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.

中文翻译：

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具有任意滚降因子的奈奎斯特激光器的广义锁模理论第一部分：奈奎斯特激光器中的主方程和光学滤波器

在本文（PART I）中，我们描述了主方程和特定的光学滤波器，这些滤波器设计为生成具有任意滚降因子的周期性Nyquist脉冲序列 $ \ alpha $ 可以从锁模奈奎斯特激光器发出的光。在第一部分中，我们推导了带有“单”的奈奎斯特脉冲激光器的摄动主方程。 $ \ alpha \ ne 0 $ 奈奎斯特脉冲，我们在其中获得新的滤波函数 $ F（\ omega）$ 确定低频和高频边缘的滤波器形状。满足周期Nyquist脉冲序列的二阶微分方程 $ \ alpha \ ne 0 $ 在锁模奈奎斯特激光器的时域中，Δε被导出并用于在时间域中的单个奈奎斯特脉冲解的直接推导。然后，通过利用奈奎斯特电位的概念，我们描述了奈奎斯特脉冲的光谱轮廓和滤波器形状之间的差异。 $ \ alpha = 0 $ 和 $ \ alpha \ ne 0 $ 。在后一部分中，我们描述了一个非扰动的主方程，该方程为具有任意滚降因子的“周期性”奈奎斯特脉冲序列提供了解决方案。我们首先表明 $ \ alpha \ ne 0 $ 周期性的奈奎斯特脉冲序列，由周期性的 $ \ delta $ 功能，具有与单个信封不同的信封形状 $ \ alpha \ ne 0 $ 奈奎斯特脉冲。这是因为定期 $ \ alpha \ ne 0 $ 脉冲序列由两个独立的周期函数组成，这会产生不同的频谱包络。然后，通过傅立叶变换主方程，我们得出新的滤波器，包括 $ F_ {1}（\ omega）$ 在低频边缘和 $ F_ {2}（\ omega）$ 在高频边缘使我们能够产生任意的奈奎斯特脉冲。