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Physical Review A

Spectral moments in coherent resonant propagation

R. Michalska-Trautman

Published Dec 1, 1980• 6 citations
Publisher

Abstract

The conservation laws for coherent resonant propagation are studied, and an interpretation of these laws as conservation of pulse spectral moments is proposed. The nonlinear spectral density is expressed in terms of the scattering data for an initially hyperbolic secant pulse. In this way spectral moments are exactly determined at any distance and are used for a partial description of the propagating pulse. The conservation laws for the three lowest moments are investigated in detail. The evolution of the zero-order moment leads to an explicit formula for the attenuation of the electromagnetic energy. The rate at which the energy is absorbed depends critically on the pulse area. There is an essential difference between propagation of pulse with area smaller than $\ensuremath{\pi}$ and those with area larger than $\ensuremath{\pi}$. The evolution of the first moment corresponds to change of the pulse average frequency. The calculated shift of the average frequency due to detuning is compared with the experimental result of Slusher and Gibbs. The second-order moment is a measure of the pulse spectral width. The evolution of the Fourier second-order moment is used to explain the observed spectral narrowing of a $\ensuremath{\pi}$ pulse and broadening of a $3\ensuremath{\pi}$ pulse.

DOI
10.1103/physreva.22.2738

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Applied Optics•Nov 1, 1972•23 citations
The conservation laws for coherent resonant propagation are studied, and an interpretation of these laws as conservation of pulse spectral moments is proposed. The nonlinear spectral density is expressed in terms of the scattering data for an initially hyperbolic secant pulse. In this way spectral moments are exactly determined at any distance and are used for a partial description of the propagating pulse. The conservation laws for the three lowest moments are investigated in detail. The evolution of the zero-order moment leads to an explicit formula for the attenuation of the electromagnetic energy. The rate at which the energy is absorbed depends critically on the pulse area. There is an essential difference between propagation of pulse with area smaller than \ensuremathπ\ensuremath{\pi}\ensuremathπ and those with area larger than \ensuremathπ\ensuremath{\pi}\ensuremathπ. The evolution of the first moment corresponds to change of the pulse average frequency. The calculated shift of the average frequency due to detuning is compared with the experimental result of Slusher and Gibbs. The second-order moment is a measure of the pulse spectral width. The evolution of the Fourier second-order moment is used to explain the observed spectral narrowing of a \ensuremathπ\ensuremath{\pi}\ensuremathπ pulse and broadening of a 3\ensuremathπ3\ensuremath{\pi}3\ensuremathπ pulse.
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 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.
Pulse shaping is the technique which controls the ultra-short pulse shape, and it became of great technological interest because of its potential applications in laser pulse compression, digital communications, microscopy etc. We demonstrate the idea of pulse-shaping technique and pulse propagation with low energy losses in a resonant linear absorbing medium. This thesis presents the results of a study of the propagation of Gaussian and hyperbolic secant ultrashort chirped and chirp-free pulses in homogeneously and inhomogeneously broadened resonant linear absorbers. Changes to the pulse shape and energy loss factor are presented as the pulse propagates in the absorber. The Fast Fourier method is used to numerically determine both the normalized intensity profile and the pulse spectrum. Our results show that, for pulse durations shorter than the relaxation time, chirped pulses in absorbing media obey the area theorem, with their shape changing with the propagation distance. Simulation results of the spectra of chirped pulses clearly show the burning of a spectral ’hole’ as the pulse propagates, with the pulse energy pushed away towards the wings. When compared to chirp-free pulses, chirped pulses reshape faster and develop wings in their tail due to initial phase modulation. Simulation results of the energy loss factor show that chirped pulses propagating in resonant linear absorbers sustain less energy losses than do chirp-free pulses. A comparison of chirped secant and Gaussian pulses shows that secant pulses propagate with lower energy losses. Analytic solutions are presented for long-distance asymptotic expressions of initial rms spectral bandwidth as well as for the attenuation factor of chirped Gaussian pulses. These analytical results are in agreement with numerical simulations. The comparison of energy losses of short chirped Gaussian pulses and long pulses of any profile in linear absorbers is also discussed in the thesis.
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 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.