A Generalized Mode-Locking Theory for a Nyquist Laser With an Arbitrary Roll-off Factor PART I: Master Equations and Optical Filters in a Nyquist Laser

In this paper (PART I), we describe master equations and specific optical filters designed to generate a periodic Nyquist pulse train with an arbitrary roll-off factor $\alpha $ that can be emitted from a mode-locked Nyquist laser. In the first part, we derive a perturbative master equation for a Nyquist pulse laser with a “single” $\alpha \ne 0$ Nyquist pulse, where we obtain a new filter function $F(\omega )$ that determines filter shapes at both low and high frequency edges. A second-order differential equation that satisfies a periodic Nyquist pulse train with $\alpha \ne 0$ is derived and utilized for the direct derivation of a single Nyquist pulse solution in the time domain for a mode-locked Nyquist laser. Then, by employing the concept of Nyquist potential, we describe the differences between the spectral profiles and filter shapes of a Nyquist pulse when $\alpha = 0$ and $\alpha \ne 0$ . In the latter part, we describe a non-perturbative master equation that provides the solution to a “periodic” Nyquist pulse train with an arbitrary roll-off factor. We show first that the spectral profile of an $\alpha \ne 0$ periodic Nyquist pulse train, which consists of periodic $\delta $ functions, has a different envelope shape from that of a single $\alpha \ne 0$ Nyquist pulse. This is because a periodic $\alpha \ne 0$ pulse train consists of two independent periodic functions, which give rise to a different spectral envelope. Then, by Fourier transforming the master equation, we derive new filters consisting of $F_{1}(\omega )$ at a low frequency edge and $F_{2}(\omega )$ at a high frequency edge to allow us to generate arbitrary Nyquist pulses.