## Experimental generation of arbitrarily shaped diffractionless superoscillatory optical beams |

Optics Express, Vol. 21, Issue 11, pp. 13425-13435 (2013)

http://dx.doi.org/10.1364/OE.21.013425

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### 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

## Introduction

13. K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett. **36**(22), 4335–4337 (2011). [CrossRef] [PubMed]

**any**equations at all. Last, we explore the power-transmittance efficiency of the sub-diffraction limited features, as function of their spatial extent, feature size, and Fourier decomposition. We present general conditions for generating arbitrarily shaped superoscillations approximating a known function in space to finite Taylor and/or Fourier expansion order. The spatial extent of the superoscillation, and the quality of the arbitrary shaping come at the expense of the power transmitted with the diffractionless superoscillatory feature. The method we present is general, allowing straight forward engineering of diffractionless superoscillations with predesigned parameters. As examples, we design arrays of diffractionless superoscillations shaped as sinusoidal and rectangular waveforms.

## Methodology

*E(r,z)*is

*m*= 2,

*l*= 2 in Eq. (1). Their centers can be arbitrarily set on the x axis, and their relative phase can be set to

*E(x,z)*has a maximum at

*x'*can be made arbitrarily small, at the expense of the reduction in the power carried by this feature. Namely, since

*y*direction) carried by this small feature during propagation along the

*z*direction scales as

## Experimental

*z*direction, illuminating a phase mask. The mask carries the phase

*m*is generally an integer. In this example

*m*= 2. The Axicon in our system has a base angle of 2° and a refractive index ~1.48,hence a light ray incident perpendicularly exits the Axicon with an angle of ~0.96°. The numerical aperture (NA) of our system is thus 0.0168. The diffraction limit of our system, given by

*x*-axis, by choosing

*D*(Fig. 2 and inset). The peak intensity of the superoscillatory feature decays with the 4th power of its width, in accordance with the predictions above.

*x*-axis connecting the centers of the two 2nd order Bessel beams in the

*xy*plane. In Fig. 3(b)-3(c), we show that this beam is indeed shape-preserving: the Superoscillatory feature stays perfectly intact, maintaining its widths, while propagating for distances of over 250 Rayleigh lengths.

## Arbitrarily shaped superoscillations

*l*instances of a

*m*

^{th}order Bessel beam are now degenerate, and the field can now be written as

13. K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett. **36**(22), 4335–4337 (2011). [CrossRef] [PubMed]

13. K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett. **36**(22), 4335–4337 (2011). [CrossRef] [PubMed]

**without the need for equation-solving at all.**Choosing as before the axis of the superoscillation as

*x*, we approximate the field in Eq. (4) for

*x*-axis iswhere we set

^{th}order polynomial term

*m*. Numerical comparison between superoscillations designed using Eq [5

5. M. V. Berry and M. R. Dennis, “Natural superoscillations in monochromatic waves in D dimensions,” J. Phys. A **42**(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]

*f(x)*, at least on the

*x*-axis, we expand the field in a Taylor series,

*f(x)*can be generated as a superposition of Bessel beams, such as

*m*contributes the

*m*

^{th}order term of the Taylor's expansion to the desired superoscillatory field. Hence, it is clear that higher order Bessel beams are required for extending the spatial support of the superoscillatory region of these non-broadening wavepackets. However, Eq. (9) also includes multiplication of the m

^{th}order beam by

*R*is roughly a measure of the fraction of power that can be carried by the superoscillatory region of the propagation-invariant beam. It is then apparent that extending the spatial support of the superoscillation requires higher orders of the Taylor expansion for

*f(x)*, and this comes at the expense of an exponential decrease in the power the superoscillation. Furthermore, shrinking the entire superoscillatory region (approximated by the

*m*

^{th}order Taylor's series) by a factor of

*r*increases

*R*by a factor of

*m*increases. Figure 4(a) shows the generation of two sinusoidal superoscillations. Here we calculate the superposition of Bessel beams up to order 3. Extending the superoscillatory region to have 6 superoscillations (Fig. 4(b)) requires 19 orders of the Taylor series of the sinusoidal waveform, hence a Bessel beam superposition up to order 19.

*f(x)*is a rectangular waveform with a period of length

*L*. We expand this waveform in a sine-series, and each sine in a Taylor series, to obtain

*(2k + 1)*

^{th}order Bessel beam is

*n*, and the superoscillation's spatial support, indicated by the Taylor order

*k*. Namely, for spatial support given by a known highest order in the Taylor series

*k*, the power scales roughly as

## Discussion and conclusion

## Acknowledgments

## References and links:

1. | A. Lipson, S. G. Lipson, and H. Lipson, |

2. | G. T. di Francia, “Supergain antennas and optical resolving power,” Nuovo Cim. |

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. |

4. | M. V. Berry, “Faster than Fourier,” in ' |

5. | M. V. Berry and M. R. Dennis, “Natural superoscillations in monochromatic waves in D dimensions,” J. Phys. A |

6. | M. R. Dennis, A. C. Hamilton, and J. Courtial, “Superoscillation in speckle patterns,” Opt. Lett. |

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. |

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. Express |

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. |

10. | F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent Waves,” Nano Lett. |

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. |

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. |

13. | K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett. |

14. | J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” J. Opt. Soc. Am. A |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(070.3185) Fourier optics and signal processing : Invariant optical fields

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: April 25, 2013

Revised Manuscript: May 18, 2013

Manuscript Accepted: May 22, 2013

Published: May 28, 2013

**Citation**

Elad Greenfield, Ran Schley, Ilan Hurwitz, Jonathan Nemirovsky, Konstantinos G. Makris, and Mordechai Segev, "Experimental generation of arbitrarily shaped diffractionless superoscillatory optical beams," Opt. Express **21**, 13425-13435 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-11-13425

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### References

- A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics (Cambridge University Press, Cambridge; New York, 2011).
- G. T. di Francia, “Supergain antennas and optical resolving power,” Nuovo Cim.9(S3), 426–438 (1952). [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(14), 1351–1354 (1988). [CrossRef] [PubMed]
- 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).
- 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]
- 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]
- 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]
- F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent Waves,” Nano Lett.9(3), 1249–1254 (2009). [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(5), 432–435 (2012). [CrossRef] [PubMed]
- 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]
- K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett.36(22), 4335–4337 (2011). [CrossRef] [PubMed]
- J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free Beams,” J. Opt. Soc. Am. A3, 128–P128 (1986).

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