Semianalytical design of antireflection gratings for photonic crystals

This article concerns the design of antireflection structures which, placed on a photonic crystal surface, significantly diminish the fraction of energy lost to reflected waves. After a review of the classes of these structures proposed to date, a new method is presented in detail for the design of antireflection gratings operating in a wide range of angles of incidence. The proposed algorithm is illustrated by means of several examples, showing the advantages and limitations.

Figure 1

Example AR structures belonging to the three principal classes described in the text. Darker areas denote regions with higher refractive index. (a) A homogeneous-layer AR coating. (b) An inhomogeneous-layer AR coating. (c) An (binary lamellar) AR grating.

Figure 2

The dependence of the reflectance $|r_0{|}^2$ of the PC shown in the inset, placed in air, on the location of the truncation plane $z=z_0$. The PC consists of a hexagonal lattice of air holes with radius $0.365a$, where $a$ is the lattice constant, etched in a dielectric matrix with permittivity $\epsilon =10.6$. The impinging wave is normally incident and $s$ polarized and has frequency $\omega =0.311\times 2\pi c/a$. It can be seen that $|r_0{|}^2$ does not fall under 0.13 for any truncation plane.

Figure 3

Successive steps of the proposed AR grating design algorithm.

Figure 4

System considered in step 1 of the AR grating design procedure.

Figure 5

Shaded are the regions of the complex ${\tilde{Z}}_3$ plane determined by the condition (15) equivalent to the constraint (14) for $s$ polarization and (a) $n_1=1$, $n_{\text{min}}=1.5$, $n_{\text{max}}=3$, $\theta =0$, (b) $n_1=1$, $n_{\text{min}}=1$, $n_{\text{max}}=3$, $\theta ={30}^{\circ }$, and (c) $n_1=1.5$, $n_{\text{min}}=1$, $n_{\text{max}}=3$, $\theta =0$. The circles $C_{\text{min}}$ and $C_{\text{max}}$ are defined in the text after Eq. (15); note that in the case (b), $C_{\text{max}}$ degenerates into the point $(1,0)$.

Figure 6

Solid line: EFC of the PC considered in Sec. 4 at frequency $\omega =0.311\times 2\pi c/a$ and for $s$ polarization. Dashed line: EFC of air at the same frequency.

Figure 7

Modulus of the electric field generated by an $s$-polarized wire source with current 1 A located above a slab of the PC studied in Sec. 4 truncated along a plane (a) lying midway between two neighboring rows of holes and (b) crossing the centers of holes.

Figure 8

Geometry of the PC studied in Sec. 4 truncated along a plane (a) lying midway between two neighboring rows of holes and (b) crossing the centers of holes. (c) Angular dependence of the reflectance of the structures shown in parts (a) and (b).

Figure 9

Shaded circle: region of the complex ${\tilde{Z}}_3$ plane determined by the condition (15) equivalent to the constraint (14) for $s$ polarization, $n_1=n_{\text{min}}=1$ and $n_{\text{max}}=2.51$. Points $A$ and $B$: reduced impedances ${\tilde{Z}}_3$ of structure S1 calculated in two different ways described in the text.

Figure 10

(a)–(b) Geometry of AR coatings S3 and S4, characterised by refractive index $n_2$ and thickness $d_2$ specified next to the drawings. (c)–(d) Geometry of binary lamellar AR gratings S5 and S6, characterized by fill factor $f$ and thickness $d_2$ specified next to the drawings. (e) Angular dependence of the reflectance of the structures shown in parts (a)–(d).

Figure 11

Definition of the geometrical parameters $w_{\text{i}}$, $w_{\text{o}}$, $h_{\text{i}}$, and $h_{\text{o}}$ of a trapezoidal grating superposed on the surface of structure S1.

Figure 12

(a)–(d) Geometry of AR gratings S7–S10 characterized by parameters $w_{\text{i}}$, $w_{\text{o}}$, $h_{\text{i}}$, and $h_{\text{o}}$ specified next to the drawings. (e) Angular dependence of the reflectance of the structures shown in parts (a)–(d). To help visualize the details of the $|r_0\left(\theta \right){|}^2$ dependence, the $y$ axis has been truncated at $|r_0{|}^2=0.2$.

Figure 13

Modulus of the electric field generated by an $s$-polarized wire source with current 1 A located above a slab of the PC studied in Sec. 4 with S7-type gratings placed on its horizontal surfaces.

Figure 14

EFC of the PC studied in Sec. 5 at frequency $\omega =0.265\times 2\pi c/a$. The shaded region corresponds to the range $\theta ⩽{45}^{\circ }$ ($|k_x|⩽0.187\times 2\pi /a$), where the EFC is approximately flat and for which the minimization of the PC's reflectance is made.

Figure 15

(a)–(b) Geometry of the PC studied in Sec. 5 truncated along a plane (a) lying midway between two neighboring rows of holes, (b) crossing the centers of holes. (c)–(d) Geometry of AR coatings S13 and S14, characterized by refractive index $n_2$ and thickness $d_2$ specified next to the drawings. (e) Angular dependence of the reflectance of the structures shown in parts (a)–(d).

Figure 16

Shaded circle: region of the complex ${\tilde{Y}}_3$ plane determined by the condition (15) equivalent to the constraint (14) for $p$ polarization, $n_1=n_{\text{min}}=1$ and $n_{\text{max}}=3.391$. Points $A$ and $B$: reduced admittances ${\tilde{Y}}_3$ of structure S11 calculated in two different ways described in the text.

Figure 17

(a)–(c) Geometry of binary lamellar AR gratings S15, S16, and S17 characterized by fill factor $f$ and thickness $d_2$ specified next to the drawings. (d) Angular dependence of the reflectance of the structures shown in parts (a)–(c).

Figure 18

(a) $p$-polarization EFC of the magneto-optical PC shown in the inset at frequency $\omega =0.4547\times 2\pi c/a$. (b) Solid line: $k_x$ dependence of the reflectance $|r_0{|}^2$ of this crystal, placed in air and truncated in the way indicated in the inset of part (a). Dashed line: a fragment of the EFC from part (a). The shaded region indicates the range of $k_x$ for which the crystal has a nonreciprocal gap, i.e., there are no propagative modes with the $x$ component of the Bloch vector equal to $-k_x$. (c) Shaded circle: region of the complex ${\tilde{Y}}_3$ plane determined by the condition (15) equivalent to the constraint (14) for $p$ polarization, $n_1=n_{\text{min}}=1$ and $n_{\text{max}}=1.50$. The reduced admittance of the crystal, ${\tilde{Y}}_3=13.5-7.3i$, lies far beyond the range of the graph.

Figure 19

Single-interface resonances of the structure S7. The shaded areas are the bulk bands of the underlying PC.