For understanding text related to optical aberrations, it is necessary
to know the meaning of terms used in their description and calculation.
Part of the latter is sign convention, the purpose of which is to assign
to every parameter related to a final determination of the optical
path length for any point in the pupil its appropriate numerical value. This
ensures that all the contributions to the path length will be properly
combined, to result in an accurate description of the wavefront form and
directly related to it orientation of individual rays.

Basic terms and parameters used in
calculation and description of primary aberrations, including sign
convention, are given in **FIG. 25-26**.

**
**
**
FIGURE 25**: Telescope's
**aperture stop**
is either an
opening, or a surface that sets physical
boundaries determining the amount of light reaching the image. The
image of the aperture stop formed by a system element preceding it
in the optical train is **entrance pupil**,
and the image of the aperture stop formed by the element or surface
fallowing it in the optical train is **exit
pupil**. Alternately, entrance pupil is the apparent aperture
as seen from object space, and exit pupil is the apparent aperture seen
from image space (when no optical element precedes aperture stop, it
coincides with entrance pupil). The two pupils coincide with the aperture stop -
and each other - for a single mirror with the stop at the
surface (also, for all practical purposes, for a single lens
objective with the stop at the front surface). The two pupils
coincide for the stop at the mirror's center of curvature. Rays from
the boundaries of the aperture stop coming to the final focus appear
as if coming from the boundaries of the exit pupil, and the **
chief
ray** **CR** - the one passing through the center of the aperture stop
- appears to be
coming from the exit pupil center. These properties make the exit
pupil an important element in the aberration calculation, since the
cause of wavefront aberrations - optical path difference of in-phase
wavefront points with respect to the central point on the chief ray
- is directly determined by its location and position vs. optical
axis. The above image illustrates a
concave mirror with the *aperture
stop* somewhat inside the mirror focus. The mirror
images the aperture stop into the *exit pupil* **ExP** which appears to be the opening
from which the rays converge (exit) toward the image. In two-mirror
systems, secondary forms the exit pupil, as its image of the primary
mirror, with the image being smaller than the
aperture. The two pupils' size ratio is given by **
pupil magnification** **m**, as
ExP=mEnP. Actual size of the exit pupil may be a factor in some
calculations. In principle, it is irrelevant, due to the
change in the radial coordinate being offset by that in the axial
coordinate. Thus, while formally the wavefront is evaluated at the
exit pupil, the coordinates used hereafter are,
conveniently, those of the aperture stop, whether the two coincide,
or not.

**
FIGURE 26**: Basic imaging terms
and parameters defined in the 3-dimensional**
**right-hand Cartesian**
coordinate system**. All
lengths, as well as the indici of reflection and refraction, are positive in
the directions of the** z**, **x** and **y** coordinate axes
arrows, negative in the opposite direction. Angles are positive when
opening clockwise from the axis, negative when opening
counterclockwise. The **
exit pupil** **ExP** from which the wavefront,
if perfect, converges
to the **Gaussian - ** or**
paraxial - focus** **GF** in
the image plane, at the distance equal to the focal length **f** for axial objects at infinity. The
**pupil radius**
**d** is the unit length for the**
normalized
height in the pupil plane** **ρ**,
ranging from 0 to 1. The **pupil angle
** **
θ**
ranges from 0 to 2π
radians (360 degrees), measured from **y**+ axis counterclockwise; it is a factor with which optical path difference
vary for asymmetrical aberrations. The **axis of aberration**
is determined
by the spatial orientation of the **chief ray**
**CR** that
passes through the pupil
center at an inclination angle
α** **
- the **field angle** - in the plane
determined by the chief ray and optical axis, defined as **
tangential plane**. **
Sagittal plane** is orthogonal to
the tangential plane, also containing the chief ray. Point of
intersection of the chief ray and image surface determines
**
Gaussian image point**, and its height
**
h **(for simplicity, the image surface is shown coinciding
with the **xy** plane; actual Gaussian image points lie on the
Petzval surface); with the Gaussian focus
point, it determines the **
**

axis of aberration.

A quick summary of the sign convention is as follows:

▪ **optical axis** of a centered system
coincides with the horizontal (**z**) axis of the coordinate system, with
zero coinciding with the center of the aperture stop;

▪ the **object** is to the left of
the optical system so that the incident light travels from left to
right; object distance is measured from the center of the aperture
stop, thus numerically negative;

▪ distance from surface to a displaced **aperture
stop** is numerically negative for the stop to the
left, positive
for the stop to the right of the surface (for instance, it is negative for
mirror-to-stop separation in the Schmidt camera, with the stop at the
corrector, and positive for stop-to-secondary separation in a two-mirror
telescope, the primary being the aperture stop for the secondary);

▪ surface **radius of curvature** is
positive if its center lies to the right from a surface, negative if the
center is to the left

▪ distance to the **image** formed by
the optical system is positive if it is to the right of the image forming
element, and negative if it is to the left from it

▪ distance from the image to the **exit pupil**
is negative for exit pupil to the right, postive for exit pupil to the
left of the image

▪ point **height** is positive if above
the optical axis, negative if below

▪ **angle** is positive if opening
upwards from the optical axis, negative if opening down;

In short, the sign convention is consistent with the coordinate frame.
More complex, or specialized texts often find it convenient to deviate
from the sign convention consistency for one or another reason, readjusting
affected parameters accordingly with respect to the sign applied. On the
other hand, not a few readers find sign inconsistency to be the greatest
convenience.

With the general parameters numerically
determined, primary aberrations of an optical surface can be described
either in their wavefront or ray form. The former are determined by
**aberration coefficients** which, when
multiplied with surface diameter and angle of incidence (for abaxial
aberrations), specify the size of wavefront deviation. The latter are
determined by their geometric size in the image plane, or **
transverse aberration**. Just as the
wavefront and the rays themselves, the two are directly related, and are
expressed with similar groups of parameters. These parameters are based
on object properties (distance, height), surface properties (diameter,
radius of curvature, conic) and image properties, as determined by the
Gaussian approximation.

Follows more detailed overview of the usual forms of presentation of
wavefront aberrations - so called aberration function. It will first
present the general form of aberration coefficients for three
point-image quality, their relation to wavefront and transverse
aberration, and then continue to the
aggregate wavefront aberration in its general form, its relation to
Seidel aberration expressions and lower-order Zernike aberration form.

◄
3.3.2. Aberrations of the conic
surface
▐
3.5. Aberration function
►

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