Abstract

We consider a class of spatially partially coherent light beams, which are generated by passing a Gaussian Schell-model beam though a wavefront-folding interferometer. In certain cases these beams are shape-invariant on propagation and can exhibit sharp internal structure with a central peak (specular beam) or a central dip (antispecular beam) whose dimensions depend on the spatial coherence area. Such beams are demonstrated experimentally and their cross-like distributions of the complex degree of spatial coherence are measured with a digital micromirror device.

© 2015 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Coherence and Quantum Optics (Cambridge University, 1995).
    [Crossref]
  2. F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
    [Crossref]
  3. S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
    [Crossref]
  4. H. W. Wessely and J. O. Bolstadt, “Interferometric technique for measuring the spatial-correlation function of optical radiation fields,” J. Opt. Soc. Am. 60, 678–682 (1970).
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  5. J. B. Breckinridge, “Coherence interferometer and astronomical applications,” Appl. Opt. 11, 2996–2998 (1972).
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  6. Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
    [Crossref]
  7. H. Arimoto and Y. Ohtsuka, “Measurements of the complex degree of spectral coherence by use of a wave-front-folded interferometer,” Opt. Lett. 22, 958–960 (1997).
    [Crossref] [PubMed]
  8. F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
    [Crossref]
  9. J. Turunen, A. Vasara, and A. T. Friberg, “Propagation-invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
    [Crossref]
  10. S. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32, 2508–2510 (2007).
    [Crossref] [PubMed]
  11. P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
    [Crossref]
  12. G. Li, Y. Qiu, and H. Li, “Coherence theory of a laser beam passing through a moving diffuser,” Opt. Express 21, 13032–13039 (2013).
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  13. H. Partanen, J. Turunen, and J. Tervo, “Coherence measurement with digital micromirror device,” Opt. Lett. 39, 1034–1037 (2014).

2014 (1)

2013 (1)

2007 (1)

2004 (1)

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[Crossref]

1997 (1)

1991 (1)

1988 (2)

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[Crossref]

1987 (1)

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

1986 (1)

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[Crossref]

1972 (1)

1970 (1)

Agrawal, G. P.

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[Crossref]

Arimoto, H.

Bolstadt, J. O.

Breckinridge, J. B.

Cada, M.

DeSantis, P.

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[Crossref]

Friberg, A. T.

J. Turunen, A. Vasara, and A. T. Friberg, “Propagation-invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
[Crossref]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

Gori, F.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[Crossref]

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

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[Crossref]

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

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[Crossref]

He, Q.

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

Huang, W.

Li, G.

Li, H.

Mandel, L.

L. Mandel and E. Wolf, Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Ohtsuka, Y.

Padovani, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[Crossref]

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

Palma, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[Crossref]

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[Crossref]

Partanen, H.

Ponomarenko, S.

Ponomarenko, S. A.

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[Crossref]

Qiu, Y.

Tervo, J.

Turunen, J.

Vasara, A.

Wessely, H. W.

Wolf, E.

L. Mandel and E. Wolf, Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[Crossref]

Opt. Commun. (3)

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[Crossref]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

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

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. E (1)

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004).
[Crossref]

Other (1)

L. Mandel and E. Wolf, Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Supplementary Material (1)

NameDescription
» Visualization 1: MP4 (904 KB)      Theoretical and measured spectral densities and degree of coherence

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Figures (6)

Fig. 1
Fig. 1 The wavefront-folding interferometer: S is the source, BS is a non-polarizing beam splittter, P1 and P2 are right-angle prisms, and D is a detector.
Fig. 2
Fig. 2 Distributions of the spectral density S(x, y) and the complex degree of spatial coherence μ(x1, 0, x2, 0) of some specular output fields when the WFI is illuminated with Gaussian Schell-model fields. Left: an isotropic case with w0y = w0x and σ0y = σ0x = w0x/4. Center: an anisotropic case with w0y = w0x, σ0y = w0x/4, and σ0x = w0x/2. Right: another anisotropic case with w0y = w0x/2, σ0y = w0x/4, and σ 0 x = w 0 x / 79 0.11 w 0 x.
Fig. 3
Fig. 3 The used experimental setup.
Fig. 4
Fig. 4 Theoretical (subscript t) and measured (subscript m) spectral densities S(x, y), corresponding absolute values |μ(x1, x2)| and phase arg[μ(x1, x2)] of the degree of coherence, with three different values of phase difference ϕ. The maximum and minimum values of the coordinate axes are ±324 μm. A corresponding animation with 21 measured values of ϕ is presented in Visualization 1.
Fig. 5
Fig. 5 Measured and theoretical values of μ(x, 0, −x, 0) as a function of the phase difference ϕ.
Fig. 6
Fig. 6 Measured coherence function. (a) Cross-spectral density, (b) degree of coherence, (c) originally measured phase, (d) spherical phase removed, and (e) intensity.

Equations (21)

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E ( x , y ) = 1 2 [ E 0 ( x , y ) + E 0 ( x , y ) exp ( i ϕ ) ] ,
W 0 ( x 1 , y 1 , x 2 , y 2 ) = E 0 * ( x 1 , y 1 ) E 0 ( x 2 , y 2 ) ,
W ( x 1 , y 1 , x 2 , y 2 ) = 1 2 [ W 0 ( x 1 , y 1 , x 2 , y 2 ) + W 0 ( x 1 , y 1 , x 2 , y 2 ) ] + 1 2 [ W 0 ( x 1 , y 1 , x 2 , y 2 ) exp ( i ϕ ) + W 0 ( x 1 , y 1 , x 2 , y 2 ) exp ( i ϕ ) ] .
T ( k x 1 , k y 1 , k x 2 , k y 2 ) = 1 ( 2 π ) 4 W ( x 1 , y 1 , x 2 , y 2 ) × exp [ i ( k x 1 x 1 + k y 1 y 1 k x 2 x 2 k y 2 y 2 ) ] d x 1 d y 1 d x 2 d y 2 .
T ( k x 1 , k y 1 , k x 2 , k y 2 ) = 1 2 [ T 0 ( k x 1 , k y 1 , k x 2 , k y 2 ) + T 0 ( k x 1 , k y 1 , k x 2 , k y 2 ) ] + 1 2 [ T 0 ( k x 1 , k y 1 , k x 2 , k y 2 ) exp ( i ϕ ) + T 0 ( k x 1 , k y 1 , k x 2 , k y 2 ) exp ( i ϕ ) ] .
W 0 ( x 1 , y 1 , x 2 , y 2 ) = J 0 [ α ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 ] ,
W ( x 1 , y 1 , x 2 , y 2 ) = J 0 [ α ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 ] + cos ϕ J 0 [ α ( x 1 + x 2 ) 2 + ( y 1 + y 2 ) 2 ] .
S ( x , y ) = 1 + cos ϕ J 0 ( 2 α x 2 + y 2 )
W 0 ( x 1 , y 1 , x 2 , y 2 ) = exp [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ 0 2 ] ,
W ( x 1 , y 1 , x 2 , y 2 ) = exp [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ 0 2 ] + cos ϕ exp [ ( x 1 + x 2 ) 2 + ( y 1 + y 2 ) 2 2 σ 0 2 ]
S ( x , y ) = 1 + cos ϕ exp [ 2 ( x 2 + y 2 ) σ 0 2 ] .
W 0 ( x 1 , y 1 , x 2 , y 2 ) = exp ( x 1 2 + x 2 2 w 0 x 2 ) exp ( y 1 2 + y 2 2 w 0 y 2 ) exp [ ( x 1 x 2 ) 2 2 σ 0 x 2 ] exp [ ( y 1 y 2 ) 2 2 σ 0 y 2 ] ,
W ( x 1 , y 1 , x 2 , y 2 ) = exp ( x 1 2 + x 2 2 w 0 x 2 ) exp ( y 1 2 + y 2 2 w 0 y 2 ) × { exp [ ( x 1 x 2 ) 2 2 σ 0 x 2 ] exp [ ( y 1 y 2 ) 2 2 σ 0 y 2 ] + cos ϕ exp [ ( x 1 + x 2 ) 2 2 σ 0 x 2 ] exp [ ( y 1 + y 2 ) 2 2 σ 0 y 2 ] } ,
S ( x , y ) = exp ( 2 x 2 w 0 x 2 ) exp ( 2 y 2 w 0 y 2 ) [ 1 + cos ϕ exp ( 2 x 2 σ 0 x 2 ) exp ( 2 y 2 σ 0 y 2 ) ] .
W ( x 1 , y 1 , x 2 , y 2 ; z ) = S 0 ( z ) exp [ x 1 2 + x 2 2 w x 2 ( z ) ] exp [ y 1 2 + y 2 2 w y 2 ( z ) ] × { exp [ ( x 1 x 2 ) 2 2 σ x 2 ( z ) ] exp [ ( y 1 y 2 ) 2 2 σ y 2 ( z ) ] + cos ϕ 0 exp [ ( x 1 + x 2 ) 2 2 σ x 2 ( z ) ] exp [ ( y 1 + y 2 ) 2 2 σ y 2 ( z ) ] } × exp [ i k 2 R x ( z ) ( x 1 2 x 2 2 ) ] exp [ i k 2 R y ( z ) ( y 1 2 y 2 2 ) ] ,
S ( x , y ; z ) = S 0 ( z ) exp [ 2 x 2 w x 2 ( z ) ] exp [ 2 y w y 2 ( z ) ] × { 1 + cos ϕ 0 exp [ 2 x 2 σ x 2 ( z ) ] exp [ 2 y 2 2 σ y 2 ( z ) ] } .
w j ( z ) = w j 0 1 + z 2 / z R j 2 ,
σ j ( z ) = σ j 0 1 + z 2 / z R j 2 ,
R j ( z ) = z + z R j / z ,
β j = ( 1 + w 0 j 2 / σ 0 j 2 ) 1 / 2
w x 0 2 β x = w y 0 2 β y

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