Abstract

We present a method for efficiently measuring the 2 × 2 correlation matrix for paraxial partially coherent beams by using diffraction from small apertures and obstacles. Several representations for this matrix function of four spatial variables are discussed and illustrated with experimental results, including various alternative definitions of the spatial degree of coherence.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Corrections

24 September 2018: A typographical correction was made to the author affiliations.


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References

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  1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007), pp. 174–197.
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    [Crossref]
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    [Crossref]
  6. 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 of light,” J. Mod. Opt. 47, 1237–1246 (2000).
    [Crossref]
  7. G. A. Swartzlander and R. I. Hernandez-Aranda, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
    [Crossref] [PubMed]
  8. 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]
  9. 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]
  10. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic fields,” Phys. Lett. A 312, 263–267 (2003).
    [Crossref]
  11. S. A. Ponomarenko and E. Wolf, “The spectral degree of coherence of fully spatially coherent electromagnetic beams,” Opt. Commun. 227, 73–74 (2003).
    [Crossref]
  12. J. Tervo, S. Tero, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
    [Crossref] [PubMed]
  13. P. Réfrégier and A. Roueff, “Coherence polarization filtering and relation with intrinsic degrees of coherence,” Opt. Lett. 31, 1175–1177 (2006).
    [Crossref] [PubMed]
  14. F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
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  15. A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24, 1063–1068 (2007).
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  16. F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785–795 (1938).
    [Crossref]
  17. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010).
    [Crossref] [PubMed]
  18. T. G. Brown and A. M. Beckley, “Stress engineering and the applications of inhomogeneously polarized optical fields,” Frontiers of Optoelectronics 6, 89–96 (2013).
    [Crossref]
  19. D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: Coherence and correlation measurements and effects in imaging,” Opt. Express 281, 20418–20426 (2008).
    [Crossref]
  20. D. P. Brown and T. G. Brown, “Coherence measurements applied to critical and Köhler vortex illumination,” Proc. SPIE 7184, 71840B (2009).
    [Crossref]
  21. M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. 31, 861–863 (2006).
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2016 (1)

2014 (1)

2013 (1)

T. G. Brown and A. M. Beckley, “Stress engineering and the applications of inhomogeneously polarized optical fields,” Frontiers of Optoelectronics 6, 89–96 (2013).
[Crossref]

2010 (1)

2009 (1)

D. P. Brown and T. G. Brown, “Coherence measurements applied to critical and Köhler vortex illumination,” Proc. SPIE 7184, 71840B (2009).
[Crossref]

2008 (1)

D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: Coherence and correlation measurements and effects in imaging,” Opt. Express 281, 20418–20426 (2008).
[Crossref]

2007 (3)

2006 (2)

2003 (3)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic fields,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

S. A. Ponomarenko and E. Wolf, “The spectral degree of coherence of fully spatially coherent electromagnetic beams,” Opt. Commun. 227, 73–74 (2003).
[Crossref]

J. Tervo, S. Tero, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
[Crossref] [PubMed]

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 of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

1999 (1)

D. L. Marks, R. A. Stack, and D. J. Brady, “Three-Dimensional Coherence Imaging in the Fresnel Domain,” Appt. Opt. 38, 1332–1342 (1999).
[Crossref]

1998 (1)

1996 (1)

1982 (1)

1938 (1)

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785–795 (1938).
[Crossref]

Alonso, M. A.

Baltes, K. M.

Beckley, A. M.

T. G. Brown and A. M. Beckley, “Stress engineering and the applications of inhomogeneously polarized optical fields,” Frontiers of Optoelectronics 6, 89–96 (2013).
[Crossref]

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010).
[Crossref] [PubMed]

Borghi, R.

Brady, D. J.

D. L. Marks, R. A. Stack, and D. J. Brady, “Three-Dimensional Coherence Imaging in the Fresnel Domain,” Appt. Opt. 38, 1332–1342 (1999).
[Crossref]

Brown, D. P.

D. P. Brown and T. G. Brown, “Coherence measurements applied to critical and Köhler vortex illumination,” Proc. SPIE 7184, 71840B (2009).
[Crossref]

D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: Coherence and correlation measurements and effects in imaging,” Opt. Express 281, 20418–20426 (2008).
[Crossref]

Brown, T. G.

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]

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]

T. G. Brown and A. M. Beckley, “Stress engineering and the applications of inhomogeneously polarized optical fields,” Frontiers of Optoelectronics 6, 89–96 (2013).
[Crossref]

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010).
[Crossref] [PubMed]

D. P. Brown and T. G. Brown, “Coherence measurements applied to critical and Köhler vortex illumination,” Proc. SPIE 7184, 71840B (2009).
[Crossref]

D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: Coherence and correlation measurements and effects in imaging,” Opt. Express 281, 20418–20426 (2008).
[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 of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

Cho, S.

Friberg, A. T.

Gori, F.

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 of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

Hernandez-Aranda, R. I.

G. A. Swartzlander and R. I. Hernandez-Aranda, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[Crossref] [PubMed]

Iaconis, C.

Jauch, H. P.

Konforti, N.

Lohmann, A. W.

Luis, A.

Marks, D. L.

D. L. Marks, R. A. Stack, and D. J. Brady, “Three-Dimensional Coherence Imaging in the Fresnel Domain,” Appt. Opt. 38, 1332–1342 (1999).
[Crossref]

Mendlovic, D.

Ponomarenko, S. A.

S. A. Ponomarenko and E. Wolf, “The spectral degree of coherence of fully spatially coherent electromagnetic beams,” Opt. Commun. 227, 73–74 (2003).
[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 of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

Réfrégier, P.

Roueff, A.

Santarsiero, M.

Shabtay, G.

Sharma, K. A.

Stack, R. A.

D. L. Marks, R. A. Stack, and D. J. Brady, “Three-Dimensional Coherence Imaging in the Fresnel Domain,” Appt. Opt. 38, 1332–1342 (1999).
[Crossref]

Swartzlander, G. A.

G. A. Swartzlander and R. I. Hernandez-Aranda, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[Crossref] [PubMed]

Tero, S.

Tervo, J.

Walmsley, I. A.

Wolf, E.

S. A. Ponomarenko and E. Wolf, “The spectral degree of coherence of fully spatially coherent electromagnetic beams,” Opt. Commun. 227, 73–74 (2003).
[Crossref]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic fields,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007), pp. 174–197.

Wood, J. K.

Zernike, F.

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785–795 (1938).
[Crossref]

Appt. Opt. (1)

D. L. Marks, R. A. Stack, and D. J. Brady, “Three-Dimensional Coherence Imaging in the Fresnel Domain,” Appt. Opt. 38, 1332–1342 (1999).
[Crossref]

Frontiers of Optoelectronics (1)

T. G. Brown and A. M. Beckley, “Stress engineering and the applications of inhomogeneously polarized optical fields,” Frontiers of Optoelectronics 6, 89–96 (2013).
[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 of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[Crossref]

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

Opt. Commun. (1)

S. A. Ponomarenko and E. Wolf, “The spectral degree of coherence of fully spatially coherent electromagnetic beams,” Opt. Commun. 227, 73–74 (2003).
[Crossref]

Opt. Express (4)

Opt. Lett. (7)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic fields,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

Phys. Rev. Lett. (1)

G. A. Swartzlander and R. I. Hernandez-Aranda, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[Crossref] [PubMed]

Physica (1)

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785–795 (1938).
[Crossref]

Proc. SPIE (1)

D. P. Brown and T. G. Brown, “Coherence measurements applied to critical and Köhler vortex illumination,” Proc. SPIE 7184, 71840B (2009).
[Crossref]

Other (1)

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007), pp. 174–197.

Supplementary Material (2)

NameDescription
» Visualization 1       Variation of the four Cartesian components of the mutual intensity matrix and the different definitions of the degree of coherence as functions of point separation, for a 5x5 sample of centroid points over the test plane with spacing of 1.56 mm.
» Visualization 2       Variation of the four Cartesian components of the mutual intensity matrix and the different definitions of the degree of coherence as functions of point separation, for a 5x5 sample of centroid points over the test plane with spacing of 1.56 mm.

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Figures (8)

Fig. 1
Fig. 1 Diagram of the experimental setup.
Fig. 2
Fig. 2 (a) Measured components of the mutual intensity as a function of point separation (x′, y′) for all pairs of points centered at the center of the test plane. The corresponding four components of the mutual intensity in the xy basis (b) and the rl basis (c).
Fig. 3
Fig. 3 Plots of the degree of coherence according to the six definitions discussed in this work, for the same field as in Fig. 2. The plot range and color palette is the same as in Fig. 2. The line plots at the bottom correspond to slices along the positive x′ (horizontal) axis.
Fig. 4
Fig. 4 Plots of the degree of coherence according to the six definitions discussed in this work, for a field generated by illuminating a larger spot at the diffuser as for the field in Figs. 2 and 3. The plot range and color palette is the same as in Fig. 2. The line plots at the bottom correspond to slices along the positive x′ (horizontal) axis.
Fig. 5
Fig. 5 Jll as a function of x′ for a 5 × 5 array positions of the centroid x0 over the test plane with spacing of 1.56 mm, for a field generated with the SEO at the image plane of the rotating diffuser. Visualization 1 shows for this sample of centroid points the corresponding variation of the four Cartesian components of the mutual intensity matrix and the degree of coherence according to the different definitions.
Fig. 6
Fig. 6 Jll as a function of x′ for a 5 × 5 array positions of the centroid x0 over the test plane with spacing of 1.56 mm, for a field generated with the SEO at 10 mm from the image of the rotating diffuser. Visualization 2 shows for this sample of centroid points the variation of the four Cartesian components of the mutual intensity matrix and the degree of coherence according to the different definitions.
Fig. 7
Fig. 7 Plots of the expressions in Eqs. (12) for the six definitions of degree of coherence as functions of the scaled point separation ρ = |x′|/w for the theoretical model of azimuthal illumination.
Fig. 8
Fig. 8 (a) Measured and (b) theoretical components of the mutual intensity matrix in terms of point separation, for critical illumination with azimuthal vortex correlations, following Eq. (11). The color scheme is the same as in previous images, but because the correlations are essentially real, only positive (aqua) and negative (red) values are appreciable.

Tables (1)

Tables Icon

Table 1 Angles of the two QWPs used to select the desired field components.

Equations (20)

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𝕁 ( x 1 , x 2 ) = [ J x x ( x 1 , x 2 ) J x y ( x 1 , x 2 ) J y x ( x 1 , x 2 ) J y y ( x 1 , x 2 ) ] = [ E x * ( x 1 ) E x ( x 2 ) E x * ( x 1 ) E y ( x 2 ) E y * ( x 1 ) E x ( x 2 ) E y * ( x 1 ) E y ( x 2 ) ] ,
S 0 ( x 1 , x 2 ) = Tr [ 𝕁 ( x 1 , x 2 ) ] = J x x ( x 1 , x 2 ) + J y y ( x 1 , x 2 ) ,
S 1 ( x 1 , x 2 ) = J x x ( x 1 , x 2 ) J y y ( x 1 , x 2 ) ,
S 2 ( x 1 , x 2 ) = J p p ( x 1 , x 2 ) J m m ( x 1 , x 2 ) = J x y ( x 1 , x 2 ) + J y x ( x 1 , x 2 ) ,
S 3 ( x 1 , x 2 ) = J r r ( x 1 , x 2 ) J l l ( x 1 , x 2 ) = i [ J x y ( x 1 , x 2 ) J y x ( x 1 , x 2 ) ] ,
𝕁 ( x 1 , x 2 ) = 1 2 [ S 0 ( x 1 , x 2 ) + S 1 ( x 1 , x 2 ) S 2 ( x 1 , x 2 ) i S 3 ( x 1 , x 2 ) S 2 ( x 1 , x 2 ) + i S 3 ( x 1 , x 2 ) S 0 ( x 1 , x 2 ) S 1 ( x 1 , x 2 ) ] = n = 0 3 S n ( x 1 , x 2 ) 2 σ n ,
μ W ( x 1 , x 2 ) = Tr [ 𝕁 ( x 1 , x 2 ) ] Tr [ 𝕁 ( x 1 , x 1 ) ] Tr [ 𝕁 ( x 2 , x 2 ) ] = S 0 ( x 1 , x 2 ) S 0 ( x 1 ) S 0 ( x 2 ) ,
μ T ( x 1 , x 2 ) = Tr [ 𝕁 ( x 1 , x 2 ) 𝕁 ( x 2 , x 1 ) ] Tr [ 𝕁 ( x 1 , x 1 ) ] Tr [ 𝕁 ( x 2 , x 2 ) ] = n = 0 3 | S n ( x 1 , x 2 ) | 2 2 S 0 ( x 1 ) S 0 ( x 2 ) .
μ R ( 1 , 2 ) ( x 1 , x 2 ) = SV 1 , 2 [ 𝕁 ( x 1 , x 1 ) 1 / 2 𝕁 ( x 1 , x 2 ) 𝕁 ( x 2 , x 2 ) 1 / 2 ] ,
μ G ( x 1 , x 2 ) = i = 1 , 2 | SV i [ 𝕁 ( x 1 , x 2 ) ] | S 0 ( x 1 ) S 0 ( x 2 ) .
μ L ( x 1 , x 2 ) = 4 3 [ Tr ( 𝕄 2 ) ( Tr 𝕄 ) 2 1 4 ] , 𝕄 = [ 𝕁 ( x 1 , x 1 ) 𝕁 ( x 1 , x 2 ) 𝕁 ( x 2 , x 1 ) 𝕁 ( x 2 , x 2 ) ] .
μ L ( x 1 , x 2 ) = { 2 P 2 ( x 1 ) S 0 2 ( x 1 ) + P 2 ( x 2 ) S 0 2 ( x 2 ) + [ 4 μ T 2 ( x 1 , x 2 ) 2 ] S 0 ( x 1 ) S 0 ( x 2 ) 3 [ S 0 ( x 1 ) + S 0 ( x 2 ) ] 2 + 1 3 } 1 / 2 .
J j j ( x 0 x 2 , x 0 + x 2 ) ˜ [ I ( j ) ( p ) I O ( j ) ( p ; x 0 ) + I A ( j ) ( p ; x 0 ) I D ( j ) ( p ) ] exp ( i k f p x ) d x d y ,
𝕁 ( x 0 x 2 , x 0 + x 2 ) w 2 [ w 2 2 y 2 2 x y 2 x y w 2 2 x 2 ] exp ( x 2 + y 2 w 2 ) ,
μ W = ( 1 ρ 2 ) exp ( ρ 2 ) ,
μ T = 1 2 ρ 2 + 2 ρ 4 2 exp ( ρ 2 ) ,
μ R ( 1 ) = exp ( ρ 2 ) ,
μ R ( 2 ) = ( 1 2 ρ 2 ) exp ( ρ 2 ) ,
μ G = 1 + | 1 2 ρ 2 | 2 exp ( ρ 2 ) ,
μ L = 1 2 ρ 2 + 2 ρ 4 3 exp ( ρ 2 ) ,

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