Abstract

A partially coherent beam with flat-topped profile is proposed. The cross-spectral density of this beam can be expressed as a finite series of the cross-spectral density of partially coherent Gaussian-Schell-model beams with different parameters. Analytical propagation formulas for partially coherent flat-topped beams are derived through aligned and misaligned optical systems. The propagation property of partially coherent flat-topped beams in free space is illustrated numerically. The fractional Fourier transform of partially coherent flat-topped beams is also studied. Our method provides a convenient way to describe partially coherent flat-topped beams and treat their propagation and transformation.

© 2004 Optical Society of America

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References

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    [CrossRef]
  2. M. R. Perrone, A. Piegari, “On the super-Gaussian unstable resonators for high-gain short pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
    [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]
  9. D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integral intensity of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).
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    [CrossRef]
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    [CrossRef]
  12. R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 18, 1627–1633 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-top beams,” Opt. Lett. 23, 313–315 (1998).
    [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]
  28. D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, H. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424–2431 (1995).
    [CrossRef]
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    [CrossRef]
  32. J. Hua, L. Liu, G. Li, “Scaled fractional Fourier transform and its optical implementation,” Appl. Opt. 36, 8490–8492 (1997).
    [CrossRef]
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    [CrossRef]

2002 (5)

2001 (3)

2000 (1)

1999 (3)

1998 (4)

Y. Zhang, B. Dong, B. Gu, G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114–1120 (1998).
[CrossRef]

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
[CrossRef]

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integral intensity of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-top beams,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

1997 (2)

J. Hua, L. Liu, G. Li, “Scaled fractional Fourier transform and its optical implementation,” Appl. Opt. 36, 8490–8492 (1997).
[CrossRef]

M. Santarsiero, D. Alello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

1996 (3)

1995 (1)

1994 (1)

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

1993 (4)

1992 (1)

1983 (1)

1982 (1)

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian-Schell-model beams,” Opt. Commun. 41, 187–194 (1982).
[CrossRef]

1980 (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Aiello, D.

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integral intensity of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

Alello, D.

M. Santarsiero, D. Alello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

Amarande, S. A.

S. A. Amarande, “Beam propagation factor and the kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).

Bagini, V.

Belafhal, A.

M. Ibnchaikh, A. Belafhal, “Closed-term propagation expressions of flattened Gaussian beams through an apertured ABCD optical system,” Opt. Commun. 193, 73–79 (2001).
[CrossRef]

Bitran, Y.

Borghi, R.

R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 18, 1627–1633 (2001).
[CrossRef]

M. Santarsiero, R. Borghi, “Correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integral intensity of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-top beams,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
[CrossRef]

M. Santarsiero, D. Alello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

Bowers, M. S.

Bulabois, J.

Cai, Y.

Courjon, D.

Deschamps, J.

Dong, B.

Dorsch, R. G.

Erden, M. F.

Friberg, A. T.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian-Schell-model beams,” Opt. Commun. 41, 187–194 (1982).
[CrossRef]

Gori, F.

Gu, B.

Hua, J.

Ibnchaikh, M.

M. Ibnchaikh, A. Belafhal, “Closed-term propagation expressions of flattened Gaussian beams through an apertured ABCD optical system,” Opt. Commun. 193, 73–79 (2001).
[CrossRef]

Li, G.

Li, Y.

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002).
[CrossRef]

Lin, Q.

Liu, L.

Liu, S.

Lohmann, A. W.

Lu, B.

Luo, S.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995), Chap. 5.
[CrossRef]

Mendlovic, D.

Mukunda, N.

Namias, V.

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Ozaktas, H.

Ozaktas, H. M.

Pacileo, A. M.

Perrone, M. R.

M. R. Perrone, A. Piegari, “On the super-Gaussian unstable resonators for high-gain short pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Piegari, A.

M. R. Perrone, A. Piegari, “On the super-Gaussian unstable resonators for high-gain short pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Ronchi, L.

S. Wang, L. Ronchi, “Principles and design of optical arrays,” in Progress in Optics, Vol. XXV, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 1988), p. 279.

Santarsiero, M.

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

M. Santarsiero, R. Borghi, “Correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
[CrossRef]

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integral intensity of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-top beams,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

M. Santarsiero, D. Alello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

Simon, R.

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian-Schell-model beams,” Opt. Commun. 41, 187–194 (1982).
[CrossRef]

Vicalvi, S.

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integral intensity of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
[CrossRef]

M. Santarsiero, D. Alello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

Wang, S.

S. Wang, L. Ronchi, “Principles and design of optical arrays,” in Progress in Optics, Vol. XXV, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 1988), p. 279.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995), Chap. 5.
[CrossRef]

Yang, G.

Zalevsky, Z.

Zhang, Y.

Zhu, B.

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

M. R. Perrone, A. Piegari, “On the super-Gaussian unstable resonators for high-gain short pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

J. Mod. Opt. (1)

M. Santarsiero, D. Alello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (11)

B. Lu, S. Luo, “General propagation equation of flattened Gaussian beams,” J. Opt. Soc. Am. A 17, 2001–2004 (2000).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
[CrossRef]

D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, H. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424–2431 (1995).
[CrossRef]

R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 18, 1627–1633 (2001).
[CrossRef]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

M. Santarsiero, R. Borghi, “Correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
[CrossRef]

R. Simon, N. Mukunda, “Gaussian-Schell-model beams and general shape invariant,” J. Opt. Soc. Am. A 16, 2465–2475 (1999).
[CrossRef]

Y. Zhang, B. Dong, B. Gu, G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114–1120 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Twisted Gaussian-Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

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

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

Opt. Commun. (7)

M. Ibnchaikh, A. Belafhal, “Closed-term propagation expressions of flattened Gaussian beams through an apertured ABCD optical system,” Opt. Commun. 193, 73–79 (2001).
[CrossRef]

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

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

S. A. Amarande, “Beam propagation factor and the kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
[CrossRef]

Y. Cai, Q. Lin, “Propagation of partially coherent twisted anisotropic Gaussian-Schell-model beams through misaligned optical system,” Opt. Commun. 211, 1–8 (2002).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian-Schell-model beams,” Opt. Commun. 41, 187–194 (1982).
[CrossRef]

Opt. Lett. (6)

Optik (1)

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integral intensity of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

Other (2)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995), Chap. 5.
[CrossRef]

S. Wang, L. Ronchi, “Principles and design of optical arrays,” in Progress in Optics, Vol. XXV, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 1988), p. 279.

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Figures (6)

Fig. 1
Fig. 1

Intensity distribution of the flat-topped light beams with different N at z = 0. (a) N = 5, (b) N = 8, (c) N = 10, (d) N = 15.

Fig. 2
Fig. 2

Modulus of the coherence degree of the partially coherent flat-topped beam between two transverse points r 1 = (0 0) mm, r 2 = (0.2 0) mm versus the order of the partially coherent flat-topped beams.

Fig. 3
Fig. 3

Misalignment diagram for a two-dimensional forward-going system.

Fig. 4
Fig. 4

Normalized 3-D intensity distribution of the partially coherent flat-topped beam in free space propagation at different propagation distances. (a) z = 0, (b) z = 300, (c) z = 1500, (d) z = 2000.

Fig. 5
Fig. 5

Optical system for performing the FRT. (a) one-lens system, (b) two-lens system.

Fig. 6
Fig. 6

Normalized 3-D-intensity distribution of partially coherent flat-topped beam in the fractional Fourier transform plane versus different fractional order p. (a) p = 0.2, (b) p = 0.8, (c) p = 1, (d) p = 1.5.

Equations (26)

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

ENr=n=1N-1n-1NNnexp-nr2w02,
ΓNr1, r2=n=1N-1n-1NNnexp-nr12+r224σI02+r1-r222σg02,
Ir=ΓNr, r=n=1N-1n-1NNnexp-nr22σI02.
μ=Γr1, r2Γr1, r1Γr2, r21/2.
ΓNr1, r2=n=1N-1n-1NNnexp-ik2 r˜TM1n-1r˜,
M1n-1=-ni2kσI02-nikσg02Inikσg02Inikσg02I-ni2kσI02-nikσg02I.
Γρ˜=1λ2 detB¯  Γr˜exp-ikl1-l2dr˜,
l1-l2=12r˜ρ˜TB¯-1Ā-B¯-1C¯-D¯B¯-1ĀD¯B¯-1r˜ρ˜,
Ā=A00A,B¯=B00-B,C¯=C00-C,D¯=D00D.
ΓNρ˜=n=1N-1n-1NNndetĀ+B¯M1n-11/2×exp-ik2 ρ˜TM2n-1ρ˜,
M2n-1=C¯+D¯M1n-1Ā+B¯M1n-1-1.
Γρ˜=1λ2detB¯1/2  Γr˜exp-iπλr˜TB˜-1Ãr˜-2r˜TB˜-1ρ˜+ρ˜TD˜B˜-1ρ˜×exp-iπλr˜TB˜-1ēf+ρ˜TB¯-1g¯hdr˜,
e=2αTεx+βTεx,
f=2αTεy+βTεy,
g=2bγT-dαTεx+2bδT-dβTεx,
h=2bγT-dαTεy+2bδT-dβTεy.
αT=1-a, βT=l-b, γT=-c, δT=1-d,
Ã=aI00aI, B˜=bI00-bI,C˜=cI00-cI, D˜=dI00dI.
ΓNρ˜=n=1N-1n-1NNndetÃ+B˜M1n-1-1/2×exp-ik2 ρ˜TM2n-1ρ˜-ik2 ρ˜TB˜-1g¯h×exp-ik2 ρ˜TB¯-1TB˜+B˜M1n-1-1ēf×expik8 ēfTB˜-1TÃ+B˜M1n-1-1ēf,
M2n-1=C˜+D˜M1n-1Ã+B˜M1n-1-1.
A=1001,B=z00z,C=0000,D=1001.
Γpũ=1λf sin ϕ2- Γ0r˜exp-iπλr˜TN11r˜+ũTN11ũ-2r˜TN12ũd2r˜,
N11=1f tan ϕI00-I, N12=1f sin ϕI00-I.
ΓNũ=n=1N-1n-1NNndetĀ+B¯M1n-11/2×exp-ik2ũTMpn-1ũ,
Mpn-1=C¯+D¯M1n-1Ā+B¯M1n-1-1.
Ā=N12-1N11=cos ϕI00I,B¯=N12-1=f sin ϕI00-I,C¯=N11N12-1N11-N12T=sin ϕf-I00I,D¯=N11N12-1=cos ϕI00I.

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