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 α 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” α ≠ 0 Nyquist pulse, where we obtain a new filter function F(ω) that determines filter shapes at both low and high frequency edges. A second-order differential equation that satisfies a periodic Nyquist pulse train with α ≠ 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 α ≠ 0 and α ≠ 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 α ≠ 0 periodic Nyquist pulse train, which consists of periodic δ functions, has a different envelope shape from that of a single α ≠ 0 Nyquist pulse. This is because a periodic α ≠ 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 F₁(ω) at a low frequency edge and F₂(ω) at a high frequency edge to allow us to generate arbitrary Nyquist pulses.