Abstract

Correlations in the illumination field have a profound impact on the image contrast for features near the resolution limit. The pupil polarization affects these correlations. We show that a polarization vortex has a particularly dramatic effect. A theoretical model is given for the correlation matrix of a partially correlated source created by placing an azimuthal polarization vortex mode converter in the pupil plane of a critical illumination system. We then validate this model experimentally using a reversed-wavefront Young interferometer, directly show the impact that the phase of the correlation function has on image contrast.

© 2008 Optical Society of America

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References

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  7. D. P. Biss and T. G. Brown, "Polarization-vortex-driven second-harmonic generation," Opt. Lett. 28, 923-925 (2003).
    [CrossRef] [PubMed]
  8. D. P. Biss, "Focal field interactions from cylindrical vector beams," Ph.D. thesis, University of Rochester, Rochester, NY 14627 (2005).
  9. S. Quabis, R. Dorn, and G. Leuchs, "Generation of a radially polarized doughnut mode of high quality," Appl. Phys. B 81, 597-600 (2005).
  10. A. K. Spilman and T. G. Brown, "Stress birefringent, space-variant wave plates for vortex illumination," Appl. Opt. 26, 61-66 (2007).
    [CrossRef]
  11. Q1. H. H. Hopkins, "The concept of partial coherence in optics," Proc. Roy. Soc. A 208, 263-277 (1951).
    [CrossRef]
  12. Q2. H. H. Hopkins, "On the diffraction theory of optical images," Proc. Roy. Soc. A 217, 408-432 (1953).
    [CrossRef]
  13. H. H. Hopkins, "Image formation with coherent and partially coherent light," Photograph. Sci. Eng. 21 (1977).
  14. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).
  15. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263 (2003).
    [CrossRef]
  16. J. Tervo, T. Set¨al¨a, and A. T. Friberg, "Theory of partially coherent electromagnetic fields in the space-frequency domain," J. Opt. Soc. Am. A 21, 2205-2215 (2004).
    [CrossRef]
  17. T. Saastamoinen, J. T. J. Turunen, T. Set¨al¨a, and A. T. Friberg, "Electromagnetic coherence theory of laser resonator modes," J. Opt. Soc. Am. A 22, 103-108 (2005).
    [CrossRef]
  18. E. Wolf, "Coherence and polarization properties of electromagnetic laser modes," Opt. Commun. 265, 60-62 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
  20. E. Wolf, "The influence of Young’s interference experiment on the development of statistical optics," Progress in Optics, E.Wolf, ed., (Elsevier Science, 2007) Vol. 50, Chap. 7.
    [CrossRef]
  21. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, New York, 2007).
  22. F. Zernike, "The concept of degree of coherence and its application to optical problems," Physica 5, 785-795 (1938).
    [CrossRef]
  23. D. P. Brown, A. K. Spilman, T. G. Brown, M. A. Alonso, R. Borghi, and M. Santarsiero, "Calibration of a reversed-wavefront interferometer for polarization coherence metrology," Proc. SPIE 6672667207(2007).
  24. D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, "Spatial coherence properties of azimuthally polarized laser modes," Opt. Commun. 281, 5287-5290 (2008).
    [CrossRef]
  25. G. Gbur, T. D. Visser, and E. Wolf, "‘Hidden’ singularities in partially coherent wavefields," J. Opt. A: Pure and Appl. Opt. 6, S239-S242 (2004).
    [CrossRef]
  26. D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143,905 (2004).
    [CrossRef]
  27. I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation vortices in partially coherent light: theory," J. Opt. Soc. Am. B 21, 1895-1900 (2004).
    [CrossRef]
  28. I. D. Maleev and G. A. Swartzlander, Jr., "Propagation of spatial correlation vortices," J. Opt. Soc. Am. B 25, 915-922 (2008).
    [CrossRef]
  29. G. Gbur and G. A. Swartzlander, Jr., "Complete transverse representation of a correlation singularity of a partially coherent field," J. Opt. Soc. Am. B 25, 1422-1429 (2008).
    [CrossRef]
  30. R. Borghi and M. Santarsiero, "Nonparaxial propagation of spirally polarized optical beams," J. Opt. Soc. Am. A 21, 2029-2037 (2004).
    [CrossRef]
  31. M. Santarsiero and R. Borghi, "Measuring spatial coherence by using a reversed-wavefront Young interferometer," Opt. Lett. 31, 861-863 (2006).
    [CrossRef] [PubMed]

2008 (3)

2007 (2)

A. K. Spilman and T. G. Brown, "Stress birefringent, space-variant wave plates for vortex illumination," Appl. Opt. 26, 61-66 (2007).
[CrossRef]

D. P. Brown, A. K. Spilman, T. G. Brown, M. A. Alonso, R. Borghi, and M. Santarsiero, "Calibration of a reversed-wavefront interferometer for polarization coherence metrology," Proc. SPIE 6672667207(2007).

2006 (3)

2005 (2)

T. Saastamoinen, J. T. J. Turunen, T. Set¨al¨a, and A. T. Friberg, "Electromagnetic coherence theory of laser resonator modes," J. Opt. Soc. Am. A 22, 103-108 (2005).
[CrossRef]

S. Quabis, R. Dorn, and G. Leuchs, "Generation of a radially polarized doughnut mode of high quality," Appl. Phys. B 81, 597-600 (2005).

2004 (5)

2003 (2)

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263 (2003).
[CrossRef]

D. P. Biss and T. G. Brown, "Polarization-vortex-driven second-harmonic generation," Opt. Lett. 28, 923-925 (2003).
[CrossRef] [PubMed]

1999 (2)

1998 (1)

1996 (1)

1994 (1)

1953 (1)

Q2. H. H. Hopkins, "On the diffraction theory of optical images," Proc. Roy. Soc. A 217, 408-432 (1953).
[CrossRef]

1951 (1)

Q1. H. H. Hopkins, "The concept of partial coherence in optics," Proc. Roy. Soc. A 208, 263-277 (1951).
[CrossRef]

1938 (1)

F. Zernike, "The concept of degree of coherence and its application to optical problems," Physica 5, 785-795 (1938).
[CrossRef]

Alonso, M. A.

D. P. Brown, A. K. Spilman, T. G. Brown, M. A. Alonso, R. Borghi, and M. Santarsiero, "Calibration of a reversed-wavefront interferometer for polarization coherence metrology," Proc. SPIE 6672667207(2007).

Biss, D. P.

Borghi, R.

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, "Spatial coherence properties of azimuthally polarized laser modes," Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

D. P. Brown, A. K. Spilman, T. G. Brown, M. A. Alonso, R. Borghi, and M. Santarsiero, "Calibration of a reversed-wavefront interferometer for polarization coherence metrology," Proc. SPIE 6672667207(2007).

M. Santarsiero and R. Borghi, "Measuring spatial coherence by using a reversed-wavefront Young interferometer," Opt. Lett. 31, 861-863 (2006).
[CrossRef] [PubMed]

R. Borghi and M. Santarsiero, "Nonparaxial propagation of spirally polarized optical beams," J. Opt. Soc. Am. A 21, 2029-2037 (2004).
[CrossRef]

Brown, D. P.

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, "Spatial coherence properties of azimuthally polarized laser modes," Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

D. P. Brown, A. K. Spilman, T. G. Brown, M. A. Alonso, R. Borghi, and M. Santarsiero, "Calibration of a reversed-wavefront interferometer for polarization coherence metrology," Proc. SPIE 6672667207(2007).

Brown, T. G.

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, "Spatial coherence properties of azimuthally polarized laser modes," Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

A. K. Spilman and T. G. Brown, "Stress birefringent, space-variant wave plates for vortex illumination," Appl. Opt. 26, 61-66 (2007).
[CrossRef]

D. P. Brown, A. K. Spilman, T. G. Brown, M. A. Alonso, R. Borghi, and M. Santarsiero, "Calibration of a reversed-wavefront interferometer for polarization coherence metrology," Proc. SPIE 6672667207(2007).

D. P. Biss and T. G. Brown, "Polarization-vortex-driven second-harmonic generation," Opt. Lett. 28, 923-925 (2003).
[CrossRef] [PubMed]

Dorn, R.

S. Quabis, R. Dorn, and G. Leuchs, "Generation of a radially polarized doughnut mode of high quality," Appl. Phys. B 81, 597-600 (2005).

Friberg, A. T.

Gbur, G.

G. Gbur and G. A. Swartzlander, Jr., "Complete transverse representation of a correlation singularity of a partially coherent field," J. Opt. Soc. Am. B 25, 1422-1429 (2008).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, "‘Hidden’ singularities in partially coherent wavefields," J. Opt. A: Pure and Appl. Opt. 6, S239-S242 (2004).
[CrossRef]

Greene, P. L.

Hall, D. G.

Hopkins, H. H.

Q2. H. H. Hopkins, "On the diffraction theory of optical images," Proc. Roy. Soc. A 217, 408-432 (1953).
[CrossRef]

Q1. H. H. Hopkins, "The concept of partial coherence in optics," Proc. Roy. Soc. A 208, 263-277 (1951).
[CrossRef]

Jordan, R. H.

Leuchs, G.

S. Quabis, R. Dorn, and G. Leuchs, "Generation of a radially polarized doughnut mode of high quality," Appl. Phys. B 81, 597-600 (2005).

Maleev, I. D.

Marathay, A. S.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143,905 (2004).
[CrossRef]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation vortices in partially coherent light: theory," J. Opt. Soc. Am. B 21, 1895-1900 (2004).
[CrossRef]

Palacios, D. M.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143,905 (2004).
[CrossRef]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation vortices in partially coherent light: theory," J. Opt. Soc. Am. B 21, 1895-1900 (2004).
[CrossRef]

Quabis, S.

S. Quabis, R. Dorn, and G. Leuchs, "Generation of a radially polarized doughnut mode of high quality," Appl. Phys. B 81, 597-600 (2005).

Saastamoinen, T.

Saghafi, S.

Santarsiero, M.

D. P. Brown, A. K. Spilman, T. G. Brown, M. A. Alonso, R. Borghi, and M. Santarsiero, "Calibration of a reversed-wavefront interferometer for polarization coherence metrology," Proc. SPIE 6672667207(2007).

M. Santarsiero and R. Borghi, "Measuring spatial coherence by using a reversed-wavefront Young interferometer," Opt. Lett. 31, 861-863 (2006).
[CrossRef] [PubMed]

R. Borghi and M. Santarsiero, "Nonparaxial propagation of spirally polarized optical beams," J. Opt. Soc. Am. A 21, 2029-2037 (2004).
[CrossRef]

Set¨al¨a, T.

Sheppard, C. J. R.

Spilman, A. K.

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, "Spatial coherence properties of azimuthally polarized laser modes," Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

A. K. Spilman and T. G. Brown, "Stress birefringent, space-variant wave plates for vortex illumination," Appl. Opt. 26, 61-66 (2007).
[CrossRef]

D. P. Brown, A. K. Spilman, T. G. Brown, M. A. Alonso, R. Borghi, and M. Santarsiero, "Calibration of a reversed-wavefront interferometer for polarization coherence metrology," Proc. SPIE 6672667207(2007).

Swartzlander, G. A.

Takeda, M.

Tervo, J.

Turunen, J. T. J.

Visser, T. D.

G. Gbur, T. D. Visser, and E. Wolf, "‘Hidden’ singularities in partially coherent wavefields," J. Opt. A: Pure and Appl. Opt. 6, S239-S242 (2004).
[CrossRef]

Volkov, S. N.

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, "Spatial coherence properties of azimuthally polarized laser modes," Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

Wang, W.

Wolf, E.

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, "Spatial coherence properties of azimuthally polarized laser modes," Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

E. Wolf, "Coherence and polarization properties of electromagnetic laser modes," Opt. Commun. 265, 60-62 (2006).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, "‘Hidden’ singularities in partially coherent wavefields," J. Opt. A: Pure and Appl. Opt. 6, S239-S242 (2004).
[CrossRef]

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263 (2003).
[CrossRef]

Zernike, F.

F. Zernike, "The concept of degree of coherence and its application to optical problems," Physica 5, 785-795 (1938).
[CrossRef]

Appl. Opt. (1)

A. K. Spilman and T. G. Brown, "Stress birefringent, space-variant wave plates for vortex illumination," Appl. Opt. 26, 61-66 (2007).
[CrossRef]

Appl. Phys. B (1)

S. Quabis, R. Dorn, and G. Leuchs, "Generation of a radially polarized doughnut mode of high quality," Appl. Phys. B 81, 597-600 (2005).

J. Opt. A: Pure and Appl. Opt. (1)

G. Gbur, T. D. Visser, and E. Wolf, "‘Hidden’ singularities in partially coherent wavefields," J. Opt. A: Pure and Appl. Opt. 6, S239-S242 (2004).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (3)

Opt. Commun. (2)

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, "Spatial coherence properties of azimuthally polarized laser modes," Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

E. Wolf, "Coherence and polarization properties of electromagnetic laser modes," Opt. Commun. 265, 60-62 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Phys. Lett. A (1)

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143,905 (2004).
[CrossRef]

Physica (1)

F. Zernike, "The concept of degree of coherence and its application to optical problems," Physica 5, 785-795 (1938).
[CrossRef]

Proc. Roy. Soc. A (2)

Q1. H. H. Hopkins, "The concept of partial coherence in optics," Proc. Roy. Soc. A 208, 263-277 (1951).
[CrossRef]

Q2. H. H. Hopkins, "On the diffraction theory of optical images," Proc. Roy. Soc. A 217, 408-432 (1953).
[CrossRef]

Proc. SPIE (1)

D. P. Brown, A. K. Spilman, T. G. Brown, M. A. Alonso, R. Borghi, and M. Santarsiero, "Calibration of a reversed-wavefront interferometer for polarization coherence metrology," Proc. SPIE 6672667207(2007).

Other (6)

E. Wolf, "The influence of Young’s interference experiment on the development of statistical optics," Progress in Optics, E.Wolf, ed., (Elsevier Science, 2007) Vol. 50, Chap. 7.
[CrossRef]

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

H. H. Hopkins, "Image formation with coherent and partially coherent light," Photograph. Sci. Eng. 21 (1977).

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

K. S. Youngworth, "Inhomogeneous polarization in confocal microscopy," Ph.D. thesis, University of Rochester, Rochester, NY 14627 (2002).

D. P. Biss, "Focal field interactions from cylindrical vector beams," Ph.D. thesis, University of Rochester, Rochester, NY 14627 (2005).

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Figures (6)

Fig. 1.
Fig. 1.

(a) Illustration of the conjugate planes and coordinate systems used in this paper. (b) Partially correlated azimuthal vortex (PCAV) illumination system. A spatially-filtered, quasi-monochromatic laser beam is focused with a lens (focal length f 1) onto a ground glass diffuser creating an array of independent point sources. The light is then relayed to an image plane with lenses of focal length f 2 and f 3. A left-hand circular (LHC) polarizer, space-variant stress-birefringent glass window, right-hand circular (RHC) analyzer, and azimuthal analyzer create an azimuthal polarization vortex within the entire pupil. The condenser lens creates a superposition of azimuthal polarization vortices at the object plane.

Fig. 2.
Fig. 2.

Reversed-wavefront Young interferometer (RWYI) with converging illumination source. A non-polarizing 50/50 beamsplitter cube creates reversed-wavefront replica. Polarization analyzers can be changed to select the different polarization components of each beam. Optical shutters can block each beam to capture the irradiance from each pinhole separately. A Young’s double pinhole mask translates horizontally and samples each beam. Condensing and imaging optics then interfere the light from each pinhole and optically magnify the interference fringes onto a CCD camera.

Fig. 3.
Fig. 3.

Diagonal components of the correlation matrix for the illumination field at the object plane of a PCAV illumination system. The magnitude is shown on the left, and the phase is shown on the right. The symbols correspond to the experimental data, and the lines correspond to the theoretical predictions. The vertical polarization component, W yy , (red squares and magenta line) exhibits an anti-correlation property for values of Δ x larger than 100 microns, but the horizontal polarization component, W xx, (blue circles and cyan line) is partially coherent for all non-zero values of Δ x . Both components are fully coherent at Δ x = 0.

Fig. 4.
Fig. 4.

Partially correlated azimuthal illumination of 1951 USAF resolution target. NA=0.004. Top image is vertically analyzed; bottom image is horizontally analyzed. Plot on the right shows a single slice through horizontally separated features on each image as shown.

Fig. 5.
Fig. 5.

Partially correlated azimuthal illumination of 1951 USAF resolution target. NA=0.25. Top image is vertically analyzed; bottom image is horizontally analyzed. Plot on the right shows a single slice through horizontally separated features on each image as shown.

Fig. 6.
Fig. 6.

Example of use in metrology. The polarization analyzer is at the orientation shown for each image (-10° and 80° clockwise from a vertical orientation). The blue (solid) curve corresponds to a slice through the left image as shown, and the red (dashed) curve corresponds to a slice through the right image as shown.

Equations (8)

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

E i ( r ) = d 2 ρ j P ij ( r ρ ) A j ( ρ ) ,
W ij ( r 1 , r 2 ) E i * ( r 1 ) E j ( r 2 )
= d 2 ρ 1 d 2 ρ 2 k P ik * ( r 1 ρ 1 ) A k * ( ρ 1 ) P j ( r 2 ρ 2 ) A ( ρ 2 )
A k * ( ρ 1 ) A ( ρ 2 ) = S k ( ρ 1 ) δ k δ ( ρ 1 ρ 2 ) ,
W ij ( x 1 , y 1 , x 2 , y 2 ) = du dv k P ik * ( x 1 u , y 1 v ) P jk ( x 2 u , y 2 v ) S k ( u , v ) .
P ( x u , y v ) = [ ( y v ) x u x u y v ] exp { β [ ( x u ) 2 + ( y v ) 2 ] } .
W ( Δ x , Δ y ) = 1 2 [ 1 β Δ y 2 β Δ x Δ y β Δ x Δ y 1 β Δ x 2 ] exp { β 2 ( Δ x 2 + Δ y 2 ) } .
DoP ( r ) 1 4 Det W ( r , r ) [ Tr W ( r , r ) ] 2

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