In writing my previous post on The Murderer Invisible, I started thinking again about the relationship between something being “transparent” and something being truly “invisible”. Most of us can appreciate that, under the right circumstances, a transparent object like a glass window can be very hard to see, but most of us also appreciate that glass is not even close to fitting the popular perception of invisibility.  In fact, though we encounter plenty of transparent things in nature, we don’t encounter invisible things.

What’s the difference?  In this short post, I thought I’d try and clarify things.

Under most circumstances, the optical properties of a bulk material such as glass can be described by two quantities: the refractive index (labeled as n) and the attenuation coefficient (often labeled as κ).  Both of these quantities have a relatively straightforward physical meaning, but have complicated implications.

The attenuation coefficient κ is a measure of how strongly light is dissipated (or attenuated) as it travels in a medium.  Typically, the intensity of light exponentially decays as it propagates in matter, and the quantity is the distance at which the intensity has dropped to 13% of its original value, or .  This is sometimes referred to as the “skin depth”.

There are two main mechanisms by which a beam of light can be attenuated in matter: absorption and scattering.  Physically, absorption occurs because the atoms of the material suck up the photons from the light beam, a bit like little sponges sitting in the path of a stream of water.  The energy of the photons is converted into other forms, typically heat (think of your car interior getting really hot when the sun is shining through the windows).

Bulk materials in which scattering is dominant typically have weak absorption and have microscopic particles embedded within them that deflect (or scatter) photons.  Most photons in a beam of light will bounce around inside the material like balls in a pachinko machine, and are removed from the original beam.

Depending on the strength of absorption in the bulk material, these deflected photons either are eventually absorbed or leave the medium in some other direction.  Opaque glass is an example of the latter: light goes through the glass, but the beam has been distorted to such an extent that no image is transmitted.  Milk is an example of the former: it is mostly water, but has globules of fat suspended in it that act as the microscopic scatterers.

The difference between “transparent” and “opaque” depends on the strength of attenuation.  For very small values of κ, there is hardly any attenuation and the material is transparent.  So we can say that a material is transparent if we can effectively treat .  But this transparent does not mean it is invisible, as we now make clear.

The refractive index n represents the fraction by which the speed of light is reduced in the medium: that is, if the speed of light in vacuum is c, then the speed of light in the medium is c/n.  This change in speed has a dramatic effect: as I have discussed in a recent “basics” post, when light crosses a planar interface from a medium of one refractive index to another, it changes its direction of travel in accordance with Snell’s law.

Snell’s law relates the refractive indices of light in the two media to the sines of the angles of incidence and transmission,

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The phenomenon of refraction is relevant for two reasons related to the discussion of invisibility.  First and foremost, our picture of refraction above is missing something: in almost all cases of refraction, some fraction of the light is reflected.  Our picture above should more accurately look as follows.

This reflected light is one of the reasons why a transparent object is typically not an “invisible” one.  Though we can see through a pane of glass, we can also usually see some of the light reflected off of it, which alerts us that something is there.

The omnipresence of reflection in ordinary materials is a real problem for optical systems.  In eyeglasses and camera lenses, reflection from the exterior of the glass reduces the amount of light received, and multiple reflections within the glass causes glare.  This can be mitigated by using an anti-reflection coating, which consists of a thin layer of additional material on the surface of the glass.  Anti-reflection coatings take advantage of the wave properties of light: an optical coating that is a quarter-wavelength thick will result in waves reflected from the exterior and interior surface that are completely out of phase and cancel each other out, as the following figure illustrates.

The problem with such a simple anti-reflection coating is that it is tuned to a particular wavelength (color) of light and to light coming in at normal incidence (directly into the surface).  Due to the nature of interference, the coating will not perfectly cancel out reflections for light of other wavelengths or other directions of illumination.  More complicated coatings with multiple layers can be produced which will improve the anti-reflection effect, but these will also not be perfect.

Of course, there will be no observed reflected light if there is little or no light on the observer’s side of the glass! This is used to great effect in the Pepper’s ghost illusion, which I have talked about before in some detail.  In the late 1800s, theater audiences were thrilled by the appearance of ghosts on stage.  The trick was the clever placement of a slanted piece of glass between audience and performers, with a brightly-lit “ghost” in a hidden nook.

Because the audience is on the dark side of the glass, there are no reflections to reveal its presence.  When the ghost is illuminated, on the audience side of the glass, its reflection now appears as a phantasm to the shocked viewers!

Even if we could completely eliminate reflections from a transparent object, it would still not be invisible!  When light is refracted, it changes direction; in an ordinary pane of glass, this isn’t a significant effect, because the light “un-refracts” when it leaves the material.

Light rays passing through any irregularly shaped piece of glass will be in general be deflected in ways that will be observable.  For instance, if we managed to fabricate a lens with a perfect anti-reflection coating, the fact that images gets focused through the lens would be a dead giveaway to its presence, even if we can’t see the glass itself!  The effect would be very much like that of the “Predator“: we would see through it, but the objects behind it would appear distorted.

So what would it take for an ordinary material to be invisible?  Not only would it need to be transparent (κ = 0) but it would also have to produce no refraction.  This would only be possible if its refractive index were equal to that of air.  Our atmosphere hardly slows down light at all, and the refractive index is nearly equal to that of vacuum: n = 1.

When H.G. Wells wrote The Invisible Man, he was very aware of this requirement, and his character Griffin explicitly mentions matching the index of the invisible object to air:

“If a sheet of glass is smashed, Kemp and beaten into a powder, it becomes much more visible while it is in the air; it becomes at last an opaque white powder. This is because the powdering multiplies the surfaces of the glass at which refraction and reflection occur. In the sheet of glass there are only two surfaces; in the powder the light is reflected or refracted by each grain it passes through, and very little gets right through the powder. But if the white powdered glass is put into water, it forthwith vanishes. The powdered glass and water have much the same refractive index; that is, the light undergoes very little refraction or reflection in passing from one to the other.

“You make the glass invisible by putting it into a liquid of nearly the same refractive index; a transparent thing becomes invisible if it is put in any medium of almost the same refractive index. And if you will consider only a second, you will see also that the powder of glass might be made to vanish in air, if its refractive index could be made the same as that of air; for then there would be no refraction or reflection as the light passed from glass to air.”

“Yes, yes,” said Kemp. “But a man’s not powdered glass!” “No,” said Griffin. “He’s more transparent!“

There’s an additional wrinkle to the problem, as well: the attenuation coefficient and the refractive index are both quantities that generally depend on the wavelength (color) of light.  To be invisible, our material would need to be transparent and have a vacuum refractive index over all of the visible wavelengths of light.  This just doesn’t happen with ordinary materials in nature.

So how can we make something invisible, in principle?  By changing the rules described above!  The first and foremost change: most materials have the same refractive index throughout them.  A piece of crown glass, for instance, has a refractive index n =1.5 throughout its bulk.  It is possible, however, to fabricate materials that have a gradient refractive index: an index that varies throughout the bulk.  How does this help us?  With a varying refractive index, we can actually guide a ray of light to bend in any direction we like, with an appropriate gradient.  We can imagine a gradient index as a large number of very small layers of material; the picture below shows the analogy.

The first cloaks introduced by Leonhardt and by Pendry et al. are based on this idea: an interior region is “cloaked” from all outside light by a shell which consists of a gradient refractive index.  This shell is ideally non-absorbing (transparent) and guides light rays smoothly around the interior region and returns them to their original trajectory on the other side.  The problem with refraction experienced by objects such as lenses is avoided: all light rays continue on the opposite side of the cloak as if they had never been deflected in the first place!  The now-famous schematic of a Pendry cloak in action, showing the rays being guided, is shown below.

But what about the problem of reflection?  The optics gets a little complicated, but in short the gradient index is designed to decrease to a value where it is equivalent to the refractive index of air at the boundary!  Therefore there is no reflection at the surface of the cloak, and all light rays pass into and through the cloak.

There are still a number of limitations with this conception of invisibility.  The foremost of these is that, to design a gradient index with the desired properties, one must be able to control the structure of matter on nearly an atomic scale.  Though great strides have been made along these lines, fabricating a cloak of reasonable size with the desired properties is still an insurmountable technical challenge.  Another limitation is that the original cloak, as designed, only shields the object from one wavelength of light!  Like the anti-reflection coating discussed earlier, the wave properties of light are important to the functioning of the cloak, and the wave properties depend crucially on the wavelength.  Other cloak designs have been proposed to work at multiple wavelengths, but broadband invisibility is still a challenge.  A third limitation is the omnipresence of absorption: if some light is absorbed by the cloak while passing through it, as is always the case, the cloak will cast a detectable “shadow”.

As this post illustrates, achieving true invisibility is extremely challenging, and requires more than just making something “see-through”.  However, with theoretical and experimental advances in optics, we might actually see something that can’t be seen in our lifetime nevertheless!

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Here’s one more “ghost” image from L’Optique, just because it’s cool!

 

Bonus optics question: what is physically incorrect in this Pepper’s ghost illustration?