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arXiv:quant-ph/0008012

Quantum Interference Mechanism of Cooperative Optical Phenomena in Extended Media

Valery I. Rupasov

Published Aug 2, 2000• 0 citations
arXivHTMLPDF

Abstract

In the quantum process of stimulated Raman scattering (SRS), a laser photon propagating in a resonance medium undergoes multifold conversions into a Stokes photon and back. The nontrivial ``cooperative'' behavior of the Stokes component of light transmitted through the medium is proven to be completely determined by the interference of scattering amplitudes in different sub-channels of the Stokes channel, which obviously combines all the sub-channels with an odd number of photon conversions. The theory of superfluorescence is then derived as the limiting case of the SRS theory.

DOI
10.48550/arxiv.quant-ph/0008012

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Annual Meeting Optical Society of America•Jan 1, 1990•0 citations
In the quantum process of stimulated Raman scattering (SRS), a laser photon propagating in a resonance medium undergoes multifold conversions into a Stokes photon and back. The nontrivial ``cooperative'' behavior of the Stokes component of light transmitted through the medium is proven to be completely determined by the interference of scattering amplitudes in different sub-channels of the Stokes channel, which obviously combines all the sub-channels with an odd number of photon conversions. The theory of superfluorescence is then derived as the limiting case of the SRS theory.
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 pencil shaped region of a Raman-active medium forms a source for Stokes-shifted light via stimulated Raman Scattering (SRS) when driven by an intense laser pulse. Stokes photons are scattered spontaneously at first, and then amplified during propagation through the medium. Since the scattering is initiated by quantum noise, the actual number of photons produced in any given pulse will be highly uncertain, even if the pulse energy of the driving laser is constant1. As the Stokes light propagates through such an extended region, it's coherence properties will be changed. The spatial coherence is determined by the size and shape of the region, which may be characterized by its Fresnel number F, defined as F= A/λSL, where A is the cross-sectional area of the interaction region, which has length L, and λS is the Stokes wavelength. The rate at which the microscopic scatterers (atoms or molecules) are dephased by collisions affects the temporal coherence of the Stokes light, characterised by the temporal coherence parameter Γτ /gL, where Γ is the collisional dephasing rate, τ is the laser pulse duration and g is the steady-state Raman gain coefficient.
The nonlinear regime of stimulated Raman scattering (SRS) may be reached in two ways. The best known is that of depletion of the pump laser as a large number of photons are scattered into the Stokes field. There is, however, another way in which gain saturation can occur, that is, by depletion of the atomic population while the laser remains undepleted. The equations which describe SRS, including damping and population inversion,1 are remarkably similar to those of well-known theories of two-level superfluorescence, and it is possible by analogy to define a delay time τ D for cooperative Raman scattering (CRS). This delay time is inversely proportional to the spontaneous scattering rate, which depends on the pump laser intensity. A sufficient condition for CRS is that the spontaneous Stokes linewidth Γ is very much greater than τ D − 1 . The only other requirement is that there be more laser photons than scatterers, so that τ D is independent of the number of scatterers. The S(1) transition in molecular hydrogen, when excited by a short (≅ 100-ps) laser pulse, provides a possible system for observing pure CRS. Similar studies have been carried out previously2 for Γ ≅ τ D − 1 .
We present a three-dimensional theory of stimulated Raman scattering (SRS) or super-radiance. In particular we address how the spatial and temporal properties of the generated SRS beam or Stokes beam of radiation depends on the spatial properties of the gain medium. Maxwell equations for the Stokes field operators and of the atomic operators are solved analytically and a correlation function for the Stokes field is derived. In the analysis we identify a super-radiating part of the Stokes radiation that exhibit beam characteristics. We show how the intensity in this beam builds up in time and at some point largely dominates the total Stokes radiation of the gain medium. We show how the SRS depends on the Fresnel number and the optical depth and that in fact these two factors are the only factors describing the coherent radiation.
A quantum theory of formation and propagation of superfluorescence and Stokes pulses in samples of small Fresnel numbers is presented. Linearized Maxwell-Heisenberg equations for the system of two-level atoms and a quantized electromagnetic field are solved analytically in the second-order Debye approximation. Under this approximation paths of rays which undergo one reflection from the sides of the sample and those which propagate without reflections are studied. We point out that due to guiding effects, if the Fresnel number is smaller than unity, the contribution from totally reflected rays is not negligible with respect to that from nonreflected rays. This leads to the interference between reflected and nonreflected rays which affects spatial properties of superfluorescence and Stokes pulses. Moreover, as a consequence of guiding effects, the spectral density of light becomes asymmetric and shifted from the atomic resonance frequency.
We have observed fluctuations in the direction of propagation of light pulses generated by stimulated Raman scattering (SRS). This fluctuation is analogous to the pointing fluctuation of laser beams,1 and is due to quantum mechanical uncertainties in the occupation numbers of different spatial modes. When SRS is generated in an interaction region with Fresnel number less than unity, the lowest order spatial mode dominates the generation process, but higher order spatial modes are also weakly and randomly excited. This results in small fluctuations in the wavefront of the light field. The direction of propagation is characterized by the transverse component of an average wave-vector determined by the gradient of the wave-front. We report experimental results by measuring the "center of mass" of the far-field intensity pattern of a single Stokes beam, and the fringe separation of an interference pattern formed by superposing two independently generated Stokes pulses. The experimental results are compared to theory.2