Abstract

Theoretical and numerical solutions of the nonlinear coupling-wave equations of second-harmonic (SH) and third-harmonic (TH) converters are investigated for both phase-matched and phase-mismatched configurations. For phase-mismatched TH generation, several kinds of schemes [the phase mismatch in either SH generation or sum-frequency generation (SFG) as well as the phase mismatch in both SH generation and SFG] are considered and analyzed. The optimal TH converting properties for generalized phase-mismatching operation are discussed and studied. It is found that efficient TH generation depends not only on the magnitude of the coupling coefficients but also on their ratio. A 100% TH output can be obtained at a particular ratio. The physical nature of the different ratios of the coupling coefficients is discussed.

© 2003 Optical Society of America

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  1. G. D. Miller, R. G. Batchke, W. M. Tulloch, D. K. Weise, M. M. Fejer, and R. L. Beyer, “42%-efficient signal-pass cw second-harmonic generation in periodically poled lithium niobate,” Opt. Lett. 22, 1834–1836 (1997).
    [CrossRef]
  2. A. Arie, G. Rosenman, A. Korenfeld, A. Skliar, M. Oron, and D. Eger, “Efficient resonant frequency doubling of a cw NdYAG laser in bulk periodically poled KTiOPO4,” Opt. Lett. 23, 28–30 (1998).
    [CrossRef]
  3. V. Pruneri, S. D. Betterworth, and D. C. Hanna, “Highly efficient green-light generation by quasi-phase-matched frequency doubling of picosecond pulses from an amplified mode-locked Nd:YLF laser,” Opt. Lett. 21, 390–392 (1996).
    [CrossRef] [PubMed]
  4. O. Pfister, J. S. Wells, L. Hollberg, L. Zink, D. A. Van Baak, M. D. Levenson, and W. R. Bosenberg, “Continuous-wave frequency tripling and quadrupling by simultaneous three-wave mixings in periodically poled crystals: application to a two-step 1.19–10.71-μm frequency bridge,” Opt. Lett. 22, 1211–1213 (1997).
    [CrossRef] [PubMed]
  5. K. Kintaka, M. Fujimura, T. Suhara, and H. Nishihara, “Third harmonic generation of Nd:YAG laser light in periodically poled LiNbO3,” Electron. Lett. 33, 1459–1461 (1997).
    [CrossRef]
  6. S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
    [CrossRef]
  7. K. C. Burr, C. L. Tang, M. A. Arbore, and M. M. Fejer, “Broadly tunable mid-infrared femtosecond optical parametric oscillator using all-solid-state-pumped periodically poled lithium niobate,” Opt. Lett. 22, 1458–1461 (1997).
    [CrossRef]
  8. T. Kartaloglu, K. G. Koprulu, O. Aytur, M. Sundheimer, and W. P. Risk, “Femtosecond optical parametric oscillator based on periodically poled KTiOPO4,” Opt. Lett. 23, 61–63 (1998).
    [CrossRef]
  9. M. Sundheimer, P. Aschieri, P. Baldi, and J. Bierlein, “Modeling and experimental observation of parametric processes in segmented KTiOPO4,” Appl. Phys. Lett. 74, 1660–1662 (1999).
    [CrossRef]
  10. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light wave in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  11. M. M. Fejer, G. A. Mage, D. H. Jundt, and R. L. Byer, “Quasiphase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
    [CrossRef]
  12. Y. Y. Zhu and N. B. Ming, “Second-harmonic generation in a Fibonacci optical superlattice and the dispersive effect of the refractive index,” Phys. Rev. B 42, 3676–3679 (1990).
    [CrossRef]
  13. Y. Chiu, U. Gopalan, M. J. Kawas, T. E. Schlesinger, D. D. Stancil, and W. P. Risk, “Integrated optical device with second-harmonic generator, electrooptic lens, and electrooptic scanner in LiTaO3,” J. Lightwave Technol. 17, 462–468 (1999).
    [CrossRef]
  14. M. Pierrou, F. Laureu, H. Karlsson, T. Kellner, C. Czeranowsky, and G. Huber, “Generation of 740 mW of blue light by intracavity frequency doubling with a quasi-phase-matched KTiOPO4 crystal,” Opt. Lett. 24, 205–207 (1999).
    [CrossRef]
  15. Y. Y. Zhu, R. F. Xiao, J. S. Fu, and G. K. L. Wong, “Third harmonic generation through coupled second-order nonlinear optical parametric processes in quasi periodically do-main inverted Sr0.6Ba0.4Nb2O6 optical superlattices,” Appl. Phys. Lett. 73, 432–434 (1998).
    [CrossRef]
  16. S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
    [CrossRef]
  17. L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995).
    [CrossRef]

1999

1998

1997

1996

1995

1992

M. M. Fejer, G. A. Mage, D. H. Jundt, and R. L. Byer, “Quasiphase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

1990

Y. Y. Zhu and N. B. Ming, “Second-harmonic generation in a Fibonacci optical superlattice and the dispersive effect of the refractive index,” Phys. Rev. B 42, 3676–3679 (1990).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light wave in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Arbore, M. A.

Arie, A.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light wave in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Aschieri, P.

M. Sundheimer, P. Aschieri, P. Baldi, and J. Bierlein, “Modeling and experimental observation of parametric processes in segmented KTiOPO4,” Appl. Phys. Lett. 74, 1660–1662 (1999).
[CrossRef]

Aytur, O.

Baldi, P.

M. Sundheimer, P. Aschieri, P. Baldi, and J. Bierlein, “Modeling and experimental observation of parametric processes in segmented KTiOPO4,” Appl. Phys. Lett. 74, 1660–1662 (1999).
[CrossRef]

Batchke, R. G.

Betterworth, S. D.

Beyer, R. L.

Bierlein, J.

M. Sundheimer, P. Aschieri, P. Baldi, and J. Bierlein, “Modeling and experimental observation of parametric processes in segmented KTiOPO4,” Appl. Phys. Lett. 74, 1660–1662 (1999).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light wave in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bosenberg, W. R.

Burr, K. C.

Byer, R. L.

L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995).
[CrossRef]

M. M. Fejer, G. A. Mage, D. H. Jundt, and R. L. Byer, “Quasiphase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Chiu, Y.

Czeranowsky, C.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light wave in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Eckardt, R. C.

Eger, D.

Fejer, M. M.

Fu, J. S.

Y. Y. Zhu, R. F. Xiao, J. S. Fu, and G. K. L. Wong, “Third harmonic generation through coupled second-order nonlinear optical parametric processes in quasi periodically do-main inverted Sr0.6Ba0.4Nb2O6 optical superlattices,” Appl. Phys. Lett. 73, 432–434 (1998).
[CrossRef]

Fujimura, M.

K. Kintaka, M. Fujimura, T. Suhara, and H. Nishihara, “Third harmonic generation of Nd:YAG laser light in periodically poled LiNbO3,” Electron. Lett. 33, 1459–1461 (1997).
[CrossRef]

Ge, C. Z.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

Gopalan, U.

Hanna, D. C.

Hollberg, L.

Huber, G.

Jundt, D. H.

M. M. Fejer, G. A. Mage, D. H. Jundt, and R. L. Byer, “Quasiphase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Karlsson, H.

Kartaloglu, T.

Kawas, M. J.

Kellner, T.

Kintaka, K.

K. Kintaka, M. Fujimura, T. Suhara, and H. Nishihara, “Third harmonic generation of Nd:YAG laser light in periodically poled LiNbO3,” Electron. Lett. 33, 1459–1461 (1997).
[CrossRef]

Koprulu, K. G.

Korenfeld, A.

Laureu, F.

Levenson, M. D.

Mage, G. A.

M. M. Fejer, G. A. Mage, D. H. Jundt, and R. L. Byer, “Quasiphase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Miller, G. D.

Ming, N. B.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

Y. Y. Zhu and N. B. Ming, “Second-harmonic generation in a Fibonacci optical superlattice and the dispersive effect of the refractive index,” Phys. Rev. B 42, 3676–3679 (1990).
[CrossRef]

Myers, L. E.

Nishihara, H.

K. Kintaka, M. Fujimura, T. Suhara, and H. Nishihara, “Third harmonic generation of Nd:YAG laser light in periodically poled LiNbO3,” Electron. Lett. 33, 1459–1461 (1997).
[CrossRef]

Oron, M.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light wave in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Pfister, O.

Pierce, J. W.

Pierrou, M.

Pruneri, V.

Qin, Y. Q.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

Risk, W. P.

Rosenman, G.

Schlesinger, T. E.

Skliar, A.

Stancil, D. D.

Suhara, T.

K. Kintaka, M. Fujimura, T. Suhara, and H. Nishihara, “Third harmonic generation of Nd:YAG laser light in periodically poled LiNbO3,” Electron. Lett. 33, 1459–1461 (1997).
[CrossRef]

Sundheimer, M.

M. Sundheimer, P. Aschieri, P. Baldi, and J. Bierlein, “Modeling and experimental observation of parametric processes in segmented KTiOPO4,” Appl. Phys. Lett. 74, 1660–1662 (1999).
[CrossRef]

T. Kartaloglu, K. G. Koprulu, O. Aytur, M. Sundheimer, and W. P. Risk, “Femtosecond optical parametric oscillator based on periodically poled KTiOPO4,” Opt. Lett. 23, 61–63 (1998).
[CrossRef]

Tang, C. L.

Tulloch, W. M.

Van Baak, D. A.

Wang, H. F.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

Weise, D. K.

Wells, J. S.

Wong, G. K. L.

Y. Y. Zhu, R. F. Xiao, J. S. Fu, and G. K. L. Wong, “Third harmonic generation through coupled second-order nonlinear optical parametric processes in quasi periodically do-main inverted Sr0.6Ba0.4Nb2O6 optical superlattices,” Appl. Phys. Lett. 73, 432–434 (1998).
[CrossRef]

Xiao, R. F.

Y. Y. Zhu, R. F. Xiao, J. S. Fu, and G. K. L. Wong, “Third harmonic generation through coupled second-order nonlinear optical parametric processes in quasi periodically do-main inverted Sr0.6Ba0.4Nb2O6 optical superlattices,” Appl. Phys. Lett. 73, 432–434 (1998).
[CrossRef]

Zhu, S. N.

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

Zhu, Y. Y.

Y. Y. Zhu, R. F. Xiao, J. S. Fu, and G. K. L. Wong, “Third harmonic generation through coupled second-order nonlinear optical parametric processes in quasi periodically do-main inverted Sr0.6Ba0.4Nb2O6 optical superlattices,” Appl. Phys. Lett. 73, 432–434 (1998).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

Y. Y. Zhu and N. B. Ming, “Second-harmonic generation in a Fibonacci optical superlattice and the dispersive effect of the refractive index,” Phys. Rev. B 42, 3676–3679 (1990).
[CrossRef]

Zink, L.

Appl. Phys. Lett.

M. Sundheimer, P. Aschieri, P. Baldi, and J. Bierlein, “Modeling and experimental observation of parametric processes in segmented KTiOPO4,” Appl. Phys. Lett. 74, 1660–1662 (1999).
[CrossRef]

Y. Y. Zhu, R. F. Xiao, J. S. Fu, and G. K. L. Wong, “Third harmonic generation through coupled second-order nonlinear optical parametric processes in quasi periodically do-main inverted Sr0.6Ba0.4Nb2O6 optical superlattices,” Appl. Phys. Lett. 73, 432–434 (1998).
[CrossRef]

Electron. Lett.

K. Kintaka, M. Fujimura, T. Suhara, and H. Nishihara, “Third harmonic generation of Nd:YAG laser light in periodically poled LiNbO3,” Electron. Lett. 33, 1459–1461 (1997).
[CrossRef]

IEEE J. Quantum Electron.

M. M. Fejer, G. A. Mage, D. H. Jundt, and R. L. Byer, “Quasiphase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Opt. Lett.

O. Pfister, J. S. Wells, L. Hollberg, L. Zink, D. A. Van Baak, M. D. Levenson, and W. R. Bosenberg, “Continuous-wave frequency tripling and quadrupling by simultaneous three-wave mixings in periodically poled crystals: application to a two-step 1.19–10.71-μm frequency bridge,” Opt. Lett. 22, 1211–1213 (1997).
[CrossRef] [PubMed]

K. C. Burr, C. L. Tang, M. A. Arbore, and M. M. Fejer, “Broadly tunable mid-infrared femtosecond optical parametric oscillator using all-solid-state-pumped periodically poled lithium niobate,” Opt. Lett. 22, 1458–1461 (1997).
[CrossRef]

G. D. Miller, R. G. Batchke, W. M. Tulloch, D. K. Weise, M. M. Fejer, and R. L. Beyer, “42%-efficient signal-pass cw second-harmonic generation in periodically poled lithium niobate,” Opt. Lett. 22, 1834–1836 (1997).
[CrossRef]

A. Arie, G. Rosenman, A. Korenfeld, A. Skliar, M. Oron, and D. Eger, “Efficient resonant frequency doubling of a cw NdYAG laser in bulk periodically poled KTiOPO4,” Opt. Lett. 23, 28–30 (1998).
[CrossRef]

T. Kartaloglu, K. G. Koprulu, O. Aytur, M. Sundheimer, and W. P. Risk, “Femtosecond optical parametric oscillator based on periodically poled KTiOPO4,” Opt. Lett. 23, 61–63 (1998).
[CrossRef]

M. Pierrou, F. Laureu, H. Karlsson, T. Kellner, C. Czeranowsky, and G. Huber, “Generation of 740 mW of blue light by intracavity frequency doubling with a quasi-phase-matched KTiOPO4 crystal,” Opt. Lett. 24, 205–207 (1999).
[CrossRef]

V. Pruneri, S. D. Betterworth, and D. C. Hanna, “Highly efficient green-light generation by quasi-phase-matched frequency doubling of picosecond pulses from an amplified mode-locked Nd:YLF laser,” Opt. Lett. 21, 390–392 (1996).
[CrossRef] [PubMed]

Phys. Rev.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light wave in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. B

Y. Y. Zhu and N. B. Ming, “Second-harmonic generation in a Fibonacci optical superlattice and the dispersive effect of the refractive index,” Phys. Rev. B 42, 3676–3679 (1990).
[CrossRef]

Phys. Rev. Lett.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

Science

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

SH and TH conversion efficiencies as a function of the normalized length of a superlattice under the hypothesis of an undepleted fundamental field and quasi-phase matching in SHG as well as (A) a smaller SFG phase mismatch (B=0.2) and (B) a larger SFG phase mismatch (B=1.0), both with α=0.3. Solid and dashed curves, SH and TH conversion efficiencies, respectively.

Fig. 2
Fig. 2

SH and TH conversion efficiencies as a function of the normalized length of a superlattice under the hypothesis of an undepleted fundamental field and quasi-phase matching in SFG as well as (A) a smaller SHG phase mismatch (B=0.2) and (B) a larger SHG phase mismatch (B=1.0), both with α=0.3. Solid and dashed curves, SH and TH conversion efficiencies, respectively.

Fig. 3
Fig. 3

SH and TH conversion efficiencies as a function of the normalized length of a superlattice under the hypothesis of an undepleted fundamental field and quasi-phase matching in both SHG and SFG simultaneously, with α=κ1/κ2=0.3. Solid and dashed curves, SH and TH conversion efficiencies, respectively.

Fig. 4
Fig. 4

SH and TH conversion efficiencies as a function of the normalized length of a superlattice under the hypothesis of an undepleted fundamental field and quasi-phase matching Δk3=0 for (A) a smaller SFG phase mismatch (B=0.2) and (B) a larger SFG phase mismatch (B=1.0), both with α=0.3. Solid and dashed curves, SH and TH conversion efficiencies, respectively.

Fig. 5
Fig. 5

Relationships of THG to the normalized length of a superlattice for several phase mismatches (Δk30) for (A) a smaller SFG phase mismatch (B=0.2) and (B) a larger SFG phase mismatch (B=1.0), both with α=0.3. Curves show several values of constant δ.

Fig. 6
Fig. 6

Dependence of optical intensities with the depleted fundamental field of a normalized length of a superlattice on the following ratios of two coupling coefficients, κ1/κ2: (A) 0.5, (B) 1.5, and (C) 8/9. Dotted, solid, and dashed curves correspond to fundamental, SH, and TH fields, respectively.

Fig. 7
Fig. 7

Schematic of the optimum quasi-phase-matched TH output with a configuration of two superlattices.

Fig. 8
Fig. 8

Calculated TH conversion efficiency versus the ratio of two coupling coefficients, κ1 and κ2, with an asymptotic value of normalized length κ2A1(0)L=8.

Fig. 9
Fig. 9

Optical intensities as a function of the normalized length of a superlattice with a depleted fundamental field and quasi-phase matching in SHG as well as (A) a smaller SFG phase mismatch (B=0.2) and (B) a larger SFG phase mismatch (B=1.0), both with α=0.3. Dotted, solid, and dashed curves correspond to fundamental, SH, and TH fields, respectively.

Fig. 10
Fig. 10

Optical intensities as a function of the normalized length of a superlattice with a depleted fundamental field and quasi-phase matching in SFG as well as (A) a smaller SHG phase mismatch (B=0.2) with α=0.3 and (B) a larger SHG phase mismatch (B=1.0) with α=2.0. Dotted, solid, and dashed curves correspond to fundamental, SH, and TH fields, respectively.

Fig. 11
Fig. 11

Optical intensities as a function of the normalized length of a superlattice with the depleted fundamental field and quasi-phase matching in the sum of SHG and SFG as well as (A) a smaller SFG phase mismatch (B=0.2) and (B) a larger SFG phase mismatch (B=1.0), both with α=0.3. Dotted, solid, and dashed curves correspond to fundamental, SH, and TH fields, respectively.

Equations (64)

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

2E=μ002Et2+μ02Pt2.
P=0χLE+PNL.
dE1dz=-i ω1d(z)n1c [E3E2*exp(-iΔk2z)+E2E1*exp(-iΔk1z)],
dE2dz=-i ω2d(z)n2cE3E1*exp(-iΔk2z)+12 E12exp(iΔk1z),
dE3dz=-i ω3d(z)n3c [E1E2exp(iΔk2z)],
Aj=nj/ωj Ej(j=1, 2, 3),
Ij=PjS=120μ0 ωj|Aj|2=120μ0 nj|Ej|2.
ηj=j|Aj|2/|A1(0)|2,
dA1dz=-iK2A3A2*exp(-iΔk2z)-iK1A2A1*exp(-iΔk1z),
dA2dz=-iK2A3A1*exp(-iΔk2z)-i2 K1A12exp(iΔk1z),
dA3dz=-iK2A1A2exp(iΔk2z),
K1=d(z)cω2ω1ω1n2n1n11/2,
K2=d(z)cω3ω2ω1n3n2n11/2.
ddz (|A1|2+2|A2|2+3|A3|2)=0,
d(z)=d33m,n gm,nexp(-iGm,nz),
dA1dz=-iκ2A3A2*exp(-iΔk2z)-iκ1A2A1*exp(-iΔk1z),
dA2dz=-iκ2A3A1*exp(-iΔk2z)-i2 κ1A12exp(iΔk1z),
dA3dz=-iκ2A1A2exp(iΔk2z),
κ1=dm,ncω2ω1ω1n2n1n11/2,
κ2=dm,ncω3ω2ω1n3n2n11/2,
Δk1=k2-2k1-Gm,n,
Δk2=k3-k2-k1-Gm,n.
dA1dz=0,
dA2dz=-iκ2A3A1*exp(-iΔk2z)-i2 κ1A12exp(iΔk1z),
dA3dz=-iκ2A1A2exp(iΔk2z).
A2(z)=-Δk2+(Δk22+4κ22A12)1/22κ2A1 C1exp-i 12 [Δk2-(Δk22+4κ22A12)1/2]z-Δk2-(Δk22+4κ22A12)1/22κ2A1 C2exp-i 12 [Δk2+(Δk22+4κ22A12)1/2]z-(κ1A12Δk3/2)Δk1Δk3-κ22A12exp(iΔk1z),
A3(z)=C1expi 12 [Δk2+(Δk22+4κ22A12)1/2]z+C2expi 12 [Δk2-(Δk22+4κ22A12)1/2]z+(κ1κ2A13/2)Δk1Δk3-κ22A12exp(iΔk3z),
C1=-(κ1κ2A13/2)Δk1Δk3-κ22A12Δk1+Δk3+(Δk22+4κ22A12)1/22(Δk22+4κ22A12)1/2,
C2=(κ1κ2A13/2)Δk1Δk3-κ22A12Δk1+Δk3-(Δk22+4κ22A12)1/22(Δk22+4κ22A12)1/2,
η1=|A1|2|A1(0)|2,
η2=2|A2|2|A1(0)|2,
η3=3|A3|2|A1(0)|2.
A2(z)=-Δk2+(Δk22+4κ22A12)1/22κ2A1 C1×exp-i 12 [Δk2-(Δk22+4κ22A12)1/2]z-Δk2-(Δk22+4κ22A12)1/22κ2A1 C2×exp-i 12 [Δk2+(Δk22+4κ22A12)1/2]z+(κ1Δk2/2)κ22,
A3(z)=C1expi 12 [Δk2+(Δk22+4κ22A12)1/2]z+C2expi 12 [Δk2-(Δk22+4κ22A12)1/2]z-(κ1A1/2)κ2exp(iΔk2z),
C1=(κ1A1/2)κ2Δk2+(Δk22+4κ22A12)1/22(Δk22+4κ22A12)1/2,
C2=-(κ1A1/2)κ2Δk2-(Δk22+4κ22A12)1/22(Δk22+4κ22A12)1/2.
A2=-(B+1+B2)C1exp[-i(B-1+B2)X]-(B-1+B2)C2exp[-i(B+1+B2)X]+A1αB,
A3=C1exp[i(B+1+B2)X]+C2exp[i(B-1+B2)X]-A1α2exp(2iBX),
C1=A1α2B+1+B221+B2,
C2=-A1α2B-1+B221+B2,
A2=-C1exp(iX)+C2exp(-iX)-A1α B4B2-1exp(2iBX),
A3=C1exp(iX)+C2exp(-iX)+A1α214B2-1exp(2iBX),
C1=-A1α412B-1,
C2=A1α412B+1,
A2(z)=-C1exp(iκ2A1z)+C2exp(-iκ2A1z),
A3(z)=C1exp(iκ2A1z)+C2exp(-iκ2A1z)-A1α/2,
C1=C2=αA14.
η2=2α22sin2[κ2A1(0)z],
η3=3α22{cos[κ2A1(0)z]-1}2.
η2=2α22[κ2A1(0)z]2,
η3=32α22[κ2A1(0)z]4.
A2=-(B+1+B2)C1exp[-i(B-1+B2)X]-(B-1+B2)C2exp[-i(B+1+B2)X],
A3=C1exp[i(B+1+B2)X]+C2exp[i(B-1+B2)X]-A1α2,
C1=A1α2-B+1+B221+B2,
C2=A1α2B+1+B221+B2,
Δk3=Δk1+Δk2=δΔk2,
A2=-(B+1+B2)C1exp[-i(B-1+B2)X]-(B-1+B2)C2exp[-i(B+1+B2)X]-A1αBδ4δ(δ-1)B2-1exp[2i(δ-1)BX],
A3=C1exp[i(B+1+B2)X]+C2exp[i(B-1+B2)X]-A1α214δ(δ-1)B2-1exp(2iδBX),
C1=A1α214δ(δ-1)B2-1(2δ-1)B+1+B221+B2,
C2=A1α214δ(δ-1)B2-1(2δ-1)B-1+B221+B2.
dA1dz=-iκ2A3A2*-iκ1A2A1*,
dA2dz=-iκ2A3A1*-i2 κ1A12,
dA3dz=-iκ2A1A2.
dm,ndm,n=893n1n3.

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