METATOY

Close-up view of a METATOY formed by an array of upright Dove prisms, seen from above. The view through the METATOY is shown in the previous image.

Telescopic optical components that have been used as the unit cell of two-dimensional arrays, and which therefore form homogeneous METATOYs, include a pair of identical lenses (focal length ${\displaystyle f}$) that share the same optical axis (perpendicular to the METATOY) and that are separated by ${\displaystyle 2f}$, that is they share one focal plane (a special case of a refracting telescope with angular magnification -1);[2] a pair of non-identical lenses (focal lengths ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$) that share the same optical axis (again perpendicular to the METATOY) and that are separated by ${\displaystyle f_{1}+f_{2}}$, that is they again share one focal plane (a generalization of the former case, a refracting telescope with any angular magnification);[3] a pair of non-identical lenses (focal lengths ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$) that share one focal plane, that is, they share the direction of the optical axis, which is not necessarily perpendicular to the METATOY, and they are separated by ${\displaystyle f_{1}+f_{2}}$ (a generalization of the former case);[4] a prism;[5] and a Dove prism[6][7][8][9]

Examples of inhomogeneous METATOYs include the moiré magnifier,[10] which is based on deliberately "mis-aligned" pairs of confocal microlens arrays; Fresnel lenses, which can be seen as non-homogeneous METATOYs made from prisms; and frosted glass, which can be seen as an extreme case of an inhomogeneous, random METATOY made from prisms.

Examples of METATOYs as defined above have existed long before analogies with metamaterials were noted and it was recognized that METATOYs can perform wave-optically forbidden ray-direction mappings (in pixellated form).[1]

Wave optics describes light at a more fundamental level than geometrical optics. In the ray-optics limit (in which the optical wavelength tends towards zero) of scalar optics (in which light is described as a scalar wave, an approximation that works well for paraxial light with uniform polarization), the light-ray field r${\displaystyle (x,y,z)}$ corresponding to a light wave ${\displaystyle u(x,y,z)}$ is its phase gradient,[11]

${\displaystyle \mathbf {r} (x,y,z)=\nabla \phi (x,y,z),}$

where ${\displaystyle \phi (x,y,z)}$ is the phase of the wave ${\displaystyle u(x,y,z)=A(x,y,z)\exp(i\phi (x,y,z))}$. But according to vector calculus, the curl of any gradient is zero, that is

${\displaystyle \nabla \times \nabla \phi (x,y,z)=0,}$

and therefore

${\displaystyle \nabla \times \mathbf {r} (x,y,z)=0.}$

This last equation is a condition, derived from wave optics, on light-ray fields. (Each of the three equations that makes up this vector equation expresses the symmetry of the second spatial derivatives, which is how the condition was initially formulated.[1])

Using the example of ray-rotation sheets,[12] it was shown that METATOYs can create light-ray fields that do not satisfy the above condition on light-ray fields.[1]

METATOYs are not metamaterials in the standard sense. The acronym "metamaterial for rays" was chosen because of a number of similarities between METATOYs and metamaterials,[1] which are discussed below, along with the differences. In addition, metamaterials provided the inspiration for early METATOYs research, as summarized in the following quote:[1]

Motivated by the desire to build optical elements that have some of the visual properties of metamaterials on an everyday size scale and across the entire visible wavelength spectrum, we recently started to investigate sheets formed by miniaturized optical elements that change the direction of transmitted light rays.

• Fresnel lens
• Zone plate

• Didactic article on METATOYs