Abstract

On the basis of the fact that a hard-edged aperture function can be expressed as finite matrices with different weighting coefficients, we obtain the analytical formula for the propagation of the broadband Gaussian Schell- model (BGSM) beam through the apertured fractional Fourier transformation (AFrFT) system. It is shown by numerical examples that the intensity distribution in the plane of a small fractional order is obviously influenced by the bandwidth when the BGSM beams propagate through the AFrFT system. Further extensions are also pointed out.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  18. L. Z. Pan and B. D. Lu, “Spectral switches of polychromatic Gaussian beams passing through an astigmatic aperture lens,” Opt. Commun. 234, 13–22 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2010

2008

2006

2005

2004

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

L. Z. Pan and B. D. Lu, “Spectral switches of polychromatic Gaussian beams passing through an astigmatic aperture lens,” Opt. Commun. 234, 13–22 (2004).
[CrossRef]

2003

2002

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[CrossRef]

2000

F. Simin and G. W. Herbert, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[CrossRef]

1999

1997

Q. Cao and X. M. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

1993

1988

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

1985

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

1983

X. Deng, W. Yu, S. Chen, L. Ding, and W. Tan, “Output power increase of high power Nd: glass laser by bandwidth,” Acta Optica Sinica 3, 97–101 (1983) (in Chinese).
[CrossRef]

1970

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1170(1970).
[CrossRef]

1965

Breazeale, M. A.

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

Cai, Y. J.

Cao, Q.

Q. Cao and X. M. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Chen, S.

X. Deng, W. Yu, S. Chen, L. Ding, and W. Tan, “Output power increase of high power Nd: glass laser by bandwidth,” Acta Optica Sinica 3, 97–101 (1983) (in Chinese).
[CrossRef]

Christov, I. P.

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

Collins, S. A.

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1170(1970).
[CrossRef]

Deng, X.

X. Deng, W. Yu, S. Chen, L. Ding, and W. Tan, “Output power increase of high power Nd: glass laser by bandwidth,” Acta Optica Sinica 3, 97–101 (1983) (in Chinese).
[CrossRef]

Deng, X. M.

Q. Cao and X. M. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Ding, L.

X. Deng, W. Yu, S. Chen, L. Ding, and W. Tan, “Output power increase of high power Nd: glass laser by bandwidth,” Acta Optica Sinica 3, 97–101 (1983) (in Chinese).
[CrossRef]

Du, X. Y.

Erdelyi, A.

A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, 1954).

Fan, D. Y.

Gao, Y. Q.

Goodman, J. W.

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

Herbert, G. W.

F. Simin and G. W. Herbert, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[CrossRef]

Jing, F.

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

Lin, Q.

Liu, D. Z.

Liu, H. J.

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

Liu, Z. Q.

Lohmann, A. W.

Lu, B. D.

L. Z. Pan and B. D. Lu, “Spectral switches of polychromatic Gaussian beams passing through an astigmatic aperture lens,” Opt. Commun. 234, 13–22 (2004).
[CrossRef]

Malitson, I. H.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Mao, H. D.

H. D. Mao and D. M. Zhao, “Second-order intensity-moment characteristics for broadband partially coherent flat-topped beams in atmospheric turbulence,” Opt. Express 18, 1741–1755(2010).
[CrossRef] [PubMed]

H. D. Mao and D. M. Zhao, “Parametric characteristics for a broadband Gaussian beam in free space,” Appl. Phys. B 100, 611–616 (2010).
[CrossRef]

H. D. Mao and D. M. Zhao, “Three 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]

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

Pan, L. Z.

L. Z. Pan and B. D. Lu, “Spectral switches of polychromatic Gaussian beams passing through an astigmatic aperture lens,” Opt. Commun. 234, 13–22 (2004).
[CrossRef]

Peng, R. W.

Porras, M. A.

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[CrossRef]

M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16, 1468–1474 (1999).
[CrossRef]

Simin, F.

F. Simin and G. W. Herbert, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[CrossRef]

Tan, W.

X. Deng, W. Yu, S. Chen, L. Ding, and W. Tan, “Output power increase of high power Nd: glass laser by bandwidth,” Acta Optica Sinica 3, 97–101 (1983) (in Chinese).
[CrossRef]

Tang, Z. X.

Wang, F.

Wang, S.

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

Wang, S. M.

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

Wei, X. F.

D. M. Zhao, H. D. Mao, H. J. Liu, S. M. Wang, F. Jing, and X. F. 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 a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Ye, Y. X.

Yu, W.

X. Deng, W. Yu, S. Chen, L. Ding, and W. Tan, “Output power increase of high power Nd: glass laser by bandwidth,” Acta Optica Sinica 3, 97–101 (1983) (in Chinese).
[CrossRef]

Zhao, D.

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

Zhao, D. M.

Zhu, B. Q.

Acta Optica Sinica

X. Deng, W. Yu, S. Chen, L. Ding, and W. Tan, “Output power increase of high power Nd: glass laser by bandwidth,” Acta Optica Sinica 3, 97–101 (1983) (in Chinese).
[CrossRef]

Appl. Opt.

Appl. Phys. B

H. D. Mao and D. M. Zhao, “Parametric characteristics for a broadband Gaussian beam in free space,” Appl. Phys. B 100, 611–616 (2010).
[CrossRef]

J. Acoust. Soc. Am.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

Q. Cao and X. M. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

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

L. Z. Pan and B. D. Lu, “Spectral switches of polychromatic Gaussian beams passing through an astigmatic aperture lens,” Opt. Commun. 234, 13–22 (2004).
[CrossRef]

Opt. Express

Phys. Rev. E

F. Simin and G. W. Herbert, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[CrossRef]

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[CrossRef]

Other

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

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

A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, 1954).

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Figures (6)

Fig. 1
Fig. 1

Two optical setups implementing AFrFT.

Fig. 2
Fig. 2

Intensity distribution of the BGSM beam in the input plane.

Fig. 3
Fig. 3

Intensity distributions of the BGSM beams with different bandwidths propagating through the type-I AFrFT system, when a 1 / w 0 = 0.5 , σ 0 / w 0 = 0.5 .

Fig. 4
Fig. 4

Intensity distributions of the BGSM beams with different bandwidths propagating through the type-II AFrFT system, when a 1 / w 0 = 0.5 , σ 0 / w 0 = 0.5 .

Fig. 5
Fig. 5

Effect of different spatial correlation lengths on the intensity distributions in the case p = 0.2 , when the BGSM beam with different bandwidths propagates through the type-II AFrFT system with a 1 / w 0 = 0.5 .

Fig. 6
Fig. 6

Effect of different sizes of aperture on the intensity distributions in the case p = 0.2 , when the BGSM beam with different bandwidth propagates through the type-II AFrFT system. σ 0 / w 0 = 0.5 .

Equations (17)

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A p ( x ) = n = 1 N A 1 ( n ) exp [ B 1 ( n ) a 1 2 x 2 ] ,
[ A ( n ) B ( n ) C ( n ) D ( n ) ] = [ 1 0 2 B 1 ( n ) / ( i k a 1 2 ) 1 ] ,
d 1 = f 0 tan ( p π / 4 ) ,
f 1 = f 0 / sin ( p π / 2 ) ,
d 2 = f 0 sin ( p π / 2 ) ,
f 2 = f 0 / tan ( p π / 4 ) .
[ A ( n ) B ( n ) C ( n ) D ( n ) ] = ( 1 d 1 0 1 ) ( 1 0 1 / f 1 1 ) [ 1 0 2 B 1 ( n ) / ( i k a 1 2 ) 1 ] ( 1 d 1 0 1 ) = [ 1 d 1 / f 1 2 d 1 B 1 ( n ) / ( i k a 1 2 ) 2 d 1 d 1 2 / f 1 2 d 1 2 B 1 ( n ) / ( i k a 1 2 ) 1 / f 1 2 B 1 ( n ) / ( i k a 1 2 ) 1 d 1 / f 1 2 d 1 B 1 ( n ) / ( i k a 1 2 ) ] .
[ A ( n ) B ( n ) C ( n ) D ( n ) ] = ( 1 0 1 / f 2 1 ) ( 1 d 2 0 1 ) ( 1 0 1 / f 2 1 ) ( 1 0 2 B 1 ( n ) / ( i k a 1 2 ) 1 ) = [ 1 d 2 / f 2 2 d 2 B 1 ( n ) / ( i k a 1 2 ) d 2 2 / f 2 + d 2 / f 2 2 2 B 1 ( n ) / ( i k a 1 2 ) + 2 d 2 B 1 ( n ) / ( i k a 1 2 f 2 ) 1 d 2 / f 2 ] .
E ( x , ω ) = exp ( i k z ) n = 1 N A 1 ( n ) i k 2 π B ( n ) + E ( x ) exp { i k 2 B ( n ) [ A ( n ) x 2 2 x x + D ( n ) x 2 ] } d x .
W ( x 1 , x 2 , ω ) = E * ( x 1 , ω ) E ( x 2 , ω ) .
W ( x 1 , x 2 , ω ) = k 2 π n = 1 N m = 1 N A 1 ( n ) A 1 * ( m ) 1 B ( n ) B * ( m ) exp { i k 2 [ D * ( m ) B * ( m ) x 1 2 + D ( n ) B ( n ) x 2 2 ] } × + + W ( x 1 , x 2 , ω ) exp { i k 2 [ A * ( m ) B * ( m ) x 1 2 + A ( n ) B ( n ) x 2 2 ] i k [ 1 B * ( m ) x 1 x 1 + 1 B ( n ) x 2 x 2 ] } d x 1 d x 2 .
W ( x 1 , x 2 , z = 0 , ω ) = S 0 ( ω ) exp ( x 1 2 + x 2 2 w 0 2 ) exp [ ( x 1 x 2 ) 2 2 σ 0 2 ] ,
f ( λ ) = n ( λ 0 ) 1 n ( λ ) 1 f ( λ 0 ) ,
n 2 ( λ ) = 1 + i = 1 3 B i 1 λ i 2 / λ 2 ,
W ( x 1 , x 2 , ω ) = S 0 ( ω ) n = 1 N m = 1 N A 1 * ( m ) A 1 ( n ) ω 2 c B * ( m ) B ( n ) 1 E ( n ) 1 E * ( m ) 1 / [ 4 σ 0 4 E ( n ) ] × exp { i ω 2 c [ D ( n ) B ( n ) x 2 2 D * ( m ) B * ( m ) x 1 2 ] } exp [ ω 2 x 2 2 4 c 2 B 2 ( n ) E ( n ) ] exp ( ω 2 4 c 2 { x 1 / B * ( m ) + x 2 / [ 2 σ 0 2 B ( n ) E ( n ) ] } 2 E * ( m ) 1 / [ 4 σ 0 4 E ( n ) ] ) ,
E ( j ) = 1 / w 0 2 + 1 / ( 2 σ 0 2 ) i ω A ( j ) / [ 2 c B ( j ) ] , j = m or n .
I ( x ) = W ( x , x , ω ) d ω .

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