Abstract

We present a model of a cw Raman laser that includes thermo-optic effects that are due to the heating that is inherent in Raman conversion. Thermal lensing and thermal index gratings at high output powers are addressed. With a quadratic duct model we show that broadening of the spatial modes is evident at low Stokes output powers and that accounting for thermal lensing in the laser design can significantly enhance the conversion efficiency. The model agrees with experimental results from a cw H2 Raman laser and allows for the design of high-power and solid-state cw Raman lasers.

© 2002 Optical Society of America

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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. 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]
  3. L. S. Meng, P. A. Roos, K. S. Repasky, and J. L. Carlsten, “High conversion efficiency, diode-pumped continuous-wave Raman laser,” Opt. Lett. 26, 426–428 (2001).
    [CrossRef]
  4. P. A. Roos, J. K. Brasseur, and J. L. Carlsten, “Intensity dependent refractive index in a non-resonant cw Raman laser that is due to thermal heating of the Raman-active gas,” J. Opt. Soc. Am. B 17, 758–763 (2000).
    [CrossRef]
  5. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), pp. 115–121.
  6. L. Casperson and A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
    [CrossRef]
  7. 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 in optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]
  8. K. S. Repasky, L. Meng, J. K. Brasseur, and J. L. Carlsten, “High-efficiency, continuous-wave Raman lasers,” J. Opt. Soc. Am. B 16, 717–721 (1999).
    [CrossRef]
  9. G. D. Boyd, W. D. Johnston, and I. P. Kaminow, “Optimization of the stimulated Raman scattering threshold,” IEEE J. Quantum Electron. QE-5, 203–206 (1969).
    [CrossRef]
  10. Y. S. Choi, “Asymmetry of the forward and backward Raman gain coefficient at 1.54 μm in methane,” Appl. Opt. 40, 1925–1930 (2001).
    [CrossRef]
  11. J. P. Gordon, R. C. C. Leite, R. S. Moore, P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
    [CrossRef]
  12. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, 1986), pp. 260–261.
  13. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 228–229.
  14. P. G. Zverev, T. T. Basiev, V. V. Osiko, A. M. Kulkov, V. N. Voitsekhovskii, and V. E. Yakobson, “Physical, chemical, and optical properties of barium nitrate Raman crystal,” Opt. Mater. 11, 315–334 (1999).
    [CrossRef]
  15. P. A. Roos, L. S. Meng, J. L. Carlsten, “Using an injection-locked diode laser to pump a cw Raman laser,” IEEE J. Quantum Electron. 36, 1280–1283 (2000).
    [CrossRef]
  16. 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]
  17. D. E. Gray, ed., American Institute of Physics Handbook, 2nd ed. (McGraw-Hill, New York, 1963).
  18. D. R. Lide, ed., Handbook of Chemistry and Physics, 80th ed. (CRC Press, New York, 1999), p. 6–171.
  19. M. M. Audibert, C. Joffrin, and J. Ducuing, “Vibrational relaxation of H2 in the range 500–40 K,” Chem. Phys. Lett. 25, 158–163 (1974).
    [CrossRef]
  20. Y. Suzaki and A. Tachibana, “Measurement of the micron-sized radius of Gaussian laser beam using the scanning knife-edge,” Appl. Opt. 14, 2809–2810 (1975).
    [CrossRef] [PubMed]
  21. H. J. Eichler, P. Gunter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 101–107.

2001

2000

1999

1998

1988

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]

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 in optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

1975

1974

M. M. Audibert, C. Joffrin, and J. Ducuing, “Vibrational relaxation of H2 in the range 500–40 K,” Chem. Phys. Lett. 25, 158–163 (1974).
[CrossRef]

1969

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

1968

L. Casperson and A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

1965

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

Audibert, M. M.

M. M. Audibert, C. Joffrin, and J. Ducuing, “Vibrational relaxation of H2 in the range 500–40 K,” Chem. Phys. Lett. 25, 158–163 (1974).
[CrossRef]

Basiev, T. T.

P. G. Zverev, T. T. Basiev, V. V. Osiko, A. M. Kulkov, V. N. Voitsekhovskii, and V. E. Yakobson, “Physical, chemical, and optical properties of barium nitrate Raman crystal,” Opt. Mater. 11, 315–334 (1999).
[CrossRef]

Boyd, G. D.

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

Brasseur, J. K.

Carlsten, J. L.

Casperson, L.

L. Casperson and A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Choi, Y. S.

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 in optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Ducuing, J.

M. M. Audibert, C. Joffrin, and J. Ducuing, “Vibrational relaxation of H2 in the range 500–40 K,” Chem. Phys. Lett. 25, 158–163 (1974).
[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 in optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, 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 in optical resonator,” Appl. Phys. B 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 in optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Joffrin, C.

M. M. Audibert, C. Joffrin, and J. Ducuing, “Vibrational relaxation of H2 in the range 500–40 K,” Chem. Phys. Lett. 25, 158–163 (1974).
[CrossRef]

Johnston, W. D.

G. D. Boyd, W. D. Johnston, and I. P. Kaminow, “Optimization of the stimulated Raman scattering threshold,” IEEE J. Quantum Electron. QE-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. QE-5, 203–206 (1969).
[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 in optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Kulkov, A. M.

P. G. Zverev, T. T. Basiev, V. V. Osiko, A. M. Kulkov, V. N. Voitsekhovskii, and V. E. Yakobson, “Physical, chemical, and optical properties of barium nitrate Raman crystal,” Opt. Mater. 11, 315–334 (1999).
[CrossRef]

Leite, R. C. C.

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

Meng, L.

Meng, L. S.

L. S. Meng, P. A. Roos, K. S. Repasky, and J. L. Carlsten, “High conversion efficiency, diode-pumped continuous-wave Raman laser,” Opt. Lett. 26, 426–428 (2001).
[CrossRef]

P. A. Roos, L. S. Meng, J. L. Carlsten, “Using an injection-locked diode laser to pump a cw Raman laser,” IEEE J. Quantum Electron. 36, 1280–1283 (2000).
[CrossRef]

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, 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 in optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Osiko, V. V.

P. G. Zverev, T. T. Basiev, V. V. Osiko, A. M. Kulkov, V. N. Voitsekhovskii, and V. E. Yakobson, “Physical, chemical, and optical properties of barium nitrate Raman crystal,” Opt. Mater. 11, 315–334 (1999).
[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]

Porto, P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, 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.

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.

Suzaki, Y.

Tachibana, A.

Voitsekhovskii, V. N.

P. G. Zverev, T. T. Basiev, V. V. Osiko, A. M. Kulkov, V. N. Voitsekhovskii, and V. E. Yakobson, “Physical, chemical, and optical properties of barium nitrate Raman crystal,” Opt. Mater. 11, 315–334 (1999).
[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 in optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Whinnery, J. R.

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

Yakobson, V. E.

P. G. Zverev, T. T. Basiev, V. V. Osiko, A. M. Kulkov, V. N. Voitsekhovskii, and V. E. Yakobson, “Physical, chemical, and optical properties of barium nitrate Raman crystal,” Opt. Mater. 11, 315–334 (1999).
[CrossRef]

Yariv, A.

L. Casperson and A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Zverev, P. G.

P. G. Zverev, T. T. Basiev, V. V. Osiko, A. M. Kulkov, V. N. Voitsekhovskii, and V. E. Yakobson, “Physical, chemical, and optical properties of barium nitrate Raman crystal,” Opt. Mater. 11, 315–334 (1999).
[CrossRef]

Appl. Opt.

Appl. Phys. B

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 in optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Appl. Phys. Lett.

L. Casperson and A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Chem. Phys. Lett.

M. M. Audibert, C. Joffrin, and J. Ducuing, “Vibrational relaxation of H2 in the range 500–40 K,” Chem. Phys. Lett. 25, 158–163 (1974).
[CrossRef]

IEEE J. Quantum Electron.

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

P. A. Roos, L. S. Meng, J. L. Carlsten, “Using an injection-locked diode laser to pump a cw Raman laser,” IEEE J. Quantum Electron. 36, 1280–1283 (2000).
[CrossRef]

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]

J. Appl. Phys.

J. P. Gordon, R. C. C. Leite, R. S. Moore, 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.

Opt. Mater.

P. G. Zverev, T. T. Basiev, V. V. Osiko, A. M. Kulkov, V. N. Voitsekhovskii, and V. E. Yakobson, “Physical, chemical, and optical properties of barium nitrate Raman crystal,” Opt. Mater. 11, 315–334 (1999).
[CrossRef]

Other

H. J. Eichler, P. Gunter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 101–107.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), pp. 115–121.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, 1986), pp. 260–261.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 228–229.

D. E. Gray, ed., American Institute of Physics Handbook, 2nd ed. (McGraw-Hill, New York, 1963).

D. R. Lide, ed., Handbook of Chemistry and Physics, 80th ed. (CRC Press, New York, 1999), p. 6–171.

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Figures (5)

Fig. 1
Fig. 1

The cw Raman laser cavity. The entire cavity is filled with the Raman-active material. The energy lost by a pump photon in Raman conversion heats the medium and sets up a radial temperature profile that broadens the pump and the Stokes waists, wp and wp, respectively.

Fig. 2
Fig. 2

Steady-state temperature profile ΔT(r) and the least-squares quadratic fit from 0r/wp4/3, ΔTquad(r).

Fig. 3
Fig. 3

Photon conversion efficiency versus incident pump power for several thermo-optic responses (δn/δT)/KT of 10 atm of H2 gas. Thermally loading the cavity (enlarging the incident pump beam such that optimal coupling requires some thermal lensing) is shown to enhance the efficiency of a laser with a strong thermo-optic response [4(δn/δT)/KT].

Fig. 4
Fig. 4

Pump spot-size measurements outside the cavity relative to its initial value. Data taken without Stokes generation demonstrate the accuracy of the measurement. The decrease in spot size outside the HFC when Stokes light is being generated indicates that the mode inside the HFC is expanding. Results from the theory are also shown.

Fig. 5
Fig. 5

Transmitted pump and Stokes powers versus incident pump power.

Equations (32)

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dPpdt=2τcJ(ξ)-LpPp-g(ξ)ωpωsPpPs=0,
dPsdt=2τc[g(ξ)PpPs-LsPs]=0,
ξ=l2n2/n1.
ρ(ξ)wj2w0j21-ξξ02-1/2,
ξ0i122s12-s.
a(ξ)= d2rφ0p(r)φp(r)=2ρ(ξ)1+ρ(ξ),
J(ξ)=a(ξ)(T1pPinc Pp)1/2,
dIs(r, z)dz=αIs(r, z)Ip(r, z),
Ej(r, z, t)=Re[E˜j(r, z)exp(-iωjt)],
E˜j(r, z)=2Ej cos(kjz)1+τ2exp-r2wj2(1+τ2),
ΔPs=g(ξ)PpPs,
g(ξ)8n1αλp+λsarctanlb0ρ(ξ).
Pp=Lsg(ξ),
Ps=ωsωpg(ξ)[a(ξ)(T1pPinc/Pp)1/2-Lp].
Q(r, z)=ΔkksdIs(r, z)dz,
Q(r, z)=Q0(r)[1+cos(2kpz)+cos(2ksz)+cos(2kpz)cos(2ksz)]=Q0(r)+Qosc(r, z).
1w¯2=2wp2+2ws2.
wth2w¯21+4Dgτvibw¯2.
Q0(r)=q0 exp-r2wth2,
q016απ2ΔkksPpPswp2ws21+4Dgτvibw¯2-1.
ΔT(r)=q0wth24KTlnRc2wth2-lnr2wth2-E1r2wth2,
ΔTquad(r)=q0wth24KTγ+lnRc2wth2-(0.518)r2wth2,
n(r)=n0-δnδTΔTquad(r),
n(r)=n0-δnδTq0wth24KTγ+lnRc2wth2+δnδT(0.518)q04KTr2.
ξ=ilρ(ξ)δnδT8(0.518)αPpPsπ2n0KTΔkks1w0p2w0s2×1+4Dgτvibρ(ξ)2w0p2+2w0s2-11/2.
ηph=ωpωs(T1s+T2s)PsPinc
aTL(ξ)2ρ(ξ)w0p2wINp21/21+ρ(ξ)(w0p2/wINp2),
Qosc(r, z)=Q0(r)n=14 an cos(2κnz),
Tosc(r=0, z)=4αPpPsπ2KTwp2ws2Δkksn=14anκn2cos(2κnz).
ηp=tanh24αPpPsπ2KTwp2ws2Δkksln0kpδnδT.
ρ(ξ)=-22s-1-1Reξ(ξ-s2ξ-1)sin(2ξ)+2s[cos(2ξ)+cos(ξ)]-2s2ξ-1 sin(ξ)(ξ-s2ξ-1)sin(2ξ)+2s[cos(2ξ)-cos(ξ)]+2s2ξ-1 sin(ξ)1/2-1.
wj2w0j2=1+δnδT8(0.518)αPpPsπ2n0KTξ02Δkksl2w0p2w0s21/2.

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