🔬 physics

Propagation of perfect optical vortex beams in a misaligned optical system

This paper derives an analytical expression for the propagation of perfect optical vortex beams through misaligned optical systems, revealing that misalignment transforms single-ring vortices into controllable multi-ring Bessel-Gauss beams at the focal plane, with specific structural characteristics dependent on topological charge and geometric ratios, before eventually reverting to a single-ring profile over longer distances.

Original authors: Bikash Kumar Das, Camilo Granados, Wenlong Gao, Stephan Fritzsche

Published 2026-07-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bikash Kumar Das, Camilo Granados, Wenlong Gao, Stephan Fritzsche

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 magical flashlight that shoots out a beam of light shaped like a perfect, glowing donut. This isn't just any donut; it's a "Perfect Optical Vortex" (POV). The coolest part about this donut is that no matter how many times you twist the light (a property scientists call "topological charge"), the size of the hole in the middle and the thickness of the ring stay exactly the same. It's like a donut that refuses to change its shape, even if you try to stretch or squeeze it with invisible hands.

But here's the catch: in the real world, nothing is ever perfectly straight. Lenses get tilted, and mirrors get bumped. This paper asks a simple question: What happens to our perfect light donut when it travels through a wobbly, misaligned optical system?

The authors, using a mix of math and computer simulations, found that when this perfect beam hits a slightly crooked lens, things get a little messy—but in a predictable way.

The Great Transformation: From Donut to Bunch of Rings

When the perfect donut beam passes through a misaligned thin lens (think of a lens that is shifted to the side or tilted slightly), it doesn't stay a single ring. Instead, at the focal point (the spot where the lens tries to focus the light), the beam transforms into something that looks like a Bessel-Gauss (BG) beam.

If you picture the original beam as a single, crisp donut, the beam coming out of the misaligned lens looks like a donut surrounded by several fainter, ghostly rings. It's like a donut that decided to grow a halo of extra rings around it.

The paper explicitly rules out the idea that the beam stays a single, perfect ring at this focal point. Instead, the simulations show it always becomes this multi-ring structure when it hits the lens.

The Drifting Center

One of the most fun discoveries is that the beam doesn't just change shape; it moves! If the lens is tilted or shifted, the center of the new, multi-ring beam drifts away from the center of the optical system.

The authors simulated this with specific numbers:

If they shifted the lens by 1 mm to the side, the beam moved 1 mm.
If they tilted the lens by 0.01 radians, the beam moved an extra 3 mm.
If they tilted it by 0.05 radians, the beam moved 16 mm.

It's like a game of pool where hitting the cue ball at a slight angle sends the whole rack of balls sliding to a new spot on the table. The paper shows that you can actually control exactly where this beam lands just by tweaking how much the lens is tilted or shifted. This isn't just a theory; the math predicts it, and the computer simulations confirm it.

The Magic of the "Self-Focusing"

The paper also explains a weird behavior called "self-focusing." Normally, light beams spread out as they travel. But these POV beams have a secret superpower: they can actually squeeze themselves tighter before spreading out again.

The authors found that the number of those extra "ghost" rings depends on a specific ratio: the size of the ring compared to its thickness.

If the ring is much bigger than it is thick (a ratio of 6.67), you get lots of extra rings.
If you make the ring thinner (a ratio of 2), you only get one extra ring.
If the ring and its thickness are exactly the same size (a ratio of 1), all the extra rings vanish! The beam becomes a single, bright ring again, but this time it's not the original perfect donut; it's a result of the self-focusing effect disappearing.

The Long Journey: Becoming Perfect Again

Here is the most surprising part of the story. The beam doesn't stay a messy, multi-ring Bessel-Gauss beam forever. The authors simulated what happens as the beam travels further away from the lens.

They tracked the beam at distances of 50 mm, 100 mm, 300 mm, and up to 4 times the focal length (4f) of the lens.

At the start (near the lens), it's a multi-ring mess.
As it travels, the inner rings fade away.
Finally, at a distance of 4f (which, in their setup, was 2000 mm away from the lens focal point), the beam transforms back into a single-ring Perfect Optical Vortex.

But wait, isn't that just the original beam? Not quite. The paper proves this new single-ring beam is "perfect" because, unlike the messy multi-ring version, its size does not change even if you change the "twist" (topological charge) of the light. Whether the twist is 1 or 2, the size of the ring and the hole in the middle stay exactly the same at this 4f distance.

What This Means (and What It Doesn't)

The authors are very clear about what they have done. They have simulated this behavior using a mathematical model. They have not built a physical experiment to prove this in a lab yet (though the math is solid).

They also rule out a few things:

They show that misalignment does not destroy the "twist" (topological charge) of the light. The beam keeps its spin, even if it moves to a new spot.
They show that the beam does not stay a single ring at the focal point of a misaligned lens; it must become multi-ring first.

So, the story is this: A perfect light donut gets bumped by a crooked lens, turns into a multi-ring halo that drifts off-center, travels a long way, and then magically reforms into a perfect single-ring donut again, ready for whatever comes next. It's a journey of distortion and recovery, all controlled by the tilt and shift of a simple lens.

Technical Summary: Propagation of Perfect Optical Vortex Beams in a Misaligned Optical System

Problem Statement
Perfect Optical Vortex (POV) beams are increasingly utilized in advanced applications such as particle trapping, optical communication, and microscopy due to their unique property: the ring radius and width remain independent of the topological charge (TC). However, the deployment of these beams in real-world scenarios is subject to inevitable optical imperfections, specifically misalignment (transverse offsets and tilting angles) of optical components. While the propagation of POV beams in ideal or free-space conditions has been studied, the influence of misalignment on their propagation dynamics and beam quality remains underexplored. Understanding how these imperfections affect the beam's integrity, centroid position, and structural evolution is critical for the robust implementation of POV-based technologies.

Methodology
The authors investigate the paraxial propagation of POV beams through misaligned optical systems using a generalized diffraction integral formalism.

Theoretical Framework: Starting with the complex field amplitude of a POV beam at the source plane (defined by ring radius $R$ , half-ring width $w_0$ , and TC $l$ ), the authors derive an analytical expression for the beam's complex field amplitude after passing through a misaligned system. This derivation utilizes a misaligned ABCD matrix approach, incorporating parameters for transverse offsets ( $\epsilon_x, \epsilon_y$ ) and tilting angles ( $\epsilon'_x, \epsilon'_y$ ).
Analytical Derivation: By solving the radial and angular integrals within the generalized diffraction integral, the authors obtain a closed-form solution (Eq. 14) describing the output field. This solution accounts for the shift in the beam's centroid and the modification of its spatial profile.
Numerical Simulation: To validate the analytical model, the authors simulate the propagation of POV beams through a specific case: a misaligned thin lens. They analyze the intensity distributions at the focal plane and at various distances beyond the focal plane, varying key parameters including TC, the ratio of ring radius to half-ring width ( $R/w_0$ ), and misalignment parameters.

Key Contributions and Results

Analytical Model for Misaligned Systems: The paper provides a rigorous analytical expression for the complex field amplitude of a POV beam propagating through a misaligned optical system. This model explicitly links system misalignment parameters to the output beam's field distribution.
Beam Displacement: The study confirms that misalignment causes the output beam to de-center. Specifically, the center of the output beam shifts from the optical axis by amounts proportional to the transverse offsets and tilting angles. The authors demonstrate that the beam's centroid position can be controlled linearly by adjusting these misalignment parameters.
Transformation at the Focal Plane: When a POV beam passes through a misaligned thin lens, it transforms into a multi-ring Bessel-Gauss (BG) beam at the focal plane. Unlike the input POV, the output BG beam's central dark core size and the radius of its maximum intensity ring expand as the input TC increases.
Role of the $R/w_0$ Ratio: The ratio of the ring radius to the half-ring width ( $R/w_0$ ) significantly influences the beam profile. A high $R/w_0$ ratio results in a multi-ring BG profile with distinct secondary rings. As this ratio decreases, the number of secondary rings reduces due to weaker self-focusing effects. When $R/w_0$ reaches unity, the self-focusing effect vanishes, and the output beam exhibits a single-ring intensity distribution.
Propagation Beyond the Focal Plane: The study reveals a dynamic evolution of the beam profile as it propagates beyond the focal plane. The multi-ring BG structure observed at the focal plane gradually transforms. At a specific propagation distance of $4f$ (four times the focal length) beyond the lens, the beam reverts to a single-ring intensity distribution.
Restoration of "Perfectness": Crucially, at the $4f$ distance, the output beam exhibits the defining characteristic of a POV: its ring radius and central dark core size become independent of the topological charge. The authors verify this by showing that beams with different TCs ( $l=1$ and $l=2$ ) possess identical beam sizes and dark core dimensions at this specific distance, confirming the beam has transformed back into a POV.

Significance
The paper establishes a fundamental understanding of how real-world optical misalignments impact the propagation of POV beams. By deriving an analytical model and demonstrating the specific transformation dynamics (POV $\to$ BG $\to$ POV) in misaligned systems, the work bridges the gap between theoretical ideal performance and practical application. The findings are significant for designing robust optical systems that utilize POV beams, as they provide the necessary theoretical tools to predict beam behavior under imperfect conditions and potentially correct for misalignment-induced aberrations in applications ranging from optical trapping to high-precision laser processing.

The Great Transformation: From Donut to Bunch of Rings

The Drifting Center

The Magic of the "Self-Focusing"

The Long Journey: Becoming Perfect Again

What This Means (and What It Doesn't)

More like this