Abstract

The recent interpretation of experiments on the nonlinear non-resonant birefringence induced in a weak probe beam by a high intensity pump beam in air and its constituents has stimulated interest in the non-resonant birefringence due to higher-order Kerr nonlinearities. Here a simple formalism is invoked to determine the non-resonant birefringence for higher-order Kerr coefficients. Some general relations between nonlinear coefficients with arbitrary frequency inputs are also derived for isotropic media. It is shown that the previous linear extrapolations for higher-order birefringence (based on literature values of n 2 and n 4) are not strictly valid, although the errors introduced in the values of the reported higher- order Kerr coefficients are a few percent.

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References

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  1. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17(16), 13429–13434 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. For example,R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, “Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm,” J. Opt. A, Pure Appl. Opt. 6(2), 282–287 (2004).
    [CrossRef]
  6. For example,S. Wu, X.-C. Zhang, and R. L. Fork, “Direct experimental observation of interactive third and fifth order nonlinearities in a time- and space-resolved four-wave mixing experiment,” Appl. Phys. Lett. 61, 1919–1921 (1992).
    [CrossRef]
  7. E. J. Canto-Said, D. J. Hagan, J. Young, and E. W. Van Stryland, “Degenerate four-wave mixing measurements of high order nonlinearities in semiconductors,” IEEE J. Quantum Electron. 27(10), 2274–2280 (1991).
    [CrossRef]
  8. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, Amsterdam, 2008).
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    [CrossRef]
  10. S. V. Popov, Y. P. Svirko, and N. I. Zheludev, Susceptibility Tensors for Nonlinear Optics (Taylor and Francis, 1995)
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    [CrossRef]
  12. M. Kolesik, D. Mirell, J.-C. Diels, and J. V. Moloney, “On the higher-order Kerr effect in femtosecond filaments,” Opt. Lett. 35(21), 3685–3687 (2010).
    [CrossRef] [PubMed]
  13. M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1972).
  14. S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Efficient third-harmonic generation through tailored IR femtosecond laser pulse filamentation in air,” Opt. Express 17(5), 3190–3195 (2009).
    [CrossRef] [PubMed]
  15. S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A 81(3), 033817 (2010).
    [CrossRef]

2010 (3)

2009 (2)

2007 (1)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007).
[CrossRef]

2006 (1)

Y.-F. Chen, K. Beckwitt, F. W. Wise, B. G. Aitken, J. S. Sanghera, and I. D. Aggarwal, “Measurement of fifth- and seventh-order nonlinearities in glasses,” J. Opt. Soc. Am. 23(2), 347–352 (2006).
[CrossRef]

2004 (1)

For example,R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, “Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm,” J. Opt. A, Pure Appl. Opt. 6(2), 282–287 (2004).
[CrossRef]

1993 (1)

1992 (1)

For example,S. Wu, X.-C. Zhang, and R. L. Fork, “Direct experimental observation of interactive third and fifth order nonlinearities in a time- and space-resolved four-wave mixing experiment,” Appl. Phys. Lett. 61, 1919–1921 (1992).
[CrossRef]

1991 (1)

E. J. Canto-Said, D. J. Hagan, J. Young, and E. W. Van Stryland, “Degenerate four-wave mixing measurements of high order nonlinearities in semiconductors,” IEEE J. Quantum Electron. 27(10), 2274–2280 (1991).
[CrossRef]

1975 (1)

V. I. Zavelishko, V. A. Martynov, S. M. Saltiel, and V. G. Tunkin, “Optical nonlinear fourth- and fifth-order susceptibilities,” Sov. J. Quantum Electron. 5(11), 1392–1393 (1975).
[CrossRef]

Abdollahpour, D.

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A 81(3), 033817 (2010).
[CrossRef]

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Efficient third-harmonic generation through tailored IR femtosecond laser pulse filamentation in air,” Opt. Express 17(5), 3190–3195 (2009).
[CrossRef] [PubMed]

Aggarwal, I. D.

Y.-F. Chen, K. Beckwitt, F. W. Wise, B. G. Aitken, J. S. Sanghera, and I. D. Aggarwal, “Measurement of fifth- and seventh-order nonlinearities in glasses,” J. Opt. Soc. Am. 23(2), 347–352 (2006).
[CrossRef]

Aitken, B. G.

Y.-F. Chen, K. Beckwitt, F. W. Wise, B. G. Aitken, J. S. Sanghera, and I. D. Aggarwal, “Measurement of fifth- and seventh-order nonlinearities in glasses,” J. Opt. Soc. Am. 23(2), 347–352 (2006).
[CrossRef]

Arabat, J.

Baba, M.

For example,R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, “Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm,” J. Opt. A, Pure Appl. Opt. 6(2), 282–287 (2004).
[CrossRef]

Beckwitt, K.

Y.-F. Chen, K. Beckwitt, F. W. Wise, B. G. Aitken, J. S. Sanghera, and I. D. Aggarwal, “Measurement of fifth- and seventh-order nonlinearities in glasses,” J. Opt. Soc. Am. 23(2), 347–352 (2006).
[CrossRef]

Canto-Said, E. J.

E. J. Canto-Said, D. J. Hagan, J. Young, and E. W. Van Stryland, “Degenerate four-wave mixing measurements of high order nonlinearities in semiconductors,” IEEE J. Quantum Electron. 27(10), 2274–2280 (1991).
[CrossRef]

Chen, Y.-F.

Y.-F. Chen, K. Beckwitt, F. W. Wise, B. G. Aitken, J. S. Sanghera, and I. D. Aggarwal, “Measurement of fifth- and seventh-order nonlinearities in glasses,” J. Opt. Soc. Am. 23(2), 347–352 (2006).
[CrossRef]

Couairon, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007).
[CrossRef]

Diels, J.-C.

Etchepare, J.

Faucher, O.

Fork, R. L.

For example,S. Wu, X.-C. Zhang, and R. L. Fork, “Direct experimental observation of interactive third and fifth order nonlinearities in a time- and space-resolved four-wave mixing experiment,” Appl. Phys. Lett. 61, 1919–1921 (1992).
[CrossRef]

Ganeev, R. A.

For example,R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, “Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm,” J. Opt. A, Pure Appl. Opt. 6(2), 282–287 (2004).
[CrossRef]

Hagan, D. J.

E. J. Canto-Said, D. J. Hagan, J. Young, and E. W. Van Stryland, “Degenerate four-wave mixing measurements of high order nonlinearities in semiconductors,” IEEE J. Quantum Electron. 27(10), 2274–2280 (1991).
[CrossRef]

Hertz, E.

Kolesik, M.

Kuroda, H.

For example,R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, “Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm,” J. Opt. A, Pure Appl. Opt. 6(2), 282–287 (2004).
[CrossRef]

Lavorel, B.

Loriot, V.

Martynov, V. A.

V. I. Zavelishko, V. A. Martynov, S. M. Saltiel, and V. G. Tunkin, “Optical nonlinear fourth- and fifth-order susceptibilities,” Sov. J. Quantum Electron. 5(11), 1392–1393 (1975).
[CrossRef]

Mirell, D.

Moloney, J. V.

Morita, M.

For example,R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, “Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm,” J. Opt. A, Pure Appl. Opt. 6(2), 282–287 (2004).
[CrossRef]

Mysyrowicz, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007).
[CrossRef]

Papazoglou, D. G.

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A 81(3), 033817 (2010).
[CrossRef]

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Efficient third-harmonic generation through tailored IR femtosecond laser pulse filamentation in air,” Opt. Express 17(5), 3190–3195 (2009).
[CrossRef] [PubMed]

Ryasnyansky, A. I.

For example,R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, “Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm,” J. Opt. A, Pure Appl. Opt. 6(2), 282–287 (2004).
[CrossRef]

Saltiel, S. M.

V. I. Zavelishko, V. A. Martynov, S. M. Saltiel, and V. G. Tunkin, “Optical nonlinear fourth- and fifth-order susceptibilities,” Sov. J. Quantum Electron. 5(11), 1392–1393 (1975).
[CrossRef]

Sanghera, J. S.

Y.-F. Chen, K. Beckwitt, F. W. Wise, B. G. Aitken, J. S. Sanghera, and I. D. Aggarwal, “Measurement of fifth- and seventh-order nonlinearities in glasses,” J. Opt. Soc. Am. 23(2), 347–352 (2006).
[CrossRef]

Suntsov, S.

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A 81(3), 033817 (2010).
[CrossRef]

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Efficient third-harmonic generation through tailored IR femtosecond laser pulse filamentation in air,” Opt. Express 17(5), 3190–3195 (2009).
[CrossRef] [PubMed]

Suzuki, M.

For example,R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, “Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm,” J. Opt. A, Pure Appl. Opt. 6(2), 282–287 (2004).
[CrossRef]

Tunkin, V. G.

V. I. Zavelishko, V. A. Martynov, S. M. Saltiel, and V. G. Tunkin, “Optical nonlinear fourth- and fifth-order susceptibilities,” Sov. J. Quantum Electron. 5(11), 1392–1393 (1975).
[CrossRef]

Turu, M.

For example,R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, “Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm,” J. Opt. A, Pure Appl. Opt. 6(2), 282–287 (2004).
[CrossRef]

Tzortzakis, S.

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A 81(3), 033817 (2010).
[CrossRef]

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Efficient third-harmonic generation through tailored IR femtosecond laser pulse filamentation in air,” Opt. Express 17(5), 3190–3195 (2009).
[CrossRef] [PubMed]

Van Stryland, E. W.

E. J. Canto-Said, D. J. Hagan, J. Young, and E. W. Van Stryland, “Degenerate four-wave mixing measurements of high order nonlinearities in semiconductors,” IEEE J. Quantum Electron. 27(10), 2274–2280 (1991).
[CrossRef]

Wise, F. W.

Y.-F. Chen, K. Beckwitt, F. W. Wise, B. G. Aitken, J. S. Sanghera, and I. D. Aggarwal, “Measurement of fifth- and seventh-order nonlinearities in glasses,” J. Opt. Soc. Am. 23(2), 347–352 (2006).
[CrossRef]

Wu, S.

For example,S. Wu, X.-C. Zhang, and R. L. Fork, “Direct experimental observation of interactive third and fifth order nonlinearities in a time- and space-resolved four-wave mixing experiment,” Appl. Phys. Lett. 61, 1919–1921 (1992).
[CrossRef]

Young, J.

E. J. Canto-Said, D. J. Hagan, J. Young, and E. W. Van Stryland, “Degenerate four-wave mixing measurements of high order nonlinearities in semiconductors,” IEEE J. Quantum Electron. 27(10), 2274–2280 (1991).
[CrossRef]

Zavelishko, V. I.

V. I. Zavelishko, V. A. Martynov, S. M. Saltiel, and V. G. Tunkin, “Optical nonlinear fourth- and fifth-order susceptibilities,” Sov. J. Quantum Electron. 5(11), 1392–1393 (1975).
[CrossRef]

Zhang, X.-C.

For example,S. Wu, X.-C. Zhang, and R. L. Fork, “Direct experimental observation of interactive third and fifth order nonlinearities in a time- and space-resolved four-wave mixing experiment,” Appl. Phys. Lett. 61, 1919–1921 (1992).
[CrossRef]

Appl. Phys. Lett. (1)

For example,S. Wu, X.-C. Zhang, and R. L. Fork, “Direct experimental observation of interactive third and fifth order nonlinearities in a time- and space-resolved four-wave mixing experiment,” Appl. Phys. Lett. 61, 1919–1921 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. J. Canto-Said, D. J. Hagan, J. Young, and E. W. Van Stryland, “Degenerate four-wave mixing measurements of high order nonlinearities in semiconductors,” IEEE J. Quantum Electron. 27(10), 2274–2280 (1991).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

For example,R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, “Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm,” J. Opt. A, Pure Appl. Opt. 6(2), 282–287 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

Y.-F. Chen, K. Beckwitt, F. W. Wise, B. G. Aitken, J. S. Sanghera, and I. D. Aggarwal, “Measurement of fifth- and seventh-order nonlinearities in glasses,” J. Opt. Soc. Am. 23(2), 347–352 (2006).
[CrossRef]

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

Opt. Express (3)

Opt. Lett. (1)

Phys. Rep. (1)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007).
[CrossRef]

Phys. Rev. A (1)

S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A 81(3), 033817 (2010).
[CrossRef]

Sov. J. Quantum Electron. (1)

V. I. Zavelishko, V. A. Martynov, S. M. Saltiel, and V. G. Tunkin, “Optical nonlinear fourth- and fifth-order susceptibilities,” Sov. J. Quantum Electron. 5(11), 1392–1393 (1975).
[CrossRef]

Other (3)

S. V. Popov, Y. P. Svirko, and N. I. Zheludev, Susceptibility Tensors for Nonlinear Optics (Taylor and Francis, 1995)

M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1972).

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, Amsterdam, 2008).

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Figures (2)

Fig. 1
Fig. 1

The pump-probe interaction geometry in reference 1. The angle between the beams was 4°.

Fig. 2
Fig. 2

Comparison between the expansion coefficients estimated by the two models. (a) Coefficients corresponding to χ(m) terms. (□) analytical model, (•) Loriot et al. estimation, dotted/dashed lines are a guide to the eye. (b) Relative error for the various coefficients of the χ(m) terms. (Dotted lines are guides to the eye).

Equations (53)

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n 2 ( ω ; ω ) = 1 4 n x 2 ε 0 c eal { χ x x x x ( 3 ) ( ω ; ω , ω , ω ) + χ x x x x ( 3 ) ( ω ; ω , ω , ω )                      + χ x x x x ( 3 ) ( ω ; ω , ω , ω ) }
n 2 ( ω p ; ω ) = 1 4 n 0 2 ε 0 c eal { χ x x x x ( 3 ) ( ω p ; ω p , ω , ω ) + χ x x x x ( 3 ) ( ω p ; ω p , ω , ω )                     + χ x x x x (3) ( ω ; ω , ω p , ω ) + χ x x x x (3) ( ω p ; ω , ω p , ω )                     + χ x x x x (3) ( ω p ; ω , ω , ω p ) + χ x x x x (3) ( ω p ; ω , ω , ω p )}
E ( r , t ) = 1 2 E x ( ω ) e i ω t + c . c .
E x p ( r , t ) = 1 2 E x p ( ω p ) e i ω p t + c . c . ;      E y p ( r , t ) = 1 2 E y p ( ω p ) e i ω p t + c . c .
P x p N L ( r , t ) = 1 2 P x p N L ( ω p ) e i ω p t + c . c . ;      P y p N L ( r , t ) = 1 2 P y p N L ( ω p ) e i ω p t + c . c .
P x p ( 3 ) ( ω p ) = 1 4 ε 0 { χ ¯ x x x x ( 3 ) ( ω p ; ω p , ω , ω ) + χ ¯ x x x x ( 3 ) ( ω p ; ω , ω , ω p )                + χ ¯ x x x x ( 3 ) ( ω p ; ω , ω p , ω ) + χ ¯ x x x x ( 3 ) ( ω p ; ω , ω , ω p )                + χ ¯ x x x x ( 3 ) ( ω p ; ω , ω p , ω ) + χ ¯ x x x x ( 3 ) ( ω p ; ω p , ω , ω ) } E x p ( ω p ) E x * ( ω ) E x ( ω ) .
χ ¯ x x x x ( 3 ) ( ω p ; ω p , ω , ω ) = q w q χ x x x x , q ( 3 ) ( ω p ; ω p , ω , ω )  etc .
χ ¯ ˜ x x x x ( 3 ) ( ω p ; ω p , ω , ω ) = χ ¯ ˜ x x x x ( 3 ) ( ω p ; ω , ω p , ω ) = χ ¯ ˜ x x x x ( 3 ) ( ω p ; ω p , ω , ω ) = χ ¯ ˜ x x x x ( 3 ) ( ω p ; ω , ω , ω p ) = χ ¯ ˜ x x x x ( 3 ) ( ω p ; ω , ω , ω p ) = χ ¯ ˜ x x x x ( 3 ) ( ω p ; ω , ω p , ω ) .
P x p ( 3 ) ( ω p ) = 6 4 ε 0 χ ¯ ˜ x x x x ( 3 ) ( ω p ; ω p , ω , ω ) E x p ( ω p ) E x * ( ω ) E x ( ω ) .
P x p ( 3 ) ( ω p ) = 1 4 3 ! 1 ! 1 ! 1 ! ε 0 χ ¯ ˜ x x x x ( 3 ) ( ω p ; ω p , ω , ω ) E x p ( ω p ) | E x ( ω ) | 2 .
P y p ( 3 ) ( ω p ) = 1 4 ε 0 { χ ¯ y y x x ( 3 ) ( ω p ; ω p , ω , ω ) + χ ¯ y y x x ( 3 ) ( ω p ; ω p , ω , ω ) + χ ¯ y x y x ( 3 ) ( ω p ; ω , ω p , ω ) + χ ¯ y x y x ( 3 ) ( ω p ; ω , ω p , ω ) + χ ¯ y x x y ( 3 ) ( ω p ; ω , ω , ω p ) + χ ¯ y x x y ( 3 ) ( ω p ; ω , ω , ω p ) } E y p ( ω ) E x * ( ω ) E x ( ω ) .
P y p ( 3 ) ( ω p ) = 1 4 3 ! 1 ! 1 ! 1 ! ε 0 χ ¯ ˜ y y x x ( 3 ) ( ω p ; ω p , ω , ω ) E y p ( ω p ) | E x ( ω ) | 2 .
P x ( 3 ) ( ω ) = 1 4 3 ! 2 ! 1 ! ε 0 χ ˜ x x x x ( 3 ) ( ω ) E x ( ω ) | E x ( ω ) | 2 = 1 2 P x p ( 3 ) ( ω p ) ,
P y ( 3 ) ( ω ) = 1 4 3 ! 1 ! 1 ! 1 ! ε 0 χ ¯ ˜ y y x x ( 3 ) ( ω ) E y ( ω ) | E x ( ω ) | 2 = P y p ( 3 ) ( ω p ) .
P x p (2 m + 1) ( ω p ) = [ 1 4 m ( 2 m + 1 ) ! m ! m ! ε 0 χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω p ) | E x ( ω ) | 2 m ] E x p ( ω p ) = ( m + 1 ) P x (2 m + 1) ( ω ) ,
P y p (2 m + 1) ( ω p ) = [ 1 4 m ( 2 m + 1 ) ! m ! m ! ε 0 χ ¯ ˜ ( 2 ) y , ( 2 m ) x ( 2 m + 1 ) ( ω p ) | E x ( ω ) | 2 m ] E y p ( ω p ) = P y (2 m + 1) ( ω ) .
P x p (1) ( ω p ) + P x p N L ( ω p ) = ε 0 [ n x 2 ( ω p ) 1 ] E x p ( ω p ) = ε 0 [ ( n 0 2 1 )                                     + m = 1 1 4 m ( 2 m + 1 ) ! m ! m ! χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω p ) | E x ( ω ) | 2 m ] E x p ( ω p ) ;
P y p (1) ( ω p ) + P y p N L ( ω p ) ε 0 [ n y 2 ( ω p ) 1 ] E y p ( ω p ) = ε 0 [ ( n 0 2 1 )                                         + m = 1 1 4 m ( 2 m + 1 ) ! m ! m ! χ ¯ ˜ y y , ( 2 m ) x ( 2 m + 1 ) ( ω p ) | E x ( ω ) | 2 m ] E y p ( ω p ) .
P x (1) ( ω ) + P x N L ( ω ) = ε 0 [ n x 2 ( ω ) 1 ] E x ( ω ) = ε 0 [ ( n 0 2 1 )                               + m = 1 1 4 m ( 2 m + 1 ) ! ( m + 1 ) ! m ! χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω ) | E x ( ω ) | 2 m ] E x ( ω ) .
P y (1) ( ω ) + P y ( N L ) ( ω ) = ε 0 [ n y 2 ( ω ) 1 ] E y ( ω ) = ε 0 [ ( n 0 2 1 )                                     + m = 1 1 4 m ( 2 m + 1 ) ! m ! m ! χ ¯ ˜ y y , ( 2 m ) x ( 2 m + 1 ) ( ω ) | E x ( ω ) | 2 m ] E y ( ω ) .
χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω p ) = ( 2 m + 1 ) χ ¯ ˜ y y , ( 2 m ) x ( 2 m + 1 ) ( ω p ) ;    χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω ) = ( 2 m + 1 ) χ ¯ ˜ y y , ( 2 m ) x ( 2 m + 1 ) ( ω ) .
P y p (1) ( ω p ) + P y p N L ( ω p )   = ε 0 [ n y 2 ( ω p ) 1 ] E y p ( ω p ) = ε 0 [ ( n 0 2 1 )                                        + m = 1 1 4 m ( 2 m + 1 ) ! ( 2 m + 1 ) m ! m ! χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω p ) | E x ( ω ) | 2 m ] E y p ( ω p ) ,
P y (1) ( ω ) + P y ( N L ) ( ω ) = ε 0 [ n y 2 ( ω ) 1 ] E y ( ω ) = ε 0 [ ( n 0 2 1 )                                    + m = 1 1 4 m ( 2 m + 1 ) ! ( 2 m + 1 ) m ! m ! χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω ) | E x ( ω ) | 2 m ] E y ( ω ) ,
n x 2 ( ω p ) = n 0 2 { 1 + [ m = 1 A ¯ m I m ] } ;     n y 2 ( ω p ) = n 0 2 { 1 + [ m = 1 1 ( 2 m + 1 ) A ¯ m I m ] } , n x 2 ( ω ) = n 0 2 { 1 + [ m = 1 1 m + 1 A ¯ m I m ] } ;     n y 2 ( ω ) = n 0 2 { 1 + [ m = 1 1 ( 2 m + 1 ) A ¯ m I m ] } ,
                                                A ¯ m = 2 n ¯ 2 m ( ω p ; ω ) n 0 , A ¯ m = 1 2 n 0 1 2 m n 0 m c m ε 0 m ( 2 m + 1 ) ! m ! m ! 1 ! χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω p ) = 1 2 n 0 ( m + 1 ) 2 m n 0 m c m ε 0 m ( 2 m + 1 ) ! m ! m ! 1 ! χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω ) .
Δ n ¯ x ( m ) ( ω p ) = n ¯ 2 m ( ω p ; ω ) I m .
n ¯ x ( ω p ) = n 0 1 + [ m = 1 A ¯ m I m ] ;   n ¯ y ( ω p ) = n 0 1 + [ m = 1 1 ( 2 m + 1 ) A ¯ m I m ] ;                                             Δ n b i r N L ( ω p ) = n x ( ω p ) n y ( ω p ) .
1 + b = s = 0 ( 1 ) s ( 2 s ) ! ( 1 2 s ) ( s ! ) 2 4 s b s = 1 + 1 2 b 1 8 b 2 + 1 16 b 3 5 128 b 4 + 7 256 b 5 ..
Δ n ¯ b i r N L ( ω p ) = n 0 s = 0 ( 1 ) s ( 2 s ) ! ( 1 2 s ) ( s ! ) 2 4 s ( [ m = 1 A ¯ m I m ] s [ m = 1 A ¯ m 2 m + 1 I m ] s ) .
Δ n ¯ b i r s = 1 ( ω p ) = n 0 { 1 3 A ¯ 1 I + 2 5 A ¯ 2 I 2 + 4 9 A ¯ 3 I 3 + 8 17 A ¯ 4 I 4 + 16 33 A ¯ 4 I 4 + 32 65 A ¯ 5 I 5                    = 2 3 n ¯ 2 ( ω p ; ω ) I + 4 5 n ¯ 4 ( ω p ; ω ) I 2 + 8 9 n ¯ 6 ( ω p ; ω ) I 3                       + 16 17 n ¯ 8 ( ω p ; ω ) I 4 + 32 33 n ¯ 10 ( ω p ; ω ) I 5
Δ n ¯ b i r N L = 2 3 n ¯ 2 ( ω p ; ω ) I + [ 4 5 n ¯ 4 ( ω p ; ω ) 4 9 n 0 n ¯ 2 2 ( ω p ; ω ) ] I 2 + [ 8 9 n ¯ 6 ( ω p ; ω ) 14 15 n 0 n ¯ 2 ( ω p ; ω ) n ¯ 4 ( ω p ; ω ) + 13 27 n 0 2 n ¯ 2 3 ( ω p ; ω ) ] I 3 + [ 16 17 n ¯ 8 ( ω p ; ω ) 26 27 n 0 n ¯ 2 ( ω p ; ω ) n ¯ 6 ( ω p ; ω ) 12 25 n 0 n ¯ 4 2 ( ω p ; ω ) + 22 15 n 0 2 n ¯ 2 2 ( ω p ; ω ) n ¯ 4 ( ω p ; ω ) 50 81 n 0 3 n ¯ 2 4 ( ω p ; ω ) ] I 4 + [ 32 33 n ¯ 10 ( ω p ; ω ) 50 51 n 0 n ¯ 2 ( ω p ; ω ) n ¯ 8 ( ω p ; ω ) 44 45 n 0 n ¯ 4 ( ω p ; ω ) n ¯ 6 ( ω p ; ω ) + 40 27 n 0 2 n ¯ 2 2 ( ω p ; ω ) n ¯ 6 ( ω p ; ω ) + 37 25 n 0 2 n ¯ 2 ( ω p ; ω ) n ¯ 4 2 ( ω p ; ω ) 67 27 n 0 3 n ¯ 2 3 ( ω p ; ω ) n ¯ 4 ( ω p ; ω ) + 847 972 n 0 4 n ¯ 2 5 ( ω p ; ω ) ] I 5 .
Δ n ¯ b i r N L ( ω p ) = m = 1 2 m 2 m + 1 n ¯ 2 m ( ω p ; ω ) I m    .
Δ n ¯ b i r N L ( ω p ) = m = 1 2 m 2 m + 1 n ¯ 2 m ( ω p ; ω ) I m
P x ( 3 ) ( ω 4 ) = 1 4 ε 0 χ x x x x ( 3 ) ( ω 4 ; ω 3 , ω 2 , ω 1 ) E 1 E 2 E 3 .
E 1 x ' = 1 2 E 1 ; E 2x' = 1 2 E 2 ; E 3 x ' = 1 2 E 3 ; E 1 y ' = 1 2 E 1  ; E 2 y ' = 1 2 E 2 ; E 3 y ' = 1 2 E 3 .
P x ' ( 3 ) ( ω 4 ) = 1 4 ε 0 [ χ x x x x ( 3 ) ( ω 4 ; ω 3 , ω 2 , ω 1 ) E 1 x ' E 2 x ' E 3 x ' + χ x x y y ( 3 ) ( ω 4 ; ω 3 , ω 2 , ω 1 ) E 1 y ' E 2 y ' E 3 x '                     + χ x y y x ( 3 ) ( ω 4 ; ω 3 , ω 2 , ω 1 ) E 1 x ' E 2 y ' E 3 y ' + χ x y x y ( 3 ) ( ω 4 ; ω 3 , ω 2 , ω 1 ) E 1 y ' E 2 x ' E 3 y ' ] P x ' ( 3 ) ( ω 4 ) = 1 4 ε 0 1 2 2 [ χ x x x x ( 3 ) ( ω 4 ; ω 3 , ω 2 , ω 1 ) + χ x x y y ( 3 ) ( ω 4 ; ω 3 , ω 2 , ω 1 )                        + χ x y y x ( 3 ) ( ω 4 ; ω 3 , ω 2 , ω 1 ) + χ x y x y ( 3 ) ( ω 4 ; ω 3 , ω 2 , ω 1 ) ] E 1 E 2 E 3 .
P x ' ( 3 ) ( ω 4 ) = 1 4 ε 0 1 2 χ x x x x ( 3 ) ( ω 4 , ω 3 , ω 2 , ω 1 ) E 1 E 2 E 3 .
χ x x x x ( 3 ) ( ω 4 ; ω 3 , ω 2 , ω 1 ) = χ x x y y ( 3 ) ( ω 4 ; ω 3 , ω 2 , ω 1 ) + χ x y y x ( 3 ) ( ω 4 ; ω 3 , ω 2 , ω 1 )                                  + χ x y x y ( 3 ) ( ω 4 ; ω 3 , ω 2 , ω 1 ) .
χ ˜ x x y y ( 3 ) ( ω p ) = χ ˜ x y y x ( 3 ) ( ω p ) = χ ˜ x y x y ( 3 ) ( ω p )       χ ˜ x x x x ( 3 ) ( ω p ) = 3 χ ˜ x x y y ( 3 ) ( ω p ) .
χ ¯ ˜ x x x x ( 3 ) ( ω p ) = 3 χ ¯ ˜ x x y y ( 3 ) ( ω p )         χ ¯ ˜ x x x x ( 3 ) ( ω ) = 3 χ ¯ ˜ x x y y ( 3 ) ( ω ) .
χ ¯ ˜ x x x x ( 3 ) ( ω p ) = 3 ! 2 ! 1 ! χ ¯ ˜ x x y y ( 3 ) ( ω p ) ;      χ ¯ ˜ x x x x ( 3 ) ( ω ) = 3 ! 2 ! 1 ! χ ¯ ˜ x x y y ( 3 ) ( ω )
P x ( 5 ) ( ω 4 ) = 1 16 ε 0 χ x x x x x x ( 5 ) ( ω 6 ; ω 5 , ω 4 , ω 3 , ω 2 , ω 1 ) E 1 E 2 E 3 E 4 E 5 .
P x ' ( 5 ) ( ω 6 ) = 1 16 ε 0 1 4 2 [ χ ¯ ˜ x x x x x x ( 5 ) ( ω 6 ; ω 5 , ω 4 , ω 3 , ω 2 , ω 1 ) + 5 ! 3 ! 2 ! χ ¯ ˜ x x y y x x ( 5 ) ( ω 6 ; ω 5 , ω 4 , ω 3 , ω 2 , ω 1 )                 + 5 ! 4 ! 1 ! χ ¯ ˜ x x y y y y ( 5 ) ( ω 6 ; ω 5 , ω 4 , ω 3 , ω 2 , ω 1 ) ] E 1 E 2 E 3 E 4 E 5 .
P x ' N L ( ω 6 ) = 1 16 ε 0 1 2 χ ˜ x x x x x x ( 5 ) ( ω 6 ; ω 5 , ω 4 , ω 3 , ω 2 , ω 1 ) E 1 E 2 E 3 E 4 E 5 .
χ ¯ ˜ x x x x x x ( 5 ) ( ω p ) = 5 χ ¯ ˜ y y x x x x ( 5 ) ( ω p ) ;     χ ¯ ˜ x x x x x x ( 5 ) ( ω ) = 5 χ ¯ ˜ y y x x x x ( 5 ) ( ω ) .
χ x x y y x x x x ( 7 ) ( ω 8 ; ω 7 , ω 6 , ω 5 , ω 4 , ω 3 , ω 2 , ω 1 ) , χ x x y y y y x x ( 7 ) ( ω 8 ; ω 7 , ω 6 , ω 5 , ω 4 , ω 3 , ω 2 , ω 1 )
P x ' N L ( ω 8 ) = 1 64 ε 0 1 8 2 [ χ ¯ ˜ x x x x x x ( 7 ) ( ω 8 ) + 7 ! 5 ! 2 ! χ ¯ ˜ x x y y x x x x ( 7 ) ( ω 8 )                   + 7 ! 4 ! 3 ! χ ¯ ˜ x x y y y y x x ( 7 ) ( ω 8 ) + 7 ! 6 ! 1 ! χ ¯ ˜ x x y y y y y y ( 7 ) ( ω 8 ) ] E 1 E 2 E 3 E 4 E 5 E 6 E 7                 = 1 64 ε 0 1 2 χ ¯ ˜ x x x x x x ( 7 ) ( ω 8 ) E 1 E 2 E 3 E 4 E 5 E 6 E 7
χ ¯ ˜ x x x x x x x x ( 7 ) ( ω p ) = 9 χ ¯ ˜ y y x x x x x x ( 7 ) ( ω p ) ;     χ ¯ ˜ x x x x x x x x ( 7 ) ( ω ) = 9 χ ¯ ˜ y y x x x x x x ( 7 ) ( ω ) .
P x ( 9 ) ( ω 10 ) = 1 256 ε 0 1 16 2 [ χ ¯ ˜ x x x x x x x x ( 9 ) ( ω 10 ) + 9 ! 7 ! 2 ! χ ¯ ˜ x x y y x x x x x x ( 9 ) ( ω 10 ) + 9 ! 5 ! 4 ! χ ¯ ˜ x x y y y y x x x x ( 9 ) ( ω 10 )                    + 9 ! 3 ! 6 ! χ ¯ ˜ x x y y y y y y x x ( 9 ) ( ω 10 ) + 9 ! 1 ! 8 ! χ ¯ ˜ x x y y y y y y y y ( 9 ) ( ω 10 ) ] E 1 E 2 E 3 E 4 E 5 E 6 E 7 E 8 E 9                 = 1 256 ε 0 1 2 χ ¯ ˜ x x x x x x x x ( 9 ) ( ω 10 ) E 1 E 2 E 3 E 4 E 5 E 6 E 7 E 8 E 9 .
χ ¯ ˜ x x x x x x x x x x ( 9 ) ( ω 10 ) = 17 χ ¯ ˜ y y x x x x x x x x ( 9 ) ( ω 10 )
χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω p ) = ( 2 m + 1 ) χ ¯ ˜ y y , ( 2 m ) x ( 2 m + 1 ) ( ω p ) ;    χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω ) = ( 2 m + 1 ) χ ¯ ˜ y y , ( 2 m ) x ( 2 m + 1 ) ( ω ) .
χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω 2 m + 2 ) = 1 2 m [ χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω 2 m + 2 ) + ( 2 m + 1 ) ! ( 2 m ) ! 1 ! χ ¯ ˜ ( 2 ) y , ( 2 m ) x ( 2 m + 1 ) ( ω 2 m + 2 )                    + ( 2 m + 1 ) ! ( 2 m 2 ) ! 3 ! χ ¯ ˜ ( 4 ) y , ( 2 m 2 ) x , ( 2 m + 1 ) ( ω 2 m + 2 )   + ( 2 m + 1 ) ! ( 2 m 4 ) ! 5 ! χ ¯ ˜ ( 6 ) y , ( 2 m 4 ) x ( 2 m + 1 ) ( ω 2 m + 2 ) ...                    + ( 2 m + 1 ) ! 2 ! ( 2 m 1 ) ! χ ¯ ˜ ( 2 m 1 ) y , ( 2 ) x ( 2 m + 1 ) ( ω 2 m + 2 ) ] .
χ ¯ ˜ ( 2 m + 2 ) x ( 2 m + 1 ) ( ω 2 m + 2 ) = 1 2 m 1 [ ( 2 m + 1 ) ! ( 2 m ) ! 1 ! χ ¯ ˜ ( 2 ) y , ( 2 m ) x ( 2 m + 1 ) ( ω 2 m + 2 )                  + ( 2 m + 1 ) ! ( 2 m 2 ) ! 3 ! χ ¯ ˜ ( 4 ) y , ( 2 m 2 ) x , ( 2 m + 1 ) ( ω 2 m + 2 )   + ( 2 m + 1 ) ! ( 2 m 4 ) ! 5 ! χ ¯ ˜ ( 6 ) y , ( 2 m 4 ) x ( 2 m + 1 ) ( ω 2 m + 2 ) ...                  + ( 2 m + 1 ) ! 2 ! ( 2 m 1 ) ! χ ¯ ˜ ( 2 m 1 ) y , ( 2 ) x ( 2 m + 1 ) ( ω 2 m + 2 ) ] .

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