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Journal of Physics B Atomic and Molecular Physics

Two-photon resonant four-wave mixing with non-monochromatic waves

NILOY KUMAR DUTTA

Published Jan 28, 1980• 10 citations
Publisher

Abstract

The two-photon resonant four-wave mixing processes are investigated when they are excited by non-monochromatic pump waves. Expressions for the intensity and linewidth of the generated wave are obtained. It is shown that the intensity of the generated wave is a Lorentzian about the two-photon resonance. The width of the Lorentzian varies linearly with the pump linewidth. The maximum and minimum in the coherent anti-Stokes Raman scattering (CARS) lineshape are shifted farther when excited by non-monochromatic beams. The shift varies linearly with the pump linewidth. In the incoherent regime, the intensity of the wave generated is directly proportional to the interaction length (L) and its linewidth Delta omega approximately 1/L.

DOI
10.1088/0022-3700/13/2/027

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The two-photon resonant four-wave mixing processes are investigated when they are excited by non-monochromatic pump waves. Expressions for the intensity and linewidth of the generated wave are obtained. It is shown that the intensity of the generated wave is a Lorentzian about the two-photon resonance. The width of the Lorentzian varies linearly with the pump linewidth. The maximum and minimum in the coherent anti-Stokes Raman scattering (CARS) lineshape are shifted farther when excited by non-monochromatic beams. The shift varies linearly with the pump linewidth. In the incoherent regime, the intensity of the wave generated is directly proportional to the interaction length (L) and its linewidth Delta omega approximately 1/L.
The theory of coherent anti-Stokes Raman scattering (CARS) is extended to include the effect of pump fluctuations. The intensities and spectra of lines in resonant CARS are calculated to all orders in fields assuming a phase-diffusion model for waves at the two pump frequencies. The bandwidth of the two lasers enters in a much more complicated way than following a simple scaling of T1{T}_{1}T1​ or T2{T}_{2}T2​. Various resonances in CARS spectra due to dynamic splitting of the energy levels are discussed for a range of detunings, field intensities, and bandwidths. In contrast to the usual spectra in strong fields, the Rabi sidebands appear as dispersion-shaped structures. The laser linewidth is shown to change dramatically the CARS line shape. The case of no saturation is also treated, thus allowing for the inclusion of more general line shapes and fluctuations of the pump waves and the nonlinear susceptibility tensor ${\ensuremath{\chi}}^{(3)}$. Gaussian statistics of the pump field are shown to lead to enhancement factors in CARS intensity, similar to those appearing in the context of multiphoton absorption processes.
When nonlinear four-wave mixing (FWM) is resonant with a Raman transition, it causes coupling between lightwaves separated by two Raman frequency shifts. This coupling is manifested in effects such as Raman gain suppression, higher-order Stokes generation, and coherent Raman spectroscopies (e.g., CARS). An understanding of these effects, however, has been impeded by a widespread misconception that Raman-resonant FWM does not transfer energy between the lightwaves and the medium. The equations for the photon densities at each light frequency show that, in actuality, two medium excitations are created or destroyed for each photon produced by FWM. This point may be further clarified by consideration of the driving nonlinear polarization and by comparisons with broadband Raman conversion and nonresonant FWM. The Raman gain suppression effect may be understood qualitatively in terms of destructive interference between FWM and the usual stimulated Raman process.
A theoretical analysis is made of the simultaneous generation of Stokes and anti-Stokes waves initiated by two-photon additional illumination. This is done allowing for Raman and parametric processes, saturation effects, and also for a possible difference between the Raman polarizabilities for the Stokes and anti-Stokes waves. The conditions for the establishment of various asymptotic regimes are determined and general expressions are derived for the ultimate conversion efficiencies, assuming that pump and additional illumination are fed into a medium (inverted or uninverted) under exact resonance and phase-matching conditions. The spatial evolution of the amplitudes of the interacting waves is investigated by an accurate analytic solution of the problem in an important particular case and using solutions in the approximation of a constant pump wave or constant additional illumination.
The nonlinear response of sodium vapor with a laser tuned near the 4D two-photon resonance is investigated over a large range of sodium concentrations and pump intensities with emphasis on the influence of coherent cancellation effects on the induced polarizations and the primary mechanism for the generation of the new frequencies. Over a wide range of circumstances, the response of the medium is dominated by parametric four-wave mixing as compared to amplified spontaneous emission or stimulated electronic hyper-Raman scattering. Secondary four-wave-mixing fields are generated by stimulated Raman scattering of one parametric four-wave-mixing field and a coupling of these fields with the driving laser field. A sharp onset of saturation of the parametric four-wave-mixing signals with sodium number density and a transition from a quadratic to a linear power dependence is observed. Both effects are explained in terms of the two-photon cancellation effect.
We consider effects which occur when four-wave sum frequency generation and multiphoton ionization are induced by lasers tuned near a three-photon resonance and simultaneously near or at a dipole allowed four-photon resonance. In studies with unfocused laser beams, if the phase mismatch of the generated four-wave-mixing field is large and the related two-photon resonance for the absorption of a four-wave-mixing photon and a laser photon results in strong absorption of the four-wave-mixing field, a coherent cancellation occurs between the pumping of the resonance by two- and four-photon processes. This interference effect occurs when the first laser is tuned on either side of the three-photon resonance and \ensuremath{\Vert}\ensuremath{\Delta}kr{\mathit{k}}_{\mathit{r}}kr​L\ensuremath{\Vert}\ensuremath{\gg}1, where \ensuremath{\Delta}kr{\mathit{k}}_{\mathit{r}}kr​ is the mismatch and L is the length of the path of the laser beams in the gas. With focused laser beams large differences occur between ionization with unidirectional beams and with counterpropagating laser beams when \ensuremath{\Vert}\ensuremath{\Delta}kr{\mathit{k}}_{\mathit{r}}kr​b\ensuremath{\Vert}\ensuremath{\gg}1, where b is the confocal parameter of the focused laser beams. Strong absorption of the four-wave-mixing field is shown not to be necessary for strong destructive interference with focused laser beams when the phase mismatch is large. This work also suggests an explanation for earlier experiments where the presence of a four-photon resonance enabled the generation of third-harmonic light in a positively dispersive wavelength region. We argue that this process can occur when the laser used to achieve the four-photon resonance is focused on the small z (z is the coordinate in the direction of propagation) side of the focal point of the laser responsible for the third-harmonic generation. In this way the absorption of the third-harmonic light on one side of the focal point is made much stronger than on the other side. Correspondingly, contributions to the third-harmonic field from the two sides of the focal point are unequal and cannot undergo complete destructive interference as would otherwise occur in a positively dispersive region.
Two-color excitation by spatially coherent and incoherent laser beams has been used to study three-photon-resonant excitation and subsequent ionization of xenon in conditions, when internally generated sum-frequency field plays an important role in excitation of atomic resonances through interfering one-photon excitation pathway. We show that the incoherence in one of the pumping fields reduces the efficiency of generated sum-frequency field, and thus suppresses the interference between the three- and the one-photon excitation channels. The degree of suppression is controlled by varying the crossing angle between coherent and incoherent laser beams. We show that ionization profiles can be analyzed on the basis of the well-studied interference of one- and three-photon transition amplitudes, but with pumping field decomposed into multiple small-scale uncorrelated domains where coherent process of four-wave mixing occurs. The gain length for a coherent process in these domains depends on the coherence degree and excitation geometry. It gives a possibility of controlling the contribution of coherent processes to the excitation of multiphoton resonances.