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Physical Review A

Relaxation Oscillations in Stimulated Ruman and Brillouin Scattering

Roosevelt Johnson, John H. Marburger

Published Sep 1, 1971• 91 citations
Publisher

Abstract

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 ${\mathrm{N}}_{2}$ and ${\mathrm{H}}_{2}$ 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.

DOI
10.1103/physreva.4.1175

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Physical Review Letters•Mar 16, 1992•18 citations
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.
Stimulated Raman and Brillouin scattering (SRS and SBS) phenomena have been investigated analytically with use of density-matrix formulations and assuming parity indefiniteness for the eigenfunctions of the energy eigenstates of a two-level system in noncentrosymmetric (NCS) crystals. In crystals, when a molecule is excited, the disturbance travels through the crystal as a vibrational wave (phonon mode). We have considered the scattering of the incident electromagnetic wave by phonon modes on a common footing, assuming that vibrations should correspond to optical phonons for SRS and to acoustic phonons for SBS. The retardation effect due to finite ${\ensuremath{\chi}}^{(2)}$ in NCS crystals is found to play a more significant role in the case of SRS than for SBS. The classical differential polarizability (\ensuremath{\partial}\ensuremath{\alpha}/\ensuremath{\partial}u)0{)}_{0})0​ that causes SRS has been determined from the present formulations in terms of the material constants of the crystal and the photon pump energy. Applying Heisenberg's uncertainty relation, we have also obtained a general formula for the determination of the electrostrictive coefficient \ensuremath{\gamma} that is responsible for SBS in crystals. We find (\ensuremath{\partial}\ensuremath{\alpha}/\ensuremath{\partial}u)0{)}_{0})0​/\ensuremath{\gamma}\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}8}$ with \ensuremath{\gamma}\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}10}$--${10}^{\mathrm{\ensuremath{-}}11}$ mks units for important III-V semiconductors. Both Raman and Brillouin gain constants are studied over a wide frequency spectrum in the semiconductors. The ratio between the two gain constants indicates that for the same pump intensity SBS exhibits higher gain than SRS by a magnitude \ensuremath{\sim}${\ensuremath{\omega}}_{\mathrm{OP}/{\ensuremath{\omega}}_{\mathrm{AP}}}$. .AE
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).
We are experimentally investigating Stimulated Brillouin Scattering (SBS) and Stimulated Raman Scattering (SRS) in the transient regime in acetone. While measuring the scattered light as a function of the input energy, we noticed an unexpected “roll-off” in the scattered light as pump energy was increased. Theory predicts as input energy increases scattered energy increases until saturation is reached. We think that the observed roll-off is related to the coherence length of the pump, and we are investigating that relationship.
The stochastic nature of initial stimulated Brillouin scattering (SBS) can lead to a deep modulation of Stokes intensity in fibers, even under conditions of strong pump depletion. It is known that at a moderate pump depletion, when parameter T1/T2 ≫ 1 (T1 is the light propagation time through the effective amplification length and T2 is the hypersound relaxation time), fluctuations correspond to the Gaussian statistics of temporal field distribution. As T1/T2 → 1 the statistics of fluctuations can change. Gaeta and Boyd1 were the first to show that at a sufficient pump depletion (by our estimate, parameter T1/T2 in Ref. 1 was ~4) the correlation function of the Stokes intensity c(τ) = (〈I(t)I(t + τ)〉 - 〈I(t))2)/(/(t)〉2 differs from the Gaussian by a more rapid initial fall and the presence of the region in which it is negative.