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

3.4. Terms and conventions

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 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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