🔢 mathematics

Waves 2026 Book of Abstracts

This abstract announces WAVES 2026, the 17th biennial international conference on mathematical and numerical aspects of wave propagation, which will be held in Montreal, Canada, from June 22–26, 2026, to serve as a leading forum for advancing theory, computation, and applications across diverse wave phenomena.

Original authors: Lise-Marie Imbert-Gerard, Chrysoula Tsogka, Daniel Appelo, Thomas Hagstrom, Yann-Meing Law, Serge Prudhomme, Behrooz Yousefzadeh

Published 2026-07-23

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Lise-Marie Imbert-Gerard, Chrysoula Tsogka, Daniel Appelo, Thomas Hagstrom, Yann-Meing Law, Serge Prudhomme, Behrooz Yousefzadeh

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

Imagine trying to listen to a friend's voice across a crowded, windy stadium. If the air is calm, you hear them clearly. But if the wind is howling and the air is turbulent, the sound gets scrambled, bouncing off invisible pockets of hot and cold air. In the world of light, this is exactly what happens when a laser beam travels through a "turbulent atmosphere," like the air above a hot road or the sky on a stormy day. The light doesn't just travel in a straight line; it gets jumbled up, creating a speckled, glittering pattern of bright and dark spots called "speckle." This is a bit like looking at sunlight reflecting off a rippling pool of water—the light is constantly shifting and dancing.

Scientists have long wanted to predict exactly how this light behaves, especially for things like free-space laser communication (sending data through the air instead of cables). The big question has been: as the light travels further and further through this messy air, does it become completely random, or is there a hidden order? For a long time, a famous idea called the "Gaussian conjecture" suggested that after traveling a long distance, the light's intensity would settle into a very specific, predictable pattern of randomness known as a "complex circular Gaussian" distribution. Think of it like rolling a die millions of times; eventually, the results average out in a way that is mathematically perfect and predictable, even if any single roll is random.

This paper, presented by Guillaume Bal and Anjali Nair from the University of Chicago, takes a deep dive into this mystery. They didn't just guess; they used advanced math to prove that the "Gaussian conjecture" is actually true for laser beams traveling through weak turbulence. They showed that as the beam goes further, the chaotic speckle pattern does indeed settle into that specific, predictable random state. Furthermore, they confirmed that a popular computer simulation method called the "phase screen method"—which breaks the journey into slices to calculate the light's path—works incredibly well, even when the slices are much larger than the tiny ripples in the air. This is a big deal because it means we can trust our computer models to predict how laser signals will behave in the real world, helping engineers design better systems for sending information through the sky without needing to simulate every single tiny molecule of air.

The Story of the Scrambled Laser

The Problem: The Twinkling Laser
Imagine you are shining a laser pointer at a friend. If the air between you is perfectly still, the dot is steady. But if the air is turbulent (like heat rising off a sidewalk), the dot starts to dance and flicker. Sometimes it's super bright, and sometimes it's almost invisible. This flickering is called "speckle," and it happens because the light waves are interfering with each other in a chaotic way.

For decades, scientists have tried to write a math rule that predicts exactly how this speckle behaves. The "Gaussian conjecture" was the leading theory. It claimed that if you let the laser travel far enough through the turbulence, the randomness of the light would smooth out into a specific, well-known mathematical shape (the Gaussian distribution). It's like saying that if you shake a box of marbles long enough, they will eventually settle into a perfectly predictable pile, even though you can't predict where any single marble will land.

The Breakthrough: Proving the Theory
Bal and Nair set out to prove this conjecture. They looked at two different mathematical models used to describe how light moves through turbulent air: the "paraxial model" (a simplified version for beams that don't spread out too much) and the "Itô-Schrödinger model" (a more complex version that treats the turbulence as random noise).

They found that in the "weak-coupling regime" (which basically means the turbulence isn't too crazy, just enough to cause some scrambling), the conjecture holds up. They proved that as the light travels a long distance, the wave field does indeed become "complex circular Gaussian." In plain English, this means the light's intensity at any given point follows a specific rule: it's most likely to be a medium brightness, but it can occasionally be very bright or very dark, and the chance of it being any specific brightness follows a neat, exponential curve.

Why This Matters for Communication
Why should you care? This result is actually a bit of "bad news" for engineers trying to send laser messages through the air. Because the light intensity follows this specific random rule, the signal can fluctuate wildly. One moment the receiver gets a strong signal, and the next, it gets almost nothing. This "scintillation" (flickering) can mess up data transmission.

However, the paper also offers a silver lining. While the light at a single point is wild, if you average the light over a slightly larger area, the fluctuations smooth out. It's like how a single raindrop hitting a puddle makes a big splash, but if you look at the whole puddle, the water level rises steadily. This suggests that by designing receivers that look at a slightly larger patch of light, we can still get a stable signal even in bad weather.

The Computer Trick: The Phase Screen
To study this, scientists use computer simulations. One of the most popular tools is the "phase screen method." Imagine the turbulent air as a stack of glass sheets. As the laser passes through each sheet, the glass bends the light a little bit. The computer calculates the bend for one sheet, moves to the next, and repeats.

Usually, for these simulations to be accurate, the sheets need to be incredibly thin—thinner than the wavelength of the light itself. This makes the calculations take forever. But Bal and Nair showed something surprising: even if you use "thick" sheets (much larger than the wavelength), the phase screen method still gives the right answer for the statistics of the light. It's like taking a photo of a fast-moving car; usually, you need a super-fast shutter speed to see it clearly. But their math shows that for this specific type of problem, even a slightly blurry photo (a coarser simulation) captures the essential "shape" of the motion perfectly.

The Bottom Line
This paper doesn't just guess; it provides a rigorous mathematical proof. It confirms that the "Gaussian conjecture" is real for laser beams in turbulent air and that our computer tools are more robust than we thought. This gives scientists and engineers a solid foundation to build better communication systems, knowing exactly how the light will behave when the atmosphere gets messy. It turns a chaotic, twinkling mess of light into a predictable, mathematically beautiful pattern.

Technical Summary: Optical Wave Reconstruction in Highly Turbulent Random Media

Problem Statement
The paper addresses the challenge of reconstructing a wavefront from noisy phase gradient data, a central task in adaptive optics. Standard algorithms, typically based on solving a linear inverse problem for the phase $\phi$ given gradient measurements $g = \nabla \phi$ , rely on the assumption that the phase function is smooth and single-valued. However, these methods fail when waves propagate through strongly turbulent media. In such regimes, intensity fluctuations cause the wave field to develop zeros, leading to phase singularities (branch points). Under these conditions, the phase reconstruction problem becomes ill-posed because standard approaches attempt to reconstruct a singular field directly, resulting in significant inaccuracies.

Methodology
The authors propose a reformulation of the problem that circumvents the difficulties associated with singular phase functions by focusing on the reconstruction of the entire complex wave field $u(x)$ , which remains well-behaved even in the presence of turbulence.

Formulation: Instead of solving for the phase $\phi$ directly, the method utilizes the polar representation of the optical field, $u(x) = \rho(x) \exp(i\phi(x))$ . The gradient of the field is expressed as:
$\nabla u = u \left( \frac{\nabla(\rho^2)}{2\rho^2} + i\nabla\phi \right)$
By substituting the measured intensity $I$ (where $I = \rho^2$ ) and the measured phase gradient $g$ into this relation, the problem is transformed into finding a field $u$ that satisfies:
$\nabla u - \left( \frac{\nabla I}{2I} + ig \right) u \approx 0$
Optimization: The authors formulate a weighted least-squares minimization problem to find the minimizer of the functional:
$\hat{J}[u] = \frac{1}{2} \| I^p \nabla u - h u \|^2$
where $h$ is a vector-valued function derived from the data ( $h = I^{p-1}(\frac{1}{2}\nabla I + iIg)$ ) and $p \geq 0$ is a weighting parameter.
Regularization: Since the minimization of $\hat{J}[u]$ is ill-posed due to non-uniqueness (the zero solution and scaled multiples are also solutions), an additional constraint is introduced. To account for measurement noise, the solution is fixed at a single point $x_0$ via a penalty term, yielding the modified functional:
$J[u] = \frac{1}{2} \| I^p \nabla u - h u \|^2 + \frac{c_0}{2} |u(x_0) - \sqrt{I(x_0)}|^2$
where $c_0 > 0$ is a penalty constant.
Numerical Implementation: The problem is discretized on a Chebyshev-Gauss-Lobatto grid. Spectral filtering is applied to handle noise. The resulting overdetermined linear system $Av = f$ (where $v$ is the vectorized field) is solved using standard direct methods based on SVD and QR decomposition.

Key Results
The paper presents a numerical simulation of a Gaussian beam propagating through strong turbulence to validate the approach. The simulation parameters include a wavelength of $1\mu m$ , a propagation distance of $1.2 km$, and a turbulent refractive index modeled by a Von-Karman spectrum. Using a $100 \times 100$ subaperture array and a $64 \times 64$ Chebyshev grid, the method achieved a relative $L^\infty$ error of $E = 2.6 \times 10^{-3}$ for the optical field. The reconstructed phase profiles demonstrate significant accuracy improvements over traditional methods in strong turbulence conditions.

Significance and Claims
The authors claim that their framework overcomes the limitations of standard phase reconstruction algorithms in highly turbulent media. By reconstructing the full complex wave field rather than the singular phase directly, the method avoids the inherent difficulties of reconstructing singular functions. The paper asserts that this variational approach, which minimizes the error between the field's gradient and the measured data, results in "significant accuracy improvements" in regimes where traditional methods fail due to branch point singularities. The reconstructed complex field allows for the recovery of the singular phase (if required) via the imaginary part of the logarithm of the field, but the primary contribution is the robust reconstruction of the field itself.

The Story of the Scrambled Laser

More like this