Abstract

The refractive index of H2 is shown to decrease linearly as a function of Stokes power and, to a much lesser extent, pump power in a nonresonant cw Raman laser. The dominant source of the index shift is shown to be thermal and significantly larger than dispersion associated with the Raman resonance. A steady-state theoretical model based on internal heating that is due to inelastic Raman scattering events accurately describes the observed behavior. With this model, frequency pulling of the Raman cavity resonance and phase distortions of the intracavity Gaussian beam are predicted for various levels of generated Stokes power.

© 2000 Optical Society of America

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References

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  1. J. K. Brasseur, K. S. Repasky, and J. L. Carlsten, “Continuous-wave Raman laser in H2,” Opt. Lett. 23, 367–369 (1998).
    [CrossRef]
  2. P. A. Roos, J. K. Brasseur, and J. L. Carlsten, “Diode-pumped nonresonant continuous-wave Raman laser in H2 with resonant optical feedback stabilization,” Opt. Lett. 24, 1130–1132 (1999).
    [CrossRef]
  3. 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]
  4. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
    [CrossRef]
  5. K. E. Rieckhoff, “Self-induced divergence of CW laser beams in liquids—a new nonlinear effect in the propagation of light,” Appl. Phys. Lett. 9, 87–88 (1966).
    [CrossRef]
  6. R. L. Carman and P. L. Kelley, “Time dependence in the thermal blooming of laser beams,” Appl. Phys. Lett. 12, 241–243 (1968).
    [CrossRef]
  7. S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
    [CrossRef]
  8. F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
    [CrossRef]
  9. M. M. Audibert, C. Joffrin, and J. Ducuing, “Vibrational relaxation in hydrogen—rare-gas mixtures,” Chem. Phys. Lett. 1, 26–28 (1973).
    [CrossRef]
  10. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), p. 140.
  11. P. M. Morse and H. Feshback, Methods of Theoretical Physics (McGraw Hill, New York, 1953), p. 1191.
  12. W. H. Beyer, CRC Standard Mathematical Tables and Formulae, 29th ed. (CRC Press, Boca Raton, Fla., 1991), p. 373.
  13. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983).
    [CrossRef]
  14. D. E. Gray, American Institute of Physics Handbook, 2nd ed. (McGraw Hill, New York, 1963). Adjusted for pressure of 12 atm and temperature of 293 K.

1999

1998

1983

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983).
[CrossRef]

1973

M. M. Audibert, C. Joffrin, and J. Ducuing, “Vibrational relaxation in hydrogen—rare-gas mixtures,” Chem. Phys. Lett. 1, 26–28 (1973).
[CrossRef]

1969

F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
[CrossRef]

1968

R. L. Carman and P. L. Kelley, “Time dependence in the thermal blooming of laser beams,” Appl. Phys. Lett. 12, 241–243 (1968).
[CrossRef]

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

1966

K. E. Rieckhoff, “Self-induced divergence of CW laser beams in liquids—a new nonlinear effect in the propagation of light,” Appl. Phys. Lett. 9, 87–88 (1966).
[CrossRef]

1965

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

Audibert, M. M.

M. M. Audibert, C. Joffrin, and J. Ducuing, “Vibrational relaxation in hydrogen—rare-gas mixtures,” Chem. Phys. Lett. 1, 26–28 (1973).
[CrossRef]

Boyko, R. W.

F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
[CrossRef]

Brasseur, J. K.

Carlsten, J. L.

Carman, R. L.

R. L. Carman and P. L. Kelley, “Time dependence in the thermal blooming of laser beams,” Appl. Phys. Lett. 12, 241–243 (1968).
[CrossRef]

Dabby, F. W.

F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
[CrossRef]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983).
[CrossRef]

Ducuing, J.

M. M. Audibert, C. Joffrin, and J. Ducuing, “Vibrational relaxation in hydrogen—rare-gas mixtures,” Chem. Phys. Lett. 1, 26–28 (1973).
[CrossRef]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983).
[CrossRef]

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983).
[CrossRef]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983).
[CrossRef]

Joffrin, C.

M. M. Audibert, C. Joffrin, and J. Ducuing, “Vibrational relaxation in hydrogen—rare-gas mixtures,” Chem. Phys. Lett. 1, 26–28 (1973).
[CrossRef]

Kelley, P. L.

R. L. Carman and P. L. Kelley, “Time dependence in the thermal blooming of laser beams,” Appl. Phys. Lett. 12, 241–243 (1968).
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983).
[CrossRef]

Krindach, D. P.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Migulin, A. V.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983).
[CrossRef]

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Repasky, K. S.

Rieckhoff, K. E.

K. E. Rieckhoff, “Self-induced divergence of CW laser beams in liquids—a new nonlinear effect in the propagation of light,” Appl. Phys. Lett. 9, 87–88 (1966).
[CrossRef]

Roos, P. A.

Shank, C. V.

F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
[CrossRef]

Sukhorukov, A. P.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983).
[CrossRef]

Whinnery, J. R.

F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
[CrossRef]

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Appl. Phys. B: Photophys. Laser Chem.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983).
[CrossRef]

Appl. Phys. Lett.

K. E. Rieckhoff, “Self-induced divergence of CW laser beams in liquids—a new nonlinear effect in the propagation of light,” Appl. Phys. Lett. 9, 87–88 (1966).
[CrossRef]

R. L. Carman and P. L. Kelley, “Time dependence in the thermal blooming of laser beams,” Appl. Phys. Lett. 12, 241–243 (1968).
[CrossRef]

Chem. Phys. Lett.

M. M. Audibert, C. Joffrin, and J. Ducuing, “Vibrational relaxation in hydrogen—rare-gas mixtures,” Chem. Phys. Lett. 1, 26–28 (1973).
[CrossRef]

IEEE J. Quantum Electron.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[CrossRef]

F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
[CrossRef]

J. Appl. Phys.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Other

D. E. Gray, American Institute of Physics Handbook, 2nd ed. (McGraw Hill, New York, 1963). Adjusted for pressure of 12 atm and temperature of 293 K.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), p. 140.

P. M. Morse and H. Feshback, Methods of Theoretical Physics (McGraw Hill, New York, 1953), p. 1191.

W. H. Beyer, CRC Standard Mathematical Tables and Formulae, 29th ed. (CRC Press, Boca Raton, Fla., 1991), p. 373.

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Figures (7)

Fig. 1
Fig. 1

Energy-level diagram for the vibrational Raman process. Decay from the first vibrational state (c) to the ground state (a), represented by a dashed line, is nonradiative and is the source for internal thermal heating of the Raman gas.

Fig. 2
Fig. 2

Experimental setup for measuring refractive-index changes in H2. The Pound–Drever–Hall method is used to lock the laser’s frequency to a resonance of the high-finesse cavity (HFC) filled with H2 to stimulate the Raman transition. Refractive-index changes in the gas are deduced by monitoring the voltage applied to the piezoelectric transducer (PZT) as a function of pump and Stokes optical powers. EOM, electro-optic modulator; AOM, acousto-optic modulator; PBS, polarizing beam splitter; MML, mode-matching lenses; λ/2, half-wave plate; λ/4, quarter-wave plate.

Fig. 3
Fig. 3

PZT voltage change as a function of pump optical power coupled into the HFC. Note the behavioral change at the Stokes laser threshold (∼850 µW).

Fig. 4
Fig. 4

Refractive-index change as a function of Stokes optical power. Solid circles, Raman line center data (ΔStokes=0); open circles, positive half-width detuned case (ΔStokes=305 MHz). The two solid lines are linear fits to the measured data, whereas the two dashed lines are theoretical predictions based on Eq. (8) for two different radii of the constant-temperature surface. The slopes of the linear fits do not depend on detuning, indicating that the source of index change is thermal.

Fig. 5
Fig. 5

Predicted refractive-index change as a function of radial distance from the beam axis for two constant-temperature surface radii. This scale illustrates the effect of thermal heating throughout the Raman laser cavity.

Fig. 6
Fig. 6

Predicted optical phase shift in the Raman laser resonator relative to the on-axis value as a function of radial distance from the beam axis out to the beam waist for three different levels of generated Stokes power. The predictions indicate that self-defocusing may be a consideration for higher-power systems.

Fig. 7
Fig. 7

Refractive index as a function of input pump power below the Stokes laser threshold. Solid line, linear fit to the line center data (solid circles); dashed line, linear fit to the half-width detuned data (open circles). These slopes do not depend on detuning, indicating a thermal source for the index change.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

Pheat=Ptrans1+AStokesTStokesνvibνStokes,
Q(r)=B exp-2r2ω12.
ω1=ω01+8Dτvibω021/2,
B2Pheatπω12L.
 ΔT(r, Pheat)=Pheat4πLK2 lnar+Ei-2r2ω12-Ei-2a2ω12,
Δn(r, Pheat)=-(n0-1)T0ΔT(r, Pheat),
Δn(r, Ptrans)
=Ptrans4πLK1-n0T01+AStokesTStokes
×νvibνStokes2 lnar+Ei-2r2ω12-Ei-2a2ω12.
Δn(r, Ptrans)Ptrans4πLK1-n0T01+AStokesTStokesνvibνStokes×ln2γa2ω12-2r2ω12,
Δϕ(r, Ptrans)2πLλpumpΔn(r, Ptrans),
Δν(r, Ptrans)cλpumpΔn(r, Ptrans),
Δn=-ΔVmeasΔVFSRλpump2L,

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