Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Phase-space measurement and coherence synthesis of optical beams


Phase-space optics allows the simultaneous visualization of both space (x) and spatial frequency (k) information. Previous experiments have focused mainly on coherent beams, which are fully described by a two-dimensional complex field (amplitude and phase). In contrast, partially coherent beams inherently have more degrees of freedom and require a four-dimensional description. This description is particularly important for propagation, in which coherence properties determine the intensity evolution. Despite this, most measurements in linear optics, and all those in nonlinear optics, have recorded only the intensity and power spectrum projections (x-space or k-space only). Measuring local coherence remains a challenging problem, especially in the full four-dimensional phase space. In turn, the recording difficulty has limited efforts to generate arbitrary, spatially varying patterns of coherence, despite their usefulness for imaging, illumination and display. We remedy both problems here, using spatial light modulators to create beams with locally varying spatial coherence and to measure their phase-space properties.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Set-up for phase-space synthesis and measurement of nonlinear propagation.
Figure 2: Linear propagation results in a shear of phase space for both coherent and partially coherent light.
Figure 3: Nonlinear propagation depends on the entire phase space.
Figure 4: Experimental phase-space measurement of modulation instability.
Figure 5: Experimental measurement of a coherence wave localized in phase space.
Figure 6: Experimental measurements of four-dimensional coherence synthesis.


  1. 1

    Barsi, C., Wan, W. & Fleischer, J. W. Imaging through nonlinear media using digital holography. Nature Photon. 3, 211–215 (2009).

    ADS  Article  Google Scholar 

  2. 2

    Tsang, M., Psaltis, D. & Omenetto, F. Reverse propagation of femtosecond pulses in optical fibers. Opt. Lett. 28, 1873–1875 (2003).

    ADS  Article  Google Scholar 

  3. 3

    Testorf, M., Hennelly, B. & Ojeda-Castaneda, J. Phase-Space Optics (McGraw-Hill, 2009).

  4. 4

    Hall, B., Lisak, M., Anderson, D., Fedele, R. & Semenov, V. Statistical theory for incoherent light propagation in nonlinear media. Phys. Rev. E 65, 035602 (2002).

    ADS  Article  Google Scholar 

  5. 5

    Walther, A. Radiometry and coherence. J. Opt. Soc. Am. 58, 1256–1259 (1968).

    ADS  Article  Google Scholar 

  6. 6

    Bastiaans, M. Applications of the Wigner distribution function to partially coherent light beams. Proc. SPIE 3729, 114–128 (1999).

    ADS  Article  Google Scholar 

  7. 7

    Brenner, K. & Ojeda-Castaneda, J. Ambiguity function and Wigner distribution function applied to partially coherent imagery. J. Mod. Opt. 31, 213–223 (1984).

    Google Scholar 

  8. 8

    Alonso, M. Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles. Adv. Opt. Photon. 3, 272–365 (2011).

    Article  Google Scholar 

  9. 9

    Dragoman, D. Phase-space interferences as the source of negative values of the Wigner distribution function. J. Opt. Soc. Am. A 17, 2481–2485 (2000).

    ADS  MathSciNet  Article  Google Scholar 

  10. 10

    Wax, A. & Thomas, J. E. Optical heterodyne imaging and Wigner phase space distributions. Opt. Lett. 21, 1427–1429 (1996).

    ADS  Article  Google Scholar 

  11. 11

    Marks, D., Stack, R., Brady, D., Munson, D. & Brady, R. Visible cone-beam tomography with a lensless interferometric camera. Science 284, 2164–2166 (1999).

    Article  Google Scholar 

  12. 12

    Marks, D., Stack, R. & Brady, D. Astigmatic coherence sensor for digital imaging. Opt. Lett. 25, 1726–1728 (2000).

    ADS  Article  Google Scholar 

  13. 13

    Raymer, M., Beck, M. & McAlister, D. Complex wave-field reconstruction using phase-space tomography. Phys. Rev. Lett. 72, 1137–1140 (1994).

    ADS  MathSciNet  Article  Google Scholar 

  14. 14

    Cámara, A., Alieva, T., Rodrigo, J. & Calvo, M. Phase-space tomography with a programmable Radon–Wigner display. Opt. Lett. 36, 2441–2443 (2011).

    ADS  Article  Google Scholar 

  15. 15

    Tran, C. et al. X-ray imaging: a generalized approach using phase-space tomography. J. Opt. Soc. Am. A 22, 1691–1700 (2005).

    ADS  MathSciNet  Article  Google Scholar 

  16. 16

    Flewett, S., Quiney, H., Tran, C. & Nugent, K. Extracting coherent modes from partially coherent wavefields. Opt. Lett. 34, 2198–2200 (2009).

    ADS  Article  Google Scholar 

  17. 17

    Schafer, B. & Mann, K. Determination of beam parameters and coherence properties of laser radiation by use of an extended Hartmann–Shack wave-front sensor. Appl. Opt. 41, 2809–2817 (2002).

    ADS  Article  Google Scholar 

  18. 18

    Lindlein, N., Pfund, J. & Schwider, J. Algorithm for expanding the dynamic range of a Shack–Hartmann sensor by using a spatial light modulator array. Opt. Eng. 40, 837–840 (2001).

    ADS  Article  Google Scholar 

  19. 19

    Bartelt, H., Brenner, K. & Lohmann, A. The Wigner distribution function and its optical production. Opt. Commun. 32, 32–38 (1980).

    ADS  Article  Google Scholar 

  20. 20

    Brenner, K. H. & Lohmann, A. W. Wigner distribution function display of complex 1D signals. Opt. Commun. 42, 310–314 (1982).

    ADS  Article  Google Scholar 

  21. 21

    Bastiaans, M. Uncertainty principle and informational entropy for partially coherent light. J. Opt. Soc. Am. A 3, 1243–1246 (1986).

    ADS  MathSciNet  Article  Google Scholar 

  22. 22

    Accardi, A. & Wornell, G. Quasi light fields: extending the light field to coherent radiation. J. Opt. Soc. Am. A 26, 2055–2066 (2009).

    ADS  Article  Google Scholar 

  23. 23

    Christodoulides, D., Eugenieva, E., Coskun, T., Segev, M. & Mitchell, M. Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media. Phys. Rev. E 63, 035601 (2001).

    ADS  Article  Google Scholar 

  24. 24

    Shkunov, V. & Anderson, D. Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media. Phys. Rev. Lett. 81, 2683–2686 (1998).

    ADS  Article  Google Scholar 

  25. 25

    Bastiaans, M. The Wigner distribution function applied to optical signals and systems. Opt. Commun. 25, 26–30 (1978).

    ADS  Article  Google Scholar 

  26. 26

    Christodoulides, D., Coskun, T., Mitchell, M. & Segev, M. Theory of incoherent self-focusing in biased photorefractive media. Phys. Rev. Lett. 78, 646–649 (1997).

    ADS  Article  Google Scholar 

  27. 27

    Dylov, D. V., Waller, L. & Fleischer, J. W. Nonlinear restoration of diffused images via seeded instability. IEEE J. Sel. Top. Quant. Electron. 916–925 (2012).

  28. 28

    Mitchell, M., Segev, M., Coskun, T. & Christodoulides, D. Theory of self-trapped spatially incoherent light beams. Phys. Rev. Lett. 79, 4990–4993 (1997).

    ADS  Article  Google Scholar 

  29. 29

    Mitchell, M., Chen, Z., Shih, M. & Segev, M. Self-trapping of partially spatially incoherent light. Phys. Rev. Lett. 77, 490–493 (1996).

    ADS  Article  Google Scholar 

  30. 30

    Soljacic, M., Segev, M., Coskun, T., Christodoulides, D. & Vishwanath, A. Modulation instability of incoherent beams in noninstantaneous nonlinear media. Phys. Rev. Lett. 84, 467–470 (2000).

    ADS  Article  Google Scholar 

  31. 31

    Kip, D., Soljacic, M., Segev, M., Eugenieva, E. & Christodoulides, D. Modulation instability and pattern formation in spatially incoherent light beams. Science 290, 495–498 (2000).

    ADS  Article  Google Scholar 

  32. 32

    Sheppard, C. J. R. Defocused transfer function for a partially coherent microscope and application to phase retrieval. J. Opt. Soc. Am. A 21, 828–831 (2004).

    ADS  Article  Google Scholar 

  33. 33

    Shirai, T. & Wolf, E. Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space. J. Opt. Soc. Am. A 21, 1907–1916 (2004).

    ADS  MathSciNet  Article  Google Scholar 

  34. 34

    Ostrovsky, A. & Hernández García, E. Modulation of spatial coherence of optical field by means of liquid crystal light modulator. Rev. Mex. Fis. 51, 442–446 (2005).

    Google Scholar 

  35. 35

    Funamizu, H. & Uozumi, J. Generation of fractal speckles by means of a spatial light modulator. Opt. Express 15, 7415–7422 (2007).

    ADS  Article  Google Scholar 

  36. 36

    Betancur, R. & Castañeda, R. Spatial coherence modulation. J. Opt. Soc. Am. A 26, 147–155 (2009).

    ADS  Article  Google Scholar 

  37. 37

    Takeda, M., Wang, W., Duan, Z. & Miyamoto, Y. Coherence holography. Opt. Express 13, 9629–9635 (2005).

    ADS  Article  Google Scholar 

  38. 38

    Mendlovic, D., Shabtay, G. & Lohmann, A. Synthesis of spatial coherence. Opt. Lett. 24, 361–363 (1999).

    ADS  Article  Google Scholar 

  39. 39

    Erden, M., Ozaktas, H. & Mendlovic, D. Synthesis of mutual intensity distributions using the fractional Fourier transform. Opt. Commun. 125, 288–301 (1996).

    ADS  Article  Google Scholar 

  40. 40

    Zalevsky, Z., Medlovic, D. & Ozaktas, H. Energetic efficient synthesis of general mutual intensity distribution. J. Opt. A 2, 83–87 (2000).

    ADS  Article  Google Scholar 

  41. 41

    Santis, P., Gori, F., Santarsiero, M. & Guattari, G. Sources with spatially sinusoidal modes. Opt. Commun. 82, 123–129 (1991).

    ADS  Article  Google Scholar 

  42. 42

    Lohmann, A., Shabtay, G. & Mendlovic, D. Synthesis of hybrid spatial coherence. Appl. Opt. 38, 4279–4280 (1999).

    ADS  Article  Google Scholar 

  43. 43

    Lajunen, H. & Saastamoinen, T. Propagation characteristics of partially coherent beams with spatially varying correlations. Opt. Lett. 36, 4104–4106 (2011).

    ADS  Article  Google Scholar 

  44. 44

    Santis, P., Gori, F., Guattari, G. & Palma, C. Synthesis of partially coherent fields. J. Opt. Soc. Am. A 3, 1258–1262 (1986).

    ADS  Article  Google Scholar 

  45. 45

    Turunen, J., Vasara, A. & Friberg, A. Propagation invariance and self-imaging in variable-coherence optics. J. Opt. Soc. Am. A 8, 282–289 (1991).

    ADS  Article  Google Scholar 

  46. 46

    Gbur, G. & Visser, T. D. Coherence vortices in partially coherent beams. Opt. Commun. 222, 117–125 (2003).

    ADS  Article  Google Scholar 

  47. 47

    Wang, W., Duan, Z., Hanson, S. G., Miyamoto, Y. & Takeda, M. Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function. Phys. Rev. Lett. 96, 073902 (2006).

    ADS  Article  Google Scholar 

  48. 48

    Anderson, D., Helczynski-Wolf, L., Lisak, M. & Semenov, V. Features of modulational instability of partially coherent light: importance of the incoherence spectrum. Phys. Rev. E 69, 025601 (2004).

    ADS  Article  Google Scholar 

  49. 49

    Dylov, D. V. & Fleischer, J. W. Observation of all-optical bump-on-tail instability. Phys. Rev. Lett. 100, 103903 (2008).

    ADS  Article  Google Scholar 

  50. 50

    Dylov, D. V. & Fleischer, J. W. Nonlinear self-filtering of noisy images via dynamical stochastic resonance. Nature Photon. 4, 323–328 (2010).

    Article  Google Scholar 

Download references


The authors thank L. Tian and S. Muenzel for valuable discussions. This work was supported by the Department of Energy and the Air Force Office of Scientific Research.

Author information




L.W. and J.W.F. conceived and designed the experiments. L.W. performed the experiments and simulations. G.S. helped with the set-up of the experiments. All authors analysed the data and contributed to the preparation of the manuscript.

Corresponding author

Correspondence to Jason W. Fleischer.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary information

Supplementary information (PDF 595 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Waller, L., Situ, G. & Fleischer, J. Phase-space measurement and coherence synthesis of optical beams. Nature Photon 6, 474–479 (2012).

Download citation

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing