JCMT Optical Constants
JCMT Optical Constants
The general optical layout of a cassegrain telescope
is illustrated in the figure below.
Notation
Dp - diameter of primary mirror
Ds - diameter of secondary mirror
Fp - distance between the vertex (pole) of the primary mirror
and its focus
Fc - distance between real foci of primary and secondary
mirrors
Fv - distance between pole of secondary mirror and its virtual
focus (which coincides with the prime focus)
phir - angle between optical axis and the line joining the edge
of the secondary mirror and the real focus
phiv - angle between optical axis and the line joining the edge
of the secondary mirror and the prime focus
In a rectilinear x-y coordinate system having x parallel to the incoming
radiation, the form of the paraboloidal primary is
xp = yp2 / 4 Fp
Additional or alternative notation is given in the following diagram :
f1 - focal length of primary = FP above.
f2 - focal length of secondary
The form of the hyperboloidal secondary in the same coordinate system is
xs =
a * ( ( 1 + ys2 / b2 )0.5 - 1 )
The dimensions of the telescope were fixed (as described in
ASR/MT/T/634/REH(84) ) by choosing values for
Dp, Ds, f1=FP and
2c=Fc.
Other parameter values follow from these and from standard equations for
paraboloids, hyperboloids and lenses; they and other dimensions
of the JCMT are given in the tables below.
Dimensions from the primary mirror to the focus 'in' the receiver cabin
| Dimension |
Symbol |
Derivation |
Value |
Diameter of
primary mirror |
Dp |
chosen |
15 m |
Diameter of
secondary mirror |
Ds |
chosen |
0.75 m |
Focal length
of primary |
Fp = f1 |
chosen |
5.4 m |
distance between
primary focus and
real
focus |
c = Fc / 2 |
chosen |
4568 mm |
|
p1 |
p1 = (Dp/2)2/4f1
|
2604.167 mm |
|
p2 |
p2 =
(f1-p1) * (Ds/Dp)
|
139.792 mm |
| Secondary mirror mass |
|
|
7.5 kg |
| Secondary mirror thickness |
|
|
3mm |
Secondary mirror
surface roughness |
|
|
9 microns
rms |
| hyperboloid constant |
a |
2a = sqrt[(2c-p2)2 + Ds2/4] -
sqrt[(p2)2 + Ds2/4]
|
4301.906 mm |
| hyperboloid constant |
b |
b2 = c2 - a2
|
1536.304 mm |
| hyperboloid eccentricity |
e |
e = c/a = sqrt(1 + b2/a2)
|
1.061855 |
| hyperboloid conic constant |
K |
K = - e2 |
-1.127536 |
hyperboloid virtual
object distance |
q |
q = c - a |
266.094 mm |
hyperboloid real
image distance |
s + g |
s + g = c + a |
8869.906 mm |
distance of real focus
behind primary |
g |
g = 2c - f1 |
3736 mm |
distance between vertices
of primary & secondary
mirrors |
s |
s = c + a - g |
5133.906 mm |
| hyperboloid magnification |
M |
M = (s + g) / q
M = (c + a) / (c - a)
M = (e + 1) / (e - 1)
|
33.3337 |
Focal length of
whole telescope |
F |
F = M * f1 |
180.00 m |
| f-ratio of whole telescope |
f# |
F / Dp |
12
16 for SCUBA
see below * |
focal length of
hyperbolic secondary |
f2 = fa |
f2 = (a + c) / (M - 1)
fa-1 = (c - a)-1 - (c + a)-1
|
274.324 mm |
paraxial radius of curvature
of hyperbolic secondary
|
r2 |
r2 = 2 * f2 |
548.648 mm |
|
phiv |
phiv = 2 * tan-1(Dp / 4 * Fp )
|
69.5557o |
|
phir |
(tan phir)-1 =
2 Fc / Ds
- 0.5 * (4 Fp / Dp - Dp / 4 fp)
|
2.387o |
| Plate scale at first real focus |
|
= 2*phir / Dp = 206265 / ( Dp * f# )
|
1.146 "/mm |
width of beam as
it passes through
hole in primary
|
|
2 * g * tan phir |
311 mm
|
| hole in primary |
|
|
~ 1 m |
beamsize
at 850microns |
|
~1.2 * lambda / Dp |
~ 14 arcsec |
beamsize
at 450microns |
|
~1.2 * lambda / Dp |
~ 7.5 arcsec |
The hole in the primary is designed to allow access to the surface,
not to just contain the f/12 beam.
The Tertiary Mirror Unit
This f/12 beam is intercepted by the tertiary mirror unit (TMU) which
can send it to any of the receivers inside the cassegrain cabin
(RxA3i, RxB, RxW), or to the instruments on the Nasmyth platforms
(SCUBA on the left; FTS, SPIFI, HARP on the right).
* - Note re f-ratio in table above
In order to ensure that the beam headed towards SCUBA is narrow enough to
pass through the elevation bearing unvignetted
a "small shift" (Holland et al 1999 MNRAS 303, 659)
is applied to the secondary mirror position for SCUBA.
This shift is about -4mm (i.e. towards the primary) as
is seen by comparing the SMU parameter SCUBA_Z_OFFSET with Rx_Z_OFFSET for
the other receivers in the
cabin. The beam is also stretched to f/16 by this same action, although
once beyond the elevation bearing the SCUBA beam is folded and refocussed
and enters the SCUBA cryostat at f/4.
A polyethylene lens relay deployed in front of SPIFI on the
right Nasmyth provides an input beam of f/15.
Tertiary mirror to receivers
| Dimension |
Symbol |
Derivation |
Value |
distance of real focus
behind primary |
g |
from table above |
3736 mm |
Distance from centre of tertiary
to feedhorn of receivers |
R |
ASR/MT/T/1061/RJSG(87) |
1500 mm
nominal, but
confirmed by direct measure
|
Distance from primary
to center of tertiary |
h |
subtraction of
above two terms |
~2236 mm
~2250+10 by direct measure |
| Tertiary dimensions |
|
|
600 mm x 425mm |
| minor axis of f/12 beam on TMU |
|
2 * R * tan phir |
125 mm
|
major axis of f/12 beam on TMU
assuming TMU at 45o to beam
|
|
R * sin phir *
((sin(45 - phir)-1 -
(sin(45 + phir)-1)
= ~ minor axis * sqrt(2)
|
177 mm
|
Distance from centre of receiver cabin
(and tertiary mirror)
to ends of elevation bearing
|
|
|
3.0m and 4.5m
(guesses as yet !) |
max diameter of f/12 beam within
elevation bearing
|
|
2 * (4.5m - (g-R)) * tan phir |
189 mm
|
diameter of elevation bearing
|
|
|
350 mm |
diameter of hole
through elevation encoder |
|
|
200 mm |
Model
These dimensions were incorporated into a software model of the
JCMT optics. Some output from this model is shown below :
The model confirms, for instance, that the shapes of the primary and
secondary, as specified, generate a tight focus at R=1499mm
(see the 'Tertiary' table - and the detail below), and that the f/12 beam
is a tight fit through the (left) elevation bearing.
The -4mm adjustment to the
position of the SMU certainly narrows the beam as it passes through
the bearing and does generate a f/16 beam (graphics not shown).
The focus of this beam is somewhere near the end of the elevation bearing
tunnel, but it is not exactly 'focussed' in the way the f/12 beam is.
I haven't included SCUBA fore-optics yet, nor X & Y focus adjustments
to the SMU, nor chopping . . . .
The program that generated this plot is /home/imc/optics/optics.exe
with an input file of optics.in.
Acknowledgements : Much of the first table combines the formalism of
Herman van de Stadt's JCMT document ASR/MT/T/342/HvdS(82) with
the amended dimensions described in Richard Hills' ASR/MT/T/634/REH(84).
Private communications with Richard Hills in September 2002 helped
clarify some issues.
Iain Coulson
Last Updated: 07 Oct 2002
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