3.3 Electromagnetic Cloaking
65
ε r = μ r =
r − a
r
(3.7)
ε φ = μ φ =
r
r − a
(3.8)
ε z = μ z =
b
b − a
2 r − a
r
(3.9)
The transformation equations for Cartesian coordinate system are given as [141]
ε xx = ε r cos
2
φ + ε φ sin
2
φ
(3.10)
ε xy = ε yx = (ε r − ε φ )sinφ cosφ
(3.11)
ε yy = ε r sin
2
φ + ε φ cos
2
φ
(3.12)
and
¯ ¯
ε = ¯ ¯
μ
(3.13)
Using the above equations, we numerically analyzed the phenomenon and the
obtained results have been shown in Fig. 3.8. Figure 3.8a illustrates the computational cell used, in which various domains and boundaries have been highlighted.
The material parameters ( and μ) of the cloak were set according to Eqs. 3.7—3.12.
The boundary of the embedded object was made a perfect electric conductor (PEC),
while the top and the bottom boundaries of the computational cell were made perfect magnetic conductor (PMC). A z-polarized plane wave of wavelength 3µm was
launched from the input port and received at the output as shown. The results of the
computation are shown in Fig. 3.8b, c, as the steady-state electric field distribution
throughout the cell. It can be seen that inside the cloak the wavefront is not broken but only gets deformed a bit and regains its original planar shape on emerging
out of it. Hence, no light gets blocked or reflected, no shadow is cast, and the object
remains absolutely cloaked. On the other hand, in the absence of the cloak, the object
substantially reflects the impinging wave, the wavefront gets broken, and a shadow
region gets formed. This illustrates the potential of the metamaterial of making a
science fiction concept a reality.
3.3.2 Electromagnetic Cloaking by Zero Refractive Index
There is another, a newer method of achieving electromagnetic cloaking using zeroindex metamaterials, popularized by Huang et al. in 2011 [27, 142–147]. They used
the rods-in-air-type photonic crystal working at Dirac frequency as the zero-index
metamaterial. The working principle of this type of cloaking is that the phase and
magnitude of the fields remain constant throughout the zero-index medium. The
phase and the magnitude of the wave at the boundary of emergence are the same
65
ε r = μ r =
r − a
r
(3.7)
ε φ = μ φ =
r
r − a
(3.8)
ε z = μ z =
b
b − a
2 r − a
r
(3.9)
The transformation equations for Cartesian coordinate system are given as [141]
ε xx = ε r cos
2
φ + ε φ sin
2
φ
(3.10)
ε xy = ε yx = (ε r − ε φ )sinφ cosφ
(3.11)
ε yy = ε r sin
2
φ + ε φ cos
2
φ
(3.12)
and
¯ ¯
ε = ¯ ¯
μ
(3.13)
Using the above equations, we numerically analyzed the phenomenon and the
obtained results have been shown in Fig. 3.8. Figure 3.8a illustrates the computational cell used, in which various domains and boundaries have been highlighted.
The material parameters ( and μ) of the cloak were set according to Eqs. 3.7—3.12.
The boundary of the embedded object was made a perfect electric conductor (PEC),
while the top and the bottom boundaries of the computational cell were made perfect magnetic conductor (PMC). A z-polarized plane wave of wavelength 3µm was
launched from the input port and received at the output as shown. The results of the
computation are shown in Fig. 3.8b, c, as the steady-state electric field distribution
throughout the cell. It can be seen that inside the cloak the wavefront is not broken but only gets deformed a bit and regains its original planar shape on emerging
out of it. Hence, no light gets blocked or reflected, no shadow is cast, and the object
remains absolutely cloaked. On the other hand, in the absence of the cloak, the object
substantially reflects the impinging wave, the wavefront gets broken, and a shadow
region gets formed. This illustrates the potential of the metamaterial of making a
science fiction concept a reality.
3.3.2 Electromagnetic Cloaking by Zero Refractive Index
There is another, a newer method of achieving electromagnetic cloaking using zeroindex metamaterials, popularized by Huang et al. in 2011 [27, 142–147]. They used
the rods-in-air-type photonic crystal working at Dirac frequency as the zero-index
metamaterial. The working principle of this type of cloaking is that the phase and
magnitude of the fields remain constant throughout the zero-index medium. The
phase and the magnitude of the wave at the boundary of emergence are the same