## Tuesday, June 30, 2015

### Six Optical Miracles

 Paul Peter Ewald (1888-1985) and Carl Wilhelm Oseen (1879-1944)
Exactly 100 years ago a remarkable paper appeared in Annalen der Physik, a prestigious German physics journal in which both Max Planck and Albert Einstein published their findings. This paper, by German physicist Paul Peter Ewald explained how light behaves when it strikes a sheet of glass. Ewald, in 1915, explained, in effect, how a window works. A few year later, Swedish physicist Carl Wilhelm Oseen extended these findings to explain how light behaves when it strikes a crystal. Together the work of these two men is known as the Ewald-Oseen Extinction Theorem.

(A new proof of the extinction theorem by Mansur Mansuripur has recently been published here,)

What's so mysterious about how a window works. Isn't a window merely a sheet of glass?

 A light wave traveling through a sheet of glass
When light strikes a window some of it bounces off (about 10%) and the remainder is refracted (bent) into the glass at an angle that depends on a number "n" called the refractive index which is different for different materials. If n is greater than 1, the light bends deeper into the material; if n is less than 1, the light bends towards the surface.

In a vacuum the speed of light is equal to a constant c. But in material media, the velocity v of light is equal to: v = c/n. For visible light in glass, the refractive index is about 1.5, so light travels at about 70% of its vacuum speed: inside glass, light travels SLOW. On the other hand, for X-rays in many materials, the refractive index is less than 1, so X-ray light travels FAST -- faster than light in a vacuum.

But those "in the know" realize that not only did Einstein prohibit anything from traveling faster than light, he also showed that light always travels at the same speed no matter who's looking at it. So both the notion of fast X-ray light and slow light in glass would seem to be outlawed by relativity. However this notion of fast and slow light in material media is being taught to high school students. It's called Snell's Law (and seems to have been discovered first by Arab physicist ibn Sahl during the Islamic Golden Age.

Snell's Law (Ibn Sahl's Law) does not violate relativity, because Einstein and Snell (ibn Sahl) are talking about two different ways of measuring velocity. Einstein's prohibition refers to group velocity, how fast a packet of light can move, while Snell's Law refers to phase velocity, how fast the peaks of a wavelet can move. The difference between group and phase velocity is often illustrated by the difference between how fast a caterpillar moves (group velocity) and how fast one of his humps moves (phase velocity). It is clear from this analogy that the phase velocity can be faster or slower than whatever speed limit the caterpillar must obey. Here's a nice animation showing the difference between group and phase velocity for deep ocean waves. (In this ocean wave case, phase velocity is faster than group velocity.)

To fully appreciate the miraculousness of the Ewald-Oseen Extinction Theorem, we must first realize that glass consists of a random arrangement of electrically-excitable molecules. And that light is a traveling electromagnetic wave that will excite each of the molecules it encounters. Each excited molecule will emit a omni-directional light wave (of the same frequency as the incident light) in somewhat the same manner as an ocean buoy will emit a scattered wave when struck by an incident wave train.

 Incident ocean wave striking a buoy produces scattered wave.
But light approaching a sheet of glass is not just going to strike one buoy, but approximately 10^24 buoys or about 1 million billion billion buoys (electrical excitable molecules). And these buoys (in glass) are arranged in a random fashion. What is the first thing you would imagine happening if a wave of light impinged on a medium composed of zillions of randomly arranged excitable molecules? Total chaos. That's what I would guess. The window will turn into some sort of deeply frosted glass that will randomly scatter light in every direction.

If windows are made of zillions of randomly arranged excitable molecules, then windows can not be transparent. That's what I would predict.

The first optical miracle is that windows are transparent. But the second, third and fourth miracle are even better.

Ewald proves for random media like glass (and Oseen does the same for ordered media such as crystals), that the net result of all of those zillions of little excited light waves add up to zero in every direction but three.

Only three big waves survive the fierce destructive interference between zillions of little waves. The first surviving wave travels in the same direction as the incident wave but vibrates exactly 180 degrees out of phase with the incident wave. Thus the incident wave is extinguished as it passes into the glass over a length of a few tenths of a micron (the extinction length). This exceedingly unlikely mechanism for extincting the incident wave gives the extinction theorem its odd name.

Two more waves survive the Great Destruction -- the reflected wave that seems to bounce off the glass but in reality is created by zillions of tiny molecules all radiating with the proper timing to direct a beam only in the reflected direction (and no other) much in the manner of phased-array radar antennas which do not physically move but are aimed by changing the phase relation between separate fixed transmitters. The glass molecules (like the phased-radars) are radiating in all directions but in both radar and window pane destructive interference removes light from every direction except one.

The third wave that survives the Great Cancellation is the internally transmitted wave that travels into the glass at "Snell's angle" with a "slow" phase velocity of c/n. Everywhere inside the glass the little wavelets travel at velocity c, but the net product of all their activity is an effective slow-phase wave, produced, like the reflected wave, by a kind of internal phased-antenna array consisting of zillions of glass molecules.

Here then are four optical miracles:

1. That glass is transparent despite its consisting of zillions of randomly situated electrically-excitable molecules;

2. That the excitable medium, all by itself, completely extinguishes the incident wave;

3. That the excitable medium, all by itself, synthesizes a "reflected wave" traveling in exactly the right direction and no other.

4. That the excitable medium, all by itself, synthesizes an "internal wave" traveling in exactly the right direction (Snell's Law) and no other.

These four miracles apply not just to windows but to anything made of glass, or to anything transparent for that matter, such as the lenses in telescopes, the lenses in your cameras, your eyeglasses, your contact lenses, the living lens and the clear fluid that fills your eye. The inner operation of each of these optical devices is explained by the four-fold miraculous Ewald-Oseen Extinction Theorem. (Happy Hundreth Anniversary!)

But wait. There's more. The EOET was derived in 1915 before quantum mechanics was devised. And quantum mechanics has one final miracle to add to the wonder of classical window panes and lenses.

Classical physics considered all of its waves to be "real", that is, made of something actually vibrating at every location where the wave exists. Quantum physics, on the other hand, considers its waves to be "unreal", that is, made only of sheer possibility. A light wave, for instance, represents only the possibility for a "photon" (or quantum of light) to be observed.

Quantum waves only become real in the act of observation, in a still mysterious process called by some "the collapse of the wave function".

Now imagine the situation of a single photon striking a window pane. I remind you that a window pane consists of a million billion billion electrically excitable molecules. And photons like to excite molecules.

What do you suppose the odds are for the photon to exit the window pane without collapsing its wave function? Because there seem to be a zillion ways for that photon to lose its phase, to shed its dignity, to become entangled with the randomly arrayed electrically-excitable mess that we call "a pane of glass", I would naively guess that a pane of glass must collapse the photon wave function with 100% certainty.

But this conclusion is wrong. For exactly the same reason that the window pane is transparent (namely the Ewald-Oseen Extinction Theorem), the window pane has a lot of fun with the photon but does not actually collapse its wave function.

This is the fifth optical miracle: Window panes do not collapse the photon wave function.

Lenses do not collapse the photon wave function. Your contact lens does not collapse the photon wave function The living lens in your eye does not collapse the photon wave function. Your aqueous and vitreous humours do not collapse the photon wave function.

But inside your retina the photon wave function does somehow finally collapse into an actual event.

Which gives rise to this marvelous visual experience.

The sixth optical miracle.

OrangeAnalytic said...

Very nice and precise summary of the nature of light interacting with transparent optic medium, Nick!

AgentProvocateur said...

beauty-fully observed...wonder-fully presented...
photon-absorbing, waveform-collapsing rhodopsin,
living pigment transducing
visual images inducing...
a million waveforms comingling -
gets me tingling...
phases interfering as
extinction is almost certain...
but three waves survive to tell
the noble laws of Snell...
and behold - we can see the
men behind the curtain!

Graeme Thomson said...

Marvellous piece of writing. Even a layman like myself can catch a glimpse of the quantum miracle.