Segmented Mirror Phasing

What is modelled

The mirror is cut along a line into N equal segments across a width D. Each segment is raised by a piston error and rotated by a tilt error. A reflected wavefront is wrong by twice the surface error, so a 50 nm step becomes a 100 nm wavefront step. The image of a distant point star is the squared Fourier transform of the pupil field, and every number on the page comes from adding that sum directly.

Equations

• Pupil phase φ(x) = 4π [ pi + ti (x − xi) ] / λ for a point x on segment i.
• Far field I(u) = | (1/D) ∫ exp(i φ(x) + 2π i u x / D) dx |2, with u the angle in units of λ/D.
• Ideal width FWHM = 0.886 λ / D.
• Marechal estimate S ≈ exp(−σφ2) for small phase errors.
• A tilt t of the whole mirror moves the image by 2 t.

Checks run in the page

• The perfectly phased mirror has nulls at u = 1 and 2 and a half-power width of 0.886 λ/D.
• Parseval: the total light in the image equals the light entering the pupil.
• Piston only: the Fourier sum equals the closed form |mean of exp(iφ)|².
• A single tilted aperture moves the image peak by 2 t D / λ.

Honest simplifications

A real mirror is a hexagonal tiling and its image is two dimensional with a hexagonal pattern; here the aperture is a line, which keeps the sums exact and the picture readable. Gaps, edge roll-off and atmospheric seeing are ignored.

• Press Visible, 60 nm piston and then Near infrared, same error: the same steps hurt far less at long wavelength.
• Raise the tilt error and watch the whole image slide sideways and broaden.
• Press Breathing mirror and follow the Strehl ratio in the time plot.