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Physical Review Letters

Phase Variation in Coherent-Optical-Pulse Propagation

George L. Lamb

Published Jul 23, 1973• 118 citations
Publisher

Abstract

It is shown that the stipulation of lossless propagation provides sufficient asymptotic information on the pulse-propagation process to permit determination of both amplitude and phase of a class of coherent optical pulses ($2n\ensuremath{\pi}$ pulses, $n=0, 1, 2, \dots{}$) by the inverse method. Results for $n=1 \mathrm{and} 2$ are summarized.

DOI
10.1103/physrevlett.31.196

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It is shown that the stipulation of lossless propagation provides sufficient asymptotic information on the pulse-propagation process to permit determination of both amplitude and phase of a class of coherent optical pulses (2n\ensuremathπ2n\ensuremath{\pi}2n\ensuremathπ pulses, n=0,1,2,…n=0, 1, 2, \dots{}n=0,1,2,…) by the inverse method. Results for n=1and2n=1 \mathrm{and} 2n=1and2 are summarized.
Various coherent-optical-pulse profiles are known to exhibit lossless propagation during transmission through a resonant atomic medium. It is shown that the requirement of lossless propagation provides asymptotic information that is sufficient to determine these profiles by the inverse method. Pulse profiles are obtained as solutions to the Marchenko equation. The analysis is applied explicitly to the propagation of $2\ensuremath{\pi}$, $4\ensuremath{\pi}$, and $0\ensuremath{\pi}$ pulses in an inhomogeneously broadened medium.
Higher conservation laws associated with the phase-dependent Maxwell-Bloch equations are derived and employed to determine both amplitudes and frequency shifts of the final-state solitons produced in a resonant medium by an initially phase-modulated optical pulse of area $<5\ensuremath{\pi}$. The higher conservation laws are shown to provide a relatively efficient method of relating the parameters of the final pulses to those of the incident pulse. Connection is made to the computer work of Diels and Hahn.
The amplification of a coherent optical pulse is analyzed by employing an inverse scattering method. In the extreme Doppler limit, one obtains an analytical expression that exhibits the expected evolution toward a $\ensuremath{\pi}$ pulse with pulse amplification and compression. In the sharp-line limit, one obtains justification for previous usage of a similarity solution to describe $\ensuremath{\pi}$-pulse propagation in an amplifier. The analysis is carried out for an initial pulse profile (truncated hyperbolic secant) which yields a sufficiently simple scattering problem that the Marchenko equation of inverse scattering theory may be solved analytically.
The initial value problem for the propagation of a pulse through a resonant two-level optical medium is solved by the inverse scattering method. In general, an incident pulse decomposes not only into a special class of pulses to which the medium is transparent but also yields radiation which is absorbed by the medium. In this respect ``this problem'' has properties markedly different from other dispersive and reversible wave phenomena some of which are tractable by the inverse scattering method. Indeed, it is remarkable that in the present case the method still applies. In particular, we show that, while there are an infinite number of local conservation laws, the integrated densities, and in particular the energy, are only conserved for a very special class of initial conditions. The theoretical results obtained are in close agreement with all the qualitative features observed in the experiments on coherent pulse propagation. Finally, we also show that causality is preserved. Two new and novel features are introduced and briefly discussed. First, we show that if the homogeneous broadening effect is a function of position in the medium, the pulses may speed up and slow down accordingly, without losing their permanent identities. Second, we have found a new kind of solution mode corresponding to a proper eigenvalue of the scattering problem which is not a bound state.
The description of coherent optical pulse propagation in terms of coupled Maxwell and Schroedinger- Bloch equations in slowly varying envelope approximation possesses conservation laws in addition to those of field energy and field momentum. Comparison with preliminary numerical results has shown that these higher conservation laws can be used to predict the amplitude of each of the 2pi pulses into which an intense coherent pulse may decompose as it propagates through an attenuator. In the present paper a comparison with more extensive numerical results is carried out. It is shown that over a wide range of variation of the parameters involved the conservation laws yield results in agreement with those obtained through numerical computation.
The inverse scattering method is used to investigate the resonant propagation of an optical pulse assumed to be initially unchirped and to have a hyperbolic-secant shape. An exact solution is found for the two-component initial eigenvalue problem. The discrete spectrum consists of a finite set of purely imaginary, equidistant values. The number of discrete eigenvalues (number of solitons) is determined by the initial pulse area AAA. The scattering amplitudes are expressed in terms of Euler gamma functions and hyperbolic functions. In addition to poles in the upper half-plane, the function b‾a\frac{\overline{b}}{a}ab​ has a pole on the real axis for $A=(2n+1)\ensuremath{\pi}$.