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Relative Throughput Tests Using Focal Reducers with a Santa Barbara Instrument Group Self-Guiding Spectrograph

A set of throughput tests was conducted using a Meade 0.63x focal reducer and a Celestron 0.63x focal reducer in concert with a Santa Barbara Instrument Group (SBIG) Self-Guiding Spectrograph (SGS) and a Celestron 14-inch Schmidt-Caseagrain telescope.  The results of this analysis would suggest that one should NOT use focal reduction that would produce an effective f-ratio of less than about f9.5.

1) Introduction

The Santa Barbara Instrument Group (SBIG) Self-Guiding Spectrograph (SGS) is a compact Czerny-Turner style spectrograph which uses a single small mirror that serves as both collimator and camera objective.  The mirror is masked with a vertical bar that separates the visible surface area into two equal-sized sections shaped like a “D” (hence often referred to as the “D-shaped” mirror).  A member of the Yahoo SGS discussion group recently posted the observation that if one measured the size of the first, or collimation, side of the mirror and measured it’s distance from the slit, it would appear that any light cone faster than about f9.5 would overfill the mirror and that light would be lost, thus reducing the instrument’s throughput efficiency.   This paper presents the results of my own very crude measurements of the SGS, some rough approximations of what the relative throughput should be given different seeing conditions and f-ratios, as well as results from actual data acquired using the SGS both with and without focal reduction.

2) Motivation

I purchased a used SGS early in 2009 with the intention of participating in the campaign to observe the recent eclipse of epsilon Aurigae.  The StarFarm has two Celestron 14-inch Schmidt-Casesegrain telescopes (SCTs) and the plan was to use one for photometry, mostly of cataclysmic variable stars, and the other for spectroscopy.  The SGS has a fixed entrance slit which is 18-microns in width.  At it’s 3900mm native focal length, the C14 has an image scale of 52.9 arcseconds per mm, or about 0.95 arcseconds across an 18-micron slit.  The seeing here is rarely better than 2 arcseconds full-width-half-max (FWHM); a more typical value would be around 2.75 to 3 arcseconds.  Thus a majority of the light from a star’s image never makes it through the spectrograph slit.  As an example, with 2.75 arcsecond FWHM seeing only about 34% of the light in the star’s image is sampled.  Add to that any errors in guiding and it quickly becomes apparent that it would be very beneficial to lower the effective focal length of the telescope in order to get more light through the slit.  According to the documentation provided with the SGS, both that supplied with the instrument, and that on SBIG’s website, the SGS will work with telescopes from f/6 to f/10.  So from the very first evening of use in January of 2009, I had been using a Meade 0.63x focal reducer/ field flattener with the SGS in order to get what I though would be a roughly f/7 beam to the slit.  Given the same 2.75 arcsecond FWHM seeing, at f/7 a little over 51% of the star’s light would make it through the slit, a gain of about 50% when compared to f11.  The post I’d read on the Yahoo SGS group, however, indicated there would likely be a trade-off as some of that extra light would never make it past an over-filled collimation mirror.  I wanted to know what that trade off might be.

3) Measurements

Collimating morror for SGS

Figure 1: This shows the approximate shape and size of the “D mirror” in the SGS. Light from the slit is collimated by the left side and sent towards the grating. Light returning from the grating is focused by the right side onto the camera chip.

 

The first thing I did was to actually measure the size of the “D” mirror and its distance from the slit.  My measurements were very crude, done with a ruler and compass, so should not be considered accurate to anything better than a few mm.  In essence I was repeating the same measurements that SGS Yahoo group member Roy Tucker had made and had reported in the post that precipitated the current investigation.  His results showed that any light cone emerging from the slit with a focal ratio faster that about f/9.5 would over-fill the mirror.  The results of my own measurements showed essentially the same thing.

 

Basically, the light passing through the slit first encounters a small mirror about 1.4 inches from the slit which directs the expanding light cone toward the D mirror, which is an additional 4 inches away.  So the distance from the slit to the D mirror is about 5.4 inches.  Figure 1 shows an approximation of the size and shape of the D mirror.

 

Shown in Figure 2 are scale drawings of the D mirror with the illumination pattern from a star’s image centered and focused on the slit for f/11 and  f/7 beams.  Approximately 61% of the light from the f/7 system falls on the reflective surface of the D mirror – the other 39% is lost.

figure2

Figure 2: Illustrating what the defocused image of an ideal point source centered in the middle of the slit looks like at the distance of the “D” mirror from the slit for focal ratios of an f/11 (left) and an f/6.3 (right). The black circle is the shadow of the SCT secondary. Yellow indicates light that is captured by the mirror;

 

As a result, based on these simple measurements and the seeing that is typical here, I concluded that I would actually loose about 9% by using a focal reducer.  The actual amount would be roughly 1.49 * 0.61 = 0.908, or about 91% of what I’d get without the focal reducer and just running everything at f/11!  I have to say that came as a bit of a surprise.

 

Furthermore, when you consider this analysis only included the geometry of one of the five reflective elements within the instrument, each of which could possibly be over-filled by a steeper-than-f/9.5 light cone, I began to realize that the total internal light loss could be much greater.

4: The Experiment

Theory is all well and good, but nothing beats actual empirical data.  So on the night of January 26/27, 2010 I obtained spectra of the 4th magnitude star Giausar (Lambda Draconis), both with and without a Meade 0.63x focal reducer.  The first set of three images was taken with the focal reducer and the second set without.  A set of Neon calibration lamp images was taken between the two sequences for wavelength calibration.  When the first spectrum at f/11 appeared I thought I’d centered on the wrong star.  There was so much more light being registered I could not believe it.  So I ran another sequence of images and got roughly the same result.  It appeared, just looking at the raw images, that the relative throughput was about six times greater at f/11.  See Figure 3, below.

Figure 3:  Spectrae of 4th-mag Giausar, each a 60-second integration.  The net sky-subtracted flux was more than six times greater when NOT using the focal reducer.  All spectra in this article were reduced in IRAF using optimal (variance-weighted) extraction and are sky-subtracted.

Figure 3: Spectrae of 4th-magnitude star Giausar, each a 60-second integration. The net sky-subtracted flux was more than six times greater when NOT using the focal reducer. All spectra in this article were reduced in IRAF using optimal (variance-weighted) extraction and are sky-subtracted.

There were several things that I immediately thought of which were external to the SGS that might have caused some level of flux difference from one image set to the next.  Maybe the focus was bad on the f/7 data – or maybe there was a change in extinction, perhaps the guiding was bad, maybe the seeing had changed, and so on.  I fully expected the f/11 data to show the larger relative throughput, just not by a factor of six!  So very late that evening I ran the experiment for a third time, this time on SAO 64841, an A0 star of about 7th magnitude. Throughout the duration of the observations of this star I was simultaneously running a sequence of 1-minute images of AM CVn, using the observatory’s other C14, which I used as a monitor for seeing and transparency.  I was also very careful to make sure the star was focused as accurately as possible on the slit for each set.  The two resulting spectra are shown in Figure 4.

Figure 4

Figure 4: Note that no attempt was made to wavelength-calibrate these data so the X-axis values are simply the pixel number.

Note that the ratio of fluxes shown in Figures 3 and 4 is slightly different.  I would tend to believe the results shown in Figure 4, which indicate a reduction in throughput of between 75% and 80% when using the focal reducer.  Not only was I much more careful taking the data shown in Figure 4, but each spectrum in Figure 4 is, additionally, based on a mean of five 300-second exposures. Thus guiding errors and short-term seeing differences should be pretty much averaged out over the length of the total exposure times.

5) About Focal Reducers

Since acquiring the SGS last winter I had been just “blindly” using a Meade 0.63x focal reducer in order to increase the effective size of the slit’s projection on the sky.  As mentioned earlier, with a reduction of 0.63x I should be able to get a roughly 50% boost in light that gets past the slit and into the spectrograph.  But just how much reduction was I getting?  That was something I needed to determine.

Focal reducers are positive lens systems; that is, you can use them by themselves as a camera lens.  Thus it’s pretty easy to measure their focal length.  In my case what I did was to focus an image of the sun on a piece of cardboard and just measure the distance between cardboard and lens center with a ruler.  Surely not the most accurate setup for measuring but I really only needed an accuracy good to a quarter inch or so anyway.

The net focal reduction for such a lens is:

$latex ~~(1)~~~~~~~~ Fr = 1.0 – frac{S}{F}&s=3$

where:

Fr = focal reduction ratio
f = focal length of lens
s = distance from center of lens to focal plane

Given that I’d often read that the optimal spacing between camera and reducer was anywhere between 85 and 105mm, I figured the lens would have a focal length of something in the range of 240 to 280mm.  But during my reading I also discovered another interesting fact specifically regarding the Meade 0.63x focal reducer.  It seems that Meade changed the focal length of this lens in about 2006, perhaps to better match their then-new DSI cameras.  Where their older focal reducers had focal lengths of about 280mm, the newer models were about half of that.

I measured the approximate distance from the lens center (approximately where the reducer’s barrel meets the threads on its back side) to the slit to be about 62mm.  I then measured the focal length of the reducer and found that it was, indeed, about half of 280mm, or about 140mm.  Putting those numbers into the equation, above, I found that I was getting about 0.56x!  That was a significant bit more focal reduction than I’d expected, and meant that instead of f/7, the effective focal ratio when using the lens was about f/6.3.  I’d also read that Celestron’s equivalent 0.63x focal reducer was designed for an optimal distance “s” equal to 85mm, which, again using the equation, above,  indicated a focal length of something on the order of 230mm.  So I made a trip to a local telescope retailer and measured one and indeed, found its focal length to be about 235mm.  Given the 62mm distance from lens to slit, the Celestron reducer should work at 0.74x – giving me an effective focal ratio of f/8.3.  That is still too low an f-ratio, but would be better than f/6.3, and would provide a second data point.

Once again, all this calculation and speculation is fine, but truth would only come by actually testing each focal reducer on the telescope with the SGS.  So on the nights of January  27/28, and January 28/29, I re-ran the experiment.

6) The Experiment II

This time the star I used was Epsilon Aurigae.  On January 27/28 I obtained two sets of spectra of Epsilon Aurigae, each set comprised of three 300-second integrations, one set with the Meade 0.63x reducer, and the other set without any reduction.  Images of a Neon calibration lamp were also taken to allow wavelength calibration.  Then, on the night of January 28/29 I obtained two more sets of spectra, essentially identical to those of the previous evening, but this time one set of three 300-second images was taken using the Celestron 0.63x reducer while the second set was again done with no focal reduction.  The results are presented in Figures 5 and 6.

 

Figure 5:  Two spectra of epsilon Aurigia taken on the night of January 27/28.  Each spectrum is a 900-second integration.  The red profile shows total "counts" using no focal reduction, the blue profile shows counts using the Meade 0.63x focal reducer.

Figure 5: Two spectra of epsilon Aurigia taken on the night of January 27/28. Each spectrum is a 900-second integration. The red profile shows total “counts” using no focal reduction (f10.5), the blue profile shows counts using the Meade 0.63x focal reducer (final f-ratio of f/6).

Figure 6

Figure 6: Epsilon Aurigae on January 28/29. The red line is the spectrum obtained without reduction, the blue line is with the Celestron 0.63x reducer. The actual focal ratios are f/10.5 and f/7.9, respectively

 

 7) Determining the Image Scale

I also needed to know the true image scale on the imaging chip, so I used the autoguider to image a field containing several moderately bright stars.  The idea was to use the stars’ known astrometric positions along with their measured x,y position in the autoguider to determine the scale.  Images were taken using each of the focal reducers as well as without any reduction.    While the physical pixel sizes for the imaging and guide chips are known (7.4 microns for the guider, 9.0 microns for the imager) there is a complicating factor in that the SGS uses a small camera lens to focus the guider on the reflective surface of the slit jaws.  Since I didn’t know what that lens might be doing to the image scale for the guide camera I also had to measure the image scale ratio between guider and imager.  To do that I obtained spectrae of a star taken at two different locations along the slit, separated by 100 pixels as viewed in the autoguider.  If the camera lens was exactly 1.0 power then a shift of 100 pixels in the autoguider should result in an 82 pixel shift (100 * (7.4/9.0)) in the imager.  In fact it turns out the camera lens provides a mild bit of image reduction for the guider; the actual shift in the imager was 102 pixels.  Thus the actual image scale in arcseconds per pixel as seen by the autoguider and the main camera is almost the same.  The camera lens in the autoguider reduces the image scale of the guider by about 0.81x.

Figure 7: Images from the SGS guide camera showing the relative scales for each of the focal reducers.  "A" is a chart of the region, "B" shows the view using no focal reduction, "C" was taken using the Celestron 0.63x reducer, and "D" taken using the Meade 0.63x focal reducer.

Figure 7: Images from the SGS guide camera showing the relative scales for each of the focal reducers. “A” is a chart of the region, “B” shows the view using no focal reduction, “C” was taken using the Celestron 0.63x reducer, and “D” taken using the Meade 0.63x focal reducer.

 

Table 1 shows the results of measuring the image scale for the guider and the resulting image scale for the main camera along with effective system focal length and f-ratio for each of the three optical configurations.  Also shown are the calculated f-ratio values based on my measurements of the focal length of each reducer lens.

Configuration

Guider Scale

(arcsec/pix)

Imager Scale

(arcsec/pix)

Effective Focal Length (mm)

Effective Focal ratio

Measured

Fr

Calculated

Fr

Meade 0.63x

0.88

0.86

2159

6.1

0.58

0.56

Celestron 0.63x

0.67

0.66

2813

7.9

0.75

0.76

No reducer

0.51

0.50

3713

10.5

n/a

n/a

Table 1:  Results from measuring the image scale using images taken with the auto guider and then applied to the imaging camera.  The image scale shown here for the telescope itself, without reduction, agrees very well with plate solutions for the camera placed at about the same back-focus as where the SGS sits.

8) Conclusions

It is clear that the SGS should, if at all possible, be used with optical systems no “faster” than about f9.5.  Using the SGS at f/6 results in an approximately 80% reduction in throughput.  Using the SGS at f/8 causes about 45% of the light to be lost.  Note that these measurements are full-system throughput numbers that include the fact that some extra light from the star does enter the spectrograph by effectively increasing the slit’s projected width on the sky via use of a reducer.  So the actual internal losses are significantly greater.  Smaller telescopes with smaller image scales will be even more severely impacted since the percentage of the star’s light passing through the slit will be less affected by focal reduction.

It’s also become clear to me that some way needs to be found to more securely fix the grating turret.  Even the slightest bump can cause it to move which obviously ruins any chance at wavelength calibration.  There is also the fact that if the grating is moving slowly during an exposure it is causing the spectrum to be blurred in the wavelength direction.

Another thing I noticed during these experiments is that the SGS tends to loose focus fairly easily – likely due to temperature differences.  I focused the unit indoors as per the instructions in the user’s guide but when I began using it at the telescope I noticed that the line widths of comparison spectra had increased a lot.  Since there was a roughly 60 degree F difference in temperatures it isn’t too surprising that the focus changed.  Thankfully that can easily be remedied by moving the “D” mirror position while at the telescope.

Finally, lest anyone thinks I am dissatisfied in any way with the performance of the SGS let me say that I find it a very useful tool and believe its design is quite ingenious.  Like any instrument to be employed in any meaningful quantitative endeavor it needs to be carefully and frequently adjusted and calibrated.  Knowing its limitations and capabilities improves one’s ability to make sensible choices regarding possible applications.  Currently I am investigating the overall dimensional stability of the device in the hopes of improving it.  As it is I don’t think it is capable of it’s theoretical 2-Angstrom resolution for long exposures due to movement, either by individual internal optical components caused by changing the unit’s position with respect to earth’s gravity (most likely flexure at the camera/SGS coupling), movement in the grating, or by expansion and contraction of the unit overall due to temperature differences.

 

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