y = A x with A a fixed matrix and x a vector of N complex numbers written on N pixels (amplitude and phase). Two matrices are
built from existing elements: A = the DFT matrix, done by one lens and a gap of its focal length, and A = a circulant matrix
(a convolution), done by a 4f system whose filter is the FFT of the matrix's first column. The camera sees |y|², compared with
|A @ x|².
The animation feeds x = cos(2π m j / N) for m = 0 … 20: the output of the lens is two lines that move out from the centre, the rows of the DFT matrix at work.
The field is constant along y on a grid of 4 rows, so the system is one-dimensional.
python matrix_vector.pyvector of N = 128 complex numbers on N pixels of 20 um; wavelength 800 nm; lens f = N dx^2 / lambda = 64.0 mm
y = A x, A = DFT matrix (one lens): relative l2 error of |y|^2 below 0.02, correlation above 0.999
y = C x, C = circulant matrix (4f system): relative l2 error of |y|^2 below 0.005, correlation above 0.999
multiply-accumulates done by the optics in one pass: N^2 = 16384 for each product
checks passed
What it shows. The lens is the DFT matrix to about 0.9 % in |y|² (a random vector with sharp pixel edges); the 4f circulant matrix to about 0.3 % (the printed values are bounds, 2 % and 0.5 %). Each product is N² = 16 384 multiply-accumulates done in one pass of light.
What it does not show. A general matrix needs the anamorphic layout (cylindrical lenses and a mask, Goodman's vector-matrix
multiplier). The engine's lens has one focal length for both axes, so a general A is not buildable with today's elements; the two
matrices above are the ones a round-lens system gives. Phase is not read: a camera records intensity only.