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▪ ** **CONTENTS
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8.2.3. Two-mirror telescopes
▐
8.3.1. Paul-Baker, flat-field
anastigmatic aplanats
►
#
**8.3. Three-mirror
telescopes**
While amateur
telescopes and cameras with more than
two power mirrors are relatively rare, there are three-mirror designs that deserve
mention, being relatively easy to make and offering exceptional
performance. Addition of the third mirror imposes additional demands in
regard to fabrication and makes alignment more difficult. On the other
hand, it allows for better correction of aberrations, sometimes with
easier to make surfaces.
The presence of a third mirror also makes the aberration
relations more complex. In effect,
aberrations of the third mirror are added to those of the primary and
secondary. This means that the starting point are the two-mirror system
aberration relations. For the first two mirrors in a three-mirror
system, they are identical to Eq. 78-78.1
given for two-mirror systems. Aberration relation for the tertiary is
identical to that for the secondary in its general form, but there are
differences in the parameter values, such as the actual angle of incidence
for off-axis points, effective stop displacement, sign, and others.
For the primary mirror, the P-V wavefront error
at the best focus for spherical aberration,
coma and astigmatism
is given by:
respectively, with **K1**
being the primary conic, **D1**
the aperture diameter, **F** the focal ratio, **α**
the field angle, and **h** the height in the focal plane.
For the secondary,
**
**
and for the tertiary
** **
in the same order, with **K**2/3
being the secondary/tertiary conic, **D2/3**
the secondary/tertiary minimum aperture (the cone width at the mirror),
**R2/3**
the secondary/tertiary radius of curvature, **Ω2/3**
the relative inverse of the object distance **ℓ2/3**
in units of the radius of curvature (**Ω2=R2**/**ℓ2**,
and **Ω3=R3**/**ℓ3**, with object, as before, being the image formed by preceding surface)
for the secondary/tertiary, and **
σ2/3**
the secondary/tertiary to the aperture stop separation in units of
secondary/tertiary radius of curvature. The angle
**α'**=[(R**1**/R**2**)+(1/m**2**)]α,
at the tertiary is the magnified field angle
**α** after reflection from the secondary.
For the tertiary, the
aperture stop location is the image of the primary mirror formed by the
secondary, which is at the distance I**2**=s**2**+s**2**R**2**/(2s**2**+R**2**)
from the primary, **s2**
being the primary-to-secondary separation. For the Cassegrain
arrangement (secondary inside the primary focus), both **s2**
and **R2**
are numerically negative, and so is **I2**,
which means that secondary's image of the primary forms to the left of
primary, farther away from it than the secondary; since the tertiary is
usually (in axial systems) between the primary and secondary,
tertiary-to-stop separation is given by S**3**=s**2**-s**3**,
with **s3**
being tertiary-to-primary separation (also numerically negative); from
this,
σ**3**=s**3**/R**3**.
The combined system P-V
wavefront error for each aberration is the sum of aberrations on each
mirror, thus W=W**1**+W**2**+W**3**. Corresponding RMS wavefront errors are given in sections
covering specific aberrations.
With the system astigmatism
corrected, best image surface coincides with the system's Petzval
surface, which is given by r**P**=(2/R**1**)-(2R**2**)+(2/R**3**),
with R**1/2/3**
being the radii of curvature for the primary, secondary and tertiary
mirror, respectively.
**THREE MIRROR SYSTEMS EXAMPLES**
It is well known that the Gregorian can be folded, in order to reduce its length,
by placing a flat at a location of the Cassegrain secondary, which then sends the
final beam through the Gregorian secondary located in front of the primary. Such system
can be made aplanatic by putting a mild curve on the front mirror, which then becomes the secondary, while
the Gregorian secondary becomes the tertiary (with the secondary being also the quarternary, being the place of last reflection).
Shown is a 300mm f/2/9.5 system, with parabolic primary, spherical concave secondary, and ellipsoidal (K=-0.465) tertiary. Due to the final beam passing through the tertiary, the field is limited
to about 0.5°. Diffraction images show a comparison with the standard Gregorian of the same primary focal ratio and secondary size
(right, a f/9.75 system with a similar back focus). In comparison with
the aplanatic Casseigrain (Ritchey-Chretien) with f/2 primary and identical secondary separation,
the Cassegrain-Gregorian is slower (f/9.5 vs. f/6), with slightly lower off axis error (0.135 vs. 0.16 wave RMS at 0.3° off axis)
and stronger field curvature (180 vs. 210mm).
Follow two examples of
three-mirror systems that I haven't seen elsewhere. One is an aplanat
with spherical primary and secondary, and the other is anastigmatic
aplanat with two moderately strong prolate ellipsoids (including primary) and a
sphere
(**FIG.128**).
The two are folded **three-mirror Cassegrain-Gregorian** systems,
3A and 3AA (for "aplanat" and "anastigmatic aplanat", respectively, with some unique qualities.
The former is aplanatic system consisting from three concave mirrors,
spherical primary and secondary, with the tertiary being a weak but
strongly curved ellipsoid (**FIG. 128 **A). Mirrors are easy to make,
and the only disadvantage is comparatively more difficult collimation.
The tertiary is significantly more sensitive to misalignment than
secondary; in the f/4.25 3A system
**FIGURE 128**: ** A **
- Aplanatic three-mirror telescope 3A with spherical
concave primary and secondary, and with
ellipsoidal (prolate ellipsoid) concave tertiary. Small diagonal flat (**DF**)
placed in the focal zone of the secondary mirror directs the light
out to the side, where is located tertiary mirror; it forms the
final focus out at the opposite side. The size of available field is
directly proportional to the size of the flat, which also determines
the effective central obstruction of the system. That gives an
opportunity to manipulate obstruction/field size via interchangeable
flat. The astigmatism is strong, but of the sign opposite to that in
conventional eyepieces. With the eyepiece astigmatism reduced
significantly by the opposite astigmatism of this mirror objective,
the actual visual field quality would be better than a field
produced by a conventional eyepiece and most other objective types.
The main drawback is collimation complexity, which would require sophisticated mechanism.
** B ** - Three-mirror anastigmatic aplanat 3AA with ellipsoidal
primary and tertiary, and spherical convex secondary. This system is
quite flexible with respect to the size and location of the mirrors,
with nearly ideal configuration for larger telescopes. Since the
primary is not spherical, it is more suitable for somewhat slower
systems ( approximately
f/8 to
f/12 with an
f/3 primary). Obviously, this doesn't apply to The European
Extremely Large Telescope (E-ELT, 42m diameter), which may be using this
type of mirror arrangement, but with f/1 primary. The flat is also
larger, with a center opening and the tertiary remaining on axis.
Small size of the flat vs. secondary mirror in both, 3A and 3AA
allows for nearly eliminating stray light from reaching the final
image (ray
spot plots for 3A and 3AA).
described below it is about 2/3 as sensitive to angular tilt as the
300mm primary. It is the level of tilt sensitivity of a 300mm f/4
paraboloid. Also, the tertiary is more than twice as sensitive to
decenter than secondary in an f/8 Ritchey-Chretien of equal
aperture (despace is comparatively forgiving). This level of
miscollimation sensitivity would demand very accurate and stable alignment
mechanism.
Various arrangements are possible, differing in mirror size, location,
conic (for instance, larger secondary can be aspherised to correct coma,
with less aspherised secondary correcting for spherical aberration,
resulting in significantly lower astigmatism), or relative aperture of
the system. Prescription for an actual f/4.25 system with an f/3 primary,
that can be scaled to both aperture and primary focal length is as follows:
**
σ1**
= 0.4583,** σ2**
= -0.1889,**
ρ2** =
-**ρ3**=-0.2111,
K**1,2** = 0,
K**3**=-0.396
where
**
σ1** and
**
σ2**
are, as before, primary-to-secondary and secondary-to-tertiary separation,
respectively, and **ρ2** and
**ρ3**
are secondary and tertiary radius of curvature, respectively, all
expressed in units of the primary radius of curvature. This determines
relative marginal ray height on the secondary k2=0.083,
in units of the aperture radius, and secondary magnification m2=ρ2/(ρ2-k2)=0.72.
With the actual cone angle being expanded by a factor of 1/m2
toward tertiary, marginal ray height at the tertiary k3=(2σ2/m2)-k2=-0.44.
With **ρ**3
also effectively greater by a factor of 1/m2,
ρ3e=ρ3/m2=-0.293,
and the tertiary magnification m3=ρ3e/(ρ3e-k3)=-2.
Best (median) field curvature is ~0.2f,
**f** being the system focal length. Effective central obstruction
**
υ** in units of the
aperture (linearly), is given by:** **
**
υ** = 2F**1**m**2**m**3**α/(m**3**-1)k**3**,
with **F1**
being
the primary focal ratio, **α**
the field angle in radians (**m****3**
and **k****3**
as
absolute values). Alternatively, the angular field radius in terms of
the relative obstruction radius **o** in units of the aperture radius is given by
α**=o(****m3-****1)k****3****/2F****1****m****2m3**.
For F**1**~3,
m2~0.72,
|m3|~2,
and |k3|~0.44,
an effective 0.2D obstruction would produce ~° field radius.
Diffraction limited field is set by astigmatism. For the above f/4.25
system, it is at the level of a f/5.6 paraboloid. Despite its magnitude in the
outer field of the
objective's image, astigmatism is not likely to be objectionable in visual use, due to
the offset with even stronger astigmatism of conventional eyepieces.
However, due to large relative apertures, good field definition here
would require eyepieces well corrected for higher-order astigmatism.
The other folded Cassegrain-Gregorian system, anastigmatic aplanat 3AA (**FIG.
81B**), can be
made in various arrangements. One example with the minimum secondary
size of 0.25D and that of the tertiary of 0.2D has the following
parameters:
**
σ1**
= 0.375,** σ2**
= -0.3,**
ρ2 **
= 1,**
ρ3** = 0.19375,
K**1 **
= -0.727,
K**2 **
= 0**,
**and** **
K**3** = -0.576
with best field curvature r**c**~0.1R**1**.
An f/3/10.6 D=400mm 3AA system still has as high as 0.998 Strehl near
the limits of the standard visual field, at 0.3°
(22mm) off-axis. Since the field curvatures of the primary and secondary
exactly offset each other, the system curvature is given by rC=R3/2.
For the lower system curvature with given primary, the secondary needs
to be more curved, and the tertiary less. Sensitivity to decenter and
tilt of the tertiary in the 3AA is several times lower than in the 3A
system.
Forfeiting visually
accessible final focus allows to design three-mirror systems that in
addition to being anastigmatic aplanats also have - or can have - flat
field. Those include Paul-Baker curved and flat-field variants, as well
as two other less known three-mirror flat-field anastigmatic aplanats that I came up
with not long ago. I haven't searched them for originality, so let's
term them as FAA1 and FAA2.
◄
8.2.3. Two-mirror telescopes
▐
8.3.1. Paul-Baker, flat-field
anastigmatic aplanats
►
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