Abstract

In this work we describe theoretical and experimental physical aspects of high-resolution imaging polarimetry and its application to polarization-multiplexed encoding. We theoretically demonstrate that it is possible to resolve the orientation of two fixed dipole-like emitters placed significantly below the resolution limit if their emission is uncorrelated. Furthermore, we experimentally demonstrate this phenomenon by illuminating closely spaced asymmetric nanopits with unpolarized light and subsequently determining their individual orientation and position from the measured spatial distributions of the azimuth angle of the polarization and degree of polarization, respectively. Reduction of the optical resolution of the imaging system is also shown to only weakly affect resolution obtainable via polarization measurements.

© 2014 Optical Society of America

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References

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  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (Elsevier, 1987), p. 539.
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    [CrossRef]
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    [CrossRef]
  4. K. Lindfors, T. Setälä, M. Kaivola, A. T. Friberg, and T. Seta, J. Opt. Soc. Am. A 22, 561 (2005).
    [CrossRef]
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    [CrossRef]
  7. C. Macias-Romero, M. R. Foreman, and P. Török, Opt. Express 19, 25066 (2011).
    [CrossRef]
  8. J. M. Bueno and M. C. W. Campbell, Opt. Lett. 27, 830 (2002).
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  9. D. Lara and C. Dainty, Opt. Lett. 30, 2879 (2005).
    [CrossRef]
  10. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, Appl. Opt. 45, 5453 (2006).
    [CrossRef]
  11. C. Macías-Romero, R. Lim, M. R. Foreman, and P. Török, Opt. Lett. 36, 1638 (2011).
    [CrossRef]
  12. W. Urbaczyk, Opt. Acta 33, 53 (1986).
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    [CrossRef]
  15. D. Goldstein, Polarized Light, 3rd ed. (Taylor & Francis, 2011), p. 770.

2011

2010

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, Phys. Rev. Lett. 104, 253601 (2010).
[CrossRef]

2008

2007

A. De Martino, S. Ben Hatit, and M. Foldyna, Proc. SPIE 6518, 65180X (2007).
[CrossRef]

2006

2005

2002

1998

P. Török, P. Higdon, and T. Wilson, Opt. Commun. 148, 300 (1998).
[CrossRef]

1986

W. Urbaczyk, Opt. Acta 33, 53 (1986).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (Elsevier, 1987), p. 539.

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (Elsevier, 1987), p. 539.

Ben Hatit, S.

A. De Martino, S. Ben Hatit, and M. Foldyna, Proc. SPIE 6518, 65180X (2007).
[CrossRef]

Bliokh, K. Y.

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, Phys. Rev. Lett. 104, 253601 (2010).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Cambridge University, 1999), p. 952.

Bueno, J. M.

Campbell, M. C. W.

Chenault, D. B.

Dainty, C.

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, Phys. Rev. Lett. 104, 253601 (2010).
[CrossRef]

D. Lara and C. Dainty, Opt. Lett. 30, 2879 (2005).
[CrossRef]

De Martino, A.

A. De Martino, S. Ben Hatit, and M. Foldyna, Proc. SPIE 6518, 65180X (2007).
[CrossRef]

Foldyna, M.

A. De Martino, S. Ben Hatit, and M. Foldyna, Proc. SPIE 6518, 65180X (2007).
[CrossRef]

Foreman, M. R.

Friberg, A. T.

Goldstein, D.

D. Goldstein, Polarized Light, 3rd ed. (Taylor & Francis, 2011), p. 770.

Goldstein, D. L.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2004).

Higdon, P.

P. Török, P. Higdon, and T. Wilson, Opt. Commun. 148, 300 (1998).
[CrossRef]

Kaivola, M.

Kriezis, E.

Lara, D.

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, Phys. Rev. Lett. 104, 253601 (2010).
[CrossRef]

D. Lara and C. Dainty, Opt. Lett. 30, 2879 (2005).
[CrossRef]

Lim, R.

Lindfors, K.

Macias-Romero, C.

Macías-Romero, C.

Munro, P.

Ostrovskaya, E. A.

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, Phys. Rev. Lett. 104, 253601 (2010).
[CrossRef]

Rodríguez-Herrera, O. G.

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, Phys. Rev. Lett. 104, 253601 (2010).
[CrossRef]

Seta, T.

Setälä, T.

Shaw, J. A.

Török, P.

Tyo, J. S.

Urbaczyk, W.

W. Urbaczyk, Opt. Acta 33, 53 (1986).
[CrossRef]

Wilson, T.

P. Török, P. Higdon, and T. Wilson, Opt. Commun. 148, 300 (1998).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Cambridge University, 1999), p. 952.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Acta

W. Urbaczyk, Opt. Acta 33, 53 (1986).
[CrossRef]

Opt. Commun.

P. Török, P. Higdon, and T. Wilson, Opt. Commun. 148, 300 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, Phys. Rev. Lett. 104, 253601 (2010).
[CrossRef]

Proc. SPIE

A. De Martino, S. Ben Hatit, and M. Foldyna, Proc. SPIE 6518, 65180X (2007).
[CrossRef]

Other

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (Elsevier, 1987), p. 539.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Cambridge University, 1999), p. 952.

J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2004).

D. Goldstein, Polarized Light, 3rd ed. (Taylor & Francis, 2011), p. 770.

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Figures (4)

Fig. 1.
Fig. 1.

Resolving the orientation of two dipole-like emitters placed below the diffraction limit. The imaging system is shown in (a) and polarization imaging simulations are shown in (b)–(f). Two fixed dipole-like emitters are placed below the resolution limit and illuminated with correlated (b), (c) and uncorrelated light (d)–(f). The dipoles are depicted by the black and the red arrows. The dotted circles depict the optical resolution of the system. The grayscale in (b) and (d) denotes irradiance, while the blue, black, and red line segments depict the polarization state of the field position-wise. (c) and (e) show the spatial distribution of the azimuth angle of the polarization given in (b) and (d), respectively. The spatial distribution of the DOP in (d) is shown in (f). It is possible to resolve the orientation of two fixed dipole-like emitters placed below the resolution limit in the case of uncorrelated illumination.

Fig. 2.
Fig. 2.

Resolving the orientation of nanostructures with different numerical apertures. (a) Scanning electron microscope (SEM) image of a track of the nanostructures ( 200 nm × 50 nm × 100 nm , 200 nm apart) milled into an Si substrate. The dashed lines in a depict, as a reference, the sizes of the pinhole (scaled to high-NA space) and diffraction limits for 0.95 and 0.55 NA. The spatial distribution of the measured azimuth angle with 0.95 NA is shown in (b), from which a line scan is taken and shown in (c) (red, left axis = 0.95 NA ; black, right axis = 0.55 NA ). The dashed red line is from rigorous simulations of high-NA imaging systems. The blue markers depict the orientation of the nanostructures obtained from the SEM image. Accompanying error bars are shown magnified in the inset. Reducing the NA of the optical system as significant as to 0.55 yields an optical resolution 215 nm above that required to resolve the nanostructures. Such a reduction increases the disparity in the vertical axis, however, without a total loss of information in the orientation.

Fig. 3.
Fig. 3.

Effect of imaging with different NA on the azimuth angle of the polarization and irradiance. (a) shows an example of the geometrical vector sum of six Stokes vectors with a weight (given by a PSF) and resultant S ( x ) , as proposed by Eq. (1). Changing the NA modifies the weighting factor of the Stokes vectors and hence affects the azimuth angle of the S ( x ) . (b) and (c) show line scans of the measured azimuth angle and irradiance of a track in the set of nanostructures with different NAs. Decreasing the NA affects drastically the irradiance compared to the azimuth angle.

Fig. 4.
Fig. 4.

Resolution in polarization imaging. (a) SEM image of the nanostructures. (b) Spatial distribution of the DOP and (c) irradiance obtained when imaging the nanostructures with unpolarized light. (d) Comparison of the experimental PSF of the optical system (red) and a cross section from a feature in (b) (blue); the respective full width at half-maxima are displayed in the figure. It is possible to resolve the position of the nanostructures using the spatial distribution of the DOP, while the irradiance remains resolution limited.

Equations (1)

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S ( x ) = i n PSF ( x x i ) · S i ,

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