Computational Imaging and Adaptive Optics

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.

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.

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).

$\blacksquare$

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: $\Delta\lambda = \lambda/R$.

$\blacksquare$

• 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.

• 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 ($I = I_0 \cos^2 \theta$).

• Assuming compressed sensing works for any undersampled signal. CS requires sparsity in some known basis and incoherence of the measurement matrix. Fix: Verify the signal is sparse in e.g. wavelet or DCT basis and that $\Phi$ satisfies the restricted isometry property.
• Confusing the Strehl ratio with resolution. Strehl ratio measures image quality relative to diffraction-limited; a low Strehl ratio means aberrations spread energy, not necessarily lower resolution. Fix: Strehl $> 0.8$ is diffraction-limited; $S < 0.1$ indicates severe aberrations.
• Neglecting anisoplanatism in AO. The turbulence correction is only valid within the isoplanatic angle $\theta_0$; stars far from the guide star are poorly corrected. Fix: Use multiple guide stars (laser tomography AO) or MOAO for wider fields.
• Forgetting the wavefront sensor latency. AO correction must be applied faster than the Greenwood frequency; otherwise, the atmosphere changes before the mirror updates. Fix: AO loop bandwidth must exceed $f_G$.

• Astronomical imaging: Adaptive optics on 8-10 m telescopes (VLT, Keck, Gemini) enables diffraction-limited imaging in the near-infrared, resolving exoplanets and stellar surfaces.
• Microscopy: Adaptive optics corrects for tissue aberrations in deep two-photon and confocal microscopy, recovering image contrast at depths $> 100\,\mu$m.
• Medical imaging: Compressed sensing accelerates MRI acquisition by undersampling $k$-space and reconstructing using total variation minimisation, reducing scan times by 2-4$\times$.
• Laser communications: Atmospheric turbulence distorts free-space optical links; AO pre-compensation at the transmitter improves coupling efficiency into single-mode fibres.
• Terahertz imaging: Compressed sensing with a single-pixel detector enables THz imaging with sparse detector arrays, useful for security screening and non-destructive testing.

Problem. An AO system reduces the wavefront RMS error to $\Delta\phi_{\rm rms} = \lambda/20$ at $\lambda = 500$ nm. Compute the Strehl ratio. Is the system diffraction-limited?

Solution. The Strehl ratio is:

$S = \exp\left[-\left(\frac{2\pi}{\lambda}\right)^2\langle\Delta\phi^2\rangle\right] = \exp\left[-\left(\frac{2\pi}{\lambda}\right)^2\left(\frac{\lambda}{20}\right)^2\right]$

$S = \exp\left[-\left(\frac{2\pi}{20}\right)^2\right] = \exp\left[-\left(\frac{\pi}{10}\right)^2\right] = \exp(-\pi^2/100) \approx \exp(-0.0987) \approx 0.906$

Since $S > 0.8$, the system is diffraction-limited. This meets the Marechal criterion ($S > 0.8$), corresponding to $\Delta\phi_{\rm rms} < \lambda/14$. Our $\lambda/20$ exceeds this requirement.

$\blacksquare$