Optical Mode Solver Research Articles

Modeling the anisotropic electro-optic interaction in hybrid silicon-ferroelectric optical modulator.

We present a numerical method to accurately model the electro-optic interaction in anisotropic materials. Specifically, we combine a full-vectorial finite-difference optical mode solver with a radio-frequency solver to analyze the overlap between optical modes and applied electric field. This technique enables a comprehensive understanding on how electro-optic effects modify individual elements in the permittivity tensor of a material. We demonstrate the interest of this approach by designing a modulator that leverages the Pockels effect in a hybrid silicon-BaTiO3 slot waveguide. Optimized optical confinement in the active BaTiO3 layer as well as design of travelling-wave index-matched electrodes is presented. Most importantly, we show that the overall electro-optic modulation is largely governed by off-diagonal elements in the permittivity tensor. As most of active electro-optic materials are anisotropic, this method paves the way to better understand the physics of electro-optic effects and to improve optical modulators.

Solving leaky modes on a dielectric slab waveguide involving materials of arbitrary dielectric anisotropy with a finite-element formulation

A planar optical waveguide mode solver is established based on a finite-element (FE) formulation for determining the guided and leaky modes that exist on waveguides made of anisotropic materials with an arbitrary permittivity tensor, for example, with arbitrary optic-axis orientation in the uniaxially anisotropic material case. Correct numerical determination of the complex effective index, especially its imaginary part that gives the modal leakage, is particularly emphasized referring to available analytical solutions. For the situation when the optic axis changes its direction only in the plane parallel to the waveguide interface planes, analytical characteristic equations for solving purely guided and leaky modes are separately derived in a more systematic manner compared with prior analytical formulae reported more than three decades ago, with the obtained complex effective indices agreeing excellently with FE solutions. It is found that in the FE analysis of leaky modes, the thickness of the perfectly matched layer (PML) and the PML theoretical reflection coefficient should be properly chosen. The FE formulation is based on either the three electric-field components or the three magnetic-field components using quadratic nodal bases, resulting in a quadratic eigenvalue equation that is then solved by the shift-and-invert Arnoldi method.

Finite element simulation of a perturbed axial-symmetric whispering-gallery mode and its use for intensity enhancement with a nanoparticle coupled to a microtoroid

We present an optical mode solver for a whispering gallery resonator coupled to an adjacent arbitrary shaped nano-particle that breaks the axial symmetry of the resonator. Such a hybrid resonator-nanoparticle is similar to what was recently used for bio-detection and for field enhancement. We demonstrate our solver by parametrically studying a toroid-nanoplasmonic device and get the optimal nano-plasmonic size for maximal enhancement. We investigate cases near a plasmonic resonance as well as far from a plasmonic resonance. Unlike common plasmons that typically benefit from working near their resonance, here working far from plasmonic resonance provides comparable performance. This is because the plasmonic resonance enhancement is accompanied by cavity quality degradation through plasmonic absorption.

A high-accuracy pseudospectral full-vectorial leaky optical waveguide mode solver with carefully implemented UPML absorbing boundary conditions

The previously developed full-vectorial optical waveguide eigenmode solvers using pseudospectral frequency-domain (PSFD) formulations for optical waveguides with arbitrary step-index profile is further implemented with the uniaxial perfectly matched layer (UPML) absorption boundary conditions for treating leaky waveguides and calculating their complex modal effective indices. The role of the UPML reflection coefficient in achieving high-accuracy mode solution results is particularly investigated. A six-air-hole microstructured fiber is analyzed as an example to compare with published high-accuracy multipole method results for both the real and imaginary parts of the effective indices. It is shown that by setting the UPML reflection coefficient values as small as on the order of 10(-40) ∼ 10(-70), relative errors in the calculated complex effective indices can be as small as on the order of 10(-12).

Design of photosensitive microstructured polymer optical fibers

We propose a new hole-assisted polymer optical fiber design to eliminate the influence of dopant diffusion and to increase the ultra violet (UV) writing efficiency in fiber Bragg grating inscription. The optical waveguide is formed inside a solid core polymethyl methacrylate (PMMA) doped with photosensitive trans-4-stilbenemethanol, surrounded by a ring of three large air holes with double cladding. We determined a map of the single-mode and multi-mode phase transitions using a finite-element-based vectorial optical mode solver. A wide range of geometrical configurations for the single-transverse-mode (HE11) propagation in the visible was obtained. The design is optimized to operate at the low optical loss wavelengths of 580 and 770 nm.

Optimization of 3-hole-assisted PMMA optical fiber with double cladding for UV-induced FBG fabrication

We propose a new hole-assisted polymer optical fiber design to eliminate the influence of dopant diffusion and to increase the UV writing efficiency in fiber Bragg grating inscription. The optical waveguide is formed inside a solid core surrounded by a ring of 3 large air holes in enhanced UV photosensitive PMMA with double-cladding. We determined a map of the single-mode and multi-mode phase transitions using a finite-element- based vectorial optical mode solver. We obtained a wide range of geometrical configuration for the single-transverse-mode (HE11) propagation in the visible. The design is optimized to operate at the minimum optical loss wavelengths of 580 nm and 770 nm.

Efficient and spurious-free integral-equation-based optical waveguide mode solver

Modal analysis of waveguides and resonators by integra-lequation formulations can be hindered by the existence of spurious solutions. In this paper, spurious solutions are shown to be eliminated by introduction of a Rayleigh-quotient based matrix singularity measure. Once the spurious solutions are eliminated, the true modes may be determined efficiently and reliably, even in the presence of degeneracy, by an adaptive search algorithm. Analysis examples that demonstrate the efficacy of the method include an elliptical dielectric waveguide, two unequal touching dielectric cylinders, a plasmonic waveguide, and a realistic micro-structured optical fiber. A freely downloadable version of an optical waveguide mode solver based on this article is available.

Mode competition in multi-core fiber amplifier

3D beam propagation numerical code and a newly developed optical mode solver were implemented for studies on the recently experimentally realized Yb fiber amplifier with 7-core hexagonal structure. It was found that the launched seven single-frequency beams matched with the cores converge to a wave field with permanently diminishing space beating patterns, but the result is sensitive to phases of separate beams. Namely, at a spread of phases of individual beams within a few tenths of radian, the convergence of the wave field to the in-phase mode is observed, while at phases random spread on the order of π the out-of-phase mode is dominating. Such behavior is observed at strong gain saturation in the cores. Both effects take place even if the guiding index of the cores fluctuates within a limited range. The mode solver predicts that the in-phase mode has the minimal small signal gain, and the gain of the out-of-phase mode is maximal one. This fact proves that there is no strict correlation between modal gains and amplification in strong saturated multi-core fiber. The smaller core index step significantly improves the convergence of the amplified radiation to the in-phase mode. An approach to the waveguide structure tailoring is described which assures the single in-phase mode stability.

Wave Equation-Based Semivectorial Compact 2-D-FDTD Method for Optical Waveguide Modal Analysis

A wave equation-based semivectorial compact 2-D finite-difference time-domain (2-D-FDTD) method is developed and validated for optical waveguide modal analysis. This approach is a combination of the Maxwell's equation-based compact 2-D-FDTD and the wave equation-based semivectorial FDTD methods. Perfectly matched layer (PML) absorbing boundary condition (ABC) is also extended to this approach. Excellent accuracy is achieved for the entire spectrum even in the region near the cutoff. Through extensive study on the excitation conditions, it indicates that this method, when used as an explicit optical mode solver, is extremely robust.

Modelling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions

A finite-element-based vectorial optical mode solver is used to analyze microstructured optical waveguides. By employing 1st-order Bayliss-Gunzburger-Turkel-like transparent boundary conditions, both the real and imaginary part of the modal indices can be calculated in a relatively small computational domain. Results for waveguides with either circular or non-circular microstructured holes, solid- or air-core will be presented, including the silica-air Bragg fiber recently demonstrated by Vienne et al. (Post-deadline Paper PDP25, OFC 2004). The results of solid-core structures are in good agreement with the results of other methods while the results of air-core structure agree to the experimental results.

A comprehensive VCSEL device simulator

This paper deals with the design and implementation of a self-consistent electrothermooptical device simulator for vertical-cavity surface-emitting lasers (VCSELs). The model is based on the photon rate equation approach. For the bulk electrothermal transport, a thermodynamic model is employed in a rotationally symmetric body. Heterojunctions are modeled using a thermionic emission model and quantum wells are treated as scattering centers for carriers. The optical field is expanded into modes that are eigensolutions of the vectorial electromagnetic wave equation with an arbitrary, complex dielectric function. The open nature of the VCSEL cavity is treated by employing perfectly matched layers. The optical gain and absorption model in the quantum-well active region is based on Fermi's Golden Rule. The subbands in the quantum well are determined by solving the stationary Schrodinger equation and using a parabolic band approximation for the electrons, light and heavy holes. The photon rate equation is fully integrated into the Newton-Raphson scheme used to solve the system of nonlinear device equations. An efficient numerical optical mode solver is used, that is based on a Jacobi-Davidson type iterative eigensolver. The latter combines a continuation scheme with preconditioner recycling. The practical relevance of the implementation is demonstrated with the simulation of a realistic etched-mesa VCSEL device.

Comparison of wavelength splitting for selectively oxidized, ion implanted, and hybrid vertical-cavity surface-emitting lasers

The wavelength splitting between the LP/sub 01/ and LP/sub 11/ modes of selectively oxidized, ion implanted, and hybrid ion implanted/selectively oxidized vertical-cavity surface-emitting lasers is studied by experiment and theory. Measured splittings at threshold show marked differences between the different laser structures due to the effects of index guiding and thermal lensing. Theoretical results were obtained using a vector optical mode solver and show good agreement with experimental results. The hybrid lasers exhibited behavior intermediate between the ion implanted and selectively oxidized lasers and could be optimized for high power single transverse mode emission.

Numerical simulation of vertical cavity surface emitting lasers

The semiconductor laser simulator MINILASE is being extended to simulate vertical cavity surface emitting lasers (VCSELs). The electronic system analysis for VCSELs is identical to that for edge emitting lasers. A brief discussion of the capabilities of MINILASE in this domain will be presented. In order to simulate VCSELs, the optical mode solver in MINILASE must be extended to handle the reduced index guiding and significant gain guiding typical of many VCSEL structures. A new approach to solving the optical problem which employs active cavity modes rather than the standard passive cavity modes is developed. This new approach results in an integral eigenvalue equation in required gain amplitudes and corresponding modal fields. Sample results from an early implementation of a gain eigenvalue solver are shown to clarify the possibilities of this approach.