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▪ CONTENTS
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3.4. Terms and conventions
▐
3.5.1. Seidel aberrations
► 3.5. OPTICAL 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 light falling onto optical surface are determined by:
Properties of the surface are determined by its:
Properties of the aberration are 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 aberration coefficient. Thus, it in effect expresses the relative form of aberration for unit pupil radius and angle. The expression that includes the actual pupil parameters is the peak aberration coefficient. In this section, focus is on single-surface aberration coefficients for the three primary aberrations affecting point-image quality - spherical, coma and astigmatism, as well as generalized aberration coefficient for two or more surfaces combined; following page presents a common form of the aggregate wavefront aberration function, in terms of aberration coefficients and pupil coordinates, 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 beginning with these three 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 object at infinity,
L=∞ and L'=f (focal length),
thus N=n/f and J=-(1/R), with
Q unchanged.
The three aggregate parameters N, J and Q fully describe
these three aberration coefficients. As mentioned, an aberration
coefficient only includes aberration parameters related to the
properties of incident light (including medium), and surface shape,
while omitting those related to the pupil and angle of incidence. As
such, it is meaningless as an indicator of the magnitude of aberration;
it can be used as a measure of its magnitude only if associated with the
pupil radius parameter d, for radially symmetrical aberrations,
like spherical and defocus, and for radially asymmetric aberrations with
specified pupil angle θ; for off-axis aberrations like coma
and astigmatism, it also requires inclusion of the incident angle α.
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 (identical to the chief ray angle for that
point height) 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.
Relations for c and a imply that, with the aperture stop at the surface, coma and
astigmatism are independent of the surface conic.
If aspheric coefficient
b is present, it is added to the sum in the brackets in (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 m being the mirror image
magnification factor. For object at infinity, m=0
and J=1/R.
Table below summarizes the three aberration coefficients for the stop at
the surface and object at infinity. TABLE 2: Aberration coefficients for the
three image quality aberrations (as opposed to the image form
aberrations, field curvature and distortion) and the corresponding
P-V wavefront error relations
Likewise, transverse aberrations, also in the paraxial image space, are
given by:
ST = -(NJ2+Q)d3L'/2n'
for primary 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. For object
at infinity, L'=f; substituting
-1 for n' gives transverse aberrations for mirror surface and object at
infinity as TS=-4sfd3=-2sRd3=-(1+K)d3/2R2
for spherical aberration, TC=-3fαd2/R2
for coma and TA=dα2
for astigmatism.
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).
EXAMPLE: A 150mm diameter f/10
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=f=-1500mm.
This gives N=-1/L'=-2/R, J=-1/R2, and Q=0, resulting in
s=2/8R3=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 focus 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,
the Gaussian focus image is not the best image; the P-V wavefront
aberrations are reduced 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 the best focus is
reduced 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 below 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.
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
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 aperture 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).
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 T,
and the new chief ray angle
αc.
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 d) is:
with new parameters, in addition to N, J
and Q, defined by Eq. (i), being Y=[(1/T)-(1/R)], and
h=Tαc,
the height of the chief ray at the surface, where
T is
the stop to surface separation and
αc
=α/[1-(T/L)] the
chief ray angle (by definition, chief ray passes through the center of
the aperture stop) with
α
being the field angle at
the surface with the stop at it)
and L, as before, the object distance.
It is immediately apparent from Eq. (l)
and (m) that now coma and astigmatism too are dependant on the
surface conic. The corresponding P-V wavefront error is obtained by
substituting these coefficients for the zero-shift coefficients in the
right most column of Table 2 (again, it is the error at
paraxial/Gaussian focus, not best focus location). If aspheric coefficient b
is present, it is
added to the sum in the brackets for all three, spherical aberration,
coma and astigmatism (not only for spherical aberration, as with the
stop at the surface).
Obviously, it is the stop-to-surface separation
T (according to sign convention positive for stop to the left of
surface, negative when at the right) that creates the difference vs.
stop at the surface geometry. When T=0,
Y=1, Q=0, αc=α,
and the stop-shift relations reduce to those with the stop at the
surface. When the stop precedes the first surface, T is simply
the stop-to-surface separation; for the following surface, the stop is
at the location of the image of aperture stop formed by the preceding
surface, hence T is the separation between this image (usually
virtual) and the surface, and so on. Similarly, when the stop is at the
first surface, like in two-mirror systems, aperture stop for the
secondary is the primary. In a three-mirror system, aperture stop for the
tertiary is the image of primary formed by the secondary.
So, in general, stop separation, based on
Eq. (1) is given by T=n'IPR/[(n'-n)IP+nR]
for general surface, with n', IP,
R and n being the refractive index of incident medium,
separation to the image formed by preceding surface, surface radius of
curvature and refractive index of transmitting medium, respectively. For
mirror surface, n'=-n and, for surface in air oriented to left, n=1,
with T=-IPR/(R-2IP).
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. |
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