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Quantum Electronics

Effect of quantum fluctuations of medium polarisation on stimulated Raman scattering in a photonic crystal

R. G. Zaporozhchenko

Published Mar 31, 2004• 1 citation
Publisher

Abstract

Stimulated Raman scattering of femtosecond pulses in a photonic crystal consisting of alternating quarter-wave dielectric layers and layers of a nonlinear-scattering medium is calculated taking into account the quantum fluctuations of the medium polarisation. Analysis of the calculations showed that the Stokes pulse duration and intensity depend substantially on the statistic properties of polarisation. The efficient SRS conversion in a photonic crystal occurs under certain dispersion conditions providing the minimum difference between dispersions of the pump and Stokes frequencies, which are determined by the length and the number of periods of the photonic crystal.

DOI
10.1070/qe2004v034n03abeh002619

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Stimulated Raman scattering of femtosecond pulses in a photonic crystal consisting of alternating quarter-wave dielectric layers and layers of a nonlinear-scattering medium is calculated taking into account the quantum fluctuations of the medium polarisation. Analysis of the calculations showed that the Stokes pulse duration and intensity depend substantially on the statistic properties of polarisation. The efficient SRS conversion in a photonic crystal occurs under certain dispersion conditions providing the minimum difference between dispersions of the pump and Stokes frequencies, which are determined by the length and the number of periods of the photonic crystal.
Manifestations of quantum fluctuations in stimulated Raman scattering ( SRS ) have been intensively studied lately1. Quantum noise present in spontaneous scattering stimulates fluctuations both in phase and in amplitude characteristics of output Stokes emission. At present statistical properties of amplitude fluctuations are studied in detail and it is shown that in linear transient regime of SRS without pump depletion statistical scatter of Stokes energy Es has negative exponential distribution2. In saturation regime of SRS the amplitude fluctuations are almost suppressed simultaneously with pump depletion. The phase fluctuations manifest both in linear and in nonlinear SRS and produce beam pointing fluctuations3, a change in Stokes emission spectra from pulse to pulse4 and cause soliton appearance in Rayleigh component of SRS5. Phase noise is observed in saturation regime due to relation between phases of Stokes and phonon waves which is preserved in processes of stationary and quasistationary SRS amplification. Spectrum phase fluctuations manifest as fine structure of SRS spectrum4, fluctuations of average Stokes frequency and Stokes linewidth fluctuations. The evolution of phase fluctuations in time is determined by the nature of spontaneous scattering and polarization of Raman media and is of great interest as one can follow its development from vacuum modes level to saturation regime. However up to now this problem has not been studied yet.
We study the stimulated Raman scattering (SRS) of light from an atomic system embedded in a photonic crystal and coherently pumped by a laser field. In our study, the electromagnetic field is treated classically and the atomic system is described quantum mechanically. Considering a decomposition of the pump and Stokes fields into the Bloch modes of the photonic crystals and using a multiscale analysis, we derive the Maxwell-Bloch equations for SRS in photonic crystals. These equations contain effective parameters that characterize the SRS gain, the nonlinear atomic response to the electromagnetic field, and the group velocity and that can be calculated in terms of the Bloch modes of the unperturbed photonic crystal. We show that if the pump laser frequency is tuned near a photonic band edge and the atomic system is carefully chosen such that the Stokes mode matches another photonic band edge, low-threshold, enhanced Raman amplification is possible. Possible physical realizations of SRS in photonic crystals are also discussed.
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.
From a classical point of view the generation of Stokes light by stimulated Raman scattering (SRS) is not possible unless a Stokes input signal is supplied. On the other hand, the gain in SRS can be so large that without input the spontaneous build-up of strong Stokes waves can be observed. A proper quantum theoretical description1 shows that this spontaneous initiation of Stokes generation is due to microscopic random fluctuations originating from quantum noise associated with the dynamical variable Q(t) which is active in the Raman process, e.g., a molecular normal mode coordinate. In highly transient SRS -when the duration of the excitation pulse is shorter than the dephasing time- the Stokes field is proportional to the initial Heisenberg operator Q(0) describing the state of the system just before the excitation pulse is turned on (t=0). In this case the statistics of the final Stokes pulse energy is determined by quantum mechanical uncertainties associated with the random Raman polarization of the initial state2. On the other hand, in a steady state situation the Stokes pulse fluctuations are substantially reduced and the Stokes pulse energy is stabilized as a result of damping processes.
An analysis is made of parametric excitation of anti-Stokes components of stimulated Raman scattering in an optical resonator filled with a medium whose refractive index is dispersion-free. The case of steady generation of the first Stokes component is considered. Expressions are obtained for the amplitudes of steady-state fluctuations of the anti-Stokes components in the case of a weak reaction of these components on the pump radiation and on the first Stokes component. Various cases of frequency dependences of the intrinsic resonator losses are considered. Conditions are found under which a large number of anti-Stokes components with similar intensities is excited so that the anti-Stokes radiation leaving the resonator is a periodic sequence of ultrashort (femtosecond) pulses with one pulse per period.
An experimental study of stimulated Raman scattering (SRS) has been performed using a subnanosecond pulse to investigate the statistical properties of Stokes light generation.1 There are macroscopic fluctuations of Stokes pulse energy as well as random generation of spatial mode patterns. When the scattering volume has a low Fresnel number, the spatial patterns of the SRS resemble cavity modes. When the Fresnel number is large, there are irregular structures appearing across the Stokes profile in a random fashion. A theoretical study of the spatial and temporal coherence properties of Raman generation using a Gaussian-profile pump pulse has also been carried out. Using a Gaussian-Laguerre mode expansion technique,2 it is found that the higher-order spatial coherence modes1 experience different gain than the lower-order spatial coherence modes, and consequently the temporal coherence associated with each individual spatial coherence mode is different.