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408 Artificial materials or metamaterials

∗
∗ When the change in the relative dielec- were powerful enough to solve the problem numerically. And that leads us
tric constant is small, as in volume holo- to the early 1990s. The pioneers were Eli Yablonovitch and Sajeev John. The
graphy, an analytical approach might favoured technological solution was to drill holes in a dielectric rather than
be successful. It turned out, however,
that even to get close to the perfect- to put together a structure of rods. Holes of submicrometre dimensions had
reflection-from-all-directions condition, to be drilled. Half a million holes later there was still no success. But suc-
the contrast in dielectric constant had to cess eventually came in the form of the diamond structure that was shown
be large, by a factor of 2 or 3. There was
no chance of an analytical solution. in Fig. 5.3. The holes had to be drilled so as to follow the directions of the
chemical bonds.
What are the applications of photonic bandgap materials? They can be used
whenever there is a need for electromagnetic waves, propagating in any dir-
ection, to be reflected. They are singularly suitable for constructing resonant
cavities. Replace a few elements of a photonic bandgap material by one cap-
able of lasing, pump the laser at a wavelength for which the bandgap material
is transparent, and the whole laser device is ready. This is actually the way to
produce very small lasers, where very small means that its dimensions are sub-
micrometre. Another application is for guiding light. If we have a cylindrical
photonic bandgap material and we clear the area around the axis, then an op-
tical wave can propagate there without being able to spread outwards in the
radial direction. This is because a wave propagating in any but the axial dir-
ection will be reflected. These waveguides are known as holey fibres. Their
advantage in applications is that one can put anything (well, nearly anything)
in the central hollow core. For example, they may be filled with nonlinear
gases, leading to stimulated Raman scattering or frequency multiplication. Or,
thinking of something more esoteric, they may be suitable for guiding atoms
and small particles along. In that application, the optical dipole forces of a
co-guided laser beam prevent adhesion to the glass surfaces and provide the
acceleration needed to overcome viscosity.

15.4 Equivalent plasma frequency of a wire medium
The properties of wire media were investigated as early as the 1950s, but
they still could be our first example of metamaterials. A wire medium is
the man-made equivalent of certain materials available in nature, materials
which exhibit plasma phenomena. Remember, these materials were discussed
in Chapter 1. We derived there an effective dielectric constant in the form,

ω 2
ε eff = ε 0 1– , (15.1)
ω 2 p

where ω p is the critical frequency at which metals become transparent. Later
we rechristened it as the plasma frequency. Interestingly enough, wire media
have similar properties. No transmission up to a certain frequency (let’s call
it ω p also) and high transmission above that frequency. Such a structure is
shown schematically in Fig. 15.4(a), and experimental results on transmission
are shown in Fig. 15.4(b), where the parameter is the number of layers. It may
be seen that above a certain frequency, which is 9.5 GHz in the present case,
there is good transmission, but reduced transmission below that frequency. As