## 3 Results
### Optical Characterization  
The power spectrum of the monolithic laser diode is measured with an optical spectrum analyzer at a laser temperature of \(T\) = 20.10 \({}^{\circ}\)C and an injection current of \(I\) = 487 mA. The trace is depicted in Figure 2a. From the spectrum, the mode spacing was calculated to be \(\Delta f_{\text{FSR}}=43.47\) GHz. If the laser is mode-locked, the optimal dispersion for chirp compensation can be estimated from the \(-10\) dB bandwidth, \(\Delta f_{\text{10 dB}}\), and \(\Delta f_{\text{FSR}}\)[28], as follows:  
\[|B|=(2\pi\Delta f_{\text{10 dB}}\Delta f_{\text{FSR}})^{-1}=2.274\text{ ps}^{2}, \tag{1}\]  
with \(\Delta f_{\text{10 dB}}=1.61\) THz. From the group velocity dispersion, \(\text{GVD}=0.023\text{ ps}^{2}/\text{m}\) of an SMF at 1550 nm follows a predicted fiber length of 99 m.  
To check whether the laser is mode-locked, the second harmonic generation (SHG) ACF is measured after the beam splitter. First, we measured the ACF without additional fiber. In this case, the total fiber length is 14 m due to the fiber from the laser, the fiber-based isolator, and the polarization controller. The ACF can be seen as a flat gray line in Figure 2b. The SMF length that provides the best chirp compensation is determined experimentally by changing the total fiber length from 14 m to 84 m in steps of 5 m and finer steps close to the optimum point. At a total fiber length of 84 m, the full-width at half maximum (FWHM) of the measured ACF reaches a minimum of 762 fs. The measured ACF is depicted in black in Figure 2b and clearly demonstrates that the laser diode is mode-locked. The mismatch of 15 m for the optimal fiber length between the estimated and the experimentally found fiber length originates from the phenomenological nature of Equation (1). The \(-10\) dB bandwidth is especially highly dependent on the shape of the optical spectrum. Figure 2: **(a)** Shows the spectrum of the FPL1009P. The total fiber length in **(a,b)** is 84 m. **(b)** Shows the autocorrelation function after dispersion compensation with 84 m SMF (black) and 14 m (gray), the simulated autocorrelation function (ACF) with a residual chirp (red), and with perfect chirp compensation (green). The measured FWHM is 762 fs, which corresponds to a deconvoluted pulse duration of 544 fs. To verify the chirp compensation, the ACF can be simulated from the measured power spectrum as follows: The electric field of the laser signal is given by the sum of those of the individual modes,  
\[\underline{E}_{\text{LD}}(t)=\sum_{k=0}^{N-1}E_{\text{k}}e^{\|2\pi\cdot(f_{0}+k \Delta f_{\text{PSK}})t+\varphi_{\text{k}}\|}\, \tag{2}\]  
where \(E_{\text{k}}\) and \(\varphi_{\text{k}}\) denote the amplitude and phase of mode \(k\), respectively. The corresponding instantaneous optical power, i.e., the optical pulse envelope, is  
\[P_{\text{opt}}(t)=\big{|}\underline{E}_{\text{LD}}(t)\big{|}^{2}, \tag{3}\]  
and the ACF, \(R(\tau)\), can be calculated according to  
\[R(\tau)=\ \int_{-\infty}^{\infty}P_{\text{opt}}(t+\tau)\cdot P_{\text{opt}}(t)\ \text{dt}. \tag{4}\]  
The chirp-free case is approximated by a linear phase relationship between the modes, i.e.,  
\[\varphi_{k,\text{lin}}=\varphi_{0,\text{lin}}\cdot k\, \tag{5}\]  
with the arbitrary phase slope \(\varphi_{0,\text{lin}}\). The residual chirp in our measurement is approximated by assuming a square phase profile, i.e.,  
\[\varphi_{k,\text{squ}}=\varphi_{0,\text{squ}}\bigg{(}\frac{k}{N}\bigg{)}^{2}. \tag{6}\]  
The parameter \(\varphi_{0,\text{squ}}\) is optimized so that the FWHM of the simulated ACF matches the FWHM of the measured ACF. The chirp-free case is depicted as the green line in Figure 2b with an FWHM of the ACF of 656 fs. The simulated ACF for the case of a residual chirp is depicted as the red line in Figure 2b and exhibits a FWHM of 762 fs. Its shape closely matches the shape of the measured ACF (black). Further, a conversion factor can be calculated for the ACF of the measured pulse with the simulated data. The ACF is always broader than the underlying pulse.