Showing papers on "Coupled mode theory published in 1999"

Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators

Abstract: Coupling from tapered fibers and polished half-block couplers into the high-Q whispering gallery modes of microsphere resonators is investigated analytically. Numerous formulas are derived to predict the external coupling Q values, and intrinsic whispering gallery loss, for arbitrary structures, and for any sphere mode. Phase-mismatch due to the differences in propagation constants between input and sphere modes is taken into account. These formulas are strictly mechanical once a simple characteristic equation is solved which relates the spherical mode orders to the resonant wave vector. Results are in very good agreement with values that are calculated by different numerical methods.

High-density integrated optics

Abstract: This paper presents two dimensional (2-D) finite difference time domain (FDTD) simulations of low-loss right-angle waveguide bends, T-junctions and crossings, based on high index-contrast waveguides. Such structures are essential for the dense integration of optical components. Excellent performance characteristics are obtained by designing the waveguide intersection regions as low-Q resonant cavities with certain symmetries and small radiation loss. A simple analysis, based on coupled mode theory in time, is used to explain the operation principles and agrees qualitatively with the numerical results.

A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions

Abstract: Extended mild-slope equations for the propagation of small-amplitude water waves over variable bathymetry regions, recently proposed by Massel (1993) and Porter & Staziker (1995), are shown to exhibit an inconsistency concerning the sloping-bottom boundary condition, which renders them non-conservative with respect to wave energy. In the present work, a consistent coupled-mode theory is derived from a variational formulation of the complete linear problem, by representing the vertical distribution of the wave potential as a uniformly convergent series of local vertical modes at each horizontal position. This series consists of the vertical eigenfunctions associated with the propagating and all evanescent modes and, when the slope of the bottom is different from zero, an additional mode, carrying information about the bottom slope. The coupled-mode system obtained in this way contains an additional equation, as well as additional interaction terms in all other equations, and reduces to the previous extended mild-slope equations when the additional mode is neglected. Extensive numerical results demonstrate that the present model leads to the exact satisfaction of the bottom boundary condition and, thus, it is energy conservative. Moreover, it is numerically shown that the rate of decay of the modal-amplitude functions is improved from O(n−2), where n is the mode number, to O(n−4), when the additional sloping-bottom mode is included in the representation. This fact substantially accelerates the convergence of the modal series and ensures the uniform convergence of the velocity field up to and including the boundaries.

Experimental observation of mode evolution in single-mode tapered optical fibers

Abstract: Step-index single-mode optical fibers were tapered and then cleaved by ion beam milling to allow direct examination of the mode evolution in the taper with a microscope objective and CCD camera. Observation of HE/sub 1m/ modes is reported in tapered fibers cleaved at diameters ranging from 8 to 35 /spl mu/m. Theoretical scalar mode fields are fit to the experimental data and the mode transitions are explained by considering coupled mode theory.

Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides

Abstract: Fibre-Bragg gratings are currently showing enormous potential as wavelength filters for both the sensor and optical communications communities.

Supermodes of grating-coupled multimode waveguides and application to mode conversion between copropagating modes mediated by backward Bragg scattering

Abstract: An analysis of multimode waveguides where several modes are coupled via quasiperiodic perturbations is presented. The supermodes (or eigenmodes) of the structure are derived and orthonormality considerations are discussed. In addition, a new type of mode converter between copropagating modes is proposed, where mode conversion is mediated by a backward propagating mode. Adiabatic and nonadiabatic coupling coefficients are considered and the supermode formalism is used to conveniently describe the mode of operation of the device.

Transfer-matrix method based on perturbation expansion for periodic and quasi-periodic binary long-period gratings

Abstract: A transfer-matrix method based on perturbation expansion is proposed as an alternative way of simulating the transmission spectrum of a binary long-period grating (LPG). We first generalize the concept of transfer matrices for a heterojunction waveguide. For the couplings among copropagating modes, forward transfer matrices are used to describe the evolution of mode amplitudes along the grating. We show that these elements are related to the well-known coupling coefficients. The method is then used for the study of ideal two-mode grating couplers, and analytic solutions are obtained. We also use the matrix method to study multimode couplings in a LPG and compare the results with those obtained by using the coupled-mode theory. To further demonstrate its usefulness, we apply the method to a special quasi-periodic LPG, the Fibonacci grating. The results show that each cladding mode contributes to several transmission dips and that the dips of different cladding modes are grouped according to the special resonance conditions.

Phase reconstruction from reflectivity in fiber bragg gratings

Abstract: It is shown that a recently proposed phase-retrieval technique for Bragg gratings [Opt.??Lett.??22, 93 (1997); J.??Lightwave Technol.?? 15, 1314 (1997)] is not well suited for gratings with imperfections. The reconstructed group delay is in many cases not a more accurate estimate than the simulated group delay of the perfect, designed grating, independently of how small the errors in the grating structure are. The error in the group delay may be especially large near the zeros in the reflection spectrum.

Nonlinear property analysis of long-period fiber gratings using discretized coupled-mode theory

Abstract: We present generalized mathematical expressions for coupled-mode equations for nonlinear pulse propagation in fiber gratings using discretized coupled-mode theory and quantitatively analyze the nonlinear properties of long-period fiber gratings, considering multimode coupling between the core mode and several cladding modes. The calculations yield nonlinear responses for the case of long-period fiber gratings, including pulse shaping and all-optical switching in the self- and cross-phase modulation regimes. In addition, we briefly discuss the group delay properties of long-period fiber gratings and present several numerical examples of nonlinear pulse compression, which is related to strong dispersion and soliton-like behavior in fiber gratings.

Transmissive tilted gratings for LP/sub 01/-to-LP/sub 11/ mode coupling

Abstract: In this letter, we describe the properties of broadband single-sided and double-sided transmissive tilted gratings for LP/sub 10/-to-LP/sub 11/ mode coupling. Some of the tilted gratings show nearly complete coupling over a 30-nm wavelength range, and appreciable coupling over a spectral width as wide as 160 nm. Coupling properties of single-sided and double-sided tilted gratings agree well with coupled mode theory predictions.

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

Abstract: 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.

Analysis of grating-assisted backward coupling employing the unified coupled-mode formalism

Abstract: Grating-assisted backward coupling between two parallel waveguides is analyzed with a set of four coupled equations. The equations were recently derived by a unified approach that is appropriate for other coupling problems as well. Examples include three different operation regions and provide for the field variation along the guides as well as for the reflectivity and transmission coefficients. It is shown that the reduction of the four-wave coupling problem to a set of two coupled equations is possible only for some specific cases and at the cost of reduced accuracy. The model may be especially useful for the design of optical components for various applications such as optical switching and multiplexing.

Modeling tools for integrated and fiber optical devices

Abstract: This presentation will emphasize the current status of advanced design and simulation tools in photonics technology. The focus will be on Wavelength Division Multiplexing (WDM) component and integrated optic circuits modeling, although some aspects of optical link simulations will also be discussed. A wide variety of numerical methods such as the Beam Propagation Method (BPM), the Coupled Mode Theory (CMT), the Transfer Matrix Method (TMM), and the Finite-Difference Time Domain Method (FDTDM) in their state-of-the-art implementation will be presented. The results from simulating selected photonic components will be discussed.

Effects of hydrogen molecule diffusion on LP0m mode coupling of long-period gratings

Abstract: We analyzed the effects of hydrogen molecule diffusion on LP 0 m mode coupling of long-period gratings with two assumptions. The fibers were treated in a H 2 chamber at a temperature 90°C and pressure around 107 atm for five days. And then we fabricated long-period gratings with these fibers after holding them at room temperature and one atmosphere pressure for 15 min to 55 h and measured coupling of LP 0 m mode. We used diffusion coefficient of hydrogen molecule, which considered the fiber as a perfect cylinder, and assumed that the core and the cladding have the same diffusion coefficient. Corning ‘Flexcor1060’ was used as a photosensitive fiber which has cutoff wavelength, 920 ± 50 nm, and the cladding/coating diameter, 125 ± 2.0 μm/250 ± 15 μm. Core diameter is 5.0 μm and numerical aperture is 0.14. The hydrogen molecule concentration is averaged in the region of the core and the cladding with time and LP 0 m mode coupling was simulated with coupled mode theory and two assumptions. We were able to predict the coupling peak wavelengths that change with time. These results were useful for designing the transmission spectrum of long-period gratings.

Long-Period Fiber Grating Analysis Using Generalized N×N Coupled-Mode Theory by Section-Wise Discretization

Abstract: For the precise analysis and design of LPFG's, a new method of generalized N $\times$ N coupled-mode theory by section-wise discretization was proposed This is applicable to the analysis for arbitrary grating structures, which can readily take grating nonuniformities and multimode couplings into account Utilizing the method, several analyses of LPFG's were presented, and relationships between the grating structures and their spectral responses were discussed

Exact analysis of cascaded second-order nonlinearity in rotated Ti:LiNbO3 Couplers

Abstract: The effects of anisotropic coupling on the nonlinear behavior of different LiNbO3 waveguides and couplers employing cascaded second-order nonlinearity are analyzed and the importance of leaky mode effect is shown. A set of practical coupling structures made of both symmetric and asymmetric waveguides is investigated and their performance is compared. The home-made computer-code, based on coupled wave theory and anisotropic mode propagation in the complex domain, allows an exact electromagnetic investigation of devices, taking into account both guided and leaky propagation in structures which exhibit losses, the generated second harmonic wave having leaky nature. As an example, for the well-investigated slab three layer waveguide, by taking into account the leaky nature of the generated second harmonic, the modal mismatch Δβ = 333 m−1 is obtained at the azimuthal angle θm ≅ 61.9∘ (between the crystal optical c-axis and the propagation direction) while the angle found via the conventional approach is θm ≅ 63.5∘. Moreover, the calculated second harmonic wave exhibits an attenuation equal to 7.44 × 10−1dB/cm that strongly influences the whole cascaded second-order phenomenon. The simulation results, for those structures in which it is possible to neglect the hybrid and leaky nature of the propagation modes, are compared with the literature data and an excellent agreement is found.

Design of optical biosensor using polymeric waveguide

Abstract: We present a detailed design of an optical biosensor using polymeric waveguides intended for blood glucose concentration measurements. Our proposed device is based on a vertical two-waveguide ARROW structure, simplifying fabrication and avoiding post tuning of sensor parameters. The theoretical analysis based on coupled mode theory and transfer matrix techniques is presented whereby all device dimensions and parameters are calculated. For a 2 cm sensor, output optical power in one guide can swing by approximately 17% depending on glucose concentration. Such results represent a significant improvement in performance when compared with conventional Mach-Zehnder sensors.

ARROW-type vertical coupler filter: design and fabrication

Abstract: Vertical coupler filters (VCF) exhibiting narrow bandwidth and low sidelobe levels have been designed and demonstrated. Narrow bandwidth filter response is achieved due to the strong asymmetry between the waveguides of the filter and the nondispersive characteristics of the anitresonant reflecting optical waveguide (ARROW) structure. An ARROW-type VCF with a conventional parallel coupled directional coupler configuration with a full width at half-maximum (FWHM) of 1.36 nm and a maximum sidelobe level of -8.5 dB was fabricated using a compound glass consisting of SiO/sub 2/ and SiO/sub 2/-Ta/sub 2/O/sub 5/. The filter sidelobe levels were then further suppressed by using an X-crossing arrangement to provide coupling strength apodization along the device. The sidelobe levels of this modified X-crossing filter were suppressed to below -23 dB and the measured FWHM was 3.9 nm. The central wavelength of the reported filters are in the 1.55 /spl mu/m region. The measured results are in good agreement with theoretical results from an analysis procedure that combines the coupled mode theory with the finite difference complex mode solver.

Transmissive Tilted Gratings for LP -to-LP Mode Coupling

Abstract: In this letter, we describe the properties of broad- band single-sided and double-sided transmissive tilted gratings for LP -to-LP mode coupling. Some of the tilted gratings show nearly complete coupling over a 30-nm wavelength range, and appreciable coupling over a spectral width as wide as 160 nm. Coupling properties of single-sided and double-sided tilted grat- ings agree well with coupled mode theory predictions.

Large spatial mode, single frequency semiconductor lasers using two dimensionalgratings

Abstract: We demonstrate single mode operation of laser diode cavities up to 200|iim wide using 2-D Bragg gratings for simultaneous longitudinal and lateral mode control.

Far Field Calculations and Measurements of an S-Bend Deformed Oversized HE11-Waveguide Antenna

Abstract: The mode conversion in an s-bend deformed HE11 waveguide is calculated. With the mode mixture at the end of the waveguide, the corresponding far-fields for different waveguide lengths are calculated and compared with measurements. Optimisations of the total waveguide length and diameter are performed.

Theoretical analysis of optimal quasi-phase matched second harmonic generation waveguide structure in LiTaO3 substrates

Abstract: Based on the QPM structure and the coupled-mode theory, the maximum SHG conversion efficiency of QPM-SHG waveguides in LiTaO3 substrates is analyzed by optimizing the mode confinement and the mode overlap factors. For TM00(ω)-to-TM00(2ω) conversion with a non-square shaped domain inversion QPM structure, the mode confinement and mode overlap factor are dependent not only on the optical field distribution for the fundamental and the SH waves in the waveguide, but also on the domain inversion shape. With the assumption that the refractive index profile of an annealed, proton exchanged LiTaO3 channel waveguide is an exponential decay function in the depth direction, and a Gaussian function in the width direction, the analytical expression of the optical field distribution for the lowest order mode can be obtained. By considering both a non-Gaussian field distribution and a non-square shaped domain inversion structure, the theoretical values for achieving the optimal QPM-SHG waveguide structure are determined.

Reflection characteristics of guided-wave Bragg gratings using the collocation method

Abstract: We present the reflection characteristics of Bragg gratings using the collocation method and the effect of various parameters such as grating profile, duty cycle of periodic variation, index change and strain on the spectra. Comparisons with the coupled mode theory have also been included.

Wavelength- and Polarization-Insensitive Integrated Directional Couplers using Mach-Zehnder Structures

Abstract: Directional couplers are essential components in a wide variety of important devices, including modulators, switches, and Bragg-grating add/drop filters [1, 2].

Characterization and performance evaluation of coupled multiwaveguide arrays

Abstract: A new and effective nonintrusive method for characterization of N coupled waveguides is presented. The method is able to furnish readily the coupling parameters of most significance, and furnishes in addition, a way of assessment of overall device quality and performance. The procedure is based on a semi-empirical implementation of coupled mode theory, by means of which different functions are defined for different input configurations. In a well-functioning device, all these function should attain a common single minimum, out of which the coupling coefficient and additional parameters of the device are deduced. Devices were fabricated on Z-cut LiNbO/sub 3/ crystals by in-diffusion of titanium. The method was applied in order to measure the wavelength and polarization dependence of the coupling coefficient.

Obliquely incident wave coupling on finite-aperture waveguide gratings

Abstract: The problem of oblique incidence of plane waves and Gaussian beams on finite-aperture gratings (the number of grooves and their length and depth are all finite) in slab waveguides is analyzed by means of a four-wave two-dimensional coupled-mode theory (2D-CMT). This model considers the finite aperture of the gratings and the correct simultaneous interaction among all four relevant waves (TE+,TE-,TM+, and TM-) by means of Bragg diffraction at oblique incidence. The grating’s geometry and boundary conditions are properly accounted for, and the problem is solved numerically by a finite-difference method. Near-field and far-field distributions, as well as reflection and transmission (power) coefficients (as functions of the plane-wave incidence angle), are calculated. The model is compared with the degenerate case of two-wave coupling that considers interaction only between pairs (e.g., TE+↔TE-), and significant differences may be observed. Compatibility and differences between the 2D-CMT and the one-dimensional CMT (grooves with infinite length) are also presented, in addition to the influence of the beam width and the groove length on the emerging waves. The analysis is general and can be performed on many kinds of realistic beams, grating shapes, and applications.

Section-wise-exact coupled-mode theory of waveguide quasi-phase-matched second-harmonic generation

Abstract: We derive exact mathematical expressions for both the amplitude and phase evolution of optical waves due to linear and nonlinear interactions in a waveguide device and propose a section-wise-exact coupled-mode theory analysis for quasi-phase-matched second-harmonic generation in the waveguide. Section-wise-exact coupled-mode theory is readily applicable to nonuniform quasi-phase-matched gratings and can be useful as a quantitative method in their analysis and design. Using the method, several numerical results are presented for second-harmonic generation in periodically poled LiNbO/sub 3/ channel waveguides. In addition, we introduce a split-step method which takes propagation losses into account and which can include linear and two-photon absorption and waveguide losses.

Coupled-Mode Analysis of an NRD-Guide Coupler Based on Singular Perturbation Technique

Abstract: A coupled-mode formulation for an NRD-guide coupler is presented using the singular perturbation technique. The first-order and second-order perturbations are taken into account in the analysis and the coupled-mode equations based on the eigenmodes of each waveguide in isolation are derived. The propagation constants obtained by these equations are compared with those by the exact theory, conventional coupled-mode theory, and improved coupled-mode theory. The numerical results of present formulation are in good agreement with the exact theory and superior to those of the other formulations.

On the radiation mode effects in integrated optical directional couplers

Abstract: In this work we study the effect of radiation modes generated at the input of a directional coupler on its performance. We focus on the splitting ratio and the coupling length of the coupler. The Beam Propagation Method (BPM) is used for numerical simulation. Numerical results are compared to analytical expectations of the coupled mode theory (CMT). We show that, when the coupler is excited by a single mode fibre, ideal 3 dB couplers could be obtained by a simple adjustment of the exciting fibre. On the other hand, when the coupler has input and output bend structures, its coupling length as well as its splitting ratio are affected by the geometry of this bend.