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▪ CONTENTS ◄ 2.3. Telescope magnification ▐ 3.2. Ray (geometric) aberrations ► 3. TELESCOPE ABERRATIONS: Types and causes
PAGE HIGHLIGHTS Any deviation of the wavefront formed by a telescope from perfect spherical - for wavefronts formed by the objective - or from perfect flat for wavefronts formed by the eyepiece, results in an optical aberration. Aberrations disturb optimum convergence of the energy to a point-image, with the result being degradation of image quality. The two main forms of measuring an aberration are:
(1) at the wavefront itself, as a deviation from the perfect reference
sphere, and
The former are known as wavefront aberrations; the later as ray, or geometric (ray) aberrations. Either aberration form has its purpose. While the wavefront aberration form is more directly related to the physical fundamentals determining image quality, ray aberration form offers more convenient graphical interface for the initial evaluation of the quality level of optical systems. Since the wavefront and the rays emerging from it are directly related, there is a constant relationship between the size of wavefront aberration, and that of corresponding transverse ray aberration relative to the size of the Airy disc. This is true for any given relative aperture; obviously, change in relative aperture for any given size of the transverse aberration relative to the Airy disc (i.e. for any given wavefront error) requires the relative longitudinal aberration to change inversely. Unrelated to the form of presentation of aberrations, it is useful to make a distinction between aberrations that are intrinsic to optical surfaces in their proper alignment, and those induced by external factors. Intrinsic telescope aberrations are those inherent to conical surfaces, to glass medium, and those resulting from fabrication errors.
Externally induced
telescope aberrations are
caused by: (1) alignment errors, (2)
forced surface deformations
caused by
thermal variations, gravity and improper mounting, and (3)
air currents/turbulence.
3.1. WAVEFRONT ABERRATIONS
As described in previous chapters, imaging
quality of a telescope rely on optical surfaces capable of producing
spherical wavefronts for the image formed by objective, then transformed
into flat wavefronts by the eyepiece. The final wavefront is formed by the
eye, ideally of spherical shape. Spherical wavefront ensures tightest possible energy concentration in the image of a point-source
and, consequently, highest contrast and resolution. In other words, the
effect of
diffraction,
which causes the point-object image to form as a bright central disc
surrounded by a number of fainter concentric rings of rapidly decreasing
intensity, is at its minimum for perfectly spherical (aberration-free)
wavefront.
Thus perfect telescope is the one that
produces flat wavefronts exiting the eyepiece. While any combination of
aberrated wavefronts at the objective and eyepiece that cancel each
other out will do the trick, it is preferred to have the objective
producing a near perfect spherical wavefront, and the eyepiece turning it
into near perfectly flat. After that, it is up to the eye how accurate will
be the final wavefront: the closer to spherical, the better.
For most people, the wavefront formed by
the eye becomes nearly spherical at ~2mm pupil diameter, and practically
spherical at ~1mm pupil. The larger eye pupil, the greater wavefront
deviations from spherical, due to eye's optical imperfections. This, in
general, has less of an effect, with larger exit pupil sizes being
associated with low-power observing, when
wavefront imperfections are in general more forgiving. Wavefront quality
is critical for high-magnification observing at small pupil sizes, when
the eye, as mentioned, produces near-spherical wavefronts, provided it
is supplied with near-perfect flat wavefront by the telescope. Any
significant deviation from
spherical in the shape of the wavefront formed by telescope's objective results in
lower quality of its image. Assuming no aberration contribution from the
eyepiece, this wavefront deformation will be transferred to the eye as
an imperfectly flat wavefront coming out of the eyepiece, passing the deviation to the wavefront formed by the eye. Since the path
length of a wave from any deviant, or aberrated point on the wavefront differs from
the wavefront radius' length, it arrives at the focal point out of
phase with the waves coming from the spherical portion of the wavefront.
The greater wave path difference, the greater
its phase difference,
and the lower wave energy contribution at the focal point. The more such
points on the wavefront, the more energy transferred to the outer
portion of diffraction pattern, and the lower image quality.
The existence of path
difference at the focal point implies that there is a point - or points
- farther off in the image space for which the wave path difference from
aberrated points at the wavefront is now smaller, and
constructive energy interference greater, than in a perfect system. In other
words, that the energy lost from the proximity of the focal point due to wavefront
aberrations will be effectively transferred toward outer area of
the diffraction pattern. Deviation of any single
point on the wavefront will not cause measurable effect on image
quality, regardless of its optical path difference; however, if an area
of the wavefront deviates from spherical, it will negatively affect
image quality, the larger area, the more so. Energy concentration at the center of diffraction pattern
becomes noticeably lower relative to energy contained in the diffraction
rings, blurring
the point image. The image quality of point-source deteriorates, and
with it image quality of extended objects (the latter being merely point-image conglomerates). In terms of loss of resolution,
expectedly, low-contrast details are affected more than those of high
inherent contrast.
Wavefront deviation is commonly presented
with the reference sphere as a straight line. This wavefront deviation
form shouldn't be confused with the wavefront itself. Picture at left
illustrates the relation between the wavefront and wavefront deviation
for three common aberrations, defocus, primary spherical and coma.
The actual wavefront converging toward focal point is always for all
practical purposes spherical, with the deviations from spherical usually
being a small fraction of micron, with the wavefront deviation form
always greatly exaggerated.
The values of maximum positive and
negative wavefront deviation from the reference sphere combined determine peak-to-valley (P-V)
wavefront error. This figure alone is meaningless with respect to the damage it
causes to image quality, unless related to a known
form of wavefront deformation. In other words, unless both maximum
wavefront deviation from spherical, as well as the form and areal extent of
its deformation are known (FIG. 21). An example of such forms of
wavefront deformations are those characteristic of the typical optical surface,
conic of revolution - spherical aberration, coma, astigmatism, field
curvature and distortion. The only useful input
from the P-V figure alone is that it approximates the worst case
scenario; that is, if the specified P-V error affects most or all of the
wavefront area, it cannot be significantly worse than a wavefront with
this level of P-V error of spherical aberration (assuming that
smaller-scale surface roughness is not significant). So 1/10 wave P-V
mirror with reasonably smooth surface cannot be significantly bellow the
quality level of 1/10 wave P-V of spherical aberration; on the other
hand, it is possible that 1/4 wave P-V mirror, with the deviation
limited to a very small wavefront area, performs as well, or even
better.
The sign of the P-V wavefront aberration is
determined by the optical path length: if it is larger than a perfect
reference path (i.e. if the wave has to travel an extra length to reach
the focus), the P-V error is positive, and vice versa. The term "optical
path length" refers to the path length that the light wave travels in a
given time, determined by the optical path length of the chief ray (the
central ray of the wavefront); therefore, it is directly dependant on the speed of light
through optical media and may differ from the geometric path length.
This is why the error on, say, mirror surface (the medium is air),
results in different optical path length - and error magnitude - than
nominally identical error on the lens surface (the mediums are air and
glass).
P-V WAVEFRONT ERROR SPHERICAL ABERRATION 95% TURNED EDGE RMS (ω) Strehl
φ RMS (ω) Strehl
φ λ/8 0.037 0.95 0.036 0.0167 0.993 0.0133 λ/4 0.0745 0.80 0.075 0.0334 0.963 0.0307 λ/2 0.149 0.39 0.0153 0.063 0.886 0.055 λ 0.296 0.09 0.245 0.127 0.861 0.061 2λ 0.59 0.05 0.273 0.254 0.845 0.065 Direct
dependence of the RMS wavefront error on the P-V error
magnitude causes it to become less reliable indicator of the
wavefront quality, especially with larger P-V errors. The
phase analog of the RMS wavefront error (ω),
denoted by
φ
(estimated from the Strehl
value, by reversing Mahajan's Strehl approximation, thus not
accurate for very low Strehl values, but suitable for illustration) is the actual
determinant of image quality. As the table shows, it is
nearly identical to the RMS wavefront error for aberrations
smoothly affecting most of all of wavefront area, as long as
their magnitude is low to moderate. For P-V errors significantly larger
than λ/2, the RMS wavefront error tends to become
significantly larger than its phase analog. For aberrations
affecting small wavefront area, the differential is larger
at both, low and high P-V error levels, widening as the
P-V/RMS errors increase; when they are very large, the
resulting phase error - and image quality - remain nearly
unchanged, despite further increase in the nominal P-V/RMS
error.
The extent of image deterioration caused
by wavefront deformations is much more reliably determined by its deviation from spherical averaged over the
entire wavefront. It is the so called
root-mean-square (RMS) wavefront error, usually expressed in
units of the wavelength of light. The RMS wavefront error is given by a
square root of the difference between the average of squared wavefront
deviations minus the square of average wavefront deviation, or RMS=
<(W-<W>)2>1/2 = (<W2>-<W>2)1/2,
with the <...>
brackets indicating an average value. For instance, if we measure wavefront
deviations at three points, for simplicity, as 0.5, 0.2 and 0.1, the
average of their squared values
<W2>=0.1,
while the square of their average value
<W>2=0.071.
The RMS error would be given as RMS=√0.1-0.071=0.17.
In more general terminology, the RMS error is what is statistically
known as standard deviation,
which is given as the square root of
variance, defined
as the average of the squared differences from the mean value. In the
case of a wavefront, the mean value is that of the deviations from
perfect at all measured points. Taking the same three values from above,
the mean is (0.5+0.2+0.1)/3=0.2667, and the variance is
[(0.5-0.2667)2+(0.2-0.2667)2+(0.1-0.2667)2)]/3=0.02889
and the standard deviation - or RMS wavefront error - is, again, √0.02889=0.17.
Throughout this site, the RMS wavefront error is denoted by ω, and
the P-V wavefront error (occasionally also peak wavefront error) is denoted by W.
The RMS value expresses statistical deviation from
perfect reference sphere, averaged over the entire wavefront. Since derived from
squared values, the RMS error is independent of the sign of P-V
wavefront deviations, thus always given as an absolute (positive)
number. To be meaningful,
the RMS wavefront error must be calculated for a large number of
points on the wavefront (or optical surface, for surface RMS).
By being an indicator of the average
optical path deviation over the entire wavefront, the RMS wavefront error is
closely related to the cumulative phase loss at
the center of diffraction pattern and, hence, to its peak intensity.
Phase variance over the pupil can be written as
φ2
= <(Φ-<Φ>)2>=(<Φ2>-<Φ>2)=(2πφ)2,
where Φ is the phase
deviation, in radians, varying over the pupil, and
φ
is the phase analog to the RMS wavefront error (with the former in units
of full phase, or 2π,
and the latter usually in units of the wavelength). Also, the phase deviation Φ
is analog to the OPD. The significance of
phase variance is that the image point at which it is at its minimum
determines so called best focus
(also, diffraction focus). While best focus often practically
coincides with the point of minimum OPD (wavefront) RMS, it is possible
that the two differ significantly, as explained ahead.
Table below summarizes the meaning of
wavefront vs. phase aberration, including basic related terms. WAVEFRONT ERROR
(ABERRATION), W PHASE ERROR
(ABERRATION), Φ=2[(W/λ)-i]π
► OPD (optical path
difference) with respect to
the reference sphere
Peak aberration (denoted by W, or WP)
► based on OPD as Φ=(OPD/λ)-i,
hence given
Despite being directly related nominally,
the two forms of deviation (OPD/wavefront/RMS and phase) are not
necessarily commensurate. The reason is that both OPD and RMS derived from it
are based on the nominal linear
For that reason, a
nominally large deviation affecting relatively small wavefront area (for
instance, a narrow zone, turned edge, tube currents or seeing error)
will have disproportionately larger effect on the OPD/P-V/RMS deviation
value, than on the phase deviation value. Since it is the cumulative phase
deviation that determines central diffraction intensity and the overall
intensity distribution within diffraction pattern, both P-V and RMS
values of this type of errors disassociate from the phase error,
indicating larger than actual damage to the image quality - the
larger magnitude of such aberrations, the more so.
As a result, the
Strehl value calculated from RMS-based
approximations will be lower, possibly significantly, than the actual
Strehl, determined by the cumulative phase deviation. Discrepancy
between the wavefront and phase RMS becomes potentially significant as
the former exceeds 0.15λ. At the RMS wavefront values of about 0.25λ and
larger, it is quite possible that a significantly higher RMS wavefront
error produces better Strehl value (although still in the single ratio
points at best). For instance, at 1 wave P-V of primary spherical
aberration at mid focus, diffraction maximas occur at the points 33%
closer and 33% farther away from paraxial focus, despite the P-V/RMS
wavefront error being still the lowest at the mid focus (FIG.
36B).
For instance, image deterioration due to
turned edge will be increasing with the TE magnitude only up to a
certain level, after which further increase in the nominal RMS
wavefront error will have little or no effect. For
mirror edge, and 0.95% TE, this level is at about 1 wave P-V, or 0.13
wave RMS. At this point, central diffraction intensity is reduced to
~0.92. Increasing the error to 2 waves P-V, or 0.26 wave RMS, causes
near negligible drop in the central intensity, down to ~0.91.
Further error increase has practically no effect, as the ratio of
constructive vs. destructive wave interference at the focus remains
nearly unchanged.
Similar effect will be observed with narrow zones (which can't cause
more of diffraction disturbance than a matching ring-like obstruction,
no matter how many waves RMS deep), or any other type of local wavefront
deviation.
However, for relatively small deviations -
generally less than
λ/2 - smoothly
distributed over all, or most of the wavefront, the cumulative OPD/RMS
and phase deviations will be closely related, indicating very similar to
nearly identical level of aberration in the common range of their
magnitudes in telescopes.
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FIGURE 20: Ray and
wavefront geometry in a perfect (top) and aberrated (bottom) telescope.
Ray aberrations - longitudinal, transverse and angular - result from
wavefront deformation, but their numerical values have no inherent
relation to the determinant of energy (re)distribution: optical path
difference (OPD), creating phase differential between waves interfering
in the image space. The efficacy of wave interference varies with cos
FIGURE
21: Two nominally identical
peak-to-valley 
deviation, while the corresponding phase deviation
effectively only
varies between 0 and 2π,
regardless of the size of linear wavefront deviation (plots at left).
For instance, a 1.5 waves OPD still causes a half of the full cycle (π
radians) phase error, same as at 0.5 waves OPD, despite the nominal
phase error being also 1.5 cycles, or 3π
radians. And same will repeat at 2.5, 3.5, and so on, waves OPD.