Abstract

Theoretically designed and experimentally realized simultaneous perfect phase matching of second and third harmonic generations were demonstrated in a one-dimensional ZnS/YF3 photonic crystal (PC) structure. Dramatic enhancement of second harmonic generation (SHG) and third harmonic generation (THG) in forward and backward directions near the photonic band edge were observed. This enhancement came from a combination of large ZnS nonlinear susceptibility coefficients, high density of optical modes and perfect phase matching of the fundamental and the harmonic waves near the photonic band edge due to modification of the dispersion curve by the PC structure. Total SHG and THG conversion efficiency over 4% is measured in only six micrometers length of photonic crystal. Theoretical calculations show good agreement with experimental measurements.

© 2006 Optical Society of America

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    [Crossref] [PubMed]
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  9. N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–486 (1970).
    [Crossref]
  10. J. P. Van der Ziel and M. Ilegem, “Optical second harmonic generation in periodic multilayer GaAs-Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  23. Refractive index (n) for YF3 is the proprietary information of the manufacturer. However, according to the manufacturer, the n of SiO2 is almost the same as the YF3 in the wavelength region of our interest, and indeed, experimental measured transmission curve is in excellent agreement with our simulation (Fig. 1a).
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2006 (1)

Hui Yang, Ping Xie, S. K. Chan, Weixin Lu, Zhao-Qing Zhang, I. K. Sou, George K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semicounductor photonic crystals”, IEEE Quantum Electronic,  42, 447–451 (2006)
[Crossref]

2005 (1)

H. Yang, P. Xie, S. K. Chan, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Efficienct second harmonic generation form large band gap II–VI semiconductor photonic crstal”, Appl. Phys. Lett. 97, 131106 (2005).
[Crossref]

2004 (4)

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, Nick N. Lepeshkin, and Robert W. Boyd, “Dramatic Enhancement of Third-Harmonic Generation in Three-Dimensional Photonic Crystals,” Phys. Rev. Lett. 92, 083903 (2004).
[Crossref] [PubMed]

P. Xie and Z.Q. Zhang, “Optical phase conjugation in third-order nonlinear photonic crystals,” Phys. Rev. A 69, 053806 (2004).
[Crossref]

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature,  432, 374–376, (2004).
[Crossref] [PubMed]

M. G. Martemyanov, T. V. Dolgova, and A. A. Fedyanin, “Optical third-harmonic generation in one-dimensional photonic crystals and microcavities,” J. of Exp. And Theor. Phys. 98, 463–477 (2004).
[Crossref]

2001 (2)

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D. Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[Crossref]

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Simultaneously phase-matched enhanced second and third harmonic generation,” Phys. Rev. E 64, 046606 (2001).
[Crossref]

2000 (1)

M. Centini, M. Scalora, C. Sibilia, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Dispersive properties of one-dimensional photonic band gap structures for second harmonic generation,” J. Opt. A: Pure Appl. Opt. 2, 121–126 (2000).
[Crossref]

1999 (2)

1989 (1)

1987 (2)

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

1976 (1)

J. P. Van der Ziel and M. Ilegem, “Optical second harmonic generation in periodic multilayer GaAs-Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[Crossref]

1970 (1)

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–486 (1970).
[Crossref]

1964 (1)

H. Cheng and P. B. Miller, “Nonlinear Optical Theory in Solids,” Phys. Rev. 134, A683–A687 (1964).
[Crossref]

1963 (1)

P. A. Franken and J. F. Ward, “Optical Harmonics and Nonlinear Phenomena,” Rev. Mod. Phys. 35, 23–39 (1963).
[Crossref]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962)
[Crossref]

Aguanno, G. D.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D. Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[Crossref]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962)
[Crossref]

Balakin, A. V.

Baudrier-Raybaut, M.

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature,  432, 374–376, (2004).
[Crossref] [PubMed]

Bertolotti, M.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Simultaneously phase-matched enhanced second and third harmonic generation,” Phys. Rev. E 64, 046606 (2001).
[Crossref]

M. Centini, M. Scalora, C. Sibilia, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Dispersive properties of one-dimensional photonic band gap structures for second harmonic generation,” J. Opt. A: Pure Appl. Opt. 2, 121–126 (2000).
[Crossref]

Bethune, D. S.

Bloembergen, N.

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–486 (1970).
[Crossref]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962)
[Crossref]

Bloemer, M. J.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Simultaneously phase-matched enhanced second and third harmonic generation,” Phys. Rev. E 64, 046606 (2001).
[Crossref]

M. Centini, M. Scalora, C. Sibilia, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Dispersive properties of one-dimensional photonic band gap structures for second harmonic generation,” J. Opt. A: Pure Appl. Opt. 2, 121–126 (2000).
[Crossref]

Boucher, D.

Bowden, C. M.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Simultaneously phase-matched enhanced second and third harmonic generation,” Phys. Rev. E 64, 046606 (2001).
[Crossref]

M. Centini, M. Scalora, C. Sibilia, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Dispersive properties of one-dimensional photonic band gap structures for second harmonic generation,” J. Opt. A: Pure Appl. Opt. 2, 121–126 (2000).
[Crossref]

Boyd, Robert W.

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, Nick N. Lepeshkin, and Robert W. Boyd, “Dramatic Enhancement of Third-Harmonic Generation in Three-Dimensional Photonic Crystals,” Phys. Rev. Lett. 92, 083903 (2004).
[Crossref] [PubMed]

Centini, M.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D. Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[Crossref]

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Simultaneously phase-matched enhanced second and third harmonic generation,” Phys. Rev. E 64, 046606 (2001).
[Crossref]

M. Centini, M. Scalora, C. Sibilia, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Dispersive properties of one-dimensional photonic band gap structures for second harmonic generation,” J. Opt. A: Pure Appl. Opt. 2, 121–126 (2000).
[Crossref]

Chan, S. K.

Hui Yang, Ping Xie, S. K. Chan, Weixin Lu, Zhao-Qing Zhang, I. K. Sou, George K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semicounductor photonic crystals”, IEEE Quantum Electronic,  42, 447–451 (2006)
[Crossref]

H. Yang, P. Xie, S. K. Chan, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Efficienct second harmonic generation form large band gap II–VI semiconductor photonic crstal”, Appl. Phys. Lett. 97, 131106 (2005).
[Crossref]

Cheng, H.

H. Cheng and P. B. Miller, “Nonlinear Optical Theory in Solids,” Phys. Rev. 134, A683–A687 (1964).
[Crossref]

D’Aguanno, G.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Simultaneously phase-matched enhanced second and third harmonic generation,” Phys. Rev. E 64, 046606 (2001).
[Crossref]

M. Centini, M. Scalora, C. Sibilia, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Dispersive properties of one-dimensional photonic band gap structures for second harmonic generation,” J. Opt. A: Pure Appl. Opt. 2, 121–126 (2000).
[Crossref]

Dolgova, T. V.

M. G. Martemyanov, T. V. Dolgova, and A. A. Fedyanin, “Optical third-harmonic generation in one-dimensional photonic crystals and microcavities,” J. of Exp. And Theor. Phys. 98, 463–477 (2004).
[Crossref]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962)
[Crossref]

Dumeige, Y.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D. Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[Crossref]

Fedyanin, A. A.

M. G. Martemyanov, T. V. Dolgova, and A. A. Fedyanin, “Optical third-harmonic generation in one-dimensional photonic crystals and microcavities,” J. of Exp. And Theor. Phys. 98, 463–477 (2004).
[Crossref]

Franken, P. A.

P. A. Franken and J. F. Ward, “Optical Harmonics and Nonlinear Phenomena,” Rev. Mod. Phys. 35, 23–39 (1963).
[Crossref]

Haïdar, R.

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature,  432, 374–376, (2004).
[Crossref] [PubMed]

Ilegem, M.

J. P. Van der Ziel and M. Ilegem, “Optical second harmonic generation in periodic multilayer GaAs-Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[Crossref]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

Konotop, V. V.

Koroteev, N. I.

Kupecek, Ph.

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature,  432, 374–376, (2004).
[Crossref] [PubMed]

Kuzmiak, V.

Lemasson, Ph.

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature,  432, 374–376, (2004).
[Crossref] [PubMed]

Lepeshkin, Nick N.

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, Nick N. Lepeshkin, and Robert W. Boyd, “Dramatic Enhancement of Third-Harmonic Generation in Three-Dimensional Photonic Crystals,” Phys. Rev. Lett. 92, 083903 (2004).
[Crossref] [PubMed]

Levenson, J. A.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D. Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[Crossref]

Lu, Weixin

Hui Yang, Ping Xie, S. K. Chan, Weixin Lu, Zhao-Qing Zhang, I. K. Sou, George K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semicounductor photonic crystals”, IEEE Quantum Electronic,  42, 447–451 (2006)
[Crossref]

Mantsyzov, B. I.

Markowicz, P. P.

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, Nick N. Lepeshkin, and Robert W. Boyd, “Dramatic Enhancement of Third-Harmonic Generation in Three-Dimensional Photonic Crystals,” Phys. Rev. Lett. 92, 083903 (2004).
[Crossref] [PubMed]

Martemyanov, M. G.

M. G. Martemyanov, T. V. Dolgova, and A. A. Fedyanin, “Optical third-harmonic generation in one-dimensional photonic crystals and microcavities,” J. of Exp. And Theor. Phys. 98, 463–477 (2004).
[Crossref]

Masselin, P.

Miller, P. B.

H. Cheng and P. B. Miller, “Nonlinear Optical Theory in Solids,” Phys. Rev. 134, A683–A687 (1964).
[Crossref]

Ozheredov, I. A.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962)
[Crossref]

Prasad, P. N.

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, Nick N. Lepeshkin, and Robert W. Boyd, “Dramatic Enhancement of Third-Harmonic Generation in Three-Dimensional Photonic Crystals,” Phys. Rev. Lett. 92, 083903 (2004).
[Crossref] [PubMed]

Pudavar, H.

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, Nick N. Lepeshkin, and Robert W. Boyd, “Dramatic Enhancement of Third-Harmonic Generation in Three-Dimensional Photonic Crystals,” Phys. Rev. Lett. 92, 083903 (2004).
[Crossref] [PubMed]

Rosencher, E.

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature,  432, 374–376, (2004).
[Crossref] [PubMed]

Sagnes, I.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D. Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[Crossref]

Sauvage, S.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D. Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[Crossref]

Scalora, M.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Simultaneously phase-matched enhanced second and third harmonic generation,” Phys. Rev. E 64, 046606 (2001).
[Crossref]

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D. Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[Crossref]

M. Centini, M. Scalora, C. Sibilia, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Dispersive properties of one-dimensional photonic band gap structures for second harmonic generation,” J. Opt. A: Pure Appl. Opt. 2, 121–126 (2000).
[Crossref]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (John Wiley & Sons, New York1984)

Shkurinov, A. P.

Sibilia, C.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Simultaneously phase-matched enhanced second and third harmonic generation,” Phys. Rev. E 64, 046606 (2001).
[Crossref]

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D. Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[Crossref]

M. Centini, M. Scalora, C. Sibilia, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Dispersive properties of one-dimensional photonic band gap structures for second harmonic generation,” J. Opt. A: Pure Appl. Opt. 2, 121–126 (2000).
[Crossref]

Sievers, A. J.

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–486 (1970).
[Crossref]

Sou, I. K.

Hui Yang, Ping Xie, S. K. Chan, Weixin Lu, Zhao-Qing Zhang, I. K. Sou, George K. L. Wong, and K. S. Wong, “Simultaneous enhancement of the second- and third-harmonic generations in one-dimensional semicounductor photonic crystals”, IEEE Quantum Electronic,  42, 447–451 (2006)
[Crossref]

H. Yang, P. Xie, S. K. Chan, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Efficienct second harmonic generation form large band gap II–VI semiconductor photonic crstal”, Appl. Phys. Lett. 97, 131106 (2005).
[Crossref]

Tiryaki, H.

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, Nick N. Lepeshkin, and Robert W. Boyd, “Dramatic Enhancement of Third-Harmonic Generation in Three-Dimensional Photonic Crystals,” Phys. Rev. Lett. 92, 083903 (2004).
[Crossref] [PubMed]

ushuev, V. A.

Van der Ziel, J. P.

J. P. Van der Ziel and M. Ilegem, “Optical second harmonic generation in periodic multilayer GaAs-Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[Crossref]

Vidakovic, P.

Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D. Aguanno, and M. Scalora, “Enhancement of second-harmonic generation in a one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. 78, 3021–3023 (2001).
[Crossref]

Ward, J. F.

P. A. Franken and J. F. Ward, “Optical Harmonics and Nonlinear Phenomena,” Rev. Mod. Phys. 35, 23–39 (1963).
[Crossref]

Weber, Marvin J.

Marvin J. Weber, CRC Handbook of Laser Science and Technology, Volume III Optical Materials: Part 1 (CRC Press, 1986).

Marvin J. Weber, CRC Handbook of Laser Science and Technology, Supplement 2: Optical Materials (CRC Press, 1995).

Wong, G. K. L.

H. Yang, P. Xie, S. K. Chan, Z. Q. Zhang, I. K. Sou, G. K. L. Wong, and K. S. Wong, “Efficienct second harmonic generation form large band gap II–VI semiconductor photonic crstal”, Appl. Phys. Lett. 97, 131106 (2005).
[Crossref]

Wong, George K. L.

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Refractive index (n) for YF3 is the proprietary information of the manufacturer. However, according to the manufacturer, the n of SiO2 is almost the same as the YF3 in the wavelength region of our interest, and indeed, experimental measured transmission curve is in excellent agreement with our simulation (Fig. 1a).

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Figures (3)

Fig. 1.
Fig. 1.

(a) The experimentally measured (dotted line) and the corresponding theoretically calculated (solid line) transmission. (b) Effective refractive index (solid line) and density of modes (DOM) (dashed line) as a function of wavelengths of the same PC.

Fig. 2.
Fig. 2.

The experimentally (solid circle) measured and theoretically (dash line without and solid line with the fluctuation of layer thickness) calculated SHG (a) and THG (b) conversion efficiency in the forward direction at excitation intensity of 3 GW/cm2.

Fig. 3.
Fig. 3.

Conversion efficiency of experimentally (solid circle) measured and theoretically calculated SHG(a) and THG(b) without (dash line) and with (solid line) fluctuation in layer thickness in the backward direction at excitation intensity of 3 GW/cm2.

Equations (3)

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d B 1 ± d z = ± i 4 π ω n ( ω ) c χ ( 2 ) ( B 1 ± * B 2 ± e ± i Δ k 1 z + B 2 ± * B 3 ± e ± i Δ k 2 z ) ± i 6 π ω n ( ω ) c χ ( 3 ) ( B 1 ± * 2 B 3 ± e ± i Δ k 3 z + B 2 ± 2 B 3 ± * e i Δ k 4 z ) ,
d B 2 ± d z = ± i 4 π ω n ( 2 ω ) c χ ( 2 ) ( B 1 ± 2 e i Δ k 1 z + 2 B 1 ± * B 3 ± e ± i Δ k 2 z ) ± i 24 π ω n ( 2 ω ) c χ ( 3 ) B 1 ± B 2 ± * B 3 ± e ± i Δ k 4 z ,
d B 3 ± d z = ± i 12 π ω n ( 3 ω ) c χ ( 2 ) B 1 ± B 2 ± e i Δ k 2 z ± i 6 π ω n ( 3 ω ) c χ ( 3 ) ( B 1 ± 3 e i Δ k 3 z + 3 B 1 ± * B 2 ± 2 e i Δ k 4 z ) ,

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