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Pure-amplitude holograms for high-efficiency generation of phase radial grating based radial carpet beams: Theory and experiments under plane-wave and Gaussian illumination

This study proposes and validates a pure-amplitude hologram technique that generates high-efficiency radial carpet beams under both plane-wave and Gaussian illumination, offering a cost-effective alternative to spatial light modulators with approximately five times greater useful power for applications like optical trapping and free-space communication.

Original authors: Saifollah Rasouli, Somaye Fathollazade, Mohammad Mohammadi

Published 2026-02-26

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Saifollah Rasouli, Somaye Fathollazade, Mohammad Mohammadi

Original paper licensed under CC BY 4.0 (http://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

The Big Idea: A "Magic Stencil" for Light

Imagine you are a chef trying to bake a very specific, intricate cake. Usually, to get a perfect shape, you need a high-tech, expensive 3D printer (in the world of light, this is a device called a Spatial Light Modulator or SLM). These printers are slow, costly, and often waste a lot of ingredients (light energy) in the process.

The researchers in this paper, led by Saifollah Rasouli, asked: "Can we bake this same fancy cake using a simple, cheap cookie cutter instead?"

Their answer is yes. They invented a new type of "cookie cutter" for light called a Pure-Amplitude Hologram (PAH). It's a simple piece of plastic with a pattern printed on it that can create complex beams of light much more efficiently than the expensive high-tech machines.

The Problem: The "Leaky Bucket"

For years, scientists have wanted to create a special type of light beam called a Radial Carpet Beam (RCB).

What is it? Imagine a beam of light that looks like a circular carpet with spokes (like a wheel) or a galaxy spinning in the air. These beams are super tough; they don't spread out easily and can heal themselves if blocked by dust or obstacles. They are great for trapping tiny particles (like cells) or sending data through the air.
The Old Way: To make these beams, scientists used "Phase Radial Gratings" (PRGs). Think of this as a glass plate that twists the light waves.
- The Catch: Making a perfect glass plate that twists light exactly the right way is incredibly hard and expensive.
- The Workaround: Most people use an SLM (the expensive 3D printer). But the SLM is like a leaky bucket. It scatters about 99% of the light into useless directions, leaving less than 1% to actually form the beam you want. It's very inefficient.

The Solution: The "Patterned Stencil"

The team came up with a clever trick. Instead of trying to twist the light (phase), they decided to block and let through light (amplitude) in a very specific way.

The Analogy: The Music Box and the Drum
Imagine you have a simple drum (a linear grating) that makes a steady beat. Now, imagine you want that drum to play a complex melody that looks like a spinning galaxy.

The Old Method: You hire a virtuoso drummer (the SLM) to hit the drum in complex patterns. It's expensive, and the drummer gets tired (wastes energy).
The New Method (PAH): You take a simple drum and paint a complex pattern on its surface. When you hit it, the pattern itself forces the sound to change.
- The researchers took a simple, striped pattern (like a barcode) and "hid" the complex galaxy pattern inside the stripes.
- When light hits this new "stencil," it doesn't just pass through; it gets sorted. The light that goes straight through stays normal. But the light that bends to the side (the first "diffraction order") instantly transforms into the perfect, spinning galaxy beam.

Why is this a Big Deal?

1. It's a Powerhouse (Efficiency)
The most exciting part of this paper is the energy savings.

The SLM (Old Way): If you put in 100 units of light, you get about 1 unit of useful beam. The rest is lost.
The PAH (New Way): If you put in 100 units of light, you get about 5 units of useful beam.
The Result: This new method is 5 times more efficient. It's like getting 5 cups of coffee for the price of 1. This is huge for applications like optical tweezers (trapping particles with light) or free-space communication, where every bit of power counts.

2. It Creates "Double" Magic
Because of the way they designed the pattern, the light doesn't just make one beam. It creates different versions of the beam in different directions.

Imagine shining a flashlight through a prism. You get a rainbow.
Here, shining a light through their hologram creates the "galaxy beam" in the +1 direction, and a slightly different, even more complex version of it in the -1 direction. It's like getting two different toys for the price of one.

3. The "Gaussian" vs. "Plane Wave" Twist
The researchers tested this with two types of light sources:

Plane Wave (A perfect, flat sheet of light): This creates a perfect, unchanging "Radial Carpet Beam" that stays the same forever. It's like a perfect, rigid sculpture.
Gaussian Beam (A flashlight beam that is bright in the middle and fades at the edges): This creates a "Radial Carpet-Like Beam." It looks like the perfect sculpture at first, but as it travels, the edges start to wiggle and change shape.
- The Analogy: Think of the Plane Wave beam as a laser pointer dot that stays sharp forever. The Gaussian beam is like a ripple in a pond; it looks nice near the center, but the ripples spread out and change as they travel further. The researchers figured out exactly how far you can go before the shape gets too distorted.

Real-World Impact

Why should we care?

Cheaper Tech: You don't need a $50,000 machine to do this. You can print this hologram on a piece of plastic using standard printing techniques.
Better Trapping: Because it's 5 times brighter, scientists can trap and move more particles at once, which is great for biology and medicine.
Faster Internet: For sending data through the air (free-space optical communication), having a beam that is brighter and doesn't waste energy means faster, more reliable connections.

Summary

The researchers took a complex, expensive problem (making special light beams) and solved it with a simple, cheap, and highly efficient trick. They turned a "leaky bucket" of light into a "powerful hose" by printing a clever pattern on a piece of plastic. It's a brilliant example of how a little bit of smart math and a simple design can outperform expensive high-tech machinery.

1. Problem Statement

Radial Carpet Beams (RCBs) are a distinct class of structured light fields known for accelerating, non-diffracting, self-healing, and local self-amplification properties. These features make them ideal for applications like optical trapping and free-space optical communication.

However, generating high-quality RCBs typically requires Pure-Phase Radial Gratings (PRGs) with a specific phase modulation depth (often $\pi$ radians).

Fabrication Challenges: Creating physical phase-only gratings with a full $\pi$ -radian phase swing is technically difficult and expensive.
SLM Limitations: Spatial Light Modulators (SLMs) are commonly used to impose the required phase profile, but they suffer from low diffraction efficiency. Due to their pixelated 2D lattice structure, SLMs scatter approximately 1% of incident power into unwanted diffraction orders even without a pattern, and the efficiency for generating RCBs directly from pure PRGs remains below 1%.
Gaussian Beam Issues: When using Gaussian beams (common in lasers) with standard amplitude radial gratings, the resulting patterns are not propagation-invariant; they revert to a Gaussian profile in the far field, losing their radial structure.

2. Methodology

The authors propose a novel generation method using Pure-Amplitude Holograms (PAHs) that embed the transmission function of a binary PRG into the phase argument of a linear amplitude grating.

Hologram Design:
- The transmission function is constructed by combining a linear amplitude grating (sinusoidal or binary) with a binary PRG.
- Mathematically, the transmission function $t(r', \theta')$ is defined as:
  $t(r', \theta') = \sum_{n=-\infty}^{\infty} t_n \exp\left[ -in \left( \frac{2\pi r' \cos \theta'}{d} + \gamma \text{sign}(\cos(m\theta')) \right) \right]$
  Where $t_n$ are Fourier coefficients of the linear grating, $d$ is the period, $m$ is the number of spokes, and $\gamma$ is the phase modulation amplitude of the embedded PRG.
- The binary PRG phase profile is "embedded" into the argument of the amplitude grating's cosine term.
Illumination Scenarios:
1. Plane-Wave Illumination: Generates ideal, propagation-invariant RCBs in non-zero diffraction orders.
2. Gaussian Beam Illumination: Generates Radial Carpet Beam-Like Patterns (RCBLPs). These are not perfectly propagation-invariant but maintain a stable structure over a specific range before the outer regions distort.
Theoretical Framework:
- The authors derived a comprehensive diffraction theory using the Fresnel-Kirchhoff integral.
- They introduced a phase-amplitude enhancement mechanism: In the $n$ -th diffraction order, the effective phase amplitude becomes $n\gamma$ . This allows the generation of distinct RCBs/RCBLPs in different orders (e.g., the second order has twice the phase modulation depth of the first).
- A similarity metric was developed to quantify the propagation invariance of RCBLPs, defining a critical distance ( $z_{in,m}$ ) where the inner structure remains stable.
Experimental Setup:
- A binary amplitude hologram was fabricated on a plastic substrate using lithographic printing.
- The duty cycle was experimentally determined (0.27) via microscopic imaging to accurately model Fourier coefficients.
- Experiments were conducted using a 532 nm Nd:YAG laser under both plane-wave and Gaussian beam illumination.

3. Key Contributions

High-Efficiency Generation: The PAH method eliminates the need for expensive SLMs and complex phase-only fabrication. It achieves a diffraction efficiency of ~4.38% in the first order, which is approximately 4.6 times higher than SLM-based generation (~0.95%).
Phase-Amplitude Enhancement: The study demonstrates that multiplying the diffraction order number ( $n$ ) by the embedded phase amplitude ( $\gamma$ ) creates scaled versions of the RCB in higher orders. This allows for the simultaneous generation of multiple distinct beam types from a single hologram.
Gaussian Beam Stability Analysis: The paper provides the first theoretical treatment of Gaussian beam diffraction from PRGs. It defines the RCBLP concept, characterizing the propagation distance over which the beam retains its structure based on the incident beam waist ( $w_0$ ) and the number of spokes ( $m$ ).
Experimental Validation: The authors successfully fabricated the hologram and verified that the experimental intensity patterns match theoretical simulations and numerical predictions for both plane-wave and Gaussian illumination.

4. Results

Efficiency Comparison:
- PAH Method: 0.44 mW output power (4.38% of 10.03 mW input) in the first order.
- SLM Method: 0.096 mW output power (0.95% of input) in the first order.
- Conclusion: The PAH method yields nearly 5 times more useful power than SLMs.
Beam Quality:
- Under plane-wave illumination, the generated beams are ideal RCBs that remain invariant during propagation.
- Under Gaussian illumination, the generated RCBLPs show a stable central region (containing $2m$ intensity spots) that persists over a calculated distance (e.g., ~74 m for $w_0=8$ mm, $m=10$ ). The outer regions gradually distort, but the core structure remains usable for practical distances.
Order Multiplicity: The experiment confirmed that different diffraction orders produce different beam profiles. For instance, with $\gamma = \pi/4$ , the second order ( $n=2$ ) exhibited an effective phase of $\pi/2$ , resulting in a beam with twice the number of intensity spots compared to the first order.
Similarity Metric: The similarity function $S(z)$ confirmed that wider incident Gaussian beams preserve the RCBLP structure over longer distances.

5. Significance

This work offers a low-cost, high-efficiency, and experimentally accessible alternative to SLM-based beam shaping.

Applications: The high power efficiency makes the technique particularly suitable for multi-particle optical trapping (where power is critical) and free-space optical communication (requiring robust, structured beams).
Scalability: The ability to generate multiple beam types simultaneously in different diffraction orders without additional hardware adds significant versatility to optical systems.
Practicality: By using simple amplitude holograms (which are easier and cheaper to fabricate than phase-only elements) and standard laser sources, this method lowers the barrier for deploying advanced structured light technologies in real-world scenarios.