## 3 Experimental Results
### RF Beating Note and Timing Jitter  
To verify the \(\Delta v_{\text{RF1}}\) obtained using the optical technique described in the above section, we performed a direct measurement of the RF linewidth using a high-_speed_ photodetector. We focused all modes of the laser onto a 45 GHz IR photodetector and monitored the electrical output using a 50 GHz PXA signal analyzer. The RF signal analyzer was set to a 51 Hz resolution bandwidth (RBW) and 100 kHz span. Figure 5a shows the measured normalized first harmonic RF power spectral density (PSD) and Lorentzian curve fit for the laser with the frequency offset for clarity. A peak frequency f0 of 28.427 GHz was obtained, where f0 represents the pulse repetition rate, which is consistent with the measured mode spacing from the optical spectrum. The Lorentzian line shape provide good fit for the measured RF PSD curve. The extracted FWHM Lorentzian linewidths, \(\Delta f_{\text{FR1}}\), of 0.654 kHz is obtained. Since the properties of intrinsic phase noise from relatively broadband spontaneous emission in the passively ML laser leads to a Lorenzian-shaped PSD of photocurrent RF phase noise, an expression of RMS pulse-to-pulse time jitter estimated from a Lorenzian fitted RF linewidth was proposed in [18] Equation (3). The pulse-to-pulse timing jitter of 2.13 and 2.09 fs are obtained from the direct RF linewidth measurement (\(\Delta v_{\text{RF1}}=0.654\) kHz) and the parabolic curve fit of phase noise vs. mode number through RF linewidth (\(\Delta v_{\text{RF1}}=0.631\) kHz), respectively. Figure 4: Phase noise vs. mode lasing wavelength (**a**), and mode number with parabolic fit (**b**), measured using an OEWaves OE4000 automated laser linewidth/phase noise measurement system. The laser was driven at 400 mA at a temperature of 18 \({}^{\circ}\)C.  
In the case of a semiconductor passively mode-locked laser, it is shown that due to material nonlinearities and saturated absorber in the gain area, the phase correlation of the optical longitudinal modes can take place resulting in a strong four wave mixing [39]. This is especially true for an active region consisting of QDot or QDash structures. The common pulse to clock RMS time jitter, also called integrated timing jitter, can be obtained from integration of the single side band (SSB) phase noise spectrum _L(f)_ by [40; 41]  
\[\sigma_{int}=\frac{1}{2\pi f_{0}}\sqrt{2\int_{f_{low}}^{f_{high}}L(f)df}=\frac{ 1}{2\pi f_{0}}\sqrt{2(A_{12}+A_{23}+A_{34}+A_{45})} \tag{1}\]  
where \(f_{high}\) and \(f_{low}\) are the upper and lower integration frequencies, respectively; _L(f)_ is equal to the normalized RF PSD for a one Hz bandwidth in the interested range; and \(A_{12}\), \(A_{23}\), \(A_{34}\), and \(A_{45}\) are corresponding to the areas of \(1/f^{3}\) flicker frequency modulation (FFM), \(1/f^{2}\) white flicker frequency noise (WFF), \(1/f^{1}\) flicker phase modulation (FPM), and \(1/f^{0}\) white (shot) noise regions, respectively [40]. A typical _L(f)_ for a passively MLL is shown in Figure 5b where \(f_{1},f_{2},f_{3}\), and \(f_{4}\) indicate the frequency boundaries of the different noise mechanisms with corresponded slopes. As pointed out in [41], this phase noise estimation method is only valid for passively MLLs at frequency offsets well above the carrier peak linewidth. Figure 5c shows measured SSB _L(f)_ which is obtained from the normalized measured RF PSD for a one Hz bandwidth in the offset frequency range from center (as origin) to 20 MHz at the RBW setting of 10 kHz for the laser. From the colored dashed lines with negative slopes corresponding to the different noise mechanisms, it is observed that \(1/f^{2}\) WFF noise is dominated at the low offset frequency from 35 kHz to 250 kHz, which is the common type of noise found in a passively mode-locked laser. It intersects \(1/f^{4}\) random walk at the phase noise level of \(-79.8\) dBc/Hz, which is caused by external environment effects on the laser. The random-walk noise near carrier peak contributes to the laser's pulse-to pulse timing jitter.