Abstract

We have measured local electric field polarization vectors in 3-dimensional space on the nanoscale. A radial polarized light is generated by using a radial polarization converter and focused by an objective lens. Gold nanoparticle functionalized tips are used to scatter the focused field into the far-field region. Two different methods, rotational analyzer ellipsometry and Stokes parameters, are used in determining the polarization state of the scattered light. Two methods give consistent results with each other. Three dimensional local polarization vectors could be reconstructed by applying back transformation of the fully characterized polarizability tensor of the tip.

© 2009 Optical Society of America

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  1. S. Yan and B. Yao, "Accurate description of a radially polarized Gaussian beam," Phys. Rev. A 77, 023827 (2008).
    [CrossRef]
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  7. M. Stalder and M. Schadt, "Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters," Opt. Lett. 21, 1948-1950 (1996).
    [CrossRef] [PubMed]
  8. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
    [CrossRef]
  9. K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000), http://www.opticsinfobase.org/abstract.cfm?URI=oe-7-2-77.
    [CrossRef] [PubMed]
  10. 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] [PubMed]
  11. B. Sick, B. Hecht, U. P. Wild, and L. Novotny, "Probing confined fields with single molecules and vice versa," J. Microsc. 202, 365-373 (2001).
    [CrossRef] [PubMed]
  12. 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] [PubMed]
  13. K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
    [CrossRef]
  14. K. G. Lee, H. W. Kihm, K. J. Ahn, J. S. Ahn, Y. D. Suh, C. Lienau, and D. S. Kim, "Vector field mapping of local polarization using gold nanoparticle functionalized tips: independence of the tip shape," Opt. Express 15, 14993-15001 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-23-14993.
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  17. R. Greef, "An automatic ellipsometer for use in electrochemical investigations," Rev. Sci. Instrum. 41, 532-538 (1970).
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    [CrossRef] [PubMed]

2008 (1)

S. Yan and B. Yao, "Accurate description of a radially polarized Gaussian beam," Phys. Rev. A 77, 023827 (2008).
[CrossRef]

2007 (3)

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, "Electron acceleration to GeV energy by a radially polarized laser," Phys. Lett. A 368, 402-407 (2007).
[CrossRef]

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

K. G. Lee, H. W. Kihm, K. J. Ahn, J. S. Ahn, Y. D. Suh, C. Lienau, and D. S. Kim, "Vector field mapping of local polarization using gold nanoparticle functionalized tips: independence of the tip shape," Opt. Express 15, 14993-15001 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-23-14993.
[CrossRef] [PubMed]

2005 (1)

J. Ellis and A. Dogariu, "Optical polarimetry of random fields," Phys. Rev. Lett. 95, 203905 (2005).
[CrossRef] [PubMed]

2002 (1)

2001 (3)

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] [PubMed]

B. Sick, B. Hecht, U. P. Wild, and L. Novotny, "Probing confined fields with single molecules and vice versa," J. Microsc. 202, 365-373 (2001).
[CrossRef] [PubMed]

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] [PubMed]

2000 (2)

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000), http://www.opticsinfobase.org/abstract.cfm?URI=oe-7-2-77.
[CrossRef] [PubMed]

1996 (1)

1993 (1)

1990 (1)

1973 (2)

D. E. Aspnes, "Foulier transformation detection system for rotating-analyzer ellipsometers," Opt. Commun. 8, 222-225 (1973).
[CrossRef]

P. S. Hauge and F. H. Dill, "Design and operation of ETA, an Automated Ellipsometer," IBM J. Res. Dev. 17, 472-489 (1973).
[CrossRef]

1971 (1)

M. Schadt and W. Helfrich, "Voltage-dependent optical activity of a twisted nematic liquid crystal," Appl. Phys. Lett. 18127-128 (1971).
[CrossRef]

1970 (1)

R. Greef, "An automatic ellipsometer for use in electrochemical investigations," Rev. Sci. Instrum. 41, 532-538 (1970).
[CrossRef]

Ahn, J. S.

Ahn, K. J.

Aspnes, D. E.

D. E. Aspnes, "Foulier transformation detection system for rotating-analyzer ellipsometers," Opt. Commun. 8, 222-225 (1973).
[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] [PubMed]

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Bomzon, Z.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

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] [PubMed]

K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000), http://www.opticsinfobase.org/abstract.cfm?URI=oe-7-2-77.
[CrossRef] [PubMed]

Choi, S. B.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

Choi, W. J.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

Davidson, N.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Dill, F. H.

P. S. Hauge and F. H. Dill, "Design and operation of ETA, an Automated Ellipsometer," IBM J. Res. Dev. 17, 472-489 (1973).
[CrossRef]

Dogariu, A.

J. Ellis and A. Dogariu, "Optical polarimetry of random fields," Phys. Rev. Lett. 95, 203905 (2005).
[CrossRef] [PubMed]

Ellis, J.

J. Ellis and A. Dogariu, "Optical polarimetry of random fields," Phys. Rev. Lett. 95, 203905 (2005).
[CrossRef] [PubMed]

Ford, D. H.

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Greef, R.

R. Greef, "An automatic ellipsometer for use in electrochemical investigations," Rev. Sci. Instrum. 41, 532-538 (1970).
[CrossRef]

Gupta, D. N.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, "Electron acceleration to GeV energy by a radially polarized laser," Phys. Lett. A 368, 402-407 (2007).
[CrossRef]

Hasman, E.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Hauge, P. S.

P. S. Hauge and F. H. Dill, "Design and operation of ETA, an Automated Ellipsometer," IBM J. Res. Dev. 17, 472-489 (1973).
[CrossRef]

Hecht, B.

B. Sick, B. Hecht, U. P. Wild, and L. Novotny, "Probing confined fields with single molecules and vice versa," J. Microsc. 202, 365-373 (2001).
[CrossRef] [PubMed]

Helfrich, W.

M. Schadt and W. Helfrich, "Voltage-dependent optical activity of a twisted nematic liquid crystal," Appl. Phys. Lett. 18127-128 (1971).
[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] [PubMed]

Kant, N.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, "Electron acceleration to GeV energy by a radially polarized laser," Phys. Lett. A 368, 402-407 (2007).
[CrossRef]

Kihm, H. W.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

K. G. Lee, H. W. Kihm, K. J. Ahn, J. S. Ahn, Y. D. Suh, C. Lienau, and D. S. Kim, "Vector field mapping of local polarization using gold nanoparticle functionalized tips: independence of the tip shape," Opt. Express 15, 14993-15001 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-23-14993.
[CrossRef] [PubMed]

Kihm, J. E.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

Kim, D. E.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, "Electron acceleration to GeV energy by a radially polarized laser," Phys. Lett. A 368, 402-407 (2007).
[CrossRef]

Kim, D. S.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

K. G. Lee, H. W. Kihm, K. J. Ahn, J. S. Ahn, Y. D. Suh, C. Lienau, and D. S. Kim, "Vector field mapping of local polarization using gold nanoparticle functionalized tips: independence of the tip shape," Opt. Express 15, 14993-15001 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-23-14993.
[CrossRef] [PubMed]

Kim, G. H.

Kim, H.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

Kim, J.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

Kimura, W. D.

Lee, B.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

Lee, K. G.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

K. G. Lee, H. W. Kihm, K. J. Ahn, J. S. Ahn, Y. D. Suh, C. Lienau, and D. S. Kim, "Vector field mapping of local polarization using gold nanoparticle functionalized tips: independence of the tip shape," Opt. Express 15, 14993-15001 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-23-14993.
[CrossRef] [PubMed]

Leger, J.

Lienau, C.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

K. G. Lee, H. W. Kihm, K. J. Ahn, J. S. Ahn, Y. D. Suh, C. Lienau, and D. S. Kim, "Vector field mapping of local polarization using gold nanoparticle functionalized tips: independence of the tip shape," Opt. Express 15, 14993-15001 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-23-14993.
[CrossRef] [PubMed]

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] [PubMed]

Novotny, L.

B. Sick, B. Hecht, U. P. Wild, and L. Novotny, "Probing confined fields with single molecules and vice versa," J. Microsc. 202, 365-373 (2001).
[CrossRef] [PubMed]

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] [PubMed]

Oron, R.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Park, D. J.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

Park, Q. H.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[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] [PubMed]

Ropers, C.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

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] [PubMed]

Schadt, M.

M. Stalder and M. Schadt, "Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters," Opt. Lett. 21, 1948-1950 (1996).
[CrossRef] [PubMed]

M. Schadt and W. Helfrich, "Voltage-dependent optical activity of a twisted nematic liquid crystal," Appl. Phys. Lett. 18127-128 (1971).
[CrossRef]

Sick, B.

B. Sick, B. Hecht, U. P. Wild, and L. Novotny, "Probing confined fields with single molecules and vice versa," J. Microsc. 202, 365-373 (2001).
[CrossRef] [PubMed]

Stalder, M.

Suh, Y. D.

Suk, H.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, "Electron acceleration to GeV energy by a radially polarized laser," Phys. Lett. A 368, 402-407 (2007).
[CrossRef]

Tidwell, S. C.

Wild, U. P.

B. Sick, B. Hecht, U. P. Wild, and L. Novotny, "Probing confined fields with single molecules and vice versa," J. Microsc. 202, 365-373 (2001).
[CrossRef] [PubMed]

Woo, D. H.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

Yan, S.

S. Yan and B. Yao, "Accurate description of a radially polarized Gaussian beam," Phys. Rev. A 77, 023827 (2008).
[CrossRef]

Yao, B.

S. Yan and B. Yao, "Accurate description of a radially polarized Gaussian beam," Phys. Rev. A 77, 023827 (2008).
[CrossRef]

Yoon, Y. C.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, "Vector field microscopic imaging of light," Nature Photon. 1, 53-56 (2007).
[CrossRef]

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] [PubMed]

K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000), http://www.opticsinfobase.org/abstract.cfm?URI=oe-7-2-77.
[CrossRef] [PubMed]

Zhan, Q.

Appl. Opt. (2)

Appl. Phys. Lett. (2)

M. Schadt and W. Helfrich, "Voltage-dependent optical activity of a twisted nematic liquid crystal," Appl. Phys. Lett. 18127-128 (1971).
[CrossRef]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

IBM J. Res. Dev. (1)

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Figures (5)

Fig. 1.
Fig. 1.

RAE vs Stokes parameters. (a) The outer-plot (black line) results from a polar plot of the squared-rooted intensities for every detecting polarizer angle. The angle (θmax ) of the measured intensity maximum corresponds to the major axis angle and the square-rooted maximum (minimum) intensity is proportional to the major (minor) axis length. (b) Parameters of an ellipse. Major axis angle ψ defines the orientation of an ellipse and the angle χ characterizes the ellipticity and the rotational sense.

Fig. 2.
Fig. 2.

(a) Experimental setup. Square-rooted intensities of the scattered light after a linear polarizer are plotted for every polarization angle and also the elliptical polarization state is determined from the Stokes parameters. (b) Polarization state of a partially polarized field. Black solid circles are the polar plots of the square rooted scattered light intensity after passing through a polarizer. The Stokes parameters give the inner red diagram which is a mixture of an ellipse (polarized part) and a circle (unpolarized part).

Fig. 3.
Fig. 3.

(a) Three dimensional tip characterization. The tip end is illuminated by Ti:Sapphire laser beams in three orthogonal directions in sequence varying the incident beam polarization. The scattered electric field is detected in the direction of incidence and also in (1 ±1 0). (b) Polarization vector mapping of a focused radially polarized light. A radially polarized beam generated by using a radial polarization converter is focused by an objective lens. A GNP functionalized tip is scanned the focusing area in three dimensional space using a 3-axes nano positioner (Nano Cube, Physik Instrumente). The polarizaton state of the scattered light is determined by applying the RAE and measuring the Stokes parameters in two orthogonal axes.

Fig. 4.
Fig. 4.

(upper) Experimentally measured field intensity distribution profiles for three orthogonal axes. (below) Numerically calculated field intensity distribution at the focus of a radial polarization of corresponding field component.

Fig. 5.
Fig. 5.

Polarization vector mapping in the focus plane (z=0) by the (a) RAE and the (b) Stokes measurement. (c) Top view of (b). (d) Side view (y=0) of (b) for several tip height (z) values. The vertical field amplitude (Ez ) is 5 times multiplied in (d) for a better visualization.

Equations (13)

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E Local ( r ) = ( E x , E z ) = ( a 1 e i ω t + i δ 1 , a 2 e i ω t + i δ 2 ) , ( a 1 , a 2 > 0 ) .
s 0 = I ( 0 ° , 0 ) + I ( 90 ° , 0 ) = a 1 2 + a 2 2 ,
s 1 = I ( 0 ° , 0 ) I ( 90 ° , 0 ) = a 1 2 a 2 2 = s 0 cos 2 χ cos 2 ψ ,
s 2 = I ( 45 ° , 0 ) I ( 135 ° , 0 ) = a 1 a 2 cos ( δ 1 δ 2 ) = s 0 cos 2 χ 2 ψ ,
s 3 = I ( 45 ° , π 2 ) I ( 135 ° , π 2 ) = a 1 a 2 sin ( δ 1 δ 2 ) = s 0 sin 2 χ .
Radial polarization = H G 10 x ̂ + H G 01 y ̂
E ( ρ , φ , z ) = ( E x E y E z ) = i k f 2 2 w 0 E 0 e ikf ( i ( I 11 I 12 ) cos φ i ( I 11 I 12 ) sin φ 4 I 10 ) ,
I 10 = 0 θ max f w ( θ ) cos θ sin 3 θ J 0 ( k ρ sin θ ) e ikz cos θ d θ
I 11 = 0 θ max f w ( θ ) cos θ sin 2 θ ( 1 + 3 cos θ ) J 1 ( k ρ sin θ ) e ikz cos θ d θ
I 12 = 0 θ max f w ( θ ) cos θ sin 2 θ ( 1 cos θ ) J 1 ( k ρ sin θ ) e ikz cos θ d θ
f w ( θ ) = exp ( f 2 sin 2 θ / w 0 2 ) .
E sca = α · E inc = i , j = 1 3 α i j E inc , j
α = ( 1.01 0.18 0.21 0.18 0.70 0.12 0.21 0.12 1 )

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