Abstract

Information on the degree of coherence of electromagnetic optical waves that is contained both in intensity modulation and in spatial polarization modulation of the resulting distribution of superposing waves is considered. Such an experimental situation is often realized in near-field optics. The possibility of experimental estimation of the degree of mutual coherence of waves polarized at the incidence plane is shown.

© 2009 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  2. T. Setala, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young's interference experiment,” Opt. Lett. 31, 2208-2210 (2006).
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    [CrossRef] [PubMed]
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    [PubMed]
  5. J. M. Schurr, A. N. Naimushin, and B. S. Fujimoto, “Measurement of the dynamic bending rigidity of DNA by a transient polarization grating method,” presented at the 7th International Conference on Laser Applications in Life Sciences, Bratislava, Slovak Republic, 24-28 August 1998, paper PL2.
  6. A. Apostol and A. Dogariu, “Non-Gaussian statistics of optical near-fields,” Phys. Rev. E 72, 025602 (2005).
    [CrossRef]
  7. N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 480-484 (2001).
    [CrossRef]
  8. B. Sick, B. Hecht, and L. Novotny, “Orientation imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482-4485 (2000).
    [CrossRef] [PubMed]
  9. J. T. Fourkas, “Rapid determination of the three-dimensional orientation of single molecules,” Opt. Lett. 26, 211-214(2001).
    [CrossRef]
  10. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7(2000).
    [CrossRef]
  11. K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75-85(2000).
    [CrossRef]
  12. S. M. Rytov, Yu. A. Kravtsov, and B. I. Tatarsky, Introduction to Statistical Radiophysics, Part II, Random Fields (Nauka, 1978) (in Russian).
  13. T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
    [CrossRef]
  14. T. Setala, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
    [CrossRef] [PubMed]
  15. J. S. Nye, “Polarization effects in the waves: the role of declination,” Proc. R. Soc. London Ser. A 387, 105-132(1983).
    [CrossRef]
  16. T. Tudor, “Waves, amplitude waves, intensity waves,” J. Opt. (Paris) 22, 291-296 (1991).
  17. O. V. Angelsky, N. N. Dominikov, P. P. Maksimyak, and T. Tudor, “Experimental revealing of polarization waves,” Appl. Opt. 38, 3112-3117 (1999).
    [CrossRef]
  18. J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137-1143(2003).
    [CrossRef] [PubMed]
  19. T. Tudor, “Polarization waves as observable phenomena,” J. Opt. Soc. Am. A 14, 2013-2020 (1997).
    [CrossRef]

2008 (1)

2006 (2)

2005 (1)

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

2003 (1)

2002 (2)

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

T. Setala, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

2001 (2)

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 480-484 (2001).
[CrossRef]

J. T. Fourkas, “Rapid determination of the three-dimensional orientation of single molecules,” Opt. Lett. 26, 211-214(2001).
[CrossRef]

2000 (3)

B. Sick, B. Hecht, and L. Novotny, “Orientation imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7(2000).
[CrossRef]

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75-85(2000).
[CrossRef]

1999 (1)

1997 (1)

1991 (1)

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

1983 (1)

J. S. Nye, “Polarization effects in the waves: the role of declination,” Proc. R. Soc. London Ser. A 387, 105-132(1983).
[CrossRef]

Angelskaya, A. O.

Angelsky, O. V.

Apostol, A.

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Brown, T. G.

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75-85(2000).
[CrossRef]

Dogariu, A.

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

Dominikov, N. N.

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7(2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7(2000).
[CrossRef]

Fourkas, J. T.

Friberg, A. T.

Fujimoto, B. S.

J. M. Schurr, A. N. Naimushin, and B. S. Fujimoto, “Measurement of the dynamic bending rigidity of DNA by a transient polarization grating method,” presented at the 7th International Conference on Laser Applications in Life Sciences, Bratislava, Slovak Republic, 24-28 August 1998, paper PL2.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7(2000).
[CrossRef]

Hecht, B.

B. Sick, B. Hecht, and L. Novotny, “Orientation imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Hell, S. W.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 480-484 (2001).
[CrossRef]

Huse, N.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 480-484 (2001).
[CrossRef]

Kaivola, M.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

T. Setala, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, and B. I. Tatarsky, Introduction to Statistical Radiophysics, Part II, Random Fields (Nauka, 1978) (in Russian).

Leuchs, G.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7(2000).
[CrossRef]

Maksimyak, P. P.

Naimushin, A. N.

J. M. Schurr, A. N. Naimushin, and B. S. Fujimoto, “Measurement of the dynamic bending rigidity of DNA by a transient polarization grating method,” presented at the 7th International Conference on Laser Applications in Life Sciences, Bratislava, Slovak Republic, 24-28 August 1998, paper PL2.

Novotny, L.

B. Sick, B. Hecht, and L. Novotny, “Orientation imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Nye, J. S.

J. S. Nye, “Polarization effects in the waves: the role of declination,” Proc. R. Soc. London Ser. A 387, 105-132(1983).
[CrossRef]

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7(2000).
[CrossRef]

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, and B. I. Tatarsky, Introduction to Statistical Radiophysics, Part II, Random Fields (Nauka, 1978) (in Russian).

Schönle, A.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 480-484 (2001).
[CrossRef]

Schurr, J. M.

J. M. Schurr, A. N. Naimushin, and B. S. Fujimoto, “Measurement of the dynamic bending rigidity of DNA by a transient polarization grating method,” presented at the 7th International Conference on Laser Applications in Life Sciences, Bratislava, Slovak Republic, 24-28 August 1998, paper PL2.

Setala, T.

Shevchenko, A.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Sick, B.

B. Sick, B. Hecht, and L. Novotny, “Orientation imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Tatarsky, B. I.

S. M. Rytov, Yu. A. Kravtsov, and B. I. Tatarsky, Introduction to Statistical Radiophysics, Part II, Random Fields (Nauka, 1978) (in Russian).

Tervo, J.

Tudor, T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Yermolenko, S. B.

Youngworth, K. S.

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75-85(2000).
[CrossRef]

Zenkova, C. Yu.

Appl. Opt. (2)

J. Biomed. Opt. (1)

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 480-484 (2001).
[CrossRef]

J. Opt. (Paris) (1)

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

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

Opt. Commun. (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7(2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. E (2)

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

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Phys. Rev. Lett. (2)

T. Setala, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

B. Sick, B. Hecht, and L. Novotny, “Orientation imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

J. S. Nye, “Polarization effects in the waves: the role of declination,” Proc. R. Soc. London Ser. A 387, 105-132(1983).
[CrossRef]

Proc. SPIE (1)

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75-85(2000).
[CrossRef]

Other (3)

S. M. Rytov, Yu. A. Kravtsov, and B. I. Tatarsky, Introduction to Statistical Radiophysics, Part II, Random Fields (Nauka, 1978) (in Russian).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

J. M. Schurr, A. N. Naimushin, and B. S. Fujimoto, “Measurement of the dynamic bending rigidity of DNA by a transient polarization grating method,” presented at the 7th International Conference on Laser Applications in Life Sciences, Bratislava, Slovak Republic, 24-28 August 1998, paper PL2.

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Figures (3)

Fig. 1
Fig. 1

(upper) Polarization modulation scheme (W1, W2 are obliquely incident waves, RW is a reference wave). (lower) Illustration of the spatial polarization modulation of the resulting field from linear polarization state through elliptical, circular, and finally to linear polarization state

Fig. 2
Fig. 2

Dependences of visibilities V of the interference patterns resulting from three-beam superposition on the phase of the reference wave φ: (curve 1) for the case of complete coherent waves | η ( 1 , 2 ) | = 1 , the VMD corresponds to M = 1 ; (curve 2) | η ( 1 , 2 ) | = 0 , the VMD corresponds to M = 0 ; (curve 3) | η ( 1 , 2 ) | = 0.25 , the VMD corresponds to M = 0.25 ; (curve 4) | η ( 1 , 2 ) | = 0.5 , the VMD corresponds to M = 0.5 ; (curve 5) | η ( 1 , 2 ) | = 0.75 , the VMD corresponds to M = 0.75 .

Fig. 3
Fig. 3

Optical arrangement for holographic experiment: Bs1, Bs2, beam splitters; M1, M2, M3, mirrors; P1, P2, P3, polarizers; PR, prism; IL, immersion liquid; H, hologram; RW, reference wave; W1, W2, object waves.

Equations (12)

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W ( Q 1 , Q 2 , t ) = E i ( Q 1 , t ) E j * ( Q 2 , t ) ,
γ 2 ( Q 1 , Q 2 , t ) = tr [ W ( Q 1 , Q 2 , t ) W ( Q 2 , Q 1 , t ) ] tr [ W ( Q 1 , Q 1 , 0 ) ] tr [ W ( Q 2 , Q 2 , 0 ) ] = i j | W i j ( Q 1 , Q 2 , t ) | 2 i j W i i ( Q 1 , Q 1 , 0 ) W j j ( Q 2 , Q 2 , 0 ) ,
η i j ( Q 1 , Q 2 , t ) = W i j ( Q 1 , Q 2 , t ) tr [ W ( Q 1 , Q 1 , 0 ) ] tr [ W ( Q 2 , Q 2 , 0 ) ] = W i j ( Q 1 , Q 2 , t ) i j W i i ( Q 1 , Q 1 , 0 ) W j j ( Q 2 , Q 2 , 0 ) .
E ( r , t ) = E ( Q 1 , t ) exp ( i k R 1 ) R 1 + E ( Q 2 , t ) exp ( i k R 2 ) R 2 + E ( Q 3 , t ) exp ( i k R 3 ) R 3 ,
Φ i j ( r , t ) = φ i j ( 1 ) ( r ) + φ i j ( 2 ) ( r ) + φ i j ( 3 ) ( r ) + 2 tr [ W ( Q 1 , Q 1 , 0 ) ] tr [ W ( Q 2 , Q 2 , 0 ) ] | η i j ( 1 , 2 ) | cos [ α i j ( 1 , 2 ) ] cos [ δ 1 ] + 2 tr [ W ( Q 1 , Q 1 , 0 ) ] tr [ W ( Q 3 , Q 3 , 0 ) ] | η i j ( 1 , 3 ) | cos [ α i j ( 1 , 3 ) ] cos [ δ 2 ] + 2 tr [ W ( Q 2 , Q 2 , 0 ) ] tr [ W ( Q 3 , Q 3 , 0 ) ] | η i j ( 2 , 3 ) | cos [ α i j ( 2 , 3 ) ] cos [ δ 3 ] ,
I ( r ) = i j Φ i j ( r , t ) , i , j = x , z .
V = max [ I ( r ) ] min [ I ( r ) ] max [ I ( r ) ] + min [ I ( r ) ] = m n i j tr [ W ( Q m , Q m , 0 ] tr [ W ( Q n , Q n , 0 ) ] φ i j ( 1 ) ( r ) + φ i j ( 2 ) ( r ) + φ i j ( 3 ) ( r ) | η i j ( m , n ) | ,
V φ = 2 i j tr [ W ( Q 1 , Q 1 , 0 ] tr [ W ( Q 2 , Q 2 , 0 ) ] φ i j ( 1 ) ( r ) + φ i j ( 2 ) ( r ) + φ i j ( 3 ) ( r ) | η i j ( 1 , 2 ) | + 2 i j tr [ W ( Q 1 , Q 1 , 0 ] tr [ W ( Q 3 , Q 3 , 0 ) ] φ i j ( 1 ) ( r ) + φ i j ( 2 ) ( r ) + φ i j ( 3 ) ( r ) | η i j ( 1 , 3 ) | cos [ φ 1 ] + 2 i j tr [ W ( Q 2 , Q 2 , 0 ] tr [ W ( Q 3 , Q 3 , 0 ) ] φ i j ( 1 ) ( r ) + φ i j ( 2 ) ( r ) + φ i j ( 3 ) ( r ) | η i j ( 2 , 3 ) | cos [ φ 2 ] ,
M = max [ V φ ] min [ V φ ] = 2 m i j tr [ W ( Q m , Q m , 0 ] tr [ W ( Q 3 , Q 3 , 0 ) ] φ i j ( m ) ( r ) + φ i j ( 3 ) ( r ) | η i j ( m , 3 ) | , m = 1 , 2 , i , j = x , z .
| η ( 1 , 2 ) | = V cos [ Δ θ ] ,
Φ i j ( r , t ) = E i ( r , t ) E j * ( r , t ) = φ i j ( 1 ) ( r ) + φ i j ( 2 ) ( r ) + φ i j ( 3 ) ( r ) + tr [ W ( Q 1 , Q 1 , 0 ) ] tr [ W ( Q 2 , Q 2 , 0 ) ] { η i j ( Q 1 , Q 2 ) exp ( i k ( R 1 R 2 ) ) + η i j ( Q 2 , Q 1 ) exp ( i k ( R 1 R 2 ) ) } + tr [ W ( Q 1 , Q 1 , 0 ) ] tr [ W ( Q 3 , Q 3 , 0 ) ] { η i j ( Q 1 , Q 3 ) exp ( i k ( R 1 R 3 ) ) + η i j ( Q 3 , Q 1 ) exp ( i k ( R 1 R 3 ) ) } + tr [ W ( Q 2 , Q 2 , 0 ) ] tr [ W ( Q 3 , Q 3 , 0 ) ] { η i j ( Q 2 , Q 3 ) exp ( i k ( R 2 R 3 ) ) + η i j ( Q 3 , Q 2 ) exp ( i k ( R 2 R 3 ) ) } .
Φ i j ( r , t ) = φ i j ( 1 ) ( r ) + φ i j ( 2 ) ( r ) + φ i j ( 3 ) ( r ) + 2 tr [ W ( Q 1 , Q 1 , 0 ) ] tr [ W ( Q 2 , Q 2 , 0 ) ] η i j ( 1 , 2 ) cos ( k ( R 1 R 2 ) ) + 2 tr [ W ( Q 1 , Q 1 , 0 ) ] tr [ W ( Q 3 , Q 3 , 0 ) ] η i j ( 1 , 3 ) cos ( k ( R 1 R 3 ) ) + 2 tr [ W ( Q 2 , Q 2 , 0 ) ] tr [ W ( Q 3 , Q 3 , 0 ) ] η i j ( 2 , 3 ) cos ( k ( R 2 R 3 ) ) ,

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