Abstract

Based on the fact that a hard-edged elliptical aperture can be expanded approximately as a finite sum of complex Gaussian functions in tensor form, an analytical expression for an elliptical Gaussian beam (EGB) truncated by an elliptical aperture and passing through a fractional Fourier transform system is derived by use of vector integration. The approximate analytical results provide more convenience for studying the propagation and transformation of truncated EGBs than the usual way by using the integral formula directly, and the efficiency of numerical calculation is significantly improved.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. Namias, "The fractional order Fourier transform and its applications to quantum mechanics," J. Inst. Math. Appl. 25, 241-265 (1980).
    [CrossRef]
  2. A. C. McBride and F. H. Kerr, "On Namias's fractional Fourier transform," IMA J. Appl. Math. 39, 159-175 (1987).
    [CrossRef]
  3. D. Mendlovic and H. M. Ozaktas, "Fractional Fourier transforms and their optical implementation: I," J. Opt. Soc. Am. A 10, 1875-1881 (1993).
    [CrossRef]
  4. H. M. Ozaktas and D. Mendlovic, "Fractional Fourier transforms and their optical implementation: II," J. Opt. Soc. Am. A 10, 2522-2531 (1993).
    [CrossRef]
  5. A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181-2186 (1993).
    [CrossRef]
  6. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, "Fractional transformations in optics," Prog. Opt. 38, 263-342 (1998).
    [CrossRef]
  7. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  8. J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
    [CrossRef]
  9. S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).
  10. Y. Cai and Q. Lin, "Fractional Fourier transform for elliptical Gaussian beams," Opt. Commun. 217, 7-13 (2003).
    [CrossRef]
  11. J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
    [CrossRef]
  12. Z. Mei and D. Zhao, "Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems," J. Opt. Soc. Am. A 21, 2375-2381 (2004).
    [CrossRef]
  13. X. Du and D. Zhao, "Propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems," J. Opt. Soc. Am. A 23, 625-631 (2006).
    [CrossRef]
  14. X. Du and D. Zhao, "Propagation of elliptical Gaussian beams in apertured and misaligned optical systems," J. Opt. Soc. Am. A 23, 1946-1950 (2006).
    [CrossRef]
  15. H. Mao and D. Zhao, "Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system," J. Opt. Soc. Am. A 22, 647-653 (2005).
    [CrossRef]

2006 (2)

2005 (1)

2004 (1)

2003 (1)

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

1998 (1)

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

1993 (3)

1991 (1)

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

1988 (1)

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

1987 (1)

A. C. McBride and F. H. Kerr, "On Namias's fractional Fourier transform," IMA J. Appl. Math. 39, 159-175 (1987).
[CrossRef]

1980 (1)

V. Namias, "The fractional order Fourier transform and its applications to quantum mechanics," J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

Alda, J.

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

Bernabeu, E.

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Cai, Y.

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

Du, X.

Kerr, F. H.

A. C. McBride and F. H. Kerr, "On Namias's fractional Fourier transform," IMA J. Appl. Math. 39, 159-175 (1987).
[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, "Fractional Fourier transform for elliptical Gaussian beams," Opt. Commun. 217, 7-13 (2003).
[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.

McBride, A. C.

A. C. McBride and F. H. Kerr, "On Namias's fractional Fourier transform," IMA J. Appl. Math. 39, 159-175 (1987).
[CrossRef]

Mei, Z.

Mendlovic, D.

Namias, V.

V. Namias, "The fractional order Fourier transform and its applications to quantum mechanics," J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

Ozaktas, H. M.

Wang, S.

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).

Wen, J. J.

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[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.

IMA J. Appl. Math. (1)

A. C. McBride and F. H. Kerr, "On Namias's fractional Fourier transform," IMA J. Appl. Math. 39, 159-175 (1987).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, "The fractional order Fourier transform and its applications to quantum mechanics," J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Commun. (2)

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

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

Prog. Opt. (1)

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

Other (2)

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

S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).

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 (2)

Fig. 1
Fig. 1

Intensity distribution of an EGB truncated by an elliptical aperture a = 1   mm , b = 2   mm , θ = 0 , and passing through a FRFT system on the plane of p = 0.5 : (a) 3D plot of the intensity distribution obtained by Eq. (10); (b) 2D plot of the intensity profile across the plane of y = 0 obtained by the integral formula Eq. (2) (solid curves) and the analytical expression Eq. (10) (dotted curves).

Fig. 2
Fig. 2

Intensity distributions of an EGB truncated by different elliptical apertures: (a) a = 10   mm , b = 10   mm , θ = 0 , (b) a = 1   mm , b = 2   mm , θ = π / 6 , and passing through a FRFT system on the plane of p = 0.5 .

Equations (16)

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

A p ( r 1 ) = { 1 , 0 , inside   the   elliptical   aperture outside   the   elliptical   aperture ,
E 2 ( r 2 ) = i λ f   sin   ϕ A p ( r 1 ) E 1 ( r 1 ) × exp [ - i k 2 ( r 1 T N 1 r 1 2 r 1 T N 2 r 2 + r 2 T N 1 r 2 ) ] d r 1 ,
N 1 = 1 f   tan   ϕ [ 1 0 0 1 ] , N 2 = 1 f   sin   ϕ [ 1 0 0 1 ] ,
E 1 ( r 1 ) = exp ( - i k 2 r 1 T Q - 1 r 1 ) ,
Q - 1 = [ q x x 1 q x y 1 q x y 1 q y y 1 ] = i λ π [ w 0 x 2 w 0 x y 2 w 0 x y 2 w 0 y 2 ] .
A p ( r 1 ) = n = 1 N A n   exp ( - r 1 T R T P n R r 1 ) ,
R = [ cos   θ - sin   θ sin   θ cos   θ ] ,
P n = [ B n / a 2 0 0 B n / b 2 ] ,
A p ( r 1 ) E 1 ( r 1 ) = n = 1 N A n   exp ( - r 1 T R T P n R r 1 ) × exp ( - i k 2 r 1 T Q - 1 r 1 ) .
E 2 ( r 2 ) = n = 1 N A n f   sin   ϕ [ det ( Q - 1 + P + N 1 ) ] - 1 / 2 × exp { - i k 2 r 2 T [ N 1 N 2 T ( Q - 1 + P + N 1 ) - 1 N 2 ] r 2 } ,
A ¯ = N 2 1 N 1 = cos   ϕ [ 1 0 0 1 ] ,
B ¯ = N 2 1 = f   sin   ϕ [ 1 0 0 1 ] ,
C ¯ = N 1 N 2 1 N 1 N 2 T = sin   ϕ f [ 1 0 0 1 ] ,
D ¯ = N 1 N 2 1 = cos   ϕ [ 1 0 0 1 ] ,
E 2 ( r 2 ) = n = 1 N A n [ det ( A ¯ + B ¯ Q i 1 ) ] 1 / 2 × exp ( - i k 2 r 2 T Q f - 1 r 2 ) ,
Q f 1 = ( C ¯ + D ¯ Q i 1 ) ( A ¯ + B ¯ Q i 1 ) - 1 , with   Q i - 1 = Q - 1 + P .

Metrics