◄
3.4. Terms and conventions
▐
3.5.1. Seidel aberrations
► ## 3.5. Aberration function, Introduction
PAGE HIGHLIGHTS Aberration function, as the term implies, describes the size of aberration as a function of its determining factors. Since aberrations result from the interaction of light with optical surface, these determining factors belong to these three main groups:
(1) properties of light
falling onto optical surface,
Properties of
Properties of the
Properties of the An aberration can be expressed in various forms; longitudinal and transverse aberration forms are ray based (geometrically), while wavefront and phase deviations are OPD-related (optical path difference, also ray based, but expressing different aspects of deviation, measured in units of wave and phase deviation, respectively). The most important aspect of aberration is the optical path difference (OPD) between the chief ray and other rays originating from the same object point at the focus location. Optical paths and OPD values are calculated using Snell's law of refraction, and/or Fermat's principle, details of which are omitted, being requiring extensive calculations. Once the optical path lengths, i.e. specific optical path differences for sufficient number of rays are known, the form of the wavefront can also be determined. With the form of wavefront known to a sufficient accuracy, phase deviations in the image space - determining diffraction effect, i.e. change in energy distribution - can be specified as well. A complete expression for any aberration form includes all the relevant parameters listed above; however, parameters related to the aberration properties - i.e. its change with the pupil coordinate and angle of incidence - can be, and routinely are omitted for simplicity and/or ease of calculation when multiple contributing surfaces are present (the radial pupil coordinate has to be compensated for in some form if marginal ray height varies from one surface to another but, since surface contributions are commonly calculated for each specific aberration alone, angular pupil coordinate and angle of incidence factors can be applied only to the coefficient sum). Also, since image vs. object magnification depends on some of these same basic parameters (object and image distance, surface radius of curvature, medium), it can be used for the alternative forms of expression, particularly for mirror surface.
An aberration expression omitting both, pupil and angle of incidence
factors, is called
Wavefront aberrations at a single optical surface with the stop at the
surface, either reflecting or
refracting (so called
s = -(NJ
for
for
for
The three aggregate parameters
Note that the P-V wavefront deviations are those for
classical primary aberrations, at
paraxial focus;
Relations for
If aspheric coefficient
(f).
For mirror surface, one of the three basic parameters can be written in
a different form, simplifying the coefficient expressions: (1/n'L')-(1/nL)=2/nR,
hence N=n/R or, for mirror in air oriented to the left, N=1/R. Also,
J=(1/L)-(1/R)=R(m+1)/(m-1), with
Table below summarizes the three aberration coefficients for the stop at
the surface and object at infinity.
Likewise, transverse aberrations, also in the paraxial image space, are given by:
ST = -(NJ2+Q)d3L'/2n'
for Transverse ray aberration can also be expressed in terms of the aberration coefficients, as:
for spherical aberration, coma and astigmatism, respectively. For object
at infinity, L'=ƒ; substituting
-1 for
The corresponding longitudinal aberration is
simply larger by a L'/d factor, or SL=4sd2L'2/n',
and AL=2aα2L'2/n'
(coma does not have the longitudinal component).
The corresponding transverse aberrations are For two-mirror system, the object is image formed by the secondary, thus its axial vertex separation from secondary's surface is numerically positive for the Gregorian, and negative for the Cassegrain. Refractive indici for the secondary are n=-1 and n'=1.
For refractive surface, n=1 and n'=nl,
The combined aberration for two or more surfaces is given as a sum of
(wavefront) aberration coefficients at each surface, as s=s1+s2+...,
c=c1+c2+...
and a=a1+a2+...,
assuming equal aperture radius at each surface. In practice, telescope systems regularly
handle converging - occasionally also diverging - cones, resulting in
unequal marginal ray height (i.e. effective aperture radius) at individual surfaces. Since the final size
of aberration always depends on the size of aperture - expressed as
For instance, system coefficient for spherical aberration for a
two-mirror system with the marginal ray heights For off-axis aberrations, however, an additional parameter needs to be accounted for: the stop position. It applies either to a single surface with a separated aperture stop, or to any multi-surface system, whether the aperture stop for the first surface coincides with it (when applies to the rest of the surfaces), or not.
With the stop displaced from the surface, the ray geometry changes (Fig. 151), with the chief ray passing not through the surface vertex, but through the center of the aperture stop. In terms of wavefront, it is relevant for off-axis points displaced laterally with respect to the surface, which results in a change of the aberration values.
For the axial point (spherical aberration), there is no change in aberration coefficient as long as the effective aperture remains unchanged. If displaced stop changes what would be the effective aperture without it (normally, determined by the clear diameter of the first optical surface), the aberration coefficient is calculated for the effective aperture.
The additional factors for calculating aberration coefficient for
off-axis image point are, then, the surface-to-stop separation
General form of surface aberration coefficient for
the three Seidel aberrations affecting point-image quality when the stop
is displaced from surface (hence with subscript
with new parameters, in addition to
It is immediately apparent from
Obviously, it is the stop-to-surface separation
So, in general, stop separation, based on
Eq. (1) is given by T=n'IPR/[(n'-n)IP+nR]
for general surface, with
Aberration coefficient are the first level
of aberration function. They only describe a single aberration, and can
be used to calculate wavefront and phase deviation resulting from it.
More often, telescope systems generate combined aberrations, which
require more complex aberration function to describe. Follows detailed description of the general form of aberration function
for the five classical, or Seidel aberrations. |