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Physical Review A

Very short optical pulses: A new approximation method

Robert A. Marth, Dale A. Holmes, Joseph H. Eberly

Published Jun 1, 1974• 19 citations

Abstract

Very short optical pulses, traveling in sharp-line resonant absorbers without change of shape, are investigated with a new approximation technique. Instead of employing the long-pulse or slowly varying envelope assumption, we have developed a method of approximation based on a series expansion in powers of a small parameter which is related to the electric field strength. The principal advantages of the method are its ability to treat very short pulses, and the simplicity with which it can be extended, both from lower to higher levels of approximation, and from one type of steady-state pulse to another. In this paper we use the approximation method to study very short pulses which require retention of the second-derivative terms in the Maxwell envelope and phase equations. Corrections to the well-known McCall-Hahn results are easily found to first order in the small parameter. As expected, the corrections begin to be important only for picosecond and shorter pulses.

DOI: 10.1103/physreva.9.2733

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Very short optical pulses, traveling in sharp-line resonant absorbers without change of shape, are investigated with a new approximation technique. Instead of employing the long-pulse or slowly varying envelope assumption, we have developed a method of approximation based on a series expansion in powers of a small parameter which is related to the electric field strength. The principal advantages of the method are its ability to treat very short pulses, and the simplicity with which it can be extended, both from lower to higher levels of approximation, and from one type of steady-state pulse to another. In this paper we use the approximation method to study very short pulses which require retention of the second-derivative terms in the Maxwell envelope and phase equations. Corrections to the well-known McCall-Hahn results are easily found to first order in the small parameter. As expected, the corrections begin to be important only for picosecond and shorter pulses.

A systematic perturbation expansion with respect to the small parameter ${(\ensuremath{\Delta}\ensuremath{\tau}\ensuremath{\omega})}^{\ensuremath{-}1}$, $\ensuremath{\Delta}\ensuremath{\tau}$ being the duration of the pulse and $\ensuremath{\omega}$ the optical frequency, is introduced. This method is applied to the problem of self-induced transparency for sharp-line absorbers at exact resonance. Corrections to the McCall and Hahn solution are obtained which describe the propagation of the form-preserving pulse with modulated frequency and slightly changed shape, group velocity, and wave vector. Earlier results by Marth, Holmes, and Eberly on phase modulation and wave-vector shift are confirmed.

Effects of propagation of an extremely short (of one or several oscillation periods) electromagnetic pulse in a medium whose resonance transition is characterized by diagonal as well as nondiagonal matrix elements of the dipole moment operator have been studied numerically. The Maxwell-Bloch system of equations is employed without using the approximation of slowly varying envelopes. An analog of the McCall and Hahn area theorem is discussed as applied to the division of the initial extremely short pulse into subpulses. The solution is obtained in the form of a solitary stable bipolar signal with a nonzero pulse area (nonzero breather).

Recent progress in ultrafast technology has led to the generation of optical pulses as short as 8 fs. The propagation of such short pulses in nonlinear optical media requires new computational techniques in which the standard approximations such as the slowly-varying envelope approximation (SVEA) and rotating-wave approximation (RWA) are not made while using the Maxwell-Bloch equations [1]. Previous calculations, which used the finite-difference time-domain (FDTD) method, found that for a a Kerr-like instantaneous nonlinearity [2] significant differences can occur when the full vector Maxwell equations are used instead of the nonlinear Schrodinger equation. Recently, the FDTD method has been extended to the case of a homogeneously broadened two-level atomic system [3].

Resonant absorption in a laser pulse can be reduced to zero if the trailing part induces stimulated emission and returns the absorbed energy back to the pulse. This kind of effect is called Self-Induced Transparency (SIT). Analytic descriptions of short pulse propagation in a resonant medium, based on Slowly Varying Envelope Approximation (SVEA) and in linear media, have been extended in studies of SIT along two principal directions: First, to improve the SVEA by introducing Short Pulse Approximation (SPA); and second, to consider the effect of nonlinear refractive index.