On phase matching and power coupling in grating-assisted couplers

Two formulas of phase-matching condition for grating-assisted couplers in the literature are compared. The power exchange between the waveguides is examined by a numerical solution to the coupled-mode equations and a direct simulation by the beam propagation method. It is demonstrated that the phase-matching of the normal modes of the coupler (array modes) maximizes the power coupling. The approximate phase-matching condition for the modes of the individual waveguides (waveguide modes), however, leads to only partial power exchange     

Improved coupled-mode formulation based on composite modes forparallel 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     

Low-loss optical branching waveguides consisting of anisotropicmaterials

Low-loss branching waveguides of the mode-conversion type consisting of anisotropic materials are proposed, and their basic wave-guiding characteristics are studied by means of coupled-mode theory. Two mode-conversion sections are introduced on both input and output sides of a conventional symmetric branching waveguide. Each arm of the branching waveguides is assumed to be a single-mode slab waveguide except for the tapered section. A coupled-mode system of equations describing mode-conversion phenomena with respect to the transverse magnetic (TM) mode in the branching waveguides is derived from the field expansion in terms of local normal modes. A Runge-Kutta-Gill method is used to numerically solve the coupled-mode equations. It is found that the proposed branching waveguides suffer mode-conversion losses to a much lesser extent than conventional branching waveguides     

Self-consistent vector coupled-mode theory for tapered opticalwaveguides

A nonorthogonal coupled-mode theory for nonparallel waveguides is derived. An orthogonal coupled-mode theory is developed based on the exact local array modes. The relation between the two formulations is established. Closed-form solutions are obtained for the two-guide synchronous tapered coupler. The formalism obeys power conservation     

A new approach to grating-assisted couplers

A novel approach to the analysis and design of grating-assisted directional couplers is proposed. Power exchange between the waveguides is maximized through phase matching of two power-orthogonal modes of the parallel coupler. It is shown that either complete power transfer or zero crosstalk can be achieved at two different coupling lengths even when the two wavelengths are strongly coupled and/or close to synchronism. An analytical solution to the coupled-mode equations is obtained for the grating of rectangular shape. A grating-assisted coupler made of two slab waveguides is examined as an example to illustrate the salient features of the scheme     

About coupled-mode theories for dielectric waveguides

A critical examination is made of recent works on coupled-mode theory for dielectric waveguides with strongly overlapping fields. It is shown that there is no best formulation. In each case, explicit or implicit approximations lead to errors that are difficult to estimate. An investigation is made of the accuracy of coupled-mode formulas, in the case of strong guidance, for TM waves along coupled slabs or for HE11 modes along circular rods. Contrary to a previous prediction, there is no breakdown of coupled-mode formulas when the guidance is increased. The coupled-mode equations are applied to the problem of nonparallel waveguides in optical directional couplers. Comparison between coupled-mode predictions and beam-propagation-method simulations shows that bending effects in converging and diverging sections affect the accuracy. An improved coupled-mode theory is proposed in order to take these effects into account     

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     

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     

Vector coupled-mode theory of dielectric waveguides

A consistent derivation of a system of vector coupled-mode (VCM) equations for parallel dielectric waveguides is presented and compared with earlier versions of the improved coupled-mode theory (ICMT). As a validity test, it is shown that the effectively scalar transverse electric and transverse magnetic (TM) coupled-mode (CM) equations are direct limits of our full VCM formulation. In particular, our formulation does not lead to the fundamental error found with earlier coupled-mode theories (CMTs) in a case of TM fields. Functional equations of our VCMT are consistent with Maxwell's equations and lead to higher precision. They can be applied to complicated arrays of strongly coupled parallel dielectric waveguides with true vectorial behavior.     

Coupling Between Dissimilar Waveguides

Investigators have used coupled-mode theory to analyze the coupling between identical waveguides; in such cases the coupling coefficients are found to be identical. If the waveguides differ, the coupling coefficients are asymmetrical and difficult to evaluate by strictly theoretical methods. An alternate approach to this case is considered in the present work. A pair of coupled-mode equations is first developed from a consideration of the permissible fields within the device. This clarifies the relationship between the coupled-mode theory and the more general classical electromagnetic theory by giving a careful definition of the coupled and the normal modes of a coupled structure. It is shown that the coupled-mode equations are an exact representation of the waveguide fields, although for engineering purposes it is often convenient to use approximate values of the coefficients of these equations. The mutual coupling coefficients are obtained from a two transmission-line model of the structure, with the actual coupling mechanism represented by a mutual impedance common to the two lines. For dissimilar lines, the ratio of the coupling coefficients is found to be equat to the ratio of the characteristic impedances. For the cases considered, this is the same as the ratio of the propagation constants of the uncoupled lines, which permits the coupling coefficients to be determined from relatively simple measurements. The adequacy of the theory has been confirmed by a series of experiments.     

A unified approach to coupled-mode phenomena

A unified approach is presented for the treatment of various coupled-mode phenomena in two parallel waveguides. This approach is summarized in a set of four coupled equations, which is derived directly from Maxwell's equations. The equations are further simplified when applied to special cases such as evanescent coupling and grating-assisted coupling between parallel waveguides [e.g., reduced to a set of two equations]. In particular, for evanescently coupled waveguides, the equations reduce to the familiar vectorial coupled-mode equations. For grating-assisted waveguides the equations agree with earlier treatments, although, in some cases, may include extra terms which were omitted previously. Considering the special case of perturbations in a single waveguide, the equations in the examples coincide with those given elsewhere in earlier works. The reduction to scalar equations or extension to multiwaveguide systems is straightforward     

Coupled-mode theory for guided-wave optics

The problem of propagation and interaction of optical radiation in dielectric waveguides is cast in the coupled-mode formalism. This approach is useful for treating problems involving energy exchange between modes. A derivation of the general theory is followed by application to the specific cases of electrooptic modulation, photoelastic and magnetooptic modulation, and optical filtering. Also treated are nonlinear optical applications such as second-harmonic generation in thin films and phase matching.     

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.     

Coupled-zigzag-wave theory for guided waves in slab waveguidearrays

The guided wave in a slab waveguide array is treated as a set of coupled zigzag waves that propagate in the individual waveguides. This treatment leads to an exact dispersion relation for the TE and TM modes of the array, which can be expressed in a recurrence form and is easy to evaluate. Approximations for weakly coupled waveguides are discussed and compared with the coupled-mode theory. It is shown that the present theory can be cast in a form that closely resembles the coupled-mode theory     

Improved coupled-mode theory for anisotropic waveguide modulators

An improved coupled-mode theory for the propagation of modulated light waves in anisotropic dielectric waveguides is presented. Starting from Maxwell's equations, a partial differential equation is derived to describe mode-coupling between two normal modes which may propagate in anisotropic waveguide systems under modulation. The theory is applied to the analysis of typical waveguide modulators; examples for LiNbO₃ phase modulator, Mach-Zehnder, and directional coupler modulator are presented. The theory is applicable to both bulk and waveguide modulators/switches from DC to the high-frequency band. The only limitation is that the modulating wave has to propagate collinearly to the light waves     

Uniform bend loss of the TE0q mode in circular waveguides with small surface impedance and large surface admittance

The bending losses of the TE0q mode in circular hollow waveguides are obtained with small surface impedance and large admittance based on the exact vector analyses for Maxwell's equations. Considerable difference is found for a special case between the present result and that obtained previously by using the coupled-mode theory.     

Coupled-mode analysis of two-parallel circular dielectricwaveguides using a singular perturbation technique

The evanescent coupling between two parallel circular dielectric waveguides is analyzed using a singular perturbation technique. The analysis is based on the vectorial wave formulation. The first-order coupled-mode equations are derived in an analytically closed form, which are rigorous in the sense that they satisfy the Maxwell equations and the boundary conditions for the composite waveguide system within the first-order perturbation. It is shown, in a general manner, that the two orthogonally polarized modes of the isolated waveguides yield the different coupling coefficients and the polarization effect is in proportion to the relative difference of permittivities of the core and cladding regions     

A unified approach for the coupled-mode analysis of nonlinearoptical couplers

A unified approach for the coupled-mode analysis of nonlinear optical couplers is proposed. This approach is basically an extension of the work of [Haus, vol. 5, p. 16, 1987] to include optical nonlinearity. After the nonlinear coupled-mode theory is established, various basis functions are used as trial fields to derive coupled-mode equations. It is found that two published coupled-mode theories can in fact be deduced from the proposed one with individual linear or nonlinear waveguide modes serving as trial fields, thus making the coupled-mode equations from variational principle and reciprocity theorem equivalent. Coupled-mode equations based on system modes are also presented. Furthermore, analytical and/or numerical methods for solving coupled-mode equations are included. More research and discussion can be conducted based on the knowledge addressed in this paper     

Coupled-mode theory for LP modes

A generalized definition of LP modes is introduced for an optical fiber with arbitrary refractive-index profile and arbitrary cross section, The orthogonality property of these LP modes has been considered. Based on a complete set of LP modes in a chosen reference fiber, a set of coupled-mode equations describing a practical fiber are established. Possible applications of these coupled-mode equations are demonstrated.     

Analysis of optical fiber directional coupling based on theHE11 modes. II. The nonidentical-core case

For pt.I see ibid., vol.8, no.6, p.823-31 (1990). The analysis of the directional coupling between two single-mode optical fiber cores based on the exact HE₁₁ modes is extended to the cases involving nonidentical fibers. The coupled-mode theory in the vectorial form is used, and analytical expressions for the coupling coefficients and the butt coupling coefficient appearing in the conventional and the coupled-mode equations are presented. The accuracy of the conventional and new theories as applied to the two-core system is examined by comparing the coupled-mode predictions with the exact numerical values for different core-radius ratios, waveguiding strengths, polarization states, and core separations. It is shown that the errors in the coupled-mode theories increase as the two cores become more dissimilar. As long as the dissimilarity between the cores is kept small, the coupled-mode calculations using the HE₁₁ modes in predicting the coupling strength can be of satisfactory accuracy even when the individual guides are not weakly guiding ones. It is found that the new theory may give relatively large errors in the touching-core case with distinctly different core radii