Abstract

We consider theoretically the detection of the spectral polarization characteristics of random, partially polarized optical beams and near fields by probing them with a dipolar nanoparticle. We show that measuring the polarization state of the scattered far field with a conventional waveplate–polarizer setup, possibly in several directions, results in the full 3×3 polarization matrix at the probe site. This allows us to deduce the distributions of the degree of polarization of the field and the Stokes parameters of the polarized part of the field with a resolution limited by the probe size. Regarding random near fields we show that, in analogy with a known result on beam fields, a degree of polarization of three-component light fields put forward in recent literature can in some cases be interpreted as a ratio of the intensity in the polarized part of the light to that of the total field. We demonstrate the technique by considering the probing of a Gaussian–Schell model beam and a thermally excited near field. The method extends the current scanning-probe techniques to the detection of partial polarization of random light fields and can find applications in nanophotonics and polarization optics.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).
  2. M. Ohtsu, ed., Near-Field Nano/Atom Optics and Technology (Springer, 1998).
  3. D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College, 2003).
  4. T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
    [CrossRef]
  5. J.-J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
    [CrossRef]
  6. A. Madrazo, R. Carminati, M. Nieto-Vesperinas, and J.-J. Greffet, “Polarization effects in the optical interaction between a nanoparticle and a corrigated surface: implications for apertureless near-field microscopy,” J. Opt. Soc. Am. A 15, 109–119 (1998).
    [CrossRef]
  7. L. Aigouy, A. Lahrech, S. Grésillon, H. Cory, A. C. Boccara, and J. C. Rivoal, “Polarization effects in apertureless scanning near-field optical microscopy: an experimental study,” Opt. Lett. 24, 187–189 (1999).
    [CrossRef]
  8. K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68, 245405 (2003).
    [CrossRef]
  9. Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunneling microscopy,” Nature 444, 740–743 (2006).
    [CrossRef]
  10. S. Grésillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. 82, 4520–4523 (1999).
    [CrossRef]
  11. V. Emiliani, F. Intonti, M. Cazayous, D. S. Wiersma, M. Colocci, F. Aliev, and A. Lagendijk, “Near-field short range correlation in optical waves transmitted through random media,” Phys. Rev. Lett. 90, 250801 (2003).
    [CrossRef]
  12. A. Apostol and A. Dogariu, “Spatial correlations in the near field of random media,” Phys. Rev. Lett. 91, 093901 (2003).
    [CrossRef]
  13. J. Laverdant, S. Buil, B. Bérini, and X. Quélin, “Polarization dependent near-field speckle of random gold films,” Phys. Rev. B 77, 165406 (2008).
    [CrossRef]
  14. R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. Pure Appl. Opt. 6, S18–S23 (2004).
  15. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [CrossRef]
  16. K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
    [CrossRef]
  17. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
    [CrossRef]
  18. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  19. R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
    [CrossRef]
  20. P. C. Chaumet, A. Rahmani, F. de Fornel, and J.-P. Dufour, “Evanescent light scattering: the validity of the dipole approximation,” Phys. Rev. B 58, 2310–2315 (1998).
    [CrossRef]
  21. I. U. Vakarelski and K. Higashitani, “Single-nanoparticle-terminated tips for scanning probe microscopy,” Langmuir 22, 2931–2934 (2006).
    [CrossRef]
  22. T. Hakkarainen, T. Setälä, and A. T. Friberg, “Near-field imaging of interacting nano objects with metal and metamaterial superlenses,” New J. Phys. 14, 043019 (2012).
    [CrossRef]
  23. C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, 1971).
  24. T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
    [CrossRef]
  25. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  26. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
  27. N. Feth, “Degree of polarization in random electromagnetic fields,” M.Sc. thesis (Kungliga tekniska högskolan, 2004).
  28. M. Alonso and E. J. Finn, Fundamental University Physics: Fields and Waves, 2nd ed. (Addison-Wesley, 1983).
  29. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
  30. T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of pulsed optical beams,” J. Opt. Soc. Am. A 30, 71–81 (2013).
    [CrossRef]
  31. T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
    [CrossRef]
  32. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).
  33. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
    [CrossRef]
  34. T. Saastamoinen and J. Tervo, “Geometric approach to the degree of polarization for arbitrary fields,” J. Mod. Opt. 51, 2039–2045 (2004).
    [CrossRef]
  35. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  36. R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
    [CrossRef]
  37. A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
    [CrossRef]
  38. T. Setälä, 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]
  39. J.-J. Greffet and C. Henkel, “Coherent thermal radiation,” Contemp. Phys. 48, 183–194 (2007).
    [CrossRef]
  40. J. M. Auñón and M. Nieto-Vesperinas, “On two definitions of the three-dimensional degree of polarization in the near field of statistically homogeneous partially coherent sources,” Opt. Lett. 38, 58–60 (2013).
    [CrossRef]

2013 (2)

2012 (1)

T. Hakkarainen, T. Setälä, and A. T. Friberg, “Near-field imaging of interacting nano objects with metal and metamaterial superlenses,” New J. Phys. 14, 043019 (2012).
[CrossRef]

2008 (1)

J. Laverdant, S. Buil, B. Bérini, and X. Quélin, “Polarization dependent near-field speckle of random gold films,” Phys. Rev. B 77, 165406 (2008).
[CrossRef]

2007 (2)

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[CrossRef]

J.-J. Greffet and C. Henkel, “Coherent thermal radiation,” Contemp. Phys. 48, 183–194 (2007).
[CrossRef]

2006 (3)

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunneling microscopy,” Nature 444, 740–743 (2006).
[CrossRef]

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

I. U. Vakarelski and K. Higashitani, “Single-nanoparticle-terminated tips for scanning probe microscopy,” Langmuir 22, 2931–2934 (2006).
[CrossRef]

2005 (1)

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

2004 (2)

T. Saastamoinen and J. Tervo, “Geometric approach to the degree of polarization for arbitrary fields,” J. Mod. Opt. 51, 2039–2045 (2004).
[CrossRef]

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. Pure Appl. Opt. 6, S18–S23 (2004).

2003 (4)

V. Emiliani, F. Intonti, M. Cazayous, D. S. Wiersma, M. Colocci, F. Aliev, and A. Lagendijk, “Near-field short range correlation in optical waves transmitted through random media,” Phys. Rev. Lett. 90, 250801 (2003).
[CrossRef]

A. Apostol and A. Dogariu, “Spatial correlations in the near field of random media,” Phys. Rev. Lett. 91, 093901 (2003).
[CrossRef]

T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68, 245405 (2003).
[CrossRef]

2002 (2)

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

T. Setälä, 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]

2001 (2)

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
[CrossRef]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef]

2000 (2)

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[CrossRef]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef]

1999 (3)

L. Aigouy, A. Lahrech, S. Grésillon, H. Cory, A. C. Boccara, and J. C. Rivoal, “Polarization effects in apertureless scanning near-field optical microscopy: an experimental study,” Opt. Lett. 24, 187–189 (1999).
[CrossRef]

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[CrossRef]

S. Grésillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. 82, 4520–4523 (1999).
[CrossRef]

1998 (2)

1997 (1)

J.-J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Aigouy, L.

S. Grésillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. 82, 4520–4523 (1999).
[CrossRef]

L. Aigouy, A. Lahrech, S. Grésillon, H. Cory, A. C. Boccara, and J. C. Rivoal, “Polarization effects in apertureless scanning near-field optical microscopy: an experimental study,” Opt. Lett. 24, 187–189 (1999).
[CrossRef]

Aliev, F.

V. Emiliani, F. Intonti, M. Cazayous, D. S. Wiersma, M. Colocci, F. Aliev, and A. Lagendijk, “Near-field short range correlation in optical waves transmitted through random media,” Phys. Rev. Lett. 90, 250801 (2003).
[CrossRef]

Alonso, M.

M. Alonso and E. J. Finn, Fundamental University Physics: Fields and Waves, 2nd ed. (Addison-Wesley, 1983).

Apostol, A.

A. Apostol and A. Dogariu, “Spatial correlations in the near field of random media,” Phys. Rev. Lett. 91, 093901 (2003).
[CrossRef]

Auñón, J. M.

Bergman, J.

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[CrossRef]

Bérini, B.

J. Laverdant, S. Buil, B. Bérini, and X. Quélin, “Polarization dependent near-field speckle of random gold films,” Phys. Rev. B 77, 165406 (2008).
[CrossRef]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef]

Blomstedt, K.

T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

Boccara, A. C.

S. Grésillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. 82, 4520–4523 (1999).
[CrossRef]

L. Aigouy, A. Lahrech, S. Grésillon, H. Cory, A. C. Boccara, and J. C. Rivoal, “Polarization effects in apertureless scanning near-field optical microscopy: an experimental study,” Opt. Lett. 24, 187–189 (1999).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef]

Buil, S.

J. Laverdant, S. Buil, B. Bérini, and X. Quélin, “Polarization dependent near-field speckle of random gold films,” Phys. Rev. B 77, 165406 (2008).
[CrossRef]

Carminati, R.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunneling microscopy,” Nature 444, 740–743 (2006).
[CrossRef]

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68, 245405 (2003).
[CrossRef]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef]

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[CrossRef]

A. Madrazo, R. Carminati, M. Nieto-Vesperinas, and J.-J. Greffet, “Polarization effects in the optical interaction between a nanoparticle and a corrigated surface: implications for apertureless near-field microscopy,” J. Opt. Soc. Am. A 15, 109–119 (1998).
[CrossRef]

J.-J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Carozzi, T.

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[CrossRef]

Cazayous, M.

V. Emiliani, F. Intonti, M. Cazayous, D. S. Wiersma, M. Colocci, F. Aliev, and A. Lagendijk, “Near-field short range correlation in optical waves transmitted through random media,” Phys. Rev. Lett. 90, 250801 (2003).
[CrossRef]

Chaumet, P. C.

P. C. Chaumet, A. Rahmani, F. de Fornel, and J.-P. Dufour, “Evanescent light scattering: the validity of the dipole approximation,” Phys. Rev. B 58, 2310–2315 (1998).
[CrossRef]

Chen, Y.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunneling microscopy,” Nature 444, 740–743 (2006).
[CrossRef]

Colocci, M.

V. Emiliani, F. Intonti, M. Cazayous, D. S. Wiersma, M. Colocci, F. Aliev, and A. Lagendijk, “Near-field short range correlation in optical waves transmitted through random media,” Phys. Rev. Lett. 90, 250801 (2003).
[CrossRef]

Cory, H.

Courjon, D.

D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College, 2003).

Dändliker, R.

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. Pure Appl. Opt. 6, S18–S23 (2004).

de Fornel, F.

P. C. Chaumet, A. Rahmani, F. de Fornel, and J.-P. Dufour, “Evanescent light scattering: the validity of the dipole approximation,” Phys. Rev. B 58, 2310–2315 (1998).
[CrossRef]

De Wilde, Y.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunneling microscopy,” Nature 444, 740–743 (2006).
[CrossRef]

Desmarest, C.

S. Grésillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. 82, 4520–4523 (1999).
[CrossRef]

Dogariu, A.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

A. Apostol and A. Dogariu, “Spatial correlations in the near field of random media,” Phys. Rev. Lett. 91, 093901 (2003).
[CrossRef]

Dufour, J.-P.

P. C. Chaumet, A. Rahmani, F. de Fornel, and J.-P. Dufour, “Evanescent light scattering: the validity of the dipole approximation,” Phys. Rev. B 58, 2310–2315 (1998).
[CrossRef]

Ellis, J.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

Emiliani, V.

V. Emiliani, F. Intonti, M. Cazayous, D. S. Wiersma, M. Colocci, F. Aliev, and A. Lagendijk, “Near-field short range correlation in optical waves transmitted through random media,” Phys. Rev. Lett. 90, 250801 (2003).
[CrossRef]

Feth, N.

N. Feth, “Degree of polarization in random electromagnetic fields,” M.Sc. thesis (Kungliga tekniska högskolan, 2004).

Finn, E. J.

M. Alonso and E. J. Finn, Fundamental University Physics: Fields and Waves, 2nd ed. (Addison-Wesley, 1983).

Formanek, F.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunneling microscopy,” Nature 444, 740–743 (2006).
[CrossRef]

Friberg, A. T.

T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of pulsed optical beams,” J. Opt. Soc. Am. A 30, 71–81 (2013).
[CrossRef]

T. Hakkarainen, T. Setälä, and A. T. Friberg, “Near-field imaging of interacting nano objects with metal and metamaterial superlenses,” New J. Phys. 14, 043019 (2012).
[CrossRef]

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[CrossRef]

T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

T. Setälä, 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]

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

Gadenne, P.

S. Grésillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. 82, 4520–4523 (1999).
[CrossRef]

Gralak, B.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunneling microscopy,” Nature 444, 740–743 (2006).
[CrossRef]

Greffet, J.-J.

J.-J. Greffet and C. Henkel, “Coherent thermal radiation,” Contemp. Phys. 48, 183–194 (2007).
[CrossRef]

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunneling microscopy,” Nature 444, 740–743 (2006).
[CrossRef]

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68, 245405 (2003).
[CrossRef]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef]

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[CrossRef]

A. Madrazo, R. Carminati, M. Nieto-Vesperinas, and J.-J. Greffet, “Polarization effects in the optical interaction between a nanoparticle and a corrigated surface: implications for apertureless near-field microscopy,” J. Opt. Soc. Am. A 15, 109–119 (1998).
[CrossRef]

J.-J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Grésillon, S.

L. Aigouy, A. Lahrech, S. Grésillon, H. Cory, A. C. Boccara, and J. C. Rivoal, “Polarization effects in apertureless scanning near-field optical microscopy: an experimental study,” Opt. Lett. 24, 187–189 (1999).
[CrossRef]

S. Grésillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. 82, 4520–4523 (1999).
[CrossRef]

Hakkarainen, T.

T. Hakkarainen, T. Setälä, and A. T. Friberg, “Near-field imaging of interacting nano objects with metal and metamaterial superlenses,” New J. Phys. 14, 043019 (2012).
[CrossRef]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

Henkel, C.

J.-J. Greffet and C. Henkel, “Coherent thermal radiation,” Contemp. Phys. 48, 183–194 (2007).
[CrossRef]

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

Higashitani, K.

I. U. Vakarelski and K. Higashitani, “Single-nanoparticle-terminated tips for scanning probe microscopy,” Langmuir 22, 2931–2934 (2006).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Intonti, F.

V. Emiliani, F. Intonti, M. Cazayous, D. S. Wiersma, M. Colocci, F. Aliev, and A. Lagendijk, “Near-field short range correlation in optical waves transmitted through random media,” Phys. Rev. Lett. 90, 250801 (2003).
[CrossRef]

Joulain, K.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunneling microscopy,” Nature 444, 740–743 (2006).
[CrossRef]

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68, 245405 (2003).
[CrossRef]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef]

Kaivola, M.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[CrossRef]

T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

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

T. Setälä, 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]

Kalkbrenner, T.

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
[CrossRef]

Karlsson, R.

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[CrossRef]

Lagendijk, A.

V. Emiliani, F. Intonti, M. Cazayous, D. S. Wiersma, M. Colocci, F. Aliev, and A. Lagendijk, “Near-field short range correlation in optical waves transmitted through random media,” Phys. Rev. Lett. 90, 250801 (2003).
[CrossRef]

Lahrech, A.

Laverdant, J.

J. Laverdant, S. Buil, B. Bérini, and X. Quélin, “Polarization dependent near-field speckle of random gold films,” Phys. Rev. B 77, 165406 (2008).
[CrossRef]

Lemoine, P.-A.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunneling microscopy,” Nature 444, 740–743 (2006).
[CrossRef]

Lindfors, K.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[CrossRef]

Madrazo, A.

Mandel, L.

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

Mlynek, J.

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
[CrossRef]

Mulet, J.-P.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunneling microscopy,” Nature 444, 740–743 (2006).
[CrossRef]

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68, 245405 (2003).
[CrossRef]

Nesci, A.

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. Pure Appl. Opt. 6, S18–S23 (2004).

Nieto-Vesperinas, M.

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef]

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

Ponomarenko, S.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

Priimagi, A.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[CrossRef]

Quelin, X.

S. Grésillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. 82, 4520–4523 (1999).
[CrossRef]

Quélin, X.

J. Laverdant, S. Buil, B. Bérini, and X. Quélin, “Polarization dependent near-field speckle of random gold films,” Phys. Rev. B 77, 165406 (2008).
[CrossRef]

Rahmani, A.

P. C. Chaumet, A. Rahmani, F. de Fornel, and J.-P. Dufour, “Evanescent light scattering: the validity of the dipole approximation,” Phys. Rev. B 58, 2310–2315 (1998).
[CrossRef]

Ramstein, M.

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
[CrossRef]

Rivoal, J. C.

L. Aigouy, A. Lahrech, S. Grésillon, H. Cory, A. C. Boccara, and J. C. Rivoal, “Polarization effects in apertureless scanning near-field optical microscopy: an experimental study,” Opt. Lett. 24, 187–189 (1999).
[CrossRef]

S. Grésillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. 82, 4520–4523 (1999).
[CrossRef]

Saastamoinen, T.

T. Saastamoinen and J. Tervo, “Geometric approach to the degree of polarization for arbitrary fields,” J. Mod. Opt. 51, 2039–2045 (2004).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Sandoghdar, V.

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
[CrossRef]

Sarychev, A. K.

S. Grésillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. 82, 4520–4523 (1999).
[CrossRef]

Setälä, T.

T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of pulsed optical beams,” J. Opt. Soc. Am. A 30, 71–81 (2013).
[CrossRef]

T. Hakkarainen, T. Setälä, and A. T. Friberg, “Near-field imaging of interacting nano objects with metal and metamaterial superlenses,” New J. Phys. 14, 043019 (2012).
[CrossRef]

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[CrossRef]

T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

T. Setälä, 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]

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

Shalaev, V. M.

S. Grésillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. 82, 4520–4523 (1999).
[CrossRef]

Shchegrov, A. V.

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef]

Shevchenko, A.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[CrossRef]

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

Shubin, V. A.

S. Grésillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. 82, 4520–4523 (1999).
[CrossRef]

Tai, C.-T.

C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, 1971).

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Tervo, J.

T. Saastamoinen and J. Tervo, “Geometric approach to the degree of polarization for arbitrary fields,” J. Mod. Opt. 51, 2039–2045 (2004).
[CrossRef]

Tortora, P.

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. Pure Appl. Opt. 6, S18–S23 (2004).

Vaccaro, L.

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. Pure Appl. Opt. 6, S18–S23 (2004).

Vakarelski, I. U.

I. U. Vakarelski and K. Higashitani, “Single-nanoparticle-terminated tips for scanning probe microscopy,” Langmuir 22, 2931–2934 (2006).
[CrossRef]

Vigoureux, J. M.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

Voipio, T.

Wiersma, D. S.

V. Emiliani, F. Intonti, M. Cazayous, D. S. Wiersma, M. Colocci, F. Aliev, and A. Lagendijk, “Near-field short range correlation in optical waves transmitted through random media,” Phys. Rev. Lett. 90, 250801 (2003).
[CrossRef]

Wolf, E.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

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

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef]

Contemp. Phys. (1)

J.-J. Greffet and C. Henkel, “Coherent thermal radiation,” Contemp. Phys. 48, 183–194 (2007).
[CrossRef]

J. Microsc. (1)

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
[CrossRef]

J. Mod. Opt. (1)

T. Saastamoinen and J. Tervo, “Geometric approach to the degree of polarization for arbitrary fields,” J. Mod. Opt. 51, 2039–2045 (2004).
[CrossRef]

J. Opt. Pure Appl. Opt. (1)

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. Pure Appl. Opt. 6, S18–S23 (2004).

J. Opt. Soc. Am. A (2)

Langmuir (1)

I. U. Vakarelski and K. Higashitani, “Single-nanoparticle-terminated tips for scanning probe microscopy,” Langmuir 22, 2931–2934 (2006).
[CrossRef]

Nat. Photonics (1)

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[CrossRef]

Nature (1)

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunneling microscopy,” Nature 444, 740–743 (2006).
[CrossRef]

New J. Phys. (1)

T. Hakkarainen, T. Setälä, and A. T. Friberg, “Near-field imaging of interacting nano objects with metal and metamaterial superlenses,” New J. Phys. 14, 043019 (2012).
[CrossRef]

Opt. Commun. (2)

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. B (3)

J. Laverdant, S. Buil, B. Bérini, and X. Quélin, “Polarization dependent near-field speckle of random gold films,” Phys. Rev. B 77, 165406 (2008).
[CrossRef]

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68, 245405 (2003).
[CrossRef]

P. C. Chaumet, A. Rahmani, F. de Fornel, and J.-P. Dufour, “Evanescent light scattering: the validity of the dipole approximation,” Phys. Rev. B 58, 2310–2315 (1998).
[CrossRef]

Phys. Rev. E (3)

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

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[CrossRef]

T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

Phys. Rev. Lett. (7)

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[CrossRef]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef]

T. Setälä, 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]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef]

S. Grésillon, L. Aigouy, A. C. Boccara, J. C. Rivoal, X. Quelin, C. Desmarest, P. Gadenne, V. A. Shubin, A. K. Sarychev, and V. M. Shalaev, “Experimental observation of localized optical excitations in random metal-dielectric films,” Phys. Rev. Lett. 82, 4520–4523 (1999).
[CrossRef]

V. Emiliani, F. Intonti, M. Cazayous, D. S. Wiersma, M. Colocci, F. Aliev, and A. Lagendijk, “Near-field short range correlation in optical waves transmitted through random media,” Phys. Rev. Lett. 90, 250801 (2003).
[CrossRef]

A. Apostol and A. Dogariu, “Spatial correlations in the near field of random media,” Phys. Rev. Lett. 91, 093901 (2003).
[CrossRef]

Prog. Surf. Sci. (1)

J.-J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Other (12)

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

M. Ohtsu, ed., Near-Field Nano/Atom Optics and Technology (Springer, 1998).

D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College, 2003).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

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

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

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

N. Feth, “Degree of polarization in random electromagnetic fields,” M.Sc. thesis (Kungliga tekniska högskolan, 2004).

M. Alonso and E. J. Finn, Fundamental University Physics: Fields and Waves, 2nd ed. (Addison-Wesley, 1983).

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, 1971).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Illustration of the spherical polar coordinates (r,θ,φ) and the related unit vectors (u^r,u^θ,u^φ) used to describe the far field scattered by a point dipole located at the origin.

Fig. 2.
Fig. 2.

Waveplate–polarizer system used to measure the far-field polarization matrix. The fast and slow axes of the waveplate are along the θ and φ axes, respectively. The polarization axis of the polarizer is at an angle β with respect to the θ axis.

Fig. 3.
Fig. 3.

Poincaré-sphere representation of the polarization states related to s(r,ω) and s(p)(r,ω). The length of s(r,ω) is P2, whereas s(p)(r,ω) is a unit vector.

Fig. 4.
Fig. 4.

Illustration of directions in which the polarization matrix of the far field scattered by a dipole is measured (a) for two-component fields and (b) for three-component fields.

Fig. 5.
Fig. 5.

Behavior of the intensities I(δ,β) in the direction (θ,φ)=(π/4,0) as a function of ρ for (a) fully polarized, (b) partially polarized, and (c) unpolarized GSM beams. The beam parameters are given in the text, while δ and β can be found in the insets. (d) shows the polarimetric information related to the beam in (b). The solid line refers to the degree of polarization P2, whereas the ellipsoids illustrate the polarization state of the fully polarized part at some specific points.

Fig. 6.
Fig. 6.

Behavior of the far-field intensities I(0,0) and I(0,π/4) in the direction (θ,φ)=(π/4,0) (left arrow) and the degree of polarization P3(z,ω) (right arrow) as a function of z/λ.

Equations (75)

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

Φ(r,ω)=E*(r,ω)ET(r,ω),
p(r0,ω)=α(ω)E(r0,ω),
α(ω)=α(ω)I,
α(ω)=αs(ω)1ik3αs(ω)/6πϵ0,
αs(ω)=4πϵ0a3ϵ(ω)1ϵ(ω)+2,
u^r=[sinθcosφ,sinθsinφ,cosθ]T,
u^θ=[cosθcosφ,cosθsinφ,sinθ]T,
u^φ=[sinφ,cosφ,0]T,
Ef(ru^r,ω)=μ0ω2α(ω)eikr4πr(Iu^ru^rT)E(r0,ω),
Φf(ru^r,ω)=c(r,ω)(Iu^ru^rT)Φ(r0,ω)(Iu^ru^rT),
c(r,ω)=μ02ω4|α(ω)|216π2r2,
ϕ(ru^r,ω)=[ϕθθ(ru^r,ω)ϕθφ(ru^r,ω)ϕφθ(ru^r,ω)ϕφφ(ru^r,ω)],
ϕ(ru^r,ω)=UT(θ,φ)Φf(ru^r,ω)U(θ,φ)=c(r,ω)UT(θ,φ)Φ(r0,ω)U(θ,φ),
Tw(δ)=(100eiδ),
Tp(β)=(cos2βsinβcosβsinβcosβsin2β),
E(δ,β;ru^r,ω)=Tp(β)Tw(δ)Ef(ru^r,ω),
I(δ,β;ru^r,ω)=tr[Tp(β)Tw*(δ)ϕ(ru^r,ω)Tw(δ)Tp(β)],
ϕθθ(ru^r,ω)=I(0,0;ru^r,ω),
ϕφφ(ru^r,ω)=I(0,π/4;ru^r,ω)+I(0,3π/4;ru^r,ω)I(0,0;ru^r,ω),
ϕφθ(ru^r,ω)=ϕθφ*(ru^r,ω),=12{[I(0,π/4;ru^r,ω)I(0,3π/4;ru^r,ω)]+i[I(0,π/4;ru^r,ω)+I(0,3π/4;ru^r,ω)2I(π/2,π/4;ru^r,ω)]},
ξ(Ωd;u^r,ω)=μ02ω4|α(ω)|2Ωd16π3a2·tr[(Iu^ru^rT)Φ(r0,ω)]tr[Φ(r0,ω)],
S0(r,ω)=Φxx(r,ω)+Φyy(r,ω),
S1(r,ω)=Φxx(r,ω)Φyy(r,ω),
S2(r,ω)=Φxy(r,ω)+Φyx(r,ω),
S3(r,ω)=i[Φyx(r,ω)Φxy(r,ω)].
si(r,ω)=Si(r,ω)S0(r,ω),i(1,2,3),
P22(r,ω)=i=13si2(r,ω).
si(p)(r,ω)=si(r,ω)P2(r,ω),i(1,2,3),
s(p)(r,ω)=s(r,ω)P2(r,ω);
Λ0(r,ω)=Φxx(r,ω)+Φyy(r,ω)+Φzz(r,ω),
Λ1(r,ω)=32[Φxy(r,ω)+Φyx(r,ω)],
Λ2(r,ω)=32i[Φxy(r,ω)Φyx(r,ω)],
Λ3(r,ω)=32[Φxx(r,ω)Φyy(r,ω)],
Λ4(r,ω)=32[Φxz(r,ω)+Φzx(r,ω)],
Λ5(r,ω)=32i[Φxz(r,ω)Φzx(r,ω)],
Λ6(r,ω)=32[Φyz(r,ω)+Φzy(r,ω)],
Λ7(r,ω)=32i[Φyz(r,ω)Φzy(r,ω)],
Λ8(r,ω)=32[Φxx(r,ω)+Φyy(r,ω)2Φzz(r,ω)],
λi(r,ω)=13Λi(r,ω)Λ0(r,ω),i(1,,8),
P32(r,ω)=i=18λi2(r,ω),
Φ(r,ω)=Φ(pp)(r,ω)+Φ(unpol)(r,ω),
Φ(pp)(r,ω)=(ξ1ξ2)Vdiag[1,0,0]V+(ξ2ξ3)Vdiag[1,1,0]V,
Φ(unpol)(r,ω)=ξ3Vdiag[1,1,1]V.
λi(pp)(r,ω)=13Λi(pp)(r,ω)Λ0(pp)(r,ω),=λi(r,ω)Λ0(r,ω)Λ0(pp)(r,ω),i(1,,8).
P3(r,ω)=P3(pp)(r,ω)Λ0(pp)(r,ω)Λ0(r,ω).
s(pp)(r,ω)=s(r,ω)P3(pp)(r,ω)P3(r,ω).
s(p)(r,ω)=s(r,ω)P3(r,ω),
Φ(r0,ω)=1c(r,ω)(2ϕθθ(ru^r,ω)2ϕθφ(ru^r,ω)02ϕθφ*(ru^r,ω)ϕφφ(ru^r,ω)0000).
Φij(ρ,ω)=[Si(ρ,ω)Sj(ρ,ω)]1/2μij(ω),
Si(ρ,ω)=Ai2(ω)exp{ρ2/[2σi2(ω)]},i(x,y),
I(0,0)=c(r,ω)2Ax2(ω)exp{ρ2/[2σx2(ω)]},
I(0,3π/4)=c(r,ω)4{Ax2(ω)exp{ρ2/[2σx2(ω)]}+2Ay2(ω)exp{ρ2/[2σy2(ω)]}22Ax(ω)Ay(ω)exp{ρ2/[4σx2(ω)]}×exp{ρ2/[4σy2(ω)]}Re{μxy(ω)}},
I(0,π/4)=c(r,ω)4{Ax2(ω)exp{ρ2/[2σx2(ω)]}+2Ay2(ω)exp{ρ2/[2σy2(ω)]}+22Ax(ω)Ay(ω)exp{ρ2/[4σx2(ω)]}×exp{ρ2/[4σy2(ω)]}Re{μxy(ω)}},
I(π/2,π/4)=c(r,ω)4{Ax2(ω)exp{ρ2/[2σx2(ω)]}+2Ay2(ω)exp{ρ2/[2σy2(ω)]}22Ax(ω)Ay(ω)exp{ρ2/[4σx2(ω)]}×exp{ρ2/[4σy2(ω)]}Im{μxy(ω)}},
Φxx(r0,ω)=1c(r,ω)ϕφφ(r,π/4,π/2,ω),
Φyy(r0,ω)=1c(r,ω)ϕφφ(r,π/2,0,ω),
Φzz(r0,ω)=1c(r,ω)ϕθθ(r,π/2,0,ω),
Φxy(r0,ω)=1c(r,ω)[2ϕφθ(r,π/2,π/4,ω)2ϕφθ(r,π/4,π/2,ω)ϕφθ(r,π/2,0,ω)],
Φxz(r0,ω)=1c(r,ω)[2ϕφθ(r,π/2,π/4,ω)ϕφθ(r,π/2,0,ω)],
Φyz(r0,ω)=1c(r,ω)ϕφθ(r,π/2,0,ω),
j*(r,ω)jT(r,ω)=ωπϵ0Im[ϵ(ω)]Θ(ω,T)δ(rr)δ(ωω)I,
E(r,ω)=iμ0ωVG(r,r,ω)j(r,ω)d3r,
G(r,r,ω)=i8π21γ2(s^tss^T+p^1tpp^2T)eik||·(RR)×ei(γ1zγ2z)d2k||,
γ1=k2|k|||2,Imγ1>0,
γ2=ϵ(ω)k2|k|||2,Imγ2>0.
ts=2γ2γ1+γ2,tp=2γ2ϵ(ω)ϵ(ω)γ1+γ2,
Φxx(r,ω)=Φyy(r,ω)=ζ(ω,T)0|k||||γ2|2Imγ2e2zImγ1×[|ts|2+|γ1|2|tp|2(|γ2|2+|k|||2)|ϵ(ω)|k4]dk||,
Φzz(r,ω)=2ζ(ω,T)0|k||||γ2|2Imγ2e2zImγ1×[|k|||2|tp|2(|γ2|2+|k|||2)|ϵ(ω)|k4]dk||,
Φij(r,ω)=0,ij,
Φ(r,ω)=[Φzz(r,ω)Φxx(r,ω)]M+Φxx(r,ω)I,
Λ0(p)(r,ω)=Φzz(r,ω)Φxx(r,ω),
Λn(p)(r,ω)=0,n(1,,7),
Λ8(p)(r,ω)=3[Φxx(r,ω)Φzz(r,ω)],
I(0,0)=c(r,ω)2[Φxx(r0,ω)+Φzz(r0,ω)],
I(0,π/4)=c(r,ω)4[3Φxx(r0,ω)+Φzz(r0,ω)].

Metrics