Light with tunable non-Markovian phase imprint

Designing the spatial coherence properties of light is fundamentally important to most applications in optics
and photonics. Usually, efforts are made to increase the light's spatial coherence, through spatial filtering, for
instance. However, for some applications, like the generation of random numbers or patterns, the opposite
is desired. When the light spatial coherence length is smaller than the detection unit (e. g. the pixel size
of a CCD camera), then the intensity measured at each pixel is uncorrelated to any other pixels. This is the
case of thermal light, usually modeled as delta-correlated. Moreover this is the spatial analogue of a temporal
Markovian process. In this work we explore for the first time a spatial phase imprint analogue of a non-
Markovian process inspired in the Sudoku puzzle.

Sudoku puzzles are a suitable basis for such conditional randomness in 2D: the fact that only $6.67\times 10^{21}$ out
of the total $9^{9\times 9}\simeq 1.96\times 10^{77}$ configurations are possible solutions provides a measure of randomness, while
the Sudoku rules (in each column, row and $9\times 9$ block the numbers $1$ to $9$ can only be used once) provide strict
constraints to this randomness.

Sudoku light can be experimentally generated by phase imprinting a conditional random pattern on a planar
wavefront via a spatial light modulator, for which the phases of different points in the beam profile are generated
using a Sudoku solver.

We solve overlapping Sudoku puzzles for a large area (e.g. $1024\times 1024$) and apply the resulting pattern as
phase imprint on a Gaussian light beam. Simulations show that such light, when focused, displays a symmetric
pattern with a centric cross-shaped minima. Furthermore, we note that the diffraction pattern of the light
after a double slit does neither correspond to coherent nor to incoherent thermal light, but rather shows fringes
in the outer lobes while having a minimum at the place of the zero's order. These counterintuitive properties
of Sudoku light, while violating the spatial equivalent of the Markovian condition, may have some applications
in the context of decay and initialization of quantum states.
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References:

Robert Fischer, Itamar Vidal, Doron Gilboa, Ricardo R. B. Correia, Ana Carolina Ribeiro-Teixeira, Sandra D. Prado, Jandir M. Hickman, Yaron Silberberg, accepted for publication in Physical Review Letters.