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Journal of the Physical Society of Japan

Quantum Theory of Stimulated Raman Effect

Kyoji Nishikawa, Fumihiko Takano

Published Jun 1, 1967• 5 citations
Publisher

Abstract

A photon theory of the stimulated Raman effect is developed. A system of coupled equations for the amplitudes of scattered photons and molecular vibrations are derived. The molecular vibration coordinates are then eliminated using the random phase approximation. The equations which result assume the form of coupled equations for the first Stokes and anti-Stokes lights. The secular equation which characterizes the solution of these coupled equations is shown to be equivalent to that of Shen and Bloembergen. Using this secular equation, the condition for temporal growth of the scattered lights is obtained and is compared with that for the spatial growth, which was obtained previously by Shen and Bloembergen. It is shown that both types of growth (temporal and spatial) proceed with comparable rates if the damping of the molecular vibration is large. If its damping is small, however, the temporal growth prevails over the spatial growth. Typical line-shape of the anti-Stokes line is obtained in the case of small damping. The result shows a strong directionality, as is the case of large damping.

DOI
10.1143/jpsj.22.1446

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A photon theory of the stimulated Raman effect is developed. A system of coupled equations for the amplitudes of scattered photons and molecular vibrations are derived. The molecular vibration coordinates are then eliminated using the random phase approximation. The equations which result assume the form of coupled equations for the first Stokes and anti-Stokes lights. The secular equation which characterizes the solution of these coupled equations is shown to be equivalent to that of Shen and Bloembergen. Using this secular equation, the condition for temporal growth of the scattered lights is obtained and is compared with that for the spatial growth, which was obtained previously by Shen and Bloembergen. It is shown that both types of growth (temporal and spatial) proceed with comparable rates if the damping of the molecular vibration is large. If its damping is small, however, the temporal growth prevails over the spatial growth. Typical line-shape of the anti-Stokes line is obtained in the case of small damping. The result shows a strong directionality, as is the case of large damping.
The semiclassical theory of spontaneous Raman scattering is reviewed. From the semiclassical theory we identify the parameters and dynamical variables involved in Raman scattering. A classical theory of stimulated Raman scattering is constructed. It is an extension of the theory of Shen and Bloembergen. We show that the system is a weakly coupled system. Because the driving term contains dynamical variables, the linewidth of the stimulated Stokes wave should be smaller than that of the spontaneous Stokes wave. Using Riemann's method, we obtain Kroll's solution rigorously, in a more general form. The steady-state limit is also derived rigorously. The conditions for the transient and steady-state gains are discussed. It is shown that for the transient case one may have an abrupt change of the Stokes gain versus incident laser power.
A quantum theory of stimulated Raman scattering is presented that takes into account three-dimensional propagation and collisional dephasing, allowing the study of the spatial and temporal coherence properties of the generated Stokes light. Maxwell-Bloch equations for the Stokes field operator and the collective atomic operators are solved analytically under low-signal-gain conditions, where the laser field and the atomic ground states remain undepleted. The intensity and the space-time autocorrelation function of the Stokes field are calculated. The Stokes field is expanded into a set of statistically independent ``coherence modes,'' which are determined explicitly for the case of a cylindrically shaped pumped volume. The Stokes pulse energy W is found to fluctuate from pulse to pulse. The probability distribution function for pulse energies P(W) is calculated for a range of Fresnel numbers of the excited volume and collisional dephasing rates. For small values of Fresnel number and dephasing rate, P(W) is a negative exponential distribution. For large values of either, P(W) narrows and approaches a Gaussian-shaped distribution. This occurs because many independent modes contribute to the Stokes emission, making it spatially and/or temporally incoherent.
A simple description of the stimulated Raman emission of Stokes radiation, and of the induced absorption found by Stoicheff at the anti-Stokes frequency, is based on the semiclassical theory of radiation. This theory gives the well-known formulas for absorption and stimulated emission when carried to the first order in the time-dependent perturbation due to the interaction of the light beam with the molecules. If there are two light beams of frequencies ωL and ω, the second-order perturbation may be associated with stimulated emission at the Stokes frequency ω=ω−1=ωL—ωR and with induced absorption at the anti-Stokes frequency ω=ω1=ωL+ωR, where ωL is the laser frequency and ωR a Raman-active vibrational frequency; in both cases, there is a molecular transition to the excited vibrational state. The highly directional and very sharp anti-Stokes emission at ω1 arises from the conversion by molecules in intense beams at ωL and ω−1 of two ωL photons into an ω−1, ω1 pair. Like spontaneous emission from excited states, the production of this anti-Stokes radiation requires the full quantum theory of radiation for an adequate description. The emission of ω1 and the observed directional absorption of ω−1 due to the reverse conversion ω−1+ω1→2ωL are explained. The anti-Stokes emission is proportional to the Stokes intensity and to the square of that of the laser, and to the square of the molecular number density; its sharpness is due to the absence of a molecular transition. The generation of photon pairs ω−n+ωn, where ωn=ωL+nωR, can occur for all values of n, but it is most efficient for n=1, due to resonance in the appropriate molecular hyperpolarizability and to the intensity of the Stokes beam at ω−1.
Stimulated Raman scattering of a randomly modulated pump is investigated by the method of successive approximations. This involves expanding solutions in terms of small parameters, which are ratios of the correlation scales of random effects to other characteristic dynamic scales of the problem. Systems of closed equations are obtained for the moments of the amplitudes of the Stokes and pump waves and of the molecular vibrations. These describe the dynamics of the process allowing for changes in the pump intensity and statistics due to a three-wave interaction. By analyzing equations in higher-order approximations, it is possible to establish the conditions of validity of the first (Markov) and second approximations. In particular, it is found that these are valid for pump intensities JL both above and below the critical value Jcr near which the gain begins to increase rapidly and reproduction of the pump spectrum by the Stokes wave is initiated. Solutions are obtained for average intensities of the Stokes wave and molecular vibrations in the first approximation in a constant pump field. It is established that, for JLJcr, the Stokes wave undergoes rapid nonsteady-state amplification which is associated with an increase in the amplitude of the molecular vibrations. The results of the calculations show good agreement with known experimental data.
A quantum theory of stimulated Raman scattering is established that takes into account the time-dependence of pump laser pulse, two-photon detuning and atomic populations, allowing the general transient and spatial propagation study of Raman memory. Heisenberg–Langevin and Maxwell equations for the collective atomic operators are solved analytically under adiabatic condition , where the electric field envelopes and the collective atomic operators vary slowly. The analytical solution of Stokes/anti-Stokes signal electric field operator is obtained. The spatial–temporal intensity of quantum signal field is studied numerically and analytically for different two-photon detunings and initial populations.
Abstract We present a generalized quantum theory of stimulated Raman scattering, which is based on coupled Heisenberg–Langevin equations. Analytical solutions in previous models neglect the effects of the frequency dispersion, assume far-off resonance and only consider slowly varying terms in the atomic operator. In obtaining the more general theory, we have derived partial integro-differential equation containing polarization kernels that include all the temporal nonlocal effects using systematic integral method over time. Analytical solutions of the Stokes field with greater generality for arbitrary spatial-temporal laser fields have been obtained using Laplace transform in the retarded space frame with minimal approximations made. The solutions are very general and potentially useful for studying systems in which highly accurate modeling is required such as quantum memories of a wide range of Raman systems.