In this paper (PART II), we present output waveforms and the corresponding spectrum of a periodic Nyquist pulse train with a roll-off factor $\alpha $ emitted from a mode-locked Nyquist laser. In the first part, the relationship between the optical filter amplitudes ${H} _{a}$ and ${H} _{b}$ installed in a Nyquist laser cavity is derived by using the inverse Fourier transformation of filter ${F} _{1}$ ( $\omega $ ) at a low frequency edge and ${F} _{2}$ ( $\omega $ ) at a high frequency edge. We found that the relationship ${H} _{b} =$ (4/3) ${H} _{a}$ for $\alpha = 0$ is changed into the relationship ${H} _{b} =$ (1/cos( $\beta ~\Omega _{m}$ /2)) ${H} _{a}$ for $\alpha ~\ne ~0$ , where $\beta =\pi $ /( $2\alpha \omega _{N}$ ), $\omega _{N}$ is the zero-crossing frequency and $\Omega _{m}$ is the modulation frequency. This relationship is important for describing the entire spectral profile of the optical filter installed in the laser cavity. In the latter part, we report how we succeeded in generating a Nyquist pulse train with an arbitrary $\alpha $ value by employing computer simulations with analytically derived optical filters consisting of ${F} _{1}$ ( $\omega $ ) and ${F} _{2}$ ( $\omega $ ), ${H} _{a}$ , and ${H} _{b}$ . We found that a Nyquist laser cannot always generate an isolated ideal Nyquist pulse train because there is interference between the wings of adjacent Nyquist pulses. We clarify the differences and similarities as regards filter shape and the corresponding waveform in the time domain of a single Nyquist pulse and a periodic Nyquist pulse train in terms of differences in power ${P}$ , time-domain distribution $\tau $ , spectrum ${S}$ , and filter shape ${F}$ . We show that a pure Nyquist pulse train can be obtained with the condition $\alpha ~{N} >10$ , where differences in ${P}$ , ${S}$ , and ${F}$ are less than 1 %, and we present a useful chart showing how to generate a Nyquist pulse train in the GHz region. ${N}$ is the number of modes in the low or high frequency region. We investigated the time domain orthogonality ${g} _{m,n}$ of the Nyquist pulse train from the laser and found that the orthogonality can be maintained although there is a small interference effect on the wing of the Nyquist pulse.

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二部分：振荡波形和光谱特性

在本文（PART II）中，我们给出了具有滚降因子的周期性Nyquist脉冲序列的输出波形和相应频谱 $ \ alpha $ 锁模奈奎斯特激光器发出的光。在第一部分中，滤光片振幅之间的关系 $ {H} _ {a} $ 和 $ {H} _ {b} $ 通过使用滤波器的傅立叶逆变换可以得出安装在Nyquist激光腔中的传感器 $ {F} _ {1} $ （ $ \ omega $ ）在低频边缘，并且 $ {F} _ {2} $ （ $ \ omega $ ）。我们发现这种关系 $ {H} _ {b} = $ （4/3） $ {H} _ {a} $ 为了 $ \ alpha = 0 $ 变成了恋爱关系 $ {H} _ {b} = $ （1 / cos（ $ \ beta〜\ Omega _ {m} $ / 2）） $ {H} _ {a} $ 为了 $ \ alpha〜\ ne〜0 $ ， 在哪里 $ \ beta = \ pi $ /（ $ 2 \ alpha \ omega _ {N} $ ）， $ \ omega _ {N} $ 是零交叉频率， $ \ Omega _ {m} $ 是调制频率。该关系对于描述安装在激光腔中的滤光器的整个光谱轮廓很重要。在后一部分中，我们报告了如何成功生成具有任意 $ \ alpha $ 通过使用计算机模拟和分析得出的光学滤光片来实现价值 $ {F} _ {1} $ （ $ \ omega $ ） 和 $ {F} _ {2} $ （ $ \ omega $ ）， $ {H} _ {a} $ ， 和 $ {H} _ {b} $ 。我们发现，奈奎斯特激光器不能总是产生孤立的理想奈奎斯特脉冲序列，因为相邻的奈奎斯特脉冲的翼之间存在干扰。我们根据功率的差异阐明单个Nyquist脉冲和周期性Nyquist脉冲序列在时域上的滤波器形状和相应波形的差异和相似性 $ {P} $ ，时域分布 $ \ tau $ ， 光谱 $ {S} $ ，以及滤镜形状 $ {F} $ 。我们证明在以下条件下可以获得纯的奈奎斯特脉冲串 $ \ alpha〜{N}> 10 $ ，其中的差异 $ {P} $ ， $ {S} $ ， 和 $ {F} $ 小于1％，我们提供了一个有用的图表，显示了如何在GHz区域生成奈奎斯特脉冲序列。 $ {N} $ 是低频或高频区域中的模式数。我们研究了时域正交性 $ {g} _ {m，n} $ 对来自激光器的奈奎斯特脉冲序列进行了分析，发现尽管对奈奎斯特脉冲的机翼有很小的干扰影响，但仍可以保持正交性。