🔬 optics

Studying thermal radiation with T-matrices

This paper introduces a T-matrix-based formalism for computing thermal radiation across various scales, demonstrating its application in analyzing circular polarization imbalances in chiral structures and its potential for studying relativistic astronomical objects.

Original authors: Juan Diego Mazo-Vásquez, Markus Nyman, Marjan Krstić, Lukas Rebholz, Carsten Rockstuhl, Ivan Fernandez-Corbaton

Published 2026-02-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Juan Diego Mazo-Vásquez, Markus Nyman, Marjan Krstić, Lukas Rebholz, Carsten Rockstuhl, Ivan Fernandez-Corbaton

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 you are standing in a dark room with a hot cup of coffee. Even though you can't see the heat, the coffee is constantly "shouting" invisible light waves (thermal radiation) into the room. Usually, this shouting is chaotic and symmetrical—it sends out light in all directions equally, and the light waves spin randomly.

But what if you could build a tiny object that, when heated, doesn't just shout randomly, but shouts in a specific direction, or even shouts in a "left-handed" spin versus a "right-handed" spin? That is the big question this paper answers.

Here is a simple breakdown of what the scientists did, using some everyday analogies.

1. The Problem: Calculating the "Heat Shout"

Scientists have known for a long time how to calculate how a simple, round ball (like a marble) glows when hot. But modern technology lets us build weird shapes: spirals, twisted chains, and even complex molecules. Calculating how these weird shapes glow is like trying to predict the sound of a jazz band playing in a cave with a million echoes. It's incredibly hard.

Usually, to solve this, you need to simulate every single wave of light hitting the object and bouncing off. It's like trying to count every single raindrop hitting a complex sculpture.

2. The Solution: The "T-Matrix" Translator

The authors introduced a new way to do this math using something called a T-matrix.

Think of the T-matrix as a universal translator or a recipe card.

Instead of simulating every single raindrop (light wave) hitting the sculpture, you just need the "recipe card" (the T-matrix) for that specific shape.
This card tells you: "If a wave comes from the left, it bounces back like this. If it comes from the top, it bounces like that."
Once you have this card, you don't need to do the hard work again. You can instantly calculate how the object will glow, how it absorbs heat, and how it interacts with light, no matter how complex the shape is.

The paper combines this "recipe card" method with a clever algebraic trick (using "Hilbert spaces," which is just a fancy way of organizing all possible light waves) to make the math much faster and cleaner.

3. The Experiment: Testing the "Heat Shout"

To prove their new method works, they tested it on three very different "instruments":

The Chain of Beads: Imagine a string of four silicon carbide beads. When heated, they glow mostly in a specific color (infrared). Because they are lined up, they glow more strongly sideways than up and down, like a flashlight beam. This was a simple test to make sure their math matched reality.
The Silver Helix (The Twist): This is the star of the show. Imagine a tiny, 200-nanometer-wide silver spring (like a Slinky). Because it is twisted, it is "chiral" (handed).
- The Result: When they heated this silver spring, it didn't just glow; it glowed with a preference for spinning. It emitted much more "left-spinning" light than "right-spinning" light (or vice versa, depending on the twist).
- The Analogy: It's like a hot spring that doesn't just release steam, but releases steam that is all spinning in the same direction, like a tornado. The imbalance was huge—almost 90% more of one spin than the other!
The Chiral Molecule (The Tiny Twist): They also looked at a single molecule called R-BINOL. This is a tiny, twisted chemical structure.
- The Result: It also tried to spin its light, but because it is so small, the effect was tiny—like a whisper compared to the silver spring's shout. However, if you have a whole solution of these molecules, their tiny whispers add up to a loud, measurable signal.

4. Why Does This Matter?

This isn't just about making pretty heat maps. This has real-world applications:

Better Energy Harvesting: If we can control exactly how heat is radiated (and in which "spin" direction), we can build better solar cells or cooling systems that are more efficient.
Detecting Molecules: Since chiral molecules (like many drugs and biological substances) have a specific "handedness," this method could help scientists detect them by looking at the unique "spin" of the heat they emit.
Space Travel: The math they developed is so flexible that it could eventually be used to calculate the heat signatures of asteroids or spaceships moving at near-light speeds, helping astronomers understand what they are looking at.

The Bottom Line

The authors built a powerful new "calculator" (the T-matrix formalism) that lets us predict how complex, weirdly shaped objects glow when they are hot. They showed that by twisting these objects (like a silver spring), we can make them emit light that spins in a specific direction. It's like turning a chaotic heat radiator into a directional, spinning laser of heat.

1. Problem Statement

Thermal radiation is ubiquitous and critical for applications like thermophotovoltaics and radiative cooling. While controlling thermal radiation via micro- and nanofabrication is a major goal, existing computational tools often struggle with complex geometries or lack a unified framework for diverse objects (from macroscopic clusters to single molecules).

Limitations of current methods: Green tensor formalisms are often semi-analytical and limited to simple geometries (e.g., planar layers). Numerical methods like Boundary Element Methods (BEM) exist but can be computationally intensive or less flexible for specific scattering problems.
The Gap: There is a need for a general, algebraic formalism that can compute thermal radiation spectra for arbitrary shapes, material parameters, and even molecular systems, while also accommodating advanced theoretical extensions (e.g., non-reciprocal objects, relativistic motion, or non-monochromatic emission modes).

2. Methodology

The authors propose a novel formalism that combines Waterman's T-matrix method with an algebraic approach to light-matter interactions.

Algebraic Framework: The electromagnetic field is treated within a Hilbert space ( $\mathcal{M}$ ). Light-matter interactions are modeled using a scattering operator $S$ , related to the transition matrix (T-matrix) $T$ by $S = I + T$ .
Absorption Operator: The number of absorbed photons is derived using the absorption operator $Q = -(T + T^\dagger + T^\dagger T)$ . By performing a Singular Value Decomposition (SVD) of the monochromatic absorption matrix $Q(k)$ , the authors identify singular modes ( $|ks\rangle$ ) and singular values ( $q_s(k)$ ) that characterize the object's absorption properties.
Directional Kirchhoff Law: The formalism utilizes the directional Kirchhoff law, which equates the thermal emission of an object in a specific direction and polarization to its absorption of radiation from a thermal bath coming from the opposite direction.
Thermal Bath Statistics: The random thermal sources are modeled as a Planckian bath. The authors assume the bath is uncorrelated in frequency, direction, and polarization.
Derivation of Emission Rate: By combining the absorption operator with the statistical properties of the thermal bath, the authors derive a compact expression for the photon emission rate density per angular wavenumber ( $k$ ), solid angle, and polarization (Eq. 18). This expression depends on the T-matrix elements and Wigner D-matrices (for spatial rotations).
Computational Strategy:
1. Compute monochromatic T-matrices ( $\tilde{T}(k)$ ) for the object.
2. Construct the absorption matrix $Q(k)$ using Eq. (4).
3. Perform SVD on $Q(k)$ to extract modes and singular values.
4. Apply Eq. (18) to calculate the thermal emission spectrum.
5. The method tracks singular vectors continuously across frequencies to ensure physical consistency.

3. Key Contributions

Unified Formalism: The paper establishes a single theoretical framework applicable to nanoparticles, clusters of particles, and individual molecules.
- For macroscopic objects, T-matrices are derived from Maxwell solvers.
- For molecules, T-matrices are mapped from quantum-chemical dynamic polarizabilities (dipolar approximation).
Algebraic Efficiency: The use of Hilbert space notation and SVD provides a compact, computationally efficient way to handle random thermal fields, avoiding the need to simulate random source realizations directly.
Extensibility: The formalism is designed to be a foundation for more nuanced theories beyond the standard directional Kirchhoff law (e.g., emission via non-monochromatic modes) and for calculating thermal radiation from objects moving at relativistic speeds (via Lorentz boosts in the algebraic setting).
Open Source Implementation: The authors provide a publicly available Python implementation using the treams package and a database of T-matrices, facilitating reproducibility and community adoption.

4. Results and Exemplary Applications

The authors validated their theory against known semi-analytical results (Mie theory for spheres) and demonstrated it on three distinct cases:

A. Chain of Dielectric Spheres (SiC):
- A chain of four SiC spheres ( $R=250$ nm) at $T=500$ K.
- Result: The emission spectrum peaks around $780$ cm $^{-1}$ , matching the absorption resonance. The emission pattern is dipole-shaped due to the chain geometry. Since the material is achiral, emission is identical for both left- and right-circular polarizations.
B. Metallic Helix (Chiral Structure):
- A silver helix ( $\sim 200$ nm size) optimized for strong electromagnetic chirality.
- Result: The structure exhibits a significant imbalance in the emission of left- vs. right-circularly polarized light.
- Metric: The authors define a thermal g-factor ( $g_{th}$ ), analogous to the dissymmetry factor in chiro-optics. The helix achieved a maximum spectral $g_{th} \approx 1.8$ (theoretical max is 2), indicating a massive imbalance where one helicity dominates the other. The total emission power difference is on the order of the emission power itself, making it experimentally observable.
C. Chiral Molecule (R-BINOL):
- A single R-BINOL molecule in Chloroform at $T=300$ K.
- Result: The molecule shows a small but non-zero thermal g-factor, with peaks in the vibrational (IR) and electronic (UV) regions.
- Scaling: While the imbalance for a single molecule is tiny ( $N_{+1} \approx 1.116001$ vs $N_{-1} \approx 1.115987$ photons/s), the formalism allows for scaling this spectrum by the number of molecules in a solution (e.g., $\sim 10^{21}$ molecules/cm $^3$ ), predicting measurable macroscopic effects.

5. Significance and Outlook

Bridging Scales: This work successfully bridges the gap between macroscopic scattering theory (T-matrices) and molecular quantum chemistry, allowing for the prediction of thermal radiation properties of molecular solutions.
Chiral Thermal Radiation: It provides a rigorous theoretical basis for understanding and designing chiral thermal emitters, which could be used for circularly polarized thermal sources without external magnetic fields.
Future Applications: The framework is poised to tackle complex systems such as:
- Relativistic thermal radiation (e.g., astronomical objects moving at high speeds).
- Non-equilibrium and time-varying media.
- Near-field radiative heat transfer between closely spaced objects (provided they do not overlap their circumscribed spheres).

In summary, the paper presents a robust, algebraic, and computationally efficient method for calculating thermal radiation spectra using T-matrices, validated by demonstrating significant circular polarization imbalances in both optimized nanostructures and chiral molecules.