## 2 In-Cavity Pulse Shaping
### Bandwidth Programmable Optical Nyquist Pulse Generation
Note also that the sinc function in Equation (3) defines a time unlimited pulse and, as a consequence of causality, is therefore only a theoretical construct. Instead, a pulse train as generated in the laser would be used in an experimental demonstration of Nyquist pulse shaping. However, given that the pulse repetition period would be significantly larger than the characteristic pulse width \(\tau_{p}\), only a negligibly small side-band field would be created by the sinc function between adjacent pulses. The solutions obtained for a flat-top spectral profile (\(\alpha=0\)) applied on the filter with different related zero-crossing pulse durations are shown in Figure 4, and compared to the solutions obtained with a convex Gaussian-top profile. The shape correction factor \(\alpha\) was optimised for each pulse duration, and no in-cavity dispersion control was used in these simulations. It is seen that the use of the basic rectangular shape for the filter's spectral response already allows us to obtain sinc-shaped pulses at the output of the laser that coincide well with the theoretical ones described by Equation (3) and feature a spectrum that is fairly close to the ideal rectangular case. When a convex spectral profile is applied to the filter, the correction of the non-perfectly-flattened pulse spectrum after the gain fibre by the filter brings about sinc pulses of enhanced quality and with an almost ideal rectangular spectrum at the laser output for all of the pulse durations being considered. The example of pulse evolution shown in Figure 3b when the basic rectangular filtering method is employed demonstrates that the pulses are shaped by the same mechanism as that governing the pulse generation regimes described in the previous section (Figure 3a). The initial slight decrease of the spectral bandwidth of the pulse in the gain fibre originates from the reshaping of the spectrum from a rectangular-like profile at the entrance of the fibre to a parabolic-like shape near the peak with a transition to a steep decay in the first stage of evolution in the fibre. After this stage, the spectrum broadens significantly and eventually develops the steep and structured edges characteristic of SPM. We also note that while the FWHM pulse duration decreases owing to spectral filtering, the root-mean-square (rms) duration is increased by the filter owing to the specific ringing feature (or ripples) on the pulse wings accompanied by the generated Nyquist pulse. The SA decreases the pulse duration and attenuates the pulse ripples. The evolution in the laser exhibits a spectral breathing factor of approximately two, which accommodates a \(B\)-integral of approximately 3 rad in the example of Figure 3b. The reduced spectral breathing and nonlinear phase shift accumulation compared to the results of the previous section stem from the shorter length of fibre used in the laser cavity for the Nyquist mode of operation. Figure 4: Temporal (**left**) and spectral (**right**) intensity profiles of the output pulse from the laser for different filter bandwidths: (**a**) \(B=400\:\mathrm{GHz}\) (\(\tau_{p}=2.5\:\mathrm{ps}\)); (**b**) \(B=1\:\mathrm{THz}\) (\(\tau_{p}=1\:\mathrm{ps}\)); and (**c**) \(B=2\:\mathrm{THz}\) (\(\tau_{p}=0.5\:\mathrm{ps}\)). The profiles obtained with a flat-top (\(\alpha=0\)) spectral pulse shaper (black) are compared to those obtained with a convex Gaussian-top spectral shaper (blue). The correction factor is \(\alpha=6.1,\,1.1,\,0.18\) in (**a–c**), respectively. Also shown are the calculated waveforms according to (3) (red dotted). A laser without dispersion control in the cavity is modelled. Adapted from [36]. Copyright 2015, IEEE. In order to illustrate the flexibility of the proposed scheme in terms of output spectral bandwidth, we have changed the zero-crossing duration parameter \(\tau_{p}\) in the filter's impulse response over a wide range and assessed the quality of the obtained pulses with the metric \(M^{2}=\int\!\mathrm{d}t\,(|\psi|^{2}-|\chi|^{2})^{2}/\int\!\mathrm{d}t\,|\psi|^ {4}\).