Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 338 / 2 / b

The projected actuator pitch is $p=D/N$ in the question's sampling convention. A sinusoidal wavefront error needs at least two samples per spatial period, by the Nyquist–Shannon sampling theorem. Thus the shortest limiting correctable scale at the primary isThe Nyquist spatial frequency is $N/(2D)$ cycles per unit length. Exactly at that limiting frequency some phases are poorly sampled; actual deformable mirror performance also depends on actuator influence functions, so this is an ideal bandwidth limit.

Count independent sinusoidal ripples by their spatial Fourier series indices. A mode with $n$ cycles across $D$ has $\Lambda =D/n$, so the given half-wave index is $j=2n$. The Nyquist spatial frequency permits $n\le N/2$. In two dimensions, a circular cutoff therefore contains approximately $\pi (N/2{)}^2$ integer wavevectors.

For a real wavefront error, the wavevectors $\mathbf{n}$ and $-\mathbf{n}$ describe the same ripple with conjugate coefficients, so count each pair once. The continuum mode-counting approximation givesThis counts one ripple with amplitude and phase per conjugate pair, not one real coefficient. Exact finite-grid counts are integers and have boundary corrections; the formula is the circular area estimate implicit in the question, with piston excluded.

The supplied speckle contrast of a sinusoidal wavefront error gives $h_0=\lambda \sqrt{C}/\pi$. Each sinusoid has root mean square error $h_0/\sqrt{2}$. For orthogonal modes, or for uncorrelated phases so that cross terms average to zero, the total mean square adds:Therefore the assumed equal-contrast modal model givesIf the individual contrasts are not equal, the corresponding formula is $h_{\mathrm{rms}}^2={\lambda }^2{\sum }_i{C}_i/(2{\pi }^2)$. The error here is optical-path wavefront error; for a near-normal reflecting mirror, physical surface error is half the optical-path error.

A wavefront error ripple of period $\Lambda$ generates a pair of speckles at angular displacement $\lambda /\Lambda$. Combining this with ${\Lambda }_{\min }=2D/N$ gives the deformable-mirror control radius along an actuator row or column:A square actuator lattice has a square ideal frequency region: $|{\theta }_x|,|{\theta }_y|\le N\lambda /(2D)$. Its full width is $N\lambda /D$, approximately $N$ diffraction-limited resolution elements per side, or $N^2$ in area. The circular subset within the row-direction radius contains approximately $\pi N^2/4$ such elements. The primary aperture shapes each speckle's point spread function; it does not turn the square sampling limit into a circular one.

This full square describes phase-error control. Simultaneous amplitude and phase correction with a single pupil-plane deformable mirror generally requires restricting the dark region to a half-plane; it is a different constraint from the sampling bandwidth.