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▪ CONTENTS
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3.4. Terms and conventions
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3.5.1. Seidel aberrations
► 3.5. Aberration function, IntroductionAberration function, as the term implies, describes the size of aberration as a function of its determining factors. It is usually given for the wavefront aberration form, and consists of two components: one is the aberration coefficient, determining the aberration level, and the other is pupil coordinate factor, specifying how the size of aberration varies with pupil coordinates (i.e. for different points on the wavefront). The latter defines the aberration as a function of the height in the pupil alone - for rotationally symmetrical aberrations, such as spherical - or for both, height in the pupil and pupil angle for rotationally asymmetrical aberrations (coma, astigmatism). Numerical value of aberration coefficients is determined by the object and surface properties. It is determined by the optical path difference between the chief ray and other rays originating from the same object point. Optical path difference values are calculated using Snell's law of refraction, combined with Fermat's principle, which states that in traveling from a point in front to a point past an optical surface, a light ray follows the path of minimum time. Once the ray optical path lengths, i.e. size of optical path differences are known, a form of the wavefront can also be determined. This page gives the relations describing aberration coefficients for the three point-image quality primary aberrations - spherical, coma and astigmatism - for a single surface, as well as two or more surfaces combined; following page presents a common form of the aggregate wavefront aberration function, in terms of the coefficients, and its relation to the original aberration expressions formulated by von Seidel. Stop at the surface Wavefront aberrations at a single optical surface with the stop at the surface, either reflecting or refracting (so called general surface), for the three primary aberrations affecting point-image quality, can be expressed with these three generalized aberration coefficients:
s = -(NJ2+Q)/8
(f)
for spherical aberration, with the
peak-to-valley wavefront error
at paraxial focus
Ws=sd4
for
coma, with the peak-to-valley wavefront
error at Gaussian
image point Wc=2cαd3cosθ,
for θ=0,
and
for astigmatism, with the peak-to-valley
wavefront error at Gaussian image point Wa=aα2d2cos2θ, for
θ=0,
n, n' being the refractive indici of the incident and
refractive/reflecting media,
and L,
L' the object and image
separation from the surface (the latter found using
Gaussian approximation), respectively, R being the surface radius of
curvature, and K being the surface conic (for coma and astigmatism, valid
for the aperture stop at the surface).
Note that the P-V wavefront deviations are those for
classical primary aberrations, at
paraxial focus; d is the aperture radius, α
the field angle in radians and θ the pupil angle (describes the
form of rotationally asymmetrical aberrations). By setting θ to
zero for coma and astigmatism, the relations give the maximum, P-V
error, along the axis of aberration.
If aspheric coefficient
b is present, it is added to the sum in the brackets in (f).
Relations for c and a imply that, with the aperture stop at the surface, coma and
astigmatism are independent of the surface conic.
Likewise, transverse aberrations, also in the paraxial image space, are
given by:
TS = -(NJ2+Q)d3L'/2n'
for spherical aberration, as the radius of paraxial blur,
Transverse ray aberration can also be expressed in terms of the aberration
coefficients, as:
for spherical aberration, coma and astigmatism, respectively.
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. In practice, telescope systems regularly
handle converging - occasionally also diverging - cones, resulting in
unequal marginal ray height at individual surfaces. Since the final size
of aberration always depends on the size of aperture - expressed as
peak aberration coefficients S=sd4,
C=cαd3
and A=a(αd)2
for spherical aberration, coma and astigmatism, respectively -
either representing the peak-to-valley wavefront error (for spherical
aberration and astigmatism), or directly related to it (one half of the
P-V wavefront error for coma) - combining aberration coefficients
requires factoring in the radius difference. It can be done either
directly, by multiplying aberration coefficient with the appropriate
radius term before combining aberration contribution from different
surfaces into a system sum, or at the level of aberration coefficient,
by correcting it for the radius term difference.
For instance, system coefficient for spherical aberration for a
two-mirror system with the marginal ray heights D1
and D2
on the primary and secondary, respectively, is given by ss=s1+(D2/D1)4s2,
with s1
and s2
being the individual aberration coefficients for the primary and
secondary (the D2/D1
ratio, representing the relative aperture of the secondary mirror -
specifically, its so called "minimum size" - in
units of the primary aperture, is used in calculating two-mirror system
aberrations as the k parameter).
EXAMPLE: Concave spherical mirror, oriented
to the left, d=75mm, R=-3000mm, n=1, n'=-1 (since light reverses
direction, but the medium is unchanged), for object at infinity, so L=-∞
and L'=R/2=ƒ=-1500mm.
This gives N=-1/L', J=-1/R, and Q=0, resulting in
s=9.26-12
and S=Ws=sd4=-0.000293mm,
c=1.1-7
(or 1.1/107=0.00000011)
and C=Wc/2=cαd3=0.047α,
a=0.00067 and A=Wa=a(αd)2=1.88α2
(Ws,Wc,
and Wa
are peak-to valley wavefront error of spherical
aberration, coma and astigmatism, respectively).
In units of 0.00055mm wavelength, the mirror aberrations in the
paraxial image space are 0.53 waves P-V of spherical aberration, 0.3
waves of coma and 0.01 wave of astigmatism, both
for α=0.1°
(as explained in more details with each specific aberration,
paraxial image space is not the best image space; the P-V wavefront
aberrations are smaller at their respective best, or diffraction focus
location by a factor of 0.25 for spherical aberration, 0.33 for coma, and unchanged for astigmatism
- however, the RMS wavefront error for astigmatism at best focus is
smaller by a factor of
1/√1.5).
The corresponding transverse aberrations are TS=-0.0234mm as the
radius of paraxial blur (the minus
sign indicates it is oriented bellow the axis for marginal ray above the
axis), TC=0.0049mm for tangential coma, and TA=0.00023mm
for the smallest astigmatic blur diameter. Expressed in Airy disc diameters
for 0.00055mm wavelength (F/745 in mm), the blur is 1.74, 0.36 and 0.017
for spherical aberration, coma and astigmatism, respectively.
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,
nl being the glass index of refraction.
For lens, the object for the rear lens surface is the image formed by
the front
surface, thus distance to it with respect to rear surface is positive with
a convex front surface and first surface object farther than the first
surface focal point; the indici for the rear surface are n=nl
and n'=1. The rest of parameters can be found using
basic imaging relations.
With the stop displaced from the surface,
the ray geometry changes (Fig. 102),
with the chief ray passing not through the surface vertex, but through
the center of the aperture stop. In terms of wavefront, it is 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 surface aperture remains identical to the stop
aperture. In converging or diverging light, the effective surface
aperture may be reduced or increased, respectively, in which case 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 T,
and the new chief ray angle
αc.
Resulting relations are given with field
flattener lens example.
Follows detailed description of the general form of aberration function
for the five classical, or Seidel aberrations. |