2.4 Zero-Index Metamaterials
31
(a) Unit cell of fishnet metamaterials
(b) Effective refractive index
Fig. 2.3 Fishnet metamaterial
How to determine effective material parameters?
A metamaterial is mostly a periodic array of metallic or dielectric or composite
resonators. A periodic structure exhibits periodic distribution of the fields.
Hence, to simulate the operation of the metamaterial, one need not to create
and computed a large array. Rather, a unit cell with appropriate boundary
conditions represents the whole array, and serves the purpose of computation of
the fields, s-parameters, current density, etc. The s-parameters used in Eqs. 2.6–
2.7 have been computed using COMSOL Multiphysics® [47]. The technique
of computation has been described below in detail.
Figure 2.4 shows the computational cell containing the unit cell of the fishnet
metamaterial used for the computation of s-parameters. The dimensions and
the materials of the unit cell are described in Fig. 2.3a, and the surrounding
medium is air. The values such as permittivity () for metallic and dielectric
domains were adapted from Palik (1999) [105]. The top surface of the cell was
made the input port (port 1), while the bottom surface was made the output
port (port 2), and d is the distance between the two ports. The incident wave
was excited at the port 1 and received at port 2. All the four lateral boundaries
were made periodic boundaries by imposing the periodic boundary condition
to induce a periodic array like behavior. The orientation of the electric (E)
and magnetic (H ) fields and the direction of propagation (k) were as shown
in the figure. The computation was run in frequency domain for the desired
range of frequencies, using the parametric sweep feature. As the simulation
runs, several global parameters gets evaluated, including the s-parameters.
Since two ports were involved, four types of s-parameters were obtained, viz.,
S 11 , S 12 , S 21 , and S 22 . Out of these, S 11 and S 21 are the most important, since
they yield the reflection and the transmission coefficient of the metamaterial,
respectively. Furthermore, they are also used to evaluate the effective material
parameters (n, z, , and μ) of the metamaterial, using Eqs. 2.6–2.9.
31
(a) Unit cell of fishnet metamaterials
(b) Effective refractive index
Fig. 2.3 Fishnet metamaterial
How to determine effective material parameters?
A metamaterial is mostly a periodic array of metallic or dielectric or composite
resonators. A periodic structure exhibits periodic distribution of the fields.
Hence, to simulate the operation of the metamaterial, one need not to create
and computed a large array. Rather, a unit cell with appropriate boundary
conditions represents the whole array, and serves the purpose of computation of
the fields, s-parameters, current density, etc. The s-parameters used in Eqs. 2.6–
2.7 have been computed using COMSOL Multiphysics® [47]. The technique
of computation has been described below in detail.
Figure 2.4 shows the computational cell containing the unit cell of the fishnet
metamaterial used for the computation of s-parameters. The dimensions and
the materials of the unit cell are described in Fig. 2.3a, and the surrounding
medium is air. The values such as permittivity () for metallic and dielectric
domains were adapted from Palik (1999) [105]. The top surface of the cell was
made the input port (port 1), while the bottom surface was made the output
port (port 2), and d is the distance between the two ports. The incident wave
was excited at the port 1 and received at port 2. All the four lateral boundaries
were made periodic boundaries by imposing the periodic boundary condition
to induce a periodic array like behavior. The orientation of the electric (E)
and magnetic (H ) fields and the direction of propagation (k) were as shown
in the figure. The computation was run in frequency domain for the desired
range of frequencies, using the parametric sweep feature. As the simulation
runs, several global parameters gets evaluated, including the s-parameters.
Since two ports were involved, four types of s-parameters were obtained, viz.,
S 11 , S 12 , S 21 , and S 22 . Out of these, S 11 and S 21 are the most important, since
they yield the reflection and the transmission coefficient of the metamaterial,
respectively. Furthermore, they are also used to evaluate the effective material
parameters (n, z, , and μ) of the metamaterial, using Eqs. 2.6–2.9.