Foundations — Resolving power — Rayleigh criterion

This page assumes you have seen none of the symbols in the parent note. We build each one — plain words, a picture, and why the topic needs it — in an order where every idea leans on the one before it.

1. Angle measured in radians — $\theta$

But we do not measure it in degrees here. We measure it in radians.

Why radians and not degrees? Because of one clean fact you can read straight off the picture: $\text{arc length }s=R\theta (\theta \text{ in radians})$ The arc length is just the radius times the angle — no messy conversion factor. Degrees would force an ugly $\pi /180$ into every equation.

2. Distance to the object — $L$, and turning an angle into a length

Before we go further we need one more everyday symbol.

Why the topic needs it: an angle on its own does not tell you a real gap in millimetres. But if you know both the opening angle $\theta$ and how far away things are ($L$), the picture of the radian gives you the gap directly. Treat the aperture as the centre of the circle and the far plane as the rim at radius $L$; then the arc-length rule $s=R\theta$ becomes $s=\theta L(\theta \text{ in radians}).$ So $s$ is the real separation (say, in millimetres) between two points that subtend the tiny angle $\theta$ at distance $L$. This single line is how every "how far apart can two dots be?" answer is produced.

3. Wavelength — $\lambda$

Why the topic needs it: the amount a wave spreads when it squeezes through a hole depends on how the wavelength compares to the hole size. A long-wavelength wave (big $\lambda$) fans out a lot; a short-wavelength wave fans out little. Visible light has $\lambda$ between about $400\text{ nm}$ (violet) and $700\text{ nm}$ (red), where $1\text{ nm}=10^{-9}\text{ m}$ (a billionth of a metre). Green light near $\lambda =550\text{ nm}$ is a common stand-in for "average" visible light.

See Huygens principle for why a wave spreads at all, and Single-slit diffraction for the full spreading pattern.

4. Aperture and its size — $D$ (or slit width $a$)

Two shapes matter:

• A circular aperture — diameter $D$ (the full width straight across the circle). Eyes, lenses, telescopes.
• A slit — a long thin gap of width $a$. Used in textbook diffraction.

Why the topic needs it: diffraction spreading is $\propto 1/D$. A wide hole spreads the light less and gives a sharper spot; a narrow hole spreads it more and blurs. The whole resolving-power story is a tug-of-war between $\lambda$ (wants to spread) and $D$ (wants to keep it tight).

5. Diffraction and the Airy disc

For a circular hole, the spread-out light does not make a uniform smear — it makes a bright central disc surrounded by faint rings. That central bright disc is the Airy disc, and its first dark ring sits at a specific angle from the centre.

The detailed shape of this disc — and where the dark rings fall — comes from a special function; see Airy disc and Bessel functions.

6. The factor 1.22 and the first minimum

Where does $1.22$ come from? For a slit the first dark fringe sits at angle $\lambda /a$. For a circle, the geometry redistributes the light and pushes the first dark ring slightly further out — the exact amount is set by the first zero of a mathematical object called the Bessel function $J_1$, which happens to give the number $1.22$.

7. The Rayleigh criterion and the symbol ${\theta }_R$

Everything above sets up the one rule the parent note is built on.

Combine ${\theta }_R$ with Section 2: two dots at distance $L$ are just separable when their gap is $s={\theta }_{R}L=1.22\lambda L/D$.

8. Numerical aperture — N.A. $=n\sin \beta$

For microscopes the object is close, so the relevant size is not the diameter but the cone of light the objective collects.

Why the topic needs it: for a near object the smallest resolvable distance is $d_{\min }=\frac{0.61\lambda }{\text{N.A.}}.$

9. The small-angle shortcut

Why the topic needs it: stars are unimaginably far, so the angles are microscopic ($10^{-4}$ rad or less). This is what lets us use $s=\theta L$ (Section 2) directly instead of dragging trigonometric functions around. See Small angle approximation.

How these foundations feed the topic

The chain is short and each link is a symbol you now own:

1. $\lambda$ (Section 3) and $D$ (Section 4) together cause diffraction (Section 5).
2. Diffraction makes the Airy disc, whose first dark ring sits at ${\theta }_{\min }=1.22\lambda /D$ — the 1.22 coming from the Bessel function (Section 6).
3. The Rayleigh criterion lines up one disc's centre with the other's first minimum, giving the limiting angle ${\theta }_R={\theta }_{\min }$ and the resolving power $1/{\theta }_R$ (Section 7).
4. Radians ($s=R\theta$, Section 1), the distance $L$ (Section 2) and the small-angle shortcut (Section 9) turn that angle into a real gap $s={\theta }_{R}L$.
5. For close objects, numerical aperture $n\sin \beta$ (Section 8) replaces $D$ and the Rayleigh $1.22$ becomes $0.61$, giving the microscope limit $d_{\min }$.

Return to the parent whenever you are ready: Resolving power — Rayleigh criterion. See also Diffraction grating — resolving power for the same idea applied to spectral lines.

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