Abstract

The modulation transfer function (MTF) is calculated for imaging with linearly, circularly and radially polarized light as well as for different numerical apertures and aperture shapes. Special detectors are only sensitive to one component of the electric energy density, e.g. the longitudinal component. For certain parameters this has advantages concerning the resolution when comparing to polarization insensitive detectors. It is also shown that in the latter case zeros of the MTF may appear which are purely due to polarization effects and which depend on the aperture angle. Finally some ideas are presented how to use these results for improving the resolution in lithography.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]

2006

2005

2004

2003

R. Dorn, S. Quabis, and G. Leuchs, "The focus of light - linear polarization breaks the rotational symmetry of the focal spot," J. Mod. Opt. 50, 1917-1926 (2003).

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Formation of linearly polarized light with axial symmetry by use of space-variant subwavelength gratings," Opt. Lett. 28, 510-512 (2003).
[CrossRef] [PubMed]

2002

2001

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light-theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B B72, 109-113 (2001).
[CrossRef]

2000

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

R. Oldenbourg and P. Török, "Point-spread functions of a polarizing microscope equipped with high-numerical-aperture lenses," Appl. Opt. 39, 6325-6331 (2000).
[CrossRef]

1996

J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, "Imaging and time-resolved Spectroscopy of single molecules at an interface," Science 272, 255-2586 (1996).
[CrossRef]

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]

1993

1992

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, and N. Streibl, "Form birefringence of surface relief gratings and its angular dependence," Opt. Commun. 89, 173-177 (1992).
[CrossRef]

1987

1986

1983

D. C. Flanders, "Submicrometer periodicity gratings as artificial anisotropic dielectrics," Appl. Phys. Lett. 42, 492-494 (1983).
[CrossRef]

1959

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. A 253, 358-379 (1959).
[CrossRef]

Biener, G.

Billy, L.

Bokor, N.

Braat, J. J. M.

Brus, L. E.

J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, "Imaging and time-resolved Spectroscopy of single molecules at an interface," Science 272, 255-2586 (1996).
[CrossRef]

Davidson, N.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, "The focus of light - linear polarization breaks the rotational symmetry of the focal spot," J. Mod. Opt. 50, 1917-1926 (2003).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light-theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B B72, 109-113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light-theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B B72, 109-113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Fainman, Y.

Flanders, D. C.

D. C. Flanders, "Submicrometer periodicity gratings as artificial anisotropic dielectrics," Appl. Phys. Lett. 42, 492-494 (1983).
[CrossRef]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light-theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B B72, 109-113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Gluch, E.

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, and N. Streibl, "Form birefringence of surface relief gratings and its angular dependence," Opt. Commun. 89, 173-177 (1992).
[CrossRef]

Haidner, H.

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, and N. Streibl, "Form birefringence of surface relief gratings and its angular dependence," Opt. Commun. 89, 173-177 (1992).
[CrossRef]

Harris, T. D.

J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, "Imaging and time-resolved Spectroscopy of single molecules at an interface," Science 272, 255-2586 (1996).
[CrossRef]

Hasman, E.

Kipfer, P.

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, and N. Streibl, "Form birefringence of surface relief gratings and its angular dependence," Opt. Commun. 89, 173-177 (1992).
[CrossRef]

Kleiner, V.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, "The focus of light - linear polarization breaks the rotational symmetry of the focal spot," J. Mod. Opt. 50, 1917-1926 (2003).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light-theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B B72, 109-113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Levy, U.

Macklin, J. J.

J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, "Imaging and time-resolved Spectroscopy of single molecules at an interface," Science 272, 255-2586 (1996).
[CrossRef]

Mansuripur, M.

Matthews, H. J.

Munro, P. R. T.

Niv, A.

Oldenbourg, R.

Pang, L.

Pereira, S. F.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, "The focus of light - linear polarization breaks the rotational symmetry of the focal spot," J. Mod. Opt. 50, 1917-1926 (2003).

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light-theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B B72, 109-113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[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. A 253, 358-379 (1959).
[CrossRef]

Schadt, M.

Sheppard, C. J. R.

Sheridan, J. T.

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, and N. Streibl, "Form birefringence of surface relief gratings and its angular dependence," Opt. Commun. 89, 173-177 (1992).
[CrossRef]

Stalder, M.

Streibl, N.

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, and N. Streibl, "Form birefringence of surface relief gratings and its angular dependence," Opt. Commun. 89, 173-177 (1992).
[CrossRef]

Török, P.

Trautman, J. K.

J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, "Imaging and time-resolved Spectroscopy of single molecules at an interface," Science 272, 255-2586 (1996).
[CrossRef]

Tsai, C.

van de Nes, A. S.

S. F. Pereira and A. S. van de Nes, "Superresolution by means of polarization, phase and amplitude pupil masks," Opt. Commun. 234, 119-124 (2004).
[CrossRef]

van de Nes, A.S.

Wolf, E.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. A 253, 358-379 (1959).
[CrossRef]

Appl. Opt.

Appl. Phys. B

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light-theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B B72, 109-113 (2001).
[CrossRef]

Appl. Phys. Lett.

D. C. Flanders, "Submicrometer periodicity gratings as artificial anisotropic dielectrics," Appl. Phys. Lett. 42, 492-494 (1983).
[CrossRef]

J. Mod. Opt.

R. Dorn, S. Quabis, and G. Leuchs, "The focus of light - linear polarization breaks the rotational symmetry of the focal spot," J. Mod. Opt. 50, 1917-1926 (2003).

J. Opt. Soc. Am. A

Opt. Commun.

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, and N. Streibl, "Form birefringence of surface relief gratings and its angular dependence," Opt. Commun. 89, 173-177 (1992).
[CrossRef]

S. F. Pereira and A. S. van de Nes, "Superresolution by means of polarization, phase and amplitude pupil masks," Opt. Commun. 234, 119-124 (2004).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Proc. R. Soc. A

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. A 253, 358-379 (1959).
[CrossRef]

Science

J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, "Imaging and time-resolved Spectroscopy of single molecules at an interface," Science 272, 255-2586 (1996).
[CrossRef]

Other

K. Kamon, "Projection exposure apparatus," United States Patent 5365371 (filed 1993).

K.-H. Schuster, "Radial polarisationsdrehende optische Anordnung und Mikrolithographie-Projektionsbelichtungsanlage damit," European Patent 0 764 858 A2 (filed 1996) and K.-H. Schuster, "Radial polarization-rotating optical arrangement and microlithographic projection exposure system," United States Patent 6885502 (filed 2002).

M. Born and E. Wolf, Principles of Optics, 6th Edition. (Cambridge University Press, Cambridge New York Oakleigh, 1997).

J. W. Goodman, Introduction to Fourier optics, 2nd. Edition (McGraw--Hill, New York, 1996).

E. Hasman, G. Biener, A. Niv, and V. Kleiner, "Space-variant polarization manipulation," Progress in Optics, E. Wolf, ed., (Elsevier Amsterdam 2005) Vol. 47, 215-289 .
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

MTF curves for a full circular aperture and sinφ ranging from 0.2 to 1.0 for different polarization states (linear, circular and radial).

Fig. 2.
Fig. 2.

MTF curves for an annular aperture (inner radius is 90% of the outer radius) and sinφ ranging from 0.2 to 1.0 for different polarization states (linear, circular and radial).

Fig. 3.
Fig. 3.

MTF curves for the case (i) of section 3, i.e. radial polarization and different annular apertures.

Fig. 4.
Fig. 4.

MTF curves for the case (iia) of section 3, i.e. radial polarization and different apodization of the aperture with the parameter n ranging from n=0 to n=9. The respective amplitude function |E(r)| is displayed in the figure captions.

Fig. 5.
Fig. 5.

MTF curves for the case (iib) of section 3, i.e. radial polarization and different apodization of the aperture with the parameter n ranging from n=1 to n=9. The respective amplitude function |E(r)| is displayed in the figure captions.

Fig. 6.
Fig. 6.

Comparison of some MTF curves using either a “hard mask” apodization via an annular aperture (green curve for r in/r aperture=0.6) or a smooth apodization via different amplitude functions (red and black curves with the amplitude function |E| in the figure caption).

Fig. 7.
Fig. 7.

Optical system for the imaging of an extended object which consists of small particles which emit like a dipole with its axis parallel to the optical axis.

Fig. 8.
Fig. 8.

Optical system for optical lithography at high aperture angles. The mask emits nearly linearly polarized light which is collimated by the first lens. The polarization converting element forms radially polarized light from the linearly polarized plane wave coming from one of the object points. The high NA objective behind the polarization converter focuses the light to a tight spot.

Fig. 9.
Fig. 9.

Scheme of a lens fulfilling the sine condition.

Equations (11)

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MTF ( ν x , ν y ) = ʃ + ʃ + PSF ( x , y ) exp ( 2 πi ( ν x x + ν y y ) ) dx dy + + PSF ( x , y ) dx dy
ν cut = 2 NA λ = 2 n sin φ λ
E ˉ ( x , y ) = E 0 ( x y 0 ) exp ( x 2 + y 2 w 0 2 )
E ˉ ( x , y ) { E 0 x 2 + y 2 ( x y 0 ) 0 otherwise for r in x 2 + y 2 r aperture
E ˉ ( x , y ) = E′ 0 ( x 2 + y 2 ) n 1 2 ( x y 0 ) exp ( ( x 2 + y 2 ) w 0 2 ) E ˉ ( r ) = E′ 0 r n exp ( r 2 w 0 2 ) and r = x 2 + y 2
d E ˉ ( r ) dr = ( nr n 1 2 r n + 1 w 0 2 ) exp ( r 2 w 0 2 ) = 0 w 0 r = r aperture = 2 n r aperture
NA image = 1 β NA Obj 1 ,
h = f sin ϑ
P′ j = g ( ϑ ) P j
g ( ϑ ) = 1 cos ϑ
E total ( r ) = α j P′ j exp ( i 2 π λ e′ j r + i 2 π λ OP D j )

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