Programmable on-chip synthesis and reconstruction of partially coherent two-mode optical fields

TL;DR

This work demonstrates the first on-chip synthesis and characterization of two-mode partially coherent optical fields using a hexagonal Mach-Zehnder interferometer mesh. By applying a non-unitary pre-processing to tune the degree of spatial coherence $D_s$, followed by a general $2\times2$ unitary $\hat{U}$ and a Stokes-unitary $\hat{U}_{s}$, the authors realize prescribed coherence matrices $\textbf{G}$ and reconstruct them through spatial Stokes measurements. They validate both coherent and partially coherent regimes with high fidelity ($F>0.98$) and show tunable interference visibility $V=D_s|\sin\delta|$, confirming controllable coherence manipulation on-chip. The results lay the groundwork for scalable, programmable structured coherence in on-chip photonics, with potential impact on communications, cryptography, computation, and spectroscopy, and point toward future multi-mode extensions and two-chip coherence-transceiver systems.

Abstract

Partially coherent light is typically studied in the context of freely propagating continuous fields. Recent developments have indicated the existence of a `coherence advantage' in multimode optical communications, where partially coherent light outperforms coherent light. However, exploiting partial coherence in such applications requires manipulating multimode field coherence in programmable on-chip platforms. We present here the first example of on-chip synthesis and characterization of two-mode optical fields in an integrated on-chip hexagonal mesh of Mach-Zehnder interferometers. Starting with incoherent two-mode light, we adjust the degree of coherence on the chip with non-unitary transformations, construct $2\times2$ unitary transformations to synthesize prescribed coherence matrices, and reconstruct the coherence matrices via measurements of the spatial Stokes parameters. These results indicate the possibility of deploying programmable photonics for producing large-dimensional structured partially coherent light for applications in communications, cryptography, sensing, and spectroscopy.

Figures (8)

• Figure 1: (a) Two-mode optical field comprising the modes $|a\rangle$ and $|b\rangle$ in single-mode waveguides. Two detectors obtain the modal weights: $I_{a}=G_{aa}$ and $I_{b}=G_{bb}$. (b) An arbitrary unitary $\hat{U}$ (Eq. \ref{['eq:GeneralUnitary']}) is constructed from a phase operator $\hat{S}(\alpha)$, followed by a variable coupler $\hat{R}(\theta)$, and then a second phase operator $\hat{S}(\beta)$. (c-f) Configurations to measure the spatial Stokes parameters and thus reconstruct the coherence matrix: (c) $s_{0}=I_{a}+I_{b}$; (d) $s_{1}=I_{a}-I_{b}$; (e) $s_{2}=I_{a}'-I_{b}'$ after the unitary $\hat{U}_{s2}$; and (f) $s_{3}=I_{a}"-I_{b}"$ after the unitary $\hat{U}_{s3}$.
• Figure 2: (a) A variable coupler $\hat{U}_{\mathrm{MZI}}$ (Eq. \ref{['eq:MZI']}) is constructed by sandwiching a phase operator $\hat{S}(\varphi_{1},\varphi_{2})$ between two symmetric couplers represented by the operator $\hat{B}$. (b) A general on-chip unitary (Eq. \ref{['eq:GeneralOnChip']}) is constructed by sandwiching a variable coupler $\hat{U}_{\mathrm{MZI}}$ between two phase operators $\hat{S}(\alpha)$ and $\hat{S}(\beta)$. (c) Schematic of the on-chip hexagonal mesh utilized in our experiments. Each of the 72 colored rectangles in the mesh is an MZI, and the lines are single-mode waveguides. The inset highlights the structure of an MZI, which corresponds to the MZI in (a). (d) The layout of the MZIs used to construct a general unitary $\hat{U}$, followed by the Stokes unitary $\hat{U}_{\mathrm{s}}$. At the entrance to $\hat{U}$ is a non-unitary transformation that adjusts the degree of spatial coherence of the diagonal coherence matrix delivered to the unitary. The outputs are then routed to detectors at the edge of the chip to record the modal weights $I_{a}$ and $I_{b}$. (e) A key to the MZIs of different coupling parameter $\Gamma$ depicted in different colors in (d). From left to right: a relay with $\Gamma=0$ corresponds to an MZI that routes $|a\rangle\rightarrow|a\rangle$ and $|b\rangle\rightarrow|b\rangle$; a switch with $\Gamma=1$ that routes $|a\rangle\rightarrow|b\rangle$ and $|b\rangle\rightarrow|a\rangle$; an attenuating switch that reduces the amplitude of the field $|a\rangle\rightarrow\gamma|b\rangle$, with $\gamma<1$, and there is no input field at $|b\rangle$; and, finally, a variable coupler given by Eq. \ref{['eq:MZI']}.
• Figure 3: (a) The on-chip configuration to calibrate any relative on-chip phases between $|a\rangle$ and $|b\rangle$ en route to $\hat{U}$. Preceding $\hat{U}$ is a variable coupler that splits a coherent input to two paths with different relative amplitudes, so that $|E\rangle=E_{a}|a\rangle+E_{b}|b\rangle$. (b) Measurement results for $I_{a}$ and $I_{b}$ to calibrate the phase offset between $|a\rangle$ and $|b\rangle$ from the input ports to $\hat{U}$. The phase $\alpha$ in $\hat{U}$ is tuned for three different settings of the modal weights.
• Figure 4: (a) Schematic of the source of incoherent light used in our experiments. LD: Laser diode; PS: phase shifter; PC: polarization controller. Black lines are all single-mode fibers. (b) Measured temporal noise signal with amplitudes from 0 to 10 v. In the right panel we plot a histogram of the noise signal showing a root-mean-square width $\approx5.5$ v. (d) Spectrum of the noise signal, showing a cutoff at $\approx100$ Hz. (d) Setup for observing the interference fringes produced by overlapping the two fields in free space prior to coupling to the chip. BS: Symmetric beam splitter; C: fiber collimator. (e) Images acquired by the camera of the intensity resulting from superposing the two fields with a slight angular tilt. When the phase shifter is turned off, high-visibility fringes are observed ($V\approx1$; left panel). When the phase shifter is fed with the noise signal from (b), the fringes are eliminated, signifying loss of coherence ($V\approx0$, middle panel). In the right panel we overlap sections through the two intensity distributions.
• Figure 5: Reconstruction of the spatial coherence matrix $\mathbf{G}=\hat{U}\mathbf{G}^{\mathrm{D}}\hat{U}^{\dagger}$ for a coherent field, $\mathbf{G}^{\mathrm{D}}=\left(1000\right)=|E\rangle\langle E|$, with $|E\rangle=\left(10\right)$. The columns correspond to $\mathbf{G}$ after the unitaries provided at the top, which are those given in Eq. \ref{['eq:6Unitaries']} after dropping the overall phases, and the expected $\mathbf{G}$ is provided in each column. We plot the real and imaginary parts of the coherence matrices, $\mathrm{Re}\{\mathbf{G}\}$ and $\mathrm{Im}\{\mathbf{G}\}$, respectively.