A Non-Perturbative Mode-Locking Theory of the Nyquist Laser with a Dirichlet Kernel Solution

A neatly repetitive train of sinc function pulses can be expressed as a Dirichlet kernel solution. By using a non-perturbative approach to derive the master equation of a Nyquist pulse laser, we succeeded in obtaining a repetitive sinc function solution with a Dirichlet kernel. A method employing non-perturbative expressions consisting of gain, loss, amplitude modulation, and a flat-top optical filter with edge enhancement was used to derive the master equation. The master equation consists of a set of integrations. We derived a new differential equation that satisfies a Dirichlet kernel function. We introduced the differential equation into the master equation as a new operator, and directly derived a Dirichlet function solution. We developed a new series method to describe the non-perturbative master equation, in which we derived the same constraints for successful mode locking as those for the integral master equation.