Abstract

In the absence of a lens to form an image, incoherent or partially coherent light scattering off an obstructive or reflective object forms a broad intensity distribution in the far field with only feeble spatial features. We show here that measuring the complex spatial coherence function can help in the identification of the size and location of a one-dimensional object placed in the path of a partially coherent light source. The complex coherence function is measured in the far field through wavefront sampling, which is performed via dynamically reconfigurable slits implemented on a digital micromirror device (DMD). The impact of an object – parameterized by size and location – that either intercepts or reflects incoherent light is studied. The experimental results show that measuring the spatial coherence function as a function of the separation between two slits located symmetrically around the optical axis can identify the object transverse location and angle subtended from the detection plane (the ratio of the object width to the axial distance from the detector). The measurements are in good agreement with numerical simulations of a forward model based on Fresnel propagators. The rapid refresh rate of DMDs may enable real-time operation of such a lensless coherency imaging scheme.

© 2017 Optical Society of America

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References

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  1. T. S. Huang, ed., Picture Processing and Digital Filtering (Springer, 1979).
    [Crossref]
  2. A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).
  3. B. Jähne, Digital Image Processing: Concepts, Algorithms, and Scientific Applications (Springer, 1997).
  4. N. George, “Lensless electronic imaging,” Opt. Commun. 133, 22–26 (1997).
    [Crossref]
  5. J. R. Fienup, “Phase retrieval algorithms: A comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  6. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  7. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [Crossref] [PubMed]
  8. H. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust. Speech Signal Process. 28, 672–680 (1980).
    [Crossref]
  9. H. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Iterative procedure for signal reconstruction from Fourier transform phase,” Opt. Eng. 21, 122–127 (1982).
  10. D. Kermisch, “Image reconstruction from phase information only,” J. Opt. Soc. Am. 60, 15–17 (1970).
    [Crossref]
  11. T. S. Huang, J. W. Burnett, and A. G. Jeczky, “The importance of phase in image processing filters,” IEEE Trans. Acoust. Speech Signal Process. 23, 529–542 (1975).
    [Crossref]
  12. A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
    [Crossref]
  13. Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Whole optical wave field reconstruction from double or multi in-line holograms by phase retrieval algorithm,” Opt. Express 11, 3234–3241 (2003).
    [Crossref] [PubMed]
  14. G. Pedrini, W. Osten, and Y. Zhang, “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. 30, 833–835 (2005).
    [Crossref] [PubMed]
  15. A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Large-scale optical-field measurements with geometric fibre constructs,” Nat. Mater. 5, 532–536 (2006).
    [Crossref] [PubMed]
  16. F. Sorin, O. Shapira, A. F. Abouraddy, M. Spencer, N. D. Orf, J. D. Joannopoulos, and Y. Fink, “Exploiting collective effects of multiple optoelectronic devices integrated in a single fiber,” Nano Lett. 9, 2630–2635 (2009).
    [Crossref] [PubMed]
  17. T. Young, “Experimental demonstration of the general law of interference of light,” Philos. Trans. Roy. Soc. London 94, 1–16 (1804).
    [Crossref]
  18. F. Zernike, “The concept of the degree of coherence and its application to optical problems,” Physica 5, 785 (1938).
    [Crossref]
  19. B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. 47, 895–902 (1957).
    [Crossref]
  20. M. Françon and S. Mallick, “Measurement of the second order degree of coherence,” in “Progress in Optics,”, vol. VIE. Wolf, ed. (North-Holland, 1967), pp. 73–104.
  21. C. Iaconis and I. A. Walmsley, “Direct measurement of the two-point correlation function,” Opt. Lett. 21, 1783–1785 (1996).
    [Crossref] [PubMed]
  22. C.-C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
    [Crossref]
  23. M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront young interferometer,” Opt. Lett. 31, 861–863 (2006).
    [Crossref] [PubMed]
  24. Y. Mejía and A. I. Gonzáles, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. 273, 428–434 (2007).
    [Crossref]
  25. A. I. Gonzáles and 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]
  26. S. Divitt, Z. J. Lapin, and L. Novotny, “Measuring coherence functions using non-parallel double slits,” Opt. Express 22, 8277–8290 (2014).
    [Crossref] [PubMed]
  27. S. Divitt and L. Novotny, “Spatial coherence of sunlight and its implications for light management in photovoltaics,” Optica 2, 95–103 (2015).
    [Crossref]
  28. B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
    [Crossref] [PubMed]
  29. S. Cho, M. A. Alonso, and T. G. Brown, “Measurement of spatial coherence through diffraction from a transparent mask with a phase discontinuity,” Opt. Lett. 37, 2724 (2012).
    [Crossref] [PubMed]
  30. J. K. Wood, K. A. Sharma, S. Cho, T. G. Brown, and M. A. Alonso, “Using shadows to measure spatial coherence,” Opt. Lett. 39, 4927–4930 (2014).
    [Crossref] [PubMed]
  31. K. A. Sharma, T. G. Brown, and M. A. Alonso, “Phase-space approach to lensless measurements of optical field correlations,” Opt. Express 24, 16099–16110 (2016).
    [Crossref] [PubMed]
  32. D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
    [Crossref]
  33. S. Tripathi and J. K. C. Toussaint, “Harnessing randomness to control the polarization of light transmitted through highly scattering media,” Opt. Express 22, 4412–4422 (2014).
    [Crossref] [PubMed]
  34. N. A. Riza, “Coherent techniques and deformable-mirror device (DMD)-based switched photonic time delay control for high frequency antenna arrays,” Proc. SPIE 3160, 97–105 (1997).
    [Crossref]
  35. M. Sheikh and N. A. Riza, “Demonstration of pinhole laser beam profiling using a digital micromirror device,” IEEE Photon. Technol. Lett. 21, 666–668 (2009).
    [Crossref]
  36. Y. X. Ren, R. D. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527, 447–470 (2015).
    [Crossref]
  37. J. P. L. Torre, M. J. Amin, and N. A. Riza, “Agile wavefront splitting interferometry and imaging using a digital micromirror device,” SPIE 9896, 98960B (2016).
  38. G. Vdovin, H. Gong, O. Soloviev, P. Pozzi, and M. Verhaegen, “Lensless coherent imaging by sampling of the optical field with digital micromirror device,” J. Opt. 17, 1–5 (2015).
    [Crossref]
  39. H. Partanen, J. Turunen, and J. Tervo, “Coherence measurement with digital micromirror device,” Opt. Lett. 39, 1034–1037 (2014).
    [Crossref] [PubMed]
  40. A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187–188 (1967).
    [Crossref]
  41. D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional coherence imaging in the Fresnel domain,” Appl. Opt. 38, 1332–1342 (1999).
    [Crossref]
  42. B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2007), 2nd ed.
  43. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
    [Crossref]
  44. J. Deschamps, D. Courjon, and J. Bulabois, “Gaussian Schell-model sources: an example and some perspectives,” J. Opt. Soc. Am. 73, 256–261 (1983).
    [Crossref]
  45. Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 15, 695–705 (1998).
  46. A. El Halawany, A. Beckus, H. E. Kondakci, M. Monroe, N. Mohammadian, G. K. Atia, and A. F. Abouraddy, “Incoherent lensless imaging via coherency back-propagation,” arXiv:1705.03993 [physics.optics].
  47. K. H. Kagalwala, G. D. Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
    [Crossref]
  48. A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
    [Crossref] [PubMed]
  49. K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
    [Crossref] [PubMed]

2016 (2)

J. P. L. Torre, M. J. Amin, and N. A. Riza, “Agile wavefront splitting interferometry and imaging using a digital micromirror device,” SPIE 9896, 98960B (2016).

K. A. Sharma, T. G. Brown, and M. A. Alonso, “Phase-space approach to lensless measurements of optical field correlations,” Opt. Express 24, 16099–16110 (2016).
[Crossref] [PubMed]

2015 (4)

S. Divitt and L. Novotny, “Spatial coherence of sunlight and its implications for light management in photovoltaics,” Optica 2, 95–103 (2015).
[Crossref]

G. Vdovin, H. Gong, O. Soloviev, P. Pozzi, and M. Verhaegen, “Lensless coherent imaging by sampling of the optical field with digital micromirror device,” J. Opt. 17, 1–5 (2015).
[Crossref]

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref] [PubMed]

Y. X. Ren, R. D. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527, 447–470 (2015).
[Crossref]

2014 (6)

2013 (1)

K. H. Kagalwala, G. D. Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

2012 (1)

2011 (1)

2009 (2)

M. Sheikh and N. A. Riza, “Demonstration of pinhole laser beam profiling using a digital micromirror device,” IEEE Photon. Technol. Lett. 21, 666–668 (2009).
[Crossref]

F. Sorin, O. Shapira, A. F. Abouraddy, M. Spencer, N. D. Orf, J. D. Joannopoulos, and Y. Fink, “Exploiting collective effects of multiple optoelectronic devices integrated in a single fiber,” Nano Lett. 9, 2630–2635 (2009).
[Crossref] [PubMed]

2007 (1)

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

2006 (2)

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Large-scale optical-field measurements with geometric fibre constructs,” Nat. Mater. 5, 532–536 (2006).
[Crossref] [PubMed]

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

2005 (1)

2003 (2)

2000 (1)

C.-C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

1999 (1)

1998 (1)

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 15, 695–705 (1998).

1997 (2)

N. A. Riza, “Coherent techniques and deformable-mirror device (DMD)-based switched photonic time delay control for high frequency antenna arrays,” Proc. SPIE 3160, 97–105 (1997).
[Crossref]

N. George, “Lensless electronic imaging,” Opt. Commun. 133, 22–26 (1997).
[Crossref]

1996 (1)

1983 (1)

1982 (2)

J. R. Fienup, “Phase retrieval algorithms: A comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref] [PubMed]

H. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Iterative procedure for signal reconstruction from Fourier transform phase,” Opt. Eng. 21, 122–127 (1982).

1981 (1)

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[Crossref]

1980 (1)

H. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust. Speech Signal Process. 28, 672–680 (1980).
[Crossref]

1978 (1)

1975 (1)

T. S. Huang, J. W. Burnett, and A. G. Jeczky, “The importance of phase in image processing filters,” IEEE Trans. Acoust. Speech Signal Process. 23, 529–542 (1975).
[Crossref]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1970 (1)

1967 (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187–188 (1967).
[Crossref]

1957 (1)

1938 (1)

F. Zernike, “The concept of the degree of coherence and its application to optical problems,” Physica 5, 785 (1938).
[Crossref]

1804 (1)

T. Young, “Experimental demonstration of the general law of interference of light,” Philos. Trans. Roy. Soc. London 94, 1–16 (1804).
[Crossref]

Abouraddy, A. F.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref] [PubMed]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref] [PubMed]

K. H. Kagalwala, G. D. Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

F. Sorin, O. Shapira, A. F. Abouraddy, M. Spencer, N. D. Orf, J. D. Joannopoulos, and Y. Fink, “Exploiting collective effects of multiple optoelectronic devices integrated in a single fiber,” Nano Lett. 9, 2630–2635 (2009).
[Crossref] [PubMed]

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Large-scale optical-field measurements with geometric fibre constructs,” Nat. Mater. 5, 532–536 (2006).
[Crossref] [PubMed]

A. El Halawany, A. Beckus, H. E. Kondakci, M. Monroe, N. Mohammadian, G. K. Atia, and A. F. Abouraddy, “Incoherent lensless imaging via coherency back-propagation,” arXiv:1705.03993 [physics.optics].

Alonso, M. A.

Amin, M. J.

J. P. L. Torre, M. J. Amin, and N. A. Riza, “Agile wavefront splitting interferometry and imaging using a digital micromirror device,” SPIE 9896, 98960B (2016).

Arnold, J.

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Large-scale optical-field measurements with geometric fibre constructs,” Nat. Mater. 5, 532–536 (2006).
[Crossref] [PubMed]

Atia, G. K.

A. El Halawany, A. Beckus, H. E. Kondakci, M. Monroe, N. Mohammadian, G. K. Atia, and A. F. Abouraddy, “Incoherent lensless imaging via coherency back-propagation,” arXiv:1705.03993 [physics.optics].

Bayindir, M.

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Large-scale optical-field measurements with geometric fibre constructs,” Nat. Mater. 5, 532–536 (2006).
[Crossref] [PubMed]

Beckus, A.

A. El Halawany, A. Beckus, H. E. Kondakci, M. Monroe, N. Mohammadian, G. K. Atia, and A. F. Abouraddy, “Incoherent lensless imaging via coherency back-propagation,” arXiv:1705.03993 [physics.optics].

Borghi, R.

Brady, D. J.

Brown, T. G.

Bulabois, J.

Burnett, J. W.

T. S. Huang, J. W. Burnett, and A. G. Jeczky, “The importance of phase in image processing filters,” IEEE Trans. Acoust. Speech Signal Process. 23, 529–542 (1975).
[Crossref]

Cheng, C.-C.

C.-C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

Cho, S.

Courjon, D.

Deschamps, J.

Divitt, S.

Dudley, D.

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
[Crossref]

Duncan, W. M.

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
[Crossref]

Fienup, J. R.

Fink, Y.

F. Sorin, O. Shapira, A. F. Abouraddy, M. Spencer, N. D. Orf, J. D. Joannopoulos, and Y. Fink, “Exploiting collective effects of multiple optoelectronic devices integrated in a single fiber,” Nano Lett. 9, 2630–2635 (2009).
[Crossref] [PubMed]

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Large-scale optical-field measurements with geometric fibre constructs,” Nat. Mater. 5, 532–536 (2006).
[Crossref] [PubMed]

Françon, M.

M. Françon and S. Mallick, “Measurement of the second order degree of coherence,” in “Progress in Optics,”, vol. VIE. Wolf, ed. (North-Holland, 1967), pp. 73–104.

Friberg, A. T.

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 15, 695–705 (1998).

George, N.

N. George, “Lensless electronic imaging,” Opt. Commun. 133, 22–26 (1997).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Giuseppe, G. D.

K. H. Kagalwala, G. D. Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

Gong, H.

G. Vdovin, H. Gong, O. Soloviev, P. Pozzi, and M. Verhaegen, “Lensless coherent imaging by sampling of the optical field with digital micromirror device,” J. Opt. 17, 1–5 (2015).
[Crossref]

Gong, L.

Y. X. Ren, R. D. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527, 447–470 (2015).
[Crossref]

Gonzáles, A. I.

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

Halawany, A. El

A. El Halawany, A. Beckus, H. E. Kondakci, M. Monroe, N. Mohammadian, G. K. Atia, and A. F. Abouraddy, “Incoherent lensless imaging via coherency back-propagation,” arXiv:1705.03993 [physics.optics].

Hayes, H. M.

H. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Iterative procedure for signal reconstruction from Fourier transform phase,” Opt. Eng. 21, 122–127 (1982).

H. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust. Speech Signal Process. 28, 672–680 (1980).
[Crossref]

He, Q.

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 15, 695–705 (1998).

Heier, H.

C.-C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

Hinczewski, D. S.

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Large-scale optical-field measurements with geometric fibre constructs,” Nat. Mater. 5, 532–536 (2006).
[Crossref] [PubMed]

Hradil, Z.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref] [PubMed]

Huang, T. S.

T. S. Huang, J. W. Burnett, and A. G. Jeczky, “The importance of phase in image processing filters,” IEEE Trans. Acoust. Speech Signal Process. 23, 529–542 (1975).
[Crossref]

Iaconis, C.

Jähne, B.

B. Jähne, Digital Image Processing: Concepts, Algorithms, and Scientific Applications (Springer, 1997).

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).

Jeczky, A. G.

T. S. Huang, J. W. Burnett, and A. G. Jeczky, “The importance of phase in image processing filters,” IEEE Trans. Acoust. Speech Signal Process. 23, 529–542 (1975).
[Crossref]

Joannopoulos, J. D.

F. Sorin, O. Shapira, A. F. Abouraddy, M. Spencer, N. D. Orf, J. D. Joannopoulos, and Y. Fink, “Exploiting collective effects of multiple optoelectronic devices integrated in a single fiber,” Nano Lett. 9, 2630–2635 (2009).
[Crossref] [PubMed]

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Large-scale optical-field measurements with geometric fibre constructs,” Nat. Mater. 5, 532–536 (2006).
[Crossref] [PubMed]

Kagalwala, K. H.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref] [PubMed]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref] [PubMed]

K. H. Kagalwala, G. D. Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

Kermisch, D.

Kondakci, H. E.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref] [PubMed]

A. El Halawany, A. Beckus, H. E. Kondakci, M. Monroe, N. Mohammadian, G. K. Atia, and A. F. Abouraddy, “Incoherent lensless imaging via coherency back-propagation,” arXiv:1705.03993 [physics.optics].

Lapin, Z. J.

Lim, J. S.

H. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Iterative procedure for signal reconstruction from Fourier transform phase,” Opt. Eng. 21, 122–127 (1982).

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[Crossref]

H. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust. Speech Signal Process. 28, 672–680 (1980).
[Crossref]

Lu, R. D.

Y. X. Ren, R. D. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527, 447–470 (2015).
[Crossref]

Mallick, S.

M. Françon and S. Mallick, “Measurement of the second order degree of coherence,” in “Progress in Optics,”, vol. VIE. Wolf, ed. (North-Holland, 1967), pp. 73–104.

Mandel, L.

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

Marks, D. L.

Mejía, Y.

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

Mohammadian, N.

A. El Halawany, A. Beckus, H. E. Kondakci, M. Monroe, N. Mohammadian, G. K. Atia, and A. F. Abouraddy, “Incoherent lensless imaging via coherency back-propagation,” arXiv:1705.03993 [physics.optics].

Monroe, M.

A. El Halawany, A. Beckus, H. E. Kondakci, M. Monroe, N. Mohammadian, G. K. Atia, and A. F. Abouraddy, “Incoherent lensless imaging via coherency back-propagation,” arXiv:1705.03993 [physics.optics].

Motka, L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref] [PubMed]

Novotny, L.

Oppenheim, A. V.

H. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Iterative procedure for signal reconstruction from Fourier transform phase,” Opt. Eng. 21, 122–127 (1982).

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[Crossref]

H. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust. Speech Signal Process. 28, 672–680 (1980).
[Crossref]

Orf, N. D.

F. Sorin, O. Shapira, A. F. Abouraddy, M. Spencer, N. D. Orf, J. D. Joannopoulos, and Y. Fink, “Exploiting collective effects of multiple optoelectronic devices integrated in a single fiber,” Nano Lett. 9, 2630–2635 (2009).
[Crossref] [PubMed]

Osten, W.

Partanen, H.

Pedrini, G.

Pozzi, P.

G. Vdovin, H. Gong, O. Soloviev, P. Pozzi, and M. Verhaegen, “Lensless coherent imaging by sampling of the optical field with digital micromirror device,” J. Opt. 17, 1–5 (2015).
[Crossref]

Raymer, M. G.

C.-C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

Rehacek, J.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref] [PubMed]

Ren, Y. X.

Y. X. Ren, R. D. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527, 447–470 (2015).
[Crossref]

Riza, N. A.

J. P. L. Torre, M. J. Amin, and N. A. Riza, “Agile wavefront splitting interferometry and imaging using a digital micromirror device,” SPIE 9896, 98960B (2016).

M. Sheikh and N. A. Riza, “Demonstration of pinhole laser beam profiling using a digital micromirror device,” IEEE Photon. Technol. Lett. 21, 666–668 (2009).
[Crossref]

N. A. Riza, “Coherent techniques and deformable-mirror device (DMD)-based switched photonic time delay control for high frequency antenna arrays,” Proc. SPIE 3160, 97–105 (1997).
[Crossref]

Sáchez-Soto, L. L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref] [PubMed]

Saleh, B. E.

B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2007), 2nd ed.

Saleh, B. E. A.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref] [PubMed]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref] [PubMed]

K. H. Kagalwala, G. D. Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

Santarsiero, M.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schell, A. C.

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187–188 (1967).
[Crossref]

Shapira, O.

F. Sorin, O. Shapira, A. F. Abouraddy, M. Spencer, N. D. Orf, J. D. Joannopoulos, and Y. Fink, “Exploiting collective effects of multiple optoelectronic devices integrated in a single fiber,” Nano Lett. 9, 2630–2635 (2009).
[Crossref] [PubMed]

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Large-scale optical-field measurements with geometric fibre constructs,” Nat. Mater. 5, 532–536 (2006).
[Crossref] [PubMed]

Sharma, K. A.

Sheikh, M.

M. Sheikh and N. A. Riza, “Demonstration of pinhole laser beam profiling using a digital micromirror device,” IEEE Photon. Technol. Lett. 21, 666–668 (2009).
[Crossref]

Slaughter, J.

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
[Crossref]

Soloviev, O.

G. Vdovin, H. Gong, O. Soloviev, P. Pozzi, and M. Verhaegen, “Lensless coherent imaging by sampling of the optical field with digital micromirror device,” J. Opt. 17, 1–5 (2015).
[Crossref]

Sorin, F.

F. Sorin, O. Shapira, A. F. Abouraddy, M. Spencer, N. D. Orf, J. D. Joannopoulos, and Y. Fink, “Exploiting collective effects of multiple optoelectronic devices integrated in a single fiber,” Nano Lett. 9, 2630–2635 (2009).
[Crossref] [PubMed]

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Large-scale optical-field measurements with geometric fibre constructs,” Nat. Mater. 5, 532–536 (2006).
[Crossref] [PubMed]

Spencer, M.

F. Sorin, O. Shapira, A. F. Abouraddy, M. Spencer, N. D. Orf, J. D. Joannopoulos, and Y. Fink, “Exploiting collective effects of multiple optoelectronic devices integrated in a single fiber,” Nano Lett. 9, 2630–2635 (2009).
[Crossref] [PubMed]

Stack, R. A.

Stoklasa, B.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref] [PubMed]

Teich, M. C.

B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2007), 2nd ed.

Tervo, J.

Thompson, B. J.

Tiziani, H. J.

Torre, J. P. L.

J. P. L. Torre, M. J. Amin, and N. A. Riza, “Agile wavefront splitting interferometry and imaging using a digital micromirror device,” SPIE 9896, 98960B (2016).

Toussaint, J. K. C.

Tripathi, S.

Turunen, J.

H. Partanen, J. Turunen, and J. Tervo, “Coherence measurement with digital micromirror device,” Opt. Lett. 39, 1034–1037 (2014).
[Crossref] [PubMed]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 15, 695–705 (1998).

Vdovin, G.

G. Vdovin, H. Gong, O. Soloviev, P. Pozzi, and M. Verhaegen, “Lensless coherent imaging by sampling of the optical field with digital micromirror device,” J. Opt. 17, 1–5 (2015).
[Crossref]

Verhaegen, M.

G. Vdovin, H. Gong, O. Soloviev, P. Pozzi, and M. Verhaegen, “Lensless coherent imaging by sampling of the optical field with digital micromirror device,” J. Opt. 17, 1–5 (2015).
[Crossref]

Walmsley, I. A.

Wolf, E.

Wood, J. K.

Young, T.

T. Young, “Experimental demonstration of the general law of interference of light,” Philos. Trans. Roy. Soc. London 94, 1–16 (1804).
[Crossref]

Zernike, F.

F. Zernike, “The concept of the degree of coherence and its application to optical problems,” Physica 5, 785 (1938).
[Crossref]

Zhang, Y.

Ann. Phys. (1)

Y. X. Ren, R. D. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527, 447–470 (2015).
[Crossref]

Appl. Opt. (2)

IEEE Photon. Technol. Lett. (1)

M. Sheikh and N. A. Riza, “Demonstration of pinhole laser beam profiling using a digital micromirror device,” IEEE Photon. Technol. Lett. 21, 666–668 (2009).
[Crossref]

IEEE Trans. Acoust. Speech Signal Process. (2)

H. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust. Speech Signal Process. 28, 672–680 (1980).
[Crossref]

T. S. Huang, J. W. Burnett, and A. G. Jeczky, “The importance of phase in image processing filters,” IEEE Trans. Acoust. Speech Signal Process. 23, 529–542 (1975).
[Crossref]

IEEE Trans. Antennas Propag. (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187–188 (1967).
[Crossref]

J. Mod. Opt. (1)

C.-C. Cheng, M. G. Raymer, and H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

J. Opt. (1)

G. Vdovin, H. Gong, O. Soloviev, P. Pozzi, and M. Verhaegen, “Lensless coherent imaging by sampling of the optical field with digital micromirror device,” J. Opt. 17, 1–5 (2015).
[Crossref]

J. Opt. Soc. Am. (3)

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

Nano Lett. (1)

F. Sorin, O. Shapira, A. F. Abouraddy, M. Spencer, N. D. Orf, J. D. Joannopoulos, and Y. Fink, “Exploiting collective effects of multiple optoelectronic devices integrated in a single fiber,” Nano Lett. 9, 2630–2635 (2009).
[Crossref] [PubMed]

Nat. Commun. (1)

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sáchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref] [PubMed]

Nat. Mater. (1)

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Large-scale optical-field measurements with geometric fibre constructs,” Nat. Mater. 5, 532–536 (2006).
[Crossref] [PubMed]

Nat. Photon. (1)

K. H. Kagalwala, G. D. Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

Opt. Commun. (3)

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 15, 695–705 (1998).

N. George, “Lensless electronic imaging,” Opt. Commun. 133, 22–26 (1997).
[Crossref]

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

Opt. Eng. (1)

H. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Iterative procedure for signal reconstruction from Fourier transform phase,” Opt. Eng. 21, 122–127 (1982).

Opt. Express (4)

Opt. Lett. (8)

Optica (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Philos. Trans. Roy. Soc. London (1)

T. Young, “Experimental demonstration of the general law of interference of light,” Philos. Trans. Roy. Soc. London 94, 1–16 (1804).
[Crossref]

Physica (1)

F. Zernike, “The concept of the degree of coherence and its application to optical problems,” Physica 5, 785 (1938).
[Crossref]

Proc. IEEE (1)

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[Crossref]

Proc. SPIE (2)

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
[Crossref]

N. A. Riza, “Coherent techniques and deformable-mirror device (DMD)-based switched photonic time delay control for high frequency antenna arrays,” Proc. SPIE 3160, 97–105 (1997).
[Crossref]

Sci. Rep. (1)

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref] [PubMed]

SPIE (1)

J. P. L. Torre, M. J. Amin, and N. A. Riza, “Agile wavefront splitting interferometry and imaging using a digital micromirror device,” SPIE 9896, 98960B (2016).

Other (7)

M. Françon and S. Mallick, “Measurement of the second order degree of coherence,” in “Progress in Optics,”, vol. VIE. Wolf, ed. (North-Holland, 1967), pp. 73–104.

T. S. Huang, ed., Picture Processing and Digital Filtering (Springer, 1979).
[Crossref]

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).

B. Jähne, Digital Image Processing: Concepts, Algorithms, and Scientific Applications (Springer, 1997).

A. El Halawany, A. Beckus, H. E. Kondakci, M. Monroe, N. Mohammadian, G. K. Atia, and A. F. Abouraddy, “Incoherent lensless imaging via coherency back-propagation,” arXiv:1705.03993 [physics.optics].

B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2007), 2nd ed.

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

Supplementary Material (1)

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» Visualization 1: MP4 (2682 KB)      Movie associated with Figure 7

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Figures (7)

Fig. 1
Fig. 1 Complex-coherence measurements of incoherent light scattered from an object. (a) Schematic of the experimental setup. Incoherent light (an LED) is incident on a 1D object and the scattered light impinges on computer-controlled double slits realized via a digital micromirror device (DMD). We depict the object obstructing the field (we also perform the experiment with only the reflected light reaching the detector). A spatial Fourier transform is implemented between the double slits and the CCD camera. In the experiment, the slits are realized in reflection mode, and is followed by a pair of lenses for optical relay and a cylindrical lens that produces the Fourier transform (omitted from the schematic for simplicity). The inset depicts the effect of the phase φ=Arg{G(x1x2)} on the interference pattern. When φ = π, the intensity at the center of the interferogram is a minimum. (b) Interference patterns produced at the CCD while increasing the slit separation. The period of the interferogram decreases with increased separation, and the visibility drops with potential revivals at larger separations. When only one slit is activated, a diffraction pattern with no interference is registered (top panel). (c) 1D sections I(x) through interferograms (vertically displaced for clarity) with decreasing period as the slit separation is increased. The phase Arg{G(x1x2)} is measured relative to the dotted vertical reference. The shifts between the dotted and dashed lines corresponds to a π-phase. (d) From each interferogram in (c) we obtain the visibility V as a function of slit separation (each colored circle is extracted from one interferogram). The visibility is related to the amplitude of the degree of coherence | g ( x 1 x 2 ) | = V ( I 1 + I 2 ) / ( 2 I 1 I 2 ). (e) The phases Arg{G(x1x2)} are extracted from the displacement of the local maxima and minima around the center fringe, with phase 0 assigned to the first interferogram.
Fig. 2
Fig. 2 (a) The measured magnitude of the degree of spatial coherence |g(x1x2)| (diamonds) at the detector plane in absence of an object (uninterrupted propagation from the source to the detector). The solid line is a theoretical fit. The measured and theoretical phase Arg{g(x1, x2)} is zero over this range [see Fig. 3(b)]. The schematic above the panel depicts the measurement geometry. The red circle is the LED source and the DMD implementing the double slits is placed at the detection plane. (b) A plot of the spatial coherence function magnitude of the source Gs(x′, x″) based on Eq. 5 that makes use of the parameters extracted from the measurements in (a).
Fig. 3
Fig. 3 Impact of the object size w on the degree of spatial coherence g(x1x2) when the object location (x0, d1) is held fixed for two source-to-object distances; (a, b) d1 = 0.5 cm and (c,d) d1 = 24 cm. (a) The measured magnitude of the degree of coherence g(x1x2)| in three cases: unimpeded propagation from the source to the detection plane (no object, w →0), w =0.5 mm, and w =1 mm. In all cases x0 =0, d1 =0.5 cm, and d =125 cm. The schematic above the panel depicts the measurement geometry (the gray circle is the object). The data points are plotted as diamonds, and the solid lines are theoretical predictions based on Eqs. 2, 3, and 5. (b) The measured phases Arg{g(x1x2)} corresponding to the three cases plotted in (a). The diamonds are data points and the solid lines are theoretical predictions. (c,d) Same as (a,b) except that d1 =24 cm; that is, the object is placed farther away from the source and closer to the detection plane (the total distance from source to the DMD is held fixed at d =125 cm).
Fig. 4
Fig. 4 Change in the degree of spatial coherence as the object is displaced in the transverse plane with respect to the optical axis, indicated by the coordinate x0. The experiments are repeated twice, corresponding to the left and right columns. Each experiment has a different source-to-object distance d1. In the left column (a,b), d1 =0.5 cm, whereas in the left column (c,d) d1 = 24 cm. In all cases, the object width is w = 0.5 mm and the total distance from source to the detection plane is d =125 cm. (a) Measured coherence magnitude |g(x1x2)| and (b) phase Arg{g(x1x2)} are shown for an object while varying x0 from −100 µm to 100 µm. In this experiment, the object is placed at d1 =0.5 cm. (c,d) Same as (a,b) except that the object is placed at d1 =24 cm from the source.
Fig. 5
Fig. 5 Change in the degree of spatial coherence with source-to-object axial distances d1. The experiments are repeated twice, corresponding to the left and right columns. Each experiment has a different transverse displacements x0 of the object. In the left column (a,b) we have x0 = 0, whereas in the right column (c,d) x0 = 100 µm. The object width is w = 0.5 mm and the total distance from source to the detection plane is d = 1.25 m. (a) Measured coherence magnitude |g(x1x2)| and (b) phase Arg{g(x1 x2)} are shown while varying d1. In this experiment, the object was placed on the optical axis x0 =0. (c,d) Same as (a,b) except that the object is displaced from the optical axis to x0 =100 µm.
Fig. 6
Fig. 6 Comparison of the degree of spatial coherence for reflective objects of varying widths w. The distance from the source to the object is d1 = 65 cm, and the total distance from source to DMD is d = 1:45 m. The objects are placed on the optical axis defines by the source x0 =0. (a) The coherence magnitude |g(x1x2)| and (b) phase Arg{g(x1−x2)} are shown while varying w. Experimental results are plotted with diamonds and theoretical predictions are solid lines. The infinite width case is equivalent to free space propagation. The phase Arg{g(x1x2)} is set to zero when |g(x1x2)| ≤ 0.05 to avoid errors stemming from the low signal level.
Fig. 7
Fig. 7 Video showing the variations in intensity and coherence as an object is moved transversely in front of the light source (see Visualization 1). Each frame in the video corresponds to the object displaced to a different transverse position x0. (a) Image of the shadow cast by a 0.5-mm-wide metal wire (object) moving in front of the LED source in the direct vicinity of the object (d2 ≈ 0). A geometric cast shadow is clear. Image size is 25 × 25 mm2. (b) Image of the shadow cast by the object at d2 ≈ 12 cm. The shadow is now blurred. Image size is 25×25 mm2. (c) Image of the shadow cast by the object at d2 ≈ 100 cm. The shadow is no longer discernible. Image size is 25 × 25 mm2. (d) Measured intensity profile I(x1; d2 = 100 cm) along the center of the beam in (c) for different transverse positions x0. Inset shows a schematic of the setup and the transverse motion of the object in the range −0.5 ≤ x0 ≤ 0.5 mm. (e) Measured |g(x1x2)| corresponding to the displaced positions of the object. When the object is on the optical axis, nulls develop in the coherence function. Lifting the ambiguity with respect to the position of the object on the left or the right of the optical axis requires measuring the phase Arg{g(x1x2)} [see Fig. 4].

Equations (5)

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h F ( x 1 , x ; d ) = exp ( i k d ) i λ d exp { i k 2 d ( x 1 x ) 2 } ,
h ( x 1 , x ) = d x ˜ h F ( x 1 , x ˜ ; d 2 ) t ( x ˜ ) h F ( x ˜ , x ; d 1 ) .
G ( x 1 , x 2 ) = d x d x h ( x 1 , x ) h * ( x 2 , x ) G s ( x , x ) .
I ( x ) sinc 2 ( k x 2 π M f ) { I 1 + I 2 + 2 | G ( x 1 x 2 ) | cos ( k x M f ( x 1 x 2 ) φ ) } ,
G s ( x , x ; λ ) = exp ( ( x + x ) 2 2 α 2 ) exp ( ( x + x ) 2 2 σ 2 ) rect ( x L ) rect ( x L ) rect ( λ λ 0 Δ λ ) .

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