Transformation optics can do the impossible—invisibility and perfect imaging—
by using the idea that optical materials change the geometry of space.
In Maxwell’s fish eye lens, light rays from any
point faithfully meet at a corresponding image
point. All light rays from the object make it to
the image. If light consisted of particles that follow the
ray trajectories, it should form a perfect image. However, light
is also a wave; and we know from Abbe that it is the waviness
of light that limits the resolution of lenses. It was assumed
that, in reality, the fish eye lens resolution would be about
half the wavelength. In 2009, I predicted that this assumption
was wrong and that Maxwell’s lens should image waves with
To understand why Maxwell’s fish eye lens works, you need
a bit of Einstein. Optical materials change the perception of
space for light. ;ey conjure up a virtual space—the space as
seen by light—that may be very di;erent from physical space.
;e virtual space can be curved in a manner similar to the way
space-time is curved by gravity, according to Einstein’s general
relativity. Curved space is not di;cult to imagine. ;e simplest
curved space is the surface of a sphere. Coincidentally, a sphere’s
surface is exactly the virtual space of Maxwell’s fish eye lens.
Now, imagine light is confined to propagate on the sphere.
(One can demonstrate this in physical reality by using a glass
sphere where light clings to the surface.) On the sphere, light no
longer propagates in straight lines. It follows the shortest path
between two points—curved geodesics. ;ese are the great
circles (circles with centers in the middle of the sphere). All
great circles starting from one point on the sphere end up at an
antipodal, conjugate point. ;is can be viewed on a globe.
Great circles are also the paths of light rays on the sphere.
All the rays emitted from a given point must meet again at
the antipodal point. ;is is true for all points of emission. ;e
antipodal points form a perfect image of the original points.
Light waves are as perfectly focused as light rays because of
the sphere’s symmetry: A wave emitted from any point on the
sphere will focus at the corresponding antipodal point. ;erefore, as Maxwell’s fish eye lens implements the geometry of the
sphere, it creates perfect images.
Transformation optics can do the impossible—invisibility
and perfect imaging—by using the idea that optical materials
change the geometry of space. In many cases, these materials
must be advanced metamaterials, but in some cases ordinary
materials may work. For example, Maxwell’s fish eye could
be made by etching a planar lens out of a silicon layer—like a
miniature contact lens. Unlike a contact lens, it creates images
inside the lens, not through it.
Some valuable applications of invisibility science may well
remain invisible not because they cloak objects, but because they
themselves hide behind the scenes in the everyday sense. Invisibility is about how to predictably manipulate light. Principles
Courtesy of Ulf Leonhardt
In Maxwell’s ;sh eye lens, electromagnetic waves propagate
in a plane in physical space (wave pattern below) as if they
were con;ned to the surface of a sphere (above). A wave
emitted from any point on the virtual sphere is focused at the
antipodal point. In physical space, waves are as perfectly
focused as in virtual space.
learned from studying invisibility could be useful in designing
new types of optical instruments. ;ese instruments may provide
a more sophisticated version of the optics integrated onto chips
that make the broadband optical connections for computers.
;ey could also serve as spectrometers for the chemical analysis
in labs on chips or perhaps lead to more e;cient solar cells.
Transformation optics may produce a plethora of spin-o; technologies besides cloaking and perfect imaging. ;e
resulting devices will serve alongside conventionally designed
bits of optics, as modest but useful invisible helpers. Whatever the final application will be, it is probably fair to say that
we have already transformed our understanding of optics. t
Ulf Leonhardt ( firstname.lastname@example.org) is the chair in theoretical physics at the
University of St. Andrews, U.K.
[ References and Resources ]
>> U. Leonhardt. Science 312, 1777 (2006).
>> R.F. Service and A. Cho. Science 330, 1622 (2010).
>> A. Greenleaf et al. Math. Res. Lett. 10, 685 (2003).
>> J.B. Pendry et al. Science 312, 1780 (2006).