Scalar computational primitives with perturbative phase interferometry

TL;DR

This work introduces a perturbative, phase-encoded optical computing approach that uses phase-parametrized interferometers to enact primitive scalar operations. By reading out changes in optical power from small phase perturbations around carefully chosen bias points, the authors realize addition, subtraction, multiplication via finite differences, and nonlinear operations such as scalar inversion, all within linear optics. Cascaded (nested) interferometer schemes extend these primitives to composite functions, with attention to calibration, readout linearity, and dynamic range. The method leverages the nonlinear geometry of phase parametrizations to achieve nonlinear-like computations while remaining in the linear-optics regime, offering a path toward more flexible analog photonic computation and potential extensions to matrix/vector operations and optical neural networks.

Abstract

We describe how weak phase modulations applied to classical coherent light in specially modified linear interferometers can be used to perform primitive computational tasks. Instead of encoding operations within a fixed unitary state, the operations are enacted by moving from one state to another. This harnesses the particular phase parametrization of an interferometer, allowing entirely linear optics to produce nonlinear operations such as division and powers. This is due to the nonlinear structure of the underlying phase parametrizations. The realized operations are approximate but can be made more accurate by decreasing the size of the input perturbations. For each operation, the inputs and outputs are changes in phase relative to a fixed bias point. The output phase is ultimately read out as a change in optical power.

Figures (13)

• Figure 1: A comparison of interferometric encoding schemes. The traditional approach (top) assigns information values to individual points in the parameter space. This ties information to the optical state (i.e. photon number distribution among various spatial modes) but discards the geometric information in the parameter space. The proposed approach (bottom) ties the information to a change in the optical state induced by a change in one or more phases. In coupled-phase interferometers, changing one phase can deform the curve which is defined by spanning another phase. In this encoding approach, the nature of the deformation can now be used to enact certain information processing operations. That is, $\Delta P$ represents a computed output for a given set of inputs $(\Delta \phi_1, \Delta \phi_2)$. In particular, $\Delta P$ is intended to directly read out the change in phase $\Delta \gamma(\Delta \phi_1, \Delta \phi_2)$, so all computations are entirely encoded in the optical phase. Examples derived in the text include addition and subtraction (Sec. 3.1), multiplication (Sec. 3.2), nonlinear operations such as squaring or scalar inversion (Sec. 3.3), and cascaded application of these (Sec. 3.4).
• Figure 2: Geometric objects such as the circle plotted on the top may be parametrized in different ways. The horizontal component of two example parametrizations of this circle, $\mathbf{f}_1(t) = (\cos(2\pi t), \sin(2\pi t))$ and $\mathbf{f}_2(t) = (\cos(2\pi t^2), \sin(2\pi t^2))$ for $0\leq t \leq 1$ are shown on the right guggenheimer. In the same sense, optical interferometers can parametrize a given scattering state space in different ways. This is exemplified in the Michelson and Grover-Michelson interferometers compared in Fig. \ref{['fig:migmi']}.
• Figure 3: The traditional Michelson interferometer (left) is formed with a beam splitter $B$ and two end-mirrors. When light scatters into $B$, all of the light passes from the input ports 1 and 2 to the mirror-arms; there is no back-scatter or light coupled between ports 1 and 2. Thus the light enters through the mirror-arms, acquires an optical phase of $\phi_j$ in arm $j$, and exits the device in one pass. However, replacing the beam-splitter with a four-port Grover coin $G$ now allows light to scatter from one port to another with an equal probability of $25\%$. The resulting device, the Grover-Michelson interferometer (right), is now a multi-pass system of coupled resonator mirror-arms, and is governed by an effective phase $\gamma(\phi_1, \phi_2)$PhysRevA.107.052615. This mapping $\gamma$ is a nonlinear reparametrization of the physical phases $(\phi_1, \phi_2)$. It is explicitly given by $\gamma(\phi_1, \phi_2) = \arctan((\sin(\phi_1 + \phi_2) - \sin\phi_1 - \sin\phi_2 )/(\cos(\phi_1 + \phi_2) - \cos\phi_1 - \cos\phi_2 + (1 + \cos^2(\phi_1 - \phi_2))/2))$. Additional plots of these different parametrizations can be found in Fig. 5 of Ref. PhysRevA.107.052615 and experimentally in Fig. 6 of Ref. Schwarze:24.
• Figure 4: Different forms of a looped beam-divider, including the three configurations on the left, produce an output equivalent to the same, common reparametrization of the loop phase $\phi \rightarrow \gamma(\phi)$, with $\gamma$ given by Eq. (\ref{['eq:gamma1']}). With lossless components, all energy entering the device must exit from the same port, so the net scattering action is the acquisition of a phase factor $e^{i\gamma(\phi)}$. These elementary "phase amplifiers" can thus be viewed as a phase-controlled source of optical phase. The configuration on the right is a traditional Sagnac interferometer, and only produces the phase factor of $e^{i\phi}$, unless a non-reciprocal phase is inserted in the loop. In that case, energy is split between the two output ports depending on the strength of the non-reciprocal phase.
• Figure 5: (top) Output probability $P(\gamma(\phi, \delta))$ vs. phase $\phi$ of a lossless, phase-amplified Mach-Zehnder interferometer for various phases $\delta$. $P = \sin^2((\gamma(\phi, \delta) - 3\pi/2)$ is the probability of a single photon entering a given input port emerging at one of the two output ports, with that of the other output port being $1 - P$. The particular amplification function $\gamma$ used here is defined in Eq. (\ref{['eq:gamma2']}). (bottom) Percent error in treating $P$ as a linear function about the origin for the same $\delta$ in the above case. If shifted by the value $-1/2$, the target function and approximant would be odd functions about $\phi = 0$. This implies that the absolute error is an even function about $\phi = 0$, since the mutual offset cancels there. However, the offset in the target function remains in the calculation of relative (percent) error, breaking that symmetry, as evident in the plot.