58
3 Applications of Zero-Index Metamaterials
side. In other words, the incoming radiation is capable of getting tunneled through
a narrow channel of the subwavelength thickness, if the channel is filled with an
epsilon-near-zero material, irrespective of how thin or irregularly shaped the channel
may be (see Fig. 3.1) [91, 131].
Silveirinha et al. have presented a rigorous analysis of this phenomenon and have
formulated a simple yet accurate relation for reflection coefficient of an electromagnetic wave impinging on an epsilon-near-zero-material-filled narrow channel, which
is given by
ρ =
(a 1 − a 2 ) + ik 0 μ r, p A p
(a 1 + a 2 ) − ik 0 μ r, p A p
(3.1)
where a 1 and a 2 are the widths of the waveguide on the two sides of the ENZfilled distorted channel (as shown in Fig. 3.1a), k 0 is the free-space wave vector
of the impinging wave, μ r, p is the permeability of the narrow channel, and A p =
wt is its area of cross section. It is understood that to maximize the tunneling (or
transmission), the reflection coefficient ρ should be minimized. In Eq. 3.1, it has
been assumed that the imaginary parts are very small compared to the real parts,
i.e., (k 0 μ r, p A p )/(a 1 + a 2 ) << 1, then the reflection coefficient gets reduced to its
minimum value given by
ρ =
a 1 − a 2
a 1 + a 2
(3.2)
or
|ρ| =
|a 1 − a 2 |
a 1 + a 2
(3.3)
According to Eq. 3.3, the reflection coefficient can be suppressed to almost zero by
making a 1 ≈ a 2 , thereby achieving 100% transmission. In the numerical analysis
presented here, a 1 = a 2 = 2λ has been assumed. It is important to note that such
easy control over ρ in terms of geometrical parameters a 1 & a 2 could be possible
because of the condition (k 0 μ r, p A p )/(a 1 + a 2 ) << 1. The condition can be satisfied
in two ways—(1) if the permeability of the ENZ region is close to zero (μ r, p ≈ 0)
and (2) if the area of cross section ( A p ) of the ENZ channel is very small, which is
already true since the channel is narrow. Now, to examine qualitatively how small the
area A p should be, we deduce that for a non-magnetic medium, the above condition
gets reduced to
k 0 A p
2a
>> 1
( 3 . 4 )
=⇒
A p
2a
>> 1 =⇒ A p <<
λ 0 a
π
(3.5)
or simply
A p << λ 0 a
(3.6)
3 Applications of Zero-Index Metamaterials
side. In other words, the incoming radiation is capable of getting tunneled through
a narrow channel of the subwavelength thickness, if the channel is filled with an
epsilon-near-zero material, irrespective of how thin or irregularly shaped the channel
may be (see Fig. 3.1) [91, 131].
Silveirinha et al. have presented a rigorous analysis of this phenomenon and have
formulated a simple yet accurate relation for reflection coefficient of an electromagnetic wave impinging on an epsilon-near-zero-material-filled narrow channel, which
is given by
ρ =
(a 1 − a 2 ) + ik 0 μ r, p A p
(a 1 + a 2 ) − ik 0 μ r, p A p
(3.1)
where a 1 and a 2 are the widths of the waveguide on the two sides of the ENZfilled distorted channel (as shown in Fig. 3.1a), k 0 is the free-space wave vector
of the impinging wave, μ r, p is the permeability of the narrow channel, and A p =
wt is its area of cross section. It is understood that to maximize the tunneling (or
transmission), the reflection coefficient ρ should be minimized. In Eq. 3.1, it has
been assumed that the imaginary parts are very small compared to the real parts,
i.e., (k 0 μ r, p A p )/(a 1 + a 2 ) << 1, then the reflection coefficient gets reduced to its
minimum value given by
ρ =
a 1 − a 2
a 1 + a 2
(3.2)
or
|ρ| =
|a 1 − a 2 |
a 1 + a 2
(3.3)
According to Eq. 3.3, the reflection coefficient can be suppressed to almost zero by
making a 1 ≈ a 2 , thereby achieving 100% transmission. In the numerical analysis
presented here, a 1 = a 2 = 2λ has been assumed. It is important to note that such
easy control over ρ in terms of geometrical parameters a 1 & a 2 could be possible
because of the condition (k 0 μ r, p A p )/(a 1 + a 2 ) << 1. The condition can be satisfied
in two ways—(1) if the permeability of the ENZ region is close to zero (μ r, p ≈ 0)
and (2) if the area of cross section ( A p ) of the ENZ channel is very small, which is
already true since the channel is narrow. Now, to examine qualitatively how small the
area A p should be, we deduce that for a non-magnetic medium, the above condition
gets reduced to
k 0 A p
2a
>> 1
( 3 . 4 )
=⇒
A p
2a
>> 1 =⇒ A p <<
λ 0 a
π
(3.5)
or simply
A p << λ 0 a
(3.6)