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Quantum Electronics

Spatial coherence of the Stokes component of stimulated Raman scattering excited in a long multimode fibre

Maksim Kitsak, Anatoly I. Kitsak

Published Jul 31, 2008• 0 citations
Publisher

Abstract

The mode structure of the Stokes component of stimulated Raman scattering (SRS) in a multimode fibre excited by a pump field with statistics corresponding to a narrowband Gaussian noise model is theoretically analysed. The relation is obtained between the number of spatially coherent radiation modes of the Stokes wave and fibre parameters and SRS excitation conditions. It follows from this relation that the degree of spatial coherence of the Stokes radiation at the fibre output is determined by the number of spatial modes of pump radiation and the averaged gain of the Stokes radiation over the interaction length. It is established that there exists the threshold gain at which the number of spatially coherent radiation modes of the Stokes component increases infinitely and, therefore, its spatial coherence tends to zero. The estimates of the number of spatially coherent radiation modes of the Stokes component showed that already at the Raman scattering excitation threshold their number is so great that the degree of spatial coherence of the Stokes component radiation corresponds to the degree of coherence of a thermal source radiation. The measured dispersion of spatial intensity fluctuations of the Stokes component sepa-rated by using an interference filter (with the transmission bandwidth ≈1 nm at 620 nm) proved to be approximately three times higher than the dispersion of radiation intensity fluctuations of a luminescent lamp. The large dispersion of spatial intensity fluctuations of the Stokes component is caused most likely by individual intensity spikes in a small-scale (unresolved by a detector with a spatial resolution of ≈14 μm) speckle pattern produced in the detection plane by monochromatic Stokes radiation.

DOI
10.1070/qe2008v038n07abeh013597

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The mode structure of the Stokes component of stimulated Raman scattering (SRS) in a multimode fibre excited by a pump field with statistics corresponding to a narrowband Gaussian noise model is theoretically analysed. The relation is obtained between the number of spatially coherent radiation modes of the Stokes wave and fibre parameters and SRS excitation conditions. It follows from this relation that the degree of spatial coherence of the Stokes radiation at the fibre output is determined by the number of spatial modes of pump radiation and the averaged gain of the Stokes radiation over the interaction length. It is established that there exists the threshold gain at which the number of spatially coherent radiation modes of the Stokes component increases infinitely and, therefore, its spatial coherence tends to zero. The estimates of the number of spatially coherent radiation modes of the Stokes component showed that already at the Raman scattering excitation threshold their number is so great that the degree of spatial coherence of the Stokes component radiation corresponds to the degree of coherence of a thermal source radiation. The measured dispersion of spatial intensity fluctuations of the Stokes component sepa-rated by using an interference filter (with the transmission bandwidth ≈1 nm at 620 nm) proved to be approximately three times higher than the dispersion of radiation intensity fluctuations of a luminescent lamp. The large dispersion of spatial intensity fluctuations of the Stokes component is caused most likely by individual intensity spikes in a small-scale (unresolved by a detector with a spatial resolution of ≈14 μm) speckle pattern produced in the detection plane by monochromatic Stokes radiation.
We conduct a theoretical analysis of the Stokes component mode structure of the Raman scattering. The Raman scattering is excited in a multimode waveguide with pump radiation having statistics corresponding to the model of the narrow-band Gaussian noise. We obtain an analytical relation connecting the number of spatially coherent modes of the Stokes component, characteristics of the waveguide and conditions of the Raman scattering excitation. It follows from the obtained relation that the Raman scattering spatial coherency degree at the waveguide's exit is determined by the number of the spatial modes of pump radiation and the amplification of the Stokes radiation throughout the waveguide's length. There exists a threshold in the amplification corresponding to the unlimited increase of the number of the spatially coherent modes of the Raman scattering and therefore to a zero of spatial coherency. Conducted estimations show that at the Raman scattering threshold value of the radiation intensity the number of the spatially coherent modes of the Raman scattering Stokes component should be comparable to that of the thermal source. Experimentally, measured dispersion of the spatial intensity fluctuations of the Stokes component isolated with an interference filter (spectral bandwidth ~ 1nm, λ = 620nm) is three time smaller than that of the luminescent lamp radiation.
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.
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.
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
A quantum theory of stimulated Raman scattering is presented that takes into account three-dimensional propagation and collisional dephasing, allowing the study of the spatial and temporal coherence properties of the generated Stokes light. Maxwell-Bloch equations for the Stokes field operator and the collective atomic operators are solved analytically under low-signal-gain conditions, where the laser field and the atomic ground states remain undepleted. The intensity and the space-time autocorrelation function of the Stokes field are calculated. The Stokes field is expanded into a set of statistically independent ``coherence modes,'' which are determined explicitly for the case of a cylindrically shaped pumped volume. The Stokes pulse energy W is found to fluctuate from pulse to pulse. The probability distribution function for pulse energies P(W) is calculated for a range of Fresnel numbers of the excited volume and collisional dephasing rates. For small values of Fresnel number and dephasing rate, P(W) is a negative exponential distribution. For large values of either, P(W) narrows and approaches a Gaussian-shaped distribution. This occurs because many independent modes contribute to the Stokes emission, making it spatially and/or temporally incoherent.
A theoretical and experimental investigation was made of the amplification of a Stokes signal in a multimode pump field when the ratio of the spectral widths of the interacting fields was varied. In the pump intensity range I p < I cr ( I cr was a critical intensity) the gain increment was considerably less than the static value and it was governed entirely by the spectral distributions of the pump and Stokes signal. In the range I p > I cr the gain increment tended to the static value. An anomalous scatter of the Raman amplifier gain was observed experimentally for Δ ν s ≈Δ ν p ≫Δ ν l (Δ ν s ,Δ ν p , and Δ ν l were the line widths of the Stokes signal, pump, and spontaneous stimulated Raman scattering). A theory developed for the purpose indicated that the scatter was observed because the phase relationships between the interacting waves at the input into the medium varied from one measurement to another.