A finite difference BPM analysis of bent dielectric waveguides

A new version of the scalar TE wave equation is introduced, one that is particularly useful in bent waveguide analysis. The slowly varying envelope equation in cylindrical coordinates for the field amplitude E is finite-differenced, with no other approximations made to it. It will be shown that this version of the equation has several advantages over other forms and gives good results for the power loss rates even at radii ⩽100 μm, as well as being useful for the study of curved structures with varying radii of curvature. Evidence is provided to show that the loss rates calculated from this equation using the two dimensional finite difference beam propagation method compare favorably with other numerical and analytical results found in the literature. Special care must be taken when applying transparent boundary conditions as the curvature increases     

The unitarity of split-operator finite difference andfinite-element methods: Applications to longitudinally varyingsemiconductor rib waveguides

Split-operator finite-difference and finite-element techniques are applied to the calculation of losses in semiconductor rib waveguide Y junctions. It is shown that, unlike the finite-difference procedure, which is unitary for both uniform and nonuniform grid-point spacings, considerable care must be exercised in formulating the split-operator finite-element method in order to preserve the power in the propagating electric field. The calculations are performed in the context of the Fresnel approximation to the scalar Helmholtz equation, which yields accurate solutions for transverse electric (TE)-polarized electric fields in rib waveguides far from cutoff     

Analysis of finite rib waveguides by matrix methods

The modal structure of finite lateral extent waveguides differs from that of the associated infinite structures by the presence of additional guided modes arising from the boundary conditions and by the discrete representation of the continuum modes. These differences are analyzed for a rib waveguide using matrix methods. The origin of the additional modes and their evolution with changing system parameters are discussed. The development of localized modes from delocalized modes is of particular interest, as is the evolution of radiation modes into discrete, localized modes of higher order as the rib increases in height. These developments are followed in detail and the changes in modal character are related to the properties of the finite system's dispersion curves. Illustrations of the effect of increasing system size on the localized and delocalized modes are given     

Rigorous analysis of multiple coupled rib waveguides

The general problem of multiple coupled rib waveguides, where energy may be leaked from one guide to the other via the substrate and radiation mode, is treated. Rigorous results including substrate and air modes coupling are sought for the general case of coupling of two or more different guides. The analysis, developed in terms of a cascade of the transverse steps, utilizing a variational solution with a single trial function and making explicit use of edge singularities at the dielectric corners in order to produce an effective and rigorous solution, is presented. Multiple coupled rib guides are then reduced to a cascade of interacting step discontinuities in the transverse direction. When comparison is possible, the numerical results obtained by the method are seen to be as accurate as those obtained by the finite element method (FEM)/finite difference method (FDM), but with a fraction of the computer time and memory involved     

A new semirigorous analysis of rib waveguides

A semirigorous analysis of rib waveguides embedded in air is proposed. The analysis is based upon a scalar analysis and two different vectorial correction schemes, through either a perturbation formula or a scalar calculation with modified effective dimensions. The scalar field is computed with a very efficient spectral technique. The method converges rapidly, and it compares favorably with recent work on rib waveguides. It can be extended to other planar structures, such as ridge waveguides     

Explicit finite difference beam propagation method: application to semiconductor rib waveguide Y-junction analysis

A novel explicit finite difference beam propagation method is developed and applied to the analysis of a strongly guiding semiconductor rib waveguide Y-junction. Its accuracy is confirmed by comparison with reported results. This method is much more efficient than the standard beam propagation method for the rib waveguide analysis.<>     

Full-vectorial mode analysis of rib waveguides by iterative Lanczosreduction

A highly efficient computer simulation code is developed to perform full-vectorial mode analysis of rib or channel waveguides. On a workstation, the mode profile and index can be calculated in 410 s if a 192×192 grid is used. The code is based on the iterative Lanczos reduction technique. Although the matrix representation of the transverse Helmholtz operator is slightly asymmetric in the vectorial mode analysis, the Lanczos reduction procedure can still converge to the eigenmode. Results for several rib waveguides are presented. They are compared to the results of the vectorial finite-element method and semivectorial analysis     

Analysis of rib dielectric waveguides

An analysis of rib dielectric waveguides based on a mode matching technique is presented. When the constituent slab guides support only one guided mode, it is shown that the cutoff condition for the higher order modes is the same as the result of the effective dielectric constant method. However, when the rib region is thick enough to support two guided slab modes, the cutoff conditions are significantly different. For this case universal design curves are produced and plotted. The results are also compared with another theory that exists to analyze such structures.     

Crosstalk reduction in intersecting rib waveguides

A numerical study of the throughput and crosstalk in intersecting semiconductor rib waveguides was carried out using the beam propagation method. The fraction of the optical power that couples out of one input waveguide and into the crossed waveguide falls to below two percent for crossing angles greater than four degrees. A simple modification to the waveguide shape at the X-crossing region was found to reduce the crosstalk for crossing angles between four and ten degrees. The crosstalk to throughput ratio is reduced by up to a factor of nine while the throughput is reduced by, at most, a few percent. In device structures that combine X-crossings with curved waveguide sections this permits greater design flexibility and improved overall loss and crosstalk performance     

Single-mode condition for silicon rib waveguides

The geometrical conditions required for single-mode propagation in large cross-section silicon-on-insulator (SOI) waveguides were investigated using the beam propagation method (BPM). The cases of both vertical-walled and sloping-walled (trapezoidal) rib structures were considered. In the trapezoidal case some approximate methods were compared with the simulated results. Expressions for the single-mode condition for both types of waveguide were defined.     

GaAs/GaAlAs curved rib waveguides

Curved dielectric optical waveguides suffer from radiation loss due to bending. To minimize the bending loss and reduce the radius of curvature, it is necessary to fabricate guides which provide strong optical confinement. This paper gives a brief review of curved waveguide analysis and presents some experimentally measured loss values for GaAs/GaAlAs curved rib waveguides. The rib waveguides, fabricated using ion beam milling, have a large rib height and are tightly guided structures. When corrected for reflection losses and input coupling efficiency, a minimum loss of approximately 3 dB has been achieved for a multimode 90° curved guide with a radius of curvature of 300 μm, and 8.5 dB for a single-mode curved guide with a radius of curvature of 400 μm. It is believed that most of this residual loss is not radiation loss due to bending, but rather scattering loss due to rib wall imperfections.     

Birefringence characteristics of benzocyclobutene rib optical waveguides

Zero-birefringence benzocyclobutene polymer rib waveguides as well as waveguides that exhibit low birefringence over a wide range of temperature are demonstrated. Experimental results agree well with theoretical calculations that take into account stress-induced birefringence.     

Polarization rotation in periodic loaded rib waveguides

Polarization rotation in an asymmetric periodic loaded rib waveguide is analyzed by a scalar coupled-mode theory with vector correction. The scalar modes of the dielectric loaded rib waveguide are calculated by using a simple variational method. The polarization rotation is found to be caused by a geometric effect due to the asymmetric loading of the waveguide. By phase-matching the two polarized modes through the periodic loading, complete polarization rotation can be achieved     

Low transition losses in bent rib waveguides

We reduce the transition losses that occur between curved and straight rib waveguides, by offsetting the guides and placing isolation trenches. Using a three-dimensional (3-D) semivectorial beam propagation program, we also can investigate more novel effects such as rib heights and sidewall slopes. Our consideration of all these effects results in an optimized design     

Finite difference methods for the analysis of integrated optical waveguides

A finite difference method based on a Crank-Nicolson discretisation procedure has been implemented and applied to the analysis of propagation of optical beams in dielectric waveguides. It is based on a new formula which takes into account the correct boundary conditions. Comparisons with the well known BPM technique show a definite improvement in computer time saving, accuracy and ease of use.<>     

Improved design of magnetooptic rib waveguides for opticalisolators

Garnet films of the following compositions (Lu,Bi)₃(Fe,Ga,Al)₅O₁₂ and (Tm,Bi)₃(Fe,Ga)₅O₁₂ are grown by liquid-phase epitaxy on [111]-oriented substrates of gadolinium gallium garnet. Double layers with opposite signs of Faraday rotation and single layers are prepared. Optical monomode rib waveguides are realized using such films. The nonreciprocal phase shift of the fundamental TM-mode is studied both theoretically and experimentally at a wavelength of 1.3 μm. The maximum nonreciprocal effect of double layer films is about 1.7 times larger than that of similar single layer waveguides. Agreement between experiments and calculations is excellent     

An iterative finite difference beam propagation method for modelingsecond-order nonlinear effects in optical waveguides

An iterative finite difference beam propagation method based on the Crank-Nicholson scheme is presented to simulate continuous wave (CW) second-order nonlinear effects in optical waveguides with the depletion of the pump wave taken into account. This method is an extension of the linear finite difference beam propagation method and preserves the same order of accuracy. Comparisons with the previously published explicit finite difference beam propagation method and the rectangular approximation method are presented. Quasi-phase matched difference frequency generation in AlGaAs and quasi-phase-matched second harmonic generation in LiNbO₃ are considered in the evaluation, showing that one iteration for the IFD-BPM is sufficient for the simulation with good accuracy and without increasing much computation time     

Extremely low loss InP/GaInAsP rib waveguides

The authors have fabricated InP/GaInAsP rib waveguides with losses as low as 0.18 dB/cm for the TE polarisation and 0.26 dB/cm for the TM polarisation. The length of the waveguides was 28 mm. These losses are measured with the Fabry-Perot method and repeated for different lengths of the waveguide for determining the reflection coefficient.<>     

A simple variational approach to optical rib waveguides

A simple method for the calculation of the mode properties of optical rib waveguides is presented. The method is based on a scalar variational principle combined with a vector perturbation analysis for the polarization correction. It is demonstrated that, with the proper trial functions, the approach can predict very accurately the dispersion characteristics with much less effort (in terms of formulation and computation) than the existing numerical techniques. The field patterns of the modes can be computed from the scalar analysis. The vector property of the mode can be taken into account by a perturbation formula. The computation is carried out on an IBM PC. The user-friendly computer program is written in Fortran     

Grafted GaAlAs rib waveguides on an InP substrate

Rib waveguides were fabricated on a 1.4 mu m thick GaAlAs epilayer granted on the surface of a semi-insulating InP substrate by epitaxial lift-off. Single-mode waveguides with propagation losses (<7 dB/cm) lower than heteroepitaxially grown counterparts have been achieved. TEM analysis on the GaAlAs/InP interface indicates surface scattering as one of the main loss mechanisms.<>