Abstract

The fractional Fourier transform (FRFT) of the flat-topped multi-Gaussian beam (FMGB) is investigated based on the three kinds of FRFT optical systems: Lohmann I, Lohmann II, and quadratic graded-index systems. The analytical expressions for the FRFT of the FMGB are derived based on the propagation of the FMGB through the three systems. By introducing a hard-edge aperture function, the analytical expressions for the FRFT of the FMGB carried out by the apertured FRFT optical systems are presented. The FRFT characteristics of the FMGB for the three kinds of FRFT optical systems with and without apertures are discussed in detail. Results show that the three types of FRFT optical systems have the same function when the apertures are ignored but that significantly different characteristics are exhibited when the apertures appear.

© 2010 Optical Society of America

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References

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2009 (6)

2008 (1)

2007 (4)

2005 (1)

2002 (1)

2001 (1)

1995 (1)

1994 (4)

1993 (3)

1988 (2)

S. D. Silvestri, P. Laporta, V. Magni, O. Svelto, and B. Majocchi, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201-203 (1988).
[CrossRef] [PubMed]

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]

1986 (1)

1980 (1)

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

1970 (1)

Alieva, T.

Bandres, M. A.

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.

Calvo, M. L.

Chen, J. N.

Collins, S. A.

Cui, Y. F.

Gao, Q.

Gao, Y. Q

Gao, Y. Q.

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335-341 (1994).
[CrossRef]

Gutierrez-Vaga, J. C.

Laporta, P.

Li, Y.

Liao, T. H.

Lin, Z. Q.

Liu, D. Z.

Liu, S. T.

Liu, Z. J.

Lohmann, A. W.

Magni, V.

Majocchi, B.

Mao, H.

McMullin, J. N.

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.

Pellat-Finet, P.

Rodrigo, J. A.

Silvestri, S. D.

Svelto, O.

Tang, B.

B.Tang and M. H. Xu, “Fractional Fourier transform for beams generated by Gaussian mirror resonator,” J. Mod. Opt. 56, 1276-1282 (2009).
[CrossRef]

Tovar, A. A.

Wang, F.

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]

Xu, M. H.

B.Tang and M. H. Xu, “Fractional Fourier transform for beams generated by Gaussian mirror resonator,” J. Mod. Opt. 56, 1276-1282 (2009).
[CrossRef]

Zhao, D.

Zhou, G. Q.

G. Q. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 2009. 56, 886-892 (2009).
[CrossRef]

G. Q. Zhou, “Fractional Fourier transform of Lorentz-Gauss beams,” J. Opt. Soc. Am. A 26, 350-355 (2009).
[CrossRef]

Zhu, B. Q.

Appl. Opt. (2)

J. Acoust. Soc. Am. (1)

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. 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. Mod. Opt. (2)

G. Q. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 2009. 56, 886-892 (2009).
[CrossRef]

B.Tang and M. H. Xu, “Fractional Fourier transform for beams generated by Gaussian mirror resonator,” J. Mod. Opt. 56, 1276-1282 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (10)

J. N. Chen, “Propagation and transformation of flat-topped multi-Gaussian beams in a general nonsymmetrical apertured double-lens system,” J. Opt. Soc. Am. A 24, 84-92 (2007).
[CrossRef]

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]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, and Z. Q. Lin, “Propagation of flat-topped multi-Gaussian beams through an apertured ABCD optical system,” J. Opt. Soc. Am. A 26, 2139-2146 (2009).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743-751 (1995).
[CrossRef]

G. Q. Zhou, “Fractional Fourier transform of Lorentz-Gauss beams,” J. Opt. Soc. Am. A 26, 350-355 (2009).
[CrossRef]

F. Wang and Y. J. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24, 1937-1944 (2007).
[CrossRef]

A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A 18, 1897-1904 (2001).
[CrossRef]

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

D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875-1881 (1993).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522-2531 (1993).
[CrossRef]

Opt. Commun. (1)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335-341 (1994).
[CrossRef]

Opt. Express (3)

Opt. Lett. (6)

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Figures (11)

Fig. 1
Fig. 1

Three kinds of FRFT systems: (a) Lohmann I system, (b) Lohmann II system, (c) GRIN system

Fig. 2
Fig. 2

On-axis intensity distribution of the FMGB versus FRFT order: (a) the beam width W 1 = 0.4 mm and the FMGB order changes; (b) the beam order N = M = 8 and the beam width changes.

Fig. 3
Fig. 3

1-D normalized intensity distributions of the FMGB in the different FRFT planes with different FMGB orders: (a) N = M = 1 , (b) N = M = 4 , (c) N = M = 8 , (d) N = M = 12 .

Fig. 4
Fig. 4

Normalized intensity distributions of the FMGB in different FRFT planes (a) p = 0 , (b) p = 0.2 , (c) p = 0.4 , (d) p = 0.6 , (e) p = 0.8 , (f) p = 1 when N = M = 8 and W 1 = 0.4 mm .

Fig. 5
Fig. 5

Normalized intensity distribution of the FMGB in the GRIN medium: (a) 2-D intensity distributions, (b) cross section at different distances z when N = M = 10 and a = 10 .

Fig. 6
Fig. 6

On-axis intensity distributions of the FMGB versus FRFT order with different aperture sizes W 2 when FMGB order is 4, FMGB width W 1 = 0.4 mm , and aperture order is 8.

Fig. 7
Fig. 7

1-D normalized intensity distributions of the FMGB in different FRFT planes when FMGB order is 4, FMGB width W 1 = 0.4 mm , aperture order is 8, and aperture sizes W 2 are (a) 0.2 mm , (b) 0.4 mm , (c) 1 mm , (d) 10 mm

Fig. 8
Fig. 8

Normalized intensity distributions of the fourth-order FMGB in the different-order FRFT planes: (a) p = 0.2 , (b) p = 0.4 , (c) p = 0.8 , (d) p = 1.2 , (e) p = 1.6 , (f) p = 1.8 when aperture size W 2 = 1 mm and aperture order is 8.

Fig. 9
Fig. 9

On-axis intensity distributions of the FMGB versus FRFT order with different aperture size W 2 when FMGB order is 4, FMGB width W 1 = 0.4 mm , and aperture order is 8.

Fig. 10
Fig. 10

Normalized intensity distributions of the FMGB in the different-order FRFT planes: (a) p = 0.2 , (b) p = 0.4 , (c) p = 0.6 , (d) p = 0.8 when beam size W 1 = 0.5 mm , aperture size W 2 = 0.4 mm , and aperture order N 2 is different ( M 2 = N 2 )

Fig. 11
Fig. 11

Normalized intensity distribution of the fourth-order FMGB in the GRIN medium with a tenth-order aperture in the input plane when beam size W 1 = 0.4 mm and aperture size W 2 = 0.3 mm

Equations (19)

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( A B C D ) = ( cos ϕ f sin ϕ 1 f sin ϕ cos ϕ ) .
( A B C D ) = ( cos ( z a ) a sin ( z a ) 1 a sin ( z a ) cos ( z a ) ) .
E p ( x 3 , y 3 ) = i k 2 π B E ( x 1 , y 1 ) exp { i k 2 B [ A ( x 1 2 + y 1 2 ) 2 ( x 1 x 3 + y 1 y 3 ) + D ( x 3 2 + y 3 2 ) ] } d x 1 d y 1 .
E ( x 1 , y 1 ) = n 1 = N 1 N 1 m 1 = M 1 M 1 exp { 1 ω 1 2 [ ( x 1 m 1 ω 1 ) 2 + ( y 1 n 1 ω 1 ) 2 ] } n 1 = N 1 N 1 m 1 = M 1 M 1 exp [ ( m 1 2 + n 1 2 ) ] ,
W 1 = ω 1 × ( N 1 + { 1 ln [ n 1 = N 1 N 1 exp ( n 1 2 ) ] } 1 2 ) .
exp ( α 2 x 2 ± q x ) d x = exp ( q 2 4 α 2 ) π α ( Re α 2 > 0 ) ,
E p ( x 3 , y 3 ) = i k 2 π f sin ϕ π P 1 n 1 = N 1 N 1 m 1 = M 1 M 1 exp [ ( n 1 2 + m 1 2 ) ] n 1 = N 1 N 1 m 1 = M 1 M 1 exp [ i k cos ϕ 2 f sin ϕ ( x 3 2 + y 3 2 ) m 1 2 n 1 2 + 1 4 P 1 ( 2 m 1 ω 1 + i k x 3 f sin ϕ ) 2 + 1 4 P 1 ( 2 n 1 ω 1 + i k y 3 f sin ϕ ) 2 ] ,
P 1 = 1 ω 1 2 + i k cos ϕ 2 f sin ϕ .
E p ( x 3 ) = i k 2 π a sin ( z a ) π P 2 m 1 = M 1 M 1 exp ( m 1 2 ) m 1 = M 1 M 1 exp { i k x 3 2 cos ( z a ) 2 a sin ( z a ) m 1 2 + 1 4 P 2 [ 2 m 1 ω 1 + i k x 3 a sin ( z a ) ] 2 } ,
P 2 = 1 ω 1 2 + i k cos ( z a ) 2 a sin ( z a ) .
A P ( x 2 , y 2 ) = n 2 = N 2 N 2 m 2 = M 2 M 2 exp { 1 ω 2 2 [ ( x 2 m 2 ω 2 ) 2 + ( y 2 n 2 ω 2 ) 2 ] } n 2 = N 2 N 2 m 2 = M 2 M 2 exp [ ( n 2 2 + m 2 2 ) ] .
( A 1 B 1 C 1 D 1 ) = ( 1 f tan ( ϕ 2 ) 0 1 ) , ( A 2 B 2 C 2 D 2 ) = ( cos ( ϕ ) f tan ( ϕ 2 ) sin ( ϕ ) f 1 ) ,
E p ( x 3 , y 3 ) = i k 2 π B 2 i k 2 π B 1 A P ( x 2 , y 2 ) E ( x 1 , y 1 ) exp { i k 2 B 1 [ A 1 ( x 1 2 + y 1 2 ) 2 ( x 1 x 2 + y 1 y 2 ) + D 1 ( x 2 2 + y 2 2 ) ] } d x 1 d y 1 exp { i k 2 B 2 [ A 2 ( x 2 2 + y 2 2 ) 2 ( x 2 x 3 + y 2 y 3 ) + D 2 ( x 3 2 + y 3 2 ) ] } d x 2 d y 2 .
E p ( x 3 , y 3 ) = i k 2 π f tan ( ϕ 2 ) i k 2 π f tan ( ϕ 2 ) 1 n 1 = N 1 N 1 m 1 = M 1 M 1 exp [ ( n 1 2 + m 1 2 ) ] 1 n 2 = N 2 N 2 m 2 = M 2 M 2 exp [ ( n 2 2 + m 2 2 ) ] π P 3 π P 4 n 2 = N 2 N 2 m 2 = M 2 M 2 n 1 = N 1 N 1 m 1 = M 1 M 1 exp { m 2 2 n 2 2 + ( 1 P 3 ω 1 2 1 ) ( m 1 2 + n 1 2 ) i k 2 f tan ( ϕ 2 ) ( x 3 2 + y 3 2 ) + 1 4 P 4 [ 2 m 2 ω 2 + i k m 1 P 3 ω 1 f tan ( ϕ 2 ) + i k x 3 f tan ( ϕ 2 ) ] 2 + 1 4 P 4 [ 2 n 2 ω 2 + i k n 1 P 3 ω 1 f tan ( ϕ 2 ) + i k y 3 f tan ( ϕ 2 ) ] 2 } ,
P 3 = 1 ω 1 2 + i k 2 f tan ( ϕ 2 ) , P 4 = 1 ω 2 2 + i k 2 f tan ( ϕ 2 ) 1 4 P 3 [ i k f tan ( ϕ 2 ) ] 2 + i k cos ( ϕ ) 2 f tan ( ϕ 2 ) .
E p ( x 3 , y 3 ) = i k 2 π f sin ϕ 1 n 1 = N 1 N 1 m 1 = M 1 M 1 exp [ ( n 1 2 + m 1 2 ) ] 1 n 2 = N 2 N 2 m 2 = M 2 M 2 exp [ ( n 2 2 + m 2 2 ) ] π P 5 n 2 = N 2 N 2 m 2 = M 2 M 2 m 1 = N 1 N 1 m 1 = M 1 M 1 exp [ i k cos ϕ 2 f sin ϕ ( x 3 2 + y 3 2 ) m 1 2 n 1 2 m 2 2 n 2 2 + 1 4 P 5 ( 2 m 1 ω 1 + 2 m 2 ω 2 + i k x 3 f sin ϕ ) 2 + 1 4 P 5 ( 2 n 1 ω 1 + 2 n 2 ω 2 + i k y 3 f sin ϕ ) 2 ] ,
P 5 = 1 ω 1 2 + 1 ω 2 2 + i k cos ( ϕ ) 2 f sin ( ϕ ) .
E p ( x 3 ) = i k 2 π a sin ( z a ) 1 m 1 = M 1 M 1 exp ( m 1 2 ) 1 m 2 = M 2 M 2 exp ( m 2 2 ) π P 6 m 2 = M 2 M 2 m 1 = M 1 M 1 exp { i k cos ( z a ) x 3 2 2 a sin ( z a ) m 1 2 m 2 2 + 1 4 P 6 [ 2 m 1 ω 1 + 2 m 2 ω 2 + i k x 3 a sin ( z a ) ] 2 } ,
P 6 = 1 ω 1 2 + 1 ω 2 2 + i k cos ( z a ) 2 a sin ( z a ) .

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