Abstract

Extension of coherence holography to vectorial regime is investigated. A technique for controlling and synthesizing optical fields with desired elements of coherence-polarization matrix is proposed and experimentally demonstrated. The technique uses two separate coherence holograms, each of which is assigned to one of the orthogonal polarization components of the vectorial fields.

© 2011 OSA

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References

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  1. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-23-9629 .
    [CrossRef] [PubMed]
  2. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-13-10633 .
    [CrossRef] [PubMed]
  3. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39(23), 4107–4111 (2000).
    [CrossRef]
  4. P. Pavliček, M. Halouzka, Z. Duan, and M. Takeda, “Spatial coherence profilometry on tilted surfaces,” Appl. Opt. 48(34), H40–H47 (2009).
    [CrossRef] [PubMed]
  5. Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express 14(25), 12109–12121 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12109 .
    [CrossRef] [PubMed]
  6. W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
    [CrossRef] [PubMed]
  7. E. Baleine and A. Dogariu, “Variable coherence tomography,” Opt. Lett. 29(11), 1233–1235 (2004).
    [CrossRef] [PubMed]
  8. H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. Lond. A Math. Phys. Sci. 208(1093), 263–277 (1951).
    [CrossRef]
  9. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. A Math. Phys. Sci. 217(1130), 408–432 (1953).
    [CrossRef]
  10. H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photograph. Sci. Eng. 21, 114–122 (1977).
  11. A. W. McCollough and G. M. Gallatin, “Illumination system with spatially controllable partial coherence compensation for line width variance in a photolithographic system,” US Patent 6628370 B1 (2003).
  12. D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express 16(25), 20418–20426 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-25-20418 .
    [CrossRef] [PubMed]
  13. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
  14. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts-Company, 2006).
  15. J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Lett. 34(16), 2429–2431 (2009).
    [CrossRef] [PubMed]
  16. J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express 18(19), 20105–20113 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-19-20105 .
    [CrossRef] [PubMed]
  17. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
    [CrossRef]
  18. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
    [CrossRef]
  19. J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-24-4-1063 .
    [CrossRef] [PubMed]
  20. O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
    [CrossRef] [PubMed]
  21. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29(6), 536–538 (2004).
    [CrossRef] [PubMed]
  22. F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25(17), 1291–1293 (2000).
    [CrossRef]
  23. A. S. Ostrovsky, G. Martínez-Niconoff, P. Martínez-Vara, and M. A. Olvera-Santamaría, “The van Cittert-Zernike theorem for electromagnetic fields,” Opt. Express 17(3), 1746–1752 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-3-1746 .
    [CrossRef] [PubMed]
  24. T. Shirai, “Some consequences of the van Cittert-Zernike theorem for partially polarized stochastic electromagnetic fields,” Opt. Lett. 34(23), 3761–3763 (2009).
    [CrossRef] [PubMed]
  25. G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
    [CrossRef]
  26. M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 26(6), 1437–1443 (2009).
    [CrossRef]
  27. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002).
    [CrossRef]
  28. D. N. Naik, R. K. Singh, T. Ezawa, Y. Miyamoto, and M. Takeda, “Photon correlation holography,” Opt. Express 19(2), 1408–1421 (2011).
    [CrossRef] [PubMed]
  29. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. 35(10), 1728–1730 (2010).
    [CrossRef] [PubMed]
  30. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
    [CrossRef]
  31. D. N. Naik, R. K. Singh, H. Itou, Y. Miyamoto, and M. Takeda, “A highly stable interferometric technique for polarization measurement,” 156, Photonics (December 2010), IIT Guwahati, India.

2011

2010

2009

2008

2006

Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express 14(25), 12109–12121 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12109 .
[CrossRef] [PubMed]

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
[CrossRef] [PubMed]

2005

2004

2003

2002

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002).
[CrossRef]

2000

1998

1982

1977

H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photograph. Sci. Eng. 21, 114–122 (1977).

1953

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. A Math. Phys. Sci. 217(1130), 408–432 (1953).
[CrossRef]

1951

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. Lond. A Math. Phys. Sci. 208(1093), 263–277 (1951).
[CrossRef]

Amra, C.

Baleine, E.

Borghi, R.

Broky, J.

Brown, D. P.

Brown, T. G.

Dennis, M. R.

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002).
[CrossRef]

Dogariu, A.

Duan, Z.

Ellis, J.

Ezawa, T.

Friberg, A. T.

Gori, F.

Halouzka, M.

Hanson, S. G.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
[CrossRef] [PubMed]

Hopkins, H. H.

H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photograph. Sci. Eng. 21, 114–122 (1977).

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. A Math. Phys. Sci. 217(1130), 408–432 (1953).
[CrossRef]

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. Lond. A Math. Phys. Sci. 208(1093), 263–277 (1951).
[CrossRef]

Ina, H.

Kobayashi, S.

Korotkova, O.

Martínez-Niconoff, G.

Martínez-Vara, P.

Miyamoto, Y.

Mondello, A.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

Naik, D. N.

Olvera-Santamaría, M. A.

Ostrovsky, A. S.

Pavlicek, P.

Piquero, G.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25(17), 1291–1293 (2000).
[CrossRef]

Ramírez-Sánchez, V.

Romanini, P.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

Rosen, J.

Santarsiero, M.

Setala, T.

Shirai, T.

Singh, R. K.

Sorrentini, J.

Takeda, M.

D. N. Naik, R. K. Singh, T. Ezawa, Y. Miyamoto, and M. Takeda, “Photon correlation holography,” Opt. Express 19(2), 1408–1421 (2011).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. 35(10), 1728–1730 (2010).
[CrossRef] [PubMed]

P. Pavliček, M. Halouzka, Z. Duan, and M. Takeda, “Spatial coherence profilometry on tilted surfaces,” Appl. Opt. 48(34), H40–H47 (2009).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-13-10633 .
[CrossRef] [PubMed]

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
[CrossRef] [PubMed]

Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express 14(25), 12109–12121 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12109 .
[CrossRef] [PubMed]

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-23-9629 .
[CrossRef] [PubMed]

J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39(23), 4107–4111 (2000).
[CrossRef]

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
[CrossRef]

Tervo, J.

Wang, W.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
[CrossRef] [PubMed]

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-23-9629 .
[CrossRef] [PubMed]

Wolf, E.

O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
[CrossRef] [PubMed]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[CrossRef]

Zerrad, M.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002).
[CrossRef]

Opt. Express

J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-24-4-1063 .
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-13-10633 .
[CrossRef] [PubMed]

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-23-9629 .
[CrossRef] [PubMed]

Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express 14(25), 12109–12121 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12109 .
[CrossRef] [PubMed]

D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express 16(25), 20418–20426 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-25-20418 .
[CrossRef] [PubMed]

A. S. Ostrovsky, G. Martínez-Niconoff, P. Martínez-Vara, and M. A. Olvera-Santamaría, “The van Cittert-Zernike theorem for electromagnetic fields,” Opt. Express 17(3), 1746–1752 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-3-1746 .
[CrossRef] [PubMed]

J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express 18(19), 20105–20113 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-19-20105 .
[CrossRef] [PubMed]

D. N. Naik, R. K. Singh, T. Ezawa, Y. Miyamoto, and M. Takeda, “Photon correlation holography,” Opt. Express 19(2), 1408–1421 (2011).
[CrossRef] [PubMed]

Opt. Lett.

Photograph. Sci. Eng.

H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photograph. Sci. Eng. 21, 114–122 (1977).

Phys. Lett. A

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[CrossRef]

Phys. Rev. Lett.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci.

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. Lond. A Math. Phys. Sci. 208(1093), 263–277 (1951).
[CrossRef]

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. A Math. Phys. Sci. 217(1130), 408–432 (1953).
[CrossRef]

Other

A. W. McCollough and G. M. Gallatin, “Illumination system with spatially controllable partial coherence compensation for line width variance in a photolithographic system,” US Patent 6628370 B1 (2003).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts-Company, 2006).

D. N. Naik, R. K. Singh, H. Itou, Y. Miyamoto, and M. Takeda, “A highly stable interferometric technique for polarization measurement,” 156, Photonics (December 2010), IIT Guwahati, India.

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Figures (5)

Fig. 1
Fig. 1

(a) Holographic recording of E x component of the object field: (b) Holographic recording of E y component of the object field. Two holograms are used to reconstruct desired images in the components of a coherence-polarization matrix.

Fig. 2
Fig. 2

Experimental set-up for generation of random electromagnetic field and its detection using interferomteric method. O1 stands for microscope objective, S spatial filter and other terms have their usual meaning.

Fig. 3
Fig. 3

Triangular Sagnac geometry used to image orthogonal polarization components on ground glass plane.

Fig. 4
Fig. 4

Elements of coherence-polarization matrix: amplitude distribution of (a) W x x ( Δ r ˜ ) (b) W y y ( Δ r ˜ ) (c) W x y ( Δ r ˜ ) (d) W y x ( Δ r ˜ ) ; phase distribution of (e) W x x ( Δ r ˜ ) (f) W y y ( Δ r ˜ ) (g) W x y ( Δ r ˜ ) , and (h) W y x ( Δ r ˜ ) .

Fig. 5
Fig. 5

Elements of coherence-polarization matrix: amplitude distribution of (a) W x x ( Δ r ˜ ) (b) W y y ( Δ r ˜ ) (c) W x y ( Δ r ˜ ) (d) W y x ( Δ r ˜ ) ; phase distribution of (e) W x x ( Δ r ˜ ) (f) W y y ( Δ r ˜ ) (g) W x y ( Δ r ˜ ) , and (h) W y x ( Δ r ˜ ) .

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

E i ( r ^ ) = H i ( r ^ ) exp [ i ϕ i ( r ^ ) ] ( Subscript:  i = x , y ) ,
W i j ( Δ r ˜ ) = W i j H ( r ^ ) exp ( i 2 π λ f Δ r ˜ r ^ ) d r ^ = H i * ( r ^ ) H j ( r ^ ) exp ( i 2 π λ f Δ r ˜ r ^ ) d r ^ ,
W i j ( Δ r ˜ ) = E ˜ i * ( r ˜ 1 ) E ˜ j ( r ˜ 1 + Δ r ˜ ) R = E ˜ i * ( r ˜ 1 ) E ˜ j ( r ˜ 1 + Δ r ˜ ) d r ˜ 1 = H i * ( r ^ 1 ) exp [ i ϕ i ( r ^ 1 ) ] H j ( r ^ 2 ) exp [ i ϕ j ( r ^ 2 ) ] × { exp [ i 2 π λ f ( r ^ 2 r ^ 1 ) r ˜ 1 ] d r ˜ 1 } exp ( i 2 π λ f Δ r ˜ r ^ 2 ) d r ^ 1 d r ^ 2         , = H i * ( r ^ 1 ) H j ( r ^ 1 ) exp ( i 2 π λ f Δ r ˜ r ^ 1 ) d r ^ 1 = W i j H ( r ^ 1 ) exp ( i 2 π λ f Δ r ˜ r ^ 1 ) d r ^ 1
exp [ i 2 π λ f ( r ^ 2 r ^ 1 ) r ˜ 1 ] d r ˜ 1 = δ ( r ^ 2 r ^ 1 )
W i i H ( r ^ 1 ) = H i ( x ^ 1 , y ^ 1 ) H i * ( x ^ 1 , y ^ 1 ) = | G i ( x ^ 1 , y ^ 1 ) | + ( 1 / 2 ) [ G i ( x ^ 1 , y ^ 1 ) + G i * ( x ^ 1 , y ^ 1 ) ] ,
W x y H ( r ^ 1 ) = H x ( x ^ 1 , y ^ 1 ) H y * ( x ^ 1 , y ^ 1 ) = | H x ( x ^ 1 , y ^ 1 ) | | H y ( x ^ 1 , y ^ 1 ) | exp { i [ φ x ( x ^ 1 , y ^ 1 ) φ y ( x ^ 1 , y ^ 1 ) ] } ,
φ x ( x ^ 1 , y ^ 1 ) φ y ( x ^ 1 , y ^ 1 ) = k { [ ( x ^ 1 + Δ x ^ 1 ) 2 + ( y ^ 1 + Δ y ^ 1 ) 2 ] [ ( x ^ 1 Δ x ^ 1 ) 2 + ( y ^ 1 Δ y ^ 1 ) 2 ] } / 2 f k ( x ^ 1 Δ x ^ 1 + y ^ 1 Δ y ^ 1 ) / f

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