Rayleigh criterion and resolving power

Diffraction is not just a curious phenomenon: it is what sets a fundamental limit on the sharpness of any optical instrument. When light from a point source passes through the circular aperture of a lens, it does not form a point but a blurred disc (the Airy disc), owing to diffraction. Two nearby sources produce two discs that, if too close together, merge and can no longer be told apart.

The Rayleigh criterion establishes the minimum separation at which two sources are still just resolvable.

Key formula — Rayleigh criterion

Two point sources seen through a circular aperture of diameter $D$ are just distinguishable when their angular separation is: $\ev{\theta_\text{min} \approx 1.22\,\frac{\lambda}{D}}$ The factor $1.22$ comes from the Fraunhofer diffraction of a circular aperture.

The message is clear: to resolve finer details (small $\theta_\text{min}$) you need either a large aperture $D$ or a small wavelength $\lambda$. This is why large telescopes have enormous mirrors and the most powerful microscopes use short-wavelength light.

Example — Resolution limit of the eye (and of Webb)

The human iris has diameter $D \approx 3\;\text{mm}$. For average visible light ($\lambda \approx 550\;\text{nm}$): $\theta_\text{min} \approx 1.22\cdot 550\cdot 10^{-9}/(3\cdot 10^{-3}) \approx 2.2\cdot 10^{-4}\;\text{rad} \approx 45''$ At $10\;\text{m}$ distance we can thus distinguish objects about $2\;\text{mm}$ apart. Telescopes do far better: with the James Webb ($D = 6.5\;\text{m}$) one gets $\theta_\text{min} \approx 10^{-7}\;\text{rad}$, enough to resolve a coin at $30\;\text{km}$ distance — were there no atmosphere in the way.

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Topics: Onde Concepts: Diffrazione

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