Telescope Mirror: Aberrations and Spot Diagram

What is modelled

Parallel light from a star reaches a concave mirror, and every ray is reflected by the law of reflection at the surface point it hits. A paraboloid sends all rays that are parallel to its axis exactly to one point. A sphere does not: rays far from the centre cross the axis too early, which is spherical aberration. A star off the axis is imaged as a small comet with a tail, which is coma, and it grows with the angle and shrinks with a slower focal ratio. The model traces a bundle of rays through the pupil and reports the size of the spot, the wavefront error from the optical path lengths, and the Strehl ratio, which tells how much of the peak brightness of a perfect image is left.

Equations

• Conic mirror z = c r2 / (1 + √(1 − (1 + k) c2 r2)) with c = 1 / (2f); k = 0 is a sphere and k = −1 a paraboloid.
• Path error of a sphere at the edge D / (512 N3); longitudinal aberration h2 / (4 R).
• Coma flare of a paraboloid 3 θ / (16 N2); Airy radius 1.22 λ N; Maréchal estimate S ≈ exp(−(2π σ)2).

Checks run in the page

• Equal optical paths for a paraboloid on axis.
• The path error of a sphere against its formula.
• The axis crossing of the marginal ray against the exact and small-angle forms.
• The tangential coma against 3 theta / (16 N squared).

Honest simplifications

A single mirror with the focus on the axis, no diagonal mirror, no central obstruction, no diffraction in the ray trace (the Airy disc is only drawn for comparison) and a Maréchal estimate of the Strehl ratio that is only good for small errors. Real telescopes also suffer from mirror figure errors, thermal effects and collimation errors.

• Press Spherical mirror, fast and see the spot diagram become a large blur; then press Spherical mirror, slow to see it shrink.
• Press Paraboloid at the field edge and see the comet shape of coma.
• Move the focus shift slider on the sphere preset to find the best focus, which is not the paraxial focus.