Foundations — Thin film interference — reflected and transmitted

Before you can read the parent note Thin film interference — reflected and transmitted, you must own every symbol it throws at you. Below, each symbol is built from plain words → a picture → why the topic needs it, in an order where each one rests on the previous.

1. What a wave even is: $\lambda$ (wavelength) and "in step"

Picture a rope you shake up and down. A travelling wave is a repeating pattern of crests (high points) and troughs (low points) moving along.

Look at the figure: the horizontal red arrow spans exactly one $\lambda$. Notice the green wave: it is an identical wave shifted sideways. Two waves are:

• in step (in phase) if their crests line up — they add to make a bigger wave (bright),
• out of step (out of phase) if one's crest sits on the other's trough — they cancel (dark).

Cover and recall:

2. Interference and coherence

When two waves overlap and their heights add point-by-point, we call the result interference. This is the beating heart of Interference of light and you meet it first in Young's double slit.

In a thin film the two rays came from the same original wave (it split), so they are automatically coherent — that is why films show steady colours and don't just flicker into grey.

3. Path difference $\Delta$

Now the key measurable quantity.

In the figure the lower wave takes a detour and travels an extra length marked $\Delta$. The rule that ties $\Delta$ to bright/dark is the entire game:

Why does the topic need $m$? Because many thicknesses give bright — one for each whole number of wavelengths that fit. $m$ counts which one.

4. Refractive index $n$, and why light "costs more" inside glass

Inside a medium light slows down, so its crests bunch up closer — the wavelength inside shrinks:

${\lambda }_{\text{film}}=\frac{\lambda }{n}$

Look at the figure: same wave, but in the teal medium ($n>1$) the crests are squeezed together — more crests fit in the same physical distance.

Recall:

5. The angles: ${\theta }_i$, ${\theta }_r$, and Snell's law

Light rarely hits straight on. We need to measure directions from the normal — an imaginary line perpendicular to the surface.

The rule linking them is Snell's law:

$\sin {\theta }_i=n\sin {\theta }_r$

Why the topic needs $\cos {\theta }_r$: the film thickness $t$ is measured straight across (along the normal), but the ray travels slanted. The projection of the slanted trip onto the normal direction is what counts, and projecting = multiplying by $\cos {\theta }_r$. That is exactly why the extra path is $2t\cos {\theta }_r$ and not $2t/\cos {\theta }_r$.

6. Thickness $t$

The ray goes down and back up, so it crosses the thickness twice — that's where the factor 2 in $2nt\cos {\theta }_r$ comes from: 2 crossings × index $n$ × depth $t$ × the projection $\cos {\theta }_r$.

7. The phase change on reflection: the $\lambda /2$ "hiccup"

This is the trickiest symbol, so build it slowly. See Phase change on reflection.

In the figure the top wave (rarer→denser) comes back inverted; the bottom one (denser→rarer) comes back the same way up.

Recall:

8. Reflected vs transmitted, and conservation of energy

Some light bounces back (reflected), some passes through (transmitted). By Conservation of energy, light that isn't reflected must be transmitted — so the two patterns are complementary (photographic negatives). Where reflection is bright, transmission is dark. Keep this fact ready; it's why the parent gives two swapped formula boxes.

How it all feeds the topic

Equipment checklist

Read each; only click when you can answer without hesitation.