Superlens

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A superlens is a lens which is capable of subwavelength imaging. Conventional lenses have a resolution on the order of one wavelength due to the so called diffraction limit. This makes it impossible to image very small objects such as individual atoms which have sizes many times smaller than the wavelength of visible light. A superlens is able to beat the diffraction limit. A very well-known superlens is the perfect lens described by John Pendry, which uses a slab of material with a negative index of refraction as a flat lens. In theory, Pendry's perfect lens is capable of perfect focusing—meaning that it can perfectly reproduce the electromagnetic field of the source plane at the image plane.

For the argumentation, plane waves are used. Let us start from a beam splitter using frustrated total internal reflection. Then we add a second layer of a material, which has a refractive index with the same absolute value, but the opposite sign of the first layer. If both layers are as thick, the beam splitter does not split anymore, but passes, for example, all energy and maybe the temporal evolution of a laser pulse. In this theory, the material with the high index of refraction can have a very high refractive index (absolute value), which means very small objects immersed into it are still resolved. This resolution is also sustained through the low refractive index material. The Kramers-Kronig relation tells us that this is only possible for a narrow range of frequencies (the material has to be tuned).

The performance limitation of conventional lenses is due to the diffraction limit. Following Pendry (Pendry, 2000), the diffraction limit can be understood as follows. Consider an object and a lens placed along the z-axis so the rays from the object are traveling in the +z direction. The field emanating from the object can be written in terms of its angular spectrum, as a superposition of plane waves:

$E(x,y,z,t) = \sum_{k_x,k_y} A(k_x,k_y) e^{i\left(k_z z + k_y y + k_x x - \omega t\right)}$

where $k z$ is a function of $k x, k y$ as:

$k_z = \sqrt{\frac{\omega^2}{c^2}-\left(k_x^2 + k_y^2\right)}$

Only the positive square root is taken as the energy is going in the +z direction. All of the components of the angular spectrum of the image for which $k z$ is real are transmitted and re-focused by an ordinary lens. However, if

$k_x^2+k_y^2 > \frac{\omega^2}{c^2}$ ,

then $k z$ becomes imaginary, and the wave is an evanescent wave whose amplitude decays as the wave propagates along the z-axis. This results in the loss of the high angular frequency components of the wave, which contain information about the high frequency (small scale) features of the object being imaged. The highest resolution that can be obtained can be expressed in terms of the wavelength:

$k_{max} \approx \frac{\omega}{c} = \frac{2 \pi}{\lambda}$

$\Delta x_{min} \approx \lambda$

So, how does a superlens get around this? A Pendry-type superlens has an index of $n = - 1$ ( $ε = - 1,μ = - 1$ ), and in such a material, transport of energy in the +z direction requires the z-component of the wavevector to have opposite sign:

$k'_z = -\sqrt{\frac{\omega^2}{c^2}-\left(k_x^2 + k_y^2\right)}$

For large angular frequencies, the evanescent wave now grows, so with proper lens thickness, all components of the angular spectrum can be transmitted through the lens undistorted. There are no problems with conservation of energy, as evanescent waves carry none (Poynting vector is 0).

In 2005, the first optical superlens was constructed by physicists at University of California, Berkeley. This allowed scientists to observe objects as small as 40 nm—one-tenth the size of objects which can be viewed with conventional optical microscopes. The superlens constructed at UC Berkeley did not employ a negative refractive index metamaterial as in Pendry's perfect lens. Instead a thin film of silver was used to amplify the evanescent waves.