7.1 Introduction

The inclusion of flexure compensation (FC) in DEIMOS is driven mainly by the desire to control flat-fielding errors. We have until now adopted the commonly held view that such errors are the result of wavelength shifts of the detector relative to the spectrum. Under that assumption, our working goal has been stability to 0.1 px, a very stringent criterion. Our most recent thinking has caused us to reconsider whether the wavelength-shift model is accurate and, even if it is, whether our stability criterion of 0.1 px is unnecessarily tight. We also need to consider other factors that may cause flat-field errors that we have so far ignored.

The current chapter reviews our current situation and sets some goals for the future. We outline evidence for and against the wavelength shift theory and suggest further factors that might also play a role in flat-fielding. Then, under the assumption that the wavelength shift theory is correct, we review the pixel stability requirement of 0.1 px and show that it probably can be relaxed slightly. With the new criterion in mind, we define a coordinate system for tilts and tips of the optical elements and derive maximum shifts allowed for the passive support system of each element both with and without flexure compensation. We discuss the theory of beam-steering under optical distortions and show that an FC system is optically feasible. Finally we consider the functional requirements of the system and discuss the hardware needed to implement it.

7.2 Causes of CCD Flat-Field Errors

The prevailing explanation for CCD flat-field errors is that they are due to wavelength shifts of the spectrum on the detector. Fringing in a CCD is an interference phenomenon and thus must be due to the confluence of two factors -- the intrinsic thickness of the CCD at any location and the exact wavelength of light falling on that spot. When we illuminate a CCD in broadband light that is constant over the CCD, we observe a weak fringing pattern (due to the fact that the spectral purity is low). Since the light is everywhere the same, this pattern must reflect the intrinsic variations in detector thickness with position.

The same device illuminated in a spectrograph shows a different pattern. Now wavelength is also varying over the device as well as thickness, so the detailed location of the fringes is different. Also the fringe pattern has a higher amplitude owing to the higher spectral purity of spectroscopic light.

The point is that there are two factors, thickness and &Lambda, in a spectroscopic fringe pattern. One is attached to the detector and moves around with it (thickness), while the other is attached to the spectrum (wavelength). If the fringe pattern were due to thickness alone, then translating the detector would not change the flat-field pattern, and we would not have difficulty flat-fielding CCDs in spectrographs. The spectroscopic flat-field problem, it is argued, must be due to the influence of varying &Lambda. If &Lambda could be kept fixed, i.e., if the spectrum could be kept from moving on the detector, the flat-fielding problem would be solved.

If this picture is correct, it suggests that not all displacements of the detector are equally serious in generating flat-fielding errors. Motions of the detector or other parts of the optical train that preserve wavelength on a pixel should be benign from the standpoint of flat-fielding errors. Motions that change wavelength are dangerous and should be minimized. In general, motions that displace the spectrum along the dispersion are worse than motions perpendicular to the dispersion.

The analysis below of a potential FC system for DEIMOS follows this logic. Basically, the approach attempts to stabilize the spectrograph to a fraction of a pixel along the dispersion but takes a somewhat looser view towards motions perpendicular to the dispersion. The basic conclusion is that such a system is mechanically and optically feasible and could be executed at reasonable effort.

However, a deeper question is: Will such a system solve the problem? Where does the spectroscopic flat-fielding problem for thinned CCDs really come from? Surprisingly, this question does not seem to be very well studied. Some evidence supports the common theory, described above, that displacements of the wavelength of the illuminating light are responsible. For example, on LRIS and many other spectrographs, spectral shifts are of order a few angstroms due to flexure, and there is a general sense that spectrographs with worse flexure have worse flat-fielding problems. Furthermore, on LRIS it can be seen that the fringe-error pattern is just the derivative of the fringe pattern itself. That is what one gets if the fringes were "phase-wrapping", not shifting bodily. Such a phase wrap could be caused over the whole CCD simply by shifting the illuminating wavelength on each pixel by a fixed percentage of &Lambda. To first order, over a limited spectral range that is what a shift of the spectrum due to flexure would do.

However, there is also counter evidence that suggests that wavelength shifts are not the whole story. An experiment with the Kast spectrograph at Lick (whose fringes are illustrated in Figure 1.2) showed that simply switching dome illumination lamps altered fringe phases. This act did not significantly change the wavelength on any pixel, yet the fringes shifted markedly. The same series of experiments revealed a bewildering variety of fringe motions on different parts of the chip that could not be correlated simply with telescope or spectrograph position, suggesting that the phenomenon was more complex than simple flexure. Finally there is the fact that fringe shifts seem to be somewhat larger than flexure alone would predict. For example, in typical LRIS data the phase shift is about 0.2 radian for a wavelength shift of 5 Å (4 px) at 9000 Å. If this shift were caused purely by color, the interference order would have to be (0.2 x 9000/2&pi x 5) ~ 60, which seems rather high.

If wavelength shifts are not the culprit, then what is? An alternative view is that fringe phase depends crucially on how a pixel is illuminated. For example, flexure due to variable gravity or pinching by the CCD package might change the shape of the CCD membrane. However, if this were the cause, fringes should be more stable on CCDs mounted on rigid substrates. LRIS is so mounted yet still has very unstable fringes. Changing CCD temperature might also pinch the package, but fringe shifts are seen on timescales that seem too short to be attributed to temperature changes.

A final idea is that fringe phase depends sensitively on the effective f/ratio of the incoming beam, and perhaps on depth of focus within a pixel as well. The effective f/ratio might vary if pupil illumination or vignetting were varying. Such a model might explain the strong variations in Kast and LRIS fringes because these tests have all been done using rather crude dome flat-field schemes, which could well illuminate the pupil differently at different telescope positions. The same reasoning would naturally explain the big change in Kast fringes simply by switching dome lamps. The magnitude of fringe shifts also seems more plausible with this model. For example, in a 15 µ-thick chip at 1 µ wavelength, the expected interference order is about 30. A fast f/2.0 camera beam has a mean entrance angle of roughly ~10 degrees Changing the mean angle by just 0.3 degrees would shift an LRIS fringe by about 0.2 radian, as observed. Such shifts in mean angle do not seem impossible with poor dome flat-field geometry.

If variable pixel illumination is really the culprit, it suggests that CCD fringe errors are partly an illusion due to poor dome flat-fielding techniques and that to cure them we should put more effort into improving the flat-fielding scheme. It also suggests that pupil integrity is very important and that variable pupil vignetting in spectrographs should be avoided. Variable vignetting is unavoidable at some points in DEIMOS owing to the fact that the pupil is oddly-shaped and rotates as the alt-az telescope tracks, causing variable obscuration. In practice the Keck segments will also not be equally reflective as they will be realuminized in batches, which will also cause variable vignetting. Although some variable vignetting is unavoidable, it should be minimized according to this argument. For example, this might pressure us to use large gratings in all grating slots, not just the 1200-line slot, as we are assuming now.

The foregoing ruminations indicate that we simply do not understand the flat-fielding problem well enough to make sensible design tradeoffs at this point. For example, the money currently budgeted for a flexure compensation system (roughly $50,000 plus software) might be better spent on bigger gratings, or on reducing native CCD fringing, or on a fancy flat-fielding scheme. Therefore, the first order of business after this PDR is to make tests using the Kast spectrograph that should definitively pin down the relative importance of wavelength shifts versus other factors for this detector at least. With those results in hand, we can decide whether the FC system described below should actually be implemented.

7.3 Image Stability Requirements on the Detector

The following discussion assumes the wavelength shift model for flat-field errors and derives the resulting stability requirements. The model takes most of its parameters from LRIS, which has been quantitatively studied. The desired wavelength stability for DEIMOS is set by the minimum tolerable flat-fielding errors, for which our goal is +/- 0.1%. LRIS' flat-field errors are typically of order 1%, which works out to a 0.2 radian phase shift with fringes of 5% amplitude. However, LRIS' fringes are unusually low in amplitude because of excellent AR coatings. If the goal is to reduce flat-fielding errors to +/- 0.1% in the presence of fringes that might be as much as three times stronger than LRIS, a stability of 30 times better, or 0.007 radian, is required. If LRIS wavelength shifts are of order 5 Å, DEIMOS' wavelengths should therefore be stable to 0.17 Å. For a 600-line mm-1 grating (the lowest contemplated dispersion), that means stability in the wavelength direction to +/- 0.25 px. The requirement we have been working to so far is +/- 0.1 px, which we now think is likely to be a little too stringent, though certainly safe.

Define the image stability tolerance, S, to be the maximum allowable image shift in pixels on the detector. In general, S is defined in two directions, Sx and Sy, along and perpendicular to the dispersion. We wish to keep the spectrum tightly locked in the direction parallel to the dispersion, but displacements perpendicular to the dispersion can be larger. In no case should displacements be allowed to degrade image quality.

With these ground rules, there are two types of stability required, each with its own timescale:

1) Stability required to maintain constancy of wavelength on a pixel. As noted, the criteria are different along the dispersion and the slit.

a. Along the dispersion: The stringent spec Sx = +/- 0.25 px applies. The need is to keep the flat-field calibration and the object spectrum aligned, so this tolerance must be met over a night at least, and longer if flat-field calibrations are to be longlived.

b. Along the slit: Displacements that are exactly parallel to the slit (i.e., following the slit curvature) have no effect on the wavelength of the illuminating light, and technically therefore no specification is needed to limit the effect of such displacements. However, the actual stability specs are stated in terms of rectilinear coordinates along and perpendicular to the dispersion. A purely vertical (perpendicular) motion of the detector follows the slit closely near the middle but not at the top and bottom, due to slit curvature. Let t = dy/dx describe the local tilt of the slit. The specification on the allowed vertical displacement of the spectrum is then Sy = t x 0.25 px, where t is the minimum value obtained over the image.

2) Stability required to maintain image quality. Again there are technically two requirements, one along the slit, which affects angular resolution on the sky, and one perpendicular to the slit, which affects wavelength resolution. In practice they are comparable, and we shall set them equal to a 1 px (0.119") total smear during an exposure, or +/- 0.5 px (+/- 0.060") relative to the center of an exposure. The image quality specification is easier than the wavelength-constancy specification on two counts: it is 2 times looser, and it lasts over only a single exposure.

Note that DEIMOS' position angle will in general vary by 360 degrees during a night, and, because Keck is an alt-az telescope, by 180 degrees during a single exposure. The image stability specifications must therefore be met through a 180 degrees rotation, while the wavelength stability specifications must be met through all position angles.

7.4 Dynamic Range and Coordinate System

It is our intention that the FC system should be capable of steering out translational motions of the image on the detector but will not be capable of steering out either rotations or distortions. Rotations must be controlled passively, and distortions cannot be dealt with at all. Distortions come from two sources. One is the native distortions of the optical system, but it can be shown that these are negligible. A more important source is that, in the presence of a grating tilting certain optical elements does not merely translate the image but also distorts it. Such distortions limit the amount by which a given element can be tipped for purposes of beam steering during spectrocopy. They also limit the amount by which that element can be allowed to sag passively, as sags over this level will produce distortions that cannot be corrected by beam steering. The first goal in analyzing a proposed flexure correction system is therefore to analyze the expected optical distortions to verify that they will be within a tolerable value.

The above introduction leads naturally to the concept of "dynamic range" for the motion of an optical element. This quantity is defined as:

Rx(&Theta) = T(&Theta) / Dx(&Theta),

where T is the average translational motion induced by a tilt &Theta of that element over the image, and Dx is the maximum deviation (in x) from simple translation over the whole image. An analogous quantity Ry exists for the y coordinate. (Note that each element has four values of dynamic range, two each for x and y on the detector, times two again for tip and tilt). If distortion to second order is considered, then T and D both increase linearly with &Theta, and R's are therefore constant.

Let us now consider the tolerances for the passive support of elements in tip and tilt. These are all expressed in angular units. There are two values, the tolerances needed to meet the image stability specifications (Sx, Sy above) entirely passively (i.e., with no flexure compensation), and those with flexure compensation in operation. In general, the second set of values is larger than the first by the relevant factor R. In words, flexure compensation allows one to loosen the passive structural tolerances by a factor equal to the dynamic range. R is also important in selecting which element(s) to tip and tilt for beam steering. In general, R should be large for a beam-steering element, as otherwise objectionable distortions will be introduced as a result of beamsteering translations.

In practice these simple statements need modification by considering in detail how the distortion patterns of the various elements interact, both with one another and with the requirements. These issues are discussed below.

The following analysis adopts a simplified coordinate system for the tips and tilts of elements. Let the middle of the slit, the middle of the grating, and the middle of the first element of the camera define the plane of the perfectly aligned spectrograph. Tilts of all elements in this plane tend to move the spectrum along the dispersion, so we will call such tilts &Theta. Tilts perpendicular to this plane are &Thetay. With this notation, the list of relevant tips and tilts is as follows:

1) Group the tent mirror and collimator mirror into a common system and denote the orientation of the parallel beam that they produce by the angles &Thetacoll,x and &Thetacoll,y. In Epps' notation, &Thetacoll,x is called the azimuth of the collimator, and &Thetacoll,y is the altitude of the collimator. Varying &Thetacoll,x is optically like varying the tilt of the grating, while varying &Thetacoll,y moves objects along the slit. Since the camera and collimator are simply two mirrors that together produce a parallel beam, &Thetacoll for the joint system will be an appropriate sum over the two elements. There is no need to decompose them here.

2) Group the camera and detector together as a rigid body (we will build them this way), and denote the tip and tilt of the camera axis by &Thetacam,x and &Thetacam,y. Because distortions in the camera are low, varying &Thetacam,x and &Thetacam,y simply translates the image on the detector (i.e., Rx and Ry for the camera are infinite).

3) The final unit is the grating. &Thetag,x is grating tilt, which is optically degenerate with &Thetacoll,x. &Thetag,y is grating pitch. In addition to these two, we also have &Thetag,roll. This is the deviation of the grating grooves from perpendicularity to the spectrograph plane. This produces both a displacement and a tilt of the spectrum on the detector. The displacement can be taken out by flexure compensation, but the tilt cannot, so a tolerance for it is needed.

To summarize, the final list of tips and tilts is:

Collimated Beam: &Thetacoll,x and &Thetacoll,y

Camera/detector: &Thetacam,x and &Thetacam,y

Grating: &Thetag,coll,x, &Thetag,coll,y, and &Thetag,roll.

7.5 Optical Feasibility and Options to Steer the Beam

The optical effects of tilting and tipping in these quantities are now considered. The results are shown in Figure 7.1. Each panel is a schematic map of the long-slit spectrum produced by the 1200-line mm-1 grating. The central wavelength is 8078 Å, and the total spectrum spans 2600 Å. The 1200-line mm-1 grating was chosen because the distortion effects are worst with it, and the 8078 Å wavelength is probably going to be the most heavily used tilt for this grating (see Table 1.5). Distortions generally worsen with longer wavelength, so these calculations should be repeated at the reddest grating setting, however.

Each picture shows these three wavelengths -- central, maximum, and minimum -- with slit curvature schematically superimposed. At each wavelength, three positions along the slit have been analyzed: middle, top, and bottom. Actually, top and bottom refer to the full 11.4 degree radius of the camera, which overhangs the true slit slightly (see Figure 1.6). The actual slit length is only 0.82 times the radius of the camera. Since all distortions increase linearly with distance from the center of the image, the present calculations overestimate the distortions in the slit direction by 1/0.82, or 21%.

At each analyzed location, small vectors show the motion of the image for a particular optical tip or tilt. Vectors are labelled with their lengths in pixels. The quantities shown are differences between the "base position" (all optical elements perfectly aligned) and the tipped or tilted position. The size of the tips and tilts is 50" in all cases.

There are seven possible motions in the preceding table. Of these, the two camera motions can be ignored because they produce simple translations with no distortions. Likewise we can lump together &Thetacoll,x and &Thetag,x since their effects are optically identical. That leaves four distinct motions, which are shown in the four panels of the figure. These are considered in turn.

7.5.1 X Coordinate

The x coordinate is in the plane of the spectrograph and includes the grating rotation (&Thetag,x), collimator azimuth (&Thetacoll,x), and camera (&Thetacam,x). The first two are optically identical and are shown in Figure 7.1d (which happens to show &Thetacoll,x, but &Thetag,x would be the same). The passive support requirements without flexure compensation are extremely tight in this mode. To keep the spectrum fixed to +/- 0.25 px in x requires holding the grating and/or collimator fixed to +/- 3.4", according to Figure 7.1d.

If this flexure were corrected, say, by steering the tent mirror (collimator azimuth), distortions would be introduced as shown in Figure 7.1d. Such a tilt would yield a horizontal translation in the direction of dispersion, plus a slight stretch. The dynamic range of this motion, Rx, is about 20. That is good, as it means that, with flexure compensation, the passive support spec of +/- 0.25 px in this direction can be relaxed to +/- 5 px, i.e., from +/- 3.4" to +/- 68".

This statement needs some elaboration. There are four total elements that affect the x coordinate position: the collimator, tent mirror, grating, and camera. All of them may flex, but only two of them can potentially be used to steer. The collimator cannot steer because it is common to both beams. The grating could in principle steer, but we would then have to place motions on both grating locations and on the flat mirror, which we wish to avoid. That leaves the tent mirror, already considered, and also the camera/detector unit as possible x-actuator sites.

There is a difference between them in that the camera/detector system yields a pure translation of the image, whereas the tent mirror yields the distortions shown in Figure 7.1d. If the grating, collimator, and tent mirror are suspected to be the main flexure sources in x, which is probably the case, then there is an advantage in using the tent mirror to correct, as its distortions exactly cancel theirs (the elements are optically degenerate). There is then in principle no limit on how large the flexure in this system can be, provided it remains within the range of the tent mirror actuators.

Under that strategy, we must then calculate an allowable &Thetacam,x for the camera/detector unit. If up to +/- 5 px of motion in the camera is correctable (limited by the dynamic range of 20), we find that a &Thetacam,x of +/- 41" is allowed. This should be feasible based on the structural stiffness estimates in Chapter 5.

The conclusion is that the critical x coordinate, which must be held constant to within 0.25 px, can feasibly be corrected by tilting the tent mirror, or possibly by tilting the camera/detector unit. The tent mirror would appear to offer significant optical advantages.

7.5.2 Y Coordinate

The y coordinate is perpendicular to the plane of the spectrograph and includes grating pitch (&Thetag,y), collimator altitude (&Thetacoll,y), and camera pitch (&Thetacam,y). The first two are not optically identical, as shown in panels a and c of Figure 7.1. Both produce a vertical translation of the image and both also tilt the spectral lines, but the degree of tilt is different in the two cases.

We start with the simpler case of grating pitch. If no flexure compensation is assumed, the limiting motion will be the +/- 0.5 px image tolerance in y. Scaling to the motion in Figure 7.1 yields a passive tolerance for &Thetag,y without FC of +/- 2.6" during an exposure. This tolerance seems rather hard. If FC is used, the allowable range of motion is now limited by the dynamic range in the x coordinate (the dynamic range in y is very large, see Figure 7.1.a). The net result is now a total correctable range of +/- 18" in &Thetag,y. This is now a longtimescale requirement, but the task is clearly much easier with FC.

The case of collimator altitude is more complicated, although the basic effect is simple. Changing &Thetacoll,y is equivalent to moving along the slit, so the vectors plotted in Figure 7.1c are parallel to the slit curvature. At first look, the distortion terms produced by &Thetacoll,y seem serious -- the formal dynamic range is only 4.1. However, since the motions are parallel to the slit, they do not change wavelength on a pixel. Hence, according to our fringing model they are completely benign, and the looser specification of 0.5 px for image quality in the y direction is the appropriate limit (we do not want the images to smear too much along the slit). This actually produces a rather tight spec. With no flexure compensation, the total allowed motion for &Thetacoll,y is only +/- 4" (during an exposure). This tolerance applies to the combined tent mirror/collimator system.

If this tolerance cannot be met passively, it will have to be corrected by steering either the camera or by steering the tent mirror itself. The tent mirror can obviously correct itself or the collimator perfectly, but using it to correct other elements (such as the grating pitch or the camera) is not advisable as its dynamic range is poor, only 4.1. and it introduces distortions not showed by these other elements. However, the camera can be used profitably to correct &Thetacoll,y, even though the design of improvement is limited. From Figure 7.1.c, the total correctable motion in &Thetacoll,y using the camera is found to be 8", limited by the motion of +/- 0.25 px in x. The tolerance on &Thetacoll,y with FC thus becomes two times looser than the passive tolerance. This is not a huge gain, but it is valuable.

We are assuming that a passive support tolerance of 8" on &Thetacoll,y can be met and that steering the system in y using only the camera/detector system will satisfactorily correct both &Thetacoll,y and &Thetag,y. It should therefore not be necessary to mount a separate y actuator on the tent mirror. We further note that for direct imaging, the grating is replaced by a flat mirror, and the geometry of the tent mirror/collimator correction is once again that of pure translation. In general, any two-axis correction scheme will work perfectly for direct imaging regardless of what elements are steered.

7.5.3 Grating Roll

Distortions due to grating roll are shown in Figure 7.1b. There is a bulk motion downwards of 6 px, plus a roll of the spectrum of almost exactly 50", the inserted value. With no flexure correction, the limit of motion is set by the +/- 0.5 px image smear in y. The net result is a limit of +/- 4" on &Thetag,roll during an exposure, which appears difficult. If FC is used, the dynamic range factors in both x and y are comparable and increase this limit to +/- 24". Thus the FC system is useful for mitigating the effects of grating roll, which was not obvious to us a priori.

7.6 Summary of Options and Strategy

The above discussion of optical distortions has led to the following conclusions:

1) To meet the stringent requirement of 0.25 px in x will probably require some form of active control in this coordinate. Without FC the tolerance on grating tilt and collimator azimuth is 3.4", which is quite hard. Fortunately the system has high dynamic range in x. If the major sources of flexure are the collimator/tent mirror/grating system, there is a benefit to putting the x correction on the tent mirror. With that done, the maximum correctable flexure in the camera/detector axis is 41", which is feasible. This tolerance must be maintained over long timescales.

2) In x, tilting the camera/detector unit can easily correct grating pitch. The total correctable range for pitch is 18", which must be met over long timescales. The tolerance without FC would be 2.6", which is quite hard. A y-actuator on the camera/ detector is therefore probably required.

3) The same camera/detector y-motion can probably correct the tent mirror/collimator system, but the tolerance on their support remains stringent owing to the rather poor dynamic range. Without FC, the allowable motion for &Thetacoll,y is 4", set by image motion along the slit. With FC, this increases to 8", set by wavelength stability in x. This is the most stringent tilt tolerance in the spectrograph. If it cannot be attained, a separate y actuator on the tent mirror would be needed. We are hoping to avoid this.

Note that this last tolerance is expressed in terms of &Thetacoll,x. As this quantity is a function of both the collimator and the tent mirror, it needs to be broken down into appropriate limits for the two elements separately.

4) Without FC, the tolerance on grating roll is 4", which would be difficult. With FC, this increases to 24".

5) Any two-coordinate actuator system will correct fully for direct imaging.

6) Strawman actuator plan:

In x: One, on the tent mirror (camera/detector is a second choice).

In y: One, on the camera/detector. Second one on tent mirror hopefully not needed.

7.7 Overview of System Components

Conceptually the FC system is a closed-loop system that comprises a light source at the focal plane that is rigidly coupled to the mask assembly, a detector that is rigidly linked to the science CCD, algorithms that evaluate any drifts in image location as a function of time, and optical element(s) that can be moved with actuators to compensate those drifts.

The proposed plan for the detector loop utilizes two extra FC CCDs just off the main mosaic at the ends of the slit (see Figure 1.6). These are 600 x 1200 CCDs but would be used in frame-transfer mode with active areas of 300 x 1200. The basic idea is to feed these CCDs with light introduced by two optical fibers at the edges of the focal plane just off the slit. The light through the fibers would come from an arc-lamp, or perhaps an etalon. In direct imaging mode, the fibers would make small dots of light on the FC CCDs. In spectroscopic mode, they would produce a line of small dots along the dispersion direction. The FC CCDs would be located so as to intercept one or more of these dots regardless of grating choice and tilt. During a long exposure with the main detectors, the FC CCDs would be read out at intervals and their image centroids calculated. Any deviations from the start of the exposure would be corrected by the FC actuators. In addition to providing guiding during an exposure, the FC system would also provide an absolute reference from the telescope focal plane to the detector. It would therefore be easy to set up the detector absolutely with respect to the focal plane in a repeatable way, which might be advantageous for calibration and astrometric purposes.

Information from the FC CCDs is limited to the mean x, y centroid of the image plus information about slit rotation near the center of the image. The default plan is simply to use the x, y centroid to correct translation errors. The rotation datum is affected by grating and detector roll, collimator altitude, and, to a lesser extent, by grating pitch. Figure 7.1 shows that the most important of these is collimator altitude. Hence, rotation would be useful in controlling a second y-actuator on the tent mirror if needed.

The system components include:

The final location, number, and nature of the actuators are still TBD.

7.8 Functional Requirements

We have shown that an FC system is optically feasible and have outlined the hardware components of such a system. The current functional specifications are as follows:

1) Range of actuator motion: still TBD but roughly 50" in both coordinates, amounting to a few pixels of motion on the CCD.

2) Accuracy of correction: should be high enough that errors do not contribute significantly to image error budget or flatielding errors. Goal in x is +/- 0.1 px, in y is +/- 0.2 px (.02"). For a typical element, this means holding a tip/tilt accuracy of about +/- 1".5.

3) Fiber diameter: ~0.10 mm = 0.14".

4) Stability of fiber mount in focal plane relative to masks and to each other: +/- 5 µ = +/- 0.0002-in = +/- 0.007" = +/- 0.06 px.

5) Image width of fiber on FC CCD: 2.3 px (includes aberrations).

6) Required number of detected photons per exposure for 0.1 px accuracy: 530 photons.

7) Correction rate: set by maximum position-angle drive rate coupled with expected passive flexures. Assume 5 px total passive flexure for a rotation through 180 degrees and a maximum drive rate of 180 degrees in 10 minutes (passing near zenith). Then accuracy of 0.25 px means correcting once every 30 sec.

8) Readout time for FC CCDs: CCDs operate in frame transfer mode to facilitate rapid readout. Active area contains 360,000 px. At a readout speed of 30 microseconds per px, the CCD can be readout in 11 sec with one amplifier, which is adequate.

9) CCDs: Front-illuminated CCD detectors. Thinning not required. Focus location in mosaic (z-direction) to 25 µ. Tolerances in x and y are loose. Stably mounted with respect to science detector to 0.07 px (1 µ).

10) Light source: should be broadband (for direct imaging through color filters). Brightness modulation is desirable. Spectral lines should be densely spaced. For both reasons, a broadband continuum source through an etalon would seem a good choice.

11) A function monitor should provide a history file and error alarms if unusual behavior is detected. Various quantities should be displayed and graphed, including the current x, y centroids, the moves, total sum of moves to date during an exposure, and the FWHM of the images.

12) The observer should be able to capture an image from the FC CCDs and analyze it as a science image.

7.9 Refinements and Future Plans

The above specifications reflect simple estimates that show that the basic concept of an FC system is feasible. However, in practice we would want to implement several refinements. A major concern is that the FC fibers will flood the spectrograph with too much scattered light. The ends of the fibers will be baffled to minimize this, but further reduction could be achieved by noting that the correction signals will be smoothly varying functions of time, except perhaps near the zenith. Hence it would be desirable to modulate the brightness of the light source as a function of the correction rate. When rates are low, the light source could be quite dim, and a predictor-corrector scheme could be developed in software that would effectively average information from the fibers over long timescales.

A further refinement would utilize the fact that the FC system does not need to read out when the main CCDs are reading out, and vice versa. It is therefore possible to save components by using the main CCD signal chain to read out the FC CCDs during an exposure. This will save money.

The notion of a simple predictor-corrector scheme can be carried a step further by considering a truly intelligent system that first "notices" what the local flexure rate is and then tailors its commands to match. This is again feasible because the FC system can be working whenever the main CCDs are not erasing or reading out. Hence the FC system can always be monitoring its own behavior and tuning its commands to suit, before an actual exposure starts.

7.10 Summary

After this PDR, the first need is to undertake measurements of the Kast spectrograph to ascertain the sensitivity of flat-fielding errors to wavelength shifts. These results will be used to evaluate the need for FC in the x-coordinate and to refine the +/- 0.25 px spec for it if needed. It appears that a y-coordinate actuator will be needed in any case, to prevent image blur. Finally the analysis in Section 5.25 of the slitmask error budget showed that there are very strong reasons to have the 4 fiber-optic feeds to provide a fiducial ruler in the focal plane, even if they are not used to control an active FC system. Therefore it looks highly probable that the fiber-optics and FC CCDs will be provided and more than likely that actuators of some sort will be built as well.