# Effects of Timing Noise on Square-Wave
## 4 Conclusions  
To conclude, we presented an analytically tractable model for a NS OEO, considering three cases: (A) a reference oscillator is absent, in which case the NS OEO is a self-sustaining  
Figure 4: Single-sideband spectrum spectrum at \(n=1\) peak for \(\Lambda=10^{-5}\), \(\sigma_{\varphi}=0.1\), \(\tilde{\tau}=10\) (case (C), realistic stabilized oscillator) as in Figure 3.  
oscillator; (B) an ideal reference oscillator is present, and the NS OEO is synchronized to the reference oscillator, maintaining the timing, to be close to the absolute time standard; and (C) a reference oscillator is present that has finite bandwidth, and the NS OEO is synchronized to the reference oscillator, though the timing is not locked to the absolute time.  
We assume a single primary noise source, and do not account for complete filtering or nonlinear distortion of the noise in leading to \(\varphi(t)\). Time-delayed feedback is not taken into account, though many OEOs rely on feedback. This might lead to temporal correlations in \(\varphi(t)\) on the timescale of the feedback time neglected here [31]. The key results are those presented in Figures 3 and 4. For case (A), the PSD is composed of a sequence of spectral lines centered at frequencies \(n\omega_{0}\) of Lorentzian shape. The line widths scale with \(n\) since the phase noise associated with the spectral component \(n\) is \(n\varphi(t)\). At the same time, the maxima decrease in height is \(n^{-1}\). Thus, as \(n\rightarrow\infty\), the PSD washes out. For case (B), the spectral line shape is more complex (non-Lorentzian); as \(n\) increases, the line widths increase and maxima decrease in height, and eventually merge to form a continuum. For case (C), the most general case, the line shape is composed of a narrow Lorentzian peak whose widths are associated with the reference oscillator atop a broad pedestal associated with the SW OEO itself. Figure 4 provides an example of the line shape for case (C) in the form of the SSB spectrum. Here, we see the \(\omega_{0}^{-2}\) dependence at a small frequency offset due to the Lorentzian component and a more complex dependence at larger \(\omega_{\text{offset}}\).  
What we wish to emphasize is the following. In the PSD, the various Fourier peaks associated with the discrete frequency components of the periodic signal are broadened with two distinct contributions. First, there is a narrow central feature that is Lorentzian broadened due to the wandering of the stabilizing oscillator. Presumably, this broadening is not severe since we assume we have chosen a high spectral-purity oscillator specifically for this purpose. This narrow central line lies on top of a broad pedestal whose shape is dominated by the fluctuations inherent in the OEO, rather than in the stabilizing oscillator. The shape of the pedestal is complex. As the frequency of the Fourier component increases, so does the broadening. That is because, for a given timing noise \(\varphi(t)\), the _phase noise_ associated with \(\varphi(t)\) increases linearly with frequency. Thus, the pedestals ultimately merge at a high frequency to form a continuum.  
This study provides basic analytic results for the stabilized OEO. The model is illustrated for parameters selected to highlight the effects, and is not motivated by any experimental results; however, the treatment is general under the assumptions of the model, and can be applied not only to SW OEOs, but also to any other example of modulation that is non-sinusoidal.  
We gratefully acknowledge the support from Conseil Regional Grand Est and CPER SusChemProc.