54
2 Zero-Index Metamaterials
node. It is a manifestation of zero-index nature, that is, as the wavelength becomes
very large, the field inside the device remains almost uniform. Whereas, in all other
cases, the black regions mark the actual nodes. Now, as one moves beyond 1550–
1625 nm and 1700 nm wavelength, nodes reappear in the device. As the refractive
index increases (though in a negative scale), the inter-nodal separation decreases, due
to the shortening of the effective wavelength. It is a simplistic yet beautiful method to
illustrate the zero-index nature of the metamaterial and also put forward phase-free
propagation as a useful application of the zero-index phenomenon.
2.8 Tunability of a Zero-Index Metamaterial
The position of the Dirac cone, and hence the corresponding zero-index frequency
depends on the geometrical and the material parameters. As a result, there can be
two types of tuning of the metamaterial structure—static tuning and dynamic tuning.
Let us take the example of the commonly used square lattice of rods. The static
tuning of the structure can be done by changing the diameter and pitch of the rods,
and/or by changing the material of the rods. The explanation about this tunability
lies in the scalability of Maxwell’s equation [37]. From our knowledge about the
photonic bandgap materials, we know that, in the case of a square array of dielectric
columns, if the radius of the columns and their periodicity is doubled (or halved),
without changing the r/a ratio, the corresponding bandgap wavelengths will be
doubled (or halved) as well. The same is true for zero-index metamaterials and the
corresponding zero-index wavelength, provided the formation of Dirac cone is not
compromised with scaling. This type of tuning is called static tuning. A drawback of
this method is that one needs to fabricate different structures for different wavelengths
of operation (i.e., the zero-index wavelengths). As once fabricated, a structure can
provide zero-index only at a single wavelength, and the index remains close to zero
within a very narrow spectral region around it. A more convenient method of tuning
the metamaterial to a different operational frequency would be by modifying the
surrounding conditions without altering its geometry. Such tuning is called dynamic
tuning. However, to the best of our knowledge, no such technique has been reported
until the drafting of this book. Nonetheless, we have discussed below two probable
routes of achieving dynamic tunability.
One may argue that by using liquids (n > 1) as surrounding media which can
be removed or replaced as per desire (Fig. 2.27a), a dynamic control over the wavelength of operation can be achieved. We examined the feasibility of this technique
numerically and the results have been shown in Fig. 2.27c–d. The results show that
dynamic tunability is not easy to achieve. Figure 2.27c shows the required value r/a
to achieve a well-defined Dirac cone (red curve) and the corresponding Dirac point
frequency (a/λ) of the Dirac cone thus obtained (blue curve), with respect to the
refractive index (n s ) of the surrounding medium. It implies that if the metamaterial
is submerged in a liquid medium, it cannot immediately start operating at a different
wavelength. Change in the refractive index of the surrounding calls for the change
2 Zero-Index Metamaterials
node. It is a manifestation of zero-index nature, that is, as the wavelength becomes
very large, the field inside the device remains almost uniform. Whereas, in all other
cases, the black regions mark the actual nodes. Now, as one moves beyond 1550–
1625 nm and 1700 nm wavelength, nodes reappear in the device. As the refractive
index increases (though in a negative scale), the inter-nodal separation decreases, due
to the shortening of the effective wavelength. It is a simplistic yet beautiful method to
illustrate the zero-index nature of the metamaterial and also put forward phase-free
propagation as a useful application of the zero-index phenomenon.
2.8 Tunability of a Zero-Index Metamaterial
The position of the Dirac cone, and hence the corresponding zero-index frequency
depends on the geometrical and the material parameters. As a result, there can be
two types of tuning of the metamaterial structure—static tuning and dynamic tuning.
Let us take the example of the commonly used square lattice of rods. The static
tuning of the structure can be done by changing the diameter and pitch of the rods,
and/or by changing the material of the rods. The explanation about this tunability
lies in the scalability of Maxwell’s equation [37]. From our knowledge about the
photonic bandgap materials, we know that, in the case of a square array of dielectric
columns, if the radius of the columns and their periodicity is doubled (or halved),
without changing the r/a ratio, the corresponding bandgap wavelengths will be
doubled (or halved) as well. The same is true for zero-index metamaterials and the
corresponding zero-index wavelength, provided the formation of Dirac cone is not
compromised with scaling. This type of tuning is called static tuning. A drawback of
this method is that one needs to fabricate different structures for different wavelengths
of operation (i.e., the zero-index wavelengths). As once fabricated, a structure can
provide zero-index only at a single wavelength, and the index remains close to zero
within a very narrow spectral region around it. A more convenient method of tuning
the metamaterial to a different operational frequency would be by modifying the
surrounding conditions without altering its geometry. Such tuning is called dynamic
tuning. However, to the best of our knowledge, no such technique has been reported
until the drafting of this book. Nonetheless, we have discussed below two probable
routes of achieving dynamic tunability.
One may argue that by using liquids (n > 1) as surrounding media which can
be removed or replaced as per desire (Fig. 2.27a), a dynamic control over the wavelength of operation can be achieved. We examined the feasibility of this technique
numerically and the results have been shown in Fig. 2.27c–d. The results show that
dynamic tunability is not easy to achieve. Figure 2.27c shows the required value r/a
to achieve a well-defined Dirac cone (red curve) and the corresponding Dirac point
frequency (a/λ) of the Dirac cone thus obtained (blue curve), with respect to the
refractive index (n s ) of the surrounding medium. It implies that if the metamaterial
is submerged in a liquid medium, it cannot immediately start operating at a different
wavelength. Change in the refractive index of the surrounding calls for the change