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Amplification of coherent optical pulses and "non-bound-state" solitons

Published Jan 1, 1987• 0 citations
OpenAlex

Abstract

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 $>\pi$ 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 $\leq$ 3. In addition to the expected portion of area $\pi$ 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$\pi$ (or 0$\pi$) solitons. Pulses whose final area is $\pi$, 3$\pi$, and 5$\pi$ 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 $\leq$ 3 that for each 2$\pi$ soliton there is a purely imaginary zero of the left reflection coefficient

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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\sbb≤\sb{\rm b}\leq\sbb≤ N bound states, and employing Lamb's one-component inverse scattering method*, pulses whose initial area can be >π>\pi>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 ≤\leq≤ 3. In addition to the expected portion of area π\piπ 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 2π\piπ (or 0π\piπ) solitons. Pulses whose final area is π\piπ, 3π\piπ, and 5π\piπ 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 ≤\leq≤ 3 that for each 2π\piπ soliton there is a purely imaginary zero of the left reflection coefficient
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 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 $A=(2n+1)\ensuremath{\pi}$.
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.
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.
The dynamics of pulses with durations comparable to the inverse transition frequency that propagate in an optical medium is studied in terms of two integrable systems of Maxwell-Bloch equations. The first model describes the field interaction with a nondegenerate medium with a permanent dipole moment and permanent external pumping. A general formula is derived for the N -soliton solution. Particular solutions are used as examples to investigate the effect of permanent dipole moment and pumping on the soliton dynamics. The second model describes the interaction between two-component electric-field pulses and a two-level degenerate medium with permanent upper-level pumping. For different initial magnetic-sublevel populations, soliton solutions are used as examples to show that pumping causes a change in polarization dynamics. A two-soliton solution is used to analyze the interaction of solitons in a two-level medium with external pumping.
If an incident pulse is chirped, the critical parameters for self-induced transparency to occur in coherent pulse propagation can no longer be obtained from the well-known McCall-Hahn area theorem. We have been able to obtain these parameters by solving the Zakharov-Shabat eigenvalue equation for the bound-state eigenvalues. We find that critical (threshold) areas will be increased for a chirped incident pulse in almost all cases, except for a box profile or for a pulse that is approximately box-like in shape. In these latter cases, the chirped critical areas will instead decrease for the second and all higher branches. The first branch’s critical area is always increased due to chirping.