2.1.

Physical Thickness and Individual Crystal Measurements at Meadowlark Optics

In this section, we show a sample of metrology performed by Meadowlark Optics as part of acceptance testing as well as internal DKIST metrology. We used a wide variety of metrology tools to verify the as-built retarders achieve the required performance. This ensures our subsequent performance models are valid as we have accurate knowledge of the components inside each assembly. We examine one of our six retarder optics and then in later sections show the impact of some manufacturing errors on the DKIST instrument performance. We denote the Pancharatnam theoretical $A-B-A$ design using the DKIST optic labeling convention for the CryoNIRSP SAR components: $G-H-G$. The numbering of each pair was sequential so the optics contains six crystals in the ordering $2G-3H-4G$. Within each compound zero-order retarder, the two crystals are oriented with fast axes within $\pm 0.3\mathrm{deg}$ of crossed. The G retarder is nominally two crystal retarders roughly 2-mm thick each representing 40 waves net retardance in each crystal. DKIST used the convention, where the plate with the larger desired net retardance was called the subtraction crystal, and the plate with the lower net retardance was called the bias crystal. For example, the $G$ compound zero-order retarder has a net retardance of 2.2 waves at 633-nm wavelength after the two crystals are assembled in subtraction with fast axes crossed. This design creates a compound true zero-order retarder at wavelengths $>2000\mathrm{nm}$. For the $H$ pair, the two retarders subtract to 3.46 waves net retardance at 633-nm wavelength.

Meadowlark Optics provided DKIST with several types of measurements to show the crystals meet manufacturing tolerances. The physical thickness was measured by a Heidenheim MT 60M metrology system with $\sim 0.5\text{-}\mu \mathrm{m}$ thickness accuracy. The individual crystals that make up the compound retarders were also tested spectrally to derive a net retardance for the assembled crystal pairs. This measurement was derived by fitting broadband spectra collected with a Varian spectrophotometer and mounting the crystal between crossed polarizers. Once the individual crystals were measured, they were assembled into compound retarders. These compound retarders were measured for net retardance at one wavelength.

In Table 3, we show a comparison of the design and measured parameters for the six crystals in this optic measured at the center of the CA. The first column shows the name of the crystal, whether $G$ or $H$ and whether bias or subtraction. The second column shows the nominal physical thickness in millimeters with all crystals. The third column shows the physical thickness measured with the Heidenheim tooling at accuracies of better than $\pm 0.001\mathrm{mm}$. The fourth column shows the design net retardance in waves.

Table 3

Measured CryoNIRSP SAR crystal properties.

The manufacturing tolerance was $\pm 1$ wave net for each individual crystal but with a much stricter requirement after the compound zero-order $G$ and $H$ pairs were assembled. The fifth column shows the measured net retardance of the crystal at 633-nm wavelengths. It was also supported by later measurements retardance with a spatial scanning system using a 599-nm central wavelength filter having a 10-nm FWHM bandpass. The sixth column of Table 3 shows the desired net retardance of the crystal pairs when mounted in subtraction. The $G$ pairs should have 2.230 waves net retardance, and the $H$ pair should be 3.346. The polishing tolerance was specified at $\pm 0.01$ waves net. The seventh column shows the actual measured net retardance. The last column shows the orientation (clocking) of each crystal.

2.2.

Meadowlark Optics Compound Retarder Metrology at Five Aperture Locations

In this section, we show how similar crystals can have substantially different uniformities when two crystals were mounted in subtraction to form compound zero-order retarders at their specific design bandpasses. We also show the polishing and mounting process for the pairs of ${\mathrm{SiO}}_2$ crystals that had very different retardance uniformities than for the pairs of ${\mathrm{MgF}}_2$ crystals. We were provided high-accuracy linear retardance measurements at five spatial locations for the two-crystal subtraction pairs manufactured by Meadowlark Optics. One point was at the center of the aperture while four other points were evenly distributed at a radius of 30 mm from the optical center representing the decentered edge of the 2.8-arc min field beam near Gregorian focus. These five-location tests were done after oiling and aligning of the crystal pairs but before final assembly and bonding of the six crystal stacks. Table 4 shows the metrology results for the final as-built parts when a 3-mm diameter test beam spot-checked the retardance.

Table 4

Compound zero-order achromat spatial polishing metrology results.

As part of final acceptance of the optics in 2017, we received Meadowlark uniformity data for the ViSP quartz crystal subtraction pairs. The retarder pairs had nominal targets of $A=0.476$ waves retardance at 633.443-nm wavelength. The $B$-type crystal pairs were designed as 0.328 waves net retardance. The superachromatic calibration retarder (SAR) for ViSP was configured $B-A-B$, and the modulator was $A-B-A$. Similarly, the DL-NIRSP retarders were designed as $D=1.000$ waves retardance and $C=0.683$ waves retardance. The SAR was configured with retarders in $D-C-D$ ordering, and the modulator was designed as $C-D-C$. For the CryoNIRSP retarders made of ${\mathrm{MgF}}_2$ crystal, the design did not utilize identical crystal thicknesses for calibration and modulation retarders. The modulator used 1.893 waves retardance for $E$ and 1.282 waves retardance for $F$ in an $E-F-E$ style design. The calibration retarder used 2.230 waves retardance for $G$ and 3.346 waves retardance for $H$ in a $G-H-G$ style design. We summarize the polishing nonuniformity for all six DKIST retarder designs in Table 5. The first column shows the optic name. The second column shows the sum of the peak-to-peak variations from Table 4 above. This sum is the maximum possible polish error, should the bicrystalline plates have the errors linearly stacked. The third column shows the sum of the RMS variations in Table 4.

Table 5

As-built summary.

Table 5 shows the peak-to-peak (P-P) and RMS polishing errors in hundredths of a wave retardance. For instance, the ViSP SAR has 0.0105 waves retardance variation peak-to-peak and is listed as 1.05 in Table 5. There are significant variations in delivered polish between the various optic designs. The ViSP designs came in with very similar peak-to-peak as well as RMS values. For the DL-NIRSP, the calibration retarder (SAR) has more than double the peak-to-peak variation and closer to triple the RMS variation. Both the ViSP and DL-NIRSP designs use ${\mathrm{SiO}}_2$ crystals at 2.1-mm physical thickness, but we see spatial variation properties varying significantly between the as-built optics. The CryoNIRSP designs used ${\mathrm{MgF}}_2$ crystals at roughly 2.2-mm physical thickness. A different polishing process was used on the softer ${\mathrm{MgF}}_2$ crystals, which resulted in polishing performance roughly three to eight times worse when summing the individual P-P and RMS crystal pair measurements.

We also need to ensure that the crystals were cut with the crystal axes parallel and perpendicular to the optical propagation direction. Crystal axis tilt errors were measured to be $<0.06\mathrm{deg}$. Given that the polishing errors can create spatially variable retardance, we neglect this slight spatial offset in the crystal axis.

For the DKIST retarders, the ${\mathrm{SiO}}_2$ crystal optics were a factor of roughly five less spatially variable in both peak-to-peak and RMS retardance variations than the ${\mathrm{MgF}}_2$ crystals. In the next section, we take these magnitudes of spatial retardance variation and compute simulations of polishing error sensitivity in some DKIST retarder designs and compare with example metrology results for spatial retardance mapping.

2.3.

Mueller Matrix Measurements with NSO Lab Spectropolarimeter

The National Solar Observatory Laboratory Spectropolarimeter (NLSP) uses two spectrographs to measure polarized spectra with rotating polarization optics a wire grid polarizing beam splitter as an analyzer. We use a fiber-coupled collimated light source stopped to a circular beam of 4-mm diameter using laser cut metal masks. A polarization state generator consists of a rotating wire grid polarizer and rotating third-wave achromatic linear retarder mounted upstream of the sample location. After the sample, a rotating third-wave linear retarder is mounted as a modulator. The final optic is a fixed orientation analyzing wire grid polarizer, which is also used as a polarizing beam splitter.

As detectors, we use visible and near-infrared spectrographs from Avantes. The visible spectrograph covers 380- to 1200-nm wavelength, and the NIR spectrograph covers 900- to 1650-nm wavelength. The beam transmitted through the wire grid polarizer feeds the visible spectrograph via filters, aperture stops, and a lens. At the lens focus, a fiber couples light to the spectrograph. The beam reflected off the wire grid polarizer is passed through a separate set of filter, aperture stop, and lens optics into the near-infrared (NIR) spectrograph. This NIR arm has an additional polarizer with wires parallel to the analyzer to remove the fresnel reflection component of the glass and maintain high contrast. We achieve reasonable signal-to-noise ratio from roughly 400- to 1600-nm wavelength with a single exposure. We achieve simultaneous measurements in both visible and near-infrared systems in the 950- to 1100-nm bandpass with reasonable signal-to-noise ratio. This allows us to estimate systematic effects and ensure stable results.

Figure 2 shows the NLSP measurements of the CryoNIRSP SAR along with the theoretical elliptical retarder model. The measurements have a spectral resolving power of roughly 290 at 380-nm wavelength rising to 800 at 1050-nm wavelength on the visible spectrograph. With the NIR spectrograph, we obtain resolving power of 200 at 950-nm wavelength rising to 460 at 1530-nm wavelength. With the as-built thicknesses of Table 3, we compare the theoretical predicted Mueller matrix shown in blue to the measurements shown in black. The optic was mounted in a rotary stage and the best-fit orientation of the theoretical model to the measurements gave an orientation of 155.5 deg.

Fig. 2

The NLSP measured CryoNIRSP SAR Mueller matrix and associated fits. The VIS and NIR spectrograph measurements have been spliced together at 1020-nm wavelength. The transmission is roughly 94% to 95% and all other Mueller matrix elements were normalized by this II term. The black curve shows the NLSP data. The green curve shows the theoretical Mueller matrix computed as a stack of six ideal linear retarders rotated to match the measured NLSP data set using the as-built crystal thicknesses. The blue curve shows the same theoretical calculation but with the design crystal physical thicknesses

Figure 2 shows excellent agreement between the measurements in black and theoretical as-built Mueller matrix from six ideal linear retarders in blue curve. The first row and column of the Mueller matrix show some small artifacts and depolarization at levels below 1%. The transmission is above 93.5% as expected for the refractive index matching oil between the ${\mathrm{MgF}}_2$ crystals and the Fresnel losses at the air interfaces. The retardance shows strong variation in the lower $3\times 3$ matrix elements. The Mueller matrix approaches an identity matrix representing full-wave integer retardance magnitude for wavelengths around 700 nm per both the design and measured crystal thicknesses. With the theoretical Pancharatnam equations and the as-built crystal thicknesses, we can construct a bounded fit for elliptical retardance parameters. We use the standard axis-angle formalism for retarders as rotation matrices. In this representation, the magnitude of the retardance is the root sum square (RSS) of the three individual retardance components. Each component of the retardance represents a rotation on the Poincaré sphere about a $QUV$ coordinate axis. We outline details of this standard retarder-as-rotation-matrix model Sec. 9 Appendix A.

Figure 3 shows fits of the elliptical retarder models to NLSP measurements. We can fit the design Mueller matrix with a retardance model whose magnitude is near two waves magnitude at 380-nm wavelength dropping to 1 wave magnitude around 700 nm with a decrease to quarter-wave at the design wavelength ranging from 2500 to 5000 nm. The black curve shows the elliptical retardance magnitude as the blue and green curves show the two components of linear retardance. The spectral variation in the blue and green curves represents the orientation of the fast axis of linear retardance. The red curve shows circular retardance.

Fig. 3

The elliptical retardance model fit to the NLSP measured Cryo SAR Mueller matrix. The black curve shows the elliptical retardance magnitude as the root-sum-squared of all three retardance components. The blue and green curves show the first and second components of linear retardance. The arctangent of the two components represents the fast-axis orientation with strong spectral changes visible in the data. The red curve shows circular retardance. We included as dashed lines an elliptical retarder fit to the theoretical Mueller matrix computed from the birefringence equations for the six crystals using the design physical thickness. We rotated the optic by $155.5\mathrm{deg}\pm 0.1\mathrm{deg}$ as the least-squares-fit to the orientation of the optic in the NLSP setup.

As this optic was designed to be a linear retarder, this term is near zero with visible ripples from clocking errors. The gap near 700-nm wavelength represents the degeneracy in this particular elliptical retarder solution as a full wave of rotation can have any arbitrary pole orientation. Fits diverge rapidly for solutions close to integer multiples of full wave magnitude. If an alternate model is chosen that is driven through zero retardance at 700 nm, no such ambiguities are seen. This is explored in more detail in Sec. 9 Appendix A.

When we assess the impacts of spatial uniformity variation on DKIST system calibration, we must know the complexity of the calibration model from the standpoint of fitting errors and model degeneracy as well as the expected magnitude of the various terms. The model of the optic may introduce unexpected complexities as we make attempts to simultaneously fit for time dependence of a thermal model and spatial variation of a retardance model while trying to minimize the number of exposures when observing at many simultaneous wavelengths with reduced calibration efficiencies.

As an example of some trade-offs, this CryoNIRSP SAR has very beneficial thermal properties when illuminated with the DKIST 300 Watt optical beam.⁴⁴ This optic sees no heat when used downstream of the calibration polarizer and sees a factor of several less than our other two quartz-based calibration retarders. The thermal drifts of retardance can be neglected in calibration fitting routines for a much longer time for this retarder under illumination. The polarization fringes in this optic are also greatly reduced and temporally stabilized through the thermal behavior.^44,45 However, these benefits must be traded off against other error sources such as the exacerbated spatial variation shown later in this paper and/or complexity of the model through the larger magnitudes of circular retardance. Other considerations include the difficulty in obtaining large ${\mathrm{MgF}}_2$ crystals compared with readily available and inexpensive ${\mathrm{SiO}}_2$ crystals. When manufacturing, changing the crystal thickness often imposes unrealistically tight tolerances on other parameters such as alignment requirements, wave front errors, or methods of bonding (optical contact, epoxies, or oils).

Many calibration procedures assume or fit only for linear retardance components to reduce the number of variables. But for some optics, ignoring circular retardance is one of the bigger errors. In Fig. 4, we show the circular retardance component measured in the CryoNIRSP SAR. We again do not show fits at wavelengths near 700 nm due to the degeneracies in the elliptical retarder model. This circular component is several degrees in magnitude at visible wavelengths. The retardance spectrally oscillates from the rotational misalignments between the six crystals with a magnitude of 1 deg to 4 deg and this oscillation becomes spectrally faster at shorter wavelengths. The magnitude of circular retardance also increases very strongly for the shortest DKIST wavelength of 393 nm with a magnitude of 10 deg. We show this metrology here to motivate further measurements of elliptical retardance for all DKIST retarders and to show the necessity of including this term in DKIST calibrations.

Fig. 4

The circular retardance component from fits to the NLSP measured Cryo SAR Mueller matrix. Clocking errors cause obvious spectral ripples. The polishing and plate alignment errors both combine to create net circular retardance.

Clocking errors between the two-crystal achromatic pairs create ellipticity. Additionally, clocking errors between individual crystals add elliptical retardance oscillations at spectral periods of order nanometers. These spectral ripples behave very differently than polishing errors. Spectral periods of these oscillations are determined by the birefringence of each bias plate instead of the refractive index for a given crystal thickness. As such, these oscillations are usually two orders of magnitude slower. For a retarder to function as a calibrator, the optical properties must be known significantly better than the desired calibration accuracy.

The NLSP has been checked for accuracy using several techniques. We verified transmission measurements using uncoated bare glass and crystal substrates. We generally find agreement to simple Fresnel equation calculations to be better than 0.3% transmission for normal incidence. We similarly verified diattenuation by tilting thin uncoated glass substrates and achieve agreement with Fresnel equations within $<0.3\%$ for tilt angles $<30\mathrm{deg}$ and substrates thinner than 0.5 mm to minimize beam deviation. When comparing our retardance measurements with other vendor measurements, we often achieve retardance agreement better than 1 deg for single layer true zero-order polycarbonate parts at normal incidence. A forthcoming paper is in preparation with a detailed analysis of NLSP. As DKIST instruments will directly derive their own fits to optic retardance, these measurements provide acceptance testing and high-quality estimates of system performance.

We now have physical thickness of each individual crystal in every optic. Meadowlark Optics provided retardance measurements of individual crystals with spectral fitting techniques as well as the assessment of the retardance for each compound retarder. The compound retarders were also measured at five locations across the CA. Using NLSP, the DKIST team has also measured the full assembly Mueller matrix and derived elliptical retardance models for every optic. With this metrology package, we can now begin the measurement and simulation process for the parts across the CA. In the next section, we show the impact of spatial variation when retarders are placed near focal planes, as is common in astronomical calibration systems. With measurements of spatial and spectral variation of the elliptical retardance, we can then show impacts to DKIST calibration as a function of wavelength, field of view, and describe consequences of optical choices, such as the retarder material, CA, and bias crystal thickness.

3.1.

Mapping Retardance Uniformity across the ${\mathit{MgF}}_2$ Two-Crystal Achromats

Following the initial five-point retarder metrology and acceptance testing efforts, we improved our retardance analysis codes. We required high-accuracy retardance spatial mapping measurements across the full CA to match new simulations of the impact of polishing error. In January of 2017, Meadowlark Optics created this mapping capability on their AE four automated retardance measuring system. The system maps elliptical retardance magnitude across the CA of a retarder using computer-controlled translation stages. In this section, we show and analyze spatial retardance maps.

In our initial testing, only the full elliptical retardance magnitude was measured. We used the CryoNIRSP SAR pairs of ${\mathrm{MgF}}_2$ crystal retarders (G2-H3-G4) to demonstrate the capability and to simulate the impact of polishing errors in our six-crystal retarder designs. After initial testing, the capability to output the orientation of the linear retardance fast axis was added. We measured retardance at 415 points on a grid spacing of 5 mm with a test beam diameter of 3 mm. Machine time for this measurement is about 7 h. In this initial setup, the CA had to be scanned in two halves, leading to a slight discontinuity in the data as the part was rotated and centered.

For the G-style crystal pairs, the design provides a net retardance of 2.30 waves at a measurement wavelength of 633.443 nm. The Meadowlark Optics AE4 system uses filters for wavelength selection, and the closest available filter to that wavelength is centered at 634.0 nm and has a bandpass full width at half maximum of 10.5 nm. The ideal design retardance for this waveplate pair in subtraction at this wavelength is 2.228 waves. Similarly for the H-style retarder pairs, we anticipate 3.346 waves net retardance for the two ${\mathrm{MgF}}_2$ crystals in subtraction. The fast and slow axes of the crystals are along the $x$ and $y$ axes in these plots, respectively. For reference, the contracted five-point measurements for uniformity acceptance of this crystal pair previously supplied to DKIST were taken at the center and at 45 deg to these axis directions.

Figure 6 shows the retardance uniformity across a 90-mm CA sampled on a 5-mm spatial grid with the 3-mm probe beam. From these datasets, we can derive distributions of polishing errors as shown in Fig. 7. The first pair of plates tested are called G2 and showed a reasonably Gaussian distribution centered about the nominal retardance values. The second pair of plates called 4G was polished to the same nominal design but with a retardance uniformity distribution that was skewed and showed polishing errors within the CA over 0.02 waves. The 3H pair of plates showed a non-Gaussian profile but with polishing errors mostly within 0.01 waves retardance variation.

Fig. 6

The measured CryoNIRSP SAR linear retardance uniformity maps for individual crystal pairs. The A-B-A design for this calibration retarder used ${\mathrm{MgF}}_2$ crystal subtraction plates of style names G2-H3-G4. These two-crystal plates were measured at 5-mm spatial separation between points across a 100-mm aperture. The system used a 3-mm diameter probe beam. The first two G-style pairs were designed to have 2.23 waves net retardance at 633-nm wavelength when combined in subtraction. The middle crystal pair is H-style and was designed to have 3.346 waves net retardance. The color scale shows uniformity variation of roughly $\pm 0.02$ waves from nominal design retardance across each 2-crystal stack.

Fig. 7

The histogram of retardance polishing nonuniformity values in the CryoNIRSP SAR retarder pairs. The A-B-A design for this calibration retarder used plates of styles G2-H3-G4 in the manufacturing documentation. The G plates were designed at 2.23 waves net retardance when the two ${\mathrm{MgF}}_2$ crystals subtract. The H plates were designed at 3.346 waves net retardance. The 4G pair of retarder uniformity was dominated by a large patch of above average retardance in the spatial maps skewing the distribution significantly.

We also performed a Zernike polynomial decomposition to the polishing uniformity data. There was substantial amplitude for the first five terms, where the retardance nonuniformity resembles tilt, power, and astigmatism. The higher order terms do not substantially improve the fit when using the Zernike basis. We also computed a simple two-dimensional (2-D) Fourier analysis of the retardance maps within a 60-mm rectangle sampled within the CA. The two lowest spatial frequencies account for 90% of the variation. This shows the polishing error is dominated by variation at the largest spatial scales.

The cumulative distribution function for the polishing errors shows what amplitude of error to expect across the CA. Figure 8 shows the cumulative distribution function (CDF) of retardance spatial nonuniformity for the 2G - 3H - 4G bicrystalline retarder pairs that make up the CryoNIRSP SAR. Solid lines show the polishing error distribution across the full 95-mm test aperture, whereas the dashed lines show the distributions considering only points within an 70-mm CA corresponding roughly to 3 arc min DKIST FOV at Gregorian focus.

Fig. 8

The cumulative distribution of retardance polishing nonuniformity values in the CryoNIRSP SAR retarder pairs G-H-G. The solid lines show a 95-mm CA while the dashed lines show a reduced 70-mm CA to avoid edge artifacts.

The 2G pair had a reasonably Gaussian-shaped distribution in the histogram of errors. Correspondingly, the green curve in Fig. 8 shows a CDF where roughly 80% of the CA sees spatial nonuniformity of 0.005 waves retardance following a reasonably Gaussian-style curve of growth. More than 95% of the CA has retardance deviations less than the nominal polishing specification of 0.01 waves retardance.

However, the 4G retarder pair had a section near the edge of the CA with stronger deviations from the average. The distribution of errors above in Fig. 7 showed a skewed distribution with many points lying up to 0.02 waves retardance below the average retardance value. The CDF for this 4G retarder pair shows that only 40% of the CA will see retardance errors $<0.005$ waves, and 85% of the CA will see polishing errors below the 0.01 waves retardance specification. Note that this was outside the five-point test location specified in the fabrication contract. As the spatial maps show this 4G retardance error is highly concentrated near the edge of the optic, DKIST instruments using a reduced footprint would see substantially less spatial nonuniformity. To demonstrate this, histograms and CDFs were recomputed considering a 70-mm CA. The skewed distribution on the 4G plate is still present, but over 99% of the points within the 70-mm CA have $<0.01$ waves retardance error spec.

For DKIST, observing modes and instruments using the extreme edge of the CA will be impacted by this error. These modes and instruments include the ViSP PCM, which uses a full 100-mm CA, wide field calibrations with most instruments, and scanning modes were steering mirrors that repoint the instrument to the edge of the calibration retarders at Gregorian focus.

4.1.

Sensitivity to Error Spatial Distributions: Order and Orientation Dependence

The spatial variance of the retardance errors illustrates how each design can be more or less sensitive to polishing variation. Using the spatial maps for each simulation, we take a uniformly weighted standard deviation for each elliptical retardance parameter across the CA.

Figure 10 shows this spatial variation of retardance for two of our six retarder designs. The left-hand side of Fig. 10 shows the CryoNIRSP modulator (PCM), and the right side shows the DL-NIRSP calibration retarder. There is a significant difference between the wavelength dependencies for each of the three elliptical retardance components. The CryoNIRSP modulator is an elliptical retarder by design so there is a significant circular retardance in nominal bandpass of 1000 to 5000 nm. Given the net retardance of each crystal is multiple waves for visible wavelengths, the spatial variation is up to 16 deg in each elliptical retardance component at the shortest wavelengths.

Fig. 10

The modeled standard deviation of the three elliptical retarder components spatial variation across the CA for two different designs. (a) shows the CryoNIRSP PCM with polishing error scaled to 0.02 waves error while (b) shows the DLNIRSP SAR with the polishing error was scaled to 0.005 waves retardance at 633 nm. Each component of retardance can have very different wavelength dependence and sensitivity for different designs. The CryoNIRSP PCM is designed for wavelengths longer than 1000 nm and shows low sensitivity in that bandpass. At short wavelengths, the sensitivity is an order of magnitude higher.

The DL-NIRSP calibration retarder on the right has more than an order of magnitude less spatial variation even though the retardance uniformity tolerance was only four times tighter. The CryoNIRSP PCM design includes plates of 1.9:1.3:1.9 waves net retardance, and the DL SAR is 0.7:1.0:0.7 at 633-nm wavelength, which gives a factor of roughly two in increased sensitivity if you only scale by compound retarder net magnitude. Each retarder design is sensitive not only to the spatial statistics of the polishing error, but there is also sensitivity to the order in which the errors in each crystal stack. With polishing error maps, we can apply errors with different magnitudes, orientations, and applied in different orders to each of the six crystals in each optic to simulate the impact on different designs. We created a model with different spatial errors as follows: the simulated error distribution on retarder pair 1 is the spatial transpose of the polishing error measured on retarder pair 2. The simulated error on retarder pair 2 becomes the error measured on retarder pair 1 rotated by 90 deg. The simulated error on retarder pair 3 becomes the error measured on retarder pair 3 spatially reversed (flipped) left to right.

For the various designs, the modeled spatial variation could be factors of a few greater depending on the specific orientation of the polishing errors. We recomputed models for all six DKIST retarders using the same polishing error magnitudes but with this alternate spatial variation. The spatial distributions are similar to previous results but with different magnitudes and spectral behavior. The ViSP SAR design wavelength ranges from 380 to 1000 nm. The design has several times less deviation as the DL-NIRSP SAR design at amplitudes in the range of 0.25 deg to 1.0 deg made of the same ${\mathrm{SiO}}_2$ material when simulated with the same magnitude polishing errors. The ViSP SAR is also a factor of few less in net retardance so the rough scaling of the polishing error sensitivity is expected.

Figure 11 compares the spatial variation of the three elliptical retardance components across the CA for the ViSP calibration retarder (SAR) optical model with variable orientation of polishing errors. In both retarder models, the magnitude of the polishing thickness errors was the same 0.005 waves net retardance at 633-nm wavelength applied the ${\mathrm{SiO}}_2$ crystals. The nominal polishing error spatial distributions lead to standard deviations as shown in the left plot of Fig. 11.

Fig. 11

The standard deviation of spatial variation in elliptical retardance components for the ViSP SAR. We used nominal polishing error spatial distributions in (a) and used the spatially modified polishing errors in (b). Derived spatial sensitivity is strongly tied to the distribution, orientation, and order of which crystals have which errors. Both cases used 0.005 waves net retardance magnitude error amplitude at the 633-nm wavelength. Blue and green show the first and second components of linear retardance, respectively. Red shows circular retardance.

Spatial standard deviations are in the range of 1.5 deg to over 6 deg. The circular retardance shown in red has peak spatial variance that gives very short wavelengths, whereas the linear retardance variation is significantly higher. When using the polishing errors at the alternate rotated and transposed orientations, the sensitivity of the ViSP SAR design is always $<1.5\mathrm{deg}$ as seen in the right plot of Fig. 11. The circular retardance component variation is greater than the linear retardance terms at some wavelengths. The only difference between the two models is the order of application and spatial orientation of the polishing errors.

To assess the sensitivity of each DKIST retarder design to polishing errors more generally, we compute a large grid of spectral models. The Mueller matrix of each optic was computed with polishing errors of the appropriate retardance error at 633-nm wavelength applied to each crystal independently against each other crystal. For completeness, we applied positive, negative, and no error for all crystals. This resulted in $3^6$ models for each DKIST six-crystal retarder design. An example of elliptical retardance fits is shown in Fig. 12 for the 729 Mueller matrix models for the ViSP calibration retarder (SAR) design. The blue and green curves show the two linear retardance components. Their wavelength variation shows rotation of the fast axis of linear retardance. This retarder is nominally a quarter-wave net retardation in the 380 to 1000-nm bandpass with zero circular retardance. The red curve shows that the circular retardance component has greater sensitivity to polishing errors around 400 and 900 nm with maximal sensitivity around 500-nm wavelength. The black curve of Fig. 12 shows the total elliptical retardance magnitude as the RSS of the three retardance components in the axis-angle rotation matrix formalism.

Fig. 12

The elliptical retarder model fits to polishing error simulations applied to the ViSP SAR design. Blue and green show the first and second components of linear retardance. Red shows circular retardance. Black shows the elliptical retardance magnitude as the root-sum-square of all three components. See text for details.

7.1.

Comparison of Uniformity Impacts with Continuously Rotating Modulators

When deciding on polishing retardance uniformity specifications, the discrete and continuously rotating retarder cases have very different sensitivities and calibration strategies. We need to include the depolarization caused by the rotation within a single exposure to correctly compare depolarization and field-dependent demodulation techniques as they impact the optical calibration. The continuously changing modulation matrix must be sampled within the exposure time. The flux detected on the camera represents an integral of the modulated intensity propagated through the analyzer to the sensor through the exposure time. Depending on the accuracy required, the integral of flux on the sensor while the retarder spins represents an impact on efficiency and computational complexities. We compute the equation for integration of flux by integrating the transmission term of the system Mueller matrix. This method would represent how flux propagating through the linearly polarizing analyzer is recorded on an ideal sensor:

The polarization response matrix ($X_{ij}$) is nominally the identity matrix for theoretically demodulated data, where the exact input modulation matrix is known and used to demodulate without error. Input vectors are transferred to output vectors exactly without a change in the magnitude or direction. We adopt a notation to match del Toro Iniesta and other works.⁶⁰ Computing the response as demodulation ($D_{ij}$) multiplied by modulation ($O_{ij}$) allows us to assess perturbations. The matrix $\mathrm{\Omega }$ is [1,0,0,0] and represents the sensor only recording total flux without sensitivity to transmitted polarization state when propagated through an ideal analyzer (polarizer).

We can apply perturbations to simulate either imperfect knowledge or errors in determining the demodulation matrix. Alternatively, we can assess errors in modulation within individual or coadded modulation cycles with various types of instrumental and statistical errors. In Eq. (4), we show the discrete modulation response matrix as ${\mathbf{X}}_d$ and the continuous modulation case as ${\mathbf{X}}_d$ with an integral from the starting and ending orientations of the modulator. These orientations are represented as ${\theta }_0$ and ${\theta }_i$, respectively. In the most ideal case, the Mueller matrix of the modulator can be assumed to be of an ideal linear retarder ($M_{LR}$) and the Mueller matrix of the polarizer is a perfect analyzer ($M{M}_a$).

We can approximate the impact of continuous rotation as the flux integral from ${\theta }_0$ to ${\theta }_i$ through each subframe. If we use a standard eight-state modulation, the retarder is sampled in 22.5-deg steps through a 180-deg rotation. We propagate flux through the retarder and analyzer and then average flux as seen by the sensor. If we break the subframe into 50 individual samples, we can approximate the flux as the sum of the system transmission in 0.45-deg steps. Figure 23 shows the modulation matrix elements for an eight-state modulation cycle when using a 127-deg linear retarder and an angular offset of 11.25 deg to put half the rotation within a single-modulation state centered on the 0-deg fast-axis orientation. Solid lines show the modulation matrix elements when assuming discrete sampling and no depolarization. Dashed lines show the impact of flux averaging within a subframe as the optic rotates. The net effect is a slight depolarization and a reduction of efficiency. We note that this reduction in the modulation matrix magnitudes slightly decreases the efficiency and also occurs identically for all field angles.

Fig. 23

The modulation matrix elements for the eight-state modulation cycle with integration beginning with the linear retarder fast axis at 0 deg and temporal sampling centered about the middle of each modulation state (e.g., 11.5 deg for the first state). Red shows $Q$. Blue shows $U$. Green shows $V$. The solid black line at 1.0 shows $I$. Solid lines show the instantaneous sampling. Dashed lines show the impact of flux averaging within a subframe.

We also note that this kind of simulation is also useful for establishing specifications on timing and synchronization. As a reference, the DKIST cameras should be able to synchronize the start and end of exposures with errors $<10\mu \mathrm{s}$. Additionally, the rotation stage has the following errors that can introduce rotational misalignments between the expected position of the retarder and the actual position. For the DKIST rotation stages, we typically maintain the retarder location within a few arc seconds of the expected orientation throughout the modulation cycle. Both the camera timing jitter and the rotational misalignments due to the rotation stage imperfections are tiny compared with other polarization artifacts, introducing errors in the response matrix of order ${10}^{-5}$ or less over typical modulation cycles at DKIST frame rates of 80 Hz or less.

7.2.

Errors in Modulation from Cal. Retarder Variation across the Field of View

We next make an estimate of the impact of spatially variable retardance in the calibration optic to the modulation matrix errors. We include these errors by modeling a simple procedure for calibrating an instrument. We create a model for the calibration optic including spatial variation of retardance magnitude and fast-axis orientation variation. We then propagate this retardance variation through a specific sequence of calibration optic orientations. An ideal instrument then modulates and analyzes this beam to record the perturbed intensities as a function of field. We then take these modeled intensities and assess the impact to DKIST by modeling a simple calibration fitting procedure. We first average the detected flux over some specified field angle across the slit of this simulated instrument. We then do a simultaneous fit for the three elliptical calibration retarder parameters as well as a field-averaged modulation matrix. This is a simple but demonstrative example of instrument calibration and of the application of this metrology to simulating future instrument performance:

As a first step, we must choose a specific sequence of calibration optic orientations to generate a diverse set of Stokes vectors. For operational efficiency, DKIST must calibrate as many instruments as possible in a simultaneous configuration. There is a similar analogy between modulation efficiency and calibration efficiency. In Selbing⁶¹ 2005 on the polarization calibration of the Swedish Solar Telescope, some simple optimization procedures are summarized.

In Sec. 2.5.2 of Selbing,⁶¹ the orthogonality of the Stokes vectors created by the calibration unit is assessed in a matrix form with one row per input state. Equation (5) shows the Stokes vector matrix (${\mathbf{M}}_{\text{Stokes}}$), which is used to derive the condition number and the relative calibration efficiency of the exposure sequence. The pseudoinverse of ${\mathbf{M}}_{\text{Stokes}}$ is created as $E_{i,j}={({\mathbf{M}}_{\text{Stokes}}^T{\mathbf{M}}_{\text{Stokes}})}^{-1}{\mathbf{M}}_{\text{Stokes}}^T$. The efficiencies are computed from the pseudoinverse as the usual sum of squared elements $e={(n\sum _1^n{E}^2)}^{-0.5}$, where n is the number of input states. This pseudoinverse $E$ can also assessed by its condition number. This is the same approach as for finding optimum demodulation matrices.^15,31,47,60

The condition number of a matrix and the associated linear equation $Ax=b$ give an estimate on how errors propagate from individual values (photometric measurements) to the solution (fitted modulation matrix elements). This is commonly defined as the ratio of the largest to smallest singular value in the singular value decomposition of a matrix. The condition number of a function shows how much the output of the function can change with a change in the input arguments. A matrix with a low condition number is said to be well conditioned, whereas a problem with a high condition number is said to be illconditioned and ill-conditioned fits have very different noise properties in the resulting fit values. For DKIST calibration, we want to ensure that the signal-to-noise ratios of the calibrated Stokes vectors are equal within a factor of a few and roughly proportional to the SNR of the instrument photometric measurements, hence needing a condition number below a few.

We make a simple example by optimizing an existing calibration sequence using this framework and the as-built ViSP SAR optical properties. Optimizing an entire sequence of input calibration states would require significant development and computation time and is beyond the scope of this paper. We simply adapt a nominal 12-state sequence adapted from the Advanced Stokes Polarimeter instrument.^62,63 We first consider this sequence noting that we can use 12 nominal orientations that would each create four individual $\pm \mathrm{QUV}$ inputs when using a quarter-wave linear retarder as is common in traditional calibration. We show in Table 9 the orientations of the calibration polarizer (CP) and calibration retarder (CR) in the first two columns. We note that in practice, no optic is perfectly achromatic or aligned, and we do not need to assume that perfectly pure individual states are created for calibration.

Table 9

Seq.

We first search the calibration efficiency space using the as-built ViSP SAR optics and find good efficiency across all DKIST wavelengths 380 to 1600 nm when the retarder is used with a starting orientation of 129 deg for the fast axis using the NLSP as-built data. We then consider adding two additional exposures as free variables, where the CP and the CR can be at any orientation between 0 deg and 180 deg. We searched this space with 7.5-deg steps so we have 25 possible orientations for each optic in each step. As we are searching two optic orientations for two exposures, we search ${25}^4$ models (390,625) using brute force. The last two bold entries show the optimized sequence in Table 9. The left-hand graphic of Fig. 24 shows this maximized $QUV$ efficiency as solid lines. The dashed lines show the minimum $QUV$ efficiency to demonstrate sensitivity to making the correct choice of optic orientation. A 40% difference in calibration of $V$ is seen with the variation in choosing these two exposures. The right-hand graphic of Fig. 24 shows the input Stokes parameters at 420-nm wavelength for this two-exposure optimized sequence. The inputs are diverse and irregular, and span the space.

Fig. 24

Fitting for optimized QUV calibration efficiency across all DKST wavelengths 380 to 1600 nm. (a) Sequence efficiency for best and worst choice of optic orientations for the two final exposures. The solid lines show the efficiency is maximized when the last two exposures are chosen as in Table 9. These two exposures have significant impact as the minimum efficiency shown in dashed lines is up to 40% lower when a poor choice of optic orientations is made. (b) Stokes vectors created at 420-nm wavelength as the optics are oriented in sequence. Note that the default 12 exposures never created a state near $-\mathrm{V}$. The optimization added two more states with large negative V magnitudes providing more diverse inputs.

We take the Meadowlark Optics metrology for the ViSP SAR and derive the footprint-averaged retardance at the appropriate orientation and beam decenter, similar to that shown above in Fig. 18. With this spatial retardance error, we can create a perturbation of the as-built elliptical retardance parameters at any wavelength by adding a scaled amount of retardance and fast-axis variation. Though we showed above that the scaling is not strictly inversely proportional to the wavelength, the errors do scale roughly as ${\lambda }^{-1}$ as shown above within the demonstrated design sensitivity of factors of 2 to 3. We create a model here where we take the perturbation measured at Meadowlark optics and apply a scaled version to the as-built elliptical retardance values at the appropriate wavelength. The left hand graphic of Fig. 25 shows the variation of the footprint-averaged retarder magnitude as a function of field angle for the six unique retarder orientations at 420-nm wavelength. The six unique retarder orientations are noted in the legend. The fast-axis variation is roughly similar in morphology with magnitudes generally $<0.1\mathrm{deg}$.

Fig. 25

(a) ViSP SAR retardance deviation from average as a function of field angle for the six unique retarder orientations during this calibration sequence at the 420-nm wavelength. Errors of the same orientation but at different magnitude are simulated at longer wavelengths. (b) Detected intensity variation from the field center value across the field of view for all eight modulation states and all 14 calibration inputs. The inner $\pm 0.5$ arc minute field generally shows constant intensities to better than $\pm 0.05\%$

We next simulate the simultaneous fit of a modulation matrix and the three elliptical retardance properties for the calibration retarder during the calibration process. We use a theoretical modulation matrix produced by a perfect linear retarder and discrete modulation (no depolarization from smearing during rotation). We used a magnitude of 120 deg and eight modulation states, evenly spaced by 22.5 deg as in the above section. For each calibration state as shown in Fig. 24, there are eight intensities recorded during modulation, giving 112 intensity points for fitting. The first four settings of the calibration sequence only use the calibration polarizer alone and do not show spatial variation across the field of view. Once the calibration retarder is inserted for input states number 4 to 13, the as-built retardance magnitude, the 129-deg optic orientation offset, and the elliptical retardance create uneven but efficiently diverse input Stokes vectors. The right-hand graphic of Fig. 25 shows the variation in intensity propagated through this ideal modulation process as a function of field angle. The intensities recorded account for the spatially varied retardance in the left-hand graphic at the nominal optic orientation and magnitude.

Intensity variations of magnitudes below 0.15% are seen across the ViSP slit field of view of $\pm 1\mathrm{arc}\min$. The field variation is not strictly symmetric, and we anticipate significant averaging across the field. However, significant gains from averaging spatially along the slit are seen. The change in intensity detected at the instrument starts at zero and only increases to magnitudes of 0.02% as we increase the field angle over which we sum intensities to over $\pm 1\text{arc}\text{minute}$. The ideal CalPol, modulator, and analyzer remain unchanged by the field angle. The calibration retarder has retardance varying as a function of field, which produces different modulated intensities across the slit, which are then spatially averaged. The right-hand graphic of Fig. 25 shows substantial symmetry in some cases as well as a wide range in magnitude of field dependence for different modulation states. These curves average down from magnitudes of 0.15% at any individual field angle to values closer to 0.02% when computing the field average uniformly weighted across the slit (linear 1-D weighting).

We take the intensity-perturbations to the calibration sequence and then perform a simultaneous fit to the calibration retarder and modulation. The calibration retarder has three variables for elliptical retardance at each wavelength. The modulation matrix has 32 variables from the eight modulation states multiplied by the four Stokes vector components at each wavelength. We have 112 measured intensities as 14 input states multiplied by eight modulation states. With 112 intensities and 35 variables at each wavelength, we have a highly constrained fit:

We define an error metric as the RSS of the difference in intensities as shown in Eq. (6) for each wavelength. The intensity model ($I_{\text{model}}$) is created using each iterations guessed elliptical retardance parameters and modulation matrix elements. The synthetic dataset ($I_{\mathrm{synth}}$) is the reference created using the field-dependent, perturbed calibration retarder, and ideal 120-deg modulator. These synthetic data include the intensity average along the ViSP slit at the appropriate field angle for each of the $m$ detected flux values as modulated for each of the input states. The elliptical retardance parameters are expected to change slightly from the as-built field-center values due to the added spatial variation. However, the modulation matrix derived will not exactly match that of the perfect 120-deg ideal retarder.

In the left-hand graphic of Fig. 26, we show the error computed from the response matrix ($\mathbf{DO}$) when the best fit modulation matrix is inverted and used to demodulate at each field angle and wavelength in the simulation. We only show the 420-nm simulation for clarity. We used the as-built retarder properties at each wavelength so the chromatic variation of the ViSP SAR is included in this simulation. The blue lines correspond to the off-diagonal modulation matrix elements varying at 420-nm wavelength. The black lines show Stokes I with very low sensitivity:

Fig. 26

ViSP SAR: (a) error matrix computed from the polarization response matrix (DO) in fit demodulation matrix compared with the theoretical ideal modulation matrix from a 120 deg linear retarder. The errors are of order 0.01% until field angles of 0.5-arc min are exceeded. (b) Change in best-fit elliptical retardance parameters for the calibration retarder. All fits are done with the spatially perturbed calibration retarder with the magnitude computes as the ratio of the wavelength to 420 nm.

As an example of the error matrix, we show tabulated values in Eq. (7) multiplied by 10,000 and rounded to 1 decimal place (${10}^{-5}$ magnitudes) at the 420-nm simulation wavelength and using 1-arc min field averaging. The first row varies $<{10}^{-5}$ while the first column errors are a few parts in ${10}^{-5}$. The off-diagonal elements are significant at magnitudes of a few ${10}^{-4}$. The diagonal elements show errors of a few ${10}^{-5}$.

The resulting variation of the derived elliptical retardance parameters for the calibration retarder is shown in the right-hand graphic of Fig. 26. Blue shows the first component of linear retardance. Green shows the second linear component. Third shows circular retardance. The dominant trend is the field variation scales with the magnitude of the spatial retardance errors as seen by the thickness of the lines. The error of 0.04 deg seen at 420-nm wavelength is roughly four times larger than the thin blue curve at 0.01-deg magnitude representing 1565-nm wavelength, scaling appropriately with the ratio of wavelengths.

This simple simulation shows how spatial polishing errors can be propagated through a calibration sequence as modulated by a polarimeter. The general trends are to have modulation matrix errors of order a few ${10}^{-4}$ when averaging over an arc minute field at shorter wavelengths. Longer wavelengths generally decrease in linear proportion to the spatial retardance error, which itself scales roughly inversely with wavelength as shown above. Some expected changes in the best-fit elliptical retardance parameters are found at magnitudes below 0.1 deg given that we sample the optic nonuniformly and in spatial regions that change with the field angle used in the average. There are several compounding factors we have ignored for this simple demonstration. A real modulator is spatially variable and will impart its own issues with calibration across a nonzero field of view. Each exposure will have highly variable signal to noise, which has been ignored presently. The DKIST mirrors will impart their own field dependence we outlined previously⁴³ as would any real system. This does not include the impact of thermal drifts as we have assessed previously.⁴⁴ However, with the metrology tools presented here and this demonstration of a simple system model, we can easily extend system-level models to include multiple artifacts. This then will be used to choose an appropriate FoV for each calibration optic and sequence to achieve high SNR while keeping modulation artifacts below our required calibration thresholds.

9.1.

Limiting Cases: Linear and Circular Retarders

Limiting cases illuminate the three degrees of freedom. If we set the $y$ and $z$ components of an elliptical retarder to zero ($u_y=u_z=0$), we put the axis of rotation in the $+Q$ direction ($x$) recover the equation for a simple linear retarder of Eq. (15) with the fast axis along $Q$ such that the optic rotates $U$ into $V$. With only the $x$ component in the retarder magnitude vector, $u_x=d$. We leave in the explicit normalization to demonstrate the notation:

Similarly, if we set the $x$ and $y$ components of the retardance to zero, we recover a pure circular retarder with the axis pointing along $+z$. This optic rotates Stokes $Q$ into Stokes $U$ following Eq. (16). This optic is called a circular retarder or sometimes confusingly a polarization rotator as it rotates the plane of linear polarization $Q$ into $U$ while leaving $V$ unperturbed:

There are other interesting limiting cases. Suppose we have an equal mix of linear and circular retardance magnitudes. Imagine the linear retardance is oriented about the $x$ direction. In this case, we have the retarder vector as $u_x=u_z$ and $u_y=0$. The normalization constraint simplifies many terms. The ratios $\frac{u_x{u}_y}{d^2}=\frac{u_x^2}{d^2}=\frac{u_z^2}{d^2}=\frac{1}{2}$. The other terms simplify as $\frac{u_x}{d}=\frac{u_z}{d}=\frac{1}{\sqrt{2}}$. With the explicit normalization included and the magnitude of the angle as $d$, we obtain Eq. (17) for an elliptical retarder with the rotation axis pointing equally in between $+Q$ and $+V$ ($u_x=u_z$).

9.2.

Degenerate Solutions: Sign conventions and Rotations of Multiple Waves

Rotation formalisms have ambiguities. As for any kind of rotation, there are often several substitutions that create identical mappings of unrotated to rotated coordinates. For each of these ambiguous solutions, the magnitude of the computed errors in our various simulations can be impacted deriving different sensitivities to polishing errors. Note that the ambiguous retardance components in this axis-angle formalism are not simply related by a sign change. Simply enforcing the spectral smoothness of a solution does not guarantee the correct solution.

For the DKIST six-crystal retarders, our task is also complicated by having infrared-optimized science instruments while laboratory acceptance testing is only performed at shorter wavelengths. We must compute in advance the retardance magnitude and design solutions appropriately. We show two equally accurate elliptical retarder fits to the CryoNIRSP calibration retarder (SAR) Mueller matrix in Fig. 28. In both cases, the identical Mueller matrices were fit including 0.001 waves polishing error at 633-nm wavelength. The $3^6$ models had this polishing error applied to each crystal against every other crystal. In the left-hand set of graphics, the elliptical retardance parameters were restricted to the domain $\pm \pi$ and the retardance magnitude was forced to be less than one wave.

Fig. 28

A comparison of errors when fitting the CryoNIRSP SAR Mueller matrix to different elliptical retardance models. The left side shows a model with retardance components restricted to be less than one wave. The right side shows a model where the retardance components were guided to be over 1.5 waves following the analytical Pancharatnam solution. The right-hand graphics have over twice the scale as the left. The top graphics show the three elliptical retarder components $(u_x,u_y,u_x)$. Blue shows the first linear component of retardance. Green shows the second linear component of retardance. Red shows circular retardance. The middle graphics show the magnitude of linear retardance $\left(\sqrt{u_x^2+u_y^2}\right)$ in blue and the magnitude of circular retardance in red ($u_z$). The left-hand design starts with a magnitude around 270 deg linear retardance with values increasing decreasing through zero around 700 nm then increasing toward 270 deg wave at 2500-nm wavelength. The right hand design starts near 90 deg linear retardance with the magnitude rising steadily with decreasing wavelength. The bottom panel shows the elliptical retarder component errors derived from the same polishing errors as a function of wavelength. In both cases left and right, the optic has the same Mueller matrix. The assumed elliptical retardance model does impact the derived sensitivities as seen in the lower two graphics.

The solution begins near one wave magnitude at short wavelengths, falls to zero around 700-nm wavelength, and then rises toward three-fourth wave net retardance at long wavelengths. In the right-hand set of plots, we enforce a solution that has a net retardance magnitude that rises from one-fourth wave at 5000-nm wavelength, through 1.0 waves magnitude at 700 nm reaching nearly 2.0 waves retardance magnitude at 380 nm. Notice that the right-hand plots have typically double the $y$-axis range. The derived sensitivity to polishing errors is more than double and has significantly different wavelength dependence. Given that we are fitting the identical Mueller matrix, we consider this numerical fitting issue to be critical to assessing retarder design sensitivities.

An example of the ViSP PCM is shown in Fig. 29. The solid lines show one solution where all three components of elliptical retardance are negative at visible wavelengths. The solid black line shows the elliptical retardance magnitude falling from roughly three quarter wave at 350-nm wavelength to about quarter wave at 2500-nm wavelength. This decreasing magnitude is physically realistic as the net retardance of the six quartz crystals in the ViSP PCM decreases significantly over this large wavelength range. The dashed lines Fig. 29 show an identically good fit to the optic Mueller matrix but with all components having a positive sign at visible wavelengths. The elliptical retardance magnitude in this case starts near 110 deg at 350-nm wavelength and rises toward almost a full wave at 2500-nm wavelength. The Mueller matrices are identical for these two solutions, and the individual components are not related by a simple exchange of sign. If one simply enforces spectral smoothness and/or quality of fit, identifying the physically realistic model is not guaranteed. We developed scripts to ensure that the as-built metrology informed the physical model of the optic and that the appropriate elliptical retardance parameters were fit. For the longer wavelength DKIST instruments of CryoNIRSP and DL-NIRSP, this care was required as the net retarder magnitude can be quite high.

Fig. 29

Multiple ER solutions for the ViSP PCM design. Solid lines show one family of solutions with mostly negative signs. Dashed lines show an alternative sign solution that is an equally good fit to the Mueller matrix. The two solutions have a different wavelength dependence and total magnitude trend. Using physically correct parameters is required when fitting a Mueller matrix and subsequently deriving tolerance sensitivities.

To ensure correct solutions, additional knowledge of the optic as well as physical constraints must be included. For instance, the net retardance of the quartz crystals falls with wavelength, thus the solid line solution from Fig. 29 is a more representative model as the net retardance falls with wavelength and approaches zero retardance in the near infrared. We note though that the Mueller matrices of both models are entirely equivalent and that describing retarders as rotations has inherent ambiguity.

9.3.

Euler Angles as an Equivalent Rotation Formalism

In earlier works, we have used a Euler formalism to describe rotation matrices.^43,64–70 In the Euler angle rotation formalism, there are three successive rotations about specific coordinate axes. We denote the three Euler angles as $(\alpha ,\beta ,\gamma )$ and use a short-hand notation where $\cos (\gamma )$ is shortened to $c_{\gamma }$. We had specified the rotation matrix (${\mathbb{R}}_{ij}$) using the $ZXZ$ convention, where rotations are applied in sequence, first about the $Z$ axis, second about the $X$ axis, and then third about the new $Z$ axis. Equation (18) shows this rotation matrix. Note that this rotation matrix is identical to a circular retarder followed by a linear retarder with the fast axis at 0 deg followed by another circular retarder. There are 12 independent conventions for applying a sequence of rotations about a coordinate axis by the Euler angles, such as $XZX$ and $YZY$. We note that the retarder matrix is a rotation formalism for $QUV$ rotating into $QUV$ and several models for rotation matrices are available:

In this Euler angle formalism, the rotation matrix is identical under exchange of $\alpha$ with $-180\mathrm{deg}+\alpha$, $\beta$ with -$\beta$, and $\gamma$ with $-180°+\gamma$. Solutions are also identical when adding multiples of $2\pi$ to any individual component. With this Euler angle formalism, there is no normalization and no natural relation to components of linear or circular retardance as with the axis-angle formalism.

In the left-hand graphic of Fig. 30, we show a Euler angle rotation matrix model fit to the NLSP measurements of the ViSP SAR dataset. The blue, green, and red curves show $\alpha$, $\beta$, and $\gamma$, respectively. The clocking error introduced spectral oscillations (ripples) are immediately visible as is the wavelength-dependent magnitude of the three rotation angles:

Fig. 30

(a) Euler angles derived in a Z-X-Z type rotation matrix fit to the NLSP measurements of the ViSP SAR. Blue shows $\alpha$, green shows $\beta$ and red shows $\gamma$. (b) Intrinsic Euler angles using the $Z-Y-X$ convention. Blue shows $\psi$, green shows $\theta$, and red shows $\psi$ when using the $Z-Y^{\prime }-X^{″}$ convention.

A common rotation matrix formalism often just called the rotation matrix uses an intrinsic Euler angle arrangement, where rotation components are specified in $Z-Y^{\prime }-X^{″}$ ordering. The angles are denoted in the $(z,y,x)$ ordering of $(\varphi ,\theta ,\psi )$ and the rotation matrix is defined as in Eq. (19). This Equation is just as easily described as a linear retarder with fast axis orientation at 0 deg followed by a linear retarder with a fast axis at 45 deg followed by a circular retarder.

With this particular intrinsic $Z-Y^{\prime }-X^{″}$ Euler angle-based rotation matrix, we can relate the individual matrix elements back to the axis-angle rotation matrix formalism common in retarder Mueller matrix fitting routines. Equations (20) through (23) show the analytic solution for the magnitude of the rotation $\theta$ and the three retarder components of the axis vector $(u_x,u_y,u_z)$ denoted either normalized or un-normalized as $(r_H,r_{45},r_R)$ or simply $(H,45,R)$ in Sueoka.^9,10

In the right-hand graphic of Fig. 30, we fit the ViSP SAR Mueller matrix measurements and show the intrinsic Euler angles $(\psi ,\theta ,\psi )$ when using the rotation matrix convention of $Z-Y^{\prime }-X^{″}$. These Euler angles also oscillate but with very different wavelength dependence than the $Z-X-Z$ convention of Fig. 30. When using Eqs. (20) through (23) to compare the axis-angle formalism to the $Z-Y^{\prime }-X^{″}$ version of the Euler angle rotation matrix, we completely reproduce the correct results. The NLSP Mueller matrix measurements are equally well fit within numerical settings of the axis-angle formalism. Both rotation matrix representations of Fig. 30 are equally good reproductions of the Mueller matrix, as are the axis-angle rotation terms presented above.

10.1.

Spatial Retardance Variation for a ${\mathit{MgF}}_2$ Retarder: The CryoNIRSP SAR

The CryoNIRSP superchromatic calibration retarder (SAR) was designed to be a quarter-wave linear retarder for wavelengths longer than 2500 nm. We also plan to calibrate the visible and near-infrared instrumentation with this optic given our recent thermal modeling, fringe amplitude estimates, and plans for calibrating the DKIST primary and secondary mirror.^44,45

This optic can be used in the full 300 Watt DKIST beam, without any polarizers or heat rejection filters with minimal influence of absorption on the temperature of the optic. With the benefits of this optics greatly reduced heat loads (NIR transparency), thermal stability, and reduced fringes, we examine in this subsection how spatial retardance errors will restrict the field of view for calibrations with this optic much more strongly for these visible wavelength use cases.

This retarder uses ${\mathrm{MgF}}_2$ crystals with thickness of (2273.14 and $2153.11\mu \mathrm{m}$) for $A$ pairs and (2333.21 and $2153.11\mu \mathrm{m}$) for $B$ pairs. These correspond to net retardance of 40 waves per plate. The $A-B-A$ pairs are polished to net retardances of 2.230, 3.346, and 2.230 waves retardance at 633.443-nm wavelength. The orientations for the crystal pairs are (0 deg, 107.75 deg, 0 deg). Our as-built measurements used 3-mm spatial sampling with a 3-mm diameter probe beam footprint. We covered a 108-mm diameter aperture with 1009 separate measurements in a grid. The wavelength was controlled using a 598.9-nm filter with a 10-nm FWHM band pass.

Figure 31 shows the elliptical retardance magnitude and linear retardance fast-axis orientation spatial maps. The elliptical retardance magnitude varies from under 74 deg to over 85 deg giving 11 deg peak-to-peak spatial variation. The linear retardance fast axis also rotates by over 5 deg peak-to-peak as seen by the right graphic of Fig. 31. The spatial variation is reasonably smooth over small spatial scales. No abrupt variations between individual spatial measurements are seen. The edges of the CA have stronger deviations as expected for polishing errors.

Fig. 31

The spatial measurements of retardance properties for the CryoNIRSP SAR as-built. The left graphics show maps measured at 598.9 nm using a $\sim 10\mathrm{nm}$ bandpass narrow-band filter and spatial sampling of $\sim 3\mathrm{mm}$. The part was nominally designed as a $\sim 90\mathrm{deg}$ linear retarder. (a) The spatial map of elliptical retardance magnitude with a color scale varying from 74 deg to 85 deg. (b) The fast axis of linear retardance with all points was within a range of $\pm 2.5\mathrm{deg}$.

The corresponding cumulative distribution of errors is seen in Fig. 32. We show two CAs to demonstrate how the inner part of the optic is more uniform. The blue curve shows the CDF for a 70-mm CA with 80% of the measurements within 1.2-deg retardance from average and all spatial points within 4.2 deg. The black curve shows a 108-mm CA. All spatial measurements are within 11 deg of the average, but this distribution has a very significant tail corresponding to the outer edge of the aperture seen in Fig. 31. For the 108-mm CA, 80% of those points are $<2.8\text{-}\mathrm{deg}$ retardance error, but the lower left part of the aperture shows a very strong deviation from average.

We performed a Fourier analysis of the retardance spatial variation to assess the spatial distribution of errors. Both one- and two-dimensional FFT power spectra show that 90% of the spatial variation is contained with spatial frequencies $>10\mathrm{mm}$. For our laboratory acceptance testing as well as system modeling, this shows that we can effectively sample at significantly more coarse spatial resolution and still capture the retardance spatial variation behavior.

Fig. 32

The cumulative distribution of the retardance magnitude spatial variation from average for the as-built CryoNIRSP SAR. See text for details.

Figure 33 shows the difference between 3-mm spatial sampling and 8-mm spatial sampling on a linear gray scale. The difference between maps is seen as higher spatial frequency noise at magnitudes of $<\pm 0.5\mathrm{deg}$ peak to peak variation though the retardance varied across the optic by 11 deg peak to peak as seen in the high spatial resolution map of Fig. 31. The RMS variation across the aperture of Fig. 33 is 0.14 deg. Thus, we capture 11 deg spatial variation with an RMS variation $100\times$ smaller when using 8-mm sampling.

We simulated the impact of polishing errors by fitting elliptical retardance to a large grid of toleranced models. We put errors of either none or $\pm 0.01$ waves of retardance error at 633-nm wavelength on all six of the crystals. This gave us 729 individual models (three choices for each of the six crystals gives $3^6$ models). To highlight the residual errors, we simply difference the fit elliptical retardance parameters from the nominal design. The residual errors can be over 30-deg retardance for some components for a worst-case stack up of errors. The polishing simulations shown in Fig. 34 illustrate how this design has varying sensitivity to circular retardance from polishing errors. The top left graphic shows the linear retardance magnitude computed as the root square sum of the two linear components. The green curves in the top right show the fast-axis orientation of linear retardance as the inverse tangent of the two linear components. The red curves shown at lower right are circular retardance errors with generally increasing sensitivity at shorter wavelengths combined with several sensitivity nulls. The elliptical retardance at lower left closely tracks the linear retardance magnitude errors. The issue in the degeneracy of the axis-angle rotation formalism is easily seen, where the part reaches one-full wave net retardance around 700-nm wavelength. A full wave of rotation gives identical results regardless of the axis of rotation.

Fig. 33

The spatial variation difference between retardance maps constructed at 3-mm spatial resolution and 8-mm spatial resolution. The linear grey scale covers $\pm 0.5\mathrm{deg}$_, and the magnitude measured was $\pm 7.5\mathrm{deg}$ in Fig. 32.

Fig. 34

The spectral variation of elliptical retardance errors in the polishing simulations for the CryoNIRSP SAR. The polishing error of 0.01 waves retardance was optionally placed on each of the six crystals and an elliptical retarder model fit to the resulting optic Mueller matrix. The blue curves at top left show the linear retardance magnitude error. The green curves in the top right show the fast axis of linear retardance with a discontinuity near 700-nm wavelength, where the linear retardance goes to zero and the inverse tangent of small errors is greatly amplified. The circular retardance error is shown at bottom right with several nulls. The elliptical magnitude error is shown in black at bottom left with close resemblance to the linear retardance magnitude except at the shortest wavelengths.

Many authors have shown more efficient calibration and modulation schemes based on retarders that are either elliptical or values that optimize the condition number of the demodulation matrix given an observing set.^{15,47,48,60,68,71–76} We can easily implement similar calibration schemes with our retarders when they do not happen to be the traditional quarter-wave linear retarder. As an example of the utility for this ${\mathrm{MgF}}_2$ retarder for on-axis calibration with strong thermal behavior benefits, we show the linear retardance magnitude away from half-wave in Fig. 35. The retardance magnitude has been restricted to be within the interval 0 deg to 180 deg and has had half a wave subtracted from the theoretical curve.

Fig. 35

The linear retardance difference of the CryoNIRSP ${\mathrm{MgF}}_2$ calibration optic from 180 deg. Thick black lines show the traditional $90\mathrm{deg}\pm 30\mathrm{deg}$ calibration retardance. Dashed horizontal black lines show the range 40 deg to 150 deg, where calibration is reasonably feasible for DKIST. Red vertical lines show expected wavelengths for DKIST instrumentation.

Typical calibration and modulation schemes, when using linear retardance, often find reasonable efficiency when the retardance is $>30\mathrm{deg}$ away from half-wave integer multiples. This particular ${\mathrm{MgF}}_2$ retarder has strongly varying retardance in the 380- to 2500-nm wavelength range as seen in Fig. 35. However, the retardance is near some multiple of one-fourth to one-third wave at many wavelengths throughout the bandpass. DKIST instruments work at specific narrow bandpasses configured on a daily-to-monthly basis.

As an example, Fig. 35 shows vertical red lines corresponding to typical solar observation wavelengths of 396, 486, 525, 589, 630, 789, 854, 1079, 1083, 1430, and 1565 nm. These lines correspond to atomic and molecular transitions of various constituents of the solar atmosphere useful for observation. The DKIST instruments often would be configured with the dichroic beam splitters to observe multiple simultaneous wavelengths. An example would be to configure wavelengths of 396, 615, and 630 nm with the three cameras in ViSP, 854 nm with VTF, and the 1083- to 1565-nm pair with two of the three cameras in DL-NIRSP. For this particular configuration, the retardance would likely be sufficient to calibrate all but the 1565-nm camera using a single-optic simultaneously. The optic is near 0.5 waves retardance at 1400-nm wavelength, at 1.0 waves around 700 nm, and near 1.5 waves near 470-nm wavelength. The thermal properties of this retarder are strongly advantageous to ${\mathrm{SiO}}_2$-based optics due to the greatly reduced absorption.⁴⁴ The oil layers between crystals also match refractive indices much better significantly reducing spectral fringes.

This optic can be very useful in calibration with a reduced field of view to avoid calibration issues caused by polishing error. In particular, the DKIST VTF requires fringe stability for the duration of calibration. The VTF is a Fabry–Perot (FP) imaging instrument, which steps through wavelengths centered around specific spectral lines of interest. The four main lines are 525, 630, 656, and 854 nm with the instrument having a $\sim 1\mathrm{nm}$ free spectral range about these lines. The instrument can sample in wavelength steps of roughly three picometers and has a spectral resolving power of $R=\lambda /\delta \lambda =\mathrm{100,000}$. The CryoNIRSP calibration retarder magnitude shown in Fig. 35 shows VTF can very efficiently be calibrated with this optic at all four primary wavelengths, even though the optic is many multiples of quarter-wave retardance. This instrument cannot as easily spectrally filter fringes as spectrograph instruments typically do given the different timescales for completing a spatial/spectral dataset. Not only does VTF need reduced fringe amplitudes, but it also needs stable retardance fringes for effective removal. This optic, at reduced FoV, may provide such a stable calibration.

Simulation of the impact to calibration can be done for this optic the same as for the ViSP SAR shown in the main text. This optic has significantly greater spatial variation with a spatial pattern that leads to significant differences in average retardance as a function of the field angle. Figure 36 shows the spatial variation of retardance for the eight footprints on the CryoNIRSP calibration retarder at 598.9-nm wavelength as this optic rotates during a typical calibration sequence. The center footprint remains constant for the on-axis beam and would see the same average retardance for all retarder orientations. For the footprint corresponding to the 5-arc min field edge, clear spatial separation is seen between footprints along with significant retardance variation between steps.

Fig. 36

The spatial retardance variation within footprints of the beam on the CryoNIRSP SAR during an eight-orientation calibration sequence at 598.9-nm wavelength.

We quantify the retardance variability during calibration by taking spatial averages over the footprint for varying FoV. Figure 37 shows an example of the average retardance across a footprint as the calibration retarder rotates. The various curves correspond to select field angles using the CryoNIRSP calibration retarder at 600-nm wavelength. Each individual footprint is 26.6 mm at this optical location in the converging F/13 beam. Magenta shows a very small field of view at 0.5-arc min diameter with a beam decenter of 3.8 mm and 0.5-deg peak-to-peak spatial variation. Black shows a 1-arc min field of view corresponding to a beam center 7.6 mm away from the retarder center and 1-deg peak-to-peak spatial variation. Green shows a 2.0-arc min field of view with 15.2 mm of decenter. Blue shows the edge of the 2.83-arc min field of view and a 21.5-mm beam decenter. Red corresponds to the maximal DKIST field of 5.0-arc min and a 38.0-mm beam decenter.

Fig. 37

The retardance spatial variation computed as an average over the footprint as the CryoNIRSP SAR spins during a calibration sequence at 598.9-nm wavelength.

For this ${\mathrm{MgF}}_2$ retarder, the variation of retardance as the optic rotates is roughly 2 deg peak to peak for a 2-arc min field of view at 15.2-mm radius, the green curve in Fig. 37. For larger field angles, however, the retardance variation with optical orientation rapidly increases to 7-deg peak to peak at the 5-arc min field edge. In addition, the red curve is not symmetric about the average value. The net retardance of the optic would be considered different as a function of effective field angle, complicating any calibration technique that assumes constant retardance. With these curves, we can conclude that field-dependent calibrations are required for high accuracy. We also find that the field angle for which we can ignore this spatial dependence is only for the inner half arc minute, at magnitudes below 0.5-deg retardance.

10.2.

Measuring a ${\mathit{MgF}}_2$ Elliptical Retarder: The CryoNIRSP Modulator

The CryoNIRSP PCM is designed to be an efficient modulating retarder rotating in front of an ideal Stokes Q analyzer for wavelengths between 1000 and 5000 nm. The efficiency is high and balanced across QUV when sampled with 5 or more evenly spaced images in a 180-deg rotation. We measured the spatial variation of elliptical retardance magnitude for this optic at 600-nm wavelength. The design predicts a 0.3529 wave linear retarder (127 deg) with no circular retardance component present in the design. Table 10 shows the design and Meadowlark metrology of the components used to assemble the optic. The $E-F-E$ style retarder mirrors the CryoNIRSP SAR but without the requirement that E pairs be parallel.

Table 10

Measured CryoNIRSP PCM crystal properties.

Measurements at MLO used 5-mm spatial sampling and a 3-mm beam footprint covering roughly a 90-mm area with 374 samples. Wavelength was controlled using a 598.9-nm interference filter with a 10-nm FWHM bandpass. Figure 38 shows the elliptical retardance magnitude map on the left with a scale running from 124.7 deg to 131.2 deg. This map was one of our first before the expanded capability to also output linear retardance fast-axis was available. The spatial pattern is very roughly saddle like with low values running in a line from top to bottom of the CA and high values on the left and right aperture edges. The spatial variation is reasonably smooth over small spatial scales. No abrupt variations between individual spatial measurements are seen. The edges of the CA have stronger deviations than the middle.

Fig. 38

The spatial measurements of retardance for the CryoNIRSP modulator after final mounting in the cell. A map was measured at 598.9-nm wavelength and spatial sampling of 5 mm out to 100-mm diameter CA. (a) Spatial map of elliptical retardance magnitude. (b) Cumulative distribution function of retardance errors from the spatial average. A larger 100-mm CA sees 80% of the aperture within 2.8-deg retardance of nominal, and a restricted 70-mm CA sees 80% of the aperture within 1.3 deg of nominal.

The resulting spatial error cumulative distribution functions (CDFs) are shown in the right of Fig. 38. We show two CA s to demonstrate how the inner part of the optic is more uniform. The blue curve shows the CDF for a 70-mm CA with 80% of the measurements within 1.2-deg retardance from average and all spatial points within 2.5-deg error. The black curve shows a 90-mm CA. All spatial measurements are within 4.5 deg of the average and 80% of those points are $<2.6\text{-}\mathrm{deg}$ retardance error.

This retarder has significantly smaller clocking errors than the ViSP SAR although this dataset was recorded with significantly lower spectral resolving power. Figure 39 shows the elliptical retarder model fits to our NLSP measurements of the optic Mueller matrix. The curves look incredibly smooth as the signal-to-noise ratio of NLSP data is $>\mathrm{10,000}$ for all wavelengths.

Fig. 39

The elliptical retarder model fit to the NLSP measurements of the CryoNIRSP modulator Mueller matrix at the zero incidence angle.

We simulated errors with amplitudes of the contracted 0.01 waves retardance at 633-nm wavelength. This corresponds to a physical thickness error of $0.54\mu \mathrm{m}$ on of each the $E-F-E$ crystal pairs. Given that we apply $\pm 0.02$ waves retardance error or zero error to each crystal against every other crystal, we get $3^6$ models of varying behavior. We found generally increasing sensitivity at shorter wavelengths with notable nulls around 400- and 800-nm wavelength. At these wavelengths, the retarder has a zero linear retardance component and is entirely a circular retarder.

For the CryoNIRSP modulator, the footprints are substantially bigger and average over more of the polishing nonuniformity. The optic is located 631-mm upstream of the F/18 focal plane on the spectrograph entrance slit. An individual beam illuminates a 39.2-mm circular footprint on the retarder. Figure 40 shows the footprints corresponding to the edge of the modulator CA at a field angle of 1.4 arc min as well as the footprint at field center.

Fig. 40

The spatial retardance variation within footprints of a single 1.4-arc min field angle beam on the CryoNIRSP modulator during an eight-orientation modulation sequence at 600-nm wavelength. Black dots show the footprint centers. A beam decenter of 29.1 mm is used corresponding to a 1.4-arc min radius field angle.

With an effective focal length of 72,000 mm, the beam decenter is 31.1 mm at the edge of the unvignetted field at 1.5-arc min radius. Within an $\pm 1$ arc minute radius field, we illuminate $\pm 40.4\mathrm{mm}$ of the optic with 20.8 mm of decenter and a 19.6-mm radius footprint. Even for the largest unvignetted field angles with CryoNIRSP, the beams overlap. For this optic, spatial averaging is most significant for all DKIST retarders. This reduces the variation footprint-to-footprint but increases the variance across any individual footprint. For this inner arc minute, we computed field-of-view variation from the mirror-induced polarization in Zemax following the procedures in Harrington and Sueoka.⁴³ The mirrors introduce some field dependence to the telescope Mueller matrix model but at levels significantly below one-degree retardance.

Figure 41 shows the average retardance over a single footprint in the left plot. The right plot of Fig. 41 shows the standard deviation of the spatial variation across the footprint as the optic rotates. With such a large footprint, the standard deviation of retardance variation across any individual footprint is up to 1.5-deg peak with typical values in the range of 0.5 deg to 1.0 deg. As the footprint is 39 mm, the average over the footprint does reduce the variability somewhat and smooth the variation with rotation of the optic. However, the peak-to-peak retardance variation is over 2 deg. Though these values are significantly larger than desired for calibration use cases, these variations are easily removed provided a field-dependent demodulation is accomplished.

Fig. 41

CryoNIRSP Modulator: (a) The average retardance in a footprint is shown at left for various field angles. (b) The standard deviation of the spatial variation across the footprint.

10.3.

Measuring a ${\mathit{SiO}}_2$ Elliptical Retarder: The ViSP Modulator

The ViSP PCM retarder was designed to be an efficient modulator for wavelengths between 380 and 1000 nm. Table 11 shows the Meadowlark metrology and the design for the components in this optic. The three retarder $A-B-A$ crystal compound retarders were designed to be mounted at 0 deg, 41.2 deg, and 148.2 deg orientation. The ViSP PCM is designed to be a 120 deg wave linear retarder with roughly 26 deg of circular retardance at 420-nm wavelength. The Meadowlark mapping technique measures elliptical retardance magnitude so all components are included. Spatial measurements used 3-mm spatial sampling and a 3-mm beam footprint covering 96-mm diameter aperture with 797 separate samples. Wavelength was set with a 420.0-nm interference filter with a 10-nm FWHM bandpass.

Table 11

Measured ViSP PCM crystal properties.