Abstract

On the basis of 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 propagation expression for an elliptical Gaussian beam (EGB) through a misaligned optical system with an elliptical aperture is derived by use of vector integration. The approximate analytical results provide more convenience for studying the propagation and transformation of EGBs than the usual way by using a diffraction integral directly, and the efficiency of numerical calculation is improved. Some numerical simulations are illustrated for the propagation properties of EGBs through apertured optical transforming systems with misaligned thin lenses.

© 2006 Optical Society of America

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References

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  1. S. A. Collins, "Lens-systems diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970).
    [CrossRef]
  2. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into a symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).
  3. 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]
  4. S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).
  5. D. Zhao, F. Ge, and S. Wang, "Generalized diffraction integral formula for misaligned optical systems in spatial-frequency domain," Optik (Stuttgart) 112, 268-270 (2001).
    [CrossRef]
  6. Y. Cai and Q. Lin, "Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatial-frequency domain," Opt. Laser Technol. 34, 415-421 (2002).
    [CrossRef]
  7. 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]
  8. 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).
  9. 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]
  10. 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]
  11. 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]
  12. 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]

2006 (1)

2004 (2)

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]

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)

Y. Cai and Q. Lin, "Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatial-frequency domain," Opt. Laser Technol. 34, 415-421 (2002).
[CrossRef]

2001 (1)

D. Zhao, F. Ge, and S. Wang, "Generalized diffraction integral formula for misaligned optical systems in spatial-frequency domain," Optik (Stuttgart) 112, 268-270 (2001).
[CrossRef]

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 a 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]

1970 (1)

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 a symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).

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 a 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.

Y. Cai and Q. Lin, "Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatial-frequency domain," Opt. Laser Technol. 34, 415-421 (2002).
[CrossRef]

Collins, S. A.

Du, X.

Ge, F.

D. Zhao, F. Ge, and S. Wang, "Generalized diffraction integral formula for misaligned optical systems in spatial-frequency domain," Optik (Stuttgart) 112, 268-270 (2001).
[CrossRef]

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, "Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatial-frequency domain," Opt. Laser Technol. 34, 415-421 (2002).
[CrossRef]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into a 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.

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]

D. Zhao, F. Ge, and S. Wang, "Generalized diffraction integral formula for misaligned optical systems in spatial-frequency domain," Optik (Stuttgart) 112, 268-270 (2001).
[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 a symmetric one by means of tensor ABCD law," Optik (Stuttgart) 85, 67-72 (1990).

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

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.

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]

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]

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, F. Ge, and S. Wang, "Generalized diffraction integral formula for misaligned optical systems in spatial-frequency domain," Optik (Stuttgart) 112, 268-270 (2001).
[CrossRef]

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

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. (1)

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

Opt. Commun. (3)

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]

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]

Opt. Laser Technol. (1)

Y. Cai and Q. Lin, "Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatial-frequency domain," Opt. Laser Technol. 34, 415-421 (2002).
[CrossRef]

Optik (Stuttgart) (2)

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

D. Zhao, F. Ge, and S. Wang, "Generalized diffraction integral formula for misaligned optical systems in spatial-frequency domain," Optik (Stuttgart) 112, 268-270 (2001).
[CrossRef]

Other (2)

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

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 (5)

Fig. 1
Fig. 1

Misaligned optical transforming system.

Fig. 2
Fig. 2

Apertured optical transforming system with a misaligned thin lens.

Fig. 3
Fig. 3

Intensity distribution of an EGB passing through an apertured optical transforming system with a misaligned thin lens: a = 1 mm , b = 2 mm , θ = 0 , and the thin lens is located at a distance z 1 = 1000 mm . (a) Three-dimensional plot of the intensity distribution obtained by Eq. (11); (b) two-dimensional plot of the intensity profile across the plane of y = 0 obtained by the diffraction integral formula of Eq. (7) (solid curves) and the analytical propagation of Eq. (11) (dotted curves).

Fig. 4
Fig. 4

Intensity distribution of an EGB passing through an apertured optical transforming system with a misaligned thin lens: a = 1 mm , b = 2 mm , θ = 0 . The lens is located at distances (a) z 1 = 0 , (b) z 1 = 2000 mm , (c) z 1 = 4000.001 mm , (d) z 1 = 6000 mm .

Fig. 5
Fig. 5

Intensity distribution of an EGB passing through an apertured optical transforming system with a misaligned thin lens, and the thin lens is located at the distance z 1 = 1000 mm : (a) a = 10 mm , b = 10 mm , θ = 0 ; (b) a = 1 mm , b = 2 mm , θ = π 6 .

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 .
E 2 ( r 2 ) = i λ [ det ( B ) ] 1 2 E 1 ( r 1 ) exp [ i k 2 ( r 1 T B 1 Ar 1 2 r 1 T B 1 r 2 + r 2 T DB 1 r 2 ) ] exp [ i k 2 ( r 1 T B 1 e f + r 2 T B 1 g h ) ] d r 1 ,
A = [ a 0 0 a ] , B = [ b 0 0 b ] ,
C = [ c 0 0 c ] , D = [ d 0 0 d ] ,
e = 2 ( α T ϵ x + β T ϵ x ) , f = 2 ( α T ϵ y + β T ϵ y ) ,
g = 2 ( b γ T d α T ) ϵ x + 2 ( b δ T d β T ) ϵ x ,
h = 2 ( b γ T d α T ) ϵ y + 2 ( b δ T d β T ) ϵ y ,
A p ( r 1 ) = { 1 inside the elliptical aperture 0 outside the elliptical aperture } ,
E 2 ( r 2 ) = i λ [ det ( B ) ] 1 2 A p ( r 1 ) E 1 ( r 1 ) exp [ i k 2 ( r 1 T B 1 Ar 1 2 r 1 T B 1 r 2 + r 2 T DB 1 r 2 ) ] × exp [ i k 2 ( r 1 T B 1 e f + r 2 T B 1 g h ) ] d r 1 .
A p ( r 1 ) = n = 1 N A n exp ( r 1 T R T P n Rr 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 + BQ 1 + BP ) ] 1 2 exp { i k 2 r 2 T [ DB 1 B 1 T ( Q 1 + P + B 1 A ) 1 B 1 ] r 2 } exp { i k 2 r 2 T [ B 1 g h + B 1 T ( Q 1 + P + B 1 A ) 1 B 1 e f ] } exp [ i k 8 e f T B 1 T ( Q 1 + P + B 1 A ) 1 B 1 e f ] ,
E ( r 2 ) = [ det ( A + BQ 1 ) ] 1 2 exp { i k 2 r 2 T [ DB 1 B 1 T ( Q 1 + B 1 A ) 1 B 1 ] r 2 } × exp { i k 2 r 2 T [ B 1 g h + B 1 T ( Q 1 + B 1 A ) 1 B 1 e f ] } × exp [ i k 8 e f T B 1 T ( Q 1 + B 1 A ) 1 B 1 e f ] .
A = [ 1 ( z z 1 ) f 1 0 0 1 ( z z 1 ) f 1 ] ,
B = [ z z 1 ( z z 1 ) f 1 0 0 z z 1 ( z z 1 ) f 1 ] ,
C = [ 1 f 1 0 0 1 f 1 ] , D = [ 1 z 1 f 1 0 0 1 z 1 f 1 ] .
α T = ( z z 1 ) f 1 , β T = z 1 ( z z 1 ) f 1 , γ T = 1 f 1 , δ T = z 1 f 1 ,
e = 2 ( z z 1 ) ϵ x f 1 , f = 0 , g = 2 z 1 ϵ x f 1 , h = 0 .

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