Abstract

We present modified scanning-type wavefront folding interferometers (WFIs), which allow spatial coherence measurements of non-uniformly correlated fields, where the degree of coherence is a function of two absolute spatial coordinates instead of coordinate separation only (Schell model). As an alternative to conventional prism-based WFI implementations, we introduce a scheme based on reflections by three mirrors. This setup allows us to avoid obstructions due to prism corners, and it is remarkably robust to polarization effects. Experimental results on measurement of fields that do not obey the Schell model are provided with the three-mirror WFI, and the results are compared to those obtained with a Young’s interferometer realized using a digital micromirror device.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambrigde University, 1995), Sect. 4.3.
    [Crossref]
  2. E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
    [Crossref]
  3. B. Anderson and P. Fuhr, “Twin-fiber interferometric method for measuring spatial coherence,” Opt. Eng. 32, 926–932 (1993).
    [Crossref]
  4. C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46, 1763–1774 (1999).
    [Crossref]
  5. C. M. Warnky, B. L. Anderson, and C. A. Klein, “Determining spatial modes of lasers with spatial coherence measurements,” Appl. Opt. 39, 6109–6117 (2000).
    [Crossref]
  6. M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. 31, 861–863 (2006).
    [Crossref] [PubMed]
  7. K. Saastamoinen, J. Tervo, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spatial coherence of polychromatic light,” Opt. Express 21, 4061–4071 (2013).
    [Crossref] [PubMed]
  8. H. Partanen, J. Turunen, and J. Tervo, “Coherence measurement with digital micromirror device,” Opt. Lett. 39, 1034–1037 (2014).
    [Crossref] [PubMed]
  9. H. E. Kondakci, A. Beckus, A. El Halawany, N. Mohammadian, G. K. Atia, and A. F. Abouraddy, “Coherence measurements of scattered incoherent light for lensless identification of an object's location and size,” Opt. Express 25, 13087–13100 (2017).
    [Crossref] [PubMed]
  10. 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,” Opt. Lett. 42, 3089–3092 (2017).
    [Crossref] [PubMed]
  11. Y. Mejía and A. Inés González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. 273, 428–434 (2007).
    [Crossref]
  12. 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, 2724–2726 (2012).
    [Crossref] [PubMed]
  13. C. Iaconis and I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996).
    [Crossref] [PubMed]
  14. R. R. Naraghi, H. Gemar, M. Batarseh, A. Beckus, G. Atia, S. Sukhov, and A. Dogariu, “Wide-field interferometric measurement of a nonstationary complex coherence function,” Opt. Lett. 42, 4929–4932 (2017).
    [Crossref]
  15. A. Efimov, “Lateral-shearing, delay-dithering Mach–Zehnder interferometer for spatial coherence measurement,” Opt. Lett. 38, 4522–4525 (2013).
    [Crossref] [PubMed]
  16. A. Efimov, “Spatial coherence at the output of multimode optical fibers,” Opt. Express 22, 15577–15588 (2014).
    [Crossref] [PubMed]
  17. W. H. Carter, “Measurement of second-order coherence in a light beam using a microscope and a grating,” Appl. Opt. 16, 558–563 (1977).
    [Crossref] [PubMed]
  18. J. Schwider, “Continuous lateral shearing interferometers,” Appl. Opt. 23, 4403–4409 (1984).
    [Crossref] [PubMed]
  19. M. Koivurova, H. Partanen, J. Turunen, and A. T. Friberg, “Grating interferometer for light-efficient spatial coherence measurement of arbitrary sources,” Appl. Opt. 56, 5216–5227 (2017).
    [Crossref] [PubMed]
  20. H. W. Wessely and J. O. Bolstadt, “Interferometric technique for measuring the spatial-correlation function of optical radiation fields,” J. Opt. Soc. Am. 60, 678–682 (1970).
    [Crossref]
  21. J. B. Breckinridge, “Coherence interferometer and astronomical applications,” Appl. Opt. 11, 2996–2998 (1972).
    [Crossref] [PubMed]
  22. Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
    [Crossref]
  23. H. Arimoto and Y. Ohtsuka, “Measurements of the complex degree of spectral coherence by use of a wave-front-folded interferometer,” Opt. Lett. 22, 958–960 (1997).
    [Crossref] [PubMed]
  24. T. K. Hakala, A. J. Moilanen, A. I. Väkeväinen, R. Guo, J.-P. Martikainen, K. S. Daskalakis, H. T. Rekola, A. Julku, and P. Törmä, “Bose–Einstein condensation in a plasmonic lattice,” Nat. Phys. 14, 739–744 (2018).
    [Crossref]
  25. A. Bhattacharjee, S. Aarav, and A. K. Jha, “Two-shot measurement of spatial coherence,” Appl. Phys. Lett. 113, 051102 (2018).
    [Crossref]
  26. F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
    [Crossref]
  27. H. Partanen, N. Sharmin, J. Tervo, and J. Turunen, “Specular and antispecular light beams,” Opt. Express 23, 28718–28727 (2015).
    [Crossref] [PubMed]
  28. O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
    [Crossref] [PubMed]
  29. T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006).
    [Crossref]
  30. J. Tervo, T. Setälä, A. Roueff, Ph. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
    [Crossref] [PubMed]
  31. A. T. Friberg and T. Setälä, “Electromagnetic theory of optical coherence,” J. Opt. Soc. Am. A 33, 2431–2442 (2016).
    [Crossref]
  32. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [Crossref]
  33. A. T. Friberg and E. Wolf, “Relationship between the complex degrees of coherence in the space-time and in the space-frequency domains,” Opt. Lett. 20, 623–625 (1995).
    [Crossref] [PubMed]

2018 (2)

T. K. Hakala, A. J. Moilanen, A. I. Väkeväinen, R. Guo, J.-P. Martikainen, K. S. Daskalakis, H. T. Rekola, A. Julku, and P. Törmä, “Bose–Einstein condensation in a plasmonic lattice,” Nat. Phys. 14, 739–744 (2018).
[Crossref]

A. Bhattacharjee, S. Aarav, and A. K. Jha, “Two-shot measurement of spatial coherence,” Appl. Phys. Lett. 113, 051102 (2018).
[Crossref]

2017 (4)

2016 (1)

2015 (1)

2014 (2)

2013 (2)

2012 (1)

2009 (1)

2007 (1)

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

2006 (2)

2005 (1)

2000 (1)

1999 (1)

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46, 1763–1774 (1999).
[Crossref]

1997 (1)

1996 (1)

1995 (1)

1993 (1)

B. Anderson and P. Fuhr, “Twin-fiber interferometric method for measuring spatial coherence,” Opt. Eng. 32, 926–932 (1993).
[Crossref]

1989 (1)

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

1988 (2)

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[Crossref]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

1984 (1)

1982 (1)

1977 (1)

1972 (1)

1970 (1)

Aarav, S.

A. Bhattacharjee, S. Aarav, and A. K. Jha, “Two-shot measurement of spatial coherence,” Appl. Phys. Lett. 113, 051102 (2018).
[Crossref]

Abouraddy, A. F.

Alonso, M. A.

Anderson, B.

B. Anderson and P. Fuhr, “Twin-fiber interferometric method for measuring spatial coherence,” Opt. Eng. 32, 926–932 (1993).
[Crossref]

Anderson, B. L.

Arimoto, H.

Atia, G.

Atia, G. K.

Batarseh, M.

Beckus, A.

Bhattacharjee, A.

A. Bhattacharjee, S. Aarav, and A. K. Jha, “Two-shot measurement of spatial coherence,” Appl. Phys. Lett. 113, 051102 (2018).
[Crossref]

Bolstadt, J. O.

Borghi, R.

Breckinridge, J. B.

Brown, T. G.

Carter, W. H.

Cho, S.

Danner, M.

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46, 1763–1774 (1999).
[Crossref]

Daskalakis, K. S.

T. K. Hakala, A. J. Moilanen, A. I. Väkeväinen, R. Guo, J.-P. Martikainen, K. S. Daskalakis, H. T. Rekola, A. Julku, and P. Törmä, “Bose–Einstein condensation in a plasmonic lattice,” Nat. Phys. 14, 739–744 (2018).
[Crossref]

Dogariu, A.

Drexler, W.

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46, 1763–1774 (1999).
[Crossref]

Efimov, A.

El Halawany, A.

Fercher, A. F.

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46, 1763–1774 (1999).
[Crossref]

Friberg, A. T.

Fuhr, P.

B. Anderson and P. Fuhr, “Twin-fiber interferometric method for measuring spatial coherence,” Opt. Eng. 32, 926–932 (1993).
[Crossref]

Gemar, H.

Gori, F.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[Crossref]

Guo, R.

T. K. Hakala, A. J. Moilanen, A. I. Väkeväinen, R. Guo, J.-P. Martikainen, K. S. Daskalakis, H. T. Rekola, A. Julku, and P. Törmä, “Bose–Einstein condensation in a plasmonic lattice,” Nat. Phys. 14, 739–744 (2018).
[Crossref]

Hakala, T. K.

T. K. Hakala, A. J. Moilanen, A. I. Väkeväinen, R. Guo, J.-P. Martikainen, K. S. Daskalakis, H. T. Rekola, A. Julku, and P. Törmä, “Bose–Einstein condensation in a plasmonic lattice,” Nat. Phys. 14, 739–744 (2018).
[Crossref]

He, Q.

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

Hitzenberger, C. K.

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46, 1763–1774 (1999).
[Crossref]

Iaconis, C.

Ina, H.

Inés González, A.

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

Jha, A. K.

A. Bhattacharjee, S. Aarav, and A. K. Jha, “Two-shot measurement of spatial coherence,” Appl. Phys. Lett. 113, 051102 (2018).
[Crossref]

Julku, A.

T. K. Hakala, A. J. Moilanen, A. I. Väkeväinen, R. Guo, J.-P. Martikainen, K. S. Daskalakis, H. T. Rekola, A. Julku, and P. Törmä, “Bose–Einstein condensation in a plasmonic lattice,” Nat. Phys. 14, 739–744 (2018).
[Crossref]

Klein, C. A.

Kobayashi, S.

Koivurova, M.

Kondakci, H. E.

Korotkova, O.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambrigde University, 1995), Sect. 4.3.
[Crossref]

Martikainen, J.-P.

T. K. Hakala, A. J. Moilanen, A. I. Väkeväinen, R. Guo, J.-P. Martikainen, K. S. Daskalakis, H. T. Rekola, A. Julku, and P. Törmä, “Bose–Einstein condensation in a plasmonic lattice,” Nat. Phys. 14, 739–744 (2018).
[Crossref]

Mejía, Y.

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

Mohammadian, N.

Moilanen, A. J.

T. K. Hakala, A. J. Moilanen, A. I. Väkeväinen, R. Guo, J.-P. Martikainen, K. S. Daskalakis, H. T. Rekola, A. Julku, and P. Törmä, “Bose–Einstein condensation in a plasmonic lattice,” Nat. Phys. 14, 739–744 (2018).
[Crossref]

Monroe, M.

Naraghi, R. R.

Ohtsuka, Y.

Padovani, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[Crossref]

Palma, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[Crossref]

Partanen, H.

Réfrégier, Ph.

Rekola, H. T.

T. K. Hakala, A. J. Moilanen, A. I. Väkeväinen, R. Guo, J.-P. Martikainen, K. S. Daskalakis, H. T. Rekola, A. Julku, and P. Törmä, “Bose–Einstein condensation in a plasmonic lattice,” Nat. Phys. 14, 739–744 (2018).
[Crossref]

Roueff, A.

Saastamoinen, K.

Santarsiero, M.

Schwider, J.

Setälä, T.

Sharmin, N.

Sukhov, S.

Takeda, M.

Tervo, J.

Tervonen, E.

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

Törmä, P.

T. K. Hakala, A. J. Moilanen, A. I. Väkeväinen, R. Guo, J.-P. Martikainen, K. S. Daskalakis, H. T. Rekola, A. Julku, and P. Törmä, “Bose–Einstein condensation in a plasmonic lattice,” Nat. Phys. 14, 739–744 (2018).
[Crossref]

Turunen, J.

Vahimaa, P.

Väkeväinen, A. I.

T. K. Hakala, A. J. Moilanen, A. I. Väkeväinen, R. Guo, J.-P. Martikainen, K. S. Daskalakis, H. T. Rekola, A. Julku, and P. Törmä, “Bose–Einstein condensation in a plasmonic lattice,” Nat. Phys. 14, 739–744 (2018).
[Crossref]

Walmsley, I. A.

Warnky, C. M.

Wessely, H. W.

Wolf, E.

Appl. Opt. (5)

Appl. Phys. B (1)

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

Appl. Phys. Lett. (1)

A. Bhattacharjee, S. Aarav, and A. K. Jha, “Two-shot measurement of spatial coherence,” Appl. Phys. Lett. 113, 051102 (2018).
[Crossref]

J. Mod. Opt. (1)

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46, 1763–1774 (1999).
[Crossref]

J. Opt. Soc. Am. (2)

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

Nat. Phys. (1)

T. K. Hakala, A. J. Moilanen, A. I. Väkeväinen, R. Guo, J.-P. Martikainen, K. S. Daskalakis, H. T. Rekola, A. Julku, and P. Törmä, “Bose–Einstein condensation in a plasmonic lattice,” Nat. Phys. 14, 739–744 (2018).
[Crossref]

Opt. Commun. (3)

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[Crossref]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

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

Opt. Eng. (1)

B. Anderson and P. Fuhr, “Twin-fiber interferometric method for measuring spatial coherence,” Opt. Eng. 32, 926–932 (1993).
[Crossref]

Opt. Express (4)

Opt. Lett. (12)

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

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
[Crossref] [PubMed]

T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006).
[Crossref]

J. Tervo, T. Setälä, A. Roueff, Ph. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
[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,” Opt. Lett. 42, 3089–3092 (2017).
[Crossref] [PubMed]

H. Arimoto and Y. Ohtsuka, “Measurements of the complex degree of spectral coherence by use of a wave-front-folded interferometer,” Opt. Lett. 22, 958–960 (1997).
[Crossref] [PubMed]

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

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, 2724–2726 (2012).
[Crossref] [PubMed]

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

R. R. Naraghi, H. Gemar, M. Batarseh, A. Beckus, G. Atia, S. Sukhov, and A. Dogariu, “Wide-field interferometric measurement of a nonstationary complex coherence function,” Opt. Lett. 42, 4929–4932 (2017).
[Crossref]

A. Efimov, “Lateral-shearing, delay-dithering Mach–Zehnder interferometer for spatial coherence measurement,” Opt. Lett. 38, 4522–4525 (2013).
[Crossref] [PubMed]

Other (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambrigde University, 1995), Sect. 4.3.
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Basic configuration for a scanning WFI. The input is split by a 50 : 50 beamsplitter BS and retroreflected by prisms P1 and P2 towards the imaging system D, which captures the interference pattern. In (a) the beam is centered on the prism edge, whereas in (b) the beam has been shifted along the x-axis. Dashed lines show the paths through P1 and solid ones through P2.
Fig. 2
Fig. 2 Some of the possible configurations for the WFI, (a) Michelson to WFI, and (b) triple mirror. The components use the same notations as before and F is the focal length of lens L.
Fig. 3
Fig. 3 Normalized fringe visibility, ��0, with different WFI implementations when the input Stokes vector traces a path along the surface of the Poincaré sphere. (a) ��3 = 0 (b) ��1 = 0, where cyan ellipses correspond to right handed and magenta to left handed polarization.
Fig. 4
Fig. 4 Young’s interferometer used for reference measurements. The slits are realized using a digital micromirror device (DMD).
Fig. 5
Fig. 5 Absolute value of the complex degree of coherence, |μ0(x1, x2)|, for the emission from a multimode laser diode, measured with the Young’s interferometer (left) and the modified WFI (right). The beam size was ∼ 2 mm.
Fig. 6
Fig. 6 Absolute value of the complex degree of coherence, |μ0(x1, x2)|, for emission from a multimode He-Ne laser, measured with the Young’s interferometer (left) and the modified WFI (right). The beam size was ∼ 2.5 mm.

Equations (34)

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E ( x d , y d ; ω ) = 1 2 { E 0 ( x , y ; ω ) exp [ i C x ( ω ) x ] + E 0 ( x , y ; ω ) exp [ i C y ( ω ) y ] exp [ i ϕ ( ω ) ] } ,
S ( x d , y d ; ω ) = 1 4 [ S 0 ( x , y ; ω ) + S 0 ( x , y ; ω ) ] + 1 4 W 0 ( x , y , x , y ; ω ) exp { i [ C x ( ω ) x + C y ( ω ) y ϕ ( ω ) ] } + 1 4 W 0 ( x , y , x , y ; ω ) exp { i [ C x ( ω ) x + C y ( ω ) y ϕ ( ω ) ] } .
S ( x d , y d ; ω ) = 1 4 [ S 0 ( x , y ; ω ) + S 0 ( x , y ; ω ) ] + 1 2 ( W 0 ( x , y , x , y ; ω ) exp { i [ C x ( ω ) x + C y ( ω ) y ϕ ( ω ) ] } ) ,
μ 0 ( x 1 , y 1 , x 2 , y 2 ; ω ) = W 0 ( x 1 , y 1 , x 2 , y 2 ; ω ) S 0 ( x 1 , y 1 ; ω ) S 0 ( x 2 , y 2 ; ω ) ,
S ( x d , y d ; ω ) = 1 4 [ S 0 ( x , y ; ω ) + S 0 ( x , y ; ω ) ] + 1 2 S 0 ( x , y ; ω ) S 0 ( x , y ; ω ) × | μ 0 ( x , y , x , y ; ω ) | cos [ Φ 0 ( x , y , x , y ; ω ) C x ( ω ) x C y ( ω ) y + ϕ ( ω ) ] ,
V ( x d , y d ; ω ) = 2 S 0 ( x , y ; ω ) S 0 ( x , y ; ω ) S 0 ( x , y ; ω ) + S 0 ( x , y ; ω ) | μ 0 ( x , y , x , y ; ω ) | .
Γ 0 ( x 1 , y 1 , x 2 , y 2 ; τ ) = 0 W 0 ( x 1 , y 1 , x 2 , y 2 ; ω ) exp ( i ω τ ) d ω ,
γ 0 ( x 1 , y 1 , x 2 , y 2 ; τ ) = Γ 0 ( x 1 , y 1 , x 2 , y 2 ; τ ) I 0 ( x 1 , y 1 ) I 0 ( x 2 , y 2 ) = g 0 ( x 1 , y 1 , x 2 , y 2 ; τ ) exp ( i ω 0 τ ) ,
I ( x d , y d ) = 0 S ( x d , y d ; ω ) d ω .
I ( x d , y d ) = 1 4 [ I 0 ( x , y ) + I 0 ( x , y ) ] + 1 4 Γ 0 [ x , y , x , y ; τ ( x , y ) τ ] + 1 4 Γ 0 * [ x , y , x , y ; τ ( x , y ) + τ ] ,
τ ( x , y ) = 2 ( x sin α x + y sin α y ) / c
I ( x d , y d ) = 1 4 [ I 0 ( x , y ) + I 0 ( x , y ) ] + 1 2 I 0 ( x , y ) I 0 ( x , y ) | γ 0 [ x , y , x , y ; τ ( x , y ) τ ] | × cos { ψ 0 [ x , y , x , y ; τ ( x , y ) τ ] ω 0 [ τ ( x , y ) τ ] } ,
V ( x d , y d ; ω ) = 2 I 0 ( x , y ) I 0 ( x , y ) I 0 ( x , y ) + I 0 ( x , y ) | γ 0 ( x , y , x , y ; τ ( x , y ) τ ) | .
E ( x d , y d ; ω ) = 1 2 { E 0 ( x , y ; ω ) exp { i [ C x ( ω ) x + C y ( ω ) y ϕ ( ω ) ] } + E 0 ( x , y ; ω ) } .
S ( x d , y d ; ω ) = 1 4 [ S 0 ( x , y ; ω ) + S 0 ( x , y ; ω ) ] + 1 2 S 0 ( x , y ; ω ) S 0 ( x , y ; ω ) × | μ 0 ( x , y , x , y ; ω ) | cos [ ϕ 0 ( x , y , x , y ; ω ) C x ( ω ) x C y ( ω ) y + ϕ ( ω ) ]
I ( x d , y d ) = 1 4 [ I 0 ( x , y ) + I 0 ( x , y ) ] + 1 2 I 0 ( x , y ) I 0 ( x , y ) | γ 0 [ x , y , x , y ; τ ( x , y ) τ ] | × cos { ψ 0 [ x , y , x , y ; τ ( x , y ) τ ] ω 0 [ τ ( x , y ) τ ] } ,
E ( x d , y d ; ω ) = 1 2 2 E 0 ( x , y ; ω ) exp { i [ C x ( ω ) x + C y ( ω ) y ϕ ( ω ) ] } + 1 2 E 0 ( x , y ; ω ) .
S ( x d , y d ; ω ) = 1 8 S 0 ( x , y ; ω ) + 1 4 S 0 ( x , y ; ω ) + 1 2 2 S 0 ( x , y ; ω ) S 0 ( x , y ; ω ) × | μ 0 ( x , y , x , y ; ω ) | cos [ ϕ 0 ( x , y , x , y ; ω ) C x ( ω ) x C y ( ω ) y + ϕ ( ω ) ]
I ( x d , y d ) = 1 8 I 0 ( x , y ) + 1 4 I 0 ( x , y ) + 1 2 2 I 0 ( x , y ) I 0 ( x , y ) | γ 0 [ x , y , x , y ; τ ( x , y ) τ ] | × cos { ψ 0 [ x , y , x , y ; τ ( x , y ) τ ] ω 0 [ τ ( x , y ) τ ] } ,
V ( x d , y d ; ω ) = 4 2 S 0 ( x , y ; ω ) S 0 ( x , y ; ω ) S 0 ( x , y ; ω ) + 2 S 0 ( x , y ; ω ) | μ 0 ( x , y , x , y ; ω ) |
V ( x d , y d ; ω ) = 2 S 1 ( x , y ; ω ) S 2 ( x , y ; ω ) S 1 ( x , y ; ω ) + S 2 ( x , y ; ω ) | μ 0 ( x , y , x , y ; ω ) | .
W ( x 1 , y 1 , x 2 , y 2 ; ω ) = E * ( x 1 , y 1 ; ω ) E T ( x 2 , y 2 ; ω ) = [ W x x ( x 1 , y 1 , x 2 , y 2 ; ω ) W x y ( x 1 , y 1 , x 2 , y 2 ; ω ) W y x ( x 1 , y 1 , x 2 , y 2 ; ω ) W y y ( x 1 , y 1 , x 2 , y 2 ; ω ) ] ,
𝒮 0 ( x d , y d ; ω ) = tr Φ ( x d , y d ; ω ) = Φ x x ( x d , y d ; ω ) + Φ y y ( x d , y d ; ω ) .
E x ( x d , y d ) = 1 2 [ r TE 2 E 0 x ( x , y ) exp ( i C x x ) + r TM 2 E 0 x ( x , y ) exp ( i C y y + i ϕ ) ] ,
E y ( x d , y d ) = 1 2 [ r TM 2 E 0 y ( x , y ) exp ( i C x x ) + r TE 2 E 0 y ( x , y ) exp ( i C y y + i ϕ ) ] ,
Φ x x ( x d , y d ) = 1 4 [ | r TE | 4 Φ 0 x x ( x , y ) + | r TM | 4 Φ 0 x x ( x , y ) ] + 1 2 | r TE | 2 | r TM | 2 Φ 0 x x ( x , y ) Φ 0 x x ( x , y ) | μ 0 x x ( x , y , x , y ) | × cos [ φ 0 x x ( x , y , x , y ) + C x x + C y y ϕ + 2 arg r TM 2 arg r TE ] ,
Φ y y ( x d , y d ) = 1 4 [ | r TM | 4 Φ 0 y y ( x , y ) + | r TE | 4 Φ 0 y y ( x , y ) ] + 1 2 | r TE | 2 | r TM | 2 Φ 0 y y ( x , y ) Φ 0 y y ( x , y ) | μ 0 y y ( x , y , x , y ) | × cos [ φ 0 y y ( x , y , x , y ) + C x x + C y y ϕ 2 arg r TM + 2 arg r TE ] ,
μ 0 j j ( x 1 , y 1 , x 2 , y 2 ; ω ) = W 0 j j ( x 1 , y 1 , x 2 , y 2 ; ω ) W 0 j j ( x 1 , y 1 , x 1 , y 1 ; ω ) W 0 j j ( x 2 , y 2 , x 2 , y 2 ; ω ) ,
Φ x x ( x d , y d ) = 1 8 | r TM | 4 Φ 0 x x ( x , y ) + 1 4 | r | 2 Φ 0 x x ( x , y ) + 1 2 2 | r TM | | r | Φ 0 x x ( x , y ) Φ 0 x x ( x , y ) | μ 0 x x ( x , y , x , y ) | × cos [ φ 0 x x ( x , y , x , y ) + C x x + C y y ϕ 2 arg r TM + arg r ] ,
Φ y y ( x d , y d ) = 1 8 | r TE | 4 Φ 0 y y ( x , y ) + 1 4 | r | 2 Φ 0 y y ( x , y ) + 1 2 2 | r TE | | r | Φ 0 y y ( x , y ) Φ 0 y y ( x , y ) | μ 0 y y ( x , y , x , y ) | × cos [ φ 0 y y ( x , y , x , y ) + C x x + C y y ϕ 2 arg r TE + arg r ] .
Φ x x ( 0 , 0 ) = Φ 0 x x ( 0 , 0 ) { 1 4 | r TE | 4 + 1 4 | r TM | 4 + 1 2 | r TE | 2 | r TM | 2 cos [ ζ x ( x , y ) + 2 arg r TM 2 arg r TE ] } ,
Φ y y ( 0 , 0 ) = Φ 0 y y ( 0 , 0 ) { 1 4 | r TE | 4 + 1 4 | r TM | 4 + 1 2 | r TE | 2 | r TM | 2 cos [ ζ y ( x , y ) 2 arg r TM + 2 arg r TE ] } ,
Φ x x ( 0 , 0 ) = Φ 0 x x ( 0 , 0 ) { 1 8 | r TM | 4 + 1 4 | r | 2 + 1 2 2 | r TM | | r | cos [ ζ x ( x , y ) 2 arg r TM + arg r ] } ,
Φ y y ( 0 , 0 ) = Φ 0 y y ( 0 , 0 ) { 1 8 | r TE | 4 + 1 4 | r | 2 + 1 2 2 | r TE | | r | cos [ ζ y ( x , y ) 2 arg r TE + arg r ] } .

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