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▪ ** **CONTENTS
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8.2.1. General aberrations
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8.2.4. Dall-Kirkham telescope
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#
**
8.2.2. CLASSICAL
TWO-MIRROR TELESCOPES**
While spherical aberration,
according to
Eq. 81, can be
corrected for any appropriate combination of the primary and secondary
mirror conic, the aberration coefficients for coma and astigmatism for
two-mirror system show that coma and astigmatism do vary, potentially
significantly, with the choice of conics. The coma aberration
coefficient (Eq. 82)
indicates that a system with spherical primary will have stronger coma
than one with paraboloidal primary. A particular value of **K****1**
resulting in the zero sum in the brackets, would result in corrected
Seidel coma, with needed value for the secondary conic for
corrected spherical aberration obtained from **Eq. 80**.
In the early days of telescopes,
limitations in both, calculation and mirror-making and testing skills
would not allow to determine precisely the coma-free conic combination.
However, it was determined that paraboloidal primary significantly
reduces coma, compared to spherical (Eq. 82.1
shows that astigmatism with paraboloidal primary actually increases, but
insignificantly in comparison with the reduction in coma). Thus, the
choice for primary conic was, for quite some time, paraboloid and, for
that reason, this arrangement is known as the *classical *two-mirror
telescope.
With paraboloidal primary (K1=-1),
which is corrected for spherical aberration, zero system spherical aberration
requires zero secondary
spherical aberration, for which the needed secondary mirror conic is,
from **Eq. 80**:
which makes it a hyperboloid. Since the requirement in the classical
two-mirror system is that either mirror has zero spherical aberration,
mirror conic can be alternatively determined from the fact that the
object for each has to be located at one of its
geometric foci. For
the paraboloidal primary, the object is at its infinity focus. For the
secondary, the object is the image formed by the primary, thus it has to
coincide with secondary mirror's near geometric focus (at the far focus
location, the final image wouldn't be real, that is, the rays
would be diverging from the secondary). Equaling the near focus separation
for the secondary given by R2/(1+ε2),
**ε2**
being the surface eccentricity, with the secondary-to-primary's-image
separation **i**
(FIG. 121), requires secondary
mirror eccentricity ε2=(R2/i)-1,
or the secondary conic constant K2=-[(R2/i)-1]2
where, according to the
sign convention,
for the primary oriented to the left, both secondary mirror radius of
curvature **R****2**
and image separation **i**
are negative in the Cassegrain and positive in the Gregorian.
With the object at its near focus,
secondary mirror will form the final image at its far focus, at a
distance equal to R2/(1-ε2).
After spherical aberration is corrected,
remaining Seidel
**coma** and
**astigmatism**, as the P-V
wavefront error at diffraction focus for object at infinity are given by:
respectively, with
**α**
being the field angle in radians, **D** the aperture diameter, **F** the system F-number
and **
f**
the system focal length.
The coma RMS wavefront error is related to the P-V error as:
**
ω** = W**c**/32 =
αD/272F**2** =
h/272F**3**
(84.3)
and for astigmatism:
**ωa**=W**a**/24**1/2 **
(84.4)
Plots at left show the P-V wavefront error (WFE, in
units of 550nm wavelength) for
the two aberrations, based on **Eq. 84.1** and **84.2**. Coma only
depends on the system F-ratio and aperture diameter, while the
astigmatism also depends on secondary magnification and weakly on the back focal length (BFL, assumed to be 0.25, or one
quarter of the primary's focal length). Astigmatism is nearly
proportional to secondary magnification, while inversely proportional to
the system F-ratio. In other words, for given focal ratio, it increases
with faster primary. The combined RMS wavefront error is a square root
of their respective squared RMS errors (the P-V/RMS ratio for coma is **
√**32, and for
astigmatism **√**24).
Evidently, coma is identical to that of a
paraboloid of the same
F-number, while the astigmatism exceeds that of a
paraboloid by a factor (m2+η)/(1+η)m.
Sign of astigmatism in the Gregorian and
Cassegrain is identical (both. **m** and **
f**
are numerically negative in the Gregorian, while positive in the
Cassegrain). Therefore, related geometric (ray) aberrations
can be determined from those given for a paraboloid in
2.2. Coma and
2.3 Astigmatism.
From **Eq. 84.1-84.4**, the
coma-to-astigmatism RMS wavefront error ratio in a classical Cassegrain
is closely approximated by D/5.5mh. This implies that, for given linear
height in image space, the relative astigmatism increases with secondary
magnification. For given angular height, it increases with the square of
secondary magnification.
Petzval field
curvature of any two-mirror system is given by **1/R****p=2[(1/R2)-(1/R1)]**,
R1
and R2
being the radius of curvature of primary and secondary mirror,
respectively. However, due to the presence of astigmatism,
best image
surface curvature varies. With the primary mirror astigmatism
independent of the conic (for the stop at the surface), it is the
secondary mirror conic and shape (convex/concave) that induces
variations in the system astigmatism. For classical two-mirror systems,
median field curvature is given by:
The relation shows that, for given primary f-ratio and
secondary magnification, best (median) image surface is somewhat less
curved in the Gregorian (see
graph). This is despite its considerably stronger Petzval surface curvature, due to its astigmatism, opposite in sign to
its Petzval, resulting in the astigmatic surfaces forming on the convex
side of the Petzval, thus less curved relative to it. In the Cassegrain,
astigmatic surfaces form on the concave side of its Petzval surface,
with their curvatures consequently stronger (note that Petzval surface
is concave toward secondary in the Cassegrain, and convex in the
Gregorian).
**
EXAMPLE**: 300mm
f/4/16 Classical Cassegrain,
thus with numerically positive secondary magnification **m**=4; opting for the back focal
distance in units of the primary's focal length **η**=0.2 (240mm), the height of marginal ray at the secondary, in units of the aperture
radius, **k**=(1+η)/(m+1)=0.24. With the paraboloidal primary (**K**1=-1),
the secondary conic for corrected spherical aberration is
K2=-[(m+1)/(m-1)]2=-2.777.
Alternatively, secondary conic can be found from **
K**2=-[(R2/i2)-1]2
which, for the secondary mirror radius
R2=-mkR1/(m-1)=-768mm,
and the secondary-mirror-to-primary's-image separation i2=kf1=-288,
also gives K2=-2.777.
The wavefront error of coma is identical to
that of a 300mm f/16 paraboloid;
at 0.25° (21mm, 0.00436 radians)
off-axis, it is **
W**c**=**αD/48F2=0.0001065, or 0.194 wave P-V in units of 550nm
wavelength. The RMS wavefront error of coma is smaller by a factor of
1/**√**32, or 0.034 wave. At
the same distance off-axis, astigmatism is **
W**a=-(m2+h)Dα2/8(1+h)mF=0.00015mm,
or 0.27 wave P-V in units of 550nm wavelength; the corresponding RMS
wavefront error is 0.056.
With R1=-2400mm
and R2=ρR1=mkR1/(m-1)=0.32R1=-768mm,
the Petzval curvature is **R**p=-565mm,
and the best (median) field curvature is **R**m=-315mm.
Above applies to a perfectly collimated system. Less than
perfectly aligned mirrors will induce additional aberrations,
axial and abaxial. Taking the above system, effect of secondary
tilt and/or decenter is illustrated below.
The effect of 1° secondary tilt (about 0.5mm tilt at the edge,
counterclockwise, top and right)
is very obvious. Part of it is the consequence of the axial ray bouncing
far off the physical field center - as the raytracing scheme shows -
bringing into the physical center uncorrected portions of field. Also,
tilt of the secondary vs. converging beam creates at its surface
all-field coma, i.e. coma of even magnitude imposed over entire field.
Here, tilting the secondary 1° counterclockwise throws the
optical axis - and optical field center - nearly 0.5° off the
physical system ais, i.e. physical field center. Near optical field
center the aberration is nearly eclusively the all-field coma, while
astigmatism creeps in going farther off. As the astigmatism plots
shows, image field is also tilted, which is of little significance
considering the magnitude of field aberrations.
Even this much of misalignment can be nearly cancelled out by
inducing offsetting misalignment - in this case 3.5mm decenter
(left, above). However, the cancelling takes place at the optical
axis, which is still 0.3° off the physical axis, i.e. physical
field center. Decenter of 1mm alone induces all field coma, with the
physical field center located about 0.06° above the optical axis.
In general, magnitude of aberration induced by element tilt/decenter
is in proportion to the numerical deviation from perfect.
Change in the mirror spacing induces spherical aberration. Every mm of
spacing reduction extends back focus by little over 15mm, inducing
about 0.014 wave P-V of undercorrection.
#
**
**
8.2.3. Aplanatic
two-mirror telescopes
It wasn't until 1910, when Ritchey in the U.S. and Chretien in France
arrived at needed conics for a coma-free two-mirror
(Cassegrain) system, that the "classical" two-mirror telescope finally was
upgraded to its optimal version. It took another 17 years before the
very first such telescope was successfully made (Ritchey, 1927).
Correction of coma in a two-mirror system requires an additional, relatively small
modification of both optical surfaces. In the Cassegrain configuration, secondary
needs to be more strongly aspherised in order to correct for the coma.
So does the primary, in order to compensate for additional spherical
under-correction induced by the more strongly aspherised, aplanatic secondary. In the Gregorian,
however, both mirrors need to be less strongly aspherised than in the
classical arrangement. Needed conics for an aplanatic two-mirror
telescope are:
Alternatively, the conics can be expressed
in terms of the primary focal length **
f1**,
system focal length **f**,
and mirror separation **s** as K1=-1-2f12(f1-s)/sf2
and K2=-[(f-f1)/(f+f1)]2-2f[
f1/(f+f1)]3/s.
Note that
**f1**
is always numerically negative, and
**f**
is numerically negative in the Gregorian, a result of
Eq. 76.
With Seidel spherical aberration and
coma corrected, the remaining aberrations are astigmatism, field
curvature and distortion. The P-V wavefront error of Seidel astigmatism is
given by:
This gives astigmatism in the
aplanatic Cassegrain - also known as Ritchey-Chretien - greater, and in
the aplanatic Gregorian smaller by a factor of (2m2+m+η)/(2m2+η)
compared to the classical types. Neglecting distortion, the remaining
aberration is best, or median field curvature, given as:
Again, for an equal set of parameters,
aplanatic Cassegrain has somewhat stronger best field curvature than aplanatic
Gregorian. Graphs below illustrate the P-V error of astigmatism (left)
and best image curvature (right) in the aplanatic two-mirror system. Note that the system focal length for the Gregorian is numerically
negative, thus its median surface has positive sign, i.e. it is convex
toward secondary; median surface in the Cassegrain has negative
curvature, i.e. it is concave toward secondary. When expressed in units
of primary's radius of curvature (R1,
dashed), which has a constant negative sign, plots for the two
configurations are of opposite sign (since the system focal length f=mR1/2,
the ratio of image curvature in units of **R**1
vs. that in units of the system f.l. is m/2).
Aplanatic Cassegrain has somewhat stronger, and aplanatic Gregorian
somewhat weaker than classical arrangements, but the difference is
small. Median field curvature of the aplanatic systems are nearly
identical to those of the classical Cassegrain and Gregorian for the
practical range of magnifications.
Till recent, most of large professional
telescopes were of Ritchey-Chretien type. Probably the most famous, *
Hubble Space Telescope*, has 2.4m f/2.3
primary mirror, at 4.9m from the convex secondary, and f/24
effective system focal ratio. This gives all the information needed to
specify its inherent aberrations: secondary magnification
m=24/2.3=10.435, relative minimum secondary size
k=(5.52-4.9)/5.52=0.112, and back focal distance in units of the primary
focal length η=(m+1)k-1=0.28. This gives the primary mirror conic K1=-1.0023,
secondary conic K2=-1.497,
P-V wavefront error of astigmatism at the edge of its 9 arc minutes
(0.15°) "data field" radius of 1.3 waves (λ=550nm), and the best image
surface curvature radius Rm=-633mm.
**
EXAMPLE**: 300mm
f/3/12 aplanatic
Gregorian, thus with numerically negative secondary magnification **m**=-4; opting for
the back focal distance in units of the primary's focal length
**η**=0.25 (225mm), the height of marginal ray at the secondary, in
units of the aperture radius, **k**=(1+η)/(m+1)=-0.417,
and the ratio of curvature radii for the two mirrors R2/R1=ρ=-0.3336. The aplanatic
conic for the primary, compensating for spherical aberration induced
by the aplanatic secondary **K**1=-0.963,
and the secondary conic for corrected coma is **K**2=-0.405.
The only remaining point-image aberration is
astigmatism, with the P-V wavefront error at 0.25° off-axis of
-0.000165mm,
giving
-0.3
wave in units of 550nm wavelength, with the corresponding RMS wavefront
error of 0.061 wave (the P-V error is of opposite sign to that in a
Cassegrain).
With R1=-1800mm
and R2=ρR1=mkR1/(m-1)=-0.334R1=600mm,
the Petzval curvature is **R**p=225mm,
and the best (median) field curvature is **R**m=349mm.
As with classical Cassegrain, any misalignment of the mirrors will
induce additional aberrations. The effect of the same 1° tilt and
1mm secondary decenter is shown below.
Here, 1° tilt is throwing optical axis as much as 0.75° off
the physical axis of the system. Ray spot plots (below right) show
0.5° field around physical axis. Toward optical axis, astigmatism
diminishes to zero in its proximity, but the magnitude of all-field
coma remains only moderately lower than the aberration at the physical
center. Similarly to misalignment in the classical Cassegrain,
the tilt error can be nearly cancelled out by decentering secondary.
However, best correctin is around optical axis, about 0.1 degree
above the physical field center.
Secondary decenter of 1mm alone generates all-field coma as dominant
aberration across 0.5° field, with the optical field center
about 0.1° above the physical field center. Since the aberration
induced is in proportion to the nominal misalignment, the tolerance
for decenter alone - taking 0.80 Strehl or better, i.e. coma blur
of 1.5 times the Airy disc or less, as the minimum - is 0.3mm. For
the tilt, it is 0.1°. Those two combined would effectivelly restore
correction level of the perfect alignment, except for field
asymmetry where the lower half has over 60% larger astigmatism.
Deviation from the optimal mirror separation induces spherical aberration,
about 0.06 wave P-V per mm. Since the back focal length extends a
little over 7mm for every mm of the spacing reduction, one inch of
back focus extention would require about 3.5mm spacing reduction,
inducing about 1/5 wave of undercorrection. The error is independent
of the aperture, i.e. twice smaller instrument will need as much of
spacing reduction for the same etention, which is twice as much relative
to the focal length.
◄
8.2.1. General aberrations
▐
8.2.4. Dall-Kirkham telescope
►
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