A partial variational approach for arbitrary discontinuities inplanar dielectric waveguides

A novel approach for analyzing arbitrary discontinuities in planar dielectric waveguides is proposed that uses the finite-element method along with the frontal solution technique. On the basis of the partial variational principle (PVP), the fields interior and exterior to the discontinuity finite-element region can be treated independently and eventually can be coupled. The interior fields are expanded by the finite-element nodal values and the corresponding local bases, while the exterior ones are handled by an approach combining modal expansion and Green's function. In numerical computation, the continuous spectra of the waveguide modes are discretized by the Laguerre expansion method. To check the correctness of the present analysis, two numerical results are compared to those of other methods. The scattering characteristics of several linearly tapered discontinuities, such as transformers and feed structures, are analyzed and compared to those having step junctions     

A modified ray-optic method for arbitrary dielectric waveguides

A modified ray-optic method (MRO) is proposed for evaluating the propagation characteristics of dielectric waveguides with arbitrary refractive index profile. The authors correct the error caused by assuming a constant phase shift for any mode at the turning point on the substrate side. Very simple formulas are given for calculating the dispersion characteristics including those for planar optical waveguides and single-mode fibers. The results are more accurate than those of the ordinary ray-optic method, especially in the near cutoff regions, and agree well with those of the exact numerical methods. A comparison with other analytical and the exact numerical methods shows that the proposed method is simpler than other methods. This method can be extended to other more complicated cases     

Fast method for calculating cutoff frequencies in single-mode fibres with arbitrary index profiles

A Cheby?shev power-series approach is used for cutoff frequencies of the second mode in fibres with arbitrary refractive-index profiles. The method is fast because only a small number of terms is required to give high accuracy. A feature of this approach is that measured profiles can be used directly without being first converted to any specific forms. Examples are given as illustrations.     

A method to predict the Raman gain spectra of germanosilicatefibers with arbitrary index profiles

In this letter, we present a technique for calculating the Raman gain spectra of germanosilicate fibers from their index profiles. The gain spectra depend on radial profiles of the guided optical modes and the glass composition, in particular the distribution of Si-O-Si and Ge-O-Si bridging bonds. We have obtained spectral coefficients, which represent the contributions from these bonds using Raman gain measurements made on a set of calibration fibers with well-characterized index profiles. We describe how these spectral coefficients can be used in the design of Raman gain fibers, and we demonstrate the accuracy of our technique over a broad range of fiber types     

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     

An efficient method for calculating cutoff frequencies in optical fibers with arbitrary index profiles

An accurate and simple method of calculating the cutoff frequencies of scalar modes in a fiber having an arbitrary refractive-index profile, including discrete numerical data from profile measurements, is proposed. First, the cutoff problem is transformed into a general matrix eigenvalue problem and then cutoff frequencies as the eigenvalues of a real symmetric matrix are evaluated. Representative test examples have been computed to illustrate the accuracy, reliability, and efficiency of the algorithm proposed.     

Semi-vectorial finite element method for arbitrary step-indexoptical waveguides

A simple semi-vectorial finite element method using a new formula is presented which treats only a linearly polarised transversal electric field E_t. The electric field should satisfy the conditions that either E_t or n²E_t is continuous, whether E_t is parallel or perpendicular to the refractive index discontinuous interface in an optical waveguide. This allows accurate and efficient analyses of the polarisation dependence of arbitrary step-index optical waveguides     

An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition

An accurate and efficient solution method using spectral collocation method with domain decomposition is proposed for computing optical waveguides with discontinuous refractive index profiles. The use of domain decomposition divides the usual single domain into a few subdomains at the interfaces of discontinuous refractive index profiles. Each subdomain can be expanded by a suitable set of orthogonal basis functions and patched at these interfaces by matching the physical boundary conditions. In addition, a new technique incorporating the effective index method and the Wentzel-Kramers-Brillouin method for the a-priori determination of the scaling factor in Hermite-Gauss or Laguerre-Gauss basis functions is introduced to considerably save computational time without experimenting with it. This method shares the same desirable property of the spectral collocation method of providing a fast and accurate solution but avoids the oscillatory solutions and improves the poor convergence problem of the simple spectral collocation method with single domain where regions of discontinuous refractive index profiles exist. Applications to several two- and three-dimensional waveguide structures having exact or accurate approximate solutions are given to test the accuracy and efficiency; all the results are found to be in excellent agreement.     

Effective index method for generalised waveguide dispersion characteristics analysis of optical channel waveguides

An approximate analysis of the generalised waveguide dispersion characteristics (GWDC) of optical channel waveguides based on the effective index (EI) method is presented. As an illustration of its practical applications, the possibility of achieving temperature-independence in glass rib waveguides is studied and predicted for the first time which may have important applications for integrated-optic devices requiring temperature-stabilised operations.<>     

A spectral method for calculation of rectangular shielded waveguides with an arbitrary anisotropic filling

A method for calculation of shielded guiding structures with an arbitrary anisotropic filling is proposed. The method is based on a modified Galerkin procedure. Field components are expanded into the eigenfunction spectrum of the comparison waveguide. The components are related via expansion coefficients, which are determined in the course of solution of a dispersion problem.     

A refined WKB method for symmetric planar waveguides withtruncated-index profiles and graded-index profiles

An accurate method is proposed, based on a modified version of the WKB method in quantum mechanics, for evaluating the propagation constants and field distribution of symmetric planar waveguides with a truncated-index and graded-index profile. Simple formulas are given for calculating the dispersion characteristics including those for the truncated parabolic and exponential profile, and results are much more accurate than those of the ordinary WKB method, especially in the near cutoff regions. This method also can be extended to other more complicated cases     

Norm-conserving finite-difference beam-propagation method for TMwave analysis in step-index optical waveguides

Nonconservation of power is a perplexing problem in the propagating beam analysis of transverse magnetic (TM) waves in a z-variant step-index optical waveguide. To conserve the power in terms of a squared norm, a modified finite-difference (FD) formula is introduced that allows a general position of a core-cladding interface. The use of the modified formula contributes to a reduction in a field profile error caused by a staircase approximation with subsequent conservation of power, particularly for a symmetrical waveguide. To obtain the power conservation even in the analysis of an asymmetrical waveguide, a z-derivative of the refractive index is taken into account. An asymmetrical taper and tilted waveguides placed in parallel are investigated to validate the present technique     

Investigation of 3D semivectorial finite-difference beampropagation method for bent waveguides

A three dimensions of semivectorial finite difference beam propagation method (FD-BPM) in cylindrical coordinates is investigated in order to analyze the optical wave propagation in the bent waveguides with the strong lateral confinement which will keep low radiation loss. The three dimensional (3D) semivectorial wave propagation equation is programmed in the quick and unconditionally stable procedure by using the alternating direction implicit method. This version of FD-BPM could well express the polarization characteristics of modal field near the dielectric interfaces and accurately simulate the wave propagation in bent waveguides even at a very small radius (<100 μm). Moreover, it is pointed out that the numerical precision of the algorithm is intensively affected by the propagation loss of bent waveguides along with the propagation step length of beam propagation method     

A new method for producing graded index PMMA waveguides

We report a new method for producing graded index polymethyl methacrylate (PMMA) waveguides using optical densification by angled X-ray exposure. The resulting larger refractive index near the surface of the waveguide serves to better isolate the guided light from potentially absorbing or scattering substrates than in waveguides of constant refractive index. We calculate that a 65° angle exposure in 6-μm-thick planar PMMA waveguide shifts the peak of the fundamental mode profile ~0.5 μm away from the substrate. Consistent with this result, we measure a reduction in the propagation loss of PMMA planar waveguides on a Ni substrate from 1.5 dB/cm to 0.5 dB/cm after angle X-ray exposure     

A variational coupled-mode theory for periodic waveguides

A simple variational theorem for the dispersion relations and propagation constants of periodic waveguides is derived. The trial fields used in the variational formula incorporate all the Floquet components correct to first order. This yields a propagation constant in which errors enter only to fourth order in the trial field. The analysis yields a new coupled mode formulation which is shown to yield excellent agreement with exact analytic solutions (in l-D), and numerical simulations (in 2-D), for high-index contrast structures     

Semivectorial beam-propagation method for analyzing polarized modesof rib waveguides

The authors report a semivectorial beam-propagation method which can determine the form birefringence in rib or channel waveguides. In this approach, a finite-difference equation is solved by using the alternating-implicit method to simulate the propagation of the optical beam. Since boundary conditions are incorporated into the finite-difference equation, polarized modes can be simulated. Numerical results on three well-known rib waveguides are presented with comparisons to published data     

A new equivalent network method for analyzing discontinuityproperties of open dielectric waveguides

A novel network approach is proposed for analyzing interacting discontinuities on open planar dielectric waveguides by accurately taking account of both surface modes and waves with continuous spectra. In this approach, a continuum of the radiation wave is recomposed into a set of the newly defined spectral composite modes, each of which carries a finite magnitude of radiation power. These new modes, in conjunction with surface modes, construct the complete orthonormal set for expressing an arbitrary local field on a dielectric slab waveguide. This leads to an equivalent network approach effective for solving discontinuity problems, even on an open waveguide using the conventional method for closed-waveguide problems. A number of numerical results are shown to demonstrate the usefulness of the proposed approach