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Physical Review

Generation of Stokes and Anti-Stokes Radiation in Raman Media

H. A. Haus, Paul L. Kelley, Herbert J. Zeiger

Published May 17, 1965• 11 citations
Publisher

Abstract

A three-dimensional analysis of stimulated Raman emission is presented for a medium through which a cylindrical laser beam passes. The buildup of the Stokes radiation from the spontaneous fluctuations in the Raman medium is considered. A generalization of the fluctuation-dissipation theorem is derived relating the current-current fluctuations at the Stokes frequency to the laser-induced (negative) conductance at this frequency. The anti-Stokes radiation is treated as arising from the interaction of the laser field and the molecular vibrations accompanying the Stokes radiation. The anti-Stokes radiation emerges in a narrow cone around the familiar phase-matching direction given by ${\mathbf{\ensuremath{\beta}}}_{\ensuremath{-}}=2{\mathbf{k}}_{L}\ensuremath{-}{\mathbf{\ensuremath{\beta}}}_{+}$, where ${\mathbf{\ensuremath{\beta}}}_{\ensuremath{-}}$ and ${\mathbf{\ensuremath{\beta}}}_{+}$ are the (real parts of the) propagation constants of Stokes and anti-Stokes waves in the medium, and ${\mathbf{k}}_{L}$ is the laser propagation vector. The angular width of the anti-Stokes radiation cone is evaluated.

DOI
10.1103/physrev.138.a960

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Soviet Journal of Quantum Electronics•Nov 30, 1982•2 citations
A three-dimensional analysis of stimulated Raman emission is presented for a medium through which a cylindrical laser beam passes. The buildup of the Stokes radiation from the spontaneous fluctuations in the Raman medium is considered. A generalization of the fluctuation-dissipation theorem is derived relating the current-current fluctuations at the Stokes frequency to the laser-induced (negative) conductance at this frequency. The anti-Stokes radiation is treated as arising from the interaction of the laser field and the molecular vibrations accompanying the Stokes radiation. The anti-Stokes radiation emerges in a narrow cone around the familiar phase-matching direction given by \ensuremathβ\ensuremath−=2kL\ensuremath−\ensuremathβ+{\mathbf{\ensuremath{\beta}}}_{\ensuremath{-}}=2{\mathbf{k}}_{L}\ensuremath{-}{\mathbf{\ensuremath{\beta}}}_{+}\ensuremathβ\ensuremath−​=2kL​\ensuremath−\ensuremathβ+​, where \ensuremathβ\ensuremath−{\mathbf{\ensuremath{\beta}}}_{\ensuremath{-}}\ensuremathβ\ensuremath−​ and \ensuremathβ+{\mathbf{\ensuremath{\beta}}}_{+}\ensuremathβ+​ are the (real parts of the) propagation constants of Stokes and anti-Stokes waves in the medium, and kL{\mathbf{k}}_{L}kL​ is the laser propagation vector. The angular width of the anti-Stokes radiation cone is evaluated.
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.
We have studied the diffractive coupling of Stokes, anti-Stokes, and pump fields in stimulated Raman scattering. We find that, depending on the strength of the anti-Stokes field relative to that of the Stokes, both soliton dynamics and beam-pointing fluctuations are affected for finite Fresnel numbers. The generation of the anti-Stokes waves can inhibit a gain-focusing mechanism that we predicted for the Stokes field, when a soliton propagates in the medium. Although no qualitative differences arise in the beam-pointing fluctuations of the Stokes field, we predict that the anti-Stokes beam may be much more stable than the Stokes counterpart. Our analysis shows that for a Fresnel number of 5, the difference can be as large as 70%, while for Fresnel numbers of order unity, it is nearly 10%.
Analytic and numerical solutions are obtained for the equations describing the propagation of several pulses of different frequencies through a Raman medium. The transient case is treated, and the analysis includes Stokes, pump, and anti-Stokes frequency components. The effect of phase mismatch \ensuremath{\Delta}k between these waves is investigated. An analytic solution is found that describes the phenomenon of transient Stokes--anti-Stokes gain suppression in the limit \ensuremath{\Delta}k=0. For nonzero values of \ensuremath{\Delta}k, the behavior of the transient gain coefficient is found to follow a scaling rule that is the analog of a similar result known from the steady-state limit. Calculations are performed to identify conditions that enhance conversion to anti-Stokes radiation. It is found that there is an optimum strength of the Stokes seed, and that proper phase matching of the initial waves is essential.
Coupled differential equations are formulated to model the generation of coherent anti-Stokes Raman radiation in molecular gases. They include the basic coherent interaction of incident pump and idler (Stokes) waves to generate the anti-Stokes radiation plus other radiation and absorption processes which influence saturation due to population depletion of the lowest energy level. Other factors considered are simultaneous excitation of more than one vibration frequency, a quasi-monochromatic idler beam slightly off the true Stokes frequency, and the temporal laser pulse shape. Some effects of these various physical factors on saturation are presented graphically.
A theory of stimulated Raman scattering (SRS) is presented which treats in a unified manner the buildup of SRS from spontaneous Raman scattering and the spatial propagation which leads to gain. Maxwell-Bloch equations describe the coupling between the Stokes field operator and the collective atomic operators, which are driven by a classical laser field under low-signal-gain conditions, where the atomic ground states and laser field remain undepleted. An analytic expression is derived for the forward-propagating Stokes field operator in a one-dimensional spatial approximation when the Fresnel number of the excited medium is unity. Steady-state and transient Stokes intensities are evaluated under low-gain (spontaneous) and high-gain (stimulated) conditions. The Stokes intensity is found to be exactly independent of a laser bandwidth arising solely from phase fluctuations. The power spectrum of steady-state SRS is evaluated, giving a gain-narrowed spectral profile when the laser is narrow band, and a spectrum identical to the laser spectrum when the laser is broad band. A strong analogy between SRS and two-level superfluorescence is found; in both cases the macroscopic, collective dipole moment is initiated by quantum fluctuations.
A theoretical analysis is made of generation and amplification of Stokes and anti-Stokes waves in gaseous media allowing for Raman and parametric processes and saturation effects. Allowance is also made for any difference between the Raman polarizabilities of the Stokes and anti-Stokes waves which may be caused in particular by the occurrence of one-photon quasiresonances of the interacting waves. Assuming that pump and Stokes (or anti-Stokes) waves are applied at the entrance to a medium (inverted or uninverted), a general solution is found for the amplitudes of the interacting waves under exact resonance and phase-matching conditions. This is used as the basis to investigate their spatial evolution (in particular, for inverse Raman light scattering) and expressions are found for the maximum efficiencies for conversion of the pump to Stokes and/or anti-Stokes waves. It is shown that depending on the relationship between the Raman polarizabilities and also on the input parameters of the waves, conditions of parametric bleaching of the medium with nonzero amplitudes of all the waves or conditions of complete conversion of the pump to the Stokes and anti-Stokes waves or to one of these are established asymptotically. An investigation is made of the stability of these solutions.