Abstract

The statement is substantiated that the experimental estimation of the degree of intrinsic coherence of statistical vector optical fields must include not only the measurement of the visibility of the interference pattern but also the degree of polarization in the resulting spatial distribution of a field.

© 2008 Optical Society of America

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References

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  1. P. Refregier and A. Roueff, “Intrinsic coherence: a new concept in polarization and coherence theory,” Opt. Photon. News 18(2), 30-35 (2007).
    [CrossRef]
  2. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267(2003).
    [CrossRef]
  3. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529-2539 (1963).
    [CrossRef]
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  5. J. F. de Boer and T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. 7, 359-371 (2002).
    [CrossRef] [PubMed]
  6. J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137-1143 (2003).
    [CrossRef] [PubMed]
  7. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett . 29, 536-538 (2004).
    [CrossRef] [PubMed]
  8. P. Refregier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051-6060(2005).
    [CrossRef] [PubMed]
  9. P. Refregier and J. Morio, “Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations,” J. Opt. Soc. A 23, 3036-3044 (2006).
    [CrossRef]
  10. T. Tudor, “Waves, amplitude waves, intensity waves,” J. Opt. (Paris) 22, 291-296 (1991).
  11. T. Tudor, “Polarization waves as observable phenomena,” J. Opt. Soc. Am. A 14, 2013-2020 (1997).
    [CrossRef]
  12. T. Setala, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young's interference experiment,” Opt. Express 31, 2208-2210 (2006).
  13. T. Setala, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence,” Opt. Express 31, 2669-2671(2006).
  14. A. Apostol and A. Dogariu, “Non-Gaussian statistics of optical near-fields,” Phys. Rev. E 72, 025602 (2005).
    [CrossRef]
  15. O. V. Angelsky, N. N. Dominikov, P. P. Maksimyak, and T. Tudor, “Experimental revealing of polarization waves,” Appl. Opt. 38, 3112-3117 (1999).
    [CrossRef]
  16. O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E , 65, 036602 (2002).
    [CrossRef]

2007 (1)

P. Refregier and A. Roueff, “Intrinsic coherence: a new concept in polarization and coherence theory,” Opt. Photon. News 18(2), 30-35 (2007).
[CrossRef]

2006 (3)

T. Setala, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young's interference experiment,” Opt. Express 31, 2208-2210 (2006).

T. Setala, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence,” Opt. Express 31, 2669-2671(2006).

P. Refregier and J. Morio, “Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations,” J. Opt. Soc. A 23, 3036-3044 (2006).
[CrossRef]

2005 (2)

P. Refregier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051-6060(2005).
[CrossRef] [PubMed]

A. Apostol and A. Dogariu, “Non-Gaussian statistics of optical near-fields,” Phys. Rev. E 72, 025602 (2005).
[CrossRef]

2004 (1)

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett . 29, 536-538 (2004).
[CrossRef] [PubMed]

2003 (2)

J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137-1143 (2003).
[CrossRef] [PubMed]

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

2002 (2)

J. F. de Boer and T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. 7, 359-371 (2002).
[CrossRef] [PubMed]

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E , 65, 036602 (2002).
[CrossRef]

1999 (1)

1997 (1)

1995 (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

1991 (1)

T. Tudor, “Waves, amplitude waves, intensity waves,” J. Opt. (Paris) 22, 291-296 (1991).

1963 (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

Friberg, A. T.

Angelsky, O. V.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E , 65, 036602 (2002).
[CrossRef]

O. V. Angelsky, N. N. Dominikov, P. P. Maksimyak, and T. Tudor, “Experimental revealing of polarization waves,” Appl. Opt. 38, 3112-3117 (1999).
[CrossRef]

Apostol, A.

A. Apostol and A. Dogariu, “Non-Gaussian statistics of optical near-fields,” Phys. Rev. E 72, 025602 (2005).
[CrossRef]

de Boer, J. F.

J. F. de Boer and T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. 7, 359-371 (2002).
[CrossRef] [PubMed]

Dogariu, A.

A. Apostol and A. Dogariu, “Non-Gaussian statistics of optical near-fields,” Phys. Rev. E 72, 025602 (2005).
[CrossRef]

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett . 29, 536-538 (2004).
[CrossRef] [PubMed]

Dominikov, N. N.

Ellis, J.

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett . 29, 536-538 (2004).
[CrossRef] [PubMed]

Friberg, A. T.

T. Setala, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence,” Opt. Express 31, 2669-2671(2006).

T. Setala, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young's interference experiment,” Opt. Express 31, 2208-2210 (2006).

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

Goudail, F.

Maksimyak, P. P.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Milner, T. E.

J. F. de Boer and T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. 7, 359-371 (2002).
[CrossRef] [PubMed]

Mokhun, A. I.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E , 65, 036602 (2002).
[CrossRef]

Mokhun, I. I.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E , 65, 036602 (2002).
[CrossRef]

Morio, J.

P. Refregier and J. Morio, “Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations,” J. Opt. Soc. A 23, 3036-3044 (2006).
[CrossRef]

Refregier, P.

P. Refregier and A. Roueff, “Intrinsic coherence: a new concept in polarization and coherence theory,” Opt. Photon. News 18(2), 30-35 (2007).
[CrossRef]

P. Refregier and J. Morio, “Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations,” J. Opt. Soc. A 23, 3036-3044 (2006).
[CrossRef]

P. Refregier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051-6060(2005).
[CrossRef] [PubMed]

Roueff, A.

P. Refregier and A. Roueff, “Intrinsic coherence: a new concept in polarization and coherence theory,” Opt. Photon. News 18(2), 30-35 (2007).
[CrossRef]

Setala, T.

T. Setala, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young's interference experiment,” Opt. Express 31, 2208-2210 (2006).

T. Setala, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence,” Opt. Express 31, 2669-2671(2006).

J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137-1143 (2003).
[CrossRef] [PubMed]

Soskin, M. S.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E , 65, 036602 (2002).
[CrossRef]

Tervo, J.

T. Setala, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young's interference experiment,” Opt. Express 31, 2208-2210 (2006).

T. Setala, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence,” Opt. Express 31, 2669-2671(2006).

J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137-1143 (2003).
[CrossRef] [PubMed]

Tudor, T.

Wolf, E.

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Appl. Opt. (1)

J. Biomed. Opt. (1)

J. F. de Boer and T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. 7, 359-371 (2002).
[CrossRef] [PubMed]

J. Opt. (Paris) (1)

T. Tudor, “Waves, amplitude waves, intensity waves,” J. Opt. (Paris) 22, 291-296 (1991).

J. Opt. Soc. A (1)

P. Refregier and J. Morio, “Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations,” J. Opt. Soc. A 23, 3036-3044 (2006).
[CrossRef]

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

Opt. Express (4)

J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137-1143 (2003).
[CrossRef] [PubMed]

P. Refregier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051-6060(2005).
[CrossRef] [PubMed]

T. Setala, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young's interference experiment,” Opt. Express 31, 2208-2210 (2006).

T. Setala, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence,” Opt. Express 31, 2669-2671(2006).

Opt. Lett (1)

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett . 29, 536-538 (2004).
[CrossRef] [PubMed]

Opt. Photon. News (1)

P. Refregier and A. Roueff, “Intrinsic coherence: a new concept in polarization and coherence theory,” Opt. Photon. News 18(2), 30-35 (2007).
[CrossRef]

Phys. Lett. A (1)

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

Phys. Rev. (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

Phys. Rev. E (2)

A. Apostol and A. Dogariu, “Non-Gaussian statistics of optical near-fields,” Phys. Rev. E 72, 025602 (2005).
[CrossRef]

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E , 65, 036602 (2002).
[CrossRef]

Other (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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Figures (9)

Fig. 1
Fig. 1

(a) Polarization modulation scheme: W1, W2 are obliquely incident waves; RW is the reference wave; and k 1 , k 2 are wave vectors. (b) Field distribution at the plane of superposition: (1) distribution caused by superposition of the E x , E x components; (2) distribution caused by superposition of the E z , E z components; (3) resulting distribution. (c) Illustration of the spatial polarization modulation of a field: E and E , electrical vector components of the superposing obliquely incident waves; E R , resulting electrical vector; I, schematic behavior of the field intensity between two points, A and B, where the oscillations due to superposition of the waves W1, W2, and RW are out of phase by π.

Fig. 2
Fig. 2

Optical arrangement for holographic experiment: Bs 1 and Bs 2 , beam splitters; M1, M2, and M3, mirrors; P1, P2, and P3, polarizers; PR, prism; IL, immersion liquid; H, hologram.

Fig. 3
Fig. 3

Dependence of the normalized intensity of the reconstructed signal ( I r ) against the azimuth of polarization of the reference wave ( α o ).

Fig. 4
Fig. 4

Result of the interference of two waves linearly polarized in the figure plane: (a) when the phase shift between the component oscillations E x and E z is equal to π; (b) when this phase shift differs from π.

Fig. 5
Fig. 5

Result of the interference of two linearly polarized waves E 1 and E 2 , which manifests itself in the following: (a) the intensity modulation, which is a result of the in-pair interaction of the E x 1 , E x 2 and E z 1 , E z 2 collinear components of the vectors E 1 and E 2 decomposition; (b) the polarization modulation, which can change from the linear with the resulting vector E R x to the linear with the resulting vector E R z through the corresponding elliptic polarizations.

Fig. 6
Fig. 6

Optical arrangement of the experiment: BS 1 , BS 2 , beam splitters; M1, M2, mirrors; Qp 1 , Qp 2 , Qp 3 , quarter-wave plates; P1–P3, polarizers; CCD, CCD camera.

Fig. 7
Fig. 7

Images of two resulting interferograms: (a) for two object waves with the plane of polarization in the figure plane; (b) for two object waves with the plane of polarization in the figure plane and the reference wave with the plane of polarization perpendicular to the figure plane; (c) for two object waves and the reference wave with the plane of polarization in the figure plane; (d) the result of doubling the period of an interference pattern for interference of three beams with polarization in the figure plane.

Fig. 8
Fig. 8

Images of the resulting interferograms: (a) for two object waves with the plane of polarization perpendicular to the figure plane; (b) for two object waves with the plane of polarization perpendicular to the figure plane and the reference wave with the plane of polarization in the figure plane; (c) for two object waves and the reference wave with the polarization orthogonal to the figure plane.

Fig. 9
Fig. 9

Intensity (I) and polarization distributions in the field obtaining as the result of superposition of the plane linear polarized in respect to the plane of incidence waves with angle of incidence equal 10 ° : (1) the result of interference of Ex components of interacting waves; (2) the result of interference of Ez components of interacting waves; (3) the spatial distribution of polarizations.

Equations (9)

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γ W 2 ( r 1 , r 2 , τ ) = t r [ W ( r 1 , r 2 , τ ) W ( r 2 , r 1 , τ ) ] t r W ( r 1 , r 1 , 0 ) t r W ( r 2 , r 2 , 0 ) = i , j | W i , j ( r 1 , r 2 , τ ) | 2 i , j W i i ( r 1 , r 1 , 0 ) W j j ( r 1 , r 1 , 0 ) ,
W ( r 1 , r 2 , τ ) = < E i ( r 1 , τ ) E j * ( r 2 , τ ) > ,
M ( r 1 , r 2 , t 1 , t 2 ) = W ( r 1 , r 2 , t 1 , t 2 ) Γ ( r 1 , t 1 ) Γ ( r 2 , t 2 ) ,
M ( r 1 , r 2 , t 1 , t 2 ) = N 2 * D ( r 1 , r 2 , t 1 , t 2 ) N 1 ,
D ( r 1 , r 2 , t 1 , t 2 ) = [ μ s ( r 1 , r 2 , t 1 , t 2 ) 0 0 μ I ( r 1 , r 2 , t 1 , t 2 ) ] ,
W ( r 1 , r 2 , t 1 , t 2 ) = [ < E x 1 ( r 2 , t 2 ) E x 2 * ( r 1 , t 1 ) > < E x 1 ( r 2 , t 2 ) E z 2 * ( r 1 , t 1 ) > < E z 1 ( r 2 , t 2 ) E x 2 * ( r 1 , t 1 ) > < E z 1 ( r 2 , t 2 ) E z 2 * ( r 1 , t 1 ) > ] .
M ( r 1 , r 2 , t 1 , t 2 ) = [ η x x ( r 1 , r 2 , t 1 , t 2 ) η x z ( r 1 , r 2 , t 1 , t 2 ) η z x ( r 1 , r 2 , t 1 , t 2 ) η z z ( r 1 , r 2 , t 1 , t 2 ) ] .
η x , z ( r 1 , r 2 , t 1 , t 2 ) = < E x ( r 2 , t 2 ) E z * ( r 1 , t 1 ) > I x ( r 2 , t 2 ) I z ( r 1 , t 1 ) .
P = V = I max I min I max + I min ,

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