Bending thin-shell primary mirrors to correct off-axis astigmatism

A problem in designing a telescopic heliostat is the need for the primary mirror to image the sun's disk over a wide range of incidence angles. In low latitudes the sun is often directly overhead (90° solar elevation), but useable insolation may still be available down to 15° solar elevation. Assuming the primary mirror is tracked to redirect the incident sunlight vertically, the angle of incidence that the primary mirror of a telescopic heliostat needs to accommodate runs from 0° to a maximum of about 37.5°.

The optical geometry becomes easier to visualize if we forget about tracking and redirecting the sun's light, and simply imagine the primary mirror to be stationary and horizontal while the sun's angle of incidence changes and the focussed image of the sun moves about.

The off-axis astigmatism of a spherical mirror can be corrected by thin-shell bending.

The distance between the mirror and its image of the sun is the focal length, L. The focal length is a function, L(θ), of the angle of incidence, θ. At normal incidence, the focal length of a spherical mirror is half its radius of curvature. Normalizing the mirror's radius of curvature to unity, we have:

L(0) = 1/2

At larger angles of incidence astigmatism appears, that is, sections of the mirror that lie in different planes have different focal lengths. In particular, a narrow strip of mirror lying in the meridional plane (the plane that contains both the incident ray and the mirror normal) will act like a fixed mirror solar concentrator, always bringing sunlight to a focus somewhere along a circle that contains both the mirror and the 0°-incidence focus. In contrast, a narrow strip of mirror lying in the sagittal plane (the plane that contains the incident ray but is perpendicular to the meridional plane) will act like a solar trough concentrator, always bringing sunlight to a focus somewhere along a horizontal line passing through the 0°-incidence focus.

Observing the geometry of a circle in one case, and a line in the other, we have for the meridional focal length, L_m, and the sagittal focal length, L_s:

L_m(θ) = 1/2 * cosθ

L_s(θ) = 1/2 * 1/cosθ

By bending a curved mirror we can tradeoff the two principal curvatures. In the case that the mirror is a thin shell, the product of the two principal curvatures (i.e., the Gaussian curvature,) will remain constant—and thus the product of the two radii of principal curvature will also remain constant. For the radius of curvature in the meridional plane, r_m, and the radius of curvature in the other principal plane (the vertical plane perpendicular to the meridional plane,) r_s, we have:

r_m * r_s = 1

These new radii of curvature simply re-scale the geometric expressions for the loci:


L_s(θ) = r_s/2 * 1/cosθ

L_m(θ) = r_m/2 * cosθ


We would like to bend a thin-shell spherical mirror to make the sagittal and meridional focal lengths equal:

r_s/2 * 1/cosθ = r_m/2 * cosθ

r_s = r_m * cos²θ.

Substituting r_m = 1/r_s,

r_s = 1/r_s * cos²θ

r_s = cosθ.

Substituting this value of r_s into the geometric expression above gives:

L = 1/2.

The focal length of a spherical mirror is independent of the angle of incidence after it has been corrected for astigmatism by thin-shell bending. This is just the result we would wish, as it means the separation between primary and secondary mirrors in a telescopic heliostat can remain fixed.