Abstract

We present a theoretical study of optical phase conjugation by degenerate four-wave mixing in one-dimensional photonic crystals containing two kinds of single-negative materials. Compared to usual photonic crystals with the same thickness and third-order nonlinear susceptibility (χ3), the generation efficiency of the optical phase- conjugate wave can be enhanced by nearly 3 orders of magnitude. Distinct from the band-edge state in usual photonic crystals, the enhancement is caused by the strong field localized at each interface of two types of single- negative materials. Owing to the unique properties of the band edge of the zero-effective-phase gap that is insensitive to the angle of incidence, the optical phase-conjugate wave can be generated efficiently in a wide range of angle of incidence.

© 2011 Optical Society of America

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  1. B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–113 (1972).
  2. B. Y. Zel’dovich, R. F. Pilipetskii, and V. V. Shkunov, Principles of Phase Conjugation (Springer, 1985).
  3. R. A. Fisher, ed., Optical Phase Conjugation (Academic, 1983).
  4. G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quantum Electron. 26, 131–191 (2002).
    [CrossRef]
  5. R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).
    [CrossRef]
  6. P. Xie and Z. Q. Zhang, “Optical phase conjugation in third-order nonlinear photonic crystals,” Phys. Rev. A 69, 053806 (2004).
    [CrossRef]
  7. P. Delaye, M. Astic, R. Frey, and G. Roosen, “Transfer-matrix modeling of four-wave mixing at the band edge of a one-dimensional photonic crystal,” J. Opt. Soc. Am. B 22, 2494–2504(2005).
    [CrossRef]
  8. L. Razzari, D. Träger, M. Astic, P. Delaye, R. Frey, G. Roosen, and R. André, “Kerr and four-wave mixing spectroscopy at the band edge of one-dimensional photonic crystals,” Appl. Phys. Lett. 86, 231106 (2005).
    [CrossRef]
  9. M. Astic, P. Delaye, R. Frey, and G. Roosen, “Enhancement of nonlinear effects at the degenerate band edge of two-dimensional photonic crystals,” Phys. Rev. E 79, 056608 (2009).
    [CrossRef]
  10. A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
    [CrossRef]
  11. H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
    [CrossRef]
  12. H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
    [CrossRef]
  13. S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
    [CrossRef]
  14. S. M. Wang and L. Gao, “Nonlinear responses of the periodic structure composed of single negative materials,” Opt. Commun. 267, 197–204 (2006).
    [CrossRef]
  15. H. T. Jiang, H. Chen, and S. Y. Zhu, “Localized gap-edge fields of one-dimensional photonic crystals with an ε-negative and a μ-negative defect,” Phys. Rev. E 73, 046601 (2006).
    [CrossRef]
  16. Q. G. Du, F. F. Ren, C. H. Kam, and X. W. Sun, “Second-harmonic generation in photonic crystals with a pair of epsilon-negative and mu-negative defects,” Opt. Express 17, 6682–6687 (2009).
    [CrossRef] [PubMed]

2009

M. Astic, P. Delaye, R. Frey, and G. Roosen, “Enhancement of nonlinear effects at the degenerate band edge of two-dimensional photonic crystals,” Phys. Rev. E 79, 056608 (2009).
[CrossRef]

Q. G. Du, F. F. Ren, C. H. Kam, and X. W. Sun, “Second-harmonic generation in photonic crystals with a pair of epsilon-negative and mu-negative defects,” Opt. Express 17, 6682–6687 (2009).
[CrossRef] [PubMed]

2006

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
[CrossRef]

S. M. Wang and L. Gao, “Nonlinear responses of the periodic structure composed of single negative materials,” Opt. Commun. 267, 197–204 (2006).
[CrossRef]

H. T. Jiang, H. Chen, and S. Y. Zhu, “Localized gap-edge fields of one-dimensional photonic crystals with an ε-negative and a μ-negative defect,” Phys. Rev. E 73, 046601 (2006).
[CrossRef]

2005

L. Razzari, D. Träger, M. Astic, P. Delaye, R. Frey, G. Roosen, and R. André, “Kerr and four-wave mixing spectroscopy at the band edge of one-dimensional photonic crystals,” Appl. Phys. Lett. 86, 231106 (2005).
[CrossRef]

P. Delaye, M. Astic, R. Frey, and G. Roosen, “Transfer-matrix modeling of four-wave mixing at the band edge of a one-dimensional photonic crystal,” J. Opt. Soc. Am. B 22, 2494–2504(2005).
[CrossRef]

2004

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

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

2003

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

2002

G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quantum Electron. 26, 131–191 (2002).
[CrossRef]

1977

1972

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–113 (1972).

Alù, A.

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
[CrossRef]

André, R.

L. Razzari, D. Träger, M. Astic, P. Delaye, R. Frey, G. Roosen, and R. André, “Kerr and four-wave mixing spectroscopy at the band edge of one-dimensional photonic crystals,” Appl. Phys. Lett. 86, 231106 (2005).
[CrossRef]

Astic, M.

M. Astic, P. Delaye, R. Frey, and G. Roosen, “Enhancement of nonlinear effects at the degenerate band edge of two-dimensional photonic crystals,” Phys. Rev. E 79, 056608 (2009).
[CrossRef]

L. Razzari, D. Träger, M. Astic, P. Delaye, R. Frey, G. Roosen, and R. André, “Kerr and four-wave mixing spectroscopy at the band edge of one-dimensional photonic crystals,” Appl. Phys. Lett. 86, 231106 (2005).
[CrossRef]

P. Delaye, M. Astic, R. Frey, and G. Roosen, “Transfer-matrix modeling of four-wave mixing at the band edge of a one-dimensional photonic crystal,” J. Opt. Soc. Am. B 22, 2494–2504(2005).
[CrossRef]

Chen, H.

H. T. Jiang, H. Chen, and S. Y. Zhu, “Localized gap-edge fields of one-dimensional photonic crystals with an ε-negative and a μ-negative defect,” Phys. Rev. E 73, 046601 (2006).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

Delaye, P.

M. Astic, P. Delaye, R. Frey, and G. Roosen, “Enhancement of nonlinear effects at the degenerate band edge of two-dimensional photonic crystals,” Phys. Rev. E 79, 056608 (2009).
[CrossRef]

L. Razzari, D. Träger, M. Astic, P. Delaye, R. Frey, G. Roosen, and R. André, “Kerr and four-wave mixing spectroscopy at the band edge of one-dimensional photonic crystals,” Appl. Phys. Lett. 86, 231106 (2005).
[CrossRef]

P. Delaye, M. Astic, R. Frey, and G. Roosen, “Transfer-matrix modeling of four-wave mixing at the band edge of a one-dimensional photonic crystal,” J. Opt. Soc. Am. B 22, 2494–2504(2005).
[CrossRef]

Du, Q. G.

Engheta, N.

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
[CrossRef]

Faisullov, F. S.

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–113 (1972).

Fisher, R. A.

R. A. Fisher, ed., Optical Phase Conjugation (Academic, 1983).

Frey, R.

M. Astic, P. Delaye, R. Frey, and G. Roosen, “Enhancement of nonlinear effects at the degenerate band edge of two-dimensional photonic crystals,” Phys. Rev. E 79, 056608 (2009).
[CrossRef]

L. Razzari, D. Träger, M. Astic, P. Delaye, R. Frey, G. Roosen, and R. André, “Kerr and four-wave mixing spectroscopy at the band edge of one-dimensional photonic crystals,” Appl. Phys. Lett. 86, 231106 (2005).
[CrossRef]

P. Delaye, M. Astic, R. Frey, and G. Roosen, “Transfer-matrix modeling of four-wave mixing at the band edge of a one-dimensional photonic crystal,” J. Opt. Soc. Am. B 22, 2494–2504(2005).
[CrossRef]

Gao, L.

S. M. Wang and L. Gao, “Nonlinear responses of the periodic structure composed of single negative materials,” Opt. Commun. 267, 197–204 (2006).
[CrossRef]

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
[CrossRef]

He, G. S.

G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quantum Electron. 26, 131–191 (2002).
[CrossRef]

Hellwarth, R. W.

Jiang, H. T.

H. T. Jiang, H. Chen, and S. Y. Zhu, “Localized gap-edge fields of one-dimensional photonic crystals with an ε-negative and a μ-negative defect,” Phys. Rev. E 73, 046601 (2006).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

Kam, C. H.

Li, H. Q.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

Pan, T.

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
[CrossRef]

Pilipetskii, R. F.

B. Y. Zel’dovich, R. F. Pilipetskii, and V. V. Shkunov, Principles of Phase Conjugation (Springer, 1985).

Popovichev, V. I.

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–113 (1972).

Ragulskii, V. V.

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–113 (1972).

Razzari, L.

L. Razzari, D. Träger, M. Astic, P. Delaye, R. Frey, G. Roosen, and R. André, “Kerr and four-wave mixing spectroscopy at the band edge of one-dimensional photonic crystals,” Appl. Phys. Lett. 86, 231106 (2005).
[CrossRef]

Ren, F. F.

Roosen, G.

M. Astic, P. Delaye, R. Frey, and G. Roosen, “Enhancement of nonlinear effects at the degenerate band edge of two-dimensional photonic crystals,” Phys. Rev. E 79, 056608 (2009).
[CrossRef]

L. Razzari, D. Träger, M. Astic, P. Delaye, R. Frey, G. Roosen, and R. André, “Kerr and four-wave mixing spectroscopy at the band edge of one-dimensional photonic crystals,” Appl. Phys. Lett. 86, 231106 (2005).
[CrossRef]

P. Delaye, M. Astic, R. Frey, and G. Roosen, “Transfer-matrix modeling of four-wave mixing at the band edge of a one-dimensional photonic crystal,” J. Opt. Soc. Am. B 22, 2494–2504(2005).
[CrossRef]

Shkunov, V. V.

B. Y. Zel’dovich, R. F. Pilipetskii, and V. V. Shkunov, Principles of Phase Conjugation (Springer, 1985).

Sun, X. W.

Tang, C. J.

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
[CrossRef]

Träger, D.

L. Razzari, D. Träger, M. Astic, P. Delaye, R. Frey, G. Roosen, and R. André, “Kerr and four-wave mixing spectroscopy at the band edge of one-dimensional photonic crystals,” Appl. Phys. Lett. 86, 231106 (2005).
[CrossRef]

Wang, S. M.

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
[CrossRef]

S. M. Wang and L. Gao, “Nonlinear responses of the periodic structure composed of single negative materials,” Opt. Commun. 267, 197–204 (2006).
[CrossRef]

Xie, P.

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

Zel’dovich, B. Y.

B. Y. Zel’dovich, R. F. Pilipetskii, and V. V. Shkunov, Principles of Phase Conjugation (Springer, 1985).

Zeldovich, B. Y.

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–113 (1972).

Zhang, Y. W.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

Zhang, Z. Q.

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

Zhu, S. Y.

H. T. Jiang, H. Chen, and S. Y. Zhu, “Localized gap-edge fields of one-dimensional photonic crystals with an ε-negative and a μ-negative defect,” Phys. Rev. E 73, 046601 (2006).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

Zi, J.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Appl. Phys. Lett.

L. Razzari, D. Träger, M. Astic, P. Delaye, R. Frey, G. Roosen, and R. André, “Kerr and four-wave mixing spectroscopy at the band edge of one-dimensional photonic crystals,” Appl. Phys. Lett. 86, 231106 (2005).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

IEEE Trans. Antennas Propag.

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Commun.

S. M. Wang and L. Gao, “Nonlinear responses of the periodic structure composed of single negative materials,” Opt. Commun. 267, 197–204 (2006).
[CrossRef]

Opt. Express

Phys. Lett. A

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Bistability and gap soliton in one-dimensional photonic crystal containing single-negative materials,” Phys. Lett. A 348, 424–431 (2006).
[CrossRef]

Phys. Rev. A

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

Phys. Rev. E

H. T. Jiang, H. Chen, and S. Y. Zhu, “Localized gap-edge fields of one-dimensional photonic crystals with an ε-negative and a μ-negative defect,” Phys. Rev. E 73, 046601 (2006).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

M. Astic, P. Delaye, R. Frey, and G. Roosen, “Enhancement of nonlinear effects at the degenerate band edge of two-dimensional photonic crystals,” Phys. Rev. E 79, 056608 (2009).
[CrossRef]

Prog. Quantum Electron.

G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quantum Electron. 26, 131–191 (2002).
[CrossRef]

Sov. Phys. JETP

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–113 (1972).

Other

B. Y. Zel’dovich, R. F. Pilipetskii, and V. V. Shkunov, Principles of Phase Conjugation (Springer, 1985).

R. A. Fisher, ed., Optical Phase Conjugation (Academic, 1983).

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Figures (7)

Fig. 1
Fig. 1

Schematic of a DFWM process. P f and P b are forward and backward pump waves, respectively. S f and S b are forward and backward pump waves, respectively.

Fig. 2
Fig. 2

Transmittances of a 12-period PC with normal materials (blue solid curve) and SNG materials (red dashed curve), respectively. d A = 60 nm and d B = 40 nm . The frequency of the lower band edge is 609.7 THz .

Fig. 3
Fig. 3

Distributions of normalized intensity of electric field in the PC with normal materials (blue solid curve) and SNG materials (red dashed curve) at 609.7 THz . B and D are nonlinear material (dark gray), and the total thickness of the nonlinear material is 480 nm , χ B 3 = χ D 3 = 1 × 10 16 m 2 / V 2 .

Fig. 4
Fig. 4

(a) Bistability properties of the PC with normal materials. Output versus input pump intensity at a single input beam (red solid curve with squares) and multiple input beams (blue dashed curve with circles). (b) Bistability properties of the PC with SNG materials. Output versus input pump intensity at a single input beam (red solid curve with squares) and multiple input beams (blue dashed curve with circles). The working frequency is 606.1 THz , and the intensity of signal beam is 0.1 MW / cm 2 .

Fig. 5
Fig. 5

Backward OPC conversion efficiencies versus input pump intensity in the PCs with normal (red solid curve with squares) and SNG (blue dashed curve with circles) materials, respectively. The working frequencies are (a)  606.1 THz and (b)  609.7 THz , respectively, and the intensity of signal beam is 0.1 MW / cm 2 . Both figures are plotted with a logarithmic ordinate.

Fig. 6
Fig. 6

Efficiency of (a) forward and (b) backward OPC generation as a function of the incident angle of the signal beam in the PCs with normal (red solid curve with squares) and SNG (blue dashed curve with circles) materials, respectively. The pump intensity is set to be 1 × 10 3 MW / cm 2 , and the signal intensity is 1 × 10 4 MW / cm 2 . The working frequency is 609.7 THz , and both figures are plotted with a logarithmic ordinate.

Fig. 7
Fig. 7

Efficiency of (a) forward and (b) backward OPC generation as a function of the incident angle of the signal beam in the PC with normal (red solid curve with squares) and SNG (blue dashed curve with circles) materials, respectively. The pump intensity is set to be 1 MW / cm 2 , and the signal intensity is 0.1 MW / cm 2 . The working frequency is 609.7 THz , and both figures are plotted with a logarithmic ordinate.

Equations (4)

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2 E n ( r ) + ε r n μ r n ω n 2 c 2 E n ( r ) = μ 0 μ r n ω n 2 P n NL ( r ) .
P = 3 ε 0 χ 3 ( j = 1 3 E j ) 2 ( j = 1 3 E j * ) ,
{ P 1 ( r ) = 3 ε 0 χ 3 [ | E 1 | 2 E 1 + 2 | E 2 | 2 E 1 + 2 | E 3 | 2 E 1 + 2 E 2 E 3 E 1 * ] P 2 ( r ) = 3 ε 0 χ 3 [ | E 2 | 2 E 2 + 2 | E 1 | 2 E 2 + 2 | E 3 | 2 E 2 + E 1 2 E 3 * ] P 3 ( r ) = 3 ε 0 χ 3 [ | E 3 | 2 E 3 + 2 | E 1 | 2 E 3 + 2 | E 2 | 2 E 3 + E 1 2 E 2 * ] .
{ 2 A 1 z 2 = k 1 z 2 A 1 3 ω 2 μ r χ 3 c 2 [ | A 1 | 2 A 1 + 2 | A 2 | 2 A 1 + 2 | A 3 | 2 A 1 + 2 A 2 A 3 A 1 * ] 2 A 2 z 2 = k 2 z 2 A 2 3 ω 2 μ r χ 3 c 2 [ | A 2 | 2 A 2 + 2 | A 1 | 2 A 2 + 2 | A 3 | 2 A 2 + A 1 2 A 3 * ] 2 A 3 z 2 = k 3 z 2 A 3 3 ω 2 μ r χ 3 c 2 [ | A 3 | 2 A 3 + 2 | A 1 | 2 A 3 + 2 | A 2 | 2 A 3 + A 1 2 A 2 * ] .

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