Mode locking theory of the Nyquist laser

We derive a master equation for a mode-locked Nyquist laser that can emit a sinc function pulse. To derive the master equation, we used a method involving exponential perturbative expressions for gain, loss, and amplitude modulation, and a flat-top optical filter with edge enhancement. The master equation is expressed as an integral equation, where a rectangular-like optical filter with edge enhancement plays an important role in generating a sinc function pulse with a flat-top spectral profile. It is important to note that the sinc function solution satisfies the spherical wave propagation of the Maxwell equation in the polar axis and is also the lowest order solution of the spherical Bessel equation. A differential equation was introduced as an operator into the master equation, which is different from the substitution of an assumed sinc function solution into the master equation, and we directly derived a sinc function solution. The time-independent Schrödinger equation approach in the spectral domain also proved that there is a sinc-like solution under a dual flat potential well.