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Physical Review A

Theory of transient stimulated Raman scattering and four-wave-mixing processes in multiple-pump systems: A Hamiltonian approach

Jui‐Teng Lin

Published May 1, 1986• 4 citations
Publisher

Abstract

A general theory of transient stimulated Raman scattering and the phase-matching diagrams are developed based on an interaction Hamiltonian which includes all the possible competing interaction channels. In a Raman system pumped by multiple crossing beams with multiple Stokes and anti-Stokes components, these channels are defined by the interactions among fields at different frequencies and/or k vectors of the pump pairs and Stokes and anti-Stokes fields. An ensemble-averaged equation of motion for the active phonon mode is derived within the Markovian approximation in which the phonon-phonon coupling (pressure effect) is reduced to an energy relaxation and a phase fluctuation. A complete set of transient coupled equations including phase modulation, Stark shift, medium (anharmonic) saturation, and all the possible four-wave-mixing channels is presented. In the transient regime, it is found that the effects of the Stark shift on the Raman efficiency are mainly governed by the input pump energy in contrast to the steady-state theory where the pump intensity plays the main role. Both analytic and numerical methods are developed in describing the nonlinear effects due to the energy feedback mechanism in the strong pump-depletion regime. Twelve possible four-wave-mixing channels are analyzed for nonlinear phase diagrams characterized by the angles among pump pairs, and Stokes and anti-Stokes fields. Finally, numerical examples are shown for the effects of the coherence lengths of high-order Stokes fields and pump pairs on the generation of low-order Stokes fields.

DOI
10.1103/physreva.33.3210

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Four-wave-mixing Processes in Multiple-pump Multiphonon Systems: Effects of Coherence Length

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a Theoretical Study on the Stimulated Raman Scattering Modulated by the Flopping Frequency Arising from the Interaction Between a Strong Resonant Field and the Medium.

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The stimulated Raman scattering (SRS), arising from the interaction between molecules of a medium and two strong light fields - a resonant field and a pumping field - is studied theoretically. The transitions of molecules between a given pair of levels is assumed to be both Raman active and infrared active. A semiclassical approach, in which the density matrix method is used for the quantum mechanical description for the medium, is employed. We expand the density matrix in various orders within which the zeroth-order is referred to as the solution of the equation of motion of the density matrix when only the resonant field is applied. -- We obtain a set of equations in second order of the density matrix element p₁₀⑵ (ω) from which the gain function can be derived. This set of equations in the special case of steady state is accidentally similar to those obtained by Mollow in his study of the power spectrum of a strong driven two-level system. We have shown that the SRS in the steady state under the resonant field interaction, is drastically reduced in general. When the flopping frequency is larger than the relaxation constants (Ω > r), we find there appears both weak positive and weak negative gains on both sides of the ordinary SRS components. Since we retain our second-order equations in a general form rather than the form for steady state, we are able to take account of the dependence on the transient zeroth solution. By employing the Laplace transform technique with the help of convolution products, we are able to solve for ρ₁₀⑵(ω) in the stationary flopping state. When the intensity of the resonant field is high, we have found theoretically that both the Stokes and anti-Stokes component become doublets and equally displaced by the amount of the flopping frequency, Ω, on the opposite sides of ordinary Stokes and anti-Stokes components, respectively. The gain maxima are approximately the order of Ω/4r times the ordinary gain of the Stokes component, where Ω/r >> 1 if the flopping exists. The gains are therefore large in comparison with the ordinary gain for SRS in the steady state.

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Nonlinear theory of transient stimulated Raman scattering and its application to long-pulse experiments

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The nonlinear theory of stimulated Raman scattering of high-energy long laser pulses in two-level media is constructed and provides a unified global interpretation of the experiments of Dr\"uhl, Wenzel, and Carlsten [Phys. Rev. Lett. 51, 1171 (1983)]. It is proved that the model, resulting from the slowly varying envelope approximation and including phase mismatch and detuning from Raman resonance, is integrable by the spectral transform method for arbitrary boundary values. The essential variable in the theory is the phase of the Stokes pulse relative to the pump. For long duration input pump pulses and when pump and Stokes pulses experience a rapid \ensuremath{\pi} phase shift, an anomalous spike of pump radiation is created in the pump depletion zone. It is shown in particular that the decay of Raman spikes is related to the velocity of this \ensuremath{\pi} phase shift. For fast flips, the possibility of observing spike overflow is established and, with multiple phase flips, a multiple-spike phenomenon is produced, suggesting new experiments. Lastly, the number of optical phonons participating to the interaction is explicitly calculated.

Physical Review A•Jul 1, 1995•20 citations
A general theory of transient stimulated Raman scattering and the phase-matching diagrams are developed based on an interaction Hamiltonian which includes all the possible competing interaction channels. In a Raman system pumped by multiple crossing beams with multiple Stokes and anti-Stokes components, these channels are defined by the interactions among fields at different frequencies and/or k vectors of the pump pairs and Stokes and anti-Stokes fields. An ensemble-averaged equation of motion for the active phonon mode is derived within the Markovian approximation in which the phonon-phonon coupling (pressure effect) is reduced to an energy relaxation and a phase fluctuation. A complete set of transient coupled equations including phase modulation, Stark shift, medium (anharmonic) saturation, and all the possible four-wave-mixing channels is presented. In the transient regime, it is found that the effects of the Stark shift on the Raman efficiency are mainly governed by the input pump energy in contrast to the steady-state theory where the pump intensity plays the main role. Both analytic and numerical methods are developed in describing the nonlinear effects due to the energy feedback mechanism in the strong pump-depletion regime. Twelve possible four-wave-mixing channels are analyzed for nonlinear phase diagrams characterized by the angles among pump pairs, and Stokes and anti-Stokes fields. Finally, numerical examples are shown for the effects of the coherence lengths of high-order Stokes fields and pump pairs on the generation of low-order Stokes fields.
The effects of four-way-mixing (4WM) and the generation of high-order Stokes (HOS) in a single pump single phonon system have been studied previously.1 In the present paper we shall develop a general theory which provides the equations of motion in a multiple pump multiphonon system which includes all pair field interactions via both Raman and 4WM. The effects of the coherence length of the pump pairs and of the 4WM on the generation of HOS will be investigated analytically and numerically. Pump depletion and the feedback effects due to the high-order processes will be included. The dephasing of the phonon modes will be derived via the phonon-phonon interaction in the Markovian approximation.2
Parametric instability induced by the medium saturation in stimulated Raman scattering is investigated theoretically via a generalized detuning (1) where ω p,s,o are the frequency of the pump, Stokes, and phonon, ϕ ˙ p s is the phase modulation, and the last term is the nonlinear correction of the highly excited anharmonic molecule with an amplitude Q.The steady-state gain and hence the Stokes efficiency, in the presence of the anharmonic term, are governed by a cubic equation of |Q|2which is to be solved from the Bloch and Maxwell equations including the pump depletion. Numerical results are shown for the transition diagram of Stokes efficiency vs input pump intensity at various system conditions at the steady-state. For the transient regime, with pulse duration shorter than the dephasing time, we analyze the temporal profiles of the Stokes and the depleted pump based on a transformed equation which combines Bloch and Maxwell equation as follows: (2) where Tis a nonlinear time, G s is the steady-state gain, l0is the input pump intensity, and Uis the real part of the phonon amplitude Q.For an input pulse intensity given by sech2(t/t p ), Eq. (2) is solved numerically to study the transient pulse profiles in the presence of pump depletion and medium saturation.
The theory of stimulated Raman scattering has been treated at various levels of sophistication from plane-wave steady-state models through 3-D transient quantum-electrodynamic calculations.1 Many physical effects, such as pump depletion, dispersion, level degeneracy, ac Stark shift, and parametric coupling of Stokes and anti-Stokes light, can be important in experimental setups. We describe a recently developed computer code which treats these effects in detail. The code is based on integration of the coupled Maxwell-Bloch equations of quantum optics. We discuss numerical predictions of the code relating to these effects and present applications to experimental investigations such as Raman beam cleanup and transient amplification.2
The equations for plane-wave steady-state propagation of n fields interacting via four wave processes in a Raman medium have been known since at least 1962. Complete analytic solutions have only been found for the process of harmonic generation. However, solutions have been obtained in the gain regime assuming zero pump depletion. Assuming all fields satisfy the phase matching condition we have found general analytic solutions to these equations. The solutions are complete in that they describe both pump deletion and saturation. Some example problems will be studied.
The stimulated Raman scattering (SRS), arising from the interaction between molecules of a medium and two strong light fields - a resonant field and a pumping field - is studied theoretically. The transitions of molecules between a given pair of levels is assumed to be both Raman active and infrared active. A semiclassical approach, in which the density matrix method is used for the quantum mechanical description for the medium, is employed. We expand the density matrix in various orders within which the zeroth-order is referred to as the solution of the equation of motion of the density matrix when only the resonant field is applied. -- We obtain a set of equations in second order of the density matrix element p₁₀⑵ (ω) from which the gain function can be derived. This set of equations in the special case of steady state is accidentally similar to those obtained by Mollow in his study of the power spectrum of a strong driven two-level system. We have shown that the SRS in the steady state under the resonant field interaction, is drastically reduced in general. When the flopping frequency is larger than the relaxation constants (Ω > r), we find there appears both weak positive and weak negative gains on both sides of the ordinary SRS components. Since we retain our second-order equations in a general form rather than the form for steady state, we are able to take account of the dependence on the transient zeroth solution. By employing the Laplace transform technique with the help of convolution products, we are able to solve for ρ₁₀⑵(ω) in the stationary flopping state. When the intensity of the resonant field is high, we have found theoretically that both the Stokes and anti-Stokes component become doublets and equally displaced by the amount of the flopping frequency, Ω, on the opposite sides of ordinary Stokes and anti-Stokes components, respectively. The gain maxima are approximately the order of Ω/4r times the ordinary gain of the Stokes component, where Ω/r >> 1 if the flopping exists. The gains are therefore large in comparison with the ordinary gain for SRS in the steady state.
The nonlinear theory of stimulated Raman scattering of high-energy long laser pulses in two-level media is constructed and provides a unified global interpretation of the experiments of Dr\"uhl, Wenzel, and Carlsten [Phys. Rev. Lett. 51, 1171 (1983)]. It is proved that the model, resulting from the slowly varying envelope approximation and including phase mismatch and detuning from Raman resonance, is integrable by the spectral transform method for arbitrary boundary values. The essential variable in the theory is the phase of the Stokes pulse relative to the pump. For long duration input pump pulses and when pump and Stokes pulses experience a rapid \ensuremath{\pi} phase shift, an anomalous spike of pump radiation is created in the pump depletion zone. It is shown in particular that the decay of Raman spikes is related to the velocity of this \ensuremath{\pi} phase shift. For fast flips, the possibility of observing spike overflow is established and, with multiple phase flips, a multiple-spike phenomenon is produced, suggesting new experiments. Lastly, the number of optical phonons participating to the interaction is explicitly calculated.