## Measuring coherence functions using non-parallel double slits |

Optics Express, Vol. 22, Issue 7, pp. 8277-8290 (2014)

http://dx.doi.org/10.1364/OE.22.008277

Acrobat PDF (2335 KB)

### 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

## 1. Introduction

2. E. Tervonen, J. Turunen, and 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, and 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 and 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, and H. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. **173**, 45–49 (2000). [CrossRef]

8. L. Basano, P. Ottonello, G. Rottigni, and 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, and 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, and R. Kowarschik, “Spatial coherence on micrometer scale measured by a nanohole array,” Opt. Commun. **285**, 389–392 (2012). [CrossRef]

11. Y. Mejía and A. I. González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. **273**, 428–434 (2007). [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 and 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, and A. T. Friberg, “Spatial coherence measurement of polychromatic light with modified Young’s interferometer,” Opt. Express **21**, 4061–4071 (2013). [CrossRef] [PubMed]

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

## 2. Apparatus design and calibration

*μ*m slit-to-slit separation region was used in the fitting because of the well known separation distance and high fringe-frequency which provided the highest precision. The measurement found an aperture-to-detector distance of 6.19 cm and was precise enough to detect distance changes on the order of 100

*μ*m.

## 3. Theoretical foundations

### 3.1. The spatial intensity distribution

*I*(

*x*) by a real detector will be governed by the detector’s spectral response function. Let

*I′*(

*x*) be the

*apparent*distribution sampled from

*I*(

*x*) by the detector. In order to illuminate the difference between

*I*(

*x*) and

*I′*(

*x*) we note that a real detector will not measure

*S*

_{1}(

*x*,

*ω*) but rather

*S*

_{1}(

*x*,

*ω*)

*η*(

*ω*) where

*η*(

*ω*) is the detector’s spectral response function. Neglecting measurement noise, this leads to an explicit expression for the apparent intensity distribution: where

### 3.2. Transferring to the spatial frequency domain

*I′*(

*x*). Let

*Ĩ′*(

*f*) ≡

_{x}*ℱ*{

*I′*(

*x*)},

*Ĩ′*

_{0}(

*f*) ≡

_{x}*ℱ*{

*I′*

_{0}(

*x*)},

*s*(

_{η}*f*) ≡

_{x}*s*(2

_{ηω}*πzcf*), and recall that

_{x}/d*μ*(2

_{ω}*πzcf*) =

_{x}/d*μ*(

*f*). Then the Fourier transform of Eq. (3) gives where ⊗ indicates a convolution, * indicates a complex conjugate, and we have used the fact that

_{x}*s*(

_{η}*f*) represents a power spectral density and is therefore real and non-negative. We now define the following three functions, which arise as the three terms on the right hand side of Eq. (4): Note that

_{x}*s*(

_{η}*f*) vanishes for

_{x}*f*< 0 because it is defined in the analytic signal representation. The usefulness of working in the spatial frequency domain becomes clear when (a) the pinholes are small enough, which means

_{x}*Ĩ′*

_{0}(

*f*) is narrow enough, and (b) the non-zero portion of

_{x}*s*(

_{η}*f*) is narrow enough such that Eq. (4) can be approximated as a sum of three non-overlapping functions [3

_{x}3. D. Ambrosini, G. S. Spagnolo, D. Paoletti, and 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]

26. R. A. Bartels, A. Paul, M. M. Murnane, H. C. Kapteyn, S. Backus, Y. Liu, and 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]

*c*. Importantly, this means that given a plot of

*Ĩ′*(

*f*) we can find each of

_{x}*A*,

*B*, and

*C*by looking in their respective domains. The results which follow are obtained by approximating Eq. (5) in different ways.

### 3.3. The apparent, normalized power spectral density

*s*(

_{η}*f*) can be extracted by making two assumptions. We assume that the pinholes are very small such that we can make the approximation

_{x}*Ĩ′*

_{0}(

*f*) ≈

_{x}*Ĩ*

_{1}

*δ*(

*f*), where

_{x}*δ*(

*f*) is the Dirac delta function. We further assume that the fields incident on the aperture have been prepared with a high degree of spatial coherence. This can be done, for example, by passing light from the source through a small aperture. It is important to note, however, that this or similar techniques may be inappropriate for use with fields whose statistical properties are shift-variant. This leads to the approximation |

_{x}*μ*(

*f*)| ≈ |

_{x}*μ*

_{1}|, where

*μ*

_{1}is a non-zero constant. These approximations give |

*A*(

*f*)| ≈

_{x}*s*(−

_{η}*f*)|

_{x}*Ĩ*

_{1}

*μ*

_{1}| and

*s*(

_{η}*f*) through the relations

_{x}### 3.4. The spectral degree of coherence

*s*(

_{η}*f*) is known by measurement but

_{x}*μ*(

*f*) is not constant. We proceed with the goal of extracting

_{x}*μ*(

*f*) from a measurement of

_{x}*I′*(

*x*). We return to Eq. (5), assuming

*Ĩ′*

_{0}(

*f*) ≈

_{x}*Ĩ*

_{1}

*δ*(

*f*), and find

_{x}*A*(

*f*) ≈

_{x}*Ĩ*

_{1}

*s*(−

_{η}*f*)

_{x}*μ*(−

*f*),

_{x}*B*(

*f*) ≈ 2

_{x}*Ĩ*

_{1}

*δ*(

*f*), and

_{x}*C*(

*f*) ≈

_{x}*Ĩ*

_{1}

*s*(

_{η}*f*)

_{x}*μ*(

^{*}*f*). Again, we can find each of

_{x}*A*,

*B*, and

*C*by looking at a plot of

*Ĩ′*(

*f*) in the corresponding domain. We can then extract

_{x}*μ*(

*f*), remembering that

_{x}*s*(

_{η}*f*) is known in this case, by using either of the following relations: It must be noted that these relations are only useful for extracting

_{x}*μ*(

*f*) at frequencies where

_{x}*A*(

*f*),

_{x}*B*(

*f*),

_{x}*C*(

*f*), and

_{x}*s*(

_{η}*f*) are measurable with sufficient signal-to-noise ratios.

_{x}### 3.5. The spectral degree of coherence in relation to the complex degree of coherence

27. A. T. Friberg and 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]

*ω*

_{0}is the center passband frequency and the (+) indicates that the source has been spectrally filtered. Thus, the spectral degree of coherence can be measured by using a series of optical band-pass filters with different center wavelengths.

*γ*

^{(+)}(

*x*) from

*I*

^{(+)}(

*x*). We note that, given a narrow passband, both

*η*(

*ω*) and

*K*(

_{ω}*ω*), as defined in Sec. 3.1, can be considered as constant. Then, similar to Eq. (2), we have

*at the pinholes*. The Fourier transform of this equation gives The spectral bandwidth condition (b) used in deriving Eq. (5) is somewhat relaxed because it is now fulfilled based on the width of the passband and not the width of the unfiltered spectrum. We define once again three functions:

*s*

^{(+)}(

*f*) need only be narrow enough such that

_{x}*A*

^{(+)}(

*f*),

_{x}*B*

^{(+)}(

*f*), and

_{x}*C*

^{(+)}(

*f*) do not overlap significantly. Then the three piecewise regions given in Eq. (12) can be separated by inspection.

_{x}*γ*(

*x*) in the unfiltered case. We define the following functions: The recovery of

*γ*(

*x*) from

*γ′*(

*x*) is possible in cases where the spectral response function,

*η*(

*ω*), of the detector is known and is wider than the power spectral density at the pinholes,

*S*(

_{Q}*ω*), and the apparent, normalized power spectral density,

*s*(

_{ηω}*ω*), is known by measurement. By Wolf [24], Eq. 3.1.3, we find that |

*K*(

_{ω}*ω*)|

^{2}∝

*ω*

^{2}. Let

*𝒲*be the set of all frequencies

*ω*for which

*η*(

*ω*) is measurable with an acceptable signal-to-noise ratio. We define the following function: In this case

*γ*(

*x*) can recovered: where the reason for defining

*η′*(

*ω*) becomes clear from its position in the denominator. It has thus been shown that, under certain circumstances, both the spectral and complex degrees of coherence for light at the position of the two pinhole aperture in a Thompson-Wolf experiment can be extracted from the apparent interference pattern generated by such an aperture.

## 4. Experimental details, results, and discussion

*s*(−

_{η}*f*), was thereby extracted. A set of data associated with this measurement is shown in Fig. 4 and serves as an example for other measurements.

_{x}*x*and

*τ*, diffraction from the non-parallel double slit aperture produces interference fringes whose separation depends hyperbolically on the slit separation. Importantly, the slow variation in slit separation produces an interferogram in which negligible diffraction occurs across the vertical dimension. This allows for each horizontal cross-section of the interferogram to be used in approximating the result that would be obtained in a Thompson-Wolf experiment by a pinhole pair of the same separation distance. The Fourier transform associated with each horizontal cross-section can be found by transforming the entire image across the horizontal dimension, as shown in Fig. 4(b). Thus, the equations given in Sec. 3 can be applied individually to each horizontal cross-section in order to extract the degree of coherence as a function of the slit separation distance.

*μ*m. The broadband nature of the source is visible in Fig. 4(c) as a decrease in fringe visibility with distance from the center. This is seen more easily in Fig. 4(d), where the bands to the left and right of the central band correspond to a significant power spectral density across the optical frequency range. These bands also to correspond the functions

*A*(

*f*),

_{x}*B*(

*f*), and

_{x}*C*(

*f*) as defined in Sec. 3.2. The spatially coherent nature of the source can be seen in the fact that the fringe visibility is good near

_{x}*x*= 0 in Fig. 4(c) and from the fact that the combined area under the left and right bands of Fig. 4(d) is similar in magnitude to the area under the central band.

*j*(

*P*

_{1},

*P*

_{2}) ≡

*γ*(

_{τ}*P*

_{1},

*P*

_{2},

*τ*= 0) is the equal time degree of coherence between points

*P*

_{1}and

*P*

_{2},

*I*(

*S*) is the intensity distribution as a function of position

*S*on the source

*σ*, and

*k*=

*ω/c*is the wavenumber of the light. In the set of coordinates defined by Fig. 5,

*P*

_{1}is at position (

*x″*,

*y″*,

*z″*) = (

*d*/2, 0,

*z*) and

*P*

_{2}is at position (

*x″*,

*y″*,

*z″*) = (−

*d*/2, 0,

*z*) for some positive constant

*z*, where

*z*is much larger than the linear dimensions of the source. This gives

*x″*and

*y″*become the variables of integration in Eqs. (18) and (19). In this instance, for a fixed

*z*,

*j*(

*P*

_{1},

*P*

_{2}) becomes a function of only

*d*and

*ω*. As a consequence of Eq. (10), the spectral degree of coherence,

*μ*(

_{ω}*d*,

*ω*), can be obtained directly from

*j*(

*d*,

*ω*). Thus, the two-dimensional spectral degree of coherence function expected to be measured by an ensemble of Thompson-Wolf experiments can be calculated for this particular source.

*μ*m. The deviation is somewhat worse in the unfiltered case and is due to the fact that the approximations necessary to derive Eq. (8) begin to break down for small separations while those necessary to derive Eq. (13) remain more valid.

*z″*-axis (as defined in Fig. 5) than the amplitude. In this case, a fit was made by considering a sheared set of pinhole pairs which are displaced from center on the

*x″*-axis with displacements that depend linearly on the slit separation. This shear is equivalent to a small tilt of the two-slit aperture relative to the detector. The shown fit was found by introducing the equivalent of a 0.31° aperture tilt. This tilt is within the alignment error of the apparatus. Further, this suggests a precise method with which to measure the relative tilt between such a two-slit aperture and detector in future studies. The expected amplitude, shown in Fig. 6(e), is relatively insensitive to this type of tilt misalignment.

## 5. Apparatus capabilities and limitations

*a priori*then only a single image need be recorded. This is in contrast to methods which require adjustable double slits or a monochromator, and therefore a large set of non-simultaneous measurements [3

3. D. Ambrosini, G. S. Spagnolo, D. Paoletti, and 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]

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

12. A. I. González 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]

## 6. Conclusion

## Appendix A: Mitigation of geometrical optical effects

*M*is the magnification and

*A*

_{1},

*A*

_{2},

*d*

_{1}, and

*d*

_{2}are as described in Fig. 7. During the experiments considered here, both

*d*

_{1}and

*d*

_{2}were known by measurement. In general,

*d*

_{1}may be unknown. In that case, a measurement of

*d*

_{1}may be attempted based on measured and expected quasi-monochromatic fringe separations within the interferogram itself, but this could lead to larger systematic error than if

*d*

_{1}were known. Otherwise,

*d*

_{1}may be large enough such that the magnification can be neglected.

## Acknowledgments

## References and links

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

2. | E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: Experimental results,” Appl. Phys. B |

3. | D. Ambrosini, G. S. Spagnolo, D. Paoletti, and S. Vicalvi, “High-precision digital automated measurement of degree of coherence in the Thompson and Wolf experiment,” Pure Appl. Opt. |

4. | C. Iaconis and I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. |

5. | S. Titus, A. Wasan, J. Vaishya, and H. Kandpal, “Determination of phase and amplitude of degree of coherence from spectroscopic measurements,” Opt. Commun. |

6. | H. Kandpal and J. Vaishya, “Experimental study of coherence properties of light fields in the region of superposition in Young’s interference experiment,” Opt. Commun. |

7. | L. Basano, P. Ottonello, G. Rottigni, and M. Vicari, “Degree of spectral coherence, space-frequency plots and correlation-induced spectral changes,” Opt. Commun. |

8. | L. Basano, P. Ottonello, G. Rottigni, and M. Vicari, “Spatial and temporal coherence of filtered thermal light,” Appl. Opt. |

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, and I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. |

10. | P. Petruck, R. Riesenberg, U. Hübner, and R. Kowarschik, “Spatial coherence on micrometer scale measured by a nanohole array,” Opt. Commun. |

11. | Y. Mejía and A. I. González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. |

12. | A. I. González and Y. Mejía, “Nonredundant array of apertures to measure the spatial coherence in two dimensions with only one interferogram,” J. Opt. Soc. Am. A |

13. | D. F. Siriani, “Beyond Young’s experiment: time-domain correlation measurement in a pinhole array,” Opt. Lett. |

14. | M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. |

15. | J.-M. Choi, “Measuring optical spatial coherence by using a programmable aperture,” J. Korean Phys. Soc. |

16. | K. Saastamoinen, J. Tervo, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spatial coherence measurement of polychromatic light with modified Young’s interferometer,” Opt. Express |

17. | A. Cámara, J. A. Rodrigo, and T. Alieva, “Optical coherenscopy based on phase-space tomography,” Opt. Express |

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

19. | 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. |

20. | M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photonics |

21. | E. Mukamel, K. Banaszek, I. A. Walmsley, and C. Dorrer, “Direct measurement of the spatial Wigner function with area-integrated detection,” Opt. Lett. |

22. | D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. |

23. | T. Young, “On the nature of light and colours,” in |

24. | E. Wolf, |

25. | J. Goodman, |

26. | R. A. Bartels, A. Paul, M. M. Murnane, H. C. Kapteyn, S. Backus, Y. Liu, and 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. | A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space–time and in the space–frequency domains,” Opt. Lett. |

28. | E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett. |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(050.1220) Diffraction and gratings : Apertures

(050.1940) Diffraction and gratings : Diffraction

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: January 10, 2014

Revised Manuscript: March 20, 2014

Manuscript Accepted: March 22, 2014

Published: April 1, 2014

**Citation**

Shawn Divitt, Zachary J. Lapin, and Lukas Novotny, "Measuring coherence functions using non-parallel double slits," Opt. Express **22**, 8277-8290 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8277

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### References

- B. J. Thompson, E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. 47, 895–902 (1957). [CrossRef]
- 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]
- 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]
- C. Iaconis, I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- J.-M. Choi, “Measuring optical spatial coherence by using a programmable aperture,” J. Korean Phys. Soc. 60, 177–180 (2012). [CrossRef]
- 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]
- A. Cámara, J. A. Rodrigo, T. Alieva, “Optical coherenscopy based on phase-space tomography,” Opt. Express 21, 13169–13183 (2013). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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).
- 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]

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