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▪ ** **CONTENTS
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8.2.5. Loveday telescope
▐
8.3. Three-mirror telescopes
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**
8.2.6. Two-mirror
telescopes: miscollimation, close object errors**
Miscollimation
sensitivity in two-mirror
telescopes
Whenever the primary and secondary mirror
in a two-mirror system are not optimally positioned, it induces certain
amount of axial and off-axis aberrations. In general, induced aberration
is proportional to the linear misalignment. While either of the two, or
both mirrors can be misaligned, the system sensitivity can be simply
shown as misalignment of the secondary relative to the primary. It can be expressed separately for *tilt*, *decenter* and
*despace*. By far the dominant aberration resulting from the first two forms
of misalignment is axial (independent of the field height) coma. Depending on the sign,
it can add up to the "regular" off-axis system coma, or subtract from
it, but the most troublesome part is its presence in the field center.
As the P-V wavefront error, it can be expressed separately for the *
tilt
as*:
and *
decenter:*
**
τ**
being the tilt angle in radians, **∆**
the linear decenter and **F** the system focal ratio (as before, **η**
is the back focal length in units of primary's focal length and **D**
the aperture diameter). The
RMS wavefront error is smaller by a factor 1/321/2.
Tilt and decenter are usually both present, so the two errors combine, with the final error given by their sum.
Whether they will add or subtract depends on the sign of **
τ**
and **∆**. Misaligned secondary doesn't induce coma when
τ=-∆[1-(m-1)K2/(m+1)]/R2,
**R****2**
being the secondary radius of curvature.
Since the secondary magnification m=ρ/(ρ-k),
relations expressed in terms of secondary magnification **m** reflect
the deviations in either secondary radius of curvature (or, for that
matter, primary's radius of curvature, since
ρ=R**2**/R**1**)
or secondary-to-primary separation, defining the minimum relative
secondary size **k**. Thus these relations can also be expressed
directly in terms of **ρ** and
**k**, which can be more convenient for determining the tolerances
for a specific system.
*Despace* (separation) error **s**
in two-mirror systems - positive for larger separation, and vice versa - results in change in the relative height of the
marginal ray at the secondary **k** into k'=k+(s/f**1**),
and secondary magnification **m** into m'=ρ/(ρ-k'). The
most significant aberration it induces is lower-order spherical. Resulting
wavefront error is obtained by substituting **k'** and **m'** for
**k** and **m** in
Eq. 81** **(top), or simply **k'** for **
k** (bottom). The P-V wavefront error at the best focus can be also
expressed in terms of secondary magnification alone as:
with **
F1**
being the primary's focal ratio. For given nominal value of **s** the
wavefront error is independent of the aperture; for relative **s**
value (in units of the primary's focal length), it changes in proportion
to it. Despace always causes change in the back focal length; exact
figure requires specific change in **k** to be known, but as a
practical approximation for **s** of up to a few milimeters a simple relation δBFL~-10ms/F_{1}
should suffice (the primary shift **s** is negative for the reduction
in secondary to primary separation, giving a positive change - i.e.
extension - in the back focal
length δBFL; **m** is the original secondary magnification, and F_{1}
the primary's focal ratio number f_{1}/D). New secondary magnification
is in close proportion to the length of converging beam from the secondary
to its new focus vs. original length.
Despace also induces coma, insignificant
in comparison. According to Schroeder's relation for angular transverse
coma created by despace (p114), its ratio to the coma of a classical
Cassegrain (or a paraboloid of identical F-number) is given by -[(2m2-1)(m-β)+(2m(m+1)]ms/(1+β).
The minus sign indicates that this coma adds to the system coma for
negative, and subtracts for positive values of **s**; in aplanatic
two-mirror systems it represents the amount of system coma.
**Eq. 91.3** shows that Dall-Kirkham with K2=0
has significantly lower sensitivity to despace compared to both,
classical and aplanatic Cassegrain (Ritchey-Chretien), with the latter
being the most sensitive. However, despace sensitivity is lowest in the
Gregorian arrangement, for which the sum in the brackets is smaller than
in a comparable Dall-Kirkham, due to numerically low negative secondary
conic of the former.
Note that the separation change
**s** is
negative when the mirror separation decreases, resulting in larger
minimum relative secondary size **k'** (primary's
f.l. is numerically negative, according to the
sign convention). From **Eq. 91.4** it is easy to see that larger than optimum
mirror separation (i.e. positive **s**) induces under-correction
(positive in sign wavefront error at the best focus), and that smaller
mirror separation (negative **s**) induces over-correction.
Plots below show the P-V wavefront error (λ=550nm)
due to miscollimation for selected two mirror
systems: 3/3.33/10 and
f/4/16 each, classical Cassegrain (Cass) and Dall-Kirkham (DK),
as well as for f/2.5/7.5 and
f/3.33/10 Ritchey-Chretien (RC). The
f/10 systems are for direct comparison. Tilt-induced coma (left)
is independent of the secondary conic, hence it is identical for any
two-mirror arrangement with given secondary magnification (**m**),
aperure (**D**, 200mm here), and back focal length (**η**,
0.2 for all systems).
At f/10, both decenter and despace
sensitivity are significantly lower in the DK than Cassegrain and RC,
while RC is moderately more sensitive than Cassegrain. At the typical focal
ratio, however, (~f/8 RC and ~f/15
Cassegrain) Cassegrain is significantly less sensitive. Note that,
as mentioned before, decenter and despace error are independent of
aperture for the deviation expressed in measuring units (if expressed
in mm, the resulting wavefront error is also in mm).
Close objects error
For relatively
close objects, when magnification of the primary - given by m
**1**
=-f**1**
/(o-f**1**
),
**f****1**
being the primary focal length and **o** the object distance - is
appreciably greater than zero,
Eq. 9 applies
to both mirrors. The aberration contribution of the primary changes from
(K**1**+1) into
[K**1**+(1-2ψ)
2],
with ψ=f**1**/o
being the primary focal length in units of the object distance. Also, due to the extended converging cone of the
primary, both, relative height of the marginal ray at the secondary **k** and
secondary magnification **m** increase. The height **k** becomes k'=(1-ψ)k+ψ,
and secondary magnification **m** becomes m'=ρ'/(ρ'-k'),
with ρ'=(1-ψ)ρ.
For the secondary, the effective primary focal ratio is now F**1**
/(1-ψ). With these changes,
after substituting **k'**, **ρ'**
and **m'** for **k**, **ρ**
and **m** in
Eq. 81 (bottom) the system P-V wavefront error of spherical aberration at
the best focus
becomes:
Since the secondary conic
**K****2**
is a factor for the aberration contribution of the secondary,
sensitivity of two-mirror telescopes to reduction in object distance
varies with the system type. Plots below, based on **Eq. 92**, show
spherical aberration induced by close object observing.
With classical Cassegrain (left), both primary and secondary induce zero
spherical aberration with object at infinity. As object distance
diminishes, paraboloidal primary induced overcorrection, but the
hyperboloidal secondary induces even more undercorrection, at a higher
rate, resulting in the increase of system undercorrection. The two
hyperboloidal mirrors of the Ritchey-Chretien (RC), figured to correct
coma, each induces spherical aberration that exactly offsets for object
at infinity. For closer objects, primary's overcorrection increases more
slowly than secondary's undercorrection, resulting in system
undercorrection. RC error is significantly larger than for the
Cassegrain, mainly due to the faster primary.
Similarly, undercorrection of Dall-Kirkham (DK) ellipsoidal primary is
offset by overcorrection of the secondary for object at infinity. For
closer objects, aberration contribution by both mirrors diminishes, but
more slowly for the secondary, with the system becoming increasingly
overcorrected. Error induced to the DK is significantly greater than
that in the Cassegrain, approaching 1/4 wave at the limit of useable
focus range.
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8.2.5. Loveday telescope
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8.3. Three-mirror telescopes
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