  Berry and Klein (J. Mod. Opt. (1997) 43 2139-2164) showed that the Talbot
effects in classical optics are naturally explained by Gauss sums studied in
number theory. Their result was based on Helmholtz equation. In this article,
we explain the effect based on Fresnel integral also by Gauss sums. These two
explanations are shown to agree with by the reciprocity law of Gauss sums. The
relation between this agreement and the wave-particle complementarity is also
discussed.
