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▪ ** **CONTENTS
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6.4. Diffraction
pattern and aberrations
▐
6.5. Strehl ratio
►
#
**6.4.1. S****tar testing telescope
quality**
As
FIG. 96
shows, even slight presence of any wavefront aberration changes the form
of diffraction pattern, leaving its unique fingerprint on it. With the
knowledge of
what those fingerprints look like, with some practical experience, the
aberrations can be detected and, most often, quantified with sufficient
accuracy. Thus, all that is
needed to test telescope optics for quality is a single point-source of
light. It can be either a star, or a small, man-made source of light
that fits the definition of a point-source. The telescope should be **
well collimated**, and in **
thermal equilibrium** with surrounding
air (depending on the telescope size, type and initial differential, it can
require anything from minutes to hours).
In order to visually analyze
diffraction pattern, it has to be sufficiently magnified. Minimum
**magnification** for conducting star test is considered to be 25x per inch
of aperture (or equal to the aperture diameter in mm), but more is better.
Optimum magnification level is probably between 40x and 50x per inch,
combined with moderate defocus, of between 3 to 5 waves (number of rings
in the defocused pattern roughly corresponds to the defocus in waves).
When testing
on a star, it shouldn't be too bright, nor too faint; either can hide
more intricate, yet potentially important features of the diffraction
pattern. Optimum level of brightness for a 6 inch aperture is ~2nd
magnitude star, which helps define best star magnitude for the test, in terms
of aperture diameter **D**, as m~5logD-8.9 for the aperture **D**
in mm, or m~5logD-1.9, for **D**
in inches. Test star should be close to zenith, to minimize the effect of
seeing, although for telescopes without tracking mechanism Polaris may
be the best choice most of the time. Needless to say, less than **
good seeing quality** will compromise
reliability of star test results (steady air is also required for
artificial star testing; no heat radiating object or areas should be
close to the light path).
When star
testing with an artificial star, two more requirements have to be met:
(1) **angular size of the light source** needs to be less than 1/2 of the
Airy disc size, and
(2) distance between the source and the telescope
needs to be large enough to keep spherical aberration induced by the finite
object distance negligible.
The first
requirement demands the artificial star to be what it needs to be for the
test: effectively a point source of light. Diffraction theory shows (*Optical
Imaging and Aberrations 2*, V. Mahajan, p214), that if the source
diameter exceeds 1/4 of the Airy disc's, resulting diffraction pattern begins to
change: its full width at half maximum (FWHM), first
minima radius and overall ring structure all become larger. At the source diameter of 1/2 the Airy
disc's, the FWHM is nearly 15% larger, with the first minima nearly
doubling in radius (in effect, the first dark ring vanishes, with the
central disc and first bright ring merging). At the source size equaling the Airy disc's, the FWHM is
already 60% larger, and the in-focus ring structure diminishes further. There is no
specifics on how it affects defocused patterns, but it is a common sense
to try to preserve the form of diffraction pattern produced by
artificial star closely resembling the one that we are comparing it to.
With today's availability of materials and products, this shouldn't be
an obstacle.
Obviously,
angular size of an artificial star is a function of its diameter and its
distance from a telescope. Simple visualization of the Airy disc in
the focus of a telescope, with two lines extending from its top and
bottom to cross at the aperture stop and diverge outwards, helps define
the size of Airy disc projection for any given distance from the
telescope. Knowing that the Airy disc diameter in radians is given by
2.44λ/D, which for λ=0.00055mm comes to 1/745D for the aperture **D**
in mm, linear diameter corresponding to angular Airy disc diameter at
any distance **L** is given by Ø=L/745D. Since we want to be as close
to an actual point-source pattern as possible, the target size for the
artificial star is 1/4 as much, or
Ø ~ L/3000D
(note that **L** needs
to be in the same measuring unit as **D**, here in mm). For instance,
for D=100mm telescope, an artificial star placed at 50m distance shouldn't
be significantly larger than 0.17mm in diameter. For distance **L**
in feet, aperture **D** in inches, the maximum source diameter **Ø**
in mm is Ø~L/250D.
It is a bit
more complicated when it comes to determining **artificial star distance**
that will not induce spherical aberration appreciably affecting test
results. The complexity arises mainly from the great variety of
different telescope systems in use today. While they are all optimized
for object at infinity - or at least at a large distance - the amount of
sensitivity to reduced object distance can vary significantly from one
type to another. Some systems are easier to deal with, in a sense that
they come with the basic specifications known and/or relatively
constant. Some others have undetermined optical specs, thus cannot be
predicted with respect to their specific object distance sensitivity.
The easiest
are **single mirror systems**: Newtonians
with either paraboloidal or spherical mirror (diagonal, by definition,
has no optical power, and its surface errors do not induce spherical
aberration). For object distance in
units of mirror's focal length L/ƒ,
and its reciprocal value ψ=ƒ/L,
the P-V wavefront error of spherical aberration at the best focus induced by
object distance is given by a simple relation,
**
**(a)
where **K** is the mirror conic, **D** the aperture diameter in
mm, and **F** the focal ratio. The sign of wavefront error indicates
the form of aberration; when the aberrated portion of the wavefront is
closer to best focus than the reference sphere, it is over-correction,
and vice versa. Therefore, the aberration induced is over-correction for
K<-(1-2ψ)2.
Taking relatively close object, at twenty focal lengths away (ψ=1/20),
conics smaller than -0.81 will be inducing over-correction error, those
greater than -0.81 under-correction, and the error will be cancelled for
K=-0.81.
With **spherical
mirror**, there is no need to worry about the close object
error; being under-corrected for object at infinity, and with closer
objects inducing overcorrection, it actually becomes better corrected
with reduced object distance. The closer object, the more so -
up to the object placed at mirror's very center of curvature, where
sphere provides perfect
imaging. For testing purposes, however, it is
good to know what level of error it *should* have - if it is really
spherical. By substituting 0 for **K** in the above relation, the
wavefront error is given as W**s**=D(1-2ψ)2/2048F3.
Deviation ratio **δ** of the actual vs.
implied value indicates non-spherical surface, with its conic given as
K=(δ-1)(1-2ψ)2.
In other words, it is prolate ellipsoid for 0<δ<1
and oblate ellipsoid for δ>1 (negative **
δ** value implies hyperboloid).
Taking, for instance, δ=0.6, implies
K=-0.324 for object at ten focal lengths away (ψ=1/10), and K=-0.361
for object at twenty focal lengths. The positive sign of **
δ** implies that the actual error is
under-correction, of the same sign as that of spherical mirror (which
induces under-correction with any object farther away than its center of
curvature). The same amount of over-correction error - thus resulting in
a negative ratio value - would imply the corresponding conics -1.296 and
-1.444, respectively (hyperboloids). The actual-to-ideal **
δ** value of 1.4, on
the other hand, implies K=0.324 and K=0.361, respectively (oblate
ellipsoids).
For a
**paraboloid**, which is axially perfect for object at infinity, there is a
price to pay with closer objects and the currency is over-correction.
Substituting K=-1 in **Eq. (a)** gives W**s**=(ψ-1)ψD/512F3,
and neglecting the **ψ**
term in the bracket (for any serious test, object needs to be
significantly farther away than 10 focal lengths, which makes **ψ**
significantly smaller than 0.1), it reduces to W**s**~ψD/512F3.
With ψ=ƒ/L=FD/L, this gives W**s**~D2/512LF2,
and the object (i.e. star) distance in terms of the wavefront
error as:
(all three
parameters, **D**, **W** and **L** are in the same units).
Since this value for the light source distance **L** is effectively based on the P-V wavefront error **Ws**
larger by a factor 1/(1-ψ) than the actual error, it is larger by a factor 1/(1-ψ),
or L/(L-ƒ), than a distance
corresponding to the actual **Ws**
value.
For the
wavelength in units of 0.00055mm (and **D** in mm), test distance is L~3.55D2/W**s**F2.
For **L** in feet and **D** in inches,
it is L~7.5D2/W**s**F2.
Substituting
optional maximum tolerable P-V error of spherical aberration for
**Ws**
gives the appropriate minimum distance for the artificial star. For
instance, by substituting W=λ/20=0.0000275mm for λ=550nm, gives the distance for this error level
for D=400mm,
ƒ/4, as L=710m.
As the original relation implies, the actual distance for that error
level is smaller by a factor of (1-ψ) than that given by the
approximation. That comes to 11% difference at a ten focal length
distance, and 5% at twenty focal lengths. Nearly exact expression
for the appropriate distance, for all practical purposes, is given by
L=(1-ψ')D2/512W**s**F2,
where **ψ'** is obtained from ψ'=ƒ/L,
using the distance **L** given by the approximation.
Solving **Eq. (a)** for **ψ** gives the exact value
as ψ=0.5-0.5(2048WF3/D)0.5,
which is used to plot required distance for 1/20 wave P-V of spherical
aberration as a function of mirror aperture diameter **D** and focal
ratio **F**, shown below.
Required distance ranges from 5.5m with a 100mm
ƒ/10 and ψ~0.18 (1/0.18=5.5m), to
2km (1.24 mile) with 500mm
ƒ/3 paraboloid and ψ=0.00075
(1.5/0.00075=2000m). The distance scales inversely to the wavefront
error, so half as large error (1/40 wave P-V) would require doubling the
distance.
Obviously,
required artificial star distance
becomes impractical for larger, fast paraboloids. Is 1/20 wave error
tolerance really necessary? It is; moreover, it may be still to much. For instance, if a
mirror is 1/6 wave P-V inherently over- or under-corrected, a 0.05 wave
of over-correction induced by the object distance would alter test
result into -1/4.6 wave or 1/8.6 wave P-V, respectively. For accurate
star testing, the error induced by object distance shouldn't
significantly exceed 1/50 wave P-V. This pretty much rules out an
artificial star test for larger Newtonians.
**
Catadioptric**
Newtonians with full-aperture Maksutov or Schmidt corrector are
significantly less sensitive to reduced object distance than
paraboloidal Newtonians, due to their spherical mirrors. Catadioptric
two-mirror systems are not uniform enough in their production types to
fall under some type of generalization.
For **
all-reflecting two-mirror
systems**, close object error of spherical aberration is given by
Eq. 92 (also plots for
the three most common systems), and for typical
commercial **SCT** by Eq. 120.3.
Ordinary
**doublet achromat** is very tolerant to the reduction in object distance in
focal ratios ~ƒ/10 and slower. It is in part due to
its relatively small
apertures, but even a 200mm achromat will likely generate less than 1/20
wave P-V of under-correction with the object (artificial star) as close
as 10 focal lengths away (given relative aperture, the error level is
nearly in proportion to the aperture size). On the fast end, however,
the sensitivity can be several times, or more, greater. A 4 inch
ƒ/6 achromat
can generate in excess of 1/5 P-V of under-correction with the object at
10 focal lengths away, due in part to the generated lower-order
aberration falling out of balance with the higher-order component (the
error is nearly inversely proportional to object distance). This
aberration duality makes the sensitivity of these instruments to object distance fairly
unpredictable, because the level of higher-order aberration and
proportion of
balanced lower- and higher order spherical aberration vary from one system to another. Similar applies to
apochromatic refractors.
Proficient star testing requires the ability to properly
interpret diffraction pattern seen in the eyepiece. Familiarity with the
characteristic forms associated with particular aberrations is easiest
to acquire by way of software-generated visual simulations. While the
actual testing commonly involves more complex pattern forms than those
caused by a single aberration, knowledge of the single-aberration
patterns is always the starting point. Following simulations are
generated by *Aberrator* (Cor Berrevoets), except the zonal error,
which is generated by *Aperture* (Harold Suiter). Shown are all
common conic surface aberrations, at the level of 0.80 and 0.95 Strehl,
for unobstructed and 0.3D (30% linear) obstructed aperture. Light
travels from left to right, thus defocus in the intrafocal direction is
numerically negative, and positive for extrafocal direction (patterns
are given for defocus values -4, -2, 0, 2 and for waves, since this
range of defocus is generally the most sensitive for detecting
aberrations). Axial cross section is not commonly presented, but it
helps better understand how particular patterns form.
The basic reference pattern is always
the one of aberration-free aperture, unobstructed or obstructed. The
presence of obstruction changes both, aberration-free and aberrated
pattern, but in the absence of aberrations patterns on the two sides
of defocus are identical for given defocus value.
The level of primary spherical
aberration at best focus resulting in 0.80 Strehl is 0.25λ
P-V WFE (0.0745λ
wave RMS). Shown is overcorrection, which is readily apparent from
the side at which the defocused pattern is larger and with less well
defined ring structure (the consequence of the outer portion of the
converging cone widening before the focus, since focusing behind it;
opposite for undercorrection). The tail tale of primary spherical is
that defocused patterns on one side are brighter in the central
portion, while brighter in the outer portion on the other.
Generally, it is easier to detect in the defocused patterns, with
the focused pattern differing from the aberration-free pattern
mainly for its brighter first ring. It is easily detectable at
λ/4.
and fairly easy even at λ/8
P-V wavefront error.
Similarly to the primary aberration,
shown is overcorrection, with the outer rays focusing behind
paraxial focus. For undercorrection, the patterns are reversed.
Mainly due to its somewhat different wavefront deviation form,
namely additional "wrinkle" at the edge, throwing light out at a
greater angle, the intra- and extrafocal patterns differ more than
with the primary spherical. In unobstructed aperture, the tail tale
sign can be a pair of equally bright rings in the focused pattern,
but the aberration is even easier to detect than the primary form,
either at the 0.95 Strehl level (0.20λ
P-V), or the "diffraction-limited" 0.80 Strehl (0.40λ
P-V, 0.075λ
RMS).
The 0.37λ
P-V wavefront error of primary astigmatism (0.0745λ
RMS), resulting in 0.80 Strehl, is easily detectable in both,
focused and defocused patterns. The former by the first bright ring
morphed into a cross-like pattern around the central maxima, and the
latter by elliptical elongation of defocused patterns, with the
orientation perpendicular one to another for the two opposite sides
of defocus. At twice lower nominal wavefront error it is, however,
harder to detect than spherical aberration; it could give only a
hint of the cross-like change in the ring form with bright stars.
Unlike the other three, primary coma
has identical patterns on either side of defocus. It is the
consequence of its wavefront deviation shape, which has one half of
it flatter, and the other more curved to identical degree. Hence the
former focuses as much longer, as the other focuses shorter, with
the identical angles of conversion, only of opposite sign (in this
particular pattern, the top half of the wavefront focuses farther,
and the bottom half closer). Luckily, it is the easiest to detect in
its focused pattern. Even at the 0.95 Strehl level (0.21λ
P-V), one side of the first bright ring is noticeably brighter.
These patterns illustrate the effect of
zonal error. The zone generates 0.075λ
RMS wavefront error (WFE), thus produces 0.80 Strehl. To some
extent, it resembles spherical aberration (it is a raised zone,
focusing behind paraxial focus, thus resembling overcorrection), but has a distinctive
double ring in addition to the first bright ring in the focused
pattern.
As mentioned, actual instrument will most often have a
mixture of different aberrations. Characteristic patterns for some other
aberration forms are given in
FIG. 96.
Harold
Suiter's "*Star Testing Astronomical Telescopes*" discusses in
details star testing, connecting it to the underlying optical theory. An
interesting new development (relatively speaking) is
Roddier's
test, (freeware) which uses CCD image of two defocused diffraction
patterns of a real star to determine wavefront quality (seeing effect is
averaged out through the exposure length, and also relatively
insignificant, due to the large size of defocused patterns).
Follows more detailed
description of the Strehl ratio and MTF.
◄
6.4. Diffraction
pattern and aberrations
▐
6.5. Strehl ratio
►
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