🔬 optics

Diagonalizing an optical coherence matrix via on-chip Stokes tomography

This paper demonstrates that Stokes tomography can be utilized on an integrated photonic circuit to efficiently diagonalize unknown optical coherence matrices in O(N) steps, transforming partially coherent light into uncorrelated orthogonal modes while simultaneously providing the original field.

Original authors: Amin Hashemi, Abbas Shiri, Bahaa E. A. Saleh, Andrea Blanco-Redondo, Ayman F. Abouraddy

Published 2026-08-25

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CC BY 4.0

Original authors: Amin Hashemi, Abbas Shiri, Bahaa E. A. Saleh, Andrea Blanco-Redondo, Ayman F. Abouraddy

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

Light is usually thought of as a stream of particles or waves traveling through space, but in the realm of modern optics, scientists are increasingly interested in how different "modes" of light—distinct patterns of energy—relate to one another. When these modes are perfectly synchronized, the light is coherent, like a laser beam. When they are completely random, the light is incoherent, like the glow from a light bulb. However, most real-world light exists in a middle ground known as partial coherence, where some modes are linked while others are not. This state, called structured coherence, is becoming a vital tool for new technologies in communication, computing, and security. The challenge lies in understanding and manipulating this complex light. To do so, researchers must first map out the relationships between the modes, a task akin to creating a detailed map of a hidden landscape, and then rearrange the light so that these relationships become simple and clear.

A team of researchers at the University of Central Florida has developed a new method to perform this mapping and rearrangement directly on a microscopic computer chip. They focused on a specific mathematical object called a coherence matrix, which acts as a blueprint describing how the different modes of light are connected. While scientists had previously figured out how to read this blueprint using a technique called Stokes tomography, they had not been able to use that same technique to physically reorganize the light on the chip. The researchers demonstrated that they could use this measurement process not just to understand the light, but to actively transform it. By guiding the light through a grid of tiny optical switches, they could first measure the complex connections between the modes and then immediately reconfigure the chip to untangle those connections. The result is a beam of light where the modes are completely independent of one another, a state that is much easier to work with for advanced applications.

The experiment took place on a photonic integrated circuit, a silicon chip containing a network of 72 tiny interferometers. These are devices that split and recombine light waves to control their behavior. The researchers began by generating a beam of light that was partially coherent, meaning its different parts were somewhat linked but not perfectly. They fed this light into the chip, which acted as a programmable processor. The first phase of their work involved a series of measurements. The chip was set to a specific configuration, and the light passed through it. The researchers measured the intensity of the light at the output, then changed the chip's settings and measured again. They repeated this process several times, each time with a slightly different arrangement of the internal switches. By collecting these intensity readings, they could mathematically reconstruct the full blueprint of the light's coherence, revealing exactly how the modes were interacting.

Once the blueprint was reconstructed, the researchers used that information to calculate a new setting for the chip. This final setting was designed to perform a specific transformation: it would take the complex, tangled light and convert it into a "coherent-mode representation." In this new state, the light is still made of the same modes, but they are now completely uncorrelated. Imagine a group of people who are all whispering different things to each other in a crowded room; the researchers' method effectively silences the cross-talk, leaving each person speaking only to themselves, with no one else listening. The chip then applied this transformation, and the output was a beam where the modes were perfectly orthogonal, meaning they were independent and carried no information about each other.

To prove that this process worked, the team tested the light before and after the transformation. Before the change, when they combined the modes, they saw clear interference patterns, a sign that the modes were linked. After the transformation, those patterns vanished, confirming that the modes were now truly independent. They also measured the strength of each mode in the final beam and found that these strengths matched the theoretical predictions for the original light's blueprint. The researchers tested this method on light with two modes and light with four modes, covering different levels of complexity and randomness. In every case, the chip successfully reconstructed the light's properties and then diagonalized it, leaving the modes uncorrelated.

A significant part of this discovery challenges a long-held assumption in the field. It was commonly believed that to fully map out the coherence of light with many modes, the number of measurement steps needed would grow very quickly, making the process slow and inefficient for complex systems. The researchers showed that this is not the case. By carefully recording all the necessary information at each step, they demonstrated that the number of steps required grows much more slowly, making the process far more practical for real-world use. This efficiency, combined with the ability to perform the entire process on a single chip, suggests a powerful new way to handle structured light. The work confirms that Stokes tomography is not just a tool for observation but a versatile instrument for active control, opening the door to more sophisticated optical systems that can process information using the unique properties of partially coherent light.

Technical Summary: Diagonalizing an Optical Coherence Matrix via On-Chip Stokes Tomography

Problem Statement
The manipulation of structured coherence—partially coherent light spanned by a finite number of modes—is becoming increasingly vital for optical communications, computation, cryptography, and spectroscopy. A critical challenge in this domain is the on-chip processing of the $N \times N$ Hermitian coherence matrix ( $G$ ) representing such light. Two fundamental tasks are required: reconstructing an unknown coherence matrix and diagonalizing it to obtain a "coherent-mode" representation ( $G_D$ ). In this representation, the field is supported by orthogonal, mutually uncorrelated modes, with modal weights corresponding to the eigenvalues of the original matrix.

While Stokes tomography has been established as a method for reconstructing $G$ , and variational processing has been used for diagonalization, previous approaches lacked a unified on-chip strategy that could perform both tasks efficiently. Furthermore, a prevailing assumption in the field was that reconstructing an $N \times N$ coherence matrix via Stokes tomography requires $O(N^2)$ measurement steps. This paper addresses the need for a more efficient, integrated approach to both reconstruct and physically diagonalize unknown coherence matrices on a single photonic platform.

Methodology
The authors demonstrate a unified procedure using a photonic integrated circuit (the iPronics Smartlight Processor) comprising a hexagonal mesh of 72 tunable Mach-Zehnder interferometers (MZIs). The methodology proceeds in two phases:

Stokes Tomography for Reconstruction:
Instead of the assumed $O(N^2)$ steps, the authors implement a sequence of $N_m$ predetermined unitary transformations ( $\hat{V}_j$ ) on the input field. In each step, the modal weights (diagonal elements of the transformed coherence matrix) are recorded by detectors and stored in an electronic processing unit (EPU).
- For $N=2$ modes, the procedure requires $N_m = 3$ steps to recover the generalized Stokes parameters (SPs) and reconstruct $G$ .
- For $N=4$ modes, the authors introduce a "modal switch" strategy. By reordering modes and applying pairs of $2 \times 2$ unitaries simultaneously, they acquire the real and imaginary parts of off-diagonal elements in parallel. This reduces the reconstruction of a $4 \times 4$ matrix to $N_m = 7$ steps (1 for diagonal elements, 6 for off-diagonal), rather than the expected higher count.
- The authors assert that generally, only $O(N)$ steps are required for reconstruction (specifically $2N-1$ for even $N$ and $2N+1$ for odd $N$ ) provided all output modal weights are recorded in each step.
Physical Diagonalization:
Once the coherence matrix $G$ is computationally reconstructed from the measured SPs, the diagonalizing unitary $\hat{V}$ is calculated. In a final step ( $N_m + 1$ ), the photonic circuit is reconfigured to implement $\hat{V}$ . This physically transforms the input field into the diagonalized state $G_D = \hat{V} G \hat{V}^\dagger$ . The circuit is designed to allow the original field $G$ and the diagonalized field $G_D$ to be available simultaneously at the output for further processing.
- For $N=4$ , the full procedure (reconstruction plus diagonalization) is completed in 8 steps ( $t_1$ through $t_8$ ), where steps $t_1$ to $t_7$ correspond to the Stokes tomography measurements and step $t_8$ implements the final diagonalizing unitary.

Key Contributions and Results
The paper validates this approach experimentally for both two-mode ( $N=2$ ) and four-mode ( $N=4$ ) structured coherence:

Efficiency Demonstration: The study explicitly dispels the notion that $O(N^2)$ steps are necessary for Stokes tomography, demonstrating that $O(N)$ steps suffice when utilizing parallel detection of modal weights.
Experimental Verification ( $N=2$ ): The authors synthesized 12 different $2 \times 2$ coherence matrices with varying degrees of coherence ( $D$ ) and entropy ( $S$ ). The reconstructed matrices showed fidelities ( $F$ ) $\geq 95\%$ against theoretical predictions.
Experimental Verification ( $N=4$ ): The procedure was applied to 8 different $4 \times 4$ coherence matrices with ranks ranging from 1 to 4. Reconstruction fidelities exceeded 94% in all cases.
Validation of Diagonalization: The diagonalization was verified through two independent methods:
1. Modal Weight Comparison: The measured weights of the diagonalized field matched the theoretical eigenvalues of the original reconstructed matrices.
2. Interferometric Correlation Check: The authors superposed the modes of the diagonalized field in an on-chip interferometer. Unlike the input fields, which displayed interference fringes (indicating correlation), the diagonalized fields showed no interference fringes, confirming the modes were mutually uncorrelated.

Significance
The paper claims that Stokes tomography is a versatile tool for the on-chip characterization of structured coherence. By enabling both the reconstruction and the physical diagonalization of unknown coherence matrices within the same integrated platform, the work provides a general strategy for manipulating structured light. This capability is significant for applications requiring the exploitation of the "coherence advantage," where partially coherent light outperforms coherent light in scenarios such as scattering-free communications, optical computing, and cryptography. The ability to perform these tasks in $O(N)$ steps rather than $O(N^2)$ represents a substantial reduction in experimental complexity and time, facilitating more practical implementations of structured coherence processing.

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