Abstract

We propose a cavity-based combined Raman and second harmonic generation scheme for generating hundreds of milliwatts of continuous-wave yellow radiation with a frequency linewidth suitable for spectroscopic applications. We suggest using H2 gas as the Raman medium in an external Raman ring resonator with an intracavity frequency doubling crystal. Using gas rather than a crystal allows for single-mode, narrow-linewidth operation suitable for many spectroscopic applications. In a specific numerical example, we predict the generation of more than 200 mW of narrow-linewidth 583 nm light from an input of 1 W of 785 nm light, which could be obtained from a low cost tapered amplifier diode system. Finally, we suggest methods for improving the laser performance.

© 2012 Optical Society of America

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    [CrossRef]
  7. K. Koch, G. T. Moore, and M. E. Dearborn, “Raman oscillation with intra-cavity second harmonic generation,” IEEE J. Quantum Electron. 33, 1743–1748 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. J. J. Weber, J. T. Green, and D. D. Yavuz, “17 THz continuous-wave optical modulator,” Phys. Rev. A 85, 013805 (2012).
    [CrossRef]
  21. A. J. Lee, D. J. Spence, J. A. Piper, and H. M. Pask, “A wavelength-versatile, continuous-wave self-Raman solid-state laser operating in the visible,” Opt. Express 18, 20013–20018 (2010).
    [CrossRef]
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  23. P. Lallemand, P. Simova, and G. Bret, “Pressure-induced line shift and collisional narrowing in hydrogen gas determined by stimulated Raman emission,” Phys. Rev. Lett. 17, 1239–1241(1966).
    [CrossRef]
  24. N. Bloembergen, “The stimulated Raman effect,” Am. J. Phys. 35, 989–1022 (1967).
    [CrossRef]
  25. S. E. Harris and A. V. Sokolov, “Broadband spectral generation with refractive index control,” Phys. Rev. A 55, R4019–R4022(1997).
    [CrossRef]
  26. A. C. Allison and A. Dalgarno, “Band oscillator strengths and transition probabilities for the Lyman and Werner systems of H2, HD, and D2,” At. Data 1, 289–304 (1969).
    [CrossRef]
  27. J. J. Ottusch and D. A. Rockwell, “Measurements of Raman gain coefficients in hydrogen, deuterium, and methane,” IEEE J. Quantum Electron. 24, 2076–2080 (1988).
    [CrossRef]
  28. W. K. Bischel and M. J. Dryer, “Wavelength dependence of the absolute Raman gain coefficient for the Q(1) transition in H2,” J. Opt. Soc. Am. B 3, 677–682 (1986).
    [CrossRef]
  29. G. D. Boyd, W. D. Johnston, and I. P. Kaminow, “Optimization of the stimulated Raman scattering threshold,” IEEE J. Quantum Electron. 5, 203–206 (1969).
    [CrossRef]
  30. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).
  31. W. K. Bischel and M. J. Dryer, “Temperature dependence of the Raman linewidth and line shift for the Q(1) and Q(0) transitions in normal and para-H2,” Phys. Rev. A 33, 3113–3123(1986).
    [CrossRef]
  32. J. T. Green, J. J. Weber, and D. D. Yavuz, “Continuous-wave, multiple-order rotational Raman generation in molecular deuterium,” Opt. Lett. 36, 897–899 (2011).
    [CrossRef]
  33. G. Ferrari, J. Catani, L. Fallani, G. Giusfredi, G. Schettino, F. Schäfer, and P. C. Pastor, “Coherent addition of laser beams in resonant passive optical cavities,” Opt. Lett. 35, 3105–3107 (2010).
    [CrossRef]

2012 (1)

J. J. Weber, J. T. Green, and D. D. Yavuz, “17 THz continuous-wave optical modulator,” Phys. Rev. A 85, 013805 (2012).
[CrossRef]

2011 (3)

2010 (5)

2009 (2)

J. T. Green, D. E. Sikes, and D. D. Yavuz, “Continuous-wave high-power rotational Raman generation in molecular deuterium,” Opt. Lett. 34, 2563–2565 (2009).
[CrossRef]

R. P. Mildren, “The outlook for diamond in Raman laser applications,” Mater. Res. Soc. Symp. Proc. 1203, 1203-J13-01 (2009).
[CrossRef]

2008 (1)

2007 (4)

G. Ferrari, “Generating green to red light with semiconductor lasers,” Opt. Express 15, 1672–1678 (2007).
[CrossRef]

D. D. Yavuz, “High-frequency modulation of continuous-wave laser beams by maximally coherent molecules,” Phys. Rev. A 76, 011805(R) (2007).

P. Dekker, H. M. Pask, D. J. Spence, and J. A. Piper, “Continuous-wave, intra-cavity doubled, self-Raman laser operation in Nd: GdVO4 at 586.5 nm,” Opt. Express 15, 7038–7046(2007).
[CrossRef]

A. J. Lee, H. M. Pask, T. Omatsu, P. Dekker, and J. A. Piper, “All solid-state continuous-wave yellow laser based on intra-cavity frequency-doubled self-Raman laser action,” Appl. Phys. B 88, 539–544 (2007).
[CrossRef]

2005 (1)

2004 (2)

S. G. Porsev, A. Derevianko, and E. N. Forston, “Possibility of an optical clock using the 61S0→P30 transition in Yb171,173 atoms held in an optical lattice,” Phys. Rev. A 69, 021403(R) (2004).

J. K. Brasseur, R. F. Teehan, P. A. Roos, B. Soucy, D. K. Neumann, and J. L. Carlsten, “High-power deuterium Raman laser at 632 nm,” J. Opt. Soc. Am. B 43, 1162–1166(2004).
[CrossRef]

2003 (1)

2002 (2)

1999 (1)

1997 (2)

K. Koch, G. T. Moore, and M. E. Dearborn, “Raman oscillation with intra-cavity second harmonic generation,” IEEE J. Quantum Electron. 33, 1743–1748 (1997).
[CrossRef]

S. E. Harris and A. V. Sokolov, “Broadband spectral generation with refractive index control,” Phys. Rev. A 55, R4019–R4022(1997).
[CrossRef]

1988 (1)

J. J. Ottusch and D. A. Rockwell, “Measurements of Raman gain coefficients in hydrogen, deuterium, and methane,” IEEE J. Quantum Electron. 24, 2076–2080 (1988).
[CrossRef]

1986 (2)

W. K. Bischel and M. J. Dryer, “Wavelength dependence of the absolute Raman gain coefficient for the Q(1) transition in H2,” J. Opt. Soc. Am. B 3, 677–682 (1986).
[CrossRef]

W. K. Bischel and M. J. Dryer, “Temperature dependence of the Raman linewidth and line shift for the Q(1) and Q(0) transitions in normal and para-H2,” Phys. Rev. A 33, 3113–3123(1986).
[CrossRef]

1969 (2)

G. D. Boyd, W. D. Johnston, and I. P. Kaminow, “Optimization of the stimulated Raman scattering threshold,” IEEE J. Quantum Electron. 5, 203–206 (1969).
[CrossRef]

A. C. Allison and A. Dalgarno, “Band oscillator strengths and transition probabilities for the Lyman and Werner systems of H2, HD, and D2,” At. Data 1, 289–304 (1969).
[CrossRef]

1967 (1)

N. Bloembergen, “The stimulated Raman effect,” Am. J. Phys. 35, 989–1022 (1967).
[CrossRef]

1966 (1)

P. Lallemand, P. Simova, and G. Bret, “Pressure-induced line shift and collisional narrowing in hydrogen gas determined by stimulated Raman emission,” Phys. Rev. Lett. 17, 1239–1241(1966).
[CrossRef]

Allison, A. C.

A. C. Allison and A. Dalgarno, “Band oscillator strengths and transition probabilities for the Lyman and Werner systems of H2, HD, and D2,” At. Data 1, 289–304 (1969).
[CrossRef]

Ban, H. Y.

Bienfang, J. C.

Bischel, W. K.

W. K. Bischel and M. J. Dryer, “Wavelength dependence of the absolute Raman gain coefficient for the Q(1) transition in H2,” J. Opt. Soc. Am. B 3, 677–682 (1986).
[CrossRef]

W. K. Bischel and M. J. Dryer, “Temperature dependence of the Raman linewidth and line shift for the Q(1) and Q(0) transitions in normal and para-H2,” Phys. Rev. A 33, 3113–3123(1986).
[CrossRef]

Bloembergen, N.

N. Bloembergen, “The stimulated Raman effect,” Am. J. Phys. 35, 989–1022 (1967).
[CrossRef]

Bonner, G. M.

Boyd, G. D.

G. D. Boyd, W. D. Johnston, and I. P. Kaminow, “Optimization of the stimulated Raman scattering threshold,” IEEE J. Quantum Electron. 5, 203–206 (1969).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

Brasseur, J. K.

J. K. Brasseur, R. F. Teehan, P. A. Roos, B. Soucy, D. K. Neumann, and J. L. Carlsten, “High-power deuterium Raman laser at 632 nm,” J. Opt. Soc. Am. B 43, 1162–1166(2004).
[CrossRef]

J. K. Brasseur, P. A. Roos, K. S. Repasky, and J. L. Carlsten, “Characterization of a continuous-wave Raman laser in H2,” J. Opt. Soc. Am. B 16, 1305–1312 (1999).
[CrossRef]

Bret, G.

P. Lallemand, P. Simova, and G. Bret, “Pressure-induced line shift and collisional narrowing in hydrogen gas determined by stimulated Raman emission,” Phys. Rev. Lett. 17, 1239–1241(1966).
[CrossRef]

Burns, D.

Carlsten, J. L.

Catani, J.

Chen, Y. F.

Dalgarno, A.

A. C. Allison and A. Dalgarno, “Band oscillator strengths and transition probabilities for the Lyman and Werner systems of H2, HD, and D2,” At. Data 1, 289–304 (1969).
[CrossRef]

Dalibard, J.

Dawson, M. D.

Dearborn, M. E.

K. Koch, G. T. Moore, and M. E. Dearborn, “Raman oscillation with intra-cavity second harmonic generation,” IEEE J. Quantum Electron. 33, 1743–1748 (1997).
[CrossRef]

Dekker, P.

P. Dekker, H. M. Pask, D. J. Spence, and J. A. Piper, “Continuous-wave, intra-cavity doubled, self-Raman laser operation in Nd: GdVO4 at 586.5 nm,” Opt. Express 15, 7038–7046(2007).
[CrossRef]

A. J. Lee, H. M. Pask, T. Omatsu, P. Dekker, and J. A. Piper, “All solid-state continuous-wave yellow laser based on intra-cavity frequency-doubled self-Raman laser action,” Appl. Phys. B 88, 539–544 (2007).
[CrossRef]

Derevianko, A.

S. G. Porsev, A. Derevianko, and E. N. Forston, “Possibility of an optical clock using the 61S0→P30 transition in Yb171,173 atoms held in an optical lattice,” Phys. Rev. A 69, 021403(R) (2004).

Dryer, M. J.

W. K. Bischel and M. J. Dryer, “Wavelength dependence of the absolute Raman gain coefficient for the Q(1) transition in H2,” J. Opt. Soc. Am. B 3, 677–682 (1986).
[CrossRef]

W. K. Bischel and M. J. Dryer, “Temperature dependence of the Raman linewidth and line shift for the Q(1) and Q(0) transitions in normal and para-H2,” Phys. Rev. A 33, 3113–3123(1986).
[CrossRef]

Fallani, L.

Ferrari, G.

Forston, E. N.

S. G. Porsev, A. Derevianko, and E. N. Forston, “Possibility of an optical clock using the 61S0→P30 transition in Yb171,173 atoms held in an optical lattice,” Phys. Rev. A 69, 021403(R) (2004).

Gerbier, F.

Giusfredi, G.

Green, J. T.

J. J. Weber, J. T. Green, and D. D. Yavuz, “17 THz continuous-wave optical modulator,” Phys. Rev. A 85, 013805 (2012).
[CrossRef]

J. T. Green, J. J. Weber, and D. D. Yavuz, “Continuous-wave, multiple-order rotational Raman generation in molecular deuterium,” Opt. Lett. 36, 897–899 (2011).
[CrossRef]

J. T. Green, J. J. Weber, and D. D. Yavuz, “Continuous-wave light modulation at molecular frequencies,” Phys. Rev. A 82, 011805 (2010).
[CrossRef]

J. T. Green, D. E. Sikes, and D. D. Yavuz, “Continuous-wave high-power rotational Raman generation in molecular deuterium,” Opt. Lett. 34, 2563–2565 (2009).
[CrossRef]

Hanssen, J. L.

Harris, S. E.

S. E. Harris and A. V. Sokolov, “Broadband spectral generation with refractive index control,” Phys. Rev. A 55, R4019–R4022(1997).
[CrossRef]

Hastie, J. E.

Huang, Y. C.

Jacka, M.

Jiang, M.

Johnston, W. D.

G. D. Boyd, W. D. Johnston, and I. P. Kaminow, “Optimization of the stimulated Raman scattering threshold,” IEEE J. Quantum Electron. 5, 203–206 (1969).
[CrossRef]

Kaminow, I. P.

G. D. Boyd, W. D. Johnston, and I. P. Kaminow, “Optimization of the stimulated Raman scattering threshold,” IEEE J. Quantum Electron. 5, 203–206 (1969).
[CrossRef]

Kemp, A. J.

Koch, K.

K. Koch, G. T. Moore, and M. E. Dearborn, “Raman oscillation with intra-cavity second harmonic generation,” IEEE J. Quantum Electron. 33, 1743–1748 (1997).
[CrossRef]

Kwon, T. Y.

Lallemand, P.

P. Lallemand, P. Simova, and G. Bret, “Pressure-induced line shift and collisional narrowing in hydrogen gas determined by stimulated Raman emission,” Phys. Rev. Lett. 17, 1239–1241(1966).
[CrossRef]

Lee, A. J.

Lee, S. B.

Lee, W. K.

Li, J.

Li, Z.

Lin, T. C.

Lubeigt, W.

McClelland, J. J.

Meng, L. S.

Mildren, R. P.

R. P. Mildren, “The outlook for diamond in Raman laser applications,” Mater. Res. Soc. Symp. Proc. 1203, 1203-J13-01 (2009).
[CrossRef]

Mimoun, E.

Moore, G. T.

K. Koch, G. T. Moore, and M. E. Dearborn, “Raman oscillation with intra-cavity second harmonic generation,” IEEE J. Quantum Electron. 33, 1743–1748 (1997).
[CrossRef]

Neumann, D. K.

J. K. Brasseur, R. F. Teehan, P. A. Roos, B. Soucy, D. K. Neumann, and J. L. Carlsten, “High-power deuterium Raman laser at 632 nm,” J. Opt. Soc. Am. B 43, 1162–1166(2004).
[CrossRef]

Omatsu, T.

A. J. Lee, H. M. Pask, T. Omatsu, P. Dekker, and J. A. Piper, “All solid-state continuous-wave yellow laser based on intra-cavity frequency-doubled self-Raman laser action,” Appl. Phys. B 88, 539–544 (2007).
[CrossRef]

Ottusch, J. J.

J. J. Ottusch and D. A. Rockwell, “Measurements of Raman gain coefficients in hydrogen, deuterium, and methane,” IEEE J. Quantum Electron. 24, 2076–2080 (1988).
[CrossRef]

Park, C. Y.

Park, S. E.

Pask, H.

Pask, H. M.

Pastor, P. C.

Piper, J.

Piper, J. A.

Porsev, S. G.

S. G. Porsev, A. Derevianko, and E. N. Forston, “Possibility of an optical clock using the 61S0→P30 transition in Yb171,173 atoms held in an optical lattice,” Phys. Rev. A 69, 021403(R) (2004).

Reader, J.

Repasky, K. S.

Rockwell, D. A.

J. J. Ottusch and D. A. Rockwell, “Measurements of Raman gain coefficients in hydrogen, deuterium, and methane,” IEEE J. Quantum Electron. 24, 2076–2080 (1988).
[CrossRef]

Roos, P. A.

Rudolph, W.

Sarlo, L. D.

Schäfer, F.

Schettino, G.

Sikes, D. E.

Simova, P.

P. Lallemand, P. Simova, and G. Bret, “Pressure-induced line shift and collisional narrowing in hydrogen gas determined by stimulated Raman emission,” Phys. Rev. Lett. 17, 1239–1241(1966).
[CrossRef]

Sokolov, A. V.

S. E. Harris and A. V. Sokolov, “Broadband spectral generation with refractive index control,” Phys. Rev. A 55, R4019–R4022(1997).
[CrossRef]

Soucy, B.

J. K. Brasseur, R. F. Teehan, P. A. Roos, B. Soucy, D. K. Neumann, and J. L. Carlsten, “High-power deuterium Raman laser at 632 nm,” J. Opt. Soc. Am. B 43, 1162–1166(2004).
[CrossRef]

Spence, D. J.

Teehan, R. F.

J. K. Brasseur, R. F. Teehan, P. A. Roos, B. Soucy, D. K. Neumann, and J. L. Carlsten, “High-power deuterium Raman laser at 632 nm,” J. Opt. Soc. Am. B 43, 1162–1166(2004).
[CrossRef]

Tsai, S. W.

Wang, J.

Wang, S. C.

Weber, J. J.

J. J. Weber, J. T. Green, and D. D. Yavuz, “17 THz continuous-wave optical modulator,” Phys. Rev. A 85, 013805 (2012).
[CrossRef]

J. T. Green, J. J. Weber, and D. D. Yavuz, “Continuous-wave, multiple-order rotational Raman generation in molecular deuterium,” Opt. Lett. 36, 897–899 (2011).
[CrossRef]

J. T. Green, J. J. Weber, and D. D. Yavuz, “Continuous-wave light modulation at molecular frequencies,” Phys. Rev. A 82, 011805 (2010).
[CrossRef]

Wong, B. C.

Yavuz, D. D.

J. J. Weber, J. T. Green, and D. D. Yavuz, “17 THz continuous-wave optical modulator,” Phys. Rev. A 85, 013805 (2012).
[CrossRef]

J. T. Green, J. J. Weber, and D. D. Yavuz, “Continuous-wave, multiple-order rotational Raman generation in molecular deuterium,” Opt. Lett. 36, 897–899 (2011).
[CrossRef]

J. T. Green, J. J. Weber, and D. D. Yavuz, “Continuous-wave light modulation at molecular frequencies,” Phys. Rev. A 82, 011805 (2010).
[CrossRef]

J. T. Green, D. E. Sikes, and D. D. Yavuz, “Continuous-wave high-power rotational Raman generation in molecular deuterium,” Opt. Lett. 34, 2563–2565 (2009).
[CrossRef]

D. D. Yavuz, “High-frequency modulation of continuous-wave laser beams by maximally coherent molecules,” Phys. Rev. A 76, 011805(R) (2007).

Yu, D. H.

Yu, H.

Zhang, H.

Zondy, J. J.

Am. J. Phys. (1)

N. Bloembergen, “The stimulated Raman effect,” Am. J. Phys. 35, 989–1022 (1967).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

A. J. Lee, H. M. Pask, T. Omatsu, P. Dekker, and J. A. Piper, “All solid-state continuous-wave yellow laser based on intra-cavity frequency-doubled self-Raman laser action,” Appl. Phys. B 88, 539–544 (2007).
[CrossRef]

At. Data (1)

A. C. Allison and A. Dalgarno, “Band oscillator strengths and transition probabilities for the Lyman and Werner systems of H2, HD, and D2,” At. Data 1, 289–304 (1969).
[CrossRef]

IEEE J. Quantum Electron. (3)

J. J. Ottusch and D. A. Rockwell, “Measurements of Raman gain coefficients in hydrogen, deuterium, and methane,” IEEE J. Quantum Electron. 24, 2076–2080 (1988).
[CrossRef]

G. D. Boyd, W. D. Johnston, and I. P. Kaminow, “Optimization of the stimulated Raman scattering threshold,” IEEE J. Quantum Electron. 5, 203–206 (1969).
[CrossRef]

K. Koch, G. T. Moore, and M. E. Dearborn, “Raman oscillation with intra-cavity second harmonic generation,” IEEE J. Quantum Electron. 33, 1743–1748 (1997).
[CrossRef]

J. Opt. Soc. Am. B (4)

Mater. Res. Soc. Symp. Proc. (1)

R. P. Mildren, “The outlook for diamond in Raman laser applications,” Mater. Res. Soc. Symp. Proc. 1203, 1203-J13-01 (2009).
[CrossRef]

Opt. Express (7)

Opt. Lett. (6)

Phys. Rev. A (6)

S. E. Harris and A. V. Sokolov, “Broadband spectral generation with refractive index control,” Phys. Rev. A 55, R4019–R4022(1997).
[CrossRef]

W. K. Bischel and M. J. Dryer, “Temperature dependence of the Raman linewidth and line shift for the Q(1) and Q(0) transitions in normal and para-H2,” Phys. Rev. A 33, 3113–3123(1986).
[CrossRef]

D. D. Yavuz, “High-frequency modulation of continuous-wave laser beams by maximally coherent molecules,” Phys. Rev. A 76, 011805(R) (2007).

J. T. Green, J. J. Weber, and D. D. Yavuz, “Continuous-wave light modulation at molecular frequencies,” Phys. Rev. A 82, 011805 (2010).
[CrossRef]

J. J. Weber, J. T. Green, and D. D. Yavuz, “17 THz continuous-wave optical modulator,” Phys. Rev. A 85, 013805 (2012).
[CrossRef]

S. G. Porsev, A. Derevianko, and E. N. Forston, “Possibility of an optical clock using the 61S0→P30 transition in Yb171,173 atoms held in an optical lattice,” Phys. Rev. A 69, 021403(R) (2004).

Phys. Rev. Lett. (1)

P. Lallemand, P. Simova, and G. Bret, “Pressure-induced line shift and collisional narrowing in hydrogen gas determined by stimulated Raman emission,” Phys. Rev. Lett. 17, 1239–1241(1966).
[CrossRef]

Other (1)

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

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Figures (4)

Fig. 1.
Fig. 1.

Longitudinal mode overlap with the Raman laser gain profile for an (a) Nd:GdVO4 Raman laser with a 3 cm cavity length and (b) an H2 Raman ring laser with an 84 cm cavity at a pressure of 10 atm. It is clear that the narrow Raman linewidth of H2 will allow for single-mode operation, while the crystal-based laser will be multimode. δ is the two-photon detuning from the Raman resonance (νpumpνStokesνRaman).

Fig. 2.
Fig. 2.

Basic schematic for the laser system. The mirrors of the bowtie cavity are highly reflective at both the pump and Stokes frequencies. For the example explored in this work, the cavity is completely immersed in 10 atm of H2. The pump, Stokes, and SHG wavelengths are 785 nm, 1166 nm, and 583 nm, respectively, the curvature of all the mirrors is 15 cm, the round trip path length is 84 cm, and the nonlinear crystal (NLC) is taken to have the properties of MgO:PPLN with a length of 6 mm.

Fig. 3.
Fig. 3.

Temporal evolution of circulating pump and Stokes powers and output SHG power. Pump (785 nm), green dashed line; Stokes (1166 nm), red dotted line; output (583 nm), blue solid line. The arrows indicate the appropriate axis for each curve. The numerical simulations of this system predict a steady state value of 211 mW of 583 nm output power for 1 W of input 785 nm light.

Fig. 4.
Fig. 4.

Relationship between steady state Stokes power and nonlinear crystal length at 1 W on input pump power. As the strength of the SHG process is increased by increasing the crystal length in the simulation, the overall output power reaches a maximum value where the SHG and SRS processes balance. Above this point, the depletion of the Stokes beam via SHG reduces the Raman efficiency, causing the final output SHG power to drop.

Equations (11)

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(ΔνDoppler)backward=(νp+νs)8κTlog(2)mc2,
(ΔνDoppler)forward=(νpνs)8κTlog(2)mc2,
dIsdz=αIp(r,z)Is(r,z),
Ep(s)(r,z)=E0;p(s)1+i(2z/b)exp(r2wp(s)2(1+i(2z/b))),
dPsdt=16αcl(λp+λs)tan1(l4b)PpPs
GPpPs.
Lp(s)=clln(R1p(s)R2p(s)R3p(s)R4p(s)κp(s)).
dPpdt=2LpPpωpωsGPsPp+2clT1pPp,inPp,
dPsdt=2LsPs+GPpPs.
dEsdz=i2deffωs2ksc2EshgEs*,
dEshgdz=ideffωshg2kshgc2Es2.

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