Abstract

We experimentally demonstrate the feasibility of a super-resolution technique based on eigenmode decomposition. This technique has been proposed theoretically but, to the best of our knowledge, has not previously been realized experimentally for optical imaging systems with circular apertures. We use a standard diffraction-limited 4f imaging system with circular apertures for which the radial eigenmodes are the circular prolate spheroidal functions. For three original objects with different content of angular information we achieve 45%, 49%, and 89% improvement of resolution over the Rayleigh limit. The work presented can be considered as progress towards the goal of reaching the quantum limits of super-resolution.

© 2012 OSA

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  1. P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A63, 063407 (2001).
    [CrossRef]
  2. H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011).
    [CrossRef] [PubMed]
  3. R. W. Boyd and J. P. Dowling, “Quantum lithography: status of the field,” Quant. Inf. Processing11, 891–901 (2012).
    [CrossRef]
  4. G. Toraldo and Di Francia, “Resolving power and information,” J. Opt. Soc. Am.45, 497–501 (1955).
    [CrossRef]
  5. J. L. Harris, “Diffraction and resolving power,” J. Opt. Soc. Am.54, 931–936 (1964).
    [CrossRef]
  6. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
    [CrossRef] [PubMed]
  7. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  8. D. R. Smith, “How to build a superlens,” Science308, 502–503 (2005).
    [CrossRef] [PubMed]
  9. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308, 534–537 (2005).
    [CrossRef] [PubMed]
  10. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Materials11, 432–435 (2012).
    [CrossRef]
  11. V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A79, 013827 (2009).
    [CrossRef]
  12. F. Guerrieri, L. Maccone, F. N. C. Wong, J. H. Shapiro, S. Tisa, and F. Zappa, “Sub-rayleigh imaging via N-photon detection,” Phys. Rev. Lett.105, 163602 (2010).
    [CrossRef]
  13. C. K. Rushforth, “Restoration, resolution, and noise,” J. Opt. Soc. Am.58, 539–545 (1968).
    [CrossRef]
  14. G. Toraldo and Di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am.59, 799–804 (1969).
    [CrossRef]
  15. M. Bertero and E. R. Pike, “Resolution in diffraction-limited imaging, a singular value analysis,” Opt. Acta29, 727–746 (1982).
    [CrossRef]
  16. A. C. D. Luca, S. Kosmeier, K. Dholakia, and M. Mazilu, “Optical eigenmode imaging,” Phys. Rev. A84, 021803 (2011).
    [CrossRef]
  17. M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett.85(18), 3789–3792 (2000).
    [CrossRef] [PubMed]
  18. I. V. Sokolov and M. I. Kolobov, “Squeezed-light source for superresolving microscopy,” Opt. Lett.29, 703–705 (2004).
    [CrossRef] [PubMed]
  19. V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A71(4), 043802 (2005).
    [CrossRef]
  20. V. N. Beskrovny and M. I. Kolobov, “Quantum theory of super-resolution for optical systems with circular apertures,” Opt. Commun.264(1), 9–12 (2006).
    [CrossRef]
  21. V. N. Beskrovny and M. I. Kolobov, “Quantum-statistical analysis of superresolution for optical systems with circular symmetry,” Phys. Rev. A78(4), 043824 (2008).
    [CrossRef]
  22. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty IV,” Bell Syst. Tech. J.43, 3009–3057 (1964).
  23. I. C. Moore and M. Cada, “Prolate spheroidal wave functions, an introduction to the Slepian series and its properties,” Appl. Comput. Harmon. Anal.16, 208–230 (2004).
    [CrossRef]
  24. G. Walter and T. Soleski, “A new friendly method of computing prolate spheroidal wave functions and wavelets,” Appl. Comput. Harmon. Anal.19, 432–443 (2005).
    [CrossRef]
  25. H. Xiao, V. Rokhlin, and N. Yarvin, “Prolate spheroidal wavefunctions, quadrature and interpolation,” IOP-Science17, 805–838 (2000).
  26. D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J.44, 1745–1759 (1965).
  27. B. R. Frieden, “Band-unlimited reconstruction of optical objects and spectra,” J. Opt. Soc. Am.57, 1013–1019 (1967).
    [CrossRef]
  28. 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]
  29. C.-S. Hu, “Prolate spheroidal wave functions of large frequency parameters c = kf and their applications in electromagnetic theory,” IEEE Trans. Antennas Propag.AP-34, 114–119 (1986).
  30. J. C. Heurtley, “Hyperspheroidal functions-optical resonators with circular mirrors,” Proc. Symp. Quasi-Opt.1, 367–375 (1964).

2012

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Materials11, 432–435 (2012).
[CrossRef]

R. W. Boyd and J. P. Dowling, “Quantum lithography: status of the field,” Quant. Inf. Processing11, 891–901 (2012).
[CrossRef]

2011

A. C. D. Luca, S. Kosmeier, K. Dholakia, and M. Mazilu, “Optical eigenmode imaging,” Phys. Rev. A84, 021803 (2011).
[CrossRef]

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011).
[CrossRef] [PubMed]

2010

F. Guerrieri, L. Maccone, F. N. C. Wong, J. H. Shapiro, S. Tisa, and F. Zappa, “Sub-rayleigh imaging via N-photon detection,” Phys. Rev. Lett.105, 163602 (2010).
[CrossRef]

2009

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A79, 013827 (2009).
[CrossRef]

2008

V. N. Beskrovny and M. I. Kolobov, “Quantum-statistical analysis of superresolution for optical systems with circular symmetry,” Phys. Rev. A78(4), 043824 (2008).
[CrossRef]

2007

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
[CrossRef] [PubMed]

2006

V. N. Beskrovny and M. I. Kolobov, “Quantum theory of super-resolution for optical systems with circular apertures,” Opt. Commun.264(1), 9–12 (2006).
[CrossRef]

2005

G. Walter and T. Soleski, “A new friendly method of computing prolate spheroidal wave functions and wavelets,” Appl. Comput. Harmon. Anal.19, 432–443 (2005).
[CrossRef]

D. R. Smith, “How to build a superlens,” Science308, 502–503 (2005).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308, 534–537 (2005).
[CrossRef] [PubMed]

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A71(4), 043802 (2005).
[CrossRef]

2004

I. C. Moore and M. Cada, “Prolate spheroidal wave functions, an introduction to the Slepian series and its properties,” Appl. Comput. Harmon. Anal.16, 208–230 (2004).
[CrossRef]

I. V. Sokolov and M. I. Kolobov, “Squeezed-light source for superresolving microscopy,” Opt. Lett.29, 703–705 (2004).
[CrossRef] [PubMed]

2001

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A63, 063407 (2001).
[CrossRef]

2000

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000).
[CrossRef] [PubMed]

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett.85(18), 3789–3792 (2000).
[CrossRef] [PubMed]

H. Xiao, V. Rokhlin, and N. Yarvin, “Prolate spheroidal wavefunctions, quadrature and interpolation,” IOP-Science17, 805–838 (2000).

1986

C.-S. Hu, “Prolate spheroidal wave functions of large frequency parameters c = kf and their applications in electromagnetic theory,” IEEE Trans. Antennas Propag.AP-34, 114–119 (1986).

1982

M. Bertero and E. R. Pike, “Resolution in diffraction-limited imaging, a singular value analysis,” Opt. Acta29, 727–746 (1982).
[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]

1969

1968

1967

1965

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J.44, 1745–1759 (1965).

1964

J. L. Harris, “Diffraction and resolving power,” J. Opt. Soc. Am.54, 931–936 (1964).
[CrossRef]

J. C. Heurtley, “Hyperspheroidal functions-optical resonators with circular mirrors,” Proc. Symp. Quasi-Opt.1, 367–375 (1964).

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty IV,” Bell Syst. Tech. J.43, 3009–3057 (1964).

1955

Abrams, D. S.

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A63, 063407 (2001).
[CrossRef]

Bertero, M.

M. Bertero and E. R. Pike, “Resolution in diffraction-limited imaging, a singular value analysis,” Opt. Acta29, 727–746 (1982).
[CrossRef]

Beskrovny, V. N.

V. N. Beskrovny and M. I. Kolobov, “Quantum-statistical analysis of superresolution for optical systems with circular symmetry,” Phys. Rev. A78(4), 043824 (2008).
[CrossRef]

V. N. Beskrovny and M. I. Kolobov, “Quantum theory of super-resolution for optical systems with circular apertures,” Opt. Commun.264(1), 9–12 (2006).
[CrossRef]

Beskrovnyy, V. N.

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A71(4), 043802 (2005).
[CrossRef]

Boto, A. N.

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A63, 063407 (2001).
[CrossRef]

Boyd, R. W.

R. W. Boyd and J. P. Dowling, “Quantum lithography: status of the field,” Quant. Inf. Processing11, 891–901 (2012).
[CrossRef]

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011).
[CrossRef] [PubMed]

Braunstein, S. L.

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A63, 063407 (2001).
[CrossRef]

Cada, M.

I. C. Moore and M. Cada, “Prolate spheroidal wave functions, an introduction to the Slepian series and its properties,” Appl. Comput. Harmon. Anal.16, 208–230 (2004).
[CrossRef]

Chad, J. E.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Materials11, 432–435 (2012).
[CrossRef]

Chan, K. W. C.

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011).
[CrossRef] [PubMed]

Chang, H. J.

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011).
[CrossRef] [PubMed]

Dennis, M. R.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Materials11, 432–435 (2012).
[CrossRef]

Dholakia, K.

A. C. D. Luca, S. Kosmeier, K. Dholakia, and M. Mazilu, “Optical eigenmode imaging,” Phys. Rev. A84, 021803 (2011).
[CrossRef]

Dowling, J. P.

R. W. Boyd and J. P. Dowling, “Quantum lithography: status of the field,” Quant. Inf. Processing11, 891–901 (2012).
[CrossRef]

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A63, 063407 (2001).
[CrossRef]

Fabre, C.

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett.85(18), 3789–3792 (2000).
[CrossRef] [PubMed]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308, 534–537 (2005).
[CrossRef] [PubMed]

Francia, Di

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]

B. R. Frieden, “Band-unlimited reconstruction of optical objects and spectra,” J. Opt. Soc. Am.57, 1013–1019 (1967).
[CrossRef]

Giovannetti, V.

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A79, 013827 (2009).
[CrossRef]

Guerrieri, F.

F. Guerrieri, L. Maccone, F. N. C. Wong, J. H. Shapiro, S. Tisa, and F. Zappa, “Sub-rayleigh imaging via N-photon detection,” Phys. Rev. Lett.105, 163602 (2010).
[CrossRef]

Harris, J. L.

Heurtley, J. C.

J. C. Heurtley, “Hyperspheroidal functions-optical resonators with circular mirrors,” Proc. Symp. Quasi-Opt.1, 367–375 (1964).

Hu, C.-S.

C.-S. Hu, “Prolate spheroidal wave functions of large frequency parameters c = kf and their applications in electromagnetic theory,” IEEE Trans. Antennas Propag.AP-34, 114–119 (1986).

Kok, P.

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A63, 063407 (2001).
[CrossRef]

Kolobov, M. I.

V. N. Beskrovny and M. I. Kolobov, “Quantum-statistical analysis of superresolution for optical systems with circular symmetry,” Phys. Rev. A78(4), 043824 (2008).
[CrossRef]

V. N. Beskrovny and M. I. Kolobov, “Quantum theory of super-resolution for optical systems with circular apertures,” Opt. Commun.264(1), 9–12 (2006).
[CrossRef]

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A71(4), 043802 (2005).
[CrossRef]

I. V. Sokolov and M. I. Kolobov, “Squeezed-light source for superresolving microscopy,” Opt. Lett.29, 703–705 (2004).
[CrossRef] [PubMed]

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett.85(18), 3789–3792 (2000).
[CrossRef] [PubMed]

Kosmeier, S.

A. C. D. Luca, S. Kosmeier, K. Dholakia, and M. Mazilu, “Optical eigenmode imaging,” Phys. Rev. A84, 021803 (2011).
[CrossRef]

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308, 534–537 (2005).
[CrossRef] [PubMed]

Lindberg, J.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Materials11, 432–435 (2012).
[CrossRef]

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
[CrossRef] [PubMed]

Lloyd, S.

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A79, 013827 (2009).
[CrossRef]

Luca, A. C. D.

A. C. D. Luca, S. Kosmeier, K. Dholakia, and M. Mazilu, “Optical eigenmode imaging,” Phys. Rev. A84, 021803 (2011).
[CrossRef]

Maccone, L.

F. Guerrieri, L. Maccone, F. N. C. Wong, J. H. Shapiro, S. Tisa, and F. Zappa, “Sub-rayleigh imaging via N-photon detection,” Phys. Rev. Lett.105, 163602 (2010).
[CrossRef]

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A79, 013827 (2009).
[CrossRef]

Mazilu, M.

A. C. D. Luca, S. Kosmeier, K. Dholakia, and M. Mazilu, “Optical eigenmode imaging,” Phys. Rev. A84, 021803 (2011).
[CrossRef]

Moore, I. C.

I. C. Moore and M. Cada, “Prolate spheroidal wave functions, an introduction to the Slepian series and its properties,” Appl. Comput. Harmon. Anal.16, 208–230 (2004).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000).
[CrossRef] [PubMed]

Pike, E. R.

M. Bertero and E. R. Pike, “Resolution in diffraction-limited imaging, a singular value analysis,” Opt. Acta29, 727–746 (1982).
[CrossRef]

Rogers, E. T. F.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Materials11, 432–435 (2012).
[CrossRef]

Rokhlin, V.

H. Xiao, V. Rokhlin, and N. Yarvin, “Prolate spheroidal wavefunctions, quadrature and interpolation,” IOP-Science17, 805–838 (2000).

Roy, T.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Materials11, 432–435 (2012).
[CrossRef]

Rushforth, C. K.

Savo, S.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Materials11, 432–435 (2012).
[CrossRef]

Shapiro, J. H.

F. Guerrieri, L. Maccone, F. N. C. Wong, J. H. Shapiro, S. Tisa, and F. Zappa, “Sub-rayleigh imaging via N-photon detection,” Phys. Rev. Lett.105, 163602 (2010).
[CrossRef]

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A79, 013827 (2009).
[CrossRef]

Shin, H.

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011).
[CrossRef] [PubMed]

Slepian, D.

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J.44, 1745–1759 (1965).

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty IV,” Bell Syst. Tech. J.43, 3009–3057 (1964).

Smith, D. R.

D. R. Smith, “How to build a superlens,” Science308, 502–503 (2005).
[CrossRef] [PubMed]

Sokolov, I. V.

Soleski, T.

G. Walter and T. Soleski, “A new friendly method of computing prolate spheroidal wave functions and wavelets,” Appl. Comput. Harmon. Anal.19, 432–443 (2005).
[CrossRef]

Sonnenblick, E.

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J.44, 1745–1759 (1965).

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308, 534–537 (2005).
[CrossRef] [PubMed]

Tisa, S.

F. Guerrieri, L. Maccone, F. N. C. Wong, J. H. Shapiro, S. Tisa, and F. Zappa, “Sub-rayleigh imaging via N-photon detection,” Phys. Rev. Lett.105, 163602 (2010).
[CrossRef]

Toraldo, G.

Walter, G.

G. Walter and T. Soleski, “A new friendly method of computing prolate spheroidal wave functions and wavelets,” Appl. Comput. Harmon. Anal.19, 432–443 (2005).
[CrossRef]

Williams, C. P.

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A63, 063407 (2001).
[CrossRef]

Wong, F. N. C.

F. Guerrieri, L. Maccone, F. N. C. Wong, J. H. Shapiro, S. Tisa, and F. Zappa, “Sub-rayleigh imaging via N-photon detection,” Phys. Rev. Lett.105, 163602 (2010).
[CrossRef]

Xiao, H.

H. Xiao, V. Rokhlin, and N. Yarvin, “Prolate spheroidal wavefunctions, quadrature and interpolation,” IOP-Science17, 805–838 (2000).

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
[CrossRef] [PubMed]

Yarvin, N.

H. Xiao, V. Rokhlin, and N. Yarvin, “Prolate spheroidal wavefunctions, quadrature and interpolation,” IOP-Science17, 805–838 (2000).

Zappa, F.

F. Guerrieri, L. Maccone, F. N. C. Wong, J. H. Shapiro, S. Tisa, and F. Zappa, “Sub-rayleigh imaging via N-photon detection,” Phys. Rev. Lett.105, 163602 (2010).
[CrossRef]

Zhang, X.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308, 534–537 (2005).
[CrossRef] [PubMed]

Zheludev, N. I.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Materials11, 432–435 (2012).
[CrossRef]

Appl. Comput. Harmon. Anal.

I. C. Moore and M. Cada, “Prolate spheroidal wave functions, an introduction to the Slepian series and its properties,” Appl. Comput. Harmon. Anal.16, 208–230 (2004).
[CrossRef]

G. Walter and T. Soleski, “A new friendly method of computing prolate spheroidal wave functions and wavelets,” Appl. Comput. Harmon. Anal.19, 432–443 (2005).
[CrossRef]

Bell Syst. Tech. J.

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J.44, 1745–1759 (1965).

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty IV,” Bell Syst. Tech. J.43, 3009–3057 (1964).

IEEE Trans. Antennas Propag.

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Figures (6)

Fig. 1
Fig. 1

Schematic of the 4f optical system used for eigenmode super-resolution imaging. The two lenses both have a focal length of f, and the distance between every plane and subsequent lens is f. The object plane has an aperture of radius Ro, the pupil plane has an aperture of radius Rp, and the image plane has an aperture of radius Ri = Ro.

Fig. 2
Fig. 2

Normalized intensity and phase profiles of three different prolate spheroidal modes. The dashed red circles indicate the area in which the images are contained. The red arrow indicates Ro.

Fig. 3
Fig. 3

(a) Transmission of prolate spheroidal modes through a 4f system. The intensity and phase profile for the Φ22 mode are shown. On propagation through the 4f system, the mode Φ22 becomes λ22Φ22. (b) Subset of eigenvalues λℓp for the 4f system with c = 10.

Fig. 4
Fig. 4

Schematic of the experimental setup.

Fig. 5
Fig. 5

Experimental results of eigenmode super-resolution. (a) The original images that were propagated through the system. (b) The recorded diffracted-limited images. (c) The super-resolved images. The images are contained within a 1 mm radius.

Fig. 6
Fig. 6

The normalised modulus squared of the decomposition coefficients of the images in Fig. 5. The insets indicate the appropriate image at each stage.

Equations (16)

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1.22 f λ D ,
Φ p = φ p ( r , c ) e i θ ,
c = 2 π R o R p λ f ,
0 2 π 0 R o Φ p Φ p * r d r d θ = 2 π δ δ p p .
A ( r , θ ) = = p = 0 α p Φ p ,
B ( r , θ ) = = p = 0 β p Φ p .
α p = 1 2 π 0 2 π 0 R o A ( r , θ ) Φ p * r d r d θ ,
β p = 1 2 π 0 2 π 0 R o B ( r , θ ) Φ p * r d r d θ .
S [ A ( r , θ ) ] = B ( r , θ ) .
S [ Φ p ] = λ p Φ p .
, p α p S [ Φ p ] = , p α p λ p Φ p = , p β p Φ p .
α p = β p λ p .
| C ( r , θ ) | 2 = | = p = 0 β p λ p Φ p | 2 .
B ( r , θ ) = 1 N q = 0 N 1 e i 2 π q / N I q ( r , θ ) ,
κ = 0 2 π 0 R o A ( r , θ ) * C ( r , θ ) r d r d θ ;
| κ η | 2 1 .

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