## 1 Introduction  
The Si photonics platform enables the fabrication of dense photonic integrated circuits and reduces their production cost, thanks to the strong optical confinement of high-index-contrast Si devices and the utilization of the matured complementary metal-oxide-semiconductor (CMOS) process, respectively. It is well suited for the development of components and subsystems for optical interconnects in data centers and high-performance computers. For these cases, optical modulators are key [1; 2; 3; 4; 5; 6] and a wide working spectrum and wide working temperature range are particularly required in optical interconnects, in addition to small size, low loss, low drive voltage and high speed. Previous studies on Si modulators have mainly used a phase shift generated by carrier plasma dispersion in p-n-doped rib waveguides. Resonator-type modulators such as microring devices often show good performance with a small size, but the working spectrum is very limited and cannot operate stably without complicated thermal stabilizations [7; 8; 9]. Therefore, most studies have employed Mach-Zehnder (MZ) devices, enabling a wide working spectrum and, hence, operating over a wide temperature range without thermal stabilizations when they use a symmetric circuit. However, MZ circuits typically need a long phase shifter length of several millimeters to obtain a sufficient extinction ratio (\(ER\)) [1; 2; 3; 4; 5; 6], resulting in high power consumption. To overcome this constraint, we have developed photonic crystal waveguide (PCW) MZ modulators. They are constructed via a CMOS process, with a phase shifter length \(L\) significantly reduced by the slow-light effect [10; 11; 12; 13; 14; 15]. For slow light with a low group velocity \(v_{\rm g}\), the slowdown factor is usually expressed by the group index \(n_{\rm g}=c/v_{\rm g}\), where \(c\) is the light velocity in a vacuum. Values of \(n_{\rm g}\) are observed to be between 3 and 4 for Si rib waveguides, while Si PCWs easily produce \(n_{\rm g}>20\)[10]. In a simplified model, the phase shift \(\Delta\phi\) in a waveguide with an equivalent modal index change induced by carrier plasma dispersion, \(\Delta n_{\rm eq}\), is expressed as [15; 16]  
\[\Delta\phi=k_{0}\,n_{\rm g}\frac{\Delta n_{\rm eq}}{n_{\rm eq}}\,\zeta\,L\,,\, \zeta\,=\,\frac{n_{\rm eq}}{\omega_{\rm b}}\frac{d\omega_{\rm b}}{dn_{\rm eq}} \approx 1 \tag{1}\]  
where \(k_{0}=2\pi/\lambda\) is the wave number in a vacuum and \(\omega_{\rm b}\) is the photonic band frequency. As \(\Delta\phi\) is proportional to \(n_{\rm g}\), the same value for \(\Delta\phi\) can be expected even for smaller \(L\) and \(\Delta n_{\rm eq}\) by employing a large \(n_{\rm g}\). Furthermore, we developed lattice-shifted PCWs (LSPCWs) that generate low-dispersion (LD) slow light, showing an almost constant \(n_{\rm g}\) over the wide wavelength range of 10-20 nm for transverse-electric (TE) polarization [10]. In our recent study, we observed eye opening at some selected wavelengths and the unchanged 10 Gbps operation was confirmed for a wavelength bandwidth of 16.9 nm at wavelength \(\lambda\sim\) 1550 nm and a temperature range of 19-124 \({}^{\circ}\)C in a device with \(L=90\) um [11]. Moreover, an interleaved p-n junction was employed, which increases the overlap of a depletion region with the slow-light mode and thus increases \(\Delta n_{\rm eq}\). This succeeded in the 25 Gbps modulation in a device with \(L=200\) um with a drive voltage of \(V_{\rm pp}=1.75\) V (on the panel of pulse pattern generator, PPG) [15]. In this study, we observed an \(ER\) of 3 dB, although the excess modulation loss (_ML_), which usually helps to increase the \(ER\), was moderately suppressed to 0.8 dB. A concern in using such LD slow light in practical modulators is the small fluctuations of \(n_{\rm g}\) in a working spectrum. These fluctuations are considered to be primarily caused by the reflection at the ends of the LSPCW and may be suppressed by technical improvements.