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outtakes from "Relationship with metamaterials"
METATOYs also satisfy at least the first half of the definition for metamaterials (the second half arguably makes the definition specific to metamaterials for light waves): Metamaterials are materials with designed properties that stem from structure, not substance, whereas man-made structures determine the electromagnetic properties, structures that are smaller than the electromagnetic wavelengths involved.

METATOY analogs of metamaterial concepts
It has been argued that a number of METATOY equivalents predated their metamaterial counterparts. These include However, the METATOY Without making any connection with metamaterials, the CLA equivalent of a superlens \cite{Stevens-Harvey-2002,Okano-Arai-2002} and a hyperlens has already been built \cite{Volkel-et-al-2003,Duparre-et-al-2005}.

The direction of the light rays themselves is changed; the direction change does not rely on anisotropy of the surrounding medium.

"Pixellated" refraction
With a pixellated light-ray field we mean the following: the plane immediately behind the METATOY is subdivided into small areas – the pixels –, and light-ray directions are continuous only within pixels, but generally discontinuous between pixels. METATOYs sacrifice homogeneity in favour of “pixellated homogeneity”, thereby gaining the ability to create “forbidden” light-ray fields.

functionality: needs to behave like a homogeneous material, so light-ray-direction change has to be independent of exact position where it hits the miniaturized optical element (so different from toilet window); hence "telescopic" components

Gumph
\section{Experiments}

\section{The view through METATOYs}

The view through a number of specific METATOYs has been examined in some detail. In a few cases, imaging happens, in the case of crossed Dove-prism sheets \cite{Courtial-Nelson-2008} and confocal lenslet arrays \cite{Courtial-2008a} of all space, in the case of pairs of ray-rotation sheets of one particular plane \cite{Hamilton-Courtial-2008c}. In all cases, the distortion introduced by the METATOY can be described as an apparent source direction as seen from an observer's position \cite{Hamilton-Courtial-2008a,Hamilton-et-al-2009}.

\subsection{Apparent light-source direction}

\section{Imaging}

pseudoscopic imaging with crossed Dove-prism sheets; imaging with CLAs; imaging with ray-rotation sheets; limits of imaging constrained only by geometry (i.e. not by wave optics)

\section{Alternative formal description of METATOY refraction}

Application of Fermat's principle to local light-ray rotation gives extended Snell's law in which refractive indices can be complex numbers; like in Snell's law, the modulus of the refractive-index ratio, $|n_2 / n_1|$, describes the change of the polar angle; unlike in Snell's law, the argument of refractive-index ratio, $\arg(n_2 / n_1)$, describes the change in the azimuthal angle. Bhuvanesh Sundar, Alasdair C. Hamilton, and Johannes Courtial, Fermat's principle with complex refractive indices and local light-ray rotation, Opt. Lett. 34, 374-376 (2009), doi: 10.1364/OL.34.000374 (submitted September 2008)

\section{External links}

METATOYs research at the University of Glasgow

IN FUTURE ADD...

\section{General light-ray direction changers}

% modified text from the 1, 2, 3... LAs paper:

Imaging we want to build a thin, planar, optical element -- a ``window'' -- that transmits light rays without displacing them in the transverse direction, but which also performs a mapping between the directions of the incident and transmitted light rays that is far more general that that performed by a normal glass window. In fact, ideally, we want to be able to change the light-ray direction completely arbitrarily. For example, it would be interesting to have a window could with the property of \emph{not} refracting light rays, so that distances in the water do not appear shorter (unlike when looking through the simple glass window of swimming and diving goggles).

If we want this window to work wave-optically, by which we mean that we want the window to transmit wave fronts without ``chopping them up'' or otherwise destroying them, then application of Fermat's principle \cite{Born-Wolf-1980-Fermat} leads to the conclusion that the light-ray directions have to change according to Snell's law.

So let's relax our requirements. We allow our window to change the light-ray direction by chopping up the wave fronts. Perhaps the simplest way to approximate a completely arbitrary light-ray direction change is shown in Fig.\ \ref{pinhole-and-prisms-array-figure}. Each incident light ray has to pass through one of an array of pinholes. Behind the pinhole, the light ray is allowed to travel for a small distance, after which it hits a tiny prism (or other optical component that changes its direction). Light rays incident from different directions hit different prisms, and as different prisms can be shaped differently, the direction of different of light rays incident from different directions can be changed differently.

Next step: symmetrise ``window'' by adding prisms also to the other side. Without these additional prisms, we can do a many-to-one direction transformation (many different light-ray directions from the left can result in the same light-ray direction on the right, but only one light-ray direction from the right can result in any light-ray direction to the left); with these additional prisms, we can do a many-to-many direction transformation (Fig.\ \ref{many-to-many-mapping-figure}). (In the many-to-one or many-to-many transformation, the light-ray direction of the transmitted light does depend on the precise position where the incident light ray hits.

What can we do with these completely general light-ray-direction changers? \begin{itemize} \item many-to-many mappings \item particularly simple laws of refraction, e.\ g.\ $\alpha_2 = a \alpha_1$ \item can we make the light-ray directions transform like tensors? (Is that what makes transformation optics work from all directions?) \item can we do some form of topological transformation? (M\"obius band, anything knotted, ...)? What is it that's transformed like this? \item what can we do with singularities in the transformation?  (We can describe the direction-to-direction mapping as a complex-to-complex map.) \item This is ``visual complex analysis -- literally!  (By the way, read the book called ``Visual Complex Analysis at some point for inspiration.) \end{itemize}

This light-ray-direction-changing window is very lossy. It is the light-ray-direction-changing-window equivalent of a pinhole camera, which can be made much less lossy by using a lens instead of a pinhole. Similarly, replace the pinhole array with a lenslet array, do something in the focal plane, and then have another (confocal with the first) lenslet array (so that incident parallel light rays are parallel again afterwards). It's not clear how to affect completely general changes in the focal plane (local Fourier engineering). Some changes are easy to affect; these include a shift in the focal plane (by moving the two confocal lenslet arrays sideways); scaling of Fourier space (by selecting a second lenslet array with a different focal length from the first). This is exactly what generalized confocal lenslet arrays do \cite{Hamilton-Courtial-2009b}.

Confocal lenslet arrays can refract light approximately like a refractive-index interface \cite{Courtial-2008a}. They can also bend light in more complex ways. Two confocal cylindrical lenslet arrays with the same focal length, for example, invert one of the components of the light-ray direction, just like a Dove-prism array \cite{Hamilton-Courtial-2008a} does. Two rotated confocal cylindrical lenslet arrays therefore act like two rotated Dove-prism sheets: they rotate the local light-ray direction \cite{Hamilton-et-al-2009}, a concept without wave-optical analog \cite{Hamilton-Courtial-2009}. This more complex ``refraction'' can then be used to try and realize concepts such as cloaking \cite{Hamilton-Courtial-2008e}.

generalized lenslet arrays \cite{Hamilton-Courtial-2009b}

combinations of generalized lenslet arrays \cite{Hamilton-Courtial-2009c}

confocal lenslet arrays \cite{Courtial-2008a}; limits \cite{Courtial-2009}

prism arrays \cite{Courtial-Nelson-2008,Hamilton-Courtial-2008a,Hamilton-et-al-2009}

Consider a thin, planar optical instrument that changes the direction of transmitted light rays. If we make the instrument the phase hologram of a glass wedge then we can choose the wedge orientation and angle so that one direction of incident light gets changed into an arbitrary transmitted-light direction.

We want more, though: we want to change all directions.

\cite{Hamilton-Courtial-2009}

What \emph{does} the phase-gradient vector do in ``normal'' light fields? \cite{Dennis-et-al-2008}

needs to be properly examined