optics

SPHERICAL ABERRATION
Spherical aberration - or correction error - is the only form of monochromatic axial aberration produced by rotationally symmetrical surfaces centered and orthogonal in regard to the optical axis. The attribute spherical probably originates in this aberration being inherent to the basic optical surface - spherical - for object at infinity. However, spherical aberration will appear whenever optical surface form doesn t properly match that of the incident wavefront. Thus, it is induced with the change of object distance or, with multi-surface objectives, with deviations in proper spacing. Spherical aberration affects the entire image field, including the very center. For that reason, its correction in a telescope is more important than that of other inherent conic surface aberrations, which affect the outer field.
Spherical aberration in the majority of amateur telescopes - especially more traditional ones, like Newtonian reflector or achromat refractor - is sufficiently accurately presented based on 4th order surface approximation. This aberration approximation is called lower-order, or primary spherical aberration (also, 4th order wavefront, or 3rd order transverse ray aberration). Telescope objectives with strongly curved surfaces - like Maksutov-Cassegrain or apochromatic refractors - generate significant amount of higher-order (6th on the wavefront, or 5th transverse ray) spherical aberration, which requires upgrading, or correcting 4th order surface approximation to the 6th order surface. Very rarely, higher order terms also need to be taken into account.

4.1.1. Lower-order spherical aberration
FIG. 20 illustrates under-corrected (negative) form of primary spherical aberration, characteristic of a spherical mirror for object at infinity. Due to the actual wavefront being not spherical, rays projected from it do not meet at the same point; the wavefront becoming more strongly curved toward the edge causes the foci for rays projected from its outer zones to fall progressively closer to the mirror.
FIGURE 20: Spherical aberration of a concave spherical mirror, commonly called under-correction (due to marginal rays falling short of paraxial focus). RIGHT: The actual wavefront W is increasingly more curved toward the edge than the reference sphere SP coinciding with wavefront produced by reflection from the imaginary paraboloid P (for object at infinity) and centered at the paraxial (Gaussian) focus G. As a result, its outer rays focus progressively closer to the mirror: while central rays meet at the paraxial focus, the edge rays meet at the marginal focus M. Best focus B is midway between the marginal and paraxial focus, due to the deviation of the actual wavefront from perfect reference sphere Sb centered at this point being the smallest. Best focus wavefront error peaks at 0.707 aperture radius; it is smaller from the wavefront error at either marginal or paraxial focus by a factor of 0.25. The aberration at paraxial focus is primary spherical aberration, and at best focus location it is balanced primary spherical aberration (it is balanced - or minimized - with defocus aberration). Ray geometry determines the longitudinal (L) and transverse (T) aberration, shown with respect to the Gaussian focus. LEFT: Relative deviation of the actual wavefront in regard to perfect reference spheres for the marginal (Sm), paraxial (SP) and best, or diffraction focus (Sb) is constant for any amount of aberration.
With over-corrected (positive) spherical aberration, marginal rays focus farther away than paraxial rays. In either case, geometrical structure of the defocused zone remains identical in regard to the paraxial focus. For the longitudinal aberration normalized to 2 (L?0???2), the geometric (ray) spot has a constant relative size (vs. size of longitudinal aberration) and structure for a given value of ?, as illustrated in FIG. 21. In units of the paraxial blur diameter (?=0), the blur size is 0.385 for ?=2 (marginal focus), 0.25 for ?=1.5 (circle of least confusion) and 0.5 for ?=1 (diffraction focus). In general, for 0???1.5 the relative blur diameter is given by (2-?)/2; for 1.5???2, it is closely approximated by (?-0.5)/4 (the approximation is exact for ?=1.5, wit the error increasing to a -2.6% maximum at ?=2).
The relative wavefront error - either P-V or RMS - for any point between the two foci, in units of the error at the paraxial or marginal focus, is also constant, as given by:
? = [1+0.9375?(?-2)]1/2 (6)
It gives the minimum relative aberration of 0.25 for ?=1, which is the mid point between marginal and paraxial focus.
Note that the parameter ? is related to peak aberration coefficients for spherical aberration and defocus, S and P, respectively, as ?=-P/S. Thus, in terms of defocus error, spherical aberration is minimized, or balanced, for P=-S, or for spherical aberration at paraxial focus combined with the identical P-V of defocus aberration.
FIGURE 21: Defocus caused by spherical aberration, illustrated by selected rays projected from the aberrated wavefront. Axial separation between the foci for paraxial and marginal rays determines longitudinal aberration L (?0 when normalized to 2. Note that ?=P/S, P and S being the peak aberration coefficients for defocus and spherical aberration, respectively. Both, transverse and wavefront aberration vary with the relative defocus ? within the aberrated focal zone. Transverse blurs, from left, at the paraxial, or Gaussian focus (?=0), at the best, or diffraction focus (?=1), at the location of the circle of least confusion (?=1.5), and at the marginal focus (?=2). Pupil semi-diameter is d, and arbitrary paraxial zone height (illustration only) is p. Darker blur coloration roughly indicates increased ray density. Smallest blur radius is determined by the point of intersection of marginal ray and ray originating at the 0.5d zone. The relative blur size, in units of the paraxial blur, is given by (?2-??/2) with ?=1 for ?=0, 1 and 1.5 (paraxial, best focus, and smallest circle locations, respectively), and with ?=1/31/2 for ?=2 (marginal focus, which is of opposite sign to the former three due to being formed by converging rays). Aberration shown is spherical under-correction; since optical paths of waves from aberrated wavefront portion here is shorter that from corresponding points on perfect reference sphere, the wavefront error is numerically negative. Neither blur size/structure, nor size of wavefront error for given (absolute) value of ? change for over-correction.
Follows detailed review of quantifying primary spherical aberration in both, wavefront and ray (geometric) form for reflecting surfaces and lenses.
Chromatic aberrations
In optics, chromatic aberration (also called achromatism or chromatic distortion) is a type of distortion in which there is a failure of a lens to focus all colors to the same convergence point. It occurs because lenses have a different refractive index for different wavelengths of light (the dispersion of the lens). The refractive index decreases with increasing wavelength.
Chromatic aberration manifests itself as "fringes" of color along boundaries that separate dark and bright parts of the image, because each color in the optical spectrum cannot be focused at a single common point on the optical axis.

Chromatic aberration or "color fringing" is caused by the camera lens not focusing different wavelengths of light onto the exact same focal plane (the focal length for different wavelengths is different) and/or by the lens magnifying different wavelengths diffently. These types of chromatic aberration are referred to as "Longitudinal Chromatic
chromatic Aberration in a Single Lens
Chromatic aberration or "color fringing" is caused by the camera lens not focusing different wavelengths on depends on the dispersion of the glass.



Longitudinal or Axial Chromatic Aberration
Focal length varies with color wavelength Lateral or Transverse Chromatic Aberration
Magnification varies with color wavelength
Chromatic aberration

When different colors of light propagate at different speeds in a medium, the refractive index is wavelength dependent. This phenomenon is known as dispersion. A well-known example is the glass prism that disperses an incident beam of white light into a rainbow of colors [1]. Photographic lenses comprise various dispersive, dielectric glasses. These glasses do not refract all constituent colors of incident light at equal angles, and great efforts may be required to design an overall well-corrected lens that brings all colors together in the same focus. Chromatic aberrations are those departures from perfect imaging that are due to dispersion. Whereas the Seidel aberrations are monochromatic, i.e. they occur also with light of a single color, chromatic aberrations are only noticed with polychromatic light.
Longitudinal and transverse chromatic aberration
One discriminates between two types of chromatic aberration. Longitudinal chromatic aberration, also known as axial color, is the inability of a lens to focus different colors in the same focal plane. For a subject point on the optical axis the foci of the various colors are also on the optical axis, but displaced in the longitudinal direction (i.e. along the axis). This behavior is elucidated in Fig. 1 for a distant light source. In this sketch, only the green light is in sharp focus on the sensor. The blue and red light have a so-called circle of confusion in the sensor plane and are not sharply imaged.

Figure 1. Origin of longitudinal chromatic aberration. The focal planes of the various colors do not coincide.
Obliquely incident light leads to the transverse chromatic aberration, also known as lateral color. It refers to sideward displaced foci. In the absence of axial color, all colors are in focus in the same plane, but the image magnification depends on the wavelength. This behavior is illustrated in Fig. 2. The occurrence of lateral color implies that the focal length depends on the wavelength, whereas the occurrence of axial color in a complex lens does not strictly require a variable focal length. This seems counterintuitive, but in a lens corrected for longitudinal chromatic aberration the principal planes do not need to coincide for all colors. Since the focal length is determined by the distance from the rear principal plane to the image plane, the focal length may depend on the wavelength even when all images are in the same plane.

Figure 2. Origin of transverse chromatic aberration. The size of the image varies from one color to the next.
Figures 1 and 2 distinguish two simplified cases because in practice the longitudinal and lateral components are coexistent. A polychromatic subject fills a volume in the image space, which is comprised of a continuum of monochromatic images of various sizes and positions. Lateral color is particularly manifest in telephoto and reversed telephoto (retro focus) lenses. Chromatic aberrations often limit the performance of otherwise well-corrected telephoto designs. Lateral color, and not astigmatism, is the chief cause of the separation betwee the sagittal and tangential curves in their modulation transfer functions. The archetypal manifestation of chromatic aberrations is color fringing along boundaries that separate dark and bright parts of the image. This said, descriptions of the perceptible effects of chromatic aberrations do vary in literature. It is read that lateral color is a more serious aberration than axial color, because the former gives rise to colored fringes while the latter merely reduces the sharpness [2]. Oberkochen holds a different view and points to axial color as the most conspicuous color aberration [3]. Hecht describes the cumulative effect of chromatic aberrations as a whitish blur or hazed overlay [4]. The residual color errors of an optical system with achromatic (under)correction for axial color lead to a magenta halo or blur around each image point [5,6].