◄ 4.2. Coma   ▐    4.4. Defocus ►
 

4.3. Telescope astigmatism

Similarly to coma, astigmatism is an off-axis point wavefront aberration, caused by obliquity of the incident wavefront relative to the optical surface. Astigmatism results simply from the projectional asymmetry arising from wavefront's inclination to the optical surface. The vertical and horizontal diameter of wavefront's project onto the surface radii that are effectively different, thus having different focus location for the tangential plane (the one containing optical axis and chief ray) and for the plane orthogonal to it (sagittal plane).

As illustration below shows, for an axial light pencil falling onto surface S the surface radius is of identical value for both tangential and sagittal cross section, equal to one half of the diameter of a sphere it is a part of. For an inclined pencil surface radius remains unchanged for tangential cross section, but the sagittal (horizontal) cross section projects onto the section of a smaller circle with the diameter R_i, thus focusing closer than the vertical section (both cross sections shown are in the same plane). All vertical sections of the astigmatic wavefront have common focus, different than the common focus of all horizontal sections. That constitutes geometry of astigmatism. Shown is geometry for the stop on the surface; with the chief ray defined as one passing through the center of aperture stop, it is obvious that placing stop in front of the surface here would increase, and an internal stop would decrease astigmatim.

 Note that the illustration also reflects other elements of focusing geometry for abaxial focal points, but what indicates astigmatism is inequality of surface radii onto which project tangential and sagittal cross section of the incoming wavefront.

Another peculiarity of astigmatism is that a cross-section along any horizontal or vertical wavefront diameter (i.e. in the sagittal and tangential plane) remains spherical, but with the radius of curvature varying with the pupil angle. That is what causes the wavefront form as a whole to deviate from spherical.

For displaced stop - either first optical surface significantly separated from the aperture stop, or secondary and tertiary surfaces (whose stop is formed by a preceding surface) - other surface properties, such as shape, position and conic, also can influence the size of astigmatism. This is due to the displaced stop for these surfaces changing the surface radial asymmetry profile. As illustrated at left, the only configuration that completely corrects for astigmatism is a sphere with the stop at the center of curvature (1). It is apparent from the two axes - that of the wavefront and that of the section of the sphere - coinciding, thus positioning the wavefront symmetrically vs. surface. Any other stop distance will not be fulfilling this condition, reintroducing off-axis aberrations (2). For conic forms other than the sphere, primary astigmatism can be either corrected with appropriate stop position (closer to mirror with smaller - i.e. negative - conics), or only minimized (with conics larger than zero). However, secondary astigmatism, even if low, remains. For a given stop location other than sphere's center of curvature, the negative conics will generate less astigmatism than the sphere if their shape results in less of a surface vs. wavefront positional asymmetry. On the illustration, that is the case with the stop position 2, as conics toward parabola and hyperbola open wider than the sphere for given radius of curvature (however, as the surface section outlined by the wavefront is not symmetrical about its axis, coma will be generated even if primary astigmatism is cancelled, with one half of the wavefront flatter than the other one).

FIG. 43 illustrates the form of astigmatic wavefront deformation for primary astigmatism, and the resulting geometric (ray) aberration.

 Gaussian focus for astigmatic wavefront lies on the Petzval surface of an optical surface, or a system. Balancing defocus aberration for this point - located on the opposite direction from the sagittal focus, and at identical distance from it as the best focus - is zero, and the wavefront error is largest. Between the sagittal and tangential focus, ray disturbance resulting from the astigmatic wavefront deformation takes on rather peculiar form (FIG. 44).

Aberration function for primary astigmatism given in Table 4 is for tangential focus, which is identical to that for the sagittal focus, except for the orientation. Neither coincides with the best (diffraction) focus, which is located midway between the two, hence requires correction by defocus. Since for given longitudinal aberration the P-V wavefront error for primary astigmatism equals that for defocus, the defocus added to it equals half the P-V wavefront error of astigmatism. Aberration function for the wavefront error of primary (lower-order) astigmatism at the best focus is given by:

with A being the astigmatism peak aberration coefficient, ρ the normalized (to 1) height in the pupil, and θ the pupil angle. It shows that the wavefront error peaks for ρ=1 and cos2θ=1 and 0 (that is, for θ=0, π/2, π, 3π/2 and 2π), which is, every 90 degrees (with the successive peaks being of opposite sign), and orthogonally to the orientations of the minimum wavefront deviation, occurring for cosθ=√0.5 (for θ=π/4, 3π/4, 5π/4 and 7π/4). It clearly outlines saddle-shaped wavefront deviation, as illustrated on FIG. 43 left.

Note that the maximum wavefront error given by Eq. 18 - which gives ± wavefront deviations, not the peak-to-valley error - is one half of the peak aberration coefficient, which equals the P-V error. Numerically, it is identical to the P-V error at either sagittal or tangential focus, Wa=Aρ2cos2θ, but its RMS value is smaller by a factor of 1/√1.5.

When the point of maximum deviation in the tangential (vertical) plane is closer to the center of reference sphere than its reference sphere point - i.e. when the tangential focus is precedes sagittal in the direction of light travel (or, simply put, is closer to the objective) - the wavefront error of astigmatism is negative. That is the sign of astigmatism in a concave mirror, illustrated on FIG. 28. There is no difference in appearance between positive and negative Seidel astigmatism, since the pattern is merely rotated by 90°, and has an inherent 180-degree rotational symmetry at the best focus location (FIG. 30).

The peak aberration coefficient A, which equals the peak-to-valley wavefront error, is given by:

As expected due to its uniformly dense geometric blur, the smallest RMS blur radius for astigmatism is at the location of the circle of least confusion. It is given by rRMS=FA√2, or smaller by a factor of √0.5 than the radius of the circle of least confusion. In units of the Airy disc diameter, the RMS blur diameter is RRMS=A√2/1.22, for the peak aberration coefficient A in units of the wavelength.

In terms of the RMS wavefront error of astigmatism ωa, the RMS blur diameter in units of Airy disc diameter is RRMS=4√3ωa/1.22.
 

Aberration coefficient of astigmatism doesn't change with object distance. For relatively close objects, transverse astigmatism increases as fi/f, fi being the image-to-pupil separation. However, it doesn't affect the wavefront error: since the wavefront radius is also longer by the same ratio, identical nominal wavefront error results in proportionally greater longitudinal and transverse aberration.

For the aperture stop displaced from mirror surface, the aberration coefficient of astigmatism changes in proportion to Δ2[Kσ2+(1-σ)2]/(σR)2, with σ being the mirror-to-stop separation (positive in sign) in units of the mirror radius of curvature. The astigmatism aberration coefficient is:

Needed stop separation for zero astigmatism is given by σ=[1-√|K|]/(K+1). Thus, astigmatism is canceled for σ=0.5 with a paraboloid and σ=1 with a sphere. The relation is not defined for K=-1 (parabola), but implicates σ=0.5 limit for K approaching -1. For positive values of the conic K, the aberration coefficient cannot be zero regardless of the stop position, since both Kσ2 and (1-σ)2 are positive.

Unlike coma, change in astigmatism caused by displaced aperture stop is independent of object distance.

Aberration coefficient of primary astigmatism for a lens with the aperture stop at the surface is identical to the one given for concave mirror (Eq. 19.1), with the radius of curvature R replaced by 2f (Eq. 99), f being the lens' focal length. For a contact doublet, it gives the peak aberration coefficient as a sum of the aberration coefficients at the first and second lens, respectively, as:

with α being the field angle, and f1,f2 the respective lens focal lengths (keep in mind that focal length of a negative lens is numerically negative in the left-to-right Cartesian coordinate system). Change of the stop position results in change of the aberration coefficient only with systems not corrected for spherical aberration, or coma, or both. Since modern refractor objectives commonly are aplanats, their astigmatism is not affected by the stop position. As already mentioned, wavefront error of astigmatism of the contact doublet doesn't change with object distance.

    Strongly curved surfaces give rise to higher order aberrations, and astigmatism is no exception. Although relatively rare in amateur telescopes, secondary (higher-order) astigmatism can be significant in systems with sub-aperture correctors, as well as in eyepieces. Unlike the P-V wavefront error of primary astigmatism, which is at either sagittal or tangential focus described with quadratic function (parabola, blue plot on FIG. 46 left), that of secondary astigmatism, given as Wa=Aρ4cos2θ, is quartic (4th power) function with a single turning point (red plot on FIG. 46 left). The shape of wavefront deviation is generally similar to that of primary astigmatism, but its near-cylindrical form has larger near-flat middle area and more strongly curved edges. The wavefront shape implies that secondary astigmatism forms longer sagittal/tangential focus lines when unbalanced, or balanced with defocus.

 Similarly to that of the primary, the error of secondary astigmatism at either sagittal or tangential focus (left; tangential focus has the same shape simply rotated by 90° about vertex) is reduced by refocusing to the best focus midway between them (right). In either case, the P-V error remains the same, but the RMS error is reduced, and smaller for given P-V error for secondary astigmatism at both focus location.

    The more efficient way of minimizing secondary astigmatism is, however, balancing it with the lower-order form of opposite sign. The error is minimized with adding the primary astigmatism with the peak coefficient smaller by 25%, with Was=A(ρ4-0.75ρ2)cos2θ, implying that the P-V error of of secondary astigmatism balanced with the lower-order form is smaller by a factor of 0.25 (fourfold) than at either sagittal or tangential focus. The RMS error is smaller than that of balanced primary astigmatism of identical P-V error by a factor √0.6.

    However, due to the different rates of aberration increase for the two astigmatism forms (2nd vs. 4th power of field angle for the primary and secondary form, respectively) this optimized balancing is only possible at a narrow field zone. Usually, primary astigmatism is left to be larger initially, with the two forms balancing farther off axis and secondary astigmatism quickly becoming dominant after that.

◄ 4.2. Coma   ▐    4.4.Defocus ►