## 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

Aperiodic Vogel spiral structures are rapidly emerging as a powerful nanophotonics platform with distinctive optical properties of interest to a number of engineering applications [1–7]. 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]. 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], light emission enhancement [7,9], and enhanced second harmonic generation [10]. Moreover, Vogel spiral arrays of Au nanoparticles support distinctive scattering resonances carrying orbital angular momentum (OAM) [4].

Vogel spiral arrays of dielectric pillars have also been recently investigated in relation to their omnidirectional photonic bandgaps [2], the distinctive properties of their bandedge modes [5], 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 [11–13]. 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]. While it is possible to generate OAM using conventional diffractive optical elements or plasmonic nanostructures [14], 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 [4,7]. We believe that the proposed analytical model efficiently captures the fundamental features of the complex scattering phenomena observed in Vogel spirals [4,7,8] 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$(r,\theta )$by the equations:

where${a}_{0}$is a constant scaling factor and$\alpha $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:*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,

*m*-th order Bessel function of the first kind. The variables $({\nu}_{r},{\nu}_{\theta})$ are Fourier conjugate of the direct-space variables $(r,\theta )$ used to represent the Vogel spiral density. We notice that in Eq. (4). the following expression:

*m*-th order Hankel transform of ${\rho}_{m}(r)$. The integral in Eq. (6) can be analytically evaluated using the sampling property of the Dirac delta function to obtain:

## 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:

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 $\delta (r,\theta )$ is defined as [15]:

*m*runs across the integers. Therefore, GA spirals are defined by a divergence angle $\alpha =2\pi [1-frac(\varphi )]$ where $frac(\varphi )$ is the fractional part of the golden number $\phi =(1+\sqrt{5})/2,$ which can be approximated by rational fractions that are consecutive Fibonacci numbers: $\phi \approx {F}_{n+1}/{F}_{n}.$ In fact, the product $m\alpha $ 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:

*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].

If two convergents are found, with numerators ${p}_{1},{p}_{2},\mathrm{...}$ and denominators ${q}_{1},{q}_{2},\mathrm{...}$ en the successive convergents are given by the formula:

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 $\zeta \approx {p}_{n}/{q}_{n}.$ 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 = $\left(1+\sqrt{5}\right)/2,$ τ = $\left(2+\sqrt{8}\right)/2,$ µ = $\left(5+\sqrt{29}\right)/2,$ 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.

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], 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 $\varphi =(1+\sqrt{5})/2,$ 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.

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:

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

_{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.

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).

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.

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.

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:

*m-*th order Bessel functions:

*z*=

*d*can be expressed from Eq. (29) as:

*Gegenbauer theorem*for cylindrical functions:

*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:

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.

## 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

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

## 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.

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