Abstract

We present an experimental method for the fast measurement of both the spectral (spatial) and complex degrees of coherence of an optical field using only a binary amplitude mask and a detector array. We test the method by measuring a two-dimensional spectral degree of coherence function created by a broadband thermal source. The results are compared to those expected by the van Cittert-Zernike theorem and found to agree well in both amplitude and phase.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. J. Thompson, E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. 47, 895–902 (1957).
    [CrossRef]
  2. E. Tervonen, J. Turunen, A. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: Experimental results,” Appl. Phys. B 49, 409–414 (1989).
    [CrossRef]
  3. D. Ambrosini, G. S. Spagnolo, D. Paoletti, S. Vicalvi, “High-precision digital automated measurement of degree of coherence in the Thompson and Wolf experiment,” Pure Appl. Opt. 7, 933–939 (1998).
    [CrossRef]
  4. C. Iaconis, I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996).
    [CrossRef] [PubMed]
  5. S. Titus, A. Wasan, J. Vaishya, H. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
    [CrossRef]
  6. H. Kandpal, J. Vaishya, “Experimental study of coherence properties of light fields in the region of superposition in Young’s interference experiment,” Opt. Commun. 186, 15–20 (2000).
    [CrossRef]
  7. L. Basano, P. Ottonello, G. Rottigni, M. Vicari, “Degree of spectral coherence, space-frequency plots and correlation-induced spectral changes,” Opt. Commun. 207, 77–83 (2002).
    [CrossRef]
  8. L. Basano, P. Ottonello, G. Rottigni, M. Vicari, “Spatial and temporal coherence of filtered thermal light,” Appl. Opt. 42, 6239–6244 (2003).
    [CrossRef] [PubMed]
  9. J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
    [CrossRef] [PubMed]
  10. P. Petruck, R. Riesenberg, U. Hübner, R. Kowarschik, “Spatial coherence on micrometer scale measured by a nanohole array,” Opt. Commun. 285, 389–392 (2012).
    [CrossRef]
  11. Y. Mejía, A. I. González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. 273, 428–434 (2007).
    [CrossRef]
  12. A. I. González, Y. Mejía, “Nonredundant array of apertures to measure the spatial coherence in two dimensions with only one interferogram,” J. Opt. Soc. Am. A 28, 1107–1113 (2011).
    [CrossRef]
  13. D. F. Siriani, “Beyond Young’s experiment: time-domain correlation measurement in a pinhole array,” Opt. Lett. 38, 857–859 (2013).
    [CrossRef] [PubMed]
  14. M. Santarsiero, R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. 31, 861–863 (2006).
    [CrossRef] [PubMed]
  15. J.-M. Choi, “Measuring optical spatial coherence by using a programmable aperture,” J. Korean Phys. Soc. 60, 177–180 (2012).
    [CrossRef]
  16. K. Saastamoinen, J. Tervo, J. Turunen, P. Vahimaa, A. T. Friberg, “Spatial coherence measurement of polychromatic light with modified Young’s interferometer,” Opt. Express 21, 4061–4071 (2013).
    [CrossRef] [PubMed]
  17. A. Cámara, J. A. Rodrigo, T. Alieva, “Optical coherenscopy based on phase-space tomography,” Opt. Express 21, 13169–13183 (2013).
    [CrossRef] [PubMed]
  18. L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
    [CrossRef]
  19. S. Cho, M. A. Alonso, T. G. Brown, “Measurement of spatial coherence through diffraction from a transparent mask with a phase discontinuity,” Opt. Lett. 37, 2724–2726 (2012).
    [CrossRef] [PubMed]
  20. M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photonics 3, 272–365 (2011).
    [CrossRef]
  21. E. Mukamel, K. Banaszek, I. A. Walmsley, C. Dorrer, “Direct measurement of the spatial Wigner function with area-integrated detection,” Opt. Lett. 28, 1317–1319 (2003).
    [CrossRef] [PubMed]
  22. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
    [CrossRef] [PubMed]
  23. T. Young, “On the nature of light and colours,” in A Course of Lectures on Natural Philosophy and the Mechanical Arts (J. Johnson, 1807), Vol. 1, pp. 464–465.
  24. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  25. J. Goodman, Introduction to Fourier Optics (Roberts, 2005).
  26. R. A. Bartels, A. Paul, M. M. Murnane, H. C. Kapteyn, S. Backus, Y. Liu, D. T. Attwood, “Absolute determination of the wavelength and spectrum of an extreme-ultraviolet beam by a Young’s double-slit measurement,” Opt. Lett. 27, 707–709 (2002).
    [CrossRef]
  27. A. T. Friberg, E. Wolf, “Relationships between the complex degrees of coherence in the space–time and in the space–frequency domains,” Opt. Lett. 20, 623–625 (1995).
    [CrossRef] [PubMed]
  28. E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett. 8, 250–252 (1983).
    [CrossRef] [PubMed]

2013 (3)

2012 (4)

J.-M. Choi, “Measuring optical spatial coherence by using a programmable aperture,” J. Korean Phys. Soc. 60, 177–180 (2012).
[CrossRef]

L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

S. Cho, M. A. Alonso, T. G. Brown, “Measurement of spatial coherence through diffraction from a transparent mask with a phase discontinuity,” Opt. Lett. 37, 2724–2726 (2012).
[CrossRef] [PubMed]

P. Petruck, R. Riesenberg, U. Hübner, R. Kowarschik, “Spatial coherence on micrometer scale measured by a nanohole array,” Opt. Commun. 285, 389–392 (2012).
[CrossRef]

2011 (2)

A. I. González, Y. Mejía, “Nonredundant array of apertures to measure the spatial coherence in two dimensions with only one interferogram,” J. Opt. Soc. Am. A 28, 1107–1113 (2011).
[CrossRef]

M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photonics 3, 272–365 (2011).
[CrossRef]

2007 (1)

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

2006 (1)

2003 (3)

E. Mukamel, K. Banaszek, I. A. Walmsley, C. Dorrer, “Direct measurement of the spatial Wigner function with area-integrated detection,” Opt. Lett. 28, 1317–1319 (2003).
[CrossRef] [PubMed]

L. Basano, P. Ottonello, G. Rottigni, M. Vicari, “Spatial and temporal coherence of filtered thermal light,” Appl. Opt. 42, 6239–6244 (2003).
[CrossRef] [PubMed]

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

2002 (2)

L. Basano, P. Ottonello, G. Rottigni, M. Vicari, “Degree of spectral coherence, space-frequency plots and correlation-induced spectral changes,” Opt. Commun. 207, 77–83 (2002).
[CrossRef]

R. A. Bartels, A. Paul, M. M. Murnane, H. C. Kapteyn, S. Backus, Y. Liu, D. T. Attwood, “Absolute determination of the wavelength and spectrum of an extreme-ultraviolet beam by a Young’s double-slit measurement,” Opt. Lett. 27, 707–709 (2002).
[CrossRef]

2000 (2)

S. Titus, A. Wasan, J. Vaishya, H. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
[CrossRef]

H. Kandpal, J. Vaishya, “Experimental study of coherence properties of light fields in the region of superposition in Young’s interference experiment,” Opt. Commun. 186, 15–20 (2000).
[CrossRef]

1998 (1)

D. Ambrosini, G. S. Spagnolo, D. Paoletti, S. Vicalvi, “High-precision digital automated measurement of degree of coherence in the Thompson and Wolf experiment,” Pure Appl. Opt. 7, 933–939 (1998).
[CrossRef]

1996 (1)

1995 (2)

1989 (1)

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

1983 (1)

1957 (1)

Alieva, T.

Alonso, M. A.

S. Cho, M. A. Alonso, T. G. Brown, “Measurement of spatial coherence through diffraction from a transparent mask with a phase discontinuity,” Opt. Lett. 37, 2724–2726 (2012).
[CrossRef] [PubMed]

M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photonics 3, 272–365 (2011).
[CrossRef]

Ambrosini, D.

D. Ambrosini, G. S. Spagnolo, D. Paoletti, S. Vicalvi, “High-precision digital automated measurement of degree of coherence in the Thompson and Wolf experiment,” Pure Appl. Opt. 7, 933–939 (1998).
[CrossRef]

Attwood, D. T.

Backus, S.

Banaszek, K.

Bartels, R. A.

Basano, L.

L. Basano, P. Ottonello, G. Rottigni, M. Vicari, “Spatial and temporal coherence of filtered thermal light,” Appl. Opt. 42, 6239–6244 (2003).
[CrossRef] [PubMed]

L. Basano, P. Ottonello, G. Rottigni, M. Vicari, “Degree of spectral coherence, space-frequency plots and correlation-induced spectral changes,” Opt. Commun. 207, 77–83 (2002).
[CrossRef]

Beck, M.

Borghi, R.

Brown, T. G.

Cai, Z.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Cámara, A.

Chantler, C. T.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Cho, S.

Choi, J.-M.

J.-M. Choi, “Measuring optical spatial coherence by using a programmable aperture,” J. Korean Phys. Soc. 60, 177–180 (2012).
[CrossRef]

Clarke, L.

Dorrer, C.

Fleischer, J. W.

L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

Friberg, A.

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

Friberg, A. T.

González, A. I.

A. I. González, Y. Mejía, “Nonredundant array of apertures to measure the spatial coherence in two dimensions with only one interferogram,” J. Opt. Soc. Am. A 28, 1107–1113 (2011).
[CrossRef]

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

Goodman, J.

J. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Hübner, U.

P. Petruck, R. Riesenberg, U. Hübner, R. Kowarschik, “Spatial coherence on micrometer scale measured by a nanohole array,” Opt. Commun. 285, 389–392 (2012).
[CrossRef]

Iaconis, C.

Kandpal, H.

S. Titus, A. Wasan, J. Vaishya, H. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
[CrossRef]

H. Kandpal, J. Vaishya, “Experimental study of coherence properties of light fields in the region of superposition in Young’s interference experiment,” Opt. Commun. 186, 15–20 (2000).
[CrossRef]

Kapteyn, H. C.

Kowarschik, R.

P. Petruck, R. Riesenberg, U. Hübner, R. Kowarschik, “Spatial coherence on micrometer scale measured by a nanohole array,” Opt. Commun. 285, 389–392 (2012).
[CrossRef]

Lai, B.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Lin, J. J. A.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Liu, Y.

Mancini, D. C.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Mayer, A.

McAlister, D. F.

McMahon, P. J.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

McNulty, I.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Mejía, Y.

A. I. González, Y. Mejía, “Nonredundant array of apertures to measure the spatial coherence in two dimensions with only one interferogram,” J. Opt. Soc. Am. A 28, 1107–1113 (2011).
[CrossRef]

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

Moldovan, N.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Mukamel, E.

Murnane, M. M.

Nugent, K. A.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Ottonello, P.

L. Basano, P. Ottonello, G. Rottigni, M. Vicari, “Spatial and temporal coherence of filtered thermal light,” Appl. Opt. 42, 6239–6244 (2003).
[CrossRef] [PubMed]

L. Basano, P. Ottonello, G. Rottigni, M. Vicari, “Degree of spectral coherence, space-frequency plots and correlation-induced spectral changes,” Opt. Commun. 207, 77–83 (2002).
[CrossRef]

Paoletti, D.

D. Ambrosini, G. S. Spagnolo, D. Paoletti, S. Vicalvi, “High-precision digital automated measurement of degree of coherence in the Thompson and Wolf experiment,” Pure Appl. Opt. 7, 933–939 (1998).
[CrossRef]

Paterson, D.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Paul, A.

Peele, A. G.

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Petruck, P.

P. Petruck, R. Riesenberg, U. Hübner, R. Kowarschik, “Spatial coherence on micrometer scale measured by a nanohole array,” Opt. Commun. 285, 389–392 (2012).
[CrossRef]

Raymer, M. G.

Riesenberg, R.

P. Petruck, R. Riesenberg, U. Hübner, R. Kowarschik, “Spatial coherence on micrometer scale measured by a nanohole array,” Opt. Commun. 285, 389–392 (2012).
[CrossRef]

Rodrigo, J. A.

Rottigni, G.

L. Basano, P. Ottonello, G. Rottigni, M. Vicari, “Spatial and temporal coherence of filtered thermal light,” Appl. Opt. 42, 6239–6244 (2003).
[CrossRef] [PubMed]

L. Basano, P. Ottonello, G. Rottigni, M. Vicari, “Degree of spectral coherence, space-frequency plots and correlation-induced spectral changes,” Opt. Commun. 207, 77–83 (2002).
[CrossRef]

Saastamoinen, K.

Santarsiero, M.

Siriani, D. F.

Situ, G.

L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

Spagnolo, G. S.

D. Ambrosini, G. S. Spagnolo, D. Paoletti, S. Vicalvi, “High-precision digital automated measurement of degree of coherence in the Thompson and Wolf experiment,” Pure Appl. Opt. 7, 933–939 (1998).
[CrossRef]

Tervo, J.

Tervonen, E.

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

Thompson, B. J.

Titus, S.

S. Titus, A. Wasan, J. Vaishya, H. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
[CrossRef]

Turunen, J.

K. Saastamoinen, J. Tervo, J. Turunen, P. Vahimaa, A. T. Friberg, “Spatial coherence measurement of polychromatic light with modified Young’s interferometer,” Opt. Express 21, 4061–4071 (2013).
[CrossRef] [PubMed]

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

Vahimaa, P.

Vaishya, J.

H. Kandpal, J. Vaishya, “Experimental study of coherence properties of light fields in the region of superposition in Young’s interference experiment,” Opt. Commun. 186, 15–20 (2000).
[CrossRef]

S. Titus, A. Wasan, J. Vaishya, H. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
[CrossRef]

Vicalvi, S.

D. Ambrosini, G. S. Spagnolo, D. Paoletti, S. Vicalvi, “High-precision digital automated measurement of degree of coherence in the Thompson and Wolf experiment,” Pure Appl. Opt. 7, 933–939 (1998).
[CrossRef]

Vicari, M.

L. Basano, P. Ottonello, G. Rottigni, M. Vicari, “Spatial and temporal coherence of filtered thermal light,” Appl. Opt. 42, 6239–6244 (2003).
[CrossRef] [PubMed]

L. Basano, P. Ottonello, G. Rottigni, M. Vicari, “Degree of spectral coherence, space-frequency plots and correlation-induced spectral changes,” Opt. Commun. 207, 77–83 (2002).
[CrossRef]

Waller, L.

L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

Walmsley, I. A.

Wasan, A.

S. Titus, A. Wasan, J. Vaishya, H. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
[CrossRef]

Wolf, E.

Young, T.

T. Young, “On the nature of light and colours,” in A Course of Lectures on Natural Philosophy and the Mechanical Arts (J. Johnson, 1807), Vol. 1, pp. 464–465.

Adv. Opt. Photonics (1)

M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photonics 3, 272–365 (2011).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

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

J. Korean Phys. Soc. (1)

J.-M. Choi, “Measuring optical spatial coherence by using a programmable aperture,” J. Korean Phys. Soc. 60, 177–180 (2012).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nat. Photonics (1)

L. Waller, G. Situ, J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[CrossRef]

Opt. Commun. (5)

P. Petruck, R. Riesenberg, U. Hübner, R. Kowarschik, “Spatial coherence on micrometer scale measured by a nanohole array,” Opt. Commun. 285, 389–392 (2012).
[CrossRef]

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

S. Titus, A. Wasan, J. Vaishya, H. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. 173, 45–49 (2000).
[CrossRef]

H. Kandpal, J. Vaishya, “Experimental study of coherence properties of light fields in the region of superposition in Young’s interference experiment,” Opt. Commun. 186, 15–20 (2000).
[CrossRef]

L. Basano, P. Ottonello, G. Rottigni, M. Vicari, “Degree of spectral coherence, space-frequency plots and correlation-induced spectral changes,” Opt. Commun. 207, 77–83 (2002).
[CrossRef]

Opt. Express (2)

Opt. Lett. (9)

S. Cho, M. A. Alonso, T. G. Brown, “Measurement of spatial coherence through diffraction from a transparent mask with a phase discontinuity,” Opt. Lett. 37, 2724–2726 (2012).
[CrossRef] [PubMed]

D. F. Siriani, “Beyond Young’s experiment: time-domain correlation measurement in a pinhole array,” Opt. Lett. 38, 857–859 (2013).
[CrossRef] [PubMed]

M. Santarsiero, R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. 31, 861–863 (2006).
[CrossRef] [PubMed]

C. Iaconis, I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996).
[CrossRef] [PubMed]

E. Mukamel, K. Banaszek, I. A. Walmsley, C. Dorrer, “Direct measurement of the spatial Wigner function with area-integrated detection,” Opt. Lett. 28, 1317–1319 (2003).
[CrossRef] [PubMed]

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[CrossRef] [PubMed]

R. A. Bartels, A. Paul, M. M. Murnane, H. C. Kapteyn, S. Backus, Y. Liu, D. T. Attwood, “Absolute determination of the wavelength and spectrum of an extreme-ultraviolet beam by a Young’s double-slit measurement,” Opt. Lett. 27, 707–709 (2002).
[CrossRef]

A. T. Friberg, E. Wolf, “Relationships between the complex degrees of coherence in the space–time and in the space–frequency domains,” Opt. Lett. 20, 623–625 (1995).
[CrossRef] [PubMed]

E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett. 8, 250–252 (1983).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

J. J. A. Lin, D. Paterson, A. G. Peele, P. J. McMahon, C. T. Chantler, K. A. Nugent, B. Lai, N. Moldovan, Z. Cai, D. C. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 074801 (2003).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

D. Ambrosini, G. S. Spagnolo, D. Paoletti, S. Vicalvi, “High-precision digital automated measurement of degree of coherence in the Thompson and Wolf experiment,” Pure Appl. Opt. 7, 933–939 (1998).
[CrossRef]

Other (3)

T. Young, “On the nature of light and colours,” in A Course of Lectures on Natural Philosophy and the Mechanical Arts (J. Johnson, 1807), Vol. 1, pp. 464–465.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

J. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

A diagram (not drawn to scale) of the essential elements of the apparatus. Broadband light from an extended source passes through the non-parallel double slit aperture and generates an interference pattern on the detector plane.

Fig. 2
Fig. 2

A schematic (not drawn to scale) of the two-slit aperture used to measure coherence functions. The black lines represent transparent slits and white represents areas of high opacity.

Fig. 3
Fig. 3

A diagram (not drawn to scale) of a Thompson-Wolf experiment as given in the Cartesian coordinates used in the text. The two circles represent pinholes which lie in the x′y′ (aperture) plane, and whose centers are placed at (x′, y′, z′) = (−d/2, 0, 0) and (d/2, 0, 0). The observation plane is defined by z′ = z for some positive constant z. This plane includes the x-axis, which is parallel to the x′-axis and is used in Eq. (1). The pattern of ellipses represents an intensity pattern generated on the observation plane by passing light through the two pinholes [1].

Fig. 4
Fig. 4

(a) The intensity interferogram recorded by the apparatus at a distance of 4.5 m from the fiber bundle and (b) the associated amplitude spectral density given by a Fourier transform across the horizontal dimension. Each horizontal cross-section of the interferogram represents a Thompson-Wolf experiment with a different pinhole separation. The white lines in (a) and (b) correspond to the cross-sections given in (c) and (d), respectively, for a slit separation of 50 μm. The images shown in (a) and (b) have been scaled non-linearly to enhance contrast.

Fig. 5
Fig. 5

The intensity distribution of light with 510 nm wavelength at the position of the fiber bundle source as measured by a camera. The diameter of the circular area is 0.62 cm. The shown coordinate axes were used in applying the van Cittert-Zernike theorem. The two circles represent the two pinholes at positions P1 and P2 of a Thompson-Wolf experiment where d is the center-to-center pinhole separation distance.

Fig. 6
Fig. 6

Measured and expected spectral degree of coherence for light at a distance of 105 cm from the source shown in Fig. 5. (a) Amplitude and (b) phase of the spectral degree of coherence measured by the two-slit apparatus in the unfiltered case. (c) Amplitude and (d) phase of the spectral degree of coherence expected from the source by the van Cittert-Zernike theorem assuming that the intensity distribution is invariant in wavelength and a 0.31° aperture tilt relative to the detector. (e) Amplitude on linear and logarithmic (inset) scales and (f) unwrapped phase comparison between the filtered, unfiltered, and expected measurements at 510 nm wavelength. The phase comparison (f) includes the fit which incorporates the 0.31° tilt as described in the text as well as that expected without tilt. The vertical lines shown in each of (a–d) correspond to cross sections shown in the respective lines plotted in (e) and (f).

Fig. 7
Fig. 7

Effects of geometry on the interferogram produced by a double slit. (a) A schematic of the geometrical magnification effect associated with light from any single position on the source which causes the illuminated spot produced on the detector to be larger than the aperture. The distances d1, and d2 are used in Eq. (20) to determine the magnification factor. (b) A schematic of the blurring effect associated with a finite-sized source. Light passing through one position on the slit can overlap on the detector with light passing through another position on the slit.

Fig. 8
Fig. 8

The cross-sections which were used in the blurring mitigation method. The red line segments in both (a) the spectral degree of coherence data and (b) the spectrum data correspond to the positions of the considered cross-sections in the spatial frequency domain.

Equations (22)

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

I ( x ) = 2 | K ( x , ω ) | 2 S Q ( ω ) ( 1 + Re [ μ ω ( ω ) exp ( i ω x d z c ) ] ) d ω ,
I ( x ) = 2 I 0 ( x ) ( 1 + Re [ s 1 ( ω ) μ ω ( ω ) exp ( i ω x d z c ) d ω ] ) ,
I ( x ) = 2 I 0 ( x ) + I 0 ( x ) [ s η ω ( ω ) μ ω ( ω ) exp ( i ω x d z c ) d ω + s η ω ( ω ) μ ω * ( ω ) exp ( i ω x d z c ) d ω ] ,
I ˜ ( f x ) = 2 I ˜ 0 ( f x ) + I ˜ 0 ( f x ) [ s η ( f x ) μ ( f x ) ] + I ˜ 0 ( f x ) [ s η ( f x ) μ * ( f x ) ] ,
A ( f x ) I ˜ 0 ( f x ) [ s η ( f x ) μ ( f x ) ] , B ( f x ) 2 I ˜ 0 ( f x ) , and C ( f x ) I ˜ 0 ( f x ) [ s η ( f x ) μ * ( f x ) ] .
I ˜ ( f x ) { A ( f x ) : f x < c B ( f x ) : c f x c C ( f x ) : f x > c
s η ( f x ) | A ( f x ) | / c | A ( f x ) | d f x , and
s η ( f x ) | C ( f x ) | / | C ( f x ) | d f x .
μ ( f x ) 2 A ( f x ) s η ( f x ) c c B ( f x ) d f x , and
μ * ( f x ) 2 C ( f x ) s η ( f x ) c c B ( f x ) d f x .
μ ω ( ω 0 ) = γ ( + ) ( 0 ) ,
I ˜ ( + ) ( f x ) = 2 I ˜ 0 ( + ) ( f x ) ( δ ( f x ) + 1 2 [ s ( + ) ( f x ) μ ( + ) ( f x ) + s ( + ) ( f x ) μ ( + ) * ( f x ) ] ) .
I ˜ ( + ) ( f x ) { A ( + ) ( f x ) : f x < c B ( + ) ( f x ) : c f x c C ( + ) ( f x ) : f x > c .
γ ( + ) ( x ) 1 { 2 A ( + ) ( f x ) } 1 { B ( + ) ( f x ) } , and
γ ( + ) * ( x ) 1 { 2 C ( + ) ( f x ) } 1 { B ( + ) ( f x ) } .
γ ( x ) 1 { 2 A ( f x ) } 1 { B ( f x ) } , and
γ * ( x ) 1 { 2 C ( f x ) } 1 { B ( f x ) } .
η ( ω ) { large : ω 𝒲 ω 2 η ( ω ) : ω 𝒲 .
γ ( x ) { 1 { γ ( x ) } η ( 2 π z c f x / d ) 0 s η ω ( ω ) η ( ω ) d ω } ,
j ( P 1 , P 2 ) = 1 I ( P 1 ) I ( P 2 ) σ I ( S ) e i k ( R 2 R 1 ) R 1 R 2 d S , where
I ( P n ) = σ I ( S ) R n 2 d S ( n = 1 , 2 ) .
M = A 2 A 1 = d 1 + d 2 d 1 ,

Metrics