# Deep Dive into the OPU¶

## The optical operation in detail¶

Independently of the details of the optical system, x and y should be considered as 1-D vectors.

The matrix-vector multiplication outputs a $$(1 \times m)$$ vector, complex-valued. This is followed by the element-wise non-linearity $$\lvert . \rvert^2$$ and the quantization due to analog to digital conversion. Finally, the output of the OPU is $$\mathbf{y}$$ a column vector of size $$(1 \times m)$$ of type uint8. The independence of the entries of the output vector means that the rows of the matrix R are independent.

## The distribution of the random matrix¶

It is possible to write $$r=\mathbf{R}_{ij}$$ in the following way:

$r = u + iv, u \sim \mathcal{N}(0, \sigma^2), v \sim \mathcal{N}(0, \sigma^2)$

The only parameter is a global gain: a multiplicative constant, that you may see as the variance of these distributions. It depends on many physical parameters.

## The binary input¶

The input device can only handle binary data.

For input data that is not binary to start with, you can use encoding schemes to transform any type of data into a binary array. Some of these encoding schemes are fixed, but we also have a data-adaptive scheme based on an autoencoder. Finally, there is also the possibility that users provide their own encoding function.

## Understanding the output¶

The output recorded by the output device is an interference pattern, called a speckle.

The histogram of a theoretical speckle is a decreasing exponential.