Light propagation in a fishnet metamaterial

We derive the dispersion relation of Bloch modes in a fishnet metamaterial in the optical frequency domain. The dependence of the longitudinal wave vector component on the transverse one and the frequency, which governs diffraction and dispersive spreading of localized light excitations, respectively, are analyzed. We show that this type of metamaterial exhibits an involved anisotropic behavior with diffraction changing the sign in passing even a zero diffraction point. For completeness we derive furthermore a formal angle-dependent effective refractive index which exhibits discontinuities. Thus we conclude that an effective refractive index tends to get meaningless and that only the dispersion relation predicts reliably light propagation in metamaterials. The results are double checked with those obtained from a retrieval algorithm based on angular resolved reflection/transmission data of a finite slab. Excellent agreement is observed.

Figure 1

(Color online) Geometry of the fishnet structure (top). The (a) real and (b) imaginary parts of the propagation constant $k_z$ vs wave number are shown for the two lowest-order BMs and normal incidence. Zeroth order—blue solid line; first order—black dashed line. The results retrieved from $R$ and $T$ of a five-layer slab are shown by magenta circles.

Figure 2

(Color online) (a) Amplitude distribution of the $y$ component of the electric field of the zeroth-order BM and the first-order BM in a plane between two fishnets.

Figure 3

(Color online) (a) Dispersion relation and (b) the derived effective index vs the angle of incidence of the lowest-order BM ($k_y=0$, $\overline{v}=6957{\text{cm}}^{-1}$; real part—blue solid line, imaginary part—green dashed line). In (b) the effective index retrieved from $R$ and $T$ of a finite slab is also shown (real part—magenta circles, imaginary part—black diamonds).

Figure 4

(Color online) Longitudinal wave vector $k_z$ [real part—(a), imaginary part—(b)] vs wave number for the lowest-order BM (blue solid line—$k_x=0$, red dashed line—$k_x=1.75\mu {\text{m}}^{-1}$, green dashed-dotted line—$k_x=2.62\mu {\text{m}}^{-1}$). The derived effective index is shown in (c) and (d). $k_x=2.62\mu {\text{m}}^{-1}$. The figure shows also the longitudinal wave vector and the index of the plane wave retrieved from $R$ and $T$ of a finite slab (magenta circles).