## 2 In-Cavity Pulse Shaping
### Bandwidth Programmable Optical Nyquist Pulse Generation  
In this section, we describe the application of our method for in-cavity pulse shaping to the generation of optical Nyquist pulses [36]. The rectangular spectrum of sinc-shaped Nyquist pulses enables bandwidth-efficient encoding of data, which is of key interest for high-capacity optical communication systems (see, e.g., [40] and the references therein). Besides optical communications, sinc pulses can bring benefits to many other fields, such as all-optical sampling [41], microwave photonics [42] and light storage [43]. Various techniques to optically generate Nyquist pulses have been demonstrated, including spectral reshaping of a mode-locked laser [44] or fibre optical parametric amplification [45]. Additionally, sequences of very high-quality sinc-shaped pulses were obtained by the direct synthesis of a phase-locked frequency comb [46]. Furthermore, a Nyquist laser that can directly emit an optical Nyquist pulse train was recently demonstrated [47]. In the laser design reported in [47], the generation of Nyquist pulses is made possible by a combination of parabolic time-domain shaping at the pulse peak with an optical intensity modulator and spectral-domain shaping on the pulse wings with a spectral filter. In our passively-mode-locked laser design, the incorporation of a simple flat-top spectral filter in the cavity makes it possible to generate sinc-shaped Nyquist pulses of high quality. Furthermore, the use of a filter's profile with a corrective convex top can compensate for the concavity of the pulse spectrum obtained after nonlinear expansion in the fibre, thereby relaxing the need for large nonlinear phase accumulation in the fibre. The simplicity of our fibre laser design is appealing for implementation in experiments. It is typical in optical communications that the availability of a certain technical solution refocuses research on the feasibility of simpler or more cost-efficient versions. Figure 3: Phase diagrams representing the combined evolution of the FWHM temporal and spectral widths of the pulse over one round-trip in the cavity for: (**a**) a spectral pulse shaper with a triangular impulse response and a net cavity dispersion of \(0.008\ \mathrm{ps}^{2}\); **(b)** a flat-top spectral pulse shaper with \(B=2\) THz (\(\tau_{p}=0.5\) ps) and in the absence of in-cavity dispersion control. For the demonstration of Nyquist pulse generation, the amplitude profile \(R(f)\) of the filter's spectral response is chosen to have sharply-decaying edges outside its bandwidth and a Gaussian top,  
\[R(f)= \begin{cases}\frac{\tau_{p}}{2}\exp\left(\alpha f^{2}\right),&|f|<\frac{1} {\tau_{p}}\\ \frac{\tau_{p}}{4},&|f|=\frac{1}{\tau_{p}}\\ 0&\mathrm{otherwise}\end{cases}  \tag{2}\]  
where \(B=2/\tau_{p}\) is the spectral bandwidth (defined as the width of both edges of the spectrum tail), \(\tau_{p}\) is the time interval between zero crossings of the corresponding sinc-function-like impulse response and \(\alpha\) is an adjustable parameter defining the width and concavity of the spectrum top. Note that approximating the Gaussian function by the first two terms of its Taylor series, _i.e._, by a quadratic function, in the filter's response would yield similar results to those presented hereafter. For the special value \(\alpha=0\), \(R(f)\) is the rectangular spectral profile associated with the [PERSON] sinc-function impulse response [44]:  
\[r(t)=\frac{\sin(2\pi t/\tau_{p})}{2\pi t/\tau_{p}} \tag{3}\]  
while \(\alpha>0\) (\(\alpha<0\)) yields a convex or inverted (concave or conventional) Gaussian top. A variety of filters with very steep edges and adjustable bandwidth are readily available. The corrective Gaussian-top profile can be provided by a programmable LCoS optical processor, the shape and bandwidth of which can be software configured [29]. Note that optimisation of the filter's profile is relatively easy as it requires adjustment of only one parameter (\(\alpha\)).