Microring resonators side coupled to signal waveguides provide compact, narrow band, and large free spectral range optical channel dropping filters. Higher order filters with improved passband characteristics and larger out-of-band signal rejection are realized through the coupling of multiple rings. The analysis of these devices is approached by the novel method of coupling of modes in time. The response of filters comprised of an arbitrarily large dumber of resonators may be written down by inspection, as a continued fraction. This approach simplifies both the analysis and filter synthesis aspects of these devices.

Microring resonator channel dropping filters

The authors give a brief historic perspective of the coupled mode theory. The development and applications of the theory in microwaves in early years and in optoelectronics and fiber optics in recent years are described. They then consider lossless coupling of two modes in time. Two coupled resonance circuits, or two coupled microwave or optical resonators, are the physical examples. The start-up of a parametric oscillator is another example. Then they look at the formal derivation of coupled mode theory and consider the more general case when the modes are not energy-orthogonal and the energies are not necessarily positive. A more detailed account of the nonorthogonal coupled mode theory developed in the last five years for optical waveguides is given. >

Coupled-mode theory

The coupled-mode theory (CMT) for optical waveguides is reviewed, with emphasis on the analysis of coupled optical waveguides. A brief account of the recent development of the CMT for coupled optical waveguides is given. Issues raised in the debates of the 1980’s on the merits and shortcomings of the conventional as well as the improved coupled-mode formulations are discussed. The conventional coupled-mode formulations are set up in a simple, intuitive way. The rigorous CMT is established on the basis of a linear superposition of the modes for individual waveguides. The cross-power terms appear logically as a result of modal nonorthogonality. The cross power is necessary for the self-consistency of the CMT for dissimilar waveguides. The nonorthogonal CMT, though more complicated, yields more-accurate results than the conventional orthogonal CMT for most practical applications. It also leads to the prediction of cross talk in directional couplers. The conventional orthogonal CMT is, however, reliably accurate for describing the power coupling between two weakly coupled, nearly identical waveguides. For dissimilar waveguides, a self-consistent orthogonal CMT can be derived by a redefinition of the coupling coefficients, and it predicts the coupling length and therefore the power exchange between the waveguides accurately if the two waveguides are far apart. Three typical coupler configurations—the uniform, the grating-assisted, and the tapered—are examined in detail. The accuracy, scope of validity, limitations, and extensions of the coupled-mode formulations are discussed in conjunction with each configuration. To verify the arguments in the discussions, comparisons with the exact analytical solutions and the rigorous numerical simulations are made.

Coupled-mode theory for optical waveguides: an overview

Compact optical channel dropping filters incorporating side-coupled ring resonators as small as 3 /spl mu/m in radius are realized in silicon technology. Quality factors up to 250, and a free-spectral range (FSR) as large as 24 nm are measured. Such structures can be used as fundamental building blocks in more sophisticated optical signal processing devices.

Ultra-compact Si-SiO 2 microring resonator optical channel dropping filters

We investigate the properties of high-Q, wide free-spectral-range semiconductor microcavity ring and disk resonators coupled to submicron-width waveguides. Key optical design parameters are characterized using finite-difference time-domain (FDTD) solutions of the full-wave Maxwell's equations. We report coupling efficiencies and resonant frequencies that include the effects of waveguide dispersion and bending and scattering losses. For diameters of 5 /spl mu/m, the microcavity resonators can have Q's in the several thousands and a free spectral range of 6 THz (50 nm) in the 1.55 /spl mu/m, wavelength range. Studies of the transmittance characteristics illustrate the transition from single-mode resonances to whispering-gallery-mode resonances as the waveguide width of the microring is increased to form a solid microdisk. We present nanofabrication results and experimentally measured transmission resonances of AlGaAs/GaAs microcavity resonators designed in part with this method.

FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators

An improved version of coupled-mode equations for parallel dielectric waveguides has been derived by using a newly found relationship that connects the propagation constants of the individual guides to the coupling coefficients via an overlap integral that measures the guides' proximity. The four parameters of these new coupled equations are simple functions of essentially one single quantity: the asynchronism of the individual guides properly normalized.

Improved coupled-mode equations for dielectric guides

An "exact" theory is developed for the design of single or muitimode dielectric tapers of arbitrary cross sections and straight or curved axes that have the unique property of guiding with no conversion either to other guided modes or to radiation modes. This remarkably simple theory encompasses a very wide gamut of strictly adiabatic tapers: some very simple, such as linear tapers with straight axes, some very sophisticated, such as directional couplers made of noncoplanar guides tapered and curved differently.

Dielectric tapers with curved axes and no loss

An approximate technique to solve guidance problems in dielectric waveguides having almost arbitrary cross-section index distribution is presented. The technique consists of reducing the solution of the three-dimensional guide to the solution of two unidimensional slabs. By starting with the circular-cylinder coordinate system instead of the Cartesian one, the results are found to be more precise and far more informative than those derived from any other approximate analytical technique. The ideas presented here may be easily extended to solve more sophisticated structures such as couplers and cavities. >

E. Marcatili

Papers

Improved coupled-mode equations for dielectric guides

Dielectric tapers with curved axes and no loss

The azimuthal effective-index method