Waves spread out as they pass through an opening or around an edge. That spreading is diffraction. You can hear someone around a corner because sound waves, about a meter long, bend around doorways. Light waves are a million times shorter, so light's diffraction only shows through very narrow openings. The dark bands it makes come from Superposition and Interference, and the angles come from Sine, Cosine and Tangent.
Key idea
Light of wavelength $\lambda$ through a slit of width $a$ makes dark fringes at $$a\sin\theta = m\lambda, \quad m = \pm 1, \pm 2, \dots$$ The central bright band runs between the two first dark fringes; on a screen a distance $L$ away it is about $2\lambda L/a$ wide (small angles). A narrower slit makes a wider pattern.
Why it works
Split the slit into a top half and a bottom half. Pair each point in the top half with the point $a/2$ below it. At the angle where those two are half a wavelength out of step, $\frac{a}{2}\sin\theta = \frac{\lambda}{2}$, every pair cancels and the screen is dark. That is $a\sin\theta = \lambda$. A narrow slit needs a bigger angle to make that half-wave difference, so the pattern spreads.
Common mistake
Thinking a narrower slit makes a narrower central bright band, as if the slit were a stencil. Diffraction works the other way: $\sin\theta = \lambda/a$ grows as $a$ shrinks. Check: shrink $a$ toward $\lambda$ and the central band spreads over the whole screen.
Worked example: Angle of the first dark fringe
Light of $600\ \text{nm}$ passes through a slit $0.0030\ \text{mm}$ wide. Find the angle of the first dark fringe.
- $a = 3.0 \times 10^{-6}\ \text{m}$, $\lambda = 6.00 \times 10^{-7}\ \text{m}$.
- $\sin\theta = \dfrac{\lambda}{a} = \dfrac{6.00 \times 10^{-7}}{3.0 \times 10^{-6}} = 0.200$.
- $\theta = \sin^{-1}(0.200) = 11.5^\circ$.
Answer: $11.5^\circ$
Worked example: Width on a screen
Red light ($650\ \text{nm}$) passes through a slit $0.10\ \text{mm}$ wide onto a screen $2.0\ \text{m}$ away. How wide is the central bright band?
- Small angles: $\sin\theta \approx \tan\theta = y/L$, so the first dark fringe is at $y = \dfrac{\lambda L}{a}$.
- $y = \dfrac{(650 \times 10^{-9})(2.0)}{1.0 \times 10^{-4}} = 0.013\ \text{m}$.
- The band runs from $-y$ to $+y$: width $2y = 0.026\ \text{m} = 2.6\ \text{cm}$.
Answer: About $2.6\ \text{cm}$