On-chip control of the coherence matrix of four-mode partially coherent light: rank, entropy, and modal Stokes parameters

TL;DR

The work addresses on-chip control of the coherence matrix $\mathbf{G}$ for four-mode partially coherent light, introducing rank and entropy as central figures of merit. It deploys a hexagonal Mach-Zehnder interferometer mesh to realize arbitrary $4\times4$ unitaries from 2×2 building blocks, includes non-unitary operations to tune coherence rank and entropy, and uses modal Stokes parameters via Kronecker Pauli matrices for full tomography. The study demonstrates rank control across 1–4, on-chip entropy tuning with iso-entropy families, and unitary transformations that mold the coherence matrix while achieving high reconstruction fidelity ($F \approx 0.95$–$0.99$). This establishes scalable, on-chip manipulation of massively multimoded partially coherent light with implications for optical information processing, communications, sensing, cryptography, and computation, and outlines concrete avenues for extending to larger modal spaces and faster tomography.

Abstract

Partially coherent light offers salutary capabilities in optical information processing that cannot be matched by coherent light. To date, this `coherence advantage' has been confirmed in proof-of-principle optical communications protocols using bulk optics. Taking full advantage of such opportunities necessitates processing multimode partially coherent light in integrated photonics platforms that alone provide the requisite stability for cascaded operations on a large scale. Here we demonstrate on-chip manipulation of four-mode partially coherent light described by a $4\times4$ Hermitian coherence matrix. Starting with generic maximally incoherent light, we utilize an on-chip hexagonal mesh of Mach-Zehnder interferometers to perform all the unitary and non-unitary tasks that are critical for realizing structured coherence: controlling the coherence rank (the number of non-zero eigenvalues of the coherence matrix); tuning the field entropy; molding the structure of the coherence matrix via $4\times4$ unitary transformations constructed out of sequences of $2\times2$ unitaries acting on pairs of modes; and tomographic reconstruction of the coherence matrix by measuring the modal Stokes parameters associated with Kronecker Pauli matrices. These results confirm the scalability of utilizing $2\times2$ on-chip building blocks for the synthesis and reconstruction of high-dimensional coherence matrices, and provide a decisive step towards large-scale on-chip manipulation of massively moded partially coherent light for applications in optical information processing.

Figures (6)

• Figure 1: (a) Chip layout of the MZI mesh. Each colored rectangle is an MZI, and lines are single-mode waveguides. (b) Structure of a single MZI (Eq. \ref{['eq:MZI']}). PS: Phase shifter. (c) A general $2\times2$ unitary (Eq. \ref{['eq:GeneralUnitary']}) formed of the MZIs highlighted in (a). (d) Construction of the unitary $\hat{U}_{B}$ and (e) $\hat{U}_{C}$ out of sequences of $2\times2$ unitaries. (f) Chip layouts for $\hat{U}_{B}$ and (g) for $\hat{U}_{C}$.
• Figure 2: Reconstructing a $4\times4$ coherence matrix $\mathbf{G}$ from the modal Stokes parameters. The left column highlights the structure of the relevant $4\times4$ Kronecker Pauli matrix; the middle shows the conceptual measurement scheme; and the right depicts the corresponding chip layout, with the paths followed by the modes indicated with different colors. (a) Measuring $s_{00}$, $s_{01}$, $s_{10}$, and $s_{11}$. The Kronecker Pauli matrices $\hat{\sigma}_{00}$, $\hat{\sigma}_{01}$, $\hat{\sigma}_{10}$, and $\hat{\sigma}_{11}$ are diagonal ($\hat{\sigma}_{01}=\hat{\sigma}_{0}\otimes\hat{\sigma}_{1}$ shown). Detectors record the modal weights $I_{1}^{(1)}$, $I_{2}^{(1)}$, $I_{3}^{(1)}$, and $I_{4}^{(1)}$, and the modal Stokes parameters are obtained using Eq. \ref{['eq:S_00_01_10_11']}. (b) Measuring $s_{02}$, $s_{03}$, $s_{12}$, and $s_{13}$. The Kronecker Pauli matrices $\hat{\sigma}_{02}$, $\hat{\sigma}_{03}$, $\hat{\sigma}_{12}$, and $\hat{\sigma}_{13}$ are block diagonal ($\hat{\sigma}_{02}=\hat{\sigma}_{0}\otimes\hat{\sigma}_{2}$ shown). Modal Stokes parameters $s_{02}$ and $s_{12}$ are obtained using Eq. \ref{['eq:S02_12']} after setting $\hat{U}_{12}=\hat{U}_{34}=\hat{U}_{2}$, and $s_{03}$ and $s_{13}$ using Eq. \ref{['eq:S03_13']} after setting $\hat{U}_{12}=\hat{U}_{34}=\hat{U}_{3}$. (c) Measuring $s_{20}$, $s_{21}$, $s_{30}$, and $s_{31}$. Modal Stokes parameters $s_{20}$, $s_{21}$ are obtained using Eq. \ref{['eq:S20_21']} after setting $\hat{U}_{13}=\hat{U}_{24}=\hat{U}_{2}$, and $s_{30}$, $s_{31}$ using Eq. \ref{['eq:S30_31']} after setting $\hat{U}_{13}=\hat{U}_{24}=\hat{U}_{3}$. (d) Measuring $s_{22}$, $s_{33}$, $s_{23}$, and $s_{32}$. Modal Stokes parameters $s_{22}$ and $s_{33}$ are obtained using Eq. \ref{['eq:S22_33']} after setting $\hat{U}_{23}=\hat{U}_{14}=\hat{U}_{2}$, and $s_{23}$ and $s_{32}$ using Eq. \ref{['eq:S23_32']} after setting $\hat{U}_{23}=\hat{U}_{14}=\hat{U}_{3}$.
• Figure 3: (a) Source of four-mode incoherent light. LD: Laser diode; FL: fiber loop; A: variable attenuator; PC: polarization controller. (b) The four modes $|1\rangle$, $|2\rangle$, $|3\rangle$, and $|4\rangle$ traverse relative fiber lengths of 0, $L$, $2L$, and $3L$, respectively, to reach the chip. (c) Setup for interfering a pair of modes. BS: Balanced beam splitter; C: fiber collimator. (d) Measured intensity profiles after superposing pairs of modes in absence of the fiber loops, whereupon the fields are mutually coherent and interference fringes are observed; and (e) in presence of the fiber loops, whereby the fields are mutually incoherent and the interference fringes are eliminated.
• Figure 4: Measured coherence matrices for maximum-entropy fields of different rank. (a) Rank-1 with $\mathbf{G}_{1}=\mathrm{diag}\{1,0,0,0\}$; (b) rank-2 with $\mathbf{G}_{2}=\mathrm{diag}\{\tfrac{1}{2},\tfrac{1}{2},0,0\}$; (c) rank-3 with $\mathbf{G}_{3}=\mathrm{diag}\{\tfrac{1}{3},\tfrac{1}{3},\tfrac{1}{3},0\}$; and (d) rank-4 with $\mathbf{G}_{4}=\mathrm{diag}\{\tfrac{1}{4},\tfrac{1}{4},\tfrac{1}{4},\tfrac{1}{4}\}$. We plot separately the real and imaginary parts of the coherence matrices, $\mathrm{Re}\{\mathbf{G}\}$ and $\mathrm{Im}\{\mathbf{G}\}$, respectively.
• Figure 5: On-chip tuning of the entropy of a four-mode partially coherent field. (a) Measured entropy $S(\lambda_{1})$ for rank-2 fields as we vary the eigenvalue $\lambda_{1}$ from 0 to 1 ($\lambda_{2}=1-\lambda_{1}$, $\lambda_{3}=\lambda_{4}=0$). The curve is the theoretical expectation, the points are measurements. (b) Samples of the eigenvalues of the reconstructed coherence matrices as we vary $\lambda_{1}$. (c) The measured entropy $S(\lambda_{1},\lambda_{2})$ for rank-3 fields as points overlaid on the theoretical surface. We vary $\lambda_{1}$ and $\lambda_{2}$ from 0 to 1, $\lambda_{3}=1-\lambda_{1}-\lambda_{2}$ and $\lambda_{4}=0$. (d) Measured entropy for iso-entropy rank-3 fields plotted along with the theoretical curves for $S=1$, 1.25, and 1.5 bits. (e) Samples of the extracted eigenvalues from reconstructed iso-entropy coherence matrices corresponding to (d). (f) The measured entropy for iso-entropy rank-4 fields with $S=1.9$ bits (plotted as black points) overlaid on the theoretically expected iso-entropy surface. (g) Same as (f) for $S=1.5$ bits, and (h) for $S=1.25$ bits. In the latter, the measurements points are white for clarity. (i) The measured entropy for 10 reconstructed coherence matrices selected from (f-h) compared to the target entropy values.