# An Integrated Photorefractive Analog Matrix-Vector Multiplier for Machine Learning +
## 2 Materials and Methods
### Photorefractive Effect
#### 2.1.2 Crossbar Array Architecture  
In this subsection, it is discussed how an analog crossbar array can be realized by photorefractive index gratings. Optical beams will represent the synaptic signals. Refractive index gratings will constitute synaptic coupling elements. Inside a photorefractive crystal, a source and destination laser beam intersect with field amplitudes \(S_{1}^{\text{(w)}}\) and \(D_{1}^{\text{(w)}}\), respectively (Figure 4a). A refractive index grating is written, and reciprocal coupling is established between the source and destination channel. The coupling strength \(w\) of the grating is tunable and governed by its reflection coefficient \(r\) (Figure 3f). When the write beams are enabled, the electric space charge distribution \(\rho_{\text{sc}}\) (Figure 3c) builds up approximately linearly until saturation comes into effect. Within limits, the forming rate \(\frac{\partial\rho_{\text{sc}}}{\partial t}\) is proportional to the optical intensity variation \(\Delta I\) (Figure 3b) [10]. Thus, the coupling strength \(w\) is programmed by the electric field amplitudes \(S_{1}^{\text{(w)}}\) and \(D_{1}^{\text{(w)}}\) and the write time \(t_{\text{write}}\) (Figure 4b). Figure 3: Schematic representation of the photorefractive effect and refractive index grating formation [10]. (**a,b**) Two interfering monochromatic beams with amplitudes \(S\) & \(D\) generate a sinusoidal intensity variation \(\Delta I\). (**c**) The fringe visibility \(\Delta I/I_{0}\) relative to the background intensity \(I_{0}\) causes an electric space-charge redistribution \(\rho_{\text{sc}}\). (**d**) \(\rho_{\text{sc}}\) gives rise to a local electrical space charge field \(E_{\text{sc}}\) that is \(\pi/2\) phase shifted with respect to the optical fringe pattern. (**e**) \(E_{\text{sc}}\) yields an refractive index grating \(\Delta n\) via the linear electro-optic effect, (**f**) which acts as a Bragg reflector. After the writing phase, the diffracted source signal is given by the product of the source amplitude \(S\) and the grating coefficient \(w\) (Figure 4b). Thus, the analog multiplication of the input signal and grating amplitude is performed. Negative, or in general complex, grating and beam coefficients are possible (Figure 4b). The optical signals can simply be phase shifted to rotate the respective coefficient in the complex plane. In order to perform the MVM, an array of coupling elements needs to be realized and the multiplied signals need to be accumulated (Figure 1b). In the photorefractive crossbar array, refractive index gratings are superimposed to add matrix elements (Figure 4b) [11]. For instance, another input signal coming from a different direction can be coupled to the same destination channel by repeating the above grating writing routine. Since the grating profile is sinusoidal, only one diffraction order is possible and only along the direction of the write beams. Therefore, the gratings do not generate channel crosstalk as long as the angular channel separation is sufficiently large compared to the inverse grating size. Thus far, signal channels have been discussed as angularly separated collimated beams. However, data enters and leaves the crossbar array via spatially separated signal paths, each carrying a vector coefficient that represents a synaptic signal (Figure 1b). Collimating optics are deployed to transform between the spatial and angular domain (Figure 4c). The complete synaptic weight matrix \(W\) is programmed in the photorefractive crossbar array by applying write signals to the respective input and output vector channel pairs. Then, when an input vector is applied, the superimposed diffracted signal consists of the multiplied and accumulated source amplitudes. The output signal thus represents the matrix-vector product (Figure 4c) and can be retrieved by photodetection in the output channels. We envision a coherent detection scheme in order to retrieve the complex output vector coefficients linearly and to enhance sensitivity compared to direct photodetection.