Abstract

Abstract

1. Introduction

As of April 1993, I think everyone agrees (or has been beaten into submission) that the F/15 secondaries should be undersized with respect to a secondary which captures light from the entire primary. This makes the aperture stop at the secondary with a cold sky background rather than at the primary with a warm background. The exit pupil at the secondary can be seen in the figure on page 1.

So, now there are two questions:

1. How much undersized should the F/15 secondary be?

2. Is the secondary F/15 before or after undersizing it?

The remainder of this memo is my attempt to answer these questions quantitatively. To avoid any suspense, the proposed answers are:

1. The secondary should be undersized so it forms the aperture stop for a field several arcminutes in diameter.

2. We have arbitrarily decided that the secondary should produce an F/15 beam after undersizing from a faster parent.

The size of the secondary is one of many items to be considered in the optimization of the infrared performance of a large telescope. These items include coating emissivity, optical quality, chopping frequency and throw to name a few. See the various memos listed in the Bibliography for additional discussion.

2. Undersizing Factors

1. The diffraction field angle for the longest infrared wavelength;
   The secondary size is chosen to reject diffracted background light from outside the edge of the primary (background diffracted at the edge of the secondary into the telescope beam). Blanco gives the diffraction field angle, , which describes how much a ray is bent passing a sharp-edged aperture:
   sin = ( X ) / ( D )

   where is the wavelength, D is the secondary diameter and X is a spatial variable which determines which ring of the Airy disk receives the diffracted energy. The numbers below represent a diffraction angle of ~0.01 degrees which cuts off the diffraction pattern at the third Airy ring (X = 3.7 ) at a wavelength of 40 microns. This rejects 98.5% of the 40 micron energy diffracted from the hot edge of the primary and passes the 1.5% which is beyond the third ring. Rejection is correspondingly stronger at shorter wavelengths.

2. The detector field size and chopper throw;
   The secondary size is chosen so the entrance pupil (stop) is always defined by the secondary. The secondary diameter is reduced as the field angle or chop angle increases (This is opposite from the optical secondary which uses the primary as the aperture stop and increases in diameter with field angle). The field angles quoted below should represent the sum of the field and chop angles. We are considering fields of 4 to 6 arcminutes in diameter based on considerations of detector size, chopper throw and isoplanatic patch size (Each of these factors varies with wavelength.).

3. Uncertainty in collimation and centration;
   There will always be uncertainty in collimation and centration of the secondary relative to the primary. This is simply an alignment tolerance. ± 0.7 mm has been chosen as a number representative of how well the optical surfaces are centered on the physical surfaces. The optical surfaces should always be aligned an order of magnitude better than this to give good images.

3. Secondary Options

Case A: Zero-field, full aperture F/15
This is edge diameter, not the diameter of the beam at the vertex.

Case B: Undersized, full aperture F/15, reduced for:
1.5% @ 40 microns (3rd Airy ring)
4 arcminute field
± 0.7 mm centration

Case C: Undersized, full aperture F/15, reduced for:
no diffraction correction (1.5% @ 1 micron)
6 arcminute field
no centration correction

Case D: Undersized to produce F/15, (full aperture ~F/14.7), reduced for:
1.5% @ 40 microns (3rd Airy ring)
4 arcminute field
± 0.7 mm centration

Case E: Undersized to produce F/15, (full aperture ~ F/14.7), reduced for:
no diffraction correction (1.5% @ 1 micron)
6 arcminute field
no centration correction

Case F: Oversized, full aperture F/15, for:
8 arcminute unvignetted field in the optical

Table 1: These tables show the various options for undersized secondary diameters as discussed in the text. The upper panel is for the MMT Conversion and Magellan F/15 Cassegrain secondaries and the lower panel is for the LBT Gregorian secondaries.

4 Example Calculations

4.1 LBT F/15 GREGORIAN FOCUS (infrared)

8.408	primary mirror diameter (m)
1.14177	primary focal ratio
-14.707	system focal ratio (iterated by hand)
2.5	vertex -- focus distance (m)
4.	field diameter (arcmin)
0.0898	primary obstruction (m)
40.	maximum infrared wavelength (microns)
3.7	tolerable diffraction radius
7.00E-4	infrared centration allowance (m)

solving based on fs and eve

-12.88088	magnification of secondary
-123.65650	system focal length (m)
10.61845	separation of m1 and m2 (m)
13.11845	pathlength from secondary to focus (m)
0.94507	focal length of secondary (m)
0.90434	vertex diameter of secondary (m) (full field)
0.89199	diameter of beam at secondary (m) (zero field)

iterative edge diameter, correct for sag and beam divergence

0.90062	edge diameter of secondary mirror (m) (full field)
1.699E-4	infrared diffraction angle (radians)

corrections to the secondary edge diameter (from zero field)

-0.00367	diffraction correction to m2 diameter (m)
-0.01230	field correction to m2 diameter (m)
-0.00140	centration correction to m2 diameter (m)

final iterative edge diameter of the undersized mirror

0.87120

infrared diameter of secondary mirror (m) (undersize)

other implied numbers

1.08480	secondary focal ratio
0.05037	sagitta of secondary mirror (m)
8.24372	effective primary aperture (m)
-15.00008	effective system focal ratio (Gregorian)
0.10979	infrared unused hole in secondary (m)
0.01157	fractional area of telescope obscuration
52.73248	net telescope collecting area (m**2)

5 Specific Secondaries

MMT/Magellan The MMT Conversion and the Magellan Project have decided to use a nearly identical amount of undersizing defined in a slightly different way. That is Case E to produce true F/15 Cassegrain (after undersizing) with the stop at the secondary for a field larger than 6 arcminutes in diameter. (A field of 6.05 arcminutes would be more nearly identical to case D.) Thanks to George Rieke for promoting this agreement. Note that this F/15 secondary has a back focal distance of 2.286 m (--90 inches) compared to 1.778 m (--70 inches) in some earlier documents.

All three projects are using the secondary aspheres appropriate for a purely parabolic primary and the focal ratios and back focal distances listed above.

NOTE: These are not fabrication dimensions. The diameters and aspheres still need to be verified with a ray-tracing program. Other fabrication details such as edge treatment have yet to be resolved.

5.1 MMT/Magellan F/15

MMT CONVERSION F/15 (revised to -90")

telescope input parameters

2.	number of mirrors in optical train
2.	cassegrain configuration
6.5024	primary mirror diameter (m)
1.25	primary focal ratio
14.704	system focal ratio
2.286	vertex -- focus distance (m)
INDEF	secondary focal length (m)
6.	field diameter (arcmin)
0.89	primary obstruction (m)
1.	maximum infrared wavelength (microns)
3.7	tolerable diffraction radius
0.	infrared centration allowance (m)

other telescope parameters

8.128

primary focal length (m)

solving based on fs and eve

11.7632	magnification of secondary
95.61129	system focal length (m)
0.002837207	throughput (ubar1*y1)
0.03399126	half angle of telescope light cone (rad)
7.31206	separation of m1 and m2 (m)
9.59806	pathlength from secondary to focus (m)
0.7618269	l = separation / back focal distance
0.28125	beta = vertex distance / focal length of m1
0.3515625	eta = normalized vertex back focus
0.01873409	specified central obstruction (fractional area)
0.2856359	diameter of primary hole (m)
0.001929649	obscuration by cassegrain hole (fractional area)
-0.8917478	focal length of secondary (m)
-72.83925	entrance pupil position relative to primary (m)
9.961522	entrance pupil magnification
0.6655136	vertex diameter of secondary (m)
0.0104753	obscuration by secondary (fractional area)
0.6527517	diameter of beam at secondary (m)
0.1365503	?exit pupil throughput (ubar*y) for secondary
-1.	primary asphere fixed
-1.406165	secondary asphere for cass
1.	primary eccentricity
1.185818	secondary eccentricity

normalized structural aberration coefficients

0.	sigmai
-1.	sigmaii
9.199695	sigmaiii
95.45465	sigmaiv
-21.88507	sigmav
0.32512	sagitta of primary mirror (m)
-3317.897	primary aspheric amplitude (microns)
0.6676853	edge diameter of secondary mirror (m)
-1.335581	secondary focal ratio
-0.0311347	sagitta of secondary mirror (m)
5.763644E-6	infrared diffraction angle (radians)
-8.764278E-5	diffraction correction to m2 diameter (m)
-0.0128122	field correction to m2 diameter (m)
0.	centration correction to m2 diameter (m)
0.6418097	infrared diameter of secondary mirror (m)
-1.389427	secondary focal ratio
-0.02877596	sagitta of secondary mirror (m)
6.374291	effective primary aperture (m)
14.99952	effective system focal ratio
331.8175	secondary aspheric amplitude (microns)
0.10219	infrared unused hole in secondary (m)
0.01873409	fractional area of telescope obscuration
31.28986	net telescope collecting area (m**2)

wavefront aberration coefficients

0.	w040 (microns) spherical aberration
-0.8201628	w131 (microns) coma
0.3872722	w222 (microns) astigmatism
2.009139	w220p (microns) field curvature
-0.04728613	w311 (microns) distortion

focal plane parameters

0.4635366	platescale (mm/arcsec)
8.726646E-4	field radius angle (ubar1), (rad)
0.1668732	linear diameter of focal plane (m)
0.047	rms angular image radius tolerance (arcsec)
21.78622	rms physical image radius tolerance (microns)
1.020884	fractional curved field radius
6.125301	maximum curved field diameter (arcmin)
1.001641	?petzval radius of curvature (m)
0.8397707	focal plane radius of curvature (m)
0.6527772	fractional flat field radius
3.916663	maximum flat field diameter (arcmin)
-0.9060713	height of largest flat field (mm)
1.39059	full field distortion (microns)

field focus curve for aligned system

radius		focal plane height		image size		wave aberration
(mm)	(arcmin)	(mm)	(+/-mm)	(micron)	(arcsec)	(micron rms)
0.00	0.00	0.	0.906	0.	0.	0.
4.17	0.15	-0.01036	0.905	0.985	0.00212	0.00484
8.34	0.30	-0.04145	0.902	1.97	0.00425	0.00971
12.52	0.45	-0.09326	0.898	2.96	0.00638	0.0147
16.69	0.60	-0.1658	0.891	3.95	0.00853	0.0197
20.86	0.75	-0.2591	0.882	4.95	0.0107	0.0249
25.03	0.90	-0.373	0.872	5.95	0.0128	0.0303
29.20	1.05	-0.5078	0.859	6.96	0.015	0.0358
33.37	1.20	-0.6632	0.843	7.98	0.0172	0.0417
37.55	1.35	-0.8394	0.825	9.01	0.0194	0.0477
41.72	1.50	-1.036	0.804	10.1	0.0217	0.0541
45.89	1.65	-1.254	0.78	11.1	0.024	0.0607
50.06	1.80	-1.492	0.752	12.2	0.0262	0.0677
54.23	1.95	-1.751	0.719	13.2	0.0286	0.075
58.41	2.10	-2.031	0.682	14.3	0.0309	0.0826
62.58	2.25	-2.332	0.639	15.4	0.0333	0.0907
66.75	2.40	-2.653	0.588	16.6	0.0358	0.0991
70.92	2.55	-2.995	0.527	17.7	0.0382	0.108
75.09	2.70	-3.357	0.452	18.9	0.0407	0.117
79.26	2.85	-3.741	0.352	20.1	0.0433	0.127
83.44	3.00	-4.145	0.195	21.3	0.0459	0.137

secondary alignment tolerances based on rms image radius

0.08	wavefront focus -- axial motion (micron/micron)
6.548042	m2 focus tolerance (on-axis) -- axial motion (micron)
1.408374	m2 focus tolerance (field) -- axial motion (micron)
2.192513E-7	scale change without refocus (fraction/micron)
-2.803483E-7	scale change with refocus (fraction/micron)
0.003177748	wavefront spherical aberration -- axial motion (micron/micron)
0.0623008	induced image radius (micron/micron)
1.344032E-4	induced image radius (arcsec/micron)
349.6941	tolerable secondary motion (micron)
22.96511	wavefront spherical aberration -- focal motion (micron/meter)
-0.04838818	tolerable focal plane motion (m)
0.0065024	wavefront spherical aberration -- primary asphere (micron/ppm)
0.1274817	induced image radius (micron/ppm)
2.750197E-4	induced image radius (arcsec/ppm)
1.708969E-4	tolerable primary asphere error
-5.000297E-4	wavefront spherical aberration -- secondary asphere (micron/ppm)
-0.009803246	induced image radius (micron/ppm)
-2.114881E-5	induced image radius (arcsec/ppm)
-0.002222348	tolerable secondary asphere error
-0.8159396	distance from m2 to zero-coma pivot (m)
-1.110223E-16	distance from prime focus to zero-coma pivot (m)
0.1089202	image motion from zero-coma rotation (arcsec/arcsec)
0.2007725	image motion from m2 vertex rotation (arcsec/arcsec)
-10.7632	image motion from lateral displacement (micron/micron)
-0.02321974	image motion from lateral displacement (arcsec/micron)
0.0160129	wavefront coma -- lateral motion (micron/micron)
0.3844943	induced image radius (micron/micron)
8.294799E-4	induced image radius (arcsec/micron)
56.66201	tolerable motion (micron)
-0.06334363	wavefront coma -- vertex rotation (micron/arcsec)
-1.520977	induced image radius (micron/arcsec)
-0.003281245	induced image radius (arcsec/arcsec)
-14.32383	tolerable rotation (arcsec)
-0.3154995	wavefront coma -- vertex chop angle (micron/arcsec)
-7.575625	induced image radius (micron/arcsec)
-0.0163431	induced image radius (arcsec/arcsec)
-2.875832	tolerable vertex chop throw (arcsec)
82.89701	tolerable zero coma chop throw (arcsec), astig only
0.6309989	wavefront coma -- primary rotation (micron/arcsec)
15.15125	induced image radius (micron/arcsec)
0.0326862	induced image radius (arcsec/arcsec)
1.437916	tolerable rotation (arcsec)

5.2 LBT F/15

F/15 GREGORIAN FOCUS (infrared)

telescope input parameters

2.	number of mirrors in optical train
3.	gregorian configuration
8.408	primary mirror diameter (m)
1.14177	primary focal ratio
-14.707	system focal ratio
2.5	vertex -- focus distance (m)
INDEF	secondary focal length (m)
4.	field diameter (arcmin)
0.0898	primary obstruction (m)
40.	maximum infrared wavelength (microns)
3.7	tolerable diffraction radius
7.000000E-4	infrared centration allowance (m)

other telescope parameters

9.600002

primary focal length (m)

solving based on fs and eve

-12.88088	magnification of secondary
-123.6565	system focal length (m)
0.002445788	throughput (ubar1*y1)
-0.03398433	half angle of telescope light cone (rad)
10.61845	separation of m1 and m2 (m)
13.11845	pathlength from secondary to focus (m)
0.8094286	l = separation / back focal distance
0.2604166	beta = vertex distance / focal length of m1
0.2973359	eta = normalized vertex back focus
1.140689E-4	specified central obstruction (fractional area)
0.2840938	diameter of primary hole (m)
0.001141664	obscuration by cassegrain hole (fractional area)
0.9450732	focal length of secondary (m)
100.0911	entrance pupil position relative to primary (m)
-9.426152	entrance pupil magnification
0.9043416	vertex diameter of secondary (m)
0.01156857	obscuration by secondary (fractional area)
0.8919865	diameter of beam at secondary (m)
0.1681304	?exit pupil throughput (ubar*y) for secondary
-1.	primary asphere fixed
-0.7325937	secondary asphere for cass
1.	primary eccentricity
0.8559169	secondary eccentricity

normalized structural aberration coefficients

0.	sigmai
-1.	sigmaii
-10.23558	sigmaiii
143.7241	sigmaiv
-24.87119	sigmav
0.4602503	sagitta of primary mirror (m)
-5652.75	primary aspheric amplitude (microns)
0.900618	edge diameter of secondary mirror (m)
1.049361	secondary focal ratio
0.05384596	sagitta of secondary mirror (m)
1.699050E-4	infrared diffraction angle (radians)
-0.003665692	diffraction correction to m2 diameter (m)
-0.01229653	field correction to m2 diameter (m)
-0.0014	centration correction to m2 diameter (m)
0.8712002	infrared diameter of secondary mirror (m)
1.084795	secondary focal ratio
0.05037334	sagitta of secondary mirror (m)
8.24372	effective primary aperture (m)
-15.00008	effective system focal ratio
-505.2776	secondary aspheric amplitude (microns)
0.1097871	infrared unused hole in secondary (m)
0.01156857	fractional area of telescope obscuration
52.73248	net telescope collecting area (m**2)

wavefront aberration coefficients

0.	w040 (microns) spherical aberration
-0.7067253	w131 (microns) coma
0.247573	w222 (microns) astigmatism
-1.738163	w220p (microns) field curvature
-0.02058863	w311 (microns) distortion

focal plane parameters

0.5995034	platescale (mm/arcsec)
5.817764E-4	field radius angle (ubar1), (rad)
0.1438808	linear diameter of focal plane (m)
0.024	rms angular image radius tolerance (arcsec)
14.38808	rms physical image radius tolerance (microns)
0.822485	fractional curved field radius
3.28994	maximum curved field diameter (arcmin)
-0.8603736	?petzval radius of curvature (m)
-1.003274	focal plane radius of curvature (m)
0.6020763	fractional flat field radius
2.408305	maximum flat field diameter (arcmin)
0.5985108	height of largest flat field (mm)
-0.6055941	full field distortion (microns)

field focus curve for aligned system

radius		focal plane height		image size		wave aberration
(mm)	(arcmin)	(mm)	(+/-mm)	(micron)	(arcsec)	(micron rms)
0.00	0.00	0.	0.599	0.	0.	0.
3.60	0.10	0.006448	0.597	0.849	0.00142	0.00417
7.19	0.20	0.02579	0.594	1.7	0.00283	0.00835
10.79	0.30	0.05803	0.589	2.55	0.00425	0.0126
14.39	0.40	0.1032	0.582	3.4	0.00567	0.0168
17.99	0.50	0.1612	0.572	4.26	0.0071	0.0212
21.58	0.60	0.2321	0.559	5.11	0.00853	0.0256
25.18	0.70	0.316	0.544	5.97	0.00996	0.0301
28.78	0.80	0.4127	0.527	6.84	0.0114	0.0348
32.37	0.90	0.5223	0.505	7.71	0.0129	.0395
35.97	1.00	0.6448	0.48	8.58	0.0143	0.0444
39.57	1.10	0.7802	0.451	9.46	0.0158	0.0495
43.16	1.20	0.9285	0.416	10.4	0.0173	0.0547
46.76	1.30	1.09	0.373	11.2	0.0188	0.0601
50.36	1.40	1.264	0.321	12.1	0.0203	0.0657
53.96	1.50	1.451	0.252	3.1	0.0218	0.0715
57.55	1.60	1.651	0.143	14.	0.0233	0.0775
61.15	1.70	1.864	0.	14.9	0.0249	0.0837
64.75	1.80	2.089	0.	15.8	0.0264	0.0902
68.34	1.90	2.328	0.	16.8	0.028	0.0969
71.94	2.00	2.579	0.	17.7	0.0296	0.104

secondary alignment tolerances based on rms image radius

0.09588546	wavefront focus -- axial motion (micron/micron)
3.60729	m2 focus tolerance (on-axis) -- axial motion (micron)
0.	m2 focus tolerance (field) -- axial motion (micron)
1.869410E-7	scale change without refocus (fraction/micron)
2.645298E-7	scale change with refocus (fraction/micron)
0.004568452	wavefront spherical aberration -- axial motion (micron/micron)
0.08958428	induced image radius (micron/micron)
1.494308E-4	induced image radius (arcsec/micron)
160.6094	tolerable secondary motion (micron)
27.53456	wavefront spherical aberration -- focal motion (micron/meter)
-0.02664784	tolerable focal plane motion (m)
0.01103283	wavefront spherical aberration -- primary asphere (micron/ppm)
0.2163464	induced image radius (micron/ppm)
3.608759E-4	induced image radius (arcsec/ppm)
6.650485E-5	tolerable primary asphere error
0.001464761	wavefront spherical aberration -- secondary asphere (micron/ppm)
0.02872299	induced image radius (micron/ppm)
4.791130E-5	induced image radius (arcsec/ppm)
5.009257E-4	tolerable secondary asphere error
1.018443	distance from m2 to zero-coma pivot (m)
0.	distance from prime focus to zero-coma pivot (m)
-0.09785176	image motion from zero-coma rotation (arcsec/arcsec)
-0.2121757	image motion from m2 vertex rotation (arcsec/arcsec)
13.88088	image motion from lateral displacement (micron/micron)
0.02315396	image motion from lateral displacement (arcsec/micron)
0.02098278	wavefront coma -- lateral motion (micron/micron)
0.5039314	induced image radius (micron/micron)
8.405814E-4	induced image radius (arcsec/micron)
28.55167	tolerable motion (micron)
0.1036036	wavefront coma -- vertex rotation (micron/arcsec)
2.488188	induced image radius (micron/arcsec)
0.004150415	induced image radius (arcsec/arcsec)
5.782554	tolerable rotation (arcsec)
-0.4882916	wavefront coma -- vertex chop angle (micron/arcsec)
-11.72702	induced image radius (micron/arcsec)
-0.01956122	induced image radius (arcsec/arcsec)
-1.226917	tolerable vertex chop throw (arcsec)
52.48784	tolerable zero coma chop throw (arcsec), astig only
0.9765831	wavefront coma -- primary rotation (micron/arcsec)
23.45404	induced image radius (micron/arcsec)
0.03912244	induced image radius (arcsec/arcsec)
0.6134586	tolerable rotation (arcsec)

6. Conclusion

7. Bibliography

• Blanco, D. 1990, ``Size of the Infra-Red Secondary'', Memo to the MMT Conversion Committee, 10/16/90.

• Gillett, F. et al. 1991, ``Performance Requirements for an IR-Optimized 8-m Telescope'', Report of the Gemini IR Subcommittee, 10/14/91, revised 11/92.

• Hill, J. M. 1990, ``Notes on Infrared Optimization'', Columbus Project Technical Memo UA-90-03.

• Rieke, G. H. 1987, ``Infrared Configuration for 8-meter Telescopes with Fast Primaries'', Report to the Magellan Project Science Advisory Committee, 09/11/87.

• Rieke, G. H. 1992, ``8-m Telescope Optical Design, Optimized for Imaging and IR'', Vu-graphs presented to a review committee, 11/08/92.

• Rieke, G. H. 1993, ``Preliminary Requirements for MMT F/15 Secondary Mirror'', Memo (limited circulation), 5/9/93.