Abstract

By simultaneously taking field localization and slow light effects into account, in this paper we make use of a field averaging method to calculate the effective nonlinear refractive index coefficient (n 2) of Kerr photonic crystals (PhCs) in the first band. Although the nonlinear PhC is beyond the traditional long-wavelength limit, interestingly, the theoretically calculated effective n 2 agrees well with one numerically measured via the self-phase-modulation induced spectral broadening. Moreover, due to the cooperative influence of field localization and slow light effects, the effective n 2 of the PhC decreases slowly at first and then goes up quickly with increasing frequency. This kind of dispersive nonlinearity is purely induced by the periodic nanostructures because the optical parameters of both components of the PhC we took are frequency-independent. Our results may pave the way for enhancing or limiting nonlinear effects and provide a method for producing the dispersive nonlinearity.

© 2015 Optical Society of America

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References

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2015 (1)

Z. Tang, L. Zhao, Z. Sui, Y. Zou, S. Wen, A. Danner, and C. Qiu, “Switchable self-defocusing and focusing in nearly isotropic photonic crystals via enhanced inverse diffraction,” Phys. Rev. A 91(6), 063824 (2015).
[Crossref]

2014 (1)

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9(4), 290–294 (2014).
[Crossref] [PubMed]

2013 (1)

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, “Phase mismatch-free nonlinear propagation in optical zero-index materials,” Science 342(6163), 1223–1226 (2013).
[Crossref] [PubMed]

2012 (3)

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, “Nonlinear magnetoelectric metamaterials: Analysis and homogenization via a microscopic coupled-mode theory,” Phys. Rev. A 86(3), 033816 (2012).
[Crossref]

K. Dolgaleva and R. W. Boyd, “Local-field effects in nanostructured photonic materials,” Adv. Opt. Photonics 4(1), 1–77 (2012).
[Crossref]

2010 (4)

S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283(8), 1621–1627 (2010).
[Crossref]

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(3), 036608 (2010).
[Crossref] [PubMed]

E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” New J. Phys. 12(9), 093010 (2010).
[Crossref]

C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12(10), 104003 (2010).
[Crossref]

2009 (2)

2007 (1)

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444(3-6), 101–202 (2007).
[Crossref]

2004 (1)

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004).
[Crossref] [PubMed]

2001 (1)

V. P. Pellegrini, “Self-consistent effective-medium approximation for strongly nonlinear media,” Phys. Rev. B 64(13), 134211 (2001).
[Crossref]

1994 (1)

K. W. Yu, Y. C. Chu, and E. M. Y. Chan, “Effective-medium theory for two-component nonlinear composites,” Phys. Rev. B Condens. Matter 50(11), 7984–7987 (1994).
[Crossref] [PubMed]

1989 (1)

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective-medium theory of nonlinear ellipsoidal composites,” Phys. Rev. A 40(10), 5729–5734 (1989).
[Crossref] [PubMed]

1986 (1)

Aouani, H.

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9(4), 290–294 (2014).
[Crossref] [PubMed]

Asakawa, K.

Baron, A.

Bowden, C. M.

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective-medium theory of nonlinear ellipsoidal composites,” Phys. Rev. A 40(10), 5729–5734 (1989).
[Crossref] [PubMed]

Boyd, R. W.

K. Dolgaleva and R. W. Boyd, “Local-field effects in nanostructured photonic materials,” Adv. Opt. Photonics 4(1), 1–77 (2012).
[Crossref]

Busch, K.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444(3-6), 101–202 (2007).
[Crossref]

Chan, E. M. Y.

K. W. Yu, Y. C. Chu, and E. M. Y. Chan, “Effective-medium theory for two-component nonlinear composites,” Phys. Rev. B Condens. Matter 50(11), 7984–7987 (1994).
[Crossref] [PubMed]

Chu, Y. C.

K. W. Yu, Y. C. Chu, and E. M. Y. Chan, “Effective-medium theory for two-component nonlinear composites,” Phys. Rev. B Condens. Matter 50(11), 7984–7987 (1994).
[Crossref] [PubMed]

Combrié, S.

Danner, A.

Z. Tang, L. Zhao, Z. Sui, Y. Zou, S. Wen, A. Danner, and C. Qiu, “Switchable self-defocusing and focusing in nearly isotropic photonic crystals via enhanced inverse diffraction,” Phys. Rev. A 91(6), 063824 (2015).
[Crossref]

de Rossi, A.

de Sterke, M.

C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12(10), 104003 (2010).
[Crossref]

Delaye, P.

Dolgaleva, K.

K. Dolgaleva and R. W. Boyd, “Local-field effects in nanostructured photonic materials,” Adv. Opt. Photonics 4(1), 1–77 (2012).
[Crossref]

Dubreuil, N.

Eggleton, B. J.

C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12(10), 104003 (2010).
[Crossref]

Flytzanis, C.

Frey, R.

Hache, F.

Haus, J. W.

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective-medium theory of nonlinear ellipsoidal composites,” Phys. Rev. A 40(10), 5729–5734 (1989).
[Crossref] [PubMed]

Huang, D.

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(3), 036608 (2010).
[Crossref] [PubMed]

E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” New J. Phys. 12(9), 093010 (2010).
[Crossref]

Ikeda, N.

Inguva, R.

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective-medium theory of nonlinear ellipsoidal composites,” Phys. Rev. A 40(10), 5729–5734 (1989).
[Crossref] [PubMed]

Inoue, K.

Joannopoulos, J. D.

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004).
[Crossref] [PubMed]

Kauranen, M.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

Larouche, S.

A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, “Nonlinear magnetoelectric metamaterials: Analysis and homogenization via a microscopic coupled-mode theory,” Phys. Rev. A 86(3), 033816 (2012).
[Crossref]

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(3), 036608 (2010).
[Crossref] [PubMed]

S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283(8), 1621–1627 (2010).
[Crossref]

Linden, S.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444(3-6), 101–202 (2007).
[Crossref]

Maier, S. A.

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9(4), 290–294 (2014).
[Crossref] [PubMed]

Mingaleev, S. F.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444(3-6), 101–202 (2007).
[Crossref]

Monat, C.

C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12(10), 104003 (2010).
[Crossref]

Navarro-Cía, M.

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9(4), 290–294 (2014).
[Crossref] [PubMed]

O’Brien, K.

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, “Phase mismatch-free nonlinear propagation in optical zero-index materials,” Science 342(6163), 1223–1226 (2013).
[Crossref] [PubMed]

Oda, H.

Pellegrini, V. P.

V. P. Pellegrini, “Self-consistent effective-medium approximation for strongly nonlinear media,” Phys. Rev. B 64(13), 134211 (2001).
[Crossref]

Poutrina, E.

A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, “Nonlinear magnetoelectric metamaterials: Analysis and homogenization via a microscopic coupled-mode theory,” Phys. Rev. A 86(3), 033816 (2012).
[Crossref]

E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” New J. Phys. 12(9), 093010 (2010).
[Crossref]

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(3), 036608 (2010).
[Crossref] [PubMed]

Qiu, C.

Z. Tang, L. Zhao, Z. Sui, Y. Zou, S. Wen, A. Danner, and C. Qiu, “Switchable self-defocusing and focusing in nearly isotropic photonic crystals via enhanced inverse diffraction,” Phys. Rev. A 91(6), 063824 (2015).
[Crossref]

Rahmani, M.

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9(4), 290–294 (2014).
[Crossref] [PubMed]

Ricard, D.

Roosen, G.

Rose, A.

A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, “Nonlinear magnetoelectric metamaterials: Analysis and homogenization via a microscopic coupled-mode theory,” Phys. Rev. A 86(3), 033816 (2012).
[Crossref]

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(3), 036608 (2010).
[Crossref] [PubMed]

Ryasnyanskiy, A.

Salandrino, A.

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, “Phase mismatch-free nonlinear propagation in optical zero-index materials,” Science 342(6163), 1223–1226 (2013).
[Crossref] [PubMed]

Smith, D. R.

A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, “Nonlinear magnetoelectric metamaterials: Analysis and homogenization via a microscopic coupled-mode theory,” Phys. Rev. A 86(3), 033816 (2012).
[Crossref]

E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” New J. Phys. 12(9), 093010 (2010).
[Crossref]

S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283(8), 1621–1627 (2010).
[Crossref]

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(3), 036608 (2010).
[Crossref] [PubMed]

Soljacic, M.

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004).
[Crossref] [PubMed]

Suchowski, H.

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, “Phase mismatch-free nonlinear propagation in optical zero-index materials,” Science 342(6163), 1223–1226 (2013).
[Crossref] [PubMed]

Sui, Z.

Z. Tang, L. Zhao, Z. Sui, Y. Zou, S. Wen, A. Danner, and C. Qiu, “Switchable self-defocusing and focusing in nearly isotropic photonic crystals via enhanced inverse diffraction,” Phys. Rev. A 91(6), 063824 (2015).
[Crossref]

Tang, Z.

Z. Tang, L. Zhao, Z. Sui, Y. Zou, S. Wen, A. Danner, and C. Qiu, “Switchable self-defocusing and focusing in nearly isotropic photonic crystals via enhanced inverse diffraction,” Phys. Rev. A 91(6), 063824 (2015).
[Crossref]

Tkeshelashvili, L.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444(3-6), 101–202 (2007).
[Crossref]

von Freymann, G.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444(3-6), 101–202 (2007).
[Crossref]

Vy Tran, Q.

Wegener, M.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444(3-6), 101–202 (2007).
[Crossref]

Wen, S.

Z. Tang, L. Zhao, Z. Sui, Y. Zou, S. Wen, A. Danner, and C. Qiu, “Switchable self-defocusing and focusing in nearly isotropic photonic crystals via enhanced inverse diffraction,” Phys. Rev. A 91(6), 063824 (2015).
[Crossref]

Wong, Z. J.

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, “Phase mismatch-free nonlinear propagation in optical zero-index materials,” Science 342(6163), 1223–1226 (2013).
[Crossref] [PubMed]

Yin, X.

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, “Phase mismatch-free nonlinear propagation in optical zero-index materials,” Science 342(6163), 1223–1226 (2013).
[Crossref] [PubMed]

Yu, K. W.

K. W. Yu, Y. C. Chu, and E. M. Y. Chan, “Effective-medium theory for two-component nonlinear composites,” Phys. Rev. B Condens. Matter 50(11), 7984–7987 (1994).
[Crossref] [PubMed]

Zayats, A. V.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

Zhang, X.

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, “Phase mismatch-free nonlinear propagation in optical zero-index materials,” Science 342(6163), 1223–1226 (2013).
[Crossref] [PubMed]

Zhao, L.

Z. Tang, L. Zhao, Z. Sui, Y. Zou, S. Wen, A. Danner, and C. Qiu, “Switchable self-defocusing and focusing in nearly isotropic photonic crystals via enhanced inverse diffraction,” Phys. Rev. A 91(6), 063824 (2015).
[Crossref]

Zou, Y.

Z. Tang, L. Zhao, Z. Sui, Y. Zou, S. Wen, A. Danner, and C. Qiu, “Switchable self-defocusing and focusing in nearly isotropic photonic crystals via enhanced inverse diffraction,” Phys. Rev. A 91(6), 063824 (2015).
[Crossref]

Adv. Opt. Photonics (1)

K. Dolgaleva and R. W. Boyd, “Local-field effects in nanostructured photonic materials,” Adv. Opt. Photonics 4(1), 1–77 (2012).
[Crossref]

J. Opt. (1)

C. Monat, M. de Sterke, and B. J. Eggleton, “Slow light enhanced nonlinear optics in periodic structures,” J. Opt. 12(10), 104003 (2010).
[Crossref]

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

Nat. Mater. (1)

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9(4), 290–294 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

New J. Phys. (1)

E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” New J. Phys. 12(9), 093010 (2010).
[Crossref]

Opt. Commun. (1)

S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283(8), 1621–1627 (2010).
[Crossref]

Opt. Express (2)

Phys. Rep. (1)

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444(3-6), 101–202 (2007).
[Crossref]

Phys. Rev. A (3)

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective-medium theory of nonlinear ellipsoidal composites,” Phys. Rev. A 40(10), 5729–5734 (1989).
[Crossref] [PubMed]

A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, “Nonlinear magnetoelectric metamaterials: Analysis and homogenization via a microscopic coupled-mode theory,” Phys. Rev. A 86(3), 033816 (2012).
[Crossref]

Z. Tang, L. Zhao, Z. Sui, Y. Zou, S. Wen, A. Danner, and C. Qiu, “Switchable self-defocusing and focusing in nearly isotropic photonic crystals via enhanced inverse diffraction,” Phys. Rev. A 91(6), 063824 (2015).
[Crossref]

Phys. Rev. B (1)

V. P. Pellegrini, “Self-consistent effective-medium approximation for strongly nonlinear media,” Phys. Rev. B 64(13), 134211 (2001).
[Crossref]

Phys. Rev. B Condens. Matter (1)

K. W. Yu, Y. C. Chu, and E. M. Y. Chan, “Effective-medium theory for two-component nonlinear composites,” Phys. Rev. B Condens. Matter 50(11), 7984–7987 (1994).
[Crossref] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(3), 036608 (2010).
[Crossref] [PubMed]

Science (1)

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, “Phase mismatch-free nonlinear propagation in optical zero-index materials,” Science 342(6163), 1223–1226 (2013).
[Crossref] [PubMed]

Other (9)

Dong Jun Technology, EastFDTD v4.0 (Dongjun Information Technology Co., Shanghai, 2013).

P. W. Milonni, Fast Light, Slow Light, and Left-Handed Light (Institute of Physics, 2005).

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Figures (4)

Fig. 1
Fig. 1 Sketch (a) and photonic band structure (b) of the two-dimensional Kerr PhC made of Si cylinders embedded in CS2 background. Only TM waves propagating along the direction of ГX are considered in this paper. Insets of (b) are the normalized amplitude distributions of the electric field in a unit cell at two frequencies indicated by red dashed lines in the first and the second band, respectively.
Fig. 2
Fig. 2 Spatial enhancement coefficients (a) and group velocity (b) of the PhC in the first band.
Fig. 3
Fig. 3 Comparison of effective linear refractive index of the PhC. The blue line and the red circles are calculated by using plane wave expansion method (PWEM) and Eq. (9), repectively.
Fig. 4
Fig. 4 Comparison of effective n 2 of the Kerr PhC: (a) only the background and (b) only the silicon cylinder responds nonlinearly, respectively. The blue line is obtained by using Eq. (11) and the red circles are retrieved based on the self-phase modulation effect.

Equations (12)

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R S i = E a v _ S i / E a v _ u n ,
R C S 2 = E a v _ C S 2 / E a v _ u n ,
D a v _ u n = f S i ε 0 ε S i E a v _ S i + f C S 2 ε 0 ε C S 2 E a v _ C S 2 ,
ε C S 2 = ε L _ C S 2 + χ ( 3 ) E a v _ C S 2 2 ,
ε L _ C S 2 = n 0 2 , χ ( 3 ) 2 n 0 n 2 .
D a v _ u n = ε 0 ( f S i ε S i R S i + f C S 2 ε L _ C S 2 R C S 2 + f C S 2 χ ( 3 ) R C S 2 3 E a v _ u n 2 ) E a v _ u n .
ε e f f = f S i ε S i R S i + f C S 2 ε L _ C S 2 R C S 2 + f C S 2 χ ( 3 ) R C S 2 3 E a v _ u n 2 .
n e f f = ε e f f n L _ e f f + n 2 _ e f f E a v _ u n 2 ,
n L _ e f f = f S i ε S i R S i + f C S 2 ε L _ C S 2 R C S 2 ,
n 2 _ e f f = f C S 2 χ ( 3 ) R C S 2 3 / ( 2 n L _ e f f ) .
n 2 _ e f f = f C S 2 χ ( 3 ) R C S 2 3 2 n L _ e f f ( f v g c v g ) ,
n f 0 , f v g c / v g 1 o r n f 0 , f v g v g / c

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