Abstract

We present a system built to perform measurements of scattering-angle-resolved polarization state distributions across the exit pupil of a high numerical aperture collector lens. These distributions contain information about the three-dimensional electromagnetic field that results from the interaction of a tightly focused field and a sub-resolution scatterer. Experimental evidence proving that the system allows for high polarization-dependent sensitivity to sub-resolution displacements of a sub-resolution scatterer is provided together with the corresponding numerical results.

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References

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  1. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
    [CrossRef]
  2. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
    [CrossRef]
  3. E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981).
    [CrossRef]
  4. M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives,” J. Opt. Soc. Am. A 3(12), 2086–2093 (1986).
    [CrossRef]
  5. R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40(11), 2293–2310 (1993).
    [CrossRef]
  6. D. P. Biss and T. G. Brown, “Primary aberrations in focused radially polarized vortex beams,” Opt. Express 12(3), 384–393 (2004).
    [CrossRef] [PubMed]
  7. P. Török, P. Varga, and G. Né́meth, “Analytical solution of the diffraction integrals and interpretation of wavefront distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12(12), 2660–2671 (1995).
    [CrossRef]
  8. H. Ling and S. W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1(9), 965–973 (1984).
    [CrossRef]
  9. P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12(2), 325–332 (1995).
    [CrossRef]
  10. V. Delaubert, N. Treps, G. Bo, and C. Fabre, “Optical storage of high-density information beyond the diffraction limit: A quantum study,” Phys. Rev. A 73(1), 013820 (2006).
    [CrossRef]
  11. J. M. Brok and H. P. Urbach, “Simulation of polarization effects in diffraction problems of optical recording,” J. Mod. Opt. 49(11), 1811–1829 (2002).
    [CrossRef]
  12. P. Török and M. Gu, “High-numerical-aperture optical microscopy and modern applications: introduction to the feature issue,” Appl. Opt. 39(34), 6277–6278 (2000).
    [CrossRef]
  13. P. Török and P. R. T. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express 12(15), 3605–3617 (2004).
    [CrossRef] [PubMed]
  14. Q. Zhan and J. R. Leger, “Measurement of surface features beyond the diffraction limit with an imaging ellipsometer,” Opt. Lett. 27(10), 821–823 (2002).
    [CrossRef]
  15. A. Rohrbach and E. H. K. Stelzer, “Optical trapping of dielectric particles in arbitrary fields,” J. Opt. Soc. Am. A 18(4), 839–853 (2001).
    [CrossRef]
  16. K.-H. Shuster, “Radial polarization - rotating optical arrangement and microlithographic projection exposure system incorporating said arrangement”, US Patent 6191880B1 (2001).
  17. D. McGloin, “Optical tweezers: 20years on,” Philos. Trans. R. Soc. Lond. A 364(1849), 3521–3537 (2006).
    [CrossRef]
  18. R. L. Eriksen, V. R. Daria, and J. Gluckstad, “Fully dynamic multiple-beam optical tweezers,” Opt. Express 10(14), 597–602 (2002).
    [PubMed]
  19. F. Kulzer and M. Orrit, “Single-molecule optics,” Annu. Rev. Phys. Chem. 55(1), 585–611 (2004).
    [CrossRef] [PubMed]
  20. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
    [CrossRef]
  21. J. T. Fourkas, “Rapid determination of the three-dimensional orientation of single molecules,” Opt. Lett. 26(4), 211–213 (2001).
    [CrossRef]
  22. A. De Martino, S. Ben Hatit, and M. Foldyna, “Mueller polarimetry in the back focal plane,” Proc. SPIE 6518, 65180X (2007).
    [CrossRef]
  23. S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drévillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi., A Appl. Mater. Sci. 205(4), 743–747 (2008).
    [CrossRef]
  24. In the figure, the dashed lines box represents a part of the experimental setup that is orthogonal to the plane of the optical bench.
  25. F. Delplancke, “Automated high-speed Mueller matrix scatterometer,” Appl. Opt. 36(22), 5388–5395 (1997).
    [CrossRef] [PubMed]
  26. D. Lara and C. Dainty, “Axially resolved complete mueller matrix confocal microscopy,” Appl. Opt. 45(9), 1917–1930 (2006).
    [CrossRef] [PubMed]
  27. W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices and polarized scattered light,” Am. J. Phys. 53(5), 468–478 (1985).
    [CrossRef]
  28. E. Compain, S. Poirier, and B. Drévillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 (1999).
    [CrossRef]
  29. A. De Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003).
    [CrossRef] [PubMed]
  30. To speed-up the calibration of the system, and reduce the noise in the measurements, we applied a 4 × 4 binning to the original 1024 × 768 pixel images. This reduced the time spent in the calibration from ~6 hours to ~30 minutes and increased the pixel alignment accuracy. The same binning was applied to all our experimental data.
  31. The Mueller matrices presented in this section were obtained from the average of 25 measurements, for each combination polarizer-analyzer, to minimize the effect of statistical errors.
  32. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999).
  33. K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1(4), 228–231 (2007).
    [CrossRef]
  34. This approximation is commonly used in the analysis of the image formation of a point-scatterer.
  35. C. J. R. Sheppard and T. Wilson, “The Image of a Single Point in Microscopes of Large Numerical Aperture,” Proc. R. Soc. Lond. A Math. Phys. Sci. 379(1776), 145–158 (1982).
    [CrossRef]
  36. The results for the gold nano-sphere were obtained as the average of 5 measurements for each combination polarizer-analyzer to minimize the effect of statistical errors.

2008 (1)

S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drévillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi., A Appl. Mater. Sci. 205(4), 743–747 (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(4), 228–231 (2007).
[CrossRef]

A. De Martino, S. Ben Hatit, and M. Foldyna, “Mueller polarimetry in the back focal plane,” Proc. SPIE 6518, 65180X (2007).
[CrossRef]

2006 (4)

D. McGloin, “Optical tweezers: 20years on,” Philos. Trans. R. Soc. Lond. A 364(1849), 3521–3537 (2006).
[CrossRef]

V. Delaubert, N. Treps, G. Bo, and C. Fabre, “Optical storage of high-density information beyond the diffraction limit: A quantum study,” Phys. Rev. A 73(1), 013820 (2006).
[CrossRef]

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[CrossRef]

D. Lara and C. Dainty, “Axially resolved complete mueller matrix confocal microscopy,” Appl. Opt. 45(9), 1917–1930 (2006).
[CrossRef] [PubMed]

2004 (3)

2003 (1)

2002 (3)

2001 (2)

2000 (1)

1999 (1)

1997 (1)

1995 (2)

1993 (1)

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40(11), 2293–2310 (1993).
[CrossRef]

1986 (1)

1985 (1)

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices and polarized scattered light,” Am. J. Phys. 53(5), 468–478 (1985).
[CrossRef]

1984 (1)

1982 (1)

C. J. R. Sheppard and T. Wilson, “The Image of a Single Point in Microscopes of Large Numerical Aperture,” Proc. R. Soc. Lond. A Math. Phys. Sci. 379(1776), 145–158 (1982).
[CrossRef]

1981 (1)

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981).
[CrossRef]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[CrossRef]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Bailey, W. M.

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices and polarized scattered light,” Am. J. Phys. 53(5), 468–478 (1985).
[CrossRef]

Ben Hatit, S.

S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drévillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi., A Appl. Mater. Sci. 205(4), 743–747 (2008).
[CrossRef]

A. De Martino, S. Ben Hatit, and M. Foldyna, “Mueller polarimetry in the back focal plane,” Proc. SPIE 6518, 65180X (2007).
[CrossRef]

Bickel, W. S.

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices and polarized scattered light,” Am. J. Phys. 53(5), 468–478 (1985).
[CrossRef]

Biss, D. P.

Bo, G.

V. Delaubert, N. Treps, G. Bo, and C. Fabre, “Optical storage of high-density information beyond the diffraction limit: A quantum study,” Phys. Rev. A 73(1), 013820 (2006).
[CrossRef]

Booker, G. R.

Brok, J. M.

J. M. Brok and H. P. Urbach, “Simulation of polarization effects in diffraction problems of optical recording,” J. Mod. Opt. 49(11), 1811–1829 (2002).
[CrossRef]

Brown, T. G.

Chen, W.

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[CrossRef]

Compain, E.

Dainty, C.

Daria, V. R.

De Martino, A.

S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drévillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi., A Appl. Mater. Sci. 205(4), 743–747 (2008).
[CrossRef]

A. De Martino, S. Ben Hatit, and M. Foldyna, “Mueller polarimetry in the back focal plane,” Proc. SPIE 6518, 65180X (2007).
[CrossRef]

A. De Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003).
[CrossRef] [PubMed]

Delaubert, V.

V. Delaubert, N. Treps, G. Bo, and C. Fabre, “Optical storage of high-density information beyond the diffraction limit: A quantum study,” Phys. Rev. A 73(1), 013820 (2006).
[CrossRef]

Delplancke, F.

Drévillon, B.

Eriksen, R. L.

Fabre, C.

V. Delaubert, N. Treps, G. Bo, and C. Fabre, “Optical storage of high-density information beyond the diffraction limit: A quantum study,” Phys. Rev. A 73(1), 013820 (2006).
[CrossRef]

Foldyna, M.

S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drévillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi., A Appl. Mater. Sci. 205(4), 743–747 (2008).
[CrossRef]

A. De Martino, S. Ben Hatit, and M. Foldyna, “Mueller polarimetry in the back focal plane,” Proc. SPIE 6518, 65180X (2007).
[CrossRef]

Fourkas, J. T.

Friberg, A. T.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1(4), 228–231 (2007).
[CrossRef]

Garcia-Caurel, E.

Gluckstad, J.

Gu, M.

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(4), 228–231 (2007).
[CrossRef]

Kant, R.

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40(11), 2293–2310 (1993).
[CrossRef]

Kim, Y.-K.

Kulzer, F.

F. Kulzer and M. Orrit, “Single-molecule optics,” Annu. Rev. Phys. Chem. 55(1), 585–611 (2004).
[CrossRef] [PubMed]

Laczik, Z.

Lara, D.

Laude, B.

Lee, S. W.

Leger, J. R.

Li, Y.

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981).
[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(4), 228–231 (2007).
[CrossRef]

Ling, H.

Mansuripur, M.

McGloin, D.

D. McGloin, “Optical tweezers: 20years on,” Philos. Trans. R. Soc. Lond. A 364(1849), 3521–3537 (2006).
[CrossRef]

Munro, P. R. T.

Né´meth, G.

Orrit, M.

F. Kulzer and M. Orrit, “Single-molecule optics,” Annu. Rev. Phys. Chem. 55(1), 585–611 (2004).
[CrossRef] [PubMed]

Poirier, S.

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(4), 228–231 (2007).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Rohrbach, A.

Setälä, T.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1(4), 228–231 (2007).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and T. Wilson, “The Image of a Single Point in Microscopes of Large Numerical Aperture,” Proc. R. Soc. Lond. A Math. Phys. Sci. 379(1776), 145–158 (1982).
[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(4), 228–231 (2007).
[CrossRef]

Stelzer, E. H. K.

Török, P.

Treps, N.

V. Delaubert, N. Treps, G. Bo, and C. Fabre, “Optical storage of high-density information beyond the diffraction limit: A quantum study,” Phys. Rev. A 73(1), 013820 (2006).
[CrossRef]

Urbach, H. P.

J. M. Brok and H. P. Urbach, “Simulation of polarization effects in diffraction problems of optical recording,” J. Mod. Opt. 49(11), 1811–1829 (2002).
[CrossRef]

Varga, P.

Wilson, T.

C. J. R. Sheppard and T. Wilson, “The Image of a Single Point in Microscopes of Large Numerical Aperture,” Proc. R. Soc. Lond. A Math. Phys. Sci. 379(1776), 145–158 (1982).
[CrossRef]

Wolf, E.

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[CrossRef]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Zhan, Q.

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[CrossRef]

Q. Zhan and J. R. Leger, “Measurement of surface features beyond the diffraction limit with an imaging ellipsometer,” Opt. Lett. 27(10), 821–823 (2002).
[CrossRef]

Am. J. Phys. (1)

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices and polarized scattered light,” Am. J. Phys. 53(5), 468–478 (1985).
[CrossRef]

Annu. Rev. Phys. Chem. (1)

F. Kulzer and M. Orrit, “Single-molecule optics,” Annu. Rev. Phys. Chem. 55(1), 585–611 (2004).
[CrossRef] [PubMed]

Appl. Opt. (4)

J. Mod. Opt. (2)

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40(11), 2293–2310 (1993).
[CrossRef]

J. M. Brok and H. P. Urbach, “Simulation of polarization effects in diffraction problems of optical recording,” J. Mod. Opt. 49(11), 1811–1829 (2002).
[CrossRef]

J. Opt. Soc. Am. A (5)

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(4), 228–231 (2007).
[CrossRef]

Opt. Commun. (2)

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[CrossRef]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Philos. Trans. R. Soc. Lond. A (1)

D. McGloin, “Optical tweezers: 20years on,” Philos. Trans. R. Soc. Lond. A 364(1849), 3521–3537 (2006).
[CrossRef]

Phys. Rev. A (1)

V. Delaubert, N. Treps, G. Bo, and C. Fabre, “Optical storage of high-density information beyond the diffraction limit: A quantum study,” Phys. Rev. A 73(1), 013820 (2006).
[CrossRef]

Phys. Status Solidi., A Appl. Mater. Sci. (1)

S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drévillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi., A Appl. Mater. Sci. 205(4), 743–747 (2008).
[CrossRef]

Proc. R. Soc. Lond. A Math. Phys. Sci. (3)

C. J. R. Sheppard and T. Wilson, “The Image of a Single Point in Microscopes of Large Numerical Aperture,” Proc. R. Soc. Lond. A Math. Phys. Sci. 379(1776), 145–158 (1982).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[CrossRef]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Proc. SPIE (1)

A. De Martino, S. Ben Hatit, and M. Foldyna, “Mueller polarimetry in the back focal plane,” Proc. SPIE 6518, 65180X (2007).
[CrossRef]

Other (7)

In the figure, the dashed lines box represents a part of the experimental setup that is orthogonal to the plane of the optical bench.

The results for the gold nano-sphere were obtained as the average of 5 measurements for each combination polarizer-analyzer to minimize the effect of statistical errors.

This approximation is commonly used in the analysis of the image formation of a point-scatterer.

To speed-up the calibration of the system, and reduce the noise in the measurements, we applied a 4 × 4 binning to the original 1024 × 768 pixel images. This reduced the time spent in the calibration from ~6 hours to ~30 minutes and increased the pixel alignment accuracy. The same binning was applied to all our experimental data.

The Mueller matrices presented in this section were obtained from the average of 25 measurements, for each combination polarizer-analyzer, to minimize the effect of statistical errors.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999).

K.-H. Shuster, “Radial polarization - rotating optical arrangement and microlithographic projection exposure system incorporating said arrangement”, US Patent 6191880B1 (2001).

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Figures (14)

Fig. 1
Fig. 1

Schematic diagram of the working principle of our system. The longitudinal component of the scattered field, Ez (s), is combined with the transversal components by the collector lens. The crosses indicate sample points in the exit pupil where the polarization state is analyzed.

Fig. 2
Fig. 2

Diagram of the vectorial polarimeter. The light path in green is the path followed by the light that forms the image of the exit pupil in cameras CCD1-CCD4. The shaded part of the system, which corresponds to the red light path, is the path followed by the light used to produce the auxiliary image discussed in section 5. CCD2 is at 45° from the direction of the beam incident on the polarizing beam-splitter, PBS, because the beam-splitter is designed to give this angular separation between the vertical and horizontal polarization components.

Fig. 3
Fig. 3

Measurements and operations necessary to obtain the 16 elements of the Mueller matrix for every data pixel in our system. From left to right, the first symbol, and subscript of I, represents the polarization state of the incident light whereas the second symbol, and subscript, represents the analyzer used in the corresponding measurement. The convention followed for the subscripts is the same as in section 2.1. The unpolarized component, for the incident light, and the total irradiance, for the analyzer, were obtained as the incoherent superposition of the H and V components accordingly.

Fig. 4
Fig. 4

Calibrated Mueller matrix of the horizontal polarizer, within the aperture of the incident beam, as obtained with our system.

Fig. 5
Fig. 5

Mueller matrix of the reference sphere used to characterize the polarization artifacts introduced by the high-NA microscope objective. The spot in the lower-left corner of the pupil is due to a scratch in the surface of the sphere produced accidentally during the alignment. Note that the images correspond to a zoom-in on the region of interest, i.e. the exit pupil.

Fig. 6
Fig. 6

(a) Diagram showing the preparation of the specimen and positioning with the auxiliary system; the specimen is being illuminated by the auxiliary laser, ALS. (b) Corresponding image of the nano-spheres as obtained by CCD5 with the flip-in mount mirror in the upright position.

Fig. 7
Fig. 7

Experimental Mueller matrix of the on axis 80nm gold nano-sphere measured with our system.

Fig. 8
Fig. 8

Diagram of the focal plane with marks at the three positions of the point-scatterer (gold nano-sphere) analyzed. The origin of the xy coordinate system corresponds to the on axis position. The z-axis (not shown) is the optical axis of our system.

Fig. 9
Fig. 9

(a) Experimentally obtained Stokes parameters distribution in the exit pupil of the collector lens for an on axis gold nano-sphere with incident light linearly polarized in the horizontal direction. (b) Corresponding numerical results for a point-scatterer. The Stokes parameters are normalized with respect to the maximum S0x.

Fig. 10
Fig. 10

(a) Experimentally obtained Stokes parameters in the exit pupil of the collector lens for a gold nano-sphere in the focal plane at x = -λ/3 with incident light linearly polarized in the horizontal direction. (b) Corresponding numerical results for a point-scatterer. The Stokes parameters are normalized with respect to the maximum S0x.

Fig. 11
Fig. 11

(a) Experimentally obtained Stokes parameters in the exit pupil of the collector lens for a gold nano-sphere in the focal plane at x = + λ/3 with incident light linearly polarized in the horizontal direction. (b) Corresponding numerical results for a point-scatterer. The Stokes parameters are normalized with respect to the maximum S0x.

Fig. 12
Fig. 12

(a) Experimentally obtained Stokes parameters distribution in the exit pupil of the collector lens for an on axis gold nano-sphere with incident light circularly polarized to the left. (b) Corresponding numerical results for a point-scatterer. The Stokes parameters are normalized with respect to the maximum S0r.

Fig. 13
Fig. 13

(a) Experimentally obtained Stokes parameters in the exit pupil of the collector lens for a gold nano-sphere in the focal plane at x = -λ/3 with incident light circularly polarized to the left. (b) Corresponding numerical results for a point-scatterer. The Stokes parameters are normalized with respect to the maximum S0r.

Fig. 14
Fig. 14

(a) Experimentally obtained Stokes parameters in the exit pupil of the collector lens for a gold nano-sphere in the focal plane at x = + λ/3 with incident light circularly polarized to the left. (b) Corresponding numerical results for a point-scatterer. The Stokes parameters are normalized with respect to the maximum S0r.

Tables (2)

Tables Icon

Table 1 Pairs of retardance introduced in the Pockels cells in wavelength units.

Tables Icon

Table 2 Fitted theoretical Mueller matrix and mean and standard deviation of the pixel value distribution, within the aperture of the beam, for the Mueller matrix of each calibration sample. Δ and Ψ are the ellipsometric angles in radians.

Equations (1)

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S=( S 0 S 1 S 2 S 3 )=( E x E x * + E y E y * E x E x * E y E y * E x E y * + E y E x * i( E x E y * E y E x * ) )

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