\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{figure_3_1.pdf}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{figure_3_2.pdf}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{figure_3_3.pdf}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{figure_3_4.pdf}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{figure_3_5.pdf}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\linewidth]{figure_3_6.pdf}
\end{subfigure}
\caption{(a) Contour of reflection coefficient $|s_0(50)|$ in linear scale. $N=50$ gives saturated values for $s_0$ and $s_{0,\pm1}$. (b) Reflection coefficient spectra, $r_A$, $|s_0(50)|$, and $|s_{0,\pm1}(50)|$. (c) Reflection phase spectra. (d) In-phase condition of $h_0(1)$ and $h_1(2)$ (green dotted lines) plotted with a transmittance contour in dB scale (identical to Fig.~1(f)). (e) Reflection phase spectra. (f) Reflection delay time of the entire HG structure. For $|s_0|$ and $|s_{0,\pm1}|$ in (b), $\angle h_0(1)$ and $\angle h_1(2)$ in (c), $\angle p_0$ and $\angle p_1$ in (e), and (f), a cap layer thickness of 279 nm is assumed.}
\label{figure_3}
\end{figure}

Thus, $|s_0|$ including multiple reflections of $|r_A|$ becomes high at these wavelengths, except around a wavelength of destructive interference. As, for example, shown in Fig.~\ref{figure_3}(b), the in-phase condition of $r_S$ and $h_0(1)$ around 1.36-$\mu$m and 1.80-$\mu$m wavelengths (see Fig.~\ref{figure_3}(c)) leads to high $|s_0|$ values, while their out-of-phase condition around 1.63 $\mu$m results in a local decrease.

Furthermore, the in-phase coupling of the 1-st order harmonic to the 0-th harmonic increases the reflection coefficient to be close to 1.0. As shown in Fig.~\ref{figure_3}(c), $h_1(2)$ and $h_0(1)$ are nearly in-phase over 1.30 $\mu$m to 1.80 $\mu$m wavelengths, i.e., $\angle h_0(1) - \angle h_1(2) \sim 0$ or $2\pi$. This in-phase coupling makes $|s_{0,\pm1}|$ close to 1.0, even compensating the local dip around 1.63 $\mu$m in the $|s_0|$ spectrum. Fig.~\ref{figure_3}(d) shows that the in-phase condition of $h_0(1)$ and $h_1(2)$ denoted by green dotted lines matches well with the position of transmittance dip lines within 1.30 $\mu$m to 1.80 $\mu$m wavelengths, which supports our explanation above. Outside these wavelengths, the transmittance dip position is determined mainly by the GM resonance condition since $|r_A|$ is no longer high, leading to a weak FP resonance.

In Fig.~\ref{figure_3}(d), the in-phase condition is significantly distorted within the grating stopband. Thanks to this distortion, the in-phase condition can be met over a broad wavelength range, given that the cap layer thickness is 279 nm. The phase difference, $\angle h_0(1) - \angle h_1(2)$ is given by $(2/\rho_0 + L_{rA}) - (2/\rho_1 + 2L_{p1} + L_{rB} + L_{rC} + L_{rT})$. As shown in Fig.~\ref{figure_3}(e), both $\angle r_B$ and $\angle r_C$ have a convex shape within the grating stopband, while other phases monotonically decrease with wavelength. This leads to the distortion of in-phase condition. Since $\angle p_0$ and $\angle p_1$ increase with the cap layer thickness, one may choose a certain cap layer thickness for obtaining the flattest in-phase condition, which is 279 nm for the considered case. Fig.~\ref{figure_3}(f) shows a reflection delay spectrum of the HG, which measures the time taken to reach a steady-state reflectance value when measuring the reflectance in time. The long delay time around