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Broadband Control of Light through Complex Media via Automatically Self-Referencing Transmission Matrix Characterisation

This paper presents an automatically self-referencing technique based on spatial state tomography that enables robust, broadband transmission matrix characterization of complex media without external references or iterative algorithms, thereby facilitating high-fidelity light control for applications in biomedical imaging and optical communications.

Original authors: Marcos Morote, Mickael Mounaix, Andrew Komonen, Martin Plöschner, Joel Carpenter

Published 2026-09-24

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CC BY 4.0

Original authors: Marcos Morote, Mickael Mounaix, Andrew Komonen, Martin Plöschner, Joel Carpenter

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

Light travels through the world in predictable ways when it moves through clear air or glass, but it behaves very differently when it encounters a chaotic environment. Imagine shining a flashlight through a thick fog or a piece of frosted glass; the beam scatters, breaking into a messy, random pattern of bright and dark spots. This happens because the material is full of tiny imperfections that bounce the light in every direction, scrambling the information the light was carrying. For decades, scientists have struggled to make sense of this chaos, particularly when trying to send images or data through such disorderly materials. The key to unlocking this potential lies in understanding a specific map of how the light enters and exits the material. This map, known as a transmission matrix, acts like a complete instruction manual for the light's journey, describing exactly how every possible path through the disorder connects to every possible path out. If researchers could measure this map accurately, they could reverse the scrambling effect, turning a messy blur back into a sharp image or a focused beam, even after it has passed through complex obstacles like biological tissue or tangled optical fibers.

The challenge has always been how to measure this map without getting lost in the noise. Traditional methods require a second, perfectly synchronized beam of light to act as a reference, comparing the scrambled light against this known standard to figure out the hidden details. This works well for lasers that are very steady and coherent, but it fails completely when the light source is broadband or short-lived, such as the white light used in medical imaging or high-speed communications. In these cases, the reference beam cannot stay synchronized with the messy light, making the measurement impossible. Other attempts to solve this have relied on guessing the answer through complex computer algorithms or by picking a specific part of the light to use as a reference, but these approaches often create blind spots where the light is too weak to measure or require so much computing power that the results are unreliable.

A team of researchers at The University of Queensland has now developed a new way to map this chaotic light that works without any external reference beam, regardless of how steady or broad the light source is. Instead of trying to compare the light to something outside the system, their method treats the light itself as its own guide. They used a technique called spatial-state tomography, which essentially breaks the light down into its fundamental building blocks and measures how each one interferes with every other one. By systematically shining different patterns of light into a fiber optic cable and recording the resulting intensity patterns at the other end, they were able to reconstruct the complete map of the fiber's behavior. This approach is unique because it does not need to guess which part of the light to use as a reference; instead, it uses the entire collection of light modes as a distributed network of references, ensuring that no part of the system is left unmeasured.

The researchers tested this method on a two-meter length of multimode fiber, a type of cable that can carry hundreds of different light paths simultaneously. They first demonstrated that the technique worked just as well as the traditional, gold-standard method for steady, single-color laser light, successfully recovering a map with 420 distinct channels. More importantly, they showed that the method remained robust when they switched to broadband light sources that mimic the short-coherence light used in real-world applications. By adjusting their measurements to match the specific bandwidth of the light source, they were able to create a custom map that optimized the control of light for that specific type of illumination. When they used these maps to shape the light at the far end of the fiber, they could focus it into sharp spots or form complex shapes like smiley faces, even when the light was highly disordered.

The significance of this work lies in its ability to extend precise control of light into regimes where it was previously thought to be unmanageable. The researchers found that for broadband sources, there is no single static map that works for all wavelengths; instead, the best control comes from a map that is specifically tuned to the bandwidth of the light being used. Their experiments showed that by matching the measurement to the source, they could maintain high-quality control over a much wider range of colors than traditional methods allow. This means that technologies relying on complex media, such as endoscopes for seeing inside the human body or communication systems sending data through the atmosphere, could become much more effective and versatile. The method removes the need for fragile, complex setups and opens the door to using simple, powerful light sources to see and communicate through the most difficult optical environments.

Technical Summary: Broadband Control of Light through Complex Media via Automatically Self-Referencing Transmission Matrix Characterisation

Problem Statement
Controlling light propagation through complex media, such as multimode fibres (MMFs) and turbid biological tissues, relies on accurately characterizing the transmission matrix (TM). While the TM is a deterministic linear operator, predicting it theoretically is generally impractical due to the unknown microscopic structure of the scattering system. Consequently, the TM must be measured experimentally.

Conventional methods, such as off-axis digital holography (DH), require an external phase reference beam. This approach faces significant limitations:

Coherence Constraints: For low-coherence or broadband sources, the external reference must maintain mutual coherence, spatial/polarization matching, and path-length stability within the short coherence length of the source. This is often impractical or fundamentally infeasible.
Signal Loss: In low-light conditions (e.g., single-photon measurements), splitting the signal for a reference beam renders recovery prohibitively noisy.
Limitations of Existing Self-Referencing Methods: Alternative approaches that avoid external references suffer from their own drawbacks. Correlation-filter methods incur losses scaling with the number of modes and require complex distal-end hardware. Intensity-only methods relying on global optimization or phase retrieval are non-convex, computationally demanding, and sensitive to noise. Furthermore, strategies using a pre-selected internal reference field create "blind spots" where the reference is weak or absent, a critical failure mode in MMFs with structured output fields. Machine learning approaches require extensive training datasets and stable conditions, and their performance degrades with broadband sources where interference patterns wash out.

Methodology: Spatial-State Tomography (SST)
The authors introduce an automatically self-referencing technique based on Spatial-State Tomography (SST). This method characterizes the TM using only intensity measurements at the distal output, eliminating the need for an external reference, iterative phase retrieval, or prior selection of an internal reference channel.

The core principles of the methodology are:

High-Dimensional Stokes Vector Reconstruction: The method frames the problem as a high-dimensional spatial analogue of Stokes polarimetry. For each output camera pixel $(x, y)$ , the relationship between input and output channels is encoded as an $N \times N$ density matrix $\hat{\rho}_{x,y}$ in the fibre's $N$ -mode Hilbert space.
Analyser States: A complete set of $2N^2 - N$ analyser states $|\Omega_k\rangle$ are sequentially excited at the proximal end of the fibre. These states consist of pairwise interference patterns between mode combinations at four phase offsets, plus individual modes.
Density Matrix Extraction: The intensity projections $I_{|\Omega_k\rangle}$ are processed to reconstruct the Stokes vector $S_{x,y}$ and the corresponding density matrix $\hat{\rho}_{x,y}$ for each pixel.
Eigenstate Extraction: The first eigenstate $|\phi_1\rangle$ of $\hat{\rho}_{x,y}$ identifies the input mode superposition that maximizes power delivery to that specific pixel.
Phase Locking: While the density matrix recovers the complex field up to an unknown phase factor, the relative phases between different camera pixels are recovered using an iterative algorithm. This algorithm leverages the spatial redundancy provided by oversampling modal speckle grains (requiring $M^2 \ge 4N$ pixels). By overlapping phase-corrected neighboring pixels within the same speckle grain, the global phase relationship is established, transforming the pixel-based TM into a mode-based transmission matrix (MTM).

Key Contributions

Source-Matched Self-Referencing: The technique inherently matches the coherence properties of the employed light source. It systematically leverages all propagating modes as distributed phase references without preselection, ensuring a suitable reference is always found.
Broadband Capability: Unlike methods dependent on stable interference patterns, SST functions effectively with both long-coherence sources (LCS) and short-coherence/broadband sources (SCS).
Experimental Validation: The authors experimentally demonstrated the framework on a 2m graded-index MMF (supporting 420 spatial/polarization modes) across coherent, low-coherence, and broadband regimes.

Experimental Results

Long Coherence Source (LCS): The SST-measured MTM ( $420 \times 420$ ) was benchmarked against a standard off-axis DH measurement. The SST matrix showed more channels near 0 dB attenuation and higher spatial resolution in singular value analysis. While field overlap metrics were comparable, visual evaluation suggested SST produced sharper patterns. The authors attribute minor discrepancies in overlap to mechanical drift and reference wave aberrations in the DH setup, noting that SST's self-referencing nature (quasi-common path and flat average phase) makes it more robust against such artifacts.
Short Coherence Source (SCS): The method was tested with emulated broadband sources (bandwidths $\Delta\lambda_S$ $Δ λ_{S}$ up to 30 nm, exceeding the fibre's correlation bandwidth $\Lambda \approx 10$ $Λ \approx 10$ nm).
- Eigenstate Analysis: As source bandwidth increased, spatial decoherence occurred, resulting in multiple non-zero eigenstates. SST successfully identified the mutually coherent modes, assembling an MTM ( $\text{MTM}_{\Delta\lambda_S}$ ) that maximizes power delivery for the specific source bandwidth.
- Beam Shaping Performance:
  - For complex patterns (e.g., "smiley/sad" faces) utilizing most modes ( $\delta\lambda_T \approx \Lambda$ ), the optimal control was achieved using an MTM matched to the source bandwidth ( $\Delta\lambda_S \le 2\Lambda$ ). Crucially, the monochromatic MTM ( $\Delta\lambda_S = 0$ ) was consistently outperformed by the broadband-matched MTM for $\Delta\lambda_S > 0$ .
  - A "broadest-band" MTM concept was identified, showing a consistent $\sqrt{2}$ bandwidth enhancement ( $\delta\lambda'_T \approx \sqrt{2}\delta\lambda_T$ ) specifically for patterns where $\delta\lambda_T \approx \Lambda$ when using an MTM matched to $\Delta\lambda_S = 2\Lambda$ .
  - For simple spot focusing ( $\delta\lambda_T > \Lambda$ ), performance was substantially improved, with the $\text{MTM}_{3\Lambda}$ generating spots with 80–85% overlap across a 3 $\Lambda$ bandwidth.

Significance and Claims
The paper claims that this work extends light control through complex media into broadband illumination regimes previously inaccessible to accurate TM characterization. By enabling robust, source-matched self-referencing measurement, the method facilitates high-fidelity spatial and polarization beam shaping for:

Biomedical Imaging: Particularly relevant for hair-thin multimode fibre endoscopes where broadband sources are preferred.
Optical Communications: Enabling control in disordered media with low-coherence sources.
High-Power Laser Applications: Where external references are often impractical.

The authors emphasize that unlike optimization-based or machine-learning approaches, SST provides a deterministic, non-iterative solution that does not rely on the stability of interference patterns washed out by broadband sources. The method reduces complex-field characterization to a sequence of independent intensity measurements while retaining the phase information necessary for deterministic beam shaping.

Technical Summary: Broadband Control of Light through Complex Media via Automatically Self-Referencing Transmission Matrix Characterisation

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