Abstract

Superoscillation is a phenomenon where a wave oscillates locally faster than its highest Fourier component. While previous reports have shown attractive possibilities for a superoscillation-based far-field superresolution imaging device, it has also been recognized that a high-energy “sideband” region coexists with the superresolution features. This sideband causes strong restrictions and necessitates trade-offs in achievable resolution, viewing area, and sensitivity of the imaging device. In this work, we introduce a new class of superoscillation waveform—which consists of a diffraction-limited hotspot surrounded by low-energy superoscillating sidelobe ripples. This waveform alleviates the aforementioned trade-off and enables superresolution imaging for complex objects over a larger viewing area while maintaining a practical level of sensitivity. Using this waveform as the point spread function of an imaging system, we demonstrate the successful superresolution of Latin letters without performing scanning and/or post-processing operations.

© 2017 Optical Society of America

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  49. G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2014).
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2017 (1)

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

2016 (1)

M. V. Berry, “Suppression of superoscillation by noise,” J. Phys. A Math. Theor. 50, 025003 (2016).
[Crossref]

2015 (2)

D. Tang, C. Wang, Z. Zhao, Y. Wang, M. Pu, X. Li, P. Gao, and X. Luo, “Ultrabroadband superoscillatory lens composed by plasmonic metasurfaces for subdiffraction light focusing,” Laser Photon. Rev. 9, 713–719 (2015).
[Crossref]

A. M. H. Wong and G. V. Eleftheriades, “Superoscillations without sidebands: power efficient sub-diffraction imaging with propagating waves,” Sci. Rep. 5, 8449 (2015).
[Crossref]

2014 (5)

K. Huang, H. Ye, J. Teng, S. P. Yeo, B. Luk’yanchuk, and C. Qiu, “Optimization-free superoscillatory lens using phase and amplitude masks,” Laser Photon. Rev. 8, 152–157 (2014).
[Crossref]

G. Yuan, E. T. F. Rogers, T. Roy, Z. Shen, and N. I. Zheludev, “Flat super-oscillatory lens for heat-assisted magnetic recording with sub-50 nm resolution,” Opt. Express 22, 6428–6437 (2014).
[Crossref]

G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2014).
[Crossref]

T. Roy, E. T. F. Rogers, G. Yuan, and N. I. Zheludev, “Point spread function of the optical needle super-oscillatory lens,” Appl. Phys. Lett. 104, 231109 (2014).
[Crossref]

D. G. Lee and P. J. S. G. Ferreira, “Direct construction of superoscillations,” IEEE Trans. Signal Process. 62, 3215–3224 (2014).
[Crossref]

2013 (4)

R. K. Amineh and G. V. Eleftheriades, “2d and 3d sub-diffraction source imaging with a superoscillatory filter,” Opt. Express 21, 8142–8156 (2013).
[Crossref]

M. V. Berry, “Exact nonparaxial transmission of subwavelength detail using superoscillations,” J. Phys. A Math. Theor. 46, 205203 (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]

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]

2012 (3)

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]

H. J. Hyvärinen, S. Rehman, J. Tervo, J. Turunen, and C. J. R. Sheppard, “Limitations of superoscillation filters in microscopy applications,” Opt. Lett. 37, 903–905 (2012).
[Crossref]

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11, 455–459 (2012).
[Crossref]

2011 (5)

K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett. 36, 4335–4337 (2011).
[Crossref]

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]

A. M. H. Wong and G. V. Eleftheriades, “Sub-wavelength focusing at the multi-wavelength range using superoscillations: an experimental demonstration,” IEEE Trans. Antennas Propag. 59, 4766–4776 (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, 181109 (2011).
[Crossref]

A. M. H. Wong and G. V. Eleftheriades, “Temporal pulse compression beyond the Fourier transform limit,” IEEE Trans. Microw. Theory Tech. 59, 2173–2179 (2011).
[Crossref]

2010 (2)

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

T. Brunet, J. L. Thomas, and R. Marchiano, “Transverse shift of helical beams and subdiffraction imaging,” Phys. Rev. Lett. 105, 034301 (2010).
[Crossref]

2009 (1)

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

2008 (3)

2007 (1)

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

2006 (5)

P. J. S. G. Ferreira and A. Kempf, “Superoscillations: faster than the Nyquist rate,” IEEE Trans. Signal 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 39, 6965–6977 (2006).
[Crossref]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006).
[Crossref]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (storm),” Nat. Methods 3, 793–796 (2006).
[Crossref]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

2000 (1)

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

1997 (1)

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[Crossref]

1994 (1)

1990 (2)

Y. Aharonov, J. Anandan, S. Popescu, and L. Vaidman, “Superpositions of time evolutions of a quantum system and a quantum time-translation machine,” Phys. Rev. Lett. 64, 2965–2968 (1990).
[Crossref]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[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]

1966 (1)

1961 (1)

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

1952 (1)

G. T. di Francia, “Super-gain antennas and optical resolving power,” Il Nuovo Cimento (1943-1954) 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]

1928 (1)

E. H. Synge, “A suggested method for extending microscopic resolution into the ultra-microscopic region,” Philos. Mag. 6(35), 356–362 (1928).
[Crossref]

1896 (1)

Lord Rayleigh, “On the theory of optical images, with special reference to the microscope,” Philos. Mag. 42(255), 167–195 (1896).
[Crossref]

1873 (1)

E. Abbé, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv. f. mickroskopische anat. 9, 413–418 (1873).
[Crossref]

Abbé, E.

E. Abbé, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv. f. mickroskopische anat. 9, 413–418 (1873).
[Crossref]

Adamo, G.

G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2014).
[Crossref]

Aharonov, Y.

Y. Aharonov, J. Anandan, S. Popescu, and L. Vaidman, “Superpositions of time evolutions of a quantum system and a quantum time-translation machine,” Phys. Rev. Lett. 64, 2965–2968 (1990).
[Crossref]

Alekseyev, L. V.

Amineh, R. K.

Anandan, J.

Y. Aharonov, J. Anandan, S. Popescu, and L. Vaidman, “Superpositions of time evolutions of a quantum system and a quantum time-translation machine,” Phys. Rev. Lett. 64, 2965–2968 (1990).
[Crossref]

Balanis, C. A.

C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. (Wiley, 2005).

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (storm),” Nat. Methods 3, 793–796 (2006).
[Crossref]

Baumgartl, J.

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, 181109 (2011).
[Crossref]

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]

Berry, M. V.

M. V. Berry, “Suppression of superoscillation by noise,” J. Phys. A Math. Theor. 50, 025003 (2016).
[Crossref]

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

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

M. V. Berry, “Faster than Fourier,” in Quantum Coherence and Reality: in Celebration of the 60th Birthday of Yakir Aharonov (World Scientific, 1994), pp. 55–65.

Betzig, E.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Brunet, T.

T. Brunet, J. L. Thomas, and R. Marchiano, “Transverse shift of helical beams and subdiffraction imaging,” Phys. Rev. Lett. 105, 034301 (2010).
[Crossref]

Bullkich, E.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11, 455–459 (2012).
[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. Mater. 11, 432–435 (2012).
[Crossref]

Chen, Y.

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

Cohen, O.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11, 455–459 (2012).
[Crossref]

Cohen-Hyams, T.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11, 455–459 (2012).
[Crossref]

Considine, P. S.

Courtial, J.

Dana, H.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11, 455–459 (2012).
[Crossref]

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

De Abajo, F. J. G.

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

Denk, W.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[Crossref]

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

Dennis, M. R.

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]

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Supplementary Material (1)

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Figures (10)

Fig. 1.
Fig. 1.

(a) 2D waveform of a design using 30 zeros, 10 of which are in a region of interest (ROI) of 1.4λ/NA. Major features of a superoscillating waveform are labeled. (b) ROI of the design. The main beam (green solid line) is kept the same as the diffraction-limited sinc function (black dotted line), but the ROI ripples are 6.5 times lower than the diffraction limit. As a trade-off, a sideband appears whose amplitude is an order of magnitude larger than the main beam.

Fig. 2.
Fig. 2.

Superoscillation PSF design using the Chebyshev method. (a) Distribution of 32 waveform zeros in the complex z plane for 1D superoscillation design [see Eq. (3)] with 10 zeros constrained within a ROI of half-width 1.9λ/NA. The edges of the ROI are labeled with red lines. (b) Bessel function weights for the equivalent 2D superoscillation design. The nulls of the superposition of Bessel beams are the same as (a) but in the radial direction [21]. (c) Zoomed-in view of the ROI of the resulting 2D superoscillation waveform (green solid line), compared to the diffraction-limited Airy sinc function (black dashed line). The ROI ripple amplitudes are 4 times lower, main beam is 5% wider, and the ROI ripples are oscillating faster than the diffraction-limited Airy disk. (d) Comparison of cross-sections of measured PSF (black dashed line) and designed PSF (green solid line). Good agreement is obtained.

Fig. 3.
Fig. 3.

Design and simulation of the binomial superoscillatory PSF. (a) 10 zeros are placed in an ROI with target half-width of 1.9λ/NA. The edges of the ROI are labeled with the red lines. (b) Bessel function weights for the equivalent 2D superoscillation design. (c) Designed binomial waveform (green solid line) compared with the diffraction-limited sinc function (black dashed line) in the ROI, with ripples completely suppressed in the region close to the main beam. (d) Designed binomial waveform has sidebands 1 order of magnitude larger than the main beam.

Fig. 4.
Fig. 4.

Simulation results of imaging the letters E and N using the PSF seen in Fig. 2. The letter E has the dimension 110  μm×87  μm. The letter N has the dimension 120  μm×130  μm. The diffraction-limited results are shown in (a) and (c). The superresolved results are shown in (b) and (d). Significant improvement in resolution is seen.

Fig. 5.
Fig. 5.

Simulation results of a letter E larger than ROI half-width of the PSF designed in Fig. 2 and significantly above the diffraction limit. (a) shows the diffraction limited image, which is clearly resolved. The superoscillatory image in (b) is significantly worse due to interference from the sideband of the PSF.

Fig. 6.
Fig. 6.

Schematic of the experimental setup. A 633 nm He–Ne laser is used to illuminate the object in the object plane. The wave propagates through a 4F system and into a Thorlabs DCC1240 scientific camera. A spatial light modulator is placed in the Fourier plane to allow direct access to the PSF of the imaging system.

Fig. 7.
Fig. 7.

Exact ring structure, with normalized modulation coefficients, which is displayed on the SLM. The total diameter of the ring mask is 6.9 mm.

Fig. 8.
Fig. 8.

(a) Simulated PSF intensity. (b) Measured PSF Intensity. Excellent agreement is obtained.

Fig. 9.
Fig. 9.

Experimental results of imaging a letter E of 110  μm×87  μm and a letter N of dimension 120  μm×130  μm. The total imaging system numerical aperture is 0.00864. (a) and (b) Camera images using diffraction-limited PSF. (c) and (d) Camera images using superoscillatory PSF designed as shown in Fig. 2b. (e) and (g) Zoomed-in images using the diffraction-limited PSFs. (f) and (h) Camera images zoomed into the superoscillation ROI where resolution is improved. The three horizontal bars of the letter E have become more visible. The outer rings are due to the PSF sidebands outside the ROI of superoscillation. Compared to the diffraction-limited case of the letter N, the intensities are much more evenly distributed across the letter, and the vertical bars are more visible.

Fig. 10.
Fig. 10.

Sensitivity analysis by adding Gaussian noise to the image system simulation. In (a), Gaussian noise with an SNR of 8 dB is applied to the coefficients. The resulting image is shown in (b). White noise with an SNR of 3 dB is applied directly to the PSF in (c), with the resulting image in (d).

Equations (10)

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Ui(u,v)=yminymaxxminxmaxUg(x,y)h(ux,vy)dxdy,
En(x)=ejΔkxnxz^.
E(x)=n=N/2N/2anzn=b0n=1N(zzn),
h(r)=n=0N/2bnJ0(krnr),
xn=log(zn)jΔkx.
xn=arg(zn)Δkx.
n=0N/2bnJ0(krnrm)=0
h(u)=c0+2n=1ZcnTn(u),
Tn(u)=cos(ncos1(u)).
h(u)=n=1N(zzn)=(1+z)N,