Abstract

In this paper we are introducing a new single shot method for measuring the nonlinear refractive index of materials in a simple interferometric pump-probe configuration. The theoretical model proposed for extracting the nonlinear refractive index from the experimental fringe pattern and the experimental configuration are presented and discussed. The results obtained by this method are in good agreement with that obtained on the same sample using the conventional Z-scan method.

© 2013 Optical Society of America

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  1. R. Boyd, Nonlinear Optics (Academic Press, 2008).
  2. R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker Inc., 2003).
  3. H. Eichler, P. Günter, and D. Pohl, Laser-Induced Dynamic Gratings (Springer, 1986).
  4. M. Sheik-Bahae, A. A. Said, T. H. Wei, Y. Y. Wu, D. J. Hagan, M. J. Soileau, and E. W. van Stryland, “Z-scan: a simple and sensitive technique for nonlinear-refraction measurements,” Proc. SPIE1148, 41–51 (1990).
    [CrossRef]
  5. M. R. RashidianVaziri, F. Hajiesmaeilbaigi, and M. H. Maleki, “Generalizing the Z-scan theory for nonlocal nonlinear media,” J. Opt.15(2), 025201 (2013).
    [CrossRef]
  6. Q. G. Yang, J. T. Seo, S. J. Creekmore, D. A. Temple, K. P. Yoo, S. Y. Kim, S. S. Jung, and A. Mott, “I-scan measurements of the nonlinear refraction and nonlinear absorption coefficients of some nanomaterials,” Proc. SPIE4797, 101–109 (2003).
    [CrossRef]
  7. D. V. Petrov, “Reflection Z-scan technique for the study of nonlinear refraction and absorption of a single interface and thin film,” J. Opt. Soc. Am. B13(7), 1491–1498 (1996).
    [CrossRef]
  8. R. A. Ganeev, A. I. Ryasnyanskii, A. L. Stepanov, T. Usmanov, C. Marques, R. C. da Silva, and E. Alves, “Investigation of the nonlinear optical characteristics of composite materials based on sapphire with silver, copper, and gold nanoparticles by the reflection Z-scan method,” Opt. Spectrosc.101(4), 615–622 (2006).
    [CrossRef]
  9. A. Petris, F. Pettazzi, E. Fazio, C. Peroz, Y. Chen, V. I. Vlad, and M. Bertolotti, “Strongly enhanced third order nonlinear response of periodically nano-structured silicon-on-insulator (SOI) measured by reflection Z-scan with femtosecond pulses,” J. Optoelectron. Adv. Mater.8, 1377–1380 (2006).
  10. I. Dancus, V. I. Vlad, A. Petris, N. Gaponik, V. Lesnyak, and A. Eychmüller, “Saturated near-resonant refractive optical nonlinearity in CdTe quantum dots,” Opt. Lett.35(7), 1079–1081 (2010).
    [CrossRef] [PubMed]
  11. E. L. Falcão-Filho, C. B. de Araújo, and J. J. Rodrigues, “High-order nonlinearities of aqueous colloids containing silver nanoparticles,” J. Opt. Soc. Am. B24(12), 2948–2956 (2007).
    [CrossRef]
  12. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am.72(1), 156–160 (1982).
    [CrossRef]
  13. V. I. Vlad and D. Malacara, “Direct spatial reconstruction of optical-phase from phase-modulated images,” Prog. Opt.33, 261–317 (1994).
  14. W. W. Macy., “Two-dimensional fringe-pattern analysis,” Appl. Opt.22(23), 3898–3901 (1983).
    [CrossRef] [PubMed]
  15. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A14(10), 2692–2701 (1997).
    [CrossRef]
  16. B. Zhao and A. Asundi, “Criteria for phase reconstruction using Fourier transformation method,” Proc. SPIE4123, 269–278 (2000).
    [CrossRef]
  17. M. Takeda, “Fourier fringe analysis and its application to metrology of extreme physical phenomena: a review [Invited],” Appl. Opt.52(1), 20–29 (2013).
    [CrossRef] [PubMed]
  18. N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun.270(2), 161–168 (2007).
    [CrossRef]

2013

M. R. RashidianVaziri, F. Hajiesmaeilbaigi, and M. H. Maleki, “Generalizing the Z-scan theory for nonlocal nonlinear media,” J. Opt.15(2), 025201 (2013).
[CrossRef]

M. Takeda, “Fourier fringe analysis and its application to metrology of extreme physical phenomena: a review [Invited],” Appl. Opt.52(1), 20–29 (2013).
[CrossRef] [PubMed]

2010

2007

E. L. Falcão-Filho, C. B. de Araújo, and J. J. Rodrigues, “High-order nonlinearities of aqueous colloids containing silver nanoparticles,” J. Opt. Soc. Am. B24(12), 2948–2956 (2007).
[CrossRef]

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun.270(2), 161–168 (2007).
[CrossRef]

2006

R. A. Ganeev, A. I. Ryasnyanskii, A. L. Stepanov, T. Usmanov, C. Marques, R. C. da Silva, and E. Alves, “Investigation of the nonlinear optical characteristics of composite materials based on sapphire with silver, copper, and gold nanoparticles by the reflection Z-scan method,” Opt. Spectrosc.101(4), 615–622 (2006).
[CrossRef]

A. Petris, F. Pettazzi, E. Fazio, C. Peroz, Y. Chen, V. I. Vlad, and M. Bertolotti, “Strongly enhanced third order nonlinear response of periodically nano-structured silicon-on-insulator (SOI) measured by reflection Z-scan with femtosecond pulses,” J. Optoelectron. Adv. Mater.8, 1377–1380 (2006).

2003

Q. G. Yang, J. T. Seo, S. J. Creekmore, D. A. Temple, K. P. Yoo, S. Y. Kim, S. S. Jung, and A. Mott, “I-scan measurements of the nonlinear refraction and nonlinear absorption coefficients of some nanomaterials,” Proc. SPIE4797, 101–109 (2003).
[CrossRef]

2000

B. Zhao and A. Asundi, “Criteria for phase reconstruction using Fourier transformation method,” Proc. SPIE4123, 269–278 (2000).
[CrossRef]

1997

1996

1994

V. I. Vlad and D. Malacara, “Direct spatial reconstruction of optical-phase from phase-modulated images,” Prog. Opt.33, 261–317 (1994).

1990

M. Sheik-Bahae, A. A. Said, T. H. Wei, Y. Y. Wu, D. J. Hagan, M. J. Soileau, and E. W. van Stryland, “Z-scan: a simple and sensitive technique for nonlinear-refraction measurements,” Proc. SPIE1148, 41–51 (1990).
[CrossRef]

1983

1982

Alves, E.

R. A. Ganeev, A. I. Ryasnyanskii, A. L. Stepanov, T. Usmanov, C. Marques, R. C. da Silva, and E. Alves, “Investigation of the nonlinear optical characteristics of composite materials based on sapphire with silver, copper, and gold nanoparticles by the reflection Z-scan method,” Opt. Spectrosc.101(4), 615–622 (2006).
[CrossRef]

Asundi, A.

B. Zhao and A. Asundi, “Criteria for phase reconstruction using Fourier transformation method,” Proc. SPIE4123, 269–278 (2000).
[CrossRef]

Bertolotti, M.

A. Petris, F. Pettazzi, E. Fazio, C. Peroz, Y. Chen, V. I. Vlad, and M. Bertolotti, “Strongly enhanced third order nonlinear response of periodically nano-structured silicon-on-insulator (SOI) measured by reflection Z-scan with femtosecond pulses,” J. Optoelectron. Adv. Mater.8, 1377–1380 (2006).

Chen, Y.

A. Petris, F. Pettazzi, E. Fazio, C. Peroz, Y. Chen, V. I. Vlad, and M. Bertolotti, “Strongly enhanced third order nonlinear response of periodically nano-structured silicon-on-insulator (SOI) measured by reflection Z-scan with femtosecond pulses,” J. Optoelectron. Adv. Mater.8, 1377–1380 (2006).

Creekmore, S. J.

Q. G. Yang, J. T. Seo, S. J. Creekmore, D. A. Temple, K. P. Yoo, S. Y. Kim, S. S. Jung, and A. Mott, “I-scan measurements of the nonlinear refraction and nonlinear absorption coefficients of some nanomaterials,” Proc. SPIE4797, 101–109 (2003).
[CrossRef]

da Silva, R. C.

R. A. Ganeev, A. I. Ryasnyanskii, A. L. Stepanov, T. Usmanov, C. Marques, R. C. da Silva, and E. Alves, “Investigation of the nonlinear optical characteristics of composite materials based on sapphire with silver, copper, and gold nanoparticles by the reflection Z-scan method,” Opt. Spectrosc.101(4), 615–622 (2006).
[CrossRef]

Dancus, I.

de Araújo, C. B.

Eychmüller, A.

Falcão-Filho, E. L.

Fazio, E.

A. Petris, F. Pettazzi, E. Fazio, C. Peroz, Y. Chen, V. I. Vlad, and M. Bertolotti, “Strongly enhanced third order nonlinear response of periodically nano-structured silicon-on-insulator (SOI) measured by reflection Z-scan with femtosecond pulses,” J. Optoelectron. Adv. Mater.8, 1377–1380 (2006).

Flynn, T. J.

Ganeev, R. A.

R. A. Ganeev, A. I. Ryasnyanskii, A. L. Stepanov, T. Usmanov, C. Marques, R. C. da Silva, and E. Alves, “Investigation of the nonlinear optical characteristics of composite materials based on sapphire with silver, copper, and gold nanoparticles by the reflection Z-scan method,” Opt. Spectrosc.101(4), 615–622 (2006).
[CrossRef]

Gaponik, N.

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T. H. Wei, Y. Y. Wu, D. J. Hagan, M. J. Soileau, and E. W. van Stryland, “Z-scan: a simple and sensitive technique for nonlinear-refraction measurements,” Proc. SPIE1148, 41–51 (1990).
[CrossRef]

Hajiesmaeilbaigi, F.

M. R. RashidianVaziri, F. Hajiesmaeilbaigi, and M. H. Maleki, “Generalizing the Z-scan theory for nonlocal nonlinear media,” J. Opt.15(2), 025201 (2013).
[CrossRef]

Ina, H.

Jung, S. S.

Q. G. Yang, J. T. Seo, S. J. Creekmore, D. A. Temple, K. P. Yoo, S. Y. Kim, S. S. Jung, and A. Mott, “I-scan measurements of the nonlinear refraction and nonlinear absorption coefficients of some nanomaterials,” Proc. SPIE4797, 101–109 (2003).
[CrossRef]

Kim, S. Y.

Q. G. Yang, J. T. Seo, S. J. Creekmore, D. A. Temple, K. P. Yoo, S. Y. Kim, S. S. Jung, and A. Mott, “I-scan measurements of the nonlinear refraction and nonlinear absorption coefficients of some nanomaterials,” Proc. SPIE4797, 101–109 (2003).
[CrossRef]

Kobayashi, S.

Lesnyak, V.

Macy, W. W.

Malacara, D.

V. I. Vlad and D. Malacara, “Direct spatial reconstruction of optical-phase from phase-modulated images,” Prog. Opt.33, 261–317 (1994).

Maleki, M. H.

M. R. RashidianVaziri, F. Hajiesmaeilbaigi, and M. H. Maleki, “Generalizing the Z-scan theory for nonlocal nonlinear media,” J. Opt.15(2), 025201 (2013).
[CrossRef]

Marques, C.

R. A. Ganeev, A. I. Ryasnyanskii, A. L. Stepanov, T. Usmanov, C. Marques, R. C. da Silva, and E. Alves, “Investigation of the nonlinear optical characteristics of composite materials based on sapphire with silver, copper, and gold nanoparticles by the reflection Z-scan method,” Opt. Spectrosc.101(4), 615–622 (2006).
[CrossRef]

Mott, A.

Q. G. Yang, J. T. Seo, S. J. Creekmore, D. A. Temple, K. P. Yoo, S. Y. Kim, S. S. Jung, and A. Mott, “I-scan measurements of the nonlinear refraction and nonlinear absorption coefficients of some nanomaterials,” Proc. SPIE4797, 101–109 (2003).
[CrossRef]

Ochoa, N. A.

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun.270(2), 161–168 (2007).
[CrossRef]

Peroz, C.

A. Petris, F. Pettazzi, E. Fazio, C. Peroz, Y. Chen, V. I. Vlad, and M. Bertolotti, “Strongly enhanced third order nonlinear response of periodically nano-structured silicon-on-insulator (SOI) measured by reflection Z-scan with femtosecond pulses,” J. Optoelectron. Adv. Mater.8, 1377–1380 (2006).

Petris, A.

I. Dancus, V. I. Vlad, A. Petris, N. Gaponik, V. Lesnyak, and A. Eychmüller, “Saturated near-resonant refractive optical nonlinearity in CdTe quantum dots,” Opt. Lett.35(7), 1079–1081 (2010).
[CrossRef] [PubMed]

A. Petris, F. Pettazzi, E. Fazio, C. Peroz, Y. Chen, V. I. Vlad, and M. Bertolotti, “Strongly enhanced third order nonlinear response of periodically nano-structured silicon-on-insulator (SOI) measured by reflection Z-scan with femtosecond pulses,” J. Optoelectron. Adv. Mater.8, 1377–1380 (2006).

Petrov, D. V.

Pettazzi, F.

A. Petris, F. Pettazzi, E. Fazio, C. Peroz, Y. Chen, V. I. Vlad, and M. Bertolotti, “Strongly enhanced third order nonlinear response of periodically nano-structured silicon-on-insulator (SOI) measured by reflection Z-scan with femtosecond pulses,” J. Optoelectron. Adv. Mater.8, 1377–1380 (2006).

RashidianVaziri, M. R.

M. R. RashidianVaziri, F. Hajiesmaeilbaigi, and M. H. Maleki, “Generalizing the Z-scan theory for nonlocal nonlinear media,” J. Opt.15(2), 025201 (2013).
[CrossRef]

Rodrigues, J. J.

Ryasnyanskii, A. I.

R. A. Ganeev, A. I. Ryasnyanskii, A. L. Stepanov, T. Usmanov, C. Marques, R. C. da Silva, and E. Alves, “Investigation of the nonlinear optical characteristics of composite materials based on sapphire with silver, copper, and gold nanoparticles by the reflection Z-scan method,” Opt. Spectrosc.101(4), 615–622 (2006).
[CrossRef]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Wei, Y. Y. Wu, D. J. Hagan, M. J. Soileau, and E. W. van Stryland, “Z-scan: a simple and sensitive technique for nonlinear-refraction measurements,” Proc. SPIE1148, 41–51 (1990).
[CrossRef]

Seo, J. T.

Q. G. Yang, J. T. Seo, S. J. Creekmore, D. A. Temple, K. P. Yoo, S. Y. Kim, S. S. Jung, and A. Mott, “I-scan measurements of the nonlinear refraction and nonlinear absorption coefficients of some nanomaterials,” Proc. SPIE4797, 101–109 (2003).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. H. Wei, Y. Y. Wu, D. J. Hagan, M. J. Soileau, and E. W. van Stryland, “Z-scan: a simple and sensitive technique for nonlinear-refraction measurements,” Proc. SPIE1148, 41–51 (1990).
[CrossRef]

Silva-Moreno, A. A.

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun.270(2), 161–168 (2007).
[CrossRef]

Soileau, M. J.

M. Sheik-Bahae, A. A. Said, T. H. Wei, Y. Y. Wu, D. J. Hagan, M. J. Soileau, and E. W. van Stryland, “Z-scan: a simple and sensitive technique for nonlinear-refraction measurements,” Proc. SPIE1148, 41–51 (1990).
[CrossRef]

Stepanov, A. L.

R. A. Ganeev, A. I. Ryasnyanskii, A. L. Stepanov, T. Usmanov, C. Marques, R. C. da Silva, and E. Alves, “Investigation of the nonlinear optical characteristics of composite materials based on sapphire with silver, copper, and gold nanoparticles by the reflection Z-scan method,” Opt. Spectrosc.101(4), 615–622 (2006).
[CrossRef]

Takeda, M.

Temple, D. A.

Q. G. Yang, J. T. Seo, S. J. Creekmore, D. A. Temple, K. P. Yoo, S. Y. Kim, S. S. Jung, and A. Mott, “I-scan measurements of the nonlinear refraction and nonlinear absorption coefficients of some nanomaterials,” Proc. SPIE4797, 101–109 (2003).
[CrossRef]

Usmanov, T.

R. A. Ganeev, A. I. Ryasnyanskii, A. L. Stepanov, T. Usmanov, C. Marques, R. C. da Silva, and E. Alves, “Investigation of the nonlinear optical characteristics of composite materials based on sapphire with silver, copper, and gold nanoparticles by the reflection Z-scan method,” Opt. Spectrosc.101(4), 615–622 (2006).
[CrossRef]

van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. H. Wei, Y. Y. Wu, D. J. Hagan, M. J. Soileau, and E. W. van Stryland, “Z-scan: a simple and sensitive technique for nonlinear-refraction measurements,” Proc. SPIE1148, 41–51 (1990).
[CrossRef]

Vlad, V. I.

I. Dancus, V. I. Vlad, A. Petris, N. Gaponik, V. Lesnyak, and A. Eychmüller, “Saturated near-resonant refractive optical nonlinearity in CdTe quantum dots,” Opt. Lett.35(7), 1079–1081 (2010).
[CrossRef] [PubMed]

A. Petris, F. Pettazzi, E. Fazio, C. Peroz, Y. Chen, V. I. Vlad, and M. Bertolotti, “Strongly enhanced third order nonlinear response of periodically nano-structured silicon-on-insulator (SOI) measured by reflection Z-scan with femtosecond pulses,” J. Optoelectron. Adv. Mater.8, 1377–1380 (2006).

V. I. Vlad and D. Malacara, “Direct spatial reconstruction of optical-phase from phase-modulated images,” Prog. Opt.33, 261–317 (1994).

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, Y. Y. Wu, D. J. Hagan, M. J. Soileau, and E. W. van Stryland, “Z-scan: a simple and sensitive technique for nonlinear-refraction measurements,” Proc. SPIE1148, 41–51 (1990).
[CrossRef]

Wu, Y. Y.

M. Sheik-Bahae, A. A. Said, T. H. Wei, Y. Y. Wu, D. J. Hagan, M. J. Soileau, and E. W. van Stryland, “Z-scan: a simple and sensitive technique for nonlinear-refraction measurements,” Proc. SPIE1148, 41–51 (1990).
[CrossRef]

Yang, Q. G.

Q. G. Yang, J. T. Seo, S. J. Creekmore, D. A. Temple, K. P. Yoo, S. Y. Kim, S. S. Jung, and A. Mott, “I-scan measurements of the nonlinear refraction and nonlinear absorption coefficients of some nanomaterials,” Proc. SPIE4797, 101–109 (2003).
[CrossRef]

Yoo, K. P.

Q. G. Yang, J. T. Seo, S. J. Creekmore, D. A. Temple, K. P. Yoo, S. Y. Kim, S. S. Jung, and A. Mott, “I-scan measurements of the nonlinear refraction and nonlinear absorption coefficients of some nanomaterials,” Proc. SPIE4797, 101–109 (2003).
[CrossRef]

Zhao, B.

B. Zhao and A. Asundi, “Criteria for phase reconstruction using Fourier transformation method,” Proc. SPIE4123, 269–278 (2000).
[CrossRef]

Appl. Opt.

J. Opt.

M. R. RashidianVaziri, F. Hajiesmaeilbaigi, and M. H. Maleki, “Generalizing the Z-scan theory for nonlocal nonlinear media,” J. Opt.15(2), 025201 (2013).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Optoelectron. Adv. Mater.

A. Petris, F. Pettazzi, E. Fazio, C. Peroz, Y. Chen, V. I. Vlad, and M. Bertolotti, “Strongly enhanced third order nonlinear response of periodically nano-structured silicon-on-insulator (SOI) measured by reflection Z-scan with femtosecond pulses,” J. Optoelectron. Adv. Mater.8, 1377–1380 (2006).

Opt. Commun.

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun.270(2), 161–168 (2007).
[CrossRef]

Opt. Lett.

Opt. Spectrosc.

R. A. Ganeev, A. I. Ryasnyanskii, A. L. Stepanov, T. Usmanov, C. Marques, R. C. da Silva, and E. Alves, “Investigation of the nonlinear optical characteristics of composite materials based on sapphire with silver, copper, and gold nanoparticles by the reflection Z-scan method,” Opt. Spectrosc.101(4), 615–622 (2006).
[CrossRef]

Proc. SPIE

M. Sheik-Bahae, A. A. Said, T. H. Wei, Y. Y. Wu, D. J. Hagan, M. J. Soileau, and E. W. van Stryland, “Z-scan: a simple and sensitive technique for nonlinear-refraction measurements,” Proc. SPIE1148, 41–51 (1990).
[CrossRef]

Q. G. Yang, J. T. Seo, S. J. Creekmore, D. A. Temple, K. P. Yoo, S. Y. Kim, S. S. Jung, and A. Mott, “I-scan measurements of the nonlinear refraction and nonlinear absorption coefficients of some nanomaterials,” Proc. SPIE4797, 101–109 (2003).
[CrossRef]

B. Zhao and A. Asundi, “Criteria for phase reconstruction using Fourier transformation method,” Proc. SPIE4123, 269–278 (2000).
[CrossRef]

Prog. Opt.

V. I. Vlad and D. Malacara, “Direct spatial reconstruction of optical-phase from phase-modulated images,” Prog. Opt.33, 261–317 (1994).

Other

R. Boyd, Nonlinear Optics (Academic Press, 2008).

R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker Inc., 2003).

H. Eichler, P. Günter, and D. Pohl, Laser-Induced Dynamic Gratings (Springer, 1986).

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Figures (2)

Fig. 1
Fig. 1

The experimental setup based on a Michelson interferometer (a) and an experimental fringe pattern obtained when the investigated sample is excited (b).

Fig. 2
Fig. 2

(a) 3D representation of the reconstructed phase (dots) fitted with Eq. (10) (surface). (b) The top view of the fitting surface. Inset: the reconstructed phase change corresponding to the tilted Gaussian excitation only.

Equations (19)

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

I( x,y,z=0 )= I 0 e ( ( ( x x 0 )cos( θ e,x ) ) 2 + ( ( y y 0 )cos( θ e,y ) ) 2 w 2 ) .
Δn( x,y,z )= n 2 I ex ( x,y,z ),
I ex ( x,y,z )=I( x,y,z=0 ) e α 0 z .
Δ Φ nl ( x,y,z=L )=k 0 L Δn( x,y,z ) dz.
Δ Φ nl ( x,y,z=L )=k n 2 I ex ( x,y,z=0 ) 1 e α 0 L α 0
Δ Φ nl (x,y)Δ Φ nl ( x,y,z>L )=Δ Φ nl ( x,y,z=L ).
Δ Φ nl ( x,y,z=L )=ΔΦ ( x,y,z ) ( 3 ) +ΔΦ ( x,y,z ) ( 5 ) +
ΔΦ ( x,y,z=L ) ( 3 ) =k n 2 I ex ( x,y,z=0 ) [ 1exp( α 0 L ) ] / α 0 ,
ΔΦ ( x,y,z=L ) ( 5 ) =k n 4 I ex 2 ( x,y,z=0 ) [ 1exp( 2 α 0 L ) ] / 2 α 0 .
Δn( x,y,z )= n 2 I ex ( x,y,z ) 1+ I ex ( x,y,z ) / I sat .
Δ Φ nl ( x,y,z=L )= k n 2 I sat α 0 ln[ I sat + I ex (x,y,z=0) I sat + I ex ( x,y,z=0 ) e α 0 L ]
I( x,y )= I 1 ( x,y )+ I 2 ( x,y )+2 ( I 1 ( x,y ) I 2 ( x,y ) ) cos( ΔΦ( x,y ) ).
ΔΦ( x,y )Δ Φ total ( x,y )=2Δ Φ nl ( x,y )+Δ Φ tilt ( x,y ),
Δ Φ tilt ( x,y )=k[ xsin( θ x )+ysin( θ y ) ],
I( x,y )=A(x,y)+B(x,y)cos( 2Δ Φ nl ( x,y )+ksin( θ x )x ),
A( x,y )= I 1 ( x,y )+ I 2 ( x,y ) and B( x,y )=2 I 1 ( x,y )+ I 2 ( x,y ) .
I( x,y )=A(x,y)+C(x,y) e ikxsin( θ x ) + C (x,y) e ikxsin( θ x ) ,
C( x,y )=( 1/2 )B( x,y ) e 2iΔ Φ nl ( x,y )
I ˜ ( u x , u y )= A ˜ ( u x , u y )+ C ˜ ( u x ksin( θ x ), u y )+ C ˜ ( u x +ksin( θ x ), u y ),

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