Abstract

Optical superoscillatory imaging, allowing unlabelled far-field super-resolution, has in recent years become reality. Instruments have been built and their super-resolution imaging capabilities demonstrated. The question is no longer whether this can be done, but how well: what resolution is practically achievable? Numerous works have optimised various particular features of superoscillatory spots, but in order to probe the limits of superoscillatory imaging we need to simultaneously optimise all the important spot features: those that define the resolution of the system. We simultaneously optimise spot size and its intensity relative to the sidebands for various fields of view, giving a set of best compromises for use in different imaging scenarios. Our technique uses the circular prolate spheroidal wave functions as a basis set on the field of view, and the optimal combination of these, representing the optimal spot, is found using a multi-objective genetic algorithm. We then introduce a less computationally demanding approach suitable for real-time use in the laboratory which, crucially, allows independent control of spot size and field of view. Imaging simulations demonstrate the resolution achievable with these spots. We show a three-order-of-magnitude improvement in the efficiency of focusing to achieve the same resolution as previously reported results, or a 26 % increase in resolution for the same efficiency of focusing.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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2017 (8)

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the Temporal Resolution Limit by Superoscillating Optical Beats,” Phys. Rev. Lett. 119, 1–5 (2017).
[Crossref]

A. M. Wong and G. V. Eleftheriades, “Broadband superoscillation brings a wave into perfect three-dimensional focus,” Phys. Rev. B 95, 075148 (2017).
[Crossref]

E. Katzav, E. Perlsman, and M. Schwartz, “Yield statistics of interpolated superoscillations,” J. Phys. A: Math. Theor. 50, 2 (2017).
[Crossref]

K. G. Makris, D. G. Papazoglou, and S. Tzortzakis, “Invariant superoscillatory electromagnetic fields in 3D-space,” J. Opt. 19, 014003 (2017).
[Crossref]

G. H. Yuan, E. T. F. Rogers, and N. I. Zheludev, “Achromatic super-oscillatory lenses with sub-wavelength focusing,” Light Sci. Appl. 6, e17036 (2017).
[Crossref]

M. Li, W. Li, H. Li, Y. Zhu, and Y. Yu, “Controllable design of super-oscillatory lenses with multiple sub-diffraction-limit foci,” Sci. Rep. 7, 1335 (2017).
[Crossref] [PubMed]

M. V. Berry, “Suppression of superoscillations by noise,” J. Phys. A: Math. Theor. 50, 2–11 (2017).
[Crossref]

T. Zacharias, B. Hadad, A. Bahabad, and Y. Eliezer, “Axial sub-Fourier focusing of an optical beam,” Opt. Lett. 42, 3205 (2017).
[Crossref] [PubMed]

2016 (3)

J. Diao, W. Yuan, Y. Yu, Y. Zhu, and Y. Wu, “Controllable design of super-oscillatory planar lenses for sub-diffraction-limit optical needles,” Opt. Express 24, 1924 (2016).
[Crossref] [PubMed]

G. Chen, Y. Li, A. Yu, Z. Wen, L. Dai, L. Chen, Z. Zhang, S. Jiang, K. Zhang, X. Wang, and F. Lin, “Super-oscillatory focusing of circularly polarized light by ultra-long focal length planar lens based on binary amplitude-phase modulation,” Sci. Rep. 6, 1–8 (2016).

Y. Eliezer and A. Bahabad, “Super-Oscillating Airy Pattern,” ACS Photon. 3, 1053–1059 (2016).
[Crossref]

2015 (4)

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, 1–8 (2015).

S. W. Hell, S. J. Sahl, M. Bates, X. Zhuang, R. Heintzmann, M. J. Booth, J. Bewersdorf, G. Shtengel, H. Hess, P. Tinnefeld, A. Honigmann, S. Jakobs, I. Testa, L. Cognet, B. Lounis, H. Ewers, S. J. Davis, C. Eggeling, D. Klenerman, K. I. Willig, G. Vicidomini, M. Castello, A. Diaspro, and T. Cordes, “The 2015 super-resolution microscopy roadmap,” J. Phys. D: Appl. Phys. 48, 443001 (2015).
[Crossref]

K. Huang, H. Liu, F. J. Garcia-Vidal, M. Hong, B. Luk’Yanchuk, J. Teng, and C. W. Qiu, “Ultrahigh-capacity non-periodic photon sieves operating in visible light,” Nat. Commun. 6, 1–7 (2015).
[Crossref]

T. Liu, T. Wang, S. Yang, L. Sun, and Z. Jiang, “Rigorous electromagnetic test of super-oscillatory lens,” Opt. Express 23, 32139 (2015).
[Crossref] [PubMed]

2014 (2)

D. G. Lee and P. J. S. G. Ferreira, “Superoscillations of prescribed amplitude and derivative,” IEEE Trans. Sig. Process. 62, 3371–3378 (2014).
[Crossref]

D. G. Lee and P. J. S. G. Ferreira, “Direct construction of superoscillations,” IEEE Trans. Sig. Process. 62, 3125–3134 (2014).
[Crossref]

2013 (6)

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

A. M. H. Wong and G. V. Eleftheriades, “An Optical Super-Microscope for Far-field, Real-time Imaging Beyond the Diffraction Limit,” Sci. Rep. 3, 1715 (2013).
[Crossref] [PubMed]

E. T. F. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15, 094008 (2013).
[Crossref]

E. T. F. Rogers, S. Savo, J. Lindberg, T. Roy, M. R. Dennis, and N. I. Zheludev, “Super-oscillatory optical needle,” Appl. Phys. Lett. 102, 031108 (2013).
[Crossref]

V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. 52, 330–339 (2013).
[Crossref] [PubMed]

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] [PubMed]

2012 (3)

F. Kenny, D. Lara, O. G. Rodríguez-Herrera, and C. Dainty, “Complete polarization and phase control for focus-shaping in high-NA microscopy,” Opt. Express 20, 14015–14029 (2012).
[Crossref] [PubMed]

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

D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun.  3, 1205 (2012).
[Crossref] [PubMed]

2011 (4)

S. Kosmeier, M. Mazilu, J. Baumgartl, and K. Dholakia, “Enhanced two-point resolution using optical eigenmode optimized pupil functions,” J. Opt. 13, 105707 (2011).
[Crossref]

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phys. Lett. 98, 1–4 (2011).
[Crossref]

M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express 19, 933–945 (2011).
[Crossref] [PubMed]

K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett. 36, 4335 (2011).
[Crossref] [PubMed]

2010 (1)

A. M. H. Wong and G. V. Eleftheriades, “Adaptation of Schelkunoff’s superdirective antenna theory for the realization of superoscillatory antenna arrays,” IEEE Antenn. Wireless Propag. Lett. 9, 315–318 (2010).
[Crossref]

2009 (4)

A. Karoui and T. Moumni, “Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions,” J. Comput. Appl. Math. 233, 315–333 (2009).
[Crossref]

F. M. Huang and N. I. Zheludev, “Super-Resolution without Evanescent Waves,” Nano Lett. 9, 1249–1254 (2009).
[Crossref] [PubMed]

B. R. Boruah and M. A. A. Neil, “Laser scanning confocal microscope with programmable amplitude, phase, and polarization of the illumination beam,” Rev. Sci. Instrum. 80, 013705 (2009).
[Crossref] [PubMed]

X. Wang, J. Fu, X. Liu, and L.-M. Tong, “Subwavelength focusing by a micro/nanofiber array,” J. Opt. Soc. Am. A 26, 1827–1833 (2009).
[Crossref]

2008 (2)

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

A. Grbic, L. Jiang, and R. Merlin, “Near-Field Plates: Subdiffraction Focusing with Patterned Surfaces,” Science 320, 511–513 (2008).
[Crossref] [PubMed]

2007 (2)

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia De Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

P. J. S. G. Ferreira, A. Kempf, and M. J. C. S. Reis, “Construction of Aharonov-Berry’s superoscillations,” J. Phys. A: Math. Theor. 40, 5141–5147 (2007).
[Crossref]

2006 (5)

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

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

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B - Condens. Matter Mater. Phys. 74, 1–5 (2006).
[Crossref]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247 (2006).
[Crossref] [PubMed]

V. F. Canales and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express 14, 10393–402 (2006).
[Crossref] [PubMed]

2005 (1)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited Optical Imaging with a Silver Superlens,” Science 308, 534–537 (2005).
[Crossref] [PubMed]

2003 (2)

D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607 (2003).
[Crossref] [PubMed]

T. Grosjean and D. Courjon, “Polarization filtering induced by imaging systems: effect on image structure,” Phys. Rev. E 67, 046611 (2003).
[Crossref]

2000 (1)

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

1996 (1)

W. Qiao, “A simple model of Aharonov – Berry’s superoscillations,” J. Phys. A: Math. Gen. 29, 2257–2258 (1996).
[Crossref]

1990 (1)

Y. Aharonov and L. Vaidman, “Properties of a Quantum System During the Time Interval Between Two Measurements,” Phys. Rev. A 41, 11–20 (1990).
[Crossref] [PubMed]

1988 (1)

1986 (1)

H. J. Landau, “Extrapolating a Band-Limited Function from its Samples Taken in a Finite Interval,” IEEE Trans. Inf. Theory 32, 464–470 (1986).
[Crossref]

1984 (1)

D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: Image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651–653 (1984).
[Crossref]

1977 (1)

Y. I. Khurgin and V. P. Yakovlev, “Progress in the Soviet Union on the Theory and Applications of Bandlimited Functions,” Proc. IEEE 65, 1005–1029 (1977).
[Crossref]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ and µ,” Sov. Phys. Uspekhi 10, 509–514 (1968).
[Crossref]

1964 (1)

D. Slepian, “Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - IV: Extensions to Many Dimensions; Generalized Prolate Spheroidal Functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
[Crossref]

1952 (1)

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

Aharonov, Y.

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Supplementary Material (1)

NameDescription
» Dataset 1       Data for this paper.

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Figures (6)

Fig. 1
Fig. 1 The CPSWFs as a basis set for superoscillatory spots. (a) Schematic of a sample superoscillatory spot showing the definition of the full width at half maximum (FWHM) and field of view (FoV). (b) Approximations to the target function ft(r = sinc (8) using different numbers, N, of basis functions. The approximation is very close for (N =)15. (c) Intensity profiles |ϕN(r)|2 of the basis functions and (d) their power spectra |ℱ{ϕN (r)}|2. k0 = 2π/λ is the free-space wavenumber. Inset in (c) are the equivalent electric Ffield profiles ϕN(r) inset in (d) are the field spectra ℱ{ϕN (r)}.
Fig. 2
Fig. 2 The Pareto front obtained from the genetic algorithm (dots) and two-function method (rings) for three different FoV: FoV = 1.6λ in (a), FoV = 1λ in (b) and FoV = 0.5λ in (c). The agreement is sufficiently good for the two-function method to be a practical alternative. Three sample spots created by the two-function method are shown in (d), with a FoV of 0.5λ. The corresponding IR are 0.23, 0.88 and 21 with IRinof 0, 0.08 and 0.2 respectively.
Fig. 3
Fig. 3 Demonstration of independent control of spot size and FoV. Selection of spots produced by the two-function method, with variation in spot size along the vertical direction and FoV along the horizontal direction. Any given combination can easily be produced on demand.
Fig. 4
Fig. 4 Comparison of vector and scalar simulations of focal spot formation for a two-function-optimised spot (FWHM=0.27 λ, FoV=0.3 λ). (a) Scalar simulation of the spot. (b) Fully vector simulation, showing that the spot is significantly distorted. Neglecting the longitudinal polarisation (see text), recovers the original spot with minor distortions (c), and when taking into account the vectorial nature during the spot design, a perfect recreation of the scalar approximation can be recovered in the transverse components (d).
Fig. 5
Fig. 5 Airy Disc (a) and Spots 1 R (b), 2 R (c), 3 R (d), 4 R (e), 5 R (f) of Table 1. Intensity maps (i) and profiles (ii) of the spots at the focal plane. Column (iii) shows sample (top) and its image (bottom) in each row, for the Airy Disc (a), Spot 1 R (b), Spot 2 R (c), Spot 3 R (d), Spot 4 R (e) and Spot 5 R (f). All intensity units are arbitrary. The centre–centre separation of the holes is shown. The dashed lines in column (iii) are guides to aid comparison between rows. All images in column (iii) satisfy the Rayleigh criterion of 26.5 % contrast.
Fig. 6
Fig. 6 Spot analysis results. (a) Strehl ratio S, (b) relative intensity (Irel), and (c) relative resolution (Rrel) plotted against relative spot size and relative isolation. Spots with good images are marked with open circles, those with distorted images are marked with stars. Colour shows the relevant parameter as indicated by the colour bars.

Tables (1)

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Table 1 Comparison of two-function method optimised spots (1 R … 5 R ) with optical-eigenmode-optimised spots (Table 1 of [40]; 1 K … 3 K ). Columns headed I.F. contain the improvement factor that has been achieved with the two-function method. Spots 1 R , 2 R , 3 R are compared with spots 1 K , 2 K , 3 K respectively. Spots 4 R and 5 R are both compared with spot 3 K .

Equations (3)

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f N ( r ) = i = 1 i = N A i ϕ i ( r ) where A i = 0 1 sin ( a r π ) a r π ϕ i ( r ) r d r 0 1 ϕ i ( r ) r d r .
IR = max { I ( r ) | r > D } I ( r = 0 )
IR in = max { I ( r ) | q < r D } I ( r = 0 )

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