Abstract

We present, theoretically and experimentally, diffractionless optical beams displaying arbitrarily-shaped sub-diffraction-limited features known as superoscillations. We devise an analytic method to generate such beams and experimentally demonstrate optical superoscillations propagating without changing their intensity distribution for distances as large as 250 Rayleigh lengths. Finally, we find the general conditions on the fraction of power that can be carried by these superoscillations as function of their spatial extent and their Fourier decomposition. Fundamentally, these new type of beams can be utilized to carry sub-wavelength information for very large distances.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics (Cambridge University Press, Cambridge; New York, 2011).
  2. G. T. di Francia, “Supergain antennas and optical resolving power,” Nuovo Cim.9(S3), 426–438 (1952).
    [CrossRef]
  3. 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(14), 1351–1354 (1988).
    [CrossRef] [PubMed]
  4. M. V. Berry, “Faster than Fourier,” in 'Quantum Coherence and Reality; in Celebration of the 60th Birthday of Yakir Aharonov' (J. S. Anandan and J. L. Safko, Eds.) World Scientific, Singapore, pp 55–65 (1994).
  5. M. V. Berry and M. R. Dennis, “Natural superoscillations in monochromatic waves in D dimensions,” J. Phys. A42(2), 022003 (2009).
    [CrossRef]
  6. M. R. Dennis, A. C. Hamilton, and J. Courtial, “Superoscillation in speckle patterns,” Opt. Lett.33(24), 2976–2978 (2008).
    [CrossRef] [PubMed]
  7. 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(18), 181109 (2011).
    [CrossRef]
  8. M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express19(2), 933–945 (2011).
    [CrossRef] [PubMed]
  9. F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A, Pure Appl. Opt.9(9), S285–S288 (2007).
    [CrossRef]
  10. F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent Waves,” Nano Lett.9(3), 1249–1254 (2009).
    [CrossRef] [PubMed]
  11. 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(5), 432–435 (2012).
    [CrossRef] [PubMed]
  12. 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(3), 031108 (2013).
    [CrossRef]
  13. K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett.36(22), 4335–4337 (2011).
    [CrossRef] [PubMed]
  14. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” J. Opt. Soc. Am. A3, 128–P128 (1986).

2013 (1)

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(3), 031108 (2013).
[CrossRef]

2012 (1)

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(5), 432–435 (2012).
[CrossRef] [PubMed]

2011 (3)

2009 (2)

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent Waves,” Nano Lett.9(3), 1249–1254 (2009).
[CrossRef] [PubMed]

M. V. Berry and M. R. Dennis, “Natural superoscillations in monochromatic waves in D dimensions,” J. Phys. A42(2), 022003 (2009).
[CrossRef]

2008 (1)

2007 (1)

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A, Pure Appl. Opt.9(9), S285–S288 (2007).
[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(14), 1351–1354 (1988).
[CrossRef] [PubMed]

1986 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” J. Opt. Soc. Am. A3, 128–P128 (1986).

1952 (1)

G. T. di Francia, “Supergain antennas and optical resolving power,” Nuovo Cim.9(S3), 426–438 (1952).
[CrossRef]

Aharonov, Y.

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(14), 1351–1354 (1988).
[CrossRef] [PubMed]

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(14), 1351–1354 (1988).
[CrossRef] [PubMed]

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(18), 181109 (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. Express19(2), 933–945 (2011).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry and M. R. Dennis, “Natural superoscillations in monochromatic waves in D dimensions,” J. Phys. A42(2), 022003 (2009).
[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(5), 432–435 (2012).
[CrossRef] [PubMed]

Chen, Y.

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A, Pure Appl. Opt.9(9), S285–S288 (2007).
[CrossRef]

Courtial, J.

de Abajo, F. J. G.

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A, Pure Appl. Opt.9(9), S285–S288 (2007).
[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(3), 031108 (2013).
[CrossRef]

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(5), 432–435 (2012).
[CrossRef] [PubMed]

M. V. Berry and M. R. Dennis, “Natural superoscillations in monochromatic waves in D dimensions,” J. Phys. A42(2), 022003 (2009).
[CrossRef]

M. R. Dennis, A. C. Hamilton, and J. Courtial, “Superoscillation in speckle patterns,” Opt. Lett.33(24), 2976–2978 (2008).
[CrossRef] [PubMed]

Dholakia, K.

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(18), 181109 (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. Express19(2), 933–945 (2011).
[CrossRef] [PubMed]

di Francia, G. T.

G. T. di Francia, “Supergain antennas and optical resolving power,” Nuovo Cim.9(S3), 426–438 (1952).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” J. Opt. Soc. Am. A3, 128–P128 (1986).

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” J. Opt. Soc. Am. A3, 128–P128 (1986).

Hamilton, A. C.

Huang, F. M.

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent Waves,” Nano Lett.9(3), 1249–1254 (2009).
[CrossRef] [PubMed]

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A, Pure Appl. Opt.9(9), S285–S288 (2007).
[CrossRef]

Kosmeier, S.

M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express19(2), 933–945 (2011).
[CrossRef] [PubMed]

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(18), 181109 (2011).
[CrossRef]

Lindberg, J.

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(3), 031108 (2013).
[CrossRef]

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(5), 432–435 (2012).
[CrossRef] [PubMed]

Makris, K. G.

Mazilu, M.

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(18), 181109 (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. Express19(2), 933–945 (2011).
[CrossRef] [PubMed]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” J. Opt. Soc. Am. A3, 128–P128 (1986).

Psaltis, D.

Rogers, E. T. F.

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(3), 031108 (2013).
[CrossRef]

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(5), 432–435 (2012).
[CrossRef] [PubMed]

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(18), 181109 (2011).
[CrossRef]

Roy, T.

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(3), 031108 (2013).
[CrossRef]

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(5), 432–435 (2012).
[CrossRef] [PubMed]

Savo, S.

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(3), 031108 (2013).
[CrossRef]

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(5), 432–435 (2012).
[CrossRef] [PubMed]

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(14), 1351–1354 (1988).
[CrossRef] [PubMed]

Zheludev, N. I.

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(3), 031108 (2013).
[CrossRef]

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(5), 432–435 (2012).
[CrossRef] [PubMed]

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(18), 181109 (2011).
[CrossRef]

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent Waves,” Nano Lett.9(3), 1249–1254 (2009).
[CrossRef] [PubMed]

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A, Pure Appl. Opt.9(9), S285–S288 (2007).
[CrossRef]

Appl. Phys. Lett. (2)

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(18), 181109 (2011).
[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(3), 031108 (2013).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A, Pure Appl. Opt.9(9), S285–S288 (2007).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” J. Opt. Soc. Am. A3, 128–P128 (1986).

J. Phys. A (1)

M. V. Berry and M. R. Dennis, “Natural superoscillations in monochromatic waves in D dimensions,” J. Phys. A42(2), 022003 (2009).
[CrossRef]

Nano Lett. (1)

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent Waves,” Nano Lett.9(3), 1249–1254 (2009).
[CrossRef] [PubMed]

Nat. Mater. (1)

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(5), 432–435 (2012).
[CrossRef] [PubMed]

Nuovo Cim. (1)

G. T. di Francia, “Supergain antennas and optical resolving power,” Nuovo Cim.9(S3), 426–438 (1952).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. Lett. (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(14), 1351–1354 (1988).
[CrossRef] [PubMed]

Other (2)

M. V. Berry, “Faster than Fourier,” in 'Quantum Coherence and Reality; in Celebration of the 60th Birthday of Yakir Aharonov' (J. S. Anandan and J. L. Safko, Eds.) World Scientific, Singapore, pp 55–65 (1994).

A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics (Cambridge University Press, Cambridge; New York, 2011).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Schematic of the experimental setup (not to scale). A continuous wave laser beam is spatially filtered (spatial filter 'SF') and is subsequently broadened and re-collimated, and then split in two, each separate branch acquiring a spiral phase mϕ when passing through phase plate 'VPP1',' VPP2. The lower branch passes through an attenuator ('AT') which sets the ratio of intensities between the beams. The upper branch can be laterally shifted in the xy plane using a telescopic system mounted on a 3D micrometric stage. The beams are recombined and pass through an axicon 'AX'. The resultant two Bessel beams are then superimposed to give a superoscillatory non-broadening beam. The beam is measured with a camera ('C') equipped with a microscope objective. The camera can slide along the z-axis, obtaining images of the xy shape of the beam at different propagation planes. Inset: Coordinate system used throughout this paper.

Fig. 2
Fig. 2

Experimental demonstration of optical shape-preserving beams having sub-diffraction-limited superoscillatory (SO) features. An asymmetric superposition of 2nd order Bessel beams (whose intensity distribution transverse to the propagation direction is shown at the lower right corner) exhibits line-shaped superoscillatory features. The intensity distributions of three such superoscillations are magnified and presented in 3D layouts surrounded with red rectangles on top. One dimensional cross sections of these superoscillations along the black dashed lines are presented above the 3D plots. The presented features (whose RMS width is as small as 4µm) are significantly smaller than the diffraction limit of the system, 19µm. Features as small as 2.5µm were also measured (see inset). We demonstrate control over the width of the feature, decreasing it from 50% (upper left) and down to 20% (upper right) of the diffraction limit. The intensity of the features decays accordingly. Inset: Power carried by the superoscillatory feature, exhibiting decay with the 4th power of its width, in accordance with our analytic derivation.

Fig. 3
Fig. 3

(a) Experiments displaying designed rotation of an elongated superoscillation at three different orientations, (i-iii). The black arrows mark the direction of the long dimension of the superoscillation. (b) Shape-preserving propagation of the superoscillatory feature: The feature is measured at z = 0 (left) and after propagating 250 Rayleigh lengths (ZR) (middle), maintaining its width while propagating. The lower images are 2D intensity distributions of the superoscillatory features, and the graphs above them are intensity cross-sections of those distributions, taken along the blue dashed line. The propagation dynamics show good agreement with theoretical prediction (right). (c) Detailed propagation dynamics, showing the cross-section of the superoscillation as it propagates along the propagation axis for 250 Rayleigh lengths. The beam intensity is normalized to unity at each plane for clarity of observation. Inset: width of the superoscillation between its zeros, as function of propagation distance – exhibiting almost a flat dependence. The superoscillations exhibits some minor focusing as it propagates, due to slight angular misalignment of the beams.

Fig. 4
Fig. 4

Designing non-diffraction superoscillations with pre-determined shapes. (a) Setting f(x)=sin(ax) , 'a' being a constant, we construct a beam with two sinusoidal intensity features, each of width ~ D 0 /7 , D 0 being the diffraction limit of the system. Here the superposition is of Bessel Beams up to m = 3. The lower graph shows the 2D intensity distribution, and the upper graph is its horizontal cross-section. (b) Extending the spatial support of the superoscillatory region to include 6 sinusiodal superoscillations requires superposition of Bessel beams up to order 19. (c) Generating a rectangular array of superoscillatory features through superposition of Bessel beams up to order 80, and Fourier decomposition of f(x) up to order 3. (d) Ratio between the peak intensity of the beam to the intensity of the superoscillatory region, as a function of the spatial support of the superoscillatory region. This dependence is roughly exponential, as expected.

Equations (13)

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

E(r,θ,z)= e ikz m=0 N l=0 l m a lm J m ( k r | r r lm | ) e im θ lm ,
E(x,z)= a 21 J 2 1 ( k r x) e ikz a 22 J 2 2 ( k r (xx')) e ikz ,
E(x,z) e ikz 2! [ a 21 ( k r x 2 ) 2 a 22 ( k r (xx') 2 ) 2 ].
E( r,z )= e ikz m=0 N a m J m ( k r r ) e imθ
E(x,z)= m=0 m=N a m s=0 ( 1 ) s s!( m+s )! ( k r x 2 ) m+2s = e ikz j=0 ( k=0 j/2 ( 1 ) k a j2k k!( jk )! 2 j ) ( k r x ) j ,
E(x,z) e ikz m=0 m=N a m m! [ k r x 2 ] m .
f(x)= m b m x m , m=0,1,2...
a m = m! b m 2 m [ k r ] m .
E(x,z)= e ikz m=0 m=N m! b m 2 m [ k r ] m J m ( k r x).
R I SO / I beam ~ 2 2N .
f(x)= n=1,3,5... c n sin( 2nπ L x) = n=1,3,5... 4 πn k=0 (1) k (2k+1)! ( 2nπx L ) 2k+1 = k=0 b 2k+1 x 2k+1
b 2k+1 = n=1,3,5... 4 πn (1) k (2k+1)! ( 2nπ L ) 2k+1 ,
E(x,z)= e ikz k=1,3,5... N k! b k 2 k [ k r ] k J k ( k r x)= = k=1,3,5... N k! 2 k [ k r ] k n=1,3,5,... 4 πn (1) k (2k+1)! ( 2nπ L ) 2k+1 J k ( k r x)

Metrics