Page 436 - Electrical Properties of Materials

418 Artificial materials or metamaterials

(where k x is a space harmonic of the object expanded into a Fourier series at
z = 0, and k is the imaginary component of the wave vector in the z-direction)

z

will have an amplitude of exp(–k d/2). If the lens has a width of d and the
z
image plane is a distance d/2 behind the lens, then the wave moving from the
rear surface to the image plane will also have a decay of exp(–k d/2). We can

z
have a perfect lens if the thickness of the lens is d and the wave inside the
negative-index material grows at the same rate. Then, of course,

–k d –k d
z
z

exp exp(k d)exp = 1, (15.33)
z
2 2
and the original amplitude of the evanescent wave is restored. Remarkably,
each component of the space harmonic spectrum is perfectly reproduced. In
other words, the transfer function (relating the amplitude and phase of a space
harmonic at the output to the input values) is constant, and its value is unity.
How can this happen? What is the physical mechanism behind it? For that,
we have to go back to surface plasmons, which we discussed in Chapter 1.
A surface plasmon is a wave that sticks to a metal surface. If, instead of a
single surface we have a metallic slab with two surfaces, then the waves stick
to both surfaces. Under certain conditions (when ε r = μ r = –1), it is only the
outer surface that is excited and the waves need to grow in order for this to be
possible.
So can we have a perfect lens? Not really. A limit will be set, if by nothing
else, then by the period of the negative-index material. If we can make metama-
terial elements of size 100 nm and if the distance between them is also 100 nm,
then there will be a chance of making a lens with a resolution approaching 200
nm. And there will be other imperfections caused by losses, tolerances, and
possibly long transients.
Should we conclude that the perfect lens is a humbug? That it is a theor-
etical construction based on invalid approximations? That there is no way of
realizing it? Absolutely not. We might say that the chances of producing an ar-
tificial material for the purpose of subwavelength imaging in the optical range
are rather limited, but that is only part of the story. It turns out that high res-
olution can still be obtained under circumstances when only the permittivity is
∗
∗ That negative permittivity is sufficient equal to –1 and the permeability can be +1. And that can happen in a metal.
for obtaining a high resolution follows Indeed, some modest success has been achieved with silver as the lens ma-
from the so-called electrostatic (ES) ap- terial. Its plasma frequency is very high; the permittivity is equal to –1 at a
proximation. This simplifies the problem
because there is no need then to solve wavelength of about 360 nm, so that a high resolution is indeed possible. But,
the wave equation and one can rely on you could argue, why is this mentioned in a chapter on artificial materials?
Laplace’s equation instead. The ES ap- Silver is a natural material, isn’t it? Well, yes, but for all practical purposes
proximation predicts perfect imaging of
a material in which only the permittiv- silver has been proclaimed an honorary artificial material by the community of
ity is negative; however, the approxima- metamaterialists.
tion has only a limited validity. For that Let us now see a few simple examples. The lens is made of a slab of sil-
reason, in the treatment that follows we ver, where only ε r is equal to –1. The imaginary part of the relative dielectric
shall give both the ‘full’ solution and the
r
ES solution when they differ from each constant is taken as ε = 0.4. The object consists of a pair of step functions of
other. 15 nm width at a distance 50 nm from each other. The imaging, for a 10 nm–
20 nm–10 nm lens configuration, is shown in Fig. 15.14(a). In the absence of
the lens, the power detected in the image plane is shown by a dot–dash line.
The two bars are no longer resolved. However, the resolution is very good in