## Wigner function measurement using a lenslet array |

Optics Express, Vol. 21, Issue 9, pp. 10511-10525 (2013)

http://dx.doi.org/10.1364/OE.21.010511

Acrobat PDF (1874 KB)

### Abstract

Geometrical–optical arguments have traditionally been used to explain how a lenslet array measures the distribution of light jointly over space and spatial frequency. Here, we rigorously derive the connection between the intensity measured by a lenslet array and wave–optical representations of such light distributions for partially coherent optical beams by using the Wigner distribution function (WDF). It is shown that the action of the lenslet array is to sample a smoothed version of the beam’s WDF (SWDF). We consider the effect of lenslet geometry and coherence properties of the beam on this measurement, and we derive an expression for cross–talk between lenslets that corrupts the measurement. Conditions for a high fidelity measurement of the SWDF and the discrepancies between the measured SWDF and the WDF are investigated for a Schell–model beam.

© 2013 OSA

## 1. Introduction

*e.g.*intensity, energy density, Poynting vector) to be calculated over a region of free space. For most propagating optical fields, a description on some plane transverse to the optical axis is sufficient to determine the rest of the field; we will denote position on the plane by

**r**= (

*x*,

*y*) and position along the optical axis by

*z*. For monochromatic coherent light, the complex–valued field

*U*(

**r**;

*z*) provides such a description. For a quasi–monochromatic partially coherent field, a common description is the mutual intensity,

*J*(

**r**

_{1},

**r**

_{2};

*z*),

*i.e.*the statistical correlation of the field at all possible pairs of points over a plane [1]. The mutual intensity can be written as where 〈·〉 denotes the ensemble average over statistical realizations of the field

*U*, and the intensity is simply given by

*I*(

**r**) =

*J*(

**r**,

**r**).

2. B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. **47**, 895 (1957) [CrossRef] .

3. W. Tango and R. Twiss, “Michelson stellar interferometry,” Prog. Optics **17**, 239–277 (1980) [CrossRef] .

4. K. Itoh and Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A **3**, 94–100 (1986) [CrossRef] .

5. 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] .

*J*. By measuring the intensity over many different defocused planes, these integrals may be inverted to recover

*J*through a process known as phase space tomography [6

6. M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. **72**, 1137–1140 (1994) [CrossRef] .

*z*be denoted by

*B*(

**r**,

**p**;

*z*) where

**p**= (

*p*,

_{x}*p*) is the transverse component of the unit direction vector. Propagation of radiance is based on geometric optics and is simple to calculate: The intensity at any point is given by the integral of

_{y}*B*over all directions Use of the radiance predates the wave theory of light, and it was initially described by assigning non-negative values to all trajectories coming from source points. Such descriptions are insufficient to model wave effects since these trajectories contain no information about constructive or destructive interference. However, in certain situations, these wave effects can be safely ignored, and a lenslet array can then be used to obtain an estimate for the radiance since they allow joint measurement of the spatial and directional distribution of light, as in Shack-Hartmann sensors [12

12. B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. **17**(2001) [PubMed] .

15. J.-H. Park, K. Hong, and B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt. **48**, H77–H94 (2009) [CrossRef] [PubMed] .

16. E. H. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell. **14**, 99–106 (1992) [CrossRef] .

*𝒲*

_{p}∝

*δ*(

**r**)

*δ*(

**u**), where

*δ*is the Dirac delta function in order to measure the incident WDF with perfect resolution. However it may be easily verified from Eq. (4) that the existence of this aperture WDF is physically impossible; uncertainty relationships place limits on the product of the widths of

*𝒲*

_{p}in space and spatial frequency. The finite extent of

*𝒲*

_{p}limits the resolution in space and spatial frequency of the measurement of

*𝒲*

_{i}. Despite these limitations, in many cases the SWDF itself is useful to provide a direct estimate of the WDF [24

24. A. Wax and J. E. Thomas, “Optical heterodyne imaging and Wigner phase space distributions,” Opt. Lett. **21**, 1427–1429 (1996) [CrossRef] [PubMed] .

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

25. H. N. Chapman, “Phase-retrieval X-ray microscopy by Wigner–distribution deconvolution,” Ultramicroscopy **66**, 153–172 (1996) [CrossRef] .

**r**

_{0}simultaneously and removes the need to scan. The advantage of using a lenslet array is that a detector pixel under a given lenslet at position

**r**

_{0}measures a point (

**r**

_{0},

**u**) in the SWDF domain according to the mapping shown in Fig. 1(a). Therefore the geometry of an array of lenslets can be tailored to measure a desired region of the SWDF domain. In section 2 we rigorously derive the relationship between measured intensity and the SWDF of the incident field for an array of lenslets. For simplicity, we consider scalar fields in one spatial dimension. We demonstrate that, in general, the unique mapping implied by Eq. (5) no longer holds. We show that the intensity at a detector pixel in general contains light from multiple lenslets which we call cross–talk. Accurate measurement of the SWDF requires minimizing this cross–talk. In addition, both fully incoherent and fully coherent cases can have considerable amounts of cross–talk. In Section 3, we illustrate tradeoffs between coherence and fidelity using a numerical example, showing that there exists an optimal “Goldilocks” regime for array pitch, given the the coherence width of the input light, such that cross–talk is reduced to a minimum without the need for additional barriers to block light between lenslets. It is in this optimal regime that each detector pixel corresponds to a single point in the SWDF domain, allowing lenslet array systems to measure the SWDF with high accuracy. Since our goal is direct measurement of the field’s coherence properties, we also consider the discrepancy between the SWDF and WDF for this example.

## 2. Theory

*λ*incident upon an ideal lenslet array with 100% fill factor, as illustrated in Fig. 2. A detector is placed at the back focal plane of the lenslet array, and we will refer to the region directly behind each lenslet on the detector as that lenslet’s detector cell. We assume an array of 2

*N*+ 1 identical unaberrated thin lenses, each of width

*w*and focal length

*f*. The transmittance function of such an array is given by where rect(·) denotes a rectangular function. In this configuration, we assign an integer index

*l*to each lenslet, with the center lenslet having index

*l*= 0; the

*N*lenslets above and

*N*lenslets below the center lenslet take on positive and negative values of

*l*, respectively. Thus, the center of each lenslet is located at

*x*=

*lw*. We have assumed an odd number of lenslets to simplify notation, although the results we obtain can easily be extended to an even number of lenslets.

*x̄*and

*x*′ are the center and difference coordinates, respectively,

*J*

_{i}is the mutual intensity of the illumination immediately before the lenslet array; the subscript i indicates that its associated function describes properties of the incident field at the input plane, and we will use this notation through the rest of the manuscript.

*lw*. The aperture of each lenslet is a rect function of width

*w*, and thus the aperture WDF is given by The total intensity at the detector plane is given by where

*w*, the spacing of the lenslet centers. The sampling rate along the spatial frequency axis in the SWDF is determined by both the detector pixel size and the linear mapping

*u*= (

*x*

_{o}−

*lw*)/

*λf*between detector coordinate

*x*

_{o}and spatial frequency coordinate

*u*. The mapping can be explained by the fact that (

*x*

_{o}−

*lw*)/

*f*equals to the angle between the ray reaching the detector pixel at

*x*

_{o}and the optical axis of the

*l*

^{th}lenslet under a small angle approximation. Note that if the angular spread of the SWDF is large enough, each detector cell will include contributions to intensity not only from the SWDF associated with its lenslet, but also from neighboring lenslets. This can be prevented by increasing the size of the lenslets or by decreasing the angular spread of the incident field by placing either a main lens with finite numerical aperture in front of the array [17] or physical barriers between lenslets [27

27. H. Choi, S.-W. Min, S. Jung, J.-H. Park, and B. Lee, “Multiple-viewing-zone integral imaging using a dynamic barrier array for three-dimensional displays,” Opt. Express **11**, 927–932 (2003) [CrossRef] [PubMed] .

*I*

_{SWDF}, because the intensity measured at

*x*

_{o}maps uniquely to the point [

*l̂w*, (

*x*

_{o}−

*l̂w*)/

*λf*] in the SWDF, where

*l̂*is

*x*

_{o}/

*w*rounded to the nearest integer.

^{th}order cross–talk term

*w*, 0,

*w*) is shown in Fig. 3. According to Eq. (10), three lines sampled at spatial coordinates −

*w*, 0,

*w*parallel to the

*u*–axis from the SWDF are mapped to the detector plane (marked by different colors in Fig. 3). To ensure one–to–one mapping, the maximum spatial frequency

*u*

_{m}of the

*l*

^{th}line sample cannot exceed

*w*/2

*λf*, as shown in case (a); otherwise, points at (

*lw*,

*u*

_{m}) and [(

*l*+ 1)

*w*,

*u*

_{m}−

*w*/(

*λf*)] from the SWDF domain will be measured by the same detector pixel at

*x*

_{o}=

*lw*+

*λfu*

_{m}, as shown in case (b).

*n*proportional to the lenslet separation. All possible pairs of lenslets with indices

*l*′ and

*l*″ such that |

*l*′ −

*l*″| =

*n*> 0 contribute to the

*n*

^{th}order cross–talk term

*n*is odd,

*l*takes a value halfway between two integers, and thus

*𝒲*

_{p}is centered at the edge between the (

*l*−1/2)

^{th}and (

*l*+1/2)

^{th}lenslets; when

*n*is even,

*l*takes every integer value, thus

*𝒲*

_{p}is centered at the

*l*

^{th}lenslet. We expect the

*n*= 1 term to be significant even in highly incoherent fields, since some points near the boundary between two neighboring lenslets are expected to be within the coherence width of the field.

## 3. Numerical example

*w*= 330

*μ*m and focal length

*f*= 5mm, yielding spatial frequency support of

*u*

_{0}= 0.132

*μ*m

^{−1}. The simulation results are shown in Fig. 4. For all three cases, the total output intensity in row (a) is composed of the SWDF term in row (b) and the total contribution of cross–talk in row (c). The total cross–talk is further analyzed by decomposing it as the 0

^{th}order term in row (d) and the total of higher order terms in row (e). Simulations on arrays with larger numbers of lenslets were also conducted; results are not shown here because they are very similar to the ones in Fig. 4.

*σ*

_{c}= 0.01

*w*), higher order cross–talk is minimal. However, due to the large angular spread in the incident field, the measurement is corrupted by 0

^{th}order cross–talk. The opposite is the highly coherent case, shown in the middle column (

*σ*

_{c}= 20

*w*). Here, most of the cross–talk comes from higher order terms. The results for a partially coherent field (

*σ*

_{c}= 0.1

*w*) is shown in the right column; cross–talk contributes minimally to the final intensity, although both 0

^{th}order and higher order terms are present.

*x*, the intensities measured in Fig. 4 are not identical under each lenslet. The reason for this is cross–talk. As expressed in Eq. (14) cross–talk under a given lenslet results from the light from neighboring lenslets. The number of neighboring lenslets contributing to the cross–talk depends on both the coherence width of the field and on the angular spread of the field. For a lenslet near the edge of the array, there are few neighbors in the direction of the edge, meaning less cross–talk from that direction. Since our simulated lenslet array consists of only 5 lenslets for simplicity, at least 2 of the lenslets are “edge lenslets,” and depending on the coherence width and angular spread, edge effects may influence all lenslets. Practical lenslet arrays are likely to contain considerably more lenslets, and generally most lenslets will have minimal contributions from edge effects. Because of this, in comparing measured intensity to the WDF and SWDF of the incident field, we consider only the intensity under the central lenslet in our simulated array.

*x*= 0,

*u*=

*x*

_{o}/

*λf*). In Fig. 5 we compare the total output intensity under the central lenslet (dotted green lines) to the corresponding slice (

*u*=

*x*

_{o}/

*λf*) of the SWDF (dashed blue lines) for each of the three simulated fields. The matching slice of the WDF (solid red lines) illustrates the effect of aperture convolution that generates the SWDF. In both the highly incoherent and partially coherent cases, the SWDF and WDF are very similar, since the WDF of the aperture is much smaller than any variations in the incident WDFs. In the highly coherent case, the incident WDF is narrower in

*u*than the aperture WDF, and therefore the SWDF is significantly broadened by the convolution. In order to recover the WDF from the measured intensity, deconvolution is necessary [25

25. H. N. Chapman, “Phase-retrieval X-ray microscopy by Wigner–distribution deconvolution,” Ultramicroscopy **66**, 153–172 (1996) [CrossRef] .

*x*,

*u*) coordinates using an error metric

*R*

_{error}: The total output intensity contains contributions both from the SWDF and from cross–talk. To compare the relative importance of these contributions, we quantify the cross–talk corruption in the output through the cross–talk intensity fraction

*R*

_{cross–talk}as The SWDF itself is a smoothed version of the incident WDF. To quantify the discrepancy due to this smoothing, we define the convolution error

*R*

_{conv}as

*σ*

_{c}/

*w*) in Fig. 6. As seen in the dashed green curve, the contribution from cross–talk increases quickly as the field becomes less coherent. When the field becomes more coherent, the contribution from cross–talk also increases until it saturates to the point in which the field is coherent within the whole array. There exists a partially coherent regime where the SWDF can be measured with minimal cross–talk corruption. Depending on accuracy requirements, this regime may provide acceptable measurements. For example, if less than 1% of cross–talk can be tolerated, then the coherence width should be such that 0.02

*w*<

*σ*

_{c}<

*w*. On the other hand, the convolution error increases as the field becomes more coherent, making the SWDF a less accurate estimate of the WDF in these situations.

*R*

_{error}considers artifacts from both cross–talk and convolution, has a similar shape to the cross–talk curve. The measurement deviates from the original WDF except in a partially coherent region. If error needs to be at most 1%, then we would need 0.02

*w*<

*σ*

_{c}< 0.4

*w*.

## 4. Concluding Remarks

^{th}and higher order cross–talk can be reduced by ensuring the incident illumination’s angular spread is such that each lenslet primarily illuminates only the pixels lying within its detector cell, such that there is a nearly one–to–one mapping from SWDF space to each detector pixel.

^{th}and higher order cross–talk can include contributions for which the light propagates highly non–paraxially from one lenslet to its neighbors. In these cases, we expect that a similar analysis can be performed using non–paraxial versions of the Wigner function [28

28. K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A **16**, 2476–2487 (1999) [CrossRef] .

29. S. Cho, J. Petruccelli, and M. Alonso, “Wigner functions for paraxial and nonparaxial fields,” J. Mod. Optic. **56**, 1843–1852 (2009) [CrossRef] .

30. N. Lindlein, J. Pfund, and J. Schwider, “Algorithm for expanding the dynamic range of a shack-hartmann sensor by using a spatial light modulator array,” Opt. Eng. **40**, 837–840 (2001) [CrossRef] .

31. M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan, “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy,” Appl. Opt. **45**, 2965–2974 (2006) [CrossRef] [PubMed] .

10. L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express **20**, 8296–8308 (2012) [CrossRef] [PubMed] .

32. Z. Zhang, Z. Chen, S. Rehman, and G. Barbastathis, “Factored form descent: a practical algorithm for coherence retrieval,” Opt. Express **21**, 5759–5780 (2013) [CrossRef] [PubMed] .

## Appendix B: Proof of Eq. (14)

*n*contributes a non–zero value to

*I*(

*x*

_{o}) only if the two rect–functions overlap. This implies that the separation

*x*′ between the pair of correlating points on the incident field can only take certain values, as determined by the following inequalities Eq. (42) implies that

*x*′ is bounded to a region of width 2

*w*− 4|

*x̄*−

*lw*| centered at

*nw*. Also recall that the magnitude of mutual intensity is significantly larger than zero at large separation distance

*x*′ only if the field is highly coherent. This implies that more terms in the summation over

*n*need to be considered if the field is more coherent. To simplify Eq. (40), we relate

*I*(

*x*

_{o}) to the WDF of the incident field and the WDF

*𝒲*

_{p}(

*x̄*,

*u*) of a rectangular aperture of width

*w*, by completing the integration with respect to

*x*′ to yield

*n*, we arrive at Eq. (14).

## Acknowledgments

## References and links

1. | L. Mandel and E. Wolf, |

2. | B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. |

3. | W. Tango and R. Twiss, “Michelson stellar interferometry,” Prog. Optics |

4. | K. Itoh and Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A |

5. | D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional coherence imaging in the Fresnel domain,” Appl. Opt. |

6. | M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. |

7. | K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wave fronts,” J. Opt. Soc. Am. A |

8. | D. M. Marks, R. A. Stack, and D. J. Brady, “Astigmatic coherence sensor for digital imaging,” Opt. Lett. |

9. | S. Cho and M. A. Alonso, “Ambiguity function and phase-space tomography for nonparaxial fields,” J. Opt. Soc. Am. A |

10. | L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express |

11. | L. Tian, S. Rehman, and G. Barbastathis, “Experimental 4D compressive phase space tomography,” in “ |

12. | B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. |

13. | G. Lippmann, “La photographie integrale,” Comptes-Rendus, Academie des Sciences |

14. | A. Stern and B. Javidi, “Three-dimensional image sensing and reconstruction with time-division multiplexed computational integral imaging,” Appl. Opt. |

15. | J.-H. Park, K. Hong, and B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt. |

16. | E. H. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell. |

17. | R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford (2005). |

18. | A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. |

19. | E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. |

20. | L. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz |

21. | A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. |

22. | Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in “ |

23. | H. Bartelt, K.-H. Brenner, and A. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. |

24. | A. Wax and J. E. Thomas, “Optical heterodyne imaging and Wigner phase space distributions,” Opt. Lett. |

25. | H. N. Chapman, “Phase-retrieval X-ray microscopy by Wigner–distribution deconvolution,” Ultramicroscopy |

26. | L. Waller, G. Situ, and J. Fleischer, “Phase–space measurement and coherence synthesis of optical beams,” Nat. Photonics |

27. | H. Choi, S.-W. Min, S. Jung, J.-H. Park, and B. Lee, “Multiple-viewing-zone integral imaging using a dynamic barrier array for three-dimensional displays,” Opt. Express |

28. | K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A |

29. | S. Cho, J. Petruccelli, and M. Alonso, “Wigner functions for paraxial and nonparaxial fields,” J. Mod. Optic. |

30. | N. Lindlein, J. Pfund, and J. Schwider, “Algorithm for expanding the dynamic range of a shack-hartmann sensor by using a spatial light modulator array,” Opt. Eng. |

31. | M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan, “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy,” Appl. Opt. |

32. | Z. Zhang, Z. Chen, S. Rehman, and G. Barbastathis, “Factored form descent: a practical algorithm for coherence retrieval,” Opt. Express |

**OCIS Codes**

(110.4980) Imaging systems : Partial coherence in imaging

(050.5082) Diffraction and gratings : Phase space in wave options

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: March 8, 2013

Revised Manuscript: April 12, 2013

Manuscript Accepted: April 13, 2013

Published: April 23, 2013

**Citation**

Lei Tian, Zhengyun Zhang, Jonathan C. Petruccelli, and George Barbastathis, "Wigner function measurement using a lenslet array," Opt. Express **21**, 10511-10525 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-10511

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### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
- B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am.47, 895 (1957). [CrossRef]
- W. Tango and R. Twiss, “Michelson stellar interferometry,” Prog. Optics17, 239–277 (1980). [CrossRef]
- K. Itoh and Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A3, 94–100 (1986). [CrossRef]
- 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]
- M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994). [CrossRef]
- K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wave fronts,” J. Opt. Soc. Am. A16, 1838–1844 (1999). [CrossRef]
- D. M. Marks, R. A. Stack, and D. J. Brady, “Astigmatic coherence sensor for digital imaging,” Opt. Lett.25, 1726–1728 (2000). [CrossRef]
- S. Cho and M. A. Alonso, “Ambiguity function and phase-space tomography for nonparaxial fields,” J. Opt. Soc. Am. A28, 897–902 (2011). [CrossRef]
- L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express20, 8296–8308 (2012). [CrossRef] [PubMed]
- L. Tian, S. Rehman, and G. Barbastathis, “Experimental 4D compressive phase space tomography,” in “Frontiers in Optics,” (Optical Society of America, 2012), p. FM4C.4.
- B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg.17(2001). [PubMed]
- G. Lippmann, “La photographie integrale,” Comptes-Rendus, Academie des Sciences146 (1908).
- A. Stern and B. Javidi, “Three-dimensional image sensing and reconstruction with time-division multiplexed computational integral imaging,” Appl. Opt.42, 7036–7042 (2003). [CrossRef] [PubMed]
- J.-H. Park, K. Hong, and B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt.48, H77–H94 (2009). [CrossRef] [PubMed]
- E. H. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell.14, 99–106 (1992). [CrossRef]
- R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford (2005).
- A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am.69, 192–198 (1979). [CrossRef]
- E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev.40, 0749–0759 (1932). [CrossRef]
- L. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz7, 559–563 (1964).
- A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am.58, 1256–1259 (1968). [CrossRef]
- Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in “IEEE International Conference on Computational Photography (ICCP),” (IEEE, 2009), pp. 1–10. [CrossRef]
- H. Bartelt, K.-H. Brenner, and A. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun.32, 32–38 (1980). [CrossRef]
- A. Wax and J. E. Thomas, “Optical heterodyne imaging and Wigner phase space distributions,” Opt. Lett.21, 1427–1429 (1996). [CrossRef] [PubMed]
- H. N. Chapman, “Phase-retrieval X-ray microscopy by Wigner–distribution deconvolution,” Ultramicroscopy66, 153–172 (1996). [CrossRef]
- L. Waller, G. Situ, and J. Fleischer, “Phase–space measurement and coherence synthesis of optical beams,” Nat. Photonics6, 474–479 (2012). [CrossRef]
- H. Choi, S.-W. Min, S. Jung, J.-H. Park, and B. Lee, “Multiple-viewing-zone integral imaging using a dynamic barrier array for three-dimensional displays,” Opt. Express11, 927–932 (2003). [CrossRef] [PubMed]
- K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A16, 2476–2487 (1999). [CrossRef]
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