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.
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,…) by the inverse method. Results for
n=1and2 are summarized.
A simple model for optical pulse propagation in a nondegenerate two-level amplifying medium is considered, under the assumption of extreme Doppler broadening. Starting with a "Pade approximant" left reflection coefficient with N simple poles and N$\sb{\rm b}\leq$ N bound states, and employing Lamb's one-component inverse scattering method*, pulses whose initial area can be
>π are obtained. Then, employing the asymptotic behavior of the eigenvalues of the kernel of the right Marchenko equation, the asymptotic behavior of the pulses far into the medium is analyzed in detail when N
≤ 3. In addition to the expected portion of area
π near the light cone which undergoes amplification and compression, pulses with a continuous leading edge develop forward-moving oscillations, and some pulses trail behind one or more 2
π (or 0
π) solitons. Pulses whose final area is
π, 3
π, and 5
π are obtained. Interestingly, the number of trailed solitons is in general not equal to the number of bound states. These solitons are associated with a subset of the zeroes of the transmission coefficient rather than of its poles. Conditions for the appearance of a soliton are given in terms of the poles and residues of the left reflection coefficient. A connection is established between the values of the pulse profiles and their first 2N
− 1 derivatives at the light cone, and the residue and pole parameters of the left reflection coefficient. For N = 2 and the case in N = 3 where the pulse has a continuous leading edge, simple conditions for the appearance of a soliton are obtained in terms of the values of the pulse profiles and their first N
− 1 derivatives at the light cone, and the poles of the left reflection coefficient. It is established for N
≤ 3 that for each 2
π soliton there is a purely imaginary zero of the left reflection coefficient
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 complete solution of the Maxwell-Bloch equations by the inverse scattering transform as given by Ablowitz, Kaup, and Newell is compared with the McCall-Hahn theory, and others. Not only it is shown that the earlier results are contained in the complete solution, but also several new results are obtained. Among these are an infinite set of "nonlinear moments" which evolve similar to the McCall-Hahn area, a closed-form solution for the nonlinear transmission, how one can determine absolute time delays, why the threshold area for lossless propagation is still exactly $\ensuremath{\pi}$ (even when the initial profile is off resonance but unchirped), and first-order effects of relaxation on $2\ensuremath{\pi}$ hyperbolic-secant pulse propagation. For the last-mentioned results, we find that in general if a $2\ensuremath{\pi}$ hyperbolic-secand pulse is initially off resonance, it will move away from resonance. Also, we find that the McCall-Hahn result for first-order effects on the time delay must be modified for small-amplitude $2\ensuremath{\pi}$ hyperbolic-secant pulses.
The propagation behavior of nontrivial $0\ensuremath{\pi}$ pulses [$\ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}\mathcal{E}(t,z)dt=0$, $\ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}{\mathcal{E}}^{2}(t,z)\mathrm{dt}>0$] in attenuating media is examined. It is found that under appropriate conditions $0\ensuremath{\pi}$ pulses exhibit anomalously low absorption which is a direct consequency of atomic coherence. Specific analytical results valid for an unbroadened medium are presented, while the corresponding solutions for an inhomogeneously broadened resonance are calculated numerically. These two results are consistent in the limit of ${T}_{2}^{*}\ensuremath{\rightarrow}\ensuremath{\infty}$. Furthermore, the influences of
T2,
T2∗, and level degeneracy on the characteristics of propagation are determined. Finally, the behavior of more general $0\ensuremath{\pi}$ pulses is studied. The criteria for the production of $0\ensuremath{\pi}$ pulses are qualitatively established by numerical methods, and the influence of the relative phase of pulse pairs on propagation is examined. The latter indicates a method for the measurement of relative pulse phase.