Department of Mathematics

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IIT Jodhpur

Circular integrated optical (ring or disk) microresonators are increasingly employed as compact and versatile wavelength filters. In this paper, we investigate a 2-D frequency domain model for these devices, based on spatial coupled mode theory. The microresonators are functionally represented in terms of two couplers with appropriate connections using bent and straight waveguides. The abstract scattering matrices of the couplers and the propagation constants of the cavity bends allow to compute the spectral responses of the resonators. Capitalizing on the availability of rigorous analytical modal solutions for bent waveguides, the constituent bent-straight waveguide couplers are modeled using a spatial coupled mode formalism derived by means of a variational principle. The resulting scattering matrices show reciprocity properties as expected according to the symmetry of the coupler structures. We present results for the spectral response and field examples for microresonators with mono- and multi-modal cavities for TE and TM polarizations. Comparisons with finite difference time domain simulations show very good overall agreement. © 2005 Elsevier B.V. All rights reserved.

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Optics Communications

Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory

A spatially three-dimensional (3-D), fully vectorial coupled mode theory model for the interaction between several straight or bent dielectric optical waveguides, each supporting multiple modes, is described. The frequency domain model is applied to the coupler regions of cylindrical microresonators, here considered for applications as integrated optical filters. For simple test cases, comparisons with results of beam propagation calculations and of a rigorous system mode analysis provide some validation of the approach. By combination of two coupler representations one obtains a complete 3-D vectorial microresonator description without any free parameters, that permits a convenient investigation of the influence of geometrical parameters on the spectral response. When applied to a microring resonator with pronouncedly hybrid cavity modes, the model reveals the manifold features that may appear in the spectra of these devices. © 2005 Elsevier B.V. All rights reserved.

Fulltext

Cylindrical integrated optical microresonators: Modeling by 3-D vectorial coupled mode theory

A rigorous classical analytic frequency domain model of confined optical wave propagation along 2D bent slab waveguides and curved dielectric interfaces is investigated, based on a piecewise ansatz for bend mode profiles in terms of Bessel and Hankel functions. This approach provides a clear picture of the behaviour of bend modes, concerning their decay for large radial arguments or effects of varying bend radius. Fast and accurate routines are required to evaluate Bessel functions with large complex orders and large arguments. Our implementation enabled detailed studies of bent waveguide properties, including higher order bend modes and whispering gallery modes, their interference patterns, and issues related to bend mode normalization and orthogonality properties. © Springer 2005.

Optical and Quantum Electronics

Analytic approach to dielectric optical bent slab waveguides

Analysis of cylindrical integrated optical microresonators involves the coupling between a straight waveguide and a bent waveguide. Our (2D) variant of coupled mode theory is based on analytically represented mode profiles. With the bend modes expressed in Cartesian coordinates, coupled mode equations can be derived in a classical way, and solved by numerical integration. Proper manipulation of the propagation matrix leads to stable results that satisfy the reciprocity properties for symmetric coupler devices, even in parameter domains of rather compact, radiative structures and strong interaction, which before seemed unaccessible by a perturbational approach. Comparisons with FDTD calculations show convincing agreement. © 2004 American Institute of Physics.

AIP Conference Proceedings

Comparison of Coupled Mode Theory and FDTD Simulations of Coupling between Bent and Straight Optical Waveguides

Optical ring resonators are commonly discussed on the basis of a frequency-domain model, that divides a resonator into coupler elements, ring cavity segments, and the straight port waveguides. We look at the assumptions underlying this model and at its implications, including remarks on reciprocity/symmetry arguments, the general power transfer characteristics, the resonance condition, the spectral distance and width of the resonances, the quantities that describe the resonator performance, and a few remarks about tuning. A survey of bend mode properties and a coupler description in terms of coupled mode theory fills the abstract notions of the model. As an example for devices that rely on a standing wave principle, in contrast to the traveling waves found in the microrings, we consider in less detail microresonators with square or rectangular cavity shapes. Also here a frequency domain coupled mode theory can be applied that opens up simple possibilities to characterize resonant configurations. © 2004 American Institute of Physics.

Analytical Approaches to the Description of Optical Microresonator Devices

International Journal of Dynamics & Control

Stochastically stable chaos for q-deformed unimodal maps

The Mathematics Student

Classification of extraspecial p-groups using quadratic forms

Gradient Based Line Search Techniques for Multi-objective Optimization Problems

Journal of Difference Equations and Applications

Dynamics of q deformed Ricker map

One of the widely discussed in the literature and relevant in practice shock models is the delta-shock model that is described by the constant time of a system's recovery after a shock. However, in practice, as time progresses and due to deterioration of a system, this recovery time is gradually increasing. This important phenomenon was not discussed in the literature so far. Therefore, in this paper, we are considering a time-dependent delta-shock model, i.e., the recovery time becomes an increasing function of time. Moreover, we assume that shocks occur according to the generalized Polya process that contains the homogeneous Poisson process, the non-homogeneous Poisson process and the Polya process as particular cases. For the defined survival model, we derive the corresponding survival function and the mean lifetime and study the related optimal replacement policy along with some relevant stochastic properties.