Vogel Spirals in Photonics

• Vogel spirals are deterministic aperiodic point sets derived from phyllotaxis, exhibiting non-repeating yet uniform local packing.
• They display multiple families of localized modes with exponential, power-law, and Gaussian decay, bridging periodic and random media.
• Their unique geometric and spectral properties enable tunable photonic applications, including filters, sensors, and structured light devices.

A Vogel spiral is a deterministic, aperiodic point set constructed using the polar coordinate parameterization $r_n = a_0 \sqrt{n}$, $\theta_n = n \alpha$, where $n$ is a non-negative integer, $a_0$ is a scale factor, and $\alpha$ is an irrational divergence angle—most often, the golden angle $\alpha = 2\pi(1 - 1/\varphi)$, with $\varphi = (1+\sqrt{5})/2$. Vogel spirals combine long-range order with complete absence of translational or rotational symmetry. This unique structural motif, inspired by phyllotaxis in botany, is widely explored for its exceptional uniformity, optimal local packing density, and its ability to support novel optical phenomena. Deterministic yet aperiodic, Vogel spirals underpin a rich taxonomy of electromagnetic modes, bridging the gap between periodic photonic crystals and disordered random media.

1. Mathematical Construction and Classification

The Vogel spiral is generated by the mapping

$$r_n = a_0 \sqrt{n}, \qquad \theta_n = n\alpha, \qquad n=1,2,...,N,$$

where $a_0$ sets the average spacing and $\alpha$ is chosen to ensure optimal aperiodicity. The canonical value is the golden angle $\alpha \approx 137.508^\circ$, derived from the golden ratio. Cartesian coordinates follow directly as $x_n = r_n \cos\theta_n$, $y_n = r_n\sin\theta_n$.

Unlike periodic lattices or Poissonian random arrays, Vogel spirals are deterministic but non-repetitive, characterized by:

• No translational or rotational symmetry.
• Strong local packing regularity, with a uniform density devoid of arbitrarily close point pairs (Zurita et al., 2021).
• Robustness to parameter variations in $\alpha$, yielding family variants such as τ-, π-, and μ-spirals (Prado et al., 2021).

Structural generalizations encompass higher-dimensional Vogel spirals via Markoff theory, enabling almost-uniform point distributions on Riemannian manifolds, with 2D packing density lower bounds of $\Delta'_2 = \pi/(2\sqrt{5}) \approx 0.702$ and 3D analogues at $\Delta'_3 = \sqrt{3}\pi/14 \approx 0.389$ (Zurita et al., 2021).

2. Physical Realizations and Experimental Approaches

Vogel spirals have been physically implemented in electromagnetic, optical, and photonic settings:

• Arrays of high-permittivity dielectric cylinders (e.g., $\varepsilon \simeq 45$, $R=3\,\mathrm{mm}$, $h=5\,\mathrm{mm}$) embedded in a spiral of radius up to $140\,\mathrm{mm}$, with $N \sim 390$ scatterers (Razo-López et al., 2023).
• Lumped dipole lattices used in microwave and optical wave transport experiments (Sgrignuoli et al., 2018).
• Vogel lattices comprised of super-Gaussian potential wells as guiding sites for soliton propagation studies in nonlinear optics (Kartashov et al., 2012).

Experiments are typically conducted using:

• TM-polarized excitation with monopole antennas, recording reflection $S_{11}(\nu)$ and transmission $S_{21}(\nu)$ parameters over 2D or 3D geometries (Razo-López et al., 2023).
• Dense raster-scanned probe arrays to reconstruct full spatial modal profiles.
• Systematic variation of spiral parameters and ambient media to tune spectral features and mode lifetimes.

3. Mode Structures and Spatial Decay Laws

Vogel spirals are unique among deterministic aperiodic geometries in supporting three coexisting families of localized electromagnetic modes, distinguished by their spatial decay laws:

• Exponential decay: $|E(r)| \sim \exp(-r/\xi)$, characteristic of Anderson localization in random media. The parameter $\xi$ defines the localization length.
• Power-law decay: $|E(r)| \sim r^{-p}$ (with $p > 0$), exhibiting multifractal spatial oscillations. Such “critical” modes display long-range correlation and nontrivial participation ratios, uncommon in both periodic and disordered arrays.
• Gaussian decay: $|E(r)| \sim \exp(-r^2/\sigma^2)$, a distinct signature of high spatial and temporal (long-lived) localization. These modes have minimal participation ratios, narrow linewidths, and are not observed in generic random or periodic systems (Prado et al., 2021, Razo-López et al., 2023).

Empirically, mode parameters such as $p \approx 1$, $\xi \approx 10$, and $\sigma \approx 15$ arise, depending on frequency and system configuration. All three decay types may co-occur within a single frequency band—a phenomenon unattainable in other photonic media (Razo-López et al., 2023, Prado et al., 2021).

4. Localization Metrics and Spectral Analysis

Comprehensive modal analysis in Vogel spirals employs the following tools:

Quantity	Symbol/Expression	Role
Quality Factor	$Q_j = \nu_j/\delta\nu_j$	Inverse relative linewidth (lifetime)
Inverse Participation Ratio	$$\mathrm{IPR}_j = \frac{\int|E_j|^4}{(\int|E_j|^2)^2}$$	Effective mode area
Thouless Conductance	$g_j = \delta\nu_j/\Delta\nu_j$	Isolation/localization criterion
Mean Density of States	$$\mathrm{DOS} \approx 1 - \langle |S_{11}|^2 \rangle$$	Number of available states

The appearance of $\langle g \rangle < 1$ signals the onset of localization, with high-$Q$ modes demonstrating sharply peaked time-domain decay (slow leakage). Modal spatial extent and temporal isolation are further quantified by structural entropy and participation ratio, enabling no-fitting classification into Gaussian, exponential, or critical decay through universal localization maps (Prado et al., 2021, Razo-López et al., 2023).

5. Theoretical Frameworks and Mode Computation

Analysis of light localization, mode classification, and cooperative phenomena in Vogel spirals employs several theoretical approaches:

• Maxwell’s equations (scalar and vector): Reduction to 2D scalar wave equations for TM fields or full 3D dyadic Green’s matrix formalism for dipoles, capturing all radiative and near-/intermediate-field coupling (Sgrignuoli et al., 2018, Razo-López et al., 2023).
• Green's-matrix spectral analysis: For arrays of point dipoles, the $N \times N$ Green’s matrix (scalar or dyadic) yields complex eigenvalues corresponding to resonance energies and linewidths, with eigenvectors denoting mode structures. The method isolates the effects of vector light, identifying that localization requires retention of near-field (∼1/r³), intermediate (∼1/r²), and radiative (∼1/r) interactions; scalar approximations (neglecting vector degrees of freedom) fail to capture the observed transitions (Sgrignuoli et al., 2018).
• Finite-size scaling and β-function: The β-function $\beta(\ln g) = d\ln g/d\ln L$, with system size $L$, demarcates the transition from diffusive ($\beta > 0$) to localized ($\beta < 0$) regimes. Vogel spirals display a unique intersection at $g_c \approx 1$, consistent with single-parameter scaling (Sgrignuoli et al., 2018).
• Nonlinear Schrödinger frameworks: In nonlinear optics, Vogel-lattice soliton families are modeled via a dimensionless NLS with refractive index modulations defined by the spiral geometry. Both "gap soliton" existence domains and spiraling dynamics (azimuthal motion, orbital angular momentum) are explicitly characterized (Kartashov et al., 2012).

6. Higher-Dimensional Extensions, Robustness, and Packing Theory

The concept of the Vogel spiral extends naturally to higher-dimensional analogues:

• 3D Vogel spiral constructions: Using Markoff-theoretic optimal lattices, deterministic spiral packings provide almost-uniform point sets in 3D balls, with density lower bounds $\Delta'_3 \approx 0.389$ (Zurita et al., 2021).
• Robustness to dimensionality: Experimental results demonstrate persistence of localized Gaussian, exponential, and power-law modal families even with significant out-of-plane field leakage (e.g., in thick slab or open 3D arrangements), implying that the underlying localization is not a singular feature of strict 2D confinement (Razo-López et al., 2023).
• Generalizations to Riemannian manifolds: The spiral-packing framework can be locally transplanted to any real-analytic Riemannian n-manifold (n ≤ 3) with a locally diagonalizable metric, yielding almost-uniform distributions with optimal lower-bound densities (Zurita et al., 2021).

7. Optical and Photonic Implications

Vogel spirals exhibit a suite of optical properties and application potential unmatched by conventional media:

• Simultaneous presence of exponentially, power-law, and Gaussian localized modes in identical system parameters (Razo-López et al., 2023).
• Suppression of proximity resonances due to strong local distance regularity, enabling high-$Q$, spatially isolated modes (Sgrignuoli et al., 2018).
• Highly tunable resonance spectra and field profiles via deterministic selection of spiral parameters, divergence angle, and scale factor.
• Strong and spatially multifractal light-matter interaction, facilitating enhanced nonlinear response, high Purcell-factor emission, and low-threshold lasing.
• Applications in aperiodic microcavities, filters, waveguides, sensors, and sources—where multi-mode behavior or field localization over multiple scales offers functional advantages (Razo-López et al., 2023, Sgrignuoli et al., 2018, Prado et al., 2021).

Novel device concepts include aperiodic cavity QED, structured random lasers, compact multi-frequency sources, and multi-scale sensors. Persistent open problems include systematic tuning of spiral metrics for targeted modal compositions, functionalization in three-dimensional volumes, and integration into active photonic devices.

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Vogel spirals crystallize the intersection of deterministic aperiodic order, optimal packing, and multifaceted photonic localization, enabling a distinctive modal landscape inaccessible to periodic or random structures [(Razo-López et al., 2023); (Sgrignuoli et al., 2018); (Zurita et al., 2021); (Prado et al., 2021); (Kartashov et al., 2012)].