1 Spectrograph Optical Design
1.1 Overview
NIFS is a near-infrared integral-field spectrograph. It comprises the science instrument (the spectrograph), and an On-Instrument Wave Front Sensor (OIWFS) as needed to stabilize the image. Because the OIWFS is a duplicate of that used in NIRI, it is not described here.
In basic functional terms, the spectrograph comprises:
· A focal plane unit incorporating the field mask, focal converter, cold stop, and order blocking filter. This delivers a 3″ square field image at a scale suitable for image slicing.
· An integral field unit (IFU) incorporating a 29-channel image slicer and image stacker. This reformats the 3″ square field into a “staircase” slit having a width of 0.1″.
· A spectrograph which forms a dispersed image of the slit on a 2048×2048 pixel detector.
Near-infrared integral-field spectrographs have been developed only recently. The first such instrument was 3D (Weitzel et al. 1996). This used a 16-element reflective IFU consisting of a stack of tilted, plane image slicer mirrors at focus and a hyperbolic array of plane mirrors to steer beams into the spectrograph from a virtual pupil that was coincident with the telescope exit pupil. This approach works well for small fields and small detector arrays. However, the ray footprints on the beam steering mirrors rapidly overlap when this design is scaled to the longer virtual slits required to feed the full fields of 2048×2048 pixel detectors. The solution is to use fore-optics or power on the image slicer mirrors to form an array of pupil images on the beam steering mirrors (Content 1997). This eliminates beam overlap, but requires a second array of field mirrors to reform a single grating pupil. This is the approach that is taken in the NIFS optical design.
1.2 Changes Since CoDR
A range of design options was presented at CoDR. These have been resolved in accordance with the recommendations of the review panel, as follows:
· An Offner optical relay system has not been used to form the cold stop pupil image. Rather, the incidental pupil image formed by the focal converter is used for this purpose.
· The concentric IFU configuration has been adopted in preference to the alternative linear IFU configuration.
· A resolving power of ~5300 has been adopted in preference to the alternative of ~4000.
· Focus control is not provided for the spectrograph. It has also been deleted from the OIWFS by modifying the existing design.
· Diamond machining has been adopted as the manufacturing method for all mirrors used in the system.
· To allow for the possible future use of a detector with extended wavelength sensitive, thermal radiation blocking has been provided at the detector chamber for wavelengths longer than ~ 4 μm.
The completed design is similar to the option described at CoDR for these conditions, but with considerable refinement. Significant changes have been made to the detailed IFU geometry in order to make diamond machining easier and improve the associated optical performance.
Because viability of the optical design is critically dependent on the success of the diamond machining method developed for it, manufacturing tests have already been commenced.
1.3 Layout
The optical layout is shown in Figure 10 to Figure 13. To better explain the principles, the fold mirrors needed to fit the optics in the duplicated NIRI cryostat are omitted from Figure 10 and Figure 11.
In all views of the optical layout, there is a discontinuity in the ray bundle at the image slicer. Up to the image slicer, the marginal rays are those for the ~ f/16 Cassegrain input beam provided by the telescope. Beyond the image slicer, the instrument is designed to capture all the radiation from within an enlarged rectangular aperture at the pupil images, so accounting for some of the diffractive spread caused by the narrow slitlets of the image slicer (§4.4.2). The ray bundle shown after the image slicer is for this rectangular aperture. Its width (spatial direction) matches the diameter of the round geometrical pupil, but its length (spectral direction) is enlarged relative to this. As described in the diffraction analysis (§4.5), the enlargement factor, K, is taken to be 1.6. In the spatial projection, the beam corresponds to the original ~ f/16 telescope input, but in the spectral projection it corresponds to an ~ f/10 input.
The full prescription for the optics is listed in Appendix C (§12.12).
1.3.1 Unfolded
Figure 10 shows a trimetric view of the unfolded system being traversed by the ray bundle for a star in one corner of the ~ 3″ square field. The ~ f/16 focus of the telescope is located in the top-left of the figure, where the light emerges from a ~ 2 mm square field mask. It is then reflected by a concave focal converter mirror to form an ~ f/256 beam. The focal converter mirror also forms a ~ 4 mm diameter pupil image adjacent to the field mask, where a flat mirror is placed to function as a cold stop. This cold stop mirror reflects the beam through an order blocking filter near the focal converter mirror to the image slicer shown at the right of the figure, where a field image is formed which is ~ 30 mm square.
The image slicer is a spherically concaved mirror made up of 29 horizontal slices, each ~ 1 mm thick. These slices constitute the spectrograph slit. They are slightly fanned about a vertical axis, with each reflecting the beam to a different mirror element of the pupil mirror array shown near the center of the Figure 10. The curvature of the image slicer is arranged to form pupil images on this array.
Each mirror element of the pupil mirror array is concave, and reflects the beam to the nearby field mirror array where an image of the corresponding field slice is formed at ~ f/16. Along the field mirror array, these slice images are stacked corner-to-corner, so forming a “staircase” slit image.
The beam is reflected from the mirror element of the field mirror array towards a large spherical collimator at left of figure. Each mirror element is concave so that the emerging beam appears to be coming from a virtual pupil located on the vertical fanning axis of the image slicer.
The beam is reflected by the collimator mirror, passes through the collimator corrector lens, and on to the diffraction grating located under the image slicer, where it is centered on the fanning axis of the slicer. The three optical surfaces of the collimator mirror and corrector lens are concentric and their common center is also located on the fanning axis of the image slicer. This causes an image of the pupil to be formed on the grating.
The dispersed beam coming from the grating finally passes though the 5-element refractive camera which images the spectrum of the entire “staircase” slit onto the detector array.
Although not shown in any of these figures, the spectrograph is also equipped with a deployable mirror that allows the diffraction grating to be bypassed for direct imaging purposes. This so-called flip mirror is located immediately in front of the grating turret.
Figure 10: Optical layout of the spectrograph in trimetric view with fold mirrors omitted and rays shown for the near IFU channel.
Figure 11 shows the same optical system as Figure 10, but in side view. In this case, rays are shown for a star in the center of the field, which are passed through the mid channel of the IFU.
Figure 11: Optical layout of the spectrograph in side view with fold mirrors omitted and rays shown for the mid IFU channel.
1.3.2 Folded
To accommodate the optical system shown above in the duplicated NIRI cryostat, a series of reflective folds is needed. Figure 12 and Figure 13 show the optical system with the fold mirrors included. The outer hexagonal line is the edge of the Cold Work Surface (CWS) plate, and the hexagonal band inside this is the flange footprint of the spectrograph housing. The OIWFS (not shown) is located on the far side of the CWS plate. The beam from the telescope travels downwards into the OIWFS, parallel to the CWS plate. The small part of the field used by the spectrograph is captured by a 45° pick-off mirror (not shown) protruding beyond the CWS plate.
Apart from the pick-off mirror, five fold mirrors are required. The first is used to fold the beam into the optical plane of the spectrograph. To avoid odd field rotation between the field mask and the spectrograph, this mirror must be oriented so that it reflects any ray that is perpendicular to the CWS plate in a horizontal or vertical direction in the figure. In this case, it is horizontal.
The second mirror is used to suitably position and orientate the optics within the available space. The final three mirrors, which share a common structure, are used to make the system sufficiently compact.
In these views the collimator corrector lens is shown trimmed on two sides. This is made necessary on one side because the fold geometry brings it close to the adjacent beam, and desirable on the other side to because the camera lens housing is nearby.
Figure 12 and Figure 13 also show the six gratings that are provided on an indexing turret because this accounts for a significant amount of space.
The remotely deployable grating bypass mirror which is used for target acquisition is not shown.
Figure 12: Optical layout of the spectrograph in trimetric view with fold mirrors included and rays shown for the near IFU channel.
Figure 13: Optical layout of the spectrograph in side view with fold mirrors included and rays shown for the mid IFU channel.
1.3.3 The Concentric Principle
A basic feature of the optical system is that it uses concentricity in the IFU and collimator to avoid off-axis aberrations. This makes the optical system identical for all 29 channels.
With reference to Figure 10 and Figure 11, the axis of concentricity is a vertical line passing through the center of the image slicer and grating. The image slicer is made from a stack of ~ 1 mm thick aluminum alloy plates, with the interface planes being perpendicular to the axis of concentricity. The reflective surfaces on the ends of the plates are fanned by a small amount about the axis in the manner of a spiral staircase. The mirror elements of the pupil and field mirror arrays are distributed on circular arcs that are centered on the same fanning axis, so that all IFU channels are radial.
This principle is illustrated in Figure 14. Ray bundles are shown for three of the 29 channels, traversing through to the collimator mirror. The vertical line through the image slicer is the fanning axis, or axis of concentricity.
The collimator is a Bouwers system with the center of curvature of all three optical surfaces being coincident on the fanning axis of the IFU. The exit pupil of each channel is arranged to be at the grating, where they are all coincident.
A characteristic of this geometry is that the distance between the image slicer and the field mirror array is equal to the focal length of the collimator, which is 418.32 mm. The IFU is therefore long. Likewise, the radius of curvature the collimator mirror is about twice this focal length, so the whole system shown in Figure 14 is 866 mm long.
Figure 14: The concentric IFU with rays shown for three of the 29 channels.
1.4 Design Detail
The various optics modules of the spectrograph are here described in some detail. Associated analysis is deferred to Appendix C (§12).
1.4.1 Focal Plane Unit and Filter-Fold Tower
The focal plane unit incorporates the pick-off probe mirror, field mask, focal converter, and cold stop. It also includes a test projector that delivers an artificial star to the OIWFS, but discussion of this is confined to the mechanical engineering section of the document. The adjacent filter-fold tower incorporates the order blocking filter, and the first two fold mirrors. The basic components of these modules (fold mirrors excluded) are shown in Figure 15.
Figure 15: The basic components of the focal plane unit and filter-fold tower.
The pick-off probe mirror (not shown) delivers a small area of the ~ f/16 telescope field to the field mask. In principle the need for a whole number of slitlets in the image slicer means that the field cannot necessarily be made exactly square (§4.6.8) as it is notionally meant to be. In practice the restriction leads to an almost exactly square field for the NIFS optical parameters anyway. Theoretically it is 1.856×1.857 mm (2.991″×2.993″). The field mask itself is slightly oversized (2×2 mm) relative to the active field to accommodate any misalignment. The field mask occupies one position in an indexable Focal Plane Mask Wheel. This allows the field mask to be replaced by other masks containing occulting disks of different size, calibration slits, and masks for optical testing.
The focal converter is located 68 mm beyond the field mask. It is a spherically concave, tilted mirror with a focal length of 64 mm. The main function of the focal ratio converter is to re-image the telescope field on the image slicer at 16 times larger scale (~ f/256). Fundamentally, this focal conversion factor is needed to meet certain geometrical requirements of the IFU and collimator, as described in §4.6.4. More practically, it also results in image slicer slitlet plates that are thick enough to be easily manufactured.
The focal converter mirror also produces a 4.0 mm diameter image of the pupil close to the field mask. The fold mirror needed to turn the beam back into the spectrograph is conveniently placed at this location, and so functions as the system cold stop. The region around this mirror must be carefully baffled in order for it to act as an efficient cold stop.
The final component is the order blocking filter. It is one of a set mounted in an indexable 8-station wheel. As shown, it has a diameter of 25 mm and a thickness of 6 mm, but variations to this can be accommodated. The large focal ratio of the beam passing though the filter, and the small size of its footprint mean that it has no significant effect on either image quality or focus (§4.10).
1.4.2 Image Slicer
The image slicer is the first component of the IFU. It is a stack of 29 slitlet mirrors plates, each 1.024 mm thick, as shown in Figure 16. The focal length of the system at this point is 2048 m (16×128 m), so the angular slitlet width is 0.5×10-6 rad (~ 0.103″). In general, the field captured by the image slicer is rectangular, with the aspect ratio determined by the number of slices chosen, the number of pixels in the detector, and the anamorphic factor of the grating. For NIFS, 29 slitlets are chosen to make the field as nearly square as possible. The field size is then 29.696 mm (2.991″) in the spectral direction, and 29.718 mm (2.993″) in the spatial direction. Detailed analysis of this field geometry is given in §4.6.8.
Figure 16: Image slicer comprising 29 slitlet mirrors, each ~ 1 mm wide and having an active length of ~ 29.7 mm. The mirrors are fanned about a vertical axis by ~ 0.127° per slice.
The image slicer is made by diamond machining a concaved spherical mirror on the front face of the stack, and then fanning the plates through a small angular range about the axis shown. The spherical radius of curvature is ~ 623 mm. The mirror face is tilted by 4°, with the fanning axis passing through its center. The off-axis angle of the beam is 1° for the axial ray. The fanning angles are arranged so that the reflected beam from each slitlet is directed towards a different element on the following pupil mirror array. The spherical figure, which is common to all slitlets, is chosen to produce a row of pupil images on the pupil mirror array. These pupil images are arranged to under-fill the elements of the pupil mirror array, and so provide a comfortable margin at the boundaries.
To make good use of this pupil image under-sizing, it is important that each one be accurately centered on its corresponding mirror element in the pupil mirror array. The fact that the image slicer face is tilted with respect to the fanning axis means that the angular offset of the slitlet is not simply half the angular offset of the IFU channel. In fact, when pupil image aberrations are also taken into account, the angular distribution of the slitlet offset angles is slightly non-linear and asymmetric with respect to the channel offset angles. This is explained in Appendix C (§12.4). Accordingly, the range of required image slicer fanning angle offsets is listed in Table 29.
Table 29: IFU Slitlet and Channel Offset Angles.
|
Channel Number |
Channel Offset Angle (deg) |
Slitlet Offset Angle (deg) |
|
-14 |
-3.5616 |
-1.7789 |
|
-13 |
-3.3072 |
-1.6518 |
|
-12 |
-3.0528 |
-1.5247 |
|
-11 |
-2.7984 |
-1.3977 |
|
-10 |
-2.5440 |
-1.2706 |
|
-9 |
-2.2896 |
-1.1435 |
|
-8 |
-2.0352 |
-1.0164 |
|
-7 |
-1.7808 |
-0.8894 |
|
-6 |
-1.5264 |
-0.7623 |
|
-5 |
-1.2720 |
-0.6352 |
|
-4 |
-1.0176 |
-0.5082 |
|
-3 |
-0.7632 |
-0.3811 |
|
-2 |
-0.5088 |
-0.2541 |
|
-1 |
-0.2544 |
-0.1270 |
|
0 |
0.0000 |
0.0000 |
|
1 |
0.2544 |
0.1270 |
|
2 |
0.5088 |
0.2540 |
|
3 |
0.7632 |
0.3811 |
|
4 |
1.0176 |
0.5081 |
|
5 |
1.2720 |
0.6350 |
|
6 |
1.5264 |
0.7620 |
|
7 |
1.7808 |
0.8890 |
|
8 |
2.0352 |
1.0160 |
|
9 |
2.2896 |
1.1429 |
|
10 |
2.5440 |
1.2698 |
|
11 |
2.7984 |
1.3968 |
|
12 |
3.0528 |
1.5237 |
|
13 |
3.3072 |
1.6506 |
|
14 |
3.5616 |
1.7775 |
1.4.3 Tri-Fold Mirror
The tri-fold mirror is the monolithic set of three mirrors shown in Figure 12. It forms part of the tri-fold tower that also includes the pupil and field mirror arrays, but these components are otherwise unrelated. The function of the tri-fold mirror is solely to fold the optical system so that it fits into the available space. It is made from a single piece of aluminum alloy, with the mirror surfaces being diamond machined.
1.4.4 Image Stacker
The image stacker comprises the pupil mirror array and the field mirror array, as shown in Figure 17. It is mounted in the tri-fold tower, along with the tri-fold mirror. Together with the image slicer, it makes up the IFU. Each element of both arrays is torically concaved. For the pupil mirror array, this is in order to produce a good image of its image slicer slitlet on the corresponding element of the field mirror array. For the field mirror array, it is in order to produce good pupil images on the grating. Although a spherical figure would suffice, a toric is no more difficult to produce in this case.
The images of adjacent slitlets on the field mirror array are stacked in staircase fashion, corner-to-corner, along the mirror array. Each element of the field mirror array reflects its beam to the collimator from a virtual pupil located on the fanning axis of the IFU. The fanning geometry is arranged so that the ends of the slitlet images fall on the boundaries between adjacent elements of the field mirror array.
The mirror elements on both arrays are distributed along circular arcs that are concentric with the fanning axis of the IFU. For the pupil mirror array, the radius is ideally 446.208 mm, and for the field mirror array, 418.32 mm. The arrays are separated by the difference between the two radii, or 27.888 mm. For the pupil mirror array elements, the toric radii are 52.957 mm in the vertical plane and 52.499 mm in the horizontal plane. For the field mirror array elements, the toric radii are 59.541 mm in the vertical plane and 59.776 mm in the horizontal plane.
Figure 17: Pupil (left) and field (right) mirror arrays with rays shown for mid and outer channels. Beams enter from right and exit to left.
The beams passing through the IFU image stacker must be angled off-axis to avoid obstruction. This off-axis angle, and the aberration it causes, is strongly dependent on the focal ratio employed at the field mirror array. This focal ratio is also that of the collimator, and so has a strong effect on the overall size of the instrument. To minimize this size, the focal ratio at the field mirror array has been made as small as it can be without causing excessive aberration. A suitable value is ~ f/16. The focal ratio of the telescope is also ~ f/16, so for convenience the chosen value has been matched to that exactly. The magnification in the IFU is therefore the reciprocal of that in the focal converter, or 1/16. The off-axis angle needed to accommodate this with reasonable beam clearance (1.6 mm from the active area of the element surfaces) is 5°. The third-order aberration theory used to derive the focal ratio is presented in §4.6.6. The calculation of the associated off-axis angle is shown in §4.6.7.
The arrays are made as monoliths, with the mirror elements being diamond machined using a fly-cutting technique. In principle, this does not produce toroidal surface figure, but provided certain geometrical conditions are met, the deviations are negligible. The geometry is explained in §4.9.2.
A significant change has been made to the IFU configuration since CoDR. At that time, the elements of both the pupil and field mirror arrays were tilted with respect to their body plates (which are perpendicular to the fanning axis). This tilt has now been eliminated to make diamond machining easier. A detailed explanation is given in Appendix C (§12.4).
1.4.5 Collimator
The collimator is designed in concert with the IFU as a concentric system in order to deliver good optical performance in all IFU channels through to the grating. It is a concentric Bouwers system comprising a spherical mirror and a meniscus corrector lens, as shown in Figure 18. For the sake of clarity, the fold mirror is not included. Also not shown is the lens trimming that is required for beam clearance in the folded arrangement. These details can be seen in Figure 12.
Figure 18: Collimator with rays shown for outer channels and diffraction-spread pupil.
The three optical surfaces of the collimator are spherical and concentric, with their common center lying on the fanning axis. Within channels, and from channel to channel, the entrance and exit pupils are all on the fanning axis.
A characteristic of this concentric geometry is that the distance from the slit image (on the field mirror array) to the image slicer (on the fanning axis) is equal to the focal length of the collimator. The length of the system shown is therefore about twice the focal length, or about 840 mm.
The corrector lens is a single refractive element, so the collimator is not achromatic. In principle this causes the focal length and focus position to change with wavelength. This effect has been limited to an acceptable level over the whole NIFS wavelength range by choosing a low-dispersion material, calcium fluoride, as the material.
The focal length of the collimator is 418.32 mm. The focal ratio from the slit image is matched to that of the telescope (~ f/16) and so the diameter of the geometrical pupil is ~ 26 mm. It is designed to pass a diffraction-spread rectangular pupil of ~ 26 mm × ~ 42 mm unvignetted. The length of the staircase slit image is ~ 54 mm. The theory used to determine the collimator geometry is described in Appendix C (§12.6).
1.4.6 Grating Turret
The Ebert angle at the grating is chosen to be 30° to achieve adequate clearance between the collimator and camera. The grating angle is ~ 20° for all gratings proposed (§4.7). The resolving power of the spectrograph is proportional to the geometrical diameter of the collimated beam for given values of the Ebert angle, grating angle, angular slitlet width, and telescope aperture diameter. A beam diameter of ~ 25 mm is required to achieve the desired resolving power of ~ 5000.
However, there is a further criterion for selecting the exact beam diameter, provided that this approximate resolving power requirement is met. Once the diffraction order and groove density have been chosen for the grating (in addition to the above parameters), the need to match a specific wavelength range to the detector width determines the exact beam diameter (a larger wavelength range requires a smaller beam diameter). For NIFS, it is proposed that the H band (1.49-1.80 μm) be matched to the detector width using a 400 l mm‑1 grating operating in first order. To achieve this, the geometrical diameter of the collimator beam must be 25.788 mm and the grating angle is 20.043°. The corresponding resolving power is 5290. Details of the beam diameter analysis are given in §4.6.10.
1.4.7 Camera
Consideration was originally given to a reflective camera design, but no suitable system was found. The adopted design is the five-element refractive system as shown in Figure 19. The materials employed are, from grating to detector, calcium fluoride, silica, zinc selenide, calcium fluoride, and silica. All are readily available in the required sizes. The limited range of available materials makes this design difficult. The achromatic pair, calcium fluoride and silica, are not well matched in terms of their relative partial differential dispersions, and the zinc selenide element is needed to give good correction. The required tolerances on this element are demanding, and achievement of performance goals may require adjustment of parameters for the other lenses to compensate for errors in its manufacture.
Some consideration was given to a system employing calcium fluoride and Schott IRG-2 glass because these materials are exceptionally well matched. This would allow a superior design, somewhat more compact, and probably with only four elements. Unfortunately IRG-2 is only manufactured to special order, and the preliminary quote provided by Schott amounted to US$40,000. This option was therefore abandoned.
Figure 19: Five-element camera from two viewpoints with rays from grating to detector.
The camera focal length is set to 286 mm to match the width of the monochromatic slit image to two pixels at the detector (36 μm), as derived in §4.6.11. The first camera surface is placed 140 mm from the grating center to provide adequate clearance from the collimator beam. The distance from grating center to the detector has been restricted to 510 mm so that it can be accommodated in the available cryostat space. This makes design difficult, but the achieved performance is nevertheless good (§4.8.1.4). No remotely controllable focus mechanism is provided (§4.10).
The spectrograph camera design produces sub-pixel images over the whole area of the detector for all wavelength bands without refocusing when used in combination with the rest of the optical system. Distortion is well controlled (§4.8.3).
1.4.8 Flip Mirror
The flip mirror is a direct-imaging device that can be deployed immediately in front of the grating turret. It bypasses the grating, and so allows an un-dispersed image of the “staircase” slit to be projected onto the detector. Its purpose is to allow efficient field acquisition imaging without disturbing the grating turret, and so guarantee grating angle stability.
Ideally, this mirror should be coincident with the center of the diffraction grating. It is in fact mounted ~ 27 mm from the grating to provide adequate clearance. As a result, the pupil is moved ~ 54 mm closer to the camera, and ~ 16 mm laterally. Both effects degrade the camera performance, but the absence of dispersion compensates.
Diagrammatic description of this device is deferred to §5.5.4.9.3.
1.5 Diffraction Effects
The angular width of the slitlets in the NIFS IFU is 0.10″. This is comparable to the Airy diffraction limit of 0.07″ for the telescope at a wavelength of 2.2 μm, and so the slitlets cause diffraction effects in the spectrograph. These are illustrated in Figure 20. The first of the six panels shows the telescope pupil. The second shows the diffraction-limited image of a point source formed through this pupil. The third shows this image masked by the slit. The fourth shows the effect of this masking on the reconstructed pupil image, where it is broadened in the spectral direction. The fifth shows this image projected onto the diffraction grating, where it is masked by a rectangular grating boundary. The sixth and final panel shows the reconstructed point source image on the detector, where diffractive broadening is also apparent in the spectral direction.
In consequence, both throughput and resolution are degraded. The only means of controlling these effects is to over-size the optical components in the spectral direction, and so allow some of the diffraction-spread radiation to be captured. Fast Fourier transformations have been used to determine the degree of over-sizing required. The adopted factor is K = 1.6. For the K band, this results in a throughput loss of ~ 3%, and an image profile attenuation of ~ 660 at three pixels off-center (as presented at CoDR).
|
Telescope Pupil |
Star Image |
Masked Star Image |
|
|
|
|
|
Pupil Image |
Masked Pupil Image |
Star Image at Detector |
|
|
|
|
Figure 20: Diffraction effects caused by masking at field (slit) and pupil (grating) images.
The optical system is designed to pass this diffraction-spread beam without excessive aberration.
1.6 Basic Geometrical Parameters
1.6.1 Ebert Angle
The Ebert angle of the spectrograph is the angle between the axial collimator beam approaching the grating, and the axial camera beam leaving the grating. This angle should be made as small as possible to maximize diffraction efficiency and minimize anamorphic and polarization effects. For NIFS, a lower limit is imposed by the need for clearance between the collimator and camera. In consequence, the Ebert angle is chosen to be
1.6.2 Grating Angle, Groove Density, and Diffraction Order
The width of the NIFS slit is fixed, and the width of its monochromatic image projected onto the detector is proportional to the anamorphic magnification of the grating, which is a function of the grating angle. For all gratings used, this slit image width should the same (matched to two pixels), and so all gratings should operate at about the same grating angle.
The grating equation can be expressed as
where θ is the grating angle, m is the diffraction order, A is the grating groove density, λcen is the central wavelength, and φ is the Ebert angle. The Ebert angle is already specified as 30° (§4.6.1).
To achieve good diffraction efficiency, the product mAλcen should be kept somewhat lower than its limit of 2cos(φ/2), because as that limit is approached, radiation leaks into the zero order and the grating begins to behave as a mirror. If the product Aλcen is made too small, however, the grating blaze function (which is about equal to the diffractive spread of a single groove) becomes too narrow and the grating grooves begin to behave as independent mirrors. Considering this in relation to available grating groove densities and the required central wavelengths, the chosen diffraction order and groove density are, respectively,
The corresponding grating angle is
As explained in §4.6.10, the spectrograph has been configured to fit the H band to the detector. A consequence of this is that the grating angle for that pass band must be set to accurately center the spectrum. Detailed analysis of this requirement is shown in Appendix C (§12.3). Treating this as the benchmark condition, the ideal grating angle is
1.6.3 Anamorphic Magnification and Angular Resolution
To provide suitable clearance between the collimator and camera, the Ebert angle is chosen to be φ = 30°. For the benchmark H band, the grating angle is θ = 20.043°. Given that this value is positive, the grating direction is blaze-to-collimator. The anamorphic magnification is then
Because this value is not unity, the two-pixel angular resolution is different for the spectral and spatial directions. The spectral and spatial angular resolutions are, respectively,
1.6.4 Focal Converter Magnification
The concentric optical system used in NIFS places a lower limit on the focal ratio at the image slicer. A focal converter is employed for this reason. For a field that has a square aspect ratio (as required), the focal converter ratio (magnification) can be expressed as
where the symbols are described in Appendix C (§12.1). Except for the IFU pupil fill factor, k, all of the independent parameters here are determined by other design requirements. The value of k, however, cannot be greater than 1, and so
For R3 = 1, fcol = 418.32 mm, dtel = 7891 mm, ftel = 128000 mm, and Dgy = 14.51×10-6 rad, we have
To make the pupil image comfortably smaller than the element width on the pupil mirror array (i.e., to reduce the value of k), the focal conversion ratio is chosen to be
1.6.5 Pupil Fill Factor
The geometry of the focal converter and IFU is determined to ensure that the pupil image on the pupil mirror array is suitably smaller than the width of the mirror elements on the array. The ratio between these is the pupil fill factor, which is
1.6.6 Pupil Mirror Array Focal Ratio and
Magnification
The pupil mirror array focal ratio is the focal ratio, F, of the beam forming the slitlet image on the field mirror array. It is also the focal ratio of the beam feeding the collimator from the field mirror array because that array is coincident with the slitlet image. The pupil mirror array magnification is the magnification of the slitlet images at the field mirror array with respect to the slitlets in the image slicer.
The beam passing through the IFU image stacker must be angled off-axis to avoid obstruction. This causes aberration that is a limit to the spectrograph optical performance, and which is strongly dependent on the pupil mirror array focal ratio. However, the focal length of the collimator is proportional to the focal ratio (to maintain the collimator beam diameter), so to minimize the size of the instrument, this focal ratio should be made as small as possible without causing unacceptable aberrations.
As presented in Appendix C (§12.2), third-order equations have been derived for the aberrations to facilitate selection of the focal ratio. Application of these equations shows that a suitable focal ratio for the pupil mirror array is F ≈ 16. For convenience, it is more specifically chosen to match that of the telescope. Nominally this is f/16, but with secondary mirror under-sizing, it becomes f/16.221.
From §4.6.4, the magnification of the focal converter that re-images the telescope focus onto the image slicer is R1 = 16. The pupil mirror array then re-establishes the original telescope focal ratio, and so the pupil array magnification is
The magnification of the combined focal reducer and IFU (from telescope focus to slit image focus) is
1.6.7 Image Stacker Off-Axis Angle
The off-axis angle of the image stacker is that angle required to pass the beam with adequate clearance. It is part of the geometry which causes the aberration described in §12.2. Given that the angular field size in the spectral direction is Dgx = 14.5 μrad, the telescope aperture is dtel = 8 m (nominal), the focal ratio of the pupil mirror array is F = 16 (nominal), the beam clearance is c = 1.6 mm, the diffractive pupil over-sizing factor is K = 1.6, and the distance between the pupil and field mirror arrays is s = 28 mm, then the required value of the image stacker off-axis angle is
1.6.8 Field Geometry
When the images of the slitlets are laid corner-to-corner in staircase fashion, they should just fill the detector in the spatial direction. Given also that the width of the slitlets is matched to two pixels, and that the anamorphic factor of the grating is dictated by other considerations, then the angular area of the field is fixed. Thus the number of slitlets used determines the aspect ratio of the field.
Using the nomenclature listed in Appendix C (§12.1), the angular size of the field in the spectral direction is
The angular size of the field in the spatial direction is
Combining these equations to eliminate dgx, the required number of slitlets in the image slicer is
For NIFS, the number of pixels in each direction of the detector is n = 2048, the aspect ratio of the field is Dgx/Dgy ≈ 1 (for a square field), the anamorphic magnification is M = 0.8219, and the angular slit width is dgx = 0.5 μrad. Applying the foregoing equations gives the following field geometry
The angular field cannot be exactly square
because N must be a whole number.
Given that the focal length at the image slicer
is 2048 m (matched to that at the telescope focus), the slitlet stack height is
29.696 mm, the slitlet length is 29.718 mm, and the slitlet width is 1.024 mm.
1.6.9 IFU Channel Fanning Geometry
The elements of
the pupil mirror and field mirror arrays are located around concentric arcs
centered on the fanning axis of the image slicer. The radii of these two arcs
are 446.208 mm and 418.320 mm, respectively. The circumferential length of each
element in the field mirror array must match the length of the slitlet image
projected onto it. The angular length of each slitlet, referred to sky, is the
angular field of the instrument in the spatial direction. From §4.6.8, this is 14.51 μrad. Given that the focal length of the system at the field mirror array
is 128 m, the angular channel fanning pitch of the IFU is 0.2544°, the total
fanning angle is 7.1232°, the circumferential pitch of the pupil mirror array
elements is 1.981 mm, and the circumferential pitch of the field mirror array
elements is 1.857 mm.
1.6.10 Collimator Beam Diameter and Focal Length
Amongst other parameters, the diameter of the collimated beam determines the resolving power of the spectrograph. This diameter must be chosen to give a resolving power of ~ 5000. However, this diameter also determines the length of the spectrum on the detector, and so the beam diameter can also be chosen to fit a spectral pass band to the detector. As it happens, the beam diameter required to fit the H band to the detector also satisfies the resolving power requirement, and so this is the criterion used to determine its exact value.
Detailed analysis of this fitting condition is given in Appendix C (§12.3). From this the required geometrical diameter of the collimated beam is
The focal ratio of the collimator has been matched to that of the telescope (f/16.221), and so the corresponding focal length of the collimator is
This fitting process also determines the exact H band grating angle to be 20.043°. The corresponding resolving power of the spectrograph is 5290 (§4.7.2).
1.6.11 Camera Focal Length
The spectrograph camera focal length is determined so that the angular slit width matches two pixels at the detector. For dhx = 0.018 mm, fcol = 418.32 mm, ftel = 128000 mm, M = 0.8219, R2 = 1, and dgx = 0.5 μrad, the camera focal length is
1.6.12 Spectrum Geometry
The 29 channels of spectra are imaged onto the detector as shown in Figure 21.
Figure 21: Spectrum geometry on the detector at true scale.
The monochromatic slit images are tilted by the “staircase” stepping of the channels, and curved by the diffraction of the grating. If the spectral coordinate is x and the spatial coordinate is y, the number of channels is N = 29, the number of pixels is n =2048, the grating angle is θ = 20°, the Ebert angle is φ = 30°, and the camera focal length is fcam = 286 mm, then the slope of the overall slit image at the center of the detector is
corresponding to a two pixel offset between slitlets. And the curvature of the overall slit image is
This corresponds to a tilt of approximately three pixels over the length of the extreme slitlets. The two solid markers show the points where the spectrum touches the edge of the detector array (see Appendix C, §12.3).
1.7 Gratings
1.7.1 Grating Selection Criteria
As explained in §4.6.2, all gratings are selected to operate in first order, to have a groove density of ~ 0.66 divided by the central wavelength of the pass band, and to operate at a grating angle of ~ 20°. This maximizes diffraction efficiency and allows the width of the monochromatic slit image to be matched to two pixels on the detector (using a suitable camera focal length).
To select a grating for each pass band, therefore, the required parameter specification is that the grating blaze angle be ~ 20° (to match the grating angle), and that groove density be ~ 0.66 divided by the central wavelength. If the groove density target cannot be closely met from available gratings, the blaze angle (and grating angle) should be adjusted to suit.
1.7.2 Grating Selection and Performance Data
The grating suite selected in accordance with the above criteria is listed in Table 30. For each pass band, the precise grating angle is calculated as
where m is the diffraction order, A is the grating groove density, λcen is the central wavelength, and φ is the Ebert angle. The Ebert angle is 30° (§4.6.1). The nominal value of A (as listed) is scaled by a factor of 1/0.99600 in the calculation to account for the thermal contraction of the aluminum alloy grating substrate at the operating temperature of 70 K (§4.11.2).
Resolving power is then determined as follows. For grating angles of up to the ideal value, the spectral resolving power is determined by the angular slitlet width as
For grating angles of more than the ideal value, the spectral resolving power is determined by the pixel size as
.
In these equations, dcol is the collimator beam diameter, dtel is the telescope beam diameter, θ is the grating angle, φ is the Ebert angle, dgx is the angular slit width referred to the sky, fcam is the focal length of the camera, and dhx is the pixel size in the spectral direction. The camera focal length is chosen so that both equations yield the same result at the ideal grating angle.
The only parameters not defined by other requirements are dcol in the first equation, and fcam in the second. These must therefore be determined to achieve the required resolving power of ~5000. More precisely, however, they are determined by the compatible criterion that the H band be made to exactly fill the detector, as described in §4.6.10. This also determines the exact value of the ideal grating angle.
From this fitting process, dcol = 25.788 mm and θ = 20.043°. Other parameter values are dtel = 7890.8 mm, fcam = 286.138 mm, φ = 30° and dgy = 0.5×10-6 rad. The pixel size dhx is 0.018 mm scaled by a factor of 0.99946 to account for the thermal contraction of the sapphire substrate at the operating temperature of 60 K.
With the diameter of the collimator beam so determined, the resolving power of each grating is calculated using whichever of the above two equations that gives the lowest value.
Table 30: Grating Suite.
|
Pass Band |
Central Wavelength (μm) |
Groove Density (g/mm) |
Blaze Angle (deg) |
Grating Angle (deg) |
Resolving Power |
Velocity Resolution (km/s) |
Spectral Range (μm) |
|
Z |
1.05 |
600 |
17.5 |
19.1 |
4990 |
60.1 |
0.94 - 1.15 |
|
J |
1.25 |
600 |
22.0 |
22.9 |
6050 |
49.6 |
1.15 - 1.35 |
|
H |
1.649 |
400 |
18.6 |
20.043 |
5290 |
56.8 |
1.49 - 1.80 |
|
K |
2.20 |
300 |
17.5 |
20.1 |
5290 |
56.7 |
1.99 - 2.40 |
Two gratings are required to cover the full J band (designated Z and J), with the selection being dictated by the availability of suitable blaze angles. The H band fills the detector. The K band grating delivers the spectral range 2.00–2.41 μm to the detector, which covers about 75% of the K band available from Mauna Kea.
This suite occupies four of the six disperser turret stations. Two spare positions are available for future use. These might be used to extend the K band coverage, and provide to an L band grating if a detector with extended wavelength sensitivity is installed. For the time being, one of the spare stations will be used for a mirror.
The active length of the gratings (spectral direction) is 51 mm with the diffraction-spread beam. The active width (spatial direction) is 26 mm. The substrate size is 70 mm×40 mm.
Efficiency curves for the selected gratings are shown in Figure 22.
Figure 22: Littrow relative efficiency curves for gratings in Table 30 in s-plane (solid line) and p-plane (dashed line) polarized light. The wavelength range used for each grating is shaded. The reflectivity of aluminum is plotted as a heavy solid line.
The Richardson Grating Laboratory catalog numbers and master dimensions for the selected gratings are listed in Table 31.
Table 31: Grating Catalog Numbers and Master Dimensions.
|
Pass Band |
Groove Density (g/mm) |
Blaze Angle (deg) |
Catalog Number |
Ruled Area (mm×mm) |
|
Z |
600 |
17.5 |
35-53-*-520 |
154×206 |
|
J |
600 |
22.0 |
35-53-*-560 |
154×206 |
|
H |
400 |
18.6 |
35-53-*-650 |
102×102 |
|
K |
300 |
17.5 |
35-53-*-770 |
154×206 |
1.7.2.1 OH Rejection Efficiency
The NIFS performance simulator, NIFSSIM, has
been used to determine the percentage of the wavelength range of each grating
that is occupied by OH airglow emission-lines (excluding the OH-free long
wavelength end of the K band). These
percentages are listed in Table 32. As
discussed at the NIFS CoDR, it is desirable to limit the percentage of pixels
contaminated by OH airglow emission to < 20%. All of the selected gratings
meet this condition.
Table 32: OH Airglow Contamination
|
Pass Band |
Central Wavelength (μm) |
Groove Density (g/mm) |
Resolving Power |
Velocity Resolution (km/s) |
Wavelength Range (μm) |
OH Content (%) |
|
Z |
1.05 |
600 |
4990 |
60.1 |
0.94 - 1.15 |
6.9 |
|
J |
1.25 |
600 |
6050 |
49.6 |
1.15 - 1.35 |
10.6 |
|
H |
1.649 |
400 |
5290 |
56.8 |
1.49 - 1.80 |
13.3 |
|
K |
2.20 |
300 |
5290 |
56.7 |
2.00 - 2.41 |