Revisiting the effective medium approximation in all-dielectric subwavelength multilayers: Breakdown and rebuilding

Here, we present a comprehensive analysis of the effective medium approximation (EMA) breakdown in all-dielectric deep-subwavelength multilayers made of alternating layers by means of the transfer matrix method. We demonstrated that the approximation is invalid at the vicinity of the effective medium's critical angle for total internal reflection and obtained an analytical criterion for the breakdown of the EMA, which depends on the layer thickness, the incident angle, and the permittivity difference between the alternate layers. We rebuilt the EMA by adding higher-order correction onto the traditional effective permittivity. Furthermore, we found that the EMA breakdown that arises from the boundary effect cannot be repaired in the traditional homogenization strategy with only one layer of effective medium. By adding an artificial matched layer after the conventional effective layer, the boundary effect breakdown was neatly removed.

Figure 1

(a) Schematic geometry of the multilayer structure. (b) The reflectivity discrepancy between rigorous calculation and the effective medium approximation is plotted as a function of the number of layer pairs ($\mathrm{N}f$) and the incident angle ($\theta$). (c) Amplified image in the vicinity of the critical angle. (d) Reflectivity as a function of $N$ for rigorous calculation (black line) and the effective medium approximation (red line) when the angle of ${\theta }_{\mathrm{in}}=59.{94}^{\circ }$ is close to the critical angle for the effective medium.

Figure 2

The criterion for the choice of parameters used to validate the effective medium approximation. (a) The contour of the criterion to discriminate whether EMA works while keeping $\overline{\varepsilon }=3$. The gray area is the region in which the EMA works, while the white area is the region where the EMA breaks down. The points marked in panel (a) give the parametric choices as A(2, 3.33, 2.67), B(5, 3.33, 2.67), C(25, 3.33, 2.67), D(5, 4.5, 1.5), and E(5, 3.1, 2.9), and the corresponding reflectivities calculated rigorously and with the conventional EMA as a function of $N$ are presented in panels (b)–(f). Here ${\theta }_{\mathrm{in}}=59.{94}^{\circ }$ and ${\varepsilon }_{\mathrm{in}}={\varepsilon }_{\mathrm{out}}=4$. It is clearly seen that the EMA works well for parametric points A, B, and E and breaks down for points C and D.

Figure 3

(a) and (b) The reflectivity discrepancy between rigorous calculation and the effective medium approximation is removed after introducing the fourth-order correction into the effective medium approximation. The reflectivity differences as a function of the number of layer pairs ($N$) and the incident angle are presented for (a) the conventional effective medium approximation and (b) the fourth-order effective medium approximation with the structure parameters ${\varepsilon }_a=5, {\varepsilon }_b=1, {\varepsilon }_{\mathrm{in}}={\varepsilon }_{\mathrm{out}}=4$, and $d_a=d_b=10$ nm. (c) Reflectivity as a function of $N$ for rigorous calculation (black line), the EMA with the fourth-order correction (red line), and the EMA with the nonlocal correction [18] (green line). We chose the following parameters: ${\varepsilon }_a=5, {\varepsilon }_b=1, {\varepsilon }_{\mathrm{in}}={\varepsilon }_{\mathrm{out}}=4, d_a=d_b=20$ nm with an incident angle of ${\theta }_{\mathrm{in}}=60.{33}^{\circ }$.

Figure 4

(a)–(c) Reflectivity of $N$-period multilayers calculated by the CMM (black line), the nonlocal EMA (red line), and the revised EMA with a matched layer (purple dashed line) as a function of $N(N\alpha$). The parameters were chosen as follows: (a) ${\varepsilon }_{\mathrm{out}}=3.01$ and ${\theta }_{\mathrm{in}}=59.{94}^{\circ }$, (b) ${\varepsilon }_{\mathrm{out}}=3$ and ${\theta }_{\mathrm{in}}=59.{94}^{\circ }$, and (c) ${\varepsilon }_{\mathrm{out}}=3.01$ and ${\theta }_{\mathrm{in}}={50}^{\circ }$. (d) Reflectivity of $N$-period multilayers calculated by the CMM (black line), the nonlocal EMA (red line), and the revised EMA with a matched layer (purple dashed line) as a function of ${\varepsilon }_{\mathrm{out}}-{\varepsilon }_{\mathrm{eff}}$ as $N=50$. Here ${\theta }_{\mathrm{in}}=59.{94}^{\circ }$ and the matched layer is chosen as ${\varepsilon }_l=1$ and the thickness $d_l$ is calculated according to Eq. (8). (e) The relationship between the thickness $d_l$ and the permittivity of ${\varepsilon }_l=1$ of the matched layer with ${\varepsilon }_{\mathrm{out}}=3.01$. The asterisk is the parameter we used in panels (a) and (c). Here ${\varepsilon }_{\mathrm{in}}=4, {\varepsilon }_a=5, {\varepsilon }_b=1$, and $d_a=d_b=10$ nm in all graphs.