Segmented Mirror Phasing

What is modelled

Eighteen hexagonal segments (same layout as the 3D lab, with a gap in the centre) each have a small height error h_i (nanometres). A reflected wavefront is shifted by twice the surface height, so segment i adds a phase of φi = 2π (2 hi) / λ. The image of a distant star is the squared modulus of the sum of all the segment fields, computed numerically for a cut along one axis and for a small picture.

Equations

• Diffraction limit θ = 1.22 λ / D (first dark ring of a circular aperture).
• Strehl ratio, the peak intensity relative to a perfect mirror: S = |(1/N) Σ eiφi|².
• Maréchal approximation for small errors: S ≈ exp(-σφ²) with σφ = 2π σwavefront / λ; S > 0.8 counts as diffraction limited.
• Segment errors follow the 3D lab pattern sin(2.39996 i) + 0.3 cos(1.17 i) scaled by the piston slider; the common height is irrelevant, only differences matter.

Checks run in the page

• The first dark ring of a circular aperture found from a numerical sum, against 1.22 λ / D.
• Exact segment sum against the Maréchal formula for a small error.
• Zero piston gives the global maximum and Strehl ratio exactly 1.
• The RMS of the segment phases against 2 π times the wavefront RMS over λ.

Honest simplifications

Only piston is modelled (no tip, tilt, edge roll-off or gaps between segments) and the aperture is scaled by a single factor from the 3D layout. The drift animation rotates the error pattern slowly; it is not a model of real thermal drift. The image shows a monochromatic point source without atmosphere.

• Choose Perfectly phased and compare the dashed ring with the first dark ring of the picture.
• Choose Poor alignment, 25 nm: the peak drops and light moves from the core to the surroundings.
• Switch to 900 nm: the same piston error costs much less Strehl at longer wavelength.