Abstract

Second-harmonic generation in a periodic structure made from N pairs of optically contacted, birefringence phase-matched, walk-off-compensating bulk plates is theoretically investigated. In the undepleted-pump approximation, analytical (heuristic) expressions for conversion efficiency versus N are derived for both type I and type II phase matching. An explicit split-step beam propagation scheme that solves exactly the coupled paraxial-wave equations is used to check the validity of the heuristic results. For type II, stronger conversion enhancement than for bulk crystal is predicted in the low-depletion regime, whereas for type I such structures avoid harmonic beam ellipticity. The periodic structures are found to behave as nonlinear harmonic birefringent filters because of the presence of periodic wave-vector mismatch grating ±Δk that results from walk-off compensation. The effect of periodicity imperfections, such as residual plate orientation mismatches, was found to be responsible for broadening of the tuning bandwidth in walk-off-compensating devices.

© 2003 Optical Society of America

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2003

2002

2001

S. Ducci, N. Treps, A. Maitre, and C. Fabre, “Pattern formation in optical parametric oscillators,” Phys. Rev. A 64, 023803 (2001).
[CrossRef]

2000

1999

F. Castaldo, G. Abbate, and E. Santamato, “Theory for a new vectorial beam-propagation method in anisotropic structures,” Appl. Opt. 38, 3904–3910 (1999).
[CrossRef]

J. P. Fève, J.-J. Zondy, B. Boulanger, R. Bonnenberger, X. Cabirol, B. Ménaert, and G. Marnier, “Optimized blue light generation in optically contacted walkoff-compensated RbTiOAsO4 and KTiOP1−yAsyO4,” Opt. Commun. 161, 359–369 (1999).
[CrossRef]

M. Vaupel, A. Mai⁁tre, and C. Fabre, “Observation of pattern formation in an optical parametric oscillator,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[CrossRef]

E. Roissé, E. Louradour, O. Gay, V. Couderc, and A. Barthélémy, “Walk-off and phase-compensated resonantly enhanced frequency-doubling of picosecond pulses using type-II nonlinear crystals,” Appl. Phys. B 69, 25–27 (1999).
[CrossRef]

T. Kaing, J.-J. Zondy, A. P. Yelisseyev, S. I. Lobanov, and L. Isaenko, “Improving the power and spectral performance of a 27–33 Thz AgGaS2 difference-frequency spectrometer,” IEEE Trans. Instrum. Meas. 48, 592–595 (1999).
[CrossRef]

1998

1997

1996

A. Steinbach, M. Rauner, F. C. Cruz, and J. C. Berquist, “Cw second harmonic generation with elliptical Gaussian beams,” Opt. Commun. 123, 207–214 (1996).
[CrossRef]

J.-J. Zondy, M. Abed, S. Khodja, C. Bonnin, B. Rainaud, H. Albrecht, and D. Lupinski, “Walk-off-compensated type-I and type-II SHG using twin-crystal AgGaSe2 and KTiOPO4 devices,” in Nonlinear Frequency Generation and Conversion, M. C. Gupta, W. J. Kozlovsky, and D. C. McPherson, eds., Proc. SPIE 2700, 66–72 (1996).
[CrossRef]

1995

J.-J. Zondy, “Experimental investigation of single and twin AgGaSe2 crystals for cw 10.2 μm SHG,” Opt. Commun. 119, 320–326 (1995).
[CrossRef]

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial-wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

A. V. Smith and M. S. Bowers, “Phase distortions in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995).
[CrossRef]

1994

1993

P. Plizka and P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” J. Mod. Opt. 40, 1909–1916 (1993).
[CrossRef]

P. Plizka and P. P. Banerjee, “Nonlinear transverse effects in second-harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
[CrossRef]

L. K. Samantha, T. Yanagawa, and Y. Yamamoto, “Technique for enhanced second harmonic output power,” Opt. Commun. 76, 250–252 (1993).
[CrossRef]

1991

J.-J. Zondy, “Comparative theory of walkoff-limited type-II versus type-I second-harmonic generation with Gaussian beams,” Opt. Commun. 81, 427–440 (1991). In Eq. (3.1a) of this reference exp(−x2) should read as exp(−x2/2).
[CrossRef]

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

1990

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]

M. A. Dreger and J. K. McIver, “Second-harmonic generation in a nonlinear, anisotropic medium with diffraction and depletion,” J. Opt. Soc. Am. B 7, 776–784 (1990).
[CrossRef]

B. Ya. Zeldovitch, Yu. E. Kapitskii, and A. N. Chudinov, “Interference between second-harmonics generated in two different KTP crystals,” Sov. J. Quantum Electron. 20, 1120–1121 (1990).
[CrossRef]

1989

M. Nieto-Vesperinas and G. Lera, “Solution to nonlinear optical frequency mixing equations with depletion and diffraction,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

1988

M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
[CrossRef]

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalimtsev, “Three-wave parametric processes in multicrystal nonlinear frequency converters,” Opt. Spectrosc. 65, 90–93 (1988).

1983

1980

S. C. Sheng and A. E. Siegman, “Nonlinear optical calculations using fast transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
[CrossRef]

1976

V. D. Volosov and A. G. Kalintsev, “Optimum optical second-harmonic generation in tandem crystals,” Sov. Tech. Phys. Lett. 2, 373–375 (1976).

1968

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

1966

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1328 (1966).
[CrossRef]

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–384 (1966).
[CrossRef]

Abbate, G.

Abed, M.

J.-J. Zondy, M. Abed, S. Khodja, C. Bonnin, B. Rainaud, H. Albrecht, and D. Lupinski, “Walk-off-compensated type-I and type-II SHG using twin-crystal AgGaSe2 and KTiOPO4 devices,” in Nonlinear Frequency Generation and Conversion, M. C. Gupta, W. J. Kozlovsky, and D. C. McPherson, eds., Proc. SPIE 2700, 66–72 (1996).
[CrossRef]

J.-J. Zondy, M. Abed, and S. Khodja, “Twin-crystal walkoff-compensated type-II second-harmonic generation: single-pass and cavity-enhanced experiments in KTiOPO4,” J. Opt. Soc. Am. B 11, 2368–2379 (1994).
[CrossRef]

J.-J. Zondy, M. Abed, and A. Clairon, “Type-II frequency doubling at λ=1.30 μm and λ=2.53 μm in flux-grown potassium titanyl phosphate,” J. Opt. Soc. Am. B 11, 2004–2015 (1994).
[CrossRef]

Acef, O.

Albrecht, H.

J.-J. Zondy, M. Abed, S. Khodja, C. Bonnin, B. Rainaud, H. Albrecht, and D. Lupinski, “Walk-off-compensated type-I and type-II SHG using twin-crystal AgGaSe2 and KTiOPO4 devices,” in Nonlinear Frequency Generation and Conversion, M. C. Gupta, W. J. Kozlovsky, and D. C. McPherson, eds., Proc. SPIE 2700, 66–72 (1996).
[CrossRef]

Alford, W. J.

Andreev, R. B.

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalimtsev, “Three-wave parametric processes in multicrystal nonlinear frequency converters,” Opt. Spectrosc. 65, 90–93 (1988).

Arisholm, G.

Armstrong, D. J.

Ashkin, A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–384 (1966).
[CrossRef]

Auerbach, J. M.

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial-wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

Banerjee, P. P.

P. Plizka and P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” J. Mod. Opt. 40, 1909–1916 (1993).
[CrossRef]

P. Plizka and P. P. Banerjee, “Nonlinear transverse effects in second-harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
[CrossRef]

Barthélémy, A.

E. Roissé, E. Louradour, O. Gay, V. Couderc, and A. Barthélémy, “Walk-off and phase-compensated resonantly enhanced frequency-doubling of picosecond pulses using type-II nonlinear crystals,” Appl. Phys. B 69, 25–27 (1999).
[CrossRef]

Berquist, J. C.

A. Steinbach, M. Rauner, F. C. Cruz, and J. C. Berquist, “Cw second harmonic generation with elliptical Gaussian beams,” Opt. Commun. 123, 207–214 (1996).
[CrossRef]

Bonnenberger, R.

J. P. Fève, J.-J. Zondy, B. Boulanger, R. Bonnenberger, X. Cabirol, B. Ménaert, and G. Marnier, “Optimized blue light generation in optically contacted walkoff-compensated RbTiOAsO4 and KTiOP1−yAsyO4,” Opt. Commun. 161, 359–369 (1999).
[CrossRef]

Bonnin, C.

Boulanger, B.

Bowers, M. S.

Boyd, G. D.

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

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–384 (1966).
[CrossRef]

Brown, M.

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]

Cabirol, X.

J. P. Fève, J.-J. Zondy, B. Boulanger, R. Bonnenberger, X. Cabirol, B. Ménaert, and G. Marnier, “Optimized blue light generation in optically contacted walkoff-compensated RbTiOAsO4 and KTiOP1−yAsyO4,” Opt. Commun. 161, 359–369 (1999).
[CrossRef]

B. Boulanger, J. P. Fève, G. Marnier, B. Ménaert, X. Cabirol, P. Villeval, and C. Bonnin, “Relative sign and absolute magnitude of d(2) nonlinear coefficients of KTP from phase-matched second-harmonic generation,” J. Opt. Soc. Am. B 11, 750–757 (1994).
[CrossRef]

Castaldo, F.

Chang, A.

Chudinov, A. N.

B. Ya. Zeldovitch, Yu. E. Kapitskii, and A. N. Chudinov, “Interference between second-harmonics generated in two different KTP crystals,” Sov. J. Quantum Electron. 20, 1120–1121 (1990).
[CrossRef]

Clairon, A.

Couderc, V.

E. Roissé, E. Louradour, O. Gay, V. Couderc, and A. Barthélémy, “Walk-off and phase-compensated resonantly enhanced frequency-doubling of picosecond pulses using type-II nonlinear crystals,” Appl. Phys. B 69, 25–27 (1999).
[CrossRef]

Coutts, J.

Cruz, F. C.

A. Steinbach, M. Rauner, F. C. Cruz, and J. C. Berquist, “Cw second harmonic generation with elliptical Gaussian beams,” Opt. Commun. 123, 207–214 (1996).
[CrossRef]

Dreger, M. A.

Ducci, S.

S. Ducci, N. Treps, A. Maitre, and C. Fabre, “Pattern formation in optical parametric oscillators,” Phys. Rev. A 64, 023803 (2001).
[CrossRef]

Eckardt, R. C.

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

Eidinger, E.

Eimerl, D.

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial-wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

Fabre, C.

S. Ducci, N. Treps, A. Maitre, and C. Fabre, “Pattern formation in optical parametric oscillators,” Phys. Rev. A 64, 023803 (2001).
[CrossRef]

M. Vaupel, A. Mai⁁tre, and C. Fabre, “Observation of pattern formation in an optical parametric oscillator,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[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]

Feit, M. D.

Fève, J. P.

Fleck, J. A.

Freegarde, T.

Gay, O.

E. Roissé, E. Louradour, O. Gay, V. Couderc, and A. Barthélémy, “Walk-off and phase-compensated resonantly enhanced frequency-doubling of picosecond pulses using type-II nonlinear crystals,” Appl. Phys. B 69, 25–27 (1999).
[CrossRef]

Hänsch, T. W.

Isaenko, L.

T. Kaing, J.-J. Zondy, A. P. Yelisseyev, S. I. Lobanov, and L. Isaenko, “Improving the power and spectral performance of a 27–33 Thz AgGaS2 difference-frequency spectrometer,” IEEE Trans. Instrum. Meas. 48, 592–595 (1999).
[CrossRef]

Kaing, T.

T. Kaing, J.-J. Zondy, A. P. Yelisseyev, S. I. Lobanov, and L. Isaenko, “Improving the power and spectral performance of a 27–33 Thz AgGaS2 difference-frequency spectrometer,” IEEE Trans. Instrum. Meas. 48, 592–595 (1999).
[CrossRef]

Kalimtsev, A. G.

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalimtsev, “Three-wave parametric processes in multicrystal nonlinear frequency converters,” Opt. Spectrosc. 65, 90–93 (1988).

Kalintsev, A. G.

V. D. Volosov and A. G. Kalintsev, “Optimum optical second-harmonic generation in tandem crystals,” Sov. Tech. Phys. Lett. 2, 373–375 (1976).

Kapitskii, Yu. E.

B. Ya. Zeldovitch, Yu. E. Kapitskii, and A. N. Chudinov, “Interference between second-harmonics generated in two different KTP crystals,” Sov. J. Quantum Electron. 20, 1120–1121 (1990).
[CrossRef]

Kato, K.

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

Khodja, S.

J.-J. Zondy, M. Abed, S. Khodja, C. Bonnin, B. Rainaud, H. Albrecht, and D. Lupinski, “Walk-off-compensated type-I and type-II SHG using twin-crystal AgGaSe2 and KTiOPO4 devices,” in Nonlinear Frequency Generation and Conversion, M. C. Gupta, W. J. Kozlovsky, and D. C. McPherson, eds., Proc. SPIE 2700, 66–72 (1996).
[CrossRef]

J.-J. Zondy, M. Abed, and S. Khodja, “Twin-crystal walkoff-compensated type-II second-harmonic generation: single-pass and cavity-enhanced experiments in KTiOPO4,” J. Opt. Soc. Am. B 11, 2368–2379 (1994).
[CrossRef]

Kleinman, D. A.

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

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–384 (1966).
[CrossRef]

Kogelnik, H.

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1328 (1966).
[CrossRef]

Kolker, D.

Lai, K. S.

Lau, E.

Leibfried, D.

Lera, G.

M. Nieto-Vesperinas and G. Lera, “Solution to nonlinear optical frequency mixing equations with depletion and diffraction,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

Li, T.

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1328 (1966).
[CrossRef]

Lim, Y. L.

Lobanov, S. I.

T. Kaing, J.-J. Zondy, A. P. Yelisseyev, S. I. Lobanov, and L. Isaenko, “Improving the power and spectral performance of a 27–33 Thz AgGaS2 difference-frequency spectrometer,” IEEE Trans. Instrum. Meas. 48, 592–595 (1999).
[CrossRef]

Louradour, E.

E. Roissé, E. Louradour, O. Gay, V. Couderc, and A. Barthélémy, “Walk-off and phase-compensated resonantly enhanced frequency-doubling of picosecond pulses using type-II nonlinear crystals,” Appl. Phys. B 69, 25–27 (1999).
[CrossRef]

Lupinski, D.

J.-J. Zondy, D. Kolker, C. Bonnin, and D. Lupinski, “Second-harmonic generation with monolithic walk-off-compensating periodic structures. 2. Experiments,” J. Opt. Soc. Am. B 20, 1695–1707 (2003).
[CrossRef]

R. F. Wu, P. B. Phua, K. S. Lai, Y. L. Lim, E. Lau, A. Chang, C. Bonnin, and D. Lupinski, “Compact 21-W 2-μm intracavity optical parametric oscillator,” Opt. Lett. 25, 1460–1462 (2000).
[CrossRef]

J.-J. Zondy, M. Abed, S. Khodja, C. Bonnin, B. Rainaud, H. Albrecht, and D. Lupinski, “Walk-off-compensated type-I and type-II SHG using twin-crystal AgGaSe2 and KTiOPO4 devices,” in Nonlinear Frequency Generation and Conversion, M. C. Gupta, W. J. Kozlovsky, and D. C. McPherson, eds., Proc. SPIE 2700, 66–72 (1996).
[CrossRef]

Mai?tre, A.

M. Vaupel, A. Mai⁁tre, and C. Fabre, “Observation of pattern formation in an optical parametric oscillator,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[CrossRef]

Maitre, A.

S. Ducci, N. Treps, A. Maitre, and C. Fabre, “Pattern formation in optical parametric oscillators,” Phys. Rev. A 64, 023803 (2001).
[CrossRef]

Marnier, G.

Marom, E.

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]

McIver, J. K.

Ménaert, B.

J. P. Fève, J.-J. Zondy, B. Boulanger, R. Bonnenberger, X. Cabirol, B. Ménaert, and G. Marnier, “Optimized blue light generation in optically contacted walkoff-compensated RbTiOAsO4 and KTiOP1−yAsyO4,” Opt. Commun. 161, 359–369 (1999).
[CrossRef]

B. Boulanger, J. P. Fève, G. Marnier, B. Ménaert, X. Cabirol, P. Villeval, and C. Bonnin, “Relative sign and absolute magnitude of d(2) nonlinear coefficients of KTP from phase-matched second-harmonic generation,” J. Opt. Soc. Am. B 11, 750–757 (1994).
[CrossRef]

Mendlovic, D.

Milonni, P. W.

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial-wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas and G. Lera, “Solution to nonlinear optical frequency mixing equations with depletion and diffraction,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

Phua, P. B.

Plizka, P.

P. Plizka and P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” J. Mod. Opt. 40, 1909–1916 (1993).
[CrossRef]

P. Plizka and P. P. Banerjee, “Nonlinear transverse effects in second-harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
[CrossRef]

Rainaud, B.

J.-J. Zondy, M. Abed, S. Khodja, C. Bonnin, B. Rainaud, H. Albrecht, and D. Lupinski, “Walk-off-compensated type-I and type-II SHG using twin-crystal AgGaSe2 and KTiOPO4 devices,” in Nonlinear Frequency Generation and Conversion, M. C. Gupta, W. J. Kozlovsky, and D. C. McPherson, eds., Proc. SPIE 2700, 66–72 (1996).
[CrossRef]

Rauner, M.

A. Steinbach, M. Rauner, F. C. Cruz, and J. C. Berquist, “Cw second harmonic generation with elliptical Gaussian beams,” Opt. Commun. 123, 207–214 (1996).
[CrossRef]

Raymond, T. D.

Roissé, E.

E. Roissé, E. Louradour, O. Gay, V. Couderc, and A. Barthélémy, “Walk-off and phase-compensated resonantly enhanced frequency-doubling of picosecond pulses using type-II nonlinear crystals,” Appl. Phys. B 69, 25–27 (1999).
[CrossRef]

Samantha, L. K.

L. K. Samantha, T. Yanagawa, and Y. Yamamoto, “Technique for enhanced second harmonic output power,” Opt. Commun. 76, 250–252 (1993).
[CrossRef]

Santamato, E.

Shabtay, G.

Sheng, S. C.

S. C. Sheng and A. E. Siegman, “Nonlinear optical calculations using fast transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
[CrossRef]

Siegman, A. E.

S. C. Sheng and A. E. Siegman, “Nonlinear optical calculations using fast transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
[CrossRef]

Smith, A. V.

Steinbach, A.

A. Steinbach, M. Rauner, F. C. Cruz, and J. C. Berquist, “Cw second harmonic generation with elliptical Gaussian beams,” Opt. Commun. 123, 207–214 (1996).
[CrossRef]

Stoll, K.

Torner, L.

Touahri, D.

Treps, N.

S. Ducci, N. Treps, A. Maitre, and C. Fabre, “Pattern formation in optical parametric oscillators,” Phys. Rev. A 64, 023803 (2001).
[CrossRef]

Vaupel, M.

M. Vaupel, A. Mai⁁tre, and C. Fabre, “Observation of pattern formation in an optical parametric oscillator,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[CrossRef]

Vetrov, K. V.

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalimtsev, “Three-wave parametric processes in multicrystal nonlinear frequency converters,” Opt. Spectrosc. 65, 90–93 (1988).

Villeval, P.

Volosov, V. D.

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalimtsev, “Three-wave parametric processes in multicrystal nonlinear frequency converters,” Opt. Spectrosc. 65, 90–93 (1988).

V. D. Volosov and A. G. Kalintsev, “Optimum optical second-harmonic generation in tandem crystals,” Sov. Tech. Phys. Lett. 2, 373–375 (1976).

Walz, J.

Wu, R. F.

Yamamoto, Y.

L. K. Samantha, T. Yanagawa, and Y. Yamamoto, “Technique for enhanced second harmonic output power,” Opt. Commun. 76, 250–252 (1993).
[CrossRef]

Yanagawa, T.

L. K. Samantha, T. Yanagawa, and Y. Yamamoto, “Technique for enhanced second harmonic output power,” Opt. Commun. 76, 250–252 (1993).
[CrossRef]

Yelisseyev, A. P.

T. Kaing, J.-J. Zondy, A. P. Yelisseyev, S. I. Lobanov, and L. Isaenko, “Improving the power and spectral performance of a 27–33 Thz AgGaS2 difference-frequency spectrometer,” IEEE Trans. Instrum. Meas. 48, 592–595 (1999).
[CrossRef]

Zalevsky, Z.

Zeldovitch, B. Ya.

B. Ya. Zeldovitch, Yu. E. Kapitskii, and A. N. Chudinov, “Interference between second-harmonics generated in two different KTP crystals,” Sov. J. Quantum Electron. 20, 1120–1121 (1990).
[CrossRef]

Zondy, J. J.

Zondy, J.-J.

J.-J. Zondy, D. Kolker, C. Bonnin, and D. Lupinski, “Second-harmonic generation with monolithic walk-off-compensating periodic structures. 2. Experiments,” J. Opt. Soc. Am. B 20, 1695–1707 (2003).
[CrossRef]

J. P. Fève, J.-J. Zondy, B. Boulanger, R. Bonnenberger, X. Cabirol, B. Ménaert, and G. Marnier, “Optimized blue light generation in optically contacted walkoff-compensated RbTiOAsO4 and KTiOP1−yAsyO4,” Opt. Commun. 161, 359–369 (1999).
[CrossRef]

T. Kaing, J.-J. Zondy, A. P. Yelisseyev, S. I. Lobanov, and L. Isaenko, “Improving the power and spectral performance of a 27–33 Thz AgGaS2 difference-frequency spectrometer,” IEEE Trans. Instrum. Meas. 48, 592–595 (1999).
[CrossRef]

J.-J. Zondy, “The effects of focusing in type-I and type-II difference frequency generations,” Opt. Commun. 149, 181–206 (1998).
[CrossRef]

J.-J. Zondy, D. Touahri, and O. Acef, “Absolute value of the d36 nonlinear coefficient of AgGaS2: prospect for a low-threshold doubly resonant oscillator-based 3:1 frequency divider,” J. Opt. Soc. Am. B 14, 2481–2497 (1997).
[CrossRef]

K. Stoll, J.-J. Zondy, and O. Acef, “Fourth-harmonic generation of a continuous-wave CO2 laser by use of an AgGaSe2/ZnGeP2 doubly resonant device,” Opt. Lett. 22, 1302–1304 (1997).
[CrossRef]

J.-J. Zondy, M. Abed, S. Khodja, C. Bonnin, B. Rainaud, H. Albrecht, and D. Lupinski, “Walk-off-compensated type-I and type-II SHG using twin-crystal AgGaSe2 and KTiOPO4 devices,” in Nonlinear Frequency Generation and Conversion, M. C. Gupta, W. J. Kozlovsky, and D. C. McPherson, eds., Proc. SPIE 2700, 66–72 (1996).
[CrossRef]

J.-J. Zondy, “Experimental investigation of single and twin AgGaSe2 crystals for cw 10.2 μm SHG,” Opt. Commun. 119, 320–326 (1995).
[CrossRef]

J.-J. Zondy, M. Abed, and A. Clairon, “Type-II frequency doubling at λ=1.30 μm and λ=2.53 μm in flux-grown potassium titanyl phosphate,” J. Opt. Soc. Am. B 11, 2004–2015 (1994).
[CrossRef]

J.-J. Zondy, M. Abed, and S. Khodja, “Twin-crystal walkoff-compensated type-II second-harmonic generation: single-pass and cavity-enhanced experiments in KTiOPO4,” J. Opt. Soc. Am. B 11, 2368–2379 (1994).
[CrossRef]

J.-J. Zondy, “Comparative theory of walkoff-limited type-II versus type-I second-harmonic generation with Gaussian beams,” Opt. Commun. 81, 427–440 (1991). In Eq. (3.1a) of this reference exp(−x2) should read as exp(−x2/2).
[CrossRef]

Appl. Opt.

Appl. Phys. B

E. Roissé, E. Louradour, O. Gay, V. Couderc, and A. Barthélémy, “Walk-off and phase-compensated resonantly enhanced frequency-doubling of picosecond pulses using type-II nonlinear crystals,” Appl. Phys. B 69, 25–27 (1999).
[CrossRef]

IEEE J. Quantum Electron.

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

IEEE Trans. Instrum. Meas.

T. Kaing, J.-J. Zondy, A. P. Yelisseyev, S. I. Lobanov, and L. Isaenko, “Improving the power and spectral performance of a 27–33 Thz AgGaS2 difference-frequency spectrometer,” IEEE Trans. Instrum. Meas. 48, 592–595 (1999).
[CrossRef]

J. Appl. Phys.

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

J. Mod. Opt.

P. Plizka and P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” J. Mod. Opt. 40, 1909–1916 (1993).
[CrossRef]

D. Eimerl, J. M. Auerbach, and P. W. Milonni, “Paraxial-wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields,” J. Mod. Opt. 42, 1037–1067 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

J.-J. Zondy, M. Abed, and S. Khodja, “Twin-crystal walkoff-compensated type-II second-harmonic generation: single-pass and cavity-enhanced experiments in KTiOPO4,” J. Opt. Soc. Am. B 11, 2368–2379 (1994).
[CrossRef]

J.-J. Zondy, D. Kolker, C. Bonnin, and D. Lupinski, “Second-harmonic generation with monolithic walk-off-compensating periodic structures. 2. Experiments,” J. Opt. Soc. Am. B 20, 1695–1707 (2003).
[CrossRef]

M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
[CrossRef]

M. A. Dreger and J. K. McIver, “Second-harmonic generation in a nonlinear, anisotropic medium with diffraction and depletion,” J. Opt. Soc. Am. B 7, 776–784 (1990).
[CrossRef]

P. Plizka and P. P. Banerjee, “Nonlinear transverse effects in second-harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
[CrossRef]

B. Boulanger, J. P. Fève, G. Marnier, B. Ménaert, X. Cabirol, P. Villeval, and C. Bonnin, “Relative sign and absolute magnitude of d(2) nonlinear coefficients of KTP from phase-matched second-harmonic generation,” J. Opt. Soc. Am. B 11, 750–757 (1994).
[CrossRef]

J.-J. Zondy, M. Abed, and A. Clairon, “Type-II frequency doubling at λ=1.30 μm and λ=2.53 μm in flux-grown potassium titanyl phosphate,” J. Opt. Soc. Am. B 11, 2004–2015 (1994).
[CrossRef]

A. V. Smith and M. S. Bowers, “Phase distortions in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995).
[CrossRef]

D. J. Armstrong, W. J. Alford, T. D. Raymond, and A. V. Smith, “Parametric amplification and oscillation with walk-off-compensating crystals,” J. Opt. Soc. Am. B 14, 460–474 (1997).
[CrossRef]

B. Boulanger, J. P. Fève, G. Marnier, C. Bonnin, P. Villeval, and J. J. Zondy, “Absolute measurement of quadratic nonlinearities from phase-matched second-harmonic generation in a single KTP crystal cut as a sphere,” J. Opt. Soc. Am. B 14, 1380–1386 (1997).
[CrossRef]

T. Freegarde, J. Coutts, J. Walz, D. Leibfried, and T. W. Hänsch, “General analysis of type-I second-harmonic generation with elliptical Gaussian beams,” J. Opt. Soc. Am. B 14, 2010–2016 (1997).
[CrossRef]

J.-J. Zondy, D. Touahri, and O. Acef, “Absolute value of the d36 nonlinear coefficient of AgGaS2: prospect for a low-threshold doubly resonant oscillator-based 3:1 frequency divider,” J. Opt. Soc. Am. B 14, 2481–2497 (1997).
[CrossRef]

G. Arisholm, “General numerical methods for simulating second-order nonlinear interactions in birefringent media,” J. Opt. Soc. Am. B 14, 2543–2549 (1997).
[CrossRef]

A. V. Smith, D. J. Armstrong, and W. J. Alford, “Increased acceptance bandwidths in optical frequency conversion by use of multiple walk-off-compensating nonlinear crystals,” J. Opt. Soc. Am. B 15, 122–141 (1998).
[CrossRef]

Opt. Commun.

L. K. Samantha, T. Yanagawa, and Y. Yamamoto, “Technique for enhanced second harmonic output power,” Opt. Commun. 76, 250–252 (1993).
[CrossRef]

J.-J. Zondy, “Experimental investigation of single and twin AgGaSe2 crystals for cw 10.2 μm SHG,” Opt. Commun. 119, 320–326 (1995).
[CrossRef]

J.-J. Zondy, “Comparative theory of walkoff-limited type-II versus type-I second-harmonic generation with Gaussian beams,” Opt. Commun. 81, 427–440 (1991). In Eq. (3.1a) of this reference exp(−x2) should read as exp(−x2/2).
[CrossRef]

J. P. Fève, J.-J. Zondy, B. Boulanger, R. Bonnenberger, X. Cabirol, B. Ménaert, and G. Marnier, “Optimized blue light generation in optically contacted walkoff-compensated RbTiOAsO4 and KTiOP1−yAsyO4,” Opt. Commun. 161, 359–369 (1999).
[CrossRef]

J.-J. Zondy, “The effects of focusing in type-I and type-II difference frequency generations,” Opt. Commun. 149, 181–206 (1998).
[CrossRef]

A. Steinbach, M. Rauner, F. C. Cruz, and J. C. Berquist, “Cw second harmonic generation with elliptical Gaussian beams,” Opt. Commun. 123, 207–214 (1996).
[CrossRef]

M. Nieto-Vesperinas and G. Lera, “Solution to nonlinear optical frequency mixing equations with depletion and diffraction,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Spectrosc.

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalimtsev, “Three-wave parametric processes in multicrystal nonlinear frequency converters,” Opt. Spectrosc. 65, 90–93 (1988).

Phys. Rev.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–384 (1966).
[CrossRef]

Phys. Rev. A

S. C. Sheng and A. E. Siegman, “Nonlinear optical calculations using fast transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
[CrossRef]

S. Ducci, N. Treps, A. Maitre, and C. Fabre, “Pattern formation in optical parametric oscillators,” Phys. Rev. A 64, 023803 (2001).
[CrossRef]

Phys. Rev. Lett.

M. Vaupel, A. Mai⁁tre, and C. Fabre, “Observation of pattern formation in an optical parametric oscillator,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[CrossRef]

Proc. IEEE

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1328 (1966).
[CrossRef]

Proc. SPIE

J.-J. Zondy, M. Abed, S. Khodja, C. Bonnin, B. Rainaud, H. Albrecht, and D. Lupinski, “Walk-off-compensated type-I and type-II SHG using twin-crystal AgGaSe2 and KTiOPO4 devices,” in Nonlinear Frequency Generation and Conversion, M. C. Gupta, W. J. Kozlovsky, and D. C. McPherson, eds., Proc. SPIE 2700, 66–72 (1996).
[CrossRef]

Sov. J. Quantum Electron.

B. Ya. Zeldovitch, Yu. E. Kapitskii, and A. N. Chudinov, “Interference between second-harmonics generated in two different KTP crystals,” Sov. J. Quantum Electron. 20, 1120–1121 (1990).
[CrossRef]

Sov. Tech. Phys. Lett.

V. D. Volosov and A. G. Kalintsev, “Optimum optical second-harmonic generation in tandem crystals,” Sov. Tech. Phys. Lett. 2, 373–375 (1976).

Other

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999), Chap. 15.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chaps. 4 and 12.

J.-J. Zondy and Cristal-Laser SA, “Structure monolithique obtenue par contact optique de cristaux non linéaires en compensation de walk-off,” French patent (brevet d’invention 96 01 197; April 4, 1999).

Custom KTP and RbTiOAsO4 OCWOC and diffusion-bonded WOC structures are commercially available at http://www.cristal-laser.fr.

R. F. Wu, K. S. Lai, E. Lau, H. F. Wong, W. J. Xie, Y. L. Lim, K. W. Lim, and L. Chia, “Multi-watt ZGP OPO based on diffusion-bonded walkoff compensated KTP OPO and Nd:YALO laser,” in Advanced Solid State Lasers, Vol. 34 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper TuA4.

R. Lebrun, G. Mennerat, and P. George, “High-efficiency mid-IR nanosecond cascaded optical parametric oscillators based on diffusion-bonded walkoff-compensated KTP and ZGP crystals,” Advanced Solid State Lasers, Vol. 34 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper TuA5.

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1993).

Ref. 2, Chap. 6, p. 155.

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Figures (16)

Fig. 1
Fig. 1

Schematic of 2N OCWOC periodic structures: (a) basic period (unit cell) with N=1, schematically showing walk-off compensation. The optic axes are shown by the arrows labeled (c). For a normal-incidence kω wave vector the o Poynting vector is parallel to kω and the e Poynting vector direction is normal to the index ellipsoid. (b) N=3. The o and e guided ray paths that originate from point source M are shown as seen by an observer point M outside the structure.

Fig. 2
Fig. 2

(b) Comparison of Eq. (53) (thicker solid curve) with the numerical split-step model (thinner curve with filled circles) for N=5. (c) Split-step calculation resolves the two sidelobe features and yields a larger intensity for the Φ=±5π peaks. For split-step computation the SHG of λ=1064 nm in the YZ plane of KTP is considered (ρ=1.828°): w0=650 μm and LC=10 mm (L=4.5×10-3), corresponding to β=ρ/δ0=109. Perfect agreement is found near Φ=0 [main lobe; see also blow-up inset (c)]. Dotted curve beside the thicker curve, single bulk crystal sinc2(Φ) curve. (a) Periodic function FN(Φ).

Fig. 3
Fig. 3

Optimized focusing functions hN(B, L) versus focusing parameter (nominal B=5) for type II and type I. The number of twinned pairs N varies from N=0 (single element) to N=6 (12 plates). Dashed curves, single-element aperture functions for noncritical phase matching.

Fig. 4
Fig. 4

Type II optimal hN(B) functions versus number of twinned pairs N for various values of nominal walk-off parameter B. Note the logarithmic vertical scale.

Fig. 5
Fig. 5

Type I optimal hN(B) functions versus number of twinned pairs N for various values of nominal walk-off parameter B.

Fig. 6
Fig. 6

Type II (eoo) heuristic and split-step focusing functions for B=5.17: (a) low-conversion regime; solid curve, heuristic model; open circles, split-step model; Pω=1 W and η=0.5. (b) high-depletion SHG computed with the split-step model for Pω=50 kW. Filled circles, η=0.5; solid curves, results of optimization over η. Inset: (a) heuristic function (solid curve) compared to hnum in the low-conversion case (both curves are optimized over η); (b) hnum with Pω=50 kW and η=0.5.

Fig. 7
Fig. 7

Type I (ooe) heuristic and split-step focusing functions for B=5.17. (a) N=5; low-conversion regime; solid curve, heuristic model; open circles, split-step model; Pω=1 W and η=0.5. (b) N=5; high-depletion SHG computed with the split-step model with Pω=50 kW and η=0.5. (c) N=0; solid curve, heuristic h(B, L); filled triangles, split-step curve; η=0.5. Inset, optimal phase-mismatch parameters corresponding to the two low-conversion curves in (a) derived from the optimization algorithms.

Fig. 8
Fig. 8

Type I far-field angular transverse intensity distribution (normalized to unity) in the walk-off plane for B=5: (a) at optimal focusing parameters given in Table 2, (b) for strong focusing with L=30. Far-field angle u=x/z was normalized to FF divergence angle δ0=λω/(πnωw0). Dashed curves, Gaussian exp(-4u2) pattern for comparison [see Eqs. (14) and (43)]. The curves have been shifted upward for clarity.

Fig. 9
Fig. 9

Type I (ooe) transverse sections of FF and SH beams at output (z=L) of a single bulk crystal (N=0) computed with the split-step method, corresponding to the optimal focusing of curve (c) in Fig. 7 (w0=33 μm, L=Lopt=1.80, κ=0.64). The SH y profile (y-prof) is a section at xmax=10.5 w0.

Fig. 10
Fig. 10

Type II (eoo) transverse sections of FF and SH beams at output (z=L) of a single bulk crystal (N=0; split-step calculation), corresponding to the optimal focusing of curve (b) in the inset of Fig. 6 (w0=161 μm, L=Lopt=0.073, κ=0.13). The SH y profile (y-prof) is centered on purpose at x/w0=0 (instead of at xmax/w01/3) for clarity.

Fig. 11
Fig. 11

Type I (ooe) transverse sections of the FF and SH intensities of the 10-OCWOC structure, corresponding to the optimal focusing of curve (b) in Fig. 6 (w0=52 μm, L=Lopt=0.708, κ=0.84). Solid curves, x-profile snapshots at various interface boundaries specified by z/L=n/10 (n=710). (b) Dotted curve, SH y profile at z/L=1, centered on purpose at x/w0=0 (instead of at xmax/w0-2/3) for clarity.

Fig. 12
Fig. 12

Type II transverse sections of the FF and SH beams of the 10-OCWOC structure at z=L, corresponding to the optimal focusing of curve (b) in Fig. 6 (w0=62 μm, L=Lopt=0.503, κ=0.74). Solid curves, x-profile snapshots at various interface boundaries specified by z/L=n/10 (n=3, 5, 7). Only the z=L x profile of the extraordinary FF(e) wave is shown [(a) dotted curve]. The dotted curve in (b) is the ordinary SH y profile at z/L=1, centered on purpose at x/w0=0 for clarity.

Fig. 13
Fig. 13

Phase-mismatch tuning curves versus N computed with the heuristic hN, normalized to unity. Each curve is given for the optimal focusing parameters that correspond to Fig. 3. Abscissas of the secondary peaks at ±Nπ are marked.

Fig. 14
Fig. 14

Tuning curves of the ideal 10-OCWOC structure: (a) at low depletion with h5(B, L=0.502) and with hnum(B, L=0.502). The waist location is η=0.5, and h5 is shifted upward for clarity. (b) Blow-up of the central feature of (a). (c) High-depletion curve (split-step) with the conversion efficiency as the y axis. (d) Blow-up of the central feature of (c); dotted curve, corresponding type I (ooe) curve.

Fig. 15
Fig. 15

Type II (eoo) tuning bandwidth (high depletion) in presence of plate orientation mismatches in degree: δθn=[0,0.05, 0.1, -0.05, 0.1, 0.02, -0.08, 0, -0.1, 0], to be compared with the ideal structure of Fig. 14(c). Small aperiodicities in orientation mismatches broaden the tuning bandwidth to the width of a one-plate element, as given by the sinc2(Φ/2N) dashed curve. The Rayleigh range corresponding to the waist is zR=20 mm. The tuning features are identical in the low-depletion limit.

Fig. 16
Fig. 16

Type II (eoe) tuning bandwidth in the presence of periodic plate orientation mismatches in degree: δθn=[0,0.1,, 0, 0.1], with the same parameters as in Figs. 14 and 15. Ideal structure, σn0.

Tables (2)

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Table 1 Parameter Definitions, Mathematical Expressions and Units Used in This Paper a

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Table 2 Tabulated Optimal Values of the hN(B, L) Curves of Fig. 3 with σ1=-σ2=σ and Corresponding Values of Focusing Parameters Lopt and σopt and Resulting Values Beff of the Effective Walk-off Parameter a

Equations (69)

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ρ=1ne(θ)dnedθ=[ne(θ)]221ne2-1no2sin 2θ.
B=ρkωLC2,
Γ=KLCkωh(B, L)
=KLCw02G(B, L),
Beff(N)B/2N,
ne(θ+ε)sin ε=ne(θ-r)sin r.
r=ε1+εne(θ)dnedθ=ε+ρε2.
Eω,oz=i4 t2-aω2Eω,o+i ωdzRcnω,o E2ω,oEω,e*exp[-iσ(z)z],
Eω,ez=i4 t2-βω,e(z) x-aω2Eω,e+i ωdzRcnω,e E2ω,oEω,o*exp[-iσ(z)z],
E2ω,oz=i8 t2-a2ω2E2ω,o+i ωdzRcn2ω,o Eω,oEω,eexp[+iσ(z)z].
i4 t2=i4(2/x2+2/y2)
Eω,p(x, y, z)=Aω,p1+i(z-f )exp[-(aω/2)z]×expXp2+y21+i(z-f ),
Pω,p=πε0cnωw02Aω,p4.
Γ=KLCkωexp(-α2ω LC)h(a, L, f, γ, σ),
h(a, L, f, γ, σ)=2πLexp(-2af )×-+|H(a, L, f, γ, σu)|2×exp(-4u2)du,
(γ, σu)=(0, σ+4βu)fortypeI(β/2, σ+2βu)fortypeII ,
H(a, l, f, γ, Σ)=12π-fl-fdτ1+iτ×exp[-aτ-γ2(τ+f )2-iΣτ].
E1z=i4 t2-a12E1+iχE3E2*exp(-iσz),
E2z=i4 t2-β x-a22E2+iχE2E1*exp(-iσz),
E3z=i8 t2-a32E3+iχE1E2exp(+iσz).
zV=(D+A+χN)V,
V(z+dz)=expdz2D{exp[dz(A+χN)]}×expdz2D×V(z)+O(dz3).
E1,2(z+dz)=expdz2 D1,2×exp-dz a1,22cos(χdz|E2,1|)×expdz2 D1,2E1,2(z)+expdz2 D1,2×iχdzE2,1*exp(-iσz)sinc(χdz|E2,1|)×exp-dz2a1,2+a32×expdz2 D3E3(z)+O(dz3),
E3(z+dz)=expdz2 D3 exp-dz a32cos(χdz|E1|)×expdz2 D3E3(z)+expdz2 D3×iχdzE1exp(+iσz)sinc(χdz|E1|)×exp-dz2a2+a32×expdz2 D2E2(z)+O(dz3).
E˜j(k, z)=drEj(r, z)exp(+ikr),
Ej(r, z)=dkE˜j(k, z)exp(-ikr).
expi dz2-kx2+ky24+βkx
hnum(B, L, f, σ)=P2ω[(1-κ)Pω2KLCkω]
Eω,p(x, y, z)=Aω,p1+izexp[-(aω /2-ikω,pzR)×(z+Nl)]expXp2+y21+iz,
zj=(j+εj)l,
βj=(-1)jβ,εj=0(jeven),εj=1(jodd),
βj=(-1)j+1β,εj=1(jeven),εj=0(jodd).
Pj2ω(M)=ε0dAω,pAω,p¯1+izexp[-aω(z+Nl)]×exp[-γ2(z-zj)2]×11+izexp-2 Xp2+y21+iz×exp[+i2kωzR(z+Nl)],
dE2ωdz=iωzRε0cn2ωexp-a2ω2 (Nl-z)-ik2ω(z+Nl)×P2ω(M),
E2ω(M)zNl=iωzRdAω2cn2ω(1+iz)exp(-a+L/2)j=-NN-1 Ij,
Ij=jl(j+1)ldz×exp[-a-z-iσj(z+Nl)-γ2(z-zj)2]1+iz×exp-2 X2+y21+iz.
j=-NN-1 Ij=j=0N-1 Ij+I-(j+1)=j=0N-1 Ijtwin.
σ1=σj=ΔkjzR,σ2=σ-(j+1)=Δk-(j+1)zR
A2ω(M)=i izRωdAω2cn2ω(1+iz)exp(-a+L/2)0ldz×exp(-γ2z2)×exp-2 (x+β¯z)2+y21+izTN(z),
TN(z)=n=0(N-2)/2exp-i σ1L2exp[-θ1(z+2nl)/l]1+i(z+2nl)+exp-i σ2L2exp[+θ2(z+2nl)/l]1-i(z+2nl)+n=1N/2exp-i σ1L2exp[+θ1(z-2nl)/l]1-i(z-2nl)+exp-i σ2L2exp[-θ2(z-2nl)/l]1+i(z-2nl),
TN(z)=n=0(N-1)/2exp-i σ1L2×exp{+θ1[z-(2n+1)l]/l}1-i[z-(2n+1)l]+exp-i σ2L2exp{-θ2[z-(2n+1)l]/l}1+i[z-(2n+1)l] +n=0(N-3)/2exp-i σ1L2×exp{-θ1[z+(2n+1)l]/l}1+i[z+(2n+1)l]×+exp-i σ2L2exp{+θ2[z+(2n+1)l]/l}1-i[z+(2n+1)l].
11+iz=1-izz2 (1-z-2+z-4-)1-izz2
exp-2 (x+β¯z)2+y21+iz=exp[-2(u2+v2)+i2z(u2+v2+i4β¯uz)],
ΓN=KLCkωexp(-a+L)hN(a±, L, γ, σ1,2),
hN(a±, L, γ, σ1,2)
=2πL-+|SN(u)|2exp(-4u2)du.
SN(u)=n=1N/2exp(-i4nβlu)×exp-i σ1L2H*(-a-, l, 2nl, γ, σ1-2βu)+exp-i σ2L2H(a-, l, 2nl, γ, σ2+2βu)+n=0(N-2)/2exp(+i4nβlu)×exp-i σ1L2H(a-, l, -2nl, γ, σ1+2βu)+exp-i σ2L2H*(-a-, l, -2nl, γ, σ2-2βu),
SN(u)=n=0(N-1)/2exp[-i2(2n+1)βlu]×exp-i σ1L2H*[-a-, l, (2n+1)l, γ, σ1-2βu]+exp-i σ2L2H[a-, l, (2n+1)l, γ, σ2+2βu]+n=0(N-3)/2exp[+i2(2n+1)βlu]exp-i σ1L2H[a-, l, -(2n+1)l, γ, σ1+2βu]+exp-i σ2L2H*[-a-, l, -(2n+1)l, γ, σ2-2βu],
σ1,22βuσ1,2±2βu,
exp(i4nβlu)exp(±i8nβlu),
exp[i2n(2n+1)βlu]exp[±i4(2n+1)βlu].
limN SN=-L/2+L/2dz exp(-iσz)1+iz,
limzR SN(u)=SN(σ1, σ2)=f(σ1)exp-i 3σ1L4+f(σ2)exp-i σ2L4,
f(σ)=12πL2NsincσL4N sin[(N+1)(σL/4N)]sin(σL/2N)+sin[(N-1)(σL/4N)]sin(σL/2N),
f(σ)=12πL2NsincσL4N 2 sin(σL/4)cos(σL/4N)sin(σL/2N).
ΓNPW=KLCw02sinc2Φ2Ncos2(Φ)FN(Φ),
FN=1N2sin[(N+1)Φ/2N]+sin[(N-1)Φ/2N]sin(Φ/N)2;
FN=4N2cos(Φ/2N)sin(Φ/2)sin(Φ/N)2.
P2ω=KLCkωh(B, L)Pω21+KLCkωh(B, L)Pω.
E1z=i4 t2-a12E1+iχE3E1*exp(-iσz),
E2z=i8 t2-β x-a22E2+iχE12exp(+iσz).
exp[(A+χN)dz]=expdz2Aexp(χdzN)expdz2A+O(dz3),
exp(χdzN)=n=0(χdzN)nn!,
N=0B-B*0,
B=iE1*exp(-iσz)00-iE1exp(+iσz).
N2n=(-1)n|E1|2nI,N2n+1=(-1)n|E1|2nN,
exp(χdzN)=Icos(χdz|E1|)+χdz sinc(χdz|E1|)N.
E1(z+dz)=expdz2 D1 exp-dz a12cos(χdz|E1|)×expdz2 D1E1(z)+expdz2 D1×iχdzE1*exp(-iσz)sinc(χdz|E1|)×exp-dz2a1+a22expdz2 D2E2(z)+O(dz3),
E2(z+dz)=expdz2 D2 exp-dz a22cos(χdz|E1|)×expdz2 D2E2(z)+expdz2 D2×iχdzE1exp(+iσz)sinc(χdz|E1|)×exp-dz2a1+a22expdz2 D1E1(z)+O(dz3).

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