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Canadian Journal of Physics

Numerical solution of the transient gain equations for stimulated backward scattering in absorbing fluids

S. S. Rangnekar, Richard H. Enns

Published Aug 1, 1976• 3 citations
Publisher

Abstract

The transient gain equations for stimulated backward scattering in absorbing fluids are solved numerically by the method of characteristics for an amplifier setup. In addition to the 'spiking' of the signal pulse previously noted by Maier, Kaiser, and Giordmaine, the existence of a characteristic dip in the amplification of the signal pulse in nonabsorbing fluids when the interaction region ceases to increase is reported. The effect of increasing absorption is to wipe out both the spike and the dip.

DOI
10.1139/p76-185

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Physical Review A•Sep 1, 1971•91 citations
The transient gain equations for stimulated backward scattering in absorbing fluids are solved numerically by the method of characteristics for an amplifier setup. In addition to the 'spiking' of the signal pulse previously noted by Maier, Kaiser, and Giordmaine, the existence of a characteristic dip in the amplification of the signal pulse in nonabsorbing fluids when the interaction region ceases to increase is reported. The effect of increasing absorption is to wipe out both the spike and the dip.
A general transient theory of light amplification in absorbing media is developed for small amplification and for small signal. The amplification by stimulated thermal Rayleigh scattering has been studied in detail for various shapes of the laser pulse. A transient gain is predicted when the frequencies of the pump and signal pulse are the same. The dependence of the amplification from the scattering angle was investigated. For narrow angles (Θ < 2 x 10-2 rad) scattering of higher order in angle was calculated. A method of measuring the Rayleigh linewidth with one laser pulse is proposed
Using well defined input pulses the amplification by stimulated thermal Rayleigh scattering was investigated experimentally as a function of time (within the pulse), of absorption coefficient, and of interaction length. The frequency difference between the two interacting light pulses was varied with an accuracy better than Δν/ν = 10-8. Amplification was measured in the backward direction and at several degrees from the forward direction. The various experimental observations are in good agreement with theory. The possibility of determining material parameters in this amplifier system is discussed
Characteristics of backward, stimulated light pulses are analyzed in the limit where rate equation approximation breaks down, and fluctuation dynamics (oscillator displacements for Raman and phonon amplitudes for Brillouin) must beconsidered. In this limit, the scattered photon flux N s is N s = a(dN l /dt), (d/dt + γ f )A f = bN l /Af,(d/dt + γ l ) = dN l /Af, where f,/referto fluctuations and pump, respectively; A f the fluctuation intensity, and constants a, b, and d depend on the medium characteristics. An analytical solution to this set of equations has been given by Maier et al.1 in the limit γ f →0 and γ l = 0. With these restrictions removed, analytical solution is difficult. Here, we are considering an analytical solution by reducing the set of equations to that of the cubic Schrodinger equation with real amplitude and by using Backlund transform. Preliminary calculations show that the solitary wave solutions are possible (a purely soliton solution is not admissible2 in the form of hyperbolic secant square).
Signal amplification is analyzed in strongly pumped fiber amplifiers both for three-level and four-level transitions, including scattering loss and excited state absorption. Simple analytic expressions are derived for the signal amplification and pump attenuation, along the fiber length, when either the signal and the pump are injected at the same fiber end (forward pumping) or when they are injected at the two opposite ends (backward pumping). Numerical examples for rare-earth doped fibers show excellent agreement between the analytical solution and the exact numerical solutions of the rate-equations for practical operating conditions.
The equations describing the transfer of intensity from a pump beam simultaneously to forward and backward stimulated scattered beams in an infinite medium are shown to lead to pulsations similar to relaxation oscillations in the scattered intensities. The period of the pulsations is simply related to the gain length for the forward-scattered beam, and may be much longer than the ordinary transient time (related to the damping of material excitations) usually associated with stimulated scattering. Similar oscillations occur for backward scattering alone (stimulated Brillouin scattering) if the scattering medium is finite. In this case the period equals the photon round-trip time in the medium if the latter exceeds the backward gain length. Both phenomena should be observable in scattering from gases such as N2{\mathrm{N}}_{2}N2​ and H2{\mathrm{H}}_{2}H2​ at high pressure, and may also play a role in determining the temporal structure of light scattered from the region of a self-focus in liquids.