## Analytical light scattering and orbital angular momentum spectra of arbitrary Vogel spirals |

Optics Express, Vol. 20, Issue 16, pp. 18209-18223 (2012)

http://dx.doi.org/10.1364/OE.20.018209

Acrobat PDF (2084 KB)

### Abstract

In this paper, we present a general analytical model for light scattering by arbitrary Vogel spiral arrays of circular apertures illuminated at normal incidence. This model suffices to unveil the fundamental mathematical structure of their complex Fraunhofer diffraction patterns and enables the engineering of optical beams carrying multiple values of orbital angular momentum (OAM). By performing analytical Fourier-Hankel decomposition of spiral arrays and far field patterns, we rigorously demonstrate the ability to encode specific numerical sequences onto the OAM values of diffracted optical beams. In particular, we show that these OAM values are determined by the rational approximations (i.e., the convergents) of the continued fraction expansions of the irrational angles utilized to generate Vogel spirals. These findings open novel and exciting opportunities for the manipulation of complex OAM spectra using dielectric and plasmonic aperiodic spiral arrays for a number of emerging engineering applications in singular optics, secure communication, optical cryptography, and optical sensing.

© 2012 OSA

## 1. Introduction

1. A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, “Golden spiral photonic crystal fiber: polarization and dispersion properties,” Opt. Lett. **33**(22), 2716–2718 (2008). [CrossRef]

7. N. Lawrence, J. Trevino, and L. Dal Negro, “Aperiodic arrays of active nanopillars for radiation engineering,” J. Appl. Phys. **111**(11), 113101 (2012). [CrossRef]

6. J. Trevino, S. F. Liew, H. Noh, H. Cao, and L. Dal Negro, “Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals,” Opt. Express **20**(3), 3015–3033 (2012). [CrossRef] [PubMed]

8. J. Trevino, C. Forestiere, G. Di Martino, S. Yerci, F. Priolo, and L. Dal Negro, “Plasmonic-photonic arrays with aperiodic spiral order for ultra-thin film solar cells,” Opt. Express **20**(S3), A418–A430 (2012). [CrossRef] [PubMed]

7. N. Lawrence, J. Trevino, and L. Dal Negro, “Aperiodic arrays of active nanopillars for radiation engineering,” J. Appl. Phys. **111**(11), 113101 (2012). [CrossRef]

9. E. F. Pecora, N. Lawrence, P. Gregg, J. Trevino, P. Artoni, A. Irrera, F. Priolo, and L. Dal Negro, “Nanopatterning of silicon nanowires for enhancing visible photoluminescence,” Nanoscale **4**(9), 2863–2866 (2012). [CrossRef] [PubMed]

10. A. Capretti, G. F. Walsh, S. Minissale, J. Trevino, C. Forestiere, G. Miano, and L. Dal Negro, “Multipolar second harmonic generation from planar arrays of Au nanoparticles with aperiodic order,” Opt. Express **20**(14), 15797–15806 (2012). [CrossRef] [PubMed]

4. J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett. **11**(5), 2008–2016 (2011). [CrossRef] [PubMed]

2. M. E. Pollard and G. J. Parker, “Low-contrast bandgaps of a planar parabolic spiral lattice,” Opt. Lett. **34**(18), 2805–2807 (2009). [CrossRef] [PubMed]

5. S. F. Liew, H. Noh, J. Trevino, L. D. Negro, and H. Cao, “Localized photonic bandedge modes and orbital angular momenta of light in a golden-angle spiral,” Opt. Express **19**(24), 23631–23642 (2011). [CrossRef] [PubMed]

6. J. Trevino, S. F. Liew, H. Noh, H. Cao, and L. Dal Negro, “Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals,” Opt. Express **20**(3), 3015–3033 (2012). [CrossRef] [PubMed]

*et al.*[5

5. S. F. Liew, H. Noh, J. Trevino, L. D. Negro, and H. Cao, “Localized photonic bandedge modes and orbital angular momenta of light in a golden-angle spiral,” Opt. Express **19**(24), 23631–23642 (2011). [CrossRef] [PubMed]

*Trevino et al.*[6

6. J. Trevino, S. F. Liew, H. Noh, H. Cao, and L. Dal Negro, “Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals,” Opt. Express **20**(3), 3015–3033 (2012). [CrossRef] [PubMed]

5. S. F. Liew, H. Noh, J. Trevino, L. D. Negro, and H. Cao, “Localized photonic bandedge modes and orbital angular momenta of light in a golden-angle spiral,” Opt. Express **19**(24), 23631–23642 (2011). [CrossRef] [PubMed]

13. M. Naylor, “Golden, √2, and π flowers: a spiral story,” Math. Mag. **75**(3), 163–172 (2002). [CrossRef]

7. N. Lawrence, J. Trevino, and L. Dal Negro, “Aperiodic arrays of active nanopillars for radiation engineering,” J. Appl. Phys. **111**(11), 113101 (2012). [CrossRef]

14. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science **334**(6054), 333–337 (2011). [CrossRef] [PubMed]

## 2. Fraunhofer diffraction by Vogel spiral point patterns

*N*is the total number of scatterers in the spiral array and the

*m*-th order Bessel function of the first kind. The variables

*m*-th order Hankel transform of

## 3. Analytical diffraction by Vogel arrays of circular apertures

*a,*the Fraunhofer diffraction pattern is therefore given by:where the square brackets in Eq. (13) contains an Airy factor, which is the Fourier transform of the circular function

## 4. Analytical Fourier-Hankel decomposition of Vogel spiral geometry

*m*runs across the integers. Therefore, GA spirals are defined by a divergence angle

*convergents*of the continued fraction, and it can be shown that even-numbered convergents are smaller than the original number ζ while odd-numbered ones are bigger [16,17]. Once the continued fraction expansion of ζ has been obtained, well-defined recursion rules exist to quickly generate the successive convergents. In fact, each convergent can be expressed explicitly in terms of the continued fraction as the ratio of certain multivariate polynomials called

*continuants*[16,17].

*m*(i.e, Bessel order

*m*) will appear in its FHD due to the denominators

^{o}) listed in Table 1 . The table also shows their rational approximations and, in its last column, the parameter E/M.

## 4. Analytical Fourier-Hankel decomposition of diffracted far field

*a*under uniform illumination can be expressed in closed form by:We will now perform the FHD of the field in Eq. (19) in order to establish a connection with the spiral geometry and the OAM spectra of far field scattered radiation. The FHD of the complex far field in Eq. (19) is given by: Now we can eliminate the integral over

_{r}-dependent coefficients, Eq. (24) displays the same azimuthal structure of Eq. (16) characterizing the FHD of the spiral geometry. In particular, we see from Eq. (24) that the complex far field scattered by Vogel spirals carries well-defined values of discrete OAM uniquely determined by the irrationality of the angle α in the exponential sum. This demonstrates analytically that arbitrary aperiodic Vogel spiral arrays can serve to encode specific numerical information about the rational approximants of ζ into the discrete OAM spectra, provided by the FHD of the far fields, of the scattered radiation.

_{r}for visualization convenience. Figure 3 demonstrates that very rich structures of discrete OAM spectra can be carried by the scattered far fields of Vogel spirals.

## 5. Analytical scattering calculation of diffracted field at a general observation plane

*z*=

*d*can be written as:where

*m-*th order Bessel functions:where

*z*=

*d*can be expressed from Eq. (29) as:Using Eq. (33), Eq. (34) can be reduced to:Now we can further simplify the integral in Eq. (34) by using the

*Gegenbauer theorem*for cylindrical functions:In fact, continuing from Eq. (35) we have: which is the main closed form expression for the scattered field obtained after the following identifications have been made:Equations (38) and (39) are the analytical solutions for the transverse field probed at

*z = d*scattered by an arbitrary Vogel spiral of point scatterers. If a spiral pattern of circular apertures of radius

*a*is considered instead, the general solution can be easily modified to obtain:Equation (40) explicitly contains the multiplicative Airy factor

## Comment on the numerical implementation of near and intermediate field solutions

## 6. Conclusions

## Acknowledgments

*Deterministic Aperiodic Structures for On-chip Nanophotonic and Nanoplasmonic Device Applications*,” under Award FA9550-10-1-0019, by the NSF Career Award No. ECCS-0846651.

## References and links

1. | A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, “Golden spiral photonic crystal fiber: polarization and dispersion properties,” Opt. Lett. |

2. | M. E. Pollard and G. J. Parker, “Low-contrast bandgaps of a planar parabolic spiral lattice,” Opt. Lett. |

3. | L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev. |

4. | J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett. |

5. | S. F. Liew, H. Noh, J. Trevino, L. D. Negro, and H. Cao, “Localized photonic bandedge modes and orbital angular momenta of light in a golden-angle spiral,” Opt. Express |

6. | J. Trevino, S. F. Liew, H. Noh, H. Cao, and L. Dal Negro, “Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals,” Opt. Express |

7. | N. Lawrence, J. Trevino, and L. Dal Negro, “Aperiodic arrays of active nanopillars for radiation engineering,” J. Appl. Phys. |

8. | J. Trevino, C. Forestiere, G. Di Martino, S. Yerci, F. Priolo, and L. Dal Negro, “Plasmonic-photonic arrays with aperiodic spiral order for ultra-thin film solar cells,” Opt. Express |

9. | E. F. Pecora, N. Lawrence, P. Gregg, J. Trevino, P. Artoni, A. Irrera, F. Priolo, and L. Dal Negro, “Nanopatterning of silicon nanowires for enhancing visible photoluminescence,” Nanoscale |

10. | A. Capretti, G. F. Walsh, S. Minissale, J. Trevino, C. Forestiere, G. Miano, and L. Dal Negro, “Multipolar second harmonic generation from planar arrays of Au nanoparticles with aperiodic order,” Opt. Express |

11. | J. A. Adam, |

12. | E. Macia, |

13. | M. Naylor, “Golden, √2, and π flowers: a spiral story,” Math. Mag. |

14. | N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science |

15. | K. IIzuka, |

16. | M. R. Schroeder, |

17. | G. H. Hardy and E. M. Wright, |

18. | J. Havil, |

**OCIS Codes**

(350.4600) Other areas of optics : Optical engineering

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: June 4, 2012

Revised Manuscript: July 18, 2012

Manuscript Accepted: July 20, 2012

Published: July 24, 2012

**Citation**

Luca Dal Negro, Nate Lawrence, and Jacob Trevino, "Analytical light scattering and orbital angular momentum spectra of arbitrary Vogel spirals," Opt. Express **20**, 18209-18223 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18209

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### References

- A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, “Golden spiral photonic crystal fiber: polarization and dispersion properties,” Opt. Lett.33(22), 2716–2718 (2008). [CrossRef]
- M. E. Pollard and G. J. Parker, “Low-contrast bandgaps of a planar parabolic spiral lattice,” Opt. Lett.34(18), 2805–2807 (2009). [CrossRef] [PubMed]
- L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev.6(2), 178–218 (2012). [CrossRef]
- J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett.11(5), 2008–2016 (2011). [CrossRef] [PubMed]
- S. F. Liew, H. Noh, J. Trevino, L. D. Negro, and H. Cao, “Localized photonic bandedge modes and orbital angular momenta of light in a golden-angle spiral,” Opt. Express19(24), 23631–23642 (2011). [CrossRef] [PubMed]
- J. Trevino, S. F. Liew, H. Noh, H. Cao, and L. Dal Negro, “Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals,” Opt. Express20(3), 3015–3033 (2012). [CrossRef] [PubMed]
- N. Lawrence, J. Trevino, and L. Dal Negro, “Aperiodic arrays of active nanopillars for radiation engineering,” J. Appl. Phys.111(11), 113101 (2012). [CrossRef]
- J. Trevino, C. Forestiere, G. Di Martino, S. Yerci, F. Priolo, and L. Dal Negro, “Plasmonic-photonic arrays with aperiodic spiral order for ultra-thin film solar cells,” Opt. Express20(S3), A418–A430 (2012). [CrossRef] [PubMed]
- E. F. Pecora, N. Lawrence, P. Gregg, J. Trevino, P. Artoni, A. Irrera, F. Priolo, and L. Dal Negro, “Nanopatterning of silicon nanowires for enhancing visible photoluminescence,” Nanoscale4(9), 2863–2866 (2012). [CrossRef] [PubMed]
- A. Capretti, G. F. Walsh, S. Minissale, J. Trevino, C. Forestiere, G. Miano, and L. Dal Negro, “Multipolar second harmonic generation from planar arrays of Au nanoparticles with aperiodic order,” Opt. Express20(14), 15797–15806 (2012). [CrossRef] [PubMed]
- J. A. Adam, A Mathematical Nature Walk (Princeton University Press, 2009).
- E. Macia, Aperiodic Structures in Condensed Matter: Fundamentals and Applications (CRC Press Taylor & Francis, 2009).
- M. Naylor, “Golden, √2, and π flowers: a spiral story,” Math. Mag.75(3), 163–172 (2002). [CrossRef]
- N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science334(6054), 333–337 (2011). [CrossRef] [PubMed]
- K. IIzuka, Elements of Photonics (John Wiley, 2002).
- M. R. Schroeder, Number Theory in Science and Communication (Springer Verlag, 1985).
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Oxford University Press, 2008).
- J. Havil, The Irrationals: A Story of the Numbers You Can't Count On (Princeton University Press, 2012).

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