Optical properties of strongly anisotropic metamaterials

Abstract

We study the behavior of wave propagation in strongly anisotropic metamaterials with different principal dielectric constants. Analytical expressions of the dispersion relation, Poynting vector, and group velocity are derived theoretically. The light propagation properties for TM and TE waves in these metamaterials are explored, in which the backward wave, negative refraction, phase-compensation effects, and far-field propagation wave carrying subwavelength information are discussed. These theoretical results are supported by numerical simulation.

Appendix

1.1 A.1 Elliptic dispersion relation

As shown in Fig. 2, a TM wave is incident into a strongly anisotropic medium II from a medium I (such as air). The boundary conditions at the interface require k _x >0, s _z >0 and there are two possible refraction directions (k _z >0 and k _z <0). According to the relationship of s _z ∝k _z /ε _x (see Eq. (9)), ε _x >0 results in upward refraction, k _z >0, and ε _x <0 results in downward refraction, k _z <0, which show that the sign of ε _x is determined by the sign of k _z .

Then how to determine the sign of ε _z ? The Poynting vector $\vec {s}$ can be described as the normal direction of the dispersion curve with the aid of the group velocity $\vec {v}_{\mathrm{g}}$ in Eq. (14). When k _z >0, the incident electromagnetic wave corresponds to upward refraction, and s _z >0 indicates that $\vec {s}$ is pointing in the outer normal direction (see the red arrows in Fig. 2). This corresponds to s _x ∝k _x /ε _z >0 from Eq. (9), and k _x >0 requires ε _z >0. When k _z <0, the incident electromagnetic wave corresponds to downward refraction; then s _z >0 requires $\vec {s}$ to be pointing in the inner normal direction; this corresponds to s _x ∝k _x /ε _z <0, and k _x >0 requires ε _z <0. So, the sign of ε _z is determined by the sign value of s _x .

Finally, the sign of μ _y is deduced from Eq. (12) in terms of the angle between the vectors $\vec {s}$ and $\vec {k}$. Figure 2 shows that when k _z >0, the angle between $\vec {s}$ and $\vec {k}$ is less than π/2. The value of the expression (12) is bigger than zero, and $\mu_{y}{\kern 1pt} > 0$. On the contrary, when k _z <0, μ _y <0 since the angle between $\vec {s}$ and $\vec {k}$ is bigger than π/2.

All of the analysis also shows that when k _z >0, the incident wave is forward refracting into the strongly anisotropic medium. All PDCs take positive values as ε _x >0, ε _z >0, μ _y >0. And, the dispersion relation of the TM wave in medium II is an elliptic equation: $\frac{k_{x}^{2}}{\varepsilon _{x}} + \frac{k_{z}^{2}}{\varepsilon _{z}} = \mu_{y}\frac{\omega ^{2}}{c^{2}}$. When k _z <0, all PDCs take negative values, ε _x <0, ε _z <0, μ _y <0. This kind of metamaterial is an anisotropic left-handed system medium. The corresponding dispersion relation is also elliptical: $- \frac{k_{x}^{2}}{|\varepsilon _{z}|} - \frac{k_{z}^{2}}{|\varepsilon _{x}|} = - |\mu_{y}|\frac{w^{2}}{c^{2}}$.

We transform the elliptic dispersion equation into standard form as

$$ \frac{k_{x}^{2}}{a^{2}} + \frac{k_{z}^{2}}{b^{2}} = 1, $$

(22)

where $a^{2} = |\varepsilon_{z}\mu_{y}|k_{0}^{2}$, $b^{2} = |\varepsilon_{x}\mu_{y}|k_{0}^{2}$, and a and b are the semiaxes of the ellipse. When a TM wave propagates in the strongly anisotropic medium, it requires

$$ k_{z}^{2} = \frac{b^{2}}{a^{2}}\bigl(a^{2} - k_{x}^{2}\bigr) > 0. $$

(23)

$k_{x}^{2} < a^{2}$ shows that a TM wave propagating in a strongly anisotropic medium with elliptic dispersion relationship is low-pass filter, and there is a cutoff wave vector k _c=a, which separates the propagating wave from the evanescent wave. When $k_{x}^{2} > a^{2}$, the TM wave is an evanescent wave propagating along the interface.

1.2 A.2 Hyperbolic dispersion relation

Figure 3 shows the hyperbolic dispersion relationship in Eq. (11). When a TM wave refracts into medium II from medium I, the boundary condition s _z >0 (s _z ∝k _z /ε _x ) implies that ε _x >0 when k _z >0, and ε _x <0 when k _z <0.

s _z >0 also requires $\vec {s}$ pointing in the outer normal direction of the hyperbolic curve (here $\vec {s}$ is antiparallel to $\vec {v}_{\mathrm{g}}$). It corresponds to s _x (∝k _x /ε _z )<0, and ε _z <0 due to k _x >0.

On the contrary, when k _z <0, $\vec {s}$ points in the inner normal direction of the hyperbolic curve; it corresponds to s _x (∝k _x /ε _z )>0, and ε _z >0 due to k _x >0. The sign value of μ _y is still decided by Eq. (12).

On the whole, when k _z >0, and PDCs satisfy ε _x >0, ε _z <0, μ _y <0, the hyperbolic dispersion equation is $- \frac{k_{x}^{2}}{|\varepsilon _{z}|} + \frac{k_{z}^{2}}{\varepsilon _{x}} = - |\mu_{y}|\frac{w^{2}}{c^{2}}$; when k _z <0 and ε _x <0, ε _z >0, μ _y >0, the corresponding same hyperbolic dispersion equation is $\frac{k_{x}^{2}}{\varepsilon _{z}} - \frac{k_{z}^{2}}{|\varepsilon _{x}|} = \mu_{y}\frac{w^{2}}{c^{2}}$.

1.3 A.3 Conjugate hyperbolic dispersion relation

Figure 4 shows the conjugate hyperbolic dispersion relationship in Eq. (11). When a TM wave refracts into medium II from medium I, when k _z >0, the boundary condition s _z >0 requires ε _x >0 due to s _z ∝k _z /ε _x , and the Poynting vector $\vec {s} (//\vec {v}_{\mathrm{g}} )$ points in the inner normal direction of the conjugate hyperbolic curve; this corresponds to $s_{x}(\propto\frac{k_{x}}{\varepsilon _{z}})< 0$, and ε _z <0 due to k _x >0. Furthermore, when k _z <0, $s_{z}(\propto\frac{k_{z}}{\varepsilon _{x}}) > 0$ implies that ε _x <0. $\vec {s}$ points in the outer normal direction of the conjugate hyperbolic curve ($\vec {s}$ antiparallel to $\vec {v}_{g}$), it corresponds to $s_{x}(\propto\frac{k_{x}}{\varepsilon _{z}}) > 0$, and ε _z >0 due to k _x >0. The sign value of μ _y is also determined by Eq. (12).

In summary, when PDCs of the medium are satisfied with ε _x >0, ε _z <0, μ _y >0, the corresponding conjugate hyperbolic dispersion equation is $\frac{k_{z}^{2}}{\varepsilon _{x}} - \frac{k_{x}^{2}}{|\varepsilon _{z}|} = \mu_{y}\frac{w^{2}}{c^{2}}$. When k _z <0, PDCs are satisfied with ε _x <0,ε _z >0, μ _y <0; the corresponding same conjugate hyperbolic dispersion equation is $\frac{k_{x}^{2}}{\varepsilon _{z}} - \frac{k_{z}^{2}}{|\varepsilon _{x}|} = - |\mu_{y}|\frac{w^{2}}{c^{2}}$.

We transform the hyperbolic and conjugate hyperbolic dispersion equations into standard form as

$$ \pm\frac{k_{x}^{2}}{a^{2}} \mp\frac{k_{z}^{2}}{b^{2}} = 1. $$

(24)

In Eq. (24), the upper signs correspond to hyperbolic dispersion while the lower signs correspond to conjugate hyperbolic dispersion.

For the hyperbolic dispersion relationship, the TM propagation wave satisfies the following relation:

$$ k_{z}^{2} = \frac{b^{2}}{a^{2}}\bigl(k_{x}^{2} - a^{2}\bigr) > 0. $$

(25)

When $k_{x}^{2} > a^{2}$, a TM wave propagating in a strongly anisotropic medium with hyperbolic dispersion relationship is high-pass filter, and k _x can be big enough to form a far-field propagation wave carrying subwavelength information. If $k_{x}^{2} < a^{2}$, it corresponds to an evanescent wave and only propagates along the surface. k _c=a is the cutoff wave vector.

1.4 A.4 Surface wave

In eight combinations of three PDC signs, two other possible combinations ε _x >0, ε _z >0, μ _y <0 and ε _x <0, ε _z <0, μ _y >0 do not satisfy the dispersion equation (11) with $k_{z}^{2} > 0$, except that when $k_{z}^{2} < 0$ and k _z is a pure imaginary number (k _z =−iq), it corresponds to the TM evanescent wave propagating along the interface.

1.5 A.5 TE wave

The discussion of the TE wave is similar to the above discussion of the TM wave. The corresponding signs of μ _x ,μ _z , and ε _y can be obtained by simply interchanging ε and μ.