to-TE(mu) mode coupling near 1550 nm.

Large magnetooptical Kerr rotation with high reflectivity from photonic bandgap structures with defects

We perform a theoretical study of enhancement of magnetooptical rotation on reflection of light from a periodic system with a defect. Using calculations based on a coupled mode approach and the transfer matrix method we demonstrate that an asymmetric placing of a single defect allows arbitrary Kerr rotations with better than 99% reflectivity from very short devices.

Injection-locked coupled microstrip leaky-mode antenna array

The paper presents a novel design for an injection-locked microstrip leaky-mode antenna array. It demonstrates that the coupling parameters can be obtained numerically, leading to the analysis of the steady-state phase relationships of the coupled oscillators based on Van der Pol equations for determining the excitation signals of the array. In particular, it applies the coupled-mode approach to investigating the clear and considerable mutual coupling effect on the radiation characteristics of the microstrip leaky-mode array. This, in turn, produces a very efficient and accurate assessment of the radiation far-field patterns. Finally, a proof-of-concept design using a two-element injection-locked microstrip leaky-mode array is presented for experimental verification, showing excellent agreement between theoretical data and measured results.

Fast coupled-mode approximation for broad-band pulse propagation in a range-dependent ocean

An approximation method is proposed for the prediction of broad-band acoustic signals propagating in range-dependent ocean environments. The method is based on the one-way coupled-mode approach and uses second-order frequency expansions of eigenvalues, eigenfunctions, and coupling coefficients about the central frequency of the source. Exploiting analytic expressions for the frequency derivatives of eigenvalues and eigenfunctions, their frequency behavior over the source bandwidth is described from the solution of the vertical eigenvalue problem at the central (reference) frequency. The expansion obtained for the coupling coefficients enables their approximate evaluation at frequencies other than central in terms of reference quantities and, thus, bypasses the need for evaluative eigenfunctions in the intermediate range segments for these frequencies. The method is applied to a 600-km range-dependent section from the Thetis-2 tomography experiment in the western Mediterranean sea. A sufficient degree of accuracy is obtained, particularly for the low- and intermediate-order modes, corresponding to the intermediate and late part of the arrival pattern in the time domain. The approximation is faster by two orders of magnitude than the exact broad-band coupled-mode calculation.

Radiatively coupled waveguide polarization splitter simulated by wave-matching-based coupled mode theory

Coupled mode theory is applied to an arrangement of three raised strip waveguides with a multimode central strip. We use semivectorial numerically computed modes of the three single isolated waveguides as a basis for propagating supermode analysis of the entire structure. The pronounced polarization dependence of the raised strip guides allows for the design of a conveniently short polarization splitter. We discuss design guidelines and estimate the fabrication tolerances. The accuracy of the coupled mode approach is assessed by comparison with rigorously computed supermodes for comparable two waveguide couplers. Both types of structures indicate the limits in the applicability of the coupled mode model.

On the use of stair steps to approximate bathymetry changes in ocean acoustic models

Stair-step discretization criteria have been established for accurately representing smoothly varying bathymetry in numerical models. It is shown that the strictest criterion applies to backscatter calculations, where the horizontal stair-step size must be a small fraction of an acoustic wavelength (Δx⩽λ/4). The forward scatter problem—assuming that backscatter is weak—can be accurately solved with an order-of-magnitude larger step sizes. A coupled-mode approach is used to illustrate solution convergence by computing backscattering from a single bottom facet as well as backscattering in the benchmark wedge used for checking ocean acoustic models in the 1987 benchmark session of the Acoustical Society of America (ASA). Published field solutions for the ASA wedge are used to illustrate forward scatter results.

Simulation of predicted performance and interpretation of radiation measurements on the Israeli tandem free-electron laser

The experimental results of radiated power and its spectral parameters as measured on the Israeli electrostatic accelerator tandem FEL (EA-FEL) are interpreted and numerically simulated. Two complementary simulation codes, FEL3D and MALT1D show good agreement with experimental results. The FEL3D code (a non-linear three-dimensional ‘amplifier’ code, based on a coupled-mode approach), was employed for calculation of the small-signal gain, extraction efficiency and saturation power. This code (FEL3D) allows one to simulate FEL operation of multiple transverse modes at a single frequency taking into account space-charge effects. Multi-frequency simulation of the mode competition process was carried out using the MALT1D code based on a space–time multi-longitudinal mode simulation of a FEL oscillator.

Propagation characteristics of a novel complementary-structure planar segmented waveguide

A new waveguide structure with periodically segmented cover and substrate regions and a uniform film region is proposed. This complementary segmented waveguide is analysed in terms of the effective index and the spot size of the guided mode, by expressing the propagating field as a linear combination of the local normal modes of the corresponding uniform slab waveguides in different propagation regions. The results match very well with those obtained using the equivalent waveguide model. The propagation loss of the mode of the proposed structure is calculated using a coupled mode approach, and the results are similar to those of a normal segmented waveguide. This waveguide structure should prove useful in the realization of novel segmented waveguide devices.

Modeling of large-bandwidth parametric amplification in the Čerenkov- idler configuration in planar waveguides

Parametric amplification in the Cerenkov-idler configuration, in which the pump and the signal fields form guided waves of a planar waveguide while the idler is radiated into the substrate, is analyzed by a coupled-mode approach. Using the quasi-phase-matching scheme for periodically domain-reversed proton-exchanged LiNbO3 waveguides upon a Z-cut substrate, we have shown that this novel configuration has a large signal gain bandwidth and is more tolerant of variations in pump frequency than the all-guided configuration is. One can compensate for the relatively lower gain in the proposed configuration by increasing the interaction length. For typical values of various parameters, numerical results for optimum waveguide parameters (corresponding to maximum signal gain) for a given pump and signal wavelength are presented. Similarly, for a given waveguide the optimum combination of the pump and the signal wavelengths has been estimated. This configuration is expected to find applications in the realization of practical parametric amplifiers and oscillators.

Coupled-mode theory of resonant-grating filters

An approximate closed-form expression for the loss in a planar phase grating is derived by using coupled-mode theory. It is shown that this loss expression can be used to determine the spectral and angular width of a resonant-grating filter. A resonant-grating filter is a free-space optic that takes advantage of grating resonances to create narrow-band reflection peaks. Design characteristics, such as bandwidth, have previously been determined by profiling the resonance in reflectivity with the use of numerically intensive vector-diffraction methods such as rigorous coupled-wave analysis. The coupled-mode approach described here, however, gives the resonant-filter width directly, without the need to profile the resonance. Therefore computation time and hence design time are reduced. In addition, it is shown that the coupled-mode approach provides physical insights into the factors contributing to filter bandwidth.

Power Exchange in Coupled Optical Waveguides

Analytical expressions for power exchange in two waveguide directional couplers have been obtained using the modified scalar coupled mode approach that takes into account the non-orthogonality of the fields and also the perturbation correction terms to the effective indices of individual waveguides which were ignored in the old Coupled Mode Theory. Our results show considerable difference when compared with old Coupled Mode Theory, specially for non-identical waveguides. However, they are consistent with results obtained by the exact modal analysis.

Modeling of parametric amplification in the Čerenkov-idler configuration in planar waveguides

We present theoretical modeling of a novel configuration for quasi-phase-matched parametric amplification in which the pump and the signal fields form guided waves of a planar waveguide while the idler is radiated into the substrate. We follow a coupled-mode approach to study the parametric amplification in periodically domain-reversed proton-exchanged LiNbO(3) waveguides upon a Z-cut substrate and present numerical results relevant to the amplification process. In particular, for given pump and signal wavelengths we have studied the dependence of the gain coefficient on the periodicity of a grating formed by domain inversion. This configuration, which is referred to as the Cerenkov-idler configuration, is shown to provide a large signal gain bandwidth and is more tolerant to variations in pump frequency.

Measurements and simulation of the radiation build-up process in a prebunched free-electron maser oscillator

Experiments on the radiation build-up process obtained using a prebunched e-beam free-electron maser at Tel-Aviv University (TAU) are compared to results of theoretical studies carried out at TAU and at the University of Maryland (UMD). Two computer codes were developed and employed for simulation of FEM operation. A non-linear three-dimensional amplifier' ' code based on a coupled-mode approach was employed for calculations of small-signal gain, extraction efficiency and saturation power. This code (FEM3D) allows simulation of FEL operation taking into account space-charge effects. The power evolution of several longitudinal modes was also studied numerically using a one-dimensional, multi-frequency simulation code MALTlD; it was also observed in initial experiments. Numerical calculations of extraction efficiency for the TAU-FEM were made for all resonator eigen-frequencies lying under the FEM net gain curve. It was found that for a constant set of parameters the maximum efficiency is obtained at an eigen-frequency, which differs from the maximum gain frequency. Prebunching of the e-beam provides a unique opportunity to choose any desired oscillator eigen-frequency and thus to select the highest efficiency mode. This makes it possible to obtain efficiency enhancement of the oscillator by a factor of about 2.

Coupled-mode analysis of power-transfer characteristics in a three-waveguide nonlinear directional coupler

A planar three-waveguide nonlinear directional coupler (NLDC) is analyzed by the use of a coupled-mode approach based on the singular perturbation technique. The self-consistent first-order coupled-mode equations are derived in an analytically closed form, which demonstrates that the power transfer in three-waveguide NLDC is described by linear-coupling terms and nonlinear self-modulation terms. The optical switching characteristics predicted by the coupled-mode theory are discussed and shown to be in good agreement with those obtained from a numerical analysis with the finite-difference beam-propagation method.