1 Appendix C: Optical Design Detail

1.1 Nomenclature

Symbols used in the optical analysis are defined as follows. Some additional symbols are defined locally.

n Number of detector pixels in each direction (square array), or refractive index

N Number of field slitlets

δγ_x Angular slitlet width

Δγ_x Angular size of field in spectral direction

Δγ_y Angular size of field in spatial direction

F Focal ratio at field mirror array

s Distance between pupil mirror array and field mirror array

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)

θ Grating angle

φ Ebert angle, or off-axis angle of collimator

f_col Focal length of collimator

f_cam Focal length of camera

m Grating diffraction order

A Grating groove density

λ_min Minimum wavelength

λ_max Maximum wavelength

λ_cen Central wavelength

d_tel Diameter of telescope aperture

d_col Diameter of collimator beam (without diffractive spread)

δh_x Pixel size

R Spectral resolving power

R₁ Focal converter magnification

R₂ Pupil mirror array magnification

R₃ Combined focal reducer and pupil mirror array magnification (= R₁R₂)

M Anamorphic magnification caused by the grating

r Radius of curvature of pupil mirror array elements

e Eccentricity of collimator ray

f Focal length of telescope at field mirror array

α_xAngular size of pupil on pupil mirror array about center of mirror curvature, along array

α_yAngular size of pupil on pupil mirror array about center of mirror curvature, across array

β_xAngular offset of pupil on pupil mirror array about center of mirror curvature, along array

β_yAngular offset of pupil on pupil mirror array about center of mirror curvature, across array

γ_xAngular deviation of ray caused by fly-cutter generation of pupil mirror array, along array

γ_yAngular deviation of ray caused by fly-cutter generation of pupil mirror array, across array

Eγ_xAngular aberration caused by fly-cutter generation of pupil mirror array, along array

Eγ_yAngular aberration caused by fly-cutter generation of pupil mirror array, across array

T Absolute temperature

ξ Thermal strain

ε Surface irregularity error

σ Wavefront error

δγ_sky Angular aberration referred to sky

d_faceFace diameter of an optical surface

d_beam Point-source beam footprint diameter on an optical surface

I IFU channel number (ranging from -14 for bottom slitlet to +14 for top slitlet)

The pupil mirror array of the integral field unit causes image aberrations because it must be operated off-axis. These aberrations are modelled as follows using third order theory in order to facilitate selection the off-axis angle. The 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 maximum angular tangential coma, ATC_max, occurs at the most off-axis point in the field, and is given by

Maximum angular astigmatism, AAS_max, also occurs at the most off-axis point in the field, and is given by

The minimum angular astigmatism, AAS_min, 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 AAS_tor being half of the difference between the uncorrected maximum and minimum values. AAS_tor 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 as the focal ratio at the field mirror array F is reduced. Large values of the pupil mirror fill factor k, and small values of clearance c are moderately desirable. The use of a toric rather than a spherical figure for the pupil mirror elements gives considerable image improvement.

The predetermined parameter values for NIFS are d_tel = 8000 mm (nominal), Δγ_x = 14.5 μrad, and K = 1.6. To give good aberration control, the chosen parameter values are F = 16, k = 0.87 mm, and c = 1.6 mm. The resulting aberrations are as listed in Table 77.

Table 77: IFU aberration values

ATC_max

(μrad)

ATC_min

(μrad)

AAS_max

(μrad)

AAS_min

(μrad)

AAS_tor

(μrad)

ASA

(μrad)

0.049

0.021

0.167

0.033

0.067

0.002

1.3 Fitting the Spectrum to the Detector

A diagram of a spectrum (not drawn to scale) is shown in Figure 165, with the two monochromatic slit images being the ends of the pass band that just fits onto the detector. These slit images are tilted by the "staircase" stacking of the slitlets, and curved by the diffraction of the grating. The two solid markers show the extent of the spectrum in the spectral direction. The separation between these points is a function of collimator beam diameter and grating angle, and so the values of these parameters can be chosen to fit a pass band to the detector.

Figure 165: The spectrum on the detector, not drawn to scale. The two solid markers show the extent of the spectrum in the spectral direction.

Taking account of slit image stepping and curvature, the conditions for this to occur are

Solving these three equations simultaneously for the H band with input parameters m = 1, A = 0.40161 g/μm (with thermal contraction), φ = 30°, λ_max = 1.80 μm, λ_min = 1.49 μm, d_tel = 7890.8 mm, N = 29 slices, n = 2048 pixels, δγ_x = 0.5 μrad and cos β ≈ 1 yields

1.4 IFU Configuration

In the ideal concentric system, the channels of the IFU are fanned about a common axis that passes through the center of the image slicer in a way that makes the channels optically identical. This common channel geometry should also be arranged to produce good image quality. The geometrical principles required to achieve this ideal can be explained as follows.

· The curved surface of the image slicer should be tangential to the fanning axis at its center. This is the condition for which all channels have the same set altitude angles, and the mirror arrays are circularly symmetric. An additional desirable feature is that the azimuth angle of each slice is exactly half the azimuth angle of the corresponding channel.

· The aluminum alloy plates onto which the pupil and field mirror elements are machined should be perpendicular to the fanning axis. This allows all the mirror elements of the circularly symmetric arrays to be mechanically cantered in the plates.

· The off-axis angles for all three mirror elements in the channel (slicer, pupil mirror, and field mirror) should be made no larger than is needed to avoid beam interference with adjacent components. This minimizes the astigmatism produced by the elements in either the field or pupil images. Suitable off-axis angles are, for the image slicer, 1°, and for the pupil and field mirrors, 5°. The image slicer and field mirrors only produce aberration in the pupil images, and so this requirement is less stringent for these than for the pupil mirror.

The configuration proposed at CoDR complied with all of these principles. Unfortunately, however, this involved manufacturing difficulties, and a new configuration has been developed. The new design violates the first of the above principles, but the consequences are easily accommodated. The change is described as follows.

1.4.1 CoDR Configuration

This is the baseline configuration presented in the CoDR documents (Vol. 1, §4), and is shown here in Figure 166. It complies with all the principles listed above, and is in this respect the optimal system. Nevertheless, it has certain practical problems.

Figure 166: CoDR configuration shown with anamorphic distortion.

A characteristic of the configuration is that the pupil and field mirror elements are tilted with respect to plates on which they are machined. This means that the relatively fast beam emerging from the field mirror has a very shallow angle with respect to the pupil mirror array plate, and there is a clearance problem for the beams coming from the lower elements of the image slicer. This would require that the back end of the pupil mirror array plate be relieved.

More importantly, however, there are repercussions for the diamond machining of the arrays. As explained in §4.9.2, it is intended that a flycutting technique be used to generate the toroidal elements on both the pupil mirror array and the field mirror array. The surfaces that this produces are not true toroids, but a good approximation is made if the vertex of the actual figure is at or near the center of the element. To achieve this, the array plates must be tilted in the diamond-machining mill so that the normal to the element surface at the element center is perpendicular to the spindle axis of the flycutter.

In principle, this correction leaves aberrations in all but the central element of each array, because the required tilt differs slightly for each element. This effect is small, however.

A further complication is that, because the array plates are tilted, the toolpath has to be adjusted in height for each element of the array.

None of these difficulties is intolerable, but their avoidance is very desirable. Given that the production of the IFU is seen as one of the most demanding aspect of the NIFS project, it is important that it be simplified as much as possible.

1.4.2 Adopted Configuration

The adopted configuration is shown in Figure 167. It complies with the principles listed in the introduction, except that the center of the image slicer is not tangential to the fanning axis. Rather, the curved image slicer face (but not the mechanical stack) is tilted downwards by 4°. The pupil and field mirror elements can then be square to the array plates, and the off-axis angle of the image slicer can be its desirable 1°. This avoids both the clearance and array manufacturing problems of the CoDR configuration.

Figure 167: Adopted configuration shown with anamorphic distortion.

The penalty of this arrangement is that the fanning geometry is no longer ideal. When the center of the image slicer is not tangential to the fanning (azimuth) axis of the system, the altitude angle of the reflected ray varies as the azimuth angle of the slice changes from channel to channel, and the azimuth angle of the reflected ray is no longer simply twice the azimuth angle of the slice.

The azimuth deviation is easily corrected by adjusting the angular spacing of the slices, but the varying altitude angle causes the pupil images to be located at different heights on the pupil mirror array. The azimuth angle correction and the altitude angle error are determined with reference to Figure 168.

Figure 168: Alt-Az reflection geometry.

In this, the reflecting surface of the slicer is at point 0. Points 0, 1, 2, and 3 lie in the reference plane perpendicular to the azimuth (fanning) axis of the IFU. Points 0, 4, 5, and 6 lie in the ray plane. Line 0-5 is the normal to the reflecting surface. Any three of the angular parameters shown fully define the geometry. For NIFS, the pre-defined parameters are the input ray altitude angle A = 3°, the image slicer altitude angle B = 4° and the variable channel offset angle D+E. If the angles are small (as they are), then the ratio of slicer offset angle to channel offset angle is

The corresponding value for the CoDR configuration is exactly 0.5.

For the outer channels, the channel offset angle (§4.6.9) is

and so the corresponding image slicer offset angle is

The deviation in altitude angle of the outer channels from that of the central channel is

Given that the distance between the image slicer and the pupil mirror array is 446.208 mm, the maximum deviation in the position of the pupil images on the array is 30 μm. In relation to the pupil image aberration, this error is negligible. In fact, sharing it equally between the central and outer channels could halve the maximum pupil deviation, but the error is so small that this additional correction is not warranted.

The analysis shown above places the axial ray in the center of the pupil mirror array element. In fact, an additional correction is made. Because the rays producing the pupil image come from a slitlet that is offset from the axial ray, and because there are aberrations in that image, the centroid of the pupil image is also displaced from the axial ray. To correct this, non-linear adjustments are made to the angular offsets of the image slicer slitlets, as determined by ray tracing. If the channel number is I (ranging from -14 for the bottom slitlet to +14 for the top slitlet), and the channel offset angular increment is 0.2544°, then the corrected image slicer offset angle can be determined as

1.5 Fly-Cutter Aberrations

The fly-cut pupil mirror array produces aberrations because it is not truly toric, as explained in §4.9.2. Figure 169 shows the geometry of the array elements. Here, the surface is only coincident with the intended toric along the α_x and α_yaxes.

Figure 169: Fly-cutter generation geometry.

In general, the error introduced to a ray reflected off the surface corresponds to an angular deviation on the sky of

The angular size of the rectangular pupil (diffraction spread) on the mirror element is

Substituting F = 16 and K = 1.6 gives β_x = 0.03125 rad and β_y = 0.05000 rad. Aberration is worst for a ray coming from the corner of the pupil image, for which α_x = β_x/2 and α_y = β_y/2. For r = 53 mm and f = 128000 mm, the ray deviation is γ_x = 0.002 μrad (0.01 pixel) and γ_y = 0.001 μrad (0.01 pixel). These values are negligible, especially as they apply to the corner of the diffraction-spread pupil where there is very little radiation energy. Within the geometrical pupil the aberration is considerably less.

An alternative approach is to examine wavefront error. This is twice the surface error, or

within the geometrical pupil. Again, this is negligible.

1.6 Collimator Geometry

Applying the sine condition to the system shown in Figure 170 gives

Combining this with Snell’s law, the focal length at the eccentric ray is

For the axial ray this reduces to

Figure 170: Bouwers concentric collimator geometry.

In principle, r₁, r₂, and r₃ can be determined to correct the focal length at three ray eccentricities (one of which might be zero for the axial ray). This however results in an inconveniently thick corrector lens with radii comparable to the focal length. Restricting the lens radii to about half the focal length results in a more reasonable lens thickness for which the focal length can be corrected at two ray eccentricities.

For NIFS, the focal length was set to 418.32 mm, the refractive index set to 1.42871 (calcium fluoride at a wavelength of 1.38 μm and temperature of 70 K), r₁ set to r₃/4, and φ set to 5° for the chief ray. Applying the first of the above equations yielded a chief ray eccentricity of e ~ 36.5 mm. The second of the above equations was then solved simultaneously to give the correct focal length at the chief ray, and to make the focal length the same for e values of ~26.5 and ~46.5 mm. This yielded surface radii of r₁ ~ 215.8 mm, r₂ ~ 227.7 mm and r₃ ~ 863.0 mm. These were subsequently adjusted slightly in the final optimization.

1.7 Collimator Chromatic Defocus

For the wavelength range 0.95 μm to 2.4 μm, the refractive index of the calcium fluoride corrector changes from 1.43119 to 1.42340 at 70 K. The focal length of the collimator is

For r₁ ≈ 215.8 mm, r₂ ≈ 227.7 mm, and r₃ ≈ 863.0 mm the corresponding change in focal length is ±0.085 mm.

1.8 Surface Irregularity and Optical Aberration

Angular aberration caused by surface irregularity is proportional to the slope error of the surface. In principle, this is dependent on both the form and magnitude of the irregularity. For small surfaces, however, polishing methods ensure that the irregularities are smooth and of low spatial frequency. The form of such irregularities is therefore independent of face size, and their effects can be determined by their magnitude alone. To characterize this, surface irregularity is here assumed to take its smoothest possible form, namely, astigmatism. For this, the surface height error is a second-order function of radial position on the surface. The curvature error varies with azimuth on the surface, but is independent of position on the surface. Alternative low-order functions are not applicable. A first-order function is simply a tilt error, which causes lateral image displacement, not blurring. A circularly symmetrical second-order function is simply curvature error, which causes axial image displacement (defocus), not blurring.

Note that this modelling does not imply that all surface irregularity is astigmatic. Rather, it assumes that astigmatism having the same magnitude as real surface irregularity produces optical aberration of the same magnitude as well.

The adoption of this astigmatism model allows determination of the relationship between surface irregularity and angular aberration. Importantly, the diameter of a point-source beam footprint on a surface can be smaller than the diameter of the surface, and the resulting attenuation of the angular aberration can also be determined.

Likewise, the relationship between surface irregularity and wavefront error can also be determined for any level of under-filling of the surface by the beam footprint.

Useful relationships derived from this model are

Where the surface is a mirror, n = -1.

The second of the foregoing equations relates angular aberration to wavefront error. Oschmann (SPE-S-G0041) has defined a corresponding relationship for the Gemini telescope. According to this, 50 nm RMS wavefront error corresponds approximately to a 0.01″ degradation in 50% encircled energy diameter at 2.2 μm. Assuming that 50% encircled energy is equivalent to the RMS angular diameter, these two relationships agree. This suggests that the astigmatism model is reasonable.

1.9 Lens Tilt

The lenses in NIFS are mounted as shown in Figure 171. The housing and lens diameter tolerances are arranged so that the minimum clearance is zero. The maximum clearance is then the quadrature sum of the tolerances. Lateral displacement of the lens within its clearance zone then causes the un-supported lens surface to tilt.

Figure 171: Lens mounting geometry.

Applying conventional precision engineering tolerances (IT7 for the housing and IT6 for the lens), the maximum eccentricity of the lens is

The corresponding tilt of the un-supported lens surface is

This is independent of which face is supported, and proportional to the lens power.

1.10 Refractive Index Models

The dispersion equation used here (Sellmeier 1) is

Where n is the refractive index, λ is the wavelength (μm), and K_i and L_i are fitted constants or functions of temperature T (K). This model is physically meaningful, and so is capable of producing accurate results over the entire transparent wavelength range of the material to which it is applied. As applied here, the results are accurate to five decimal places within the specified temperature and wavelength ranges. Numerical values are shown in Table 38 for selected temperatures and wavelengths.

1.10.1 Calcium Fluoride

Tropf (1995) has fitted the Sellmeier dispersion model to data for calcium fluoride over the temperature range 93.2 - 473.2 K. From this

The accuracy of these constants is uncertain at the spectrograph body temperature (70 K) because this is somewhat outside the specified fitting range (93.2 - 473.2 K). No data have been found which are more relevant.

1.10.2 Fused Silica

Insufficient data have been found to allow the Sellmeier constants to be determined as functions of temperature. The constants for ambient and cryogenic temperatures are therefore determined from separate data.

1.10.2.1 Laboratory Temperature

From Malitson (1965), the Sellmeier constants for fused silica at a temperature of 20 C (293.15 K) are:

1.10.2.2 Detector Enclosure Temperature

The only refractive index data found for fused silica at appropriate wavelengths and temperatures were those included in §3.10.2 of the Gemini Near Infrared Imager (NIRI) Critical Design Review document. It is not known how these data were derived. Although the NIRI document sites several references, none of these seems relevant. Nevertheless, the data are duplicated here in Table 78.

Table 78: Refractive Index Data for Fused Silica at a Temperature of 60 K.

Wavelength (μm)	Refractive Index
1.0	1.44906
1.1	1.44783
1.2	1.44666
1.3	1.44551
1.4	1.44436
1.5	1.44317
1.75	1.44002
2.0	1.43649
2.2	1.43335
2.4	1.42988
2.6	1.42605

Fitting the Sellmeier model to this data gives the following constants:

1.10.2.3 Spectrograph Body Temperature

In the absence of specific data, the Sellmeier constants were derived for a temperature of 70 K by fitting the model to a set of refractive indices estimated by linear interpolation between the values shown in Table 78 and the corresponding values at ambient temperature. The resulting constants are:

1.10.3 Zinc Selenide

Tropf (1995) has fitted the Sellmeier dispersion model to data for zinc selenide over the temperature range 93.2 - 473.2 K. From this

1.10.4 Vacuum

The refractive index of a vacuum (relative to standard air) is simply the reciprocal of the absolute refractive index of standard air. Edlén (1953) has modeled the absolute refractive index of standard air over UV, visible, and near-infrared wavelengths as

Fitting the Sellmeier equation to the reciprocal of this function gives the constants for a vacuum as

1.11 Thermal Strain Models

Thermal strain models have been constructed for the various structural and optical materials used in NIFS by fitting polynomials to published data. For a temperature T (K), the thermal strain is

The number of terms used and the resolution of the coefficients are chosen to provide a fitting accuracy of five decimal places over the temperature range specified. The resulting coefficients are listed in Table 79. The data sources and applicable temperature ranges are listed in Table 80.

Numerical values of thermal strain are listed in Table 39 for relevant operating temperatures. The accuracy of these for zinc selenide is uncertain because the operating temperatures are outside the fitting range. No better data have been found for this material.

Table 79: Thermal Strain Polynomial Coefficients.

Material	Polynomial Coefficient
Material	k₀	k₁	k₂	k₃	k₄	k₅	k₆
Grade 6061 Aluminum Alloy	-0.004197	-1.64e-07	-1.477e-09	9.6465e-10	-5.6149e-12	1.3533e-14	-1.2222e-17
Grade 304 Stainless Steel	-0.002818	-8.217e-06	1.4105e-07	-4.7114e-10	8.6678e-13	-6.594e-16	-
OFHC Copper	-0.003253	-6.10e-07	7.621e-09	8.1588e-10	-5.2462e-12	1.3269e-14	-1.2228e-17
Calcium Fluoride	-0.003082	8.95e-07	-5.4373e-08	1.1580e-09	-5.7739e-12	1.2987e-14	-1.1275e-17
Fused Silica	0.000102	-1.344e-06	4.145e-09	-2.65e-12	-	-	-
Zinc Selenide	-0.001000	-7.437e-06	8.4670e-08	-2.7462e-10	4.8475e-13	-3.5086e-16	-
Sapphire	-0.000538	6.0e-08	-5.555e-09	6.202e-11	-7.651e-14	-	-

Table 80: Thermal Strain Data Sources.

Material	Temperature Range (K)	Data Source
Grade 6061 Aluminum Alloy	6 - 350	Infrared & Electro-Optical Systems Handbook, Vol 3, Table 6.15
Grade 304 Stainless Steel	23.15 - 373.15	American Institute of Physics Handbook, Table 4f-3
OFHC Copper	6 - 350	Infrared & Electro-Optical Systems Handbook, Vol 3, Table 6.17
Calciun Fluoride	6.4 - 321.93	Journal of Chemical Physics, Vol 41, No 8, 2324
Fused Silica	23.15 - 373.15	American Institute of Physics Handbook, Table 4f-3
Zinc Selenide	93.15 - 393.15	NSB Technical Note 993, US Gov Printing Office, Washington DC, 1979
Sapphire	23.15 - 373.15	American Institute of Physics Handbook, Table 4f-3

1.12 Optical Prescription

The optical prescriptions for cryogenic (cold) and manufacturing (hot) conditions are listed in Table 81 and Table 82, respectively. A global coordinate system is used, with the first surface (the field mask) being located at the origin. The optical axis is coincident with the Z-axis until it is deviated by the focal converter mirror. The location of each surface is specified by the X-Y-X coordinated of its vertex. The orientation of each surface is specified by its Alt-Az coordinates. The Az-axis is parallel to the X-axis, and when Alt and Az are both zero, the surface is perpendicular to the Z-axis at its vertex. Surface radius is positive if the surface is convex when viewed in the +Z direction with Alt and Az both zero.

The prescriptions are for the mid IFU channel only. Other channels are displaced about the fanning axis as described in §4.6.9. Likewise, the image slicer slitlets are displaced about the fanning axis as described in §4.4.2. With respect to Table 81 and Table 82, the fanning axis of the image slicer passes through the vertex of the image slicer and is displaced from the X-axis by Az = -90° and Alt = -4°.

Where double entries are shown, the first value is in or about the X-direction, and the second in or about the Y-direction (without Alt-Az offset).

For reference purposes, the Zemax file is listed in §12.12.3.

1.12.1 Cold Configuration

The optical design is optimized for cryogenic conditions, taking account of refractive index changes described in §4.11.1. Except for camera lens 5 and the detector, all components are operated at 70 K. Camera lens 5 operates at 65 K, and the detector at 60 K. No particular grating is listed in Table 81, but they are all replicated onto aluminum alloy substrates, and the thermal effect on groove density has been accounted for. Likewise, thermal strain in the sapphire substrate of the detector has also been accounted for.

Table 81: Cold Optical Prescription of Spectrograph.

Surface Description	Leading Medium	Surface Radius (mm)	X (mm)	Y (mm)	Z (mm)	Az (deg)	Alt (deg)	Size (mm)
Field Mask	Vacuum	∞	0.0000	0.0000	0.0000	0	0	1.992 1.922
Focal Converter Mirror	Vacuum	-127.4880	0.0000	0.0000	67.7280	0	-3	Ø 9.960
Cold Stop Mirror	Vacuum	∞	6.6998	0.0000	3.9840	0	-1.5	Ø 3.984
Fold Mirror 1	Vacuum	∞	15.3433	0.0000	168.9116	45	0	19.920 19.920
Fold Mirror 2	Vacuum	∞	18.5276	60.7594	168.9116	-90	15.193	19.920 19.920
Fold Mirror 3	Vacuum	∞	-125.1869	-216.6597	168.9116	90	-13.193	26.892 99.600
Image Slicer Mirror	Vacuum	-622.8586	-133.6141	266.1394	168.9116	-90	0	1.024 29.718
Pupil Mirror Array	Vacuum	52.4992 52.9573	-141.4313	-181.7049	168.9116	90	4	4.980 1.918
Field Mirror Array	Vacuum	-59.7759 -59.5409	-145.8106	-154.0550	168.9116	-90	-4	4.980 1.857
Fold Mirror 4	Vacuum	∞	-146.8698	-214.7372	168.9116	90	5	17.928 99.600
Collimator Mirror	Vacuum	863.0399	-220.6180	162.9942	168.9116	-90	-8.6	53.784 149.400
Fold Mirror 5	Vacuum	∞	-178.9271	-220.4664	168.9116	90	12	53.784 99.600
Collimator Lens A	Vacuum	227.7467	-255.7825	19.1943	168.9116