Abstract

We derive the spectral dependence of the non-linear susceptibility of any order, generalizing the common form of Sellmeier equations. This dependence is fully defined by the knowledge of the linear dispersion of the medium. This finding generalizes the Miller formula to any order of non-linearity. In the frequency-degenerate case, it yields the spectral dependence of non-linear refractive indices of arbitrary order.

© 2010 Optical Society of America

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  1. R. W. Boyd, Nonlinear optics (Academic Press, 2008).
  2. J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135 (2006).
    [CrossRef]
  3. M. Soljačić and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonics crystals," Nature Mater. 3, 211 (2004).
    [CrossRef]
  4. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, "Measurement of high-order Kerr refractive index of major air components," Opt. Express 16, 13429 (2009).
    [CrossRef]
  5. P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, "Higherorder Kerr terms allow ionization-free filamentation in air," submitted to Phys. Rev. Lett.
  6. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, "Self-channeling of high-peak-power femtosecond laser pulses in air," Opt. Lett. 20, 73 (1995).
    [CrossRef] [PubMed]
  7. S. L. Chin, S. A. Hosseini,W. Liu, Q. Luo, F. Theberge, N. Akózbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863 (2005).
    [CrossRef]
  8. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, "Ultrashort filaments of light in weakly-ionized, optically-transparent media," Rep. Prog. Phys. 70, 1633 (2007).
    [CrossRef]
  9. A. Couairon and A. Mysyrowicz, "Femtosecond filamentation in transparent media," Phys. Rep. 441, 47 (2007).
    [CrossRef]
  10. J. Kasparian and J.-P. Wolf, "Physics and applications of atmospheric nonlinear optics and filamentation," Opt. Express 16, 466 (2008).
    [CrossRef] [PubMed]
  11. R. C. Miller, "Optical second harmonic generation in piezoelectric crystals," Appl. Phys. Lett. 5, 17 (1964).
    [CrossRef]
  12. V. Mizrahi and D. P. Shelton, "Dispersion of Nonlinear Susceptibilities of Ar, N2, and O2 Measured and Compared," Phys. Rev. Lett. 55, 696 (1985).
    [CrossRef] [PubMed]
  13. A. Owyoung, Ph.D. thesis, California Institute of Technology, 1972 (unpublished).
  14. J. Zhang, Z. H. Lu, and L. J. Wang, "Precision refractive index measurements of air, N2, O2, Ar, and CO2 with a frequency comb," Appl. Opt. 47, 3143 (2008).
    [CrossRef] [PubMed]
  15. B. Boulanger, Y. Petit, P. Segonds, C. Félix, B. Ménaert, J. Zaccaro, and G. Aka, "Absorption and fluorescence anisotropies of monoclinic crystals : the case of Nd:YCOB," Opt. Express 16, 7997 (2008).
    [CrossRef] [PubMed]
  16. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918 (1962).
    [CrossRef]

2009 (1)

V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, "Measurement of high-order Kerr refractive index of major air components," Opt. Express 16, 13429 (2009).
[CrossRef]

2008 (3)

2007 (2)

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, "Ultrashort filaments of light in weakly-ionized, optically-transparent media," Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

A. Couairon and A. Mysyrowicz, "Femtosecond filamentation in transparent media," Phys. Rep. 441, 47 (2007).
[CrossRef]

2006 (1)

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

2005 (1)

S. L. Chin, S. A. Hosseini,W. Liu, Q. Luo, F. Theberge, N. Akózbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863 (2005).
[CrossRef]

2004 (1)

M. Soljačić and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonics crystals," Nature Mater. 3, 211 (2004).
[CrossRef]

1995 (1)

1985 (1)

V. Mizrahi and D. P. Shelton, "Dispersion of Nonlinear Susceptibilities of Ar, N2, and O2 Measured and Compared," Phys. Rev. Lett. 55, 696 (1985).
[CrossRef] [PubMed]

1964 (1)

R. C. Miller, "Optical second harmonic generation in piezoelectric crystals," Appl. Phys. Lett. 5, 17 (1964).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918 (1962).
[CrossRef]

Aka, G.

Akózbek, N.

S. L. Chin, S. A. Hosseini,W. Liu, Q. Luo, F. Theberge, N. Akózbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863 (2005).
[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 (1962).
[CrossRef]

Becker, A.

S. L. Chin, S. A. Hosseini,W. Liu, Q. Luo, F. Theberge, N. Akózbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863 (2005).
[CrossRef]

Béjot, P.

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, "Higherorder Kerr terms allow ionization-free filamentation in air," submitted to Phys. Rev. Lett.

Bergé, L.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, "Ultrashort filaments of light in weakly-ionized, optically-transparent media," Rep. Prog. Phys. 70, 1633 (2007).
[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 (1962).
[CrossRef]

Boulanger, B.

Braun, A.

Chin, S. L.

S. L. Chin, S. A. Hosseini,W. Liu, Q. Luo, F. Theberge, N. Akózbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863 (2005).
[CrossRef]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

Couairon, A.

A. Couairon and A. Mysyrowicz, "Femtosecond filamentation in transparent media," Phys. Rep. 441, 47 (2007).
[CrossRef]

Du, D.

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 (1962).
[CrossRef]

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

Faucher, O.

V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, "Measurement of high-order Kerr refractive index of major air components," Opt. Express 16, 13429 (2009).
[CrossRef]

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, "Higherorder Kerr terms allow ionization-free filamentation in air," submitted to Phys. Rev. Lett.

Félix, C.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

Henin, S.

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, "Higherorder Kerr terms allow ionization-free filamentation in air," submitted to Phys. Rev. Lett.

Hertz, E.

V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, "Measurement of high-order Kerr refractive index of major air components," Opt. Express 16, 13429 (2009).
[CrossRef]

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, "Higherorder Kerr terms allow ionization-free filamentation in air," submitted to Phys. Rev. Lett.

Hosseini, S. A.

S. L. Chin, S. A. Hosseini,W. Liu, Q. Luo, F. Theberge, N. Akózbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863 (2005).
[CrossRef]

Joannopoulos, J. D.

M. Soljačić and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonics crystals," Nature Mater. 3, 211 (2004).
[CrossRef]

Kandidov, V. P.

S. L. Chin, S. A. Hosseini,W. Liu, Q. Luo, F. Theberge, N. Akózbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863 (2005).
[CrossRef]

Kasparian, J.

J. Kasparian and J.-P. Wolf, "Physics and applications of atmospheric nonlinear optics and filamentation," Opt. Express 16, 466 (2008).
[CrossRef] [PubMed]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, "Ultrashort filaments of light in weakly-ionized, optically-transparent media," Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, "Higherorder Kerr terms allow ionization-free filamentation in air," submitted to Phys. Rev. Lett.

Korn, G.

Kosareva, O. G.

S. L. Chin, S. A. Hosseini,W. Liu, Q. Luo, F. Theberge, N. Akózbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863 (2005).
[CrossRef]

Lavorel, B.

V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, "Measurement of high-order Kerr refractive index of major air components," Opt. Express 16, 13429 (2009).
[CrossRef]

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, "Higherorder Kerr terms allow ionization-free filamentation in air," submitted to Phys. Rev. Lett.

Liu, W.

S. L. Chin, S. A. Hosseini,W. Liu, Q. Luo, F. Theberge, N. Akózbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863 (2005).
[CrossRef]

Liu, X.

Loriot, V.

V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, "Measurement of high-order Kerr refractive index of major air components," Opt. Express 16, 13429 (2009).
[CrossRef]

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, "Higherorder Kerr terms allow ionization-free filamentation in air," submitted to Phys. Rev. Lett.

Lu, Z. H.

Luo, Q.

S. L. Chin, S. A. Hosseini,W. Liu, Q. Luo, F. Theberge, N. Akózbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863 (2005).
[CrossRef]

Ménaert, B.

Miller, R. C.

R. C. Miller, "Optical second harmonic generation in piezoelectric crystals," Appl. Phys. Lett. 5, 17 (1964).
[CrossRef]

Mizrahi, V.

V. Mizrahi and D. P. Shelton, "Dispersion of Nonlinear Susceptibilities of Ar, N2, and O2 Measured and Compared," Phys. Rev. Lett. 55, 696 (1985).
[CrossRef] [PubMed]

Mourou, G.

Mysyrowicz, A.

A. Couairon and A. Mysyrowicz, "Femtosecond filamentation in transparent media," Phys. Rep. 441, 47 (2007).
[CrossRef]

Nuter, R.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, "Ultrashort filaments of light in weakly-ionized, optically-transparent media," Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

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 (1962).
[CrossRef]

Petit, Y.

Schroeder, H.

S. L. Chin, S. A. Hosseini,W. Liu, Q. Luo, F. Theberge, N. Akózbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863 (2005).
[CrossRef]

Segonds, P.

Shelton, D. P.

V. Mizrahi and D. P. Shelton, "Dispersion of Nonlinear Susceptibilities of Ar, N2, and O2 Measured and Compared," Phys. Rev. Lett. 55, 696 (1985).
[CrossRef] [PubMed]

Skupin, S.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, "Ultrashort filaments of light in weakly-ionized, optically-transparent media," Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

Soljacic, M.

M. Soljačić and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonics crystals," Nature Mater. 3, 211 (2004).
[CrossRef]

Squier, J.

Theberge, F.

S. L. Chin, S. A. Hosseini,W. Liu, Q. Luo, F. Theberge, N. Akózbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863 (2005).
[CrossRef]

Vieillard, T.

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, "Higherorder Kerr terms allow ionization-free filamentation in air," submitted to Phys. Rev. Lett.

Wang, L. J.

Wolf, J.-P.

J. Kasparian and J.-P. Wolf, "Physics and applications of atmospheric nonlinear optics and filamentation," Opt. Express 16, 466 (2008).
[CrossRef] [PubMed]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, "Ultrashort filaments of light in weakly-ionized, optically-transparent media," Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, "Higherorder Kerr terms allow ionization-free filamentation in air," submitted to Phys. Rev. Lett.

Zaccaro, J.

Zhang, J.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. C. Miller, "Optical second harmonic generation in piezoelectric crystals," Appl. Phys. Lett. 5, 17 (1964).
[CrossRef]

Can. J. Phys. (1)

S. L. Chin, S. A. Hosseini,W. Liu, Q. Luo, F. Theberge, N. Akózbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863 (2005).
[CrossRef]

Nature Mater. (1)

M. Soljačić and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonics crystals," Nature Mater. 3, 211 (2004).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rep. (1)

A. Couairon and A. Mysyrowicz, "Femtosecond filamentation in transparent media," Phys. Rep. 441, 47 (2007).
[CrossRef]

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918 (1962).
[CrossRef]

Phys. Rev. Lett. (1)

V. Mizrahi and D. P. Shelton, "Dispersion of Nonlinear Susceptibilities of Ar, N2, and O2 Measured and Compared," Phys. Rev. Lett. 55, 696 (1985).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, "Ultrashort filaments of light in weakly-ionized, optically-transparent media," Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

Other (3)

R. W. Boyd, Nonlinear optics (Academic Press, 2008).

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, "Higherorder Kerr terms allow ionization-free filamentation in air," submitted to Phys. Rev. Lett.

A. Owyoung, Ph.D. thesis, California Institute of Technology, 1972 (unpublished).

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Figures (1)

Fig. 1.
Fig. 1.

Spectral dependence of the non-linear refractive indices (a) n 2, (b) n 4, (c) n 6, and (d) n 8 of O2, N2, air and Ar at atmospheric pressure.

Equations (32)

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

χ ( 2 ) ( ω 0 ; ω 1 , ω 2 ) χ ( 2 ) ( ω 0 ; ω 1 , ω 2 ) = χ ( 1 ) ( ω 0 ) χ ( 1 ) ( ω 1 ) χ ( 1 ) ( ω 2 ) χ ( 1 ) ( ω 0 ) χ ( 1 ) ( ω 1 ) χ ( 1 ) ( ω 2 )
V ( r ) = V ( 0 ) + i + j + k 2 x i y j z k i ! j ! k ! [ d i + j + k V x i y j z k ] r = 0
F ( r ) = V ( r ) = q 2 x i 1 y j z k ( i 1 ) ! j ! k ! x i y j 1 z k i ! ( j 1 ) ! k ! x i y j z k 1 i ! j ! ( k 1 ) ! [ q V x i y j z k ] r = 0
F ( r ) = q 2 ( 1 Ne ) q 1 P x i 1 P y j P z k ( i 1 ) ! j ! k ! P x i P y j 1 P z k i ! ( j 1 ) ! k ! P x i P y j P z k 1 i ! j ! ( k 1 ) ! [ q V x i y j z k ] r = 0
F ( r ) = q 1 ( 1 Ne ) q P x i P y j P z k i ! j ! k ! [ q + 1 V x i + 1 y j z k ] r = 0 [ q + 1 V x i y j + 1 z k ] r = 0 [ q + 1 V x i y j z k + 1 ] r = 0
Q x , x i , y ( j ) , z ( k ) ( q ) Q y , x ( i ) , y ( j ) , z ( k ) ( q ) Q z , x ( i ) , y ( j ) , z ( k ) ( q ) ( 1 ) q m × i ! j ! k ! [ q + 1 V x i + 1 y j z k ] r = 0 [ q + 1 V x i y j + 1 z k ] r = 0 [ q + 1 V x i y j z k + 1 ] r = 0
d 2 P d t 2 + γ ̿ d 2 P d t + ω ̿ e 2 P + Ne q = 1 Q ( q ) : l = 1 q P Ne = Ne 2 m E ( t )
P = l = 1 α l P ( l )
d 2 P ( 1 ) d t 2 + γ ̿ d P ( 1 ) d t + ω ̿ e 2 P ( 1 ) = N e 2 m E ( t )
d 2 P ( q ) d t 2 + γ ̿ d P ( q ) d t + ω ̿ e 2 P ( q ) = Ne Q ( q ) : u = 1 q P ( 1 ) Ne
+ Ω ( ω ) P 0 ( 1 ) ( ω ) e iωt = N e 2 m + E 0 ( ω ) e iωt
P 0 ( 1 ) ( ω 0 ) = N e 2 m E 0 ( ω 0 ) Ω ( ω 0 )
χ ( 1 ) ( ω 0 ) = 1 Ω ( ω 0 ) N e 2 m ε 0 = N e 2 m ε 0 ( ω e 2 ω 0 2 + i ω 0 γ )
d 2 P ( q ) d t 2 + γ d P ( q ) d t + ω e 2 P ( q ) = Ne Q ( q ) ( P ( 1 ) Ne ) q
+ Ω ( ω ) P 0 ( q ) ( ω ) e iωt = Q ( q ) ( Ne ) q 1 ( + P 0 ( 1 ) ( ω ) e iωt d ω ) q
= Q ( q ) ( Ne ) q 1 Π l = 1 q ( P 0 ( 1 ) ( ω l ) e i ω l t d ω l )
P 0 ( q ) ( ω 0 ) = Q ( q ) Ω ( ω 0 ) ( Ne ) q 1 δ ( l = 0 q ω l = 0 ) × Π l = 1 q ( P ( 1 ) ( ω l ) d ω l )
P 0 ( q ) ( ω 0 ) = Ne ( e m ) q Q ( q ) Ω ( ω 0 ) l = 0 q ω l = 0 ( Π l = 1 q E 0 ( ω l ) Ω ( ω l ) )
P 0 ( q ) ( ω 0 ) = Ne ( e m ) q Q ( q ) Ω ( ω 0 ) C q u 1 , , u q Π l = 1 q ( E 0 ( ω l ) Ω ( ω l ) ) u l
P 0 ( q ) ( ω 0 ) = ε 0 C q u 1 , , u q χ ( q ) ( ω 0 ; ω 1 , , ω q ) Π l = 1 q E 0 u l ( ω l )
χ ( q ) ( ω 0 ; ω 1 , , ω q ) = - Ne ( e m ) q Q ( q ) ε 0 Π l = 0 q Ω ( ω l )
= m ε 0 q N q e q + 1 Q ( q ) Π l = 0 q χ ( 1 ) ( ω l )
χ ( q ) ( ω 0 ; ω 1 , , ω q ) χ ( q ) ( ω 0 ; ω 1 , , ω q ) = Π l = 0 q Ω ( ω l ) Π l = 0 q Ω ( ω l ) = Π l = 0 q χ ( 1 ) ( ω l ) Π l = 0 q χ ( 1 ) ( ω l )
Ω = ( ω 0 ) p 0 ( q ) ( ω 0 ) = Ne ( e m ) q Q ( q ) : ( l = 1 q P ( 1 ) ( ω l ) Ne )
P 0 ( q ) ( ω 0 ) = ε 0 χ ( q ) : l = 1 q E ( ω l )
χ v , x ( i ) , y ( i ) , z ( k ) ( q ) ( ω 0 ; ω 1 , , ω q ) χ v , x ( i ) , y ( i ) , z ( k ) ( q ) ( ω 0 ; ω 1 , , ω q ) = Π l = 0 q Ω v l ( ω ' l ) Π l = 0 q Ω v l ( ω l )
= Π l = 0 q χ v l ( 1 ) ( ω l ) Π l = 0 q χ v l ( 1 ) ( ω l )
χ ( 2 p + 1 ) ( ω 0 ) χ ( 2 p + 1 ) ( ω 0 ) = ( Ω ( ω 0 ) Ω ( ω 0 ) ) 2 p + 2 = ( χ ( 1 ) ( ω 0 ) χ ( 1 ) ( ω 0 ) ) 2 p + 2
n 2 p ( ω 0 ) n 2 p ( ω ' 0 ) = ( n 0 2 ( ω 0 ) 1 n 0 2 ( ω 0 ) 1 ) 2 p + 2
10 8 ( n 0 1 ) = 8015.514 + 2368616 128.7459 1 / λ 2 + 19085.73 50.01974 1 / λ 2
10 8 ( n 0 , N 2 1 ) = 8736.28 + 2398095.2 128.7 1 / λ 2
10 8 ( n 0 , o 2 1 ) = 15532.45 + 456402.97 50.0 1 / λ 2

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