Effective-medium theory for multilayer metamaterials: Role of near-field corrections

Although many effective-medium theories have been proposed for studying metamaterials, most of them do not work well for multilayer metamaterials with small interlayer distances. Based on rigorous mode-expansion analyses on a model system consisting of multiple layers of subwavelength gratings, we identify that the failures of conventional effective-medium theories are caused by neglecting strong near-field couplings in homogenizing such systems. These understandings motive us to propose an alternative homogenization approach for strongly coupled multilayer metamaterials, in which predominant near-field corrections have been considered automatically. Our effective-medium theory can well describe multilayer metamaterials with arbitrary interlayer distances including particularly those systems for which conventional effective-medium theories fail. We finally extend our theory to multilayer metamaterials with complicated microstructures and validate the theory by full-wave simulations as well as microwave experiments. Our theory not only well complements the available effective-medium theory formalisms, but also provides a powerful tool to study the properties of strongly coupled metamaterials, which may find many applications in practice.

Figure 1

(a) Schematics of the conventional effective-medium theory to determine the effective parameters of a multilayer metamaterial. Transmission spectra obtained by finite-element-method simulations (open circles) and the conventional effective-medium theory (red solid lines) for a series of N-layer metamaterials with (b) N changing from 1 to 7 with $h_d=3\mathrm{mm}$ fixed and for a series of N-layer metamaterials with (c) $h_d$ changing from 9 to 3 mm with $N=5$ fixed. A metallic layer contains a periodic array (with lattice constant 15 mm along two directions) of metallic crosses, each formed by two 0.018-mm-thick bars with width 5 mm and length 10 mm.

Figure 2

(a) Schematics of the three-layer model metamaterial under study with geometric parameters given by $a=0.2\mathrm{mm}$, $P=3\mathrm{mm}$, $h_m=0.018\mathrm{mm}$, $h_d=0.4\mathrm{mm}$. The permittivity of the spacer is ${\varepsilon }_d=4.3$. (b) Spectra of transmission amplitude calculated by finite-element-method simulations (blue open circles) and rigorous mode-expansion method (red solid line). (c) Calculated ${\sigma }_{\mathrm{MSE}}$ vs $N_{\mathrm{spacer}}$ (blue triangles), $N_{\mathrm{air}}$ (green squares), and $N_{\mathrm{slit}}$ (red circles), respectively. (d) Spectra of transmission amplitude calculated by finite-element-method simulations (blue open circles) and mode-expansion method with $N_{\mathrm{spacer}}=1\mathrm{and}{N}_{\mathrm{air}}=N_{\mathrm{slit}}=201$ (green line) and $N_{\mathrm{air}}=N_{\mathrm{slit}}=1\mathrm{and}{N}_{\mathrm{spacer}}=201$ (red line).

Figure 3

(a) $|H_y|$ field distribution on an arbitrary $x-z$ plane inside the metamaterial, calculated by finite-element-method simulations at 23.02 GHz. (b) Computed $|H_y|$ field distributions along the central line in the spacer region (blue solid line) and along the central line across the metallic layer (red dashed line). (c) The ratios between energy flows carried by all high-order modes and that carried by the fundamental mode in the spacer (blue line) and the slit region (red line), calculated by the mode-expansion method for different values of $h_d$. (d) Spectra of transmission amplitude calculated by finite-element-method simulations (blue open circles) and the conventional effective-medium theory (red solid line).

Figure 4

(a)–(d) Schematics of retrieving the effective parameters of a multilayer metamaterial in the near-field-corrected effective-medium theory. (e),(g) Effective parameters retrieved by (e) the near-field-corrected effective-medium theory and (g) the conventional effective-medium theory. (f),(h) Transmission spectra computed by finite-element-method simulations (open circles), (f) the near-field-corrected effective-medium theory (red curves), and (h) the conventional effective-medium theory (red curves).

Figure 5

(a),(b) ${\sigma }_{\mathrm{MSE}}$ calculated with the conventional effective-medium theory (red squares) and the near-field-corrected effective-medium theory (blue circles) for a series of N-layer metamaterials with (a) N changing from 3 to 15 with $h_d=0.4\mathrm{mm}$ fixed and another series of nine-layer metamaterials with (b) $h_d$ changing from 0.02 to 2 mm. (c),(d) ${\sigma }_{\mathrm{MSE}}$ as functions of $h_d$ and ${\lambda }_d$ for the nine-layer metamaterials computed by (c) the near-field-corrected effective-medium theory and (d) the conventional effective-medium theory. Here, the white solid line denotes the wavelengths of the magnetic resonance for different $h_d$.

Figure 6

(a),(b) Schematics of retrieving the effective parameters of an arbitrary multilayer metamaterial in the near-field-corrected effective-medium theory. Here, ${\varepsilon }_d=3.5,{\mu }_d=1$, and the metallic layer contains a periodic array (with lattice constant 6.5 mm) of metallic cross with geometric parameters: bar length 6 mm, bar width 3.2 mm, thickness 0.018 mm. The spacer thickness is 1.6 mm. (c) Finite-element-method-calculated E-field distributions (with color/arrow representing the amplitude/direction of the field and the yellow dashed line representing the profile of the microstructure) of the fundamental modes inside metallic regions. (d),(e) Frequency-dependent effective parameters calculated by (d) the near-field-corrected effective-medium theory and (e) the conventional effective-medium theory, respectively. (f),(h),(j) Transmission spectra computed by finite-element-method simulations (open circles) and the near-field-corrected effective-medium theory (solid lines) for metamaterials with $N=5,7,9$. (g),(i),(k) Transmission spectra computed by finite-element-method simulations (open circles) and the conventional effective-medium theory (solid lines) for metamaterials with $N=5,7,9$.

Figure 7

(a) Top-view and (b) side-view pictures of the fabricated five-layer sample. Here, a metallic layer contains an array (with periodicity 6.5 mm) of metallic crosses with geometric parameters: bar length 6 mm, bar width 3.2 mm, and thickness 0.018 mm. The spacer thickness is 1.6 mm. (c) Spectra of transmission amplitude of the five-layer metamaterial, obtained by finite-element-method simulations (blue solid line), the near-field-corrected effective-medium theory (red dashed line), and experimental measurements (open circles). (d) Spectra of transmission amplitude of the five-layer metamaterial, obtained by finite-element-method simulations (blue solid line), the conventional effective-medium theory (red dashed line), and experimental measurements (open circles).

Figure 8

${\sigma }_{\mathrm{MSE}}$ and $\chi$ calculated with the near-field-corrected effective-medium theory (blue circles) and the effective-medium theory (red squares) on multilayer metamaterials consisting of cross-shaped metallic microstructures with varying spacer thickness $h_d$. Insets schematically depict the shapes of metallic resonators, with bar width being (a) 5.0 mm, (b) 3.2 mm, (c) 1.6 mm, respectively.

Figure 9

Effective-medium parameters calculated by NFC-EMT for subsystems with (a) one layer, (b) two layers, and (c) four layers with different branches. Effective-medium parameters calculated by (d) NFC-EMT and (e) conventional EMT for structures with different layers with proper branches selected. (f) ${\sigma }_{n_{\mathrm{eff}}}$ calculated with NFC-EMT (blue circles) and conventional EMT (red squares) for a nine-layer structure vs number of layers (M) taken into account for homogenization.