Abstract
We report the first experimental demonstration of a prime number sieve via linear optics. The prime numbers distribution is encoded in the intensity zeros of the far field produced by a spatial light modulator hologram, which comprises a set of diffraction gratings whose periods correspond to all prime numbers below 149. To overcome the limited far field illumination window and the discretization error introduced by the spatial light modulator finite spatial resolution, we rely on additional diffraction gratings and sequential recordings of the far field. This strategy allows us to optically sieve all prime numbers below 1492 = 22201.
Highlights
The study of prime numbers, millenia old, is still a trending topic in mathematics
The only remaining limitation associated with the spatial light modulator (SLM) pixel resolution is the number of gratings that can be fitted in the phase mask
By generating high resolution and on-demand phase masks via an SLM, we prove the effectiveness of the sieve by correctly classifying integers up to 22201 as either primes or composites
Summary
The study of prime numbers, millenia old, is still a trending topic in mathematics. Interferometric [10,11] and optoelectronically assisted algorithms [12] schemes have been proposed, and experimental demonstrations were carried out based on the Talbot effect [13,14] and polychromatic interference in a multipath interferometer [15]. The only remaining limitation associated with the SLM pixel resolution is the number of gratings that can be fitted in the phase mask. Capped by this limitation, we demonstrate an optical sieve capable of identifying all prime numbers below 1492 = 22201
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