SDN 0006.02

Summary of Preliminary Flexure Analysis

1. Introduction

This design note summarizes the results of the calculations of the structural deflections and their opto-mechanical effects. The details of these calculations will (might) be presented elsewhere, nominally in SDN 006.01.

The present version of the note is based on results from analysis of the preliminary structural design, completed in early October. Modifications made to the design since then are not included (but will be added when re-analysis is complete). As discussed below, these changes will not qualitatively impact the results.

2. Overview

2.1 Requirements

The analysis was intended to determine whether GNIRS meets two flexure requirements:

·        The wavefront sensor must keep the object centered on the slit. The most critical aspect of this is the centering perpendicular to the slit, since a significant decenter will cause loss of light. For a narrow (0.1 arcsec) slit, the decenter required to cause a 5% light loss is roughly 10 mm for excellent seeing, and significantly more for more typical site seeing (see SDN 003.17). Decenter along the slit matters for the integral field units, since the way the image is sliced up depends on the centering. For an IFU with 0.12 arcsec samples the decenter required to produce about 5% cross-talk is about 12 mm for excellent seeing. For conventional slits, the decenter along the slit propagates to the detector (see below).

·        The motion of the slit must keep the spectrum from moving around on the detector. In the dispersion direction, the shift is specified as not more than 0.1 pixel in 1 hour; along the slit it is less clearly specified, but we would like similar performance. In the dispersion direction, for almost all cases the slit defines what is imaged on the detector since it is usually narrower than the object image. Along the slit it is the object itself that defines the spectrum position.

2.2 Calculations

The calculations consist of three main parts.

The first of this is a NASTRAN analysis of the GNIRS structure. The analysis presented here is based on a detailed preliminary design; while modifications have been made in the course of completing the design, they have not been radical changes. The mechanisms and optics have been modeled as rigid bodies.

The second part of a complete analysis would consist of inclusion of flexure effects for the mechanisms. This has not been done explicitly in the present analysis. Our measurements of the prototype mechanisms indicate, consistent with NIRI lab and telescope tests, that elastic deformations of the mechanisms will be very small – the designs are compact and well-mounted. The predominant form of flexure will be non-linear effects – in other words, things that go “clunk”. These effects must be minimized by design and during assembly and test of the mechanisms; the requirements have been discussed elsewhere (SDN002.XX sequence). They will obviously add to the effects discussed here; as these mechanism shifts are non-linear and not particularly repeatable, neither effect will compensate efficiently for the other.

The third part of the analysis consists of determining the effect on the image locations of the calculated deflections. We have taken two approaches to this. For the first approach, we have used an opto-mechanical analysis program written by Earl Pearson, while for the second Ming Liang has used a commercial program, TracePro. In both cases the optical elements and their deflections are modeled in absolute coordinates. This approach differs from the local coordinates used by programs such as Zemax, which are more convenient for optical design, but which complicate applying deflections specified in absolute coordinates.

The results below are based on the deflections from the NASTRAN model applied to Ming’s calculations. Earl’s calculations give similar results, usually differing by no more than a micron.

For both programs, we ran tests using a rigid-body rotation of the instrument to confirm that there was no net shift of the image of the slit on the detector.

3 Preliminary Results

In the discussion that follows, coordinates are defined as follows:

·        The z-axis is along the initial optical axis, pointing into the instrument at the entrance window.

·        The y-axis is vertical in the standard instrument layout. Positive y is up.

·        The x-axis is horizontal. The positive direction points toward the side opposite the camera (i.e., the detector coordinates are negative in x).

It is also necessary to understand how the critical locations are oriented.

·        At the pick-off mirror (or the entrance to the WFS field lens), the direction along the slit is in x, and the direction perpendicular to the slit is y.

·        At the slit, the long axis runs in y, and the x axis is now perpendicular. Note the 90-degree rotation.

·        The science detector is almost but not quite in the yz plane; it is rotated about 4 degrees about the y-axis. For the purposes of measuring shifts displacements in z are assumed to be close enough to shifts in the local plane. The y axis corresponds to points along the slit, while the z axis corresponds to points along the spectrum.

·        The WFS detector is rotated about the y axis by 50 degrees, so it is not close to either the xy or zy planes. For the WFS displacements, the shifts in absolute coordinates were transformed to the local coordinates, defined as x’ and y’ (note y’ = y). The WFS x’ axis is corresponds to correction along the slit (slit y axis) and its y’ axis corresponds to correction perpendicular to the slit (slit x axis)

For each location, the analysis determined how much the physical element moved (displacement in xyz) and then how much the axial ray moved; the quantity of interest is the difference between the two.

The coordinate system described above and its relation to “up” and “down” refer to the standard orientation we have been using for the bench layout. The instrument can be mounted at both a “side-looking” and “up-looking” position on the instrument rotator, and may then be oriented over a large range of positions. GNIRS can therefore have almost any possible orientation on the telescope (the exception is pointing straight down, or nearly so).

3.1 WFS to Slit

The shift of the axial ray relative to the WFS detector and to the spectrograph slit are given below. Only shifts in the plane of the detector and slit are given, as focus shifts at both locations were well under a micron. All shifts are in microns. The sense is that a shift with a positive sign means that the star image moves in a positive sense in the coordinate system of the slit/detector.

Table 1 – WFS and Slit Shifts

If the WFS is being used to track a guide star, it will cause the telescope to move to keep the star centered on its detector – in other words, it will remove the deflections tabulated in Table 1, and this “correction” will propagate to the slit.

The relative magnifications at the two focal planes is a factor of 5.7 (WFS has more demagnification). The shift of the axial ray at the slit produced by the WFS recentering is tabulated below, together with the net displacement of the ray (object) on the slit.

It is then possible to calculate the worst-case 1-hour shift for a given axis (x or y in this case) using the procedure described in the Appendix.

Table 2 – WFS Correction and Net Slit Shifts

The largest changes in deflection during 1 hour for Dx and Dy are 4.7 mm and 2.7 mm respectively. Both of these are well under the 10 mm limit discussed above for light losses or cross-talk. Even though the orientations required to produce the two worst cases are somewhat different, we can simply add them (rms) to get a limit on the total worst case displacement of 5.4 mm.

3.2 Slit to Science Detector

Similar calculations for the science detector give the results listed in Table 3 (the values at the slit are taken from Table 1). The values for the slightly rotated detector axis are considered to be the same as along the z axis; differences due to the 4-degree rotation will be less than 0.1 micron. Focus shifts appear to be 1 micron or less, and therefore are neglected. All values are for the long cameras. The short cameras are less sensitive; the deflections will be roughly 3 times smaller.

Table 3 – Science Detector and Slit Shifts

If we consider that the slit is what is re-imaged on the detector, then the values in the first two columns should be used to correct the shifts at the detector. This assumption is certainly reasonable for the dispersion direction, where the long-camera slit is almost always significantly narrower than a stellar image. Along the slit, this assumption is less reasonable, but for Table 4, below, we assume that the slit defines the image in both axes. (This would be valid if the slit were a pinhole.)

The correction at the detector needs to allow for demagnification onto the detector (factor of 0.87) and coordinate transformations.

Table 4 – Slit Shifts at Detector and Net Shift

Calculations similar to those above (see Appendix A) give a worst case shift in 1 hour of 5.9 mm in y and 7.2 mm in z. These occur for change-in-gravity vectors that are almost orthogonal; a close estimate of the worst combined case is probably obtained by applying the worst-case vector for z to the y deflection, which gives a total deflection of 7.6 mm. The specification of 0.1 pixel shift is 2.7 mm, so these values are a factor of not quite 3 higher than desired.

In the y axis, one can argue that what matters is not the slit position, since the slit edges are parallel, but rather the star position, as determined by the flexure plus any shifts due to the wavefront sensor. One should use the corrections from column 3 of Table 2 (WFS y-correction), rather than those from Table 3, to calculate a “star shift” at the detector, as given below in Table 5. This shift is presented for the y-axis only (that is, along the slit).

Table 5: Star Shift at Detector

These values are somewhat less than those determined using Table 4 (worst case 4.0 mm), but the critical shift (and larger value) is in z.

4. Discussion

4.1 WFS to Slit

The data in Table 2 show that the structure is rigid enough to allow the WFS to keep the star centered on the slit or IFU. The results do not rely on two large shifts canceling each other, so it is reasonable to suppose that things will not change significantly when the final design is analyzed.

The GNIRS mechanisms that might affect this specification will not in fact have much influence. Only the slit slide could have a significant effect, and in order to do so, its flexure would have to be ~20 mm under 1 g. Based on our tests, this should not happen.

The OIWFS mechanisms could in principle have a greater influence; the NIRI commissioning tests suggest that this is probably not the case.

4.1 Slit/Star to Science Detector

The data in Tables 4 and 5 show that the structure is not rigid enough to meet the flexure specification “as is”. Examination of the data in the tables suggests a solution, however: making provision for a correction using gravity deflection of the collimator. Specifically, with gravity loading in the –y direction (down) the slit image moves down 22 mm on the detector and the star image moves down 15 mm. Assuming the latter is what we want to correct, a slight change in tilt about the x-axis (about 1.2 arcsec) needs to be produced by gravity loading of the collimator mount. If we assume the correction is good to 5 mm, this will reduce the worst-case 1-hour y-deflection to about 1.8 mm, which is satisfactory.

In the z-direction, there is deflection for gravity in both y and z. Correction by the collimator requires a change in tilt about the y axis. It is likely to be difficult to do this for both gravity directions. If one assumes that only the z-gravity effect can be corrected (again, to about 5 mm: change in tilt about 2 arcsec), the worst case 1-hour shift is about 3.7 mm. This is somewhat worse (40%) than the specification, but close enough that attempting to introduce active compensation seems unjustified. Analysis of the final design may show some improvement, and design of the collimator flexure might allow partial compensation.

Appendix A: Calculation of Deflection Changes

In general we can think of the deflections discussed above as a vector D, due to a gravity vector G, where the relation between them is

D = M * G

M  is a 3 x 3 matrix whose components correspond to the entries in the tables (plus values near zero that specify the terms that determine D_focus).

The change in deflection due to a change in gravity is then given by

D’ = M * G’

where G’ is the vector change in gravity (difference between the two vectors over an hour).

With the instrument rotator in operation, the instrument’s orientation is identical to what it would be if Gemini were an equatorial telescope (unlike the situation at Nasmyth). Hence any 1-hour observation consists of a rotation of the gravity vector through 15 degrees. The magnitude of this vector must be 0.26, and almost any orientation is possible (the calculations were done for 1 g, so the units of g drop out).

The rows of M, which correspond to the columns of the tables above, can each be thought of as vectors, e.g, M_x, so that

D_i’ = M_i * G’

This is just a scalar product of vectors, so

D_i’ = |M_i| |G’| cos(q)

and the largest value for Di is just the product of the magnitudes of the 2 vectors, or 0.26|M_i| . Note that the largest deflection in one axis doesn’t necessarily occur at the same orientation as the largest deflection in the orthogonal axis.

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