Abstract

The forward problem of focusing light using a high numerical aperture lens can be described using the Debye-Wolf integral, however a solution to the inverse problem does not currently exist. In this work an inversion formula based on an eigenfunction representation is derived and presented which allows a field distribution in a plane in the focal region to be specified and the appropriate pupil plane distribution to be calculated. Various additional considerations constrain the inversion to ensure physicality and practicality of the results and these are also discussed. A number of inversion examples are given.

© 2008 Optical Society of America

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2005

S.-S. Yu, B. J. Lin, A. Yen, C.-M. Ke, J. Huang, B.-C. Ho, C.-K. Chen, T.-S. Gau, H.-C. Hsieh, and Y.-C. Ku, "Thin-film optimization strategy in high numerical aperture optical lithography I - Principles," J. Microlith. Microfab. Microsyst. 4, 043003 (2005).
[CrossRef]

K. C. ToussaintJr., S. Park, J. E. Jureller, and N. F. Scherer, "Generation of optical vector beams with a diffractive optical element interferometer," Opt. Lett 30, 2846-2848 (2005).
[CrossRef]

2004

S. S. Sherif and P. Török, "Pupil plane masks for super-resolution in high numerical aperture focussing," J. Mod. Opt. 51, 2007-2019 (2004).

2002

2000

K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical vector beams," Opt. Express 7, 77-87 (2000).
[CrossRef] [PubMed]

M. A. A. Neil, T. Wilson and R. Juškaitis, "A wavefront generator for complex pupil function synthesis and point spread function engineering," J. Microsc. 197, 219-223 (2000).
[CrossRef] [PubMed]

B. Sick, B. Hecht, and L. Novotny "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

1999

U. Brand, G. Hester, J. Grochmalicki, and R. Pike "Super-resolution in optical data storage," J. Opt. A: Pure Appl. Opt. 1, 794-800 (1999).
[CrossRef]

1998

M. A. Neifeld, "Information, resolution, and space bandwidth product," Opt. Lett. 18, 1477-1479 (1998).
[CrossRef]

P. Török, P. D. Higdon, and T. Wilson, "On the general properties of polarised light conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
[CrossRef]

1996

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, "Single molecule dynamics studied by polarization modulation," Phys. Rev. Lett. 77, 3979-3982 (1996).
[CrossRef] [PubMed]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky and C. Ferreira, "Space bandwidth product of optical signals and systems," J. Opt. Soc. Am. A 13, 470-473 (1996).
[CrossRef]

1995

1990

1987

1986

1985

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Spatial filter for increasing the depth of focus," Opt. Lett 10, 520-522 (1985).
[CrossRef] [PubMed]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Spatial filter for increasing the depth of focus," Opt. Lett. 10, 520-523 (1985).
[CrossRef] [PubMed]

1983

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by Simulated Annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

D. Slepian, "Some comments on Fourier analysis, uncertainty and modeling," SIAM Review 25, 379-393 (1983).
[CrossRef]

1978

W. H. Lee, "Computer-generated holograms: techniques and applications," Prog. Opt. 16, 119232 (1978).

1972

Y. Mushiake, K. Matsumura, and N. Nakajima, "Generation of radially polarized optical beam mode by laser oscillation," Proc. IEEE. 60, 1107-1109 (1972).
[CrossRef]

1971

B. R. Frieden "Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions," Prog. Opt. 9, 311-407 (1971).
[CrossRef]

1966

B. Karczewski and E. Wolf, "Comparison of three theoreis of electromagnetic difftraction at an aperture Part I: coherence matrices, Part II: The far field," J. Opt. Soc. Am 56, 1207-19 (1966).
[CrossRef]

D. J. Innes and A. L. Bloom, "Design of optical systems for use with laser beams," Spectra-Phys. Laser Tech. Bull. 5, 1-10 (1966).

1964

C. W. McCutcheon,"Generalised Aperture and the Three-Dimensional Diffraction Image," J. Opt. Soc. Am. 54, 240-244 (1964).
[CrossRef]

D. Slepian "Prolate spheroidal wave functions, Fourier analysis and uncertainty IV Extensions to many dimensions; Generalised prolate spheroidal functions," Bell Syst. Tech. J. 43, 3009-3057 (1964).

1962

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty III The dimension of the space of essentially time- and band-limited signals," Bell Syst. Tech. J. 41, 1295-1336 (1962).

1961

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty I," Bell Syst. Tech. J. 40, 43-64 (1961).

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty II," Bell Syst. Tech. J. 40, 65-84 (1961).

1960

1959

E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. R. Soc. London, Ser. A 253, 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. London, A 253, 358-379 (1959).
[CrossRef]

1952

T. di Francia, "Super-gain antennas and optical resolving power," Nuovo Cimento, Suppl. 9, 426-435 (1952).
[CrossRef]

T. di Francia, "Super-gain antennas and, optical resolving power," Nuovo Cimento 9, 426-438 (1952).
[CrossRef]

1948

D. Gabor, "A new microscopic principle," Nature (London) 161, 777-778 (1948).
[CrossRef]

Berriel-Valdos, L. R.

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Spatial filter for increasing the depth of focus," Opt. Lett. 10, 520-523 (1985).
[CrossRef] [PubMed]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Spatial filter for increasing the depth of focus," Opt. Lett 10, 520-522 (1985).
[CrossRef] [PubMed]

Bhattacharya, K.

D. R. Chowdhury, K. Bhattacharya, S. Sanyal, and A. K. Chakraborty, "Performance of a polarization-masked lens aperture in the presence of spherical aberration," J. Opt. A: Pure Appl. Opt. 4, 98-104 (2002).
[CrossRef]

Biener, G.

Bloom, A. L.

D. J. Innes and A. L. Bloom, "Design of optical systems for use with laser beams," Spectra-Phys. Laser Tech. Bull. 5, 1-10 (1966).

Bomzon, Z.

Brand, U.

U. Brand, G. Hester, J. Grochmalicki, and R. Pike "Super-resolution in optical data storage," J. Opt. A: Pure Appl. Opt. 1, 794-800 (1999).
[CrossRef]

Brown, T. G.

Cathey, W. T.

Chakraborty, A. K.

D. R. Chowdhury, K. Bhattacharya, S. Sanyal, and A. K. Chakraborty, "Performance of a polarization-masked lens aperture in the presence of spherical aberration," J. Opt. A: Pure Appl. Opt. 4, 98-104 (2002).
[CrossRef]

Chemla, D. S.

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, "Single molecule dynamics studied by polarization modulation," Phys. Rev. Lett. 77, 3979-3982 (1996).
[CrossRef] [PubMed]

Chen, C.-K.

S.-S. Yu, B. J. Lin, A. Yen, C.-M. Ke, J. Huang, B.-C. Ho, C.-K. Chen, T.-S. Gau, H.-C. Hsieh, and Y.-C. Ku, "Thin-film optimization strategy in high numerical aperture optical lithography I - Principles," J. Microlith. Microfab. Microsyst. 4, 043003 (2005).
[CrossRef]

Chowdhury, D. R.

D. R. Chowdhury, K. Bhattacharya, S. Sanyal, and A. K. Chakraborty, "Performance of a polarization-masked lens aperture in the presence of spherical aberration," J. Opt. A: Pure Appl. Opt. 4, 98-104 (2002).
[CrossRef]

di Francia, T.

T. di Francia, "Super-gain antennas and, optical resolving power," Nuovo Cimento 9, 426-438 (1952).
[CrossRef]

T. di Francia, "Super-gain antennas and optical resolving power," Nuovo Cimento, Suppl. 9, 426-435 (1952).
[CrossRef]

Dorsch, R. G.

Dowski, E. R.

Enderle, T.

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, "Single molecule dynamics studied by polarization modulation," Phys. Rev. Lett. 77, 3979-3982 (1996).
[CrossRef] [PubMed]

Ferreira, C.

Ford, D. H.

Foreman, M. R.

S. S. Sherif, M. R. Foreman, and P. Török, "Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system," Opt. Express (to be published).
[PubMed]

Frieden, B. R.

B. R. Frieden "Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions," Prog. Opt. 9, 311-407 (1971).
[CrossRef]

Gabor, D.

D. Gabor, "A new microscopic principle," Nature (London) 161, 777-778 (1948).
[CrossRef]

Gau, T.-S.

S.-S. Yu, B. J. Lin, A. Yen, C.-M. Ke, J. Huang, B.-C. Ho, C.-K. Chen, T.-S. Gau, H.-C. Hsieh, and Y.-C. Ku, "Thin-film optimization strategy in high numerical aperture optical lithography I - Principles," J. Microlith. Microfab. Microsyst. 4, 043003 (2005).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by Simulated Annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Grochmalicki, J.

U. Brand, G. Hester, J. Grochmalicki, and R. Pike "Super-resolution in optical data storage," J. Opt. A: Pure Appl. Opt. 1, 794-800 (1999).
[CrossRef]

Ha, T.

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, "Single molecule dynamics studied by polarization modulation," Phys. Rev. Lett. 77, 3979-3982 (1996).
[CrossRef] [PubMed]

Hasman, E.

Hecht, B.

B. Sick, B. Hecht, and L. Novotny "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Hegedus, Z. S.

Hester, G.

U. Brand, G. Hester, J. Grochmalicki, and R. Pike "Super-resolution in optical data storage," J. Opt. A: Pure Appl. Opt. 1, 794-800 (1999).
[CrossRef]

Higdon, P. D.

P. Török, P. D. Higdon, and T. Wilson, "On the general properties of polarised light conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
[CrossRef]

Ho, B.-C.

S.-S. Yu, B. J. Lin, A. Yen, C.-M. Ke, J. Huang, B.-C. Ho, C.-K. Chen, T.-S. Gau, H.-C. Hsieh, and Y.-C. Ku, "Thin-film optimization strategy in high numerical aperture optical lithography I - Principles," J. Microlith. Microfab. Microsyst. 4, 043003 (2005).
[CrossRef]

Hsieh, H.-C.

S.-S. Yu, B. J. Lin, A. Yen, C.-M. Ke, J. Huang, B.-C. Ho, C.-K. Chen, T.-S. Gau, H.-C. Hsieh, and Y.-C. Ku, "Thin-film optimization strategy in high numerical aperture optical lithography I - Principles," J. Microlith. Microfab. Microsyst. 4, 043003 (2005).
[CrossRef]

Huang, J.

S.-S. Yu, B. J. Lin, A. Yen, C.-M. Ke, J. Huang, B.-C. Ho, C.-K. Chen, T.-S. Gau, H.-C. Hsieh, and Y.-C. Ku, "Thin-film optimization strategy in high numerical aperture optical lithography I - Principles," J. Microlith. Microfab. Microsyst. 4, 043003 (2005).
[CrossRef]

Innes, D. J.

D. J. Innes and A. L. Bloom, "Design of optical systems for use with laser beams," Spectra-Phys. Laser Tech. Bull. 5, 1-10 (1966).

Jureller, J. E.

K. C. ToussaintJr., S. Park, J. E. Jureller, and N. F. Scherer, "Generation of optical vector beams with a diffractive optical element interferometer," Opt. Lett 30, 2846-2848 (2005).
[CrossRef]

Juškaitis, R.

M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson "Method for the generation of arbitrary complex vector wave fronts," Opt. Lett. 27, 1929-1931 (2002).
[CrossRef]

M. A. A. Neil, T. Wilson and R. Juškaitis, "A wavefront generator for complex pupil function synthesis and point spread function engineering," J. Microsc. 197, 219-223 (2000).
[CrossRef] [PubMed]

Karczewski, B.

B. Karczewski and E. Wolf, "Comparison of three theoreis of electromagnetic difftraction at an aperture Part I: coherence matrices, Part II: The far field," J. Opt. Soc. Am 56, 1207-19 (1966).
[CrossRef]

Ke, C.-M.

S.-S. Yu, B. J. Lin, A. Yen, C.-M. Ke, J. Huang, B.-C. Ho, C.-K. Chen, T.-S. Gau, H.-C. Hsieh, and Y.-C. Ku, "Thin-film optimization strategy in high numerical aperture optical lithography I - Principles," J. Microlith. Microfab. Microsyst. 4, 043003 (2005).
[CrossRef]

Kimura, W. D.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by Simulated Annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Kleiner, V.

Ku, Y.-C.

S.-S. Yu, B. J. Lin, A. Yen, C.-M. Ke, J. Huang, B.-C. Ho, C.-K. Chen, T.-S. Gau, H.-C. Hsieh, and Y.-C. Ku, "Thin-film optimization strategy in high numerical aperture optical lithography I - Principles," J. Microlith. Microfab. Microsyst. 4, 043003 (2005).
[CrossRef]

Landau, H. J.

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty III The dimension of the space of essentially time- and band-limited signals," Bell Syst. Tech. J. 41, 1295-1336 (1962).

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty II," Bell Syst. Tech. J. 40, 65-84 (1961).

Lee, W. H.

W. H. Lee, "Computer-generated holograms: techniques and applications," Prog. Opt. 16, 119232 (1978).

Lin, B. J.

S.-S. Yu, B. J. Lin, A. Yen, C.-M. Ke, J. Huang, B.-C. Ho, C.-K. Chen, T.-S. Gau, H.-C. Hsieh, and Y.-C. Ku, "Thin-film optimization strategy in high numerical aperture optical lithography I - Principles," J. Microlith. Microfab. Microsyst. 4, 043003 (2005).
[CrossRef]

Lohmann, A. W.

Massoumian, F.

Matsumura, K.

Y. Mushiake, K. Matsumura, and N. Nakajima, "Generation of radially polarized optical beam mode by laser oscillation," Proc. IEEE. 60, 1107-1109 (1972).
[CrossRef]

McCutcheon, C. W.

Mendlovic, D.

Montes, E.

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Spatial filter for increasing the depth of focus," Opt. Lett 10, 520-522 (1985).
[CrossRef] [PubMed]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Spatial filter for increasing the depth of focus," Opt. Lett. 10, 520-523 (1985).
[CrossRef] [PubMed]

Motamedi, M.

Mushiake, Y.

Y. Mushiake, K. Matsumura, and N. Nakajima, "Generation of radially polarized optical beam mode by laser oscillation," Proc. IEEE. 60, 1107-1109 (1972).
[CrossRef]

Nakajima, N.

Y. Mushiake, K. Matsumura, and N. Nakajima, "Generation of radially polarized optical beam mode by laser oscillation," Proc. IEEE. 60, 1107-1109 (1972).
[CrossRef]

Neifeld, M. A.

M. A. Neifeld, "Information, resolution, and space bandwidth product," Opt. Lett. 18, 1477-1479 (1998).
[CrossRef]

Neil, M. A. A.

M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson "Method for the generation of arbitrary complex vector wave fronts," Opt. Lett. 27, 1929-1931 (2002).
[CrossRef]

M. A. A. Neil, T. Wilson and R. Juškaitis, "A wavefront generator for complex pupil function synthesis and point spread function engineering," J. Microsc. 197, 219-223 (2000).
[CrossRef] [PubMed]

Novotny, L.

B. Sick, B. Hecht, and L. Novotny "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Ojeda-Castañeda, J.

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Spatial filter for increasing the depth of focus," Opt. Lett 10, 520-522 (1985).
[CrossRef] [PubMed]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Spatial filter for increasing the depth of focus," Opt. Lett. 10, 520-523 (1985).
[CrossRef] [PubMed]

Park, S.

K. C. ToussaintJr., S. Park, J. E. Jureller, and N. F. Scherer, "Generation of optical vector beams with a diffractive optical element interferometer," Opt. Lett 30, 2846-2848 (2005).
[CrossRef]

Pike, R.

U. Brand, G. Hester, J. Grochmalicki, and R. Pike "Super-resolution in optical data storage," J. Opt. A: Pure Appl. Opt. 1, 794-800 (1999).
[CrossRef]

Pollak, H. O.

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty III The dimension of the space of essentially time- and band-limited signals," Bell Syst. Tech. J. 41, 1295-1336 (1962).

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty I," Bell Syst. Tech. J. 40, 43-64 (1961).

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty II," Bell Syst. Tech. J. 40, 65-84 (1961).

Poon, T. C.

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. London, A 253, 358-379 (1959).
[CrossRef]

Sanyal, S.

D. R. Chowdhury, K. Bhattacharya, S. Sanyal, and A. K. Chakraborty, "Performance of a polarization-masked lens aperture in the presence of spherical aberration," J. Opt. A: Pure Appl. Opt. 4, 98-104 (2002).
[CrossRef]

Sarafis, V.

Scherer, N. F.

K. C. ToussaintJr., S. Park, J. E. Jureller, and N. F. Scherer, "Generation of optical vector beams with a diffractive optical element interferometer," Opt. Lett 30, 2846-2848 (2005).
[CrossRef]

Selvin, P. R.

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, "Single molecule dynamics studied by polarization modulation," Phys. Rev. Lett. 77, 3979-3982 (1996).
[CrossRef] [PubMed]

Sherif, S. S.

S. S. Sherif and P. Török, "Pupil plane masks for super-resolution in high numerical aperture focussing," J. Mod. Opt. 51, 2007-2019 (2004).

S. S. Sherif, M. R. Foreman, and P. Török, "Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system," Opt. Express (to be published).
[PubMed]

Sick, B.

B. Sick, B. Hecht, and L. Novotny "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Slepian, D.

D. Slepian, "Some comments on Fourier analysis, uncertainty and modeling," SIAM Review 25, 379-393 (1983).
[CrossRef]

D. Slepian "Prolate spheroidal wave functions, Fourier analysis and uncertainty IV Extensions to many dimensions; Generalised prolate spheroidal functions," Bell Syst. Tech. J. 43, 3009-3057 (1964).

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty I," Bell Syst. Tech. J. 40, 43-64 (1961).

Tidwell, S. C.

Török, P.

S. S. Sherif and P. Török, "Pupil plane masks for super-resolution in high numerical aperture focussing," J. Mod. Opt. 51, 2007-2019 (2004).

P. Török, P. D. Higdon, and T. Wilson, "On the general properties of polarised light conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
[CrossRef]

S. S. Sherif, M. R. Foreman, and P. Török, "Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system," Opt. Express (to be published).
[PubMed]

Toussaint, K. C.

K. C. ToussaintJr., S. Park, J. E. Jureller, and N. F. Scherer, "Generation of optical vector beams with a diffractive optical element interferometer," Opt. Lett 30, 2846-2848 (2005).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by Simulated Annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Weiss, S.

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, "Single molecule dynamics studied by polarization modulation," Phys. Rev. Lett. 77, 3979-3982 (1996).
[CrossRef] [PubMed]

Welford, W. T.

Wilson, T.

M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson "Method for the generation of arbitrary complex vector wave fronts," Opt. Lett. 27, 1929-1931 (2002).
[CrossRef]

M. A. A. Neil, T. Wilson and R. Juškaitis, "A wavefront generator for complex pupil function synthesis and point spread function engineering," J. Microsc. 197, 219-223 (2000).
[CrossRef] [PubMed]

P. Török, P. D. Higdon, and T. Wilson, "On the general properties of polarised light conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
[CrossRef]

Wolf, E.

B. Karczewski and E. Wolf, "Comparison of three theoreis of electromagnetic difftraction at an aperture Part I: coherence matrices, Part II: The far field," J. Opt. Soc. Am 56, 1207-19 (1966).
[CrossRef]

E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. R. Soc. London, Ser. A 253, 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. London, A 253, 358-379 (1959).
[CrossRef]

Yen, A.

S.-S. Yu, B. J. Lin, A. Yen, C.-M. Ke, J. Huang, B.-C. Ho, C.-K. Chen, T.-S. Gau, H.-C. Hsieh, and Y.-C. Ku, "Thin-film optimization strategy in high numerical aperture optical lithography I - Principles," J. Microlith. Microfab. Microsyst. 4, 043003 (2005).
[CrossRef]

Youngworth, K. S.

Yu, S.-S.

S.-S. Yu, B. J. Lin, A. Yen, C.-M. Ke, J. Huang, B.-C. Ho, C.-K. Chen, T.-S. Gau, H.-C. Hsieh, and Y.-C. Ku, "Thin-film optimization strategy in high numerical aperture optical lithography I - Principles," J. Microlith. Microfab. Microsyst. 4, 043003 (2005).
[CrossRef]

Zalevsky, Z.

A

E. Wolf, "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. R. Soc. London, Ser. A 253, 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. London, A 253, 358-379 (1959).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

D. Slepian "Prolate spheroidal wave functions, Fourier analysis and uncertainty IV Extensions to many dimensions; Generalised prolate spheroidal functions," Bell Syst. Tech. J. 43, 3009-3057 (1964).

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty I," Bell Syst. Tech. J. 40, 43-64 (1961).

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty II," Bell Syst. Tech. J. 40, 65-84 (1961).

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty III The dimension of the space of essentially time- and band-limited signals," Bell Syst. Tech. J. 41, 1295-1336 (1962).

J. Microlith. Microfab. Microsyst.

S.-S. Yu, B. J. Lin, A. Yen, C.-M. Ke, J. Huang, B.-C. Ho, C.-K. Chen, T.-S. Gau, H.-C. Hsieh, and Y.-C. Ku, "Thin-film optimization strategy in high numerical aperture optical lithography I - Principles," J. Microlith. Microfab. Microsyst. 4, 043003 (2005).
[CrossRef]

J. Microsc.

M. A. A. Neil, T. Wilson and R. Juškaitis, "A wavefront generator for complex pupil function synthesis and point spread function engineering," J. Microsc. 197, 219-223 (2000).
[CrossRef] [PubMed]

J. Mod. Opt.

S. S. Sherif and P. Török, "Pupil plane masks for super-resolution in high numerical aperture focussing," J. Mod. Opt. 51, 2007-2019 (2004).

J. Opt. A: Pure Appl. Opt.

D. R. Chowdhury, K. Bhattacharya, S. Sanyal, and A. K. Chakraborty, "Performance of a polarization-masked lens aperture in the presence of spherical aberration," J. Opt. A: Pure Appl. Opt. 4, 98-104 (2002).
[CrossRef]

U. Brand, G. Hester, J. Grochmalicki, and R. Pike "Super-resolution in optical data storage," J. Opt. A: Pure Appl. Opt. 1, 794-800 (1999).
[CrossRef]

J. Opt. Soc. Am

B. Karczewski and E. Wolf, "Comparison of three theoreis of electromagnetic difftraction at an aperture Part I: coherence matrices, Part II: The far field," J. Opt. Soc. Am 56, 1207-19 (1966).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature (London)

D. Gabor, "A new microscopic principle," Nature (London) 161, 777-778 (1948).
[CrossRef]

Nuovo Cimento

T. di Francia, "Super-gain antennas and, optical resolving power," Nuovo Cimento 9, 426-438 (1952).
[CrossRef]

Nuovo Cimento,

T. di Francia, "Super-gain antennas and optical resolving power," Nuovo Cimento, Suppl. 9, 426-435 (1952).
[CrossRef]

Opt. Commun.

P. Török, P. D. Higdon, and T. Wilson, "On the general properties of polarised light conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
[CrossRef]

Opt. Express

S. S. Sherif, M. R. Foreman, and P. Török, "Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system," Opt. Express (to be published).
[PubMed]

K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical vector beams," Opt. Express 7, 77-87 (2000).
[CrossRef] [PubMed]

Opt. Lett

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Spatial filter for increasing the depth of focus," Opt. Lett 10, 520-522 (1985).
[CrossRef] [PubMed]

K. C. ToussaintJr., S. Park, J. E. Jureller, and N. F. Scherer, "Generation of optical vector beams with a diffractive optical element interferometer," Opt. Lett 30, 2846-2848 (2005).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, "Single molecule dynamics studied by polarization modulation," Phys. Rev. Lett. 77, 3979-3982 (1996).
[CrossRef] [PubMed]

B. Sick, B. Hecht, and L. Novotny "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Proc. IEEE.

Y. Mushiake, K. Matsumura, and N. Nakajima, "Generation of radially polarized optical beam mode by laser oscillation," Proc. IEEE. 60, 1107-1109 (1972).
[CrossRef]

Prog. Opt.

B. R. Frieden "Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions," Prog. Opt. 9, 311-407 (1971).
[CrossRef]

W. H. Lee, "Computer-generated holograms: techniques and applications," Prog. Opt. 16, 119232 (1978).

Science

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by Simulated Annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

SIAM Review

D. Slepian, "Some comments on Fourier analysis, uncertainty and modeling," SIAM Review 25, 379-393 (1983).
[CrossRef]

Spectra-Phys. Laser Tech. Bull.

D. J. Innes and A. L. Bloom, "Design of optical systems for use with laser beams," Spectra-Phys. Laser Tech. Bull. 5, 1-10 (1966).

Other

S. S. Sherif and W. T. Cathey, "Depth of field control in incoherent hybrid imaging systems," in Optical Imaging and Microscopy - Techniques and Advanced Systems, P. Török and F.-J. Kao, eds., (Springer, New York 2007).

R. Pike, D. Chana, P. Neocleous, and S. Jiang, "Superresolution in scanning optical systems," in Optical Imaging and Microscopy - Techniques and Advanced Systems, P. Török and F.-J. Kao, eds., (Springer, New York 2007).

T. Zolezzi "Well-posedness criteria in optimization with application to the calculus of variations," in Nonlinear Analysis: Theory, Methods and Applications, 25, 437-453 (1995).

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, (SIAM, Philadelphia, PA. 1997).

M. Endo, "Pattern formation method and exposure system," Patent No. 7094521 (2006).

A. Rohrbach, J. Huisken, and E. H. K. Stelzer, "Optical trapping of small particles," in Optical Imaging and Microscopy - Techniques and Advanced Systems, P. Török and F.-J. Kao eds., (Springer, New York 2007).

S. Inoué, "Exploring living cells and molecular dynamics with polarized light microscopy," in Optical Imaging and Microscopy - Techniques and Advanced Systems, P. Török and F.-J. Kao, eds., (Springer, New York 2007).

J. C. Heurtley, "Hyperspheroidal functions - optical resonators with circular mirrors," Proc. Symp. on Quasi Optics, New York p. 367 (1964).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed., (McGraw-Hill 1996).

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Figures (8)

Fig. 1.
Fig. 1.

Circular prolate spheroidal eigenvalues for different orders (N and n) and space bandwidth products c.

Fig. 2.
Fig. 2.

Circular prolate spheroidal functions for different orders n for N=0 and for different space bandwidth products c. Prolate functions plotted have been normalised so that Φ0,0(0)=1. Note the different vertical scales between plots.

Fig. 3.
Fig. 3.

Coordinate system and geometry of Debye-Wolf diffraction integral.

Fig. 4.
Fig. 4.

When trying to produce a structure smaller than the diffraction limit as shown in (a) significant energy is pushed outside the specification area (b) as bounded by the dashed line. The solid line shows the size of the Airy disc. This can be understood from the high order contributions to the specified field as shown in (c) which plots ln(|A N,n |2λ N,n ) to allow comparison between all modes

Fig. 5.
Fig. 5.

Polarisation structure of illuminating beam required to give zero longitudinal field component in the focus of a high aperture lens as found by inversion (NA=0.966, c=20). Arrows indicate plane of polarisation of field at each point.

Fig. 6.
Fig. 6.

(a) Comparison between the desired axial field profile and that found from a simulated annealing optimisation algorithm. (b) Apodisation mask required to produce the optimised distribution (NA=0.966). (c) Resulting intensity distribution on the focal plane showing sidelobe pattern.

Fig. 7.
Fig. 7.

(a) Colour plot showing the transmittance of the apodising mask in the pupil plane, whilst white lines represent the plane of oscillation of the electric field vector. (b) Variation of the Strehl intensity ratio and the encircled energy for the Ex and Ey components as the mask order n 0 is increased.

Fig. 8.
Fig. 8.

Optical distribution in the focal plane for an apodised and polarised structured beam with truncation point n 0=1 (top) for NA=0.966 and c=4. White circles again denote the extent of the Airy disc. Variation of the central intensity focal spot over the Airy disc as mask order n 0 is increased (bottom). Note the intensity scales differ with each plot, but have been equalised for easy comparison.

Equations (39)

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

ψ N , n ( c , r , θ ) = Φ N , n ( c , r ) cos N θ sin N θ N = 0 , 1 , 2 , . . . , n = 0 , 1 , 2 , . . .
0 r 0 J N ( ω r ) Φ N , n ( c , r ) rdr = ( 1 ) n ( r 0 Ω ) λ N , n 1 2 Φ N , n ( c , ω r 0 Ω )
Φ N , n ( c , r ) = ( λ N , n rr 0 ) 1 2 φ N , n ( c , r r 0 )
0 r 0 Φ N , n ( c , r ) Φ N , m ( c , r ) rdr = λ N , n δ nm
n = 0 λ N , n 1 Φ N , n ( c , r ) Φ N , n ( c , r ) = δ ( r r ) r for 0 r , r r 0
f ( r , ϕ ) = n = n = 0 A N , n Φ N , n ( c , r ) exp ( i N ϕ )
A N , n = 1 2 π λ N , n 0 2 π 0 r 0 f ( r , ϕ ) Φ N , n ( c , r ) exp ( i N ϕ ) rdrd ϕ .
I enc = 0 2 π 0 r 0 f ( r , ϕ ) 2 rdrd ϕ 0 2 π 0 f ( r , ϕ ) 2 rdrd ϕ
I enc = N = n = 0 A N , n 2 λ N , n N = n = 0 A N , n 2
f ( r , ϕ ) = A 0,0 Φ 0,0 ( c , r )
E ( r p ) = i λ Θ a ( s x , s y ) s z exp ( ik s · r p ) ds x ds y
a u ϕ = g u ϕ e i Ψ u ϕ u ϕ e u ϕ
e = ( e x ( u , ϕ ) e y ( u , ϕ ) )
= ( ( 1 + 1 u 2 ) ( 1 1 u 2 ) cos 2 ϕ ( 1 1 u 2 ) sin 2 ϕ ( 1 1 u 2 ) sin 2 ϕ ( 1 + 1 u 2 ) + ( 1 1 u 2 ) cos 2 ϕ 2 u cos ϕ 2 u sin ϕ )
E j ( ρ p , ϕ p , z p ) =
        2 iA ( α k ρ p max ) m = N = n = 0 i N A m , N , n j ( 1 ) n λ N , n J m ( kz p ) exp ( i N ϕ p ) Φ N , n ( α ρ p ρ p max )
x p = ρ p cos ϕ p y p = ρ p sin ϕ p z p = z p .
0 2 π 0 ρ p max E j ( ρ p , ϕ p , z p ) Φ Q , q ( α ρ p ρ p max ) exp ( i Q ϕ p ) ρ p d ρ p d ϕ p
                        = 2 i A ( α k ρ p ) m = N = n = 0 i N A m , N , n j ( 1 ) n λ N , n 3 2 J m ( k z p ) δ QN δ qn
E N , n j = i A π ( α k ρ p max ) ( 1 ) n i N λ N n 1 2 m = J m ( k z p ) A m , N , n j
E N , n j = 1 2 π λ N , n 0 2 π 0 ρ p max E j ( ρ p , ϕ p , z p ) Φ N , n ( α ρ p ρ p max ) exp ( i N ϕ p ) ρ p d ρ p d ϕ p
m = J m ( kz p ) A m , N , n j = i π A ( k ρ p max α ) ( 1 ) n i N λ N , n 1 2 E N , n j
J m ( kz p ) z p = 0 = { 1 for m = 0 0 otherwise
A N , n j = i π A ( k ρ p max α ) ( 1 ) n i N λ N , n 1 2 E N , n j
f ( r ) = n = 0 λ N , n 1 Φ N , n ( r ) 0 r 0 f ( r ) Φ N , n ( r ) r d r     for  
r > 0 if N > 0   r 0 if N = 0
κ = λ 0 , 0 λ N , n min 1 λ N , n min
e j k = 1 S j k Π j k e ( u , ϕ ) u d u d ϕ
E x ( 0 , 0 , z p ) = E 0 rect ( z p w )
m = J m ( kz p ) A m , 0 , n j = i π A ( k ρ p max α ) ( 1 ) n λ 0 , n 1 2 E 0 , n j
cos ψ H = E x ( 0 , 0 , z p ) 2 , E x o p t ( 0 , 0 , z p ) 2 E x ( 0 , 0 , z p ) 2 1 2 E x o p t ( 0 , 0 , z p ) 2 1 2
E x ( 0 , 0 , z p ) 2 , E x opt ( 0 , 0 , z p ) 2 = E x ( 0 , 0 , z p ) 2 E x opt ( 0 , 0 , z p ) 2 dz p
E x ( 0 , 0 , z p ) 2 = E x ( 0 , 0 , z p ) 4 d z p
E x ( ρ p , ϕ p , 0 ) = 1 ρ p δ ( α ρ p ρ p max ) δ ( ϕ p )
= N = n = 0 λ N , n 1 Φ N , n ( c , α ρ p ρ p m a x ) Φ N , n ( c , 0 ) exp ( i N ϕ )
A N , n x = { i 2 A ( k ρ p m a x α ) ( 1 ) n λ 0 , n 3 2 Φ 0 , n ( c , 0 )       for     N = 0 0                                                                                                         for   N 0
e x ( u , ϕ ) = 1 ( 1 u 2 ) 21 22 11 22 12 21 n = 0 A 0 , n x Φ 0 , n ( c , u )
e y ( u , ϕ ) = 1 ( 1 u 2 ) 12 11 11 22 12 21 n = 0 A 0 , n x Φ 0 , n ( c , u )
g ( u , ϕ ) = ( e x 2 + e y 2 ) 1 2 = 2 u 2 ( 1 sin 2 ϕ ) 2 ( 1 u 2 ) n = 0 A 0 , n x Φ 0 , n ( c , u )

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