Abstract

We report on the low-threshold operation of a KTP optical parametric oscillator (OPO) pumped by a diode-pumped, Q-switched Nd:YLF laser. Thresholds below 0.5 mJ have been obtained for pumping at 1.047 μm, with internal conversion efficiencies approaching 50%. A threshold model is presented and compared with previous models. Reasonable agreement is observed between experiment and theory. Examination of the spectral properties of the OPO suggests a method of obtaining consistent single-axial-mode operation without insertion of intracavity elements or use of a seed laser.

© 1994 Optical Society of America

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  1. F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, "KxRb1-xTiOPO4: a new nonlinear optical material," J. Appl. Phys. 47, 4980–4985 (1976).
    [CrossRef]
  2. J. L. Nightingale, W. J. Silva, G. E. Reade, W. J. Kozlovsky, and R. L. Byer, "Second harmonic generation in MgO doped lithium niobate," in Lasers and Nonlinear Optical Materials, L. G. DeShazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 681, 20–26 (1986).
    [CrossRef]
  3. C. Chen, B. Wu, A. Jiang, and G. You, "A new type of ultraviolet SHG crystal β-BaB2O4," Sci. Sin. Ser. B 28, 235–243 (1985).
  4. C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, "New nonlinear-optical crystal: LiB3O5," J. Opt. Soc. Am. B 6, 612–621 (1989).
  5. R. L. Byer, M. M. Choy, R. L. Herbst, D. S. Chemla, and R. S. Feigelson, "Second harmonic generation and infrared mixing in AgGaSe2," Appl. Phys. Lett. 24, 65–68 (1974).
    [CrossRef]
  6. A recent review of diode-pumped solid state lasers is D. W. Hughes and J. R. M. Barr, "Lasers diode pumped solid state lasers," J. Phys. D 25, 563–586 (1992); see also T. Y. Fan and D. G. Welch, eds., special issue on diode pumped solid state lasers, IEEE J. Quantum Electron. 28, 940–1209 (1992).
    [CrossRef]
  7. S. J. Brosnan and R. L. Byer, "Optical parametric oscillator threshold and linewidth studies," IEEE J. Quantum Electron. QE-15, 415–431 (1979).
    [CrossRef]
  8. S. Guha, F.-J. Wu, and J. Falk, "The effects of focusing on parametric oscillation," IEEE J. Quantum Electron. QE-18, 907–912 (1982).
    [CrossRef]
  9. H. Kogelnik, "Coupled and conversion coefficients for optical modes," Proc. Symp. Quasi-Opt. 14, 333–347 (1964).
  10. M. Zahler and Y. Ben-Aryeh, "Loss effects in classical non-degenerate parametric amplifier," Opt. Commun. 79, 361–365 (1990).
    [CrossRef]
  11. R. Asby, "Theory of optical parametric amplification from a focused Gaussian beam," Phys. Rev. B 2, 4273–4282 (1970).
    [CrossRef]
  12. G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597–3639 (1968).
    [CrossRef]
  13. D. A. Kleinman, A. Ashkin, and G. D. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338–379 (1966).
    [CrossRef]
  14. J. C. Jacco and G. M. Laoiacono, "Nature of the infrared spectrum in band-edge region of KTiOPO4," Appl. Phys. Lett. 58, 560–561 (1991).
    [CrossRef]
  15. K. Kato, "Parametric oscillation at 3.2 µm in KTP pumped in 1.064 µm," IEEE J. Quantum Electron. 27, 1137–1140 (1991).
    [CrossRef]
  16. D. W. Anthon and C. D. Crowder, "Wavelength dependent phase matching in KTP," Appl. Opt. 27, 2650–2652 (1988).
    [CrossRef] [PubMed]
  17. T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, "Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOPO4," Appl. Opt. 26, 2390–2394 (1987).
    [CrossRef] [PubMed]
  18. V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of KTP crystal frequency converters in the 0.4-4.0 µm range," Sov. J. Quantum Electron. 18, 1059–1060 (1988).
    [CrossRef]
  19. H. Vanherzeele, J. D. Bierlein, and F. C. Zumsteg, "Index of refraction measurements and parametric generation in hydrothermally grown KTiOPO4," Appl. Opt. 27, 3314–3316 (1988).
    [CrossRef] [PubMed]
  20. D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LiIO3 and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
    [CrossRef]
  21. D. A. Kleinman, "Nonlinear dielectric polarization in optical media," Phys. Rev. 126, 1977–1979 (1962).
    [CrossRef]
  22. R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, "Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3 and KTP measured by phase-matched second-harmonic generation," IEEE J. Quantum Electron. 26, 922–933 (1990).
    [CrossRef]
  23. H. Vanherzeele and J. D. Bierlein, "Magnitude of the nonlinear-optical coefficient of KTiOPO4," Opt. Lett. 17, 982–984 (1992).
    [CrossRef] [PubMed]
  24. C. F. Rae, J. A. C. Terry, B. D. Sinclair, M. H. Dunn, and W. Sibbett, "Single-frequency, end-pumped Nd:YLF laser excited by a 12 mJ diode-laser array," Opt. Lett. 17, 1673–1675 (1992).
    [CrossRef] [PubMed]
  25. K. Kato and M. Masutani, "Widely tunable 90° phase-matched KTP parametric oscillator," Opt. Lett. 17, 178–179 (1992).
    [CrossRef]
  26. T. R. Steele, D. C. Gerstenberger, A. Drobshoff, and R. W. Wallace, "Broadly tunable high-power operation of an all solid state titanium doped sapphire laser system," Opt. Lett. 16, 399–401 (1991).
    [CrossRef] [PubMed]
  27. J. E. Bjorkholm, "Some effects of spatially nonuniform pumping in pulsed optical parametric oscillators," IEEE J. Quantum Electron. QE-7, 109–118 (1971).
    [CrossRef]
  28. G. Hilber, D. J. Brink, A. Lago, and R. Wallenstein, "Optical frequency conversion in gases using Gaussian laser beams with different confocal parameters," Phys. Rev. A 38, 6231–6239 (1988).
    [CrossRef] [PubMed]
  29. G. Robertson, A. Henderson, and M. H. Dunn, "Efficient, single axial mode operation of a beta barium borate optical parametric oscillator pumped by an excimer laser," Appl. Phys. Lett. 62, 123–125 (1993).
    [CrossRef]
  30. W. R. Bosenberg and D. R. Guyer, "Single-frequency optical parametric oscillator," Appl. Phys. Lett. 61, 387–389 (1992).
    [CrossRef]
  31. L. R. Marshall, A. Kaz, and R. L. Burnham, "Eyesafe laser with 2% electrical efficiency," in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 370–371.
  32. B. I. Denker, V. V. Osiko, S. E. Sverchkov, Yu. E. Sverchkov, A. P. Fefelov, and S. I. Khomenko, "Highly efficient erbium glass lasers with Q-switching based on frustrated total internal reflection," Sov. J. Quantum Electron. 22, 500–503 (1992).
    [CrossRef]
  33. J. A. Hutchison and T. H. Allik, "Diode array-pumped Er, Yb:phosphate glass laser," Appl. Phys. Lett. 60, 1424–1426 (1992).
    [CrossRef]

1993 (1)

G. Robertson, A. Henderson, and M. H. Dunn, "Efficient, single axial mode operation of a beta barium borate optical parametric oscillator pumped by an excimer laser," Appl. Phys. Lett. 62, 123–125 (1993).
[CrossRef]

1992 (6)

W. R. Bosenberg and D. R. Guyer, "Single-frequency optical parametric oscillator," Appl. Phys. Lett. 61, 387–389 (1992).
[CrossRef]

B. I. Denker, V. V. Osiko, S. E. Sverchkov, Yu. E. Sverchkov, A. P. Fefelov, and S. I. Khomenko, "Highly efficient erbium glass lasers with Q-switching based on frustrated total internal reflection," Sov. J. Quantum Electron. 22, 500–503 (1992).
[CrossRef]

J. A. Hutchison and T. H. Allik, "Diode array-pumped Er, Yb:phosphate glass laser," Appl. Phys. Lett. 60, 1424–1426 (1992).
[CrossRef]

H. Vanherzeele and J. D. Bierlein, "Magnitude of the nonlinear-optical coefficient of KTiOPO4," Opt. Lett. 17, 982–984 (1992).
[CrossRef] [PubMed]

C. F. Rae, J. A. C. Terry, B. D. Sinclair, M. H. Dunn, and W. Sibbett, "Single-frequency, end-pumped Nd:YLF laser excited by a 12 mJ diode-laser array," Opt. Lett. 17, 1673–1675 (1992).
[CrossRef] [PubMed]

K. Kato and M. Masutani, "Widely tunable 90° phase-matched KTP parametric oscillator," Opt. Lett. 17, 178–179 (1992).
[CrossRef]

1991 (3)

T. R. Steele, D. C. Gerstenberger, A. Drobshoff, and R. W. Wallace, "Broadly tunable high-power operation of an all solid state titanium doped sapphire laser system," Opt. Lett. 16, 399–401 (1991).
[CrossRef] [PubMed]

J. C. Jacco and G. M. Laoiacono, "Nature of the infrared spectrum in band-edge region of KTiOPO4," Appl. Phys. Lett. 58, 560–561 (1991).
[CrossRef]

K. Kato, "Parametric oscillation at 3.2 µm in KTP pumped in 1.064 µm," IEEE J. Quantum Electron. 27, 1137–1140 (1991).
[CrossRef]

1990 (2)

M. Zahler and Y. Ben-Aryeh, "Loss effects in classical non-degenerate parametric amplifier," Opt. Commun. 79, 361–365 (1990).
[CrossRef]

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, "Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3 and KTP measured by phase-matched second-harmonic generation," IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

1989 (1)

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, "New nonlinear-optical crystal: LiB3O5," J. Opt. Soc. Am. B 6, 612–621 (1989).

1988 (5)

D. W. Anthon and C. D. Crowder, "Wavelength dependent phase matching in KTP," Appl. Opt. 27, 2650–2652 (1988).
[CrossRef] [PubMed]

V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of KTP crystal frequency converters in the 0.4-4.0 µm range," Sov. J. Quantum Electron. 18, 1059–1060 (1988).
[CrossRef]

H. Vanherzeele, J. D. Bierlein, and F. C. Zumsteg, "Index of refraction measurements and parametric generation in hydrothermally grown KTiOPO4," Appl. Opt. 27, 3314–3316 (1988).
[CrossRef] [PubMed]

D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LiIO3 and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
[CrossRef]

G. Hilber, D. J. Brink, A. Lago, and R. Wallenstein, "Optical frequency conversion in gases using Gaussian laser beams with different confocal parameters," Phys. Rev. A 38, 6231–6239 (1988).
[CrossRef] [PubMed]

1987 (1)

1985 (1)

C. Chen, B. Wu, A. Jiang, and G. You, "A new type of ultraviolet SHG crystal β-BaB2O4," Sci. Sin. Ser. B 28, 235–243 (1985).

1982 (1)

S. Guha, F.-J. Wu, and J. Falk, "The effects of focusing on parametric oscillation," IEEE J. Quantum Electron. QE-18, 907–912 (1982).
[CrossRef]

1979 (1)

S. J. Brosnan and R. L. Byer, "Optical parametric oscillator threshold and linewidth studies," IEEE J. Quantum Electron. QE-15, 415–431 (1979).
[CrossRef]

1976 (1)

F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, "KxRb1-xTiOPO4: a new nonlinear optical material," J. Appl. Phys. 47, 4980–4985 (1976).
[CrossRef]

1974 (1)

R. L. Byer, M. M. Choy, R. L. Herbst, D. S. Chemla, and R. S. Feigelson, "Second harmonic generation and infrared mixing in AgGaSe2," Appl. Phys. Lett. 24, 65–68 (1974).
[CrossRef]

1971 (1)

J. E. Bjorkholm, "Some effects of spatially nonuniform pumping in pulsed optical parametric oscillators," IEEE J. Quantum Electron. QE-7, 109–118 (1971).
[CrossRef]

1970 (1)

R. Asby, "Theory of optical parametric amplification from a focused Gaussian beam," Phys. Rev. B 2, 4273–4282 (1970).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

1966 (1)

D. A. Kleinman, A. Ashkin, and G. D. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338–379 (1966).
[CrossRef]

1964 (1)

H. Kogelnik, "Coupled and conversion coefficients for optical modes," Proc. Symp. Quasi-Opt. 14, 333–347 (1964).

1962 (1)

D. A. Kleinman, "Nonlinear dielectric polarization in optical media," Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

Kogelnik, H.

H. Kogelnik, "Coupled and conversion coefficients for optical modes," Proc. Symp. Quasi-Opt. 14, 333–347 (1964).

Allik, T. H.

J. A. Hutchison and T. H. Allik, "Diode array-pumped Er, Yb:phosphate glass laser," Appl. Phys. Lett. 60, 1424–1426 (1992).
[CrossRef]

Anthon, D. W.

Asby, R.

R. Asby, "Theory of optical parametric amplification from a focused Gaussian beam," Phys. Rev. B 2, 4273–4282 (1970).
[CrossRef]

Ashkin, A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Barnes, N. P.

D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LiIO3 and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
[CrossRef]

Barr, J. R. M.

A recent review of diode-pumped solid state lasers is D. W. Hughes and J. R. M. Barr, "Lasers diode pumped solid state lasers," J. Phys. D 25, 563–586 (1992); see also T. Y. Fan and D. G. Welch, eds., special issue on diode pumped solid state lasers, IEEE J. Quantum Electron. 28, 940–1209 (1992).
[CrossRef]

Ben-Aryeh, Y.

M. Zahler and Y. Ben-Aryeh, "Loss effects in classical non-degenerate parametric amplifier," Opt. Commun. 79, 361–365 (1990).
[CrossRef]

Bierlein, J. D.

Bjorkholm, J. E.

J. E. Bjorkholm, "Some effects of spatially nonuniform pumping in pulsed optical parametric oscillators," IEEE J. Quantum Electron. QE-7, 109–118 (1971).
[CrossRef]

Bosenberg, W. R.

W. R. Bosenberg and D. R. Guyer, "Single-frequency optical parametric oscillator," Appl. Phys. Lett. 61, 387–389 (1992).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

D. A. Kleinman, A. Ashkin, and G. D. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Brink, D. J.

G. Hilber, D. J. Brink, A. Lago, and R. Wallenstein, "Optical frequency conversion in gases using Gaussian laser beams with different confocal parameters," Phys. Rev. A 38, 6231–6239 (1988).
[CrossRef] [PubMed]

Brosnan, S. J.

S. J. Brosnan and R. L. Byer, "Optical parametric oscillator threshold and linewidth studies," IEEE J. Quantum Electron. QE-15, 415–431 (1979).
[CrossRef]

Burnham, R. L.

L. R. Marshall, A. Kaz, and R. L. Burnham, "Eyesafe laser with 2% electrical efficiency," in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 370–371.

Byer, R. L.

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, "Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3 and KTP measured by phase-matched second-harmonic generation," IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, "Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOPO4," Appl. Opt. 26, 2390–2394 (1987).
[CrossRef] [PubMed]

S. J. Brosnan and R. L. Byer, "Optical parametric oscillator threshold and linewidth studies," IEEE J. Quantum Electron. QE-15, 415–431 (1979).
[CrossRef]

R. L. Byer, M. M. Choy, R. L. Herbst, D. S. Chemla, and R. S. Feigelson, "Second harmonic generation and infrared mixing in AgGaSe2," Appl. Phys. Lett. 24, 65–68 (1974).
[CrossRef]

J. L. Nightingale, W. J. Silva, G. E. Reade, W. J. Kozlovsky, and R. L. Byer, "Second harmonic generation in MgO doped lithium niobate," in Lasers and Nonlinear Optical Materials, L. G. DeShazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 681, 20–26 (1986).
[CrossRef]

Chemla, D. S.

R. L. Byer, M. M. Choy, R. L. Herbst, D. S. Chemla, and R. S. Feigelson, "Second harmonic generation and infrared mixing in AgGaSe2," Appl. Phys. Lett. 24, 65–68 (1974).
[CrossRef]

Chen, C.

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, "New nonlinear-optical crystal: LiB3O5," J. Opt. Soc. Am. B 6, 612–621 (1989).

C. Chen, B. Wu, A. Jiang, and G. You, "A new type of ultraviolet SHG crystal β-BaB2O4," Sci. Sin. Ser. B 28, 235–243 (1985).

Choy, M. M.

R. L. Byer, M. M. Choy, R. L. Herbst, D. S. Chemla, and R. S. Feigelson, "Second harmonic generation and infrared mixing in AgGaSe2," Appl. Phys. Lett. 24, 65–68 (1974).
[CrossRef]

Crowder, C. D.

Denker, B. I.

B. I. Denker, V. V. Osiko, S. E. Sverchkov, Yu. E. Sverchkov, A. P. Fefelov, and S. I. Khomenko, "Highly efficient erbium glass lasers with Q-switching based on frustrated total internal reflection," Sov. J. Quantum Electron. 22, 500–503 (1992).
[CrossRef]

Drobshoff, A.

Dunn, M. H.

G. Robertson, A. Henderson, and M. H. Dunn, "Efficient, single axial mode operation of a beta barium borate optical parametric oscillator pumped by an excimer laser," Appl. Phys. Lett. 62, 123–125 (1993).
[CrossRef]

C. F. Rae, J. A. C. Terry, B. D. Sinclair, M. H. Dunn, and W. Sibbett, "Single-frequency, end-pumped Nd:YLF laser excited by a 12 mJ diode-laser array," Opt. Lett. 17, 1673–1675 (1992).
[CrossRef] [PubMed]

Dyakov, V. A.

V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of KTP crystal frequency converters in the 0.4-4.0 µm range," Sov. J. Quantum Electron. 18, 1059–1060 (1988).
[CrossRef]

Eckardt, R. C.

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, "Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3 and KTP measured by phase-matched second-harmonic generation," IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, "Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOPO4," Appl. Opt. 26, 2390–2394 (1987).
[CrossRef] [PubMed]

Falk, J.

S. Guha, F.-J. Wu, and J. Falk, "The effects of focusing on parametric oscillation," IEEE J. Quantum Electron. QE-18, 907–912 (1982).
[CrossRef]

Fan, T. Y.

Fan, Y. X.

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, "Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3 and KTP measured by phase-matched second-harmonic generation," IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, "Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOPO4," Appl. Opt. 26, 2390–2394 (1987).
[CrossRef] [PubMed]

Fefelov, A. P.

B. I. Denker, V. V. Osiko, S. E. Sverchkov, Yu. E. Sverchkov, A. P. Fefelov, and S. I. Khomenko, "Highly efficient erbium glass lasers with Q-switching based on frustrated total internal reflection," Sov. J. Quantum Electron. 22, 500–503 (1992).
[CrossRef]

Feigelson, R. S.

T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, "Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOPO4," Appl. Opt. 26, 2390–2394 (1987).
[CrossRef] [PubMed]

R. L. Byer, M. M. Choy, R. L. Herbst, D. S. Chemla, and R. S. Feigelson, "Second harmonic generation and infrared mixing in AgGaSe2," Appl. Phys. Lett. 24, 65–68 (1974).
[CrossRef]

Gerstenberger, D. C.

Gettemy, D. J.

D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LiIO3 and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
[CrossRef]

Gier, T. E.

F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, "KxRb1-xTiOPO4: a new nonlinear optical material," J. Appl. Phys. 47, 4980–4985 (1976).
[CrossRef]

Guha, S.

S. Guha, F.-J. Wu, and J. Falk, "The effects of focusing on parametric oscillation," IEEE J. Quantum Electron. QE-18, 907–912 (1982).
[CrossRef]

Guyer, D. R.

W. R. Bosenberg and D. R. Guyer, "Single-frequency optical parametric oscillator," Appl. Phys. Lett. 61, 387–389 (1992).
[CrossRef]

Harker, W. C.

D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LiIO3 and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
[CrossRef]

Henderson, A.

G. Robertson, A. Henderson, and M. H. Dunn, "Efficient, single axial mode operation of a beta barium borate optical parametric oscillator pumped by an excimer laser," Appl. Phys. Lett. 62, 123–125 (1993).
[CrossRef]

Herbst, R. L.

R. L. Byer, M. M. Choy, R. L. Herbst, D. S. Chemla, and R. S. Feigelson, "Second harmonic generation and infrared mixing in AgGaSe2," Appl. Phys. Lett. 24, 65–68 (1974).
[CrossRef]

Hilber, G.

G. Hilber, D. J. Brink, A. Lago, and R. Wallenstein, "Optical frequency conversion in gases using Gaussian laser beams with different confocal parameters," Phys. Rev. A 38, 6231–6239 (1988).
[CrossRef] [PubMed]

Hu, B. Q.

Huang, C. E.

Hughes, D. W.

A recent review of diode-pumped solid state lasers is D. W. Hughes and J. R. M. Barr, "Lasers diode pumped solid state lasers," J. Phys. D 25, 563–586 (1992); see also T. Y. Fan and D. G. Welch, eds., special issue on diode pumped solid state lasers, IEEE J. Quantum Electron. 28, 940–1209 (1992).
[CrossRef]

Hutchison, J. A.

J. A. Hutchison and T. H. Allik, "Diode array-pumped Er, Yb:phosphate glass laser," Appl. Phys. Lett. 60, 1424–1426 (1992).
[CrossRef]

Jacco, J. C.

J. C. Jacco and G. M. Laoiacono, "Nature of the infrared spectrum in band-edge region of KTiOPO4," Appl. Phys. Lett. 58, 560–561 (1991).
[CrossRef]

Jiang, A.

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, "New nonlinear-optical crystal: LiB3O5," J. Opt. Soc. Am. B 6, 612–621 (1989).

C. Chen, B. Wu, A. Jiang, and G. You, "A new type of ultraviolet SHG crystal β-BaB2O4," Sci. Sin. Ser. B 28, 235–243 (1985).

Kato, K.

K. Kato and M. Masutani, "Widely tunable 90° phase-matched KTP parametric oscillator," Opt. Lett. 17, 178–179 (1992).
[CrossRef]

K. Kato, "Parametric oscillation at 3.2 µm in KTP pumped in 1.064 µm," IEEE J. Quantum Electron. 27, 1137–1140 (1991).
[CrossRef]

Kaz, A.

L. R. Marshall, A. Kaz, and R. L. Burnham, "Eyesafe laser with 2% electrical efficiency," in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 370–371.

Kholodnykh, A. I.

V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of KTP crystal frequency converters in the 0.4-4.0 µm range," Sov. J. Quantum Electron. 18, 1059–1060 (1988).
[CrossRef]

Khomenko, S. I.

B. I. Denker, V. V. Osiko, S. E. Sverchkov, Yu. E. Sverchkov, A. P. Fefelov, and S. I. Khomenko, "Highly efficient erbium glass lasers with Q-switching based on frustrated total internal reflection," Sov. J. Quantum Electron. 22, 500–503 (1992).
[CrossRef]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

D. A. Kleinman, A. Ashkin, and G. D. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338–379 (1966).
[CrossRef]

D. A. Kleinman, "Nonlinear dielectric polarization in optical media," Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

Kozlovsky, W. J.

J. L. Nightingale, W. J. Silva, G. E. Reade, W. J. Kozlovsky, and R. L. Byer, "Second harmonic generation in MgO doped lithium niobate," in Lasers and Nonlinear Optical Materials, L. G. DeShazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 681, 20–26 (1986).
[CrossRef]

Krasnikov, V. V.

V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of KTP crystal frequency converters in the 0.4-4.0 µm range," Sov. J. Quantum Electron. 18, 1059–1060 (1988).
[CrossRef]

Lago, A.

G. Hilber, D. J. Brink, A. Lago, and R. Wallenstein, "Optical frequency conversion in gases using Gaussian laser beams with different confocal parameters," Phys. Rev. A 38, 6231–6239 (1988).
[CrossRef] [PubMed]

Laoiacono, G. M.

J. C. Jacco and G. M. Laoiacono, "Nature of the infrared spectrum in band-edge region of KTiOPO4," Appl. Phys. Lett. 58, 560–561 (1991).
[CrossRef]

Li, R.

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, "New nonlinear-optical crystal: LiB3O5," J. Opt. Soc. Am. B 6, 612–621 (1989).

Lin, S.

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, "New nonlinear-optical crystal: LiB3O5," J. Opt. Soc. Am. B 6, 612–621 (1989).

Lindholm, G.

D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LiIO3 and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
[CrossRef]

Marshall, L. R.

L. R. Marshall, A. Kaz, and R. L. Burnham, "Eyesafe laser with 2% electrical efficiency," in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 370–371.

Masuda, H.

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, "Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3 and KTP measured by phase-matched second-harmonic generation," IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

Masutani, M.

Nightingale, J. L.

J. L. Nightingale, W. J. Silva, G. E. Reade, W. J. Kozlovsky, and R. L. Byer, "Second harmonic generation in MgO doped lithium niobate," in Lasers and Nonlinear Optical Materials, L. G. DeShazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 681, 20–26 (1986).
[CrossRef]

Osiko, V. V.

B. I. Denker, V. V. Osiko, S. E. Sverchkov, Yu. E. Sverchkov, A. P. Fefelov, and S. I. Khomenko, "Highly efficient erbium glass lasers with Q-switching based on frustrated total internal reflection," Sov. J. Quantum Electron. 22, 500–503 (1992).
[CrossRef]

Pryalkin, V. I.

V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of KTP crystal frequency converters in the 0.4-4.0 µm range," Sov. J. Quantum Electron. 18, 1059–1060 (1988).
[CrossRef]

Pshenichnikov, M. S.

V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of KTP crystal frequency converters in the 0.4-4.0 µm range," Sov. J. Quantum Electron. 18, 1059–1060 (1988).
[CrossRef]

Rae, C. F.

Razumikhina, T. B.

V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of KTP crystal frequency converters in the 0.4-4.0 µm range," Sov. J. Quantum Electron. 18, 1059–1060 (1988).
[CrossRef]

Reade, G. E.

J. L. Nightingale, W. J. Silva, G. E. Reade, W. J. Kozlovsky, and R. L. Byer, "Second harmonic generation in MgO doped lithium niobate," in Lasers and Nonlinear Optical Materials, L. G. DeShazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 681, 20–26 (1986).
[CrossRef]

Robertson, G.

G. Robertson, A. Henderson, and M. H. Dunn, "Efficient, single axial mode operation of a beta barium borate optical parametric oscillator pumped by an excimer laser," Appl. Phys. Lett. 62, 123–125 (1993).
[CrossRef]

Sibbett, W.

Silva, W. J.

J. L. Nightingale, W. J. Silva, G. E. Reade, W. J. Kozlovsky, and R. L. Byer, "Second harmonic generation in MgO doped lithium niobate," in Lasers and Nonlinear Optical Materials, L. G. DeShazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 681, 20–26 (1986).
[CrossRef]

Sinclair, B. D.

Solomatin, V. S.

V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of KTP crystal frequency converters in the 0.4-4.0 µm range," Sov. J. Quantum Electron. 18, 1059–1060 (1988).
[CrossRef]

Steele, T. R.

Sverchkov, S. E.

B. I. Denker, V. V. Osiko, S. E. Sverchkov, Yu. E. Sverchkov, A. P. Fefelov, and S. I. Khomenko, "Highly efficient erbium glass lasers with Q-switching based on frustrated total internal reflection," Sov. J. Quantum Electron. 22, 500–503 (1992).
[CrossRef]

Sverchkov, Yu. E.

B. I. Denker, V. V. Osiko, S. E. Sverchkov, Yu. E. Sverchkov, A. P. Fefelov, and S. I. Khomenko, "Highly efficient erbium glass lasers with Q-switching based on frustrated total internal reflection," Sov. J. Quantum Electron. 22, 500–503 (1992).
[CrossRef]

Terry, J. A. C.

Vanherzeele, H.

Wallace, R. W.

Wallenstein, R.

G. Hilber, D. J. Brink, A. Lago, and R. Wallenstein, "Optical frequency conversion in gases using Gaussian laser beams with different confocal parameters," Phys. Rev. A 38, 6231–6239 (1988).
[CrossRef] [PubMed]

Wu, B.

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, "New nonlinear-optical crystal: LiB3O5," J. Opt. Soc. Am. B 6, 612–621 (1989).

C. Chen, B. Wu, A. Jiang, and G. You, "A new type of ultraviolet SHG crystal β-BaB2O4," Sci. Sin. Ser. B 28, 235–243 (1985).

Wu, F.-J.

S. Guha, F.-J. Wu, and J. Falk, "The effects of focusing on parametric oscillation," IEEE J. Quantum Electron. QE-18, 907–912 (1982).
[CrossRef]

Wu, Y.

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, "New nonlinear-optical crystal: LiB3O5," J. Opt. Soc. Am. B 6, 612–621 (1989).

You, G.

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, "New nonlinear-optical crystal: LiB3O5," J. Opt. Soc. Am. B 6, 612–621 (1989).

C. Chen, B. Wu, A. Jiang, and G. You, "A new type of ultraviolet SHG crystal β-BaB2O4," Sci. Sin. Ser. B 28, 235–243 (1985).

Zahler, M.

M. Zahler and Y. Ben-Aryeh, "Loss effects in classical non-degenerate parametric amplifier," Opt. Commun. 79, 361–365 (1990).
[CrossRef]

Zumsteg, F. C.

Appl. Opt. (3)

Appl. Phys. Lett. (5)

G. Robertson, A. Henderson, and M. H. Dunn, "Efficient, single axial mode operation of a beta barium borate optical parametric oscillator pumped by an excimer laser," Appl. Phys. Lett. 62, 123–125 (1993).
[CrossRef]

W. R. Bosenberg and D. R. Guyer, "Single-frequency optical parametric oscillator," Appl. Phys. Lett. 61, 387–389 (1992).
[CrossRef]

J. A. Hutchison and T. H. Allik, "Diode array-pumped Er, Yb:phosphate glass laser," Appl. Phys. Lett. 60, 1424–1426 (1992).
[CrossRef]

J. C. Jacco and G. M. Laoiacono, "Nature of the infrared spectrum in band-edge region of KTiOPO4," Appl. Phys. Lett. 58, 560–561 (1991).
[CrossRef]

R. L. Byer, M. M. Choy, R. L. Herbst, D. S. Chemla, and R. S. Feigelson, "Second harmonic generation and infrared mixing in AgGaSe2," Appl. Phys. Lett. 24, 65–68 (1974).
[CrossRef]

IEEE J. Quantum Electron. (6)

S. J. Brosnan and R. L. Byer, "Optical parametric oscillator threshold and linewidth studies," IEEE J. Quantum Electron. QE-15, 415–431 (1979).
[CrossRef]

S. Guha, F.-J. Wu, and J. Falk, "The effects of focusing on parametric oscillation," IEEE J. Quantum Electron. QE-18, 907–912 (1982).
[CrossRef]

K. Kato, "Parametric oscillation at 3.2 µm in KTP pumped in 1.064 µm," IEEE J. Quantum Electron. 27, 1137–1140 (1991).
[CrossRef]

D. J. Gettemy, W. C. Harker, G. Lindholm, and N. P. Barnes, "Some optical properties of KTP, LiIO3 and LiNbO3," IEEE J. Quantum Electron. 24, 2231–2237 (1988).
[CrossRef]

J. E. Bjorkholm, "Some effects of spatially nonuniform pumping in pulsed optical parametric oscillators," IEEE J. Quantum Electron. QE-7, 109–118 (1971).
[CrossRef]

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, "Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3 and KTP measured by phase-matched second-harmonic generation," IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

J. Appl. Phys. (2)

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, "KxRb1-xTiOPO4: a new nonlinear optical material," J. Appl. Phys. 47, 4980–4985 (1976).
[CrossRef]

J. Opt. Soc. Am. B (1)

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, "New nonlinear-optical crystal: LiB3O5," J. Opt. Soc. Am. B 6, 612–621 (1989).

Opt. Commun. (1)

M. Zahler and Y. Ben-Aryeh, "Loss effects in classical non-degenerate parametric amplifier," Opt. Commun. 79, 361–365 (1990).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. (2)

D. A. Kleinman, "Nonlinear dielectric polarization in optical media," Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

D. A. Kleinman, A. Ashkin, and G. D. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Phys. Rev. A (1)

G. Hilber, D. J. Brink, A. Lago, and R. Wallenstein, "Optical frequency conversion in gases using Gaussian laser beams with different confocal parameters," Phys. Rev. A 38, 6231–6239 (1988).
[CrossRef] [PubMed]

Phys. Rev. B (1)

R. Asby, "Theory of optical parametric amplification from a focused Gaussian beam," Phys. Rev. B 2, 4273–4282 (1970).
[CrossRef]

Proc. Symp. Quasi-Opt. (1)

H. Kogelnik, "Coupled and conversion coefficients for optical modes," Proc. Symp. Quasi-Opt. 14, 333–347 (1964).

Sci. Sin. Ser. B (1)

C. Chen, B. Wu, A. Jiang, and G. You, "A new type of ultraviolet SHG crystal β-BaB2O4," Sci. Sin. Ser. B 28, 235–243 (1985).

Sov. J. Quantum Electron. (2)

V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of KTP crystal frequency converters in the 0.4-4.0 µm range," Sov. J. Quantum Electron. 18, 1059–1060 (1988).
[CrossRef]

B. I. Denker, V. V. Osiko, S. E. Sverchkov, Yu. E. Sverchkov, A. P. Fefelov, and S. I. Khomenko, "Highly efficient erbium glass lasers with Q-switching based on frustrated total internal reflection," Sov. J. Quantum Electron. 22, 500–503 (1992).
[CrossRef]

Other (3)

L. R. Marshall, A. Kaz, and R. L. Burnham, "Eyesafe laser with 2% electrical efficiency," in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 370–371.

J. L. Nightingale, W. J. Silva, G. E. Reade, W. J. Kozlovsky, and R. L. Byer, "Second harmonic generation in MgO doped lithium niobate," in Lasers and Nonlinear Optical Materials, L. G. DeShazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 681, 20–26 (1986).
[CrossRef]

A recent review of diode-pumped solid state lasers is D. W. Hughes and J. R. M. Barr, "Lasers diode pumped solid state lasers," J. Phys. D 25, 563–586 (1992); see also T. Y. Fan and D. G. Welch, eds., special issue on diode pumped solid state lasers, IEEE J. Quantum Electron. 28, 940–1209 (1992).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Schematic of experimental setup. The λ/2 plate and the first polarizer act as a variable attenuator; the combination of the two polarizers and the Faraday rotator provides isolation; the f = 30 cm lens acts as a mode-matching lens and is AR coated at 1.06 μm; M1 and M2 are the input and the output mirrors of the OPO, highly transmitting at 1 μm, with varied signal reflections (see text) and highly transmitting at the idler wavelength, with RofC 100 or 500 mm for both faces, 5 mm thick; the Si filters block the pump. Ms’s, signal mirrors, 98% reflecting at 1.54 μm, high transmission at pump and idler wavelengths; SM’s, aluminized steering mirrors. The Si CCD camera monitors the pump spectrum in conjunction with the F–P (Fabry–Perot) étalon; the IR vidicon monitors the signal spectral content; the signal and the pump are measured with pyroelectric energy meter PEM. FSR, free spectral range.

Fig. 2
Fig. 2

(a) Signal energy versus pump energy for various pump spot sizes, mirror radii of curvature, and 18–20% output coupling. (b) Signal energy versus pump energy for various output couplings. The pump waist is elliptical, 160–180 μm, and the mirror Rof C’s are 100 mm.

Fig. 3
Fig. 3

Comparison of measured pump depletions with the curves calculated from Eq. (17) and either the cold-cavity (upper curves) or the gain-guided (lower curves) spot sizes: (a) 210-μm pump spot size, 500-mm RofC mirrors; (b) 230-μm pump spot size, 100-mm RofC mirrors; (c) 160–180-μm pump spot size, 500 mm RofC mirrors; (d) 160–180 μm pump spot size, 100-mm RofC mirrors.

Fig. 4
Fig. 4

Measured and calculated threshold energies for the 100- and 500-mm mirrors and 20% output coupling. In (a)–(c) E_BB is the value calculated with the BB theory, and E_G_cc and E_G_scf are the values obtained with the Guha model with the cold-cavity and self-consistent-field spot sizes, respectively. In (a)–(c) the two curves are calculated from the Guha model, for a self-consistent-field spot, with a parasitic loss of either 0% or 4% per crystal face. Note that the parasitic loss (per crystal face) varies from point to point as indicated. (b) Measured and calculated threshold energies for 500-mm mirrors and 18% (pump waist, 160–180 mm) or 20% (pump waist, 210 mm) output coupling. (c) Measured and calculated threshold energies as a function of output coupling for the 100-mm mirrors and a 160–180-μm pump spot.

Fig. 5
Fig. 5

(a) Signal spectrum observed from the OPO, showing signal clusters and individual axial modes. The wavelength scale is horizontal. (b) Almost single-mode operation observed when the integration time of the vidicon is overcome by pulsing at 1 pulse/s.

Fig. 6
Fig. 6

Position of the central resonance moving from slightly right of center, to center, to left of center as the mirror substrate and the mount are heated and then allowed to cool.

Tables (3)

Tables Icon

Table 1 Comparison of Published Sellmeier Equations: Predictions of Signal and Idler Wavelengths

Tables Icon

Table 2 Summary of Measured Thresholds, Slope Efficiencies, and Maximum Conversion Efficiencies

Tables Icon

Table 3 Comparison of Measured and Calculated Slope Efficiencies

Equations (22)

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G = exp ( - 2 α l c ) cosh 2 ( Γ L ) ,
G = exp ( - α ¯ z ) 2 { [ 1 + ( Δ α / 2 ) 2 + ( Δ k ) 2 2 b 2 ] cosh ( β z ) + [ 1 - ( Δ α / 2 ) + ( Δ k ) 2 2 b 2 ] cos ( δ z ) + 2 2 b 2 × [ ( β Δ α / 2 + Δ Δ k ) sinh ( β z ) + ( δ Δ α / 2 - β Δ k ) sin ( δ z ) ] } ,
2 b 2 = { [ ( 2 γ ) 2 + ( Δ α / 2 ) 2 - ( Δ k ) 2 ] 2 + ( Δ k ) 2 ( Δ α ) 2 } 1 / 2 , β = { [ ( 2 γ ) 2 + ( Δ α / 2 ) 2 - ( Δ k ) 2 ] 2 + ( Δ k ) 2 ( Δ α ) 2 } 1 / 4 × cos { ½ arg [ ( 2 γ ) 2 + ( Δ α / 2 ) 2 - ( Δ k ) 2 + i Δ k Δ α ] } , δ = { [ ( 2 γ ) 2 + ( Δ α / 2 ) 2 - ( Δ k ) 2 ] 2 + ( Δ k ) 2 ( Δ α ) 2 } 1 / 4 × sin { ½ arg [ ( 2 γ ) 2 + ( Δ α / 2 ) 2 - ( Δ k ) 2 + i Δ k Δ α ] } , α ¯ = ½ ( α s + α i ) ,             Δ α = α i - α s ,             γ = κ g s I p , κ = 2 ω s ω i d eff 2 n s n i n p 0 c 3 ,             g s = w p ( w p 2 + 2 s s 2 ) 1 / 2 w s 2 + w p 2 , I p = 1 2 n p c 0 E p 2 ,
Δ P s = - 1 2 ω s Im E s * P s d x d y d z .
E p , s = E 0 , p , s 1 + i τ p , s exp ( i k p , s z ) exp [ - ( x 2 + y 2 ) w 0 p , s 2 ( 1 + i τ p , s ) ] ,
E 0 p , s 2 = 16 P p , s n p , s c w 0 p , s 2 ,             τ p = 2 z - f p l b p , τ s = 2 z - l / 2 b s ,             b p , s = w 0 p , s 2 k p , s .
P s , i = 2 d eff E p E i , s * .
E i ( r ) = 2 π i ω i c P i ( r ) G e ( r , r ) d r ,
G e ( r , r ) = ω i 2 π i c exp [ - α i 2 ( z - z ) ] exp [ i k i ( z - z ) ] z - z × exp { i k i [ x - ρ i ( l - z ) - x ] 2 + ( y - y ) 2 2 ( z - z ) }
Δ P s = K 3 P s P p h s ,
K 3 = 128 ω s ω i ( 2 d eff ) 2 π 2 l n s n i n p c 3 k s k p k s + k p , h s = 1 4 ( 1 + k ) ξ p + k ξ s Re ( I 1 ) , I 1 = 0 1 0 z 2 d z 1 d z 2 × exp [ ( - α i / 2 + i Δ k l ) ( z 1 - z 2 ) ] exp ( ρ i 2 f w i ) γ , γ = - i 4 ( z 1 - z 2 ) ( 1 ξ 5 + 1 ξ 6 ) + z 1 z 2 + 1 4 ξ 5 ξ 6 - 1 4 ξ p ξ s ξ 6 [ ( z 1 - z 2 ) g 1 i - g 2 + ( z 1 + z 2 ) g 3 ] , g 1 = k 2 - k ( 2 f p + 1 ) + 2 f p , g 2 = 2 f p ξ 5 + ( ξ s - 2 f p ξ p ξ p ξ s ) k , g 3 = 1 + 2 f p ξ 5 + k ( ξ s - ξ p ) ξ s ξ p - ξ s - 2 f p ξ p ξ s ξ p , k = k - 2 f p k - 1 ,             f w i = ( z - z ) 2 [ ( a 3 * ) 2 i b 1 + a 3 * ] ,
P s ( l ) = P s ( 0 ) + Δ P s = G P s ,
G = 1 + γ ,             γ = Δ P s P s = K 3 P p h s .
K 3 = ( 2.383 × 10 - 3 ) d eff 2 l λ s λ i n i ( n s λ p + n p λ s ) ,
N m + 1 = [ R 1 R 2 ( 1 - R c ) 4 exp ( - 2 α s l ) ] G ( t ) N m ,
N e = N m + 1 { 1 + R 2 [ ( 1 - R 1 ) ( 1 - R c ) 2 exp ( - α s l ) - 1 ] R 1 R 2 ( 1 - R c ) 3 exp ( - α s l ) } ,
P p = P p 0 exp - ( 2 t / τ p ) 2 ,
E s = η ( E p - E pt ) .
η = g s T p ( T T + L ) ( ω s ω p ) ,
E s = ( T T + L ) ( ω s ω p ) δ E p .
δ E p E p = g s ( 1 - 1 N ) ,
Δ λ s = λ 2 l [ ( n s - n i ) + λ i n i λ i + λ s n s λ s ] - 1 ,

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