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Journal of the Optical Society of America B

Optical pulse propagation in dynamic Fabry–Perot resonators

Yuzhe Xiao, Drew N. Maywar, Govind Prasad Agrawal

Published Jun 13, 2011• 18 citations
Publisher

Abstract

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.

DOI
10.1364/josab.28.001685

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arXiv:2004.03000•Apr 6, 2020•1 citation
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.
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 show that the temporal analog of a Fabry–Perot resonator (FPR) can be realized by using two moving temporal boundaries, formed by intense pump pulses inside a dispersive medium (such as an optical fiber). We analyze such FPRs using a transfer-matrix method, similar to that used for spatial structures containing multiple thin films. We consider a temporal slab formed using a single square-shape pump pulse and find that the resonance of such an FPR has transmission peaks whose quality ( Q ) factors decrease rapidly with an increasing velocity difference between the pump and probe pulses. We propose an improved design by using two pump pulses. We apply our transfer-matrix method to this design and show considerable improvement in the Q factors of various peaks. We also show that such FPRs can be realized in practice by using two short pump pulses that propagate as solitons inside a fiber. We verified the results of the transfer-matrix method by directly solving the pulse propagation equation with the split-step Fourier method.
We study the dynamics of a stable Fabry–Perot resonator in which one mirror is a phase-conjugate mirror (PCM) with instantaneous response. The temporal variations in the model are caused by round-trip effects. An exact solution is given in a form of an infinite series. The step-function response is summed exactly. Numerical studies with Gaussian input pulses and PCM reflectivity variations are presented, and their behavior is explained. Approximate formulas for responses to slowly varying pulses are derived. Noise effects are simulated with a model in which the phases of the probe and the amplitude reflection coefficient of the PCM make uncorrelated jumps with Poissonian statistics. The theoretical model is aimed for optimizing experiments on phase-conjugate resonators involving transients and fluctuating fields.
Control over the time dependence of the refractive index of a material allows one to modify and manipulate the properties of light propagating through it. While metamaterials provide a promising avenue in this context, another route has been extensively explored by the ultrafast community - the so-called molecular modulators. Indeed, impulsively-aligned diatomic molecules provide a unique medium, where periodic rotational revivals induced by a pump pulse persist for tens of picoseconds, offering an excellent opportunity for the controlled modification of the refractive index and, therefore, of femtosecond laser pulses propagating through these media. Here we present an analytical theory which describes this process and stumble across a novel mechanism revealing exponential transformations of the probe pulse - its compression, amplification and frequency up-conversion. In particular, our analytical results predict the generation of amplified ultrashort (about 20 fs) ultraviolet pulses centered around 550 nm, starting with near infrared input pulses centered on 1 μm of about 30 fs duration, under very realistic experimental conditions.
A possible device concept for photonic switching is the crossed resonator in which two Fabry-Perot resonators with perpendicular cavity axes enclose a nonlinear medium. Light entering one of the cavities can modify the light exiting the other cavity by altering the refractive index of the medium. This configuration has been proposed for use as an optical logic gate.1 In this work we also demonstrate its potential as an optical switch and transistor. We model the device following the single Fabry-Perot cavity work for steady-state fields of Miller2 and for pulsed inputs following Bischofberger and Shen.3 We assume the medium has a Kerr nonlinearity with an intensity-independent background absorption. Using parameters for the organic polymer polydiacetylene (PTS) etched into an optical waveguide, we find differential gains of greater than three for steady-state input fields of reasonable powers and that this device in certain regimes behaves like an optical transistor. For the time-dependent case, we model the medium temporal response by the Debye equation using the medium response time for PTS. We present results for input pulses of varying intensity and duration.
Light interaction with optical cavities is of fundamental interest to enhance the light-matter interaction and to shape the spectral features of the electromagnetic fields. Important efforts have been carried out to develop modal theories of open optical cavities relying on an expansion of the fields on the eigen-fields of the cavity. Here, we show how such an expansion predicts the temporal dynamics of optical resonators. We consider a Fabry-Perot cavity to derive the full analytical expressions of the internal and scattered field on the quasi-normal modes basis together with the complex eigen-frequencies. We evince the convergence and accuracy of this expansion before deriving the impulse response function (IRF) of the open cavity. We benefit from this modal expansion and IRF to demonstrate that the eigen-modes of the open cavity impact the signals only during the transient regimes and not in the permanent regime.