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

doi:10.1364/OE.20.018209

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

Luca Dal Negro, Nate Lawrence, and Jacob Trevino »View Author Affiliations

Optics Express, Vol. 20, Issue 16, pp. 18209-18223 (2012)
http://dx.doi.org/10.1364/OE.20.018209

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▸ Article Outline
– Abstract
1. Introduction
2. Fraunhofer diffraction by Vogel spiral point patterns
3. Analytical diffraction by Vogel arrays of circular apertures
4. Analytical Fourier-Hankel decomposition of Vogel spiral geometry
4. Analytical Fourier-Hankel decomposition of diffracted far field
5. Analytical scattering calculation of diffracted field at a general observation plane
Comment on the numerical implementation of near and intermediate field solutions
6. Conclusions
– References and links

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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.

1. Introduction

Aperiodic Vogel spiral structures are rapidly emerging as a powerful nanophotonics platform with distinctive optical properties of interest to a number of engineering applications [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]

]. This fascinating class of deterministic aperiodic media possess circularly symmetric scattering rings in Fourier space entirely controlled by simple generation rules inducing a very rich structural complexity with a degree of local order in between amorphous and random systems and described by multi-fractal geometry [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]

]. We have recently shown that Vogel spiral arrays of metallic nanoparticles give rise to polarization-insensitive light diffraction across a broad spectral range and are ideally suited to enhance light-matter coupling on a planar substrate, leading to thin-film solar cell enhancement [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]

], light emission enhancement [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]

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]

], and enhanced second harmonic generation [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]

]. Moreover, Vogel spiral arrays of Au nanoparticles support distinctive scattering resonances carrying orbital angular momentum (OAM) [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]

Vogel spiral arrays of dielectric pillars have also been recently investigated in relation to their omnidirectional photonic bandgaps [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]

], the distinctive properties of their bandedge modes [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]

], and their multifractal local density of states (LDOS) [6

]. In particular, Seng et al. [5

] and Trevino et al. [6

] have shown that the golden angle (GA) and generalized Vogel spiral arrays of air cylinders in a dielectric medium support large photonic gaps and radially-localized bandedge modes that are absent in both photonic crystals and quasicrystal structures. Using Fourier-Hankel decomposition (FHD) analysis, it was also shown theoretically [5

] that well-defined OAM values are encoded into the near field mode patterns of the localized bandedge modes of GA spirals. This remarkable property of GA spirals follows from the geometrical fact that consecutive families of dielectric rods (i.e., parastichies arms) are arranged in Fibonacci sequences [11

J. A. Adam, A Mathematical Nature Walk (Princeton University Press, 2009).

–13

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

]. Direct excitation of GA spiral optical resonances, carrying geometry-induced OAM values discretized in consecutive Fibonacci numbers, has been recently achieved at 1.54 μm using the Erbium emission from active arrays of dielectric nanopillars [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]

]. While it is possible to generate OAM using conventional diffractive optical elements or plasmonic nanostructures [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]

], most approaches are generally limited to creating only few OAM azimuthal values.

The controlled generation and manipulation of well-defined OAM sequences carrying large values of azimuthal numbers using planar Vogel arrays could lead to novel nanophotonic active and passive structures that leverage the geometry of isotropic Fourier space for secure optical communication, optical sensing, imaging, and light sources on a chip. However, a general analytical model capable of providing insights into the fundamental optical behavior and design principles for the manipulation of scattered radiation by general aperiodic Vogel spirals is still missing, confining their engineering analysis to the often unattainable numerical simulation of large-scale aperiodic systems.

In this paper, we present a general analytical model for light scattering by arbitrary aperiodic Vogel arrays of circular apertures illuminated at normal incidence. In particular, we show closed form solutions for both the Fraunhofer diffraction (far field) and the field scattered at an arbitrary distance from the Vogel array. Moreover, by performing analytical FHD of spiral arrays and corresponding far field patterns, we rigorously demonstrate the ability to encode specific numerical sequences, not limited to Fibonacci ones, in the OAM values of diffracted optical beams. These results unveil the fundamental mathematical structure of the complex diffraction patterns of Vogel spirals and their connection with the OAM values coded in the far fields. Specifically, we will demonstrate mathematically that the OAM values are directly related to distinctive number-theoretic properties of the Vogel spiral geometry. Our analysis, which is generally valid within single-scattering scalar diffraction theory, well describes the robust optical behavior of large-scale Vogel spiral arrays, which is largely independent of material composition and illumination conditions [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]

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

]. We believe that the proposed analytical model efficiently captures the fundamental features of the complex scattering phenomena observed in Vogel spirals [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]

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

] and can therefore be utilized to rapidly design optical beams carrying multiple values of OAM states for emerging applications in singular optics, secure communication, and optical cryptography.

2. Fraunhofer diffraction by Vogel spiral point patterns

A point pattern with general Vogel spiral geometry can be represented in polar coordinates

(r, θ)

by the equations:

r_{n} = \sqrt{n} a_{0}

(1)

θ_{n} = n α

(2)

where

a_{0}

is a constant scaling factor and

α

is an irrational number known as the divergence angle. This gives the constant angle between successive particles in the spiral array. A spiral array of point scatterers is represented by the following density function:

ρ (r, θ) = \sum_{n = 1}^{N} \frac{1}{r} δ (r - \sqrt{n} a_{0}) δ (θ - n α)

(3)

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

1 / r

factor comes from the definition of the Delta function in cylindrical coordinates. If a plane wave is normally incident on the array, the field at the plane of the array (z = 0) is given by,

E_{z = 0} (r, θ) = \frac{E_{0}}{r} \sum_{n = 1}^{N} δ (r - \sqrt{n} a_{0}) δ (θ - n α)

(4)

Within the validity limits of scalar diffraction theory, the Fraunhofer far field of a general Vogel spiral density function is obtained by the evaluation of its cylindrical Fourier transform (i.e., Fourier-Hankel transform):

E_{\infty} (ν_{r}, ν_{θ}) = \sum_{m = - \infty}^{+ \infty} {(j)}^{m} e^{j m ν_{θ}} 2 π \int_{0}^{\infty} r ρ_{m} (r) J_{m} (2 π ν_{r} r) d r

(5)

where:

ρ_{m} (r) = \frac{1}{2 π} \int_{- π}^{π} E_{z = 0} (r, θ) e^{- j m θ} d θ

(6)

and

J_{m} (2 π ν_{r} r)

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

(ν_{r}, ν_{θ})

are Fourier conjugate of the direct-space variables

(r, θ)

used to represent the Vogel spiral density. We notice that in Eq. (4). the following expression:

H_{m} {ρ_{m} (r)} = 2 π \int_{0}^{\infty} r ρ_{m} (r) J_{m} (2 π ν_{r} r) d r

(7)

is the m-th order Hankel transform of

ρ_{m} (r)

. The integral in Eq. (6) can be analytically evaluated using the sampling property of the Dirac delta function to obtain:

ρ_{m} (r) = \frac{E_{0}}{2 π r} \sum_{n = 1}^{N} δ (r - \sqrt{n} a_{0}) e^{- j m n α}

(8)

Now we can calculate the far field diffraction pattern of the general Vogel spiral point pattern by inserting Eq. (8) into the general expression in Eq. (5) as follows:

E_{\infty} (ν_{r}, ν_{θ}) = \sum_{m = - \infty}^{+ \infty} {(j)}^{m} e^{j m ν_{θ}} \int_{0}^{\infty} E_{0} \sum_{n = 1}^{N} δ (r - \sqrt{n} a_{0}) e^{- j m n α} J_{m} (2 π ν_{r} r) d r

(9)

By exchanging the order of the summation under the integral in Eq. (9) and after using again the sampling property of the delta function we obtain:

E_{\infty} (ν_{r}, ν_{θ}) = E_{0} \sum_{n = 1}^{N} \sum_{m = - \infty}^{+ \infty} {(j)}^{m} J_{m} (2 π \sqrt{n} a_{0} ν_{r}) e^{j m (ν_{θ} - n α)}

(10)

Using the well-known Bessel identity (i.e., cylindrical wave expansion of a plane wave):

e^{j z \cos φ} = \sum_{m = - \infty}^{+ \infty} {(j)}^{m} J_{m} (z) e^{j m φ}

(11)

we can finally obtain the closed form analytical solution for the Fraunhofer far field pattern of a point pattern with an arbitrary Vogel spiral geometry:

E_{\infty} (ν_{r}, ν_{θ}) = E_{0} \sum_{n = 1}^{N} e^{j 2 π \sqrt{n} a_{0} ν_{r} \cos (ν_{θ} - n α)}

(12)

3. Analytical diffraction by Vogel arrays of circular apertures

The analytical diffraction solution for a Vogel spiral array of identical circular apertures of finite radius is obtained using the general convolution theorem of Fourier transforms. This is accomplished multiplying the point pattern solution in Eq. (12) by the Fourier transform (evaluated in cylindrical coordinates) of a circular aperture. In the case of diffraction of normal incident light by Vogel spiral arrays of circular apertures of radius a, the Fraunhofer diffraction pattern is therefore given by:

E_{\infty} (ν_{r}, ν_{θ}) = E_{0} [(a / ν_{r}) J_{1} (2 π ν_{r} a)] \sum_{n = 1}^{N} e^{j 2 π \sqrt{n} a_{0} ν_{r} \cos (ν_{θ} - n α)}

(13)

where the square brackets in Eq. (13) contains an Airy factor, which is the Fourier transform of the circular function

c i r c (r / a)

that represents the individual circular apertures.

Now that we have identified the mathematical structure of the diffracted far field by general Vogel spirals we can proceed with the decomposition of the spiral geometry into its radial and azimuthal components via the analytical evaluation of the Fourier-Hankel transform of the spiral geometry and of the far field diffraction pattern. This analysis, which will be carried on in sections 3 and 4, will show the fundamental connection between the FHD of the spiral geometry and of the diffracted far field, providing analytical insights into the design of aperiodic structures capable to generate scattered beams carrying OAM with multiple azimuthal numbers.

4. Analytical Fourier-Hankel decomposition of Vogel spiral geometry

We will now proceed with the analytical calculation of the FHD of an arbitrary Vogel spiral geometry and show its relation with the Fibonacci numbers in the particular case of the GA spiral. Moreover, using continued fraction expansion of irrational numbers we will generalize the results to arbitrary Vogel spirals. This will enable the design of spiral geometries featuring well-defined peaks in their FHD not necessarily limited to the Fibonacci series. The Fourier-Hankel transform of the spiral density function

δ (r, θ)

is defined as [15

K. IIzuka, Elements of Photonics (John Wiley, 2002).

f (m, k_{r}) = \frac{1}{2 π} \int_{0}^{\infty} \int_{0}^{2 π} r d r d θ ρ (r, θ) J_{m} (k_{r} r) e^{i m θ}

(14)

where:

k_{r} = 2 π ν_{r}

and

ρ (r, θ) = \sum_{n = 1}^{N} \frac{1}{r} δ (r - \sqrt{n} a_{0}) δ (θ - n α) .

Inserting the density function into Eq. (14) we obtain:

f (m, k_{r}) = \frac{1}{2 π} \sum_{n = 1}^{N} \int_{0}^{\infty} \int_{0}^{2 π} d r d θ δ (r - \sqrt{n} a_{0}) δ (θ - n α) J_{m} (k_{r} r) e^{i m θ}

(15)

Using the sampling property of the delta functions into Eq. (15) we can readily obtain:

f (m, k_{r}) = \frac{1}{2 π} \sum_{n = 1}^{N} J_{m} (k_{r} a_{0} \sqrt{n}) e^{i n m α}

(16)

We notice that for the GA spiral, the exponential factor

e^{i m n α}

in the summation of Eq. (16) generates peaks for all the Fibonacci numbers when the azimuthal number m runs across the integers. Therefore, GA spirals are defined by a divergence angle

α = 2 π [1 - f r a c (ϕ)]

where

f r a c (ϕ)

is the fractional part of the golden number

φ = (1 + \sqrt{5}) / 2,

which can be approximated by rational fractions that are consecutive Fibonacci numbers:

φ \approx F_{n + 1} / F_{n} .

In fact, the product

m α

in the exponential sum of Eq. (16) will contribute strongly to the sum when

m = F_{n},

which is a Fibonacci number. Under this condition, the exponents are approximately all integer multiples of 2π (i.e., in-phase phasors) and the FHD will feature strong peaks.

It is very important to notice that an identical analysis also applies to Vogel spiral structure not necessarily generated by the GA. This is so because an arbitrary divergence angle α is directly determined by an arbitrary irrational number ζ, which admits precisely one infinite continued fraction representation (and vice versa) of the form:

ζ = [a_{0}; a_{1}, a_{2}, a_{3}, ...] = a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2} + \frac{1}{a_{3} + \frac{1}{a_{4} + ...}}}}

(17)

An infinite continued fraction representation of an irrational number is very useful because its initial segments provide excellent rational approximations to that number. The rational approximations (i.e., fractions) are called the 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

M. R. Schroeder, Number Theory in Science and Communication (Springer Verlag, 1985).

,17

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Oxford University Press, 2008).

]. 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

M. R. Schroeder, Number Theory in Science and Communication (Springer Verlag, 1985).

,17

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Oxford University Press, 2008).

If two convergents are found, with numerators

p_{1}, p_{2}, ...

and denominators

q_{1}, q_{2}, ...

en the successive convergents are given by the formula:

\frac{p_{n}}{q_{n}} = \frac{a_{n} p_{n - 1} + p_{n - 2}}{a_{n} q_{n - 1} + q_{n - 2}}

(18)

Thus to generate new terms into a rational approximation only the two previous convergents are necessary. The initial or seed values required for the evaluation of the first two terms are (0,1) and (1,0) for (

p_{- 2}, p_{- 1}

) and (

q_{- 2}, q_{- 1}

), respectively.

It is clear from the discussion above that for spirals generated using an arbitrary irrational number ζ, azimuthal peaks of order m (i.e, Bessel order m) will appear in its FHD due to the denominators

q_{n}

of the rational approximations (i.e., the convergents) of

ζ \approx p_{n} / q_{n} .

In fact, for all integer Bessel orders

m = q_{n}

the exponential sum in Eq. (16) gives in-phase contributions to the FHD creating strong peaks. Therefore, once the rational approximations of the irrational number ζ have been identified based on continued fractions, we can easily design a Vogel spiral geometry that “codes” in its FHD peaks the numeric sequence associated to the denominators

q_{n}

of successive rational approximations of ζ, generalizing the Fibonacci series previously obtained for GA spirals.

In order to better illustrate these important aspects of our analysis we will discuss the scattering properties of four different Vogel spirals generated by the irrational numbers φ =

(1 + \sqrt{5}) / 2,

τ =

(2 + \sqrt{8}) / 2,

µ =

(5 + \sqrt{29}) / 2,

the mathematical constant π and their corresponding aperture angles (α^o) listed in Table 1 . The table also shows their rational approximations and, in its last column, the parameter E/M.

Table 1 Listing of irrational angles as well as their corresponding rational approximations (p/q). E/M is a measure of the difficulty to approximate the irrational number with a given a set of rational approximates, where E is the absolute difference from the irrational value for a given p/q and M is the Hurwitz bound.

Irrational Number	Aperture Angle (α°)	Rational Approximations (p/q)		E/M (q)
φ	137.51…	$\frac{1}{1}, \frac{2}{1}, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \frac{13}{8}, \frac{21}{13}, \frac{34}{21}, \frac{55}{21}, \frac{89}{55}, \frac{144}{89}, \frac{233}{144}$		0.99999 (144)
τ	210.88…	$\frac{1}{1}, \frac{5}{2}, \frac{12}{5}, \frac{29}{12}, \frac{70}{29}, \frac{169}{70}, \frac{408}{169}$		0.79057 (169)
µ	290.67…	$\frac{5}{1}, \frac{26}{5}, \frac{135}{26}, \frac{701}{135}, \frac{3640}{701}$		0.41523 (135)
π	309.03…	$\frac{3}{1}, \frac{22}{7}, \frac{355}{113}$	0.00762 (113)

This parameter, which ranges from 0 to 1, is a measure of the “difficulty” to approximate irrational numbers using rationals. The parameter E/M is defined as the ratio of the approximation error E to the Hurwitz’s bound

M = 1 / \sqrt{5} q^{2} .

This definition follows from Hurwitz’s theorem [18

J. Havil, The Irrationals: A Story of the Numbers You Can't Count On (Princeton University Press, 2012).

], which states that every number has infinitely many rational approximations of the form p/q with an approximation error less than M. Table 1 shows that, in this precise sense, the golden number

ϕ = (1 + \sqrt{5}) / 2,

utilized to generate the golden angle α, is the most irrational number because its approximation error gets as large as possible compatible to the Hurwitz’s bound.

This slow convergence is contrasted with π, whose rational approximation error is much better than the Hurwitz theoretical bound. The values of the two other irrational numbers in Table 1 are chosen at intermediate E/M values in order to study Vogel spirals obtained by irrational numbers equally sampled across the [0,1] interval.

Vogel spirals generated using divergence angles determined by the irrational numbers in Table 1 are shown in Figs. 1 and 2 together with the analytically calculated far fields obtained by Eq. (12). Interestingly, we see in Figs. 1 and 2 that the degree of rotational symmetry and the complexity of the azimuthal structure of the far field patterns decrease for Vogel spirals obtained using irrational numbers that can be efficiently approximated in rationals, in the precise sense of the Hurwitz’s bounds listed in Table 1.

Fig. 1 (a,b) GA and τ Vogel spirals with 2000 particles. (c,d) Analytically calculated far field radiation pattern of arrays in (a,b) respectively at a wavelength of 633nm for a structure with a₀ = 14.5μm. The far field radiation pattern has been truncated with an angular aperture of 4°.

Fig. 2 (a,b) µ and π Vogel spirals with 2000 particles. (c,d) Analytically calculated far field radiation pattern of arrays in (a,b) respectively at a wavelength of 633nm for a structure with a₀ = 14.5μm. The far field radiation pattern has been truncated with a circular aperture.

In fact, the GA and τ spirals feature an almost continuous rotational symmetry in their far field patters, while less defined azimuthal structure is evident for the other two spiral structures. Therefore, “the degree of irrationality” of Vogel spirals is a key parameter that determines the azimuthal complexity of the scattered far fields, and the richness of their OAM spectra. Mathematically, the link between the spiral geometry and the azimuthal structure of the scattered far fields can be clearly understood by Fourier-Hankel analysis. In particular, we will demonstrate in the next section by performing FHD analysis that a direct connection exists between the continued fraction expansion of ζ, which yields the specific spiral geometry, and the azimuthal structure of the scattered far fields, which carry well defined OAM values.

4. Analytical Fourier-Hankel decomposition of diffracted far field

We have demonstrated before that the far field of an arbitrary Vogel spiral array of apertures of width a under uniform illumination can be expressed in closed form by:

E_{\infty} (ν_{r}, ν_{θ}) = E_{0} [(a / ν_{r}) J_{1} (2 π ν_{r} a)] \sum_{n = 1}^{N} e^{j 2 π \sqrt{n} a_{0} ν_{r} \cos (ν_{θ} - n α)}

(19)

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:

f (m, 2 π r) = \frac{E_{0} a}{2 π} \int_{0}^{\infty} \int_{0}^{2 π} d ν_{r} d ν_{θ} [\sum_{n = 1}^{N} e^{j 2 π \sqrt{n} a_{0} ν_{r} \cos (ν_{θ} - n α)}] J_{m} (2 π r ν_{r}) J_{1} (2 π ν_{r} a) e^{i m ν_{θ}}

(20)

= \frac{E_{0} a}{2 π} \sum_{n = 1}^{N} \int_{0}^{\infty} \int_{0}^{2 π} d ν_{r} d ν_{θ} e^{j 2 π \sqrt{n} a_{0} ν_{r} \cos (ν_{θ} - n α)} J_{m} (2 π r ν_{r}) J_{1} (2 π ν_{r} a) e^{i m ν_{θ}}

(21)

Now we can eliminate the integral over

ν_{θ}

by using the following representation for the Bessel function:

J_{- m} (k_{ρ} ρ) e^{- i m φ} = \frac{1}{2 π} \int_{0}^{2 π} d β e^{i k_{ρ} ρ \cos (β - φ) + i m β + i m π / 2}

(22)

after the formal identifications:

β \to ν_{θ},

ϕ \to n α,

ρ \to ν_{r},

k_{ρ} \to 2 π \sqrt{n} a_{0} .

Therefore, Eq. (21) reduces after few simple steps to:

f (m, 2 π r) = \frac{E_{0} a}{2 π} {(- i)}^{m} \sum_{n = 1}^{N} e^{- i n m α} \int_{0}^{\infty} d ν_{r} J_{m} (2 π r ν_{r}) J_{- m} (2 π \sqrt{n} a_{0} ν_{r}) J_{1} (2 π ν_{r} a)

(23)

The analytical closed form solution in Eq. (23) has the general mathematical structure:

f (m, β) \propto \sum_{n = 1}^{N} A_{m β} e^{i n m α}

(24)

We believe that this is a central result in our analytical study. In fact, apart from the different intensity of the ν_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.

These unique features of Vogel spiral order provide exciting new opportunities for OAM engineering and the manipulation of optical vortices using dielectric photonic structures. In this work, we demonstrate that the design of nanostructured surfaces with Vogel spiral geometry can in fact provide a very large spectrum of OAM values relying uniquely on light scattering phenomena. This can be achieved in principle using low-loss dielectric structures forming tailored Vogel spiral arrays, as we will show below. In Fig. 3 , we show the calculated Furrier-Hankel transforms of the investigated Vogel spirals, summed over the radial wavenumber k_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.

Fig. 3 Fourier-Hankel transforms of far field scattered radiation from GA (Fig. 1(c)), τ (Fig. 1(d)), µ (Fig. 2(c)) and π (Fig. 2(d)) Vogel spirals summed over the radial wavenumber k_r.

We also notice in Fig. 3 a decreasing number of peaks, associated to reduced structural complexity, for structures generated by divergence angles that are “less irrational” (i.e., that are better approximated in rationals in the Hurwitz’s sense explained above). Therefore, Vogel spiral structures with large E/M values are ideal to transmit very rich spectra of OAM values in the far field.

In order to understand the role played by the spiral dimension in the far field OAM spectra, we show in Fig. 4 the calculated far fields and the corresponding OAM spectra (i.e., FHD of the far fields) for GA spirals of increasing particle numbers, ranging from 500 to 4000. We can see in Fig. 4 that azimuthal peaks at larger Bessel orders start to appear as the size of the spirals increase and the azimuthal structure of the scattered far fields becomes richer. However, the effect is not dramatic and the OAM spectra are quite stable with respect to the particle number in the arrays, as long as they are sufficiently large (i.e., few hundreds in this case).

Fig. 4 (a-c) Far field radiation pattern of GA spirals with 500, 1000 and 4000 particles respectively. (d-f) Fourier-Hankel transform of (a-c), respectively, summed over the radial wavenumber k_r at a wavelength of 633nm for a structure with a₀ = 14.5μm. The far field radiation pattern has been truncated with an angular aperture of 4°.

On the other hand, a much more dramatic effect occurs when sampling the far field spectra over larger angular windows, corresponding to larger cross sections of scattered beams in the far field. This is demonstrated, in the case of the GA spiral, by Fig. 5 . We can see in Fig. 5 that many more azimuthal peaks in the OAM spectra of the scattered radiation are obtained as we use different apertures in the far field. The spectra in Figs. 4(d)–4(f) become much more densely populated by peaks at azimuthal orders corresponding to larger Fibonacci and Lucas numbers, which are found among the denominators of the rational approximations of the golden angle. The effect of the angular size on the diffracted beam in the far field is therefore very important and directly affects the range of discrete OAM values that can be obtained in the far field zone.

Fig. 5 (a,b) Far field radiation pattern of GA spirals truncated with different apertures, (a) 8°, (b) 4°. (c,d) Fourier-Hankel transform of (a,b), respectively, summed over the radial wavenumber k_r at a wavelength of 633nm for a structure with a₀ = 14.5μm and 2000 particles.

The positions of all the peaks in the OAM spectra of the structures analyzed in Figs. 3 and 5 are summarized in Table 2 . As predicted by our analytical model, Table 2 demonstrates that all the OAM peak positions correspond to azimuthal orders that coincide with the denominators of the rational approximations of the irrational numbers used to generate the different spirals. These results therefore demonstrate the general ability of Vogel spirals to encode pre-defined numerical sequences in the OAM spectra of far field scattered radiation.

Table 2 Listing of irrational angles and the peak positions in the Fourier-Hankel transform of the spirals in Figs. 3 and 5. The peaks correspond to the denominators of the rational approximations of the irrational aperture angles of the spirals.

Spiral Aperture Angle (Fig.)	Fourier-Hankel Transform Peaks
GA (3a)	2, 3, 5, 8, 13, 21, 34, 55, 89, 144
Τ (3b)	2, 5, 7, 12, 17, 29, 41, 70, 99, 169
µ (3c)	5, 26, 135
π (3d)	7, 14, 113
Golden Angle, Large Aperture (5a)	2, 3, 5, 8, 13, 21, 26, 29, 34, 42, 47, 55, 68, 76, 89, 110, 123, 144, 178, 199
Golden Angle, Small Aperture (5b)	2, 3, 5, 8, 13, 21, 34, 55, 89, 144

In what follows, we will extend our analytical theory to the calculation of the scattered field at finite distance (i.e., near and intermediate field zones) from the scattering Vogel spiral. The asymptotic convergence of the calculated intermediate fields towards the analytical far field solution provided in section 2 will also be demonstrated for consistency.

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

In this last section, we will analytically calculate the field scattered by a general aperiodic Vogel spiral at a finite observation plane in the near and intermediate field zones using the general angular spectrum representation approach. In a Cartesian coordinate system, the transverse field sampled at an arbitrary plane z = d can be written as:

E (x, y; d) = F^{- 1} {H (2 π ν_{x}, 2 π ν_{y}; d) \hat{E} (2 π ν_{x}, 2 π ν_{y}; 0)}

(25)

where

H (2 π ν_{x}, 2 π ν_{y}; d) = e^{j 2 π d \sqrt{1 / λ^{2} - ν_{x}^{2} - ν_{y}^{2}}}

is the free-space propagator in reciprocal space,

\hat{E} (2 π ν_{x}, 2 π ν_{y}; 0)

is the Fourier transform of the field at the object plane, and

F^{- 1}

denotes the inverse Fourier transform operation. For our purpose, we would need to evaluate Eq. (24) in cylindrical coordinate, using the free-space propagator given below:

H (ν_{r}; d) = e^{j 2 π d \sqrt{1 / λ^{2} - ν_{r}^{2}}}

(26)

and the previously derived Fourier spectrum of the spiral:

E_{\infty} (ν_{r}, ν_{θ}) = E_{0} \sum_{n = 1}^{N} e^{j 2 π \sqrt{n} a_{0} ν_{r} \cos (ν_{θ} - n α)}

(27)

Therefore, in order to obtain the field we need to evaluate Eq. (25) in cylindrical coordinates, which is:

E (r, θ; d) = F^{- 1} {E_{0} \sum_{n = 1}^{N} e^{j 2 π \sqrt{n} a_{0} ν_{r} \cos (ν_{θ} - n α)} e^{j 2 π d \sqrt{1 / λ^{2} - ν_{r}^{2}}}} = F^{- 1} [G (ν_{r}, ν_{θ})]

(28)

Using the cylindrical representation of the inverse Fourier transform, we can rewrite Eq. (27) as follows:

E (r, θ; d) = \sum_{m = - \infty}^{+ \infty} {(j)}^{m} e^{j m θ} 2 π \int_{0}^{\infty} ν_{r} G_{m} (ν_{r}) J_{m} (2 π ν_{r} r) d ν_{r}

(29)

where:

G_{m} (ν_{r}) = \frac{1}{2 π} \int_{- π}^{π} G (ν_{r}, ν_{θ}) e^{- j m ν_{θ}} d ν_{θ}

(30)

Using the definition of

G (ν_{r}, ν_{θ})

into Eq. (30) we obtain:

G_{m} (ν_{r}) = \frac{E_{0}}{2 π} \int_{- π}^{π} d ν_{θ} \sum_{n} e^{j [a_{0} 2 π \sqrt{n} ν_{r} \cos (ν_{θ} - n α) + 2 π d \sqrt{1 / λ^{2} - ν_{r}^{2}} - m ν_{θ}}

(31)

which can be expressed as:

G_{m} (ν_{r}) = \frac{E_{0}}{2 π} e^{j 2 π d \sqrt{1 / λ^{2} - ν_{r}^{2}}} \sum_{n = 1}^{N} \int_{- π}^{π} d ν_{θ} e^{j [2 π a_{0} \sqrt{n} ν_{r} \cos (ν_{θ} - n α) - m ν_{θ}]}

(32)

The definite integral in Eq. (32) can be evaluated by recalling the following representation of the m-th order Bessel functions:

J_{m} (k_{ρ} ρ) e^{j m φ} = \frac{1}{2 π} \int_{0}^{2 π} d α e^{j k_{ρ} ρ \cos (α - φ) - j m α - j m π / 2}

(33)

where

(k_{ρ} ρ, ω, φ)

are replaced with

(2 π a_{0} \sqrt{n} ν_{r}, ν_{θ}, n α) .

Therefore the scattered field at a given plane z = d can be expressed from Eq. (29) as:

E (r, θ; d) = E_{0} \sum_{m = - \infty}^{+ \infty} {(j)}^{m} e^{j m θ} \int_{0}^{\infty} ν_{r} e^{j 2 π d \sqrt{1 / λ^{2} - ν_{r}^{2}}} \sum_{n = 1}^{N} \int_{- π}^{π} d ν_{θ} e^{j [2 π a_{0} \sqrt{n} ν_{r} \cos (ν_{θ} - n α) - m ν_{θ}]} J_{m} (2 π ν_{r} r) d ν_{r}

(34)

Using Eq. (33), Eq. (34) can be reduced to:

E (r, θ; d) = 2 π E_{0} \sum_{m = - \infty}^{+ \infty} {(j)}^{m} e^{j m θ} e^{j m π / 2} \sum_{n = 1}^{N} e^{- j m n α} \int_{0}^{\infty} ν_{r} e^{j 2 π d \sqrt{1 / λ^{2} - ν_{r}^{2}}} J_{m} (2 π a_{0} \sqrt{n} ν_{r}) J_{m} (2 π ν_{r} r) d ν_{r}

(35)

Now we can further simplify the integral in Eq. (34) by using the Gegenbauer theorem for cylindrical functions:

J_{0} (k_{r} R) = \sum_{m = - \infty}^{+ \infty} J_{m} (k_{r} r_{1}) J_{m} (k_{r} r_{2}) e^{i m θ}

(36)

In fact, continuing from Eq. (35) we have:

= 2 π E_{0} \sum_{n = 1}^{N} \int_{0}^{\infty} ν_{r} e^{j 2 π d \sqrt{1 / λ^{2} - ν_{r}^{2}}} \sum_{m = - \infty}^{+ \infty} e^{j m (θ + π - n α)} J_{m} (2 π a_{0} \sqrt{n} ν_{r}) J_{m} (2 π ν_{r} r) d ν_{r}

(37)

E (r, θ; d) = 2 π E_{0} \sum_{n = 1}^{N} \int_{0}^{\infty} ν_{r} e^{j 2 π d \sqrt{1 / λ^{2} - ν_{r}^{2}}} J_{0} (k_{r} R) d ν_{r}

(38)

which is the main closed form expression for the scattered field obtained after the following identifications have been made:

r_{1} = a_{0} \sqrt{n}, r_{2} = r, k_{r} = 2 π ν_{r}, R = \sqrt{r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos (β)}, β = θ + π - n α

(39)

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:

E (r, θ; d) = 2 π E_{0} a \sum_{n = 1}^{N} \int_{0}^{\infty} e^{j 2 π d \sqrt{1 / λ^{2} - ν_{r}^{2}}} J_{0} (k_{r} R) J_{1} (2 π ν_{r} a) d ν_{r}

(40)

Equation (40) explicitly contains the multiplicative Airy factor

(a / ν_{r}) J_{1} (2 π ν_{r} a)

that accounts for the finite size of the apertures.

We validate the analytical result of Eq. (40) by calculating the transverse electric field patterns at increasing distances from a 100-particles GA spiral array (Fig. 6 ). Figure 6(a) corresponds to a sub-wavelength distance from the array and correctly reproduces the GA spiral geometry in the near field zone, since no Fresnel approximation was introduced in our general diffraction model. As generally expected, we can appreciate from Fig. 6 that when the scattered fields propagate away from the array plane, the spatial resolution of the fields is gradually lost and the field profiles approach the analytically calculated far field solution shown in Fig. 6(d). This result completes our analytical model for light scattering by arbitrary Vogel spiral arrays. In the next section, we will briefly discuss an alternative analytical formulation that improves the efficiency in the numerical implementation of the proposed model.

Fig. 6 (a,b,c) Electric field pattern of radiation at 633nm from a GA spiral array (100 particles, a₀ = 1.2μm) of circular apertures of radius 100nm at propagation distances of 0.2 λ, 100 λ and 400 λ, respectively. (d) Far field radiation pattern of the same GA spiral array shown in (a,b,c) for comparison.

Comment on the numerical implementation of near and intermediate field solutions

It is worth noticing that the computational complexity of Eq. (40) can be further reduced by expressing the Bessel function

J_{0}

in terms of the generalized hypergeometric series

J_{α} (x) = \frac{{(x / 2)}^{α}}{Γ (α + 1)} {}_{0}F_{1} (α + 1; - x^{2} / 4)

(41)

with:

{}_{p}F_{q} (a_{1}, ..., a_{p}; b_{1}, ..., b_{q}; z) = \sum_{n = 0}^{\infty} \frac{{(a_{1})}_{n} ... {(a_{p})}_{n}}{{(b_{1})}_{n} ... {(b_{q})}_{n}} \frac{z^{n}}{n!}

(42)

where we introduced the Pochhammer symbol:

{(a)}_{n} = a (a + 1) (a + 2) ... (a + n - 1), (a_{0}) = 1

(43)

for the rising factorial.

Using the hypergeometric representation, the expression for the scattered field becomes:

E (r, θ; d) = 2 π a E_{0} \sum_{n = 1}^{N} \sum_{m = 1}^{\infty} \frac{R^{2 m}}{{(m!)}^{2}} \int_{0}^{\infty} e^{j 2 π d \sqrt{1 / λ^{2} - ν_{r}^{2}}} {(- 1)}^{m} {[{(2 π ν_{r})}^{2} / 4]}^{m} J_{1} (a 2 π ν_{r}) d ν_{r}

(44)

We notice that the integrand in Eq. (44) depends now only on

ν_{r} .

This fact speeds up significantly the evaluation of the transverse field profile with respect to the expression in Eq. (40), since the integrand factor in Eq. (40) is also dependent on the spatial

(r, θ)

grid values through the dependence on the parameter

R

6. Conclusions

In conclusion, we have established a general analytical diffraction model for light scattering by arbitrary Vogel spirals illuminated at normal incidence. Our analysis has unveiled the fundamental mathematical structure of the complex Fraunhofer diffraction patterns of Vogel spirals. Moreover, we performed analytical Fourier-Hankel decomposition of different spiral arrays and far field patterns and we demonstrated that scattered radiation features discrete OAM values determined by the rational approximations of the continued fraction expansions of the irrational spiral angles. Finally, we extended our model to the analytical calculation of the scattered field in the near and intermediate zones using the general angular spectrum approach. These findings open novel and exciting opportunities for the manipulation of complex OAM spectra exploiting light scattering by aperiodic spiral arrays for a number of emerging engineering applications in singular optics, secure communication, optical cryptography, and optical sensing.

Acknowledgments

This work was supported by the AFOSR program “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. 33(22), 2716–2718 (2008). [CrossRef]
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]
3.	L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev. 6(2), 178–218 (2012). [CrossRef]
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]
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]
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]
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]
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]
11.	J. A. Adam, A Mathematical Nature Walk (Princeton University Press, 2009).
12.	E. Macia, Aperiodic Structures in Condensed Matter: Fundamentals and Applications (CRC Press Taylor & Francis, 2009).
13.	M. Naylor, “Golden, √2, and π flowers: a spiral story,” Math. Mag. 75(3), 163–172 (2002). [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]
15.	K. IIzuka, Elements of Photonics (John Wiley, 2002).
16.	M. R. Schroeder, Number Theory in Science and Communication (Springer Verlag, 1985).
17.	G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Oxford University Press, 2008).
18.	J. Havil, The Irrationals: A Story of the Numbers You Can't Count On (Princeton University Press, 2012).

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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