Factorization of Numbers with the Temporal Talbot Effect: Optical Implementation by a Sequence of Shaped Ultrashort Pulses

We report on the successful operation of an analogue computer designed to factor numbers. Our device relies solely on the interference of classical light and brings together the field of ultrashort laser pulses with number theory. Indeed, the frequency component of the electric field corresponding to a sequence of appropriately shaped femtosecond pulses is determined by a Gauss sum which allows us to find the factors of a number.

Figure 1

Spectra of a $M+1=4$ pulse sequence ($N=105$). $l=3$ (a) and $l=9$ (b). The vertical line represents ${\omega }_L$.

Figure 2

Experimental realization of factoring using a sequence of shaped ultrashort pulses: (a) $N=105=3\times 5\times 7$ with 4 pulses, (b) $N=15251=101\times 151$ with $M+1=9$ pulses. Experiment (dots); theory (crosses).

Figure 3

Suppression of the ghost factor $l=138$ of $N=19043=137\times 139$ for increasing number of pulses. Inset : Experimental realization of factoring $N=19043$ with $M+1=9$ pulses. Experiment (dots); theory (crosses).