# Magic Mirror on the Wall, How Do You Even Work at All?

**Bernard Murphy, PhD, Chief Technology Officer, Atrenta Inc.**

12/1/2014 05:45 PM EST

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The more one looks into it, the more one realizes that aspects of mirrors that initially appear to be intuitive are, in fact, extremely hard to explain.

EE Times editor Max Maxfield recently offered this challenge in a comment on a previous post: "I still cannot wrap my brain around how mirrors work -- from simple things like why is the angle of incidence equal to the angle of reflection, all the way up to how the photons 'bounce' off the atoms forming the mirror without being scattered to the four winds, as it were."

He's not looking for an easy answer using basic optics or even Maxwell's equations. His question is based on Richard P. Feynman's 1990 book *QED: The Strange Theory of Light and Matter* (where QED stands for Quantum Electrodynamics). I thought that I would knock out a quick response with a few examples, but this has turned into one of the harder questions I have attempted. Getting to a reasonable answer has made me reset my own understanding.

In fairness to anyone who hasn't read the book, here is a highly condensed summary of how Feynman explains reflection. The idea is to sum components of reflection over all conceivable paths. We want to prove that the angle of incidence is equal to the angle of reflection (AOI=AOR), but we can't start with that assumption. Instead, we have to consider all paths. Feynman does this considering the experiment below -- looking at the various possible paths from the source reflecting off each part of the mirror and ending at the detector.

We sum contributions at the detector by considering each contribution as an amplitude with an associated phase (shown by the arrows below the mirror). We assume the only difference in phase between the paths is due to the lengths of the paths (more on this later), which results in phase shifts between contributions at the detector. The phase shift changes slowly around the center line (at which point AOI=AOR), where the path length varies slowly. The path length (and therefore the phase) changes faster as we move away from the center. When we add these contributions together, they add constructively near the center but increasingly cancel through phase mismatch as we move away from that center. As a result, we obtain a peak around AOI=AOR and very little intensity as we move away from the peak on either side.

All of this is understandable, but what does it have to do with QED? In researching this blog, I first thought Feynman was using creative license to keep his explanation simple. Then I decided he was bending the truth just a bit. Finally, I realized his explanation -- apart from minor details -- is completely accurate and is the most intuitive explanation of QED I can imagine. Thus, the best I can hope for is to add some color to that explanation.

Let's start by saying that we believe photons are real, because we can reduce light intensity until we see single flashes at the detector, and the flashes always have the same intensity for a given frequency of light. So light is quantized, but whatever behavior we invent for this new model, it must still correspond at a macro scale with everything we expect about light behaving as a wave. We also need to double-check what has to be new and what is really just unexpected classical behavior.

An apparent problem emerges in imagining the experiment being performed using a laser as illustrated below.

The light isn't going all over the place, so what gives? In fact, this experiment is a little deceptive. If we look at the mirror from behind the laser, we can see a light spot, which means that light is reflected back toward the laser. This means that, even at the macro level and even for a laser, light is scattered in all directions at reflection. On this point, Feynman's explanation is completely classical, though not the way we normally think about light. Scattering in this way also corresponds with Huygens' principle (1678) that a light wavefront advances by treating each point on the wavefront as a new wavefront, which expands in all directions.

Given this, summing up the paths accounting for phase is also completely classical. That's what you do with waves. There are just two problems. The first is how all this applies to photon "particles"; the second concerns the assumption about phase differences. On the first point, my reading shows two lines of thinking. The most heavily represented is what I'll call the "mystery and imagination" track. Quantum behavior is weird, and we can't really understand what is happening, but the math works. In the meantime, we wrestle with how to imagine a photon particle behaving like a wave. I think most of us are secretly attracted to this track, because it gives us exotic behaviors as fuel for philosophizing about exotic possible causes. Perhaps photons are extended wave packets and behave as waves. Perhaps the universe splits into multiple universes at each event such as reflection, and so on.

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Max The Magnificent 12/2/2014 6:32:02 PM

even after you rotate it from portrait to landscape?ROFLOL

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Some Guy 12/2/2014 6:18:56 PM

"... all in your head!"

I've suspected this for some time now. And my kids will tell you they knew it all along.

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mhrackin 12/2/2014 5:22:54 PM

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ccorbj 12/2/2014 4:58:54 PM

And then there's the measurement problem, which is only for advanced insanity...

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Some Guy 12/2/2014 4:36:53 PM

Why does a mirror swap left and right,

but not up and down?

even after you rotate it from portrait to landscape?

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Max The Magnificent 12/2/2014 3:47:20 PM

and then it starts to get complicated and confusing...I'm laughing through my tears :-)

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GSKrasle 12/2/2014 3:34:22 PM

It gets worser!

Back in my MatSci/Metallurgy days, I did a big research project on why metals are, well, metallic. In particular, why Cu and Au are coloured, and the chemically/electronically very similar Ag is not.

There are a lot of presumably equivalent models that apply to light/EM reflection (and transmission), some particularly applicable to metals, some to anything metallic-looking, and some that are more general. Modeling metals as a conductive 'gas' of electrons with a periodic (or for liquids, a statistically-distributed) lattice of positive charges works, and gives a 'plasma temperature' for the 'gas' which is very high and an amusing thing to contemplate. Impinging EM fields induce currents, which can (depending on the local resistance, inductance, capacitance [which are affected by the relativistically-adjusted electron{/hole} mass and the positive lattice]), produce a new outgoing EM field with, you guessed it, the characteristics we expect in a metallic reflection. But that model is inapplicable to non-metals, where reflections also have a peculiar dependency on the polarization of the incident EM field. But for metals, this model makes light reflection similar to antenna reflection. Modeling the surface as a mismatch in the complex electromagnetic impedance works too (and was the most successful for my purposes). (The complex refractive index is a function of λ, ε, μ, and all sorts of other parameters.)

Long story short, Ag IS coloured... if you're an insect. Cu, Ag and Au basically have a low-pass in their reflection characteristics, as does everything, but Cu is in the green, Au in the blue, and Ag in the near-UV. On a larger-scale than the narrow window of human visual sensitivity, they look very very similar. Li also has interesting characteristics, and both the colours of the transmitted light (if you have a thin enough layer) and of the liquid metals are interesting. A practical consequence is that UV and a UV-sensitive camera can be used to distinguish Ag- from Al-plated chips (which was once important to me when some dice got mixed and had to be non-contaminatedly segregated).

Oh, and your concern about 'bumpiness' of the surface applies only if that 'bumpiness' is comparable-to or larger than the incident wavelength (and THAT may depend on incident angle)....

Which brings me to my image-processing/software days: When rendering an image, the brightness of the light from a point on a surface varies according to how close (angularly) the eye is to the perfect specular direction. For matte objects, that specular reflection brightness may be modeled as proportional to (cos(θ)), where θ is the angle from the viewing direction to the ideal specular direction. For shinier objects, there is an exponent: (cos(θ))^n, with n=>∞ for metallic reflection. (Phong Shading in Raytracing). So 'reflections' from your bumpy surfaces will actually have an exponent n which is a function of λ, just as the components of the complex index of refraction (n, κ) are.

And then it starts to get complicated and confusing; I doubt this site would be happy if I dumped a bunch of equations on it....

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Max The Magnificent 12/2/2014 2:24:33 PM

I'd prefer port and brandy.I'll drink to that! LOL

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RichQ 12/2/2014 2:22:40 PM

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Max The Magnificent 12/2/2014 2:06:48 PM

It seems to me that the problem is our use of the terms "left" and "right" which are relative to the observer.You'd prefer "Port" and "Starboard"?