# Effects of Timing Noise on Square-Wave
## 3 Results  
In this section, we present numerical results for \(S_{f}(\omega)\) for the three cases, _viz._, (A), (B), and (C). We consider an even unit-amplitude square wave with 50% duty cycle, 0 dc offset, and period \(t_{0}\) for which  
\[c_{n}=\frac{4}{n\pi}\sin\frac{n\pi}{2}.\]  
For simplicity, we measure time in units of \(2\pi/\omega_{0}\) and frequency in units of \(\omega_{0}\), and take \(t_{0}=2\pi\omega_{0}^{-1}\). We consider \(\mathcal{F}(\tau)\) and \(S_{f}(\omega)\). First, in Figure 1a, take the unstabilized oscillator [case (A) with a substantial value of \(\Lambda=0.03\)] and in Figure 1b when timing noise is absent. With increasing \(\tau\), as the timing wanders, the acf peaks decrease while the troughs increase, the widths of the peaks scale with \(n\) (with \(\tau=nt_{0}\)), and the acf washes out, tending to zero, as seen in Figure 1a. In Figure 1b we see the case in which timing noise is absent. The acf is a sequence of triangle functions (the acf of a rectangle function) centered at \(nt_{0}\). We now compare cases (A), (B), and (C). Figure 2 shows \(\mathcal{F}(\tau)\) for (a) \(\Lambda=0.005\), \(\sigma_{\varphi}=0\) [case (A)], (b) \(\Lambda=0\), \(\sigma_{\varphi}=0.1\), \(\tilde{\tau}=10\) [case (B)], and (c) \(\Lambda=0.005\), \(\sigma_{\varphi}=0.1\), \(\tilde{\tau}=10\) [case (C)]. In Figure 2a, we assume the reference oscillator is absent, but take a very small value of \(\Lambda\) for the SW OEO itself for comparison with cases (B) and (C). As noted above, the widths of the autocorrelation peaks increase, the baseline rises, and the peak heights fall off with increasing \(\tau\) as more distant peaks become less correlated as the timing error eventually wanders from its initial value over the timescale (\(\omega_{0}\Lambda\)). Figure 2b corresponds to case (B), in which an ideal reference oscillator is present \(\Lambda=0\); however, \(\sigma_{\varphi}=0.1\) and \(\tau=10\). For \(\tau\gtrsim\tilde{\tau}\), \(\mathcal{F}(t)\) tends to periodic (see Equation (11)). In this case, there is some smoothening of the triangular acf seen in Figure 2b. Figure 2c includes a reference oscillator that is allowed to wander, albeit slowly (\(\Lambda=0.005\)), and the SW OEO itself also has timing noise (\(\sigma_{\varphi}=0.1\), \(\tilde{\tau}=10\)). Here we see a combination of features noted for cases (A) and (B), _viz._, an initial falling off of the acf on timescale \(\tilde{\tau}\) and a tendency of the acf to wash out at larger \(\tau\) ith a decrease in the peak height, an increase of the troughs, and a broadening of the peaks. We now turn to the PDF \(S_{f}(\omega)\). Figure 3 shows \(S_{f}(\omega)\) for (a) \(\Lambda=10^{-5}\), \(\sigma_{\varphi}=0\) [case (A), unstabilized oscillator], (b) \(\Lambda=0\), \(\sigma_{\varphi}=0.1\), \(\bar{\tau}=10\) [case (B), ideally stabilized oscillator], and (c) \(\Lambda=10^{-5}\), \(\sigma_{\varphi}=0.1\), \(\bar{\tau}=10\) [case (C), realistic stabilized oscillator]. We have chosen parameters different from those in Figure 2 to emphasize the expected features. Figure 3a consists of a sequence of narrow Lorentzian peaks. The line width increases with \(n\) and they are modulated in amplitude by \(|c_{n}|^{2}\) (not visible on the plot due to the limited vertical range displayed). In Figure 3b, we see a sequence of peaks that, for increasing \(n\), depart increasingly from Gaussian in shape (see Figure 4 below). These features provide the pedestals we shall see in Figure 3c. Both contributions to the spectral lines are included in Figure 3c. Here, we see the pedestals seen in Figure 3b, on top of which are narrow Lorentzian peaks, as in Figure 3a. Such line shapes are known from injection-locked mode-locked laser [17]. In Figure 4, we explore the line shape in more detail. Shown is the single-sideband (SSB) spectrum of the \(n=1\) line from Figure 3c.