Abstract

An alternative method to generate J0 Bessel beams with controlled spatial partial coherence properties is introduced. Far field diffraction from a discrete number of source points on an annular region is calculated. The average for different diffracted fields produced at several rotation angles is numerically calculated and experimentally detected. Theoretical and experimental results show that for this particular case, the J0 Bessel beam is a limit when the number of points tends towards infinity and the associated complex degree of coherence is also a function of the number of points.

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References

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    [CrossRef]
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2009

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80(5), 053804 (2009).
[CrossRef]

2008

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).

H. T. Eyyuboglu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).
[CrossRef]

2007

G. J. Williams, H. M. Quiney, A. G. Peele, and K. A. Nugent, “Coherent diffractive imaging and partial coherence,” Phys. Rev. B 75(10), 104102 (2007).
[CrossRef]

G. M. Niconoff, J. Ramírez San Juan, J. M. López, and P. M. Vara, “Incoherent convergence of diffraction free fields,” Opt. Commun. 275(1), 10–13 (2007).
[CrossRef]

2006

2004

2002

Z. Bouchal and J. Perina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49(10), 1673–1689 (2002).
[CrossRef]

1998

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

1987

Bouchal, Z.

Z. Bouchal and J. Perina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49(10), 1673–1689 (2002).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

Callens, N.

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

Dubois, F.

Durnin, J.

Eyyuboglu, H. T.

H. T. Eyyuboglu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).
[CrossRef]

Friberg, A.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80(5), 053804 (2009).
[CrossRef]

Gao, C.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).

Gao, M.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).

Hoyos, M.

Kurowski, P.

Li, F.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).

Liu, Y.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).

López, J. M.

G. M. Niconoff, J. Ramírez San Juan, J. M. López, and P. M. Vara, “Incoherent convergence of diffraction free fields,” Opt. Commun. 275(1), 10–13 (2007).
[CrossRef]

Monnom, O.

Niconoff, G. M.

G. M. Niconoff, J. Ramírez San Juan, J. M. López, and P. M. Vara, “Incoherent convergence of diffraction free fields,” Opt. Commun. 275(1), 10–13 (2007).
[CrossRef]

Nugent, K. A.

G. J. Williams, H. M. Quiney, A. G. Peele, and K. A. Nugent, “Coherent diffractive imaging and partial coherence,” Phys. Rev. B 75(10), 104102 (2007).
[CrossRef]

Peele, A. G.

G. J. Williams, H. M. Quiney, A. G. Peele, and K. A. Nugent, “Coherent diffractive imaging and partial coherence,” Phys. Rev. B 75(10), 104102 (2007).
[CrossRef]

Perina, J.

Z. Bouchal and J. Perina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49(10), 1673–1689 (2002).
[CrossRef]

Quiney, H. M.

G. J. Williams, H. M. Quiney, A. G. Peele, and K. A. Nugent, “Coherent diffractive imaging and partial coherence,” Phys. Rev. B 75(10), 104102 (2007).
[CrossRef]

Ramírez San Juan, J.

G. M. Niconoff, J. Ramírez San Juan, J. M. López, and P. M. Vara, “Incoherent convergence of diffraction free fields,” Opt. Commun. 275(1), 10–13 (2007).
[CrossRef]

Saastamoinen, K.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80(5), 053804 (2009).
[CrossRef]

Tao, S. H.

Turunen, J.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80(5), 053804 (2009).
[CrossRef]

Vahimaa, P.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80(5), 053804 (2009).
[CrossRef]

Vara, P. M.

G. M. Niconoff, J. Ramírez San Juan, J. M. López, and P. M. Vara, “Incoherent convergence of diffraction free fields,” Opt. Commun. 275(1), 10–13 (2007).
[CrossRef]

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

Williams, G. J.

G. J. Williams, H. M. Quiney, A. G. Peele, and K. A. Nugent, “Coherent diffractive imaging and partial coherence,” Phys. Rev. B 75(10), 104102 (2007).
[CrossRef]

Yourassowsky, C.

Yuan, X.

Appl. Opt.

J. Mod. Opt.

Z. Bouchal and J. Perina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49(10), 1673–1689 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281, 1968–1975 (2008).

G. M. Niconoff, J. Ramírez San Juan, J. M. López, and P. M. Vara, “Incoherent convergence of diffraction free fields,” Opt. Commun. 275(1), 10–13 (2007).
[CrossRef]

Opt. Laser Technol.

H. T. Eyyuboglu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).
[CrossRef]

Phys. Rev. A

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80(5), 053804 (2009).
[CrossRef]

Phys. Rev. B

G. J. Williams, H. M. Quiney, A. G. Peele, and K. A. Nugent, “Coherent diffractive imaging and partial coherence,” Phys. Rev. B 75(10), 104102 (2007).
[CrossRef]

Other

R. Bracewell, The Fourier Transform & Its Applications, (McGraw-Hill Science/Engineering/Math, 1999).

J. W. Goodman, Introduction to Fourier Optics, (Roberts & Company Publishers, 2004).

M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover Publications, 1965).

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

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, (Cambridge University Press, 1992).

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Figures (5)

Fig. 1
Fig. 1

Geometrical parameters. a) Original angular position depending on the number of points . b) Each point is rotated to angle Δ, but all points remain on a circular trajectory because the same radius R is maintained constant.

Fig. 2
Fig. 2

Plot of Eq. (5) for 2, 4, 10 and 50 points plus J0 function. As the number of points increases, the function tends towards the Bessel function, as can be seen for the case of 50 points.

Fig. 3
Fig. 3

Plot of γ for different numbers of points. For n 20 the complex degree of coherence is very near to 1. The asymptotic value obtained is 0.994 with a standard error of 5 × 10 3 .

Fig. 4
Fig. 4

Numerical results for different cases. a) 4 points case, b) 10 points case, c) 20 points case. The associated transversal normalized intensity distribution is plotted above. As the number of coherent points increases, the image contrast increases and the intensity distribution tends towards a squared J0 Bessel function. The behavior of the lobes coincide with Fig. 2.

Fig. 5
Fig. 5

Experimental results for different cases a) 4 points b) 10 points c) 50 points. The intensity distributions are in good agreement with results obtained numerically in Fig. 4, the number of minima and maxima of intensity are the same. The corresponding γ values are: 0.444, 0.649 and 1.

Equations (7)

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t ( r , θ ; Δ ) = δ ( r R ) π r p = 1 n δ [ θ ( 2 π p n Δ ) ] ,
A ( ρ , ϕ ) = 1 π 0 2 π 0 δ ( r R ) p = 1 n δ [ θ ( 2 π p n Δ ) ] exp [ i 2 π r ρ cos ( θ ϕ ) ] d r d θ .
A ( ρ , ϕ ) = 1 π p = 1 n q = i q J q ( 2 π R ρ ) exp [ i q ( 2 π p n Δ ϕ ) ] ,
φ ( r , z ; Δ ) = 0 2 π 0 A ( ρ , ϕ ) J 0 ( 2 π r ρ ) exp [ i 2 π z 1 / λ 2 ρ 2 ] ρ d ρ d ϕ ,
I ¯ ( Δ ) = I ( r , z ) Δ = 1 π 2 { J 0 ( 2 π R r ) 2 + 2 k = 1 J k ( 2 π R r ) 2 p = 1 n q = 1 n cos [ 2 π k n ( p q ) ] } ,
Γ [ I ¯ ( r ; v ) , J 0 ( r ) ] = 0 I ¯ ( r ; n ) J 0 ( r ) d r .
γ = 0 I ¯ ( r ; n ) J 0 ( r ) d r ( 0 I ¯ ( r ; n ) 2 d r ) 1 / 2 ( 0 J 0 ( r ) 2 d r ) 1 / 2 .

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