Section 02
The Idea, Step by Step
Drop two pebbles into a still pond and the ripples spread out and cross. Where two crests meet, the water heaves up extra high; where a crest meets a dip, the surface barely moves. Light does exactly the same thing. Shine a single colour of light through two very thin, closely spaced slits and the screen beyond does not show two bright lines — it shows a row of bright and dark stripes. Those stripes are the proof that light travels as a wave.
Naming the pieces
Three things set the pattern: the wavelength $\lambda$ (how long one ripple of light is — this is its colour), the slit separation $d$ (how far apart the two openings are), and the screen distance $L$. Light from the two slits reaches a given spot on the screen having travelled slightly different distances. That gap is the path difference. When the path difference is a whole number of wavelengths, crest lands on crest and you get a bright spot; when it is off by half a wavelength, crest lands on trough and they cancel into darkness.
Bright fringes & their spacing
$$d\sin\theta=m\lambda,\qquad \Delta y=\frac{\lambda L}{d}$$
1
One worked number. Take red light, $\lambda=650$ nm, slits $d=0.5$ mm apart, screen $L=1$ m away. The stripes are spaced $\Delta y=\lambda L/d=(650\times10^{-9}\times1)/(0.5\times10^{-3})\approx1.3$ mm apart — wide enough to see with your own eye.
The precise picture
Add two waves of the same frequency with a phase difference $\phi=2\pi\,\delta/\lambda$, where the path difference is $\delta=d\sin\theta$. Their combined brightness follows $I=I_0\cos^2(\phi/2)$ — smoothly bright, dark, bright across the screen. A single slit is not a point either: every part of its width $a$ radiates a wavelet (Huygens), and those wavelets cancel in pairs when $a\sin\theta=m\lambda$, carving the pattern into a broad central peak of angular half-width $\lambda/a$ with fainter side peaks. Notice the twist: a narrower slit spreads light more. Squeezing the light in space forces its direction to fan out — the same trade-off that appears in the uncertainty principle.
2
How the sliders map. $d$ sets the fringe spacing (larger $d$ → tighter stripes), $\lambda$ sets colour and overall scale, $L$ stretches the whole pattern, $a$ controls how much a single slit spreads, and $t$ is the film thickness that tints a soap bubble.
3
Try this in the sim above. In Double Slit, raise $d$ and watch the bright stripes crowd together, just as $\Delta y=\lambda L/d$ predicts. Then sweep $\lambda$ from 400 to 700 nm and watch the spacing grow as the colour reddens. Switch to Single Slit and shrink $a$ — the central bright band gets wider, not narrower.
Section 03
Equations & Derivation
Young's Double Slit — Bright Fringes
$$d\sin\theta = m\lambda,\quad y_m = \frac{m\lambda L}{d},\quad m=0,\pm1,\pm2,\ldots$$
Single Slit — Minima
$$a\sin\theta = m\lambda,\quad m=\pm1,\pm2,\ldots\quad\text{(dark fringes)}$$
Diffraction Grating
$$d\sin\theta = m\lambda,\quad \text{resolving power }R=mN$$
Thin Film Interference (one phase reversal, e.g. soap bubble)
$$2nt=(m+\tfrac{1}{2})\lambda\;\text{(constructive)},\quad 2nt=m\lambda\;\text{(destructive)}$$
Symbol Definitions
| Symbol | Quantity | SI Unit |
|---|
| $d$ | Slit separation (double slit) or grating spacing | m |
| $a$ | Single slit width | m |
| $\lambda$ | Wavelength in vacuum | m |
| $m$ | Order of interference (integer) | dimensionless |
| $L$ | Distance to screen | m |
| $n$ | Refractive index of film | dimensionless |
| $t$ | Film thickness | m |
1
Double slit. Light from two slits arrives at a screen with path difference $\delta=d\sin\theta$. Bright fringes when $\delta=m\lambda$ (constructive), dark when $\delta=(m+\tfrac{1}{2})\lambda$ (destructive). Fringe spacing $\Delta y=\lambda L/d$.
2
Single slit diffraction. Each point within the slit acts as a Huygens source. Minima occur when $a\sin\theta=m\lambda$ — pairs of sources cancel. Central maximum is twice as wide as secondary maxima.
3
Diffraction grating. $N$ slits — extremely sharp maxima when all $N$ sources add constructively: $d\sin\theta=m\lambda$. Resolving power $R=mN$ — ability to separate close wavelengths. Used in spectroscopy.
4
Thin film interference. A reflection off a denser (higher-index) medium flips the wave by $\pi$ (half a wavelength); a reflection off a less-dense medium does not. For a soap bubble (air–soap–air) only the top reflection flips, so the two reflected rays begin half a wavelength out of step. Combined with the round-trip path $2nt$, constructive interference needs $2nt=(m+\tfrac{1}{2})\lambda$ and destructive needs $2nt=m\lambda$. Anti-reflection coatings sit between air and glass so both reflections flip — the extra shift cancels, and minimum reflection (destructive) then occurs at $2nt=(m+\tfrac{1}{2})\lambda$.
Ref: Halliday, Resnick & Walker 10th Ed., Ch. 35–36; Serway & Jewett 8th Ed., Ch. 27; Hecht — Optics (5th Ed.).
Section 05
Common Misconceptions
❌ Destructive interference destroys photons.
✅ No photons are created or destroyed. Energy is redistributed spatially — where one wave interferes destructively, another interferes constructively. The total energy in constructive fringes equals what would be uniform if there were no interference. Conservation of energy is always satisfied.
📖 HRW 10th Ed., §35-2.
❌ Diffraction only occurs with waves — not particles.
✅ Matter waves (electrons, neutrons, atoms) also diffract. Davisson-Germer (1927) observed electron diffraction from nickel crystal, confirming de Broglie's hypothesis. Neutron diffraction is now a standard technique for determining crystal structures. Even large molecules have been diffracted.
📖 HRW 10th Ed., §38-4.
❌ A narrower slit produces a narrower diffraction pattern.
✅ The opposite is true: narrower slit → more diffraction → wider central maximum. The angular width of the central maximum is $2\lambda/a$. This inverse relationship is fundamental and manifests in the Heisenberg uncertainty principle.
📖 HRW 10th Ed., §36-2.
Misconception research: Arons — A Guide to Introductory Physics Teaching; Driver et al. — Making Sense of Secondary Science.