Coupled-mode Formalism Research Articles

Geometrical approach to stationary waves in a shallow grating

A geometrical analysis of the monochromatic solutions near the first band gap for a shallow Kerr grating is presented.The analysis is based on the coupled mode formalism and on Stokes variables. We investigate the electric field for nonzero energy flow, in particular we consider the phase difference between the counter-propagating coupled modes. Phase portraits for zero and nonzero energy flow are topologically different, and we clarify the way in which they are connected, thus identifying families of trajectories for nonzero flow that disappear when the flow goes to zero.

A COUPLED-MODE THEORY-BASED ANALYSIS OF COUPLED MICROSTRIP LINES ON A FERRITE SUBSTRATE - ABSTRACT

Presented herein is a coupled-mode formulation for coupled microstrip lines on a magnetized ferrite substrate. The formulation discussed here is an extension of the coupled-mode theory for microstrip lines on an isotropic substrate. Since the magnetized ferrite exhibits a biaxial anisotropy in its permeability, the guided-wave fields in the magnetized ferrite are not subject to the conventional reciprocity relation for fields in an isotropic medium. Thus, a generalized reciprocity relation is first derived from two sets of guided-wave fields, which propagate in ferrite magnetized transversely along the strip surface. The reciprocity relation is then used to derive coupled-mode equations for coupled microstrip lines on a ferrite substrate. As a basic numerical example, the new formulation is applied to two coupled microstrip lines on a ferrite substrate.

Optical discrete solitons in waveguide arrays I Soliton formation

We investigate the generation of discrete spatial solitons in arrays of coupled waveguides. Light was launched into the center of the array, and different beam sizes and array geometries were tested. At low power, the propagating field spreads as it couples to more waveguides. When the intensity is increased, localization is observed around the input waveguides, leading to the formation of a discrete soliton. For wide input beams, exciting a few waveguides, soliton splitting, which is due to instability induced by multiphoton absorption, is observed. All of these effects are described well by a coupled-mode formalism.

Modelling of finite-size grating waveguide filters on the basis of a free-space diffraction approach

The field reflection coefficientR of a high strength and short length waveguide grating section is obtained by resorting to a free-space diffraction approach whereby the waveguide grating is excited by means of a virtual high index prism. The effective index of the guided mode, the coupling coefficient between contradirectionally propagating modes of the grating waveguide as well as the modal field are first obtained in the case of an infinite grating by interpreting the results given by an exact diffraction solution by means of a three-wave coupled mode formalism. These results are then introduced in the coupled wave analysis of the structure of finite extent to provide the field at the entrance of the waveguide grating section where the usual overlap integral with the incident field delivers the reflection coefficient. We neglect only the result of radiation mode propagation. This assumption can be made when the grating section is longer than the propagating length of all the leaky modes.

Optical Properties of a Chiral Liquid Crystal in a Generalized Coupled Mode Formalism

We propose a general formalism suitable for calculating the propagation of the optical normal modes through a chiral liquid crystal as one of the characteristic parameters varies. Based on the fact that the normal modes are the eigenstates of the generator of translation (the wave number operator K), the change in K is evaluated by means of a characteristic parameter such as chirality. The numerical calculations are in good agreement with the analytic results.

ANALYSIS OF THE OPTICAL PROPERTIES OF A HELICOIDAL LIQUID CRYSTAL IN A GENERAL COUPLED MODE FORMALISM

The optical properties of a chiral liquid crystal (CiLC) are analyzed in terms of the rotation of the principal axes (RP) together with the one-dimensional periodicity (DP) along the helicoidal axis. Within a general coupled mode formalism, the helicoidal nature of a CiLC is represented by both RP and DP. The formalism developed here is a powerful tool for calculating the optical properties of a CiLC for oblique incidence. Our numerical calculations show an exact agreement with the previous analytic results.

Design and modeling of passive optical switches and power dividers using non-planar coupled fiber arrays

Arrays of coupled waveguides such as the ubiquitous directional coupler are used extensively in optoelectronic devices, with demonstrated applications to optical communications networks, fiber interferometers, and optical homodyne receivers. In order to analyze the transmission characteristics of circular arrays of coupled optical fibers, we have developed a matrix representation of the coupled-mode formalism, allowing for varying fiber diameters and differing coupling strengths between the fibers in the array. The model is used to identify design criteria for application of such arrays as passive optical switches and power dividers.

Coupled perturbed modes

Conventional coupled modes are combined with perturbation theory to allow fast range-dependent normal mode calculations in deep water. The mode-coupling method follows R. B. Evan’s two-way coupled mode formulation [R. B. Evans, J. Acoust. Soc. Am. 74, 188–195 (1983)], where conservation of pressure and particle displacement along the z axis yields an algorithm to couple the modes as sound propagates in range. The conventional approach requires calculation of a complete set of normal modes at each step in range and is computationally intensive. Here perturbation theory is used to find the normal modes and wave numbers of each step in terms of linear combinations of the modes of the previous step. The process can be repeated for many steps and allows rapid calculation to long ranges. The method of coupled perturbed modes is applied to benchmark and canonical problems.

Relation between coupled-mode theory and equivalent layers for multilayer interference coatings

The method of equivalent layers is a commonly used technique for designing optical multilayer interference coatings. Herpin's theorem [C. R. Acad. Sci. 225, 182 (1947)] states that every symmetrical multilayer structure is equivalent, at one arbitrary wavelength, to a single homogeneous layer. The Herpin equivalent layer is described by two design parameters, the equivalent index and the equivalent thickness. Alternatively, we recently developed an exact coupled-mode analysis for the description of multilayer interference coatings composed of a symmetrical combination of layers. The design parameters of the coupled-mode theory are the exact coupling coefficient and the exact detuning coefficient. Recently we used this method in the design of chirped mirrors for dispersion compensation. We prove that the two methods are equivalent and derive relations that link the design parameters of both formalisms. By use of these relations it is possible to translate between the coupled-mode formalism and the method of equivalent layers. The simultaneous availability of both design methods gives a new perspective on the analytical design of optical interference coatings with challenging spectral response characteristics.

Characteristics of grating-assisted couplers

Characteristics of grating-assisted coupling between two parallel waveguides are analyzed. The influence of the grating parameters, such as groove depth, duty cycle, and refractive indices is considered. Chirped and parallel gratings as well as gratings with sinusoidal envelope periodicity are also addressed. The analysis is based on a unified coupled-mode formalism, with the transfer-matrix method as a general solution technique. It is shown how to modify the grating parameters to provide a specific spectral response (reflectivity and transmission coefficients). As an example, two parallel gratings are used to obtain a similar response to a single grating of double length. The location of the grating between the two waveguides as well as the light-wave injection direction are very important. The presented methods and effects may be useful for design and analysis in the fields of optical communications, sensing, and processing.

A one-way coupled mode solution, Huygens’ principle and mode coupling coefficients

The normal mode formulation for waveguide propagation has physical appeal since sound propagates in discrete entities—namely modes—and an understanding of the progression of modes sheds light on arrival times of pulses and features of ocean ducting. In the development of the range-dependent version several researchers employ the conservation of normal pressure and normal particle velocity along the z axis in the mean as the conserved quantity that allows for an algorithm to couple the modes as sound progresses in range. The usual derivations are based on R. Evan’s seminal work on the two-way coupled mode formulation. Here it is shown that Huygens’ principle and an approximate form of the Green’s function in the waveguide leads to a suitable algorithm as long as reflected or backward going flux is small relative to forward flux. The derivation, moreover, leads to expressions for mode coupling amplitudes that allow one to follow the distribution of flux along specific modal tracts. [Work supported by the Office of Naval Research and Naval Research Laboratory.]

Results from a simulation study of mode coupling due to acoustic wave-soliton interactions

There is a strong interest in understanding the interactions between ocean acoustic waves and shallow water solitons. Central to these interactions are the acoustic mode conversions caused by the solitons. Such mode conversions potentially occur each time the acoustic wave and soliton interact; however, at certain combinations of acoustic-oceanographic wavelengths, the interactions can produce large losses in acoustic signal. Mode coupling results will be discussed from simulation studies that examined the signal loss due to interactions with solitons in the Strait of Messina. Coupled mode formulations were avoided so that interpretations of results were not dependent upon any particular mode coupling scheme. Rather, the full-wave, two-way, finite-element model, FFRAME, was used to produce numerically accurate predictions of the complex pressure field as it passed through the soliton. This complex field was then decomposed into its component wave numbers via an FFT to determine the mode structure. This simulation included the entire acoustic field (forward propagation, backscatter, multiple scatter, etc.). The results were realistic and difficult to interpret. Results will be shown at various points along the way as the acoustic wave passed thorugh the soliton. [Work supported by ONR/NRL and by a High Performance Computing DoD grant.]

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

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.

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

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.

Nonlinear pulse propagation in birefringent fiber Bragg Gratings

We present two sets of equations to describe nonlinear pulse propagation in a birefringent fiber Bragg grating. The first set uses a coupled-mode formalism to describe light in or near the photonic band gap of the grating. The second set is a pair of coupled nonlinear Schroedinger equations. We use these equations to examine viable switching experiments in the presence of birefringence. We show how the birefringence can both aid and hinder device applications.

Coupling of parallel-cavity laser arrays with high-contrast-reflectivity mirrors

A coupled-mode formulation is presented for parallel cavities of a laser array. The interaction between neighboring elements is via diffraction resulting from perturbation of the boundary conditions. Each high-reflectivity cavity mirror is surrounded by a much-lower-reflectivity background, which, for practical purposes, is considered to be zero. The formulation applies both to ordinary laser arrays and to vertical-cavity laser arrays with high-contrast laterally patterned mirrors. Some examples are considered and compared against exact numerical calculations.

Analysis of mode coupling on guided-wave structures using Morse critical points

New insight on mode coupling in waveguiding structures is obtained from the theory of Morse critical points (MCP's). It is shown that the traditional coupled-mode formalism has a clear analytical connection with functional properties of the characteristic determinant in the vicinity of the Morse critical point, which determines the minimum of coupling. The relationship between perturbed and independent modes in the mode-coupling region is obtained using the Taylor polynomial of order two about the Morse critical point, and it is found that the coupling factor is proportionally related to the value of a characteristic function at this point. The qualitative modal behaviour in the mode-interaction region is predicted by a simple normal form, which can be geometrically interpreted as a result of the intersection of a saddle surface and a plane corresponding to the minimum of the coupling factor. Numerical results for a variety of guided-wave structures, including printed-circuit transmission lines, planar-slab waveguides, and shielded microstrip-like lines demonstrate the efficiency of the proposed approach for the rapid identification of mode-coupling regions, and for reconstruction of dispersion behaviour in those regions via simple analytic (normal) forms.

Coupled-Mode Analysis of Bent Planar Waveguides with Finite Claddings

Coupled-mode analysis of uniformly bent slab waveguides with finite claddings is demonstrated. Using the method of conformal transformation, a curved waveguide structure is translated into an equivalent straight waveguide with a modified asymmetric index profile. The wave propagation through this structure is described by coupled-mode formalism with ideal mode expansion. The step-index profiles as well as graded-index profiles are considered. The numerical results of transmitted power, wave propagation along the bend and pure bend loss are presented for TE0-mode excitation.

Numerical Analysis of Nonlinear Directional Couplers

The original coupled-mode theory for the nonlinear directional coupler (NLDC) and two later improved theories are reviewed and compared with the analysis using the segmentation method proposed in this work. The segmentation method is more accurate in analyzing the NLDC because it takes into account the real situation that the local nonlinear guided modes in the NLDC will vary along the propagation direction due to the accompanying variation in the local power distribution. In the coupled-mode formulations reviewed, however, either linear guided modes or nonlinear modes at a fixed power level are employed when evaluating related coupling quantities.

Coupled-mode analysis of an ideal microstrip six-port directional coupler on an anisotropic substrate

The coupled-mode formulation has been applied to simplify the analysis of a symmetrical three-line microstrip coupler (TMC) on an anisotropic substrate. The existence of an ideal six-port directional coupler has been proved under the following conditions: the coupling coefficients of the system are equalized using substrate anisotropy and the lines are matched to their characteristic impedances. The singular behaviour of the normal-mode parameters that occur at the critical equalization point of k1 = kc has been examined. The numerical and simulation results confirm the developed analysis.