Rayleigh Criterion
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The Rayleigh criterion is a widely used standard for determining when two point sources can be distinguished (resolved) by an optical system.
Statement
Two point sources are considered just resolved when the central maximum of one source’s Airy pattern falls on the first minimum (first dark ring) of the other’s.
For a circular aperture of diameter $D$:
$$ \theta_{\text{min}} = 1.22 \frac{\lambda}{D} $$where:
- $\theta_{\text{min}}$ = minimum resolvable angular separation (radians)
- $\lambda$ = wavelength of light
- $D$ = aperture diameter
Derivation
The angular spectrum of a circular aperture gives the Airy pattern:
$$ I(\theta) \propto \left[\frac{2J_1(\pi D \sin\theta / \lambda)}{\pi D \sin\theta / \lambda}\right]^2 $$The first zero of $J_1(x)$ (see Bessel function) occurs at $x \approx 3.83$, giving:
$$ \sin\theta_{\text{first zero}} = \frac{3.83 \lambda}{\pi D} \approx 1.22\frac{\lambda}{D} $$For small angles, $\sin\theta \approx \theta$.
Visual Interpretation
| Separation | Appearance |
|---|---|
| $\theta > 1.22\lambda/D$ | Clearly resolved (distinct peaks) |
| $\theta = 1.22\lambda/D$ | Just resolved (central dip visible) |
| $\theta < 1.22\lambda/D$ | Unresolved (single merged peak) |
At the Rayleigh limit, the combined intensity shows a dip of about 26% between the two peaks.
Applications
Telescopes
$$ \theta_{\text{min}} = 1.22 \frac{\lambda}{D} $$Larger aperture → better angular resolution. This drives the desire for large telescope mirrors.
| Telescope | Diameter | Resolution at 550 nm |
|---|---|---|
| Human eye | 7 mm | 1 arcminute |
| Hubble | 2.4 m | 0.05 arcsec |
| James Webb | 6.5 m | 0.07 arcsec (at 2 μm) |
Microscopes
For a microscope with numerical aperture NA:
$$ d_{\text{min}} = 0.61 \frac{\lambda}{\text{NA}} $$This is the minimum resolvable distance between two points.
Photography
The f-number relates to resolution. Higher f-numbers (smaller apertures) reduce resolution due to diffraction:
$$ \theta_{\text{min}} = 1.22 \lambda \cdot (f/\#) / f $$Limitations
The Rayleigh criterion is somewhat arbitrary:
- Other criteria exist (Sparrow, Houston)
- Signal processing can improve resolution beyond this limit (super-resolution)
- It assumes incoherent illumination and no noise
However, it remains the standard benchmark for optical system resolution.