## 1.

## Predicting DKIST Polarization

Predicting the Mueller matrix of a many mirror system with highly powered optics across the field of view (FoV) is an important tool for the design and use of large astronomical telescopes. The Daniel K. Inouye Solar Telescope (DKIST) on Haleakalā, Maui, Hawai’i has a 4.2-m off-axis $f/2$ primary mirror (4.0 m illuminated) and a suite of polarimetric instrumentation in a coudé laboratory.^{1}^{–}^{3} The telescope uses seven mirrors to feed light to the coudé lab.^{1}^{,}^{4}^{–}^{8} Operations involve four polarimetric instruments presently spanning the 380- to 5000-nm wavelength range. A train of dichroic beam splitters allows for rapid changing of instrument configurations and simultaneous operation of three polarimetric instruments covering 380 to 1800 nm.^{7}^{–}^{10} Complex modulation and calibration strategies are required for such a mulitinstrument system.^{7}^{,}^{8}^{,}^{11}^{–}^{14} The planned 4-m European Solar Telescope (EST), though on-axis, will also require similar calibration considerations.^{15}^{–}^{18} Many solar and night-time telescopes are calibrating complex optical pathways.^{19}^{–}^{42}

Several other large astronomical telescopes are in development and include plans for polarimeters. For many years, a night-time spectropolarimeter on the 4-m Advanced Electro-Optical System (AEOS) Telescope on Maui has been pursuing a campaign of polarization calibration.^{43}^{–}^{47} We have developed Zemax modeling tools to compute the polarization of an optical system provided the optical model and the coating prescription for the optics. These Zemax modeling tools have been used on the AEOS telescope and the HiVIS spectropolarimeter. We also apply the tools here to the DKIST telescope and a predict Mueller matrices for two of the first light polarimetric instruments. We refer the reader to recent papers outlining the various capabilities of the first-light instruments.^{1}^{,}^{3}^{,}^{7}^{,}^{8}

In this work, we follow standard notation. The Stokes vector is denoted as $\mathbf{S}={[I,Q,U,V]}^{\mathrm{T}}$. The Mueller matrix is the $4\times 4$ matrix that transfers Stokes vectors. Each element of the Mueller matrix is denoted as the transfer coefficient.^{48}^{–}^{50} For instance, the coefficient (1,0) in the first row transfers $Q$ to $I$ and is denoted $QI$. The first row is denoted $II$, $QI$, $UI$, $VI$. The first column of the Mueller matrix is thus $II$, $IQ$, $IU$, $IV$.

## 1.1.

### DKIST Optics Overview

The DKIST optical train includes an off-axis 4-m diameter parabolic primary mirror (M1) that creates an $f/2$ prime focus. At prime focus, there is a heat stop, which limits the FoV to roughly 5-arc min and reduces the heat load on all downstream optics. The secondary mirror (M2) is also an off-axis ellipse (conic $-0.54$), which relays this beam to an $f/13$ Gregorian focus. Just above the Gregorian focus, there is an optical station for insertion and removal of several masks, targets, artificial light sources, and a set of polarization calibration optics. There are also field stops for 2.8-arc min diameter and 5-arc min diameter at Gregorian focus. The third mirror (M3) is a flat fold mirror at 45-deg incidence angle that directs the light toward the off-axis ellipse (M4, conic $-0.37$). This parabolic mirror changes the diverging $f/13$ beam to a converging $f/53$ beam and creates a pupil conjugate plane near the next flat steering mirror (M5). Between M4 and M5 is also the elevation axis of rotation for the telescope. M5 folds at 30 deg and also functions as a fast steering mirror with tip/tilt capability. The sixth mirror in the system (M6) is also a flat and directs the beam vertically downward toward the coudé lab folding at a 60-deg angle. The seventh mirror is another flat fold mirror that levels the beam into the coudé laboratory folding at 90-deg incidence angle. The eighth mirror (M8) is an off-axis parabola that collimates the beam. The ninth mirror is a flat fold that directs the beam toward the deformable mirror (DM) as part of the integrated adaptive optics system. The pupil of the system is conjugated near the flat DM and this represents the 10th optic in the system (M10).

Figure 1 shows the optical concept for the system. There are four powered optics that perform the relays. In order, the beam has an $f/2$ prime focus, an $f/13$ Gregorian focus, an $f/53$ intermediate focus, and a collimated coudé laboratory.

The coudé lab was designed to allow simultaneous operation of many instruments observing different wavelengths by using several custom dichroic beam splitters. An adaptive optics system was also integrated into the design. The wave front sensor (WFS) of the adaptive optics system is fed by a reflection off the uncoated front surface of a window, denoted WFS-BS1 that is mounted after the DM. There are three polarimetric instruments that use the adaptive optics system: the visible spectro-polarimeter (ViSP), the visible tunable filter (VTF), and the diffraction limited near-infrared spectro-polarimeter (DL-NIRSP). In addition to these polarimetric instruments, there are two arms of a high-resolution imaging system that also use the AO feed. The visible broadband imager (VBI) is essentially two separate instruments, one a red imager (VBI-red) and the other a blue imager (VBI-blue).

Another first light instrument was designed to include infrared capabilities at wavelengths as long as 5000 nm and was optimized for seeing-limited science cases. This instrument, the cryogenic near-infrared spectro-polarimeter (CryoNIRSP) does not use the adaptive optics system and has a separate optical path after M9. For CryoNIRSP, an additional fold mirror (M9a) is inserted into the beam after M9 to direct light to the system feed optics.

For calibration purposes, we will describe a possible configuration using the AO system and the several simultaneous channels of the various instruments. Note that both ViSP and DL-NIRSP have three separate cameras that can record three separately configured wavelengths each. Many configurations are possible and the spectrographs are designed to be reconfigurable in minutes with substantial automation. As an example, DL-NIRSP configuration of the three spectrographs could be (789 or 854.2 nm) on camera 1, (1074.7 or 1083 nm) on camera 2, and (1430 or 1565 nm) on camera 3. At the same time, ViSP could be configured to a vast array of possible spectral lines covering 380 to 1100 nm. Depending on how the dichroics are arranged, the various instruments could be sent limited ranges of wavelengths permitting only some of the cameras to be used.

One setup could configure the first two dichroics CL2 and CL2a to send the VBI-blue camera wavelengths shorter than 430 nm, ViSP wavelengths to 660 nm, VTF to 860 nm, and DL-NIRSP the long wavelength bandpass. With this setup, ViSP could be configured to use at least two of the cameras between 430 and 660 nm, and the DL-NIRSP could use two of the three cameras working at wavelengths longer than 1000 nm. We would need to have polarization calibrations for 5 polarimetric channels (2 on ViSP, VTF, 2 on DL-NIRSP) with calibrations done after the dichroics are installed.

Figure 2 shows a conceptual optical schematic along with Zemax and solid models of the system. The telescope is a classical elevation over azimuth design. In addition to the usual azimuth and elevation degrees of freedom, the entire laboratory floor rotates freely. This coudé rotator is one of the major construction components and gives independent optical control of the field rotation without requiring the use of a three-mirror derotator (K cell)^{1} or more optics such as THEMIS^{51} or the EST.^{16}^{,}^{17} Figure 3 shows the optical beam on the lab floor, feeding all instruments. The optional CryoNIRSP feed is shown as a semitransparent line. All other instruments can be operated simultaneously with the adaptive optics.

In the optical modeling efforts, all three degrees of freedom (azimuth, elevation, coudé table angle) impact polarization calibration plans through the rotation of the projected image against the solar disk and subsequent rotation of polarization calibrations when tracking with images fixed to parallactic or solar coordinates. The azimuth and coudé table angles are redundant optically but do repoint the system celestially. We must consider the relative image rotation angle and time-dependent Mueller matrices when tracking the sun under a variety of use cases that either use or do not use the table angle.

When considering polarization performance of the optical system, the AOI variation as well as the variation across the FoV are both important considerations. As an example, the DKIST primary mirror converts a collimated beam from a single field angle to an $f/2$ converging beam. The effective fold angle for M1 is roughly 28.1 deg, but the bundle of rays exiting the optic sees fold angles between 14.3 deg and 41.1 deg across the beam footprint. The polarization properties vary strongly with AOI and this imparts polarization variation as a function of position in the beam leaving M1. Table 1 shows the variation in incidence angle for the on-axis (zero-field) beam in the design. The first column “optic name” lists the optic. The next four columns show the AOI for the chief ray of the beam, the marginal ray with maximum and minimum AOI, and the range of incidence angles. Subsequent columns of Table 1 show the effective $f/\text{number of the incoming and outgoing}$ beams as well as the coating on the optic. The primary mirror is coated with bare aluminum, which quickly forms a thin oxide layer. All other mirrors M2 to the DM are coated with enhanced protected silver down to the coudé lab. These multilayer coatings tend to have stronger dependence on polarization properties with incidence angle and hence are important to model accurately across the full FoV. The wave front sensor beam splitter (WFS-BS1) has an uncoated front surface to feed the Fresnel reflection to the adaptive optics WFS. The coudé lab dichroic beam splitters CL $N$ all have custom coatings to reflect some wavelengths while efficiently transmitting all other wavelengths. All beam splitters have an antireflection coating on the back surface optimized for their specific transmission wavelength region.

## Table 1

Optical properties of the DKIST on-axis beam with AOI.

Optic | AOI | AOI | AOI | AOI | f/# | f/# | Coating |
---|---|---|---|---|---|---|---|

Name | Chief | Max | Min | Range | In | Out | |

M1 | 14.04 | 20.56 | 7.13 | 13.4 | $\infty $ | 2 | $\mathrm{Al}-{\mathrm{Al}}_{2}{\mathrm{O}}_{3}$ |

M2 | 11.84 | 17.27 | 6.03 | 11.3 | 2 | 13 | Enh. Ag |

M3 | 45 | 47.19 | 42.81 | 4.4 | 13 | 13 | Enh. Ag |

M4 | 1.76 | 2.57 | 0.93 | 1.6 | 13 | 53 | Enh. Ag |

M5 | 15 | 15.53 | 14.47 | 1.1 | 53 | 53 | Enh. Ag |

M6 | 30 | 30.53 | 29.47 | 1.1 | 53 | 53 | Enh. Ag |

M7 | 45 | 45.53 | 44.47 | 1.1 | 53 | 53 | Enh. Ag |

M8 | 5.33 | 5.60 | 5.06 | 0.5 | 53 | $\infty $ | Enh. Ag |

M9 | 10 | 10 | 10 | 0 | $\infty $ | $\infty $ | Enh. Ag |

DM | 15 | 15 | 15 | 0 | $\infty $ | $\infty $ | Enh. Ag |

BS1 | 15 | 15 | 15 | 0 | $\infty $ | $\infty $ | None |

CL2 | 15 | 15 | 15 | 0 | $\infty $ | $\infty $ | Dichroic |

CL2a | 15 | 15 | 15 | 0 | $\infty $ | $\infty $ | Dichroic |

CL3 | 15 | 15 | 15 | 0 | $\infty $ | $\infty $ | Dichroic |

CL3a | 15 | 15 | 15 | 0 | $\infty $ | $\infty $ | Dichroic |

CL4 | 15 | 15 | 15 | 0 | $\infty $ | $\infty $ | Dichroic |

Whenever the pupil is demagnified, the field variation of incidence angles is increased accordingly. The incidence angle variation on the primary mirror is also the FoV angle. The optics demagnify the entrance pupil onto the DM, which is only 0.2 m across giving a $20\times $ demagnification from the 4-m entrance aperture. We trace the incidence angle variation in Table 2 for the chief ray for every field point in the 2.8-arc min diameter field. As powered optics change the relationship between angles across the beam, we list the input and output variation for each powered optic where changes occur. The *Beam Loc* column in Table 2 shows the surface where AOI variation with FoV is computed. The next four columns show the field center incidence angle and the Min/Max incidence angles at the FoV edge. The FoV of the primary mirror is the nominal 2.8 arc min (0.05 deg). However, as the optics demagnify the beam, the incidence angles increase across the field. In the coudé lab, the pupil on the DM sees an incidence angle variation with FoV of 0.9 deg, roughly $20\times $ the original 2.8-arc min FoV. It is apparent from Table 2 to see that the field variation is roughly one degree in all optics of the $f/53$ beam and in the collimated beam, which is the dominant source of field variation effects for polarization calibration.

## Table 2

Properties of the chief ray with FoV.

Beam | AOI | AOI | AOI | AOI | Beam |
---|---|---|---|---|---|

Loc. | (0,0) | Max | Min | w/FoV | f/# |

M1 in | 14.00 | 14.02 | 13.98 | 0.05 | $\infty $ |

M2 in | 11.84 | 11.93 | 11.76 | 0.17 | 2 |

M2 out | 4.39 | 4.57 | 4.20 | 0.38 | 13 |

M3 | 45 | 45.19 | 44.81 | 0.38 | 13 |

M4 in | 1.76 | 1.89 | 1.63 | 0.38 | 13 |

M4 out | 3.52 | 3.98 | 3.07 | 0.90 | 53 |

M5 | 15 | 15.38 | 14.63 | 0.90 | 53 |

M6 | 30 | 30.38 | 29.63 | 0.90 | 53 |

M7 | 45 | 45.45 | 44.56 | 0.90 | 53 |

M8 out | 10.66 | 11.11 | 10.22 | 0.90 | $\infty $ |

M9 | 10 | 10.45 | 9.54 | 0.90 | $\infty $ |

DM | 15 | 15.45 | 14.55 | 0.90 | $\infty $ |

BS1 | 15 | 15.45 | 14.55 | 0.90 | $\infty $ |

CL2 | 15 | 15.45 | 14.55 | 0.90 | $\infty $ |

## 2.

## Zemax Computations

Zemax traces individual rays in the Jones formalism through a geometric model. Zemax can propagate rays from any position in the entrance pupil at any field angle through the optical design. In Zemax, we have developed a script to trace polarized rays across the pupil and field while specifying a series of wavelengths, polarization states, and system optical configurations. We have adapted scripts initially developed by Harrington et al.^{46}

The new scripts can change fold angles, rotation angles, wavelengths, etc., in order to provide the ability to simulate a wide range of optical system configurations. With this functionality, we can derive polarization properties of the system across the beam footprint for any desired setting on any desired surface in the system. We have recently applied this analysis to the 4-m AEOS telescope and compared the predictions to polarization calibrations derived from the daytime sky.^{52}

Depending on the sensitivity and computational speed required, the pupil sampling, field sampling, wavelength coverage, and telescope pointing step size can be increased to sample the beam to a desired accuracy. Typically, sampling the footprint in a $20\times 20$ grid of rays achieves $<0.0001$ level numerical precision, consistent with our simulation needs and is a good compromise between computation speed and numerical accuracy. For other systems, more fine sampling may be desired when confronting more complex situations such as assessing vignetting at each field angle and, in general, the symmetries in the polarization properties of the final beam footprint when polarization analysis is performed.

The Zemax scripts output over 30 electric field vector components for every ray traced. The optical surface can also be specified to examine polarization properties on any optic in the design. To compute a Muller matrix from the Jones formalism, we independently trace a set of the six purely polarized inputs $\pm Q$, $\pm U$, and $\pm V$ through the system. For each of the six inputs, Zemax calculates the electric field properties in 3-D at the specified surface.

These electric field calculations are turned into Stokes vector formalism for each of the pure input states. The computed intensity is the square of the $XY$ components of electric field amplitude ($ExEx+EyEy$). Stokes $Q$ is the $X$ and $Y$ amplitude difference: ($ExEx-EyEy$). The term $\delta $ represents the phase difference and is one of the Zemax outputs but can also be computed directly as $\delta ={\phi}_{x}-{\phi}_{y}$. Stokes $U$ is computed from $X$ and $Y$ electric field amplitudes accounting for coherent phase variations: $2ExEy\text{\hspace{0.17em}}\mathrm{cos}(\delta )$. Stokes $V$ is similarly computed with both $XY$ field amplitudes and phases: $2ExEy\text{\hspace{0.17em}}\mathrm{sin}(\delta )$.

We demonstrate Zemax polarization properties in this paper by computing the electric field distributions, Stokes vectors across various footprints and Mueller matrices for various optical systems. Figure 4 shows the articulation of the DKIST optical design from the primary mirror to the sixth mirror in the system (M6), which represents the optical configurations determining the azimuth–elevation pointing of the system as traced by our scripts. In the coming sections, we will show the Zemax computations for a simple flat fold mirror in a powered beam, and then a variety of surfaces within the DKIST design.

## 3.

## Coating Formulations

The complex refractive index and thickness of each layer in a dielectric coating impact the polarization performance. For typical enhanced-protected metal coatings, there are one to several dielectric layers coated on top of the metal layer. Coatings are typically optimized for reflectivity over certain wavelength ranges but also be optimized for retardation and diattenuation.

Multilayer coatings can create two or more wavelengths where the retardation near the theoretical 180 deg for a perfect reflection. They can also introduce substantial retardation and diattenuation at other wavelengths, which depends strongly on incidence angles. The DKIST calibration plan presently groups the telescope feed optic Mueller matrices and reduces the number of variables required to predict the telescope Mueller matrix for all wavelengths, fields, and telescope pointings.^{7}^{,}^{8}^{,}^{52} To create estimates of the likely DKIST Mueller matrix dependencies on field, telescope pointing, and wavelength, we need a model for the coating formula that captures the relevant dependencies on incidence angle and wavelength.

DKIST internal studies reported measurements of the retardation and reflectivity for witness samples across the 400- to 900-nm wavelength range. To estimate DKIST polarization performance, we needed our model coating formula to be representative of the expected retardance, diattenuation, and reflectivity.

Without access to a manufacturer-provided formula, we found a simple search of standard coating materials identified a reasonable formula for the coating that matched reflectivity, diattenuation, and retardance. For our polarization performance calculations, the retardance and diattenuation are important to match than the overall reflectivity. With an approximate coating formula, we can to estimate the amplitude of several polarization effects expected in DKIST. Having a formula allows us to estimate the expected dependence on incidence angle or FoV with reasonable amplitudes.^{53} We wrote a Zemax script to output a coating polarization report for many combinations of material thicknesses allowing an efficient search of several possible coating formulas. For enhanced and/or protected silver formulas, fused silica (${\mathrm{SiO}}_{2}$), zinc sulfide (ZnS), magnesium flouride (${\mathrm{MgF}}_{2}$), and aluminum oxide (${\mathrm{Al}}_{2}{\mathrm{O}}_{3}$) can be used as the protected layer. An example coating model run at 570-nm wavelength is shown in Fig. 5. Several additional coating formulas are shown in our recent publication.^{52}

Often the harder materials (sapphire, fused silica) are used as the durable protective over-coating while other layers or materials are included to minimize retardance or maximize reflectivity at particular wavelengths with thickness tolerances of a few nanometer. All of the two-layer protective coating formulas shown by Harrington et al.^{54} and the searches reported here do have two separate 180-deg retardance crossing wavelengths around 400 and 850 nm as was desired by the DKIST project.

We searched the common materials of ZnS, ${\mathrm{SiO}}_{2}$, ${\mathrm{MgF}}_{2}$, and ${\mathrm{Al}}_{2}{\mathrm{O}}_{3}$ in two-layer protective coatings over the metal layer. As an example of one of these searches, Fig. 6 shows a search of up to 200-nm aluminum oxide over a layer of up to 120 nm of zinc sulfide. Figure 7 shows variations in retardance for the reected beam with 5 nm changes in thickness of two dielectric layer thicknesses (ZnS and ${\mathrm{Al}}_{2}{\mathrm{O}}_{3}$). We identified a coating formula that has similar retardation, reflectivity, and diattenuation to our witness samples for the DKIST mirrors. This coating formula was not an exhaustive search of possible design space but simply a few iterations to achieve a reasonable match. For modeling efforts presented here, this coating formula will be useful to predict the system Mueller matrix for the CryoNIRSP instrument. We show the 1000- to 5000-nm wavelength range and select wavelengths for the ViSP instrument where the model coating formula matches the witness sample retardance.

Figure 8 shows the reflectivity, retardance, and diattenuation for this enhanced protected silver coating formula. For this coating design, there are two wavelengths where the 180-deg phase change from reflection is exactly met, but these points are functions of incidence angle. At lower incidence angles, the 180-deg retardance points shift to longer wavelengths. For the DKIST design, not all feed mirrors share the same incidence angle, so there will be no one wavelength where the telescope Mueller matrix is free of cross talk. There is a strong dependence on wavelength with 20-deg retardation amplitudes seen in the visible and near infrared at 45-deg incidence angles.

In addition, there are two wavelengths where the diattenuation is zero, but these two wavelengths are also functions of incidence angle. There will also not be any one wavelength where the telescope Mueller matrix is free of induced polarization. The actual coating formula from the various vendors providing the mirrors for all telescope and instrument optics are proprietary to the manufacturers. However, many enhanced dielectric protective coatings have at least 2 and in some cases many layers of material deposited on top of the metal. Additional adhesion layers also complicate the formula. We rely on these simple models to represent a close approximation to the polarization behavior as functions of the relevant variables.

## 4.

## Flat Mirrors

## Eq. (1)

$${\mathbf{M}}_{ij}=\left(\begin{array}{cccc}\frac{1}{2}(1+\frac{{R}_{p}}{{R}_{s}})& \frac{1}{2}(1-\frac{{R}_{p}}{{R}_{s}})& 0& 0\\ \frac{1}{2}(1-\frac{{R}_{p}}{{R}_{s}})& \frac{1}{2}(1+\frac{{R}_{p}}{{R}_{s}})& 0& 0\\ 0& 0& \sqrt{\frac{{R}_{p}}{{R}_{s}}}{C}_{\delta}& \sqrt{\frac{{R}_{p}}{{R}_{s}}}{S}_{\delta}\\ 0& 0& -\sqrt{\frac{{R}_{p}}{{R}_{s}}}{S}_{\delta}& \sqrt{\frac{{R}_{p}}{{R}_{s}}}{C}_{\delta}\\ & & & \end{array}\right).$$DKIST staff and others in the literature have used simple Mueller matrix formulas based on a single ray at a single-fold angle to estimate Mueller matrix properties.^{55} In Zemax, a flat mirror in a collimated beam represents that approximation and should reproduce the simple Mueller matrix dependencies found with simple theory.

## 4.1.

### Converting Electric Fields to Mueller Matrices in Converging Beams

A flat fold mirror model was created in Zemax. A paraxial 200-mm focal length lens was inserted at the 100-mm diameter system stop. This 100-mm diameter beam comes to focus after 200 mm of propagation and is an $f/2$ system. As the entire system is paraxial, the rms spot size at the focal plane is zero within numerical accuracy. There are no geometrical aberrations in the system. The fold mirror was set to an incidence angle of 45 deg corresponding to a 90-deg fold. The mirror was placed half way between the stop and focal plane, corresponding to 100 mm of propagation between the stop and the focus. Figure 9 shows the system layout. For an $f/2$ fold, the incidence angles vary from 30.96 deg to 59.04 deg.

We demonstrate the conversion between electric field vectors and Mueller matrices by using a converging $f/2$ beam reflecting of a flat fold mirror at 45-deg incidence angle.

We computed Mueller matrices for this fold mirror with one of our enhanced protected silver formulas. The $II$ and $QQ$ elements of the Mueller matrix were 0.9513. When the Mueller matrix was normalized such that $II=1$, the $IQ$ and $QI$ terms were 0.0513, the $UV$ and $VU$ terms had an amplitude of 0.3238 with the $UU$ and $VV$ terms at 0.9447. The coating analysis in Zemax agreed with these electric field calculations. A fold at 45-deg incidence angle with this coating gave a retardance of 161.1 deg, a diattenuation of 0.0513 and ${R}_{s}$ was 93.42%, whereas ${R}_{p}$ was 84.33%.

Following the formula in Fig. 1 and using the computed reflectivities from the Zemax coating file, we recover the same Mueller matrix computed from the electric fields. This shows consistency between the mathematics used in the Zemax coating computations and our electric field calculations. Note that with incidence angles varying from 30.96 deg to 59.04 deg across the footprint, there will be strong variation in the derived Mueller matrices.

The scripts produce the electric field real and imaginary components from which we compute the $XYZ$ components of the electric field across the footprint. In our case for this $f/2$ converging beam, the rays all converge to the focal plane and as such are spatially overlapping. Figure 10 shows the $XYZ$ electric field amplitudes when all pure Jones vectors are input (representing the six fully polarized pure Stokes inputs).

Zemax propagates rays in the Jones formalism by a specified optical path length along a computed propagation direction. When the electric field distribution is computed, the $xyz$ coordinates represent that of the global $xyz$ coordinates computed for the system at that location in space. In Fig. 10, there are substantial $Z$ components to the electric field for all nonchief rays.

In any real polarimeter, the performance of the analyzing polarizer, beam splitter transmission functions, reflection coefficients, etc., will all be substantially different than if one simply simulates the behavior of the chief ray. As a simple computational aide, we collimate the incoming $f/2$ beam with a paraxial lens. When collimated, the $Z$ component of the electric field went to zero. The electric field computation we outlined above, the Mueller matrix is derived from the Stokes parameters considering only the $XY$ components of the electric field. In most situations, the $Z$ component of the field is small as the $f/\text{number of the beam}$ in the polarimeter is typically greater than 10. A more detailed computation with the full 3-D electric field distribution is possible and will be explored in future works.

The properties of the Mueller matrix change substantially with the $f/\text{number of the beam}$. When considering the normalized Mueller matrix with only the fold at 45 deg, the $IQ$ and $QI$ terms were 0.0513, the $UV$ and $VU$ terms had an amplitude of 0.3238 with the $UU$ and $VV$ terms at 0.9447. When summing over the footprint in the $f/2$ beam, we see a mild increase in the $IQ$ and $QI$ terms to 0.0517. The $UV$ and $VU$ terms change by about 5% to 0.3065 amplitude. The $UU$ and $VV$ terms also change to 0.9492.

In addition, the $QQ$ term is no longer identical with the $II$ term. In the normalized Mueller matrix, all terms are divided by $II$ but the $QQ$ term is 0.9968 after normalization, showing some depolarization. The diattenuation, depolarization, and retardance of the system can become much more complex than the formula for a simple single fold mirror.

We should note that the variation in electric field properties seen in Fig. 10 is dominated by AOI variation across the mirror. We have run calculations where we use a second paraxial lens near the focal plane to collimate the system. As expected, the $Z$ component of the electric field goes to zero. Figure 11 shows the amplitude of the $X$ and $Y$ electric field components when the converging beam is collimated near focus. The $Z$ component of the electric field is zero. With the symmetries of the $Z$ field component and the use of both the $+$ and $-$ Stokes inputs to derive the system Mueller matrix, the numerical values of the Mueller matrix are the same whether the system is collimated or converging $f/2$.

Figure 12 shows the derived Mueller matrix from the $f/2$ fold computed after a collimating paraxial lens, which sets the $Z$ component of the electric field to zero for all rays. Note when the system is not collimated on the optical surface evaluated, there is an asymmetry between the top row of I to $QUV$ terms and the first column of $QUV$ to $I$ terms in the derived Mueller matrix. As an example, this $f/2$ fold gives the Mueller matrix in Fig. 12 when collimated but when at $f/2$, the $VI$ term ranges about $\pm 1\%$ while the $IV$ term is unchanged and is zero with numerical precision limits.

As we use only the $XY$ components of the electric field to compute the Stokes vectors from the Jones formalism, we make an error in steeply converging beams. A real polarimeter uses an optic to analyze the beam, propagating some components of the electric field vector that also vary strongly with incidence angle and the type of optic used. An open question to be investigated in future work is what the limits are as a function of $f/\text{number on a real analyzer}$. For the time being, we will only convert from Jones to Stokes in slow systems such as $f/200$ for AEOS, $f/40$ for HiVIS, $f/26$ for ViSP presented below.

## 4.2.

### Aluminum Oxide Coatings and Comparisons with Flat Mirrors Reflections in the DST

The Zemax model predictions change drastically in response to changing coating formulas. Small changes in the refractive index, absorption coefficients, or thicknesses of protective layers can change the diattuenation by $>1\%$ and retardance by many degrees. If the retardation and reflectivity of the coating formula for the DKIST mirrors are not matched in detail, the system performance predictions will be inaccurate. For off-axis, high AOI systems like DKIST, the coating performance must be accurately measured with the incidence angle to be modeled correctly as a formula.

Modeling of aluminum metal coating and the aluminum oxide layer that forms over top is important for computing system reflectivity and polarization performance. Aluminum oxide has the same chemical formula as sapphire (${\mathrm{Al}}_{2}{\mathrm{O}}_{3}$) but with an amorphous (noncrystalline) structure and different birefringence properties. Various studies have been done on the polarization and reflective properties of aluminum and aluminum oxide compared to standard optical constants handbooks.^{34}^{,}^{46}^{,}^{47}^{,}^{55}^{,}^{56}

From the Dunn Solar Telescope (DST) staff, a formula was derived as 872 nm of aluminum over 40 nm of aluminum oxide by fitting the telescope polarization model (courtesy of David Elmore, private communication). Models from Socas-Navarro et al.^{34} derive the mirror optical constants along with other properties from fits to the telescope Mueller matrix.

In this DST coating formula, the aluminum has a complex refractive index specified at many wavelengths. Certainly many other formulas are easily considered in response to other studies and with our own ellipsometer.^{55} An early DKIST study we performed also used (0.667, $-5.57$) and (0.7 $-7.0$) for the aluminum refractive index. Note that in studies by Harrington on the AEOS telescope,^{46}^{,}^{47} the aluminum index of refraction was shown to have strong polarimetric impact. For DKIST as shown later, the aluminum oxide is only on M1, whereas the enhanced protected silver coatings dominate the system polarization behavior.

Van Harten et al.^{55} found thickness of 0.5 to 4 nm of oxide. This is in contrast to the 40 to 50 nm used in studies at the Dunn Solar Telescope.^{34} We note that the wire grid polarizers we use have a wire thickness of $<40\text{\hspace{0.17em}}\mathrm{nm}$ and a pitch of 80 nm such that the entire wire would oxidize and cease to function if values like 40 nm were realistic.

For this study, we are simply demonstrating the impact of coatings on polarization performance predictions. The predicted aluminum reflectivity versus wavelength is roughly similar to other models when using the refractive indices and coating layer thicknesses reported here. The interpolation between wavelengths in this DKIST coating file is also apparent due to the coarse wavelength sampling, but the overall behavior shows the expected reflectivity of 82% to 87% in the 380- to 900-nm wavelength range.

To test this work against earlier DKIST performance predictions, we use 500-nm wavelength, 45-deg AOI (a 90-deg fold angle), 1.625 real index for the oxide, and (0.6667, $-5.5726$) refractive index for the aluminum following internal DKIST reports. The resulting transmission is an exact match at 87.22%. The reflectance for ${R}_{s}$ is 88.90% and ${R}_{p}$ is 85.53% with the phase ($\delta $) of 2.53 deg, which also exactly matches Eq. (1). The derived Mueller matrices follow Eq. (1) to many decimal places showing that we do reproduce the theoretical equation within good limits. For an 80-deg fold at a wavelength of 500 nm, the oxidized aluminum gives 86.85% transmission, $IQ$ and $QI$ terms of 1.54%, $UU$ and $VV$ terms of 0.8918 amplitude, $UV$ and $VU$ terms of 0.4522 amplitude and follows the theoretical equation. The intensity to linear polarization and linear polarization to intensity terms are a few percent. Properties of aluminum-coated fold mirror pairs can be compared with the common *turret* style solar telescopes of the DST, the German Vacuum Tower Telescope^{29}^{,}^{41}^{,}^{42} and are common in designs of other solar telescope optical relays.^{16}^{,}^{36}^{,}^{57}

## 5.

## DKIST Gregorian Focus

Zemax calculations were performed to compare the baseline DKIST coating files against early reports and predictions for the Gregorian focus Mueller matrix. The oxidized aluminum formula is coated on M1, whereas an enhanced protected silver formula is coated on M2. The total intensity is around 0.87- at 500-nm wavelength, which includes reflection losses by coatings (mostly from aluminum at that wavelength). Figure 13 shows the layout of the first two DKIST mirrors and the beam to Gregorian focus.

An internal 2002 DKIST report showed calculations and trade-offs for the polarization of prime and Gregorian focus concluding that the field dependence was well below calibration limits and was thus negligible. This working assumption was carried forward in all DKIST documentation and is also supported here. The 2002 report was based on an $f/30$ Gregorian focus design, though the present DKIST design is now at $f/13$. However, the report concluded that variation across the Gregorian FoV was negligible. “The off-axis elements are all well below ${10}^{-5}$, and therefore, no calibration would be needed even at 2.5 arc min away from the center of the field-of-view.”

The properties of the current $f/13$ Gregorian focus are investigated below. Table 3 shows the Mueller matrix computed by the scripts for an $f/13$ beam and a slow beam (effective $f/30$) at 400-nm wavelength. The total transmission ranges from 72.05% to 72.76% across the footprint. The Mueller matrix elements in the $f/13$ beam have some small amplitude elements ($UI$, $IU$, $QV$, $VQ$, $QU$, $UQ$) that are not present in the theoretical formula for a reflected Mueller matrix. However, neglecting the $Z$ electric field component influences some components of the Mueller matrix at $f/13$. For the $f/30$ beam calculation in the lower half of Table 3, these elements go to zero. There are still some depolarization terms even when the calculations are done in a paraxially collimated system. The $IQ$ and $QI$ terms are 0.45%, the $QQ$ term is not 1, and the $UV$, $VU$ terms have 0.0847 amplitude. This depolarization originates in the averaging over the aperture.^{49}^{,}^{58}^{–}^{62} Figure 14 shows the Mueller matrices varying across the footprint of the Gregorian focus beam. Substantial variation is seen along with some asymmetries.

## Table 3

Gregorian focus Mueller matrix.

1.0000 | 0.0044 | 0.0001 | 0.0000 |

0.0044 | 0.9991 | $-0.0028$ | 0.0007 |

0.0001 | 0.0028 | 0.9956 | $-0.0841$ |

0.0000 | $-0.0004$ | 0.0842 | 0.9946 |

1.0000 | 0.0045 | 0.0000 | 0.0000 |

0.0045 | 0.9991 | 0.0000 | 0.0000 |

0.0000 | 0.0000 | 0.9955 | $-0.0847$ |

0.0000 | 0.0000 | 0.0848 | 0.9946 |

The Mueller matrix computed for Gregorian focus at 400-nm wavelength, $f/13$ above, $f/30$, and collimated below. Depolarization is seen $\sim 0.1\%$ in the diagonal as $QQ\ne 1$ and $UU\ne VV<0.9964$ as ${\mathrm{cos}[\mathrm{sin}}^{-1}(0.0847)]$.

We compute Mueller matrix variation for both the on-axis footprint and the 5-arc min field. As expected for an $f/2$ off-axis system, there are substantial asymmetries in the footprints for all field angles. However, most of the Mueller matrix terms are symmetric and largely average to zero after reflecting off the M1–M2 mirror pair. As concluded in a 2002 DKIST report, the Mueller matrix variation across the 5-arc min FoV is at the ${10}^{-5}$ amplitude level and is well below other DKIST calibration issues.

We show the expected amplitude of some Mueller matrix elements in Fig. 15. The $IQ$ and $QI$ terms show a higher amplitude at short wavelengths, caused by the silver, and a peak at 800 nm of about 0.5%, caused by the aluminum. Similar effects are seen in the amplitude of the $UV$ and $VU$ elements on the right side of Fig. 15. The matrix element is above 0.08 at short wavelengths and falls toward the visible band but again rises in the NIR, mostly caused by the silver coating.

## 6.

## CryoNIRSP Predictions with Pointing, Wavelength, and Field

We created polarization models for all optics feeding the CryoNIRSP instrument as functions of wavelength, telescope pointing, and FoV. Figure 16 shows top and side views of the coudé lab solid model and the optical design for the coudé lab optics between the seventh mirror (M7) and the CryoNIRSP modulator. We used the enhanced protected silver coating on all the DKIST optics from M2 to the CryoNIRSP modulator. As the CryoNIRSP instrument requires inserting a fold mirror, the beam path to CryoNIRSP is all-reflective. Though the coatings for CryoNIRSP optics are not yet completed, we anticipate that the coating formulas presented here will be used on all relevant CryoNIRSP optics. The telescope azimuth, elevation, and wavelength were varied in the scripts to articulate the system as in Fig. 4.

The Zemax models were computed at wavelengths of 500 to 5000 nm in steps of 500 nm. The telescope azimuth and elevation were computed over the full hemisphere in steps of 2 deg to capture the complete (Az, El) behavior of the system Mueller matrix.

The computation was done in a coordinate reference system that is tied to the entrance pupil of the optical train. This causes the definition of the $QU$ coordinate grid to rotate as seen from the perspective of a fixed $XYZ$ frame where the Mueller matrix is computed. This means that, in addition to any circular retardance in the system, there is a purely geometrical rotation from a $QU$ reference frame at a downstream optic to a $QU$ input frame in the entrance aperture of the Zemax optical design.

Figure 17 shows the system Mueller matrices as functions of azimuth and elevation at the CryoNIRSP modulator for wavelength of 2500 nm for the on-axis beam. All 16 elements of the Mueller matrix are shown with azimuth on the $X$-axis and elevation on the $Y$-axis in their own box. Each box is linearly scaled to highlight the azimuth–elevation dependence of the individual Mueller matrix elements. Each Mueller matrix has been normalized by the transmission at each individual telescope pointing so the (0,0) element is always 1.

As the DKIST relay is a series of fold mirrors articulated in azimuth and elevation, there is a simple functional form of the Mueller matrix. Each Mueller matrix element can be represented as sin and cos functions of azimuth, $2*\text{azimuth}$, elevation, and $2*\text{elevation}$. This is caused by the crossing and uncrossing of the incidence planes of the various fold mirrors between the azimuth and elevation rotation axes. This is also the same behavior as shown for the AEOS telescope.^{52} In that paper, on-sky calibrations were used to derive the system Mueller matrix, and the functional dependence was easily fit with simple trigonometric functions.

In Fig. 17, the $QUV$ to $QUV$ terms are linearly scaled to amplitudes of $\pm 1$. The $QU$ to $QU$ terms contain both the geometric rotation from the coordinate system and also any possible circular retardance. The intensity to $QU$ terms and the $QU$ to intensity terms ($IQ$, $QI$, $IU$, $UI$) are scaled to $\pm 1\%$ and show strong $\mathrm{cos}\text{\hspace{0.17em}}(2*\mathrm{Az})$ dependence along with elevation variation. The $IV$ and $VI$ terms are scaled to $\pm 0.2\%$ and show similar functional dependence. These terms reach their maximum amplitudes at some (azimuth, elevation) combinations with zero amplitude seen in specific elements only at specific telescope pointings.

Given the slowly varying retardation and diattenuation of the coating formula in the 1500- to 5000-nm wavelength range from Fig. 8, the differences between wavelengths are mostly seen as an amplitude change of the induced polarization terms. Figure 17 shows representative of the azimuth–elevation behavior for the Mueller matrix at all near-infrared wavelengths coming into the CryoNIRSP modulator.

## 6.1.

### Field and Wavelength Variation

Select Cryo-NIRSP Mueller matrix elements at a telescope azimuth, elevation combinations of 0 deg and 45 deg are shown in Fig. 18. The wavelength dependence is dominated by the model coating formula and the relative geometry between the groups of mirrors in the DKIST design.

The predictions are generally limited by the wavelength sampling of the vendor-supplied coating data. The coating files specified have coarse sampling in the near-infrared spectral region, leading to some linear behavior with wavelength in Fig. 18. Since mirror pairs can rotate an incoming $Q$ signal into the $UV$ cross-talk term of a subsequent mirror, Fig. 18 shows that some azimuth, elevation combinations have minimal $UV$ dependence regardless of the coating retardation.

If the FoV dependence is above calibration requirement amplitudes, we have to add additional variables to the calibration plans. Figure 19 shows the variation from field edge to field center of the $QUV$ to $QUV$ Mueller matrix elements for a full 5-arc min FoV. Azimuth, elevation combinations of (0 deg, 45 deg) are used at left and (45 deg, 0 deg) are used on the right. The variation reaches amplitudes of up to 0.02 with a strong dependence on telescope pointing. The wavelength dependence generally follows the retardance formula for the coating combined with the geometric effects of one mirror rotating a linear polarization signal into the $UV$ cross-talk axis of another mirror. In general, this causes strong $QUV$ to $QUV$ rotations.

## 7.

## Visible Spectro-Polarimeter

The ViSP uses several reflections and transmissions through a train of beam splitters. The beam splitters can be antireflection coated windows, mirrors, and/or dichroic beam splitters. We chose here to show the Mueller matrix of the beam delivered to the modulator inside this instrument. The beam path on the coudé lab floor is shown for ViSP in Fig. 20. We use the DKIST feed optics with an all-mirror feed to ViSP with azimuth and elevation and leave analysis of the dichric beam splitters to a future paper. We can examine a case where the reflections are enhanced protected silver mirrors and the transmissions are simple uncoated window substrates. We are in the process of modeling formulas and testing dichroic beam splitter coating samples to assess the polarization performance through many layer coatings.

There are a few wavelengths in the visible region where the retardance of the model coating formula matches the witness sample to better than 1 deg. We choose three wavelengths to model the DKIST telescope optics and the ViSP instrument optics to the modulator. At 400-nm wavelength, the model coating formula matches to 0.26 deg. At 600 nm, the retardance matches to 0.80 deg. At 800-nm wavelength, the retardance matches to 0.09 deg. As of this time, ViSP has not yet selected an actual vendor to coat their mirrors so these results are approximate and can easily be rerun once we have more information about the actual coatings chosen by the team.

Figure 21 shows the computed Mueller matrix elements while articulating the telescope Zemax design in azimuth from 0 deg to 360 deg and elevation from 0 deg to 90 deg pointing range, well beyond the actual capabilities of the telescope mechanical structure. As expected, there is a large amplitude $QU$ to $QU$ term variation that represents the geometric rotation between the coordinates of the modulator and the coordinates of the primary mirror in addition to any circular birefringence causing $QU$ to $QU$ polarization effects. As Zemax uses local mirror coordinates, this geometric rotation is present in all models where mirrors are articulated via Zemax coordinate breaks.

As expected, the linear to circular polarization terms are present but are nowhere near as large an amplitude as for the AEOS telescope where we derived similar predictions.^{43}^{,}^{52} The DKIST telescope feed optics have a much more benign polarization behavior with azimuth and elevation due to the reduced incidence-angle folds. The AEOS beam has five mirrors at 45-deg incidence angle whereas DKIST has [45 deg, 15 deg, 30 deg, 45 deg]. The AEOS beam has 45-deg incidence before the elevation axis, 135-deg incidence between elevation and azimuth axes, and another 45 deg to level the beam on the coudé floor. For DKIST, these numbers are 45 deg, 45 deg, and 45 deg.

With this coating formula at 400-nm wavelength, the first four mirrors in the DKIST train have a diattenuation less than 5% with ($UU$, $VV$) terms of 0.97 and ($UV$, $VU$) terms of 0.24. Though the primary and secondary mirrors have substantial incidence angles and variation across the beam, the primary $UV$ term comes from the 45-deg incidence angle of M3. The second group of mirrors is the two flat fold mirrors M5 and M6, which are at incidence angles of 15 deg and 30 deg, respectively. This group of mirrors has essentially the same linear to circular cross talk. The diattenuation is always less than 4% as the telescope is articulated with ($UU$, $VV$) terms of 0.96 and ($UV$, $VU$) terms of 0.27. This compares quite favorably with a configuration of three separate flat folds working at 45-deg incidence angles articulated in (azimuth, elevation) where the linear to circular terms ($UV$, $VU$) can be above 0.85 at certain pointings.^{43}

For the remaining ten mirrors on the coudé lab feeding light to the ViSP modulator, the diattenuation is about 4% with ($UU$, $VV$) terms of 0.91 and ($UV$, $VU$) terms of 0.40. There is a 45-deg incidence angle mirror (M7), two 15-deg incidence angle mirrors (M10, BS2), a powered feed mirror at 28 deg, and four other mirrors below 12 deg.

## 8.

## Summary

We presented Zemax optical models and performance predictions for the DKIST telescope feed optics and two of the first light polarimetric instruments, the CryoNIRSP and the ViSP. Simple flat fold mirror in a powered beam was studied to demonstrate the sensitivity to Mueller matrix elements to the $f/\text{number of the beam}$. We also explored the limitations inherent in converting the Jones matrices for individual rays to the Stokes vector for the optical model using only the $X$ and $Y$ components of the electric field. The Mueller matrix calculations match the theoretical formula for a flat mirror based on retardance and diattenuation in agreement with theory and previous studies in the literature.

We will be assessing models for grouping the DKIST mirrors together to predict telescope polarization as functions of field, wavelength, and configuration in forthcoming publications. Beam splitter coating models are in progress and typically require 30 to 90 layers. With these coating models, we can assess the performance of the instruments in both reflection and transmission through the beam splitters. Most of the beams fed to the first light instruments interact with the dichroic beam splitter optics 2 to 4 times. In a future work, we will present polarization models of the other DKIST instruments accounting for the many layer dichroic coated optics. Other ray trace programs could potentially be explored in the future. Polaris-M is an in house polarization ray tracing software developed at the University of Arizona Polarization Laboratory.^{63}^{–}^{67} The DKIST project has used this to model the polarimetric calibration optics at the Gregorian focus.^{53} A forthcoming paper will describe the polarization performance of the calibration optics and polarization fringe predictions.

The Mueller matrix for the Gregorian focus of the DKIST primary and secondary does not have substantial FoV variation at the ${10}^{-5}$ amplitudes, agreeing with previous DKIST design studies. The Mueller matrices vary substantially across any individual footprint from a single FoV due to the incidence angle variation across the highly powered primary and secondary mirrors. These variations are substantially reduced when averaging over the footprint of the beam.

A model coating formula for an enhanced protected silver coating was derived to match witness sample data on coating retardance and diattenuation for the DKIST optics. With this model formula, we showed the azimuth–elevation dependence for the system Mueller matrix for CryoNIRSP and ViSP instruments as functions of field and wavelength. The Mueller matrix elements showed 2% variations in the linear to circular polarization terms for CryoNIRSP across a 5-arc min field. We now have a modeling tool that allows us to compute polarization across the FoV as the telescope moves in azimuth and elevation. With these computational tools, we can assess the quality of simple models of grouped mirrors as a way to calibrate the functional dependence of the system Mueller matrix under a wide variety of configurations and variables.

## Acknowledgments

This work was supported by the DKIST project. The DKIST is managed by the National Solar Observatory (NSO), which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under a cooperative agreement with the National Science Foundation (NSF). We thank Dr. David Elmore for his assistance, guidance, and insight into the long history of work on the DKIST project. We thank Dr. Don Mickey for sharing the ZPL scripts in 2003 as part of early work on the AEOS telescope. This work made use of the Dave Fanning and Markwardt IDL libraries. We also thank a few anonymous vendors for guidance on typical formulas for protected metal coatings. We thank Infinite Optics for witness samples, coating formulas, and assistance from knowledgeable staff.