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Quadrifocal Diffractive Lenses based onAperiodically Structured Lucas Sequences

This paper introduces a new family of quadrifocal diffractive lenses based on aperiodically structured Lucas sequences that offer enhanced axial light control and high energy efficiency, demonstrating significant potential for applications in optical trapping, multifocal imaging, and intraocular lenses.

Original authors: Arlen B. Pérez-Hernández, Adrián Garmendia-Martínez, Francisco M. Muñoz-Pérez, Walter D. Furlan, Vicente Ferrando, Juan A. Monsoriu

Published 2026-06-24

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CC BY 4.0

Original authors: Arlen B. Pérez-Hernández, Adrián Garmendia-Martínez, Francisco M. Muñoz-Pérez, Walter D. Furlan, Vicente Ferrando, Juan A. Monsoriu

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a flashlight. Usually, a lens in that flashlight focuses all the light into a single, bright spot on the wall. But what if you wanted that light to create four distinct, bright spots at different distances from the flashlight, all at the same time?

This paper introduces a new type of "smart lens" that does exactly that. Instead of using a traditional curved piece of glass, these lenses use a special pattern of microscopic rings to split the light into four specific focal points.

Here is how they did it, explained simply:

1. The Secret Recipe: A Mathematical "DNA"

To design these lenses, the researchers didn't just guess where to put the rings. They used a mathematical recipe called the Lucas Sequence.

Think of the Lucas Sequence like a family tree of numbers. It starts with two specific numbers (2 and 1), and every new number is just the sum of the two numbers before it (2, 1, 3, 4, 7, 11, 18...). This is very similar to the famous Fibonacci sequence, but it starts with different "parents."

The researchers used this number family to decide how to arrange the rings on the lens. Because the Lucas Sequence has a unique "personality" compared to the Fibonacci sequence, it naturally creates four focal points instead of the usual two.

2. The Golden Ratio Connection

There is a famous number in nature called the Golden Ratio (roughly 1.618), which you can find in the spirals of seashells and the arrangement of sunflower seeds.

The paper claims that the spacing between these four new light spots follows this Golden Ratio perfectly. It's as if the lens is whispering a mathematical secret: "The distance between spot 1 and spot 2 is related to the distance between spot 2 and spot 3 by this special golden number." This proves the lens is working exactly as the math predicted.

3. Two Ways to Build the Lens

The team built two versions of this lens to see which worked better:

The "On/Off" Switch Version (Binary-Phase): Imagine a lens where each ring is either "flat" or "stepped up" by a specific amount. It's like a staircase with only two heights. This works, but it wastes some light energy.
The "Smooth Slide" Version (Kinoform): This version is like a smooth, continuous ramp rather than a staircase. It guides the light more efficiently.

The Result: The "Smooth Slide" (Kinoform) version is much better. It captures about 44% more light for the four focal spots than the "On/Off" version. It's the difference between a leaky bucket and a perfectly sealed one.

4. Testing the Lens

To prove it worked, the researchers didn't just do math on a computer. They built a real setup using a laser and a special digital screen (called a Spatial Light Modulator) that acted as the lens.

They shone a laser through this digital lens and measured the light. The results were a perfect match with their computer predictions. The lens successfully created four bright spots at the exact distances the Lucas Sequence numbers said they would be.

5. What Can This Lens Do?

The paper suggests this technology is ready for three main jobs:

Optical Trapping: Imagine using light to hold tiny particles (like cells or dust) in mid-air. This lens could hold four different particles at four different distances simultaneously, like a set of invisible tweezers working in parallel.
Multifocal Imaging: In cameras or microscopes, you usually have to move the lens back and forth to focus on different layers of a 3D object. This lens could take a picture of all four layers at once, making the process much faster.
Intraocular Lenses (Eye Implants): For people needing cataract surgery, this could lead to a new type of artificial eye lens that gives clear vision at four different distances (near, intermediate, far, and very far) without needing to switch glasses.

Summary

In short, the researchers took a specific mathematical pattern (the Lucas Sequence), turned it into a physical lens design, and proved that it can split a single beam of light into four perfectly spaced focal points. By using a "smooth ramp" design instead of a "staircase," they made it highly efficient, opening the door for faster imaging and better control over light in medical and scientific tools.

Technical Summary: Quadrifocal Diffractive Lenses based on Aperiodically Structured Lucas Sequences

Problem Statement
Conventional diffractive optical elements (DOEs), such as Fresnel zone plates, rely on periodic radial arrangements to concentrate light. While effective for primary focusing, their periodic nature inherently generates secondary foci associated with higher diffraction orders, limiting independent control over axial energy distribution. Although phase-only and kinoform implementations improve energy efficiency by directing power into the principal order, strictly periodic designs lack the flexibility to precisely dictate the number, position, and relative intensity of resulting foci. While deterministic aperiodic structures (e.g., based on Fibonacci or Silver Mean sequences) have been explored to generate multifocal profiles, there remains a need for alternative mathematical frameworks that offer unique axial irradiance distributions and focal-plane ratios not accessible through existing sequences.

Methodology
The authors propose a new family of aperiodic diffractive lenses, termed Lucas Diffractive Lenses (LDLs), where the zone distribution is governed by the Lucas number sequence. The Lucas sequence ( $L_k$ ) shares the same recurrence relation as the Fibonacci sequence ( $L_k = L_{k-1} + L_{k-2}$ ) but utilizes different initial conditions ( $L_0 = 2, L_1 = 1$ ), generating the sequence $\{2, 1, 3, 4, 7, 11, 18, \dots\}$ .

The design process involves two main steps:

Sequence Generation: Two initial "seed" sequences, $S_0 = \{B, A\}$ and $S_1 = \{A\}$ , are defined. Higher-order sequences ( $S_n$ ) are generated recursively by concatenating the two previous sequences ( $S_n = S_{n-1} || S_{n-2}$ ). The length of the $n$ -th sequence corresponds to the $n$ -th Lucas number ( $L_n$ ).
Lens Implementation: The lens is divided into $L_n$ $L_{n}$ rings of equal area. Two phase profiles are constructed based on the sequence elements:
- Binary-Phase Lucas Zone Plate (LZP): Elements of type 'A' correspond to a phase of 0, and 'B' to a phase of $\pi$ .
- Kinoform Lucas Zone Plate (KLZP): A linear phase distribution is applied in regions where the sequence presents the pair "AB" (creating a phase "tooth"), while remaining regions have a phase of 0. This design aims to concentrate energy into the first diffraction order.

The focusing properties were analyzed numerically using the Fresnel diffraction integral to compute axial irradiance profiles. Experimental validation was performed using a phase-only Spatial Light Modulator (SLM) illuminated by a He–Ne laser ( $\lambda = 632.8$ nm), with a telescopic system and spatial filtering to isolate the first diffraction order.

Key Contributions

Novel Sequence-Based Design: The paper introduces the application of the Lucas sequence to diffractive lens design, demonstrating that distinct initial conditions in a linear recurrence relation can yield qualitatively different optical behaviors compared to Fibonacci-based designs.
Quadrifocal Generation: Unlike Fibonacci-based lenses which typically produce bifocal outputs, the proposed LDLs generate four distinct focal planes along the optical axis with a partially balanced energy distribution.
Analytical Predictability: The study establishes that the axial positions of the four focal planes are deterministically linked to Fibonacci numbers ( $F_{n-1}, F_n, 2F_{n-1}, F_{n+1}$ ) and that the ratios between these positions converge to the golden ratio ( $\phi$ ) and its powers, reflecting the intrinsic mathematical properties of the Lucas sequence.
Efficiency Comparison: A comparative analysis confirms that while both binary-phase and kinoform implementations produce identical focal positions, the kinoform configuration significantly enhances energy efficiency.

Results

Focal Positions: Numerical simulations for orders $n=7, 8,$ and $9$ revealed four well-defined focal peaks. The positions ( $u_{An}, u_{Bn}, u_{Cn}, u_{Dn}$ ) converged to values related to Fibonacci numbers with relative errors below 1% for higher orders. Specifically, the ratios between focal positions approached $u_{Bn}/u_{An} \approx \phi$ , $u_{Cn}/u_{An} \approx 2$ , and $u_{Dn}/u_{An} \approx \phi^2$ .
Energy Efficiency: The kinoform implementation demonstrated superior performance, concentrating most incident light into the first diffraction order. Quantitative comparison showed that the normalized peak intensities of the binary-phase design were approximately 44% lower on average across the four focal planes compared to the kinoform design.
Experimental Validation: Experimental results using an SLM-based setup for an order $n=7$ lens (radius $a=1.5$ mm) confirmed the generation of four focal planes. The discrepancy between experimental and numerical focal positions was less than 1%. Measured focal ratios ( $z_A/z_B \approx 1.61$ , $z_A/z_C \approx 2.01$ , $z_A/z_D \approx 2.59$ ) closely matched the theoretical golden ratio predictions.

Significance and Claims
The authors claim that this work demonstrates the versatility of Lucas-sequence-based diffractive lenses for advanced optical technologies. The primary significance lies in the ability to generate a quadrifocal system with a partially balanced energy distribution and predictable axial spacing governed by the golden ratio, without relying on iterative numerical optimization.

The paper highlights the potential application of these lenses in:

Optical Trapping: Simultaneous confinement of multiple particles along the optical axis.
Multifocal Imaging: Parallel acquisition to reduce scanning time and maintain uniform resolution across planes.
Quadrifocal Intraocular Lenses (IOLs): Offering an extended depth-of-focus concept for cataract and presbyopia surgery, potentially providing better depth coverage than current bifocal or trifocal designs.
Artificial Vision and Robotics: Simultaneous imaging of different axial planes without mechanical scanning.
Optical Tomography: Discrete axial sampling of a volume within a single exposure.

The study concludes that while the current work focuses on monochromatic performance and moderate sequence orders, the deterministic nature of the Lucas sequence provides a robust framework for designing multifocal elements with controlled axial light distribution.