Fibonacci Helices Drilling in Nanofabrication

• Fibonacci helices drilling is a nanofabrication process that creates spiral micro- and nano-structures using Fibonacci sequences and the golden angle for precise material ablation.
• It employs laser-pumped CNT vortex dynamics to generate hypersonic shock waves, resulting in patterned ablation and nanoparticle deposition that follow Fibonacci scaling laws.
• This method enables advanced applications in fiber optics, sensing, and neuro-ultrasonics by integrating structured spiral channels to modulate optical vortices and enhance device performance.

Fibonacci helices drilling refers to the fabrication of helical or spiral micro- and nano-structures whose geometry is governed by Fibonacci sequences and the golden ratio, producing spatial arrangements observed in both natural phyllotaxis and advanced nanofabrication contexts. In experimental implementations, such as laser-pumped drilling in carbon nanotube (CNT)-infused optical fibers, these patterns arise due to the interaction of vortex shock waves and the intrinsic physics of dense packing, yielding structured ablation and nanoparticle deposition that closely follow Fibonacci spiral laws (Silva et al., 9 Dec 2025, Mughal et al., 2016).

1. Geometric and Mathematical Fundamentals

Fibonacci helices are characterized by spatial distributions of points or material ablation along helicoidal trajectories where the scaling and angular divergence adhere to the Fibonacci sequence and the golden ratio $\varphi=(1+\sqrt{5})/2$. The canonical form models a series of points or holes—representing ablation loci or discrete nanoparticles—placed on or within a cylindrical or planar substrate. The positions $(x_n, y_n, z_n)$ of the $n$-th site are determined using the following parametric relations:

$$\theta_n = 2\pi\,\alpha\,n, \quad z_n = n\,\Delta z, \quad x_n = R\cos\theta_n, \quad y_n = R\sin\theta_n,$$

where $\alpha$ is the divergence fraction, ideally chosen as $\alpha=1/\varphi^2 \approx 0.381966$, corresponding to the golden angle $\delta\approx137.51^\circ$ (Mughal et al., 2016). The sequence of visible helices—parastichies—follows the recurrence $[l, m, n]\rightarrow[l+m, l, m]$ with $l, m, n$ the parastichy counts, iterating through consecutive Fibonacci numbers.

When scaling is introduced in both radius and pitch in accordance with the golden ratio, the three-dimensional parametric equations for helix $k$ in cylindrical coordinates $(r, \theta, z)$ become:

$$r(\theta) = R_0\,\varphi^{\theta/(2\pi)},\quad z(\theta) = P_0\,\varphi^{\theta/(2\pi)},$$

$$x_k(\theta) = r(\theta)\cos(\theta + 2\pi k/3),\quad y_k(\theta) = r(\theta)\sin(\theta + 2\pi k/3),\quad z_k(\theta) = h_0\,\varphi^{\theta/(2\pi)},$$

with $k=0,1,2$ for a set of three co-rotating helices, $R_0$ the initial radius, and $P_0, h_0$ scale and pitch constants (Silva et al., 9 Dec 2025).

2. Physical Mechanisms in Laser-Pumped CNT Vortex Drilling

In the context of laser-pumped CNT vortex drilling in optical fibers, the formation of Fibonacci helices is dictated by the interplay of fluid dynamics, shock physics, and CNT-material interaction. The experimental protocol employs a CNT-methanol suspension at $\approx2.5\,\mathrm{mg/mL}$, injected into a syringe containing single-mode optical fibers. A 980 nm laser (161 mW) initiates absorption-driven thermal expansion of CNTs, launching overlapping thermoelastic and tensile waves. These coalesce within the cylindrical confinement into hypersonic vortex shock waves, driving CNT bundles at axial velocities $V_p\approx 725\,\textrm{m/s}$, with internal shock waves propagating at $$U_s = \tau V_p + c_\mathrm{CNT}\approx 5742\,\textrm{m/s}$$, where $\tau=1.3$ is the CNT Grüneisen parameter and $c_\mathrm{CNT}\approx 4800\,\textrm{m/s}$ is the bundle sound velocity (Silva et al., 9 Dec 2025).

The resultant shock pressure of $P_s \approx 6.7\,\mathrm{GPa}$ exceeds the tensile strength of silica ($\sigma\approx 5.7\,\mathrm{GPa}$), ablating the fiber core and redistributing ablated material into spiral channels and layered deposits exhibiting Fibonacci scaling.

3. Analytical and Experimental Characterization

Scanning electron microscopy (SEM), 3D profilometry, and numerical fluid dynamics model the geometry and spatial structure of ablated features. After discrete laser exposures (5, 10, 20 minutes):

• A $\sim10\,\mu\mathrm{m}$ diameter ring (thickness $\sim850\,\mathrm{nm}$) forms at the core (5 min).
• Concentric rings with diameters $\sim13\,\mu\mathrm{m}$ and $\sim31\,\mu\mathrm{m}$, with pronounced peak and trough profile structure following $\varphi$-scaling (10 min).
• Expansion to $\sim125\,\mu\mathrm{m}$ diameter and depths up to $5\,\mu\mathrm{m}$ (20 min), with both radial and thickness dimensions progressing in powers of $\varphi$.

Simulations using the Navier–Stokes equations, with an ablation rate proportional to $(P_s-\sigma)$, reliably predict that material removal tracks the high-pressure spiral (Fibonacci) ridges and low-pressure valleys, aligning with measured channel positions and widths, and preserving the threefold symmetry intrinsic to the triple-helix structure. Fast Fourier Transform (FFT) analysis of binary-processed cross-sections further corroborates the Fibonacci spiral spacings and symmetries (Silva et al., 9 Dec 2025).

4. Generative Recipes and Implementation in Phyllotactic Drilling

Dense packing models from phyllotaxis yield a generalizable "recipe" for generating Fibonacci drilling patterns not only in materials science, but also in biological and engineered contexts. The algorithm proceeds via:

1. Defining Parameters:
   • Cylinder radius $R$, hole diameter $d$, target height $H$, desired hole count $N$.
2. Divergence Angle:
   • $\alpha=F_j/F_{j+2}$ for integer $j$ (finite Fibonacci), or $\alpha=1/\varphi^2$ for ideal limit.
   • $\delta=2\pi \alpha$ divergence angle.
3. Axial Spacing:
   • For dense six-neighbour packing: $\Delta z = \sqrt{d^2 - 4R^2\sin^2(\delta/2)}$.
   • For fixed fill: $\Delta z = H/N$.
4. Coordinate Generation:

$$\theta_n = n\,\delta; \quad x_n = R \cos \theta_n; \quad y_n = R \sin \theta_n; \quad z_n = n \Delta z; \quad n = 0,1,\ldots,N-1$$

This configurational output is used as direct input for CNC, milling, or laser-drilling apparatus (Mughal et al., 2016).

Table: Key Parameters in Fibonacci Helices Drilling

Parameter	Typical Value/Formula	Description
Divergence	$\delta=2\pi/\varphi^2 \approx 137.51^\circ$	Golden-angle per Fibonacci packing
Ring Radius	$R_0\varphi^n$	Radial scaling with $n$-th spiral arm
Thickness	$h, h\varphi, h\varphi^2$	Spiral thickness progression
Core Depth	$d, d\varphi, d\varphi^2$	Helical ablation depth scaling

5. Applications in Photonics, Sensing, and Ultrasound

Fibonnaci helices drilling enables functional material surfaces with quasi-periodic, threefold (or higher) spiral symmetry. In fiber optics, the fabricated helical microstructures on fiber tips serve as integrated vortex phase modulators: incident light acquires quantized orbital angular momentum, with the spatial phase evolution determined by the topological charge of the ablated helical geometry. The Fibonacci scaling confers broadband, low-loss performance and supports higher-order optical vortex generation. Further applications include:

• Fiber-based sensors: Spiral microstructures offer enhanced sensitivity via tailored modal distributions.
• High-power pulsed lasers: Helical ablation structures facilitate robust, integrated vortex generation for advanced laser architectures.
• Biomedical neuro-ultrasonics: Deeply buried helical channels function as ultrasonic emitters with spiral wavefronts, promising for stimulation or sensing applications in neural tissues (Silva et al., 9 Dec 2025).

6. Connections to Phyllotaxis and Disk Packing

The organization of ablation or deposition in Fibonacci helices directly parallels mathematical models of plant phyllotaxis and optimal disk packing. Under the constraints of local hexagonal coordination, homogeneity, and configurational continuity, Fibonacci spiral counts and golden-angle divergences emerge in both natural and engineered systems (Mughal et al., 2016). Theoretical frameworks established by dense disk packing on cylinders explain the persistence and optimality of these patterns, while laser-induced vortex mechanisms realize these geometries at the micro- and nanoscale in synthetic materials.

7. Control, Limitations, and Future Directions

Control over Fibonacci helices drilling is achieved by tuning key experimental variables: laser power, CNT concentration, exposure time, and geometry of confinement. These parameters govern the number and scale of spiral arms, the depth and profile of drilled channels, and the density of CNT–silica deposited layers.

Limitations—and areas of current research—include precision in aligning the ablation pattern with analytical predictions in regimes of extreme miniaturization or for workpieces of non-uniform cross-section, as well as the integration of such structures into scalable fiber-optic and sensing device manufacturing. A plausible implication is that further refinement of fluid dynamic control and nanoparticle chemistry may enable custom phase and wavefront engineering for quantum optics, advanced telecommunications, and bio-interfacing platforms (Silva et al., 9 Dec 2025).