1 Spectrograph Optical Design

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					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 1 to Figure 4. To better explain the principles, the fold mirrors needed
to fit the optics in the duplicated NIRI cryostat are omitted from Figure 1 and Figure 2.
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 (§1.4.2). 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 (§1.5), 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 (§Error! Reference source not found.12.12).

1.3.1 Unfolded
Figure 1 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 1. 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 1: Optical layout of the spectrograph in trimetric view with fold mirrors omitted and rays
                                   shown for the near IFU channel.

Figure 2 shows the same optical system as Figure 1, 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 2: 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 3 and Figure 4 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 3 and Figure 4 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 3: Optical layout of the spectrograph in trimetric view with fold mirrors included and rays
                                   shown for the near IFU channel.
Figure 4: 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 1 and Figure 2, 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 5. 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 5 is 866 mm long.




              Figure 5: 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 (§Error! Reference source not found.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 6.
              Figure 6: 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 (§1.6.8) 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 §1.6.4. 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 (§1.10).

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 7. 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 §1.6.8.




Figure 7: 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 (§Error! Reference source not found.12.4). Accordingly, the range of
required image slicer fanning angle offsets is listed in Table 1.
                              Table 1: IFU Slitlet and Channel Offset Angles.
                                   Channel      Channel            Slitlet
                                   Number      Offset Angle     Offset Angle
                                                   (deg)            (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 3. 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 8. 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 8: 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 §1.6.6. The calculation of the
associated off-axis angle is shown in §1.6.7.

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 §1.9.2.

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 (§Error! Reference source not found.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 9. 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 3.




        Figure 9: 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
(§Error! Reference source not found.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 (§1.7). 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 §1.6.10.

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 10. 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 10: 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 §1.6.11. 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 (§1.8.1.4). No remotely
controllable focus mechanism is provided (§1.10).

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 (§1.8.3).

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 §Error! Reference source not found.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 11. 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 11: 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

                                                    30

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

                                                        mA  cen
                                            sin  
                                                      2 cos 2 
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° (§1.6.1).

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,

                                                      m 1

                                                  A  0.66  cen

The corresponding grating angle is

                                                       20
As explained in §1.6.10, 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 (§Error! Reference source not
found.12.3). Treating this as the benchmark condition, the ideal grating angle is

                                                    20.043
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

                                                    cos   2 
                                               M 
                                                    cos   2 
                                                   0.8219

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,

                                     x  0.50 rad        (0.103 arcsec)

                                       y    0.41 rad    (0.085 arcsec)

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
                                                     1 f col d tel
                                             R1 
                                                    kR3 f tel 2  y

where the symbols are described in Appendix C (§Error! Reference source not found.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

                                                     1 f col d tel
                                             R1 
                                                     R3 f tel 2  y

For R3 = 1, fcol = 418.32 mm, dtel = 7891 mm, ftel = 128000 mm, and y = 14.51×10-6 rad, we have

                                                    R1  13 .9

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

                                                     R1  16

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     f col d tel
                                            k
                                                  R1 R3 f tel 2  y
                                                0.868

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 (§Error! Reference source not found.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 §1.6.4, 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

                                                             1
                                                    R2 
                                                            16
The magnification of the combined focal reducer and IFU (from telescope focus to slit image focus) is

                                                   R3  R1 R2
                                                       1

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 §Error! Reference source not found.12.2.
Given that the angular field size in the spectral direction is x = 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

                                                x d tel F  c K
                                          E                   
                                                     2s          4F
                                              5

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 (§Error! Reference source not found.12.1), the angular size
of the field in the spectral direction is

                                                   x  N     x


The angular size of the field in the spatial direction is

                                                       1 n
                                               y         x M
                                                       2N

Combining these equations to eliminate x, the required number of slitlets in the image slicer is

                                                       n  x
                                               N             M
                                                       2  y

For NIFS, the number of pixels in each direction of the detector is n = 2048, the aspect ratio of the field is
x/y ≈ 1 (for a square field), the anamorphic magnification is M = 0.8219, and the angular slit width is
x = 0.5 μrad. Applying the foregoing equations gives the following field geometry
                                                   N  29

                                   x  14.50 rad (2.991 arcsec)

                                   y  14 .51 rad       (2.993 arcsec)

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 §1.6.8, 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 (§Error! Reference source not
found.12.3). From this the required geometrical diameter of the collimated beam is

                                           d col  25 .788    mm

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

                                            f col  418 .32 mm

This fitting process also determines the exact H band grating angle to be 20.043°. The corresponding
resolving power of the spectrograph is 5290 (§1.7.2).

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 hx = 0.018 mm, fcol = 418.32 mm, ftel = 128000 mm, M = 0.8219, R2 = 1, and x = 0.5 μrad,
the camera focal length is
                                                       f col 1 1
                                        f cam  2hx
                                                       f tel MR2  x
                                              286 mm

1.6.12 Spectrum Geometry
The 29 channels of spectra are imaged onto the detector as shown in Figure 12.




                       Figure 12: 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

                                          dx     N
                                             2
                                          dy     n
                                              0.0283 (1.62)

corresponding to a two pixel offset between slitlets. And the curvature of the overall slit image is

                              d 2x     2         tan 
                                   
                              dy 2
                                     f cam tan  tan  2  1
                                    0.00232 mm-1         (radius  431 mm)

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,
§Error! Reference source not found.12.3).

1.7 Gratings

1.7.1 Grating Selection Criteria
As explained in §1.6.2, 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 2. For each pass band, the
precise grating angle is calculated as

                                                       mAcen 
                                           arcsin              
                                                       2 cos 2

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° (§1.6.1). 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 (§1.11.2).

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

                                              d col sin  cos 2 1
                                       R2
                                              d tel cos   2  x

For grating angles of more than the ideal value, the spectral resolving power is determined by the pixel size
as

                                               f cam sin  cos 2 
                                         R                          .
                                               hx cos   2 

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, x is the angular slit width referred to the sky, fcam is the focal length of the
camera, and hx 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 §1.6.10. 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 y = 0.5×10-6 rad. The pixel size hx 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 2: Grating Suite.
Pass           Central         Groove         Blaze         Grating      Resolving      Velocity       Spectral
Band          Wavelength       Density        Angle          Angle        Power        Resolution       Range
                (μm)           (g/mm)         (deg)          (deg)                       (km/s)         (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 13.




Figure 13: Littrow relative efficiency curves for gratings in Table 2 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 3.
                      Table 3: Grating Catalog Numbers and Master Dimensions.
                Pass              Groove           Blaze         Catalog          Ruled
                Band              Density          Angle         Number           Area
                                  (g/mm)           (deg)                        (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 4. 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 4: OH Airglow Contamination
  Pass             Central         Groove        Resolving       Velocity      Wavelength        OH
  Band            Wavelength       Density        Power         Resolution      Range           Content
                    (μm)           (g/mm)                         (km/s)         (μm)            (%)
  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         9.8


1.7.3 Scattered Light
Scattering at the grating may prove to contribute significantly to near-angle scattering of OH airglow line
emission into the adjacent continuum. This would degrade the efficiency with which airglow emission-lines
can be rejected. The Richardson Grating Laboratory was asked to quantify this effect, but no data were
provided. However, replicated gratings may actually have lower scattered light levels due to the smoothing
of surface defects by the epoxy and inversion of the grooves. In any event, short of ruling new master
gratings, there is nothing that can be done to reduce this scatter.

1.7.4 Grating Manufacture
Richardson Grating Laboratories can produce replica gratings on client-supplied substrates. The
recommended substrate for cryogenic applications is grade 6061 aluminum alloy. The surface roughness
tolerance is 25 μm RMS. The perpendicularity tolerance between adjacent sides is 0.1°. The grating side of
the substrate must have a 45° bevelled edge, with 1.5 mm face width.

1.8 Image Quality
The first parts of this section present spot diagrams demonstrating the performance of the idealized design.
Later, system performance is investigated giving a quantitative account of design, figure, dimensional, and
alignment errors. Finally, distortion is considered.
1.8.1 Spot Diagrams
Spot diagrams are presented for the whole system (including the telescope) using the geometric pupil. They
are also presented for the spectrograph alone, with point sources at the image slicer and diffraction-spread
pupils, and for the collimator and camera with collimated beams and diffraction-spread pupils.

1.8.1.1 For Geometric Pupil
For the assembled system, image quality is examined at relevant surfaces through the optical system. Both
field and pupil images are considered.

For the field images, the entrance pupil is taken to be the secondary mirror of the telescope, projected
through the system without the diffractive spread in the spectral direction that is caused by the narrow
slitlets of the image slicer. This diffractive effect is considered separately in §1.8.1.2.

For the pupil images, masks at preceding field images form the pupil.

It will be seen that the image quality at all three field images (image slicer, field mirror array, and detector)
deteriorates for the near channel (top slice) of the IFU. Despite its appearance, this aberration is not
defocus. It is largely astigmatism seen at the circle-of-least-confusion, and is caused by the off-axis
spherical mirror of the focal converter. Replacing this sphere with a toroid would considerably reduce
astigmatism, but this has been avoided because it would require rotational registration of the mirror, and the
aberration is not serious in any case.

1.8.1.1.1 At Cold Stop
The entrance pupil of the telescope is the secondary mirror. An image of this mirror is formed by the focal
ratio converter mirror in NIFS, and is cast onto the 4 mm diameter cold stop mirror. The pupil of the
telescope becomes the field for this imaging process, and the square field mask at the telescope focus
becomes the pupil. This mask is made 2 mm square, which is slightly larger than the 1.86 mm (3″) square
field.

This masking imposes a diffraction limit on the quality of the pupil image, against which the geometrical
aberration must be judged. In linear terms, the diffraction limit for a square aperture is the product of the
focal ratio and the wavelength. Given that the focal length of the focal converter mirror is 64 mm, the focal
ratio producing the pupil image is f/32. The diffraction limit is ~ 32 μm even for the shortest wavelength of
~ 1 μm.

The purpose of the cold stop (apart from folding the beam) is to cleanly pass light that comes from within
the boundary of the secondary mirror, but mask light that does not. The aberration of interest, therefore, is
that for the edge of the pupil in the tangential (but not sagittal) direction. For simplicity the figure of the
focal ratio converter mirror is made spherical. The dominant aberration is astigmatism because this mirror
is tilted, so the pupil image is worst at its most off-axis point (i.e., the top edge shown in Figure 6). At this
point, the tangential spread of the image is 9 μm (Figure 14), which is small compared to the diffraction
limit. In fact, much of the aberration is caused by de-focus, because the pupil mirror has been positioned for
an infinitely distant secondary mirror. This approximation results in a blur of about 8 μm. The geometrical
image quality could be dramatically improved by correct focus positioning, and by using a toric figure for
the focal ratio converter chosen to optimise the field image on the image slicer, but the aberration is not
significant in any case.
Figure 14: Spot diagram for the most off-axis point on the cold stop perimeter. Box is 10 μm square.



1.8.1.1.2 At Image Slicer
The focal converter re-images the field at the field mask onto image slicer with a magnification of 16. This
field is ~ 30 mm square and is sliced into 29 slitlet strips, each with a width of ~ 1 mm. Image aberration
should be small compared to this width.

The focal ratio converter is a tilted mirror with spherical figure. This and the changing field angles around
the image slicer result in image aberration (mostly astigmatism). Spot diagrams of images at the image
slicer are shown in Figure 15 for the center and ends of the top, mid, and bottom slitlets. In general, the full
extent of the blurring is ~ 0.10 mm, which is considerably smaller than the slitlet width (and the
corresponding pixel size). The geometrical image quality could be improved somewhat by using a toric
figure on the focal ratio converter mirror, but there is no need for this complication.
                                                 Far End
                             Position in Slice
                                                 Center
                                                 Near End




                                                            Top   Mid     Bottom
                                                                  Slice

     Figure 15: Spot diagrams at various points on the image slicer. Boxes are 0.512 mm square,
                         corresponding to one pixel in the spectral direction.



1.8.1.1.3 At Pupil Mirror Array
The cold stop is imaged onto each element of the pupil mirror array by the corresponding curved slitlet
mirror in the image slicer. The IFU geometry is arranged so that the pupil image under-fills the mirror
element in the fanning direction by a factor of 0.868 (§1.6.5). Given that the circumferential pitch of the
pupil mirror elements is 1.981 mm, the margin is 0.131 mm. Geometrical aberrations in the pupil image,
and the diffraction limit, should be small compared to this.
The slitlets of the image slicer become the pupil for this re-imaging process. The linear diffraction limit
they impose is the product of the focal ratio with which they form the pupil images and the wavelength.
The separation between the image slicer and the pupil mirror array is ~ 446 mm, and the slitlet mirrors are
~ 1 mm thick and ~ 30 mm long. The spectral and spatial focal ratios are therefore ~ f/440 and f/15,
respectively. For the longest wavelength applicable (2.5 μm), the diffraction limit is ~ 1.1 mm
perpendicular to the pupil mirror array, and ~ 38 μm along the array. This latter value is small compared to
the clearance margin of 0.131 mm.

The large diffraction limit in the spectral direction has already been accounted for elsewhere with the
adoption of the pupil aperture enlargement factor of K = 1.6 (§1.5).

The geometrical aberration in the pupil image varies from channel to channel (slitlet to slitlet). For the
central channel, the extent of the blur in the direction of the array is 14 μm. For the far channel (bottom
slitlet), it is 9 μm. For the near channel (top slitlet), it is 18 μm. This is small compared to both the
clearance margin and the diffraction limit. Aberration perpendicular to the direction of the array is
comparable, but of no importance. These values are for the center of the pupil image, but because the angle
subtended by the image about the image slicer is small, quality does not vary greatly around the pupil.




                                       Top              Mid            Bottom
                                                       Slice

  Figure 16: Spot diagrams for the center of the pupil on the pupil mirror array. Array direction is
                                 vertical. Boxes are 40 μm square.



1.8.1.1.4 At Field Mirror Array
For each channel of the IFU, a slitlet in the image slicer is re-imaged onto an element of the field mirror
array by an element in the pupil mirror array at a focal ratio of ~ f/16. The width of the slitlet image is 64
μm, and one pixel corresponds to 32 μm in the spectral direction. Aberrations in the star images on the field
mirror array should be small compared to these values. To control astigmatism, the pupil mirror array
elements have a toric figure.

The image quality varies a little with position in each slitlet, and with the position of the slitlet. It is worst at
the ends of the top slitlet, but the blur is always considerably less than the projected pixel size. Much of this
is astigmatism caused by the focal converter. Representative spot diagrams are shown in Figure 17. Third
order analysis of the IFU contribution is presented in Appendix C (§Error! Reference source not
found.12.2). This shows its dependency on the various IFU parameters.
                                                 Far End
                             Position in Slice
                                                 Center
                                                 Near End



                                                            Top   Mid     Bottom
                                                                  Slice

      Figure 17: Spot diagrams at various positions on the field mirror array. Boxes are 32 μm,
                         corresponding to one pixel in the spectral direction.



1.8.1.1.5 At Grating
The pupil images projected onto the pupil mirror array are re-imaged onto the grating by the field mirror
array and collimator. In this case the pupils for the re-imaging process are the slitlet images on the field
mirror array. These slitlet pupils are 64 μm wide and 1.857 mm long, and they are located ~ 418 mm from
the grating (on the fanning axis). The spatial and spectral focal ratios forming the pupil images on the
grating are therefore ~ f/225 and ~ f/6500, respectively. At a wavelength of 2.5 μm the diffraction limit of
the pupil images on the grating is therefore ~ 0.6 mm in the spatial direction and ~ 16 mm in the spectral
direction. The geometrical aberrations shown in Figure 18 are overwhelmed by this diffractive spread in the
spectral direction. The relationship to the spot diagrams for the pupil mirror array (Figure 16) is clear. The
pupil image magnification between the grating and the pupil mirror array is 15:1.




  Figure 18: Spot diagram for the center of the pupil on the grating as formed through the top, mid
                          and bottom IFU channels. Box is 1 mm square.



1.8.1.1.6 At Detector
Spot diagrams are shown for the Z, H, and K pass bands in Figure 19, Figure 20, and Figure 21,
respectively, for representative wavelengths and positions on the image slicer. Figure 22 shows the
corresponding spot diagrams with the flip mirror deployed. All cases use the same focus setting, and for the
most part, blurring is less than a pixel. The larger aberration for the near channel of the IFU (top slice of the
image slicer) is mostly astigmatism caused by the focal converter. The larger aberration at long wavelength
is mostly defocus caused by the collimator corrector.
                                      Far End
                                      Center




                                                                        0.95
                                      Near End
                                      Far End




                                                                               Wavelength (μm)
                  Position in Slice
                                      Center




                                                                        1.05
                                      Near End
                                      Far End
                                      Center




                                                                        1.15
                                      Near End




                                                 Top   Mid     Bottom
                                                       Slice

Figure 19: Spot diagrams at the detector for the Z band. Boxes are 18 μm square, or one pixel.
                                      Far End
                                      Center




                                                                        1.49
                                      Near End
                                      Far End




                                                                                Wavelength (μm)
                  Position in Slice
                                      Center




                                                                        1.645
                                      Near End
                                      Far End
                                      Center




                                                                        1.80
                                      Near End




                                                 Top   Mid     Bottom
                                                       Slice

Figure 20: Spot diagrams at the detector for the H band. Boxes are 18 μm square, or one pixel.
                                      Far End
                                      Center




                                                                        2.0
                                      Near End
                                      Far End




                                                                              Wavelength (μm)
                  Position in Slice
                                      Center




                                                                        2.2
                                      Near End
                                      Far End
                                      Center




                                                                        2.4
                                      Near End




                                                 Top   Mid     Bottom
                                                       Slice

Figure 21: Spot diagrams at the detector for the K band. Boxes are 18 μm square, or one pixel.
                                        Far End
                                        Center




                                                                          1.05
                                        Near End
                                        Far End




                                                                                  Wavelength (μm)
                    Position in Slice
                                        Center




                                                                          1.645
                                        Near End
                                        Far End
                                        Center




                                                                          2.20
                                        Near End




                                                   Top   Mid     Bottom
                                                         Slice

Figure 22: Spot diagrams at the detector for the flip mirror. Boxes are 18 μm square, or one pixel.
1.8.1.2 For Diffraction-Spread Pupil
The spectrograph has been designed to cope with a pupil spread by diffraction at the image slicer, as
explained in §1.5. To demonstrate this, spot diagrams are shown in Figure 23. For this, point sources on the
image slicer are imaged through the system onto the detector using a rectangular pupil. While the input to
the image slicer is a round ~ f/16 beam, the output here is taken to f/16 in the spatial direction and f/10 in
the spectral direction (K = 1.6). In reality, the pupil spread will be heavily apodized, but even illumination
is assumed here. The spot diagrams are shown only for the H band because this is adequately
representative.
                                     Far End
                                     Center




                                                                       1.49
                                     Near End
                                     Far End




                                                                               Wavelength (μm)
                 Position in Slice
                                     Center




                                                                       1.645
                                     Near End
                                     Far End
                                     Center




                                                                       1.80
                                     Near End




                                                Top   Mid     Bottom
                                                      Slice

Figure 23: Spot diagrams with diffraction-spread pupil. Boxes are 18 μm square, or one pixel.
1.8.1.3 For Collimator
Figure 24 shows spot diagrams for the collimator alone, operating in reverse with a perfectly collimated
beam and diffraction-spread pupil. The dominant aberration is a slight chromatic de-focus. In this regard,
performance is biased towards the shorter wavelength limit to accord with the diffraction limit
characteristics. Although not shown, there is also some lateral image displacement with wavelength
because the collimator operates off-axis.




                                                           1.05           1.65          2.20
                                                                     Wavelength (μm)

Figure 24: Spot diagram for collimator alone with diffraction-spread pupil. Boxes are 32 μm square
                        (corresponding to one pixel in spectral direction).



1.8.1.4 For Camera
Figure 25 shows spot diagrams for the camera alone, operating with a perfectly collimated beam and
diffraction-spread pupil. As with the collimator, performance is biased towards the shorter wavelength limit
to accord with the diffraction limit characteristics.
                                                  Corner
                           Position on Detector
                                                  Edge
                                                  Center




                                                                  1.05           1.65          2.20
                                                                         Wavelength (μm)

  Figure 25: Spot diagram for camera alone with diffraction-spread pupil. Boxes are 18 μm square
                                           (one pixel).
1.8.2 System Performance

1.8.2.1 Required Performance
From the Functional and Performance Requirements Document (FPRD), clause REQ-OCD-0005, the total
wavefront error introduced by the NIFS spectrograph optical system will be no greater than 120 nm RMS
over the wavelength range 0.95-2.5 μm. This corresponds to a Strehl ratio of > 0.8 at a wavelength of 1.6
μm.

While the specified performance level is achieved by the adopted design, it is observed that it is
unnecessarily stringent because the angular pixel size (and two-pixel slit width) of NIFS is too large to take
advantage of it. In effect, NIFS does not sample the telescope image at the diffraction limit, but the
specification assumes that it does.

To investigate this, it is instructive to examine the relationship between wavefront error, ζ, and angular
aberration. Using the surface irregularity model (astigmatism) described in Appendix C (§Error!
Reference source not found.12.8)

                                                    1
                                                 d tel sky
                                                    8

where dtel is the telescope aperture diameter (8 m) and sky is the angular diameter of the aberration. If the
angular aberration is set to match two pixels in the spectral direction (the slit width), then  = 0.5 μrad and
ζ = 500 nm. If the angular aberration is set to match two pixels in the spatial direction, then  = 0.41 μrad
and ζ = 410 nm. Clearly, the specified limit of 120 nm is small.

1.8.2.2 Achieved Performance
As dealt with in the following sections, the total wavefront error is made up of contributions from design
defects, surface irregularities, component dimension errors, alignment errors, and refractive index model
errors. The combined effects are shown in Table 9.

In determining wavefront errors, diffractive spread of the beam pupil caused by the image slicer, and
geometrical clipping of the beam pupil caused by the under-sizing of the telescope secondary mirror, are
both ignored. This is justified because both effects are small, and they counteract each other.

1.8.2.2.1 Design Defects
Ray tracing the idealized optical system from the field mask to the detector with an f/16 entrance beam has
determined design defects. The wavefront error generated varies somewhat with field position and
wavelength, as shown in Table 5.
             Table 5: Wavefront error at various wavelengths and field positions for idealized design.
                                                              Field Position
         Wavefront
                                   Top Slice                    Mid Slice                Bottom Slice
           Error
                          Near                    Far        Both                 Near                    Far
         (nm RMS)                   Center                             Center               Center
                          End                     End        Ends                 End                     End
                 0.95      51         43           49         32          27       24          16          23
             Z   1.05      44         37           44         30          22       21          13          22
                 1.15      51         43           45         31          25       20          15          23
Wavelength




                 1.49      42         43           57         28          20       43          30          26
             H   1.65      45         39           47         29          20       23          14          22
                 1.80      59         45           40         27          18       28          30          39
                 2.00      38         46           65         25          16       50          34          24
             K   2.20      58         53           58         31          24       19          13          20
                 2.40      79         62           47         48          45       25          37          49

A representative value for the whole field and wavelength range is determined by taking the RMS of all the
values in the table. This is ζ = 37 nm RMS.

1.8.2.2.2 Surface Irregularities
Surface irregularity, and the wavefront error it causes, has been estimated for all optical elements of the
spectrograph. The results are shown in Table 6. The listing is of components rather than surfaces, and
where relevant, double entries are used to specify the two surfaces of the component. Combined effects are
determined by adding in quadrature.

Of the transmissive elements, all but the order blocking filter have been specified to have a surface
irregularity of 20 nm RMS. This corresponds to ~ λ/30 RMS (~ λ/8 p-v) at visible wavelength, and can be
achieved by both the RSAA optics workshop and commercial suppliers. The filters have already been
purchased and have λ/4 RMS flatness at the central wavelength of 1.6 μm, or 400 nm RMS.

Diffraction gratings replicated onto glass substrates usually achieve a figure accuracy of ~ λ/30 RMS (~ λ/8
p-v) at visible wavelength. This corresponds to a surface irregularity of 20 nm RMS. For NIFS, aluminum
alloy is used as the substrate, and this is subject to distorting load for mounting purposes. Analysis of this is
presented in Appendix D (§Error! Reference source not found.13.5.8.4), and predicts a surface distortion of
~ 50 nm p-v. To account for the expected distortion and allow for other sources of error, a surface
irregularity of 50 nm RMS is assigned.

Except for the gratings, all the reflective elements in the spectrograph take the form of diamond machined
aluminum alloy. The contractor for the diamond machining work, LFM at the University of Bremen, have
advised that surface irregularity will vary with the size of the surface. For the smallest surfaces, the 4.5 mm
long elements of the pupil and field mirror arrays, they expect it to be ~ 100 nm p-v. We interpret this to be
~ 25 nm RMS. For the largest surface, the 140 mm long Bouwers collimator mirror, they expect it to be ~
300 nm p-v. We interpret this to be ~ 80 nm RMS.

A well-established and empirically determined relationship in mechanical engineering is that, for any given
machining technology, the surface precision achieved is proportional to the 1/3 power of its linear size.
This is consistent with the forgoing data, and in combination with it, allows the following relationship to be
specified. If dface is the envelope diameter of the face, then the surface irregularity is

                                                rms  15 106 d face1 3

This equation is used to nominate the surface irregularity of all the mirrors listed in Table 6. For the pupil
and field mirror arrays, there is in principle some additional figure error contributed by the fly-cutter
generation geometry. However, this effect is shown to be negligible in Appendix C (§Error! Reference
source not found.12.5).

The wavefront error caused by surface irregularity is dependent on both the refractive index of the surface
material, and the size of the point-source beam footprint on the surface. For the limiting case of a beam
being focused on a surface, no wavefront error is caused. Using the surface irregularity model
(astigmatism) described in Appendix C (§Error! Reference source not found.12.8), the following
relationship can be defined to account for these effects. If n is the refractive index across the surface, dbeam
is the diameter of the point-source beam footprint, dface is the envelope diameter of the surface, and ε is the
surface irregularity, then the wavefront error is

                                                                 2
                                              n  1  d beam 
                                                             
                                           
                                                n  d face 
                                                            
This equation is used to nominate all the wavefront errors listed in Table 6. Where the surface is reflective,
n = -1.

                        Table 6: Wavefront errors caused by surface irregularity.
                                                  Point Source
                                     Face
                                                     Beam            Refractive       Surface      Wavefront
                                   Envelope
Component                                          Footprint          Index at     Irregularity      Error
                                   Diameter
                                                   Diameter           1.65 μm       (nm RMS)       (nm RMS)
                                    (mm)
                                                     (mm)
Cryostat Window                      180           21.6 / 20.7         1.43          20 / 20            0
Pick-Off Mirror                       14               6.3              -1             36              15
Focal Converter Mirror                10               4.2              -1             32              11
Cold Stop Mirror                       4               4.0              -1             24              24
Filter                                25            3.7 / 3.6          1.44         400 / 400           4
Fold Mirror 1                         20               4.7              -1             41               4
Fold Mirror 2                         26               3.1              -1             44               1
Fold Mirror 3                         90               1.9              -1             67               0
Image Slicer Mirror                   30                0               -1             47               0
Pupil Mirror Array Element            4.5             1.75              -1             25               7
Field Mirror Array Element            4.5               0               -1             25               0
Fold Mirror 4                         90               3.8              -1             67               0
Collimator Mirror                    140              28.0              -1             78               6
Fold Mirror 5                         90              27.3              -1             67              12
Collimator Corrector Lens           81 / 71        26.7 / 26.2         1.43          20 / 20            1
Grating                               70              32.0              -1             50              21
Camera Lens 1                       95 / 95        31.8 / 30.9         1.43          20 / 20            1
Camera Lens 2                       85 / 95        30.1 / 29.6         1.44          20 / 20            1
Camera Lens 3                       89 / 79        23.8 / 22.1         2.44          20 / 20            1
Camera Lens 4                       96 / 96        13.9 / 12.0         1.43          20 / 20            0
Camera Lens 5                       55 / 65         3.9 / 3.4          1.44          20 / 20            0
Total (RSS)                            -                -                -              -              40


1.8.2.2.3 Component Dimension Errors
The only dimension errors that cause image degradation are those for parameters that are part of corrected
systems. For NIFS, the corrected systems are the collimator and the camera. Mirrors employed outside
these systems function as single surfaces for which the dimension errors (surface curvature or tilt) are small
relative to the parent dimensions. The variations in image quality produced by these errors are therefore
small relative to those produced by the idealized design.

Component dimension errors for the collimator and camera are listed in Table 7, along with the resulting
wavefront errors. The tolerances shown assume a high but achievable degree of precision using standard
optical manufacturing methods. The radii errors correspond to curvature errors of 0.00001 mm -1. Wedge
errors correspond to edge thickness run-out of 0.010 mm TIR.

                  Table 7: Wavefront errors caused by component dimension errors.
                                     Radius 1    Radius 2     Thickness               Wavefront
                                                                            Wedge
            Component                 Error       Error        Error                   Error
                                                                            (deg)
                                      (mm)        (mm)          (mm)                    (nm)
            Collimator Mirror          ±6            -            -           -           1
            Collimator Corrector      ± 0.5       ± 0.5        ± 0.10       0.008         5
            Camera Lens 1             ± 0.25      ± 0.6        ± 0.10       0.006         2
            Camera Lens 2             ± 0.4        ± 12        ± 0.10       0.006         6
            Camera Lens 3             ± 0.08      ± 0.06       ± 0.10       0.006        15
            Camera Lens 4             ± 0.10      ± 0.06       ± 0.10       0.006         3
            Camera Lens 5             ± 0.08      ± 0.4        ± 0.10       0.010         1
            Total (RSS)                  -           -            -           -          17

The total wavefront error for component dimension errors is taken to be twice that accounted for above, or
34 nm.

1.8.2.2.4 Alignment Errors
Like component dimension errors, alignment errors only cause image degradation when the components are
part of corrected systems. For NIFS, only the collimator and camera components are relevant. Alignment
tolerances for these are listed in Table 8, along with the resulting wavefront errors.

All lenses employed in NIFS are located by relying on precision fits between the lens and its housing. No
adjustment is provided. The fit is arranged to provide a minimum clearance of zero, and so the maximum
clearance is half the quadrature sum of the two tolerances. This eccentricity causes a tilt of the un-supported
lens surface that is independent of which surface is supported, and proportional to the lens power.

The equations for determining this tilt are shown in Appendix C (§Error! Reference source not
found.12.9), and are used to compile Table 8. An additional 0.010° is included to account for tilt resulting
from machining errors. An estimated axial displacement error is also applied to each component, based on
experience.

                        Table 8: Wavefront errors caused by lens misalignment.
                                                             Displacement      Wavefont
                                             Surface Tilt
                    Component                                   Error           Error
                                                (deg)
                                                                 (mm)         (nm RMS)
                    Collimator Mirror           0.010            ±0.10            0
                    Collimator Corrector        0.010            ±0.10            0
                    Camera 1                    0.021            ±0.05            4
                    Camera 2                    0.015            ±0.05            5
                    Camera 3                    0.012            ±0.10            7
                    Camera 4                    0.026            ±0.10            4
                    Camera 5                    0.026            ±0.10            0
                    Total (RSS)                   -                -             10
The total wavefront error for system misalignment is taken to be twice that accounted for above, or 20 nm.

1.8.2.2.5 Refractive Index Model Errors
The refractive indices used for design (§1.11.1) are thought to be accurate to five decimal places. At this
level, ray tracing shows that the errors are not significant.

1.8.2.2.6 Total Errors
The total wavefront error caused by the spectrograph is estimated in Table 9. The resulting value of 67 nm
RMS is less than the allowance of 120 nm RMS.

                                      Table 9: Total Wavefront Error.
                                                                   Wavefront
                                 Error Source                        Error
                                                                   (nm RMS)
                                 Design Defects                       37
                                 Surface Irregularities               40
                                 Component Dimension Errors           34
                                 Alignment Errors                     20
                                 Refractive Index Errors               0
                                 Total (RSS)                          67

The estimated wavefront error corresponds to an image blur of ~ 0.27 pixels in the spectral direction when
interpretted in the context of the surface irregularity model (§1.8.2.1). The spectral resolution achieved is
the quadrature sum of this blur with the two pixel wide slit image. This causes a ~ 1% degradation of the
two pixel velocity resolutions listed in Table 2; e.g., the 56.7 km s-1 two pixel resolution of the K grating is
degraded to ~ 57.3 km s-1.

1.8.3 Distortion
Distortion control is important to ensure that spectra for all object points on the reformatted spectrograph
slit align accurately with the detector pixel array. Ray tracing of the system shows that spectrum centroid
deviation for the two outer IFU channels is less that 0.1 pixel.

1.9 IFU Optics Manufacture
It has been decided that diamond machining is the most suitable method of producing the optical surfaces
required for all three IFU components (image slicer, pupil mirror array, and field mirror array). Grade 6061
aluminum alloy is the chosen material. Discussions held with two diamond-machining companies, P-O E in
England and LFM at the University of Bremen, have concluded that the proposed machining methods are
feasible, and manufacture has started at LFM.

1.9.1 Image Slicer
The image slicer is made from a stack of aluminum alloy plates ~ 1 mm thick with spherically concaved
reflective surfaces on one side, as shown in Figure 7. The mirror stack is fanned at a rate of ~ 0.127° per
plate. The geometry of the surfaces is such that, without fanning, they would all be part of a single sphere.
This is a feature that facilitates manufacture. The plate stack is equipped with an accurate dual-mode
registration system, one corresponding to the un-fanned configuration, and the other to the fanned
configuration. The face of the stack will be diamond turned to the common spherical figure while clamped
in the first mode, and then re-assembled in the second mode.
The angular error in the fanning of the stack must not cause displacements of the pupil images on the pupil
mirror array that are significant in relation to the margin between the image and the element boundary. At
ambient temperature, the array elements are pitched at 0.2544° increments on a radius of 448 mm, and so
have a width of 1.99 mm. The pupil fill factor is 0.868 (§1.6.5), and so the margin is 0.13 mm. The dowel
pins that register the image slicer stack have a center distance of 100 mm, and it is anticipated that their
positions can be controlled to an accuracy of ~ 1 μm. The pupil image placement errors should therefore be
~ 0.009 mm, or ~ 7% of the margin. This is adequate.

To avoid manufacturing complexity, it is intended that the image slicer stack be made from stock aluminum
alloy sheet, without any surface modification. The thickness tolerance on stock sheet of this thickness is
±0.1 mm, or ±10%. To avoid significant change in field size, the tolerance should be restricted to, say,
±0.01 mm, or ±1%. To achieve this, a sheet of suitable material has already been selected from a large
volume of stock held by a merchant, purchased, and set aside for the NIFS image slicer. Its thickness has
been measured to be 1.032 mm at ambient temperature. This corresponds to 1.028 mm at the operating
temperature of 70 K, which is 4 μm or 0.4% larger than the ideal.

1.9.2 Mirror Arrays
The alignment of the array elements must be accurate. The tilt error corresponding to one pixel of image
shift is ~ 570 μrad for the pupil mirror array elements. It is proposed that each mirror array be machined as
a monolith to achieve adequate alignment control of the mirror array elements. By machining in a single
set-up, the accuracy that can be achieved across a single optical element can be applied to the whole array,
and so relative misalignment within the array is effectively eliminated.

An additional consideration in adopting a monolith is that the depth of the array elements must be well
controlled. The position of the boundary between adjacent elements is very sensitive to depth differences
because the curved elements are shallow. For the field mirror array, the depth error corresponding to a
boundary displacement of one pixel (18 μm) is only 0.64 μm. Again, however, this is not a problem if the
machining is done in one set-up.

The monolith would take the form of a rectangular plate with the array surfaces machined into one of the
long narrow faces. Such plates can be mounted and adjusted relatively easily.

For both arrays, a series of identical concaved toroidal mirror elements must be distributed along circular
arcs that are centered on the common fanning axis of the IFU. The method devised to do this uses a fly-
cutting technique that is illustrated in Figure 26. The machine employed, a Nanotech 500 FG Free Form
Generator, has the configuration of a conventional milling machine and is able to translate a fly-cutter along
any three-dimensional locus. The fly-cutter has a single cutting tip spinning rapidly about the spindle axis
to describe the fly-cutter circle. In the plane of the array plate, the curvature radius of the generated surface
is that of the fly-cutter circle. In the perpendicular projection, it is determined by the locus of the traverse.
                  Figure 26: Fly-cutter method used for pupil and field mirror arrays.

A limitation of this fly-cutting method is that it does not produce truly spherical (or toric) surfaces. Rather,
the surface is tangential to the intended sphere along two orthogonal lines on the surface. Elsewhere the
deviation follows a “quad-foil” pattern, somewhat like the 17th or 18th order Zernike polynomial terms. A
diagram of the whole fly-cutter envelope is shown in Figure 27. The shape is only spherical at the two
vertices. The generation process described places one of these vertices at the center of the mirror element.
This makes the deviations negligible over the active area of the element. The detailed investigation of this
is presented in Appendix C (§Error! Reference source not found.12.5).




                           Figure 27: Fly-cutter envelope from two viewpoints.
1.10 Chromatic and Filter-Change Defocus
The spectrograph optical system is able to operate over the whole wavelength range without re-focus. The
only refractive components in the system are the camera, the corrector lens of the collimator, and the order
blocking filter. The performance requirement is that changes in wavelength or between filters do not cause
defocus corresponding to more than the resolution of the instrument.

Ray tracing shows that chromatic defocus in the camera is negligible by this criterion.

Filters are positioned between the focal converter and the image slicer, where the system focal length is
2048 m and the focal ratio is ~ f/256. It is assumed that the variation in filter thickness is up to 6 mm. The
range of focus displacement at the image slicer will be about one third of this, or ±1 mm. The
corresponding angular blur referred to the sky is 0.002 μrad, or 0.008 pixels in the spectral direction. This is
negligible.

Defocus caused by the collimator corrector lens is more prominent, necessitating the selection of a low
dispersion material. As shown in Appendix C (§Error! Reference source not found.12.7), the defocus in
the collimator corresponds to ±0.085 mm at the field mirror array, where the system focal length is 128 m
and the focal ratio is ~ f/16. The corresponding angular blur referred to the sky is 0.042 μrad, or 0.17 pixels
in the spectral direction. This is adequate.

1.11 Cryogenic Compensation
NIFS operates at cryogenic temperature, but is manufactured at laboratory temperature. Moving between
these conditions causes thermal strain in the instrument, and changes to the refractive properties of the
lenses used. Design compensation is therefore required. Amongst other things, this is the reason for the
different “hot” and “cold” optical prescriptions shown in Appendix C (§Error! Reference source not
found.12.12).

1.11.1 Refractive Properties
NIFS makes use of three refractive optical materials: calcium fluoride (CaF 2), fused silica (SiO2), and zinc
selenide (ZnSe). Calcium fluoride is used for the window in the cryostat, the corrector in the collimator,
and two lenses in the camera. Fused silica and zinc selenide are used for the remaining three lenses in the
camera. In normal operation the collimator and camera are cryogenically cooled to the spectrograph body,
and because the refractive indexes are significantly dependant on temperature, these systems must be
designed using cryogenic data. In consequence, cryogenic refractive index data are required for all three
materials.

One of the fused silica lenses is mounted immediately in front of the detector where it acts as a field
flattener. As a matter of convenience, it is also used to block any thermal radiation at wavelengths greater
than 4 μm to which a future detector may be sensitive. To ensure that it does not radiate at these
wavelengths, it must form part of the detector enclosure that is held at a lower temperature than the
spectrograph body. Hence there are two cryogenic temperatures that are relevant for fused silica.

From Gemini Near Infrared Imager (NIRI) files, the operating temperature of the calcium fluoride cryostat
window is assumed to be -13 C. Refractive index data are therefore included for calcium fluoride at this
window temperature.

Some optical testing of all components will be done at laboratory temperature, so refractive index data are
also required for all three materials at laboratory temperature.
The adopted laboratory, cryostat window, spectrograph body, and detector enclosure temperatures are,
respectively, 20 C (193.15 K), -13 C (260.15 K), 70 K, and 60 K. The corresponding refractive indexes are
listed in Table 10 for the central, maximum, and minimum wavelengths of each spectral band. Derivation
of these data is presented in Appendix C (§Error! Reference source not found.12.10).

                    Table 10: Refractive index values at selected temperatures and wavelengths.
Refractive                    Z-Band                  J-Band                  H-Band                  K-Band
Index                 0.94 m 1.05 m 1.14 m 1.16 m 1.25 m 1.35 m 1.49 m 1.65 m 1.80 m 2.00 m 2.20 m 2.42 m
          20 C        1.42922 1.42846 1.42792 1.42780 1.42731 1.42679 1.42608 1.42528 1.42451 1.42344 1.42230 1.42095
Calcium
         -13 C        1.42962 1.42887 1.42833 1.42821 1.42773 1.42721 1.42651 1.42572 1.42496 1.42391 1.42279 1.42146
Fluoride
          70 K        1.43119 1.43046 1.42993 1.42982 1.42935 1.42886 1.42819 1.42744 1.42672 1.42572 1.42465 1.42340
             20 C     1.45120 1.44980 1.44874 1.44851 1.44748 1.44635 1.44474 1.44280 1.44087 1.43809 1.43501 1.43127
Fused
             70 K     1.44991 1.44849 1.44742 1.44719 1.44615 1.44499 1.44335 1.44138 1.43941 1.43656 1.43342 1.42959
Silica
             60 K     1.44984 1.44843 1.44736 1.44713 1.44609 1.44494 1.44329 1.44132 1.43935 1.43649 1.43335 1.42951
Zinc         20 C     2.49701 2.48359 2.47559 2.47407 2.46817 2.46301 2.45749 2.45283 2.44953 2.44618 2.44365 2.44148
Selenide 70 K         2.48172 2.46896 2.46133 2.45989 2.45426 2.44932 2.44404 2.43958 2.43642 2.43321 2.43077 2.42868
Vacuum N/A            0.99973 0.99973 0.99973 0.99973 0.99973 0.99973 0.99973 0.99973 0.99973 0.99973 0.99973 0.99973


In accordance with convention, all the refractive indexes are specified relative to standard air. Allowance
must therefore be made for the fact that the cryostat is evacuated during normal operation. A convenient
means of doing this is to apply the refractive index of a vacuum (also relative to standard air) to the space
between lenses within the cryostat. This refractive index (which is less than one) is therefore also specified.

Refractive index data for the OIWFS are not included.

1.11.2 Thermal Strain
The various optical and structural materials used in NIFS all change size with temperature. This thermal
strain must be accounted for in two ways.

Firstly, the lenses in NIFS are located by precision fits in metal housings. The clearance in these is arranged
to approach zero at the cryogenic operating temperature, and so a specific clearance must be provided at the
manufacturing temperature to compensate for the differential strain.

Secondly, thermal strain changes the curvature and position of optical surfaces. In this regard, the system is
designed to perform correctly at the operating temperature, and so the engineering specifications must be
adjusted to allow manufacture at laboratory temperature.

The thermal strain of the relevant materials is listed in Table 11 for the expected operating temperatures.
For the spectrograph body, this is 70 K. The detector enclosure operates at 60 K, and houses various
components. These include the fused silica field flattener lens near the detector, the detector itself (with its
sapphire substrate), and various fittings made from the three listed metals. The tube that attaches the field
flattener to the camera body is made of stainless steel to provide low thermal conductivity, and operates at
an average temperature of 65 K.
                              Table 11: Thermal Strain Values at Selected Temperatures.
                   Thermal Strain                                      Temperature (K)
                   wrt 293.15 K (20 C)                              70       65        60
                        Grade 6061 Aluminum Alloy                -0.00400 -0.00403 -0.00407
                        Grade 304 Stainless Steel                -0.00284 -0.00287 -0.00289


                   Material
                        OFHC Copper                              -0.00308 -0.00312 -0.00314
                        Calcium Fluoride                         -0.00301 -0.00302 -0.00304
                        Fused Silica                              0.00003 0.00003 0.00004
                        Zinc Selenide                            -0.00119 -0.00119 -0.00119
                        Sapphire                                 -0.00054 -0.00054 -0.00054

These values were derived using the thermal strain models described in Appendix C (§Error! Reference
source not found.12.11).

1.11.2.1 Positional Correction
The main structure and all the mirrors in NIFS are made from grade 6061 aluminum alloy. In this regard,
the thermal strain is isotropic and a simple scale factor can be applied to provide compensation. Where
lenses are involved, however, compensation is more complex.




                               Figure 28: Positional compensation geometry for lenses.

The lens mounting method is shown in Figure 28. The anchor point, P0, is the apex of the cone that is
tangent to the lens at the circle where it contacts the housing. As temperature changes, this is the point
where there is no relative movement between the lens and housing. The relative displacement between the
lens and housing at any point, P, is the product of the distance P 0 – P and the thermal strain difference εlens -
εmount. At the vertex of the mounted lens surface, point P 1, it is

                                                   1  cos 
                                          e1  r1            lens   mount 
                                                   cos 

This principle has been applied to the cryogenic geometry to derive the ambient geometry.

1.12 Filters
Order blocking filters are required to prevent out-of-band light reaching the detector. Order blocking filters
suitable for the H and K gratings have already been purchased from NDC Infrared Engineering as part of a
joint NIRI/GNIRS/NIFS acquisition. Transmission curves are plotted in Figure 29. The J grating can use
the photometric J filter offered as a catalog item by OCLI. The filter for the Z grating will have to be
custom manufactured. Barr Associates Inc. will do this for ~ $US4,000 per filter. Parameters for these
filters are listed in Table 12.

                           Table 12: NIFS Order Blocking Filter Parameters.
                                                              Grating
           Filter Data
                                  Z                J               H                 K
           Supplier               Barr             OCLI            NDC               NDC
           50% Cut-On (μm)        0.94  0.01      1.1 ± 0.01      1.47 ± 0.015      1.92 ± 0.019
           50% Cut-Off (μm)       1.16      1.4 ± 0.01      1.80 ± 0.018      2.52 ± 0.025
           Peak Transmission      > 75%            > 60%           > 75%             > 75%
           Blocking               10-4             10-3            10-4              10-4
           Diameter (mm)          25               25              25                25
           Status                 Pending          Catalog item In-hand              In-hand




  Figure 29: Transmission curves for H (left) and K (right) grating order blocking filters purchased
       from NDC Infrared Engineering. The band passes of the H and K gratings are shaded.


1.13 Field Acquisition
The case for being able to bypass the grating with a field-viewing plane mirror is twofold; it allows the
undispersed “staircase” slit image to be recorded to aid in acquiring faint objects, and it can also be used to
check the alignment of the IFU. A field-viewing mirror is unnecessary for acquisition purposes if
sufficiently accurate object and guide star coordinates are known in advance. However, many situations can
be envisaged where it will be advantageous to image the sky directly through NIFS to confirm critical
positioning or alignments.

The data cube generated from NIFS spectral data can be collapsed in the spectral direction to produce an
image of the sky. However, this image inherits the dark current and read noise from all 2048 spectral pixels
recorded at each spatial location. It is more efficient to obtain a direct image. By making the mirror a
deployable device mounted immediately in front of the grating turret, acquisition images can be recorded
without disturbing the grating setting.

1.14 Optical Coatings

1.14.1 Mirrors and Gratings
The NIFS optical path includes a large number of mirrors, and high reflectivity mirror coatings are required
to maximize the system throughput. In making the choice of coatings, consideration has been given to the
cryogenic vacuum environment and the choice of substrate material. Diamond-machined grade 6061
aluminum alloy has been selected for all mirrors in the spectrograph.
Consideration was given to protected silver coatings where reflectivities are equivalent to fresh bare silver.
Examples of protected silver and gold coatings are the FSS-99 and FSG-98 coatings from Denton Vacuum.
Similar coatings can be applied in Australia by at least two vendors.

However, because the vacuum environment will provide protection, bare coatings are preferred, with gold
and silver being the options. Aluminum alloy is a suitable substrate for these coatings. Reflectivity for these
metals is shown in Table 13 (R. N. Wilson, Reflecting Telescopes, Optics II, Table 6.1), and from this gold
is chosen.

A suitable process is ion-assisted deposited (IAD) gold, as this will give a low scatter, durable, coated
surface with reflectance values as listed in Table 13. An alternative is the new electrochemical process
trademarked as LaserGold. This is available in the USA from Epner Co. It produces a hard, durable coating,
again with the reflectance values listed in Table 13. Whichever method is adopted, arrangements will be
made to have the coating applied as soon as possible after diamond machining to minimize surface
deterioration.

                                        Table 13: Reflectivity of Freshly Evaporated Metals.
                                          Wavelength      Ag Reflectivity     Au Reflectivity
                                            (μm)              (%)                 (%)
                                             0.9              99.3                98.4
                                             1.0              99.4                98.6
                                             1.5              99.4                99.0
                                             2.0              99.4                99.1
                                             3.0              99.4                99.3


1.14.2 Lenses
The spectrograph lenses are made from calcium fluoride, fused silica, and zinc selenide. Standard
anti-reflection (AR) coatings are planned for the calcium fluoride (96% transmission) and fused silica (96%
transmission) elements. These can be applied at Avtronics in Australia.

A special AR coating is available from Janos Technology Inc for zinc selenide lenses. Its performance is
shown in Figure 30. The plan is that Janos will both manufacture the lens and apply the coating.


                                10
                                9
                                8
              Reflectance (%)




                                7
                                6
                                5
                                4
                                3
                                2
                                1
                                0
                                  750     1000   1250   1500    1750   2000    2250    2500     2750   3000
                                                               Wavelength (nm)

     Figure 30: Reflectance of AR coating for zinc selenide available from Janos Technology Inc.
1.15 Throughput
NIFS is required to have an optical throughput of > 15% in the wavelength range 1.0-2.5 μm for the
complete optical system, including the telescope, but not including the adaptive optics system.

1.15.1 Model Assumptions
The optical throughput has been modeled by considering the reflection losses at each optical surface. It is
assumed that all mirrors outside the cryostat are coated with protected silver. Except for the ZnSe meniscus
lens in the camera, all lenses are assumed to be AR coated with a single layer of MgF2 with a design
wavelength of 1.50 μm. Reflection loss estimates for these surfaces are based on the wavelength-dependent
refractive indices of the coating and lens materials using single layer AR coating theory. The coating for the
ZnSe lens is assumed to be a proprietary system available from Janos Technology Inc. (Figure 30). The
throughput of the order blocking filters is assumed equal to the minimum peak transmission values listed in
Table 12 (75% for all three wavelengths considered). Diffraction losses at the NIFS pupil mirrors and
grating are discussed in §1.5. The adopted value is 3%.

The diffraction grating efficiencies are dealt with in §1.7. Relative efficiency curves obtained from the
Richardson Grating Laboratory are > 80% over the wavelength range of interest for NIFS. A relative
efficiency of 80% is therefore adopted for present purposes.

The quantum efficiency of the HAWAII-2 array to be used in NIFS is expected to be similar to that of
HAWAII-1 arrays. Data for this is derived from the Rockwell Science Center Focal Plane Array web pages.

Bulk absorption in the transmissive components is assumed to be negligible. No allowance is made for
absorption or scattering in the Earth’s atmosphere.

1.15.2 Performance
The NIFS system throughput estimates are summarized in Table 14.
              Table 14: NIFS System Throughput Budget for the Concentric IFU Design.
                                                                     Transmission
                 Component                     Surface
                                                               1.00μm 1.65μm 2.20μm
                 Telescope Primary             O/C Silver      0.979    0.986   0.987
                 Telescope Secondary           O/C Silver      0.979    0.986   0.987
                 ISS Fold Mirror               O/C Silver      0.979    0.986   0.987
                 Cryostat Window               CaF2            0.938    0.938   0.938
                 Pick-Off Mirror               Gold            0.986    0.990   0.991
                 Focal Converter Mirror        Gold            0.986    0.990   0.991
                 Cold Stop Mirror              Gold            0.986    0.990   0.991
                 Filter                        …               0.80     0.80    0.80
                 Fold 1 Mirror                 Gold            0.986    0.990   0.991
                 Fold 2 Mirror                 Gold            0.986    0.990   0.991
                 Fold 3 Mirror                 Gold            0.986    0.990   0.991
                 Image Slicer: Reflectivity    Gold            0.986    0.990   0.991
                 Image Slicer: Diffraction     …               0.99     0.98    0.97
                 Pupil Mirror Array Mirror     Gold            0.986    0.990   0.991
                 Field Mirror Array Mirror     Gold            0.986    0.990   0.991
                 Fold 4 Mirror                 Gold            0.986    0.990   0.991
                 Collimator Mirror             Gold            0.986    0.990   0.991
                 Fold 5 Mirror                 Gold            0.986    0.990   0.991
                 Collimator Corrector Lens     CaF2 / MgF2     0.949    0.960   0.955
                 Grating: Efficiency           …               0.80     0.80    0.80
                 Grating: Reflectivity         Gold            0.986    0.990   0.991
                 Camera Lens 1                 CaF2 / MgF2     0.949    0.960   0.955
                 Camera Lens 2                 Silica / MgF2   0.950    0.951   0.950
                 Camera Lens 3                 ZnSe / Janos    0.994    0.998   0.986
                 Camera Lens 4                 CaF2 / MgF2     0.949    0.960   0.955
                 Camera Lens 5                 Silica / MgF2   0.950    0.951   0.950
                 Detector QE                   …               0.518    0.583   0.623
                 Total (without AO)                            0.184    0.230   0.240
                 ALTAIR                                        0.773    0.825   0.843
                 Total (with AO)                               0.142    0.190   0.202


1.16 Emissivity
NIFS is required to have an instrument effective emissivity of less than 1% at a wavelength of 2.2 μm.

The cryostat window is the dominant contributor to the instrumental effective emissivity. In the near-
infrared, the absorption coefficient of crystal calcium fluoride varies from ~ 0.49×10 -3 cm-1 to ~ 5.2×10-3
cm-1 depending on supplier (Browder et al., 1991, Handbook of Infrared Optical Materials, ed. P. Klocek,
p. 231). A typical value for optical grade material is ~ 0.78×10 -3. Adopting this leads to an emissivity of ~
0.002 for the 20 mm thick cryostat window. Assuming that the contribution from other components in the
cryostat is small, the total instrument effective emissivity should be ~ 0.2%.

1.17 Ghost Images
Ghost images are caused by spurious reflections at refractive surfaces. In NIFS, the possible sources are the
camera lenses, the collimator corrector, and the cryostat window. Two different effects are dealt with here,
namely, point-source ghosts and OH background ghosts.
1.17.1 Point-Source Ghosts
The performance requirement for NIFS is that relative ghost image intensity must be < 10 -4 at distances of
> 2″ from the parent image. Relative ghost intensity has been estimated for a one-pixel on-axis source
image.

1.17.1.1 Camera
Ghost images can be a problem in axially symmetric refractive systems, such as the NIFS camera. In
general, two different effects produce ghosts. A field image ghost is produced when reflections from lens
surfaces form a roughly focused image of the field on the detector. A pupil image ghost is formed in the
center of the field when reflections from lens surfaces form a roughly focused image of the pupil on the
detector. In either case, the detector itself may be the initial source of the reflections. In fact, such cases
tend to dominate because detector reflectivity is considerably higher than lens surface reflectivity. Strong
field ghosts tend to involve two reflections, with one being from the detector.

For field image ghosts, the intensity becomes significant because they are small and therefore concentrated.
For pupil image ghosts, the intensity becomes significant because light from all field images is superposed
into one single image.

In principle, ghost images can be formed as a result any even-numbered series of reflections. For NIFS,
only two-reflection ghosts have been investigated because of the rapid attenuation that occurs as the
number of reflections is increased. The reflectivity of the detector is assumed to be 30%, and that of each
lens surface (AR-coated) is assumed to be 2%.

The results are listed in Table 15 for all cases where the relative ghost intensity was found to be greater
than 10-10. The surfaces are numbered from the grating, with surface 11 being the detector. None of these
ghosts exceeds the specification.

              Table 15: Relative intensities for two-reflection ghost images in the camera.
                                    Ghost      Reflection      Relative
                                   Number      Surfaces      Ghost Intensity
                                     1          11 - 10         4.4×10-8
                                     2           11 - 9         9.4×10-9
                                     3           10 - 9         4.5×10-9
                                     4           10 - 7         4.4×10-9
                                     5           10 - 8         4.0×10-9
                                     6           11 - 7         3.0×10-9
                                     7            6-5           2.0×10-9
                                     8            9-7           1.5×10-9

All two-reflection ghosts are shown plotted on the detector face in Figure 31 for single parent images with
angular eccentricities of 0, 1, 2, and 3 degrees. The maximum relative intensities are 8.0×10 -8, 5.5×10-8,
5.6×10-8, and 5.9×10-9, respectively.
 Figure 31: Two-reflection ghost images on the detector for single parent images at eccentricities of
                                         0, 1, 2, and 3 deg.



1.17.1.2 Collimator Corrector
There are three possible ghost images that can originate from reflection at a surface of the corrector. For the
first surface, a subsequent reflection can occur at the collimator mirror, returning the beam to the detector.
For the second surface, the subsequent reflection can occur at either the collimator mirror, or the first
corrector surface. Of these three possibilities, ray tracing shows that the latter is by far the most important.
It forms a ghost spectrum on the detector, defocused to an area of about 5 mm 2, and shifted by about 165
pixels in the direction of increasing wavelength. Assuming a reflectivity of 2%, this ghost has a relative
intensity of 2.0×10-8. It is therefore one of the strongest ghosts in the system, but it is still within
specification.

1.17.1.3 Cryostat Window
For a star image in the center of the field, reflection between the two surfaces of the cryostat window
produces a de-focused image at the field mask with a diameter of 1.46 mm. The field image is 1.86 mm
square, and so the ghost forms a background over about half of the field. The window is un-coated calcium
fluoride with a reflectivity of 3.1%. The ghost image then has a relative intensity of 4.8×10 -7. This is the
strongest ghost in the system, but is still within specification. Use of an occulting mask to remove the
parent image will not reduce the ghost intensity.

1.17.2 OH Background Ghosts
An important alternative manifestation of ghosting is that for the OH spectrum. These spectral lines are
densely distributed over the whole detector, and so ghosting produces an image of the detector on the
detector. In general there will be little spectral structure in this image because it will be smoothed by
defocus (and other aberrations). Rigorous modeling of this effect is difficult, but the relative intensity of
this integrated ghost can be roughly estimated as follows.

If the number of two-pixel wide spectral lines in the pass band is N, the number of pixels in each direction
of the array is n, the size of the pixels is x, the reflectivities of the surfaces producing a ghost are R1, R2,
etc, in order of occurrence, the side length of the detector image is a, and the diameter of the defocused
point-source spot is b, then


                                                       
                                                                        2

              Relative Ghost Intensity  2R1 R2 nN  x  if b  a (field ghost)
                                                       a 
or
                                                       
                                                                        2

              Relative Ghost Intensity  2R1 R2 nN  x  if b  a (pupil ghost)
                                                       b 

These equations assume that aberrations other than defocus are insignificant. Fixed parameter values are
n = 2048 pixels and x = 0.018 mm. The reflectivity values are specific to the ghost being considered. It is
estimated that N ≈ 100 lines for a pass band. With the values of a and b determined by ray tracing, the
relative intensities of the strongest ghosts are listed in Table 16. They are all two-reflection ghosts.

                  Table 16: Integrated Relative Ghost Intensities for the OH Spectrum.
                                                                                     Relative
                   Ghost             Reflection                      a        b
                                                     R1     R2                         Ghost
                   Identification    Surfaces                      (mm)     (mm)
                                                                                     Intensity
                                 1     11 - 10      0.30   0.02     31.0      5.2     8.3×10-4
                                 2      11 - 9      0.30   0.02    101.0     11.8     7.8×10-6
                                 3      10 - 9      0.02   0.02     99.5      4.3     5.4×10-6
                                 4      10 - 7      0.02   0.02    213.8      4.3     1.2×10-6
                   Camera
                                 5      10 - 8      0.02   0.02    239.6      4.6     9.3×10-7
                                 6      11 - 7      0.30   0.02    147.5     19.6     3.7×10-5
                                 7       6-5        0.02   0.02     42.0      6.5     3.0×10-5
                                 8       9-7        0.02   0.02     20.6      6.5     1.2×10-4
                   Collimator         Corrector     0.02   0.02     36.9      2.5     3.9×10-5
                   Total (sum)            -           -      -         -        -     1.1×10-3

The listing order for the camera ghosts is the same as that in Table 15, but the intensity ranking has changed
because of the integration effect. All eight cases can be classified as field ghosts rather than pupil ghosts
because b is always less than a. In this regard there is an anomaly for ghost 8. This is the ghost that remains
near the center of the detector for all field angles shown in Figure 31, and so it behaves more like a pupil
ghost than indicated in Table 16. The explanation of this anomaly is that the ghost image is strongly
aberrated. The theory determines the value of a for the chief ray, but in this case the marginal rays are
drawn back towards the center of the detector. The integration effect will therefore be stronger than
predicted in this region, and the total relative ghost intensity will be somewhat higher than 1.1×10 -3.

Ghosting from the collimator corrector is also included. Ghosting from the cryostat window is not relevant
here because it occurs before the image slicer, and the ghost OH spectrum is therefore coincident with the
direct OH spectrum.

The criterion for judging this relative ghost intensity is that it should be no more than comparable to the
ratio of accumulate dark current to well depth when OH lines begin to reach saturation. This is expected to
occur for exposure times of about one hour. Ideally, the dark current will be ~ 0.01 e - pix-1 s-1. For a well
depth of ~ 50,000 e-, this ratio is ~ 7×10-4. Compared to this, the ghost image intensity is marginally
acceptable.

1.18 Scattered Light
The main source of scattered light is expected to be the diamond machined mirror surfaces. Diamond
machining has been adopted as the manufacturing method for all mirrors in the spectrograph. Its use is
practically unavoidable for the three IFU components, and it also offers advantages for other mirrors in the
system. The nature of the process produces surface irregularity in the form of regularly spaced grooves.
Using a small tool feed and large tool tip radius minimizes the effect. Conventionally, the feasible limits for
these are taken to be ~ 1 μm for the feed, and ~ 1 mm for the tip radius. The cyclical profile error introduces
diffraction spikes into the scatter distribution profile.

This grooving effect tends to be reduced by burnishing because aluminum alloy is a soft material. On the
other hand, alloying elements tend to segregate and form hard inclusions, and it is these that apparently
cause much of the surface roughness seen on diamond machined aluminum alloy mirrors. The alloy grade
6061 is proposed to minimize this effect.

Precise analysis of scatter is difficult because the process is complex, and because it is affected by the
detailed structure (power spectral density function) of the surface roughness. This detail is uncertain for the
planned manufacturing method. An alternative rough analysis is as follows.

When a mirror reflects light, surface roughness causes some of it to scatter. This scatter is distributed over
the whole hemisphere above the surface, but its intensity decreases with angular deviation from the
reflected beam. Thus, scatter produces a near-angle halo around the image. Although the intensity
distribution in this halo is difficult to quantify, the theory for predicting the total integrated scatter, TIS, as a
proportion of irradiance is well established. The diamond-machining contractor has advised that surface
roughness, δ will be less than 10 nm RMS. Assuming that this upper limit is the achieved value, then for a
reference wavelength, λ of 1.65 μm, the total integrated scatter is

                                                        4 
                                                                  2

                                                 TIS       
                                                         
                                                      0.006
The so-called Bi-directional Reflectance Distribution Function (BRDF) is a measure of scatter intensity
against angle of deviation from the reflected beam. Figure 32 (Stover, Optical Scattering Measurement and
Analysis, ISBN 0-8194-1934-6) shows a measured plot of BRDF against scatter angle for a diamond-
machined mirror.
                Figure 32: Scatter versus scatter angle for a diamond-machined mirror.

If the near-reflection scattered radiation intensity is k times the average scattered radiation intensity, then it
is visually estimated from the plot that

                                                     k  103

If the angular slit width of the spectrograph is x, the diameter of the telescope aperture is dtel, and the
diameter of the point-source beam footprint on a diamond machined mirror is dbeam, and the monochromatic
point-source image occupies 2×2 pixels (ignoring anamorphic effects) on the detector, then the relative halo
intensity is

                                                                2
                                        I halo   k  d tel 
                                                           x 2TIS
                                                   d
                                       I image 2  beam  

Scatter is worst for mirrors with small beam diameters. The apparent anomaly that mirrors located at a field
focus (with a beam diameter of zero) in fact produce no halo is explained because they in effect produce an
intense halo that all falls within the image pixels. The foregoing equation is only applicable if the beam
footprint is large relative to the image size.

For NIFS, dbeam = 1.75 mm based on dtel = 7891 mm and x = 0.5×10-6 rad. Applying the foregoing
equation to each of the mirrors in the spectrograph gives the results listed in Table 17.
         Table 17: Relative halo intensity caused by scatter from diamond-machined mirrors.
                                                     Point Source            Relative
                    Mirror
                                                    Beam Footprint              Halo
                    Identification
                                                    Diameter (mm)            Intensity
                    Pick-Off Mirror                        6.3                0.0×10-5
                    Focal Converter Mirror                 4.2                0.1×10-5
                    Cold Stop Mirror                       4.0                0.1×10-5
                    Fold Mirror 1                          4.7                0.1×10-5
                    Fold Mirror 2                          3.1                0.2×10-5
                    Fold Mirror 3                          1.9                0.4×10-5
                    Image Slicer                      Field Focus             0.0×10-5
                    Pupil Mirror Array Element            1.75                0.5×10-5
                    Field Mirror Array                Field Focus             0.0×10-5
                    Fold Mirror 4                          3.8                0.1×10-5
                    Collimator Mirror                     28.0                0.0×10-5
                    Fold Mirror 5                         27.3                0.0×10-5
                    Grating                               32.0                0.0×10-5
                    Total (sum)                             -                 1.5×10-5

The specification for ghost images requires that this value be < 10-4, which it is. Given the uncertainties in
the analysis, however, the margin is small. The pupil mirror array dominates, and of those components that
contribute significantly, it is the only one for which diamond machining is essential. Replacing the others
with glass would reduce the scatter by a factor of three. In general, the relatively small performance margin
predicted is consistent with the conventional belief that diamond-machined surfaces are acceptable for use
at near-infrared wavelengths, but not visual wavelengths.

Close examination of Figure 32 suggests that there could be a very narrow but strong rise in BRDF within
about 0.1° of the reflected beam. Even if this is the case, it should not cause a problem. For the small beam
footprint diameters that make the halo intensity significant, a 0.1° spread at the mirror corresponds to the
angular resolution of the spectrograph when referred to the sky.

An alternative criterion for assessing the scatter is to consider the total background illumination it causes on
the detector. According to the specification, this should be < 10% of all radiation entering the spectrograph
for each pass band. As shown above, the total integrated scatter is dependent on wavelength, and for the
worst case of the Z band (λ = 1.05 μm), it is 0.014. If the complimentary reflection is compounded for all 12
mirrors, the total integrated scatter for the system is 16%. Clearly, only a small proportion of this will reach
the detector, and so the specification will be satisfied.

1.19 Baffling
Baffling must be provided to prevent scattered light from reaching the detector. It is proposed to baffle the
instrument chamber by partitioning it into many zones by means of thin blackened panels, with the beam
being passed though holes with no more clearance than is necessary. The rational for this is that it forms
large cavities with small apertures, and the entering radiation is widely spread to reduce flux, as for a black
body cavity at low temperature. Baffles closely conforming to the beam are avoided where possible
because stray light tends to be intercepted at grazing incidence, reflected, and contained within the beam
space. The folded nature of the spectrograph will require some close baffling between adjacent beams.

Particular attention has been given to the region containing the input field mask and the cold stop, because
the former is the source of all spurious radiation, and the latter is the main means of excluding it.

Detailed discussion of baffling is deferred to the mechanical engineering section (§Error! Reference
source not found.5.5.4.10).
1.20 Blackening
Surfaces within the cooled chamber of the instrument should have high emissivity to suppress scattered
radiation. Many proprietary infrared blacks are available for this purpose. For NIFS, the plan is to use
Chemglaze Z306. It has an emissivity of ~ 0.9 at the wavelengths of interest, low outgassing rates, and
good mechanical adhesion properties.

1.21 Thermal Radiation
The spectrograph chamber is almost a fully enclosed cavity held at a constant temperature, so the thermal
radiation flux within it will be that of a black body. Analysis on this basis indicates that the spectrograph
temperature must be held below ~ 135 K to ensure that the total radiation flux remains significantly below
the expected detector dark current for a 2.5 μm cut-off detector. The NIRI cryostat is known to hold its
instrument temperature at ~ 70 K, and so no problems are foreseen in this regard. A more stringent limit of
< 65 K would be imposed if NIFS were to use a 5.5 μm cut-off detector, but this possibility is catered for by
enclosing the detector in a chamber that is locally connected to the second stage of a cryocooler. This
enclosure includes the fused silica field flattener, which also acts as a blocking filter with a cut-off of ~ 4.5
μm wavelength. This is low enough to suppress thermal radiation from the ~ 70 K spectrograph chamber.

1.22 Alignment Stability

1.22.1 Required Performance
From the FPRD, instability in the instrument must not cause image shifts of more than 0.1 pixels (1.8 μm)
over any 15° change in attitude. This stability is applicable at the spectrograph detector with respect to an
image that is stationary on the OIWFS detector. Because the OIWFS is a duplicate of that in NIRI, it is
assumed that this part of the system is stable, and that compliance must only be demonstrated for the
spectrograph.

1.22.2 Achieved Performance
Factors affecting alignment stability are structural deformation of the spectrograph housing and
deformation in the optics modules.

1.22.2.1 Housing Deformation
As described in Appendix D (§Error! Reference source not found.13.1), the dominant structural
deformations are sag across the platform formed by the CWS plate and Optable, local sag in the CWS plate,
and shear in the spectrograph housing. As the telescope changes attitude, the first two cause differential
displacement of the optics modules perpendicular to the CWS plate, and the last causes these modules to tilt
by an angle equal to the shear. The displacement and tilt of each optics module causes image shift, as listed
in Table 18.
                 Table 18: Image Displacement Sensitivity to Structural Deformations.
                                                           dhy            dhy
                             Optics Module                 dy             d
                                                       (μm / μm)     (μm / μrad)
                             Focal Plane Unit             0.04             -
                             Image Slicer                 0.00           0.00
                             Tri-fold Mirror             -0.72           0.04
                             Collimator Mirror            0.71          -0.61
                             Collimator Corrector        -0.02           0.00
                             Grating Turret               0.00           0.53
                             Camera Lens 1                               0.00
                             Camera Lens 2                               0.00
                             Camera Lens 3            1.00 (group)       0.02
                             Camera Lens 4                               0.01
                             Camera Lens 5                               0.00
                             Detector                    -1.00           0.00
                             Total                     1.74 RSS         -0.01

The deflections calculated in Appendix D (§Error! Reference source not found.13.1) are those caused by
the full weight of the supported components. When the attitude change is restricted to 15°, the maximum
deflection change that can be experienced is reduced by a factor of

                                              2 sin 7.5  0.26
The maximum change in the sag of the CWS plate is then y ≈ 0.08 μm. The resulting displacement of the
optics modules will then be within the range 0.00-0.08 μm. Because no attempt is made to identify the
distribution of the error amongst the optics modules, the combined effect on image displacement is
estimated as the quadrature sum. The resulting image displacement is

                                                           dhy
                                               h y  y
                                                         dy
                                                     0.15m

The tilt error caused by housing shear is  ≈ 0.26 μrad. Because all the optics modules share this tilt, the
combined effect on image displacement is estimated as the arithmetic sum. The resulting image
displacement is

                                                           dh y
                                              h y  
                                                         d
                                                     0.003m

Although these estimates are rough, they are adequate to demonstrate that the effect of structural
deformation is small in relation to the allowable image displacement of 1.80 μm.

1.22.2.2 Module Deformation
Of all the optics modules in the spectrograph, the grating turret is the most critical with regard to stability.
This is because it must include a mechanism for grating rotation, and as shown in Table 18, image
displacement is relatively sensitive to its tilt error.
Analysis of this is presented in Appendix D (§Error! Reference source not found.13.5). This shows that
image displacement is small relative to the allowance of 0.1 pixels per 15° attitude change.

1.23 Lens Material Availability
Purchase of refractive elements for the cryostat and OIWFS is already underway as part of the NIRI
duplication program (§Error! Reference source not found.9.3.6.1) using the original suppliers. All lens
materials employed in the spectrograph are readily available in the required grades and sizes. Facilities
required for lens manufacture are available at RSAA, except for the ZnSe item. Suitable AR coatings are
available in all cases.

A list of the materials required and their possible suppliers is given in Table 19.

                             Table 19: Potential Optical Materials Suppliers.
  Material                                               Company
  CaF2 optical crystal lens blanks                       Optovac, Corning Inc, USA
  Silica, IR grade blanks of Inf 302                     Heraeus Amersil, USA
  ZnSe, CVD material, purchased as finished lens         Janos Technology Inc, USA


1.24 ALTAIR Compatibility
Ray tracing has been used to investigate the optical compatibility of NIFS with the ALTAIR system. The
specifications employed for ALTAIR are those described in the Zemax file gaos2_970619_tilted-bs.zmx
and the document “Optical Design of Gemini ALTAIR”. The issues considered are summarized as follows.

1.24.1 Vignetting
Vignetting of the field passed by ALTAIR to NIFS is important for accessing the availability of guide stars.
The primary source of vignetting in ALTAIR is the collimator mirror, followed by the upper fold mirror.
The clear diameters of these mirrors in the two ALTAIR references agree; 150 mm for the collimator
mirror and 240×170 mm for the upper fold mirror. The other mirrors in the Zemax file are larger than stated
in the paper. Figure 33 shows the vignetting plot of the field delivered to the NIFS OIWFS. The vignetting
starts at a field diameter of ~ 2′ and has an attenuation of < 50% at the 3′ field diameter of the NIFS cryostat
window. This means that OIWFS guide stars will be accessible over the full NIFS field, albeit with some
attenuation in the outer 1′. The spectrograph field is not vignetted.




                                    Figure 33: ALTAIR vignetting plot.
1.24.2 ALTAIR Exit Pupil
ALTAIR does not affect the apparent axial position of the telescope exit pupil. It does cause a small lateral
shift of the pupil image on the NIFS cold stop mirror. With the Atmospheric Dispersion Corrector (ADC)
inserted this is 48 μm, and with the ADC withdrawn, it is 14 μm. This is not large relative to the 32m
diffraction limit of the pupil image (§1.8.1.1.1), but could be corrected by a small tilt adjustment of the
science fold mirror.

1.24.3 Field Rotation
ALTAIR rotates the field delivered to NIFS, but this can be corrected by means of the instrument rotator.

1.24.4 Focus Position
ALTAIR does not affect the position of the focal plane at its design wavelength of 1.1 μm. However, at the
NIFS central wavelength of 1.65 μm, and with the ADC deployed, there is a focus shift of 70 μm. This is
not significant and in any case it can be corrected by focus adjustment within ALTAIR. It is not expected
that the ADC will be used in the H band with NIFS (see NIFS CoDR documentation).

1.25 Optical Design Risks

1.25.1 IFU Mirror Array Manufacture
The method developed for manufacture of the two IFU mirror arrays involves diamond machining with a
flycutter, as described in §1.9.2. Although machinery exists to produce the required geometry, the
technique is unproven with regard to surface roughness. A test program is underway to resolve this issue.
 ALTAIR Exit Pupil

ALTAIR does not affect the apparent axial position of the telescope exit pupil. It does cause a small lateral
shift of the pupil image on the NIFS co ld stop mirror. With the Atmospheric Dispersion Corrector (ADC)
inserted this is 48 μ m, and with the ADC withdrawn, it is 14 μ m. This is not large relat ive to the 32  m
diffraction limit of the pupil image (§ 1.8.1.1.1), but could be corrected by a small t ilt ad justment of the
science fold mirror.

1.24.3 Field Rotation
ALTAIR rotates the field delivered to NIFS, but this can be corrected by means of the instrument rotator.

1.24.4 Focus Position

ALTAIR does not affect the position of the focal p lane at its design wavelength of 1.1 μ m. However, at the
NIFS central wavelength of 1.65 μ m, and with the ADC deployed, there is a focus shift of 70 μ m. This is
not significant and in any case it can be corrected by focus adjustment within A LTAIR. It is not expected
that the ADC will be used in the H band with NIFS (see NIFS CoDR documentation).

1.25 Optical Design Risks

1.25.1 IFU Mirror Array Manufacture

The method developed for manufacture of the two IFU mirror arrays involves diamond machin ing with a
flycutter, as described in §1.9.2. A lthough machinery exists to produce the required geometry, the
technique is unproven with regard to surface roughness. A test program is underway to resolve this issue.