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INSTRUMENT SCIENCE REPORT
FOC­056
TITLE: Observations to Determine the FOC Focus
AUTHOR: P. Hodge DATE: 4 October 1994
ABSTRACT
This report describes Science Verification observations taken at different positions of the
refocussing mirror mechanism. Three tests of focus were used, and all three gave fairly
consistent results. Significant changes in focus quality were seen at both f/48 and f/96, with
the best focus for f/48 near the nominal position and the best focus for f/96 near or below
the lower limit of the range.
0 DISTRIBUTION:
FOC Project: D. Eaton, B. G. Taylor, R. Thomas, N. Towers
IDT: entire FOC IDT
TIB: D. Baxter, J. C. Blades, C. Cox, P. Greenfield, W. Hack, R. Jedrzejewski, A. Nota, F. Paresce,
All Instrument Scientists
SCARS: D. Bazell, D. Gilmore, P. Hodge
SESD: W. Baggett, M. Miebach
SPD: F. Macchetto
USB: B. Gillespie
ST/ECF: H.­M. Adorf, P. Benvenuti, A. Caulet, R. Fosbury, R. Hook

1 Introduction
The FOC relays can be focussed independently of the OTA by moving the mirror that folds
the light path. The mirror can be moved a total distance of about 19 mm, or 38 mm of
optical path. With perfect optics, that focussing range would be fairly broad at f/48, wide
enough to be useful at f/96, and insignificant at f/288. The spherical aberration of the OTA
primary mirror complicates the question of best focus. We do see very obvious changes in
sharpness at f/48. Quantitative measurements of focus show significant changes across the
focussing range at f/96, and qualitatively the images at smaller refoc positions look sharper.
Three sets of focus observations have been taken. All showed that the best focus for the
f/48 relay is not far from the default refoc mirror position, but the best focus for f/96 is near
or possibly below the lower limit of the focussing range. The positions of the refoc mirrors
in the f/48 and f/96 relays can be found in the UDL header file with the keyword names
XC48RFMP and XC96RFMP. Their default values up to the present time are XC48RFMP = 110,
XC96RFMP = 91.
2 Observations
Focus observations were taken on several dates. Here is a table which summarizes the
observations. The column labeled defocus gives the offset in microns of the OTA secondary
from the ``best'' position, as calculated by a program written by Bill Sparks.
Table 1. Summary of the observations.
date (UT) f/ratio proposal object filters defocus
1990 Nov 15,16 f/96 1507 NGC 104 F130M, F220W \Gamma4:3
1990 Nov 16 f/288 1507 NGC 104 F130M, F220W \Gamma4:4
1990 Nov 28 f/48 1507 NGC 104 F140W, F220W \Gamma6:8
1991 Mar 15,16 f/48 3144 NGC 6397 F140W, F220W +0.8
1991 Mar 16 f/96 3144 NGC 6397 F140W, F210M, F346M +0.8
1991 Dec 18 f/48 3384 NGC 7078 F140W, F220W \Gamma3:3
1991 Dec 18,19 f/96 3384 NGC 7078 F140W, F220W, F342W \Gamma3:3
2

The individual image names and times of observation are given below. Exposure times are
given when they differ from the nominal value, in which case the refoc position is also listed.
f/96, F130M (proposal 1507, 1990 Nov 15 23:32:48 ­ Nov 16 03:10:56):
Root names x0dw0501t through x0dw0505t. These were all blank.
f/96, F220W (proposal 1507, 1990 Nov 16 04:20:43 ­ 06:28:25):
Root names x0dw0506t through x0dw0509t. Nominal exposure time 596.625. The fifth
observation (refoc = 212) was lost. These images contain many stars.
root name refoc exptime
x0dw0509t 162 531.250
f/288, F130M (proposal 1507, 1990 Nov 16 07:34:32 ­ 09:43:12):
Root names x0dw050at through x0dw050et. These were all blank.
f/288, F220W (proposal 1507, 1990 Nov 16 10:47:46 ­ 14:24:13):
Root names x0dw050ft through x0dw050jt. Nominal exposure time 596.625. There are
only three stars visible.
root name refoc exptime
x0dw050gt 62 557.250
x0dw050it 162 420.000
f/48, F140W (proposal 1507, 1990 Nov 28 08:32:26 ­ 12:53:34):
Root names x0dw0101t through x0dw0108t. Nominal exposure time 597.625. These images
contain many stars.
root name refoc exptime
x0dw0102t 42 437.250
x0dw0105t 132 491.250
x0dw0108t 222 497.250
f/48, F220W (proposal 1507, 1990 Nov 28 14:04:56 ­ 18:01:17):
Root names x0dw0109t through x0dw010gt. Nominal exposure time 597.625. These images
contain many stars.
root name refoc exptime
x0dw010at 42 551.250
x0dw010ct 102 546.250
x0dw010dt 132 523.250
x0dw010ft 192 540.250
3

f/48, F140W (proposal 3144, 1991 Mar 15 16:17:16 ­ 21:12:13):
Root names x0i50101t through x0i50108t. Nominal exposure time 597.625. These images
contain about a dozen stars.
root name refoc exptime
x0i50101t 25 424.125
x0i50103t 72 304.875
x0i50106t 162 423.875
x0i50108t 222 264.875
f/48, F220W (proposal 3144, 1991 Mar 15 22:21:31 ­ Mar 16 02:12:21):
Root names x0i50109t through x0i5010gt. Nominal exposure time 597.625. The exposure
time is uncertain for the seventh (refoc = 192) observation. These images contain about a
dozen stars, but most of the brighter ones are saturated.
root name refoc exptime
x0i5010at 42 302.875
x0i5010dt 132 271.875
x0i5010ft 192 471.250
f/96, F140W (proposal 3144, 1991 Mar 16 13:09:35 ­ 16:16:15):
Root names x0i50501t through x0i50505t. Nominal exposure time 596.625. The first
(refoc = 12) observation was offset by about (­170,0) from the others. These images contain
about half a dozen stars.
root name refoc exptime
x0i50501t 12 248.000
x0i50503t 112 406.000
f/96, F210M (proposal 3144, 1991 Mar 16 16:22:50 ­ 18:23:14):
Root names x0i50506t through x0i5050at. Nominal exposure time 596.625. The exposure
time is uncertain for the first (refoc = 12) observation. The header gave a value of 304.25,
but a value of 213.0 is more likely to be correct. See the discussion in Section 3. These
images contain about half a dozen stars.
root name refoc exptime
x0i50506t 12 213.000
x0i50509t 162 421.000
4

f/96, F346M (proposal 3144, 1991 Mar 16 19:19:05 ­ 21:21:51):
Root names x0i5050bt through x0i5050ft. Nominal exposure time 596.625. The few stars
that show in the F140W and F210M images are very saturated in these F346M images, but
many fainter stars are now visible. Some of these fainter stars were used for focus.
root name refoc exptime
x0i5050ct 62 421.000
x0i5050ft 212 220.000
f/48, F140W (proposal 3384, 1991 Dec 18 2:43 ­ 8:55):
Root names x0rn0101t through x0rn0108t. Nominal exposure time 596.875 seconds.
f/48, F220W (proposal 3384, 1991 Dec 18 9:12 ­ 15:40):
Root names x0rn0109t through x0rn010gt. Nominal exposure time 596.875 seconds.
f/96, F140W (proposal 3384, 1991 Dec 18 20:28 ­ 23:41):
Root names x0rn0201t through x0rn0205t. Nominal exposure time 595.875 seconds. The
first observation (refoc = 12) was offset by about (­47,­50) from the others.
f/96, F220W (proposal 3384, 1991 Dec 19 1:00 ­ 4:14):
Root names x0rn0206t through x0rn020at. Nominal exposure time 595.875 seconds.
root name refoc exptime
x0rn0208t 112 401.250
f/96, F342W (proposal 3384, 1991 Dec 19 4:31 ­ 7:27):
Root names x0rn020bt through x0rn020ft. Nominal exposure time 595.875 seconds. These
images were extremely saturated with a background of hundreds of counts, making them
useless for focus measurement.
root name refoc exptime
x0rn020et 162 401.250
5

Here are the approximate X and Y coordinates of stars that were used for determining the
focus.
Proposal 1507, NGC 104:
f/48, F140W f/48, F220W f/96, F220W f/288, F220W
X Y X Y X Y X Y
347 61 80 240 65 285 342 343
260 129 263 170 67 230 386 303
63 204 300 254 65 189 232 252
211 336 223 295 170 305
343 332 59 341 168 238
436 466 336 413 171 143
321 291
303 93
387 91
Proposal 3144, NGC 6397:
f/48, F140W f/48, F220W f/96, F140W f/96, F210M f/96, F346M
X Y X Y X Y X Y X Y
32 55 333 312 272 250 40 276 212 410
375 103 257 320 385 255 118 217 332 358
339 147 336 341 449 272 167 288 339 378
380 149 270 356 275 245 157 115
330 201 206 383 389 250 341 63
320 413 452 266 287 109
458 453
6

Proposal 3384, NGC 7078:
f/48, F140W f/48, F220W f/96, F140W f/96, F220W
X Y X Y X Y X Y
194 80 25 91 341 174 295 69
308 122 308 121 77 183 214 98
328 203 239 137 264 244 119 129
114 238 424 145 276 283 409 193
321 321 79 185 234 290 81 230
371 337 113 237 332 296 420 300
311 372 122 325 261 354 131 353
119 452 371 336 260 458 262 354
144 490 95 420 243 419
441 418 260 458
142 490
3 Data Analysis and Results
Three different tests of focus were applied, although not all tests were applied to all obser­
vations. The ``sharpness criterion'' is the sum of the squares of pixel values divided by the
square of the sum of pixel values. This test was suggested by IDT members and is widely
used at ground­based sites. It is related to the detection of faint stars in the presence of
background using optimal weighting. The ``characteristic radius'' is the sum of pixel values
multiplied by the distance from a star image center to the pixel, divided by the sum of
pixel values. This just seemed like a reasonable test, and it gives results comparable to the
sharpness criterion and with similar sensitivity. The ``encircled energy'' is the number of
counts within some radius of a star image center, divided by the exposure time. Maximizing
the energy within a radius of 0.1 arcseconds has been adopted by the HST project as the
criterion for best focus. The signal­to­noise ratio for a single star image, severely affected by
spherical aberration, should correlate well with the encircled energy. Note that any of these
tests can give a biased result if the background has not been properly subtracted.
For each set of observations, a specific set of stars was chosen to be used for the deter­
mination of focus. The approximate pixel coordinates of the stars are given in the tables
above. The fextract task was used to extract a portion of the image around each star.
For both proposals 1507 and 3144, the input images were always the raw (*.d0h) images.
The fsquares, fradius and encounts tasks were then used on the extracted images to
get the sharpness, characteristic radius and encircled energy respectively. Plots of these
7

measurements are given below.
One refoc step is 0.083 mm, and the range is at least 200 steps, which is 16.6 mm, or 33.2 mm
in optical path difference. The OTA secondary magnifies by about M = 10.44, so moving
the OTA secondary by a given distance along the axis moves the secondary focus a distance
which is larger by the factor M 2 + 1 = 110. A shift of 200 refoc steps therefore corresponds
to a change of focus at f/24 or to an offset of the OTA secondary mirror as follows:
f/48 f/96 f/288
at f/24 focus: 8.3 2.075 0.2306 mm
OTA secondary: 75.5 18.86 2.096 microns
A one­micron change in the position of the OTA secondary mirror away from the OTA
primary mirror could be compensated for by the following change in refoc mirror position:
f/48 f/96 f/288
delta refoc: \Gamma2:65 \Gamma10:6 \Gamma95:4 steps
A positive value for defocus (see Table 1) means the OTA secondary is farther than nominal
from the OTA primary. Due to desorption, the distance between OTA primary and secondary
decreases with time, so the defocus in microns also decreases, and the location of the best
focus for the FOC moves toward a larger refoc position.
Dave Baxter has pointed out that the image scale changes as the refoc mirror is moved.
The exit pupil of the OTA plus FOC is roughly a meter and a half from the detector, and
we expect the image scale to be directly proportional to that distance. The scale should
therefore change by the fraction 33.2/1500, or about two percent, over the refoc range. The
distance from the exit pupil to the detector is smaller in the f/48 relay than in the f/96 relay,
so the variation in scale should be larger at f/48 than at f/96. This is in agreement with
the observations. The image is magnified at larger refoc positions relative to smaller ones,
which confirms that the refoc mirror moves away from the FOC secondary and detector as
the refoc position increases. When a filter is inserted into the light path, the refoc mirror
is moved to compensate for the change in focus, with a larger refoc position for a thicker
filter. For example, the refoc position is increased by 31 for a polarizing filter. This change,
however, does not result in any change in image scale.
For the f/96, F210M observations on 1991 Mar 16, the sharpness and radius both indicate
that refoc positions 12 and 62 are about equally good. The exposure time for refoc position 12
(x0i50506t) is given in the image header as 304.25 seconds (actually \Gamma304.25, the negative
sign indicating uncertainty). Using 304.25 seconds, the encircled energy seems low by perhaps
8

30 or 40 percent. The UDL header gives five times in keywords XEXPTM01 through XEXPTM05.
The interpretation of these UDL keywords in the case of errors is described in a memo by
Nigel Towers. These values are the start and stop times of the exposure, so in this case the
exposure was interrupted (probably by loss of lock) into at least two intervals. If the shutter
happened to be closed at the scheduled end of the exposure, an extra stop time would be
recorded in the UDL, and ignoring that final time gives the value of 304.25 sec. If the shutter
opens and then closes again within 15 seconds, the start time is recorded in the UDL, but
the stop time would not be, and an error 19 would be generated. Wayne Baggett pointed
out that an error 19 (and an error 56) did occur on that date, but the time of the errors
was closer to 18:00 than 16:23 when the observation began. If we assume that those errors
do actually apply to this observation, however, there are two reasonable possibilities for the
five UDL times. The following table indicates for each case whether the time is actually a
start or stop time, and the time interval in seconds to each UDL time is also given.
first possibility second possibility
Start 0. Start 0.
stop not recorded stop 97.25
Start 97.25 Start 273.875
stop 273.875 stop not recorded
Start 480.875 Start 480.875
stop 596.625 stop 596.625
For the first case, we would compute the exposure time by ignoring the first UDL time, which
would result in an error of up to 15 seconds. The exposure time would be (273:875 \Gamma 97:25)+
(596:625 \Gamma 480:875), which is 292.375, plus up to 15 seconds. For the second case, we would
ignore the second start time (at 273.875), giving (97:25 \Gamma 0:) + (596:625 \Gamma 480:875) = 213:,
plus up to 15 seconds. The smaller time was assumed to be correct, and is tabulated in
Section 2 above, simply because it gave a more believable encircled energy plot. If we had
the actual time when the error 19 occurred we would know which UDL time to delete, but
that information is no longer available.
The earlier f/48 images are blurrier than they should have been. Measurements of the full­
width at half­maximum of f/48 (F140W) and f/96 (F210M) images taken in November 1990
gave the results that FWHM at f/48 = 4 pixels, FWHM at f/96 = 3 pixels. Many of the f/48
images taken that month for focus suffered a loss of fine lock. To investigate the possibility
that telescope motion was responsible for the width of the f/48 images, telescope pointing
information was obtained using the OMS system for the times of the f/48 F220W focus
observations in November 1990. Most of the time the telescope pointing remained within
one f/48 pixel. Even when lock was lost, the pointing was bad for only a short time before
the FOC shutter closed. Two plots are included which show the relative telescope position
9

in X and Y at approximately one­second intervals. One of the plots shows the jitter for an
observation which ended with a loss of lock, and the other shows typically good pointing.
The f/48 detector focus was improved in the fall of 1991 (Towers and Miebach, 1991, In­
strument Science Report FOC­055). The optical focus observations taken since then, in
December, showed a FWHM as low as 2.5 pixels with the F220W filter.
The results are summarized in the following table, which gives the focus position as deter­
mined by the various methods used here. These values of focus position have been corrected
for filter thickness and the position of the OTA secondary. The uncorrected focus posi­
tions are given in the titles for most of the individual plots in the figures; the number is at
the right end of the title following the arrow. The focus correction for filters F130M and
F140W is 6 refoc steps, while the correction for F210M, F220W, F346M, and F342W is 10
steps. For example, for F220W at f/96 with defocus = \Gamma3.3 (from Table 1), the correction
\Gamma3:3 \Delta 10:6 \Gamma 10 = \Gamma45 would be added to the raw (observed) focus position to get the focus
for nominal OTA secondary position with no filter.
Table 2. Summary of the results. The column labeled refoc s is the refoc value for the focus
position as determined by the sharpness criterion, refoc r is the focus as determined by the
effective radius, and refoc ee is the focus as determined by the encircled energy. The value in
the correction column was added to the observed focus position to correct for filter thickness
and OTA secondary mirror position. The positions given in this table, therefore, are for a
defocus position of zero and no filter. A dash in the table indicates that no well defined
focus position was found.
10

date f/ratio filter refoc s refoc r refoc ee correction
1990 Nov 28 f/48 F140W --- --- --- \Gamma24
1990 Nov 28 f/48 F220W 101 98 102 \Gamma28
1991 Mar 15 f/48 F140W 68 85 81 \Gamma4
1991 Mar 15 f/48 F220W 72 79 92 \Gamma8
1991 Dec 18 f/48 F140W 70 83 66 \Gamma15
1991 Dec 18 f/48 F220W 72 73 91 \Gamma19
1990 Nov 15 f/96 F130M --- --- --- \Gamma52
1990 Nov 16 f/96 F220W --- \Gamma12 --- \Gamma56
1991 Mar 16 f/96 F140W --- --- --- +2
1991 Mar 16 f/96 F210M 2 40 56 \Gamma2
1991 Mar 16 f/96 F346M --- --- --- \Gamma2
1991 Dec 18 f/96 F140W \Gamma32 \Gamma8 17 \Gamma41
1991 Dec 19 f/96 F220W \Gamma37 \Gamma23 \Gamma6 \Gamma45
1991 Dec 19 f/96 F342W --- --- --- \Gamma45
1990 Nov 16 f/288 F130M --- --- --- \Gamma426
1990 Nov 16 f/288 F220W --- --- --- \Gamma430
4 Comparison with Simulations
It is appropriate to compare the observations with simulated images to be sure we understand
what contributes to the out­of­focus images. According to PSFs created by Chris Burrows,
the encircled energy should fall off only about 2/3 as quickly as we find that it does. The
curve from simulated images is quite broad and somewhat asymmetrical, being higher toward
larger refoc positions. The location that gives the maximum encircled energy changes with
wavelength; this can be understood as smoothing of the asymmetrical curve one obtains
using geometrical optics, with more smoothing at longer wavelengths. From 220 nm to 486
nm the peak shifts by about 3 mm at f/24, or 70 refoc steps at f/48 and 300 steps at f/96.
At f/48 a shift of order 20 refoc steps is expected between F140W and F220W. The shift
is toward larger refoc numbers at longer wavelengths. At f/96 the existing observations
longward of 220 nm are not very satisfactory for determining focus, although in the 1991
March data there is no obvious shift with wavelength. Paresce has pointed out that some
of the stars are very red, so the effective wavelength may be quite different from the filter
peak wavelength. so much of the flux may have been from the visual rather than the UV.
It would be very useful to obtain f/48 observations in the blue and UV to check this effect.
These simulations were monochromatic and were sampled rather than integrated over pixels,
so they should be redone more carefully before we get too upset about a lack of agreement
with observations.
11

5 Conclusions
At f/48 no change in the default focus position is needed. At f/96, however, all three criterea
show a very clear trend across the focussing range with a peak somewhere around 10 to 40,
or in some cases even below the low end of the range. On visual inspection of the images (see
Figures 4 and 5) it's clear that the smaller refoc positions are better. While the quantitative
differences are rather small, the data do indicate that the default focus position is not the
best. At the time of writing (i.e. considering the desorption), a focus position of 20 seems
close to optimum for f/96.
12

6 Figure Captions
Data for several stars at each focus position were combined to form each of the plots, except
the fine­focus sweep observations of Grw+70 ffi 5824. The approximate coordinates of these
stars are listed in the tables in Section 2.
Figure 1. The f/48, F140W image x0i50103t of NGC 6397 from proposal 3144 taken on
1991 March 15. This is included as an example.
Figure 2. This is an image created by the fextract task and includes portions from all of
the proposal 3384, f/48, F140W observations taken on 1991 Dec 18. Each column shows the
same star at different focus positions, with the refoc number given at the left, and each row
shows different stars at the same focus position. The star images are 15 pixels apart.
Figure 3. As in Figure 2 but for f/48, F220W. The star images are 15 pixels apart.
Figure 4. As in Figure 2 but for f/96, F140W. Only half of the eight stars are shown to
improve the scale for display, but all eight stars were used in determining the focus. The
star images are 21 pixels apart.
Figure 5. As in Figure 2 but for f/96, F220W taken on 1991 Dec 19. Only half of the ten
stars are shown. The star images are 29 pixels apart.
Figure 6. Proposal 1507, f/48 observations of NGC 104 through the F140W filter, using six
stars in each image. The upper plot show the sharpness criterion, and the lower plot show
the characteristic radius.
Figure 7. Proposal 1507, f/48 observations of NGC 104 through the F220W filter, based on
six stars in each image. The upper left shows the sharpness criterion, the upper right shows
counts per second within an 0.1 arcsec radius, and the lower left shows the characteristic
radius. The curves fit to the data are Gaussians in the upper two graphs and a hyperbola in
the lower left graph. In each case, the refoc position at the center of symmetry of the fitted
curve is displayed at the right end of the graph title. This is the raw position, not corrected
for filter thickness or desorption.
Figure 8. Proposal 1507, f/96 and f/288 observations of NGC 104 through the F220W filter.
The left hand side shows f/96. The upper plots show the sharpness criterion for f/96 and
f/288, and the lower plot shows the characteristic radius for f/96. For f/96 nine stars were
used, and for f/288 three stars were used.
13

Figure 9. Proposal 3144, f/48 observations of NGC 6397 through the F140W filter, based
on seven stars in each image. As in figure 7.
Figure 10. Proposal 3144, f/48 observations of NGC 6397 through the F220W filter, based
on nine stars in each image. As in figure 7.
Figure 11. Proposal 3144, f/96 observations of NGC 6397 through the F140W filter, based
on three stars in each image. As in figure 7.
Figure 12. Proposal 3144, f/96 observations of NGC 6397 through the F210M filter, based
on six stars in each image. As in figure 7.
Figure 13. Proposal 3144, f/96 observations of NGC 6397 through the F346M filter and mini­
sweep observations of Grw+70 ffi 5824 through the F120M filter. The left hand side shows data
for F346M, based on six stars in each image, with the sharpness criterion in the upper left
and counts per second within an 0.1 arcsec radius in the lower left. The upper right plot
shows the sharpness criterion for the F120M mini­sweep data where the focus was changed
by moving the OTA secondary mirror.
Figure 14. Proposal 3384, f/48 observations of NGC 7078 through the F140W filter, based
on nine stars in each image. As in figure 7.
Figure 15. Proposal 3384, f/48 observations of NGC 7078 through the F220W filter, based
on 11 stars in each image. As in figure 7.
Figure 16. Proposal 3384, f/96 observations of NGC 7078 through the F140W filter, based
on eight stars in each image. As in figure 7.
Figure 17. Proposal 3384, f/96 observations of NGC 7078 through the F220W filter, based
on ten stars in each image. As in figure 7.
Figure 18. Relative telescope position at approximately one­second intervals during the
x0dw010at observation of NGC 104 with f/48, F220W, showing a loss of lock.
Figure 19. Relative telescope position during the x0dw010bt observation of NGC 104 with
f/48, F220W.
14