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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Studies of the angular frequency spectrum Raman scattering Stokes component radiation excited in an extensive multimode waveguide

Maksim Kitsak, Anatoly I. Kitsak

Published Apr 18, 2008• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.797107

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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.
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.
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.
An analysis is made of the laws governing amplification of the Stokes wave in stimulated Raman scattering (STRS) in a dispersive medium for different models of wide-band pump radiation. An accurate solution of the STRS equations is obtained for pump radiation having a quadratically time-modulated phase. It is shown that for pump radiation for which the width of the instantaneous frequency spectrum is much less than the Raman scattering line width, the laws governing amplification of the Stokes wave differ substantially from those predicted by statistical STRS theory. Solutions of STRS equations in a dispersive medium are found for a model of temporally periodic pump radiation having a high intermode beat frequency.
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.
The development of the Stokes spectrum and the dependence of Raman gain upon pump laser bandwidth in a narrow linewidth Raman medium are analyzed in terms of a closed set of multimode equations. The analysis predicts that the Stokes gain coefficient takes on its maximum value, which is independent of bandwidth, when the Stokes spectrum replicates the pump spectrum. It is also shown that in the absence of dispersion the spectrum of a weak Stokes wave is driven to duplicate the pump spectrum, and that for noise input the average power generated in the Stokes wave is independent of the pump bandwidth.