44
2 Zero-Index Metamaterials
(a)
(b)
Fig. 2.17 Fate of light falling on a zero-index metamaterial slab, according to Snell’s law, for a
normal incidence b oblique incidence
Figure 2.17 shows the case of normal incidence. We see that light incidents normally on the slab and propagates normally inside it. This behavior has been verified numerically as well, and the result of the simulation has been presented in
Fig. 2.18a. It can be seen that light penetrates, as it is, into the zero-index region.
However, an interesting behavior is observed when light falls obliquely on the ZIM,
i.e., at any angle greater than zero. By pondering a little, one can easily figure
out the reason for this behavior. Since n 2 ≈ 0, the critical angle for the system
shown in Fig. 2.17 is θ c = sin
−1
(n 2 /n 1 ) ≈ 0. Thus, light is able to propagate into
the zero-index medium only if θ i ≈ 0 and for any value of θ i substantially larger
than zero, it gets total internally reflected. Mathematically, if θ i >> 0 and n 2 ≈ 0,
then sin θ r = n 1 sin θ i /n 2 >> 1, therefore θ r cannot acquire any real value. Consequently, no refraction but only total internal reflection is feasible. This prediction by
Snell’s law has been schematically shown in Fig. 2.17b and numerically confirmed
and demonstrated in Fig. 2.18b. This property of total internal reflection is very interesting and can be very useful in making on-chip waveguides with zero-index cladding.
An interesting phenomenon attached to the total internal reflection is Goos–Hänchen
shift, attributed to the slight penetration of light into the cladding. The Goos–Hänchen
shift has been deeply studied for significant optical waveguidance systems like optical fibers and planar waveguides, in order to find ways to minimize it. Moreover, a
negative Goos–Hänchen shift has been observed and studied in the negative-indexmetamaterial-based waveguidance systems. Hence, it is worthwhile to investigate the
modification of the GH shift phenomenon in zero-index-metamaterial-based devices
too. In fact, in Chap. 3, we have rigorously analyzed the Goos–Hänchen shift for a
glass–ENZ system for both s- and p-polarizations. That is all worth mentioning about
the propagation of light from a positive to a zero-index medium. Next, we consider
the reverse case, i.e., light propagating from a zero-index medium to air.
2 Zero-Index Metamaterials
(a)
(b)
Fig. 2.17 Fate of light falling on a zero-index metamaterial slab, according to Snell’s law, for a
normal incidence b oblique incidence
Figure 2.17 shows the case of normal incidence. We see that light incidents normally on the slab and propagates normally inside it. This behavior has been verified numerically as well, and the result of the simulation has been presented in
Fig. 2.18a. It can be seen that light penetrates, as it is, into the zero-index region.
However, an interesting behavior is observed when light falls obliquely on the ZIM,
i.e., at any angle greater than zero. By pondering a little, one can easily figure
out the reason for this behavior. Since n 2 ≈ 0, the critical angle for the system
shown in Fig. 2.17 is θ c = sin
−1
(n 2 /n 1 ) ≈ 0. Thus, light is able to propagate into
the zero-index medium only if θ i ≈ 0 and for any value of θ i substantially larger
than zero, it gets total internally reflected. Mathematically, if θ i >> 0 and n 2 ≈ 0,
then sin θ r = n 1 sin θ i /n 2 >> 1, therefore θ r cannot acquire any real value. Consequently, no refraction but only total internal reflection is feasible. This prediction by
Snell’s law has been schematically shown in Fig. 2.17b and numerically confirmed
and demonstrated in Fig. 2.18b. This property of total internal reflection is very interesting and can be very useful in making on-chip waveguides with zero-index cladding.
An interesting phenomenon attached to the total internal reflection is Goos–Hänchen
shift, attributed to the slight penetration of light into the cladding. The Goos–Hänchen
shift has been deeply studied for significant optical waveguidance systems like optical fibers and planar waveguides, in order to find ways to minimize it. Moreover, a
negative Goos–Hänchen shift has been observed and studied in the negative-indexmetamaterial-based waveguidance systems. Hence, it is worthwhile to investigate the
modification of the GH shift phenomenon in zero-index-metamaterial-based devices
too. In fact, in Chap. 3, we have rigorously analyzed the Goos–Hänchen shift for a
glass–ENZ system for both s- and p-polarizations. That is all worth mentioning about
the propagation of light from a positive to a zero-index medium. Next, we consider
the reverse case, i.e., light propagating from a zero-index medium to air.