Subwavelength Multilayer Dielectrics: Ultrasensitive Transmission and Breakdown of Effective-Medium Theory

We show that a purely dielectric structure made of alternating layers of deep subwavelength thicknesses exhibits novel transmission effects which completely contradict conventional effective medium theories exactly in the regime in which those theories are commonly used. We study waves incident at the vicinity of the effective medium’s critical angle for total internal reflection and show that the transmission through the multilayer structure depends strongly on nanoscale variations even at layer thicknesses smaller than $\lambda /50$. In such deep subwavelength structures, we demonstrate dramatic changes in the transmission for variations in properties such as periodicity, order of the layers, and their parity. In addition to its conceptual importance, such sensitivity has important potential applications in sensing and switching.

Figure 1

Schematics of a multilayer structure ($N=3$ periods), surrounded by homogeneous materials with permittivities ${ϵ}_{\text{in}}$ and ${ϵ}_{\text{out}}$. A plane wave is incident at angle $\theta$, at the proximity of the critical angle for total internal reflection.

Figure 2

Transmission as a function of structure width ($=Nd$) for several structures. (a) For layer thicknesses $d=9$, 10, 11 nm, ${ϵ}_{\text{in}}={ϵ}_{\text{out}}=4$. (b) For $d=10\mathrm{nm}$ and ${ϵ}_{\text{in}}=4$, ${ϵ}_{\text{out}}={ϵ}_{\text{eff}}$ with a regular layer order (${ϵ}_a$ followed by ${ϵ}_b$) or reversed (${ϵ}_b$ followed by ${ϵ}_a$). The prediction of the effective medium (EM) theory is marked by the yellow horizontal lines. All plots in (a) and (b), apart from the yellow line, are calculated via transfer-matrix formalism.

Figure 3

(a) Amplitude of the reflection and transmission coefficients from the output boundary of the multilayer stack versus ${ϵ}_{\text{out}}-\overline{\mathit{ϵ}}$ deduced from the transfer-matrix calculation (dashed line) compared against the values found from naive effective medium theory (solid line). The results from both methods coincide for ${ϵ}_{\text{out}}-\overline{\mathit{ϵ}}>0.1$, but clearly when the permittivity difference is small, effective medium yields erroneous results. (b) Transmission as a function of the width of the last layer in the structure, for two values of ${ϵ}_{\text{out}}$ (red and black lines). Near impedance matching, the transmission is extremely sensitive to nanometric changes in the width of the last layer. Here, $N=34$ and the layers are in reversed order (${ϵ}_b$ before ${ϵ}_a$).

Figure 4

(a) Amplitude of $E_{+},E_{-}$, the forward and backward propagating components, and $E_{\mathrm{TM}}=E_{+}+E_{-}$, the total field, as found through the transfer-matrix calculation exactly on the transmission resonance ($N=518$) and the field ${\mathrm{E}}_{\mathrm{EM}}$ found from the effective model. The total field found in both methods agrees closely, but the components do not. (b) A magnified portion of (a), showing the field amplitude inside the layers. (c) The sine of the relative phase between $E_{+}$ and $E_{-}$. The peaks of (c) correspond to the minima in (a).