Synthesis of transmission line models for metamaterial slabs at optical frequencies

We study the scattering parameters of typical transmission line resonant circuits and their corresponding nonlocal effective material parameters at optical frequencies using transmission line theory. We find that each circuit has specific features in terms of nonlocal permittivity and permeability and can be utilized to quantitatively synthesize transmission line circuit models for metamaterial slabs. As application examples, we construct circuit models for cut wires, cut-wire pairs, metallic meshes, and mesh pairs. Moreover, we calculate local material parameters in a simple way using fitted impedance parameters together with a transmission line homogenization procedure. Using transmission line (TL) models, we show how the material parameters can be optimized in actual metamaterials. The physical dependence of circuit parameters on structure geometries validates our TL model further.

Figure 1

(Color online) Circuit A is a dimensionless shunt admittance composed of a series combination of $L$ and $C$ loading a TL section of length $l$ at its center. (a), (b), (c), and (d) show the calculated frequency-dependent reflectance (solid line)/transmittance (dashed line), phase $S_{11}$ (solid line)/$S_{21}$ (dashed line), retrieved permittivity, and permeability, respectively. The short-dashed line in (c) is calculated using Eq. (2), a TL homogenization procedure.

Figure 2

(Color online) Influence of the TL-section length $l$ on the extracted effective ${\mu }_{rN}$ (a) and ${\epsilon }_{rN}$ (b). A shorter TL section reduces the nonphysical region (induced by the Bragg mode) of ${\mu }_{rN}^{\prime }$, which resembles ${\mu }_{rL}^{\prime }$ more, a unit constant.

Figure 3

(Color online) Circuit B is a dimensionless shunt admittance composed of a parallel combination of a capacitance and an inductance loading a TL section of length $l$ at its center. (a), (b), (c), and (d) show the calculated frequency-dependent reflectance (solid line)/transmittance (dashed line), scattering-parameter phase, retrieved permittivity, and permeability (dashed lines for the imaginary parts), respectively.

Figure 4

(Color online) Circuit C is a dimensionless series impedance composed of a parallel combination of a capacitance and an inductance loading a TL section of length $l$ at its center. (a), (b), (c), and (d) shows the calculated frequency-dependent reflectance (solid line)/transmittance (dashed line), scattering-parameter phase, retrieved permittivity, and permeability (dashed lines for the imaginary parts), respectively.

Figure 5

(Color online) Frequency-dependent ${\mu }_{rN}^{\prime }$ (a) and ${\mu }_{rL}^{\prime }$ (b) for circuit $C$ with varying values of $L$ and $C$.

Figure 6

(Color online) (a) Two cut-wire unit cells demonstrate the interactions at the defined polarization. (b) Overall structure of the 2D periodic Ag cut-wire array (purple blocks) in vacuum (gray block). (c) Circuit model for both the unit cell and the overall structure, which is identical to circuit A shown in Fig. 1. The structure parameters are $L=500\text{nm}$, $W=150\text{nm}$, $d=20\text{nm}$, and $P=600\text{nm}$. The $E$ field is along the length of the wires.

Figure 7

(Color online) (a) Frequency-dependent reflectance/transmittance and (b) the retrieved permeability from numerical simulation (solid curves) and TL calculation using model I (dashed curves) for the structures shown in Fig. 6. The circuit parameters are tabulated in Table .

Figure 8

(Color online) (a) Two cut-wire-pair unit cells to show the interaction at the defined polarization. (b) Overall structure of the 2D periodic Ag cut-wire-pair array in vacuum. (c) Circuit model for both the unit cell and the overall structure, which is a combination of circuits A and B. The geometry of the cut wires is the same as that in Fig. 6. The spacer between the cut-wire pairs is 60-nm-thick vacuum (dark blue). The $E$ field is along the length of the wires.

Figure 9

(Color online) Frequency-dependent $S$ parameters and retrieved ${\mu }_{rN}^{\prime }$ from numerical simulation (solid curves) and TL calculation (dashed curves) for the cut-wire pairs shown in Fig. 8 with $d_s=60\text{nm}$.

Figure 10

(Color online) Frequency-dependent ${\mu }_{rL}^{\prime }$ of cut-wire pairs shown in Fig. 8 with varied spacer thickness $d_s$. (b) Series capacitance $C_2$ versus spacer thickness of cut-wire pairs (scattered data), which can be well understood by using the quasistatic model for a parallel capacitor (solid line).

Figure 11

(Color online) (a) Two unit cells of the mesh to show the interaction at the defined polarization. (b) Overall structure of the 2D periodic Ag grid (gray) in vacuum. (c) Circuit model for both the unit cell and the overall structure, which is a combination of circuits B and A. The $E$ field is along the $x$ direction. The geometry parameters are $W=150\text{nm}$, $L=500\text{nm}$, $d=20\text{nm}$, and $P=600\text{nm}$.

Figure 12

(Color online) Reflectance/transmittance, $S$-parameter phase, and recovered material parameters from numerical simulations (solid curves) of the mesh structure and TL calculations using circuit B alone (dotted curves) and using model III shown in Fig. 11 (dashed curves). The vertical dashed guide lines indicate the phase-jump frequencies.

Figure 13

(Color online) (a) Two unit cells of mesh pairs and the interaction between them at the defined polarization. (b) Overall structure of the 2D periodic metallic mesh-pair array in vacuum. (c) Circuit model for both the unit cell and the overall structure. The $E$ field is along the $x$ direction.

Figure 14

(Color online) $S$ parameters and the recovered material parameters from numerical simulation (solid curves) of the mesh-pair structure and TL calculation (dashed curves) using model IV shown in Fig. 13.

Figure 15

(Color online) Local material parameters of cut-wire pairs and mesh pairs calculated using the TL models II and IV and the fitted circuit element parameters listed in Table . (a) Real parts of the local permittivity and (b) real parts of the local permeability.