Abstract

Using the nonlinear-imaging technique with a phase object (NIT-PO), we have studied third-order nonlinearities of various samples. In this work, we develop, for pure nonlinear refractive materials, an approximate method to calculate the nonlinear refractive coefficient analytically. By decomposing the object field passing through the phase object into two top-hat beams of different phases and beam radius, we acquire the approximate phase contrast, from which we extract the nonlinear refractive coefficient. This approximation is valid when the on-axis nonlinear phase shift by the sample is less than π. In addition, this approximation serves to estimate the sensitivity and monotonic interval for nonlinearity measurements more easily and thus helps us to maximize both the sensitivity and monotonic interval of measurements. We test this method with CS2, a well-characterized third-order nonlinear refractive material using 21 ps laser pulses at 532nm. We expect this method can be applied to high-order nonlinear refraction cases.

© 2010 Optical Society of America

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References

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  1. G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using a 4f coherent imaging system with phase objects,” Phys. Rev. A 69, 053813-1–053813-6 (2004).
    [CrossRef]
  2. M. Sheik-Bahae, A. A. Said, T. 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]
  3. G. Boudebs and Cid B. de Araijo, “Characterization of light-induced modification of the nonlinear refractive index using a one-laser-shot nonlinear imaging technique,” Appl. Phys. Lett. 85, 3740–3742 (2004).
    [CrossRef]
  4. J.-L. Godet, H. Derbal, S. Cherukulappurath, and G. Boudebs, “Optimization and limits of optical nonlinear measurements using imaging technique,” Eur. Phys. J. D 39, 307–312 (2006).
    [CrossRef]
  5. Y. Li, X. Zhang, Y. Wang, K. Yang, and Y. Song, “Optimization of phase objects in 4f coherent imaging system for nonlinear refraction measurements,” Opt. Commun. 266, 686–690 (2006).
    [CrossRef]
  6. Y. Li, K. Yang, X. Zhang, Q. Chang, Y. Wang, and Y. Song, “The study of the nonlinear absorption in the nonlinear-imaging technique with phase object,” Opt. Commun. 281, 3913–3918 (2008).
    [CrossRef]
  7. D. Rativa, R. E. de Araujo, A. S. L. Gomes, and B. Vohsen, “Hartmann-Shack wavefront sensing for nonlinear materials characterization,” Opt. Express 17, 22047–22053 (2009).
    [CrossRef] [PubMed]
  8. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).
  9. A. Roorda, F. Romero-Borja, W. Donnelly, H. Queener, T. Hebert, and M. Campbell, “Adaptive optics scanning laser ophthalmoscopy,” Opt. Express 10, 405–412 (2002).
    [PubMed]
  10. B. Vohnsen, I. Iglesias, and P. Artal, “Confocal scanning laser ophthalmoscope with adaptive optical wavefront correction,” Proc. SPIE 4964, 24–32 (2003).
    [CrossRef]
  11. J. Bueno, B. Vohnsen, L. Roso, and P. Artal, “Temporal wavefront stability of an ultrafast high-power laser beam,” Appl. Opt. 48, 770–777 (2009).
    [CrossRef] [PubMed]

2009 (2)

2008 (1)

Y. Li, K. Yang, X. Zhang, Q. Chang, Y. Wang, and Y. Song, “The study of the nonlinear absorption in the nonlinear-imaging technique with phase object,” Opt. Commun. 281, 3913–3918 (2008).
[CrossRef]

2006 (2)

J.-L. Godet, H. Derbal, S. Cherukulappurath, and G. Boudebs, “Optimization and limits of optical nonlinear measurements using imaging technique,” Eur. Phys. J. D 39, 307–312 (2006).
[CrossRef]

Y. Li, X. Zhang, Y. Wang, K. Yang, and Y. Song, “Optimization of phase objects in 4f coherent imaging system for nonlinear refraction measurements,” Opt. Commun. 266, 686–690 (2006).
[CrossRef]

2004 (2)

G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using a 4f coherent imaging system with phase objects,” Phys. Rev. A 69, 053813-1–053813-6 (2004).
[CrossRef]

G. Boudebs and Cid B. de Araijo, “Characterization of light-induced modification of the nonlinear refractive index using a one-laser-shot nonlinear imaging technique,” Appl. Phys. Lett. 85, 3740–3742 (2004).
[CrossRef]

2003 (1)

B. Vohnsen, I. Iglesias, and P. Artal, “Confocal scanning laser ophthalmoscope with adaptive optical wavefront correction,” Proc. SPIE 4964, 24–32 (2003).
[CrossRef]

2002 (1)

1992 (1)

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

1990 (1)

M. Sheik-Bahae, A. A. Said, T. 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]

Artal, P.

J. Bueno, B. Vohnsen, L. Roso, and P. Artal, “Temporal wavefront stability of an ultrafast high-power laser beam,” Appl. Opt. 48, 770–777 (2009).
[CrossRef] [PubMed]

B. Vohnsen, I. Iglesias, and P. Artal, “Confocal scanning laser ophthalmoscope with adaptive optical wavefront correction,” Proc. SPIE 4964, 24–32 (2003).
[CrossRef]

Boudebs, G.

J.-L. Godet, H. Derbal, S. Cherukulappurath, and G. Boudebs, “Optimization and limits of optical nonlinear measurements using imaging technique,” Eur. Phys. J. D 39, 307–312 (2006).
[CrossRef]

G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using a 4f coherent imaging system with phase objects,” Phys. Rev. A 69, 053813-1–053813-6 (2004).
[CrossRef]

G. Boudebs and Cid B. de Araijo, “Characterization of light-induced modification of the nonlinear refractive index using a one-laser-shot nonlinear imaging technique,” Appl. Phys. Lett. 85, 3740–3742 (2004).
[CrossRef]

Bueno, J.

Campbell, M.

Chang, Q.

Y. Li, K. Yang, X. Zhang, Q. Chang, Y. Wang, and Y. Song, “The study of the nonlinear absorption in the nonlinear-imaging technique with phase object,” Opt. Commun. 281, 3913–3918 (2008).
[CrossRef]

Cherukulappurath, S.

J.-L. Godet, H. Derbal, S. Cherukulappurath, and G. Boudebs, “Optimization and limits of optical nonlinear measurements using imaging technique,” Eur. Phys. J. D 39, 307–312 (2006).
[CrossRef]

G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using a 4f coherent imaging system with phase objects,” Phys. Rev. A 69, 053813-1–053813-6 (2004).
[CrossRef]

de Araijo, Cid B.

G. Boudebs and Cid B. de Araijo, “Characterization of light-induced modification of the nonlinear refractive index using a one-laser-shot nonlinear imaging technique,” Appl. Phys. Lett. 85, 3740–3742 (2004).
[CrossRef]

de Araujo, R. E.

Derbal, H.

J.-L. Godet, H. Derbal, S. Cherukulappurath, and G. Boudebs, “Optimization and limits of optical nonlinear measurements using imaging technique,” Eur. Phys. J. D 39, 307–312 (2006).
[CrossRef]

Donnelly, W.

Godet, J.-L.

J.-L. Godet, H. Derbal, S. Cherukulappurath, and G. Boudebs, “Optimization and limits of optical nonlinear measurements using imaging technique,” Eur. Phys. J. D 39, 307–312 (2006).
[CrossRef]

Gomes, A. S. L.

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T. 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]

Hebert, T.

Iglesias, I.

B. Vohnsen, I. Iglesias, and P. Artal, “Confocal scanning laser ophthalmoscope with adaptive optical wavefront correction,” Proc. SPIE 4964, 24–32 (2003).
[CrossRef]

Li, Y.

Y. Li, K. Yang, X. Zhang, Q. Chang, Y. Wang, and Y. Song, “The study of the nonlinear absorption in the nonlinear-imaging technique with phase object,” Opt. Commun. 281, 3913–3918 (2008).
[CrossRef]

Y. Li, X. Zhang, Y. Wang, K. Yang, and Y. Song, “Optimization of phase objects in 4f coherent imaging system for nonlinear refraction measurements,” Opt. Commun. 266, 686–690 (2006).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

Queener, H.

Rativa, D.

Romero-Borja, F.

Roorda, A.

Roso, L.

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. 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]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. 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]

Song, Y.

Y. Li, K. Yang, X. Zhang, Q. Chang, Y. Wang, and Y. Song, “The study of the nonlinear absorption in the nonlinear-imaging technique with phase object,” Opt. Commun. 281, 3913–3918 (2008).
[CrossRef]

Y. Li, X. Zhang, Y. Wang, K. Yang, and Y. Song, “Optimization of phase objects in 4f coherent imaging system for nonlinear refraction measurements,” Opt. Commun. 266, 686–690 (2006).
[CrossRef]

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. 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]

Vohnsen, B.

J. Bueno, B. Vohnsen, L. Roso, and P. Artal, “Temporal wavefront stability of an ultrafast high-power laser beam,” Appl. Opt. 48, 770–777 (2009).
[CrossRef] [PubMed]

B. Vohnsen, I. Iglesias, and P. Artal, “Confocal scanning laser ophthalmoscope with adaptive optical wavefront correction,” Proc. SPIE 4964, 24–32 (2003).
[CrossRef]

Vohsen, B.

Wang, Y.

Y. Li, K. Yang, X. Zhang, Q. Chang, Y. Wang, and Y. Song, “The study of the nonlinear absorption in the nonlinear-imaging technique with phase object,” Opt. Commun. 281, 3913–3918 (2008).
[CrossRef]

Y. Li, X. Zhang, Y. Wang, K. Yang, and Y. Song, “Optimization of phase objects in 4f coherent imaging system for nonlinear refraction measurements,” Opt. Commun. 266, 686–690 (2006).
[CrossRef]

Wei, T.

M. Sheik-Bahae, A. A. Said, T. 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]

Yang, K.

Y. Li, K. Yang, X. Zhang, Q. Chang, Y. Wang, and Y. Song, “The study of the nonlinear absorption in the nonlinear-imaging technique with phase object,” Opt. Commun. 281, 3913–3918 (2008).
[CrossRef]

Y. Li, X. Zhang, Y. Wang, K. Yang, and Y. Song, “Optimization of phase objects in 4f coherent imaging system for nonlinear refraction measurements,” Opt. Commun. 266, 686–690 (2006).
[CrossRef]

Zhang, X.

Y. Li, K. Yang, X. Zhang, Q. Chang, Y. Wang, and Y. Song, “The study of the nonlinear absorption in the nonlinear-imaging technique with phase object,” Opt. Commun. 281, 3913–3918 (2008).
[CrossRef]

Y. Li, X. Zhang, Y. Wang, K. Yang, and Y. Song, “Optimization of phase objects in 4f coherent imaging system for nonlinear refraction measurements,” Opt. Commun. 266, 686–690 (2006).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

G. Boudebs and Cid B. de Araijo, “Characterization of light-induced modification of the nonlinear refractive index using a one-laser-shot nonlinear imaging technique,” Appl. Phys. Lett. 85, 3740–3742 (2004).
[CrossRef]

Eur. Phys. J. D (1)

J.-L. Godet, H. Derbal, S. Cherukulappurath, and G. Boudebs, “Optimization and limits of optical nonlinear measurements using imaging technique,” Eur. Phys. J. D 39, 307–312 (2006).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T. 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]

Opt. Commun. (2)

Y. Li, X. Zhang, Y. Wang, K. Yang, and Y. Song, “Optimization of phase objects in 4f coherent imaging system for nonlinear refraction measurements,” Opt. Commun. 266, 686–690 (2006).
[CrossRef]

Y. Li, K. Yang, X. Zhang, Q. Chang, Y. Wang, and Y. Song, “The study of the nonlinear absorption in the nonlinear-imaging technique with phase object,” Opt. Commun. 281, 3913–3918 (2008).
[CrossRef]

Opt. Express (2)

Phys. Rev. A (1)

G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using a 4f coherent imaging system with phase objects,” Phys. Rev. A 69, 053813-1–053813-6 (2004).
[CrossRef]

Proc. SPIE (1)

B. Vohnsen, I. Iglesias, and P. Artal, “Confocal scanning laser ophthalmoscope with adaptive optical wavefront correction,” Proc. SPIE 4964, 24–32 (2003).
[CrossRef]

Other (1)

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

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Figures (9)

Fig. 1
Fig. 1

Arrangement of NIT-PO consists of a 4 f coherent imaging system. BS 1 is the beam splitter; M 1 - M 2 are mirrors; L 1 - L 3 are lenses; A is an aperture with phase object; tf is the neutral filter; NL is the nonlinear sample.

Fig. 2
Fig. 2

Electric field intensity and phase distribution in the Fourier plane, θ = 0.3 .

Fig. 3
Fig. 3

Comparison of amplitude and phase of the field in the image plane for TH1, TH2 and the whole field before and after phase filtering by sample. (a) amplitude of TH1, (b) phase of TH1, (c) amplitude of TH2, (d) phase of TH2, (e) amplitude of the whole field, (f) phase of the whole field.

Fig. 4
Fig. 4

(a) Amplitude and (b) phase of the on-axis field in the image plane of TH1 and TH2.

Fig. 5
Fig. 5

Comparison of results between approximation theory and numerical simulation. (a) Solid curve is the result of approximation theory, the dashed curve is the result of numerical simulation. (b) Error curve.

Fig. 6
Fig. 6

Phase contrast versus different on-axis intensity for C S 2 . Filled squares are experimental results using NIT-PO, the solid curve is linear fit.

Fig. 7
Fig. 7

Oscillation of phase contrast ( | ϕ 0 | < 12 , θ = 0.3 , φ L = 0.4 π ).

Fig. 8
Fig. 8

(a) Optimal phase shift of PO for different θ value. (b) Solid curve, maximum sensitivity of measurements at φ L M ( θ ) ; dashed curve, sensitivity of measurements with φ L = ± π 2 .

Fig. 9
Fig. 9

Monotonic interval of phase contrast for θ = 0.1 . The solid curve is the phase contrast with an optimal positive PO, φ L = 0.5 π ; the dashed curve is the phase contrast with an optimal negative PO, φ L = 0.5 π .

Equations (86)

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E i ( r ) = E 0 circ ( r R a ) + E 0 [ exp ( i φ L ) 1 ] circ ( r L p ) .
E f ( ξ ) = 1 λ f 1 B { E i ( r ) } = 2 E f 0 { J 1 ( ξ ) ξ + [ exp ( i φ L ) 1 ] θ J 1 ( θ ξ ) ξ } ,
I f ( ξ ) = | E f ( ξ ) | 2 = 4 I f 0 { ( J 1 ( ξ ) ξ ) 2 + 4 sin 2 ( φ L 2 ) [ ( θ J 1 ( θ ξ ) ξ ) 2 θ J 1 ( θ ξ ) ξ J 1 ( ξ ) ξ ] } ,
Δ n = n 2 I .
ϕ NL ( ξ ) = k n 2 I ( ξ ) L = ϕ 0 P f ( ξ ) ,
P f ( ξ ) = I ( ξ ) I f 0 = 4 { ( J 1 ( ξ ) ξ ) 2 + 4 sin 2 ( φ L 2 ) [ ( θ J 1 ( θ ξ ) ξ ) 2 θ J 1 ( θ ξ ) ξ J 1 ( ξ ) ξ ] } ,
E im ( r ) = E im 1 ( r ) + E im 2 ( r ) ,
E im 1 ( r ) = E 0 0 J 1 ( ξ ) exp [ i ϕ 0 P f ( ξ ) ] J 0 ( r ξ R a ) d ξ ,
E im 2 ( r ) = [ exp ( i φ L ) 1 ] E 0 0 θ J 1 ( θ ξ ) exp [ i ϕ 0 P f ( ξ ) ] J 0 ( r ξ R a ) d ξ ,
E im 1 ( 0 ) = E 0 ( 1 + 1 2 ! a 2 ϕ 0 2 ) exp [ i ( δ 11 ϕ 0 + 1 3 ! δ 13 ϕ 0 3 ) ] ,
E im 2 ( 0 ) = [ exp ( i φ L ) 1 ] E 0 ( 1 + 1 2 ! b 2 ϕ 0 2 ) exp [ i ( δ 21 ϕ 0 + 1 3 ! δ 23 ϕ 0 3 ) ] .
a 2 = I 12 I 11 2 ,
b 2 = I 22 I 21 2 ,
δ 11 = i I 11 ,
δ 21 = i I 21 ,
δ 13 = i [ I 13 3 I 12 I 11 + 2 I 11 3 ] ,
δ 23 = i [ I 23 3 I 22 I 21 + 2 I 21 3 ] ,
I 1 k = | 1 E 0 k E im 1 ( 0 ) ϕ 0 k | ϕ 0 = 0 = 0 J 1 ( ξ ) ( i P f ( ξ ) ) k d ξ ,
I 2 k = 1 [ exp ( i φ L ) 1 ] E 0 | k E im 2 ( 0 ) ϕ 0 k | ϕ 0 = 0 = 0 θ J 1 ( θ ξ ) ( i P f ( ξ ) ) k d ξ .
T ( 0 ) = I im ( 0 ) I 0 = { A 1 2 + 4 A 2 2 sin 2 ( φ L 2 ) + 4 A 1 A 2 sin ( φ L 2 ) sin ( δ 1 ϕ 0 + 1 3 ! δ 3 ϕ 0 3 φ L 2 ) } ,
Δ T 1 I 0 [ I ( r ) r [ 0 , L p ] I ( r ) r [ L p , R a ] ] .
π L p 2 T p + π ( R a 2 L p 2 ) T o = π R a 2 T in ,
T in = 1 + 1 2 ! c 2 ϕ 0 2 + 1 4 ! c 4 ϕ 0 4 .
Δ T = 1 1 θ 2 [ 4 sin ( φ L 2 ) A 1 A 2 sin ( δ 1 ϕ 0 + 1 3 ! δ 3 ϕ 0 3 φ L 2 ) + A 1 2 + 4 sin 2 ( φ L 2 ) A 2 2 T in ] .
Δ T ( ϕ 0 , φ L ) = Δ T ( ϕ 0 , φ L ) .
Δ T = 2 sin ( φ L ) δ 1 1 θ 2 ϕ 0 = 2 sin ( φ L ) δ 1 1 θ 2 k L Δ n 0 ,
Δ n 0 ( t ) = Δ n 0 ( t ) I 0 ( t ) d t I 0 ( t ) d t = Δ n 0 2 .
Δ T = 2 sin ( φ L ) δ 1 1 θ 2 k L n 2 I f 0 .
Θ = d Δ T d ϕ 0 = 2 sin ( φ L ) δ 1 1 θ 2 .
Θ = 2 sin ( φ L ) δ 0 sin ( φ L ) ,
cos ( φ L ) [ I 11 ( θ , φ L ) I 21 ( θ , φ L ) ] + sin ( φ L ) [ I 11 ( θ , φ L ) I 21 ( θ , φ L ) ] φ L = 0 .
Δ T ϕ 0 = 0 and ϕ 0 + > ϕ 0 .
Δ T ϕ 0 = 4 sin ( φ L 2 ) 1 θ 2 ( δ 1 + 1 2 δ 3 ϕ 0 2 ) cos ( δ 1 ϕ 0 + 1 3 ! δ 3 ϕ 0 3 φ L 2 ) .
δ 1 ϕ 0 + δ 3 3 ! ϕ 0 3 φ L 2 = ( m + 1 2 ) π ,
δ 1 ϕ 0 + + δ 3 3 ! ϕ 0 + 3 φ L 2 = 1 2 π ,
δ 1 ϕ 0 + δ 3 3 ! ϕ 0 3 φ L 2 = 1 2 π .
ϕ 0 + = φ L + π 2 δ 1 ,
ϕ 0 = φ L π 2 δ 1 .
Δ n = n 4 I 2 .
Δ T n 4 I f 0 2 .
E i ( r ) = E 0 circ ( r R a ) + E 0 [ exp ( i φ L ) 1 ] circ ( r L p ) .
E f ( ξ ) = 1 λ f 1 B { E i ( r ) } = 2 E f 0 { J 1 ( ξ ) ξ + [ exp ( i φ L ) 1 ] θ J 1 ( θ ξ ) ξ } ,
I f ( ξ ) = | E f ( ξ ) | 2 = 4 I f 0 { ( J 1 ( ξ ) ξ ) 2 + 4 sin 2 ( φ L 2 ) [ ( θ J 1 ( θ ξ ) ξ ) 2 θ J 1 ( θ ξ ) ξ J 1 ( ξ ) ξ ] } ,
P f ( ξ ) = I f ( ξ ) I f 0 = 4 { ( J 1 ( ξ ) ξ ) 2 + 4 sin 2 ( φ L 2 ) [ ( θ J 1 ( θ ξ ) ξ ) 2 θ J 1 ( θ ξ ) ξ J 1 ( ξ ) ξ ] } .
E im 1 ( r ) = E 0 0 J 1 ( ξ ) exp [ i ϕ 0 P f ( ξ ) ] J 0 ( r ξ R a ) d ξ ,
E im 2 ( r ) = [ exp ( i φ L ) 1 ] E 0 0 θ J 1 ( θ ξ ) exp [ i ϕ 0 P f ( ξ ) ] J 0 ( r ξ R a ) d ξ .
E im 1 ( 0 ) = E 0 0 J 1 ( ξ ) exp [ i ϕ 0 P f ( ξ ) ] d ξ ,
E im 2 ( 0 ) = [ exp ( i φ L ) 1 ] E 0 0 θ J 1 ( θ ξ ) exp [ i ϕ 0 P f ( ξ ) ] d ξ .
E im 1 ( 0 ) = E 0 ( 1 + 1 2 ! a 2 ϕ 0 2 ) exp [ i ( δ 11 ϕ 0 + 1 3 ! δ 13 ϕ 0 3 ) ] ,
E im 2 ( 0 ) = [ exp ( i φ L ) 1 ] E 0 ( 1 + 1 2 ! b 2 ϕ 0 2 ) exp [ i ( δ 21 ϕ 0 + 1 3 ! δ 23 ϕ 0 3 ) ] .
E ( ϕ 0 ) = 0 E f ( ξ ) exp [ i ϕ 0 P f ( ξ ) ] d ξ ,
E ( ϕ 0 ) = A exp ( i ϕ ) ,
E f ( ξ ) = { J 1 ( ξ ) for TH 1 θ J 1 ( θ ξ ) for TH 2 } ,
E ( ϕ 0 ) = { E im 1 ( 0 ) E 0 for TH 1 E im 2 ( 0 ) [ exp ( i φ L ) 1 ] E 0 for TH 2 } ,
E ( ϕ 0 ) = ( A + i A ϕ ) e i ϕ i ϕ ( 0 ) = i δ 1 ,
E ( ϕ 0 ) = ( A + 2 i A ϕ + i A ϕ A ϕ ) e i ϕ A ( 0 ) ( ϕ ( 0 ) ) 2 = a 2 δ 1 2 ,
E ( ϕ 0 ) = ( A + 3 i A ϕ + 3 i A ϕ + i A ϕ 3 A ( ϕ ) 2 3 A ϕ ϕ i A ϕ ) e i ϕ 3 i A ( 0 ) ϕ ( 0 ) + i ϕ ( 0 ) i ( ϕ ( 0 ) ) 3 = 3 i a 2 δ 1 + i δ 3 i δ 1 3 ,
A ( 0 ) = ϕ ( 0 ) = 0 ,
A ( 0 ) = 1 , A ( 0 ) = a 2 ,
ϕ ( 0 ) = δ 1 , ϕ ( 0 ) = δ 3 .
E ( k ) ( ϕ 0 ) = 0 E f ( ξ ) ( i P f ( ξ ) ) k exp [ i ϕ 0 P f ( ξ ) ] d ξ 0 E f ( ξ ) ( i P f ( ξ ) ) k d ξ .
I k 0 E f ( ξ ) ( i P f ( ξ ) ) k d ξ .
I 1 = i δ 1 ,
I 2 = a 2 δ 1 2 ,
I 3 = 3 i a 2 δ 1 + i δ 3 i δ 1 3 .
a 2 = I 2 I 1 2 ,
δ 1 = i I 1 ,
δ 3 = i [ I 3 3 I 2 I 1 + 2 I 1 3 ] .
P in = 2 π 0 R a | E im 1 ( r ) | 2 r d r .
T in = P in π R a 2 = 2 R a 2 0 R a | E im 1 ( r ) | 2 r d r .
T in 1 + 1 2 ! 2 T in ( ϕ 0 = 0 ) ϕ 0 2 ϕ 0 2 + 1 4 ! 4 T in ( ϕ 0 = 0 ) ϕ 0 4 ϕ 0 4 = 1 + 1 2 ! c 2 ϕ 0 2 + 1 4 ! c 4 ϕ 0 4 ,
k T in ϕ 0 k = k ϕ 0 k { 2 R a 2 0 R a | E im 1 ( r ) | 2 r d r } = 2 R a 2 0 R a ( k ϕ 0 k | E im 1 ( r ) | 2 ) r d r ,
k ϕ 0 k | E im 1 ( r ) | 2 = k ϕ 0 k { ( Re { E im 1 ( r ) } ) 2 + ( Im { E im 1 ( r ) } ) 2 } .
E Re ( r ) = Re { E im 1 ( r ) } = 0 J 1 ( ξ ) cos [ ϕ 0 P ( ξ ) ] J 0 ( r ξ R a ) d ξ ,
E Im ( r ) = Im { E im 1 ( r ) } = 0 J 1 ( ξ ) sin [ ϕ 0 P ( ξ ) ] J 0 ( r ξ R a ) d ξ .
2 E Re ( r ) ϕ 0 2 = 2 ( E Re ( r ) ) 2 + 2 E Re ( r ) E Re ( r ) 2 I ̃ 12 ( r ) ,
2 E Im ( r ) ϕ 0 2 = 2 ( E Im ( r ) ) 2 + 2 E Im ( r ) E Im ( r ) 2 ( I ̃ 11 ( r ) ) 2 ,
k ϕ 0 k | E im 1 ( r ) | 2 = k ϕ 0 k { ( E Re ( r ) ) 2 + ( E Im ( r ) ) 2 } 2 ( I ̃ 11 ( r ) ) 2 2 I ̃ 12 ( r ) .
E Re ( r ) ϕ 0 = 0 J 1 ( ξ ) P f ( ξ ) sin [ ϕ 0 P f ( ξ ) ] J 0 ( r ξ R a ) d ξ 0 ,
2 E Re ( r ) ϕ 0 2 = 0 J 1 ( ξ ) P f 2 ( ξ ) cos [ ϕ 0 P ( ξ ) ] J 0 ( r ξ R a ) d ξ 0 J 1 ( ξ ) P f 2 ( ξ ) J 0 ( r ξ R a ) d ξ = I ̃ 12 ( r ) ;
E Im ( r ) ϕ 0 = 0 J 1 ( ξ ) P f ( ξ ) cos [ ϕ 0 P f ( ξ ) ] J 0 ( r ξ R a ) d ξ 0 J 1 ( ξ ) P f ( ξ ) J 0 ( r ξ R a ) d ξ = I ̃ 11 ( r ) ,
2 E Im ( r ) ϕ 0 2 = 0 J 1 ( ξ ) P f 2 ( ξ ) sin [ ϕ 0 P ( ξ ) ] J 0 ( r ξ R a ) d ξ 0 ;
4 | E im 1 ( r ) | 2 ϕ 0 4 = 6 ( I ̃ 12 ( r ) ) 2 + 2 I ̃ 14 ( r ) 8 I ̃ 11 ( r ) I ̃ 13 ( r ) .
T in 1 + 1 2 ! c 2 ϕ 0 2 + 1 4 ! c 4 ϕ 0 4 ,
c 2 = 2 0 1 { 2 [ I ̃ 11 ( η ) ] 2 2 I ̃ 12 ( η ) } η d η ,
c 4 = 2 0 1 { 6 [ I ̃ 12 ( η ) ] 2 + 2 I ̃ 14 ( η ) 8 I ̃ 11 ( η ) I ̃ 13 ( η ) } η d η ,

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