Abstract

Here we report on an extension of common Z-scan method to arbitrary polarized incidence light for measurements of anisotropic third-order nonlinear susceptibility in isotropic medium. The normalized transmittance formulas of closed-aperture Z-scan are obtained for linearly, elliptically and circularly polarized incidence beam. The theoretical analysis is examined experimentally by studying third-order nonlinear susceptibility of CS2 liquid. Results show that the elliptically polarized light Z-scan method can be used to measure simultaneously the two third-order nonlinear susceptibility components χxyyx (3) and χxxyy (3). Furthermore, the elliptically polarized light Z-scan measurements of large nonlinear phase shift are also analyzed theoretically and experimentally.

© 2009 Optical Society of America

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2008 (1)

2007 (3)

Z. B. Liu, Y. L. Liu, B. Zhang, W. Y. Zhou, J. G. Tian, W. P. Zang, and C. P. Zhang, “Nonlinear absorption and optical limiting properties of carbon disulfide in a short-wavelength region,” J. Opt. Soc. Am. B 24, 1101 (2007).
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[Crossref]

2005 (1)

2004 (3)

I. Fuks-Janczarek, B. Sahraoui, I. V. Kityk, and J. Berdowski, “Electronic and nuclear contributions to the third-order optical susceptibility,” Opt. Commun. 236, 159 (2004).
[Crossref]

R.A. Ganeev, A.I. Ryasnyansky, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433 (2004).
[Crossref]

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, 282–287 (2004).
[Crossref]

2000 (2)

K. Kiyohara, K. Kamada, and K. Ohta, “Orientational and collision-induced contribution to third-order nonlinear optical response of liquid CS2,” J. Chem. Phys. 112, 6338, (2000).
[Crossref]

R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, and X. N. Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443 (2000).
[Crossref]

1997 (4)

1996 (1)

1995 (2)

1994 (3)

1993 (2)

1992 (1)

1990 (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[Crossref]

1964 (1)

P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. 12, 507–509 (1964).
[Crossref]

Aitchison, J. S.

S. J. Wagner, J. Meier, A. S. Helmy, J. S. Aitchison, D. Modotto, M. Sorel, and D. C. Hutchings, “Polarization-Dependent Nonlinear Refraction in GaAs/AlAs Superlattice Waveguides,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FWE2.

Araújo, C.B. de

Baba, M.

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, 282–287 (2004).
[Crossref]

Berdowski, J.

I. Fuks-Janczarek, B. Sahraoui, I. V. Kityk, and J. Berdowski, “Electronic and nuclear contributions to the third-order optical susceptibility,” Opt. Commun. 236, 159 (2004).
[Crossref]

Boucher, V.

R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, and X. N. Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443 (2000).
[Crossref]

Boudebs, G.

Bourdin, J. P.

Boyd, R.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, 2003).

Bridges, R.

Chapple, P. B.

P. B. Chapple, J. Straromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251–293 (1997).
[Crossref]

Chen, J.

Chen, S. Q.

Chevalier, R.

R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, and X. N. Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443 (2000).
[Crossref]

Chis, M.

Demenicis, L.

DeSalvo, J.R.

DeSalvo, R.

Ding, J.

Dorkenoo, K. D.

R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, and X. N. Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443 (2000).
[Crossref]

Fan, Y.

Fischer, G.

Fuks-Janczarek, I.

I. Fuks-Janczarek, B. Sahraoui, I. V. Kityk, and J. Berdowski, “Electronic and nuclear contributions to the third-order optical susceptibility,” Opt. Commun. 236, 159 (2004).
[Crossref]

Ganeev, R. A.

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, 282–287 (2004).
[Crossref]

Ganeev, R.A.

R.A. Ganeev, A.I. Ryasnyansky, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433 (2004).
[Crossref]

Gomes, A. S. L.

Gu, B.

Hagan, D. J.

Helmy, A. S.

S. J. Wagner, J. Meier, A. S. Helmy, J. S. Aitchison, D. Modotto, M. Sorel, and D. C. Hutchings, “Polarization-Dependent Nonlinear Refraction in GaAs/AlAs Superlattice Waveguides,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FWE2.

Hermann, J. A.

P. B. Chapple, J. Straromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251–293 (1997).
[Crossref]

J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993).
[Crossref]

Hutchings, D. C.

S. J. Wagner, J. Meier, A. S. Helmy, J. S. Aitchison, D. Modotto, M. Sorel, and D. C. Hutchings, “Polarization-Dependent Nonlinear Refraction in GaAs/AlAs Superlattice Waveguides,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FWE2.

Ishizawa, N.

R.A. Ganeev, A.I. Ryasnyansky, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433 (2004).
[Crossref]

Kamada, K.

K. Kiyohara, K. Kamada, and K. Ohta, “Orientational and collision-induced contribution to third-order nonlinear optical response of liquid CS2,” J. Chem. Phys. 112, 6338, (2000).
[Crossref]

Kityk, I. V.

I. Fuks-Janczarek, B. Sahraoui, I. V. Kityk, and J. Berdowski, “Electronic and nuclear contributions to the third-order optical susceptibility,” Opt. Commun. 236, 159 (2004).
[Crossref]

Kiyohara, K.

K. Kiyohara, K. Kamada, and K. Ohta, “Orientational and collision-induced contribution to third-order nonlinear optical response of liquid CS2,” J. Chem. Phys. 112, 6338, (2000).
[Crossref]

Kuroda, H.

R.A. Ganeev, A.I. Ryasnyansky, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433 (2004).
[Crossref]

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, 282–287 (2004).
[Crossref]

Lefkir, M.

Liang, J.

J. Liang, H. Zhao, and X. Zhou, “Polarization-dependence effects of refractive index change associated with photoisomerization investigated with Z-scan technique,” J. Appl. Phys. 101, 013106 (2007).
[Crossref]

Liu, Y. L.

Liu, Z. B.

Maker, P. D.

P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. 12, 507–509 (1964).
[Crossref]

McDuff, R. G.

P. B. Chapple, J. Straromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251–293 (1997).
[Crossref]

J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993).
[Crossref]

Mckay, T. J.

P. B. Chapple, J. Straromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251–293 (1997).
[Crossref]

Meier, J.

S. J. Wagner, J. Meier, A. S. Helmy, J. S. Aitchison, D. Modotto, M. Sorel, and D. C. Hutchings, “Polarization-Dependent Nonlinear Refraction in GaAs/AlAs Superlattice Waveguides,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FWE2.

Melo, C. P. de

Modotto, D.

S. J. Wagner, J. Meier, A. S. Helmy, J. S. Aitchison, D. Modotto, M. Sorel, and D. C. Hutchings, “Polarization-Dependent Nonlinear Refraction in GaAs/AlAs Superlattice Waveguides,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FWE2.

Morita, M.

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, 282–287 (2004).
[Crossref]

Ohta, K.

K. Kiyohara, K. Kamada, and K. Ohta, “Orientational and collision-induced contribution to third-order nonlinear optical response of liquid CS2,” J. Chem. Phys. 112, 6338, (2000).
[Crossref]

Palffy-Muhoray, P.

W. Zhao and P. Palffy-Muhoray, “Z-scan measurements of ?3 using top-hat beams,” Appl. Phys. Lett. 65,673–675 (1994).
[Crossref]

Petrov, D. V.

Phu, X. N.

R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, and X. N. Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443 (2000).
[Crossref]

Rivoire, G.

Ryasnyansky, A. I

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, 282–287 (2004).
[Crossref]

Ryasnyansky, A.I.

R.A. Ganeev, A.I. Ryasnyansky, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433 (2004).
[Crossref]

Sahraoui, B.

I. Fuks-Janczarek, B. Sahraoui, I. V. Kityk, and J. Berdowski, “Electronic and nuclear contributions to the third-order optical susceptibility,” Opt. Commun. 236, 159 (2004).
[Crossref]

Said, A. A.

Sakakibara, S.

R.A. Ganeev, A.I. Ryasnyansky, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433 (2004).
[Crossref]

Santos, C. G. dos

Savage, C. M.

P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. 12, 507–509 (1964).
[Crossref]

Sheik-Bahae, M.

Song, F.

Sorel, M.

S. J. Wagner, J. Meier, A. S. Helmy, J. S. Aitchison, D. Modotto, M. Sorel, and D. C. Hutchings, “Polarization-Dependent Nonlinear Refraction in GaAs/AlAs Superlattice Waveguides,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FWE2.

Souto-Maior, R.

Straromlynska, J.

P. B. Chapple, J. Straromlynska, J. A. Hermann, T. J. Mckay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251–293 (1997).
[Crossref]

Stryland, E. W. Van

Sutherland, R. L.

R. L. Sutherland, Handbook of Nonlinear Optics (Second Edition); (Marcel Dekker: New York, 2003).

Suzuki, M.

R.A. Ganeev, A.I. Ryasnyansky, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433 (2004).
[Crossref]

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, 282–287 (2004).
[Crossref]

Terhune, R. W.

P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. 12, 507–509 (1964).
[Crossref]

Tian, J. G.

Turu, M.

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, 282–287 (2004).
[Crossref]

R.A. Ganeev, A.I. Ryasnyansky, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433 (2004).
[Crossref]

Volle, R.

R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, and X. N. Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443 (2000).
[Crossref]

Wagner, S. J.

S. J. Wagner, J. Meier, A. S. Helmy, J. S. Aitchison, D. Modotto, M. Sorel, and D. C. Hutchings, “Polarization-Dependent Nonlinear Refraction in GaAs/AlAs Superlattice Waveguides,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FWE2.

Wang, H.

Wang, J.

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[Crossref]

Xia, T.

Yan, X. Q.

Z. B. Liu, X. Q. Yan, W. Y. Zhou, and J. G. Tian, “Evolutions of polarization and nonlinearities in an isotropic nonlinear medium,” Opt. Express 16, 8144 (2008).
[Crossref] [PubMed]

Z. B. Liu, X. Q. Yan, J. G. Tian, W. Y. Zhou, and W. P. Zang. “Nonlinear ellipse rotation modified Z -scan measurements of third-order nonlinear susceptibility tensor,” Opt.Exp. 15, 13351 (2007).
[Crossref]

Zang, W. P.

Zhang, B.

Zhang, C. P.

Zhang, C. Z.

Zhang, G. Y.

Zhao, H.

J. Liang, H. Zhao, and X. Zhou, “Polarization-dependence effects of refractive index change associated with photoisomerization investigated with Z-scan technique,” J. Appl. Phys. 101, 013106 (2007).
[Crossref]

Zhao, W.

W. Zhao and P. Palffy-Muhoray, “Z-scan measurements of ?3 using top-hat beams,” Appl. Phys. Lett. 65,673–675 (1994).
[Crossref]

Zhou, W. Y.

Zhou, X.

J. Liang, H. Zhao, and X. Zhou, “Polarization-dependence effects of refractive index change associated with photoisomerization investigated with Z-scan technique,” J. Appl. Phys. 101, 013106 (2007).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

R.A. Ganeev, A.I. Ryasnyansky, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433 (2004).
[Crossref]

Appl. Phys. Lett. (1)

W. Zhao and P. Palffy-Muhoray, “Z-scan measurements of ?3 using top-hat beams,” Appl. Phys. Lett. 65,673–675 (1994).
[Crossref]

IEEE J. Quantum Electron. (1)

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E. W. Van Stryland and M. Sheik-Bahae, “Z-scan Measurements of Optical Nonlinearities,” in Characterization Techniques and Tabulations for Organic Nonlinear Materials, M. G. Kuzyk and C. W. Dirk, Eds., page 655–692, Marcel Dekker, Inc., 1998.

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http://en.wikipedia.org/wiki/CS2

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Figures (4)

Fig. 1.
Fig. 1.

Experimental arrangements for polarization changeable Z-scan

Fig. 2.
Fig. 2.

Normalized transmittance curves for linearly, circularly and elliptically polarized (φ 1=78deg) incidence beams.

Fig. 3.
Fig. 3.

Normalized transmittance curves for different peak on-axis radiation intensities: (a) 4.26, (b) 7.93, (c) 12.13, (d) 17.59, (e) 20.02, (f) 24.83 GW/cm2 . The polarizer angle φ 1 is 78 deg. Owing to the blue line covers the red dashed line we can not watch red lines in (a) and (b).

Fig. 4.
Fig. 4.

Peak-to-valley transmittance difference (a) and B value used in above fits (b) as a function of peak on-axis radiation intensity.

Equations (33)

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E ( z , r , t ) = E 0 ( t ) w 0 w z exp ( r 2 w z 2 r 2 2 R ( z ) ( z , t ) ) .
E ( z , r , t ) = ( E 0 ( t ) cos φ 1 x ̂ + E 0 ( t ) sin φ 1 e i δ 1 y ̂ ) w 0 w z exp ( r 2 w z 2 i κ r 2 2 R ( z ) ( z , t ) ) ,
E ( z , r , t ) = E + ( z , r , t ) σ ̂ + + E ( z , r , t ) ) σ ̂
= ( E + , 0 ( t ) σ ̂ + + E , 0 ( t ) σ ̂ ) w 0 w z exp ( r 2 w z 2 i π r 2 λR ( z ) ( z , t ) ) ,
{ E + , 0 ( t ) = E 0 ( t ) ( cos φ 1 i e i δ 1 sin φ 1 ) / 2 E , 0 ( t ) = E 0 ( t ) ( cos φ 1 + i e i δ 1 sin φ 1 ) / 2 .
n ± = n 0 + 2 π n 0 [ A E ± 2 + ( A + B ) E 2 ] ,
n ± ( z , r , t ) = n 0 + Δ n ± = n 0 + 2 π n 0 [ A + 1 ± sin δ 1 sin 2 φ 1 2 B ] E ( z , r , t ) 2 .
{ d Δϕ dz' = κ Δ n ( I ) , dl dz' = α ( I ) I
{ Δ ϕ + ( z , r , t ) = Δ ϕ + , 0 ( z , t ) exp ( 2 r 2 w z 2 ) , Δ ϕ ( z , r , t ) = Δ ϕ , 0 ( z , t ) exp ( 2 r 2 w z 2 )
Δ ϕ ± , 0 ( z , t ) = Δ ϕ ± , 0 ( t ) / ( 1 + z 2 / z 0 2 ) .
Δ ϕ ± , 0 ( t ) = κ Δ n ± , 0 ( t ) L eff .
E e ( z , r , t ) = E e , + ( z , r , t ) σ ̂ + + E e , ( z , r , t ) σ ̂
= E + ( z , r , t ) e αL / 2 e i Δ ϕ + ( z , r , t ) σ ̂ + + E ( z , r , t ) e αL / 2 e i Δ ϕ ( z , r , t ) σ ̂ .
e i Δ ϕ ± ( z , r , t ) = m = 0 [ i Δ ϕ ± , 0 ( z , t ) ] m m ! exp ( 2 m r 2 w z 2 ) .
E a ( r , t ) = E a , + ( r , t ) σ ̂ + + E a , ( r , t ) σ ̂
= E + ( z , r = 0 , t ) e αL / 2 m = 0 [ i Δ ϕ + , 0 ( z , t ) ] m m ! w m 0 w m exp ( r 2 w m 2 r 2 2 R m + i θ m ) σ ̂ +
+ E ( z , r = 0 , t ) e αL / 2 m = 0 [ i Δ ϕ , 0 ( z , t ) ] m m ! w m 0 w m exp ( r 2 w m 2 r 2 2 R m + i θ m ) σ ̂
w m 0 2 = w z 1 / ( 2 m + 1 )
d m = k w m 0 2 / 2
R m = d ( 1 g g 2 + d 2 / d m 2 ) 1
θ m = tan 1 ( d / d m g )
w m 2 = w m 0 2 ( g 2 + d 2 / d m 2 )
T ( r a , z , t ) = dt 0 r a E a ( r , t ) 2 rdr S dt 0 E a ( r , t , Δ ϕ ±,0 = 0 ) 2 rdr ,
T = 1 S m = 0 n = 0 1 ( m + n + 1 ) G mn { cos ( c mn π 2 ) e b mn Y a 2 · cos ( c mn π 2 + d mn Y a 2 ) }
G mn = 1 sin δ 1 sin 2 φ 1 2 [ Δ ϕ + , 0 ( z , t ) ] m + n m ! n ! + 1 sin δ 1 sin 2 φ 1 2 [ Δ ϕ , 0 ( z , t ) ] m + n m ! n !
b mn = ( 1 + x 2 ) [ 2 m + 1 x 2 + ( 2 m + 1 ) 2 + 2 n + 1 x 2 + ( 2 n + 1 ) 2 ]
d mn = 4 x ( 1 + x 2 ) ( m n ) ( m + n + 1 ) [ x 2 + ( 2 m + 1 ) 2 ] [ x 2 + ( 2 n + 1 ) 2 ]
c mn = m n
S = 1 exp ( 2 Y a 2 )
T CA = 1 2 m = 0 n = 0 G mn ( m + n + 1 ) { b mn cos ( c mn π 2 ) + d mn sin ( c mn π 2 ) } .
T CA = 1 + 4 x ( x 2 + 1 ) ( x 2 + 9 ) [ 1 sin δ 1 sin 2 φ 1 2 Δ ϕ + , 0 ( t ) + 1 + sin δ 1 sin 2 φ 1 2 Δ ϕ , 0 ( t ) ]
= 1 + 4 x ( x 2 + 1 ) ( x 2 + 9 ) Δ ϕ eff
Δ n ± , 0 ( t ) = Δ n ± , 0 / 2

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