Computational Imaging and Adaptive Optics

19.1 Compressed Sensing

Compressed sensing (Candes, Tao, Donoho, 2006) shows that signals that are sparse in some basis can be reconstructed from far fewer measurements than Nyquist sampling requires:

$\hat{\mathbf{x}} = \arg\min_{\mathbf{x}} \|\mathbf{x}\|_1 \quad \text{subject} to \mathbf{y} = \Phi\mathbf{x}$

Where $\Phi$ is the measurement matrix and $|\cdot|_1$ is the $L^1$ norm promoting sparsity.

19.2 Adaptive Optics

Atmospheric turbulence causes phase distortions in astronomical images. Adaptive optics (AO) corrects these in real time using a deformable mirror. The Strehl ratio:

$S = \exp\left[-\left(\frac{2\pi}{\lambda}\right)^2\langle\Delta\phi^2\rangle\right]$

For diffraction-limited imaging ($S > 0.8$): $\Delta\phi_{\text{rms} < \lambda/14}$. On an 8 m telescope at visible wavelengths, the deformable mirror must update at $>500$ Hz to track the Greenwood frequency $f_G \sim 100$ Hz.

Worked Examples

Example 1: Thin film interference

Problem. A soap film ($n = 1.33$) of thickness $300 \mathrm{ nm}$ is illuminated by white light. Which wavelength is constructively reflected?

Solution. Phase change at the front surface (air to soap); no phase change at the back (soap to air). Constructive: $2nd = (m + \frac{1}{2})\lambda$. $2(1.33)(300 \times 10^{-9}) = (m + 1/2)\lambda$. For $m = 0$: $\lambda = 4 \times 1.33 \times 300 = 1596 \mathrm{ nm}$ (infrared). $m = 1$: $\lambda = 1596/3 = 532 \mathrm{ nm}$ (green, visible).

$\blacksquare$

Example 2: Diffraction grating resolution

Problem. A grating has $N = 5000$ lines illuminated. Find the resolving power in the second order.

Solution. $R = mN = 2 \times 5000 = 10\,000$. The minimum wavelength difference resolvable: $\Delta\lambda = \lambda/R$.

$\blacksquare$

Common Pitfalls

• Confusing group and phase velocity. Phase velocity $v_p = \omega/k$; group velocity $v_g = d\omega/dk$. Fix: In a dispersive medium $v_p \neq v_g$; the group velocity is the speed at which the envelope (energy) travels.
• Wrong interference condition. Constructive: path difference $= n\lambda$. Destructive: path difference $= (n + 1/2)\lambda$. Fix: For thin films, also account for the phase change on reflection ($\pi$ phase shift from denser medium).
• Confusing Fraunhofer and Fresnel diffraction. Fraunhofer: far-field (parallel rays, simpler math). Fresnel: near-field. Fix: Fraunhofer: $a \sin\theta = n\lambda$. Fresnel: requires Fresnel integrals or numerical methods.

Summary

• Phase velocity: $v_p = \omega/k$. Group velocity: $v_g = d\omega/dk$; energy/information travels at $v_g$.
• Interference: thin films, Michelson interferometer, Fabry-Pérot etalon.
• Diffraction: single slit, double slit, diffraction grating; Rayleigh criterion for resolution.
• Polarisation: Brewster”s angle, Malus’s law ($I = I_0 \cos^2 \theta$).

Cross-References