3.1.

Design

SPIFFI is equipped with four diffraction gratings. The gratings for J-, H-, and K-bands are operated in the second order and the grating for the combined $\mathrm{H}+\mathrm{K}$-band in the first order. All four diffraction gratings are directly ruled on identical blanks made of 6061 aluminum alloy. The dimensions of the grating blanks are $160\times 140\times 20{\mathrm{mm}}^3$. Seven lightweighting holes with 15-mm diameters are drilled through the blank in the $X$- and $Y$-directions (the $Z$-direction is normal to the grating surface). Figure 2 shows a CAD model of the grating blank that was used for the simulations in this paper. The aluminum blank was electroless nickel–phosphorus (NiP) plated with a layer thickness between 100 and $200\mu \mathrm{m}$ on all the surfaces. The two large faces of the NiP-plated blank were then polished symmetrically to be flat. The remaining NiP layer on these surfaces is between 50% and 90% of the original layer thickness. Finally, a 2- to $3\text{-}\mu \mathrm{m}$-thick gold layer was applied on the front surface before the gratings were ruled.

Fig. 2

CAD model of the aluminum grating blank with the lightweighting holes used for the simulations. The real blank has additional small notches and holes for fixation screws in the corners.

3.2.

Finite Element Analysis

To quantify bimetallic stresses and the resulting deformations of the grating surface when the grating is cooled down from room temperature to the operating temperature of SPIFFI at 80 K, we implemented an FEA of the aluminum grating blank with the NiP coating. Unfortunately, we only roughly know the original layer thickness and a range of possible values for the thickness postpolishing. We therefore simulated three layer thicknesses (100, 150, and $200\mu \mathrm{m}$) that span the range provided by the manufacturer, as well as polishing steps that left 50%, 75%, or 100% of the original layer thickness. Figure 3 shows the deformation resulting from the three polishing values for an original layer thickness of $200\mu \mathrm{m}$.

Fig. 3

FEA simulation of the grating surface deformation when cooling a grating blank with an original NiP layer thickness of $200\mu \mathrm{m}$ from room temperature to 80 K. From left to right, the remaining layer thickness after polishing are 100%, 75%, and 50%. The peak-to-valley in the center region ($P/V_{\text{center}}$ values) are calculated for rectangle of $82\mathrm{mm}\times 72\mathrm{mm}$ in the center of the grating corresponding to a grid of four lightweighting holes. The deviations at the edges are larger. All images are on the same color scale.

The bimetallic stresses deform the surface resulting in a regular grid structure. We define the $X$-direction to be the dispersion direction and the $Y$-direction to be the ruling direction, therefore, the period of the grid structure is $D_x=21\mathrm{mm}$ and $D_y=18\mathrm{mm}$. Of particular interest is the deformation in the central parts of the grating, since the outer parts and especially the corners of the grating are vignetted by a pupil mask. The pupil mask also covers the areas that may be affected by the small notches and holes for fixation screws in the real grating blank. The peak-to-valley ($P/V$) deformation in the central region of $82\mathrm{mm}\times 72\mathrm{mm}$ (oriented on the lightweighting holes) is listed in Table 1 for the nine gratings simulated. We chose to report the $P/V$ in this region as a proxy for the amplitude of the periodic deformation, as the deformation in this region is the most regular and is not affected by edge effects.

Table 1

FEA simulated P/V deformation of the grating in the central region of 82  mm×72  mm (oriented on the lightweighting holes) for different layer thicknesses and polishing depths. The P/V deformation at the edges is larger by up to a factor of two due to edge effects.

In general, thicker original layers result in more deformation. Additionally, for a given original layer thickness, more surface polishing results in more deformation. This indicates that mismatched layer thicknesses between the inside and outside of the lightweighting structure result in more deformation; however, given the specific geometry of our grating blank, we still found deformation even with matched layer thicknesses. Simulations with thicker NiP on the surface than inside the lightweighting holes are not presented here, as this is not a realistic scenario given the manufacturing process of the grating. However, the edges cases (NiP only inside the holes or only on the surface) are presented in Appendix B, and show that the deformations induced by these two layers oppose each other.

Given the large range in $P/V$ deformation values for relatively small changes in the amount of polishing done to the grating blanks in the FEA, and our unknown layer thicknesses, a cryogenic measurement had to be performed to get an idea of the actual deformation of the SPIFFI diffraction gratings.

3.3.

Cryogenic Measurements

To measure cryogenic bimetallic bending effects of the grating and verify our FEA analysis, we set up the SPIFFI J-band spare grating on a rotatable stage in a cryogenically cooled vacuum chamber at the Max Planck Institute for Extraterrestrial Physics (MPE). The grating was installed on a stress-free mount with heat straps from the cold plate attached to the four corners of the grating using the small holes for fixation screws. A temperature sensor was installed directly on top of the grating near one corner using another of these fixation screw holes.

For the measurements of the grating surface, we used a FISBA $\mu$Phase^® 500 interferometer together with its beam expander lens providing a 152.4-mm diameter output beam. The interferometer is a Twyman–Green phase-shifting type, operated with a stabilized He–Ne laser at a wavelength of $632.8\mathrm{n}\mathrm{m}$. The interferometer and beam expander lens were placed outside of the cryostat and we measured the grating surface in Littrow configuration (see Fig. 4) through an optical quality cryostat window. The diameter of the cryostat window is 137 mm, thus this is also the limiting size for the interferogram of the grating surface.

Fig. 4

Layout of the grating surface measurement at cryogenic temperatures. The SPIFFI J-band spare grating is mounted on a rotatable stage in a test cryostat. Through an optical window (137 mm diameter), the grating surface deformation is measured interferometrically in Littrow configuration via a static FFA.

The test cryostat at MPE could not be shielded easily against vibrations, thus it was not possible to measure the grating surface in the phase scanning mode of the interferometer. We used a static fringe analysis. Therefore, the phase value of a certain pixel cannot be measured independently, thus the phase sign gets lost, and the results are less accurate than a phase-shifting measurement. In exchange, only one image is needed to complete the measurement, so it is fast enough for us to obtain interferograms in unstable environments. We used the Fourier fringe analysis (FFA)¹⁵ that is very effective if the surface deformations are small ($<\lambda /5$). For this kind of static fringe analysis, a basic tilt has to be applied to the grating surface resulting in a carrier frequency in the interferogram. These fringes are afterward removed in the Fourier domain to obtain a surface deviation map. The filtering used in the Fourier domain affects the final $P/V$ value measured for higher-frequency deformations, and thus care must be taken to ensure an accurate measurement.

The result of the FFA measurement is shown in Fig. 5. The surface deviation shown is the difference between the cold measurement at 128 K and the warm measurement at room temperature, to show only the deformation due to cryogenic bimetallic stresses. The $P/V_{\text{center}}$ deformation is $0.80\mu \mathrm{m}$; however, this value is also affected by the oblique astigmatism that can be recognized in the interferogram. Without this astigmatism, the $P/V$ value is $\sim 0.60\mu \mathrm{m}$. This lies comfortably in the range of values simulated for the NiP layer and polishing thicknesses provided by the manufacturer.

Fig. 5

Image of the surface deformation from cooling the SPIFFI spare J-band grating from room temperature to 128 K. The measurement shown is (cold surface)—(warm surface) to show only the effects of cryogenic deformation. The aperture of the measurement was limited by the size of the cryostat window. The $P/V_{\text{center}}$ of $0.80\mu \mathrm{m}$ is calculated over the same area as the FEA. The color scale on this measurement is the same as in Fig. 3 for easy comparison.

In our measurement campaign, the lowest temperature reached at the grating was 103 K. The grating is heated by thermal radiation through the cryostat window, preventing the grating from reaching a lower temperature. The surface measurement taken at a temperature of 103 K was found to be identical to the one at 128 K to within the measurement uncertainty, although the quality of the 103 K measurement was not as good as the measurement at 128 K shown in Fig. 5. It should also be noted that this was a measurement of a single spare grating–each grating blank was polished separately, so it is likely that the NiP layer thicknesses are different on the different gratings.

4.1.

Setup and Parameters

In order to simplify the model, we adopt an approximate setup for simulating the spectrograph. We assume a point source at the location of the image slicer. The light is collimated by a CODE V lens module with a focal length of 2873 mm [identical to the instrument collimator focal length, although in the instrument the collimator is a three-mirror anastigmat (TMA)]. We do not simulate the full TMA, as we have measured the TMA wavefront from the location of the spectrograph pupil and include it in the simulations as an interferogram applied to the CODE V collimator lens module. The diffraction gratings for the J, H, K, and H + K bands are set up as listed in Table 2. The diffracted light is then imaged onto the focal plane by a CODE V lens module with a focal length of 342 mm, identical to the instrument camera. The setup allows us to place an interferogram of the surface deformation of the grating on the CODE V grating surface. Due to the anamorphic distortion of the diffraction gratings, the slit width geometrically imaged onto the detector is $27\mu \mathrm{m}$ using the J, H, and K band gratings and $29\mu \mathrm{m}$ using the H + K band grating.

Table 2

Table of grating parameters for the SPIFFI instrument used in our CODE V simulation. For the gratings, (reflection angle)—(incidence angle) = 45 deg. All parameters in the simulation are for the 80-K operating temperature.

Figure 6 shows the layout of the simulation for the case of the J-band grating. The figure also shows a “black box” preoptics and the instrument cold stop. The preoptics act as a scale changer for the incoming light, and for a fully illuminated slit (the case for our spectral line profile measurements) the effect is to change the numerical aperture (NA) of the light entering the spectrograph slit for the different pixel scales. We account for this effect using the CODE V two-dimensional (2-D) partial phase coherence (PPC) function, which allows the user to specify the relative NA of the incoming light beam and an effective slit size. In Fig. 6, the shaded area shows the geometric light path for the $100\mathrm{mas}/\mathrm{px}$, while the solid (outer) lines show the light path for the $250\mathrm{mas}/\mathrm{px}$ scale.

Fig. 6

Layout of the optical simulation in CODE V for the J-band grating (not to scale). All other gratings were set up in the same way. The wavelengths shown in blue, green, and red are 1.10, 1.25, and $1.40\mu \mathrm{m}$, coving the J-band. The preoptics changes the NA of the light entering the image slicer. The shaded area shows the geometric light path for the $100\mathrm{mas}/\mathrm{px}$ scale, while the solid (outer) lines show the light path for the $250\mathrm{mas}/\mathrm{px}$ scale. The $25\mathrm{mas}/\mathrm{px}$ scale is not shown, but would have an even smaller light cone than the $100\mathrm{mas}/\mathrm{px}$ scale.

Unless otherwise noted, throughout this section, the default grating deformation used in the simulation is the one calculated by the FEA with an original layer thickness of $200\mu \mathrm{m}$ and 75% of the layer thickness ($150\mu \mathrm{m}$) remaining after polishing. Additionally, throughout this section, we show results only for the J- and K-bands, as the behavior of the H-band is intermediate between the two. H-band results are discussed in Sec. 5.

4.2.

Point Spread Function

To see the diffraction effects of the grating surface deformation, we begin by calculating a PSF for a point source. On the grating, we place an interferogram of the grating surface deformation. Figure 7 shows the resulting PSFs for the deformation calculated by the FEA for the J- and K-band setups. Note that the grid structure of the PSF has a spacing that is a function of wavelength, as expected from diffraction theory. In J-band, the spacing of the peaks is approximately one detector pixel ($18\mu \mathrm{m}$) in the focal plane, while in K-band the spacing is $\sim 2$ detector pixels ($36\mu \mathrm{m}$) in the focal plane.

Fig. 7

Examples of the PSFs produced in the optical simulation for FEA calculated grating surface for wavelengths in the center of (a) J-band ($1.25\mu \mathrm{m}$) and (b) K-band ($2.2\mu \mathrm{m}$). The spectral direction is the $X$-direction, while the slit extends in the $Y$-direction.

At shorter wavelengths, diffraction orders farther from the center grow in strength relative to the central order. This is the general result expected for sinusoidal deformations as the wavelength approaches the deformation amplitude.¹⁶ Additionally, for deformation amplitudes comparable to $\sim \lambda /2$ or larger, the central order(s) can be suppressed.^16,17 For the largest deformation amplitudes tested, we see central order suppression especially in J-band. This is explored in the next section.

4.3.

Line Spread Function

The spectral line profile is the measured line spread function of the instrument. This section shows how the cryogenic deformation of the diffraction gratings affects the spectral line profile of the instrument.

4.3.1.

Basic LSF

In its most basic form, the line spread function of the instrument is simply a cross section of the convolution of the PSF with the image of the slit in the focal plane. In SPIFFI, the slit image is $27\mu \mathrm{m}$ wide in the focal plane. To build intuition about how the amplitude of the grating deformation affects the line profile of the instrument, in Fig. 8 we plot this “basic LSF” for both an infinitely thin slit (equivalent to collapsing the PSF plotted in Fig. 7 along the slit direction), and the actual SPIFFI slit size for two different amplitudes of the grating deformation. In general, as the amplitude of the deformation increases, more power shifts from the central peak to the shoulders, and for large amplitudes, the central diffraction order is suppressed.

Fig. 8

Examples of basic LSFs produced by convolving the slit image with the PSFs produced by the optical simulation for two different amplitudes of FEA calculated distortion as well as a “perfect grating.” (a) The result for an infinitely thin slit and (b) the result for the SPIFFI slit width. LSFs are shown for wavelengths at the center of J-band ($1.25\mu \mathrm{m}$, left) and K-band ($2.2\mu \mathrm{m}$, right). In general, as the amplitude of the grating deformation increases, power shifts from the central peak to the shoulders.

By construction, the LSF produced in this way does not include any diffraction effects from the slit. The “horns” that appear in the J-band LSF in the lower left plot in Fig. 8 are a result of the fact that the spacing of the diffraction orders is smaller than the slit width in J-band, and they apprear purely from the convolution of the PSF shown in Fig. 7 with the slit image. In the next section, we will include diffraction effects from the slit.

4.3.2.

Partial phase coherence LSF

As discussed in Sec. 2, the line profiles vary depending on the pixel scale selected (25, 100, or $250\mathrm{mas}/\mathrm{px}$). Since our line profile measurements are taken with the slit fully illuminated by a flat-field source, the only difference among the three pixel scales is the NA of the incoming light illuminating the slit. This has two consequences. The first effect can be seen in Fig. 6, where for smaller pixel scales the geometric light path through the instrument results in smaller beam footprints on the spectrograph optics. The second is that the incoming light at the image slicer will diffract differently at the slit edges depending on the illumination NA. We can model both effects in CODE V using the 2-D PPC function, which includes the full diffraction effects of partially coherent light propagating through the optical system.

In the setup of the 2-D PPC, we use a rectangular object that has an equivalent slit width of $27\mu \mathrm{m}$ and length of $400\mu \mathrm{m}$ in the focal plane. The actual SPIFFI slit length is 1152 mm in the focal plane, but the 400 mm length reduces simulation times and is sufficiently long to avoid edge effects. For the three pixel scales $[250,100,25]\mathrm{mas}/\mathrm{px}$, we use a relative NA of the incoming light of [1.0, 0.4, 0.1] with respect to the $250\mathrm{mas}/\mathrm{px}$ scale. The left column of Figs. 9 and 10 shows the LSF in the center of the slit that results from the CODE V PPC calculation in the three pixel scales for the case of a perfect grating (Fig. 9) and a single amplitude of the grating deformation (Fig. 10).

Fig. 9

Examples of 2-D PPC LSFs in all the three pixel scales for a perfect grating. LSFs are shown in (a) for wavelengths at the center of J-band ($1.25\mu \mathrm{m}$) and in (b) for the center of K-band ($2.2\mu \mathrm{m}$). The left column includes the effects of PPC, the middle column additionally includes the collimator wavefront error (measured on the mirrors installed in January 2016), and the right column additionally includes the effects of the detector pixel function, and thus depicts the simulated detected line profiles. The slit diffraction effects are largest in the $25\mathrm{mas}/\mathrm{px}$ scale in both J and K bands, and the collimator wavefront error shifts power to the left. The detector pixel function smooths out the detected line profile, such that not much difference is seen between J and K band simulated detected line profiles for the case of a perfect grating.

Fig. 10

Examples of 2-D PPC LSFs in all three pixel scales with a grating deformation from the $200\text{-}\mu \mathrm{m}$-thick NiP polished to 75% thickness ($150\mu \mathrm{m}$). LSFs are shown in (a) for wavelengths at the center of J-band ($1.25\mu \mathrm{m}$) and in (b) for the center of K-band ($2.2\mu \mathrm{m}$). The left column includes the effects of PPC, the middle column additionally includes the collimator wavefront error (measured on the mirrors installed in January 2016), and the right column additionally includes the effects of the detector pixel function, and thus depicts the simulated detected line profiles. The size of the shoulders is significantly smaller in the $25\mathrm{mas}/\mathrm{px}$ scale in both J and K bands, and the collimator wavefront error shifts power into the left shoulder. The detector pixel function smooths out the detected line profile.

In general, diffraction at the slit edges is largest in the smallest pixel scale due to the smaller NA of the illumination. This appears as “ringing” at the slit edges. The ringing is larger at smaller pixel scales (smaller NA) and at longer wavelengths. The behavior with NA and wavelength are both expected from the diffraction theory.¹⁶ Note that in the J-band plots of Fig. 10, the “ringing” is confused with the “horns” (which are a result of the diffraction order spacing). An additional effect that appears in the 2-D PPC simulation is that the size of the shoulders is significantly smaller in the 25 mas pixel than the larger pixel scales in both J and K bands. This is due to the smaller beam footprint on the diffraction grating, minimizing the effect of the periodic deformation of the grating surface.

4.3.5.

Effect of pupil position on the LSF

During the upgrade in 2016, we moved the pupil position on the grating slightly in the $X$-direction (the dispersion direction). Additionally, the pupil position on each individual grating is slightly different. We explored the effect of the pupil position on the line profiles by shifting the pupil and repeating the simulation. We shifted the pupil by $\pm 10\%$, which for our entrance pupil diameter of 110 mm corresponds to a shift of $\pm 11\mathrm{mm}$ on the grating surface. The periodic structure in the $y$-direction has a period of 21 mm, so this pupil shift magnitude covers all phases of the periodic structure. Figure 11 shows the line profiles for a range of pupil positions in the $25\mathrm{mas}/\mathrm{px}$ scale for the $200\text{-}\mu \mathrm{m}$-thick NiP layer polished to 50% ($100\mu \mathrm{m}$) thickness. The change in line profile with pupil position is smaller for the grating deformations with smaller amplitudes, hence, we show here the largest amplitude grating deformation to highlight the effect. The 100 and $250\mathrm{mas}/\mathrm{px}$ scales are not shown, since there is essentially zero variation of the line profiles with pupil position in these pixel scales.

Fig. 11

Line profiles in the (a) J- and (b) K- bands for a range of pupil positions in the $25\mathrm{mas}/\mathrm{px}$ scale for the grating deformation from the $200\text{-}\mu \mathrm{m}$-thick NiP layer polished to 50% ($100\mu \mathrm{m}$) thickness. The 100 and 250 mas scales are not shown, since there is essentially zero variation with pupil position in these pixel scales.

The results of the pupil shift simulation show that the amplitude and shape of the $25\mathrm{mas}/\mathrm{px}$ scale line profiles depend on the exact pupil position on the grating, while exact pupil position has very little effect on the $100\mathrm{mas}/\mathrm{px}$ lines, and almost no effect on the $250\mathrm{mas}/\mathrm{px}$ lines. This can be understood when one considers that due to the low NA of the light entering the image slicer in the smallest pixel scale, only a small portion of the grating is illuminated in the $25\mathrm{mas}/\mathrm{px}$ scale, on the order of a single period of the grating deformation (neglecting the diffraction effects of the slicer on the illumination). Figure 6 shows schematically the effect of smaller NA illumination on the beam footprint on the grating. Thus for the $25\mathrm{mas}/\mathrm{px}$ scale, the shape of the wavefront after the grating depends more strongly on the pupil position. Since the grating shows periodic surface deformations on scales smaller than the illuminated pupil size in the 100 and $250\mathrm{mas}/\mathrm{px}$ scales, for these pixel scales, the effect of a pupil shift on the line profiles is much smaller, because the Fourier transform of the surface deformation is nearly identical for small relative shifts on a periodic surface. Additionally, the effect is larger at smaller wavelengths, in our simulation, only the J-band showed very strong line profile variation with pupil position.

5.1.

K-Band

In K-band, the simulation using the grating deformation from a $200\text{-}\mu \mathrm{m}$-thick NiP layer polished to 75% ($150\mu \mathrm{m}$) thickness matches the K-band data quite well, shown in Fig. 12. The simulation with pupil centered on the grating is also adequate to describe the data in the $25\mathrm{mas}/\mathrm{px}$ scale, although the line profile variation in K-band with a shifted pupil is rather low [see Fig. 11(b)], so it would be difficult to tell if the pupil were shifted on the grating. The measured line profiles also show a low variation along the slitlet (see Ref. 7 for an example of this small variation).

Fig. 12

(a) K-band data versus (b) simulation using the grating deformation from a $200\text{-}\mu \mathrm{m}$-thick NiP layer polished to 75% ($150\mu \mathrm{m}$) thickness in all three pixel scales.

5.2.

H-Band

In H-band, the comparison is also straightforward. As in the K-band case, the simulation using the grating deformation from a $200\text{-}\mu \mathrm{m}$-thick NiP layer polished to 75% ($150\mu \mathrm{m}$) thickness matches the H-band data, shown in Fig. 13. As in K-band, having the pupil centered on the grating is also adequate to describe the data in the $25\mathrm{mas}/\mathrm{px}$ scale, although a shift left or right of a few percent would give similar results, as the profile variation with pupil shift is low in H-band with this deformation amplitude. The measured line profiles also show a low variation of the line profiles with slitlet position. (see Ref. 7 for an example of this small variation).

Fig. 13

(a) H-band data versus (b) simulation using the grating deformation from a $200\text{-}\mu \mathrm{m}$-thick NiP layer polished to 75% ($150\mu \mathrm{m}$) thickness in all three pixel scales.

5.3.

J-Band

As already mentioned in Sec. 2, the J-band line profiles appear different from the other two bands. The most obvious difference is the double-peaked structure, most visible in the 100 and $250\mathrm{mas}/\mathrm{px}$ scales. Additionally, there is an obvious asymmetry in the peak heights in the $25\mathrm{mas}/\mathrm{px}$ scale. Given the results of the simulations, it became clear that to reproduce the J-band line profiles, a larger amplitude grating deformation was required to get the central order suppression necessary to result in a double-peaked line profile. Of the layer thicknesses/polishing simulated, we found that the $200\text{-}\mu \mathrm{m}$-thick layer with 50% ($100\mu \mathrm{m}$) thickness remaining after polishing gave results that matched the simulations the best. Figure 14 shows the measured line profiles and the best-matching simulation from the 200/100 NiP layer deformation.

Fig. 14

(a) J-band data versus (b) simulation using the grating deformation from a $200\text{-}\mu \mathrm{m}$-thick NiP layer polished to 50% ($100\mu \mathrm{m}$) thickness in all three pixel scales. In the simulated line profiles, the pupil position has been shifted $\sim 4\mathrm{mm}$ in the dispersion direction to reproduce the asymmetric peak heights in the $25\mathrm{mas}/\mathrm{px}$ scale.

In addition, the J-band line profiles in the $25\mathrm{mas}/\mathrm{px}$ scale vary strongly across a single slitlet. We could reproduce this behavior by varying the pupil position on the grating. One possible source of movement of the pupil position is the image slicer, which is described in detail in Ref. 18. A torsional deformation of a slicer mirror with a measured amplitude of 2 HeNe fringes (close to what could be expected based on measurements of the slicer mirrors; see Ref. 1 for an example of a slicer interferogram) would result in the pupil shifting on the grating by $\sim \pm 10\mathrm{mm}$, which is nearly an entire period of the grating deformation function. Both the asymmetry in the $25\mathrm{mas}/\mathrm{px}$ scale line profiles and the variation in line profile along a slitlet can be explained with a shift in the pupil (due to, for example, a torsion in the slicer mirrors) in combination with the grating deformation.

Figure 15 compares the simulated and measured line profile variations in the smallest pixel scale. A shift in the central pupil position of $\sim 4\mathrm{mm}$ in the dispersion direction on the grating can provide the necessary asymmetry to reproduce the behavior of the central $25\mathrm{mas}/\mathrm{px}$ line profile (shown in Fig. 14); however, to fully explain the variation in the line profiles over the full length of the slitlets, a continuously varying pupil position is required. If a torsion in the image slicer is the reason for a continuously varying pupil position, then we expect the same pupil position variation on all diffraction gratings. This variation is only noticeable in the measurements on the J-band grating. As noted in the previous two sections, both measurements and simulations of the H- and K-band line profiles showed very low variation with pupil position on the grating.

Fig. 15

(a) J-band data showing the variation in line profile over a single slitlet versus (b) simulation showing the line profile variation with a pupil shift of $\pm 10\%$.

In the J-band, it is clear that the simulation is not a perfect reproduction of the line profiles—the 100 and $250\mathrm{mas}/\mathrm{px}$ scales simulations are slightly wider and the central dip is less pronounced, while in the $25\mathrm{mas}/\mathrm{px}$ scales the right shoulder is less obvious. To explore this, we also tested purely sinusoidal grating deformations with a period equal to the spacing of the lightweighting holes and a variety of $P/V$ values. A sinusoidal deformation with a $P/V$ of $0.58\mu \mathrm{m}$ also gives results that match the measured line profiles reasonably well, but with narrower line profiles. This indicated to us that while the primary component of the grating deformation that affects the line profiles is the amplitude of the Fourier component corresponding to the period of the lightweighting holes, the exact shape of the deformation (i.e., other Fourier components of the deformation) play a role in the width of the profiles. Without a cryogenic measurement of the gratings that are currently in use in SPIFFI, we cannot know exactly what the deformation amplitude and shape of each grating is. However, we can say with confidence that to reproduce the behavior seen, the J-band grating needs to have a different and higher deformation amplitude than the other two gratings. This is discussed more in the following section.

5.4.

Single Wavelength Measurements on Multiple Gratings

To cement our result, we attempted to reproduce the measurements that led us to suspect the grating blank was causing problems in the first place—namely, producing simulations of a single wavelength on the three diffraction gratings. We choose a wavelength at the center of J-band ($1.25\mu \mathrm{m}$), as shorter wavelengths are more sensitive to the details of the grating deformation. The simulation setup uses the K-, H-, and J-band gratings in the fourth, third, and second diffraction orders, respectively. The grating deformations used in the simulation are the ones needed to reproduce the in-band results of the previous sections, namely the deformation from NiP layer thicknesses of $200/150\mu \mathrm{m}$ for the H- and K-band gratings, and NiP layer thicknesses of $200/100\mu \mathrm{m}$ for the J-band grating.

Figure 16 shows the results for $1.25\mu \mathrm{m}$ in the $250\mathrm{mas}/\mathrm{px}$ scale. The agreement between measurement and simulation is quite good, although the line profile is clearly higher and narrower in the H- and K-band grating simulations than in the measurements.

Fig. 16

(a) J-band ($1.25\mu \mathrm{m}$) wavelength line on all three diffraction gratings versus (b) simulation using the grating deformations found in the previous three sections. A slightly higher deformation amplitude on the H- and K-band gratings results in a better match to the data.

In the smaller pixel scales, there is a similar slight disagreement between the simulation and measurements. (Figure 17 in Appendix A shows these measurements in all pixel scales and wavelengths). We were able to more closely reproduce the measured line profiles by increasing the amplitude of deformation on the H- and K-band gratings by $\sim 20\%$, which was achieved by scaling the amplitude of the $200/150\mu \mathrm{m}$ simulation by a factor of 1.2. However, here we do not attempt to fine-tune the deformation amplitudes to match the measurements exactly—we simply note that slightly higher deformation amplitudes on the H- and K-band gratings (although still smaller than the deformation of the J-band grating) reproduce the measurements better.

Fig. 17

Measured LSF at single wavelengths on multiple diffraction gratings in all three pixel scales. (a) $25\mathrm{mas}/\mathrm{px}$, (b) $100\mathrm{mas}/\mathrm{px}$, and (c) $250\mathrm{mas}/\mathrm{px}$. The wavelengths shown are at roughly the center wavelength of each band, and are selected from the bright arc-lamp calibration lines (top row: $1.249\mu \mathrm{m}$, middle row: $1.652\mu \mathrm{m}$, bottom row: $2.208\mu \mathrm{m}$). Each line was measured on as many gratings as possible.

The results of this test leave us to conclude that in order to reproduce the entire data set we have obtained from 1.0 to $2.5\mu \mathrm{m}$ on three diffraction gratings, the H- and K-gratings need to be very similar to each other, while the J-band grating needs to be different. The primary difference required is that the deformation amplitude induced by the lightweighting structure needs to be larger on the J-band grating. We found in our notes on the production of the gratings that the J-band blank was repolished at least once to achieve higher surface flatness. The results of the FEA of the grating blanks show that more polishing results in higher deformation amplitudes, so this result is consistent with the information we have on the grating manufacturing.