# iPhone under the spotlight

When writing the quantum mechanics series, I want some introduction to the interference phenomenon to motivate light and particles as waves. That is most easily observed with light, and it can be seen using just your mobile phone. I said that it is easy to see the pattern, but not so easy if you are to perform measurements. But what if you insist? Let’s try!

## Qualitative observations

Before measurements and calculations, let’s make some quick observations for sanity checks of our measurements and calculations. In the figure above, the phone is in landscape orientation. The interference pattern is asymmetric: the light extends further in the direction from top to bottom (i.e., horizontal phone direction). In portrait orientation the interference pattern is a little different. I take the photo again for consistency.

Hm… in portrait orientation, the horizontal phone direction still gets wider and stronger interference pattern, but the difference is not as pronounced as in the landscape orientation.

Perhaps it is just for one phone? Let’s try another: Samsung Galaxy Note 3.

The interference pattern is more complex: it doesn’t look rectangular any more. Horizontal interference is still stronger, and is still more pronounced in landscape orientation. And… somehow they also give different hue! It seems the red light is not well interfered.

I’ve actually tried a third phone, my HTC M8, as well. It is not shown here because I want to take photos with it, and it is tricky to use a phone to take a photo of itself.  I also have a Samsung Galaxy Tab to try. Both give interference patterns similar to the iPhone 5s above.

## Measurements

Now let’s turn to some crude measurements. We don’t need sophisticated instruments (I don’t have them anyway). I learned at school (doing astronomy) that we all have simple rulers for measuring angles: our body! If you fully extend your arms, the angle covered by your hand is fixed. Everyone has a slightly different size, so it is good to calibrate it. Simply count the number of times it takes for the width of a fully extended hand to cover 90° (mine is 4.5), which allows you to know the angle covered by a fully extended hand (90° / 4.5 = 20° for me). Then you can use this to create smaller measures: fist (10°), finger (1.7°), etc. Let’s measure! Put your phone under a spotlight so that you see it’s interference pattern, and adjust it so that it is at arm distance. Match the width of the pattern with your fingers on your second hand. That’s it! Something magical happened when I took this photo. The interference pattern became a square instead of rectangular! I estimate that the first interference pattern, at either direction, forms at around 0.4° to 0.5° from the main reflection. What I missed initially is that there are two fainter reflections before the bright reflection, so it looks rectangular of aspect ratio around 3 to 1. In fact, the 4th and 5th horizontal pattern is so faint that they are invisible above.

## What the theory tells

Time for calculations. How a photon gets reflected at an angle other than the normal reflection angle which equals the incident angle? Due to the regular pixels, the display essentially acts as a diffraction grating. This is the schematics, with two reflection centers. Because the light source and observer are very far compared to the distance between reflection centers ( $d$ below) and the wavelength, it doesn’t hurt to consider the paths to be perfectly parallel. For the photon to land at the right spot on our eyes (or cameras), it should interfere constructively (so that the probability that the photon hit is high). Whether the interference of the photon is constructive or destructive is determined by path length difference. In the figure above, we note that the incident path to $B$ is $d_i = d \sin \theta_i$ longer than the incident path to $A$. Exactly symmetric arguments tells us that the outgoing path from $A$ is $d_o = d \sin \theta_o$ longer than the outgoing path from $B$. Combining the two, the path through $A$ is longer than the path through $B$ by: $\displaystyle d_o - d_i = d(\sin \theta_o - \sin \theta_i) = 2d\sin\frac{\theta_o-\theta_i}{2}\cos\frac{\theta_o+\theta_i}{2}\quad.$

For constructive interference to occur, this should be an integer multiple of the wavelength of the photon, $\lambda$. This is always the case (path difference is 0) if $\theta_i = \theta_o$, which is the real reason why the reflection rule “incident angle equals reflected angle” worked so well. But we want the non-zero multiples to find the interference angles.

We will simplify our life with some approximations. We note that the difference between $\theta_i$ and $\theta_o$ is going to be tiny, something within 5°. So we can approximate $\sin\frac{\theta_o-\theta_i}{2}$ by just $(\theta_o-\theta_i)/2$, and $\cos\frac{\theta_o+\theta_i}{2}$ by just $\cos \theta_i$. This produces the simpler formula for interference: $\displaystyle d(\theta_o-\theta_i)\cos\theta_i = n\lambda\quad,$

or $\displaystyle \Delta \theta = \theta_o-\theta_i = \frac{n\lambda}{d\cos \theta_i}\quad.$

The wavelength of red light is around 650 nm. From the data sheet, iphone 5s display is 326 ppi (pixel per inch), so inter-pixel distance is 25.4 mm / 326 = 78 μm. Because we are measuring the horizontal pixels at center, we don’t have a $\cos \theta$ to divide. Following the formula, we have $\displaystyle \Delta \theta = \frac{\lambda}{d} = \frac{650\mbox{ nm}}{78\mbox{ \begin{math}\mu\end{math}m}}$

This works out to be 0.0083 radians, or 0.48°. A very good agreement considering how crude our estimates are!

The formula also tells us that the horizontal and vertical separation of the patterns are different, due to the $\cos \theta$ factor. It makes the vertical separation slightly larger. E.g., if the top-to-bottom incident angle is 30°, the factor is $1/\cos 30^\circ = 1.155$, so the top-to-bottom distance between interference is 15.5% larger than that in the left-to-right direction. This explains why the interference is more pronounced when the phone is in landscape orientation: the stronger horizontal direction is combined with the effect with a 15.5% enhancement. I guess other factors are also at play, though.

## One out of Three

Our next task is to understand why one every three of this horizontal interference patterns is much stronger, but the vertical direction does not show something similar. Because 3 is such a magic number, I thought of sub-pixel layout. I read the data sheets of iPhone 5s and Samsung Note 3 more carefully. The former is said to use an LED-backlit IPS LCD display, while the latter is a super AMOLED display. But it doesn’t seem to mean much about the sub-pixel layout.

Can I discover it myself? I suddenly recall that my son received a Christmas gift (thanks Ceci)… Under the little toy microscope (said to magnify by 50x to 80x), the iPhone display showing a white “a” on a black background looks like this: Yes, I’m correct! It has horizontal sub-pixel layout of red-green-blue, so the 326 ppi is actually 978 ppi horizontally. The brighter interference pattern is caused by this more dense pixel count. The slightly fainter pattern is probably caused by the perfectly repeating pattern once every 3 sub-pixels.

What about the Note 3? This is how the top part looks like under the toy microscope, when it displays a black “2” over a white background… That’s pretty… and that’s complex. That’s why we see a pretty complex interference pattern on it! Repeating the experiment, I found that regular interference pattern is seen at 8 directions instead of 4. This should be expected given the sub-pixel layout above. It is hard to make a measurement this time, unluckily. It is too complex, and the somewhat curved screen protector doesn’t help either. I just note that at low angle the interference pattern is really beautiful. There are quite a few things I don’t understand: The stronger horizontal interference and the greenish hue in landscape orientation are still complete mysteries to me. If you have an answer, please let me know!

Isaac To, January 2017

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