# 47.5 GHz Membrane-III-V-on-Si Directly Modulated Laser for Sub-pJ/bit 100-Gbps Transmission
## 2 Materials and Methods  
In conventional DMLs, the 3-dB bandwidth and ultimately the modulation speed are widely dependent on the CPR effect, typically given by the relaxation oscillation frequency (\(f_{R}\)). However, this CPR frequency is proportional to the square-root of the bias current (\(I_{b}\)) above the threshold current (\(I_{th}\)):  
\[f_{R}\propto\sqrt{I_{b}-I_{th}}. \tag{1}\]  
In other words, for a DML design with a fixed spatial structure, spatial dimensions, materials, and confinement factor, one can achieve a further increase in the 3-dB bandwidth by increasing the applied bias current. On the other hand, however, the power consumption also increases by increasing the bias current, which leads to a power-bandwidth tradeoff. As discussed in the introduction, in the recent years, emerging longitudinal cavity designs that enable the DL and PPR effects can alleviate this tradeoff [9; 10; 11; 12; 13; 14; 15; 16; 17]. In such a case, the DL and/or PPR effects effectively enhance the high-frequency components of the modulated signal, leading to an extension of the 3-dB bandwidths beyond those allowed by pure CPR dynamics. In more details, the DL effect entails an enhancement on the \(f_{R}\) by a factor \(\chi_{DL}\)[9] such that  
\[f_{R}\rightarrow\chi_{DL}\cdot f_{R}. \tag{2}\]  
A sufficient condition for \(\chi_{DL}>1\) and, hence, an enhancement of the 3-dB bandwidth, can be achieved in a longitudinal design that allows for a positive derivative of the mirror loss (\(a_{m}\)) in respect to wavelength (\(\lambda\)) within the vicinity of the lasing wavelength:  
\[\chi_{DL}>1\Leftarrow\partial a_{m}/\partial\lambda>0. \tag{3}\]  
This condition can be satisfied on the long-wavelength side of band-pass filters such as DBRs [9], or Fabry-Perot etalons induced by a passive feedback [16; 17]. Unlike the DL effect, the PPR effect does not directly enhance the \(f_{R}\), but rather generates a second resonance frequency in the electro-optic (EO) response of the DML through a photon-photon (i.e., two-mode) interaction. This can also be understood as an optical amplification mechanism of the modulated signal (or, rather, its high-frequency components) via a secondary longitudinal mode. Such PPR frequency (\(f_{PPR}\)) is proportional to the spectral separation between the main lasing mode and a nearby longitudinal mode (\(\Delta\lambda\)) via  
\[f_{PPR}=\frac{c}{\lambda_{0}^{2}}|\Delta\lambda|, \tag{4}\]  
where \(c\) is the speed of light in vacuum and \(\lambda_{0}\) is the central wavelength. By carefully designing, therefore, \(\Delta\lambda\) in order to obtain a \(f_{PPR}\) a few tens of GHz higher than the intrinsic CPR-based 3-dB bandwidth, one can achieve an increase of more than 2x times in the EO-response bandwidth. One key design consideration, though, is to achieve a relatively flat EO response, void of large dips or valleys between the \(f_{R}\) and \(f_{PPR}\), as these would cause a significant penalty on single-carrier modulation formats that typically encompass flat and continuous RF spectra. The PPR and DL can be used in combination, albeit inefficiently, since the DL and PPR optimum spectral positions are typically different. In such a case, a well-designed PPR is usually prioritized for the modulation. The DBR-based laser design that allowed us to utilize both the DL and PPR effects is shown in Figure 1. The distributed reflector (DR) longitudinal design, shown in Figure 1a, includes a distributed feedback (DFB) section with nonshifted grating, sandwiched between two DBR mirrors with unequal lengths. Similar to our previous works, the shorter, rear DBR (\"DBR-1\") section is spectrally detuned compared to the DFB in order to filter one of the two main longitudinal modes present in the DFB's transmittance spectrum [6]. This is schematically shown in Figure 1c. This design ensures a stable single-mode operation with a large side-mode suppression ratio (SMSR) at a long wavelength and temperature range.