There’s Always An Angle

“No, Moire, when I said the glasses get dark or light depending I was talking about those glasses that just block out shiny, like from windows across the street when the Sun hits ’em just wrong.”

“I got this, Sy. That’s about polarized light, Feder, and polarized sunglasses. Sy and me, we talked about that when we were thinkin’ Star Trek weapons.”

“You guys talk about everything, Vinnie.”

“Pretty much. Anyhow, it goes back to how electrons make light. Electrons got charge and that makes an electric field around them, right? When you jiggle an electron up and down the field jiggles and sooner or later that’ll make some other electron jiggle like maybe in your eye and you see that as light. How’m I doing, Sy?”

“You’re on a roll. Keep it going.”

“Okay, so the electron doesn’t have to jiggle only up and down, it can do side‑to‑side if it feels like it or anything in between and the field goes along with all of that. When you got a lot of electrons doing that together, different‑angle waves go out and that poor second electron gets shoved all around the compass, right?”

“Hey, don’t all those jiggles just cancel each other out?”

“Nah, ’cause their timing’s off. They’re not in sync or nothing so the jiggles push in every direction random‑like.”

“How about lasers? I thought their waves all marched in sync.”

“They’re in sync strong‑and‑weak, but I guess whether they’re up‑and‑down in sync depends on the technology, right, Sy?”

“Right, Vinnie. Simple diode laser beams usually aren’t polarized, but special-purpose lasers may be designed with polarization in the package. Of course, any beam can be polarized if it’s bounced off something at just the right angle.”

“What’s the angle got to do with it, Moire?”

“I bet I know. Sy. Is that bounce angle connected to the prism stuff?”

“Nice shot, Vinnie. Carry on.”

“Ok, Feder, follow me ’cause this is a little complicated. Sy, can I borrow your whiteboard?”

“Sure.”

“Thanks. All right, this thick green wiggle is a regular light ray’s electric field, coming in at a low angle and jiggling in all directions. It hits a window or something, that’s the black line, and some of it gets reflected, that’s the red wiggle, and some gets through but not as much which is why the second green line is skinny. The fast‑slow marks are about wave speeds but it’s why the skinny wiggle runs at that weird angle. We good?”

“Mostly, I guess, but where does the polarization come in?”

“I’m gettin’ there. That’s what the dots are about. I’m gonna pretend that all those different polarization directions boil down to either up‑and‑down, that’s the wiggles, and side to side, that’s the dots. Think of the dots as wiggle coming out and going back in cross‑ways to the up‑and‑down. It’s OK to do that, right, Sy?”

“Done in the best families, Vinnie. Charge on.”

“So anyway, the up‑and‑down field can sink into the window glass and mess around with the atoms in there. They pass some of the energy down through the glass but the rest of it gets gets thrown back out like I show it.”

“But there’s no dots going down.”

“Ah-HAH! The side‑to‑side field doesn’t sink into the glass at all ’cause the atoms ain’t set up right for that. That side‑to‑side energy bounces back out and hits you in the eyes which is why you use those polarizing sunglasses.”

“But how do those glasses work is what I asked to begin with.”

“That’s all I got, Sy, your turn.”

“Nice job, Vinnie. How they cut the glare, Mr Feder, is by blocking only Vinnie’s side‑to‑side waves. Glare is mostly polarized light reflected off of horizontal surfaces like water and roadway. Block that and you’re happy. How they work is by selective absorption. The lenses are made of long, skinny molecules stretched out in parallel and doped with iodine molecules. Iodine’s a big, mushy atom with lots of loosely-held electrons, able to absorb many frequencies but only some polarizations. If a light wave passes by jiggling in the wrong direction, its energy gets slurped. No more glare.”

~~ Rich Olcott

Calvin And Hobbes And i

Hobbes 2I so miss Calvin and Hobbes, the wondrous, joyful comic strip that cartoonist Bill Watterson gave us between 1985 and 1995.  Hobbes was a stuffed toy tiger — except that 6-year-old Calvin saw him as a walking, talking man-sized tiger with a sarcastic sense of humor.

So many things in life and physics are like Hobbes — they depend on how you look at them.  As we saw earlier, a fictitious force disappears when viewed from the right frame of reference.  There’s that particle/wave duality thing that Duc de Broglie “blessed” us with.  And polarized light.

In an earlier post I mentioned that light is polar, in the sense that a single photon’s electric field acts to vibrate an electron (pole-to-pole) within a single plane.
wavesIn this video, orange, green and blue electromagnetic fields shine in from one side of the box onto its floor.  Each color’s field is polar because it “lives” in only one plane.  However, the beam as a whole is unpolarized because different components of the total field direct recipient electrons into different planes giving zero net polarization.  The Sun and most other familiar light sources emit unpolarized light.

When sunlight bounces at a low angle off a surface, say paint on a car body or water at the beach, energy in a field that is directed perpendicular to the surface is absorbed and turned into heat energy.  (Yeah, I’m skipping over a semester’s-worth of Optics class, but bear with me.)  In the video, that’s the orange wave.

At the same time, fields parallel to the surface are reflected.  That’s what happens to the blue wave.

Suppose a wave is somewhere in between parallel and perpendicular, like the green wave.  No surprise, the vertical part of its energy is absorbed and the horizontal part adds to the reflection intensity.  That’s why the video shows the outgoing blue wave with a wider swing than its incoming precursor had.

The net effect of all this is that low-angle reflected light is polarized and generally more intense than the incident light that induced it.  We call that “glare.”  Polarizing sunglasses can help by selectively blocking horizontally-polarized electric fields reflected from water, streets, and that *@%*# car in front of me.

Wave_Polarisation
David Jessop’s brilliant depiction of plane and circularly polarized light

Things can get more complicated. The waves in the first video are all in synch — their peaks and valleys match up (mostly). But suppose an x-directed field and a y-directed field are headed along the same course.  Depending on how they match up, the two can combine to produce a field driving electrons along the x-direction, the y-direction, or in clockwise or counterclockwise circles.  Check the red line in this video — RHC and LHC depict the circularly polarized light that sci-fi writers sometimes invoke when they need a gimmick.

Physicists have several ways to describe such a situation mathematically.  I’ve already used the first, which goes back 380 years to René Descartes and the Cartesian x, y,… coordinate system he planted the seed for.  We’ve become so familiar with it that reading a graph is like reading words.  Sometimes easier.

In Cartesian coordinates we write x– and y-coordinates as separate functions of time t:
x = f1(t)
y = f2(t)
where each f could be something like 0.7·t2-1.3·t+π/4 or whatever.  Then for each t-value we graph a point where the vertical line at the calculated x intersects the horizontal line at the calculated y.

But we can simplify that with a couple of conventions.  Write √(-1) as i, and say that i-numbers run along the y-axis.  With those conventions we can write our two functions in a single line:
x + i y = f1(t) + i f2(t)
One line is better than two when you’re trying to keep track of a big calculation.

But people have a long-running hang-up that’s part theory and part psychology.  When Bombelli introduced these complex numbers back in the 16th century, mathematicians complained that you can’t pile up i thingies.  Descartes and others simply couldn’t accept the notion, called the numbers “imaginary,” and the term stuck.

Which is why Hobbes the way Calvin sees him is on the imaginary axis.

~~ Rich Olcott