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Physical Review A

Theory of Stokes Pulse Shapes in Transient Stimulated Raman Scattering

Robert L. Carman, Fujio Shimizu, C. S. Wang, N. Bloembergen

Published Jul 1, 1970• 491 citations
Publisher

Abstract

The theory of transient stimulated Raman scattering has been extended to include an arbitrary shape of the laser pump pulse. It is shown that the maximum Stokes gain depends on the total energy content per unit area of the pump pulse, and not on the instanteous intensity for an exciting pulse of short duration. The Stokes pulse has a leading edge which rises sharply to a maximum, where the maximum occurs with some delay with respect to the maximum of the pump pulse. The trailing edge follows the decay of the pump. In a nondispersive medium, the gain is not reduced by frequency broadening of the laser output, while in a dispersive medium, considerable gain reduction is expected. Numerical results for various laser-pulse shapes and spectral distributions are presented.

DOI
10.1103/physreva.2.60

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Physical Review A•Jul 1, 1995•20 citations
The theory of transient stimulated Raman scattering has been extended to include an arbitrary shape of the laser pump pulse. It is shown that the maximum Stokes gain depends on the total energy content per unit area of the pump pulse, and not on the instanteous intensity for an exciting pulse of short duration. The Stokes pulse has a leading edge which rises sharply to a maximum, where the maximum occurs with some delay with respect to the maximum of the pump pulse. The trailing edge follows the decay of the pump. In a nondispersive medium, the gain is not reduced by frequency broadening of the laser output, while in a dispersive medium, considerable gain reduction is expected. Numerical results for various laser-pulse shapes and spectral distributions are presented.
The results of numerical calculations of the transient stimulated Raman scattering reported previously have been verified experimentally under conditions where both linear dispersion and self-focusing effects were negligible. The existence of a delay between maxima of the laser and Stokes pulses is experimentally demonstrated for the first time, while the pulse shortening in time via Raman scattering is established more firmly than in previous work. The incident-laser-pulse duration, generated-Stokes-pulse duration, and delay between intensity maxima for the laser and Stokes pulses were measured with the two-photon absorption-fluorescence technique. The effective phonon-dephasing time is determined via spontaneous Raman scattering. By using these measured quantities, inferences are made as to the magnitude of the transient gain and the shape of the exciting picosecond laser pulse.
The authors have extended the theory of stimulated Raman scattering to include the effects of laser bandwidths, in both the transient and steady-state regimes. The case of two interacting laser beams, a pump laser and a probe (Stokes) laser, is treated. Using the phase-diffusion model for laser bandwidth, the authors demonstrate that in the absence of dispersion, the forward Raman gain is essentially independent of the laser bandwidths in the high-gain limit, while in the low-gain limit the gain coefficient is inversely proportional to the sum of the bandwidths. It is further shown that when the pump-laser bandwidth is much larger than the linewidth of the Raman medium, the stimulated Stokes output assumes the same spectrum as the pump laser in the high-gain limit. A possible interpretation of these results is discussed assuming a "phase-locking" of the Stokes phase to the fluctuations in the pump-laser phase, due to the nonlinear interaction of the two beams through the Raman medium.
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.
A study is made of the problem of stimulated Raman scattering (STRS) in a resonant medium pumped by a strong pulse satisfying the adiabaticity condition. It is shown that in the absence of relaxation the maximum STRS gain is reached during the tail (trailing edge) of the pump pulse, which is the reason for the absence of the Stark shift. If the pump pulse is considerably longer than the relaxation time, the lag of the STRS pulse behind the pump pulse is short and then the Stark shift is manifested in the spectrum and a characteristic interference structure is formed. Noise and monochromatic "seeding" variants are considered.
A three-dimensional quantum theory of stimulated Raman scattering is presented which takes into account a general transverse shape of the pump laser. The transverse spatial autocorrelation function and the intensity of the Stokes pulse are derived analytically, by use of the Maxwell-Bloch equations in low-signal-gain conditions and introduction of a many-spatial-modes approximation. We find that the spatial and temporal coherence are coupled. The transverse coherence depends on the pump-laser shape, and the coherence area increases with increasing gain. Time scales for the buildup of stimulated intensity and correlations for the various transverse parts of the Stokes pulse are obtained.
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.