Abstract

The azimuthal dependence of third-order and cascaded second-order nonlinear coupling are used to measure the relative contributions of each to direct third-harmonic generation in β-barium borate. This enabled the measurement of the values of χ10(3), χ11(3), and χ16(3) relative to the known χij(2). Finally, conversion efficiencies to 3ω of up to 6% from a single crystal were achieved with a femtosecond chirped-pulse-amplification laser with 200 GW/cm2 in collimated beams.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]

1999 (3)

1996 (6)

J. M. Burzler, S. Hughes, and B. S. Wherrett, “Split-step Fourier methods applied to model nonlinear refractive effects in optically thick media,” Appl. Phys. B 62, 389–397 (1996).
[CrossRef]

R. E. Bridges, R. W. Boyd, and G. P. Agrawal, “Multidimensional coupling owing to optical nonlinearities. I. General formulation,” J. Opt. Soc. Am. B 13, 553–559, 560–569 (1996).
[CrossRef]

T. Ditmire, A. M. Rubenchik, D. Eimerl, and M. D. Perry, “Effects of cubic nonlinearity on frequency doubling of high-power laser pulses,” J. Opt. Soc. Am. B 13, 649–655 (1996).
[CrossRef]

A. J. Taylor, G. Rodriguez, and T. S. Clement, “Determination of n2 by direct measurement of the optical phase,” Opt. Lett. 21, 1812–1814 (1996).
[CrossRef] [PubMed]

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

1995 (3)

1994 (1)

M. D. Perry and G. Mourou, “Terawatt to petawatt subpicosecond lasers,” Science 264, 917–924 (1994).
[CrossRef] [PubMed]

1993 (2)

C. Schwan, A. Penzkofer, N. J. Marx, and K. H. Drexhage, “Phase-matched third-harmonic generation of Nd: glass-laser picosecond pulses in a new cyanine-dye solution,” Appl. Phys. B 57, 203–211 (1993).
[CrossRef]

R. Danielus, P. Di Trapani, A. Dubietis, A. Piskarskas, D. Podenas, and G. P. Banfi, “Self-diffraction through cascaded second-order frequency-mixing effects in β-barium borate,” Opt. Lett. 18, 574–576 (1993).
[CrossRef]

1992 (4)

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-harmonic generation,” IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

1989 (2)

Y. X. Fan, R. C. Eckardt, R. L. Byer, C. Chen, and A. D. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron. 25, 1196–1199 (1989).
[CrossRef]

D. Eimerl, S. Velsko, L. Davis, F. Wang, G. Loiacono, and G. Kennedy, “Deuterated L-arginine phosphate: a new efficient nonlinear crystal,” IEEE J. Quantum Electron. 25, 179–193 (1989).
[CrossRef]

1988 (1)

P. Qiu and A. Penzkofer, “Picosecond third-harmonic light generation in β-BaB2O4,” Appl. Phys. B 45, 225–236 (1988).
[CrossRef]

1987 (2)

C. C. Shang and H. Hsu, “The spatial symmetric forms of third-order nonlinear susceptibility,” IEEE J. Quantum Electron. QE-23, 177–179 (1987).
[CrossRef]

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

1986 (1)

Y. Zhao, “The spatial symmetry of the third-order susceptibility tensor,” IEEE J. Quantum Electron. QE-22, 1012 (1986).
[CrossRef]

1985 (1)

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985).
[CrossRef]

1982 (1)

G. R. Meredith, “Second-order cascading in third-order nonlinear optical processes,” J. Chem. Phys. 77, 5863–5871 (1982).
[CrossRef]

1981 (2)

G. R. Meredith, “Cascading in optical third-harmonic generation by crystalline quartz,” Phys. Rev. B 24, 5522–5532 (1981).
[CrossRef]

G. R. Meredith, “Lower order effects in nonlinear two- and three-photon resonance spectroscopies,” J. Chem. Phys. 75, 4317–4325 (1981).
[CrossRef]

1980 (1)

1977 (2)

S. A. Akhmanov, L. B. Meisner, S. T. Parinov, S. M. Saltiel, and V. G. Tunkin, “Cubic nonlinear susceptibilities of crystals in the optical band; the signs and magnitudes of the susceptibilities of crystals with and without centers of inversion,” Sov. Phys. JETP 46, 898–907 (1977).

V. V. Rostovtseva, A. P. Sukhorukov, V. G. Tunkin, and S. M. Saltiel, “Higher harmonics generation by cascade processes in focused beams,” Opt. Commun. 22, 56–60 (1977).
[CrossRef]

1976 (2)

C. Flytzanis and N. Bloembergen, “Infrared dispersion of third-order susceptibilities in dielectrics: retardation effects,” Prog. Quantum Electron. 4, 271–300 (1976).
[CrossRef]

J. A. Fleck Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. B 10, 129–160 (1976).
[CrossRef]

1974 (1)

J. C. Diels and F. P. Schafer, “Phase-matched third-harmonic generation in dye solutions,” Appl. Phys. 5, 197–202 (1974).
[CrossRef]

1973 (1)

R. B. Miles and S. E. Harris, “Optical third-harmonic gen-eration in alkali metal vapors,” IEEE J. Quantum Electron. QE-9, 470–484 (1973).
[CrossRef]

1972 (1)

C. Greninger, J. Marburger, and D. R. White, “Influence of third-order nonlinearity on second-harmonic generation,” IEEE J. Quantum Electron. QE-8, 543–544 (1972).
[CrossRef]

1971 (1)

M. Okada, “Third-order nonlinear optical coefficients of LiIO3,” Appl. Phys. Lett. 18, 451–452 (1971).
[CrossRef]

1969 (1)

J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185, 57–72 (1969).
[CrossRef]

1965 (1)

J. E. Midwinter and J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order non-linear optical polarization,” Brit. J. Appl. Phys. 16, 1667–1674 (1965).
[CrossRef]

Agrawal, G. P.

Akhmanov, S. A.

S. A. Akhmanov, L. B. Meisner, S. T. Parinov, S. M. Saltiel, and V. G. Tunkin, “Cubic nonlinear susceptibilities of crystals in the optical band; the signs and magnitudes of the susceptibilities of crystals with and without centers of inversion,” Sov. Phys. JETP 46, 898–907 (1977).

Backus, S.

Banfi, G. P.

Banks, P. S.

Bloembergen, N.

C. Flytzanis and N. Bloembergen, “Infrared dispersion of third-order susceptibilities in dielectrics: retardation effects,” Prog. Quantum Electron. 4, 271–300 (1976).
[CrossRef]

Boyd, R. W.

Bridges, R. E.

Burzler, J. M.

J. M. Burzler, S. Hughes, and B. S. Wherrett, “Split-step Fourier methods applied to model nonlinear refractive effects in optically thick media,” Appl. Phys. B 62, 389–397 (1996).
[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-harmonic generation,” IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

Y. X. Fan, R. C. Eckardt, R. L. Byer, C. Chen, and A. D. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron. 25, 1196–1199 (1989).
[CrossRef]

Chen, C.

Y. X. Fan, R. C. Eckardt, R. L. Byer, C. Chen, and A. D. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron. 25, 1196–1199 (1989).
[CrossRef]

Chien, C. Y.

Clement, T. S.

Coe, J. S.

Craxton, R. S.

Danielus, R.

Davis, L.

D. Eimerl, S. Velsko, L. Davis, F. Wang, G. Loiacono, and G. Kennedy, “Deuterated L-arginine phosphate: a new efficient nonlinear crystal,” IEEE J. Quantum Electron. 25, 179–193 (1989).
[CrossRef]

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

DeSalvo, R.

Di Trapani, P.

Diels, J. C.

J. C. Diels and F. P. Schafer, “Phase-matched third-harmonic generation in dye solutions,” Appl. Phys. 5, 197–202 (1974).
[CrossRef]

Ditmire, T.

Drexhage, K. H.

C. Schwan, A. Penzkofer, N. J. Marx, and K. H. Drexhage, “Phase-matched third-harmonic generation of Nd: glass-laser picosecond pulses in a new cyanine-dye solution,” Appl. Phys. B 57, 203–211 (1993).
[CrossRef]

Dubietis, A.

Durfee III, C. G.

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]

Y. X. Fan, R. C. Eckardt, R. L. Byer, C. Chen, and A. D. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron. 25, 1196–1199 (1989).
[CrossRef]

Eimerl, D.

T. Ditmire, A. M. Rubenchik, D. Eimerl, and M. D. Perry, “Effects of cubic nonlinearity on frequency doubling of high-power laser pulses,” J. Opt. Soc. Am. B 13, 649–655 (1996).
[CrossRef]

M. S. Webb, D. Eimerl, and S. P. Velsko, “Wavelength insensitive phase-matched second-harmonic generation in partially deuterated KDP,” J. Opt. Soc. Am. B 9, 1118–1127 (1992).
[CrossRef]

D. Eimerl, S. Velsko, L. Davis, F. Wang, G. Loiacono, and G. Kennedy, “Deuterated L-arginine phosphate: a new efficient nonlinear crystal,” IEEE J. Quantum Electron. 25, 179–193 (1989).
[CrossRef]

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[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-harmonic generation,” IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

Y. X. Fan, R. C. Eckardt, R. L. Byer, C. Chen, and A. D. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron. 25, 1196–1199 (1989).
[CrossRef]

Feit, M. D.

P. S. Banks, M. D. Feit, and M. D. Perry, “High-intensity third-harmonic generation in beta berium borate through second-order and third-order susceptibilities,” Opt. Lett. 24, 4–6 (1999).
[CrossRef]

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).
[CrossRef]

M. D. Feit and J. A. Fleck, Jr., “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154–1164 (1980).
[CrossRef] [PubMed]

J. A. Fleck Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. B 10, 129–160 (1976).
[CrossRef]

Fleck Jr., J. A.

M. D. Feit and J. A. Fleck, Jr., “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154–1164 (1980).
[CrossRef] [PubMed]

J. A. Fleck Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. B 10, 129–160 (1976).
[CrossRef]

Flytzanis, C.

C. Flytzanis and N. Bloembergen, “Infrared dispersion of third-order susceptibilities in dielectrics: retardation effects,” Prog. Quantum Electron. 4, 271–300 (1976).
[CrossRef]

Gale, G. M.

Gallot, G.

Graham, E. K.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

Greninger, C.

C. Greninger, J. Marburger, and D. R. White, “Influence of third-order nonlinearity on second-harmonic generation,” IEEE J. Quantum Electron. QE-8, 543–544 (1972).
[CrossRef]

Hache, F.

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, and E. W. Van Stryland, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
[CrossRef] [PubMed]

Harris, S. E.

R. B. Miles and S. E. Harris, “Optical third-harmonic gen-eration in alkali metal vapors,” IEEE J. Quantum Electron. QE-9, 470–484 (1973).
[CrossRef]

Herman, S.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).
[CrossRef]

Hsu, H.

C. C. Shang and H. Hsu, “The spatial symmetric forms of third-order nonlinear susceptibility,” IEEE J. Quantum Electron. QE-23, 177–179 (1987).
[CrossRef]

Hughes, S.

J. M. Burzler, S. Hughes, and B. S. Wherrett, “Split-step Fourier methods applied to model nonlinear refractive effects in optically thick media,” Appl. Phys. B 62, 389–397 (1996).
[CrossRef]

Ito, R.

Jiang, A. D.

Y. X. Fan, R. C. Eckardt, R. L. Byer, C. Chen, and A. D. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron. 25, 1196–1199 (1989).
[CrossRef]

Kapteyn, H. C.

Kennedy, G.

D. Eimerl, S. Velsko, L. Davis, F. Wang, G. Loiacono, and G. Kennedy, “Deuterated L-arginine phosphate: a new efficient nonlinear crystal,” IEEE J. Quantum Electron. 25, 179–193 (1989).
[CrossRef]

Kondo, T.

Korn, G.

Kubota, S.

Loiacono, G.

D. Eimerl, S. Velsko, L. Davis, F. Wang, G. Loiacono, and G. Kennedy, “Deuterated L-arginine phosphate: a new efficient nonlinear crystal,” IEEE J. Quantum Electron. 25, 179–193 (1989).
[CrossRef]

Marburger, J.

C. Greninger, J. Marburger, and D. R. White, “Influence of third-order nonlinearity on second-harmonic generation,” IEEE J. Quantum Electron. QE-8, 543–544 (1972).
[CrossRef]

Marx, N. J.

C. Schwan, A. Penzkofer, N. J. Marx, and K. H. Drexhage, “Phase-matched third-harmonic generation of Nd: glass-laser picosecond pulses in a new cyanine-dye solution,” Appl. Phys. B 57, 203–211 (1993).
[CrossRef]

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]

Meisner, L. B.

S. A. Akhmanov, L. B. Meisner, S. T. Parinov, S. M. Saltiel, and V. G. Tunkin, “Cubic nonlinear susceptibilities of crystals in the optical band; the signs and magnitudes of the susceptibilities of crystals with and without centers of inversion,” Sov. Phys. JETP 46, 898–907 (1977).

Meredith, G. R.

G. R. Meredith, “Second-order cascading in third-order nonlinear optical processes,” J. Chem. Phys. 77, 5863–5871 (1982).
[CrossRef]

G. R. Meredith, “Cascading in optical third-harmonic generation by crystalline quartz,” Phys. Rev. B 24, 5522–5532 (1981).
[CrossRef]

G. R. Meredith, “Lower order effects in nonlinear two- and three-photon resonance spectroscopies,” J. Chem. Phys. 75, 4317–4325 (1981).
[CrossRef]

Midwinter, J. E.

J. E. Midwinter and J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order non-linear optical polarization,” Brit. J. Appl. Phys. 16, 1667–1674 (1965).
[CrossRef]

Miles, R. B.

R. B. Miles and S. E. Harris, “Optical third-harmonic gen-eration in alkali metal vapors,” IEEE J. Quantum Electron. QE-9, 470–484 (1973).
[CrossRef]

Morris, J. R.

J. A. Fleck Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. B 10, 129–160 (1976).
[CrossRef]

Mourou, G.

C. Y. Chien, G. Korn, J. S. Coe, J. Squier, G. Mourou, and R. S. Craxton, “Highly efficient second-harmonic generation of ultraintense Nd:glass laser pulses,” Opt. Lett. 20, 353–355 (1995).
[CrossRef] [PubMed]

M. D. Perry and G. Mourou, “Terawatt to petawatt subpicosecond lasers,” Science 264, 917–924 (1994).
[CrossRef] [PubMed]

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985).
[CrossRef]

Murnane, M. M.

Nakamura, H.

New, G. H. C.

J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185, 57–72 (1969).
[CrossRef]

Ohdaira, K.

Okada, M.

M. Okada, “Third-order nonlinear optical coefficients of LiIO3,” Appl. Phys. Lett. 18, 451–452 (1971).
[CrossRef]

Okamoto, T.

Parinov, S. T.

S. A. Akhmanov, L. B. Meisner, S. T. Parinov, S. M. Saltiel, and V. G. Tunkin, “Cubic nonlinear susceptibilities of crystals in the optical band; the signs and magnitudes of the susceptibilities of crystals with and without centers of inversion,” Sov. Phys. JETP 46, 898–907 (1977).

Penzkofer, A.

C. Schwan, A. Penzkofer, N. J. Marx, and K. H. Drexhage, “Phase-matched third-harmonic generation of Nd: glass-laser picosecond pulses in a new cyanine-dye solution,” Appl. Phys. B 57, 203–211 (1993).
[CrossRef]

P. Qiu and A. Penzkofer, “Picosecond third-harmonic light generation in β-BaB2O4,” Appl. Phys. B 45, 225–236 (1988).
[CrossRef]

Perry, M. D.

P. S. Banks, M. D. Feit, and M. D. Perry, “High-intensity third-harmonic generation in beta berium borate through second-order and third-order susceptibilities,” Opt. Lett. 24, 4–6 (1999).
[CrossRef]

T. Ditmire, A. M. Rubenchik, D. Eimerl, and M. D. Perry, “Effects of cubic nonlinearity on frequency doubling of high-power laser pulses,” J. Opt. Soc. Am. B 13, 649–655 (1996).
[CrossRef]

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).
[CrossRef]

M. D. Perry and G. Mourou, “Terawatt to petawatt subpicosecond lasers,” Science 264, 917–924 (1994).
[CrossRef] [PubMed]

Piskarskas, A.

Podenas, D.

Qiu, P.

P. Qiu and A. Penzkofer, “Picosecond third-harmonic light generation in β-BaB2O4,” Appl. Phys. B 45, 225–236 (1988).
[CrossRef]

Rentzepis, P. M.

Roberts, D. A.

D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

Rodriguez, G.

Rostovtseva, V. V.

V. V. Rostovtseva, A. P. Sukhorukov, V. G. Tunkin, and S. M. Saltiel, “Higher harmonics generation by cascade processes in focused beams,” Opt. Commun. 22, 56–60 (1977).
[CrossRef]

Rubenchik, A. M.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).
[CrossRef]

T. Ditmire, A. M. Rubenchik, D. Eimerl, and M. D. Perry, “Effects of cubic nonlinearity on frequency doubling of high-power laser pulses,” J. Opt. Soc. Am. B 13, 649–655 (1996).
[CrossRef]

Saltiel, S. M.

V. V. Rostovtseva, A. P. Sukhorukov, V. G. Tunkin, and S. M. Saltiel, “Higher harmonics generation by cascade processes in focused beams,” Opt. Commun. 22, 56–60 (1977).
[CrossRef]

S. A. Akhmanov, L. B. Meisner, S. T. Parinov, S. M. Saltiel, and V. G. Tunkin, “Cubic nonlinear susceptibilities of crystals in the optical band; the signs and magnitudes of the susceptibilities of crystals with and without centers of inversion,” Sov. Phys. JETP 46, 898–907 (1977).

Schafer, F. P.

J. C. Diels and F. P. Schafer, “Phase-matched third-harmonic generation in dye solutions,” Appl. Phys. 5, 197–202 (1974).
[CrossRef]

Schwan, C.

C. Schwan, A. Penzkofer, N. J. Marx, and K. H. Drexhage, “Phase-matched third-harmonic generation of Nd: glass-laser picosecond pulses in a new cyanine-dye solution,” Appl. Phys. B 57, 203–211 (1993).
[CrossRef]

Shang, C. C.

C. C. Shang and H. Hsu, “The spatial symmetric forms of third-order nonlinear susceptibility,” IEEE J. Quantum Electron. QE-23, 177–179 (1987).
[CrossRef]

Sheik-Bahae, M.

Shoji, I.

Shore, B. W.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).
[CrossRef]

Squier, J.

Stegeman, G.

Stegeman, G. I.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Strickland, D.

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985).
[CrossRef]

Stuart, B. C.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).
[CrossRef]

Sukhorukov, A. P.

V. V. Rostovtseva, A. P. Sukhorukov, V. G. Tunkin, and S. M. Saltiel, “Higher harmonics generation by cascade processes in focused beams,” Opt. Commun. 22, 56–60 (1977).
[CrossRef]

Tatsuki, K.

Taylor, A. J.

Tomov, I. V.

Torner, L.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Tunkin, V. G.

V. V. Rostovtseva, A. P. Sukhorukov, V. G. Tunkin, and S. M. Saltiel, “Higher harmonics generation by cascade processes in focused beams,” Opt. Commun. 22, 56–60 (1977).
[CrossRef]

S. A. Akhmanov, L. B. Meisner, S. T. Parinov, S. M. Saltiel, and V. G. Tunkin, “Cubic nonlinear susceptibilities of crystals in the optical band; the signs and magnitudes of the susceptibilities of crystals with and without centers of inversion,” Sov. Phys. JETP 46, 898–907 (1977).

Van Stryland, E. W.

Van Wonterghem, B.

Velsko, S.

D. Eimerl, S. Velsko, L. Davis, F. Wang, G. Loiacono, and G. Kennedy, “Deuterated L-arginine phosphate: a new efficient nonlinear crystal,” IEEE J. Quantum Electron. 25, 179–193 (1989).
[CrossRef]

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

Velsko, S. P.

Wang, F.

D. Eimerl, S. Velsko, L. Davis, F. Wang, G. Loiacono, and G. Kennedy, “Deuterated L-arginine phosphate: a new efficient nonlinear crystal,” IEEE J. Quantum Electron. 25, 179–193 (1989).
[CrossRef]

Ward, J. F.

J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185, 57–72 (1969).
[CrossRef]

Warner, J.

J. E. Midwinter and J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order non-linear optical polarization,” Brit. J. Appl. Phys. 16, 1667–1674 (1965).
[CrossRef]

Webb, M. S.

Wherrett, B. S.

J. M. Burzler, S. Hughes, and B. S. Wherrett, “Split-step Fourier methods applied to model nonlinear refractive effects in optically thick media,” Appl. Phys. B 62, 389–397 (1996).
[CrossRef]

White, D. R.

C. Greninger, J. Marburger, and D. R. White, “Influence of third-order nonlinearity on second-harmonic generation,” IEEE J. Quantum Electron. QE-8, 543–544 (1972).
[CrossRef]

Xie, S.

Yang, X.

Zalkin, A.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

Zeboulon, A.

Zhao, Y.

Y. Zhao, “The spatial symmetry of the third-order susceptibility tensor,” IEEE J. Quantum Electron. QE-22, 1012 (1986).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. (1)

J. C. Diels and F. P. Schafer, “Phase-matched third-harmonic generation in dye solutions,” Appl. Phys. 5, 197–202 (1974).
[CrossRef]

Appl. Phys. B (4)

P. Qiu and A. Penzkofer, “Picosecond third-harmonic light generation in β-BaB2O4,” Appl. Phys. B 45, 225–236 (1988).
[CrossRef]

J. M. Burzler, S. Hughes, and B. S. Wherrett, “Split-step Fourier methods applied to model nonlinear refractive effects in optically thick media,” Appl. Phys. B 62, 389–397 (1996).
[CrossRef]

J. A. Fleck Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. B 10, 129–160 (1976).
[CrossRef]

C. Schwan, A. Penzkofer, N. J. Marx, and K. H. Drexhage, “Phase-matched third-harmonic generation of Nd: glass-laser picosecond pulses in a new cyanine-dye solution,” Appl. Phys. B 57, 203–211 (1993).
[CrossRef]

Appl. Phys. Lett. (1)

M. Okada, “Third-order nonlinear optical coefficients of LiIO3,” Appl. Phys. Lett. 18, 451–452 (1971).
[CrossRef]

Brit. J. Appl. Phys. (1)

J. E. Midwinter and J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order non-linear optical polarization,” Brit. J. Appl. Phys. 16, 1667–1674 (1965).
[CrossRef]

IEEE J. Quantum Electron. (8)

Y. X. Fan, R. C. Eckardt, R. L. Byer, C. Chen, and A. D. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron. 25, 1196–1199 (1989).
[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]

D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

Y. Zhao, “The spatial symmetry of the third-order susceptibility tensor,” IEEE J. Quantum Electron. QE-22, 1012 (1986).
[CrossRef]

C. C. Shang and H. Hsu, “The spatial symmetric forms of third-order nonlinear susceptibility,” IEEE J. Quantum Electron. QE-23, 177–179 (1987).
[CrossRef]

R. B. Miles and S. E. Harris, “Optical third-harmonic gen-eration in alkali metal vapors,” IEEE J. Quantum Electron. QE-9, 470–484 (1973).
[CrossRef]

D. Eimerl, S. Velsko, L. Davis, F. Wang, G. Loiacono, and G. Kennedy, “Deuterated L-arginine phosphate: a new efficient nonlinear crystal,” IEEE J. Quantum Electron. 25, 179–193 (1989).
[CrossRef]

C. Greninger, J. Marburger, and D. R. White, “Influence of third-order nonlinearity on second-harmonic generation,” IEEE J. Quantum Electron. QE-8, 543–544 (1972).
[CrossRef]

J. Appl. Phys. (1)

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

J. Chem. Phys. (2)

G. R. Meredith, “Second-order cascading in third-order nonlinear optical processes,” J. Chem. Phys. 77, 5863–5871 (1982).
[CrossRef]

G. R. Meredith, “Lower order effects in nonlinear two- and three-photon resonance spectroscopies,” J. Chem. Phys. 75, 4317–4325 (1981).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (2)

V. V. Rostovtseva, A. P. Sukhorukov, V. G. Tunkin, and S. M. Saltiel, “Higher harmonics generation by cascade processes in focused beams,” Opt. Commun. 22, 56–60 (1977).
[CrossRef]

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985).
[CrossRef]

Opt. Lett. (7)

Opt. Quantum Electron. (1)

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Phys. Rev. (1)

J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185, 57–72 (1969).
[CrossRef]

Phys. Rev. B (2)

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).
[CrossRef]

G. R. Meredith, “Cascading in optical third-harmonic generation by crystalline quartz,” Phys. Rev. B 24, 5522–5532 (1981).
[CrossRef]

Prog. Quantum Electron. (1)

C. Flytzanis and N. Bloembergen, “Infrared dispersion of third-order susceptibilities in dielectrics: retardation effects,” Prog. Quantum Electron. 4, 271–300 (1976).
[CrossRef]

Science (1)

M. D. Perry and G. Mourou, “Terawatt to petawatt subpicosecond lasers,” Science 264, 917–924 (1994).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

S. A. Akhmanov, L. B. Meisner, S. T. Parinov, S. M. Saltiel, and V. G. Tunkin, “Cubic nonlinear susceptibilities of crystals in the optical band; the signs and magnitudes of the susceptibilities of crystals with and without centers of inversion,” Sov. Phys. JETP 46, 898–907 (1977).

Other (10)

X. Mu, M. V. Makarov, and Y. J. Ding, “Generation of blue light in Ce:KTiOPO4 crystals using subpicosecond laser pulses,” presented at the 1999 OSA Annual Meeting, Santa Clara, Calif., September 26–30, 1999.

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C. Flytzanis, in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975), Vol. I, Part A, Chap. 2.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

D. H. Jundt, Crystal Technology, Palo Alto, Calif. 94303 (personal communication, 1996).

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

S. Singh, in CRC Handbook of Laser Science and Technology, M. J. Weber, ed., (CRC Press, Boca Raton, Fla., 1986), Vol. III, Part 1.

“Standards on piezoelectric crystals,” Proc. IRE 37, 1378–1395 (1949).

“IEEE standards on piezoelectricity,” IEEE/ANSI Std. 176–1987 (Institute of Electrical and Electronics Engineers, New York, 1987).

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Figures (15)

Fig. 1
Fig. 1

Illustration of processes involved in third-harmonic generation by cascaded second-order processes. The first process is SHG (ω+ω2ω, indicated by the left brace) followed by sumfrequency generation (SFG, 2ω+ω3ω, indicated by the right brace). The space between k2ω and k1ω on the second line indicates a possible phase mismatch for SHG and SFG processes.

Fig. 2
Fig. 2

Dependence of Ceff2 for type I phase matching on azimuthal angle ϕ for second-order interactions only (solid curve) and third-order interactions only (dashed curve). The amplitudes are normalized to unity. For calculating the solid curve, it is assumed that d15/d22=0.16/2.2. 31

Fig. 3
Fig. 3

Dependence of Ceff2 for type II phase matching on azimuthal angle ϕ for second-order interactions only (solid curve) and third-order interactions only (dashed curve). The amplitudes are normalized to unity. In calculating the dashed curve, it is assumed that C11/3 cos2 θm+C16 sin2 θm=2C10 for illustrative purposes.

Fig. 4
Fig. 4

Experimental setup for measurement of THG. M1 and M2 are a concave/convex mirror pair to down collimate the beam, RS denotes a rotation stage, GON denotes a goniometer with the axis of rotation parallel to the c axis, APER denotes the 4-mm aperture, KG-3 is KG-3 filter glass, UG-5 is UG-5 filter glass, DC are four dichroic mirrors, and Meter is a pyroelectric energy meter.

Fig. 5
Fig. 5

Effective nonlinear coupling as a function of internal azimuthal angle ϕint in BBO. The solid curve is a fit with known values for second-order coefficients.

Fig. 6
Fig. 6

Effective nonlinear coupling as a function of internal azimuthal angle ϕint in BBO. The solid curve is a fit with known values for second-order coefficients.

Fig. 7
Fig. 7

Calculated amplitudes of electric field envelope functions Aj as a function of propagation distance in BBO crystal. The solid curve shows what would be expected for the third harmonic in the presence of χ(3) only, and the dashed curve shows the expected behavior for the third harmonic in the presence of χ(2) only. The dotted and dash-dotted curves are the ordinary and extraordinary second-harmonic waves, respectively.

Fig. 8
Fig. 8

Predicted conversion efficiency in a 1-mm crystal of BBO for both type I and type II phase matching. The dotted curve is for calculation done for type II phase matching with no cascaded process allowed. The dash-dotted curve is the calculated conversion efficiency with no third-order coupling, i.e., only cascaded interactions one allowed.

Fig. 9
Fig. 9

Calculated square of (a) electric field amplitude and (b) phase at the crystal exit, involving only χ(3) and no explicit SPM (CjSPM=0, dj=0). Input intensity is 800 GW/cm2. The solid curve is the ordinary fundamental, the dash-dotted curve is the extraordinary fundamental, and the dashed curve is the third harmonic (extraordinary).

Fig. 10
Fig. 10

Calculated square of (a) electric field amplitude and (b) phase at the crystal exit involving only χ(2) and χ(3) with no explicit SPM (CjSPM=0). Input intensity is 800 GW/cm2. The solid curve is the ordinary fundamental, the dash-dotted line is the extraordinary fundamental, and the dashed curve is the third harmonic (extraordinary).

Fig. 11
Fig. 11

Calculated square of (a) electric field amplitude and (b) phase at the crystal exit, involving all possible processes. Input intensity is 800 GW/cm2. The solid curve is the ordinary fundamental, the dash-dotted curve is the extraordinary fundamental, and the dashed curve is the third harmonic (extraordinary).

Fig. 12
Fig. 12

Conversion efficiency from a single BBO crystal at 351 nm along with a quadratic fit to low-drive points. Type I phase matching with ϕint=-15°.

Fig. 13
Fig. 13

Conversion efficiency from a single BBO crystal at 351 nm along with a quadratic fit to low-drive points. Type II phase matching (ϕint=0°).

Fig. 14
Fig. 14

Calculated longitudinal growth of type I and type II 3ω energies. 350-fs pulses, 200-GW/cm2 input intensity (8 mJ).

Fig. 15
Fig. 15

Measured energy generated at the third harmonic in a 1-mm crystal of d-LAP. The input energy at 200 GW/cm2 is 8 mJ. Type II phase matching. The curve is a cubic, least-squares fit to data.

Tables (11)

Tables Icon

Table 1 Possible Interactions for THGa

Tables Icon

Table 2 Effective Nonlinear Coefficient for Second-Order Interactions, Neglecting Dispersion (sss Interactions)

Tables Icon

Table 3 Effective Nonlinear Coefficient for Second-Order Interactions, Neglecting Dispersion (ssf Interactions)

Tables Icon

Table 4 Effective Nonlinear Coefficient for Second-Order Interactions, Neglecting Dispersion (sff Interactions)

Tables Icon

Table 5 Effective Nonlinear Coefficient for Second-Order Interactions, Neglecting Dispersion (fff Interactions)

Tables Icon

Table 6 Effective Nonlinear Coefficient for Third-Order Interactions, Neglecting Dispersion (ssss Interactions)

Tables Icon

Table 7 Effective Nonlinear Coefficient for Third-Order Interactions, Neglecting Dispersion (sssf Interactions)

Tables Icon

Table 8 Effective Nonlinear Coefficient for Third-Order Interactions, Neglecting Dispersion (ssff Interactions)

Tables Icon

Table 9 Effective Nonlinear Coefficient for Third-Order Interactions, Neglecting Dispersion (sfff Interactions)

Tables Icon

Table 10 Effective Nonlinear Coefficient for Third-Order Interactions, Neglecting Dispersion (ffff Interactions)

Tables Icon

Table 11 Nonzero Third-Order Tensor Elements for Crystal Class 3m

Equations (66)

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Type Islow+slow+slowfast,
Type IIslow+slow+fastfast,
Type IIIslow+fast+fastfast,
(1) 1ω (slow polarization),
(2) 1ω (fast polarization),
(3) 2ω (slow polarization),
(4) 2ω (fast polarization),
(5) 3ω (fast polarization).
A1z=-β11 A1t-i2β21 2A1t2+i2k12A1+iω1n1c 3C1SPM2|A1|2A1+3C1XPM|A5|2A1+3C2XPM|A2|2A1+iω1n1c d1A3A1* exp(iΔk1z)+d3A4A1* exp(iΔk3z)+d2A3A2* exp(iΔk2z)+d4A4A2* exp(iΔk4z)+d5A5A4* exp(iΔk5z)+d7A5A3* exp(iΔk7z)+3C1A5A2*A1* exp(iΔk11z)+3C22A5A1*2 exp(iΔk12z)+3C32A5A2*2 exp(iΔk13z),
A2z=-β12 A2t-i2β22 2A2t2+i2k22A2+iω2n2c 3C2SPM2|A2|2A2+3C3XPM|A5|2A2+3C4XPM|A1|2A2+iω2n2c d2A3A1* exp(iΔk2z)+d4A4A1* exp(iΔk4z)+d6A5A4* exp(iΔk6z)+d8A5A3* exp(iΔk8z)+d9A3A2* exp(iΔk9z)+d10A4A2* exp(iΔk10z)+3C12A5A1*2 exp(iΔk11z)+3C3A5A1*A2* exp(iΔk13z),
A3z=-β13 A3t-i2β23 2A3t2+i2k32A3+iω3n3c 3C3SPM2|A3|2A3+3C5XPM|A5|2A3+3C6XPM|A2|2A3+3C7XPM|A1|2A3+iω3n3c d12A12 exp(-iΔk1z)+d2A2A1 exp(-iΔk2z)+d92A22 exp(-iΔk9z)+d7A5A1* exp(iΔk7z)+d8A5A2* exp(iΔk8z),
A4z=-β14 A4t-i2β24 2A4t2+i2k42A4+iω4n4c 3C3SPM2|A4|2A4+3C8XPM|A5|2A4+3C9XPM|A2|2A4+3C10XPM|A1|2A4+iω4n4c d32A12 exp(-iΔk3z)+d4A2A1 exp(-iΔk4z)+d102A22 exp(-iΔk10z)+d5A5A1* exp(iΔk5z)+d6A5A2* exp(iΔk6z),
A5z=-β15 A5t-i2β25 2A5t2+i2k52A5+iω5n5c 3C5SPM2|A5|2A5+3C11XPM|A2|2A5+3C12XPM|A1|2A5+iω5n5c d5A4A1 exp(-iΔk5z)+d6A4A2 exp(-iΔk6z)+d7A3A1 exp(-iΔk7z)+d8A3A2 exp(-iΔk8z)+3C12A12A2 exp(-iΔk11z)+C22A13 exp(-iΔk12z)+3C32A1A22 exp(-iΔk13z).
Δk1=k3-2k1,Δk2=k3-k1-k2,Δk3=k4-2k1,Δk4=k4-k1-k2,Δk5=k5-k4-k1,Δk6=k5-k4-k2,Δk7=k5-k3-k1,Δk8=k5-k3-k2,Δk9=k3-2k2,Δk10=k4-2k2,Δk11=k5-2k1-k2,Δk12=k5-3k1,Δk13=k5-k1-2k2.
dj=eˆ1·χ(2) : eˆ2eˆ3/2, j=110,
Cj-10=eˆ1·χ(3) : eˆ2eˆ3eˆ4/4, j=1113,
A1z=iω1n1cd1A3A1* exp(iΔk1z)+d7A5A3* exp(iΔk7z)+3C22A5A1*2 exp(iΔk12z)0,
A3z=iω3n3c d12A12 exp(-iΔk1z)+d7A5A1* exp(iΔk7z),
A5z=iω5n5cd7A3A1 exp(-iΔk7z)+C22A13 exp(-iΔk12z).
A3(z)=iω32n3cζd1A12 exp(iΔk7z/2)sin ζz,
A5(z)=d1A122d7A1* exp(-iΔk7z/2)×cos ζz+Δk72ζ sin ζz-1,
ζ2=ω5ω3n5n3c2d72|A1|2+Δk722.
I3(z)=ω3n1cζ2 d1I120cn3 sin2 ζz,
I5(z)=d12n5I14n1d72 1+cos2 ζz+Δk724ζ2 sin2 ζz-2 cos ζz cos Δk7z2-Δk7ζ sin ζz sin Δk7z2.
η=ω5ω3n5n3c2 d72Δk7|A1|2.
I5(z)=ω3n3cΔk72 ω52(d1d7)2n5n1302c4I13z2.
I5(z)=ω52C22n5n1302c4I13z2.
Ceff=ω3n3cΔk7d1d7.
=C110000C160C11/30C100C110C16-C100C100C11/300-C10C330C160C160C100.
Ceff=2πλ2ω0d22d22 sin 6ϕ2 cos θmn3Δk7-cos3 θmn4Δk5+d15 cos 3ϕcos2 θm sin θmn4Δk5-sin θmn3Δk7+C10 cos3ϕ sin θm.
Ceff=2πλ2ω0 d222cos2 3ϕ2 cos4 θmn4Δk5-cos2 θmn3Δk8+sin2 3ϕ2 cos2 θmn3Δk7-cos4 θmn4Δk6-2d22d15 sin 3ϕsin θm cos3 θmn4Δk6+sin 2θmn3Δk7+d152 sin2 θm3 cos2 θmn4Δk6+2n4Δk5+d33n4Δk6(-d22 sin 3ϕ cos θm sin3 θm+d15 sin4θm)+3C113 cos2 θm+C16 sin2 θm+C10 sin 2θm sin 3ϕ.
ACeff2=I3ωIω3=A2πλ2ω0 d222 sin 6ϕ2 cos θmn3Δk7-cos3 θmn4Δk5+cos 3ϕd22d15cos2 θm sin θmn4Δk5-sin θmn3Δk7+C10 sin θm cos 3ϕ2
=A[-83.6 pm2/V2 sin 6ϕ+(0.61C10+9.36 pm2/V2)cos 3ϕ]2,
KCeff2=I5Iω3=KAd222 cos2 3ϕ+Bd222 sin2 3ϕ+sin 3ϕ(Dd22d15+Ed22d33+C10 sin 2θm)+Fd33d15+C113 cos2 θm+C16 sin2 θm2,
A=2π3λ2ω0 2 cos4 θmn4Δk5-cos2 θmn3Δk8=10.3,
B=2π3λ2ω0 2 cos2 θmn3Δk7-cos4 θmn4Δk6=-4.50,
D=-2 2π3λ2ω0 sin θm cos3 θmn4Δk6+sin 2θmn3Δk7=5.64,
e=-2π3λ2ω0 sin3 θm cos θmn4Δk6=-1.17,
F=2π3λ2ω0 sin4 θmn4Δk6=1.26.
Ajz=(Dj+1j)Aj+2j,
Dj=-β1j t-i2β2j 2t2+i2kj2,
11=iω1n1c 3C1SPM2|A1|2+3C1XPM|A5|2+3C2XPM|A2|2,
21=iω1n1c d1A3A1* exp(iΔk1z)+d3A4A1* exp(iΔk3z)+d2A3A2* exp(iΔk2z)+d4A4A2* exp(iΔk4z)+d5A5A4* exp(iΔk5z)+d7A5A3* exp(iΔk7z)+3C1A5A2*A1* exp(iΔk11z)+3C22A5A1*2 exp(iΔk12z)+3C32A5A2*2 exp(iΔk13z).
Ajz=DAj,
Ajz=1j Aj,
Ajz=2j.
Aj(n+1)=F-1{exp[Dj(ω, k)Δz]F[Aj(n)]},
Dj(ω, k)=iωβ1j+i2ω2β2j-i2kj(kjx2+kjy2).
y(n+1)=y(n)+Δzf[z(n)+Δz/2, y(n)+Δ22f(z(n), y(n))].
n2=3CeffSPM2n0=0cn02γ,
C1ω=12(C11+3C18)
C2ω=18(C11+3C18)+C334+3C162=14C1ω+C334+3C162
Cord=C11,
Cext=C11+C33.
(H2N)2+CNH(CH2)3CH(NH3)+COO-·H2PO4-·H2.
d1=-d22,
d4=d5=d8=-d21 cos2 θm+d36 sin 2θm-d23 sin2 θm,
C1=C18 cos2 θm+C24 sin2 θm-C15 sin 2θm.
=C110C130C15C16C17C18000C220C240000C290C310C330C35C36C37C3800,
Ceff=2πλ2ω0 d1d8n3Δk8+2 d4d5n4Δk5+3C1.
Ceffd-LAP1.52251/225CeffBBO,
d1=0,
d3=d7=-d36 sin 2ϕ sin θm,
d5=d36 cos 2ϕ sin 2θm,
C2=14(C11-3C18)sin 4ϕ cos θm.
Ceff=2πλ2ω0 d3d5n4Δk5+C2.

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