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IEEE Journal of Quantum Electronics

Exact solution of a π-pulse problem

John A. Armstrong, E. Courtens

Published Jun 1, 1968• 65 citations
Publisher

Abstract

The exact solution describing steady-state resonant pulse propagation in a two-level inverted medium is given. This steady-state pulse is often referred to as a 180° or π-pulse. No approximation of a slowly varying envelope is made, and hence the solution satisfies the second-order wave equation as well as the equations of motion of the density matrix of the system. As a result the theory is able to describe very short pulses, whose width can be of the order of the period of the optical carrier. The model for the system includes a nonresonant loss and a finite value of the transverse relaxation time T2, but neglects dispersion in the host medium. The steady-state pulse is found to be frequency modulated and asymmetric in shape.

DOI
10.1109/jqe.1968.1075345

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The exact solution describing steady-state resonant pulse propagation in a two-level inverted medium is given. This steady-state pulse is often referred to as a 180° or π-pulse. No approximation of a slowly varying envelope is made, and hence the solution satisfies the second-order wave equation as well as the equations of motion of the density matrix of the system. As a result the theory is able to describe very short pulses, whose width can be of the order of the period of the optical carrier. The model for the system includes a nonresonant loss and a finite value of the transverse relaxation time T2, but neglects dispersion in the host medium. The steady-state pulse is found to be frequency modulated and asymmetric in shape.
A steady-state solution describing\Pi-pulse propagation in a homogeneously broadened laser amplifier embedded in a dispersive lossy host is found. For a given model of dispersion, the solution satisfies exactly the second-order wave equation together with the equations of motion for the density matrix of the two-level resonant system. The salient feature of the solution is that a small amount of dispersion in the host group velocity causes a large monotonic frequency sweep during the pulse. The effects of this chirp on the pulse properties are described in detail, and the relation of the theory to experiments is discussed.
We obtain an exact linearly polarized vector solitary solution for the amplification of an optical pulse with a time width short compared with both population and polarization decay time. This dissipative soliton results from the balance between the gain from inverted resonant two-level atoms and the linear loss of the host material. We suppose that the excited state of the active centers is degenerate on the projection of the angular momentum, with a symmetric state of inversion. Quite remarkably, we find that after the passing of the pulse, the atomic medium does not return to the ground state, and a finite Zeeman coherence is generated. Numerical simulations demonstrate the stability of vector π solitons with an arbitrary state of elliptical polarization in the presence of both linear birefringence and group-velocity dispersion of the host material.
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.