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Journal of the Optical Society of America B

Dynamic mode theory of optical resonators undergoing refractive index changes

Brian A. Daniel, Drew N. Maywar, Govind Prasad Agrawal

Published Aug 22, 2011• 35 citations

Abstract

We present a theoretical model describing the dynamics of the electromagnetic field in an optical resonator undergoing refractive index changes. We use an operator formulation of Maxwell’s equations with a standard time-dependent perturbation theory to derive the dynamic mode-amplitude equations that govern the response of a resonator to a perturbing dipole-moment density. We show that in the case of time-dependent changes in the refractive index, a coupling matrix Γkm(t) that appears in the equations accounts for all novel physical processes that can be expected to occur. In particular, the phenomenon of adiabatic wavelength conversion is governed by the diagonal elements of this matrix, and the off-diagonal elements are responsible for the transfer of energy from an excited resonator mode into its neighboring modes. Our model clearly shows that the latter process can occur only when the index changes are spatially nonuniform. We discuss the spatially uniform and nonuniform cases separately and compare the predictions of our model with experimental data available in the literature. The overall good agreement suggests that this model should be useful in the study of dynamic optical resonators. Moreover, since we do not make any assumptions about the type of dielectric cavity used, the width of input pulses, or the speed with which the refractive index is changed, this model should be applicable under most experimental situations.

DOI: 10.1364/josab.28.002207

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We present a theoretical model describing the dynamics of the electromagnetic field in an optical resonator undergoing refractive index changes. We use an operator formulation of Maxwell’s equations with a standard time-dependent perturbation theory to derive the dynamic mode-amplitude equations that govern the response of a resonator to a perturbing dipole-moment density. We show that in the case of time-dependent changes in the refractive index, a coupling matrix Γkm(t) that appears in the equations accounts for all novel physical processes that can be expected to occur. In particular, the phenomenon of adiabatic wavelength conversion is governed by the diagonal elements of this matrix, and the off-diagonal elements are responsible for the transfer of energy from an excited resonator mode into its neighboring modes. Our model clearly shows that the latter process can occur only when the index changes are spatially nonuniform. We discuss the spatially uniform and nonuniform cases separately and compare the predictions of our model with experimental data available in the literature. The overall good agreement suggests that this model should be useful in the study of dynamic optical resonators. Moreover, since we do not make any assumptions about the type of dielectric cavity used, the width of input pulses, or the speed with which the refractive index is changed, this model should be applicable under most experimental situations.

We present a simple and intuitive model based on the impulse response of linear electrical systems for describing the propagation of optical pulses through a dynamic Fabry–Perot resonator whose refractive index changes with time. Our model shows that the adiabatic wavelength conversion process in resonators results from a scaling of the round-trip time with index changes. For pulses longer than the cavity round-trip time, we find that more energy can be transferred to the new wavelength when the input pulses are slightly detuned from the cavity resonance and the refractive index does not change too rapidly. In fact, the optimum duration of index changes scales with the photon lifetime of the resonator. We describe the evolution of the shape and spectrum of picosecond pulses inside a resonator under a variety of input conditions and with the magnitude and duration of index variations. We also apply our general theory to the case of pulses whose widths are shorter than the round-trip time and derive an analytical expression for the output field under quite general conditions. This analysis reveals a shifting of the spectral comb as well as compression of the temporal pulse train that depends on the both the magnitude and sign of the index change. Our results should find applications in the area of optical signal processing using resonant photonic structures.

A theory of the two-wave mixing in a photorefractive medium placed in a ring cavity is developped following the model first derived by Yariv and Kwong [1]: a steady refractive index involving the photorefractive effect is used as a non-linear source-term (PNL) in the Maxwell equations, and the cavity field is expanded on the transverse cavity modes. In this model we take into account the intensity change of both the pump and resonator beams inside the crystal due to the mutual power exchange: we have developed a multimode model which gives in the long-time limit the z-dependance of the fields (z is the optical axis) for a passage in the crystal, using the expansion in terms of the complete set of the cavity transverse modes. [2]

A dynamical analysis of nonlinear optical resonators is performed in the framework of a single-mode formalism in the plane-wave regime, taking the finite response time of the nonlinear medium into account. We demonstrate that analytically solvable generic equations can be derived for the time evolution of the output intensity, in the two limiting cases where the medium's response time is either very large or very small as compared to the cavity buildup time. The stability picture and the characteristic time scales, important for applications in the field of optical information processing, are investigated analytically. This study reveals that a surprising effect may occur where an increase in the material-response time leads to a decrease of the relaxation time of the nonlinear device. This effect is called the speeding-up phenomenon. We also present a numerical study of the full nonlinear dynamics.

A slightly refined perturbation theory is used to obtain a unitary transformation which approximately diagonalizes the total Hamiltonian for a system made up of a radiation field in interaction with gas molecules. The dependence of the energy of the total system on the number of photons present gives the usual formula for the refractive index of a gas. The transformation is also applied to the field operators and it is shown that, within the approximations made, the phenomenological Maxwell equations are obeyed in the new representation as equations between certain matrix elements of operators. The transformed representation is also used to derive the Frank-Tamm formula for \ifmmode \check{C}\else \v{C}\fi{}erenkov radiation.

We demonstrate numerically that the wavelength conversion of light is possible by the simple dynamic refractive index tuning of an optical cavity in a photonic crystal. We also clarify the mechanism and conservation rule for this conversion process. In addition, we discuss the observability of this phenomenon in realistic cavities. Our results indicate that this linear adiabatic wavelength conversion process can be observed for various high-

Q

microcavities.

A quantum-mechanical formalism is developed for representing the complex refractive index of a gas, including both dispersive and absorptive parts. In analogy with some previous work by the author, it is shown that the phenomenological Maxwell equations are obeyed as equations between certain matrix elements of operators representing the field quantities. The method makes use of a canonical transformation due to Arnous and Bleuler, and also of the "damping operators" previously introduced by the author. It is shown that the calculation of the complex refractive index reduces to the calculation of a single diagonal matrix element of a damping operator. The actual evaluation of this matrix element is taken up in the following paper.