Abstract

We develop a rigorous theory to describe coupled multiple nonlinear processes in a single nonlinear multilayer structure using iteration technique and transfer-matrix method. Pump depletion is taken into account. The validity of the theory is confirmed by analyzing coupled third-harmonic generation (CTHG). The CTHG process consists of two processes: second-harmonic generation and sum-frequency generation. These two processes are coupled together when their phase-matching conditions are satisfied simultaneously. The conversion efficiencies of second-harmonic and third-harmonic under various phase-matching conditions are studied numerically. The results demonstrate that our theory can efficiently and accurately deal with the coupled nonlinearity problem. The model can deal with periodic or aperiodic nonlinear structures. Our approach may be helpful in designing nonlinear photonic crystals and quasi-phase-matching structures.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  30. J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77, 195127 (2008).
    [CrossRef]

2010

2009

2008

Q. F. Zhu, D. Y. Wang, and Y. Zhang, “Design of defective nonlinear photonic crystals for multiple wavelengths’ second harmonic generation,” J. Opt. A: Pure Appl. Opt. 10, 025201 (2008).
[CrossRef]

L. M. Zhao, C. Li, Y. S. Zhou, and F. H. Wang, “Multiple wavelength second-harmonic generation in one-dimensional nonlinear photonic crystals,” J. Opt. Soc. Am. B 25, 2010–2014 (2008).
[CrossRef]

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77, 195127 (2008).
[CrossRef]

2007

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E 75, 056606 (2007).
[CrossRef]

J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91, 022903 (2007).
[CrossRef]

M. Lu, X. F. Chen, Y. P. Chen, and Y. X. Xia, “Algorithm to design aperiodic optical superlattice for multiple quasi-phase matching,” Appl. Opt. 46, 4138–4143 (2007).
[CrossRef]

2006

L. J. Chen, X. F. Chen, Y. P. Chen, and Y. X. Xia, “Multiple quasi-phase-matching in two-dimensional domain-inverted aperiodic optical superlattice,” Phys. Lett. A 349, 484–487 (2006).
[CrossRef]

2005

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef]

2004

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[CrossRef]

2003

J. L. He, J. Liao, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Simultaneous cw red, yellow, and green light generation, ‘traffic signal lights,’ by frequency doubling and sum-frequency mixing in an aperiodically poled LiTaO3,” Appl. Phys. Lett. 83, 228–230 (2003).
[CrossRef]

2002

2001

Y. Zhang and B. Y. Gu, “Optimal design of aperiodically poled lithium niobate crystals for multiple wavelengths parametric amplification,” Opt. Commun. 192, 417–425 (2001).
[CrossRef]

X. H. Wang and B. Y. Gu, “Nonlinear frequency conversion in 2D χ(2) photonic crystals and novel nonlinear double-circle construction,” Eur. Phys. J. B 24, 323–326 (2001).
[CrossRef]

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

1999

Y. Jeong and B. Lee, “Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion,” IEEE J. Quantum Electron. 35, 162–178 (1999).
[CrossRef]

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649–1656 (1999).
[CrossRef]

1997

1996

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

1992

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

1982

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18, 1029–1041 (1982).
[CrossRef]

1962

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]

Arbore, M. A.

Arie, A.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef]

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649–1656 (1999).
[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]

Artigas, D.

Assanto, G.

Bahabad, A.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef]

Bloembergen, N.

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]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Elsevier, 2003).

Byer, R. L.

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

Chen, L. J.

L. J. Chen, X. F. Chen, Y. P. Chen, and Y. X. Xia, “Multiple quasi-phase-matching in two-dimensional domain-inverted aperiodic optical superlattice,” Phys. Lett. A 349, 484–487 (2006).
[CrossRef]

Chen, X. F.

M. Lu, X. F. Chen, Y. P. Chen, and Y. X. Xia, “Algorithm to design aperiodic optical superlattice for multiple quasi-phase matching,” Appl. Opt. 46, 4138–4143 (2007).
[CrossRef]

L. J. Chen, X. F. Chen, Y. P. Chen, and Y. X. Xia, “Multiple quasi-phase-matching in two-dimensional domain-inverted aperiodic optical superlattice,” Phys. Lett. A 349, 484–487 (2006).
[CrossRef]

Chen, Y. H.

Chen, Y. P.

M. Lu, X. F. Chen, Y. P. Chen, and Y. X. Xia, “Algorithm to design aperiodic optical superlattice for multiple quasi-phase matching,” Appl. Opt. 46, 4138–4143 (2007).
[CrossRef]

L. J. Chen, X. F. Chen, Y. P. Chen, and Y. X. Xia, “Multiple quasi-phase-matching in two-dimensional domain-inverted aperiodic optical superlattice,” Phys. Lett. A 349, 484–487 (2006).
[CrossRef]

Du, J.

J. L. He, J. Liao, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Simultaneous cw red, yellow, and green light generation, ‘traffic signal lights,’ by frequency doubling and sum-frequency mixing in an aperiodically poled LiTaO3,” Appl. Phys. Lett. 83, 228–230 (2003).
[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]

Fejer, M. M.

Feng, S.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[CrossRef]

Fradkin-Kashi, K.

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649–1656 (1999).
[CrossRef]

Giessen, H.

Gu, B. Y.

X. H. Wang and B. Y. Gu, “Nonlinear frequency conversion in 2D χ(2) photonic crystals and novel nonlinear double-circle construction,” Eur. Phys. J. B 24, 323–326 (2001).
[CrossRef]

Y. Zhang and B. Y. Gu, “Optimal design of aperiodically poled lithium niobate crystals for multiple wavelengths parametric amplification,” Opt. Commun. 192, 417–425 (2001).
[CrossRef]

Hagan, D. J.

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

He, J. L.

J. L. He, J. Liao, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Simultaneous cw red, yellow, and green light generation, ‘traffic signal lights,’ by frequency doubling and sum-frequency mixing in an aperiodically poled LiTaO3,” Appl. Phys. Lett. 83, 228–230 (2003).
[CrossRef]

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Hebling, J.

Jennings, G.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[CrossRef]

Jeong, Y.

Y. Jeong and B. Lee, “Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion,” IEEE J. Quantum Electron. 35, 162–178 (1999).
[CrossRef]

Jundt, D. H.

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

Kuhl, J.

Lai, J. Y.

Lee, B.

Y. Jeong and B. Lee, “Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion,” IEEE J. Quantum Electron. 35, 162–178 (1999).
[CrossRef]

Li, C.

Li, J. J.

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77, 195127 (2008).
[CrossRef]

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E 75, 056606 (2007).
[CrossRef]

J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91, 022903 (2007).
[CrossRef]

Li, Z. Y.

M. L. Ren and Z. Y. Li, “Exact iterative solution of second harmonic generation in quasi-phase-matched structures,” Opt. Express 18, 7288–7299 (2010).
[CrossRef]

M. L. Ren and Z. Y. Li, “Enhanced nonlinear frequency conversion in defective nonlinear photonic crystals with designed polarization distribution,” J. Opt. Soc. Am. B 27, 1551–1560 (2010).
[CrossRef]

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77, 195127 (2008).
[CrossRef]

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E 75, 056606 (2007).
[CrossRef]

J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91, 022903 (2007).
[CrossRef]

Liao, J.

J. L. He, J. Liao, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Simultaneous cw red, yellow, and green light generation, ‘traffic signal lights,’ by frequency doubling and sum-frequency mixing in an aperiodically poled LiTaO3,” Appl. Phys. Lett. 83, 228–230 (2003).
[CrossRef]

Lifshitz, R.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef]

Liu, H.

J. L. He, J. Liao, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Simultaneous cw red, yellow, and green light generation, ‘traffic signal lights,’ by frequency doubling and sum-frequency mixing in an aperiodically poled LiTaO3,” Appl. Phys. Lett. 83, 228–230 (2003).
[CrossRef]

Liu, Y. J.

Liu, Z. W.

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Lu, M.

Luo, G. Z.

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Magel, G. A.

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

Marco, O.

Meenakshi, S.

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18, 1029–1041 (1982).
[CrossRef]

Mehendale, S. C.

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18, 1029–1041 (1982).
[CrossRef]

Meyn, J. P.

Ming, N. B.

J. L. He, J. Liao, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Simultaneous cw red, yellow, and green light generation, ‘traffic signal lights,’ by frequency doubling and sum-frequency mixing in an aperiodically poled LiTaO3,” Appl. Phys. Lett. 83, 228–230 (2003).
[CrossRef]

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[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]

Palfalvi, L.

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]

Pfister, O.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[CrossRef]

Pooser, R.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[CrossRef]

Reid, D. T.

Ren, M. L.

Rühle, W. W.

Rustagi, K. C.

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18, 1029–1041 (1982).
[CrossRef]

Sapaev, U. K.

Sheng, Y.

J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91, 022903 (2007).
[CrossRef]

Stegeman, G. I.

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

Torner, L.

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

Wang, D. Y.

Q. F. Zhu, D. Y. Wang, and Y. Zhang, “Design of defective nonlinear photonic crystals for multiple wavelengths’ second harmonic generation,” J. Opt. A: Pure Appl. Opt. 10, 025201 (2008).
[CrossRef]

Wang, F. H.

Wang, H. T.

J. L. He, J. Liao, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Simultaneous cw red, yellow, and green light generation, ‘traffic signal lights,’ by frequency doubling and sum-frequency mixing in an aperiodically poled LiTaO3,” Appl. Phys. Lett. 83, 228–230 (2003).
[CrossRef]

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Wang, X. H.

X. H. Wang and B. Y. Gu, “Nonlinear frequency conversion in 2D χ(2) photonic crystals and novel nonlinear double-circle construction,” Eur. Phys. J. B 24, 323–326 (2001).
[CrossRef]

Wu, H. Y.

Xia, J.

Xia, Y. X.

M. Lu, X. F. Chen, Y. P. Chen, and Y. X. Xia, “Algorithm to design aperiodic optical superlattice for multiple quasi-phase matching,” Appl. Opt. 46, 4138–4143 (2007).
[CrossRef]

L. J. Chen, X. F. Chen, Y. P. Chen, and Y. X. Xia, “Multiple quasi-phase-matching in two-dimensional domain-inverted aperiodic optical superlattice,” Phys. Lett. A 349, 484–487 (2006).
[CrossRef]

Xie, D.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[CrossRef]

Xu, F.

J. L. He, J. Liao, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Simultaneous cw red, yellow, and green light generation, ‘traffic signal lights,’ by frequency doubling and sum-frequency mixing in an aperiodically poled LiTaO3,” Appl. Phys. Lett. 83, 228–230 (2003).
[CrossRef]

Yang, S. D.

Zhang, C.

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Zhang, D. Z.

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77, 195127 (2008).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Schematic diagram of a one-dimensional nonlinear structure. N is the total number of layers. zn is position of the layer boundary. njn stands for the refractive index at frequency of ωj in the nth layer. χn(2) is the second-order nonlinear susceptibility of the nth layer.θj0 means the angle of electric field Ej relative to the z-axis in input background medium.

Fig. 2.
Fig. 2.

(a) Conversion efficiencies of three waves versus layer number for the case of perfect phase matching of process I. Only E1 is incident E10=20.00V/μm. The black solid, red dashed, and blue dotted curves denote ω1, ω2, and ω3, respectively. (b) Conversion efficiency of second-harmonic wave ω2 in the index-matched QPM structure obtained by our model (red dot curve) and the method in [21] (black solid curve).

Fig. 3.
Fig. 3.

Effects of phase mismatch on the conversion efficiencies of SW and TW. The departure of domain length from coherence length of SHG is 0.0001Lsh, 0.0005Lsh, 0.001Lsh, respectively.

Fig. 4.
Fig. 4.

Conversion efficiencies of three waves versus layer number for the case of perfect phase matching of process II. Only E1 is incident E10=20.00V/μm.

Fig. 5.
Fig. 5.

Process II is perfectly phase matched. Both E1 and E2 are incident having the same amplitude. E10=E20=20.00V/μm. (a) Conversion efficiency of TW versus layer number, (b) photon flux Ij/hωj versus layer number.

Fig. 6.
Fig. 6.

Photon flux Ij/hωj as a function of layer number. Process II is perfectly phase matched. Both E1 and E2 are incident having the same photon flux. E10=20.00V/μm.

Fig. 7.
Fig. 7.

Conversion efficiencies of generated waves versus layer number. Process I and process II are simultaneously phase matched. Only E1 is incident E10=10.00V/μm.

Fig. 8.
Fig. 8.

Effect of phase mismatch on the conversion efficiencies of SW and TW. The departure of domain length is (a) 0.0001Lsh, 0.0002Lsh, and 0.0004Lsh; (b) 0.0005Lsh, 0.0008Lsh, and 0.0012Lsh; (c) 0.0012Lsh, 0.002Lsh, and 0.003Lsh.

Equations (20)

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2ω1=ω2(Process I),
ω1+ω2=ω3(Process II).
d2E1ndz2+β1n2E1n=2k102χn(2)E1n*E2n2k102χn(2)E2n*E3n,
d2E2ndz2+β2n2E2n=k202χn(2)E1n22k202χn(2)E1n*E3n,
d2E3ndz2+β3n2E3n=2k302χn(2)E1nE2n.
k2nsin(θ20)=2k1nsin(θ10),k3nsin(θ30)=k1nsin(θ10)+k2nsin(θ20).
d2E1ndz2+β1n2E1n=0.
E1n(z)=Ω1n,Fexp[iβ1n(zzn1)]+Ω1n,Bexp[iβ1n(zzn1)],
E2n(z)=Ω2n,Fexp[iβ2n(zzn1)]+Ω2n,Bexp[iβ2n(zzn1)]+A2n,Fexp[2iβ1n(zzn1)]+A2n,Bexp[2iβ1n(zzn1)]+B2n,
A2n,F=k202χn(2)(Ω1n,F)2/(β2n24β1n2),A2n,B=k202χn(2)(Ω1n,B)2/(β2n24β1n2),B2n=2k202χn(2)Ω1n,FΩ1n,B/β2n2.
E3n(z)=Ω3n,Fexp[iβ3n(zzn1)]+Ω3n,Bexp[iβ3n(zzn1)]+A3n,Fexp[i(β1n+β2n)(zzn1)]+A3n,Bexp[i(β1n+β2n)(zzn1)]+B3n,Fexp[i(β1nβ2n)(zzn1)]+B3n,Bexp[i(β1nβ2n)(zzn1)]+C3n,Fexp[i(β1n+2β1n)(zzn1)]+C3n,Bexp[i(β1n+2β1n)(zzn1)]+D3n,Fexp[iβ1n(zzn1)]+D3n,Bexp[iβ1n(zzn1)],
A3n,F=2k302χn(2)Ω1n,FΩ2n,F/[β3n2(β1n+β2n)2],A3n,B=2k302χn(2)Ω1n,BΩ2n,B/[β3n2(β1n+β2n)2],B3n,F=2k302χn(2)Ω1n,FΩ2n,B/[β3n2(β1nβ2n)2],B3n,B=2k302χn(2)Ω1n,BΩ2n,F/[β3n2(β1nβ2n)2],C3n,F=2k302χn(2)Ω1n,FA2n,F/[β3n2(β1n+2β2n)2],C3n,F=2k302χn(2)Ω1n,BA2n,B/[β3n2(β1n+2β2n)2],D3n,F=2k302χn(2)(Ω1n,BA2n,F+Ω1n,FB2n)/(β3n2β1n2),D3n,B=2k302χn(2)(Ω1n,FA2n,B+Ω1n,BB2n)/(β3n2β1n2).
E1n(i)(z)=a1n,F(i)exp[iβ1n(zzn1)]+a1n,B(i)exp[iβ1n(zzn1)]+a2n,F(i)exp[i(β1n+Δβ2n)(zzn1)]+a2n,B(i)exp[i(β1n+Δβ2n)(zzn1)]+a3n,F(i)exp[i(β1n+Δβ1n)(zzn1)]+a3n,B(i)exp[i(β1n+Δβ1n)(zzn1)]+a4n,F(i)exp[i(β1n+Δβ2nΔβ1n)(zzn1)]+a4n,B(i)exp[i(β1n+Δβ2nΔβ1n)(zzn1)]+a5n,F(i)exp[i(β1n+Δβ1nΔβ2n)(zzn1)]+a5n,B(i)exp[i(β1n+Δβ1nΔβ2n)(zzn1)]+a6n,F(i)exp[i(β1n+Δβ1n+Δβ2n)(zzn1)]+a6n,B(i)exp[i(β1n+Δβ1n+Δβ2n)(zzn1)]+a7n,F(i)exp[i(β1nΔβ2n)(zzn1)]+a7n,B(i)exp[i(β1nΔβ2n)(zzn1)]+a8n,F(i)exp[i(β1nΔβ1n)(zzn1)]+a8n,B(i)exp[i(β1nΔβ1n)(zzn1)]+a9n,F(i)exp[i(β1nΔβ1nΔβ2n)(zzn1)]+a9n,B(i)exp[i(β1nΔβ1nΔβ2n)(zzn1)]+b1n,F(i)exp[i(β1n+β2n)(zzn1)]+b1n,B(i)exp[i(β1n+β2n)(zzn1)]+b2n,F(i)exp[i(β2n+β3n)(zzn1)]+b2n,B(i)exp[i(β2n+β3n)(zzn1)],
E2n(i)(z)=c1n,F(i)exp[iβ2n(zzn1)]+c1n,B(i)exp[iβ2n(zzn1)]+c2n,F(i)exp[i(β2n+Δβ2n)(zzn1)]+c2n,B(i)exp[i(β2n+Δβ2n)(zzn1)]+c3n,F(i)exp[i(β2n+Δβ1n)(zzn1)]+c3n,B(i)exp[i(β2n+Δβ1n)(zzn1)]+c4n,F(i)exp[i(β2n+Δβ2nΔβ1n)(zzn1)]+c4n,B(i)exp[i(β2n+Δβ2nΔβ1n)(zzn1)]+c5n,F(i)exp[i(β2n+Δβ1nΔβ2n)(zzn1)]+c5n,B(i)exp[i(β2n+Δβ1nΔβ2n)(zzn1)]+c6n,F(i)exp[i(β2n+Δβ1n+Δβ2n)(zzn1)]+c6n,B(i)exp[i(β2n+Δβ1n+Δβ2n)(zzn1)]+c7n,F(i)exp[i(β2nΔβ2n)(zzn1)]+c7n,B(i)exp[i(β2nΔβ2n)(zzn1)]+c8n,F(i)exp[i(β2nΔβ1n)(zzn1)]+c8n,B(i)exp[i(β2nΔβ1n)(zzn1)]+c9n,F(i)exp[i(β2nΔβ1nΔβ2n)(zzn1)]+c9n,B(i)exp[i(β2nΔβ1nΔβ2n)(zzn1)]+d1n,F(i)exp[i(β1n+β3n)(zzn1)]+d2n,B(i)exp[i(β1n+β3n)(zzn1)],
E3n(i)(z)=g1n,F(i)exp[iβ3n(zzn1)]+g1n,B(i)exp[iβ3n(zzn1)]+g2n,F(i)exp[i(β3n+Δβ2n)(zzn1)]+g2n,B(i)exp[i(β3n+Δβ2n)(zzn1)]+g3n,F(i)exp[i(β3n+Δβ1n)(zzn1)]+g3n,B(i)exp[i(β3n+Δβ1n)(zzn1)]+g4n,F(i)exp[i(β3n+Δβ2nΔβ1n)(zzn1)]+g4n,B(i)exp[i(β3n+Δβ2nΔβ1n)(zzn1)]+g5n,F(i)exp[i(β3n+Δβ1nΔβ2n)(zzn1)]+g5n,B(i)exp[i(β3n+Δβ1nΔβ2n)(zzn1)]+g6n,F(i)exp[i(β3n+Δβ1n+Δβ2n)(zzn1)]+g6n,B(i)exp[i(β3n+Δβ1n+Δβ2n)(zzn1)]+g7n,F(i)exp[i(β3nΔβ2n)(zzn1)]+g7n,B(i)exp[i(β3nΔβ2n)(zzn1)]+g8n,F(i)exp[i(β3nΔβ1n)(zzn1)]+g8n,B(i)exp[i(β3nΔβ1n)(zzn1)]+g9n,F(i)exp[i(β3nΔβ1nΔβ2n)(zzn1)]+g9n,B(i)exp[i(β3nΔβ1nΔβ2n)(zzn1)]+f1n,F(i)exp[i(β1nβ2n)(zzn1)]+f1n,B(i)exp[i(β1nβ2n)(zzn1)]+f2n,F(i)exp[iβ1n(zzn1)]+f2n,B(i)exp[iβ1n(zzn1)].
E10,F(0)=E10,E1N,B(0)=0,E20,F(0)=E20,E2N,B(0)=0,E30,F(0)=E30,E3N,B(0)=0.
amn,γ(0)=0,b1n,γ(0)=0,b2n,γ(0)=0;cln,γ(0)=0,d1n,γ(0)=0,d2n,γ(0)=0;gln,γ(0)=0,f1n,γ(0)=0,f2n,γ(0)=0.(m=29,l=19,γ=F,B).
|I3n(i+1)(z)I3n(i)(z)|I3n(i)(z)δ,
ηjF(i)=IjN,F(i)I0=njN+1|EjN,F(i)|2j=13nj0|Ej0(i)|2,ηjB(i)=Ij0,B(i)I0=nj0|Ej0,B(i)|2j=13nj0|Ej0(i)|2,(j=1,2,3).
ddz(I1ω1+I3ω3)=0,ddz(I2ω2+I3ω3)=0,ddz(I1ω1+I2ω2)=0.

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