Coupled-mode Formalism Research Articles

Assessment of coupled-mode theory for tapered optical coupler

The accuracy and the scope of validity of different coupled-mode formulations for a tapered optical waveguide coupler are assessed by comparison with the beam propagation method (BPM). It is demonstrated that the coupled-mode theory (CMT) based on local modes is valid only for small tilt angles, whereas an improved formulation that takes into account the wavefront-tilt effect yields highly accurate results for relatively large tilt angles. The radiation loss at the input and the output junctions can also be estimated by a simple mode-matching method combined with the improved CMT. The coherent coupling between two junctions,which is neglected in the coupled-mode formulations, is observed in the BPM simulations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Improved coupled-mode formulation based on composite modes for parallel grating-assisted co-directional couplers

An improved coupled-mode formulation based on the ideal modes of the coupled waveguides (ideal composite modes) is presented. In comparison with the formulation based on the ideal modes of the individual waveguides (ideal waveguide modes), the formulation in terms of composite modes is more rigorous and yields a more accurate grating period and coupling lengths. In addition, the radiation loss due to input and output junctions can in the composite-mode formulation. A new to the coupled-mode equations is derived in which all the spatial harmonics generated by the periodic grating are taken into account. The power exchange between the waveguides is examined by considering the input and the output conditions. The phase-matching conditions and the coupling lengths are calculated and compared with the analysis in terms of the waveguide modes. The grating period predicted by the waveguide-mode formulation agrees very well with that by the composite-mode formulation; however, dramatically different coupling lengths are predicted by the two formulations. >

A rigorous analysis of tunable semiconductor lasers with built-in grating-assisted vertical coupler filter

A general theoretical model is established and analyzed for a tunable semiconductor laser with a built-in co-directional grating-assisted vertical coupler filter. Possible internal reflection at the gain-coupler junction is included. A rigorous coupled-mode formulation based on the composite modes in the absence of the grating perturbation and a mode-matching method at the waveguide junctions is combined with multimode rate equation analysis. The dispersion of the semiconductor material is also included in the model. The accuracy and the scope of validity of the model are verified for both weakly and strongly coupled filter structures. Simple expressions for equivalent mirror reflectance, lasing wavelength, threshold gain and side-mode suppression ratio (SMSR) are derived and typical examples examined. The effects of the filter and the internal reflection on the longitudinal mode selectivity are discussed. >

A derivation of the differential equations of coupled-mode propagation

In this paper the differential equations of the coupled-mode theory of acoustic propagation are derived. The derivation properly accounts for the boundary conditions imposed at an interface and gives rise to terms not normally found in other coupled-mode formulations. The solutions of this couple-mode system agree very well in numerical experiments with those obtained from a boundary integral equation method.

Contra-directional coupling in grating-assisted guided-wave devices

Contradirectional power coupling in grating-assisted guided-wave devices is studied by applying a vector nonorthogonal coupled-mode formulation. The coupled-mode equations are solved by a transfer matrix method. All the space-harmonics generated by the periodic grating are considered. The coupling can be understood in terms of the interference among the normal modes of coupled waveguides with a grating perturbation. Phase-matching grating periods for maximum reflections are equal to the beat lengths between the two normal modes involved in the coupling process. The reflections are built up constructively (Bragg reflection), resulting in stopbands in the spectral response. The expressions for the grating periods are obtained and compared with those derived from conventional phase-matching conditions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Wavefront-tilt effect in nonparallel optical waveguides

The wavefront-tilt effect in nonparallel optical waveguides is studied theoretically. A self-consistent coupled-mode formalism based on guided normal modes of the individual waveguides is developed, and the wavefront-tilt effect due to the nonparallel structures is accounted for. The wavefront-tilt caused by junctions at input and output is considered by using a mode-matching method. The power exchanged between two slab waveguides separating at an angle is investigated by an analytical solution to the coupled-mode equations. >

A coupled-mode analysis of modulation instability in optical fibers

A coupled-mode formulation in the frequency domain has been developed and applied to the analysis of the modulation instability (MI) due to the self- and cross-phase modulations (SPM and XPM) in optical fibers. The modulation instability of two copropagating beams with the same polarization at different carrier frequencies is examined in detail. It is demonstrated that, by identifying the phase-matching conditions among different sideband components, the modulation instability due to the SPM and the XPM may be described by a set of reduced coupled-mode equations. Simple analytical expressions for the gain are derived under this condition. The approximate formulas are compared with the full wave solutions and their scopes of validity are discussed. >

A scalar coupled-mode theory with vector correction

A coupled-mode formulation based on scalar modes is developed for guided-wave devices in optoelectronics and fiber optics. In the formulation, the vector property of the electromagnetic field is represented as additional coupling to the coupling due to the refractive-index perturbations. The nonorthogonality of the scalar modes in the individual waveguides is taken into account. The theory is applied to directional couplers made of parallel slabs, rectangular channel waveguides, and circular fibers. Comparisons with the exact solutions available demonstrate very good agreement. Within the framework of the simple scalar analysis, the theory may be applied to the modeling and analysis of propagation and coupling of the vector electromagnetic waves in a broad range of practical optical guided-wave devices. >

Coupled‐mode formulation of parallel dielectric waveguides using singular perturbation technique

AbstractA new coupled‐mode formulation for parallel dielectric waveguides is presented by using a singular perturbation technique. The theory is applied to two parallel, planar dielectric waveguides and the coupled‐mode equations that are asymptotically correct are obtained.

Optical wavelength filter with tapered couplers

Optical wavelength filters based on tapered directional couplers are analyzed by nonorthogonal coupled-mode theory. The nonorthogonal coupled-mode formulation in general yields more accurate results than the conventional orthogonal coupled-mode theory. The focus is on the novel features of the tapered couplers used as wavelength filters. It is demonstrated that, in addition to reducing the crosstalk and suppressing the sidelobes, the tapered structure can also make the filtering response independent of the device length beyond a critical coupling length,.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Power exchange in tapered optical couplers

The power exchange between nonparallel optical waveguides or tapered couplers is analyzed by the coupled-mode theory based on local modes of the individual waveguides (local waveguide modes). A self-consistent nonorthogonal coupled-mode formulation is presented and transformed into an orthogonal form which is equivalent to the coupled-mode equations for the local modes of the coupled waveguides (local array modes). The coupled-mode equations are then solved numerically and the effect of the taper on the power exchange between the two guides is studied. >

Efficient analysis tool for coupled-SAW-resonator filters

The advantages of the coupled-mode (COM) formalism and the transmission-matrix approach are combined in order to create exact and computationally efficient analysis and synthesis tools for the design of coupled surface acoustic wave resonator filters. The models for the filter components, in particular gratings, interdigital transducers (IDTs) and multistrip couplers (MSCs), are based on the COM approach that delivers closed-form expressions. To determine the pertinent COM parameters, the COM differential equations are solved and the solution is compared with analytically derived expressions from the transmission-matrix approach and the Green's function method. The most important second-order effects, such as energy storage, propagation loss, and mechanical and electrical loading, are fully taken into account. As an example, a two-pole, acoustically coupled resonator filter at 914.5 MHz on AT quartz is investigated. Excellent agreement between theory and measurement is found.

Frequency response and tunability of grating-assisted directional couplers

The spectral properties of grating-assisted directional couplers are studied using an improved coupled mode formulation. Key parameters for the design of these structures, such as the grating period, the coupling length, and other structural parameters, are calculated. The frequency response, the filter bandwidth, and the tuning range are analyzed. The technique is used to examine a specific case of InGaAsP-InP tunable filters, and the results are compared to a prior experiment.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Coupled mode formulation for directional couplers with longitudinal perturbation

A grating-assisted directional coupler is investigated using an improved coupled-mode formulation for multi-waveguide systems with longitudinal perturbation. This approach is capable of handling three-dimensional structures as well as a complex index of refraction and nonisotropic media. The case of two coupled channel waveguides is closely examined, and a numerical analysis is carried out for several cases of slab structures. >

Structural considerations for bimodal nonlinear optical switching

Based on parameters that arise in both scalar and vector coupled-mode formalisms, the authors consider the effect of circular and planar multilayer guide structure and the related choice of modes in the design of bimodal all-optical switches. Finite-cladding fiber LP/sub 01/-LP/sub 02/ and LP/sub 01/-LP/sub 11/ modal interferometers and MQW-sandwich couplers are included as application examples. It is shown that the approach presented provides further insight into complex propagation characteristics of bimodal nonlinear waveguiding structures. This approach is expected to be useful in the design of all-photonic switching devices.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Coupled-mode equations for nonlinear directional couplers

We point out some inconsistencies in the usual coupledmode formulation employed to model the nonlinear behavior of symmetric directional couplers. In particular, we show that special care is required when neglecting some of the nonlinear terms in the equations.

Coupled-wave small-signal transient analysis of GaAs distributed amplifier

The analysis takes into consideration the effect of c/sub dg/ in the active device, leading to a coupled-mode formulation. Dispersions within the transmission lines and the presence of two normal modes make it impractical to obtain broadband matching. Numerical results for specific terminations and various degrees of passive and active coupling clearly indicate the influence of c/sub dg/ and the necessity for coupled-mode analysis. The numerical scheme, which is based on Bromwich integration, can be incorporated into CAD routines for time-domain response optimization.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

High-power TE11 and TM11 circular polarizers in oversized waveguides at 70 GHz

Elliptical deformation of oversized, smooth-wall circular waveguides can produce choosable elliptical or circular polarization from a linearly polarized TE11 or TM11 mode used as intermediate linearly polarized modes in TEO1 to HE11 mode conversion sequences in electron cyclotron resonance heating (ECRH) of magnetically confined thermonuclear fusion plasmas with high-power gyrotrons. Mode coupling in elliptically distorted overmoded circular waveguides has been studied theoretically and experimentally in order to optimize TE11 (and TM11) polarizers (I.D.=27.79 mm) for the 1 MW/70 GHz long-pulse (3s) ECRH system on the Garching Stellarator W VII-AS. Coupling coefficients for ellipticity coupling of non-degenerate modes are given (coupled-mode differential equations formalism). The polarization converters essentially consist of smooth-wall circular waveguides which are gradually squeezed. A sine-squared function of the length coordinate is used to get an almost elliptical crosssection in the middle and circular cross sections at both ends. Arbitrary elliptical polarization states can be generated introducing an extremely low level (<<1%) of undesired spurious modes. Well defined differential phase characteristics have been achieved.

Mode coupling in tapered structures

The theory of strong mode coupling involving nonorthogonal modes is extended to tapered structures. Initially, an attempt is made to show that previous formulations of the problem of nonorthogonal-mode coupling in tapered structures are incomplete. An equation is derived which is the generalization to tapered structures of the coupled-mode formalism with nonorthogonal modes. >

Combined directional and contradirectional coupling in a three-waveguide configuration

A three-waveguide coupler, combining directional and contradirectional coupling between the central and side waveguides, respectively, is discussed. Coupled-mode formalism is used to examine the superposed processes both in synchronism and near synchronism. The nature of the solution is shown to depend on the relative coupling strengths of the individual processes. For stronger contradirectional coupling, the power is divided among the three output ports, at ratios closely related to the ratio of the coupling coefficients, and with a nonperiodic, slowly varying length dependence. The coupler can thus be used as a power divider without the need for a precisely special case where the directional and contradirectional coupling coefficients are equal.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>