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3.3.2. Aberrations of the conic surfaceWhile the conics are free from spherical aberration when the object is located at either of the two specific conjugates (foci), it is not the case with the rest of aberrations. For off-axis points, incoming wavefronts are inclined to the surface, which results in the loss of symmetry needed for the formation of a perfect spherical wavefront. Thus all conic surfaces of revolution - sphere, ellipses, parabola and hyperbolas - suffer from coma and astigmatism, which are independent of the conic for the stop at the surface and object at infinity. Also, when the object is not in one of the two specific conjugates of a conic, it will also suffer from spherical aberration. Since the aberration inherent to conical surfaces vary, in general, with their eccentricity, degree and sign of curvature, object distance, as well as the entrance pupil location, two or more surfaces can be combined in order to minimize or cancel the final aberration. In all, there is five intrinsic telescope aberrations characteristic of a conic surface of revolution:
- spherical
aberration Only spherical aberration is an axial aberration; the rest of them are off-axis, or field aberrations. The first three aberrations - spherical, coma and astigmatism - are a result of wavefront deviations from spherical, hence they affect energy distribution in the diffraction image - that is, point-object image quality (FIG. 12). The last two - field curvature and distortion - are caused by variations in wavefront radius or tilt angle with the point height in the image space, thus affecting image form. Properties of each aberration are described by the aberration function, expressed as a power series, with every successive power level - or "order" - resulting in significantly lower aberration contribution. While higher-order aberrations can become significant in some optical systems, in most instances it is the first power term - so called primary, or Siedel aberrations - that practically determines level of aberrations.
FIGURE 12: The three primary, or
Seidel
aberrations affecting point-image quality.
Spherical aberration is radially symmetrical wavefront
deviation in which it becomes
progressively more strongly curved toward the edges, than a reference
sphere centered at paraxial (Gaussian) focus G. It occurs
whenever object imaged is not located at one of the two
specific foci
of a conic. Here, it is illustrated on a spherical reflecting surface
imaging distant object. As
the incoming flat wavefront interacts with mirror surface, its atoms begin to emit waves in all
directions. The change in sagitta of spherical surface (∆)
determines surface slope, which reshapes the wavefront (FIG.
3) into non-spherical form, curving more strongly toward the edge.
At the moment when the paraxial segment of the incident wavefront
reaches mirror surface, its edge portion is reflected and converging
toward the axis. However, while the paraxial mirror points (P)
form a wavefront section nearly coinciding with the sphere SP,
placed at the mirror vertex and centered at the paraxial (Gaussian)
focus (G), the edge points form wavefront section M nearly
coinciding with a smaller sphere SM,
centered at a point closer to the mirror, called marginal ray focus (CM).
Other mirror segments form wavefronts belonging to spheres centered at
intermediate points in between these two (SI/CI).
The phase difference, determined by the path length between the actual
wavefront point and corresponding one on the reference sphere, peaks at the edge
of the wavefront, with respect to paraxial focus reference sphere.
Higher- and lower-order aberrations are not separate entities; they are rather an artifact of the calculation method. Wavefront aberrations originate at the optical surface, but the general conic surface can't be mathematically described in simple terms. It requires an expansion series, with the surface sagitta (depth at the center) given by z=(r2/2R)+[(1+K)r4/8R3] +[(1+K)2r6/16R5]+..., with r being the surface aperture radius, R the radius of curvature and K the conic constant. Lower- or fourth-order (according to the exponent in r) wavefront aberrations are calculated using the first two terms in the series. In fact, based not on the actual surface, but on its close approximation. It is accurate for relatively weak and/or small size surfaces. With larger, more strongly curved surfaces, the next higher, sixth-order term may become significant, in which case the lower-order aberration alone doesn't accurately present the amount - and the form - of aberration. It has to be balanced with the higher-order form. For instance, typically strongly curved lenses of an apo refractor induce relatively significant sixth-order spherical aberration. If a system is constructed based on zero fourth-order spherical aberration, the residual sixth-order aberration is likely to compromise system performance. To prevent this, system surfaces are corrected as to allow certain amount of lower-order spherical aberration to offset its higher-order counterpart, so that the total aberration is minimized. Allowing certain amount of the lower-order aberration here is, in fact, correcting the actual surfaces beyond their fourth-order zero-aberration approximation so that they are closer to that perfect surface profile that would produce a perfect wavefront. Higher-order off-axis wavefront aberrations, coma and astigmatism, also result from this same conic surface approximation used in the aberration calculation. Similarly, higher-order ray aberrations stem from the approximated incident angle on the surface, which determines the reflected or refracted ray angle. Since either tangent or sine of the incident angle at the optical surface are a function, among other elements, of the sagitta z, they can't be expressed in simple terms. Instead, the calculation uses angle approximation from, this time, trigonometric function expansion series, such as sinα=a-(a3/3!)+(a5/5!)-(a7/7!)+..., with α being the incident angle on the surface in degrees, and a this angle in radians. For small angles (paraxial approximation) sinα is nearly identical to a; for larger angles, however, the difference becomes significant. Lower-order aberration calculation (Seidel) adds the second term, (a3/3!), which is why it is called third-order calculation. It is sufficiently accurate for weaker/smaller surfaces. Larger, more strongly curved surfaces may require adding the fifth-order, or yet higher terms to the angle approximation in order to obtain sufficiently accurate result. Higher-order terms - secondary or Schwarzschild, as well as tertiary aberrations - are negligible with most amateur telescope types, and also significantly more complex. For those reasons, higher aberrations will not be specifically addressed in this text. Follows brief overview of terms and conventions that are the basis of the aberration calculation.
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