Abstract

We report on a theoretical and experimental investigation of the spectral properties and the conversion efficiency of 355-nm-pumped pulsed optical parametric oscillators (OPO’s) of β-barium borate with noncollinear phase matching. If the noncollinear phase matching is tangential, the pump wave’s angular acceptance—which is approximately 0.3 mrad cm-1 for collinear excitation—is increased as much as 9 mrad cm-1. Because of the large angular acceptance the OPO output energy increases to a factor of 4 greater than that for conventional collinear excitation when the OPO is pumped by a laser beam with a divergence of 3 mrad. Besides an increase in efficiency, the noncollinear phase matching provides a broadening of the spectral width of the OPO signal radiation of as much as 20 nm in the whole tuning range of 420–700 nm. The measured spectral widths are in good agreement with the results of an analysis that takes into account the influence of the divergence of the resonant OPO wave.

© 1999 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  37. K.-J. Boller and T. Schröder, “Demonstration of broadband intracavity spectroscopy in a pulsed optical parametric oscillator made of β-barium borate,” J. Opt. Soc. Am. B 10, 1778 (1993).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1997

1996

1995

1994

W. R. Bosenberg, L. K. Cheng, and J. D. Bierlein, “Optical parametric frequency conversion properties of KTiOAsO4,” Appl. Phys. Lett. 65, 2765 (1994).
[CrossRef]

S. G. Dolinchuk, N. E. Kornienko, and V. I. Zadorozhnii, “Noncritical vectorial phase matchings in nonlinear optics of crystals and infrared up-conversion,” Infrared Phys. Technol. 35, 881 (1994).
[CrossRef]

L. A. W. Gloster, Z. X. Jiang, and T. A. King, “Characterization of an Nd:YAG-pumped β-BaB2O4 optical parametric oscillator in collinear and noncollinear phase-matched configurations,” IEEE J. Quantum Electron. 30, 2961 (1994).
[CrossRef]

L. A. W. Gloster, I. T. McKinnie, and T. A. King, “Noncollinear phase matching in a type I barium borate optical parametric oscillator,” Opt. Commun. 112, 328 (1994).
[CrossRef]

T. Schröder, K.-J. Boller, A. Fix, and R. Wallenstein, “Spectral properties and numerical modelling of a critically phase-matched nanosecond LiB3O5 optical parametric oscillator,” Appl. Phys. B: Laser Opt. 58, 425 (1994).
[CrossRef]

1993

1992

S. X. Dou, D. Josse, and J. Zyss, “Comparison of collinear and one-beam noncritical noncollinear phase matching in optical parametric amplification,” J. Opt. Soc. Am. B 9, 1312 (1992).
[CrossRef]

D. Brüggemann, J. Hertzberg, B. Wies, Y. Waschke, R. Noll, K.-F. Knoche, and G. Herziger, “Test of an optical parametric oscillator (OPO) as a compact and fast tunable Stokes source in coherent anti-Stokes Raman spectroscopy (CARS),” Appl. Phys. B: Photophys. Laser Chem. 55, 378 (1992).
[CrossRef]

M. J. T. Milton, T. J. McIlveen, D. C. Hanna, and P. T. Woods, “High-efficiency infrared generation by difference-frequency mixing using tangential phase matching,” Opt. Commun. 87, 273 (1992).
[CrossRef]

G. C. Bhar, P. K. Datta, and S. Das, “Tangentially phase matched second-harmonic generation in various crystals,” J. Appl. Phys. 71, 3620 (1992).
[CrossRef]

C. L. Tang, W. R. Bosenberg, T. Ukachi, R. J. Lane, and L. K. Cheng, “Optical parametric oscillators,” Proc. SPIE 80, 365 (1992), and references therein.

1991

A. Fix, T. Schröder, and R. Wallenstein, “The optical parametric oscillators of beta-barium borate and lithium borate: new sources of powerful tunable laser radiation in the ultraviolet, visible and near infrared,” Laser Optoelektron. 23, 106 (1991), and references therein.

D. N. Nikogosyan, “Beta barium borate (BBO)—a review of its properties and applications,” Appl. Phys. A: Solids Surf. 52, 359 (1991).
[CrossRef]

S. X. Dou, D. Josse, and J. Zyss, “Noncritical properties of noncollinear phase-matched second-harmonic and sum-frequency generation in 3-methyl-4-nitropyridine-1-oxide,” J. Opt. Soc. Am. B 8, 1732 (1991).
[CrossRef]

1990

M. Ebrahimzadeh, A. J. Henderson, and M. H. Dunn, “An excimer-pumped β-BaB2O4 optical parametric oscillator tunable from 354 nm to 2.370 μm,” IEEE J. Quantum Electron. 26, 1241 (1990).
[CrossRef]

S. Burdulis, R. Grigonis, A. Piskarskas, S. Sinkevicius, V. Sirutkaitis, A. Fix, J. Nolting, and R. Wallenstein, “Visible optical parametric oscillation in synchronously pumped beta-barium borate,” Opt. Commun. 74, 398 (1990).
[CrossRef]

W. R. Bosenberg and C. L. Tang, “Type II phase matching in a β-barium borate optical parametric oscillator,” Appl. Phys. Lett. 56, 1819 (1990).
[CrossRef]

1988

T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, “The application of incoherent light for the study of femtosecond-picosecond relaxation in condensed phase,” Appl. Phys. B: Photophys. Laser Chem. 47, 107 (1988), and references therein.
[CrossRef]

1986

K. Kato, “Second-harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron. QE-22, 1013 (1986).
[CrossRef]

1984

B. Schröder, “Frequency characteristic and spatial intensity distribution of light generated and amplified by parametric three-photon interaction,” Opt. Commun. 49, 75 (1984).
[CrossRef]

1979

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

1976

1974

R. L. Herbst, R. N. Fleming, and R. L. Byer, “A 1.4–4 μm high-energy angle-tuned LiNbO3 parametric oscillator,” Appl. Phys. Lett. 25, 520 (1974).
[CrossRef]

1969

J. Warner, “Phase-matching for optical up-conversion with maximum angular aperture—theory and practice,” Opto-Electron. 1, 25 (1969).
[CrossRef]

Andreoni, A.

Bahr, G. C.

G. C. Bahr, P. K. Datta, U. Chatterjee, S. Das, and H. L. Bhat, “Characterization of biaxial crystals for tangentially phase-matched frequency conversion,” Appl. Phys. B: Photophys. Laser Chem. 56, 327 (1993).
[CrossRef]

Barnes, N. P.

Bhar, G. C.

G. C. Bhar, P. K. Datta, and S. Das, “Tangentially phase matched second-harmonic generation in various crystals,” J. Appl. Phys. 71, 3620 (1992).
[CrossRef]

Bhat, H. L.

G. C. Bahr, P. K. Datta, U. Chatterjee, S. Das, and H. L. Bhat, “Characterization of biaxial crystals for tangentially phase-matched frequency conversion,” Appl. Phys. B: Photophys. Laser Chem. 56, 327 (1993).
[CrossRef]

Bierlein, J. D.

W. R. Bosenberg, L. K. Cheng, and J. D. Bierlein, “Optical parametric frequency conversion properties of KTiOAsO4,” Appl. Phys. Lett. 65, 2765 (1994).
[CrossRef]

Boller, K.-J.

T. Schröder, K.-J. Boller, A. Fix, and R. Wallenstein, “Spectral properties and numerical modelling of a critically phase-matched nanosecond LiB3O5 optical parametric oscillator,” Appl. Phys. B: Laser Opt. 58, 425 (1994).
[CrossRef]

K.-J. Boller and T. Schröder, “Demonstration of broadband intracavity spectroscopy in a pulsed optical parametric oscillator made of β-barium borate,” J. Opt. Soc. Am. B 10, 1778 (1993).
[CrossRef]

Bosenberg, W. R.

W. R. Bosenberg, L. K. Cheng, and J. D. Bierlein, “Optical parametric frequency conversion properties of KTiOAsO4,” Appl. Phys. Lett. 65, 2765 (1994).
[CrossRef]

W. R. Bosenberg and D. R. Guyer, “Broadly tunable, single-frequency optical parametric frequency-conversion system,” J. Opt. Soc. Am. B 10, 1716 (1993), and references therein.
[CrossRef]

C. L. Tang, W. R. Bosenberg, T. Ukachi, R. J. Lane, and L. K. Cheng, “Optical parametric oscillators,” Proc. SPIE 80, 365 (1992), and references therein.

W. R. Bosenberg and C. L. Tang, “Type II phase matching in a β-barium borate optical parametric oscillator,” Appl. Phys. Lett. 56, 1819 (1990).
[CrossRef]

Brosnan, S. J.

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

Brüggemann, D.

D. Brüggemann, J. Hertzberg, B. Wies, Y. Waschke, R. Noll, K.-F. Knoche, and G. Herziger, “Test of an optical parametric oscillator (OPO) as a compact and fast tunable Stokes source in coherent anti-Stokes Raman spectroscopy (CARS),” Appl. Phys. B: Photophys. Laser Chem. 55, 378 (1992).
[CrossRef]

Burdulis, S.

S. Burdulis, R. Grigonis, A. Piskarskas, S. Sinkevicius, V. Sirutkaitis, A. Fix, J. Nolting, and R. Wallenstein, “Visible optical parametric oscillation in synchronously pumped beta-barium borate,” Opt. Commun. 74, 398 (1990).
[CrossRef]

Byer, R. L.

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

R. L. Herbst, R. N. Fleming, and R. L. Byer, “A 1.4–4 μm high-energy angle-tuned LiNbO3 parametric oscillator,” Appl. Phys. Lett. 25, 520 (1974).
[CrossRef]

Cavallari, M.

Chatterjee, U.

G. C. Bahr, P. K. Datta, U. Chatterjee, S. Das, and H. L. Bhat, “Characterization of biaxial crystals for tangentially phase-matched frequency conversion,” Appl. Phys. B: Photophys. Laser Chem. 56, 327 (1993).
[CrossRef]

Cheng, L. K.

W. R. Bosenberg, L. K. Cheng, and J. D. Bierlein, “Optical parametric frequency conversion properties of KTiOAsO4,” Appl. Phys. Lett. 65, 2765 (1994).
[CrossRef]

C. L. Tang, W. R. Bosenberg, T. Ukachi, R. J. Lane, and L. K. Cheng, “Optical parametric oscillators,” Proc. SPIE 80, 365 (1992), and references therein.

Corcoran, V. J.

Danielius, R.

Das, S.

G. C. Bahr, P. K. Datta, U. Chatterjee, S. Das, and H. L. Bhat, “Characterization of biaxial crystals for tangentially phase-matched frequency conversion,” Appl. Phys. B: Photophys. Laser Chem. 56, 327 (1993).
[CrossRef]

G. C. Bhar, P. K. Datta, and S. Das, “Tangentially phase matched second-harmonic generation in various crystals,” J. Appl. Phys. 71, 3620 (1992).
[CrossRef]

Datta, P. K.

G. C. Bahr, P. K. Datta, U. Chatterjee, S. Das, and H. L. Bhat, “Characterization of biaxial crystals for tangentially phase-matched frequency conversion,” Appl. Phys. B: Photophys. Laser Chem. 56, 327 (1993).
[CrossRef]

G. C. Bhar, P. K. Datta, and S. Das, “Tangentially phase matched second-harmonic generation in various crystals,” J. Appl. Phys. 71, 3620 (1992).
[CrossRef]

Di Trapani, P.

Dolinchuk, S. G.

S. G. Dolinchuk, N. E. Kornienko, and V. I. Zadorozhnii, “Noncritical vectorial phase matchings in nonlinear optics of crystals and infrared up-conversion,” Infrared Phys. Technol. 35, 881 (1994).
[CrossRef]

Dou, S. X.

Driscoll, T. J.

Dubietis, A.

Dunn, M. H.

J. Wang, M. H. Dunn, and C. F. Rae, “Polychromatic optical parametric generation by simultaneous phase matching over a large spectral bandwidth,” Opt. Lett. 22, 763 (1997).
[CrossRef] [PubMed]

M. Ebrahimzadeh, A. J. Henderson, and M. H. Dunn, “An excimer-pumped β-BaB2O4 optical parametric oscillator tunable from 354 nm to 2.370 μm,” IEEE J. Quantum Electron. 26, 1241 (1990).
[CrossRef]

Ebrahimzadeh, M.

M. Ebrahimzadeh, A. J. Henderson, and M. H. Dunn, “An excimer-pumped β-BaB2O4 optical parametric oscillator tunable from 354 nm to 2.370 μm,” IEEE J. Quantum Electron. 26, 1241 (1990).
[CrossRef]

Ellingson, R. J.

Ferguson, A. I.

Fix, A.

R. Urschel, A. Fix, R. Wallenstein, D. Rytz, and B. Zysset, “Generation of tunable narrow-band midinfrared radiation in a type I potassium niobate optical parametric oscillator,” J. Opt. Soc. Am. B 12, 726 (1995).
[CrossRef]

T. Schröder, K.-J. Boller, A. Fix, and R. Wallenstein, “Spectral properties and numerical modelling of a critically phase-matched nanosecond LiB3O5 optical parametric oscillator,” Appl. Phys. B: Laser Opt. 58, 425 (1994).
[CrossRef]

A. Fix, T. Schröder, R. Wallenstein, J. G. Haub, M. J. Johnson, and B. J. Orr, “Tunable β-barium borate optical parametric oscillator: operating characteristics with and without injection seeding,” J. Opt. Soc. Am. B 10, 1744 (1993), and references therein.
[CrossRef]

A. Fix, T. Schröder, and R. Wallenstein, “The optical parametric oscillators of beta-barium borate and lithium borate: new sources of powerful tunable laser radiation in the ultraviolet, visible and near infrared,” Laser Optoelektron. 23, 106 (1991), and references therein.

S. Burdulis, R. Grigonis, A. Piskarskas, S. Sinkevicius, V. Sirutkaitis, A. Fix, J. Nolting, and R. Wallenstein, “Visible optical parametric oscillation in synchronously pumped beta-barium borate,” Opt. Commun. 74, 398 (1990).
[CrossRef]

Fleming, R. N.

R. L. Herbst, R. N. Fleming, and R. L. Byer, “A 1.4–4 μm high-energy angle-tuned LiNbO3 parametric oscillator,” Appl. Phys. Lett. 25, 520 (1974).
[CrossRef]

Foggi, P.

Gale, G. M.

Gloster, L. A. W.

A. L. Oien, I. T. McKinnie, P. Jain, N. A. Russell, D. M. Warrington, and L. A. W. Gloster, “Efficient, low-threshold collinear and noncollinear β-barium borate optical parametric oscillator,” Opt. Lett. 22, 859 (1997).
[CrossRef] [PubMed]

L. A. W. Gloster, I. T. McKinnie, and T. A. King, “Noncollinear phase matching in a type I barium borate optical parametric oscillator,” Opt. Commun. 112, 328 (1994).
[CrossRef]

L. A. W. Gloster, Z. X. Jiang, and T. A. King, “Characterization of an Nd:YAG-pumped β-BaB2O4 optical parametric oscillator in collinear and noncollinear phase-matched configurations,” IEEE J. Quantum Electron. 30, 2961 (1994).
[CrossRef]

Grigonis, R.

S. Burdulis, R. Grigonis, A. Piskarskas, S. Sinkevicius, V. Sirutkaitis, A. Fix, J. Nolting, and R. Wallenstein, “Visible optical parametric oscillation in synchronously pumped beta-barium borate,” Opt. Commun. 74, 398 (1990).
[CrossRef]

Guyer, D. R.

Hache, F.

Hall, G. J.

Hanna, D. C.

M. J. T. Milton, T. J. McIlveen, D. C. Hanna, and P. T. Woods, “High-efficiency infrared generation by difference-frequency mixing using tangential phase matching,” Opt. Commun. 87, 273 (1992).
[CrossRef]

Hattori, T.

T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, “The application of incoherent light for the study of femtosecond-picosecond relaxation in condensed phase,” Appl. Phys. B: Photophys. Laser Chem. 47, 107 (1988), and references therein.
[CrossRef]

Haub, J. G.

Henderson, A. J.

M. Ebrahimzadeh, A. J. Henderson, and M. H. Dunn, “An excimer-pumped β-BaB2O4 optical parametric oscillator tunable from 354 nm to 2.370 μm,” IEEE J. Quantum Electron. 26, 1241 (1990).
[CrossRef]

Herbst, R. L.

R. L. Herbst, R. N. Fleming, and R. L. Byer, “A 1.4–4 μm high-energy angle-tuned LiNbO3 parametric oscillator,” Appl. Phys. Lett. 25, 520 (1974).
[CrossRef]

Hertzberg, J.

D. Brüggemann, J. Hertzberg, B. Wies, Y. Waschke, R. Noll, K.-F. Knoche, and G. Herziger, “Test of an optical parametric oscillator (OPO) as a compact and fast tunable Stokes source in coherent anti-Stokes Raman spectroscopy (CARS),” Appl. Phys. B: Photophys. Laser Chem. 55, 378 (1992).
[CrossRef]

Herziger, G.

D. Brüggemann, J. Hertzberg, B. Wies, Y. Waschke, R. Noll, K.-F. Knoche, and G. Herziger, “Test of an optical parametric oscillator (OPO) as a compact and fast tunable Stokes source in coherent anti-Stokes Raman spectroscopy (CARS),” Appl. Phys. B: Photophys. Laser Chem. 55, 378 (1992).
[CrossRef]

Jain, P.

Jiang, Z. X.

L. A. W. Gloster, Z. X. Jiang, and T. A. King, “Characterization of an Nd:YAG-pumped β-BaB2O4 optical parametric oscillator in collinear and noncollinear phase-matched configurations,” IEEE J. Quantum Electron. 30, 2961 (1994).
[CrossRef]

Johnson, M. J.

Josse, D.

Kaino, T.

Kanbara, H.

Kasai, H.

T. Yanagawa, Y. Kurokawa, H. Kasai, and H. Nakanishi, “Degenerate four-wave mixing using an optical parametric oscillator as an incoherent light source,” Opt. Commun. 137, 103 (1997).
[CrossRef]

Kato, K.

K. Kato, “Second-harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron. QE-22, 1013 (1986).
[CrossRef]

King, T. A.

L. A. W. Gloster, Z. X. Jiang, and T. A. King, “Characterization of an Nd:YAG-pumped β-BaB2O4 optical parametric oscillator in collinear and noncollinear phase-matched configurations,” IEEE J. Quantum Electron. 30, 2961 (1994).
[CrossRef]

L. A. W. Gloster, I. T. McKinnie, and T. A. King, “Noncollinear phase matching in a type I barium borate optical parametric oscillator,” Opt. Commun. 112, 328 (1994).
[CrossRef]

Knoche, K.-F.

D. Brüggemann, J. Hertzberg, B. Wies, Y. Waschke, R. Noll, K.-F. Knoche, and G. Herziger, “Test of an optical parametric oscillator (OPO) as a compact and fast tunable Stokes source in coherent anti-Stokes Raman spectroscopy (CARS),” Appl. Phys. B: Photophys. Laser Chem. 55, 378 (1992).
[CrossRef]

Kobayashi, T.

T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, “The application of incoherent light for the study of femtosecond-picosecond relaxation in condensed phase,” Appl. Phys. B: Photophys. Laser Chem. 47, 107 (1988), and references therein.
[CrossRef]

Kornienko, N. E.

S. G. Dolinchuk, N. E. Kornienko, and V. I. Zadorozhnii, “Noncritical vectorial phase matchings in nonlinear optics of crystals and infrared up-conversion,” Infrared Phys. Technol. 35, 881 (1994).
[CrossRef]

Kurokawa, K.

T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, “The application of incoherent light for the study of femtosecond-picosecond relaxation in condensed phase,” Appl. Phys. B: Photophys. Laser Chem. 47, 107 (1988), and references therein.
[CrossRef]

Kurokawa, Y.

T. Yanagawa, Y. Kurokawa, H. Kasai, and H. Nakanishi, “Degenerate four-wave mixing using an optical parametric oscillator as an incoherent light source,” Opt. Commun. 137, 103 (1997).
[CrossRef]

Lane, R. J.

C. L. Tang, W. R. Bosenberg, T. Ukachi, R. J. Lane, and L. K. Cheng, “Optical parametric oscillators,” Proc. SPIE 80, 365 (1992), and references therein.

McIlveen, T. J.

M. J. T. Milton, T. J. McIlveen, D. C. Hanna, and P. T. Woods, “High-efficiency infrared generation by difference-frequency mixing using tangential phase matching,” Opt. Commun. 87, 273 (1992).
[CrossRef]

McKinnie, I. T.

Milton, M. J. T.

M. J. T. Milton, T. J. McIlveen, D. C. Hanna, and P. T. Woods, “High-efficiency infrared generation by difference-frequency mixing using tangential phase matching,” Opt. Commun. 87, 273 (1992).
[CrossRef]

Naganuma, K.

Nakanishi, H.

T. Yanagawa, Y. Kurokawa, H. Kasai, and H. Nakanishi, “Degenerate four-wave mixing using an optical parametric oscillator as an incoherent light source,” Opt. Commun. 137, 103 (1997).
[CrossRef]

Nikogosyan, D. N.

D. N. Nikogosyan, “Beta barium borate (BBO)—a review of its properties and applications,” Appl. Phys. A: Solids Surf. 52, 359 (1991).
[CrossRef]

Noll, R.

D. Brüggemann, J. Hertzberg, B. Wies, Y. Waschke, R. Noll, K.-F. Knoche, and G. Herziger, “Test of an optical parametric oscillator (OPO) as a compact and fast tunable Stokes source in coherent anti-Stokes Raman spectroscopy (CARS),” Appl. Phys. B: Photophys. Laser Chem. 55, 378 (1992).
[CrossRef]

Nolting, J.

S. Burdulis, R. Grigonis, A. Piskarskas, S. Sinkevicius, V. Sirutkaitis, A. Fix, J. Nolting, and R. Wallenstein, “Visible optical parametric oscillation in synchronously pumped beta-barium borate,” Opt. Commun. 74, 398 (1990).
[CrossRef]

Oien, A. L.

Orr, B. J.

Pelouch, W. S.

Piskarskas, A.

P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, R. Danielius, A. Dubietis, and A. Piskarskas, “Matching of group velocities in three-wave parametric interaction with femtosec-ond pulses and application to traveling-wave generators,” J. Opt. Soc. Am. B 12, 2237 (1995).
[CrossRef]

S. Burdulis, R. Grigonis, A. Piskarskas, S. Sinkevicius, V. Sirutkaitis, A. Fix, J. Nolting, and R. Wallenstein, “Visible optical parametric oscillation in synchronously pumped beta-barium borate,” Opt. Commun. 74, 398 (1990).
[CrossRef]

Powell, A. J.

Powers, P. E.

Rae, C. F.

Russell, N. A.

Rytz, D.

Schröder, B.

B. Schröder, “Frequency characteristic and spatial intensity distribution of light generated and amplified by parametric three-photon interaction,” Opt. Commun. 49, 75 (1984).
[CrossRef]

Schröder, T.

T. Schröder, K.-J. Boller, A. Fix, and R. Wallenstein, “Spectral properties and numerical modelling of a critically phase-matched nanosecond LiB3O5 optical parametric oscillator,” Appl. Phys. B: Laser Opt. 58, 425 (1994).
[CrossRef]

K.-J. Boller and T. Schröder, “Demonstration of broadband intracavity spectroscopy in a pulsed optical parametric oscillator made of β-barium borate,” J. Opt. Soc. Am. B 10, 1778 (1993).
[CrossRef]

A. Fix, T. Schröder, R. Wallenstein, J. G. Haub, M. J. Johnson, and B. J. Orr, “Tunable β-barium borate optical parametric oscillator: operating characteristics with and without injection seeding,” J. Opt. Soc. Am. B 10, 1744 (1993), and references therein.
[CrossRef]

A. Fix, T. Schröder, and R. Wallenstein, “The optical parametric oscillators of beta-barium borate and lithium borate: new sources of powerful tunable laser radiation in the ultraviolet, visible and near infrared,” Laser Optoelektron. 23, 106 (1991), and references therein.

Sinkevicius, S.

S. Burdulis, R. Grigonis, A. Piskarskas, S. Sinkevicius, V. Sirutkaitis, A. Fix, J. Nolting, and R. Wallenstein, “Visible optical parametric oscillation in synchronously pumped beta-barium borate,” Opt. Commun. 74, 398 (1990).
[CrossRef]

Sirutkaitis, V.

S. Burdulis, R. Grigonis, A. Piskarskas, S. Sinkevicius, V. Sirutkaitis, A. Fix, J. Nolting, and R. Wallenstein, “Visible optical parametric oscillation in synchronously pumped beta-barium borate,” Opt. Commun. 74, 398 (1990).
[CrossRef]

Solcia, C.

Tang, C. L.

P. E. Powers, R. J. Ellingson, W. S. Pelouch, and C. L. Tang, “Recent advances of the Ti:sapphire-pumped high-repetition-rate femtosecond optical parametric oscillator,” J. Opt. Soc. Am. B 10, 2162 (1993).
[CrossRef]

C. L. Tang, W. R. Bosenberg, T. Ukachi, R. J. Lane, and L. K. Cheng, “Optical parametric oscillators,” Proc. SPIE 80, 365 (1992), and references therein.

W. R. Bosenberg and C. L. Tang, “Type II phase matching in a β-barium borate optical parametric oscillator,” Appl. Phys. Lett. 56, 1819 (1990).
[CrossRef]

Terasaki, A.

T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, “The application of incoherent light for the study of femtosecond-picosecond relaxation in condensed phase,” Appl. Phys. B: Photophys. Laser Chem. 47, 107 (1988), and references therein.
[CrossRef]

Ukachi, T.

C. L. Tang, W. R. Bosenberg, T. Ukachi, R. J. Lane, and L. K. Cheng, “Optical parametric oscillators,” Proc. SPIE 80, 365 (1992), and references therein.

Urschel, R.

Wallenstein, R.

R. Urschel, A. Fix, R. Wallenstein, D. Rytz, and B. Zysset, “Generation of tunable narrow-band midinfrared radiation in a type I potassium niobate optical parametric oscillator,” J. Opt. Soc. Am. B 12, 726 (1995).
[CrossRef]

T. Schröder, K.-J. Boller, A. Fix, and R. Wallenstein, “Spectral properties and numerical modelling of a critically phase-matched nanosecond LiB3O5 optical parametric oscillator,” Appl. Phys. B: Laser Opt. 58, 425 (1994).
[CrossRef]

A. Fix, T. Schröder, R. Wallenstein, J. G. Haub, M. J. Johnson, and B. J. Orr, “Tunable β-barium borate optical parametric oscillator: operating characteristics with and without injection seeding,” J. Opt. Soc. Am. B 10, 1744 (1993), and references therein.
[CrossRef]

A. Fix, T. Schröder, and R. Wallenstein, “The optical parametric oscillators of beta-barium borate and lithium borate: new sources of powerful tunable laser radiation in the ultraviolet, visible and near infrared,” Laser Optoelektron. 23, 106 (1991), and references therein.

S. Burdulis, R. Grigonis, A. Piskarskas, S. Sinkevicius, V. Sirutkaitis, A. Fix, J. Nolting, and R. Wallenstein, “Visible optical parametric oscillation in synchronously pumped beta-barium borate,” Opt. Commun. 74, 398 (1990).
[CrossRef]

Wang, J.

Warner, J.

J. Warner, “Phase-matching for optical up-conversion with maximum angular aperture—theory and practice,” Opto-Electron. 1, 25 (1969).
[CrossRef]

Warrington, D. M.

Waschke, Y.

D. Brüggemann, J. Hertzberg, B. Wies, Y. Waschke, R. Noll, K.-F. Knoche, and G. Herziger, “Test of an optical parametric oscillator (OPO) as a compact and fast tunable Stokes source in coherent anti-Stokes Raman spectroscopy (CARS),” Appl. Phys. B: Photophys. Laser Chem. 55, 378 (1992).
[CrossRef]

Wies, B.

D. Brüggemann, J. Hertzberg, B. Wies, Y. Waschke, R. Noll, K.-F. Knoche, and G. Herziger, “Test of an optical parametric oscillator (OPO) as a compact and fast tunable Stokes source in coherent anti-Stokes Raman spectroscopy (CARS),” Appl. Phys. B: Photophys. Laser Chem. 55, 378 (1992).
[CrossRef]

Woods, P. T.

M. J. T. Milton, T. J. McIlveen, D. C. Hanna, and P. T. Woods, “High-efficiency infrared generation by difference-frequency mixing using tangential phase matching,” Opt. Commun. 87, 273 (1992).
[CrossRef]

Yanagawa, T.

T. Yanagawa, Y. Kurokawa, H. Kasai, and H. Nakanishi, “Degenerate four-wave mixing using an optical parametric oscillator as an incoherent light source,” Opt. Commun. 137, 103 (1997).
[CrossRef]

T. Yanagawa, K. Naganuma, H. Kanbara, and T. Kaino, “Optical parametric oscillator incoherent spectroscopy,” Opt. Lett. 21, 318 (1996).
[CrossRef] [PubMed]

Zadorozhnii, V. I.

S. G. Dolinchuk, N. E. Kornienko, and V. I. Zadorozhnii, “Noncritical vectorial phase matchings in nonlinear optics of crystals and infrared up-conversion,” Infrared Phys. Technol. 35, 881 (1994).
[CrossRef]

Zyss, J.

Zysset, B.

Appl. Opt.

Appl. Phys. A: Solids Surf.

D. N. Nikogosyan, “Beta barium borate (BBO)—a review of its properties and applications,” Appl. Phys. A: Solids Surf. 52, 359 (1991).
[CrossRef]

Appl. Phys. B: Laser Opt.

T. Schröder, K.-J. Boller, A. Fix, and R. Wallenstein, “Spectral properties and numerical modelling of a critically phase-matched nanosecond LiB3O5 optical parametric oscillator,” Appl. Phys. B: Laser Opt. 58, 425 (1994).
[CrossRef]

Appl. Phys. B: Photophys. Laser Chem.

G. C. Bahr, P. K. Datta, U. Chatterjee, S. Das, and H. L. Bhat, “Characterization of biaxial crystals for tangentially phase-matched frequency conversion,” Appl. Phys. B: Photophys. Laser Chem. 56, 327 (1993).
[CrossRef]

D. Brüggemann, J. Hertzberg, B. Wies, Y. Waschke, R. Noll, K.-F. Knoche, and G. Herziger, “Test of an optical parametric oscillator (OPO) as a compact and fast tunable Stokes source in coherent anti-Stokes Raman spectroscopy (CARS),” Appl. Phys. B: Photophys. Laser Chem. 55, 378 (1992).
[CrossRef]

T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, “The application of incoherent light for the study of femtosecond-picosecond relaxation in condensed phase,” Appl. Phys. B: Photophys. Laser Chem. 47, 107 (1988), and references therein.
[CrossRef]

Appl. Phys. Lett.

W. R. Bosenberg, L. K. Cheng, and J. D. Bierlein, “Optical parametric frequency conversion properties of KTiOAsO4,” Appl. Phys. Lett. 65, 2765 (1994).
[CrossRef]

R. L. Herbst, R. N. Fleming, and R. L. Byer, “A 1.4–4 μm high-energy angle-tuned LiNbO3 parametric oscillator,” Appl. Phys. Lett. 25, 520 (1974).
[CrossRef]

W. R. Bosenberg and C. L. Tang, “Type II phase matching in a β-barium borate optical parametric oscillator,” Appl. Phys. Lett. 56, 1819 (1990).
[CrossRef]

IEEE J. Quantum Electron.

L. A. W. Gloster, Z. X. Jiang, and T. A. King, “Characterization of an Nd:YAG-pumped β-BaB2O4 optical parametric oscillator in collinear and noncollinear phase-matched configurations,” IEEE J. Quantum Electron. 30, 2961 (1994).
[CrossRef]

M. Ebrahimzadeh, A. J. Henderson, and M. H. Dunn, “An excimer-pumped β-BaB2O4 optical parametric oscillator tunable from 354 nm to 2.370 μm,” IEEE J. Quantum Electron. 26, 1241 (1990).
[CrossRef]

K. Kato, “Second-harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron. QE-22, 1013 (1986).
[CrossRef]

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

Infrared Phys. Technol.

S. G. Dolinchuk, N. E. Kornienko, and V. I. Zadorozhnii, “Noncritical vectorial phase matchings in nonlinear optics of crystals and infrared up-conversion,” Infrared Phys. Technol. 35, 881 (1994).
[CrossRef]

J. Appl. Phys.

G. C. Bhar, P. K. Datta, and S. Das, “Tangentially phase matched second-harmonic generation in various crystals,” J. Appl. Phys. 71, 3620 (1992).
[CrossRef]

J. Opt. Soc. Am. B

S. X. Dou, D. Josse, and J. Zyss, “Noncritical properties of noncollinear phase-matched second-harmonic and sum-frequency generation in 3-methyl-4-nitropyridine-1-oxide,” J. Opt. Soc. Am. B 8, 1732 (1991).
[CrossRef]

S. X. Dou, D. Josse, and J. Zyss, “Comparison of collinear and one-beam noncritical noncollinear phase matching in optical parametric amplification,” J. Opt. Soc. Am. B 9, 1312 (1992).
[CrossRef]

W. R. Bosenberg and D. R. Guyer, “Broadly tunable, single-frequency optical parametric frequency-conversion system,” J. Opt. Soc. Am. B 10, 1716 (1993), and references therein.
[CrossRef]

P. E. Powers, R. J. Ellingson, W. S. Pelouch, and C. L. Tang, “Recent advances of the Ti:sapphire-pumped high-repetition-rate femtosecond optical parametric oscillator,” J. Opt. Soc. Am. B 10, 2162 (1993).
[CrossRef]

R. Urschel, A. Fix, R. Wallenstein, D. Rytz, and B. Zysset, “Generation of tunable narrow-band midinfrared radiation in a type I potassium niobate optical parametric oscillator,” J. Opt. Soc. Am. B 12, 726 (1995).
[CrossRef]

P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, R. Danielius, A. Dubietis, and A. Piskarskas, “Matching of group velocities in three-wave parametric interaction with femtosec-ond pulses and application to traveling-wave generators,” J. Opt. Soc. Am. B 12, 2237 (1995).
[CrossRef]

A. Fix, T. Schröder, R. Wallenstein, J. G. Haub, M. J. Johnson, and B. J. Orr, “Tunable β-barium borate optical parametric oscillator: operating characteristics with and without injection seeding,” J. Opt. Soc. Am. B 10, 1744 (1993), and references therein.
[CrossRef]

K.-J. Boller and T. Schröder, “Demonstration of broadband intracavity spectroscopy in a pulsed optical parametric oscillator made of β-barium borate,” J. Opt. Soc. Am. B 10, 1778 (1993).
[CrossRef]

Laser Optoelektron.

A. Fix, T. Schröder, and R. Wallenstein, “The optical parametric oscillators of beta-barium borate and lithium borate: new sources of powerful tunable laser radiation in the ultraviolet, visible and near infrared,” Laser Optoelektron. 23, 106 (1991), and references therein.

Opt. Commun.

M. J. T. Milton, T. J. McIlveen, D. C. Hanna, and P. T. Woods, “High-efficiency infrared generation by difference-frequency mixing using tangential phase matching,” Opt. Commun. 87, 273 (1992).
[CrossRef]

S. Burdulis, R. Grigonis, A. Piskarskas, S. Sinkevicius, V. Sirutkaitis, A. Fix, J. Nolting, and R. Wallenstein, “Visible optical parametric oscillation in synchronously pumped beta-barium borate,” Opt. Commun. 74, 398 (1990).
[CrossRef]

B. Schröder, “Frequency characteristic and spatial intensity distribution of light generated and amplified by parametric three-photon interaction,” Opt. Commun. 49, 75 (1984).
[CrossRef]

L. A. W. Gloster, I. T. McKinnie, and T. A. King, “Noncollinear phase matching in a type I barium borate optical parametric oscillator,” Opt. Commun. 112, 328 (1994).
[CrossRef]

T. Yanagawa, Y. Kurokawa, H. Kasai, and H. Nakanishi, “Degenerate four-wave mixing using an optical parametric oscillator as an incoherent light source,” Opt. Commun. 137, 103 (1997).
[CrossRef]

Opt. Lett.

Opto-Electron.

J. Warner, “Phase-matching for optical up-conversion with maximum angular aperture—theory and practice,” Opto-Electron. 1, 25 (1969).
[CrossRef]

Proc. SPIE

C. L. Tang, W. R. Bosenberg, T. Ukachi, R. J. Lane, and L. K. Cheng, “Optical parametric oscillators,” Proc. SPIE 80, 365 (1992), and references therein.

Other

R. Urschel, U. Bäder, A. Borsutzky, and R. Wallenstein, “Pulsed optical parametric oscillators with noncollinear phase matching,” in Advanced Solid State Lasers, C. R. Pollock and W. R. Bosenberg, eds., Vol. 10 of Trends in Optics and Photonics (Optical Society of America, Washington, D.C. 1997), pp. 94–96.

R. L. Byer, “Optical parametric oscillators,” in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975), Vol. 1, Part B, pp. 587–702.

V. G. Dmitriev, G. G. Gurzandyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, 2nd ed. (Springer-Verlag, Berlin, 1995).

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Figures (31)

Fig. 1
Fig. 1

Illustration of type I tangential phase matching (e>o+o). kp, ks, and ki are wave vectors and Sp and Si are the corresponding Poynting vectors. For further details, see text.

Fig. 2
Fig. 2

Illustration of the influence of the divergence Δθs of the OPO signal wave on the OPO bandwidth. The mismatch Δk caused by the different directions of ks (marked by dashed circles) is compensated for by the corresponding change in the direction of ki and a wavelength shift of the OPO (lengths of ks and ki). (a) collinear type I phase matching (e>o+o), (b) noncollinear type I phase matching (e>o+o), (c) collinear type II phase matching (e>o+o).

Fig. 3
Fig. 3

(a) Noncollinear angles θnc,s of the signal wave and θnc,i of the idler wave of a 355-nm-pumped tangentially phase-matched type I BBO OPO. (b) Angular pump wave acceptance of the 355-nm-pumped type I BBO OPO with tangential and collinear phase matching.

Fig. 4
Fig. 4

Dependence of the wavelength of the signal wave on the phase-matching angle calculated for a 355-nm-pumped type I BBO OPO with tangential, collinear, and noncollinear (θnc,s=-1°, -2°, -3°, -4°) phase matching.

Fig. 5
Fig. 5

Angular pump wave acceptance of the noncollinearly phase-matched type I BBO OPO. The noncollinear angle θnc,s=-2° provides tangential phase matching at signal wavelength λs=527 nm.

Fig. 6
Fig. 6

(a) Noncollinear angles θnc,s of the signal wave and θnc,i of the idler wave of a 355-nm-pumped tangentially phase-matched type II BBO OPO (e>o+e). (b) Angular pump wave acceptance of the 355-nm-pumped type II BBO OPO (e>o+e) with collinear and tangential phase matching.

Fig. 7
Fig. 7

Signal wavelength of the type I and type II 355-nm-pumped BBO OPO’s as a function of the noncollinear angle θnc,s. In each case the wavelength of the signal wave generated with collinear phase matching is 500 nm.

Fig. 8
Fig. 8

Signal bandwidth of the type I and type II 355-nm-pumped BBO OPO’s as a function of the noncollinear angle θnc,s for a signal wave with a wavelength of 500 nm and a divergence of 2 mrad.

Fig. 9
Fig. 9

Noncollinear angle θnc,s that provides a minimum bandwidth of the signal wave of the 355-nm-pumped type II BBO OPO (e>o+e).

Fig. 10
Fig. 10

Measured (filled circles) and calculated (solid curve) wavelengths of the signal radiation of the 355-nm-pumped tangentially phase-matched type I BBO OPO as a function of the phase-matching angle. For comparison, the dashed curve indicates the wavelengths of the corresponding collinearly phase-matched BBO OPO.

Fig. 11
Fig. 11

Measured (filled circles) and calculated (solid curve) noncollinear angle θnc,s of the tangentially phase-matched type I BBO OPO as a function of the signal wavelength.

Fig. 12
Fig. 12

Conversion efficiency of the 355-nm-pumped noncollinearly phase-matched type I BBO OPO as a function of the signal wavelength. The constant noncollinear angle of θnc,s=-2.1° corresponds to tangential phase matching at λs=540 nm. The divergence of the pump beam at 0.2, 1.0, and 3.0 mrad is shown; the pump energy density is 400 mJ/cm2.

Fig. 13
Fig. 13

Conversion efficiency of the type I BBO OPO as a function of noncollinear angle θnc,s measured for a signal wavelength of 540 nm. The phase matching is tangential at θnc,s=-2.1°. The divergence of the pump beam at 0.2, 1.2, and 3.2 mrad is shown; the pump energy density is 370 mJ/cm2.

Fig. 14
Fig. 14

(a) Conversion efficiency of the type I BBO OPO with tangential or collinear phase matching as a function of the pump divergence. The signal wavelength is 540 nm and the pump energy density is 510 mJ/cm2. (b) Pump energy density at threshold of the type I BBO OPO with tangential and collinear phase matching at λs=540 nm as a function of the pump divergence.

Fig. 15
Fig. 15

Conversion efficiency of the 355-nm-pumped type II BBO OPO with collinear phase matching as a function of the pump divergence. The signal wavelength is 541 nm and the pump energy density is 800 mJ/cm2.

Fig. 16
Fig. 16

Conversion efficiency of the type II BBO OPO as a function of noncollinear angle θnc,s. For the signal wavelength of 541 nm the phase matching is tangential at θnc,s=-0.18°. The divergence of the pump beam is 0.3 or 3.0 mrad and the pump energy density is 820 mJ/cm2.

Fig. 17
Fig. 17

Pulse energies of the signal and the idler wave of a tangentially phase-matched type I BBO OPO pumped by the third harmonic of a compact Nd:YAG laser (pulse energy, 10.5 mJ; beam divergence, 2 mrad). The OPO is singly resonant for the signal wave (30% output coupling) and the optical cavity length is 25 mm.

Fig. 18
Fig. 18

Bandwidth of the signal wave of the 355-nm-pumped type I BBO OPO with noncollinear and collinear phase matching as a function of the pump divergence. The signal wavelength is 450 nm, the pump energy density is 530 mJ/cm2, and the length of the optical OPO cavity is 31 mm.

Fig. 19
Fig. 19

Bandwidth of the signal wave of the type I BBO OPO with noncollinear and collinear phase matching as a function of the pump energy, given by the ratio of pump power P and pump power PT at threshold. The signal wavelength is 451 nm, the pump divergence is 0.2 mrad, and the length of the optical OPO cavity is 31 mm.

Fig. 20
Fig. 20

(a) Divergence and (b) bandwidth of the signal beam of the type I BBO OPO with noncollinear and collinear phase matching as a function of the length of the optical OPO cavity. The signal wavelength is 451 nm, the pump divergence is 0.2 mrad, and the signal pulse energy is 2 mJ.

Fig. 21
Fig. 21

Signal bandwidth of the noncollinearly phase-matched type I BBO OPO as a function of signal beam divergence. The dotted curve represents the calculated dependence from the model of Section 3. The solid curve represents the same calculations but multiplied by a factor of 1.6.

Fig. 22
Fig. 22

Measured (filled circles) and calculated (solid curve) spatial variation of the wavelength across the signal beam of the tangentially phase-matched type I BBO OPO. The wavelengths are measured behind a slit of 0.2-mm width at a distance of 3.2 m from the OPO. The error bars represent the bandwidth of the signal radiation detected behind the slit. The total signal bandwidth (represented by the dashed horizontal lines) is 3.9 nm and the total signal divergence (dashed vertical lines) 4.2 mrad. The pump divergence is 0.2 mrad; the length of the optical OPO cavity is 24 mm.

Fig. 23
Fig. 23

Dependence of the signal bandwidth measured behind the slit on the beam direction. The solid curve represents the values calculated for a beam diameter of 2.9 mm and a slit of 0.2 mm width for a distance of 3.2 m.

Fig. 24
Fig. 24

Measured (filled circles) and calculated (solid curve) spatial variation of the wavelength across the signal beam of the collinearly phase-matched type I BBO OPO measured behind a slit of 0.2 mm width at a distance of 3.2 m from the OPO. The error bars represent the bandwidth of the signal radiation measured behind the slit. The total signal bandwidth (represented by the dashed horizontal lines) is 0.16 nm and the total signal divergence (dashed vertical lines) 3.8 mrad. The pump divergence is 0.2 mrad; the OPO optical cavity length is 24 mm.

Fig. 25
Fig. 25

Measured (open and filled circles) and calculated (solid curves) signal wavelengths of the 355-nm-pumped type I BBO OPO with collinear and noncollinear (θnc,s=-1°, -2°, -3°) phase matching as a function of the phase-matching angle.

Fig. 26
Fig. 26

Bandwidth of the signal wave of the type I BBO OPO with noncollinear phase matching as a function of the signal wavelength. As in Fig. 24, the noncollinear angles θnc,s are -1°, -2°, and -3°. The pump divergence is 0.2 mrad; the OPO optical cavity length 31 mm.

Fig. 27
Fig. 27

Bandwidth of the 560-nm signal wave of a type I BBO OPO as a function of noncollinear angle θnc,s. The pump divergence is 0.2 mrad; the OPO optical cavity length is 31 mm. The dashed curve represents the signal bandwidth as determined by the amplification bandwidth.

Fig. 28
Fig. 28

Measured (filled circles) and calculated (solid curve) signal wavelength of the 355-nm-pumped type II BBO OPO (e>o+e) as a function of noncollinear angle θnc,s. The phase-matching angle is 36.2°; the corresponding signal wavelength for collinear phase matching is 504 nm.

Fig. 29
Fig. 29

Signal bandwidth of the type II BBO OPO (e>o+e) as a function of noncollinear angle θnc,s. The pump divergence is 0.2 mrad; the OPO optical cavity length 24 mm. The dashed line represents the calculated amplification bandwidth.

Fig. 30
Fig. 30

Measured (filled circles) and calculated (solid curve) spatial variation of the wavelength across the signal beam of the collinearly phase-matched type II BBO OPO as measured behind a slit of 0.2 mm width at a distance of 3.2 m from the OPO. The error bars represent the bandwidth of the signal radiation measured behind the slit. The total signal bandwidth (indicated by the dashed horizontal lines) is 1.0 nm and the total signal divergence (dashed vertical lines) 2.3 mrad. The pump divergence is 0.2 mrad; the OPO optical cavity length is 24 mm.

Fig. 31
Fig. 31

Calculated correlation time of a 355-nm-pumped type I BBO OPO with collinear and noncollinear phase matching (θnc,s=-1°, -2°). The calculations are based on the bandwidths of Fig. 25 assuming Gaussian-shaped OPO spectra.

Tables (1)

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Table 1 Noncollinear Angle Θnc,i Required for Tangential Phase Matching

Equations (9)

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ρ(θ)=arctan[(no/ne)2 tan θ]-θ,
kp=ks+ki,
ωpn(ωp, θpm)=ωsn(ωs, θpm+θnc,s)cos θnc,s+ωin(ωi, θpm+θnc,i)cos θnc, i,
0=ωsn(ωs, θpm+θnc,s)sin θnc,s+ωin(ωi, θpm+θnc,i)sin θnc,i.
Δk=kp-ks-ki.
Δkpar=1c[ωpn(ωp, θpm)-ωsn(ωs, θpm+θnc,s)×cos θnc,s-ωin(ωi, θpm+θnc,i)cos θnc,i],
Δkperp=1c[ωsn(ωs, θpm+θnc,s)sin θnc,s+ωin(ωi, θpm+θnc,i)sin θnc,i].
Δk=(Δkpar2+Δkperp2)1/2.
Δ k lc=0.886π,

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