Abstract

We show that the image quality of ghost imaging (GI) can be controlled by the use of shaped incoherent sources. The formula for the point-spread function in the GI system has been derived and is determined by the Fourier transform of the source intensity distribution. Compared with the widely used Gaussian Schell-model source, we find that using a cosine-Gaussian Schell-model source can lead to the degradation of GI quality, while the quality of GI can be increased with a cosh-Gaussian Schell-model source. Even under atmospheric turbulence, the image resolution of GI still can be improved by means of the cosh-Gaussian Schell-model source.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. 74, 3600 (1995).
    [CrossRef]
  2. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, Phys. Rev. A 52, R3429 (1995).
    [CrossRef]
  3. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. Lett. 87, 123602 (2001).
    [CrossRef]
  4. J. Cheng and S. Han, Phys. Rev. Lett. 92, 093903 (2004).
    [CrossRef]
  5. J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. G. Wang, Phys. Rev. Lett. 94, 173601 (2005).
    [CrossRef]
  6. F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, Phys. Rev. Lett. 94, 183602 (2005).
    [CrossRef]
  7. E. Baleine, A. Dogariu, and G. S. Agarwal, Opt. Lett. 31, 2124 (2006).
    [CrossRef]
  8. M. H. Zhang, Q. Wei, X. Shen, Y. F. Liu, H. L. Liu, J. Cheng, and S. S. Han, Phys. Rev. A 75, 021803 (2007).
    [CrossRef]
  9. J. Cheng, Phys. Rev. A 78, 043823 (2008).
    [CrossRef]
  10. J. H. Shapiro, Phys. Rev. A 78, 061802 (2008).
  11. B. I. Erkmen and J. H. Shapiro, Phys. Rev. A 79, 023833 (2009).
    [CrossRef]
  12. J. Cheng, Opt. Express 17, 7916 (2009).
    [CrossRef]
  13. O. Katz, Y. Bromberg, and Y. Silberberg, Appl. Phys. Lett. 95, 131110 (2009).
    [CrossRef]
  14. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, Phys. Rev. Lett. 104, 253603 (2010).
    [CrossRef]
  15. X. H. Chen, I. N. Agafonov, K. H. Luo, Q. Liu, R. Xian, M. V. Chekhova, and L. A. Wu, Opt. Lett. 35, 1166 (2010).
    [CrossRef]
  16. P. Zhang, W. Gong, X. Shen, and S. S. Han, Phys. Rev. A 82, 033817 (2010).
    [CrossRef]
  17. M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, Phys. Rev. A 87, 033813 (2013).
    [CrossRef]
  18. C. L. Luo and J. Cheng, J. Opt. Soc. Am. A 30, 1166 (2013).
    [CrossRef]
  19. J. Cheng and J. Lin, Phys. Rev. A 87, 043810 (2013).
    [CrossRef]
  20. J. C. Ricklin and F. M. Davidson, J. Opt. Soc. Am. A 19, 1794 (2002).
    [CrossRef]
  21. R. E. Meyers, K. S. Deacon, and Y. H. Shih, Appl. Phys. Lett. 98, 111115 (2011).
    [CrossRef]
  22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

2013 (3)

M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, Phys. Rev. A 87, 033813 (2013).
[CrossRef]

C. L. Luo and J. Cheng, J. Opt. Soc. Am. A 30, 1166 (2013).
[CrossRef]

J. Cheng and J. Lin, Phys. Rev. A 87, 043810 (2013).
[CrossRef]

2011 (1)

R. E. Meyers, K. S. Deacon, and Y. H. Shih, Appl. Phys. Lett. 98, 111115 (2011).
[CrossRef]

2010 (3)

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, Phys. Rev. Lett. 104, 253603 (2010).
[CrossRef]

X. H. Chen, I. N. Agafonov, K. H. Luo, Q. Liu, R. Xian, M. V. Chekhova, and L. A. Wu, Opt. Lett. 35, 1166 (2010).
[CrossRef]

P. Zhang, W. Gong, X. Shen, and S. S. Han, Phys. Rev. A 82, 033817 (2010).
[CrossRef]

2009 (3)

B. I. Erkmen and J. H. Shapiro, Phys. Rev. A 79, 023833 (2009).
[CrossRef]

J. Cheng, Opt. Express 17, 7916 (2009).
[CrossRef]

O. Katz, Y. Bromberg, and Y. Silberberg, Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

2008 (2)

J. Cheng, Phys. Rev. A 78, 043823 (2008).
[CrossRef]

J. H. Shapiro, Phys. Rev. A 78, 061802 (2008).

2007 (1)

M. H. Zhang, Q. Wei, X. Shen, Y. F. Liu, H. L. Liu, J. Cheng, and S. S. Han, Phys. Rev. A 75, 021803 (2007).
[CrossRef]

2006 (1)

2005 (2)

J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. G. Wang, Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef]

2004 (1)

J. Cheng and S. Han, Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef]

2002 (1)

2001 (1)

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef]

1995 (2)

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. 74, 3600 (1995).
[CrossRef]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, Phys. Rev. A 52, R3429 (1995).
[CrossRef]

Abouraddy, A. F.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef]

Agafonov, I. N.

Agarwal, G. S.

Bache, M.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef]

Baleine, E.

Brambilla, E.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef]

Bromberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

Cao, D. Z.

J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. G. Wang, Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef]

Chekhova, M. V.

Chen, X. H.

Cheng, J.

C. L. Luo and J. Cheng, J. Opt. Soc. Am. A 30, 1166 (2013).
[CrossRef]

J. Cheng and J. Lin, Phys. Rev. A 87, 043810 (2013).
[CrossRef]

J. Cheng, Opt. Express 17, 7916 (2009).
[CrossRef]

J. Cheng, Phys. Rev. A 78, 043823 (2008).
[CrossRef]

M. H. Zhang, Q. Wei, X. Shen, Y. F. Liu, H. L. Liu, J. Cheng, and S. S. Han, Phys. Rev. A 75, 021803 (2007).
[CrossRef]

J. Cheng and S. Han, Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef]

Davidson, F. M.

Deacon, K. S.

R. E. Meyers, K. S. Deacon, and Y. H. Shih, Appl. Phys. Lett. 98, 111115 (2011).
[CrossRef]

Dogariu, A.

Erkmen, B. I.

B. I. Erkmen and J. H. Shapiro, Phys. Rev. A 79, 023833 (2009).
[CrossRef]

Fan, H.

M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, Phys. Rev. A 87, 033813 (2013).
[CrossRef]

Ferri, F.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, Phys. Rev. Lett. 104, 253603 (2010).
[CrossRef]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef]

Gatti, A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, Phys. Rev. Lett. 104, 253603 (2010).
[CrossRef]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef]

Gong, W.

P. Zhang, W. Gong, X. Shen, and S. S. Han, Phys. Rev. A 82, 033817 (2010).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Han, S.

J. Cheng and S. Han, Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef]

Han, S. S.

P. Zhang, W. Gong, X. Shen, and S. S. Han, Phys. Rev. A 82, 033817 (2010).
[CrossRef]

M. H. Zhang, Q. Wei, X. Shen, Y. F. Liu, H. L. Liu, J. Cheng, and S. S. Han, Phys. Rev. A 75, 021803 (2007).
[CrossRef]

Huang, F.

J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. G. Wang, Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef]

Katz, O.

O. Katz, Y. Bromberg, and Y. Silberberg, Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

Klyshko, D. N.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. 74, 3600 (1995).
[CrossRef]

Li, H. G.

J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. G. Wang, Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef]

Li, M. F.

M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, Phys. Rev. A 87, 033813 (2013).
[CrossRef]

Lin, J.

J. Cheng and J. Lin, Phys. Rev. A 87, 043810 (2013).
[CrossRef]

Liu, H. L.

M. H. Zhang, Q. Wei, X. Shen, Y. F. Liu, H. L. Liu, J. Cheng, and S. S. Han, Phys. Rev. A 75, 021803 (2007).
[CrossRef]

Liu, Q.

Liu, Y. F.

M. H. Zhang, Q. Wei, X. Shen, Y. F. Liu, H. L. Liu, J. Cheng, and S. S. Han, Phys. Rev. A 75, 021803 (2007).
[CrossRef]

Lugiato, L. A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, Phys. Rev. Lett. 104, 253603 (2010).
[CrossRef]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef]

Luo, C. L.

Luo, K. H.

M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, Phys. Rev. A 87, 033813 (2013).
[CrossRef]

X. H. Chen, I. N. Agafonov, K. H. Luo, Q. Liu, R. Xian, M. V. Chekhova, and L. A. Wu, Opt. Lett. 35, 1166 (2010).
[CrossRef]

Magatti, D.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, Phys. Rev. Lett. 104, 253603 (2010).
[CrossRef]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef]

Meyers, R. E.

R. E. Meyers, K. S. Deacon, and Y. H. Shih, Appl. Phys. Lett. 98, 111115 (2011).
[CrossRef]

Pittman, T. B.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, Phys. Rev. A 52, R3429 (1995).
[CrossRef]

Ricklin, J. C.

Saleh, B. E. A.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef]

Sergienko, A. V.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. 74, 3600 (1995).
[CrossRef]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, Phys. Rev. A 52, R3429 (1995).
[CrossRef]

Shapiro, J. H.

B. I. Erkmen and J. H. Shapiro, Phys. Rev. A 79, 023833 (2009).
[CrossRef]

J. H. Shapiro, Phys. Rev. A 78, 061802 (2008).

Shen, X.

P. Zhang, W. Gong, X. Shen, and S. S. Han, Phys. Rev. A 82, 033817 (2010).
[CrossRef]

M. H. Zhang, Q. Wei, X. Shen, Y. F. Liu, H. L. Liu, J. Cheng, and S. S. Han, Phys. Rev. A 75, 021803 (2007).
[CrossRef]

Shih, Y. H.

R. E. Meyers, K. S. Deacon, and Y. H. Shih, Appl. Phys. Lett. 98, 111115 (2011).
[CrossRef]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. 74, 3600 (1995).
[CrossRef]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, Phys. Rev. A 52, R3429 (1995).
[CrossRef]

Silberberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

Strekalov, D. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, Phys. Rev. A 52, R3429 (1995).
[CrossRef]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. 74, 3600 (1995).
[CrossRef]

Sun, X. J.

J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. G. Wang, Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef]

Teich, M. C.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef]

Wang, K. G.

J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. G. Wang, Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef]

Wei, Q.

M. H. Zhang, Q. Wei, X. Shen, Y. F. Liu, H. L. Liu, J. Cheng, and S. S. Han, Phys. Rev. A 75, 021803 (2007).
[CrossRef]

Wu, L. A.

M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, Phys. Rev. A 87, 033813 (2013).
[CrossRef]

X. H. Chen, I. N. Agafonov, K. H. Luo, Q. Liu, R. Xian, M. V. Chekhova, and L. A. Wu, Opt. Lett. 35, 1166 (2010).
[CrossRef]

Xian, R.

Xiong, J.

J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. G. Wang, Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef]

Zhang, M. H.

M. H. Zhang, Q. Wei, X. Shen, Y. F. Liu, H. L. Liu, J. Cheng, and S. S. Han, Phys. Rev. A 75, 021803 (2007).
[CrossRef]

Zhang, P.

P. Zhang, W. Gong, X. Shen, and S. S. Han, Phys. Rev. A 82, 033817 (2010).
[CrossRef]

Zhang, Y. R.

M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, Phys. Rev. A 87, 033813 (2013).
[CrossRef]

Appl. Phys. Lett. (2)

O. Katz, Y. Bromberg, and Y. Silberberg, Appl. Phys. Lett. 95, 131110 (2009).
[CrossRef]

R. E. Meyers, K. S. Deacon, and Y. H. Shih, Appl. Phys. Lett. 98, 111115 (2011).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (8)

M. H. Zhang, Q. Wei, X. Shen, Y. F. Liu, H. L. Liu, J. Cheng, and S. S. Han, Phys. Rev. A 75, 021803 (2007).
[CrossRef]

J. Cheng, Phys. Rev. A 78, 043823 (2008).
[CrossRef]

J. H. Shapiro, Phys. Rev. A 78, 061802 (2008).

B. I. Erkmen and J. H. Shapiro, Phys. Rev. A 79, 023833 (2009).
[CrossRef]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, Phys. Rev. A 52, R3429 (1995).
[CrossRef]

P. Zhang, W. Gong, X. Shen, and S. S. Han, Phys. Rev. A 82, 033817 (2010).
[CrossRef]

M. F. Li, Y. R. Zhang, K. H. Luo, L. A. Wu, and H. Fan, Phys. Rev. A 87, 033813 (2013).
[CrossRef]

J. Cheng and J. Lin, Phys. Rev. A 87, 043810 (2013).
[CrossRef]

Phys. Rev. Lett. (6)

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. 74, 3600 (1995).
[CrossRef]

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, Phys. Rev. Lett. 104, 253603 (2010).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef]

J. Cheng and S. Han, Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef]

J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. G. Wang, Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, Phys. Rev. Lett. 94, 183602 (2005).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Geometry of a GI system with a turbulent atmosphere. The source field f(u) is split into two beams by the nonpolarized BS. And z0, z1, z2 are the propagation distances from the source to the object, from the source to the reference detector, and from the object to the signal detector, respectively. And u, y, x1, x2 are coordinates at the source plane, object plane, reference detector plane, and signal detector plane.

Fig. 2.
Fig. 2.

Intensity distribution of shaped Gaussian sources and the corresponding PSFs in the GI system. a0=2.5cm, λ=0.785μm, z1=1500m. (a) The normalized intensity distribution over the cosine-Gaussian sources. (b) The normalized intensity distribution over the cosh-Gaussian sources. (c) The normalized PSF of GI with the cosine-Gaussian sources. (d) The normalized PSF of GI with cosh-Gaussian sources. Red curves represent the modulation parameter ω0=0m1, black real points represent ω0=20m1, green curves represent ω0=25m1, and black curves represent ω0=30m1.

Fig. 3.
Fig. 3.

GI of a double-slit with different turbulent strengths Cn2, and modulation parameters ω. The system parameters are set as λ=0.785μm, δl=1.5cm, δg=1mm, D=1m, and the distances with z0=z1=1500m, z2=500m. Curves are shifted to be distinguished clearly. From the bottom to the top ω=0, 25, 45m1. (a) The case of weak turbulence, Cn2=1×1016m2/3. (b) The case of mediate turbulence, Cn2=5×1015m2/3. (c) The case of very strong turbulence, Cn2=2×1014m2/3. (d) The dependence of the visibility on ω in GI through the mediate turbulence, Cn2=5×1015m2/3.

Equations (14)

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

G(x1,x2)=E1(x1)E1*(x1)E2(x2)E2*(x2)E1(x1)E1*(x1)E2(x2)E2*(x2),
G(x1,x2)=|Γ(u1,u2)exp[iπλz1(u1x1)2]exp[iπλz0(u2y)2]×t*(y)exp[iπλz2(yx2)2]du1du2dy|2,
Γ(u1,u2)=Es(u1)Es*(u2)=f(u1)δ(u1u2),
G(x1,x2)=|f(u1)exp[iπλz1(u1x1)2]exp[iπλz0(u1y)2]×t*(y)exp[iπλz2(yx2)2]du1dy|2.
I(x1)=G(x1,x2)dx2=|t(y)|2dy|f(u1)exp[2iπu1λz1(x1y)]du1|2=|t(y)|2|F(x1yλz1)|2dy=|t(y)|2|F(yλz1)|2,
|F1(yλz1)|2=|F(yλz1)|2|[1+exp(2a02ω02)cosh(2πa02ω0yλz1)]|2,
|F2(yλz1)|2=|F(yλz1)|2|[1+exp(2a02ω02)cos(2πa02ω0yλz1)]|2.
E1(x1)=du1Es(u1)exp[jπλz1(x1u1)2]exp[ϕ1(u1,x1)],
E2(x2)=du2dyEs(u2)exp[jπλz0(yu2)2]exp[ϕ0(u2,y)]×t(y)exp[jπλz2(yx2)2]exp[ϕ2(x2,y)],
Es(u)Es*(u)=exp[u2+u2δl2(uu)2δg2]cosh(2ωu)cosh(2ωu),
exp[ϕi(x,y)+ϕi*(x,y)]=exp[(xx)2+(xx)(yy)+(yy)2ρi2],
G(x1,x2)=dydyt(y)t*(y)exp[(yy)2(1ρ02+1ρ22)]exp[iπλ(1z0+1z2)(y2y2)]exp[i2πλz2x2(yy)]H(y,y,x1),
H(y,y,x1)=du1du1du2du2exp[u12+u22δl2+(u1u2)2δg2+u22+u12δl2+(u2u1)2δg2]cosh(2ωu1)cosh(2ωu2)cosh(2ωu2)cosh(2ωu1)exp[(u2u2)2+(u2u2)(yy)ρ02]exp[(u1u1)2ρ12]exp[iπλz1((u12u12)2(u1u1)x1)]exp[iπλz0((u22u22)2(u2yu2y))].
V=G(x1,x2)maxI1(x1)I2(x2)max.

Metrics