Experimental set-up
The key components of our experimental set-up are shown in Fig. 1a and Extended Data Fig. 1.
Optical subsystem
The optical subsystem performs matrix–vector multiplication. The basic components are the optical sources (input vector), a system of fan-out optics to project the light onto the modulator matrix and a system of fan-in optics to project the light onto a photodetector array (output vector). The corresponding schematic is shown in Extended Data Fig. 2.
The incoherent light sources are an array of 16 independently addressable microLEDs. Each microLED is driven with a bias current and an offset voltage. The variable value is encoded by the light intensity, with a value of zero corresponding to the microLED bias point. Mathematical positive values are represented by microLED drive currents greater than the bias value. Negative values are represented by drive currents less than the bias value. The diameter of each emitter is 50 μm and the pitch is 75 μm. The sources are fabricated in gallium nitride wafers on a sapphire substrate and the die is wire-bonded onto a printed circuit board (Fig. 1c). The emission spectrum is centred at 520 nm with a full-width of half-maximum of 35 nm and the operational −3-dB bandwidth is 200 MHz at 20 mA, see Supplementary Fig. 1.
After the sources, there is a polarizing beamsplitter (PBS). From this point, there are two equivalent optical paths in this set-up. Each path performs two functions: first, they allow us to use both polarizations of the unpolarized light output; second, they allow us to perform non-negative and non-positive multiplications with only intensity modulation. Each path contains one amplitude modulator matrix and one photodetector array. The modulator matrix is a reflective parallel-aligned nematic liquid-crystal SLM. We refer to the first part of the optical system as the fan-out system. The task of this fan-out system is to image the microLEDs onto the SLM, where the weights are displayed, and to spread the light horizontally into lines. The microLEDs are arranged in a one-dimensional line (let this be the y axis) and are imaged onto the SLM using a 4F system composed of a high-numerical-aperture (Thorlabs TL10X-2P, numerical aperture 0.5, ×10 magnification, 22-mm field number) collection objective and a lower-numerical-aperture lens group composed of 2 achromatic doublets with combined focal length 77 mm. There is a cylindrical lens, Thorlabs LJ1558L1, in infinity space of this 4F system. This lens adds defocus to the image of the source array on the SLM but only in the x direction, so that the projected light pattern is a set of long horizontal lines, one per microLED. Each matrix element occupies a patch of 12 (height) × 10 (width) pixels of the modulator array. An 8-bit look-up table is used to linearize the SLM response as a function of grey level.
The SLM is imaged onto the photodetector array using a 4F system (the fan-in system). The first lens group of the fan-in is the same as the second lens group of the fan-out system as this is in double pass. From here, the light is directed towards the intended photodetector array through a second PBS. The light from each column of the SLM is collected by an array of 16 silicon photodetectors to perform the required summation operation. The active area of each element is 3.6 × 0.075 mm2. The photodetectors are on a pitch of 0.125 mm. The operation bandwidth is 490 MHz at −10 V measured at 600 nm.
Analog electronic subsystem
After the photodetector array, the signals are in the analog electronic domain. The photocurrents from each photodetector element are amplified by a linear trans-impedance amplifier (Analog Devices MAX4066). Each trans-impedance amplifier provides 25-kΩ gain and is characterized by an input referred noise of \(3\,{\rm{pA}}\,\sqrt{\text{Hz}}\) and has differential outputs. The corresponding 2 sets (1 per photodetector board) of 16 differential pairs of signals are fed to the main boards where the per-channel nonlinear operation and other analog electronic processing is carried out. Each of the 16 signals sees the following circuitry: (1) a variable gain amplifier (VGA; Texas Instruments VCA824) to allow the input signal range to be set and equalized across channels; (2) a difference amplifier to perform the operation of subtracting the negative input signal from the positive one and achieve signed voltages (signed multiplications); (3) a VGA that adds and subtracts signals from the described path, referred to as gradient term, to the annealing and momentum terms, as per equation (1), while providing a common gain control to all these paths; (4) an electronic switch (ADG659) to open and close the loop to set and reset the solving state; (5) a buffer amplifier to distribute the signal to the gradient, annealing and momentum paths; (6) a bipolar differential pair to implement the tanh nonlinearity; (7) a VGA to adjust the signal level between the nonlinearity and the required voltage and current onto the microLED alternating-current input circuit. Both the annealing and momentum paths have VGAs with a common external control so that we can implement time-varying annealing and momentum schedules.
Each channel also has an offset to the common control signal added to allow minor adjustment or correction of channel-to-channel variations. The other VGAs are set with digital-to-analog converters controlled over an inter-integrated circuit (I2C) bus. This allows slower control at per-experiment timescales.
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