Constitutive parameter retrieval for uniaxial metamaterials with spatial dispersion

We propose a constitutive parameter retrieval approach in which all electromagnetic parameters of the medium are obtained taking spatial dispersion into account. Moreover, the constraint on nonmagnetic metal/dielectric metamaterials is relaxed. This procedure is applied to metal/dielectric stacks in order to address the effects of the layer thickness, layer number, and material choice on the spatial dispersion. The results demonstrate that the investigated metal/dielectric stacks have a clear magnetic response, particularly for thicker layers. Moreover, this magnetic response is also a function of the magnitude of the $|k_x/k_0|$ ratio, where $k_x$ is the wave vector parallel to the interface planes and $k_0$ is the free-space wave number. We demonstrate that the real part of the dispersion curve flattens out (with a corresponding large imaginary part being present) as a result of the absence of propagating modes inside the metamaterial. This flat region is strongly dependent on the thickness of the layers and is a direct manifestation of spatial dispersion. Using this parameter retrieval method we calculate the Purcell factor for Rb atoms 10 nm above the surface of a ${\mathrm{Ag}/\mathrm{TiO}}_2$ stack with two filling factors $\rho$ $[\rho =l/(l+d)$ with $l$ and $d$ as the metal and dielectric layer thicknesses, respectively], $\rho =0.3$ and $\rho =0.5$, and having $N=13$ layers, for the emission wavelengths of 435 nm and 785 nm. Results are then compared with three different approaches, and we show that if spatial dispersion is not properly taken into account, then the Purcell factor is overestimated. Our approach shows excellent agreement with Purcell factors obtained from precise and accurate numerical calculations of the corresponding nonhomogenized structures.

Figure 1

(a) Metal/dielectric stack with incident wave impinging on the surface from the left. Part of the incident wave is reflected $\left(S_{11}\right)$ and part is transmitted $\left(S_{21}\right)$. The metal and dielectric thicknesses are $l$ and $d$, respectively. (b) Equivalent homogenous anisotropic medium.

Figure 2

Diagrams show field components incident on the interface between air and a homogenous, anisotropic medium for (a) TE and (b) TM polarizations.

Figure 3

Diagram shows the algorithm for extracting constitutive parameters.

Figure 4

Real (left column) and imaginary (right column) parts of the extracted electromagnetic parameters of a layered medium of ${\mathrm{Ag}/\mathrm{TiO}}_2$ with $\rho =0.1$ for dielectric thickness of 15 (squares), 30 (circles), and 45 nm (triangles) for [15] (dashed lines, hollow symbols) and the CPR approach (solid lines, full symbols). (a,b) ${\epsilon }_{\perp }$, (c,d) ${\epsilon }_{\left|\right|}$, (e,f) ${\mu }_{\perp }$, and (g,h) ${\mu }_{\left|\right|}$. The EMT (solid line, stars) results are also plotted for comparison. The inset in (c,d) shows the full magnitude of the real and imaginary parts of ${\epsilon }_{\left|\right|}$ for the 45-nm case of [15].

Figure 5

Real (left column) and imaginary (right column) parts of the extracted electromagnetic parameters of a layered medium of ${\mathrm{Ag}/\mathrm{TiO}}_2$ with $\rho =0.3$ for dielectric thickness of 15 (squares), 30 (circles), and 45 nm (triangles) for [15] (dashed lines, hollow symbols) and the CPR approach (solid lines, full symbols. (a,b) ${\epsilon }_{\perp }$, (c,d) ${\epsilon }_{\left|\right|}$, (e,f) ${\mu }_{\perp }$, and (g,h) ${\mu }_{\left|\right|}$. The EMT (solid line, stars) results are also plotted for comparison.

Figure 6

Real (left column) and imaginary (right column) parts of (a) ${\epsilon }_{\left|\right|}$ and (b) ${\epsilon }_{\perp }<0$ for 3 (squares), 15 (circles), and 27 (triangles) layers for [15] (dashed lines, hollow symbols) and the CPR approach (solid lines, full symbols). The EMT (solid line, stars) results are also plotted for comparison.

Figure 7

Real (left column) and imaginary (right column) parts of the dispersion curves of a ${\mathrm{Ag}/\mathrm{SiO}}_2$ stack with $\rho =0.3$ (a,b) and $\rho =0.5$ (c,d). The dielectric thicknesses (used as a parameter) is 8 nm (squares), 16 nm (circles), 24 nm (triangles), and 32 nm (diamonds) for [15] (dashed lines, hollow symbols) and the CPR approach (solid lines, full symbols). Results from the EMT (solid line, stars) are also shown for comparison.

Figure 8

Dispersion curves for ${\mathrm{Ag}/\mathrm{TiO}}_2$ [(a)–(d)] and Au/Si [(e)–(h)] stacks with (a,e) $\rho =0.2$, (b,f) $\rho =0.3$, (c,g) $\rho =0.4$, and (d,h) $\rho =0.5$. The dielectric thicknesses (used as a parameter) are 8 nm (squares), 16 nm (circles), 24 nm (triangles), and 32 nm (diamonds). Note that at $\rho =0.4$ and $\rho =0.5,{\mathrm{Ag}/\mathrm{TiO}}_2$ corresponds to a single-sheet hyperboloid while Au/Si corresponds to a two-sheet hyperboloid surface.

Figure 9

Purcell factor calculated using (i) the benchmark approach (squares), (ii) the EMT approach (stars), (iii) [15] (triangles), and (iv) the CPR approach (circles). The calculation is carried out for a Rb atom 10 nm above the surface of a ${\mathrm{Ag}/\mathrm{TiO}}_2$ stack with $\rho =0.3$ (hollow symbols) and 0.5 (full symbols) for $\lambda =785$ nm (a) and 435 nm (b).