Reflectivity calculated for a three-dimensional silicon photonic band gap crystal with finite support

We study numerically the reflectivity of three-dimensional (3D) photonic crystals with a complete 3D photonic band gap. We employ the finite element method to study crystals with the cubic diamondlike inverse woodpile structure. The high-index backbone has a dielectric function similar to silicon. We study crystals with a range of thicknesses up to ten unit cells $\left(L\le 10c\right)$. The crystals are surrounded by vacuum, and have a finite support as in experiments. The polarization-resolved reflectivity spectra reveal Fabry-Pérot fringes related to standing waves in the finite crystal, as well as broad stop bands with nearly $100\%$ reflectivity, even for thin crystals. The frequency ranges of the stop bands change little with angle of incidence, which is plausible since the stop bands are part of the 3D band gap. Moreover, this result supports the previous assertion that intense reflection peaks measured with a large numerical aperture provide a faithful signature of the 3D photonic band gap. For $p$-polarized waves, we observe an intriguing hybridization between the Fabry-Pérot resonances and the Brewster angle that remains to be observed in experiments. From the strong reflectivity peaks, it is inferred that the maximum reflectivity observed in experiments is not limited by finite size. The frequency ranges of the stop bands agree very well with stop gaps in the photonic band structure that pertain to infinite and perfect crystals. The angle-dependent reflectivity spectra provide an improved interpretation of the reflectivity measurements performed with a certain numerical aperture and a new insight in the crystal structure, namely unequal pore radii in $X$ and $Z$ directions. The Bragg attenuation lengths $L_B$ are found to be smaller by a factor 6 to 9 than earlier estimates that are based on the width of the stop band. Hence, crystals with a thickness of 12 unit cells studied in experiments are in the thick crystal limit $\left(L\gg L_B\right)$. Our reflectivity calculations suggest that the 3D silicon photonic band gap crystals are interesting candidates for back reflectors in a solar cell in order to enhance the photovoltaic efficiency.

Figure 1

The tetragonal primitive unit cell of the cubic inverse woodpile photonic crystal structure. (a) Perspective view of the unit cell with the $XYZ$ coordinate system. The two sets of pores are parallel to the $X$ and the $Z$ axes. (b) View of the unit cell along the $Z$ axis with the lattice parameters $a$ and $c$ and the pore radius $r$. (c) View of the unit cell along the $X$ axis.

Figure 2

(a) Photonic band structure for the 3D inverse woodpile photonic crystal with $\frac{r}{a}=0.245$ and ${\epsilon }_{\text{Si}}=11.68$. The red bar marks the 3D photonic band gap, and the yellow bars mark stop gaps in the $\mathrm{\Gamma }X$ and $\mathrm{\Gamma }Z$ directions. (b) First Brillouin zone showing the high symmetry points and the origin at $\mathrm{\Gamma }$.

Figure 3

Illustration of the computational cell for a photonic crystal with thickness $L=4c$. (a) View along the $X$ direction $\left[101\right]$. The source of plane waves is at the left, and is separated by an air layer from the crystal. The computational cell is bounded by absorbing boundaries at $-Z$ and $+Z$, and by periodic boundary conditions on $\pm X$ and $\pm Y$. The blue color represents the high-index backbone of the crystal having dielectric function similar to silicon. (b) Perspective view of the computational cell.

Figure 4

Calculated angle- and frequency- resolved reflectivity spectra in the $\mathrm{\Gamma }Z$ direction for a crystal with thickness $L=4c$ for (a) $s$ polarization and (b) $p$ polarization. The dark blue color represents high reflectivity that occurs in the stop band at all angles. The white color represents near $0\%$ reflectivity that occurs in the Fabry-Pérot fringes, at the Brewster angle, and in their hybridization in the range ${54}^{\circ }\le \theta \le {61}^{\circ }$. The brown double arrow represents the stop gap in the $\mathrm{\Gamma }Z$ direction (from Fig. 2). The black box indicates the region of high-resolution results shown in Fig. 5.

Figure 5

Hybridization of the Fabry-Pérot resonances and the Brewster angle, shown by angle- and frequency- resolved calculated reflectivity spectra in $\mathrm{\Gamma }Z$ direction for $p$ polarization. The black dashed line is a guide to the eye that connects the midpoints (circles) of the bends in the fringes. The white dotted line indicates the Brewster angle ${\theta }_B$.

Figure 6

Calculated reflectivity spectra for a Si inverse woodpile photonic band gap crystal along the $\mathrm{\Gamma }Z$ high symmetry direction in wave vector space. The red curve in (i) and the green curve in (iii) are reflectivity spectra calculated for $s$ and $p$ polarization, respectively. The corresponding band structure for the $\mathrm{\Gamma }Z$ direction is shown in (ii), where the 3D band gap is shown in orange. The wave vector is expressed as $k^{\prime }=(ka/2\pi )$. The polarization character of bands near the gap is assigned in Fig. 7. The frequency ranges of the $s$- and $p$-stop bands agree excellently with corresponding stop gaps in the photonic band structure.

Figure 7

Reflectivity near the stop band for (a) $s$- and (b) $p$-polarized light for Si inverse woodpile crystals. In (i), red dashed-dotted and green dashed curves are calculated results, as in Fig. 6. (ii) The band structures for (a) $s$- and (b) $p$-polarized light, where the 3D band gap is shown in orange. The wave vector is expressed as $k^{\prime }=(ka/2\pi )$. The vertical black dashed lines indicate the edges of the stop band (i) and the matching stop gap edges (ii). Near the stop gap edges, we identify $s$ bands (red dashed-dotted curves) and $p$ bands (green dashed curves).

Figure 8

Comparison between numerical calculations and experimental results for the reflectivity peaks near the stop band for (a) $s$- and (b) $p$-polarized light for Si inverse woodpile crystals. (i) The calculated results, where red dashed-dotted and green dashed curves are reflectivity spectra for angles of incidence from $6^{\circ }$ to ${40}^{\circ }$ off normal in the $\mathrm{\Gamma }Z$ direction as well as in the equivalent $\mathrm{\Gamma }X$ direction. In (ii), blue squares are measurements from Ref. [30] in the $\mathrm{\Gamma }Z$ direction, and magenta circles in the equivalent $\mathrm{\Gamma }X$ direction. The top ordinate shows the frequency in wave numbers $\left({\mathrm{cm}}^{-1}\right)$ for a lattice parameter $a=677$ nm as in the experiments.

Figure 9

Transmission versus thickness for a silicon inverse woodpile photonic crystal in the $\mathrm{\Gamma }Z$ direction for $s$ (top) and $p$ polarizations (bottom). Red squares, black circles, and blue triangles pertain to frequencies below, inside, and above the stop gap, respectively. The green dashed lines are the exponential decay of transmission with crystal thickness at frequencies in the stop gap [Eq. (2)]. The black dashed line is a guide to the eye that shows modulations in the stop band.

Figure 10

Analytical calculation versus numerical computation for reflection and transmission spectra of semi-infinite dielectric medium for $p$ polarization. The medium has dielectric permittivity $\epsilon =12.1$. The analytically calculated reflectivity $\left(R_A\right)$ and transmission $\left(T_A\right)$ are shown in green and black lines, respectively. Blue circles and red diamonds represent the numerically computed reflectivity $\left(R_N\right)$ and transmission $\left(T_N\right)$, respectively. ${\theta }_B$ denotes the Brewster angle.

Figure 11

Analytically calculated angle- and frequency-resolved reflectivity spectra of a thin dielectric film for $p$ polarization. The film has a dielectric permittivity $\epsilon =12.1$. The Brewster angle at ${\theta }_B={73}^{\circ }$ is constant with frequency.