Gemini Near-Infrared Integral-Field Spectrograph (NIFS)
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Appendix: Optical Design Parameters
12.1 Nomenclature
The symbols used in this section are defined as follows:
n Number of detector pixels in each direction (square array)
N Number of image slicer slitlets
dgx Angular slitlet width
Dgx Angular size of field in spectral direction
Dgy Angular size of field in spatial direction
F4 Focal ratio at field mirror array
s2 Radius of pupil mirror array arc
s4 Radius of field mirror array arc
E Tilt angle of pupil and field mirror array elements with respect to the center ray
c Clearance between image slicer output beam and image envelope on field mirror array
k Fill factor for pupil images on elements of pupil mirror array
K Pupil aperture enlargement factor in spectral direction (to account for diffraction at slitlets)
q Grating angle
f Ebert angle
fcol Focal length of collimator
fcam Focal length of camera
m Grating diffraction order
A Grating groove density
lmin Minimum wavelength
lmax Maximum wavelength
lcen Central wavelength
dtel Diameter of telescope aperture
dcol Diameter of collimator beam (without diffractive spread)
dhx Pixel size
R Spectral resolving power
M Anamorphic magnification caused by the grating
r Radius of curvature of pupil mirror array elements
f Focal length of telescope at field mirror array
ax Angular size of pupil on pupil mirror array about center of mirror curvature, along array
ay Angular size of pupil on pupil mirror array about center of mirror curvature, across array
bx Angular offset of pupil on pupil mirror array about center of mirror curvature, along array
by Angular offset of pupil on pupil mirror array about center of mirror curvature, across array
gx Angular deviation of ray caused by fly-cutter generation of pupil mirror array, along array
gy Angular deviation of ray caused by fly-cutter generation of pupil mirror array, across array
Egx Angular aberration caused by fly-cutter generation of pupil mirror array, along array
Egy Angular aberration caused by fly-cutter generation of pupil mirror array, across array
T Absolute temperature
x Thermal strain
12.2 Image Slicer Field Geometry
When the images of the slitlets are laid end-to-end 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, the angular area of the field is fixed. Thus the number of slitlets used determines the aspect ratio of the field. For NIFS, the field should be approximately square. The field geometry is determined as follows. The angular size of the field in the spectral direction is given by
where N is the number of image slicer slitlets and dgx is the angular slitlet width. The angular size of the field in the spatial direction is given by
where n is the number of detector pixels in each direction (for a square array), q is the grating angle, and f is the Ebert angle. Thus
which is rounded
to nearest whole number. For n = 2048
pixels, q = 19.919°, f = 30°, dgx = 0.5 mrad,
and Dgx = Dgy, the image slicer has N =
29 slitlets and Dgx =14.50 mrad
(~ 2.99²) and Dgy = 14.53 mrad (~3.00²). Given that the focal length at the image slicer
is 2048 m, the slitlet stack height is 29.7 mm, the slitlet length is 29.8 mm,
and the slitlet width is 1.024 mm.
12.3 IFU Fold Geometry
The
elements of the pupil and field mirror arrays must be tilted to pass the beam.
This causes the aberration that is the main limit to the spectrograph optical
performance. Third-order equations have been derived for the aberrations this
produces to facilitate design of the IFU with respect to image quality. These
equations apply to images of the slitlet centers, and assume that the
separation between the pupil and field mirror arrays is small compared to that
between the image slicer and the pupil mirror array. The aberrations are
referred to the sky.
The required tilt angle of the pupil and field mirror array elements with respect to the center ray, E, is given by
where the symbols are described in §12.1. The maximum angular tangential coma, ATCmax, occurs at the most off-axis point in the field, and is given by
.
Maximum angular astigmatism, AASmax, also occurs at the most off-axis point in the field, and is given by
.
The minimum angular astigmatism, AASmin, occurs at the least off-axis point in the field, and is given by
A fixed amount of correction can be applied across the field if a toric figure is used in place of a spherical figure for the pupil mirror array elements. In fact, the maximum and minimum corrected angular astigmatism can be made to have equal magnitudes and opposite signs with the magnitude, AAStor, being half of the difference between the uncorrected maximum and minimum values. AAStor is then given by
The angular spherical aberration (focused for minimum diameter), ASA, is given by
Application of these equations shows that astigmatism is the dominant aberration, and that it increases rapidly with the chosen values of the focal ratio at the field mirror array, F4, and the tilt angle of the pupil and field mirror array elements, E. Large values of the pupil mirror fill factor, k, are moderately desirable. The use of a toric rather than a spherical figure for the pupil mirror elements gives considerable image improvement.
The basic parameter values for NIFS are dtel = 8000 mm, Dgx = 14.5 mrad, and K = 1.6. The chosen parameter values are E = 5°, F4 = 16, s2 = 448 mm, and s4 = 420 mm. For these values, c = 1.6 mm, k = 0.88 mm, and the aberrations are as listed in Table 1.
Table 1: IFU Aberration Values
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ATCmax (mrad) |
ATCmin (mrad) |
AASmax (mrad) |
AASmin (mrad) |
AAStor (mrad) |
ASA (mrad) |
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0.049 |
0.022 |
0.165 |
0.033 |
0.066 |
0.002 |
The k value yielded is somewhat different from
that derived in the separate analysis of fanning geometry (§12.4) because of the simplifying assumptions
involved in this analysis.
12.4 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 448 mm and 420 mm, respectively. The circumferential length of each element
in the field mirror array must match the length of the slitlet image projected
onto it. Given that the angular length of each slitlet, referred to sky, is
14.53 mrad, and the focal length at the field
mirror array is 128 m, the angular fanning pitch of the IFU is 0.2537°, the total
fanning angle is 7.1036°, the circumferential pitch of the pupil mirror array
elements is 1.984 mm, and the circumferential pitch of the field mirror array elements is 1.860 mm.
The IFU geometry must be arranged so that the
width of the pupil images on the pupil mirror array is comfortably less than
the width of the mirror elements.
For the proposed design, the pupil fill factor, k, is given by
.
Then k = 0.825 for the design parameters of Dgy = 14.53 mrad, dtel =
8000 mm, F4 = 16, s2 = 448 mm, and s4 = 420 mm. A smaller value would give more clearance, but
would also increase image aberrations.
12.5 Collimator Beam Diameter, Focal Length and
Spectral Resolution
As discussed in §4.3.4, the precise criterion used to determine the collimator beam diameter is that the H band (1.49-1.80 mm) fills the detector for the selected H grating. The conditions for this to occur are
Solving simultaneously with m = 1, A = 0.4 l / mm, f = 30°, lmin = 1.49 mm, lmax = 1.80 mm, dtel = 8000 mm, n = 2048 pixels, N = 29 slices, and dgx = 0.5 mrad gives a geometrical collimated beam diameter of dcol = 26.3 mm and a grating angle of q = 19.919°.
Given that the focal ratio of the input to the collimator is 16, the collimator focal length is fcol = 421 mm. The spectral resolving power is given by
which for the parameters above gives R = 5280. The corresponding central wavelength is found via the grating equation
to be lcen = 1.645 mm.
12.6 Camera Focal Length
The spectrograph camera focal length is determined so that the angular slitlet width matches two pixels at the detector. Accordingly, the camera focal length is given by
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For dhx = 0.018 mm, dcol = 26.3 mm, dtel = 8000 mm, and dgx = 0.5 mrad, the camera focal length must be fcam = 288 mm.
12.7 Optical Specification
The optical specification for the concentric IFU design, through the central channel of the IFU and without fold mirrors, is given in Table 2. With respect to Figure 72 in §4.2, X and Z are the vertical and horizontal distances, respectively, from the center of the field mask. The non-central channels are identical except that they are fanned in different directions from the image slicer. The fanning geometry for these channels is explained in §4.3.2, §4.3.3 and §12.4.
Table 2: Optical Listing for the Central Channel of the Concentric IFU Design.
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Component Description |
Media |
Radius (mm) |
X (mm) |
Z (mm) |
Pitch (deg) |
Size (mm) |
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Field Mask |
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Flat |
0 |
0 |
0 |
2´2 |
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Focal Ratio Converter |
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128 |
0 |
68 |
3 |
Ø10 |
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Cold Stop |
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Flat |
-6.727 |
4 |
2.5 |
Ø4 |
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Image Slicer |
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-623.457 |
-24.601 |
1028 |
0 |
1.024´30 |
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Pupil Mirror Array |
|
53.135´53.548 |
-32.421 |
580 |
-4 |
4´1.984 |
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Field Mirror Array |
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-60.835 |
-36.855 |
608 |
-4 |
4´1.086 |
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Collimator Mirror |
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868.525 |
-66.225 |
159.474 |
0 |
63´160 |
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Collimator Corrector |
CaF2 |
229.231 |
-66.225 |
798.769 |
0 |
60´80 |
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217.131 |
-66.225 |
810.869 |
0 |
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Grating |
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Flat |
-103.009 |
1028 |
~31 |
60´40 |
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Camera Lens 1 |
CaF2 |
-157.368 |
-164.381 |
902.169 |
26 |
Ø96 |
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219.775 |
-173.149 |
884.193 |
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Camera Lens 2 |
Silica |
177.476 |
-174.902 |
880.598 |
26 |
Ø96 |
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756.716 |
-179.286 |
871.610 |
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Camera Lens 3 |
ZnSe |
-95.076 |
-200.590 |
827.930 |
26 |
Ø90 |
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-82.136 |
-204.973 |
818.943 |
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Camera Lens 4 |
CaF2 |
-109.230 |
-274.905 |
675.561 |
26 |
Ø94 |
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206.594 |
-283.672 |
657.585 |
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Camera Lens 5 |
Silica |
84.408 |
-309.227 |
605.191 |
26 |
Ø54 |
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-205.928 |
-311.857 |
599.798 |
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Focal Plane |
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Flat |
-325.408 |
572.015 |
26 |
36.864´36.864 |