Abstract

Lorentz–Gauss beams are introduced to describe certain laser sources that produce highly divergent beams. The fractional Fourier transform (FRFT) is applied to treat the propagation of Lorentz–Gauss beams. Based on the definition of convolution and the convolution theorem of the Fourier transform, an analytical expression for a Lorentz–Gauss beam passing through an FRFT system has been derived. By using the derived expression, the properties of a Lorentz–Gauss beam in the FRFT plane are graphically illustrated with numerical examples.

© 2009 Optical Society of America

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References

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  1. O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
    [CrossRef]
  2. A. Naqwi and F. Durst, “Focus of diode laser beams: a simple mathematical model,” Appl. Opt. 29, 1780-1785 (1990).
    [CrossRef] [PubMed]
  3. W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. QE-11, 400-402 (1975).
    [CrossRef]
  4. G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55, 993-1002 (2008).
    [CrossRef]
  5. G. Zhou, “Study on the propagation properties of Lorentz beam,” Acta Phys. Sin. 57, 3494-3498 (2008).
  6. O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).
    [CrossRef]
  7. G. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25, 2594-2599 (2008).
    [CrossRef]
  8. D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875-1881 (1993).
    [CrossRef]
  9. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522-2531 (1993).
    [CrossRef]
  10. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181-2186 (1993).
    [CrossRef]
  11. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 263-342 (1998).
    [CrossRef]
  12. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  13. Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27, 1672-1674 (2002).
    [CrossRef]
  14. Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7-13 (2003).
    [CrossRef]
  15. Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
    [CrossRef]
  16. Y. Cai and Q. Lin, “Properties of flattened Gaussian beam in the fractional Fourier transform plane,” J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
    [CrossRef]
  17. Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian-Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
    [CrossRef]
  18. Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6, 307-311 (2004).
    [CrossRef]
  19. D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
    [CrossRef]
  20. C. Zheng, “Fractional Fourier transform for off-axis elliptical Gaussian beams,” Opt. Commun. 259, 445-448 (2006).
    [CrossRef]
  21. C. Zheng, “Fractional Fourier transform for a hollow Gaussian beam,” Phys. Lett. A 355, 156-161 (2006).
    [CrossRef]
  22. X. Du and D. Zhao, “Fractional Fourier transform of truncated elliptical Gaussian beams,” Appl. Opt. 45, 9049-9052 (2006).
    [CrossRef] [PubMed]
  23. X. Du and D. Zhao, “Fractional Fourier transform of off-axial elliptical cosh-Gaussian beams,” Optik (Jena) 119, 379-382 (2008).
  24. X. Du and D. Zhao, “Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams,” Phys. Lett. A 366, 271-275 (2007).
    [CrossRef]
  25. C. Zheng, “Fractional Fourier transform of an elliptical dark-hollow beam,” Opt. Laser Technol. 40, 632-640 (2008).
    [CrossRef]
  26. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).
  27. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

2008 (5)

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55, 993-1002 (2008).
[CrossRef]

G. Zhou, “Study on the propagation properties of Lorentz beam,” Acta Phys. Sin. 57, 3494-3498 (2008).

X. Du and D. Zhao, “Fractional Fourier transform of off-axial elliptical cosh-Gaussian beams,” Optik (Jena) 119, 379-382 (2008).

C. Zheng, “Fractional Fourier transform of an elliptical dark-hollow beam,” Opt. Laser Technol. 40, 632-640 (2008).
[CrossRef]

G. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25, 2594-2599 (2008).
[CrossRef]

2007 (2)

X. Du and D. Zhao, “Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams,” Phys. Lett. A 366, 271-275 (2007).
[CrossRef]

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).
[CrossRef]

2006 (4)

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

C. Zheng, “Fractional Fourier transform for off-axis elliptical Gaussian beams,” Opt. Commun. 259, 445-448 (2006).
[CrossRef]

C. Zheng, “Fractional Fourier transform for a hollow Gaussian beam,” Phys. Lett. A 355, 156-161 (2006).
[CrossRef]

X. Du and D. Zhao, “Fractional Fourier transform of truncated elliptical Gaussian beams,” Appl. Opt. 45, 9049-9052 (2006).
[CrossRef] [PubMed]

2004 (2)

Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6, 307-311 (2004).
[CrossRef]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

2003 (4)

Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7-13 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Properties of flattened Gaussian beam in the fractional Fourier transform plane,” J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian-Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

2002 (1)

1998 (1)

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 263-342 (1998).
[CrossRef]

1993 (3)

1990 (1)

1975 (1)

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. QE-11, 400-402 (1975).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Cai, Y.

Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6, 307-311 (2004).
[CrossRef]

Y. Cai and Q. Lin, “Properties of flattened Gaussian beam in the fractional Fourier transform plane,” J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian-Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7-13 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27, 1672-1674 (2002).
[CrossRef]

Du, X.

X. Du and D. Zhao, “Fractional Fourier transform of off-axial elliptical cosh-Gaussian beams,” Optik (Jena) 119, 379-382 (2008).

X. Du and D. Zhao, “Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams,” Phys. Lett. A 366, 271-275 (2007).
[CrossRef]

X. Du and D. Zhao, “Fractional Fourier transform of truncated elliptical Gaussian beams,” Appl. Opt. 45, 9049-9052 (2006).
[CrossRef] [PubMed]

Dumke, W. P.

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. QE-11, 400-402 (1975).
[CrossRef]

Durst, F.

Gawhary, O. E.

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).
[CrossRef]

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

Ge, D.

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian-Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Jing, F.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Lin, Q.

Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6, 307-311 (2004).
[CrossRef]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian-Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Properties of flattened Gaussian beam in the fractional Fourier transform plane,” J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7-13 (2003).
[CrossRef]

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27, 1672-1674 (2002).
[CrossRef]

Liu, H.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 263-342 (1998).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181-2186 (1993).
[CrossRef]

Mao, H.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

Mendlovic, D.

Naqwi, A.

Ozaktas, H. M.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Severini, S.

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).
[CrossRef]

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Wang, S.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

Wei, X.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

Xu, Y.

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55, 993-1002 (2008).
[CrossRef]

Zalevsky, Z.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 263-342 (1998).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Zhao, D.

X. Du and D. Zhao, “Fractional Fourier transform of off-axial elliptical cosh-Gaussian beams,” Optik (Jena) 119, 379-382 (2008).

X. Du and D. Zhao, “Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams,” Phys. Lett. A 366, 271-275 (2007).
[CrossRef]

X. Du and D. Zhao, “Fractional Fourier transform of truncated elliptical Gaussian beams,” Appl. Opt. 45, 9049-9052 (2006).
[CrossRef] [PubMed]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

Zheng, C.

C. Zheng, “Fractional Fourier transform of an elliptical dark-hollow beam,” Opt. Laser Technol. 40, 632-640 (2008).
[CrossRef]

C. Zheng, “Fractional Fourier transform for off-axis elliptical Gaussian beams,” Opt. Commun. 259, 445-448 (2006).
[CrossRef]

C. Zheng, “Fractional Fourier transform for a hollow Gaussian beam,” Phys. Lett. A 355, 156-161 (2006).
[CrossRef]

Zheng, J.

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55, 993-1002 (2008).
[CrossRef]

Zhou, G.

G. Zhou, “Study on the propagation properties of Lorentz beam,” Acta Phys. Sin. 57, 3494-3498 (2008).

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55, 993-1002 (2008).
[CrossRef]

G. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25, 2594-2599 (2008).
[CrossRef]

Acta Phys. Sin. (1)

G. Zhou, “Study on the propagation properties of Lorentz beam,” Acta Phys. Sin. 57, 3494-3498 (2008).

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. QE-11, 400-402 (1975).
[CrossRef]

J. Mod. Opt. (1)

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55, 993-1002 (2008).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (4)

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

Y. Cai and Q. Lin, “Properties of flattened Gaussian beam in the fractional Fourier transform plane,” J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian-Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6, 307-311 (2004).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (4)

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274-284 (2007).
[CrossRef]

Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7-13 (2003).
[CrossRef]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

C. Zheng, “Fractional Fourier transform for off-axis elliptical Gaussian beams,” Opt. Commun. 259, 445-448 (2006).
[CrossRef]

Opt. Laser Technol. (1)

C. Zheng, “Fractional Fourier transform of an elliptical dark-hollow beam,” Opt. Laser Technol. 40, 632-640 (2008).
[CrossRef]

Opt. Lett. (1)

Optik (Jena) (1)

X. Du and D. Zhao, “Fractional Fourier transform of off-axial elliptical cosh-Gaussian beams,” Optik (Jena) 119, 379-382 (2008).

Phys. Lett. A (2)

X. Du and D. Zhao, “Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams,” Phys. Lett. A 366, 271-275 (2007).
[CrossRef]

C. Zheng, “Fractional Fourier transform for a hollow Gaussian beam,” Phys. Lett. A 355, 156-161 (2006).
[CrossRef]

Prog. Opt. (1)

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 263-342 (1998).
[CrossRef]

Other (3)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

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Figures (4)

Fig. 1
Fig. 1

Optical system for performing the fractional Fourier transform. (a) Lohmann I system. (b) Lohmann II system.

Fig. 2
Fig. 2

Normalized intensity of a Lorentz–Gauss beam with a different fractional order p in the FRFT plane.

Fig. 3
Fig. 3

Normalized intensity of a Lorentz–Guass beam with a different parameter w 0 in the FRFT plane.

Fig. 4
Fig. 4

Normalized intensity of a Lorentz–Guass beam with a different parameter w 0 x in the FRFT plane.

Equations (22)

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E ( x 0 , y 0 ) = E 0 w 0 x w 0 y [ 1 + ( x 0 w 0 x ) 2 ] [ 1 + ( y 0 w 0 y ) 2 ] exp [ i k ( x 0 2 + y 0 2 ) 2 q 0 ] ,
( A B C D ) = ( cos φ f sin φ sin φ f cos φ ) .
E ( x , y ) = exp ( i k z ) i λ B E ( x 0 , y 0 ) exp { i k 2 B [ A ( x 0 2 + y 0 2 ) 2 ( x 0 x + y y 0 ) + D ( x 2 + y 2 ) ] } d x 0 d y 0 .
E ( x , y ) = E 0 i λ f sin φ exp ( i k z ) E ( x ) E ( y ) ,
E ( j ) = 1 w 0 j [ 1 + ( j 0 w 0 j ) 2 ] exp { i k 2 f sin φ [ ( cos φ + f sin φ q 0 ) j 0 2 2 j 0 j + cos φ j 2 ] } d j 0 ,
E ( j ) = w 0 j exp ( i k ( 1 f + cot φ q 0 ) 2 ( cot φ + f q 0 ) j 2 ) 1 w 0 j 2 + j 0 2 exp [ k ( cot φ + f q 0 ) 2 i f ( j cos φ + f sin φ q 0 j 0 ) 2 ] d j 0 .
f 1 ( τ ) f 2 ( τ ) = f 1 ( η ) f 2 ( τ η ) d η ,
E ( j ) = w 0 j exp ( i k ( 1 f + cot φ q 0 ) 2 ( cot φ + f q 0 ) j 2 ) [ f 1 ( j cos φ + f sin φ q 0 ) f 2 ( j cos φ + f sin φ q 0 ) ] ,
f 1 ( τ ) = 1 w 0 j 2 + τ 2 ,
f 2 ( τ ) = exp ( k ( cot φ + f q 0 ) 2 i f τ 2 ) .
f 1 ( τ ) f 2 ( τ ) = f 1 ( ξ ) f 2 ( ξ ) exp ( i ξ τ ) d ξ ,
E ( j ) = π i f 2 k ( cot φ + f q 0 ) exp ( i k ( 1 f + cot φ q 0 ) 2 ( cot φ + f q 0 ) j 2 ) exp ( w 0 j ξ ) exp ( i f ξ 2 2 k ( cot φ + f q 0 ) ) × exp ( i ξ j cos φ + f sin φ q 0 ) d ξ .
E ( j ) = π i f 2 k ( cot φ + f q 0 ) exp ( i k ( 1 f + cot φ q 0 ) 2 ( cot φ + f q 0 ) j 2 ) { exp [ k ( cot φ + f q 0 ) 2 i f ( w 0 j + i j cos φ + f sin φ q 0 ) 2 ] × 0 exp { i f 2 k ( cot φ + f q 0 ) [ ξ + k ( cot φ + f q 0 ) i f ( w 0 j + i j cos φ + f sin φ q 0 ) ] 2 } d ξ + 0 exp { i f 2 k ( cot φ + f q 0 ) [ ξ + k ( cot φ + f q 0 ) i f ( w 0 j i j cos φ + f sin φ q 0 ) ] 2 } d ξ × exp [ k ( cot φ + f q 0 ) 2 i f ( w 0 j i j cos φ + f sin φ q 0 ) 2 ] } .
E ( j ) = π exp ( i k ( 1 f + cot φ q 0 ) 2 ( cot φ + f q 0 ) j 2 ) { exp [ k ( cot φ + f q 0 ) 2 i f ( w 0 j + i j cos φ + f sin φ q 0 ) 2 ] × [ 0 exp ( ξ 2 ) d ξ 0 T 1 exp ( ξ 2 ) d ξ ] + exp [ k ( cot φ + f q 0 ) 2 i f ( w 0 j i j cos φ + f sin φ q 0 ) 2 ] × [ 0 exp ( ξ 2 ) d ξ 0 T 2 exp ( ξ 2 ) d ξ ] } ,
T 1 = k ( cot φ + f q 0 ) 2 i f ( w 0 j + i j cos φ + f sin φ q 0 ) ,
T 2 = k ( cot φ + f q 0 ) 2 i f ( w 0 j i j cos φ + f sin φ q 0 ) .
erf ( T ) = 2 π 0 T exp ( s 2 ) d s
E ( j ) = π 2 exp ( i k ( 1 f + cot φ q 0 ) 2 ( cot φ + f q 0 ) j 2 ) [ E j + ( j ) + E j ( j ) ] ,
E j ± ( j ) = exp [ k ( cot φ + f q 0 ) 2 i f ( w 0 j ± i j cos φ + f sin φ q 0 ) 2 ] { 1 erf [ k ( cot φ + f q 0 ) 2 i f ( w 0 j ± i j cos φ + f sin φ q 0 ) ] } .
E ( x , y ) = E 0 π 2 i 4 λ f sin φ exp ( i k z ) exp ( i k ( 1 f + cot φ q 0 ) ρ 2 2 ( cot φ + f q 0 ) ) { exp [ k ( cot φ + f q 0 ) 2 i f ( w 0 x + i x cos φ + f sin φ q 0 ) 2 ] × { 1 erf [ k ( cot φ + f q 0 ) 2 i f ( w 0 x + i x cos φ + f sin φ q 0 ) ] } + exp [ k ( cot φ + f q 0 ) 2 i f ( w 0 x i x cos φ + f sin φ q 0 ) 2 ] × { 1 erf [ k ( cot φ + f q 0 ) 2 i f ( w 0 x i x cos φ + f sin φ q 0 ) ] } } { exp [ k ( cot φ + f q 0 ) 2 i f ( w 0 y + i y cos φ + f sin φ q 0 ) 2 ] × { 1 erf [ k ( cot φ + f q 0 ) 2 i f ( w 0 y + i y cos φ + f sin φ q 0 ) ] } + exp [ k ( cot φ + f q 0 ) 2 i f ( w 0 y i y cos φ + f sin φ q 0 ) 2 ] × { 1 erf [ k ( cot φ + f q 0 ) 2 i f ( w 0 y i y cos φ + f sin φ q 0 ) ] } } ,
E ( x , y ) = ( 1 ) n E 0 exp ( i k z ) w 0 x w 0 y 1 ( 1 + x 2 w 0 x 2 ) ( 1 + y 2 w 0 y 2 ) exp ( i k ρ 2 2 q 0 ) .
I = E ( x , y ) 2 .

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