# Effects of Timing Noise on Square-Wave
## 1 Introduction  
Narrow-band optoelectronic oscillators (OEO) comprise a class of optoelectronic devices that spontaneously produce a narrow-band optical signal, which is amplitude-modulated by a sinusoid of microwave frequency [1, 2, 3, 4, 5, 6, 7]. By OEO, we refer, in general, to optoelectronic-based systems that spontaneously produce microwave-modulated optical signals, i.e., without external modulation. For example, we have shown that a laser diode with time-delayed optical feedback [6] or with combined optical and optoelectronic feedback [7] can result in output of a narrow-band microwave-modulated optical signal. OEO's spontaneously producing other types of amplitude modulation [8], such as square waves (SQ) [9] and pulses trains [10], have recently been demonstrated. Phase noise is an essential figure of merit defining sinusoidal OEO performance [2, 4, 6, 7, 11, 12, 13, 14, 15, 16]. More generally, we can think of phase noise as a consequence of timing noise. While there are many underlying physical mechanisms that may give rise to timing noise [12, 13], one might still ask how timing noise, in a generic sense, affects device operation in the case of a non-sinusoidal (NS) OEO. Our aim in this study is to provide an analytically tractable framework for the theoretical description of timing noise in NS OEO producing periodically modulated optical output. NS OEOs may include optical and microwave frequency comb sources [17], OEOs who output signals consist of a select set of harmonics of the fundamental modulation frequency, of sources of pulse trains of specified shape. Square-wave (SW) generation by semiconductor-laser-based systems has been demonstrated, for example, in VCSELs [18, 19, 20, 21], edge-emitters [22, 23, 24, 25, 26, 27], semiconductor ring lasers (SRLs) [28, 29], and quantum dot lasers [30]. In the above, SW generation is achieved without external modulation, but the square waveform results from the internal dynamics of the system. Specifically, we focus here on a SW OEO [9]. Such devices may be of interest, for example, to generate optical binary bit trains for digital communications and data processing. Timing noise OEOs can have numerous origins [5]. For example, it could involve fluctuations in the optical delay in a feedback loop or in the laser cavity, or fluctuations in the electronics driving the system (so-called technical noise, e.g., in the injection current). Our approach aims to be sufficiently generic, to provide a simple and analytically tractable model for timing noise (not accounting for technical noise), though the model clearly may be simplistic and omit important noise mechanisms. To suppress phase noise in an OEO, one approach is to inject the output of a high-spectral-purity oscillator (henceforth, the reference oscillator) into an OEO, which may otherwise show considerable phase noise [31; 32]. Thus, by synchronizing the OEO to the reference oscillator, the latter may stabilize the former [33; 34; 35]. This approach can result in substantial reduction in the phase noise present in the output [36; 37; 38; 39]. This reduction is often observed as a narrowing of the power spectral density (PSD) of the oscillator output near the center of the spectral line with a line shape composed of a narrow line center superimposed on a broad pedestal. This line shape is due to the imposition of the reference oscillator of a long-time overall phase correlation on top of faster phase noise around the slowly wandering value. There are a range of theoretical viewpoints to timing noise in oscillators. One method is to describe the noise by a physical or circuit model, or to assume a system perspective [40; 41; 42; 43]. By the time the noise is manifested in the output, it has been processed (linearly or nonlinearly) and filtered. Other approaches employ stochastic techniques [44; 45; 46]. Stabilization by means of a reference oscillator (sometimes called injection locking) relying on oscillator synchronization has been employed for decades [47; 48].