Abstract

We propose to use a mask with a nonredundant array (NRA) of multiple apertures to measure spatial coherence in two dimensions. The spatial distribution of the apertures in the mask is made in such a way that we obtain a quasi-uniform sampling in the coherence domain. The spatial coherence is obtained by Fourier transform of the interferogram generated by the mask when it is illuminated by the light field under analysis.

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References

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  1. M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversal-wavefront Young interferometer,” Opt. Lett. 31, 861–863 (2006).
    [Crossref] [PubMed]
  2. Y. Mejía and A. I. González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. 273, 428–434 (2007).
    [Crossref]
  3. A. I. González and Y. Mejía, “Measuring spatial coherence with a two-dimensional aperture array,” in AIP Conference Proceedings, Vol.  992, 478–483 (2008).
    [Crossref]
  4. G. L. Rogers, “The process of image formation as the retransformation of the coherence function,” Proc. Phys. Soc. 81, 323(1963).
    [Crossref]
  5. W. T. Rhodes, “Image formation and Young’s fringes,” in The Art and Science of Holography, A Tribute to Emmett Leith and Yuri Denisyuk, H.J.Caulfield, eds. (SPIE Press, 2004), pp. 111–126.
  6. M. Michalski, E. E. Sicre, and H. J. Rabal, “Display of the complex degree of coherence due to quasi-monochromatic spatially incoherent sources,” Opt. Lett. 10, 585–587 (1985).
    [Crossref] [PubMed]
  7. J. C. Barreiro and J. Ojeda-Castañeda, “Degree of coherence: a lensless measuring technique,” Opt. Lett. 18, 302–304 (1993).
    [Crossref] [PubMed]
  8. F. Gori, “Coherence of light,” in Second Winter College on Optics (International Centre for Theoretical Physics, 20 Feb.–10 Mar. 1995), http://cdsagenda5.ictp.trieste.it/full_display.php?ida=a02292.
  9. Y. Mejía, V. B. Markov, and R. Castañeda, “Analysis of a quasi-periodical structure through its autocorrelation function,” Opt. Eng. 35, 2845–2851 (1996).
    [Crossref]
  10. E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in X- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1978).
  11. F. D. Russell and J. W. Goodman, “Nonredundant arrays and postdetection processing for aberration compensation in incoherent imaging,” J. Opt. Soc. Am. 61, 182 (1971).
    [Crossref]
  12. M. J. E. Golay, “Point arrays having compact, nonredundant autocorrelations,” J. Opt. Soc. Am. 61, 272–273 (1971).
    [Crossref]
  13. L. Baumert, Cyclic Difference Sets, Lecture Notes in Mathematics (Springer-Verlag, 1971).
  14. J. W. Goodman, Statistical Optics (Wiley, 1985).
  15. E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
    [Crossref]
  16. C. M. Warnky, B. L. Anderson, and C. A. Klein, “Determining spatial modes of lasers with spatial coherence measurements,” Appl. Opt. 39, 6109–6117 (2000).
    [Crossref]

2007 (1)

Y. Mejía and A. I. González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. 273, 428–434 (2007).
[Crossref]

2006 (1)

2000 (1)

1996 (1)

Y. Mejía, V. B. Markov, and R. Castañeda, “Analysis of a quasi-periodical structure through its autocorrelation function,” Opt. Eng. 35, 2845–2851 (1996).
[Crossref]

1993 (1)

1989 (1)

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[Crossref]

1985 (1)

1978 (1)

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in X- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1978).

1971 (2)

1963 (1)

G. L. Rogers, “The process of image formation as the retransformation of the coherence function,” Proc. Phys. Soc. 81, 323(1963).
[Crossref]

Anderson, B. L.

Barreiro, J. C.

Baumert, L.

L. Baumert, Cyclic Difference Sets, Lecture Notes in Mathematics (Springer-Verlag, 1971).

Borghi, R.

Caroli, E.

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in X- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1978).

Castañeda, R.

Y. Mejía, V. B. Markov, and R. Castañeda, “Analysis of a quasi-periodical structure through its autocorrelation function,” Opt. Eng. 35, 2845–2851 (1996).
[Crossref]

Di Cocco, G.

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in X- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1978).

Friberg, A. T.

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[Crossref]

Golay, M. J. E.

González, A. I.

Y. Mejía and A. I. González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. 273, 428–434 (2007).
[Crossref]

A. I. González and Y. Mejía, “Measuring spatial coherence with a two-dimensional aperture array,” in AIP Conference Proceedings, Vol.  992, 478–483 (2008).
[Crossref]

Goodman, J. W.

Gori, F.

F. Gori, “Coherence of light,” in Second Winter College on Optics (International Centre for Theoretical Physics, 20 Feb.–10 Mar. 1995), http://cdsagenda5.ictp.trieste.it/full_display.php?ida=a02292.

Klein, C. A.

Markov, V. B.

Y. Mejía, V. B. Markov, and R. Castañeda, “Analysis of a quasi-periodical structure through its autocorrelation function,” Opt. Eng. 35, 2845–2851 (1996).
[Crossref]

Mejía, Y.

Y. Mejía and A. I. González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. 273, 428–434 (2007).
[Crossref]

Y. Mejía, V. B. Markov, and R. Castañeda, “Analysis of a quasi-periodical structure through its autocorrelation function,” Opt. Eng. 35, 2845–2851 (1996).
[Crossref]

A. I. González and Y. Mejía, “Measuring spatial coherence with a two-dimensional aperture array,” in AIP Conference Proceedings, Vol.  992, 478–483 (2008).
[Crossref]

Michalski, M.

Natalucci, L.

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in X- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1978).

Ojeda-Castañeda, J.

Rabal, H. J.

Rhodes, W. T.

W. T. Rhodes, “Image formation and Young’s fringes,” in The Art and Science of Holography, A Tribute to Emmett Leith and Yuri Denisyuk, H.J.Caulfield, eds. (SPIE Press, 2004), pp. 111–126.

Rogers, G. L.

G. L. Rogers, “The process of image formation as the retransformation of the coherence function,” Proc. Phys. Soc. 81, 323(1963).
[Crossref]

Russell, F. D.

Santarsiero, M.

Sicre, E. E.

Spizzichino, A.

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in X- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1978).

Stephen, J. B.

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in X- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1978).

Tervonen, E.

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[Crossref]

Turunen, J.

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[Crossref]

Warnky, C. M.

Appl. Opt. (1)

Appl. Phys. B (1)

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

Y. Mejía and A. I. González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. 273, 428–434 (2007).
[Crossref]

Opt. Eng. (1)

Y. Mejía, V. B. Markov, and R. Castañeda, “Analysis of a quasi-periodical structure through its autocorrelation function,” Opt. Eng. 35, 2845–2851 (1996).
[Crossref]

Opt. Lett. (3)

Proc. Phys. Soc. (1)

G. L. Rogers, “The process of image formation as the retransformation of the coherence function,” Proc. Phys. Soc. 81, 323(1963).
[Crossref]

Space Sci. Rev. (1)

E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in X- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1978).

Other (5)

L. Baumert, Cyclic Difference Sets, Lecture Notes in Mathematics (Springer-Verlag, 1971).

J. W. Goodman, Statistical Optics (Wiley, 1985).

W. T. Rhodes, “Image formation and Young’s fringes,” in The Art and Science of Holography, A Tribute to Emmett Leith and Yuri Denisyuk, H.J.Caulfield, eds. (SPIE Press, 2004), pp. 111–126.

F. Gori, “Coherence of light,” in Second Winter College on Optics (International Centre for Theoretical Physics, 20 Feb.–10 Mar. 1995), http://cdsagenda5.ictp.trieste.it/full_display.php?ida=a02292.

A. I. González and Y. Mejía, “Measuring spatial coherence with a two-dimensional aperture array,” in AIP Conference Proceedings, Vol.  992, 478–483 (2008).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1

Crosshatch pattern shows the region of the autocorrelation that contains only half of the correlation peaks (leaving out the central peak). This region determines the maximum number of apertures that we can have in a NRA.

Fig. 2
Fig. 2

Three NRAs (upper row) and their autocorrelations (lower row). Each NRA is composed by nine apertures within a grid of size ( 6 × 6 ) intersections. Nine is the maximum number of apertures that can be placed in a nonredundant way in these grids. In the autocorrelations, filled circles represent autocorrelation peaks, all of the same height; the central peak, indicated by an open circle, shows that this peak is different in height (total number of apertures).

Fig. 3
Fig. 3

(a) Region for the minimum moment of inertia (Golay’s criterion), (b) region for the nonredundant array with the most compact autocorrelation (in our example).

Fig. 4
Fig. 4

Setup for measuring the spatial coherence. Lens L1 expands the light field; lens L2 (two identical lenses) focuses the interferogram generated by a mask with a NRA of apertures (the mask is placed just before the lens L2); lens L3 takes the image of the interference pattern using a CCD sensor. For the first experiment, the rotating ground glass is placed in the back focal plane of L1.

Fig. 5
Fig. 5

Interferograms generated with the NRA of Fig. 2c and their corresponding Fourier spectrums for the laser beam with the rotating ground glass. (a) Interferogram obtained when the center of the aperture grid is on the optical axis, (b) interferogram obtained when the center of the aperture grid is separated 8 mm from the optical axis, (c) Fourier spectrum of (a), (d) Fourier spectrum of (b).

Fig. 6
Fig. 6

(a) Modulus of the spatial coherence measured from Fig. 5c, (b) spatial coherence measured from Fig. 5d. The spatial coherence of this laser beam is shift invariant.

Fig. 7
Fig. 7

Irradiance of the cross section of a diode laser ( 635 nm , 0.6 mW ).

Fig. 8
Fig. 8

Interferograms generated with the NRA of Fig. 2c and their corresponding Fourier spectra for the diode laser, (a) Interferogram obtained when the center of the aperture grid is on the optical axis, (b) interferogram obtained when the center of the aperture grid is separated by 8 mm from the optical axis, (c) Fourier spectrum of (a), (d) Fourier spectrum of (b).

Fig. 9
Fig. 9

(a) Modulus of the spatial coherence measured from Fig. 8c; (b) modulus of the spatial coherence measured from Fig. 8d.

Equations (6)

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I ˜ ( r ) = Λ ( r ) [ n = 1 N I n δ ( r ) + n = m + 1 N m = 1 N 1 I n I m { μ n m δ ( r ( r n r m ) ) + μ n m * δ ( r + ( r n r m ) ) } ] ,
I ˜ ( r ) = Λ ( r ) [ S 0 δ ( r ) + j = 1 N S j { μ j δ ( r d j ) + μ j * δ ( r + d j ) } ] ,
| μ j | = | c j | | c 0 | S 0 S j ,
α j = φ j ,
k c = Int [ ( 1 + 1 + 8 n c a ) 2 ] ,
μ m n = exp [ i π λ z ( r n 2 r m 2 ) ] exp [ ( r n r m ) 2 ω 2 ] ,

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