Separating partially coherent light

Abstract

Recent advances in optical imaging and communication increasingly involve high-dimensional, partially coherent light, creating a growing need for scalable tools to measure and manipulate coherence. Here, we demonstrate the automatic separation of spatially partially coherent light into "coherence modes" -- its orthogonal and mutually incoherent components. To make this separation possible, we exploit variational processing in layered self-configuring interferometer architectures in a silicon photonic circuit. This process formally finds and measures the eigenvectors and eigenvalues of the coherency matrix, hence measuring the partially coherent state, while leaving it intact and separated after optimization. Furthermore, we show that mutually incoherent beams, if spatially orthogonal, can be automatically separated even if they are completely overlapped, hence separating unknown laser beams based only on their mutual incoherence. Our experiment finds and separates the two strongest coherence modes starting from a nine-mode sampling of the partially or fully overlapping fields from two independent lasers. The method requires a number of physical components that scales linearly with the rank $r$ of the coherency matrix and operates through a sequence of $r$ in situ gradient-based optimizations enabled by electronic drive frequency multiplexing of interferometer phase shifters. We benchmark its performance against a mixture-based tomographic method, also implemented on chip. These results establish a scalable framework for programmable coherence analysis and control in imaging, communication, and photonic information processing.

Figures (4)

• Figure 1: An integrated photonic circuit to process spatial partial coherence. (a) Schematic of the chip and incident beam layout. The chip consists of a 2-layer self-configuring network. (b) Optical micrograph of the whole chip. (c) Zoom-in on the two self-configuring layers. (d) Zoom-in on the grating coupler array, which acts as a free-space interface. (e) Schematic of a $2\times2$ Mach-Zehnder interferometer (MZI) circuit. (f) Schematic of the method for parallel gradient measurement via voltage dithering. (g) Evolution over an optimization run of the network parameters ($\theta$ and $\varphi$, bottom), and the sensitivity of the power output (first layer) to these parameters ($\partial P/ \partial \theta$, $\partial P/ \partial \varphi$). (h) Two-mode interferogram measured between the outputs of layers 1 and 2 (measured with MZI highlighted in green in (a,b). Interferograms corresponding to different epoch numbers are shifted along the $y$-axis for readability.
• Figure 2: Variational coherence tomography on an integrated photonic circuit. (a) PCLA tomography: each coherence mode of the coherency matrix is learned by global optimization of the corresponding PCLA layer. (b) Mixture tomography: each mixture component (a spatially coherent laser beam) is learned separately, and the full coherency matrix is reconstructed numerically. (c-e) Benchmarking PCLA and mixture tomography by performing variational coherence tomography on layer 1 (left) and 2 (right). (c) Fidelity of PCLA unitary compared to ground truth unitary matrix estimated from mixture tomography. (d) Lock-in signal (corresponding to coherency matrix eigenvalue). (e) 2-mode contrast evolution over epoch number. Left inset shows a schematic of the interferometer to measure beating between outputs of layers 1 and 2. Right inset shows the interferogram for the maximum (beginning of layer 1 optimization) and minimum (layer 2 optimization) contrast positions.
• Figure 3: Measuring two-beam entropy via coherence tomography. (a) Two-beam configuration: beam 1 is fixed and overlaps with the bottom four grating couplers; beam 2 moves along the $x$ direction. (b) Intensity distribution over a set of four grating couplers for 4 specific positions of beam 2. (c) Measured eigenvalues, output of layers 1 and 2, respectively. (d) Measured entropy. The grey shaded area corresponds to a calculated entropy from the intensity entropy, where a (fitted) phase front tilt of $4.40\deg$ is added to beam 2.
• Figure 4: Discriminating overlapping beams and coherence-based self-configuring networks with PCLA. (a) Schematic of the beam configuration showing the connection between beam position in the source plane to angle of incidence in this experiment. (b) Beams 1 and 2 are overlapping in real space (on the grating coupler array). (c) Beam 1 is incident at a fixed angle, and beam 2 is swept across Fourier space (varying position in source plane, corresponding to varying the angle of incidence on the chip). (d) Dot product metric between beam 1 and 2, after learning beam 1 with layer 1. (e) PCLA operation at the position of maximum orthogonality (blue star in (d)). The two beams are mutually incoherent and orthogonal, and are separated into two output modes of the network (with cross-talk of -13.0 dB). (f) Fourier-domain sweep. At each point, a 2-layer, sequential optimization is performed. Top row: Measured eigenvalues, output of layers 1 and 2, respectively; middle row: Entropy of the coherency matrix; bottom row: cross-talk after optimization.