Abstract

The mathematical phenomenon of superoscillation, in which a spectrally bound function oscillates locally at a rate faster than its fastest Fourier component, has found use in both theoretical and applied areas of optical research. We show the existence of a complementary phenomenon we term sub-oscillation, in which a spectrally lower bound limited function oscillates locally at an arbitrarily low frequency. The relevance of superoscillations to various fields, such as weak measurements, beam shaping, and super-resolution imaging, suggests that sub-oscillations could also find various uses. Here, we construct a spatially sub-oscillatory optical beam to experimentally demonstrate optical super defocusing, i.e., a very fast, exceptional expansion of a partially blocked light beam.

© 2017 Optical Society of America

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References

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    [Crossref]
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  9. M. Berry and P. Shukla, “Pointer supershifts and superoscillations in weak measurements,” J. Phys. A 45, 015301 (2012).
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  10. M. Berry, “Exact nonparaxial transmission of subwavelength detail using superoscillations,” J. Phys. A 46, 205203 (2013).
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    [Crossref]
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    [Crossref]
  27. R. Remez and A. Arie, “Super-narrow frequency conversion,” Optica 2, 472–475 (2015).
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    [Crossref]
  30. Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” arXiv preprint arXiv:1607.02352 (2016).
  31. R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Super-oscillating electron wave functions with sub-diffraction spots,” arXiv preprint arXiv:1604.05929 (2016).
  32. A. M. Wong and G. V. Eleftheriades, “Superoscillatory antenna arrays for sub-diffraction focusing at the multi-wavelength range in a waveguide environment,” in IEEE Antennas and Propagation Society International Symposium (IEEE, 2010), pp. 1–4.
  33. A. M. Wong and G. V. Eleftheriades, “Sub-wavelength focusing at the multi-wavelength range using superoscillations: an experimental demonstration,” IEEE Trans. Antennas Propag. 59(12), 4766–4776 (2011).
    [Crossref]
  34. M. Berry and P. Shukla, “Typical weak and superweak values,” J. Phys. A 43, 354024 (2010).
    [Crossref]
  35. M. Berry, “Quantum backflow, negative kinetic energy, and optical retro-propagation,” J. Phys. A 43, 415302 (2010).
    [Crossref]
  36. M. Berry, “Superluminal speeds for relativistic random waves,” J. Phys. A 45, 185308 (2012).
    [Crossref]
  37. A. Bracken and G. Melloy, “Probability backflow and a new dimensionless quantum number,” J. Phys. A 27, 2197–2211 (1994).
    [Crossref]
  38. K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
    [Crossref]
  39. Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “The mathematics of superoscillations,” Mem. Am. Math. Soc. 247, 5–23 (2017).
    [Crossref]
  40. E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38, 3546–3549 (2013).
    [Crossref]
  41. B. S. Kumar, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” arXiv preprint arXiv:1609.08858 (2016).

2017 (1)

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “The mathematics of superoscillations,” Mem. Am. Math. Soc. 247, 5–23 (2017).
[Crossref]

2016 (2)

Y. Eliezer and A. Bahabad, “Super-oscillating airy pattern,” ACS Photon. 3, 1053–1059 (2016).
[Crossref]

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2016).
[Crossref]

2015 (3)

2014 (1)

2013 (6)

E. Greenfield, R. Schley, I. Hurwitz, J. Nemirovsky, K. G. Makris, and M. Segev, “Experimental generation of arbitrarily shaped diffractionless superoscillatory optical beams,” Opt. Express 21, 13425–13435 (2013).
[Crossref]

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38, 3546–3549 (2013).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

A. M. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).

M. Berry, “Exact nonparaxial transmission of subwavelength detail using superoscillations,” J. Phys. A 46, 205203 (2013).
[Crossref]

E. Katzav and M. Schwartz, “Yield-optimized superoscillations,” IEEE Trans. Signal Process. 61, 3113–3118 (2013).
[Crossref]

2012 (4)

M. Berry and P. Shukla, “Pointer supershifts and superoscillations in weak measurements,” J. Phys. A 45, 015301 (2012).
[Crossref]

Y. Gorodetski, K. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. Ebbesen, “Weak measurements of light chirality with a plasmonic slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref]

M. Berry, “Superluminal speeds for relativistic random waves,” J. Phys. A 45, 185308 (2012).
[Crossref]

E. T. 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. Mater. 11, 432–435 (2012).
[Crossref]

2011 (3)

A. M. Wong and G. V. Eleftheriades, “Sub-wavelength focusing at the multi-wavelength range using superoscillations: an experimental demonstration,” IEEE Trans. Antennas Propag. 59(12), 4766–4776 (2011).
[Crossref]

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “Some mathematical properties of superoscillations,” J. Phys. A 44, 365304 (2011).
[Crossref]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

2010 (2)

M. Berry and P. Shukla, “Typical weak and superweak values,” J. Phys. A 43, 354024 (2010).
[Crossref]

M. Berry, “Quantum backflow, negative kinetic energy, and optical retro-propagation,” J. Phys. A 43, 415302 (2010).
[Crossref]

2009 (2)

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9, 1249–1254 (2009).
[Crossref]

M. Berry and M. Dennis, “Natural superoscillations in monochromatic waves in d dimensions,” J. Phys. A 42, 022003 (2009).
[Crossref]

2008 (3)

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref]

N. I. Zheludev, “What diffraction limit?” Nat. Mater. 7, 420–422 (2008).
[Crossref]

M. R. Dennis, A. C. Hamilton, and J. Courtial, “Superoscillation in speckle patterns,” Opt. Lett. 33, 2976–2978 (2008).
[Crossref]

2006 (2)

P. J. Ferreira and A. Kempf, “Superoscillations: faster than the Nyquist rate,” IEEE Trans. Signal Process. 54, 3732–3740 (2006).
[Crossref]

M. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
[Crossref]

2000 (1)

A. Kempf, “Black holes, bandwidths and Beethoven,” J. Math. Phys. 41, 2360–2374 (2000).
[Crossref]

1994 (1)

A. Bracken and G. Melloy, “Probability backflow and a new dimensionless quantum number,” J. Phys. A 27, 2197–2211 (1994).
[Crossref]

1988 (1)

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref]

1961 (1)

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–63 (1961).
[Crossref]

1952 (1)

G. T. Di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento 9, 426–438 (1952).
[Crossref]

1943 (1)

S. A. Schelkunoff, “A mathematical theory of linear arrays,” Bell Syst. Tech. J. 22, 80–107 (1943).
[Crossref]

Aharonov, Y.

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “The mathematics of superoscillations,” Mem. Am. Math. Soc. 247, 5–23 (2017).
[Crossref]

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “Some mathematical properties of superoscillations,” J. Phys. A 44, 365304 (2011).
[Crossref]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref]

Albert, D. Z.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref]

Arie, A.

R. Remez and A. Arie, “Super-narrow frequency conversion,” Optica 2, 472–475 (2015).
[Crossref]

B. K. Singh, R. Remez, Y. Tsur, and A. Arie, “Super-airy beam: self-accelerating beam with intensified main lobe,” Opt. Lett. 40, 4703–4706 (2015).
[Crossref]

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Super-oscillating electron wave functions with sub-diffraction spots,” arXiv preprint arXiv:1604.05929 (2016).

B. S. Kumar, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” arXiv preprint arXiv:1609.08858 (2016).

Bahabad, A.

Y. Eliezer and A. Bahabad, “Super-oscillating airy pattern,” ACS Photon. 3, 1053–1059 (2016).
[Crossref]

Y. Eliezer and A. Bahabad, “Super-transmission: the delivery of superoscillations through the absorbing resonance of a dielectric medium,” Opt. Express 22, 31212–31226 (2014).
[Crossref]

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” arXiv preprint arXiv:1607.02352 (2016).

Bartal, G.

Bekshaev, A. Y.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

Bent, N.

Berry, M.

M. Berry, “Exact nonparaxial transmission of subwavelength detail using superoscillations,” J. Phys. A 46, 205203 (2013).
[Crossref]

M. Berry and P. Shukla, “Pointer supershifts and superoscillations in weak measurements,” J. Phys. A 45, 015301 (2012).
[Crossref]

M. Berry, “Superluminal speeds for relativistic random waves,” J. Phys. A 45, 185308 (2012).
[Crossref]

M. Berry and P. Shukla, “Typical weak and superweak values,” J. Phys. A 43, 354024 (2010).
[Crossref]

M. Berry, “Quantum backflow, negative kinetic energy, and optical retro-propagation,” J. Phys. A 43, 415302 (2010).
[Crossref]

M. Berry and M. Dennis, “Natural superoscillations in monochromatic waves in d dimensions,” J. Phys. A 42, 022003 (2009).
[Crossref]

M. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
[Crossref]

M. Berry, “‘Faster than Fourier,’ quantum coherence and reality: in celebration of the 60th birthday of Yakir Aharonov,” in Proceedings of the International Conference on Fundamental Aspects of Quantum Theory (World Scientific, 1994), p. 55.

M. Berry, “Superoscillations, endfire and supergain in quantum theory,” in Quantum Theory: A Two-Time Success Story: Yakir Aharonov Festschrift, D. C. Struppa and J. M. Tollaksen, eds. (Springer, 2013), pp. 327–336.

Blau, Y.

Bliokh, K.

Y. Gorodetski, K. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. Ebbesen, “Weak measurements of light chirality with a plasmonic slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

Bolduc, E.

Boyd, R. W.

Bracken, A.

A. Bracken and G. Melloy, “Probability backflow and a new dimensionless quantum number,” J. Phys. A 27, 2197–2211 (1994).
[Crossref]

Braverman, B.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Chad, J. E.

E. T. 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. Mater. 11, 432–435 (2012).
[Crossref]

Colombo, F.

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “The mathematics of superoscillations,” Mem. Am. Math. Soc. 247, 5–23 (2017).
[Crossref]

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “Some mathematical properties of superoscillations,” J. Phys. A 44, 365304 (2011).
[Crossref]

Courtial, J.

David, A.

Dennis, M.

M. Berry and M. Dennis, “Natural superoscillations in monochromatic waves in d dimensions,” J. Phys. A 42, 022003 (2009).
[Crossref]

Dennis, M. R.

E. T. 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. Mater. 11, 432–435 (2012).
[Crossref]

M. R. Dennis, A. C. Hamilton, and J. Courtial, “Superoscillation in speckle patterns,” Opt. Lett. 33, 2976–2978 (2008).
[Crossref]

Di Francia, G. T.

G. T. Di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento 9, 426–438 (1952).
[Crossref]

Dolev, S.

Dunin-Borkowski, R. E.

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Super-oscillating electron wave functions with sub-diffraction spots,” arXiv preprint arXiv:1604.05929 (2016).

Ebbesen, T.

Y. Gorodetski, K. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. Ebbesen, “Weak measurements of light chirality with a plasmonic slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref]

Eleftheriades, G. V.

A. M. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).

A. M. Wong and G. V. Eleftheriades, “Sub-wavelength focusing at the multi-wavelength range using superoscillations: an experimental demonstration,” IEEE Trans. Antennas Propag. 59(12), 4766–4776 (2011).
[Crossref]

A. M. Wong and G. V. Eleftheriades, “Superoscillatory antenna arrays for sub-diffraction focusing at the multi-wavelength range in a waveguide environment,” in IEEE Antennas and Propagation Society International Symposium (IEEE, 2010), pp. 1–4.

Eliezer, Y.

Y. Eliezer and A. Bahabad, “Super-oscillating airy pattern,” ACS Photon. 3, 1053–1059 (2016).
[Crossref]

Y. Eliezer and A. Bahabad, “Super-transmission: the delivery of superoscillations through the absorbing resonance of a dielectric medium,” Opt. Express 22, 31212–31226 (2014).
[Crossref]

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” arXiv preprint arXiv:1607.02352 (2016).

Ferreira, P. J.

P. J. Ferreira and A. Kempf, “Superoscillations: faster than the Nyquist rate,” IEEE Trans. Signal Process. 54, 3732–3740 (2006).
[Crossref]

Froim, S.

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” arXiv preprint arXiv:1607.02352 (2016).

Gao, P.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2016).
[Crossref]

Genet, C.

Y. Gorodetski, K. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. Ebbesen, “Weak measurements of light chirality with a plasmonic slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref]

Gjonaj, B.

Gorodetski, Y.

Y. Gorodetski, K. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. Ebbesen, “Weak measurements of light chirality with a plasmonic slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref]

Greenfield, E.

Hamilton, A. C.

Hareli, L.

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” arXiv preprint arXiv:1607.02352 (2016).

Hasman, E.

Y. Gorodetski, K. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. Ebbesen, “Weak measurements of light chirality with a plasmonic slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref]

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref]

Huang, F. M.

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9, 1249–1254 (2009).
[Crossref]

Hurwitz, I.

Karimi, E.

Katzav, E.

E. Katzav and M. Schwartz, “Yield-optimized superoscillations,” IEEE Trans. Signal Process. 61, 3113–3118 (2013).
[Crossref]

Kempf, A.

P. J. Ferreira and A. Kempf, “Superoscillations: faster than the Nyquist rate,” IEEE Trans. Signal Process. 54, 3732–3740 (2006).
[Crossref]

A. Kempf, “Black holes, bandwidths and Beethoven,” J. Math. Phys. 41, 2360–2374 (2000).
[Crossref]

Kleiner, V.

Y. Gorodetski, K. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. Ebbesen, “Weak measurements of light chirality with a plasmonic slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref]

Kocsis, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Kofman, A. G.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

Kumar, B. S.

B. S. Kumar, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” arXiv preprint arXiv:1609.08858 (2016).

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref]

Lindberg, J.

E. T. 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. Mater. 11, 432–435 (2012).
[Crossref]

Lobachinsky, L.

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” arXiv preprint arXiv:1607.02352 (2016).

Lu, P.-H.

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Super-oscillating electron wave functions with sub-diffraction spots,” arXiv preprint arXiv:1604.05929 (2016).

Luo, X.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2016).
[Crossref]

Makris, K. G.

Melloy, G.

A. Bracken and G. Melloy, “Probability backflow and a new dimensionless quantum number,” J. Phys. A 27, 2197–2211 (1994).
[Crossref]

Mirin, R. P.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Nagar, H.

B. S. Kumar, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” arXiv preprint arXiv:1609.08858 (2016).

Nemirovsky, J.

Nori, F.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

Pollak, H. O.

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–63 (1961).
[Crossref]

Popescu, S.

M. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
[Crossref]

Pu, M.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2016).
[Crossref]

Ravets, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Remez, R.

R. Remez and A. Arie, “Super-narrow frequency conversion,” Optica 2, 472–475 (2015).
[Crossref]

B. K. Singh, R. Remez, Y. Tsur, and A. Arie, “Super-airy beam: self-accelerating beam with intensified main lobe,” Opt. Lett. 40, 4703–4706 (2015).
[Crossref]

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Super-oscillating electron wave functions with sub-diffraction spots,” arXiv preprint arXiv:1604.05929 (2016).

Rogers, E. T.

E. T. 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. Mater. 11, 432–435 (2012).
[Crossref]

Roichman, Y.

B. S. Kumar, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” arXiv preprint arXiv:1609.08858 (2016).

Roy, T.

E. T. 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. Mater. 11, 432–435 (2012).
[Crossref]

Sabadini, I.

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “The mathematics of superoscillations,” Mem. Am. Math. Soc. 247, 5–23 (2017).
[Crossref]

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “Some mathematical properties of superoscillations,” J. Phys. A 44, 365304 (2011).
[Crossref]

Santamato, E.

Savo, S.

E. T. 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. Mater. 11, 432–435 (2012).
[Crossref]

Schelkunoff, S. A.

S. A. Schelkunoff, “A mathematical theory of linear arrays,” Bell Syst. Tech. J. 22, 80–107 (1943).
[Crossref]

Schley, R.

Schwartz, M.

E. Katzav and M. Schwartz, “Yield-optimized superoscillations,” IEEE Trans. Signal Process. 61, 3113–3118 (2013).
[Crossref]

Segev, M.

Shalm, L. K.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Shitrit, N.

Y. Gorodetski, K. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. Ebbesen, “Weak measurements of light chirality with a plasmonic slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref]

Shukla, P.

M. Berry and P. Shukla, “Pointer supershifts and superoscillations in weak measurements,” J. Phys. A 45, 015301 (2012).
[Crossref]

M. Berry and P. Shukla, “Typical weak and superweak values,” J. Phys. A 43, 354024 (2010).
[Crossref]

Singh, B. K.

Slepian, D.

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–63 (1961).
[Crossref]

Stein, B.

Y. Gorodetski, K. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. Ebbesen, “Weak measurements of light chirality with a plasmonic slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref]

Steinberg, A. M.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Stevens, M. J.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Struppa, D.

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “The mathematics of superoscillations,” Mem. Am. Math. Soc. 247, 5–23 (2017).
[Crossref]

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “Some mathematical properties of superoscillations,” J. Phys. A 44, 365304 (2011).
[Crossref]

Tang, D.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2016).
[Crossref]

Tavabi, A. H.

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Super-oscillating electron wave functions with sub-diffraction spots,” arXiv preprint arXiv:1604.05929 (2016).

Tollaksen, J.

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “The mathematics of superoscillations,” Mem. Am. Math. Soc. 247, 5–23 (2017).
[Crossref]

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “Some mathematical properties of superoscillations,” J. Phys. A 44, 365304 (2011).
[Crossref]

Tsur, Y.

B. K. Singh, R. Remez, Y. Tsur, and A. Arie, “Super-airy beam: self-accelerating beam with intensified main lobe,” Opt. Lett. 40, 4703–4706 (2015).
[Crossref]

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Super-oscillating electron wave functions with sub-diffraction spots,” arXiv preprint arXiv:1604.05929 (2016).

Vaidman, L.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref]

Wang, C.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2016).
[Crossref]

Wang, J.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2016).
[Crossref]

Wang, Y.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2016).
[Crossref]

Wong, A. M.

A. M. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).

A. M. Wong and G. V. Eleftheriades, “Sub-wavelength focusing at the multi-wavelength range using superoscillations: an experimental demonstration,” IEEE Trans. Antennas Propag. 59(12), 4766–4776 (2011).
[Crossref]

A. M. Wong and G. V. Eleftheriades, “Superoscillatory antenna arrays for sub-diffraction focusing at the multi-wavelength range in a waveguide environment,” in IEEE Antennas and Propagation Society International Symposium (IEEE, 2010), pp. 1–4.

Yan, W.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2016).
[Crossref]

Zhang, Y.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2016).
[Crossref]

Zhao, Z.

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2016).
[Crossref]

Zheludev, N. I.

E. T. 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. Mater. 11, 432–435 (2012).
[Crossref]

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9, 1249–1254 (2009).
[Crossref]

N. I. Zheludev, “What diffraction limit?” Nat. Mater. 7, 420–422 (2008).
[Crossref]

ACS Photon. (1)

Y. Eliezer and A. Bahabad, “Super-oscillating airy pattern,” ACS Photon. 3, 1053–1059 (2016).
[Crossref]

Bell Syst. Tech. J. (2)

S. A. Schelkunoff, “A mathematical theory of linear arrays,” Bell Syst. Tech. J. 22, 80–107 (1943).
[Crossref]

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–63 (1961).
[Crossref]

IEEE Trans. Antennas Propag. (1)

A. M. Wong and G. V. Eleftheriades, “Sub-wavelength focusing at the multi-wavelength range using superoscillations: an experimental demonstration,” IEEE Trans. Antennas Propag. 59(12), 4766–4776 (2011).
[Crossref]

IEEE Trans. Signal Process. (2)

P. J. Ferreira and A. Kempf, “Superoscillations: faster than the Nyquist rate,” IEEE Trans. Signal Process. 54, 3732–3740 (2006).
[Crossref]

E. Katzav and M. Schwartz, “Yield-optimized superoscillations,” IEEE Trans. Signal Process. 61, 3113–3118 (2013).
[Crossref]

J. Math. Phys. (1)

A. Kempf, “Black holes, bandwidths and Beethoven,” J. Math. Phys. 41, 2360–2374 (2000).
[Crossref]

J. Phys. A (9)

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “Some mathematical properties of superoscillations,” J. Phys. A 44, 365304 (2011).
[Crossref]

M. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
[Crossref]

M. Berry and M. Dennis, “Natural superoscillations in monochromatic waves in d dimensions,” J. Phys. A 42, 022003 (2009).
[Crossref]

M. Berry and P. Shukla, “Pointer supershifts and superoscillations in weak measurements,” J. Phys. A 45, 015301 (2012).
[Crossref]

M. Berry, “Exact nonparaxial transmission of subwavelength detail using superoscillations,” J. Phys. A 46, 205203 (2013).
[Crossref]

M. Berry and P. Shukla, “Typical weak and superweak values,” J. Phys. A 43, 354024 (2010).
[Crossref]

M. Berry, “Quantum backflow, negative kinetic energy, and optical retro-propagation,” J. Phys. A 43, 415302 (2010).
[Crossref]

M. Berry, “Superluminal speeds for relativistic random waves,” J. Phys. A 45, 185308 (2012).
[Crossref]

A. Bracken and G. Melloy, “Probability backflow and a new dimensionless quantum number,” J. Phys. A 27, 2197–2211 (1994).
[Crossref]

Mem. Am. Math. Soc. (1)

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, “The mathematics of superoscillations,” Mem. Am. Math. Soc. 247, 5–23 (2017).
[Crossref]

Nano Lett. (1)

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9, 1249–1254 (2009).
[Crossref]

Nat. Mater. (2)

E. T. 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. Mater. 11, 432–435 (2012).
[Crossref]

N. I. Zheludev, “What diffraction limit?” Nat. Mater. 7, 420–422 (2008).
[Crossref]

New J. Phys. (1)

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

Nuovo Cimento (1)

G. T. Di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento 9, 426–438 (1952).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Optica (2)

Phys. Rev. Lett. (2)

Y. Gorodetski, K. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. Ebbesen, “Weak measurements of light chirality with a plasmonic slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[Crossref]

Sci. Rep. (2)

A. M. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).

C. Wang, D. Tang, Y. Wang, Z. Zhao, J. Wang, M. Pu, Y. Zhang, W. Yan, P. Gao, and X. Luo, “Super-resolution optical telescopes with local light diffraction shrinkage,” Sci. Rep. 5, 18485 (2016).
[Crossref]

Science (2)

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref]

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[Crossref]

Other (6)

M. Berry, “‘Faster than Fourier,’ quantum coherence and reality: in celebration of the 60th birthday of Yakir Aharonov,” in Proceedings of the International Conference on Fundamental Aspects of Quantum Theory (World Scientific, 1994), p. 55.

M. Berry, “Superoscillations, endfire and supergain in quantum theory,” in Quantum Theory: A Two-Time Success Story: Yakir Aharonov Festschrift, D. C. Struppa and J. M. Tollaksen, eds. (Springer, 2013), pp. 327–336.

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” arXiv preprint arXiv:1607.02352 (2016).

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Super-oscillating electron wave functions with sub-diffraction spots,” arXiv preprint arXiv:1604.05929 (2016).

A. M. Wong and G. V. Eleftheriades, “Superoscillatory antenna arrays for sub-diffraction focusing at the multi-wavelength range in a waveguide environment,” in IEEE Antennas and Propagation Society International Symposium (IEEE, 2010), pp. 1–4.

B. S. Kumar, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” arXiv preprint arXiv:1609.08858 (2016).

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Figures (4)

Fig. 1.
Fig. 1.

Continuous spectrum superoscillatory (with δ=0.25,α=2) and sub-oscillatory (δ=0.25,α=1) signals. (left) A superoscillatory signal. (right) A sub-oscillatory signal. (a) Bound frequency distribution. (b) Logarithmic scale representation of the real part of the functions (continuous blue line) and logarithmic scale of their highest (lowest) Fourier mode (dashed red line). (c) Linear scale representation of the signals (continuous blue line), with the most extreme Fourier component (the dashed red line, the highest component for the superoscillatory signal kmax, the lowest component for the sub-oscillatory signal kmin) and with a Fourier mode not in the spectrum (the dotted–dashed yellow line, ksup(iα), ksub(iα) for the two cases) that matches each function around x=0.

Fig. 2.
Fig. 2.

Discrete spectrum superoscillatory and sub-oscillatory signals. (left) Superoscillatory signal with the parameters of N=10 and a=2. (right) Sub-oscillatory signal with parameters of N=10 and a=12. (a) Log-scale representations of the functions (continuous blue line) with the most extreme Fourier component (dashed red line, fastest (slowest) for the superoscillatory (sub-oscillatory) signal). (b) Linear-scale representation of the function (continuous blue line) with the most extreme Fourier component (dashed red line, fastest (slowest) for the superoscillatory (sub-oscillatory) signal) and with a Fourier mode outside the spectrum (yellow line), which matches the superoscillation (sub-oscillation). (c) The spectrum in linear scale. The superoscillatory spectrum is in the range of [N,N], while the sub-oscillatory spectrum is in the range of [N,]. (d) The spectrum in logarithmic scale.

Fig. 3.
Fig. 3.

Experimental setup. BE, beam expander; SLM, spatial light modulator; CAM, CMOS camera; M, mirror; L1,L2,L3 are lenses. The sum of the distances d1,d2 equals lens L3 focal length. Z0 marks the location of the Fourier plane for an object in the SLM plane.

Fig. 4.
Fig. 4.

Experimental demonstration of super defocusing of a light beam. (left) Phase masks applied to the SLM, serving as the spectral distribution for sub-oscillatory signals. The dotted white line marks the signal’s spectral boundaries. (middle) Measured intensity pattern at the plane corresponding to the Fourier plane of the SLM (serving as the far-field pattern). (right) Horizontal line through the center of the measured intensity patterns (continuous blue lines), theoretically calculated waveforms (dotted black lines) overlaid with the measurement for the regular, non-sub-oscillatory case (corresponding to a=1, dashed red line). The different patterns differ by the control parameter a. As a gets smaller, the pattern sub-oscillates slower, and the beam defocuses stronger. (a) a=1, (b) a=0.9, (c) a=0.8, and (d) a=0.7. Dotted–dashed black lines mark the location of zeros around the sub-oscillation. Dotted–dashed black lines mark the sub-oscillation’s period.

Equations (20)

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

f(x,δ,α)=u1u2A(u,δ,α)exp(ik(u)x)du,
A(u,δ,α)=12πδexp([uiα]22δ2),
limδ0+f(x,δ,α)exp(ik1(iα)x)=exp(icosh(α)x),
f(x,δ,α)=12πδ22exp(i[1+u22]x)exp([uiα]22δ2)du,
f(x,δ,α)=i21iξexp(ix(2α22iξ)2(1iξ))×[erf(2+iα2iξδ22iξ)+erf(2iα2iξδ22iξ)],
f(x,δ,α)11+iξexp(ix[1α22(1+ξ2)])exp(α2ξx2(1+ξ2)).
q(ξ)=1α2(1ξ2)2(1+ξ2)2.
f(x)=[f1(x)]N=(cos(x)+iasin(x))N,
f(x)=(cos2(x)+a2sin(x))N/2exp(iNarctan[atan(x)]),
f(x)=(a+12)Nm=0N(1)mN!m!(Nm)!(a1a+1)mei(N2m)x,
kf(x)=Imddxlog(f(x))=Nacos2x+a2sin2x.
g(x)=(cos2(x)+a2sin(x))N/2exp(iNarctan[atan(x)]).
g1(x)=f11(x)=1cos(x)+iasin(x).
g1(x)=m=m=+Gm(1)exp(imx),
Gm(1)=12πππexp(imx)dxcos(x)+iasin(x)={2[(a+1)12(m1)(a1)12(m+1)],mO0else},
g(x)=(g1(x))N={ki=N}N!k1!k2!km!l=1m(Gl(1)eilx)kl,
g(x)=(G1(1))Nexp(iNx)+.
kg(x)=Imddxlog(g(x))=Nacos2x+a2sin2x,
g(k)=m=MMGm(N)(a)exp([kmΔ]22σ02),
Gm(N)(a)=1πππcos(mx)dx(cos(x)+iasin(x))N,

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