Transverse Aberration

1.0 Introduction

The first Danjon-Couder criterion, "DC-1", for a good primary [1, p. 105] is "The radius of the circle of least aberration should be comparable with that of the theoretical diffraction disk and, on the average, the transverse aberrations should not exceed the diffraction disk radius." We will show that the light doesn't go where the transverse aberration says. Thus, the DC-1 criterion is not useful for evaluating mirrors.

Three views of mirror errors: 1) cross axis Gaviola caustic test measurements, 2) the Milles-Lacroix plot of Foucault measurements, and 3) transverse aberrations, are all proportional to mirror slope errors. The caustic and Foucault test data require further processing to evaluate the mirror, which implies that the transverse aberration also would not directly reflect the mirror's performance.

We will find the surface errors of a "diffraction limited" 6" mirror, its transverse aberrations (which are surprisingly large), and its diffraction pattern. We will see that the sphere's diffraction pattern change relative to the errorless Airy pattern has no apparent relationship to the transverse aberrations.

2.0 Computations

A spherical 6" mirror (D = 152.4 mm; all units in the following are millimeters) just satisfies the Rayleigh Criterion (relative to assumed wavelength k = 0.000550) if its radius R = 2483.890243 mm. The best-fit parabola [2] is given by x = a + by², with the sphere passing through the origin of the xy-coordinate system, its center on the x-axis, and a = -0.000046, b = 2.013445204E-4. Other quantities and formulas of interest are:

Focal length of the best-fit parabola f = 1/4b = 1241.652862,

Position of the focal plane F = f + a = 1241.652816,

Radius of the Airy disk A = 1.22fk/D,

Parabola's x-coordinate ,

Sphere's x-coordinate ,

Sphere's surface error ,

Sphere's slope ,

Transverse aberration .

3.0 Results

Fig. 1. shows the sphere's surface error (displacement from the best-fit parabola) scaled by k.

Figure 1. 6"f/8.15 Sphere Surface Error.

The two peak errors in Fig. 1. are both k/12, and the valley error is -k/24, giving, as required by the Rayleigh Criterion, surface P-V = k/8.

Fig. 2. shows the transverse aberration scaled by the Airy disk radius and with reversed sign.

Figure 2. Transverse Aberration.

Several things are of interest in Fig. 2. First, the large transverse aberration at the edge of the spherical mirror would suggest that most of the scattered light would fall between 2 and 3 Airy disk radii from the center of the disk due to the large relative area of the edge zones, while the rest of the "scattered" light would still be within the Airy disk. Second, note that the transverse aberration is proportional to the slope of the surface error, as promised in the Introduction: the peaks and valleys of the surface error occur precisely at zeros of the transverse aberration.

Finally, Fig. 3. shows two diffraction patterns, the sphere's in red and the parabola's in blue.

Figure 3. Diffraction Patterns.

Fig. 3. shows the major effect of the sphere's errors on the image happens within 2 Airy disk radii, rather than between 2 and 3 as predicted by the transverse aberration.

Actually, to be fair, the change of the sphere's diffraction pattern relative to the errorless Airy pattern obviously cannot be deduced by just looking at the surface error plot either - it involves an application of wave optics theory. The important point is that the input to the theory is the mirror surface error, not the error slope, i.e., large transverse aberrations may not hurt the image.

4.0 References

[1] J. Texereau, How to make a Telescope, Second Ed., Willmann-Bell, Richmond, 1984.

[2] Mirror Mathematics on this Web site.