## Invisibility physics: “Reflectionless” objects make an appearance

(This is a continuation of my “history of invisibility physics” series of posts.  The earlier posts are: Part I, Part II, Part III.)

Up through the late 1940s, it seems that the only type of invisibility that authors were considering were “radiationless orbits”: motions of charged particles of extended size which in principle could accelerate without emitting radiation.  These are not invisible objects per se, but rather objects that should produce radiation according to conventional wisdom but in fact do not.

A truly invisible object would be one which does not scatter any radiation incident upon it; that is, light which shines on the object is not reflected or absorbed, but instead is transmitted in such a way that it appears to the outside observer as if there were no object present.  But are such invisible objects even possible?

In 1956, a paper appeared in the Journal of Applied Physics which provided at least a partial answer to this question.  In their article, “Reflectionless transmission through dielectrics and scattering potentials,” Irvin Kay and Harry Moses demonstrated theoretically that one could construct stratified media that perfectly transmit waves of a given frequency, regardless of the direction of incidence of the illuminating wave.  Light shining on their theoretical media would be completely transmitted, with no reflected light!

First, an explanation of terminology: a stratified medium is a medium that consists of multiple planar layers, just like the layers of soil in the ground.  A medium stratified in the x-direction, for instance, might appear something like this:

In optics, a stratified medium consists of layers of materials with different refractive indices.  At each interface, a light wave is partially reflected and partially refracted; refraction follows Snell’s law,

$n_1\sin\theta_1 = n_2\sin\theta_2$,

where $n_1$ and $n_2$ are the refractive indices of the media on either side of the interface, and $\theta_1$ and $\theta_2$ are the angles the light wave make with respect to the perpendicular to the surface.

Even at the time of Kay and Moses’ paper, layered media were already being used to reduce reflections, in the form of anti-reflection coatings.  Camera lenses and eyeglasses typically possess such a coating, which consists of a very thin layer of material on top of the optical lens.  Light impinging on a coated lens can be reflected from the surface of the coating or the surface of the lens itself.  Such a coating is typically made to have a thickness of a quarter of a wavelength; when light impinges normally on the surface of such a medium,  the internally-reflected wave travels a half-wavelength farther than the surface-reflected wave, and they end up canceling by destructive interference:

By an appropriate choice of the refractive index of the coating material, this cancellation can in principle be made  perfect — but only for a  normally incident wave.  For a wave which impinges upon the surface at an angle, the path length difference between the two reflected waves ends up being shorter than a half-wavelength, and the destructive interference is not perfect:

It may seem at first that the path difference should be longer than half a wavelength; in the picture, I show the extra path length from the internal reflection (yellow+purple) minus the extra length the directly reflected wave travels outside the medium (blue); the result is less than half a wavelength!  This picture is still oversimplified, because I have neglected refraction in the medium, but at least it illustrates the point!

If we want the coating to be non-reflecting for additional directions of incidence, we can add additional layers.  The math gets complicated, but essentially two additional layers are needed for each new angle of incidence, one to accommodate for each possible polarization state of the illuminating light.

Kay and Moses looked at the possibility of making this process perfect, at least in principle; that is, they looked to design a stratified medium which is perfectly non-reflecting for any direction of illumination.  They found that one could construct a continuously stratified medium (a smoothly-varying refractive index) that would have these properties; such a medium might appear as follows:

We may think of a continuously stratified medium as consisting of an infinite number of infinitely-thin layers; in this sense it is not surprising that one can find a medium which is reflectionless for all directions of incidence.

Though they had electromagnetic waves in mind when doing their research, Kay and Moses in fact solved a simpler problem: the reflection of scalar waves from a “potential function” $V(x)$, which may be thought of as representing the “hills” and “valleys” the wave must climb. A few examples of such potentials are shown below:

There is a surprising amount of freedom in the mathematical construction of reflectionless potentials, and one can design potentials with an arbitrary number of independent parameters.  The first plot above, for instance, shows potentials constructed with one free parameter, while the second plot shows a potential with three free parameters.   It is possible to construct an infinite number of distinct reflectionless potentials of increasing complexity.

It is important to note that the potentials pictured above are reflectionless for any direction of incidence, as long as the wave is at least in part propagating to the right.  A potential of this form is realizable with more or less regular optical materials, as a negative potential corresponds to a refractive index greater than unity.  Kay and Moses’ potentials can therefore be constructed without the use of optically resonant materials or fancy negative refractive index metamaterials!

There are a number of strong limitations to these reflectionless potentials.  First and foremost, they are only reflectionless for a single frequency of light; light of other frequencies will experience partial reflection.  This anticipates one of the biggest limitations of modern “cloaking” devices: the narrow range of frequencies for which they can be invisible.  Also, the potentials described here are of infinite extent; that is, $V(x)$ has nonzero values for arbitrarily large values of $x$.  Any real device would have a potential of finite width, which would in turn not be perfectly reflectionless.  Finally, plane waves illuminating from different angles will be propagated through the potential with different phases.  A beam of light illuminating a reflectionless potential might have all of its energy transmitted through the potential, but the beam shape would be highly distorted in the process, much like a light beam is distorted on passing through an irregularly shaped piece of glass.  A reflectionless potential would therefore serve very poorly as an invisibility device.

It might be possible to overcome these limitations with more sophisticated potentials; curiously, though, it seems that Kay and Moses did not follow up on their work, and neither did any other researchers.  The idea certainly caught people’s attention, however; I’ve personally had a number of researchers bring up the Kay and Moses paper when discussing ideas of invisibility.  The idea of reflectionless potentials set the stage, in a sense, for the later attempts to devise true invisible objects.

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I. Kay and H.E. Moses, “Reflectionless transmission through dielectrics and scattering potentials,” J. Appl. Phys. 27 (1956), 1503-1508.

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### 6 Responses to Invisibility physics: “Reflectionless” objects make an appearance

That’s very interesting! I wasn’t aware of the possibility of using gradient-index materials to reduce reflection — I had only seen it used in gradient-index (GRIN) lenses (to make them flat) and graded-index optical fibers (to keep the light near the fiber core).

In retrospect, I guess this makes perfect sense, since as you note, it’s the limiting case of applying anti-relflective coatings.

2. yoron says:

Thank for posting. You’re always interesting to read. As you like optics you must enjoy holographic systems too. I will put a link to a thread I posted with two new works trying to explain gravity, and would be pleased to read your response. One is from the string theorist Erik Verlinde and the other that came out almost simultaneous is from the theoretical physicist Thanu Padmanabhan. Both are dealing with gravity and seems similar.

I’m reading Erik’s work now and there he refers to ‘the holographic principle’ as described for black holes. I would really appreciate if you would take a look at it and perhaps post an explanation for how to think of it, here or there. It’s not that the idea is all together new to me, but I have severe problems equating it to a ‘materialistic universe’. But never the less he’s making perfect sense on a lot of other subjects, so I would really like to see the idea behind that. You seem to have a gift for such things 🙂

The link goes to the naked scientist.
http://www.thenakedscientists.com/forum/index.php?topic=28588.msg298507#msg298507

3. yoron says:

Lovely 🙂

Thanks.

4. David says:

So… is it possible to design a cloaking device using Kay Moses potentials?