Abstract

Analytical models for random sources producing far fields with frame-like intensity profiles are introduced. The frames can have polar and Cartesian symmetry and adjustable sharpness of the inner and outer edges. The frames are shape invariant throughout the far zone but expand due to diffraction with growing distance from the source. The generalization to multiple nested frames is also discussed. The applications of the frames are envisioned in material surface processing and particle trapping.

© 2014 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
  2. F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
    [CrossRef]
  3. Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [CrossRef] [PubMed]
  4. Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
    [CrossRef] [PubMed]
  5. H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [CrossRef] [PubMed]
  6. S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [CrossRef] [PubMed]
  7. O. Korotkova, S. Sahin, E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
    [CrossRef] [PubMed]
  8. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
    [CrossRef] [PubMed]
  9. O. Korotkova, E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in free space and atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
    [CrossRef]
  10. C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
    [CrossRef] [PubMed]
  11. S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. 24(1), 69–82 (2014).
    [CrossRef]
  12. F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
    [CrossRef] [PubMed]
  13. F. Gori, V. R. Sanchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
    [CrossRef]
  14. M. Santarsiero, G. Piquero, J. C. G. de Sande, F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
    [CrossRef] [PubMed]

2014 (5)

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[CrossRef] [PubMed]

O. Korotkova, E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in free space and atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[CrossRef]

C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. 24(1), 69–82 (2014).
[CrossRef]

M. Santarsiero, G. Piquero, J. C. G. de Sande, F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[CrossRef] [PubMed]

2013 (3)

2012 (2)

2011 (1)

2009 (1)

F. Gori, V. R. Sanchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

1987 (1)

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[CrossRef]

Avramov-Zamurovic, S.

S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. 24(1), 69–82 (2014).
[CrossRef]

Cai, Y.

de Sande, J. C. G.

Gori, F.

M. Santarsiero, G. Piquero, J. C. G. de Sande, F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[CrossRef] [PubMed]

F. Gori, V. R. Sanchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[CrossRef]

Korotkova, O.

O. Korotkova, E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in free space and atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[CrossRef]

S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. 24(1), 69–82 (2014).
[CrossRef]

S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. 24(1), 69–82 (2014).
[CrossRef]

C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[CrossRef] [PubMed]

Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[CrossRef] [PubMed]

Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[CrossRef] [PubMed]

S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[CrossRef] [PubMed]

O. Korotkova, S. Sahin, E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[CrossRef] [PubMed]

Lajunen, H.

Liang, C.

Liu, X.

Malek-Madani, R.

S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. 24(1), 69–82 (2014).
[CrossRef]

Mei, Z.

Nelson, C.

S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. 24(1), 69–82 (2014).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[CrossRef]

Piquero, G.

Saastamoinen, T.

Sahin, S.

Sanchez, V. R.

F. Gori, V. R. Sanchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

Santarsiero, M.

M. Santarsiero, G. Piquero, J. C. G. de Sande, F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[CrossRef] [PubMed]

F. Gori, V. R. Sanchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

Shchepakina, E.

O. Korotkova, E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in free space and atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[CrossRef]

O. Korotkova, S. Sahin, E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[CrossRef] [PubMed]

Shirai, T.

F. Gori, V. R. Sanchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

Wang, F.

Yuan, Y.

J. Opt. (1)

O. Korotkova, E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in free space and atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

F. Gori, V. R. Sanchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[CrossRef]

Opt. Lett. (8)

Waves in Complex and Random Media. (1)

S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. 24(1), 69–82 (2014).
[CrossRef]

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

Illustration of the frames with circular symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of δ ox , δ oy , δ ix , δ iy and M = 1.

Fig. 2
Fig. 2

Illustration of the frames with circular symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of δ ox , δ oy , δ ix , δ iy and M = 40.

Fig. 3
Fig. 3

Illustration of the frames with elliptical symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of δ ox , δ oy , δ ix , δ iy and M = 40.

Fig. 4
Fig. 4

Illustration of the frames with square symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of δ ox , δ oy , δ ix , δ iy and M = 40.

Fig. 5
Fig. 5

Illustration of the frames with rectangular symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of δ ox , δ oy , δ ix , δ iy and M = 40.

Fig. 6
Fig. 6

Illustration of the square frame combination.

Fig. 7
Fig. 7

Illustration of the combination of frames with different symmetries.

Equations (22)

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W (0) ( ρ 1 , ρ 2 ;ω)= E ( ρ 1 ;ω) E ( ρ 2 ;ω),
W (0) ( ρ 1 , ρ 2 )= p(v) H ( ρ 1 ,v )H( ρ 2 ,v) d 2 v,
H( ρ ,v)=τ( ρ )exp(2πiv ρ ),
W (0) ( ρ 1 , ρ 2 )= τ ( ρ 1 )τ( ρ 2 )μ( ρ 1 ρ 2 ),
W (0) ( ρ 1 , ρ 2 ) τ 2 ( ρ 1 + ρ 2 2 )μ( ρ 1 ρ 2 ).
S () (k s ) (2πk) 2 r 2 s z 2 μ ˜ (k s ),
p(v)= p o (p) (v) p i (p) (v),
p o (p) (v)= A C p m=1 M (1) m1 ( M m )exp[ m 2 ( δ ox 2 v x 2 + δ oy 2 v y 2 ) ] = A C p ( 1 { 1exp[ 1 2 ( δ ox 2 v x 2 + δ oy 2 v y 2 ) ] } M ),
p i (p) (v)= A C p m=1 M (1) m1 ( M m )exp[ m 2 ( δ ix 2 v x 2 + δ iy 2 v y 2 ) ] = A C p ( 1 { 1exp[ 1 2 ( δ ix 2 v x 2 + δ iy 2 v y 2 ) ] } M ),
C p = m=1 M (1) m1 m ( M m ),
δ ix > δ ox and δ iy > δ oy .
p(v)= p o (p) (v) p i (p) (v) = A C p m=1 M (1) m1 ( M m ){ exp[ m 2 ( δ ox 2 v x 2 + δ oy 2 v y 2 ) ]exp[ m 2 ( δ ix 2 v x 2 + δ iy 2 v y 2 ) ] }.
μ( ρ d )= A C p m=1 M (1) m1 ( M m ) ×{ 1 m δ ox δ oy exp[ 1 2m ( ρ ' dx 2 δ ox 2 + ρ ' dy 2 δ oy 2 ) ] 1 m δ ix δ iy exp[ 1 2m ( ρ ' dx 2 δ ix 2 + ρ ' dy 2 δ iy 2 ) ] },
μ(0)=1, μ()=0, and |μ( ρ d )|1.
1 δ ox δ oy 1 δ ix δ iy =1.
A= ( 1 δ ox δ oy 1 δ ix δ iy ) 1 .
p(v)= p o (r) (v) p i (r) (v),
p o (r) (v)= A C r 2 m=1 M (1) m1 ( M m ) exp[ m δ ox 2 v x 2 2 ] m=1 M (1) m1 ( M m ) exp[ m δ oy 2 v y 2 2 ] = 1 C r 2 { 1 ( 1exp[ δ ox 2 v x 2 2 ] ) M }{ 1 ( 1exp[ δ oy 2 v y 2 2 ] ) M },
p i (r) (v)= A C r 2 m=1 M (1) m1 ( M m ) exp[ m δ ix 2 v x 2 2 ] m=1 M (1) m1 ( M m ) exp[ m δ iy 2 v y 2 2 ] = 1 C r 2 { 1 ( 1exp[ δ ix 2 v x 2 2 ] ) M }{ 1 ( 1exp[ δ iy 2 v y 2 2 ] ) M },
C r = m=1 M (1) m1 m ( M m ) .
μ(ρ ' d )= A C r 2 δ ox δ oy m=1 M ( M m ) (1) m1 m exp[ 1 2m ρ ' dx 2 δ ox 2 ] m=1 M ( M m ) (1) m1 m exp[ 1 2m ρ ' dy 2 δ oy 2 ] A C r 2 δ ix δ iy m=1 M ( M m ) (1) m1 m exp[ 1 2m ρ ' dx 2 δ ix 2 ] m=1 M ( M m ) (1) m1 m exp[ 1 2m ρ ' dy 2 δ iy 2 ] .
μ( ρ ' d )=B n=1 N a n μ n ( ρ ' d ) , B= ( n=1 N a n ) 1 ,

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