Radiation loss of grating-assisted directional coupler

The coupling between the guided modes and between guided and radiation modes of two parallel slab waveguides forming a directional coupler in order is computed to determine the radiation losses introduced by the coupling grating. The problem is solved in two stages. First, the guided modes for each waveguide are computed separately and the radiation modes are only determined for the more complicated of the two waveguides, the one that is nearer to the grating. Modifications caused by the presence of the opposite waveguide are then taken into account by computing first-order correction terms. For a practical example of slab waveguides defined in GaInAsP, the authors find that the radiation losses per power exchange length remain below 0.02 dB for a rectangular grating depth of 0.01 μm     

Radiation modes of a five-layer symmetric slab waveguide

The radiation modes of a five-layered symmetric slab waveguide are treated in detail. Since the slab is assumed to be infinitely extended in the yz plane, all field quantities are considered to be independent of y. The analysis is based on modal expansion formulation and the modes can be classified as TE and TM as well as even and odd ones. Applying the boundary conditions at interfaces, the expansion coefficients of the field components are easily calculated. In addition, the normalization factors and the orthogonality properties are determined by deriving appropriate integrals over the cross section of the waveguide. Finally, the validity of the orthogonality relations between the radiation modes and the guided ones, which propagate in such a waveguide, is proved explicitly.     

Far-infrared guided wave optics experiments with anisotropic crystal quartz waveguides

The TE and TM modes of anisotropic dielectric waveguides of crystal quartz have been studied in the far infrared using an optically pumped waveguide laser with a hybrid output mirror. Using Si and Ge prism couplers, excitation efficiencies of 56 percent of the theoretically calculated values were obtained for the TE0mode. The measured propagation constants were found to agree closely with theory when the perturbation of the guided modes by the prism couplers was included. The attenuation constant for the TE0mode atlambda_{0} = 496.1mum was found to be as low asalpha_{TE_{0}} = 0.05cm^-1. A metal grating coupler for backward-wave coupling through a prism shaped TPX plastic substrate was also used to couple 89 percent of the guided energy in the TE0mode out of the waveguide in the reverse direction.     

Analysis of a line-defect waveguide on a silicon-on-insulator two-dimensional photonic-Crystal slab

This paper describes the investigation of the waveguide properties of a silicon-on-insulator (SOI)-based two-dimensional photonic-crystal slab. It is found that coupling between transverse-electric (TE)-like defect modes and transverse-magnetic (TM)-like slab modes occurs in some frequency range due to structural asymmetries in the vertical direction. This TE-TM coupling, together with the smaller refractive-index contrast between the slab and dielectric cladding, results in propagation losses for a line-defect waveguide in an SOI structure. The paper also presents optimization methods for obtaining a wide lossless propagation band using such a line defect.     

Hollow dielectric waveguide for distributed feedback lasers

Hollow dielectric waveguides with gas-filled core and a dielectric tube seem promising as laser structures. The tube fulfills the dual role of a guidance structure for the electromagnetic radiation as well as that of a container for the plasma of the gas laser. Feedback for laser oscillation is achieved by providing coupling between forward- and backward-traveling modes by means of periodic ripples of the inner surface of the hollow dielectric tube. This paper is devoted to the derivation of the coupling coefficients between two guided modes that are coupled by ripples in the tube wall. The calculation is based on the model of a slab waveguide. Expressions for the leaky-mode losses and the scattering losses inevitably introduced by the coupling mechanism are also presented.     

Chasi-Guided Modes and Related Radiation Losses in Optical Dielectric Waveguides with External Higher Index Surroundings

Mode filter actions are found theoretically in an optical dielectric waveguide consisting of a core and a thin cladding layer which is further surrounded by an external higher index region. The propagating waves, which are usually considered to be cutoff modes, can be guided with a small amount of loss under a certain condition. These waves are defined here as quasi-guided modes. These modes tend to the guided modes of the guide when the cladding thickness increases infinitely. A method is given to estimate the losses. As an example, the radiation losses are formulated for a symmetric slab waveguide, and are found to be approximately proportional to the cube of the mode number of the quasi-guided mode. Therefore, losses of the quasi-guided modes depend strongly on the mode number. It is suggested that fibers with large core diameters can be used as quasi-single mode fibers by covering the clad-type multimode fibers with external higher index surroundings and choosing the parameters properly.     

Analysis of discontinuities in an asymmetric dielectric slabwaveguide by combination of finite and boundary elements

An approach that combines the finite-element, and boundary-element methods is extended to the analysis of discontinuities in an asymmetric slab waveguide. The discontinuity region is divided into three regions. One is a finite region with arbitrary inhomogeneities, and the others are semi-infinite and homogeneous regions. The finite-element and boundary-element methods are applied to the former and latter regions, respectively. For uniform waveguide regions connected to discontinuities, analytical solutions are used, in which all the eigenmodes, namely, guided modes, substrate radiation modes, and substrate-cladding radiation modes, are taken into account. To show the validity and usefulness of this approach, computed results are given for three kinds of step-discontinuities with transverse electric (TE) and transverse magnetic (TM) mode incidences     

Reflection loss of laser mode from tilted end mirror

The reflection of the lowest-order guided TE mode of a slab waveguide from a tilted end mirror is discussed. The problem is applicable to injection lasers if the slab is regarded as an effective refractive index approximation of the channel or ridge waveguide of the laser structure. For a mode width of 3 μm, 5° mirror tilt results in reflection losses in excess of 25 dB; for modes of 6-μm width, 5° mirror tilt causes reflection losses in excess of 45 dB. For small tilt angles the results of this calculation agree with those based on a Gaussian approximation of the guided mode. For larger angles, and hence higher reflection losses, the results depart significantly from the Gaussian approximation     

Analysis of metallized-grating coupled twin-waveguide structures

In this paper, we present an analysis of grating coupled twin-waveguide structures consisting of a semitransparent metal-low-index waveguide on top of a corrugated high-index laser waveguide. A rigorous Floquet analysis of TE modes is performed for trapezoid shaped gratings. Starting from on-resonant Floquet solutions, we establish a relation to simple coupled-mode equations yielding results, that show excellent agreement with the rigorous Floquet solutions. Grating coupled radiation plays a crucial role in the waveguide structures under consideration: when phase matching of the guided modes occurs, narrow-band resonant suppression of radiation loss with linewidths in the subnanometer regime can be achieved. The grating geometry has an important impact on these resonances, which is made evident by various numerical examples demonstrated in this paper     

Spontaneous emission coupling to radiation and guided modes ofplanar waveguide structures

A general analysis of spontaneous emission in the vicinity of a symmetric slab dielectric waveguide is presented using explicit energy-normalized functions for the complete set of bound and radiating TE and TM modes. Expressions for the spontaneous emission rates are derived for each mode type, and calculated results are presented illustrating the emission behavior as a function of the position of the radiating atom, for various slab waveguides. The model is used to analyze the spontaneous emission coupling efficiency to guided modes in planar erbium-doped amplifier structures. Spontaneous emission factors for the TE and TM modes are calculated for the cases of erbium doping of the core and of the cladding     

Directional couplers made of nonidentical asymmetric slabs. Part I: Synchronous couplers

We discuss TE and TM mode directional couplers made of nonidentical asymmetric slab waveguides. Approximate expressions are provided for the coupling coefficients of synchronous (no grating) couplers and their accuracy is checked by comparison with exact solutions that are based on solving the guided mode problem of the total structure consisting of the two slabs considered to be a single waveguide.     

Rigorous Analysis of the Scattering of Surface Waves in an Abruptly Ended Slab Dielectric Waveguide

The reflection and the scattering properties of even TE and TM surface waves incident in an abruptly ended dielectric slab waveguide are analyzed. The discontinuity is regarded as a junction between two open waveguides namely the dielectric slab waveguide and the free space waveguide. The boundary conditions acting together with the orthogonality provide singular coupled integral equations on the discrete and the continuous wave amplitudes at the discontinuity. These singular coupled intergral equations with Cauchy kernels and infinite limits of integration are solved by iteration via the Neuman series. Numerical results are presented for the reflectivity of the even TE/sub 0/ and TM/sub 0/ fundamental modes, together with their mode conversion on even TE/sub 2/ and TM/sub 2/ in a slab where two guided modes can propagate. Reflectivity and mode conversion of higher order excitations are also investigated     

Pulse propagation in a nonlinear dielectric slab waveguide

The evolution of an optical pulse in a single-mode, step index dielectric slab waveguide which is characterized by an intensity dependent dielectric function in the core and cladding regions is treated by means of differential equation techniques. A cubic order non-linearity is considered. The electromagnetic field distribution in the slab waveguide region satisfies a non-linear wave equation. This field can be represented in terms of even TE guided modes with a slowly varying envelope amplitude function. Then using the well known approximation, based on the slowly varying character of the amplitude function, a non linear partial differential equation is obtained for the amplitude function. As the coefficients of this equation depend on the distance across the transverse direction X, an averaging technique over x is applied to reduce the nonlinear partial differential equation into a form that is easily transformed to the so-called non-linear Scroedinger differential equation. This equation is then attacked by means of the well known Inverse Scattering method in the case of reflection less potentials. The single and double soliton solutions are obtained explicitly for a single-mode slab waveguide. Finally numerical results are presented in the time domain.     

Analysis of Radiation Phenomena in a Wedge Ended Dielectric Slab Waveguide

Radiation phenomena observed in a wedge shape ended dielectric slab waveguide are analyzed using mode matching technique. The case of transverse electric polarization (TE) being parallel to dielectric slab waveguide is assumed. In order to describe the fields in the wedge region, a stack of dielectric plates is assumed and in each layer the fields are expanded in terms of the mixed spectrum of guided and radiated modes. A similar expansion is used in the constant thickness slab waveguide while in free space medium a continuous-radiation mode expansion is used. Then a mode matching approach is applied, incorporating the orthogonality properties of mixed spectrum modes, in order to compute the wave fields inside the dielectric slab waveguide and wedge medium. Mode matching is achieved by discretizing the continuous radiation mode spectrum leading into a numerically stable solution provided a sufficient large number of points are used to convert integrals into finite summations. Numerical computations are carried out for various wedge geometries and shapes including linear and exponential profiles.     

A matrix method for studying TM modes of optical planar waveguideswith arbitrary index profiles

Application of a previously proposed matrix method (which can only be used for TE mode solutions) to studying the wave characteristics of TM modes is described. To derive the matrix equation for TM-mode solutions of slab waveguides, the gradient of a continuous field is defined as the sum of a continuous function and a stepwise function so that boundary conditions are satisfied. By expanding both the index profile and mode field distribution into a truncated Fourier series, a particular matrix equation for determining mode indices and mode field distributions is obtained. Such a matrix equation can be generalized as BX = β²X, where B is a constant matrix, ß is the propagation constant, and the vector X accounts for the mode field distribution. Wave characteristics of TM modes of slab waveguides with arbitrary index profiles can thus be obtained following the solution of a linear algebra problem in a way similar to that for studying TE modes. The matrix B here, however, as indicated by this work, is different from that for TE mode solutions. Also, it is pointed out in this paper that the matrix B for TM-mode solutions has a different form as a different kind of waveguide is considered. Numerical results presented herein show that a sufficiently high accuracy can be obtained by using the proposed method     

对称单负材料平面波导的波导模

基于经典电磁理论建立了芯层为单负材料的对称三层平板波导的导波方程,采用图解法研究了该波导中的TE和TM波导模,分析了TE波导模和TM波导模会受波导参数μ1/μ2和R的影响,给出TE模和TM模解的横向场分布图.结果显示,在单负材料的对称三层平板波导中只存在慢波导模,TE慢波模只能在磁负材料中传播,TM慢波模只能在电负材料...     

Intrinsic modes of radiation in ferrite patch antennas

We have found two types of radiation modes for patch antennas loaded with ferrite materials. Each mode of radiation is a linear combination of normal modes of propagation in parallel plate waveguide separated by a slab of ferrite material. We have introduced new boundary conditions in which only TE modes of oscillation in the patch antenna cavity result. According to different propagation directions relative to the applied DC field these TE modes are distinguished as transverse modes and longitudinal modes, and they possess mutually perpendicular radiation polarizations. While the longitudinal TE modes are found to form discrete modes in the frequency domain, the radiation frequency of a transverse TE mode can be continuously tuned over a wide frequency range by varying the biasing magnetic field. Circularly polarized radiations may result from simultaneous excitations of these two modes. Ferrite patch antennas of square geometry have been fabricated and tested. The measured resonant frequencies compared very well with our theory     

Rigorous analysis of coupling between laser and passive waveguidein multilayer slab waveguide

An approach that combines the finite-element and boundary-element methods (FEM/BEM) is extended to the analysis of arbitrarily shaped discontinuities in a multilayer slab waveguide. Here all the eigenmodes, namely, guided modes, substrate radiation modes, and substrate-cover radiation modes, are taken into account and are constructed by an extended matrix method. Computed results are given for the rigorous analysis of the coupling between a five-or six-layer laser and a four-layer passive waveguide in a monolithically integrated circuit. Some of the results are for multimode incidence. Since the coupling is one of the most important discontinuity problems in practice, this result confirms the usefulness of the approach     

Analysis of Wave Propagation in Anisotropic Film Waveguides with Bent Optical Axes

We present an analytical method for studying the wave propagation in anisotropic planar optical waveguides where the oblique angle between the optical axis and the propagation axis changes arbitrarily in the film surface along the propagation length. The analysis is based on the coupled-mode theory, where the coupling between a guided mode and radiation modes is regarded to be of major importance. We apply a hypothetical boundary method to quantize the continuum of radiation modes, and replace the continuously changing oblique angle by a step approximation. It is shown that these approximations do not degrade the computational accuracy. To exemplify the wave-propagation properties, we deal with a waveguide consisting of LiNbO/sub 3/ and let the oblique angle change linearly along the propagation length. It is found that the incident guided TE mode leaks its power primarily in a very narrow region centered on the critical oblique angle, and that TE radiation modes play an important role in the power conversion, even though they carry far less power than the TM radiation modes.     

A hybrid dielectric slab-beam waveguide for the sub-millimeter waveregion

A hybrid dielectric-slab-beam waveguide is suggested which should be well suited as a transmission medium for the design of planar quasi-optical integrated circuits and devices operating in the millimeter- and sub-millimeter-wave regions. The guide consists of a grounded dielectric slab in which a sequence of equally spaced cylindrical lenses is fabricated. The center line of the slab guide is the axis of the lenses. The structure uses two distinct waveguiding principles in conjunction to guide electromagnetic waves. In the direction normal to the slab surface, the guided fields behave as surface waves of the slab guide; their energy is largely confined to the interior of the dielectric and they are guided by total reflection at the slab surface. In the lateral direction the waves behave as Gauss-Hermite-beam modes that are guided by the lenses, which periodically reconstitute their cross sectional phase distribution, resulting in a wave beam that is iterated with the lens spacing. The guided fields are in effect TE and TM modes. The analysis of the new guiding structure is analyzed, the mode spectrum is calculated, and the iteration loss due to the finite size of the lenses is estimated