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Journal of the Optical Society of America B

Temporal reflection and refraction of optical pulses inside a dispersive medium: an analytic approach

Junchi Zhang, William R. Donaldson, Govind Prasad Agrawal

Published Feb 2, 2021• 29 citations

Abstract

We develop an analytic approach for reflection of light at a temporal boundary inside a dispersive medium and derive frequency-dependent expressions for the reflection and transmission coefficients. Using the analytic results, we study the temporal reflection of an optical pulse and show that our results agree fully with a numerical approach used earlier. Our approach provides approximate analytic expressions for the electric fields of the reflected and transmitted pulses. Whereas the width of the transmitted pulse is modified, the reflected pulse is a mirrored version of the incident pulse. When a part of the incident spectrum lies in the region of total internal reflection, both the reflected and transmitted pulses are distorted considerably.

DOI: 10.1364/josab.416058

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We develop an analytic approach for reflection of light at a temporal boundary inside a dispersive medium and derive frequency-dependent expressions for the reflection and transmission coefficients. Using the analytic results, we study the temporal reflection of an optical pulse and show that our results agree fully with a numerical approach used earlier. Our approach provides approximate analytic expressions for the electric fields of the reflected and transmitted pulses. Whereas the width of the transmitted pulse is modified, the reflected pulse is a mirrored version of the incident pulse. When a part of the incident spectrum lies in the region of total internal reflection, both the reflected and transmitted pulses are distorted considerably.

We theoretically revisit the problem of the propagation of coherent light pulses through a linear medium when the carrier frequency of the pulses coincides with the minimum of a narrow dip in the medium transmission. Considering realistic contrasts between the maximum and minimum transmission of the medium and incident pulses of strictly finite duration, we combine temporal and spectral approaches to obtain analytical expressions of the transmitted pulse that reproduce the main features of the exact numerical solutions derived by fast Fourier transform. A special attention is paid to the advance of the pulse maximum over that of a pulse covering the same distance in vacuum and the ratio of this advance to the pulse duration (fractional advance) is optimized.

A study is made of the problem of transmission of light pulses by a boundary of a nonlinear medium under conditions of coherent interaction with matter. It is shown that in the case of a semiinfinite nonlinear medium the reflection coefficient is governed by the linear characteristics of the media in contact so that the boundary-value problem reduces to the Cauchy problem. In another case, when a nonlinear medium is a thin film at the interface between two linear media, it is found that such a film is an almost perfect mirror reflecting weak pulses but it becomes transparent to pulses of sufficiently high intensity and the mirror then modulates the amplitude of such pulses.

The reflection, transmission, and absorption of a symmetric electromagnetic pulse are calculated. The carrier frequency of the pulse is close to the frequency of direct interband transitions in a quantum well (QW). The QW energy levels are assumed to be discrete, with two closely spaced excited levels being taken into account. The QW width is assumed to be sufficiently large and comparable to the light wavelength corresponding to the pulse carrier frequency. In this case, the dependence of the momentum matrix element for the interband transition on the light wave vector should be taken into account. The refractive indices of the QW and barriers are assumed to be equal. The problem is solved for an arbitrary relation between the radiative and non-radiative lifetimes of the excited electronic states. It is shown that spatial dispersion considerably affects the shapes of the reflected and transmitted pulses. The greatest changes occur in the case where the inverse radiative lifetime is close to the difference between the frequencies of the interband transitions considered.