FOCUS
WAVEFRONT control
incident field E in, the resulting output field E out can be expressed as a Born series: E out = [ V + VG 0 V + V( G 0 V) 2 + …] E in, where G 0 is the free-space Green function. The relationship between E out and E in is linear, but the relationship between V( and hence ε) and E out is not linear.
This distinction is important. In conventional wavefront shaping, information is encoded onto the incident field E in while the scattering potential is fixed. The encoded information therefore undergoes a linear optical transformation. However, information can be encoded into V directly. A single interaction then produces a term linear in V, whereas repeated interactions with the same encoded potential generate terms containing higher orders of V. The resulting output therefore contains nonlinear combinations of the encoded information, despite light propagation and all light-matter interactions remaining linear. This approach is referred to as structural nonlinearity, which can be implemented using an optical cavity [ 4 ] or multiple passes on an SLM [ 5 ], and conveniently allows to provide a nonlinearity the user can exploit without the need of a high power source or a specific material with a nonlinear response.
We implement this principle experimentally using the apparatus shown in Fig. 2 [ 6 ]. The input information is
Figure 2. Structural nonlinearity for forecasting a chaotic time series.( a) Experimental apparatus( b) Mackey-Glass time series, and experimental forecast using either a conventional reservoir or the structurally nonlinear reservoir( c) Normalised mean square error of both forecasts to assess short-term forecast pointwise accuracy( Adapted from [ 6 ]).
encoded as a binary spatial pattern on a digital micromirror device( DMD). Each micromirror can be set to one of two tilt angles, directing the reflected light along two distinct optical paths. The two paths thus carry complementary versions of the same input pattern, a property that we exploit to implement the conventional linear and structurally nonlinear configurations simultaneously.
In the conventional configuration, light interacts with the encoded DMD pattern only once before propagating through a scattering medium. The field incident on the scattering medium is therefore linearly related to the pattern displayed on the DMD. The resulting speckle pattern provided the conventional linear optical features, and will be used to benchmark results. In the second path, the optical geometry redirects the beam onto the DMD, such that it interacts a second time with the same displayed image. The input information consequently acts twice on the input field. This repeated interaction generates a structurally nonlinear mapping of the input.
Both beams propagate through scattering materials to provide additional spatial mode mixing, and the resulting speckle patterns are measured on separate cameras. The two complementary outputs of the DMD thus allow the linear and structurally nonlinear mappings to be generated independently and simultaneously within the same experimental apparatus. This provides a direct comparison between the two approaches while keeping similar experimental conditions.
RESULTS We evaluate the computational benefit of structural nonlinearity on the task of forecasting a chaotic time series [ 6 ]. Chaotic systems provide a particularly demanding prediction task, as small errors rapidly accumulate and lead to diverging trajectories. Here, we consider the Mackey-Glass time series [ 3 ]. At each time step, the current state of the time series is encoded on the DMD and processed simultaneously by the linear and the structurally nonlinear reservoir architectures described in the previous section. The resulting speckle intensity patterns are measured by the cameras, and used as the reservoir states for the prediction task.
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