Abstract

We report the measurement of the absolute d36 nonlinear coefficient of silver gallium sulfide (AgGaS2) from three phase-matched nonlinear interactions spanning from 10.2 to 0.78 µm: the sum-frequency of a CO2 laser (10.2 µm) and a near-IR AlGaAs diode laser (0.842 µm), the second-harmonic generation of a KCl:Li color-center laser (2.53 µm), and the noncritically phase-matched 3ω-ω2ω difference frequency between the AlGaAs diode laser and the KCl:Li laser to generate 1.265 µm. From the theoretical evaluation of the Gaussian-beam aperture functions for these type I interactions with arbitrary focusing parameters, beam-waist locations, and absorption losses, we have deduced from the three processes the same consistent value d36=13(±2) pm/V for AgGaS2. Our value is independent of the growth origin of the material. In light of the trustworthy value of d36 measured from our experiments we analyze the feasibility of a continuous-wave doubly resonant parametric oscillator-(DRO-)based 3:1 frequency divider pumped by a near-IR diode laser. The predicted pump power threshold for the ring-resonator DRO lies in the range of 70–140 mW for a 14–18-mm-long sample with an absorption loss near the level of 1% cm-1 at the output frequencies.

© 1997 Optical Society of America

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  1. D. Touahri, O. Acef, A. Clairon, J.-J. Zondy, R. Felder, L. Hilico, B. de Beauvoir, F. Biraben, and F. Nez, “Frequency measurement of the 5S1/2(F=3)–5D5/2(F=5) two-photon transition in rubidium,” Opt. Commun. 133, 471 (1997).
    [CrossRef]
  2. J.-J. Zondy, D. Touahri, and O. Acef, “Infra-red tovisible non-linear up and down conversion processes using AgGaS2 crystals,” in Advanced Solid State Lasers, S. A. Payne and C. Pollock, eds., Vol. 1 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 164–167.
  3. J.-J. Zondy and D. Touahri, “Updated thermo-optic coefficients of AgGaS2 from temperature-tuned noncritical 3ω− ω→2ω infrared parametricamplification,” J. Opt. Soc. Am. B 14, 1331 (1997).
    [CrossRef]
  4. N. C. Wong, “Optical-to-microwave frequency chain utilizing a two-laser-based opticalparametric oscillator network,” Appl. Phys. B 61, 143 (1995).
    [CrossRef]
  5. P. Canarelli, Z. Benko, A. H. Hielscher, R. F. Curl, and F. K. Tittel, “Measurement of nonlinear coefficient and phase-matching characteristicsof AgGaS2,” IEEE J. Quantum Electron. 28, 52 (1992).
    [CrossRef]
  6. D. S. Chemla, P. J. Kupecek, D. S. Robertson, and R. C. Smith, “Silver thiogallate, a new material with potential for infrared devices,” Opt. Commun. 3, 29 (1971).
    [CrossRef]
  7. P. J. Kupecek, C. A. Schwartz, and D. S. Chemla, “Silver thiogallate (AgGaS2). Part I: nonlinear optical properties,” IEEE J. Quantum Electron. QE-10, 540 (1974).
    [CrossRef]
  8. H. Vanherzeele and J. D. Bierlein, “Magnitude of the nonlinear coefficients of KTiOPO4,” Opt. Lett. 17, 982 (1992).
    [CrossRef] [PubMed]
  9. B. Boulanger, J. Ph. Fève, G. Marnier, B. Ménaert, and X. Cabirol, “Relative sign and absolute magnitude of the χ(2) coefficients of KTP by SHG measurements,” J. Opt. Soc. Am. B 11, 750 (1994).
    [CrossRef]
  10. J.-J. Zondy, M. Abed, and A. Clairon, “Type-II frequency doubling at λ=1.30 μm and λ=2.53 μmin flux grown potassium titanyl phosphate,” J. Opt. Soc. Am. B 11, 2004 (1994).
    [CrossRef]
  11. R. S. Feigelson and R. K. Route, “Recent developments in the growth of chalcopyrite crystals for nonlinearinfrared applications,” Opt. Eng. 26, 113 (1987).
    [CrossRef]
  12. D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beam,” Phys. Rev. 145, 338 (1966).
    [CrossRef]
  13. S. Guha and J. Falk, “The effect of focusing in the three-frequency parametric upconverter,” J. Appl. Phys. 51, 50 (1980).
    [CrossRef]
  14. G. Lera and M. Nieto-Vesperinas, “Non-paraxial method for non-linear optical problems: far-infrared generationwith depletion and diffraction,” J. Opt. 20, 169 (1989).
    [CrossRef]
  15. 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 harmonicgeneration,” IEEE J. Quantum Electron. 26, 922 (1990).
    [CrossRef]
  16. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
    [CrossRef]
  17. J.-J. Zondy, “Comparative theory of walkoff-limited type-II versus type-I secondharmonic generation with Gaussian beams,” Opt. Commun. 81, 427 (1991); erratum, in Eq. (3.1.a), exp(−x2) should read exp(−x2/2).
    [CrossRef]
  18. J.-P. Fève, B. Boulanger, and G. Marnier, “Experimental study of walk-off attenuation for type-II second-harmonicgeneration in KTP,” IEEE J. Quantum Electron. 31, 1569 (1995).
    [CrossRef]
  19. B. Boulanger, J.-P. Fève, G. Marnier, C. Bonnin, P. Villeval, and J.-J. Zondy, “Absolute measurement of quadratic nonlinearities from phase-matchedsecond-harmonic generation in a single KTP crystal cut in a sphere,” J. Opt. Soc. Am. B 14, 1380 (1997).
    [CrossRef]
  20. D. Touahri, O. Acef, and J.-J. Zondy, “30-THz upconversion of an AlGaAs diode laser with AgGaS2: bridging the several-terahertzfrequency gap in the near infrared,” Opt. Lett. 21, 213 (1996).
    [CrossRef] [PubMed]
  21. T.-B. Chu and M. Broyer, “Intracavity cw difference frequency generation by mixing three photonsand using Gaussian laser beams,” J. Phys. (France) 46, 523 (1985).
    [CrossRef]
  22. P. Canarelli, Z. Benko, R. Curl, and F. K. Tittel, “Continuous-wave infrared laser spectrometer based on difference frequencygeneration in AgGaS2for high resolution spectroscopy,” J. Opt. Soc. Am. B 9, 197 (1992).
    [CrossRef]
  23. J. E. Bjorkholm, “Analysis of the doubly resonant optical parametric oscillator withoutpower-dependent reflections,” IEEE J. Quantum Electron. QE-5, 293 (1969).
    [CrossRef]
  24. S. Guha, F. Wu, and J. Falk, “The effect of focusing on parametric oscillation,” IEEE J. Quantum Electron. QE-18, 907 (1982).
    [CrossRef]
  25. The SHG sample was grown at the Institute of Mineralogy andPetrography, Russian Academy of Sciences, Siberian Branch, Novosibirsk (Russia)and manufactured by EKSMA Co., Vilnius (Lithuania).
  26. J.-J. Zondy, M. Abed, and S. Khodja, “Twin-crystal walk-off-compensated type-II second-harmonic generation:single-pass and cavity-enhanced experiments in KTiOPO4,” J. Opt. Soc. Am. B 11, 2368 (1994).
    [CrossRef]
  27. SFG and DFG samples were grown and manufactured by ClevelandCrystals, Inc., Ohio.
  28. U. Simon, C. E. Miller, C. C. Bradley, R. G. Hulet, R. F. Curl, and F. K. Tittel, “Difference-frequency generation in AgGaS2 by use of single-mode diode-laser pump sources,” Opt. Lett. 18, 1062 (1993).
    [CrossRef]
  29. U. Simon, S. Waltman, I. Loa, F. K. Tittel, and L. Hollberg, “External-cavity difference frequency source near 3.3 μm, basedon combining a tunable diode laser with a diode-pumped Nd:YAG laser in AgGaS2,” J. Opt. Soc. Am. B 12, 323 (1995).
    [CrossRef]
  30. E.-J. Canto-Said, M. P. McCann, P. G. Wigley, and G. J. Dixon, “Broadly tunable mid-infrared intracavity difference-frequency laser,” Opt. Lett. 20, 1268 (1995).
    [CrossRef] [PubMed]
  31. H.-D. Kronfeldt, G. Basar, and B. Sumpf, “Application of a cw tunable infrared spectrometer based on difference-frequencygeneration in AgGaS2for self-broadening investigations of NO at μm,” J. Opt. Soc. Am. B 13, 1859 (1996).
    [CrossRef]
  32. R. C. Miller, “Optical second-harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17 (1964).
    [CrossRef]
  33. K. Kato, “High-power difference frequency generation at 5–11 μm in AgGaS2,” IEEE J. Quantum Electron. QE-20, 698 (1984).
    [CrossRef]
  34. A. G. Yodh, H. W. K. Tom, G. D. Aumillier, and R. S. Miranda, “Generation of tunable mid-infrared picosecond pulses pulses pulsesat 76 MHz,” J. Opt. Soc. Am. B 8, 1663 (1991).
    [CrossRef]
  35. G. D. Boyd, H. Kasper, and J. H. McFee, “Linear and nonlinear optical properties of AgGaS2, CuGaS2and CuInS2 andtheory of the wedge technique for the measurement of nonlinear coefficients,” IEEE J. Quantum Electron. QE-7, 563 (1971).
    [CrossRef]
  36. E. C. Cheung, K. Koch, and G. T. Moore, “Measurement of second-order nonlinear optical coefficients from thespectral brightness of parametric fluorescence,” Opt. Lett. 19, 168 (1994).
    [CrossRef]
  37. M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurement of visible and infrarednonlinear crystals,” Phys. Rev. B 14, 1693 (1974).
    [CrossRef]
  38. K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite cross-section,” IEEE J. Quantum Electron. 31, 769 (1995).
    [CrossRef]
  39. Absorption loss as low as 0.55–0.65%/cm hasbeen measured from laser calorimetry at 1.064 μm on some specificallyhigh-quality AGS samples. G. Catella, Cleveland Crystals Inc., Cleveland, Ohio, and R. Kondrotas, EKSMA Co., Vilnius, Lithuania (personal communication).
  40. A. Haraski and K. Kato, “New data on the nonlinear optical constant, phase-matching and opticaldamage of AgGaS2,” Jpn. J. Appl. Phys. 36, 700 (1997).
    [CrossRef]

1997 (4)

D. Touahri, O. Acef, A. Clairon, J.-J. Zondy, R. Felder, L. Hilico, B. de Beauvoir, F. Biraben, and F. Nez, “Frequency measurement of the 5S1/2(F=3)–5D5/2(F=5) two-photon transition in rubidium,” Opt. Commun. 133, 471 (1997).
[CrossRef]

A. Haraski and K. Kato, “New data on the nonlinear optical constant, phase-matching and opticaldamage of AgGaS2,” Jpn. J. Appl. Phys. 36, 700 (1997).
[CrossRef]

J.-J. Zondy and D. Touahri, “Updated thermo-optic coefficients of AgGaS2 from temperature-tuned noncritical 3ω− ω→2ω infrared parametricamplification,” J. Opt. Soc. Am. B 14, 1331 (1997).
[CrossRef]

B. Boulanger, J.-P. Fève, G. Marnier, C. Bonnin, P. Villeval, and J.-J. Zondy, “Absolute measurement of quadratic nonlinearities from phase-matchedsecond-harmonic generation in a single KTP crystal cut in a sphere,” J. Opt. Soc. Am. B 14, 1380 (1997).
[CrossRef]

1996 (2)

1995 (5)

E.-J. Canto-Said, M. P. McCann, P. G. Wigley, and G. J. Dixon, “Broadly tunable mid-infrared intracavity difference-frequency laser,” Opt. Lett. 20, 1268 (1995).
[CrossRef] [PubMed]

U. Simon, S. Waltman, I. Loa, F. K. Tittel, and L. Hollberg, “External-cavity difference frequency source near 3.3 μm, basedon combining a tunable diode laser with a diode-pumped Nd:YAG laser in AgGaS2,” J. Opt. Soc. Am. B 12, 323 (1995).
[CrossRef]

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite cross-section,” IEEE J. Quantum Electron. 31, 769 (1995).
[CrossRef]

N. C. Wong, “Optical-to-microwave frequency chain utilizing a two-laser-based opticalparametric oscillator network,” Appl. Phys. B 61, 143 (1995).
[CrossRef]

J.-P. Fève, B. Boulanger, and G. Marnier, “Experimental study of walk-off attenuation for type-II second-harmonicgeneration in KTP,” IEEE J. Quantum Electron. 31, 1569 (1995).
[CrossRef]

1994 (4)

1993 (1)

1992 (3)

1991 (1)

1990 (1)

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 harmonicgeneration,” IEEE J. Quantum Electron. 26, 922 (1990).
[CrossRef]

1989 (1)

G. Lera and M. Nieto-Vesperinas, “Non-paraxial method for non-linear optical problems: far-infrared generationwith depletion and diffraction,” J. Opt. 20, 169 (1989).
[CrossRef]

1987 (1)

R. S. Feigelson and R. K. Route, “Recent developments in the growth of chalcopyrite crystals for nonlinearinfrared applications,” Opt. Eng. 26, 113 (1987).
[CrossRef]

1985 (1)

T.-B. Chu and M. Broyer, “Intracavity cw difference frequency generation by mixing three photonsand using Gaussian laser beams,” J. Phys. (France) 46, 523 (1985).
[CrossRef]

1984 (1)

K. Kato, “High-power difference frequency generation at 5–11 μm in AgGaS2,” IEEE J. Quantum Electron. QE-20, 698 (1984).
[CrossRef]

1982 (1)

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

1980 (1)

S. Guha and J. Falk, “The effect of focusing in the three-frequency parametric upconverter,” J. Appl. Phys. 51, 50 (1980).
[CrossRef]

1974 (2)

P. J. Kupecek, C. A. Schwartz, and D. S. Chemla, “Silver thiogallate (AgGaS2). Part I: nonlinear optical properties,” IEEE J. Quantum Electron. QE-10, 540 (1974).
[CrossRef]

M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurement of visible and infrarednonlinear crystals,” Phys. Rev. B 14, 1693 (1974).
[CrossRef]

1971 (2)

G. D. Boyd, H. Kasper, and J. H. McFee, “Linear and nonlinear optical properties of AgGaS2, CuGaS2and CuInS2 andtheory of the wedge technique for the measurement of nonlinear coefficients,” IEEE J. Quantum Electron. QE-7, 563 (1971).
[CrossRef]

D. S. Chemla, P. J. Kupecek, D. S. Robertson, and R. C. Smith, “Silver thiogallate, a new material with potential for infrared devices,” Opt. Commun. 3, 29 (1971).
[CrossRef]

1969 (1)

J. E. Bjorkholm, “Analysis of the doubly resonant optical parametric oscillator withoutpower-dependent reflections,” IEEE J. Quantum Electron. QE-5, 293 (1969).
[CrossRef]

1968 (1)

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

1966 (1)

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beam,” Phys. Rev. 145, 338 (1966).
[CrossRef]

1964 (1)

R. C. Miller, “Optical second-harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17 (1964).
[CrossRef]

Abed, M.

Acef, O.

D. Touahri, O. Acef, A. Clairon, J.-J. Zondy, R. Felder, L. Hilico, B. de Beauvoir, F. Biraben, and F. Nez, “Frequency measurement of the 5S1/2(F=3)–5D5/2(F=5) two-photon transition in rubidium,” Opt. Commun. 133, 471 (1997).
[CrossRef]

D. Touahri, O. Acef, and J.-J. Zondy, “30-THz upconversion of an AlGaAs diode laser with AgGaS2: bridging the several-terahertzfrequency gap in the near infrared,” Opt. Lett. 21, 213 (1996).
[CrossRef] [PubMed]

Ashkin, A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beam,” Phys. Rev. 145, 338 (1966).
[CrossRef]

Aumillier, G. D.

Basar, G.

Benko, Z.

P. Canarelli, Z. Benko, A. H. Hielscher, R. F. Curl, and F. K. Tittel, “Measurement of nonlinear coefficient and phase-matching characteristicsof AgGaS2,” IEEE J. Quantum Electron. 28, 52 (1992).
[CrossRef]

P. Canarelli, Z. Benko, R. Curl, and F. K. Tittel, “Continuous-wave infrared laser spectrometer based on difference frequencygeneration in AgGaS2for high resolution spectroscopy,” J. Opt. Soc. Am. B 9, 197 (1992).
[CrossRef]

Bierlein, J. D.

Biraben, F.

D. Touahri, O. Acef, A. Clairon, J.-J. Zondy, R. Felder, L. Hilico, B. de Beauvoir, F. Biraben, and F. Nez, “Frequency measurement of the 5S1/2(F=3)–5D5/2(F=5) two-photon transition in rubidium,” Opt. Commun. 133, 471 (1997).
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm, “Analysis of the doubly resonant optical parametric oscillator withoutpower-dependent reflections,” IEEE J. Quantum Electron. QE-5, 293 (1969).
[CrossRef]

Bonnin, C.

Boulanger, B.

Boyd, G. D.

G. D. Boyd, H. Kasper, and J. H. McFee, “Linear and nonlinear optical properties of AgGaS2, CuGaS2and CuInS2 andtheory of the wedge technique for the measurement of nonlinear coefficients,” IEEE J. Quantum Electron. QE-7, 563 (1971).
[CrossRef]

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

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beam,” Phys. Rev. 145, 338 (1966).
[CrossRef]

Bradley, C. C.

Broyer, M.

T.-B. Chu and M. Broyer, “Intracavity cw difference frequency generation by mixing three photonsand using Gaussian laser beams,” J. Phys. (France) 46, 523 (1985).
[CrossRef]

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 harmonicgeneration,” IEEE J. Quantum Electron. 26, 922 (1990).
[CrossRef]

M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurement of visible and infrarednonlinear crystals,” Phys. Rev. B 14, 1693 (1974).
[CrossRef]

Cabirol, X.

Canarelli, P.

P. Canarelli, Z. Benko, A. H. Hielscher, R. F. Curl, and F. K. Tittel, “Measurement of nonlinear coefficient and phase-matching characteristicsof AgGaS2,” IEEE J. Quantum Electron. 28, 52 (1992).
[CrossRef]

P. Canarelli, Z. Benko, R. Curl, and F. K. Tittel, “Continuous-wave infrared laser spectrometer based on difference frequencygeneration in AgGaS2for high resolution spectroscopy,” J. Opt. Soc. Am. B 9, 197 (1992).
[CrossRef]

Canto-Said, E.-J.

Chakmakjian, S. H.

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite cross-section,” IEEE J. Quantum Electron. 31, 769 (1995).
[CrossRef]

Chemla, D. S.

P. J. Kupecek, C. A. Schwartz, and D. S. Chemla, “Silver thiogallate (AgGaS2). Part I: nonlinear optical properties,” IEEE J. Quantum Electron. QE-10, 540 (1974).
[CrossRef]

D. S. Chemla, P. J. Kupecek, D. S. Robertson, and R. C. Smith, “Silver thiogallate, a new material with potential for infrared devices,” Opt. Commun. 3, 29 (1971).
[CrossRef]

Cheung, E. C.

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite cross-section,” IEEE J. Quantum Electron. 31, 769 (1995).
[CrossRef]

E. C. Cheung, K. Koch, and G. T. Moore, “Measurement of second-order nonlinear optical coefficients from thespectral brightness of parametric fluorescence,” Opt. Lett. 19, 168 (1994).
[CrossRef]

Choy, M. M.

M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurement of visible and infrarednonlinear crystals,” Phys. Rev. B 14, 1693 (1974).
[CrossRef]

Chu, T.-B.

T.-B. Chu and M. Broyer, “Intracavity cw difference frequency generation by mixing three photonsand using Gaussian laser beams,” J. Phys. (France) 46, 523 (1985).
[CrossRef]

Clairon, A.

D. Touahri, O. Acef, A. Clairon, J.-J. Zondy, R. Felder, L. Hilico, B. de Beauvoir, F. Biraben, and F. Nez, “Frequency measurement of the 5S1/2(F=3)–5D5/2(F=5) two-photon transition in rubidium,” Opt. Commun. 133, 471 (1997).
[CrossRef]

J.-J. Zondy, M. Abed, and A. Clairon, “Type-II frequency doubling at λ=1.30 μm and λ=2.53 μmin flux grown potassium titanyl phosphate,” J. Opt. Soc. Am. B 11, 2004 (1994).
[CrossRef]

Curl, R.

Curl, R. F.

U. Simon, C. E. Miller, C. C. Bradley, R. G. Hulet, R. F. Curl, and F. K. Tittel, “Difference-frequency generation in AgGaS2 by use of single-mode diode-laser pump sources,” Opt. Lett. 18, 1062 (1993).
[CrossRef]

P. Canarelli, Z. Benko, A. H. Hielscher, R. F. Curl, and F. K. Tittel, “Measurement of nonlinear coefficient and phase-matching characteristicsof AgGaS2,” IEEE J. Quantum Electron. 28, 52 (1992).
[CrossRef]

de Beauvoir, B.

D. Touahri, O. Acef, A. Clairon, J.-J. Zondy, R. Felder, L. Hilico, B. de Beauvoir, F. Biraben, and F. Nez, “Frequency measurement of the 5S1/2(F=3)–5D5/2(F=5) two-photon transition in rubidium,” Opt. Commun. 133, 471 (1997).
[CrossRef]

Dixon, G. J.

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 harmonicgeneration,” IEEE J. Quantum Electron. 26, 922 (1990).
[CrossRef]

Falk, J.

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

S. Guha and J. Falk, “The effect of focusing in the three-frequency parametric upconverter,” J. Appl. Phys. 51, 50 (1980).
[CrossRef]

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 harmonicgeneration,” IEEE J. Quantum Electron. 26, 922 (1990).
[CrossRef]

Feigelson, R. S.

R. S. Feigelson and R. K. Route, “Recent developments in the growth of chalcopyrite crystals for nonlinearinfrared applications,” Opt. Eng. 26, 113 (1987).
[CrossRef]

Felder, R.

D. Touahri, O. Acef, A. Clairon, J.-J. Zondy, R. Felder, L. Hilico, B. de Beauvoir, F. Biraben, and F. Nez, “Frequency measurement of the 5S1/2(F=3)–5D5/2(F=5) two-photon transition in rubidium,” Opt. Commun. 133, 471 (1997).
[CrossRef]

Fève, J. Ph.

Fève, J.-P.

Guha, S.

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

S. Guha and J. Falk, “The effect of focusing in the three-frequency parametric upconverter,” J. Appl. Phys. 51, 50 (1980).
[CrossRef]

Haraski, A.

A. Haraski and K. Kato, “New data on the nonlinear optical constant, phase-matching and opticaldamage of AgGaS2,” Jpn. J. Appl. Phys. 36, 700 (1997).
[CrossRef]

Hielscher, A. H.

P. Canarelli, Z. Benko, A. H. Hielscher, R. F. Curl, and F. K. Tittel, “Measurement of nonlinear coefficient and phase-matching characteristicsof AgGaS2,” IEEE J. Quantum Electron. 28, 52 (1992).
[CrossRef]

Hilico, L.

D. Touahri, O. Acef, A. Clairon, J.-J. Zondy, R. Felder, L. Hilico, B. de Beauvoir, F. Biraben, and F. Nez, “Frequency measurement of the 5S1/2(F=3)–5D5/2(F=5) two-photon transition in rubidium,” Opt. Commun. 133, 471 (1997).
[CrossRef]

Hollberg, L.

Hulet, R. G.

Kasper, H.

G. D. Boyd, H. Kasper, and J. H. McFee, “Linear and nonlinear optical properties of AgGaS2, CuGaS2and CuInS2 andtheory of the wedge technique for the measurement of nonlinear coefficients,” IEEE J. Quantum Electron. QE-7, 563 (1971).
[CrossRef]

Kato, K.

A. Haraski and K. Kato, “New data on the nonlinear optical constant, phase-matching and opticaldamage of AgGaS2,” Jpn. J. Appl. Phys. 36, 700 (1997).
[CrossRef]

K. Kato, “High-power difference frequency generation at 5–11 μm in AgGaS2,” IEEE J. Quantum Electron. QE-20, 698 (1984).
[CrossRef]

Khodja, S.

Kleinman, D. A.

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

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beam,” Phys. Rev. 145, 338 (1966).
[CrossRef]

Koch, K.

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite cross-section,” IEEE J. Quantum Electron. 31, 769 (1995).
[CrossRef]

E. C. Cheung, K. Koch, and G. T. Moore, “Measurement of second-order nonlinear optical coefficients from thespectral brightness of parametric fluorescence,” Opt. Lett. 19, 168 (1994).
[CrossRef]

Kronfeldt, H.-D.

Kupecek, P. J.

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[CrossRef]

D. S. Chemla, P. J. Kupecek, D. S. Robertson, and R. C. Smith, “Silver thiogallate, a new material with potential for infrared devices,” Opt. Commun. 3, 29 (1971).
[CrossRef]

Lera, G.

G. Lera and M. Nieto-Vesperinas, “Non-paraxial method for non-linear optical problems: far-infrared generationwith depletion and diffraction,” J. Opt. 20, 169 (1989).
[CrossRef]

Liu, J. M.

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite cross-section,” IEEE J. Quantum Electron. 31, 769 (1995).
[CrossRef]

Loa, I.

Marnier, G.

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 harmonicgeneration,” IEEE J. Quantum Electron. 26, 922 (1990).
[CrossRef]

McCann, M. P.

McFee, J. H.

G. D. Boyd, H. Kasper, and J. H. McFee, “Linear and nonlinear optical properties of AgGaS2, CuGaS2and CuInS2 andtheory of the wedge technique for the measurement of nonlinear coefficients,” IEEE J. Quantum Electron. QE-7, 563 (1971).
[CrossRef]

Ménaert, B.

Miller, C. E.

Miller, R. C.

R. C. Miller, “Optical second-harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17 (1964).
[CrossRef]

Miranda, R. S.

Moore, G. T.

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite cross-section,” IEEE J. Quantum Electron. 31, 769 (1995).
[CrossRef]

E. C. Cheung, K. Koch, and G. T. Moore, “Measurement of second-order nonlinear optical coefficients from thespectral brightness of parametric fluorescence,” Opt. Lett. 19, 168 (1994).
[CrossRef]

Nez, F.

D. Touahri, O. Acef, A. Clairon, J.-J. Zondy, R. Felder, L. Hilico, B. de Beauvoir, F. Biraben, and F. Nez, “Frequency measurement of the 5S1/2(F=3)–5D5/2(F=5) two-photon transition in rubidium,” Opt. Commun. 133, 471 (1997).
[CrossRef]

Nieto-Vesperinas, M.

G. Lera and M. Nieto-Vesperinas, “Non-paraxial method for non-linear optical problems: far-infrared generationwith depletion and diffraction,” J. Opt. 20, 169 (1989).
[CrossRef]

Robertson, D. S.

D. S. Chemla, P. J. Kupecek, D. S. Robertson, and R. C. Smith, “Silver thiogallate, a new material with potential for infrared devices,” Opt. Commun. 3, 29 (1971).
[CrossRef]

Route, R. K.

R. S. Feigelson and R. K. Route, “Recent developments in the growth of chalcopyrite crystals for nonlinearinfrared applications,” Opt. Eng. 26, 113 (1987).
[CrossRef]

Schwartz, C. A.

P. J. Kupecek, C. A. Schwartz, and D. S. Chemla, “Silver thiogallate (AgGaS2). Part I: nonlinear optical properties,” IEEE J. Quantum Electron. QE-10, 540 (1974).
[CrossRef]

Simon, U.

Smith, R. C.

D. S. Chemla, P. J. Kupecek, D. S. Robertson, and R. C. Smith, “Silver thiogallate, a new material with potential for infrared devices,” Opt. Commun. 3, 29 (1971).
[CrossRef]

Sumpf, B.

Tittel, F. K.

Tom, H. W. K.

Touahri, D.

Vanherzeele, H.

Villeval, P.

Waltman, S.

Wigley, P. G.

Wong, N. C.

N. C. Wong, “Optical-to-microwave frequency chain utilizing a two-laser-based opticalparametric oscillator network,” Appl. Phys. B 61, 143 (1995).
[CrossRef]

Wu, F.

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

Yodh, A. G.

Zondy, J.-J.

Appl. Phys. B (1)

N. C. Wong, “Optical-to-microwave frequency chain utilizing a two-laser-based opticalparametric oscillator network,” Appl. Phys. B 61, 143 (1995).
[CrossRef]

Appl. Phys. Lett. (1)

R. C. Miller, “Optical second-harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17 (1964).
[CrossRef]

IEEE J. Quantum Electron. (9)

K. Kato, “High-power difference frequency generation at 5–11 μm in AgGaS2,” IEEE J. Quantum Electron. QE-20, 698 (1984).
[CrossRef]

G. D. Boyd, H. Kasper, and J. H. McFee, “Linear and nonlinear optical properties of AgGaS2, CuGaS2and CuInS2 andtheory of the wedge technique for the measurement of nonlinear coefficients,” IEEE J. Quantum Electron. QE-7, 563 (1971).
[CrossRef]

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite cross-section,” IEEE J. Quantum Electron. 31, 769 (1995).
[CrossRef]

J. E. Bjorkholm, “Analysis of the doubly resonant optical parametric oscillator withoutpower-dependent reflections,” IEEE J. Quantum Electron. QE-5, 293 (1969).
[CrossRef]

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

P. Canarelli, Z. Benko, A. H. Hielscher, R. F. Curl, and F. K. Tittel, “Measurement of nonlinear coefficient and phase-matching characteristicsof AgGaS2,” IEEE J. Quantum Electron. 28, 52 (1992).
[CrossRef]

P. J. Kupecek, C. A. Schwartz, and D. S. Chemla, “Silver thiogallate (AgGaS2). Part I: nonlinear optical properties,” IEEE J. Quantum Electron. QE-10, 540 (1974).
[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 harmonicgeneration,” IEEE J. Quantum Electron. 26, 922 (1990).
[CrossRef]

J.-P. Fève, B. Boulanger, and G. Marnier, “Experimental study of walk-off attenuation for type-II second-harmonicgeneration in KTP,” IEEE J. Quantum Electron. 31, 1569 (1995).
[CrossRef]

J. Appl. Phys. (2)

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

S. Guha and J. Falk, “The effect of focusing in the three-frequency parametric upconverter,” J. Appl. Phys. 51, 50 (1980).
[CrossRef]

J. Opt. (1)

G. Lera and M. Nieto-Vesperinas, “Non-paraxial method for non-linear optical problems: far-infrared generationwith depletion and diffraction,” J. Opt. 20, 169 (1989).
[CrossRef]

J. Opt. Soc. Am. B (9)

B. Boulanger, J.-P. Fève, G. Marnier, C. Bonnin, P. Villeval, and J.-J. Zondy, “Absolute measurement of quadratic nonlinearities from phase-matchedsecond-harmonic generation in a single KTP crystal cut in a sphere,” J. Opt. Soc. Am. B 14, 1380 (1997).
[CrossRef]

B. Boulanger, J. Ph. Fève, G. Marnier, B. Ménaert, and X. Cabirol, “Relative sign and absolute magnitude of the χ(2) coefficients of KTP by SHG measurements,” J. Opt. Soc. Am. B 11, 750 (1994).
[CrossRef]

J.-J. Zondy, M. Abed, and A. Clairon, “Type-II frequency doubling at λ=1.30 μm and λ=2.53 μmin flux grown potassium titanyl phosphate,” J. Opt. Soc. Am. B 11, 2004 (1994).
[CrossRef]

J.-J. Zondy and D. Touahri, “Updated thermo-optic coefficients of AgGaS2 from temperature-tuned noncritical 3ω− ω→2ω infrared parametricamplification,” J. Opt. Soc. Am. B 14, 1331 (1997).
[CrossRef]

J.-J. Zondy, M. Abed, and S. Khodja, “Twin-crystal walk-off-compensated type-II second-harmonic generation:single-pass and cavity-enhanced experiments in KTiOPO4,” J. Opt. Soc. Am. B 11, 2368 (1994).
[CrossRef]

P. Canarelli, Z. Benko, R. Curl, and F. K. Tittel, “Continuous-wave infrared laser spectrometer based on difference frequencygeneration in AgGaS2for high resolution spectroscopy,” J. Opt. Soc. Am. B 9, 197 (1992).
[CrossRef]

A. G. Yodh, H. W. K. Tom, G. D. Aumillier, and R. S. Miranda, “Generation of tunable mid-infrared picosecond pulses pulses pulsesat 76 MHz,” J. Opt. Soc. Am. B 8, 1663 (1991).
[CrossRef]

U. Simon, S. Waltman, I. Loa, F. K. Tittel, and L. Hollberg, “External-cavity difference frequency source near 3.3 μm, basedon combining a tunable diode laser with a diode-pumped Nd:YAG laser in AgGaS2,” J. Opt. Soc. Am. B 12, 323 (1995).
[CrossRef]

H.-D. Kronfeldt, G. Basar, and B. Sumpf, “Application of a cw tunable infrared spectrometer based on difference-frequencygeneration in AgGaS2for self-broadening investigations of NO at μm,” J. Opt. Soc. Am. B 13, 1859 (1996).
[CrossRef]

J. Phys. (France) (1)

T.-B. Chu and M. Broyer, “Intracavity cw difference frequency generation by mixing three photonsand using Gaussian laser beams,” J. Phys. (France) 46, 523 (1985).
[CrossRef]

Jpn. J. Appl. Phys. (1)

A. Haraski and K. Kato, “New data on the nonlinear optical constant, phase-matching and opticaldamage of AgGaS2,” Jpn. J. Appl. Phys. 36, 700 (1997).
[CrossRef]

Opt. Commun. (2)

D. S. Chemla, P. J. Kupecek, D. S. Robertson, and R. C. Smith, “Silver thiogallate, a new material with potential for infrared devices,” Opt. Commun. 3, 29 (1971).
[CrossRef]

D. Touahri, O. Acef, A. Clairon, J.-J. Zondy, R. Felder, L. Hilico, B. de Beauvoir, F. Biraben, and F. Nez, “Frequency measurement of the 5S1/2(F=3)–5D5/2(F=5) two-photon transition in rubidium,” Opt. Commun. 133, 471 (1997).
[CrossRef]

Opt. Eng. (1)

R. S. Feigelson and R. K. Route, “Recent developments in the growth of chalcopyrite crystals for nonlinearinfrared applications,” Opt. Eng. 26, 113 (1987).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. (1)

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beam,” Phys. Rev. 145, 338 (1966).
[CrossRef]

Phys. Rev. B (1)

M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurement of visible and infrarednonlinear crystals,” Phys. Rev. B 14, 1693 (1974).
[CrossRef]

Other (5)

Absorption loss as low as 0.55–0.65%/cm hasbeen measured from laser calorimetry at 1.064 μm on some specificallyhigh-quality AGS samples. G. Catella, Cleveland Crystals Inc., Cleveland, Ohio, and R. Kondrotas, EKSMA Co., Vilnius, Lithuania (personal communication).

SFG and DFG samples were grown and manufactured by ClevelandCrystals, Inc., Ohio.

The SHG sample was grown at the Institute of Mineralogy andPetrography, Russian Academy of Sciences, Siberian Branch, Novosibirsk (Russia)and manufactured by EKSMA Co., Vilnius (Lithuania).

J.-J. Zondy, D. Touahri, and O. Acef, “Infra-red tovisible non-linear up and down conversion processes using AgGaS2 crystals,” in Advanced Solid State Lasers, S. A. Payne and C. Pollock, eds., Vol. 1 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 164–167.

J.-J. Zondy, “Comparative theory of walkoff-limited type-II versus type-I secondharmonic generation with Gaussian beams,” Opt. Commun. 81, 427 (1991); erratum, in Eq. (3.1.a), exp(−x2) should read exp(−x2/2).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic of the AGS-based optical synthesis chain.

Fig. 2
Fig. 2

Comparison of experimental internal angular acceptance curves (graph markers) with the calculated Gaussian beam SHG mismatch function h(σ, η=0.5) [Eq. (4), solid thick curves] and with the plane-wave mismatch function {[sinh2(α2ωlc/4) +sin2(Δklc/2)]/[(α2ωlc/4)2+(Δklc/2)2], solid thin curve}. Note the actual broadening of the bandwidth compared with the plane-wave prediction. The oscillatory structure of the plane-wave function is smeared off by the walk-off of the 2ω wave (ρ =1.26°, lc=10 mm, and B=ρ(kωlc)1/2/2).

Fig. 3
Fig. 3

Experimental SHG conversion efficiency of the CCL. The solid line is obtained from a least-squares fit.

Fig. 4
Fig. 4

Curve of optimum idler (λ1=10.2 µm) focusing parameters (l1)opt (left vertical scale), given a signal (λ2=0.842 µm) focusing parameter l2, and corresponding (hSFG)opt (right scale) for an AGS upconverter of length lc=15 mm (B=4.052). The maximum value of (hSFG)opt=0.108 corresponds to (l1, l2)opt =(5.12, 0.54) or w2=54.7 µm and w1=63.6 µm, with the maximum ΓSFG=0.673 mW/W2. Note that owing to the idler absorption (α1=0.43 cm-1), the optimum waist location is nearer to the input facet (ηopt=0.4). The filled dot and the filled square are experimental data points for (l1, l2)exp and the corresponding (hSFG)exp.

Fig. 5
Fig. 5

Crystal length optimization for CO2 laser upconversion in AgGaS2. The optimum waist curves are scaled according to the left axis. The optimum ΓSHG (in mW/W2) and waist location (η) are scaled according to the right axis. The solid curves correspond to an absorption of α1=0.43 cm-1 while the dashed curves are zero-absorption curves. In the latter cases η is 0.5 (focus location at the middle of the crystal).

Fig. 6
Fig. 6

Type I 3ωω2ω OPA small signal net gain g =ΓP3 (P3 in W) versus pump power P3. The solid line is a least-squares fit of the experimental data.

Fig. 7
Fig. 7

Optimized DFG aperture functions of the 3:1 NCPM-OPA versus the reduced focusing parameter l=lc/zd (lc =15 mm, a=0, and β=0, λ3=0.842 µm, and λ1 =2.53 µm): curve a, the case of ECP for pump (3ω) and idler (ω) input waves (z3=z1=z0); curve b, the pump waist w3 is fixed to its optimum value of curve a, and the idler waist is varied; curve c, the idler waist w1 is fixed to its optimum value of curve a, and the pump waist is varied. The filled dot represents the value of hDFG corresponding to the experimental focusing parameters (waists w3=50 µm and w1=95 µm).

Fig. 8
Fig. 8

Theoretical pump-power threshold as a function of crystal length for a doubly resonant, ring-cavity, AGS-based OPO operating as a 3:1 frequency divider in the mid-IR: curve a, α=0.8% cm-1; Curve b, α=1% cm-1; Curve c, α =1.2% cm-1. The value d36=13 pm/V is used, and δ1,2 is the fractional cavity loss owing to the resonator.

Tables (4)

Tables Icon

Table 1 Summary of Published Values on the d36 Nonlinear Coefficient of AgGaS2

Tables Icon

Table 2 Room-Temperature Refractive Indices of AgGaS2a

Tables Icon

Table 3 Characteristics of the Three AGS Samples Used for SHG, SFG, and OPA/DFG and the Related Beam Parameters and Conversion Efficienciesa

Tables Icon

Table 4 Summary of the Values of the d36 Nonlinear Coefficient of AgGaS2a

Equations (89)

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Δk=k3e(θPM)-k2o-k1o=0,
(2+kg2)Eg(r)=-ωg2c2PNL(r),
d=eg·d(2):ea·eb=d36(-ωg, ωa,±ωb)×sin(θPM+ρ)sin(2φ),
tan ρ=-ne2(θPM, λ3)2[no-2(λ3)-ne-2(λ3)]sin(2θPM).
Eg(r)=i ωgcngdrPNL(r)Gg(r, r),
Gg(r, r)=kg2πiZexp-αg2Z+ikgZ×expikg(X2+Y2)2Z,
X=(x-ρz)-(x-ρz),
Y=y-y,
Z=z-z>0.
Ej(r)=Aj1+iτjexp-αj2z+ikjz-iφj×exp-[x-ρj(z-fc)]2+y2wj2(1+iτj),
Pj=πε0cnjwj2Aj2/4.
ΓSHG=KSHGlckω exp(-α2ωlc)hSHG(a, l, f, σ),
ΓSHGKSHG(lc/w0)2 exp(-α2ωlc)GSHG(a, l, f, ,σ),
hSHG=12l×-fl-fdτdτ ×exp[-a(τ+τ+2 f)-β2(τ-τ)2-iσ(τ-τ)](1+iτ)(1-iτ),
z0=(1/2)kωw02,a=(αω-α2ω/2)z0,
β=ρz0/w0,σ=Δkz0,
KSHG=(2ω2d2/πε0c3nω2n2ω),
E3(x, y, lc)=i ω3cn3dA1A2 exp[-α3lc/2+i(k3lc-φ1-φ2)]×0lcdz exp(-αz+iΔkz)QS(τ)×exp-[x-ρ(lc-z)]2+y2wu2RS(τ),
QS(τ)=(1+iκ1τ)(1+iκ2τ)+i(1+iκeτ)(l-f-τ),
RS(τ)=QS(τ)/(1+iκeτ).
zu=12k3wu2=k1+k2k1z2+k2z1z1z2,
κ1,2=zu/z1,2=k1+k2k1z2+k2z1z2,1,
κe=k1+k2k1z2+k2z12z1z2,
l=lczu,f=fczu=ηl,
a=α1+α2-α32zu,σ=Δkzu.
ΓSFG=KSFGlc(k1-1+k2-1)-1×exp(-α3lc)hSFG(a, l, f, σ),
ΓSHGKSFG lc2w12+w22exp(-α3lc)GSFG(a, l, f, σ),
hSFG(a, l, f, σ)
=κe2l-fl-fdτdτ×exp[-a(τ+τ+2 f)+iσ(τ-τ)](1-iκeτ)QS(τ)+(1+iκeτ)QS*(τ)Fβ(τ, τ),
Fβ(τ, τ)
=exp{-β2[τ2/RS(τ)+τ2/RS*(τ)]}×exp+β2[τ/RS(τ)+τ/RS*(τ)]2×QS(τ)QS*(τ)(1-iκeτ)QS(τ)+(1+iκeτ)QS*(τ),
B=ρ2k32lc1/2,β=ρzuwu=2Bl,
KSFG=8ω32d2πε0c3n1n2n3.
Eg(x, y, lc)=i ωgcngdA3Ai* exp[-αglc/2+i(kglc-φ3+φi)]×0lcdz exp(-αz+iΔkz)QD(τ)×exp-x2+y2wd2RD(τ)+2ρx(z-fc)w32SD(τ)-ρ2(z-fc)2w32TD(τ),
QD(τ)=(1+iκ3τ)(1-iκiτ)+i(1-iκeτ)(l-f-τ),
RD(τ)=QD(τ)/(1-iκeτ),
SD(τ)=QD(τ)/(1-iκiτ),
TD(τ)=QD(τ)(1-iκiτ)+i(1-r)(l-f-τ),
r=wi2wi2+w32=k3zikiz3+k3zi.
zd=12kgwd2=k3-kikiz3+k3ziziz3,
κi,3=zd/zi,3=k3-kikiz3+k3ziz3,i,
κe=k3-kikiz3+k3zi2ziz3.
l=lczd,f=fczd=ηl,
a=α3+αi-αg2zd,σ=Δkzd,
ΓDFG=KDFG lc(ki-1-k3-1)-1×exp(-αglc)hDFG(a, l, f, σ),
ΓDFGKDFGlc2wi2+w32exp(-αglc)GDFG(a, l, f, σ),
hDFG(a, l, f, σ)
=κe2l-fl-fdτdτ 
×exp[-a(τ+τ+2 f)+iσ(τ-τ)](1+iκeτ)QD(τ)+(1-iκeτ)QD*(τ)×Fβ(τ, τ),
Fβ(τ, τ)
=exp{-rβ2[τ2/TD(τ)+τ2/TD*(τ)]}×exp+r2β2[τ/SD(τ)+τ/SD*(τ)]2×QD(τ)QD*(τ)(1+iκeτ)QD(τ)+(1-iκeτ)QD*(τ),
B=ρ2(kglc)1/2,β=ρzdwd=2Bl,
KDFG=8ωg2d2πε0c3n1n2n3.
h˜DFG(a˜, l˜, f˜, σ˜)
=14l˜1-μ1+μ-f˜l˜-f˜dτdτ×exp[-a˜(τ+τ+2 f˜)+iσ˜(τ-τ)]1-i 1+μ21-μ2(τ-τ)+ττ.
ΔP3=ε0cn32[|E3(x, y, 0)|2-|E3(x, y, lc)+E3S(x, y, lc)|2]dxdy,
E3S(x, y, lc)=i ω3cn3dA1A2×exp-α3lc2+ik3lc-i(φ1+φ2)×exp-x2+y2w32[1+i(l-f)]0lcdz×exp(-αz+iΔkz)(1+iτ)[1+i(l-f)],
ΔP3=[1-exp(-α3lc)]P3-exp(-α3lc)(k1k2/k3)×[KSFGP1P2lchSFG+(KSFGP1P2P3)1/2×[π(w12+w22)]1/2×H0 Re(i exp[-i(σf+Δφ+χ)]}),
HDRO(a, l, f, σ)H0 exp(-iχ)=1(2π)1/2-fl-fdτ×exp[-a(τ+f)-iστ]1-iτ.
ΔP3¯=ΔP1¯+ΔP2¯,
ΔP3¯ω3=ΔP1¯ω1=ΔP2¯ω2,
ΔP1,2=exp(-α1,2lc)ΔP1,2¯.
P1/P2=(ε2ω1/ε1ω2)exp[-(α1-α2)lc].
σf+Δφ+χ=-π/2.
P1=P10P3P3th-1,
P10=exp[(α2+α3)lc] ω1ω2ε2K˜lc(k1-1+k2-1)-1hSFG,
P3th=exp[(α1+α2+2α3)lc] ε1ε2K˜lc(k1-1+k2-1)-1h˜,
h˜=πlH02=12l-fl-fdτ-fl-fdτ×exp[-a(τ+τ+2 f)-iσ(τ-τ)](1-iτ)(1+iτ),
K˜=ω1ω2ω32KSFG=8ω1ω2d2πε0c3n1n2n3.
η=2 exp(-α3lc) ω1 exp(-α1lc)+ω2 exp(-α2lc)ω3×h˜hSFG1N1-1N.
T(ϕ, w0)=2πw020ϕ/2dr2πr exp-2r2w02.
w0=-ϕ22 ln(1-T)1/2.
σopt=-1.0,ηopt=0.5,
(hSHG)opt=0.1007,(GSHG)opt=0.6036.
ΓSHGI(AGS)ΓSHGII(KTP)=lc(AGS)lc(KTP)w0(KTP)w0(AGS)2×M(AGS)M(KTP)GSHG(AGS)GSHG(KTP)×exp[-α2ωlc(AGS)].
σopt=3.5,ηopt=0.5,
(hSFG)opt=0.0804,(GSFG)opt=0.1635.
σopt=-0.562,ηopt=0.5,
(hDFG)opt=0.0697,(GDFG)opt=0.4799.
g(lc)=(ε1ε2/lch˜)exp[(α1+α2+2α3)lc]
HDRO(a, l, f, σ)
H0 exp(-iχ)
=2κ3π κe-fl-fdτ exp[-a(τ+f)-iστ]Q˜(τ)×exp-κ3β2 τ2S˜(τ),
Q˜(τ)=(1-iκeτ)(1+iκ3τ)+κ3(1-iκ1τ)(1-iκ2τ),
S˜(τ)=Q˜(τ)/(1-iκeτ),
κ3=zuz3=k3k1z2+k2z1z1z2z3,
h˜=πlκeH02=2κ3κel-fl-fdτ-fl-fdτ 
×exp[-a(τ+τ+2 f)-iσ(τ-τ)]Q˜(τ)Q˜*(τ)
×exp-κ3β2τ2S˜(τ)+τ2S˜*(τ).

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