Point Spread Function (PSF), Strehl Ratio & Airy Disc: Maréchal's Criterion & Optical Diffraction Limits

For a circular entrance pupil of diameter $D$ and focal length $f'$ in a medium of refractive index $n'$, Fraunhofer diffraction yields the radial intensity distribution of the Airy Pattern:

$$I(\theta) = I_0 \left( \frac{2 J_1(\pi D \sin\theta / \lambda)}{\pi D \sin\theta / \lambda} \right)^2 \implies d_{\text{Airy}} = 2.44 \cdot \frac{\lambda \cdot f'}{D}$$

For the human eye ($f' \approx 17\ \text{mm}$) under green light ($\lambda = 555\ \text{nm}$):

• At $D = 2.0\ \text{mm}$: $d_{\text{Airy}} = 2.44 \cdot \frac{0.000555 \times 17}{2.0} \approx \mathbf{11.5\ \mu\text{m}}$ (Diffraction dominates; exceeds foveal cone diameter).
• At $D = 6.0\ \text{mm}$: $d_{\text{Airy}} \approx \mathbf{3.8\ \mu\text{m}}$ (Theoretical diffraction core is smaller than foveal cones; however, higher-order aberrations explode!).

The Strehl Ratio ($S$) is the ratio of peak on-axis intensity of the aberrated PSF compared to that of an unaberrated, diffraction-limited PSF:

$$S = \frac{I_{\text{aberrated}}(0, 0)}{I_{\text{diffraction}}(0, 0)} \approx e^{-\sigma_\phi^2} \approx 1 - \left( \frac{2\pi}{\lambda} \cdot RMS \right)^2$$

According to Maréchal's Criterion, an optical system is deemed 'diffraction-limited' (essentially optically perfect) if:

$$\mathbf{S \ge 0.80} \iff RMS_{\text{wavefront}} \le \frac{\lambda}{14} \approx 0.040\ \mu\text{m}$$

Most young human eyes achieve a Strehl ratio of only $0.10\text{ to } 0.35$ under a dilated $6\ \text{mm}$ pupil, illustrating how heavily human vision is limited by natural aberrations rather than pure diffraction.

Any real-world scene $O(x, y)$ perceived by a patient is the mathematical 2D convolution of the object with the ocular PSF: $I(x, y) = O(x, y) \otimes PSF(x, y)$. Asymmetric PSFs with coma wings generate the ghosted letter shadows seen in keratoconus and post-LASIK decentration.