Historical research guide · 1093 · 2008
Transformation Optics and the Geometry of Light
Ulf Leonhardt · Thomas G. Philbin
In one minute
Ulf Leonhardt and Thomas Philbin set out a single idea and then show four devices that are all the same device. The idea: a piece of glass is a curved space. Light takes the quickest path, and the refractive index is what sets how long a path takes, so a material with a varying index bends light exactly the way curvature does. Media look to light like geometries, and geometries look like media. That turns design inside out — draw the geometry you want first, then build a material that performs the coordinate transformation. Do it with a hole in the coordinate grid and you get an invisibility cloak, where both the object and the act of hiding vanish. Do it with a fold and you get the perfect lens. Let the material move and you are transforming time as well as space, which gives the optical Aharonov-Bohm effect around a vortex and, where the flow outruns light in the medium, an event horizon that radiates. This is the engineer’s handbook for the geometry of light.
Connection
Chapter 4 is about engineering the metric, and this is the clearest published account of doing it with materials instead of mass: write the coordinate transformation you want, then build it. Chapter 5 gets the other half of the same argument — a flowing medium is a space-time, complete with horizons that radiate — and chapter 2 gets the striking corollary that a transformation medium can turn the Casimir force around and hold a mirror up against gravity.
From the source
An optical medium is a geometry. By Fermat’s principle light follows the path of extremal optical length, and the refractive index is the measure of that length, so any medium establishes a geometry for light — the glass of a lens, the water of a river, the air of a mirage. Media appear to light as geometries, and geometries act as effective media; since 2000 it has also been understood that media perceive electromagnetic fields as geometries.Section 1, Introduction; Section 2, Fermat’s principle; Section 6, Summary
Transformation optics is the design method that follows: build a material that appears to perform a coordinate transformation from physical space to a virtual electromagnetic space in which light travels straight. Cloaking devices, perfect lenses, vortices and horizons are then four cases of one construction, differing only in the topology of the transformation, and all four have been verified experimentally at least in part.Section 1; Section 4; Section 5; Section 6, Summary
A perfect spherical cloak is a transformation with a hole in the coordinate grid: the whole shell at the inner radius corresponds to a single point of electromagnetic space, so anything inside is excluded from the field, while beyond the outer radius the two spaces agree and the waves are indistinguishable from waves in empty space. The object disappears and so does the act of hiding. The same construction sets its own limit — the volume of a single point is zero, so one eigenvalue of the dielectric tensor must vanish at the inner lining and the phase velocity there must reach infinity, which confines perfect invisibility to a single frequency. Imperfect devices inspired by the idea remain possible.Section 5.2, Eqs. 5.11 to 5.13, Figs. 12 and 13
The perfect lens is the same machinery with a multi-valued map: in the fold of the transformation each point of electromagnetic space has three faithful images in physical space, so fields at one are perfectly imaged onto the others. The fold also explains why negative permittivity and permeability are needed — inside it the coordinate system changes handedness. Sub-resolution imaging has since been confirmed in experiments.Section 5.3, Eq. 5.14 with Figs. 14 and 15
A perfect lens with gain can sustain a repulsive Casimir force across a gap with no mediating medium between the mirrors, and the repulsion could be strong enough for one mirror to overcome gravity — levitating on zero-point energy.Section 5.3, closing paragraph, citing Leonhardt and Philbin, New J. Phys. 9, 254 (2007)
Moving media perform space-time transformations rather than purely spatial ones. Where the flow speed crosses the speed of light in the medium a horizon forms, part of an escaping wave must have tunnelled through it, and the pairs created carry a Planck spectrum whose temperature is set by the velocity gradient at the horizon. Few-cycle light pulses in photonic-crystal fibre create moving refractive-index profiles that travel at the speed of light, and could give a detectable amount of that radiation in the laboratory.Sections 5.4 and 5.6, Eqs. 5.58 to 5.62 and the closing paragraph of Section 5.6
The way in
https://arxiv.org/abs/0805.4778How to cite it
Ulf Leonhardt, Thomas G. Philbin (2008) Transformation Optics and the Geometry of Light arXiv:0805.4778