Abstract

A hard-edged elliptical aperture is described approximately by a tensor form, which can be expanded as a finite sum of complex Gaussian functions. An analytical propagation expression for a decentered elliptical Gaussian beam (DEGB) through an axially nonsymmetrical optical system with an elliptical aperture is derived by using vector integration. The approximate analytical results are compared with numerically integral ones, and it is shown that this method can significantly improve the efficiency of numerical calculation. Some numerical simulations are illustrated for the propagation properties of DEGBs through apertured and nonsymmetrical optical transforming systems.

© 2006 Optical Society of America

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  1. A. R. Al-Rashed and B. E. A. Saleh, 'Decentered Gaussian beams,' Appl. Opt. 34, 6819-6825 (1995).
    [CrossRef] [PubMed]
  2. P. J. Cronin, P. Török, P. Varga, and C. Cogswell, 'High-aperture diffraction of a scalar, off-axis Gaussian beam,' J. Opt. Soc. Am. A 17, 1556-1564 (2000).
    [CrossRef]
  3. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, 'Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,' Optik (Stuttgart) 85, 67-72 (1990).
  4. 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]
  5. Y. Cai and Q. Lin, 'Decentered elliptical Gaussian beam,' Appl. Opt. 41, 4336-4340 (2002).
    [CrossRef] [PubMed]
  6. 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]
  7. J. J. Wen and M. A. Breazeale, 'Computer optimization of the Gaussian beam description of an ultrasonic field,' in Computational Acoustics, D.Lee, A.Cakmak, and R.Vichnevetsky, eds. (Elsevier, 1990).
  8. D. Zhao, H. Mao, W. Zhang, and S. Wang, 'Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,' Opt. Commun. 224, 5-12 (2003).
    [CrossRef]
  9. 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]
  10. Z. Mei and D. Zhao, 'Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transforming systems,' J. Opt. Soc. Am. A 21, 2375-2381 (2004).
    [CrossRef]
  11. Z. Mei and D. Zhao, 'Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams,' Appl. Opt. 44, 1381-1386 (2005).
    [CrossRef] [PubMed]

2005 (1)

2004 (2)

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]

Z. Mei and D. Zhao, 'Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transforming systems,' J. Opt. Soc. Am. A 21, 2375-2381 (2004).
[CrossRef]

2003 (1)

D. Zhao, H. Mao, W. Zhang, and S. Wang, 'Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,' Opt. Commun. 224, 5-12 (2003).
[CrossRef]

2002 (1)

2000 (1)

1995 (1)

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]

1990 (1)

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, 'Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,' Optik (Stuttgart) 85, 67-72 (1990).

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]

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]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, 'Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,' Optik (Stuttgart) 85, 67-72 (1990).

Al-Rashed, A. R.

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]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, 'Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,' Optik (Stuttgart) 85, 67-72 (1990).

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]

J. J. Wen and M. A. Breazeale, 'Computer optimization of the Gaussian beam description of an ultrasonic field,' in Computational Acoustics, D.Lee, A.Cakmak, and R.Vichnevetsky, eds. (Elsevier, 1990).

Cai, Y.

Cogswell, C.

Cronin, P. J.

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]

Lin, Q.

Y. Cai and Q. Lin, 'Decentered elliptical Gaussian beam,' Appl. Opt. 41, 4336-4340 (2002).
[CrossRef] [PubMed]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, 'Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,' Optik (Stuttgart) 85, 67-72 (1990).

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]

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]

D. Zhao, H. Mao, W. Zhang, and S. Wang, 'Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,' Opt. Commun. 224, 5-12 (2003).
[CrossRef]

Mei, Z.

Saleh, B. E. A.

Török, P.

Varga, P.

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]

D. Zhao, H. Mao, W. Zhang, and S. Wang, 'Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,' Opt. Commun. 224, 5-12 (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]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, 'Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,' Optik (Stuttgart) 85, 67-72 (1990).

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]

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]

J. J. Wen and M. A. Breazeale, 'Computer optimization of the Gaussian beam description of an ultrasonic field,' in Computational Acoustics, D.Lee, A.Cakmak, and R.Vichnevetsky, eds. (Elsevier, 1990).

Zhang, W.

D. Zhao, H. Mao, W. Zhang, and S. Wang, 'Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,' Opt. Commun. 224, 5-12 (2003).
[CrossRef]

Zhao, D.

Z. Mei and D. Zhao, 'Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams,' Appl. Opt. 44, 1381-1386 (2005).
[CrossRef] [PubMed]

Z. Mei and D. Zhao, 'Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transforming systems,' J. Opt. Soc. Am. A 21, 2375-2381 (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]

D. Zhao, H. Mao, W. Zhang, and S. Wang, 'Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,' Opt. Commun. 224, 5-12 (2003).
[CrossRef]

Appl. Opt. (3)

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. Opt. Soc. Am. A (2)

Opt. Commun. (3)

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]

D. Zhao, H. Mao, W. Zhang, and S. Wang, 'Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,' Opt. Commun. 224, 5-12 (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]

Optik (Stuttgart) (1)

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, 'Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,' Optik (Stuttgart) 85, 67-72 (1990).

Other (1)

J. J. Wen and M. A. Breazeale, 'Computer optimization of the Gaussian beam description of an ultrasonic field,' in Computational Acoustics, D.Lee, A.Cakmak, and R.Vichnevetsky, eds. (Elsevier, 1990).

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Figures (7)

Fig. 1
Fig. 1

(a), (c) Amplitude of the Gaussian expansion for the hard-edged elliptical aperture function. (b), (d) Projections of (a) and (c), respectively. The parameters of the elliptical aperture are a = 1 mm , b = 2 mm , θ = π 6 . (a), (b) correspond to N = 10 ; (c), (d) correspond to N = 15 .

Fig. 2
Fig. 2

Intensity sections with y = 0 of a DEGB passing through an elliptical aperture followed by a free space; a = 1 mm , b = 2 mm , θ = 0 ° , r 0 T = ( 0 0 ) . Solid curves denote results with the approximate analytical Eq. (14); circles show results with the diffraction integral Eq. (10). Different propagation distances: (a) N x = 1 , (b) N x = 0.1 .

Fig. 3
Fig. 3

Three-dimensional normalized intensity distribution of a DEGB passing through an elliptical aperture followed by a free space, a = 1 mm , b = 2 mm , θ = 0 ° , r 0 T = ( 0.5 + 0.5 i 0.5 + 0.5 i ) . Different propagation distances: (a) N x = 1 , (b) N x = 0.1 .

Fig. 4
Fig. 4

Three-dimensional normalized intensity distribution of a DEGB passing through an elliptical aperture followed by a free space; N x = 1 , θ = 0 ° , r 0 T = ( 0.5 + 0.5 i 0.5 + 0.5 i ) . The radii of the two primary axes of the elliptical aperture are (a) a = 1 mm , b = 4 mm ; (b) a = 1 mm , b = 0.5 mm .

Fig. 5
Fig. 5

Three-dimensional normalized intensity distribution of a DEGB passing through an elliptical aperture followed by a free space; N x = 1 , a = 1 mm , b = 2 mm , r 0 T = ( 0.5 + 0.5 i 0.5 + 0.5 i ) . The azimuthal angles of the elliptical aperture are (a) θ = π 4 , (b) θ = π 2 , (c) θ = 3 π 4 , (d) θ = π .

Fig. 6
Fig. 6

Three-dimensional normalized intensity distribution of DEGB passing through an elliptical aperture followed by a free space; N x = 1 , a = 1 mm , b = 2 mm , θ = 0 ° . The decentered parameters are (a) r 0 T = ( 0 0 ) , (b) r 0 T = ( 1 1 ) , (c) r 0 T = ( i i ) , (d) r 0 T = ( 1 + i 1 + i ) .

Fig. 7
Fig. 7

Three-dimensional normalized intensity distribution of a DEGB passing through an apertured focusing system, z = f = 1580 mm , θ = 0 ° , r 0 T = ( 0.5 + 0.5 i 0.5 + 0.5 i ) . The radii of the two primary axes of the elliptical aperture are (a) a = 10 mm , b = 10 mm ; (b) a = 1 mm , b = 2 mm .

Equations (20)

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E ( r 1 ) = exp ( i k 2 r 1 T Q 1 r 1 ) ,
Q 1 = [ q x x 1 q x y 1 q y x 1 q y y 1 ] , with q x y 1 = q y x 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 ] ,
E ( r 1 ) = exp [ i k 2 ( r 1 r 0 ) T Q 1 ( r 1 r 0 ) ] ,
E 2 ( r 2 ) = i n 1 λ [ det ( B ) ] 1 2 exp ( i k l 0 ) E 1 ( r 1 ) exp ( i k l 1 ) d r 1 ,
l 1 = 1 2 ( r 1 r 2 ) T L ( r 1 r 2 ) .
L = [ n 1 B 1 A n 1 B 1 n 2 ( C D B 1 A ) n 2 D B 1 ] ,
( r 2 r 2 ) = [ A B C D ] ( r 1 r 1 ) .
A p ( r 1 ) = { 1 inside the elliptical aperture 0 outside the elliptical aperture } ,
E 2 ( r 2 ) = i λ [ det ( B ) ] 1 2 exp ( i k l 0 ) E 1 ( r 1 ) A p ( r 1 ) exp ( i k l 1 ) d r 1 .
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 ] ,
E ( r 2 ) = n = 1 N A n [ det ( A + B Q 1 + B P ) ] 1 2 exp ( i k l 0 ) exp { i k 2 r 2 T [ D B 1 B 1 T ( Q 1 + P + B 1 A ) 1 B 1 ] r 2 } exp [ i k r 0 T ( B + B P Q + A Q ) 1 r 2 ] exp { i k 2 r 0 T [ Q + ( B 1 A + P ) 1 ] 1 r 0 } ,
( B 1 A ) T = B 1 A , ( B 1 ) T = C D B 1 A , ( D B 1 ) T = D B 1 .
E ( r 2 ) = [ det ( A + B Q 1 ) ] 1 2 exp ( i k l 0 ) exp { i k 2 r 2 T [ D B 1 B 1 T ( Q 1 + B 1 A ) 1 B 1 ] r 2 } exp [ i k r 0 T ( B + A Q ) 1 r 2 ] exp [ i k 2 r 0 T ( Q + A 1 B ) 1 r 0 ] .
A = [ 1 0 0 1 ] , B = [ z 0 0 z ] , C = [ 0 0 0 0 ] ,
D = [ 1 0 0 1 ] .
A = [ 1 z f 0 0 1 z f ] , B = [ z 0 0 z ] ,
C = [ 1 f 0 0 1 f ] , D = [ 1 0 0 1 ] .

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