# Raman Gain

Definition: optical gain arising from stimulated Raman scattering

German: Raman-Verstärkung

Categories: nonlinear optics, physical foundations

Units: %, dB

Formula symbol: <$g_\textrm{R}$>

Author: Dr. Rüdiger Paschotta

Raman gain is optical gain (amplification) arising from stimulated Raman scattering. It can occur in transparent solid media (e.g. optical fibers), liquids and gases under the influence of intense pump light, and is used in Raman amplifiers and Raman lasers. Its magnitude depends on the optical frequency offset (in the terahertz region) between pump and signal wave, to some smaller extent on the pump wavelength, and on material properties. Compared with laser gain e.g. in rare-earth-doped laser gain media, Raman gain requires higher pump intensities and/or longer interaction lengths, has substantially different saturation characteristics, and a gain spectrum which depends on the wavelength of the pump light.

The quasi-continuous interaction between some narrow-band pump wave and a Stokes-shifted wave (with a lower optical frequency) via stimulated Raman scattering can be described with the following coupled differential equations:

$$\begin{array}{l} \frac{{\partial {I_{\rm{s}}}}}{{\partial z}} = {g_{\rm{R}}}\;{I_{\rm{p}}}\;{I_{\rm{s}}}\\ \frac{{\partial {I_{\rm{p}}}}}{{\partial z}} = - \frac{{{\nu _{\rm{p}}}}}{{{\nu _{\rm{s}}}}}{g_{\rm{R}}}\;{I_{\rm{p}}}\;{I_{\rm{s}}} \end{array}$$where <$I_\textrm{p}$> and <$I_\textrm{s}$> are the optical intensities (with units of W/m^{2}), <$\nu_\textrm{p}$> and <$\nu_\textrm{s}$> the optical frequencies of the two beams, and <$g_\textrm{R}$> is the Raman gain coefficient, which is e.g. of the order of 10^{−13} m/W for silica fibers for the optimum frequency offset of roughly 13 THz. It is assumed that both beams fully overlap and propagate in the <$z$> direction. Of course, additional terms can be added to the equations, e.g. in order to include spontaneous Raman scattering and linear propagation losses.

The equations show that the Stokes waves experiences a local gain coefficient <$g_\textrm{R} I_\textrm{p}$>, whereas the pump wave loses some more energy than is transferred to the Stokes wave. This is because one pump photon is converted into one Stokes photon (with lower energy) and a phonon. The phonon energy corresponds to the difference of photon energies. This lost optical energy is effectively converted into heat.

Due to the rather low value of the Raman gain coefficients e.g. of typical silica fibers, strong Raman conversion requires many meters of fiber if the optical powers are limited to a few watts. For optical pulses, e.g. with nanosecond pulse durations, where the peak powers can be much higher (many kilowatts), strong Raman conversion may occur even within short lengths of fiber (see Figure 1).

The Raman gain coefficient depends mainly on the difference in optical frequencies, but also to some extent on the pump frequency and the polarization directions. Figure 2 shows the Raman gain for fused silica as a function of the frequency difference, assuming equal linear polarizations of both beams. There is a maximum Raman gain for a frequency offset of 13.2 THz. For example, a pump wave at 1064 nm leads to the largest Raman gain at a signal wavelength of 1116 nm. The peaks in the Raman spectrum correspond to certain vibration modes of the silica structure.

The quantity on the horizontal axis is the frequency offset of the signal wave with respect to the pump wave. The data are based on Ref. [4].

The modeling of stimulated Raman scattering for broadband beams and particularly for ultrashort pulses is significantly more involved. It can be based on, e.g., a Raman response function <$h(t)$>, describing the delayed temporal response of the nonlinear polarization to the electric field [2, 4, 6].

The Raman gain can be fairly strong in optical fibers, where substantial optical intensities can be maintained over long lengths. When the Raman gain corresponding to some optical wave exceeds the order of 70 dB, substantial power is transferred to a Stokes wave, even if there is no Stokes input to the fiber. This effect can limit the performance of high-power fiber lasers and amplifiers.

### Bibliography

[1] | R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguide”, Appl. Phys. Lett. 22 (6), 276 (1973), DOI:10.1063/1.1654637 |

[2] | K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers”, IEEE J. Quantum Electron. 25 (12), 2665 (1989), DOI:10.1109/3.40655 |

[3] | J. Bromage et al., “A method to predict the Raman gain spectra of germanosilicate fibers with arbitrary index profiles”, IEEE Photon. Technol. Lett. 14 (1), 24 (2002), DOI:10.1109/68.974149 |

[4] | D. Hollenbeck and C. D. Cantrell, “Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function”, J. Opt. Soc. Am. B 19 (12), 2886 (2002), DOI:10.1364/JOSAB.19.002886 |

[5] | X. Yan et al., “Raman transient response and enhanced soliton self-frequency shift in ZBLAN fiber”, J. Opt. Soc. Am. B 29 (2), 238 (2012), DOI:10.1364/JOSAB.29.000238 |

[6] | G. P. Agrawal, Nonlinear Fiber Optics, 4th edn., Academic Press, New York (2007) |

See also: nonlinearities, Raman scattering, Raman amplifiers, Raman lasers, silica fibers, spotlight 2010-03-03

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