Diffraction — single slit intensity pattern derivation: Foundations

This page builds every symbol and idea the parent derivation leans on, starting from a smart 12-year-old who has never seen any of it. Nothing below is assumed — each block earns the next.

0 — What a "wave" and a "wavelength" even are

Look at the figure: the little rulers mark off one full $λ$ — from one crest (top of a hump) to the next crest. For visible light $λ$ is tiny, around $400$ – $700$ nanometres ( $1$ nm $= 1 0^{- 9}$ m, a billionth of a metre).

1 — Phase: how far along its cycle a wave is

Two waves that are "in step" (same phase) add to make a bigger wave. Two waves exactly half a cycle apart ( $π$ out of phase) are one going up while the other goes down by the same amount — they cancel. This up-adds / opposite-cancels rule is the entire physics of the pattern.

2 — Path difference: why phases end up different

The link between extra distance and extra phase is the key sentence of the whole derivation:

3 — Angles and the slit: $θ$ , $a$ , and $sin θ$

In the figure, two parallel rays leave the top and bottom edges of the slit heading toward the same distant point at angle $θ$ . Drop a perpendicular (the dashed line) from the top ray onto the bottom ray. The little right triangle that appears has:

hypotenuse = the slit width $a$ ,
the side opposite the angle $θ$ = the extra path the bottom ray travels.

4 — Adding many waves: phasors

Once every strip has its own phase, we must add all their waves. Adding wiggly curves by hand is painful, so physicists use a picture called a phasor.

When all phases are equal, the arrows point the same way → they stack into one long straight arrow → maximum brightness.
When the phases fan out evenly and curl all the way back to the start, the arrows form a closed loop → start and end coincide → total arrow has length zero → darkness.

This is the geometric heart of Step 2 in the parent note. See Phasor Addition of Waves for the full machinery.

5 — The two starring variables: $β$ and $I$

Recall What does

$β = 0$ mean physically? Zero phase spread — every strip perfectly in step (this happens at $θ = 0$ ). The phasors stack straight, $\frac{s i n β}{β} \to 1$ , and $I = I_{0}$ : the central maximum. ::: $β = 0$ means all wavelets in phase → brightest central peak.

6 — The tool that appears in the answer: $\frac{s i n β}{β}$

Here is the geometry that produces the famous ratio, shown step by step rather than just asserted. The tiny equal phasors, each turned a little more than the last, bend into an arc of a circle. Call the radius of that circle $R$ .

Now connect these to light:

The arc length never changes as $θ$ varies — it is the total of all the little phasor lengths laid end to end, which is exactly the all-in-phase amplitude. So $arc = A_{0}$ , giving $2 R β = A_{0}$ .
The resultant amplitude $A$ is the straight chord across that arc: $A = chord = 2 R sin β$ .

Divide one by the other to eliminate the unknown radius $R$ : $\frac{A}{A _{0}} = \frac{2 R s i n β}{2 R β} = \frac{s i n β}{β} ⟹ A = A_{0} \frac{s i n β}{β} .$

At $β = 0$ this ratio is $\frac{0}{0}$ , which looks undefined — but as $β$ shrinks, $sin β \approx β$ , so the ratio approaches $1$ . That is why the centre is finite and bright, not a hole.

Prerequisite map

The parent derivation and its neighbours build on all of this: see Diffraction — single slit intensity pattern derivation, and the physics of "many sources" ideas in Huygens Principle, Young's Double Slit Experiment, and later Diffraction Grating.

Equipment checklist

Cover the right side and check you can answer each before reading the derivation:

One full up-and-down of a wave covers a distance of... ::: one wavelength $λ$ .
One full cycle of phase equals... ::: $2 π$ radians (or $36 0^{\circ}$ ).
Extra path $Δ$ turns into extra phase via the factor... ::: $\frac{2 π}{λ}$ .
Two waves exactly $π$ out of phase do what when added? ::: cancel each other completely.
The extra distance the bottom-edge ray travels compared with the top-edge ray is... ::: $a sin θ$ .
We use $sin θ$ (not $cos$ or $tan$ ) because we project the slit width onto... ::: the direction the rays travel.
The fringe order $m$ counts... ::: which dark fringe (its values are $\pm 1, \pm 2, \dots$ , never $0$ ).
The number of dark fringes that can appear is limited by the condition... ::: $∣ m ∣ \leq a / λ$ (since $∣ sin θ ∣ \leq 1$ ).
A phasor is an arrow whose length means ... and whose direction means ... ::: length = amplitude, direction = phase.
Phasors curling into a closed loop means the total field is... ::: zero → a dark fringe.
$A_{0}$ is defined as the amplitude when... ::: every strip is in phase ( $θ = 0$ ); it equals the arc length of the phasor curve.
For a circular arc of radius $R$ subtending angle $2 β$ : arc length = ... and chord = ... ::: arc $= 2 R β$ , chord $= 2 R sin β$ .
$β = \frac{π a sin θ}{λ}$ represents physically... ::: half the total phase spread across the slit.
Intensity relates to amplitude $A$ by... ::: $I \propto A^{2}$ (square it).
As $β \to 0$ , the ratio $\frac{sin β}{β} \to$ ... ::: $1$ (giving the bright central maximum).