Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit

We present an effective medium theory that simultaneously determines the effective permittivity and effective permeability of a magnetodielectric composite beyond the long-wavelength limit. In the long-wavelength limit, our theory reproduces previous results obtained in two and three dimensions. To verify our theory, we performed band-structure calculations on a periodic composite in two dimensions for a specific configuration with complicated Mie resonances in the microstructure. Our results show that our theory can be valid even when the wavelength in the background medium, ${\lambda }_0$, is as small as 1.3 times the lattice constant, $a$, whereas the long-wavelength theory breaks down when ${\lambda }_0<3.5a$.

Figure 1

Microstructure of the composite.

Figure 2

Permittivity ${\epsilon }_s$ and permeability ${\mu }_s$ of the scatterer.

Figure 3

Band structure along the $\Gamma \text{−}X$ direction calculated by the multiple-scattering method.

Figure 4

(a) The circles denote the band structures near the $\Gamma$ point in the frequency range $0⩽\tilde{f}⩽0.3$. The solid curves represent the effective medium results of Eq. (7). (b) and (c) The effective permittivity and effective permeability obtained from Eq. (7), respectively.

Figure 5

Real part (solid curve) and imaginary part (dashed curve) of the effective refractive index, $n_{\mathrm{eff}}$, obtained from Eq. (7). Squares denote the states at the $\Gamma$ point obtained from the band-structure calculations. The solid squares correspond to the $m=0$ modes and the open squares correspond to the $m=1$ modes.

Figure 6

(a) The circles denote the band structures near the $\Gamma$ point in the frequency range $0.0559⩽\tilde{f}⩽0.0565$. The solid curves represent the effective medium results of Eq. (7). (b) and (c) The effective permittivity and effective permeability obtained from Eq. (7), respectively.

Figure 7

Real part (solid curve) and imaginary part (dashed curve) of the effective refractive index, $n_{\mathrm{eff}}$, obtained from Eq. (7). Squares denote the states at the $\Gamma$ point obtained from the band-structure calculations. The solid squares correspond to the $m=0$ modes and the open squares correspond to the $m=1$ modes.

Figure 8

(a) The circles denote the band structures near the $\Gamma$ point in the frequency range $0.6⩽\tilde{f}⩽0.75$. The solid curves represent the effective medium results of Eq. (7). (b) and (c) The effective permittivity and effective permeability obtained from Eq. (7), respectively. (d) and (e) show the same variables as (a)–(c) but all solid curves are obtained from Eq. (8).

Figure 9

Real part (solid curve) and imaginary part (dashed curve) of the effective refractive index, $n_{\mathrm{eff}}$, obtained, respectively, from (a) Eq. (7) and (b) Eq. (8). Squares denote the states at the $\Gamma$ point obtained from the band-structure calculations. The solid squares correspond to the $m=0$ modes and the open squares correspond to $m=1$ modes, respectively.

Figure 10

(a) The circles denote the band structures near the $\Gamma$ point in the frequency range $0.7632⩽\tilde{f}⩽0.7645$. The solid curves represent the effective medium results of Eq. (7). (b) and (c). The effective permittivity and effective permeability obtained from Eq. (7), respectively. (d) and (e) show the same variables as (a)–(c) but all solid curves are obtained from Eq. (8).

Figure 11

Electric-field intensity distribution at different frequencies near $\tilde{f}=0.1536$.