Aspects Of Number Theory Research Articles

Theory of Talbot lasers

We provide a theoretical study of frequency-shifted feedback (FSF) lasers, i.e., lasers with an internal frequency shifter, seeded with a monochromatic wave. The resulting spectrum consists in a set of equidistant modes, labeled by $n$, whose phases vary quadratically with $n$. We prove the emergence of a temporal fractional Talbot effect, leading to generation of Fourier-transform-limited pulses at a repetition rate tunable by the parameters of the FSF cavity (cavity length and frequency shift per round trip), and limited by the spectral bandwidth of the laser. We characterize in detail the output field of this so-called ``Talbot laser'' and emphasize its specific intensity fluctuations. We evidence connections with some aspects of number theory by the appearance of Gauss sums and theta series in the expression of the laser field. Our predictions are in full agreement with the experimental results published in Guillet de Chatellus et al. [Opt. Express 21, 15065 (2013)]. Practical applications and limitations are discussed.

Aspects of Number Theory in Terms of Potential Theory

We develop the Potential Theory on the set N* of all natural numbers ≥2 associated with the kernel V (resp. V*) given by

Distributions on Toroidal groups

In recent years, it has become clear that the notion of distribution is playing a central role in certain aspects of number theory.

Partitions of Positive Integers: An Elementary Topic of Number Theory

The author presents some interesting aspects of number theory and graphing that should be appealing to students.