Optimal Control of Coherent Light Scattering for Binary Decision Problems

Because of quantum noise fluctuations, the rate of error achievable in decision problems involving several possible configurations of a scattering system is subject to a fundamental limit known as the Helstrom bound. Here, we present a general framework to calculate and minimize this bound using coherent probe fields with tailored spatial distributions. As an example, we experimentally study a target located in between two disordered scattering media. We first show that the optimal field distribution can be directly identified using a general approach based on scattering matrix measurements. We then demonstrate that this optimal light field successfully probes the presence of the target with a number of photons that is reduced by more than 2 orders of magnitude as compared to unoptimized fields.

Figure 1

(a) Representation of the experiment, which consists in optically probing the presence of a target ($3\mu \mathrm{m}$ in diameter) located between two scattering media. The system is illuminated with an incident field that is spatially modulated using a digital micromirror device (DMD). The field that comes out of the system is measured by a camera in $N$ spatial modes using a homodyne scheme. (b) The measured field spans a complex $N$-dimensional space. The expectation value of the field depends on the presence of the target, and its variance is ultimately limited by quantum noise fluctuations. The incident field optimally probes the presence of the target when the statistical distance $d_{12}$ is maximized. (c),(d) Intensity distributions experimentally measured with the optimal incident field when the hidden target is (c) absent and (d) present. Because of complex absorption and scattering processes involved within the system, these distributions appear as speckle patterns. Despite this complexity, the optimal light field is strongly affected by the presence of the target, which allows us to detect this target with a minimum rate of error.

Figure 2

(a) Eigenvalues ${\mathrm{\Lambda }}_j$ of the discrimination operator, normalized by the average value $\overline{\mathrm{\Lambda }}$ (the averaging is performed over all possible incident states). Small eigenvalues (associated with measurement noise) are closely spaced and appear as a continuum due to the finite thickness of the lines, while large eigenvalues (associated with eigenstates that significantly interact with the target) appear as discrete lines. (b)–(e) Intensity distribution in the target plane measured for a few representative eigenstates—including (b) the first eigenstate (i.e., the optimal state), (c) the second eigenstate, and (d) the eighth eigenstate—as well as for (e) the average state (defined as an equally weighted linear superposition of all eigenstates). These distributions were measured in the absence of the target, and without the scattering medium located between the target and the camera. (f) Image of the target (a single bead) measured under spatially incoherent illumination. (g)–(l) Analogous to (a)–(f) for a target composed of six beads.

Figure 3

Rate of error as a function of the number of incident photons for the average state (blue) and the optimal state (red). In both cases, the experimentally observed error rates (data points) are compared to the theoretical values associated with our homodyne setup (solid lines) and to the Helstrom bound (dashed lines). Shaded areas represent 95.4% confidence intervals, taking into account only the statistical error caused by the finite number of measurements ($N_{\mathrm{rep}}=4000$). Note that, while we illuminate the system with up to $7.4\times {10}^6$ photons, many photons are scattered out of the field of view by the diffusers. As a consequence, we only detect up to 140 photons over the area covered by the camera sensor.