Abstract

We have investigated proposed all-optical switching by use of cascading of phase-matched sum-frequency-generation (SFG) and difference-frequency-generation (DFG) processes. The switching operation is accomplished by both a π phase change and an amplitude change of the signal field in SFG–DFG processes. We theoretically confirmed that the proposed switch requires the least operation power of any switch that uses cascading SFG–DFG processes. The switching characteristics have been numerically clarified, and the experimental full-switching characteristics when a bulk-type periodically poled LiNbO3 is used correspond to the numerical analysis. An ultrafast all-optical switch with low switching power and small size can be expected from use of waveguide-type devices.

© 1997 Optical Society of America

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  1. T. Morioka, H. Takara, K. Mori, and M. Saruwatari, Electron. Lett. 28, 521 (1992).
    [CrossRef]
  2. M. Asobe, T. Ohara, I. Yokohama, and T. Kaino, Electron. Lett. 32, 1396 (1996).
    [CrossRef]
  3. H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
    [CrossRef] [PubMed]
  4. G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993).
    [CrossRef] [PubMed]
  5. Y. Baek, R. Schiek, and G. I. Stegeman, Opt. Lett. 20, 2168 (1995).
    [CrossRef]
  6. Y. Baek, R. Schiek, G. Krijnen, G. I. Stegeman, and I. Baumann, in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CWF61.
  7. G. Assanto, Z. Wang, D. J. Hagan, and E. W. Van Stryland, Appl. Phys. Lett. 67, 2120 (1995).
    [CrossRef]
  8. L. Lefort and A. Barthelemy, Opt. Lett. 20, 1749 (1995).
    [CrossRef]
  9. G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. Van Stryland, Appl. Phys. Lett. 62, 1323 (1993).
    [CrossRef]
  10. C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 2650 (1993).
    [CrossRef]
  11. A. L. Belostotosky, A. S. Leonov, and A. V. Meleshko, Opt. Lett. 19, 856 (1994).
    [CrossRef]
  12. D. C. Hutchings, J. S. Aitchison, and C. N. Ironside, Opt. Lett. 18, 793 (1993).
    [CrossRef] [PubMed]
  13. I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, in Extended Abstracts (The 56th Autumn Meeting, 1995) (Japan Society of Applied Physics, Tokyo, 1995), paper 29a-P-8 (in Japanese).
  14. M. Asobe, I. Yokohama, H. Itoh, and T. Kaino, Opt. Lett. 22, 274 (1997).
    [CrossRef] [PubMed]
  15. L. E. Myers, R. C. Edkardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, J. Opt. Soc. Am. B 12, 2102 (1995).
    [CrossRef]
  16. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).
  17. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
    [CrossRef]
  18. M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, Appl. Phys. Lett. 62, 435 (1993).
    [CrossRef]
  19. A. Harada, Y. Okazaki, K. Kamiyama, and S. Umegaki, Appl. Phys. Lett. 59, 1535 (1991).
    [CrossRef]
  20. M. Asobe, T. Kanamori, and K. Kubodera, IEEE J. Quantum Electron. 29, 2325 (1993).
    [CrossRef]

1997 (1)

1996 (1)

M. Asobe, T. Ohara, I. Yokohama, and T. Kaino, Electron. Lett. 32, 1396 (1996).
[CrossRef]

1995 (4)

1994 (1)

1993 (6)

M. Asobe, T. Kanamori, and K. Kubodera, IEEE J. Quantum Electron. 29, 2325 (1993).
[CrossRef]

G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993).
[CrossRef] [PubMed]

D. C. Hutchings, J. S. Aitchison, and C. N. Ironside, Opt. Lett. 18, 793 (1993).
[CrossRef] [PubMed]

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. Van Stryland, Appl. Phys. Lett. 62, 1323 (1993).
[CrossRef]

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, Appl. Phys. Lett. 62, 435 (1993).
[CrossRef]

1992 (2)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

T. Morioka, H. Takara, K. Mori, and M. Saruwatari, Electron. Lett. 28, 521 (1992).
[CrossRef]

1991 (1)

A. Harada, Y. Okazaki, K. Kamiyama, and S. Umegaki, Appl. Phys. Lett. 59, 1535 (1991).
[CrossRef]

1990 (1)

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef] [PubMed]

Aitchison, J. S.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

D. C. Hutchings, J. S. Aitchison, and C. N. Ironside, Opt. Lett. 18, 793 (1993).
[CrossRef] [PubMed]

Arnold, J. M.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

Asobe, M.

M. Asobe, I. Yokohama, H. Itoh, and T. Kaino, Opt. Lett. 22, 274 (1997).
[CrossRef] [PubMed]

M. Asobe, T. Ohara, I. Yokohama, and T. Kaino, Electron. Lett. 32, 1396 (1996).
[CrossRef]

M. Asobe, T. Kanamori, and K. Kubodera, IEEE J. Quantum Electron. 29, 2325 (1993).
[CrossRef]

I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, in Extended Abstracts (The 56th Autumn Meeting, 1995) (Japan Society of Applied Physics, Tokyo, 1995), paper 29a-P-8 (in Japanese).

Assanto, G.

G. Assanto, Z. Wang, D. J. Hagan, and E. W. Van Stryland, Appl. Phys. Lett. 67, 2120 (1995).
[CrossRef]

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. Van Stryland, Appl. Phys. Lett. 62, 1323 (1993).
[CrossRef]

G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993).
[CrossRef] [PubMed]

Baek, Y.

Y. Baek, R. Schiek, and G. I. Stegeman, Opt. Lett. 20, 2168 (1995).
[CrossRef]

Y. Baek, R. Schiek, G. Krijnen, G. I. Stegeman, and I. Baumann, in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CWF61.

Bakker, H. J.

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef] [PubMed]

Barthelemy, A.

Baumann, I.

Y. Baek, R. Schiek, G. Krijnen, G. I. Stegeman, and I. Baumann, in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CWF61.

Belostotosky, A. L.

Bosenberg, W. R.

Byer, R. L.

L. E. Myers, R. C. Edkardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, J. Opt. Soc. Am. B 12, 2102 (1995).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Edkardt, R. C.

Fejer, M. M.

L. E. Myers, R. C. Edkardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, J. Opt. Soc. Am. B 12, 2102 (1995).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Hagan, D. J.

G. Assanto, Z. Wang, D. J. Hagan, and E. W. Van Stryland, Appl. Phys. Lett. 67, 2120 (1995).
[CrossRef]

Harada, A.

A. Harada, Y. Okazaki, K. Kamiyama, and S. Umegaki, Appl. Phys. Lett. 59, 1535 (1991).
[CrossRef]

Hutchings, D. C.

Ironside, C. N.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

D. C. Hutchings, J. S. Aitchison, and C. N. Ironside, Opt. Lett. 18, 793 (1993).
[CrossRef] [PubMed]

Itoh, H.

M. Asobe, I. Yokohama, H. Itoh, and T. Kaino, Opt. Lett. 22, 274 (1997).
[CrossRef] [PubMed]

I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, in Extended Abstracts (The 56th Autumn Meeting, 1995) (Japan Society of Applied Physics, Tokyo, 1995), paper 29a-P-8 (in Japanese).

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Kaino, T.

M. Asobe, I. Yokohama, H. Itoh, and T. Kaino, Opt. Lett. 22, 274 (1997).
[CrossRef] [PubMed]

M. Asobe, T. Ohara, I. Yokohama, and T. Kaino, Electron. Lett. 32, 1396 (1996).
[CrossRef]

I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, in Extended Abstracts (The 56th Autumn Meeting, 1995) (Japan Society of Applied Physics, Tokyo, 1995), paper 29a-P-8 (in Japanese).

Kamiyama, K.

A. Harada, Y. Okazaki, K. Kamiyama, and S. Umegaki, Appl. Phys. Lett. 59, 1535 (1991).
[CrossRef]

Kanamori, T.

M. Asobe, T. Kanamori, and K. Kubodera, IEEE J. Quantum Electron. 29, 2325 (1993).
[CrossRef]

Krijnen, G.

Y. Baek, R. Schiek, G. Krijnen, G. I. Stegeman, and I. Baumann, in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CWF61.

Kubodera, K.

M. Asobe, T. Kanamori, and K. Kubodera, IEEE J. Quantum Electron. 29, 2325 (1993).
[CrossRef]

Kuipers, L.

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef] [PubMed]

Lagendijk, A.

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef] [PubMed]

Lefort, L.

Leonov, A. S.

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Meleshko, A. V.

Mori, K.

T. Morioka, H. Takara, K. Mori, and M. Saruwatari, Electron. Lett. 28, 521 (1992).
[CrossRef]

Morioka, T.

T. Morioka, H. Takara, K. Mori, and M. Saruwatari, Electron. Lett. 28, 521 (1992).
[CrossRef]

Myers, L. E.

Nada, N.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, Appl. Phys. Lett. 62, 435 (1993).
[CrossRef]

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Ohara, T.

M. Asobe, T. Ohara, I. Yokohama, and T. Kaino, Electron. Lett. 32, 1396 (1996).
[CrossRef]

Okazaki, Y.

A. Harada, Y. Okazaki, K. Kamiyama, and S. Umegaki, Appl. Phys. Lett. 59, 1535 (1991).
[CrossRef]

Pierce, J. W.

Planken, P. C. M.

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef] [PubMed]

Saitoh, M.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, Appl. Phys. Lett. 62, 435 (1993).
[CrossRef]

Saruwatari, M.

T. Morioka, H. Takara, K. Mori, and M. Saruwatari, Electron. Lett. 28, 521 (1992).
[CrossRef]

Schiek, R.

Y. Baek, R. Schiek, and G. I. Stegeman, Opt. Lett. 20, 2168 (1995).
[CrossRef]

Y. Baek, R. Schiek, G. Krijnen, G. I. Stegeman, and I. Baumann, in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CWF61.

Sheik-Bahae, M.

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. Van Stryland, Appl. Phys. Lett. 62, 1323 (1993).
[CrossRef]

G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993).
[CrossRef] [PubMed]

Stegeman, G. I.

Y. Baek, R. Schiek, and G. I. Stegeman, Opt. Lett. 20, 2168 (1995).
[CrossRef]

G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993).
[CrossRef] [PubMed]

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. Van Stryland, Appl. Phys. Lett. 62, 1323 (1993).
[CrossRef]

Y. Baek, R. Schiek, G. Krijnen, G. I. Stegeman, and I. Baumann, in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CWF61.

Takara, H.

T. Morioka, H. Takara, K. Mori, and M. Saruwatari, Electron. Lett. 28, 521 (1992).
[CrossRef]

Umegaki, S.

A. Harada, Y. Okazaki, K. Kamiyama, and S. Umegaki, Appl. Phys. Lett. 59, 1535 (1991).
[CrossRef]

Van Stryland, E.

G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993).
[CrossRef] [PubMed]

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. Van Stryland, Appl. Phys. Lett. 62, 1323 (1993).
[CrossRef]

Van Stryland, E. W.

G. Assanto, Z. Wang, D. J. Hagan, and E. W. Van Stryland, Appl. Phys. Lett. 67, 2120 (1995).
[CrossRef]

Wang, Z.

G. Assanto, Z. Wang, D. J. Hagan, and E. W. Van Stryland, Appl. Phys. Lett. 67, 2120 (1995).
[CrossRef]

Watanabe, K.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, Appl. Phys. Lett. 62, 435 (1993).
[CrossRef]

Yamada, M.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, Appl. Phys. Lett. 62, 435 (1993).
[CrossRef]

Yokohama, I.

M. Asobe, I. Yokohama, H. Itoh, and T. Kaino, Opt. Lett. 22, 274 (1997).
[CrossRef] [PubMed]

M. Asobe, T. Ohara, I. Yokohama, and T. Kaino, Electron. Lett. 32, 1396 (1996).
[CrossRef]

I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, in Extended Abstracts (The 56th Autumn Meeting, 1995) (Japan Society of Applied Physics, Tokyo, 1995), paper 29a-P-8 (in Japanese).

Yokoo, A.

I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, in Extended Abstracts (The 56th Autumn Meeting, 1995) (Japan Society of Applied Physics, Tokyo, 1995), paper 29a-P-8 (in Japanese).

Appl. Phys. Lett. (4)

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. Van Stryland, Appl. Phys. Lett. 62, 1323 (1993).
[CrossRef]

G. Assanto, Z. Wang, D. J. Hagan, and E. W. Van Stryland, Appl. Phys. Lett. 67, 2120 (1995).
[CrossRef]

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, Appl. Phys. Lett. 62, 435 (1993).
[CrossRef]

A. Harada, Y. Okazaki, K. Kamiyama, and S. Umegaki, Appl. Phys. Lett. 59, 1535 (1991).
[CrossRef]

Electron. Lett. (2)

T. Morioka, H. Takara, K. Mori, and M. Saruwatari, Electron. Lett. 28, 521 (1992).
[CrossRef]

M. Asobe, T. Ohara, I. Yokohama, and T. Kaino, Electron. Lett. 32, 1396 (1996).
[CrossRef]

IEEE J. Quantum Electron. (3)

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

M. Asobe, T. Kanamori, and K. Kubodera, IEEE J. Quantum Electron. 29, 2325 (1993).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (6)

Phys. Rev. A (1)

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef] [PubMed]

Other (3)

Y. Baek, R. Schiek, G. Krijnen, G. I. Stegeman, and I. Baumann, in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CWF61.

I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, in Extended Abstracts (The 56th Autumn Meeting, 1995) (Japan Society of Applied Physics, Tokyo, 1995), paper 29a-P-8 (in Japanese).

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

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Figures (12)

Fig. 1
Fig. 1

Schematic configurations of all-optical switches using cascading of phase-matched SFG and DFG processes based on (a) an ANI and (b) a PPI.

Fig. 2
Fig. 2

Loci of a signal field from the initial position (E2) to the full-switching position (-E2) on a complex plane for a third-order nonlinear optical switch (Route 1), a phase-mismatched SFG–DFG cascaded switch (ΔkL=2π; Route 2), and the proposed phase-matched SFG–DFG cascaded switch (ΔkL=0; Route 3).

Fig. 3
Fig. 3

Calculated ΓL dependence of (a) the normalized signal intensity in a second-order nonlinear medium, (b) the signal phase in a second-order nonlinear medium, and (c) the normalized signal output from an interferometer for the proposed phase-matched (ΔkL=0) ANI structure switch when the initial signal-to-gate photon intensity ratios are (λ2I2)/(λ1I1)0 (thick solid curves), (λ2I2)/(λ1I1)=0.01 (thin solid curves), (λ2I2)/(λ1I1)=0.1 (dotted–dashed curves), (λ2I2)/(λ1I1)=0.5 (triple-dotted–dashed curves), and (λ2I2)/(λ1I1)=1 (dashed curves).

Fig. 4
Fig. 4

Calculated ΓL dependence of (a) the normalized signal intensity in a second-order nonlinear medium, (b) the signal phase in a second-order nonlinear medium, and (c) the normalized signal output from an interferometer for the phase-mismatched (ΔkL=2π) ANI structure switch. Curves have the same values as for Fig. 3.

Fig. 5
Fig. 5

Calculated ΓL dependence of (a) the normalized signal intensity in a second-order nonlinear medium, (b) the signal phase in a second-order nonlinear medium, and (c) the normalized signal output from an interferometer for the phase-mismatched (ΔkL=π) PPI structure switch. Curves have the same values as in Fig. 3.

Fig. 6
Fig. 6

Calculated dependence of ΓL on (λ2I2)/(λ1I1) for full-switching operation.

Fig. 7
Fig. 7

Calculated ΔkL dependence of normalized signal output intensity under conditions of ΓL=π and (λ2I2)/(λ1I1)=0.01.

Fig. 8
Fig. 8

Experimentally measured and calculated SFG tuning curves for a 10-mm-long bulk-type PPLN with a pump wavelength of 1.319 µm.

Fig. 9
Fig. 9

Experimental configuration of the proposed phase-matched SFG–DFG cascading all-optical switch based on a polarization interferometer with bulk-type PPLN.

Fig. 10
Fig. 10

Experimentally measured intensities of a switched signal and SFG as functions of gate peak power.

Fig. 11
Fig. 11

Calculated intensities of a switched signal and SFG as functions of gate peak power for a continuous-wave (CW) gate and a Gaussian gate pulse with a width less than that of the signal pulse.

Fig. 12
Fig. 12

(a) Dependences of the measured (open circles) and the calculated (solid curve) switched signal intensity on the polarization angle of the gate light and (b) dependence of the switched signal intensity on the time delay between the gate and the signal pulses.

Tables (1)

Tables Icon

Table 1 Phase-Matched Conditions, Interferometer Structures, and Switching Gate Intensities Required for Full Switching Operation

Equations (45)

Equations on this page are rendered with MathJax. Learn more.

Isig.out=1/2[(IA-IB)2+2IAIB(1-cos ϕ)],
dE1*dz=-i 2πdn1λ1E3*E2 exp(iΔkz),
dE2*dz=-i 2πdn2λ2E3*E1 exp(iΔkz),
dE3dz=i 2πdn3λ3E1E2 exp(iΔkz).
d2E2dz2+iΔk dE2dz
+4π2d2n2n3λ2λ3|E1|2-4π2d2n1n2λ1λ2|E3|2E2=0.
E2=E2(0)exp-i Δkz2cos(Γz)+i Δk2Γsin(Γz),
Γ=[Γ2+(Δk/2)2]1/2,
Γ=2πd(n2n3λ2λ3)1/2|E1(0)|.
cos(Γz)+i Δk2Γsin(Γz)=1.
Γz=nπ(naninteger).
E2(z)=E2(0)cosΔk2z-i sinΔk2z.
(Δkz/2)=(2p+1)π(paninteger).
Γ2z2=Γ2z2+(2p+1)2π2=(2mπ)2,
Γz=[(2m)2-(2p+1)2π]1/2.
Γz=3π,
Δkz=2π.
I1(Δkz=2π)=3n1n2n3λ2λ38η0d2z2,
E2(z)=E2(0)-cosΔk2z+i sinΔk2z.
(Δkz/2)=2pπ(paninteger).
Γ2z2=Γ2z2+(2p)2π2=[(2m+1)π]2,
Γz=[(2m+1)2-(2p)2π]1/2.
Γz=π,
Δkz=0.
I1(Δkz=0)=n1n2n3λ2λ38η0d2z2.
(Δkz/2)=[1/2+(2p)]π(paninteger).
Γ2z2=Γ2z2+[1/2+(2p)]2π2=(2mπ)2,
Γz=[(2m)2-(1/2+2p)2π]1/2.
Γz=7/2 π,
Δkz=3π.
I1(Δkz=3π)=7n1n2n3λ2λ316η0d2z2.
(Δkz/2)=[±1/2+(2p)]π(paninteger).
Γ2z2=Γ2z2+[±1/2+(2p)]2π2=[(2m+1)π]2,
Γz=[(2m+1)2-(±1/2+2p)2]1/2π.
Γz=(3/2)π,
Δkz=π.
I1(Δkz=π)=3n1n2n3λ2λ316η0d2z2.
E2(z)=E2(0)cos(Γz).
E3(z)=iλ2/λ3E2(0)sin(Γz).
I2(z)=I2(0)cos2(Γz)=I2(0)cos222η0πdz(n1n2n3λ2λ3)1/2I1,
I3(z)=λ2λ3I2(0)sin2(Γz)=λ2λ3I2(0)sin222η0πdz(n1n2n3λ2λ3)1/2I1.
Isig.out(z)=Isig.in sin4Γz2=Isig.in sin42η0πdz(n1n2n3λ2λ3)1/2I1,
ηSFG=8π2η0d2z2n1n2n3λ32.
I3=λ2λ3I2(0)sin2λ3λ2ηSFGI11/2.
Isig.out(z)=Isig.in sin412λ3λ2ηSFGI11/2.

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