◄ 4.1.3. Higher-order spherical aberration   ▐    4.3. Astigmatism ►
 

4.2. ComA OPTICAL ABERRATION

In general, coma wavefront aberration occurs either due to the incident wavefront being either tilted, or decentered with respect to the optical surface. Hence, it is either an aberration affecting off-axis image points, or the  result of lateral misalignment of optical surfaces, respectively. Important difference between the two is that with the former coma takes form of an off-axis aberration increasing with the field angle, and with the latter it is of even magnitude across the field, including field center.

The form of coma most often dominating in amateur telescopes is the lower (4th) order or primary coma. As illustrated on FIG. 39, its wavefront deviation has reverse symmetry along the axis of aberration, with one side flatter, and the other more curved with respect to its perfect reference sphere. 

Mapping rays scattered by coma wavefront deformation follows simple, elegant symmetry. As illustrated on FIG. 40, every zone in the pupil focuses not into a point, but into a circle of the diameter

with h being the linear height in the image plane, z=ρd=ρD/2 the zonal height in the pupil (D is the pupil, or aperture diameter and ρ the zonal height normalized to 1), and f the focal length. The circle is formed by each pair of diametrically opposed points on a given pupil zone focusing into a single circle point. The center of each circle is also shifted along the axis of aberration, away from the Gaussian image point by a length equal to the circle diameter.

The wavefront aberration function for primary coma at the Gaussian image point is given in Table 4. But best focus location is shifted laterally from this point, i.e. requires correction for the wavefront tilt. Aberration function (i.e. the wavefront error as optical path difference with respect to a reference sphere) for coma at the best focus is given by:

with C being the coma peak aberration coefficient, ρ the pupil radius normalized to 1 and θ the pupil angle. Note that Wc at its maximum represents 1/2 of the P-V wavefront error, since it gives identical value of opposite sign for negative ρ (i.e. for the opposite half of pupil radius). As shown at left, the aberration peaks for ρ=1 and θ=0° and 180° - i.e. along the axis of aberration - with the sign of aberration changing symmetrically on both sides of the wavefront (except for θ=0, when deviation is zero across the pupil radius). Second factor in the brackets - effectively correction for wavefront tilt - makes the P-V coma error at the best focus ("balanced" coma, shown at left) three times smaller at its maximum than the P-V error at the paraxial (Gaussian) focus. The latter is referred to as "classical", or Seidel coma, where the P-V wavefront error is given by W=Cρ3cosθ (FIG. 42, middle).

The RMS wavefront error for coma at the best focus, in terms of the peak aberration coefficient C is:

ωc = C/720.5          (16)

In terms of the P-V wavefront error it is ωc=2Wc/√32, with Wc being, as given above, the coma peak wavefront error (half of the P-V error).

On FIG. 39, the peak error on the top radius (edge point) is positive, being farther to the center of a perfect reference sphere than its perfect reference point, while the peak error on the bottom radius is negative. This is so called "positive" coma, with the top section of the wavefront flattened with respect to the reference sphere, and the bottom portion more curved. Its comatic tail is oriented outward, away from field center. Presence of the pupil angle factor (cosθ) indicates that the aberration is not rotationally symmetrical. The peak aberration coefficient is given by:

with c being the coma aberration coefficient (simplified notation for 1w31 from Table 4), α the field angle (in radians) and d the pupil radius.

For general optical surface, coma aberration coefficient is given by:

c=0.5n2[(1/O)-(1/R)][(1/n'I)-(1/nO)]

where I, O are the image and object distance, and n, n' are the refractive index of incidence and refraction or reflection, respectively.

    Coma aberration coefficient c for either refractive or reflective surface, for object at infinity, and aperture stop at the surface is given by:

with f being the focal length, and n and n' being the index of incidence and refraction/reflection, respectively.

For a single thin lens with the stop at the surface, coma coefficient is:

with p, q and f being the lens position factor, shape factor and focal length, respectively. Thin lens is coma-free if its shape factor q and position factor p relate as q=-(2n+1)(n-1)p/(n+1). Lens doublet can have both, coma and spherical aberration cancelled. For given shape p and q, the sign of coma is not affected by lens power, i.e. it is the same for positive and negative lens. However, if placed one after another, their respective p values - and possibly sign as well - will be different, and so will the coma induced by each element.

    With regard to the aperture stop position, it does affect coma only with objectives not corrected for spherical aberration. With an aplanatic (corrected for spherical aberration and coma for infinity) lens system, stop position doesn't affect neither coma, nor astigmatism and field curvature. 

    For mirror surface, the coma coefficient for any object distance is given by:

with m being the transverse image magnification (defined following Eq. 9.). For distant objects m=0, n=1 for mirror in air oriented to the left, and the coefficient reduces to:

As the aberration function shows, deviation from the perfect reference wavefront is largest for ρ=1 and either cosθ = 1 or cosθ = -1 (i.e. θ=0 or θ=π radians, respectively). With θ being the position angle in the pupil, measured to the plane of the axis of aberration, the maximum deviation occurs at the marginal points of the wavefront in the plane of aberration (FIG. 25, left). The deviation is zero for cosθ=0 (i.e. θ=π/2 or θ=3π/2), which is the wavefront diameter orthogonal to the plane of aberration. For ρ smaller than |~0.8|, the deviation changes the sign, as a result of the two lobe-like deformations, illustrated on FIG. 39, left.

As mentioned, the above relations for coma aberration are for a mirror with the aperture stop at the surface. For displaced stop, the coma aberration coefficient c changes with a factor Δ[ c'=1-(1+K)σ]/σR;, with Δ being the chief ray height on the mirror (given by a product of the mirror-to-stop separation and incident angle), K being the mirror conic and σ the mirror-to-stop separation in units of the mirror radius of curvature, numerically positive. Thus, for displaced stop, the aberration coefficient for object at infinity is:

Since Δ and σ change in proportion to the stop separation, stop position doesn't influence coma of a paraboloid (K=-1), while cancels coma of any conic K>-1 for the stop separation σ=1/(1+K). On the other hand, displaced stop increases coma for K<-1 (hyperboloid).

This applies for objects at infinity; for close objects for which the mirror object-image magnification m significantly differs from zero, the change factor in the coma coefficient caused by the aperture stop displacement is obtained by replacing both unit figures in the "infinity" object distance factor c' with (m+1)/(m-1), or (1-Ω), Ω being the reciprocal of the object distance in units of the radius of curvature, numerically positive. It gives the coefficient as:

Note that (1-Ω) is just a different form of (1-2ψ) parameter, used in Eq. 9.1, ψ being the reciprocal of the object distance O in units of the focal length, ψ=f/O.

The wavefront error changes with the coefficient. Considering that the magnification m is negative, coma diminishes with object distance, dropping to zero for m=-1 (for object at the mirror center of curvature) and stop at the surface (σ=0), regardless of the conic.

Due to the peak wavefront deviations with coma affecting relatively small area, the error averaged over the entire wavefront is smaller for coma than for spherical aberration for any given P-V wavefront error (FIG. 41).

For small levels of the aberration, the shift from the Gaussian image point to the best, i.e. minimum RMS focus location is transverse, in the image plane and along the axis of aberration, given by 4FC/3=8FWc/9. That places best focus at 1/4.5 of the blur length away from the Gaussian image point (the tip). It corresponds to 2/3 of the separation between Gaussian focus and centroid, the latter being the same for the geometric image and PSF. The shift effectively corrects for the wavefront tilt element. Unlike spherical aberration, the peak aberration coefficient C for coma does not equal the p-v wavefront error. At the location of best focus, the P-V wavefront error is 2C/3, with the peak wavefront error being ±C/3. The error is smaller by a factor of 3 than at the location of Gaussian image point, for which the tilt is not corrected.

As with other aberrations, focus with the lowest RMS wavefront error produces the highest PSF peak only as long as the magnitude of aberration remains relatively small. In general, under 0.15 wave RMS. Specifically for coma, the PSF peak shifts away from the best focus (with respect to the magnitude of wavefront error), as defined above, as the aberration exceeds 0.76 wave P-V (0.134 wave RMS). For larger errors, the PSF peak shifts somewhat back, generally closer to the Gaussian focus, but its exact location varies with the error magnitude (left).

    Simulations of the actual PSF at right show the shift in 2-D. It occurs along the axis of aberration (or, in terms of the geometric ray spot, along its axis of symmetry), hence the magnitude of shift diminishes from its maximum for line of sight perpendicular to the axis, to zero for line of sight coinciding with the axis of aberration. Image below shows main characteristics of coma aberration on the ray spot plot and the corresponding PSF - as a X/Y plot and diffraction image - for a 200mm f/5 paraboloid with stop at the mirror's focus (to eliminate astigmatism) and on the best field surface (R=-1m).

The raytrace (OSLO Edu) choses as the center point (0,0) the best focus, i.e. center of a reference sphere having the lowest RMS error with respect to the actual wavefront. Peak intensity, however, is shifted along the axis of aberration (Y axis), closer to the Gaussian (geometric) focus. At this magnitude of aberration, neither the central intensity, determining the Strehl ratio, defined as a ratio of aberrated vs. unaberrated central PSF intensity (in this case about 0.05), nor the intensity corresponding to the minimum RMS focus, coincide with the PSF peak intensity.

Coma transverse aberration also can be expressed in terms of the peak aberration coefficient C. Tangential coma is given by T=6fC/n'D, f being the focal length and D the aperture diameter. For mirror oriented to the left, n'=-1, and T=6FC, with F being the focal ratio number, F=f/D. With the peak aberration coma coefficient C=αD/32F2 for distant objects, tangential coma can be expressed directly as:

Analogous to the higher-order spherical aberration, strongly curved surfaces generate higher-order (secondary) coma, similar in appearance to its lower-order form, but not identical. Unlike primary coma, where the zonal ray circles increase with the square of zonal height, and the circle center is at its diameter away from Gaussian focus, with secondary coma the zonal ray circle diameter increases with the fourth power of zonal height, while its center shifts by 1.5 the circle radius from Gaussian focus. As a consequence, the cone formed by these circles is wider, outlining a 90° angle (FIG. 42). The ray geometry indicates that secondary coma displaces relatively less energy into the outer portion of the blur, with the sagittal coma being only 1/4 of the tangential coma (blur length). Accordingly, the resulting diffraction image would have less noticeable comatic tail, and shorter, more compact sagittal image than primary coma.

    According to Wcs=Csρ5cosθ (subscript s for "secondary"), for given value of the peak aberration coefficient, the P-V wavefront of secondary coma at the Gaussian focus is identical to the P-V wavefront error of primary coma, but the RMS wavefront error is smaller by a factor of 0.78. When balanced with wavefront tilt, the P-V error Wcst=Cs(ρ5-0.5ρ)cosθ is reduced by a factor of 0.5. But it is further reduced by balancing it with primary coma, to Wcstp=Cs(ρ5-1.2ρ3+0.3ρ)cosθ, for ρ=1 (both reversal points, at ρ=0.31 and ρ=0.79 radius, have smaller deviation), or 1/10 of its magnitude at the Gaussian focus.

    In ray tracing, balancing secondary coma, if significant, is a part of optimization. After cancelling lower-order coma in the 3rd order (for transverse ray, 4th order on the wavefront) approximation, residual higher-order (5th transverse, 6th on the wavefront, or secondary) coma is minimized by reintroducing lower-order coma of opposite sign, with the coefficient (absolute value) larger by a factor of 1.2 (plus tilt factor, which determines lateral shift from Gaussian focus, i.e. point at which the wavefront error is minimized). As mentioned, the combined RMS wavefront error is smaller by a factor of 0.21 than for unbalanced secondary coma. This means that, similarly to the secondary spherical aberration, secondary coma cannot be entirely removed by balancing it with the lower-order form, only minimized. However, since the magnitude of unbalanced secondary coma is typically relatively low, and both, primary and secondary coma have the same rate of increase (with the square of field angle), the level of balanced coma is usually negligible.

◄ 4.1.3. Higher-order spherical aberration   ▐    4.3. Astigmatism ►

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