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Physical Review

Wave Propagation in a Dispersive and Emissive Medium

T. A. Weber, Dennis Benedict Trizna

Published Apr 8, 1966• 21 citations
Publisher

Abstract

Recently a measurement of the velocity of longitudinal ultrasonic waves propagating through a phonon maser amplifier showed that the pulse traveled with the maximum or cutoff velocity at resonance. With an approximate dispersion rule for a homogeneously broadened spin-resonance transition, the propagation of the wave is analyzed using a method for the asymptotic expansion of Fourier-type integrals. This analysis, contrary to earlier studies, indicates that the pulse observed, or at least the leading edge of that pulse, was an amplified precursor (transient response). For a sinusoidal imput signal, the asymptotic wave form at a given position is given for all possible values of the time.

DOI
10.1103/physrev.144.277

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The interaction between an electromagnetic (em) pulse and a maser medium is described by a general set of five equations, under the assumption of a homogeneously broadened electric-dipole transition with two Bloch relaxation times T2and T1and of a linear broadband loss mechanism. When the equations are specialized at resonance, their solutions include the results of the previous treatments on the amplifier problem obtained under particular assumptions. The steady-state pulse (S. S. P.) introduced by Wittke and Warter forT_{2}/T_{1} = 0is here generalized forT_{2}/T_{1} \neq 0and it is shown to propagate at the same velocity of the light in the medium. In the caseT_{2}/T_{1} = 0the steady state is described by exact analytical relations. For times short in comparison to the relaxation times, a solution is given which generalizes the usual interaction formula between an em field and a two-level system by introducing propagation effects. In the general case out of resonance, it is shown that an S. S. P. exists, and that its frequency coincides with the frequency of the atomic transition, independent of the frequency of the input field.

IEEE Journal of Quantum Electronics•Jul 1, 1965•268 citations
Recently a measurement of the velocity of longitudinal ultrasonic waves propagating through a phonon maser amplifier showed that the pulse traveled with the maximum or cutoff velocity at resonance. With an approximate dispersion rule for a homogeneously broadened spin-resonance transition, the propagation of the wave is analyzed using a method for the asymptotic expansion of Fourier-type integrals. This analysis, contrary to earlier studies, indicates that the pulse observed, or at least the leading edge of that pulse, was an amplified precursor (transient response). For a sinusoidal imput signal, the asymptotic wave form at a given position is given for all possible values of the time.
Pulse propagation in maser-type traveling-wave amplifiers with a homogeneously broadened transition is treated by a formalism analogous to the Bloch equations. Phenomenological dephasing (T2) and recovery (T1) times are defined, and a linear (nonsaturable) loss mechanism is included. In the numerical calculations, only the case of negligible excitation during the time it takes a pulse to pass is considered. When pulses are allowed to grow until the amplifier is ``saturated,'' steady-state pulses having a unique shape and intensity independent of the initial pulse are found. The parameters of these steady-state pulses depend only on the ratio of the linear loss and gain coefficients. Steady-state pulses have a peak intensity that decreases monotonically from infinity to zero as the linear loss coefficient varies from zero to the gain coefficient, while the pulse width correspondingly varies from zero to infinity, and the pulse energy varies from a finite value to zero. Steady-state pulses propogate at a velocity less than that of the small signal velocity in the medium.
The uniform asymptotic description of the propagation of an input Gaussian modulated harmonic signal of fixed angular frequency ω c and initial pulse width T into the half-space z > 0 that is occupied by a single resonance Lorentz medium is described. The asymptotic theory shows that the propagated field can be expressed solely in terms of a generalized Sommerfeld and Brillouin precursor field structure. If the applied signal frequency ω c is either below or near the medium resonance frequency, the propagated field is found to be dominated by the generalized Brillouin precursor field, while if ω c is well above the medium resonance frequency it is dominated by the generalized Sommerfeld precursor field. The resultant pulse distortion in the dispersive medium is shown to be due solely to the manner in which the initial Gaussian pulse envelope spectrum modifies the precursor field amplitude. Finally, the frequency dependence of the pulse velocity in the single resonance Lorentz medium is described.
The propagation of a steplike change in the amplitude of a dc pulse and a step in the envelope of a plane sinusoidal wave are investigated. It is shown that a step (e.g., the beginning or end of a pulse or wave) propagates like a pulse in the medium; i.e., its velocity is a weighted average of the group velocity, and its rate of spread is determined by the distribution of group velocities within the spectrum of the step. The results apply to a real pulse (e.g., sound) in an infinite, homogeneous, dispersive, lossless medium, and to the quantum-mechanical wave function of a free particle.
The propagation of short pulses in a dispersive medium can be described in simple physical terms by employing the concept of the group velocity provided the medium is lossless. When the medium is absorbing, the group velocity description breaks down. In a previous paper,1 we presented a similar description employing the velocity of energy of monochromatic waves in place of the group velocity which is valid for a particular dispersive medium with absorption known as the Lorentz medium. That model describes the entire dynamics of the pulse as it propagates with the exception of the early stages of the evolution of the second precursor (otherwise known as the Brillouin precursor). Since that region contains the peak values of the pulse, it is important to include it in the model. The present paper describes the arrival of the Brillouin precursor in terms of the group velocity of a propagating field of the form exp [ωttω z] where ω and kω are real and positive.
The interaction between an electromagnetic (em) pulse and a maser medium is described by a general set of five equations, under the assumption of a homogeneously broadened electric-dipole transition with two Bloch relaxation times T2and T1and of a linear broadband loss mechanism. When the equations are specialized at resonance, their solutions include the results of the previous treatments on the amplifier problem obtained under particular assumptions. The steady-state pulse (S. S. P.) introduced by Wittke and Warter forT_{2}/T_{1} = 0is here generalized forT_{2}/T_{1} \neq 0and it is shown to propagate at the same velocity of the light in the medium. In the caseT_{2}/T_{1} = 0the steady state is described by exact analytical relations. For times short in comparison to the relaxation times, a solution is given which generalizes the usual interaction formula between an em field and a two-level system by introducing propagation effects. In the general case out of resonance, it is shown that an S. S. P. exists, and that its frequency coincides with the frequency of the atomic transition, independent of the frequency of the input field.