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Journal of Physics A Mathematical and General

Propagation of the average carrier frequency of chirped pulses

R. Michalska-Trautman

Published Oct 1, 1976• 4 citations
Publisher

Abstract

The modulation of frequency in resonant propagation can be characterized by the distance-dependent average carrier frequency of the pulse. The author shows that the propagation of this average frequency is governed by a conservation law specific for chirped pulses. He finds a relation between the average frequency, the corresponding average wavevector and the energy of the pulse.

DOI
10.1088/0305-4470/9/10/023

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The modulation of frequency in resonant propagation can be characterized by the distance-dependent average carrier frequency of the pulse. The author shows that the propagation of this average frequency is governed by a conservation law specific for chirped pulses. He finds a relation between the average frequency, the corresponding average wavevector and the energy of the pulse.
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.
A study is made of the distance dependence of carrier phase and average frequency of an electromagnetic pulse propagating through a quantum two-level system. Phenomenological differential equations are introduced to describe the distance rate of change of first- and second-moment deviations of the pulse carrier frequency from its original input frequency. In equilibrium the equations predict characteristics of pulse propagation in steady state. For both absorber and amplifier cases, computer plots are presented of the distance dependence of pulse shape, carrier phase, Fourier amplitude, and pulse energy for selected types of homogeneous and inhomogeneous line broadening and for specific input-pulse conditions. Off-resonance formation of a single $2\ensuremath{\pi}$ hyperbolic secant pulse of self-induced transparency may evolve from input-pulse areas less than $\ensuremath{\pi}$, as well as slightly greater than $\ensuremath{\pi}$, accompanied by frequency modulation and pulse breakup. The amplification of pulsed carrier radiation is accompanied by frequency pulling toward resonance. With and without phase modulation, stability conditions for stable pulse propagation are obtained. Either frequency pulling toward or frequency pushing away from resonance is dominant for an absorber, according to the dominance, respectively, of homogeneous or inhomogeneous line broadening. In the slowly-varying-pulse-envelope approximation, a steady-state solution is found for a symmetrically-phase-modulated pulse propagating in an amplifier with scattering losses. The mean carrier frequency for the pulse is displaced off resonance by one linewidth above the resonance transition frequency of a homogeneously broadened system.
A theoretical analysis of the propagation of near resonant and linearly chirped light pulses through a resonant medium is presented under the restriction that the pulses are far enough off resonance or weak enough so that they do not significantly alter the atomic populations. If the pulses are of duration comparable to or less than the phase memory time T(2) of the atoms, it is found that the transient response of the medium gives rise to amplitude-and phase-modulating effects. Analytic solutions are presented to describe the propagation of an off-resonant pulse which is very short compared with T(2) and an off-resonant rectangular pulse. The evolution of a Gaussian pulse having a positive linear chirp is studied numerically.
Changes in the modulus and phase of a slowly varying envelope of an ultrashort pulse are calculated for the case of propagation across a saturable absorber. Vibronic relaxation of the optical transition in the absorber is allowed for. A study is made of the dependences of the duration of a pulse transmitted by a nonlinear absorber on the vibrational relaxation time of dye molecules and on the detuning of the carrier frequency of a pulse relative to the maximum of the equilibrium absorption spectrum. It is shown that a chirp appears even in the absence of such detuning provided the energy of lf quanta of optically active vibrations is comparable with the thermal excitation energy. This chirp originates from partial completion of the vibrational relaxation process during a pulse, so that absorption and refraction occur in a system which is not in a vibrational equilibrium.
We consider the propagation of a light pulse in a medium with a single resonance. If the frequency of the pulse is tuned far from resonance and the pulse duration is much shorter than the lifetime of the excited state of the resonant particles in the medium (atoms in a gas, impurity ions in a solid, etc.), the group velocity of the pulse is appreciably reduced. It is shown that the slowing down of the group velocity of the pulse is accompanied with a pulse chirp, which produces a pulse broadening in time. It is proposed to use two samples in sequence with opposite chirps (up chirp and down chirp or vice versa) compensating the pulse broadening. Then the pulse can be delayed with almost no losses, distortion, and broadening. However, there is a maximum distance, beyond which the pulse experiences corruption. Pumping with an auxiliary laser beam can control the delay time of the light pulse in the medium. Conditions to eliminate the contribution of the dephasing processes in the pulse propagation are considered.
We analyze the phase modulation, or chirping, of coherent lossless light pulses propagating without distortion through resonant absorbers. In order to do this, we generalize the pioneering work of McCall and Hahn in two different directions. In the first place, of course, we abandon their assumption that the phase of the pulses has no temporal dependence. We prove that for slowly varying single pulses, chirping is not possible. However, we describe many multiple-pulse trains which are necessarily chirped, even under the slowly varying envelope restriction, and also describe the envelope modulations which produce large chirps. We show that certain zero-$\ensuremath{\pi}$ chirped pulse trains are contained as special cases of our general results. Our second generalization of the McCall-Hahn work concerns the background material in which the two-level resonant atoms are suspended. We allow the host medium to possess significant nonresonant nonlinearities. We find that undistorted lossless single pulses are possible in such a medium and that they are necessarily chirped.