1.1.

Mirror Grouping Models, Polarization, and System Mueller Matrices

We create a polarization model for the telescope and the suite of instruments using our knowledge of the optical coatings and substrates. We have shown, in HS17,⁵⁸ some predictions for the polarization behavior of the telescope and instrument feed optics using nominal coating formulas derived with the Zemax-provided refractive indices. We have shown that the mirrors introduce some slight field-of-view dependence for the polarization calibration as well as a very mild depolarization from our off-axis primary and secondary mirrors. The magnitude of field-dependent variation shown in our prior work⁵⁸ is not changed by our work presented here. In this paper, we consider only the on-axis beam at the nominal instrument boresight. Field-of-view considerations represent a significant complication. A common technique for simplifying the system polarization models is to group mirrors together that maintain a fixed orientation with respect to each other. We call this the group model. For DKIST systems engineering, we also need to predict the Mueller matrix of the system while accounting for polarization properties of the many mirrors mounted in front of the polarization modulators in each instrument. The basic calibration plan is to use the DKIST calibration optics to simultaneously fit for the telescope group model, the modulation matrix of the instruments and certain properties of the calibration optics. We show here the mathematics behind some of the simplifications assumed in the group model. We then asses how to predict these terms in later sections of the paper.

In this work, we denote the Stokes vector as $\mathbf{S}={[I,Q,U,V]}^T$. In this formalism, $I$ represents the total intensity, $Q$ and $U$ are the linearly polarized intensity along polarization position angles 0 deg and 45 deg in the plane perpendicular to the light beam, and $V$ is the right-handed circularly polarized intensity. The typical convention for astronomical polarimetry by the International Astronomical Union is for the $+Q$ electric field vibration direction to be aligned to celestial North–South, while $+U$ has the electric field vibration direction aligned to North–East and South–West. The propagation axis points toward the observer. In laboratory settings, frequently, $+Q$ is defined as horizontal or vertical. For solar studies, a common definition is to have $+Q$ parallel to the solar equator or in the positive right ascension direction^59–61

The Mueller matrix is the $4\times 4$ matrix that transfers Stokes vectors.^62–64 Each element of the Mueller matrix is denoted as the transfer coefficient.^64,65 For instance, the coefficient [0, 1] in the first row transfers $Q$ to $I$ and is denoted as $QI$. The first row terms are denoted as $II$, $QI$, $UI$, and $VI$. The first column of the Mueller matrix elements are $II$, $IQ$, $IU$, and $IV$. In this paper, we will use the notation in Eq. (1). The output Stokes vector is related to the input vector via a simple transfer equation ${\mathbf{S}}_{i_{\text{output}}}={\mathbf{M}}_{ij}{\mathbf{S}}_{i_{\text{input}}}$. With this formalism, the Stokes vector from some patch of solar atmosphere would be transferred by the Mueller matrix of each optic between the Sun and the sensor.

We adopt a notation where a rotation is denoted as $\mathbb{R}$. We note that a rotation of a Mueller matrix must include rotations on both sides of the matrix to preserve input coordinate systems: $\mathbb{R}(-\theta )$ $\mathbf{M}$ $\mathbb{R}(-\theta )$. There are three main coordinate rotations in DKIST. The elevation axis is between M4 and M5 (${\mathbb{R}}_{El}$). The azimuth axis is between M6 and M7 (${\mathbb{R}}_{Az}$). The DKIST CL is on a rotating platform so that there is a separate rotational degree of freedom with the coudé angle in addition to the azimuth of the target (${\mathbb{R}}_{TA}$)

There are six mirrors that collect the solar flux and relay the beam to the CL. The seventh mirror (M7) folds the beam onto the CL floor. The eighth mirror (M8) is an off-axis collimating mirror and the ninth mirror (M9) is a coma-correcting fold mirror with a specific figure. With this notation, we can explicitly compute the transfer equations to see how Stokes vectors will behave at various locations along the optical path. In Eq. (2), we show the input Stokes vector ${\mathbf{S}}_{\text{input}}$ being transferred from the telescope primary mirror to the CL ${\mathbf{S}}_{\text{coude}}$ just before reflecting off M7

The instrument Cryo-NIRSP does not use the AO system. The system uses a pickoff flat mirror called M9a at 9-deg incidence angle that directs light to this instrument. The next mirror in the system is a flat pupil steering mirror we denote CSM working at 4-deg incidence angle. This is followed by the off-axis mirror focusing the beam at $F/18$ using a 1.1-deg fold angle-denoted CFM. The beam passes through the polarization modulator to the spectrograph entrance slit. Equation (3) shows the Mueller matrices transferring the CL Stokes vector to the Cryo-NIRSP modulator. The Cryo-NIRSP feed optics and modulator properties are described in our prior references.^16,17,58,66

For the rest of the polarimetric instruments, the beam must propagate through the AO system, as shown in Eq. (4). The DL-NIRSP, ViSP, and VTF all see M7, M8, and M9 as well as the tenth mirror as the AO system deformable mirror (DM = M10). All instruments using the AO system must also account for the WFS-BS1 transmission and diattenuation from both the uncoated front surface and the broadband antireflection-coated wedged back surface, which we denote in Eq. (4) as BS1f and BS1b, respectively.

As an example of the Mueller matrix calculation, we show in Eq. (5) the dichroic coating front surface reflection and broadband antireflection-coated back surface reflections off the interchangeable FIDO optics feeding the DL-NIRSP instrument. We explicitly call out the optical stations CL2, CL3, and CL4 along with the separate Mueller matrices for the front and back surface reflections. We show later both theoretical models and polarimetric measurements for several of the FIDO dichroic coatings used for computing the system Mueller matrices.

The mirrors included in the DL-NIRSP relay optics also need to be included to compute the expected Mueller matrix of the system to the modulator. In Eq. (6), we show the optics transferring the Stokes vector exiting the last FIDO optic through the DL-NIRSP mirrors in the $F/24$ configuration to the modulator mounted in front of the imaging fiber bundle. The Zemax optical model for the on-axis field angle is seen in Fig. 3

Fig. 3

The DL-NIRSP $F/24$ optical Zemax model from the CL3 FIDO optic station through the seven instrument reflections to the modulator and fiber bundle.

The fundamental assumption of the group model is that the static optics can all be multiplied together and fit with a greatly reduced number of variables. For the present modeling efforts, we also ignore all field-of-view dependence though we can calculate magnitudes and make the models more complex as needed. We show the group model in Eq. (7).

The primary and secondary mirrors are ahead of the calibration optics so they are fit separately using a variety of techniques. The third mirror is near a focal plane and is actively pointed to maintain optical alignment by small amounts. The third and fourth mirrors are modeled together as a group, ignoring the small angular offsets. The first four mirrors are upstream of the elevation axis. Similarly, M6 is near a pupil plane and is also tilted by small amounts to maintain optical alignment. The fifth and sixth mirrors are modeled together as a group. After the rotations about the azimuth and coudé table axes, all optics are fixed and become part of the Mueller matrix for all optics ahead of the modulator, denoted as ${\mathbf{M}}_{\mathrm{Mod}}$. This Mueller matrix is expected to be computed for all field angles and wavelengths on every sensor and it includes all optics in the relay optics, AO system, FIDO, and within the instruments

We have shown an example for DL-NIRSP with the $F/24$ configuration in Eqs. (4)–(6). The Mueller matrix combines polarization behavior of six surfaces through the feed optics and AO system, six surfaces with complex dichroic coatings in FIDO, and another seven mirror surfaces inside DL-NIRSP ahead of the modulator. For any configuration change in FIDO or an instrument, the Mueller matrix must be calculated. This Mueller matrix will be computed for several example FIDO configurations later in this paper. Later in this paper, we provide an assessment of the sensitivity to mirror coating properties on each mirror. The assumptions underlying the simplifications in the group model will also be assessed.

We will also adopt an astronomical convention for displaying Mueller matrices where we will normalize every element by the $II$ element to remove the influence of transmission on the other matrix elements, as seen in Eq. (8). Thus, subsequent figures will display a matrix that is not formally a Mueller matrix but is convenient for displaying the separate effects of transmission, retardance, and diattenuation in simple forms. The transmission is shown in the [0,0] element, while all other elements are normalized by transmission for convenient interpretation.

1.2.

Summary: Optical Path and Predicting System Polarization

We have outlined the DKIST optical path to the CL through the instruments to the modulating retarders. We have provided a model for polarization of the various optics in the system by functional groups called the group model. The six mirrors between the sky and the CL are combined into three groups that rotate with respect to one another. We have then shown how the Mueller matrices of the coudé relay optics, the AO system, the FIDO dichroics, and internal instrument optics are combined into a single Mueller matrix representing all optics ahead of the polarization modulator. We have shown an example of the DL-NIRSP instrument in the $F/24$ configuration and describe the 19 optical surfaces that will impact this system Mueller matrix. The ViSP and Cryo-NIRSP instruments are described in a prior reference.¹¹ Next, we have shown how we can measure polarization properties of coated optics in reflection and transmission. We have examined the mirror data sets and repeatability in Sec. 2. Section 3 shows examples of fitting mirror coating models to data sets along with a comparison of metrology for retardance and diattenuation from multiple instruments. We have shown broadband antireflection coatings in Sec. 4 used in DKIST beam splitters and calibration optics. Models for FIDO dichroic coatings and examples from our vendor are shown in Sec. 5. We have discussed transmission and polarization artifacts only seen at high spectral resolving power. We have then combined this coating information to show predictions of the group model parameters and the Mueller matrix from the derived optical coating properties of all optics between the sky and the polarimeter. The combined spectral behavior of the mirrors, beam splitters, and windows is assessed in a few anticipated DKIST observing configurations in Sec. 6.

3.1.

Refractive Indices, Materials Parameters, and Coating Model Interpretation

The refractive index values used in various software modeling packages vary. In common software packages such as TFCalc or Zemax, the basic software package provides some default materials and refractive index values as lookup tables or simple equations. This nominal is useful from a modeling perspective as it provides rough guides to actual coating materials. However, there are several factors that impact the actual refractive index of deposited materials in the coating process. These include the coating deposition process, temperature, final material density, varying growth styles, impurity concentrations, and others. Thus, detailed information is required from a vendor about their specific materials and known process before having confidence in modeling coatings using refractive index data. In most cases, we do not need to know and likely will not be told the actual materials and all the proprietary details of a coating. However, a useful performance model for mirror properties can be created using simple parametric curves from the public materials data as representative of common coating types. We do not need to know specific coating formulation details to estimate the polarization performance of the system. As a modeling approach for this paper, we simply assume that the refractive index formulas may change substantially, in some cases over 15%. We list examples in Table 3 for a wavelength of 850 nm drawn from TFCalc, Zemax, MHW,¹⁵ and other references. The superscript “b” denotes the Boidin reference and the “t” denotes the TFCalc default. For instance, crystal ${\mathrm{TiO}}_2$ would have an index of 2.3, while in some coating literature the index is in the range 2.0 to 2.2. Thus, a refractive index curve fit in our study is not indicative of an actual material but of an effective refractive index that may be similar to materials commonly used in the coating process.

Table 3

Indices.

Figure 9 shows example curves taken from a common website (RefractiveIndex.info), the default coating file in the software packages TFCalc and Zemax, along with vendor catalog values. On the left side we can see crystal sapphire (${\mathrm{Al}}_2{\mathrm{O}}_3$) having an index around 1.75. When used in a coating, TFCalc gives a value slightly above 1.6 constant for all wavelengths as the dashed red curve, while Zemax uses a value slightly below 1.6 that falls with wavelength as the dashed magenta curve. The website RefractiveIndex.info cites Boidin et al.⁶⁸ in the red curve from just above 1.7 falling to 1.65 at long wavelengths. We have internal DKIST engineering reports that also use slightly higher and lower values as seen by the green and purple curves.

Fig. 9

The refractive indices published for common coating materials. The RefractiveIndex.info website data are shown along with data from the Zemax coating file provided with version 16.5, 2016 and the TFCalc default material files where applicable. (a) Shows ${\mathrm{Al}}_2\mathrm{O}3$ and (b) shows ${\mathrm{MgF}}_2$ crystal and coating values. The CVI Melles Griot catalog equation is used for crystal ${\mathrm{MgF}}_2$. The difference in refractive index between amorphous coated materials and crystalline materials can be several percent, giving strong changes in predicted coating performance.

Slightly better agreement is seen between values for a common coating material, ${\mathrm{MgF}}_2$. The right-hand graphic of Fig. 9 shows the CVI Laser Optics & Melles Griot catalog equation for crystalline ${\mathrm{MgF}}_2$ as the solid and dashed magenta lines. The magenta dot-dashed line represents the weighted average of an amorphous version simply computed as twice the ordinary and once the extraordinary equations combined. Zemax provides a point-wise linearly interpolated version seen as the dashed blue line in Fig. 9 that essentially tracks the CVI catalog equation for the ordinary index of crystal ${\mathrm{MgF}}_2$. The TFCalc software package uses a wavelength independent value of 1.38.

The refractive index of the silver metal has a strong influence on polarization properties of the coatings. Literature and vendor values vary widely. Figure 10 shows an example of linearly interpolated TFCalc default values in green with limited data points and some strong changes in spectral behavior around 400- and 1000-nm wavelength. We also show simple functional fits to the real and imaginary components of silver metal used in various coating models. The solid lines show simple polynomial models for the real part of the refractive index (n) with values ranging from 0.05 to 0.5.

Fig. 10

The real and imaginary refractive index components of silver. The real component ($n$) is shown as the solid lines on the left-hand $y$ axis with values ranging from 0.05 to 0.5. The green lines show the default TFCalc refractive indices. Note some discontinuities at 400- and 1000-nm wavelengths. Different colors show simple polynomial model fits to TFCalc default values and models provided by vendors in various example coatings. The dashed lines show the imaginary component ($k$) in exponential form ($n\text{-}ki$) on the right-hand $y$ axis. The values of $k$ reflect orders of magnitude change in the wave penetration depth into the metal.

The imaginary part ($k$) is shown as dashed lines in Fig. 10 using a Zemax-style convention where the index is modeled as ($n\text{-}ki$). This imaginary index is exponential in behavior and the electromagnetic wave does not penetrate more than a few nanometers into the metal layer when the complex index is $>2$ or 3. For the silver metal coating model in Fig. 10, we see values of roughly 3 at blue wavelengths rising linearly to 11 or 12 at wavelengths of 1600 nm. Often the vendor curves are discontinuous as the green line of Fig. 10 shows for the default TFCalc values.

In prior works,^11,67 we only allowed the thickness of one or two dielectric coating layers to vary as a simple two variable optimization problem. However, the metal layer thickness and refractive index both have large polarimetric impact. We show retardance and diattenuation changes in Fig. 11 when using two different formulations for the silver metal indices. In both models we use 10 nm of ZnS with indices from RefractiveIndex.info coated over 105 nm of ${\mathrm{Al}}_2{\mathrm{O}}_3$ with refractive indices using the Boidin et al.⁶⁸ values in RefractiveIndex.info. We change only the refractive index of the silver to show the impact.

Fig. 11

The difference between retardance and diattenuation parameters computed using two different refractive index values for silver metal in the coating for a reflection off a DKIST-enhanced protected silver mirror formula at 45-deg incidence.

The black curve with the left-hand $Y$ axis shows that retardance can change by over 2-deg peak to peak for a coating with 20 deg to 40 deg, a 10% effect. Much larger changes are seen in diattenuation. The diattenuation change shown in blue using the right-hand $Y$ axis is up to 4% at short wavelengths and roughly half a percent in the visible-to-NIR wavelengths. The diattenuation for this coating is only 1.5% magnitude peak to peak. This metal index difference changes the diattenuation over 300%. The silver metal properties can dominate the fit of diattenuation. Vendors sometimes provide coating performance predictions. Infrequently, this may be accompanied by names of materials used, such as coating $X$ is ${\mathrm{SiO}}_2$ protecting the Ag. Usually, the layer thickness is not disclosed and the refractive index values for the as-coated material differ significantly from literature values. The retardance is much more sensitive to the dielectric material thicknesses as well as refractive indices. The diattenuation is very sensitive to the real and imaginary refractive index components of the metal as well as the dielectrics.

In Fig. 12, we show the reflectance predicted for this same coating model but we use a few different variations of the silver metal complex refractive index. The reflectivity data show clear discontinuities where TFCalc, as well as our Berreman code, performs a linear interpolation between points in a table of refractive index values. Solid lines shows the S-polarization state, while dashed lines show the P-polarization state. This particular coating has diattenuation change sign in the 700- to 1050-nm wavelength range. The blue curve shows a vendor-provided TFCalc model. The vendors may have their own internal materials databases, sometimes where refractive indices are adjusted to their results or at other times modified versions of literature values. We expect significant variation among any historical literature values, the vendor models, and actual coatings. This is especially true when our simple models likely do not correspond to actual materials and may not include all the layers in the actual coating. The green curve shows one of our Berreman models of silver refractive indices that overestimates the reflectivity at NIR wavelengths by 1.5%. The red curve shows a separate Berreman prediction using lower real refractive index components for the silver that underestimates the reflectivity by over 2%. The black curve is a by-hand modification of the green curve at infrared wavelengths to show that reflectivity can be met, but the diattenuation prediction is still significantly in error. The various lookup tables of silver refractive indices presented can change the reflectivity by over 3%. As we currently do not perform a simultaneous fit to reflectivity, diattenuation, and retardance, we expect our models to contain errors as presented in this section. This becomes a major limitation of polarization performance modeling, requiring future development for simultaneous fitting of many variables.

Fig. 12

The reflectivity variation when linearly interpolating tabulated data for refractive index of the silver metal layer. The dashed lines show parallel polarization state ($P$), while the solid lines show the perpendicular polarization state ($S$) at a 45-deg incidence angle. Blue shows a vendor prediction, while black, red, and green show Berreman models using various refractive index curves. Discontinuities come from interpolation between refractive index table values.

3.2.

DKIST ViSP: Enhanced Protected Silver Mirrors with 29 Dielectric Layers

We show in this section a significantly more complex enhanced silver mirror coating that was nominally going to be used in DKIST by one of our partner instruments. The ViSP team chose to use a many-layered enhancement on the feed mirrors between the FIDO dichroics and their modulating retarder after the spectrograph entrance slit. This coating is nominally 29 layers of dielectric with an oscillating high-low refractive index design. This represents roughly $3\mu \mathrm{m}$ of dielectric coated on top of the silver metal layer. Given the many layers, significant spectral variation is expected along with the presence of narrow spectral features. Though the team has stripped and recoated their mirrors with an alternate coating, we include this coating metrology here to show the impact of many-layered enhanced designs and consideration of manufacturing issues.

Figure 13 shows the ViSP mirror witness sample polarization properties measured in NLSP at 45-deg incidence. The left-hand graphic shows retardance in black and diattenuation in blue for the full NLSP measurement range. The right-hand graphic shows the shorter visible wavelengths where very rapid but well-measured spectral changes are seen. The retardance changes by over 300 deg in 20-nm wavelength range, giving a spectral gradients up to 60 deg per 1-nm wavelength. This bandpass is comparable to the full spectral range measured across the ViSP sensors. Diattenuation similarly changes from $-10\%$ to $+20\%$ in a very narrow bandpass. This kind of mirror coating has impact for DKIST as the modulation matrix must be wavelength-dependent with the assumption of variation at these magnitudes. We note that requirements against strong spectral gradient in retardance and diattenuation are not included in any specifications. Given the metrology results, the ViSP team has already stripped and recoated their mirrors. Behavior for the ViSP feed optics had they kept these coatings will be explored in later sections of this paper. This case is a good example of what happens when many-layered coatings are specified giving rise to complex, large, and spectrally narrow polarization properties.

Fig. 13

Retardance in black and diattenuation in blue measured with NLSP for the ViSP feed mirrors coating witness sample at 45-deg incidence. (a) Shows all NLSP wavelengths from 380 to 1650 nm. (b) Shows the 380- to 520-nm wavelength range where the mirrors show a very strong change in retardance of almost a complete wave over 20-nm bandpass.

3.3.

Summary of Coating Model Fits to Reflectivity, Diattenuation, and Retardance

We have presented examples of two-layer coating models fit to NLSP retardance measurements in this section. We have shown how the dielectric layer thickness and material refractive index impacts fitting measured retardance curves. In Sec. 3.1, we have shown how the complex refractive index of the metal layer strongly influences the reflectivity and diattenuation. When fitting a coating formula to measured data, the refractive indices of all components need to be assessed. For DKIST, the retardance values are critical as they determine the field dependence of the cross talk and ultimately drive requirements on how DKIST calibrates instruments and with what model for the mirror retardance. When attempting to fit retardance, diattenuation, and reflectivity simultaneously, all refractive index values become critical. In Sec. 3.1, we have shown how public literature values for refractive indices may roughly approximate coating behavior, but a detailed fit to high accuracy in all performance parameters requires substantially more detailed knowledge of the coating materials properties. Examples of coating model fits and witness sample measurements were provided for three IOI samples where we have much better refractive index information. Measurements of polarization for a more complex mirror coating with 29 dielectric layers over silver initially planned for use in the DKIST ViSP have been shown in Sec. 3.2. In Sec. 10, we show more examples of mirror polarization properties from commercial sources used in DKIST and the GST (formerly the NST) at the BBSO, along with examples of coating models and refractive index variation impacts on predicted behavior. We have samples from the GREGOR solar telescope and DKIST VTF instrument shown in Sec. 10.1. We move on from many-layered dielectric protected mirrors to many-layered dielectric coatings used as antireflection coatings in Secs. 4 and 5. Techniques for fitting properties of such coatings also become more complex.

4.1.

Antireflection and Dichroic Summary: Fitting Many-Layered TFCalc Models

In this section we show how simple antireflection coating designs can be adjusted to fit the as-built layer thicknesses by using unpolarized transmission spectra. With these adjusted models, we are able to then predict transmission, diattenuation, and retardance that closely match our NLSP measurements and vendor metrology. A wide-wavelength range antireflection coating design with 14 layers of ${\mathrm{SiO}}_2$ and ${\mathrm{HfO}}_2$ and two additional layers (top and bottom) is fit to unpolarized transmission data. Subsequent predictions are made for NLSP retardance and diattenuation with agreement better than 1-deg retardance and a fraction of a percent diattenuation. Errors increased with higher incidence angles due to the exacerbated optical misalignments in NLSP caused by the tilted glass substrate. In Sec. 11, we show coating repeatability and measurements of one-side and two-side coated samples. These samples are identical in coating design to three DKIST windows within FIDO called Coudé Windows 1, 2, and 3 (e.g., denoted as C-W1). The AO wavefront beam splitter (WFS-BS1) is only back-side coated with WBBAR1 with FIDO C-W2 being delivered similarly. The FIDO C-W1 and C-W3 will be both-side coated similar to the two-side coated windows in the DKIST calibration optic (CalPol2 window). Now that we have successfully shown this example of repeatably manufacturing and fitting relatively simple, thin coatings in transmission, we move on to the much thicker FIDO dichroic coatings working both in transmission and in reflection.

5.1.

FIDO Dichroic C-BS-465: Coating Spatial Uniformity and Model Fitting

In this section, we make a detailed analysis of the thinnest FIDO dichroic coating. We assess two separate coating shots for spatial uniformity, spectral performance, and variability of the individual coating layers. This BS-465 design uses 24 layers, has a 1502-nm physical thickness with the thinnest layer at 17.0 nm in the design. The coating follows a common design with a strippable layer as the base then alternating ${\mathrm{SiO}}_2$ and ${\mathrm{TiO}}_2$ layers. A thicker ${\mathrm{SiO}}_2$ outer layer is the air interface.

Table 7 shows the test runs for this dichroic coating. We received uniformity measurements at 15-deg incidence angle on nine samples coated in run 10-0153 around April 21, 2018. The transmission and reflection measured by IOI with their spectrograph is shown in the left-hand graphic of Fig. 18 as the solid lines. We then used this spectral data and selected wavelengths from 350 to 1200 nm to perform coating model fitting in TFCalc.

Table 7

Dichroic coating 465 testing.

Fig. 18

(a) Shows TFCalc models after fitting the IOI measurements, along with the uniformity sample measurements. (b) Shows the differences between the average and each individual curve to highlight spatial variation. Blue shows spatial variation in measurements, while black shows spatial variation derived from the TFCalc models. Dichroic 465 test run 10-1053 had transmission measured on nine witness samples at AOI = 0 deg. Transmission curves are adjusted for a theoretical $\sim 3.8\%$ reflection loss from the uncoated sample back surface.

After this uniformity testing, additional tests of the chamber for spatial uniformity and refractive index of the deposited materials were done. The tests used a standard 13-layer narrow band filter design ${(\mathrm{HL})}^3$ 2H ${(\mathrm{LH})}^3$. Table 7 shows these two tests in the same chamber (10) but with coating shot 0154 immediately after the dichroic and another test 6 days later in shot 0156. With the nominal wavelength set around 480 nm for this test, the quarter-wave optical thickness is 82 nm for the low-index material ${\mathrm{SiO}}_2$ and 55 nm for the high-index material ${\mathrm{TiO}}_2$. The variance of the filter central wavelength was used as a statistical measure of the layer-by-layer variation, showing that we did indeed pass a 1% variation of the physical thickness across the aperture. With these updated refractive indices in hand, we could fit the TFCalc models to the C-BS-465 dichroic sample data. The left-hand graphic of Fig. 18 shows the best-fit TFCalc models as dashed lines. Some of the spectral oscillations are not well fit, but the transition wavelengths and general behavior are reproduced. We can expect spatial variation in both reflection and transmission across the beam following the magnitudes shown in Fig. 18.

The right-hand graphic of Fig. 18 shows the variation between the mean and the nine individual spatial samples. The TFCalc model variation is shown in black and measurement residual errors are shown in blue. As expected, there are larger errors near wavelengths where spectral gradients are strongest. We also do not reproduce a somewhat larger spectral oscillation around 475-nm wavelength with a depth of a few percent. Note that we have adjusted the transmission to account for the $\sim 3.8\%$ Fresnel reflection loss from the uncoated sample back surface using the Fresnel equations and the refractive index data for Heraeus Infrasil provided by the manufacturer.

The statistics of layer variation in these coatings show that we pass a 1% physical coating thickness variation across the clear aperture, and that the variations essentially follow Gaussian statistics across the aperture. In the left-hand graphic of Fig. 19, we show the variation between layers in the TFCalc fitting process. The baseline design thickness is shown as the blue curve with each layer of alternating ${\mathrm{SiO}}_2$ and ${\mathrm{TiO}}_2$ having thicknesses between roughly 30 and 80 nm. Each of the nine colored curves shows a fit to the individual spectra.

Fig. 19

TFCalc fits to the nine uniformity sample measurements for dichroic 465 test run 10-1053. (a) Shows all nine of the TFCalc models. Some layers are computed to vary more than others, especially the outer layer. (b) Shows the cumulative distribution of errors for each layer in each model with respect to the average.

In the right-hand graphic of Fig. 19, we show the statistics of the variation between layers. The cumulative distribution of errors in each individual layer against the nominal design is shown as the solid black line. We did not see any evidence of any layer being particularly thicker or thinner as a function of aperture radius or layer depth. A few layers show more variation than others, but the statistics are essentially the same as other layers with radius and depth. The crosses show 1-sigma 68% and 2-sigma 95% errors for a Gaussian distribution fit to the histogram. More than 68% of the layers are within 1.15-nm layer thickness of the average. For 95% of the points, the layer variation is $<2.9\mathrm{nm}$.

Figure 20 shows the transmission and reflection data in the appropriate bandpass intended for feeding the DKIST instruments. Reflectivity is over 99% for the range 380 to 450 nm. Similarly, after compensating for the Fresnel reflection, the transmission is over 96% for all wavelengths between 480 nm and the long wavelength cutoff of the metrology, except for a small bandpass around 550 nm. The mean transmission is over 97% for this coating. The actual FIDO dichroic will have the WBBAR1 coating on the back side, minimizing the back surface reflection losses.

Fig. 20

(a) The transmission bandpass and (b) reflection bandpasses for IOI measurements of the nine samples throughout the chamber for the C-BS-465 dichroic coating uniformity test run 10-1053 at AOI = 0 deg.

Figure 21 shows the IOI transmission data around the transition wavelength at normal incidence (0 deg) corrected for the back surface Fresnel losses. The transition wavelength between reflection and transmission is slightly longer than the nominal 50% transmission at 465-nm wavelength. For this design there is also roughly 5-nm wavelength shift to the blue when used at the nominal 15-deg FIDO orientation. The transition wavelength is noted by the straight black line of 479.3 nm along with a spread of $\pm 1.3\text{-}\mathrm{nm}$ wavelength at 50% transmission across the aperture. This 20% transmission spatial variation across the aperture would raise calibration concerns if using this optic at the transition wavelength. However, the nominal dichroic coating specification of reflectivity of $>90\%$ for wavelengths short of 440 nm and $>90\%$ transmissive for wavelengths longer than 490 nm is easily met with this test. The design is also easily adjusted to slightly shorter transition wavelength following these test results.

Fig. 21

Transmission measurements at AOI = 0 deg of the nine uniformity samples for C-BS-465 dichroic run 10-1053.

We received all nine samples and performed testing in NLSP for polarization. Figure 22 shows the NLSP-measured retardance and diattenuation as a function of incidence angle. We tilted the sample between 0 deg and 45 deg. The retardance has strong spectral gradients near the transition wavelengths, as expected. Diattenuation is spectrally stable in transmission with mild oscillation in the transition band. For the beam at 15-deg incidence, the diattenuation is $<1\%$.

Fig. 22

(a) Transmission retardance and (b) diattenuation measured with NLSP at various incidence angles for the C-BS-465 dichroic samples. The solid lines show the data and dashed lines allow comparison with the best-fit TFCalc model predictions.

We note that the NLSP reflective arm currently only measures samples at 45-deg incidence so we do not have a direct measurement of polarization performance at the FIDO incidence angle of 15 deg. The TFCalc models closely match the predictions, and this dichroic has essentially negligible polarization impact in the reflection band for the reflected beam. Retardance is $<5\mathrm{deg}$ across the entire 380- to 460-nm wavelength range. Diattenuation is $<0.2\%$ across that same range. The IOI measurements of reflected diattenuation in this region are also consistent with zero, given their measurement uncertainty. This will of course not be true for the more complex dichroics discussed later, but we conclude here that we have a valid model supported by several data sets for this simple FIDO dichroic.

5.2.

Infinite Optics Dichroic A: Reflection 380 to 580 nm

When fitting significantly more complex coating designs, there is significant degeneracy. A completely free fit could adjust the thickness of every layer and the refractive index of each material to match the data. Fitting every individual spectral oscillation is usually quite difficult, resulting in unrealistic layer thickness and fits that are not close to the actual deposited coating. Often there are direct measurements and estimates of layer thicknesses recorded during the coating process providing the ability to constrain the fit. We demonstrate some constrained fitting here.

We received an Infinite Optics sample from run number 8-5652, which we labeled as Dichroic A. This dichroic reflects wavelengths shorter than roughly 580-nm wavelength. This design was not optimized for polarization or for use at high incidence angles and is expected to have strong polarization artifacts. The design uses 61 layers for a total $4.00\text{-}\mu \mathrm{m}$ thickness. Most of the layers are alternating ${\mathrm{SiO}}_2$ and ${\mathrm{TiO}}_2$ at roughly quarter-wave thickness for a 510-nm reference wavelength (87 nm/59 nm per layer). We can compare the theoretical TFCalc design files with our NLSP measurements in transmission. Figure 23 shows the measured II Mueller matrix element from NLSP along with the TFCalc predictions at the appropriate AOI. The two NLSP measurements are shown in black with normal incidence as the solid line and 45-deg incidence as the dashed line. The blue and green curves show the TFCalc design files for incidence angles 0 deg and 45 deg, respectively.

Fig. 23

NLSP transmission spectra are shown in black along with TFCalc design predictions as blue and green for IOI Dichroic A run number 8-5652.

We took the TFCalc design and adjusted the transmission to account for the Fresnel reflection off the uncoated back surface. Note that we had lower resolving power of the NLSP setup for these measurements. A 9-nm FWHM instrument profile reduced the magnitude of some of the narrow, low magnitude spectral transmission features seen in the lower left reflection band. It also mildly reduces some of the spectral ripple magnitudes at the shortest wavelengths. But there are significant amplitude differences and the instrument profile does not reduce the spectral ripple significantly. As seen in Fig. 23, the rough shape of the measured transmission curve is matched by the nominal design as is the 50/50 transition wavelength, but the stack up of thickness variation in 61 coating layers combined with variations in refractive indices can cause the detailed spectral dependence to change significantly in spectral ripple as well as transition wavelength. Fortunately, we can use TFCalc to constrain a fit of the design to the NLSP measured transmission, diattenuation, and retardance. Depending on the limits set for the allowable variation of each layer and on the refractive index values chosen, matching all spectral oscillations in a single transmission spectrum can drive the coating model far away from the actual layer thicknesses estimated by other means.

An example of the fitting process is shown in Fig. 24 for the transmitted beam at an incidence angle of 45 deg. The left graphic shows the measured transmission function along with various fits. The right-hand graphic shows retardance in green and measured diattenuation in blue as well as the same series of constrained fits. Each of the individual fitted models is computed with ever wider range allowed for the variation of each layer thickness against the design. The fits improve to this single spectrum, but some layer thickness become quite different from the estimates derived during the coating process. The retardance is nearly a full wave in the transition band from short wavelength reflection to long wavelength transmission. The diattenuation is similarly close to 100%. In the transmission band, the retardance smoothly decays from a full wave around 600-nm wavelength to under 0.1 waves (36 deg) for long wavelengths. The diattenuation, however, has a more complex spectral pattern. This filter has been nominally designed to have high transmission of only around 700-nm wavelength. Diattenuation contains many spectral oscillations of a few percent in the high transmission bandpass but up to 20% at NIR wavelengths.

Fig. 24

(a) Shows the NLSP-measured transmission for the Dichroic A, IO run number 8-5652, as dashed red. The TFCalc design is solid back, the TFCalc unconstrained fit is solid blue (matching the NLSP data well), and the constrained TFCalc fits are thin green lines. (b) Shows the NLSP-measured retardance and diattenuation in transmission as black dashed lines with the TFCalc designs as solid lines. The $Y$ axis is both percentage diattenuation and retardance in 100ths of waves. The various blue lines show TFCalc diattenuation models and green lines show retardance models.

5.3.

Infinite Optics Dichroic B: Reflection <680 nm on High Refractive Index Glass

We received another Infinite Optics sample from run number 7-2802, which we labeled as Dichroic B. This dichroic reflects wavelengths shorter than roughly 680-nm wavelength and is coated on a high refractive index SF11 glass substrate. This design was not optimized for polarization or for use at high incidence angles and is expected to have strong polarization artifacts and spectrally narrow features. The design uses 84 layers for a total $6.59\text{-}\mu \mathrm{m}$ thickness. Most of the layers are alternating ${\mathrm{SiO}}_2$ and ${\mathrm{TiO}}_2$, the same as the DKIST FIDO dichroic designs. In Fig. 25, we show the polarization properties of Dichroic B in transmission at 45-deg incidence derived from the NLSP-measured Mueller matrix. The retardance is shown in black using the left-hand $y$ axis running from 0 deg to just over 300 deg. The blue curve shows diattenuation using the right-hand $y$ axis running from 0% to 100%. As this dichroic is highly reflective for wavelengths shorter than 680 nm, the transmitted flux is very low. However, we do reproduce stable polarization measurements with NLSP even with transmission $<1\%$. We have typical signal-to-noise ratios (SNRs) over 10,000 for a transmissive optic, so we still can achieve SNRs in the range of 1000 with 1% transmission. There are narrow spectral features in diattenuation of the transmitted beam that are entirely real, as we detail in the next section. As this coating was not optimized for transmission, strong diattenuation is seen for the transmitted beam even in the transmission bandpass.

Fig. 25

The NLSP-measured polarization properties in transmission for the Infinite Optics Dichroic B sample at 45-deg incidence with run number 7-2802. Black shows retardance on the left-hand $y$ axis. Blue shows diattenuation on the right-hand $y$ axis.

This coating has been designed only for high reflectivity at low incidence angles. For coatings designed without specification of polarization, the coating can have very strong rates of spectral change in addition to large magnitudes of both retardance and diattenuation. In Fig. 26, we show the NLSP-measured retardance and diattenuation of Dichroic B in reflection at an incidence angle of 45 deg.

Fig. 26

The NLSP-measured retardance in black and diattenuation in blue for the Infinite Optics Dichroic B sample run 7-2802 at 45-deg incidence in reflection. The 440- to 655-nm wavelength region is selected as the efficient reflection bandpass.

The retardance is shown in black with multiple wavelength regions showing strong spectral changes wrapping through several waves of retardance. The diattenuation is shown in blue with large antisymmetric swings in the same wavelength regions where retardance changes quickly. Spectral changes are over $\pm 10\%$ in just a few nanometers wavelength, comparable to the spectral resolving power of NLSP. We show this dichroic as an example of how optimizing a design for only reflectivity can create narrow spectral regions where the transmission and polarization performance pose calibration challenges.

5.4.

Narrow Spectral Features: Spectral Calibration with Many-Layered Coatings

Coating models predict narrower spectral features and stronger wavelength gradients as the number of layers and coating thickness increase. Features in common astronomical dichroics can exist where the reflectivity can drop over 10% from the $>99.9\%$ nominal performance along with strong polarization-dependent response.

As an example, Fig. 27 shows the constrained fit TFCalc models to the NLSP data sets from above. Blue shows the transmission spectrum and black shows the diattenuation for a 45 deg incidence angle. We note that in the reflection band around 450 to 500 nm, this design shows almost no narrow spectral features and minimal sensitivity to the varying fit constraints. However, at shorter wavelengths of 380 to 440 nm, there are transmission features predicted at amplitudes of 5% to almost 40% depending on the model. As NLSP does not have significant signal for this dichroic at short wavelengths, we can only plot the measured transmission spectrum as the thick blue line of Fig. 27.

Fig. 27

The TFCalc models of Dichroic A run 8-5652 from a constrained fit to the NLSP data. Narrow, strong features are predicted at short wavelengths with high sensitivity to the specific constraints applied. Blue shows transmission and black shows diattenuation. The thick dashed blue line is the NLSP-measured transmission spectrum.

We measured the transmission spectra for Infinite Optics Dichroics A, run 8-5652, and B, run 7-2802, over the 370- to 600-nm wavelength range at 4 pm per step with the Meadowlark Spex instrument we previously described.⁶⁹ In the Meadowlark Spex system, their double-grating CT spectrograph used a $30\text{-}\mu \mathrm{m}$ wide slit setting and a photomultiplier as the sensor. Spex had an instrument profile of roughly 25 pm optical FWHM over visible wavelengths. Sampling was 4 pm wavelength per step or about six samples per optical FWHM. Spectral resolving power is thus roughly 20,000 sampled at one part in 120,000.⁶⁹

Figure 28 shows examples of the narrow spectral features measured in the Infinite Optics Dichroics A and B. The left-hand graphic shows a 25-nm bandpass measuring transmission for Dichroic A run 8-5652 in the wavelength region where this optic is highly reflective. The nominal reflectivity is over 99.9% as designed but there are narrow spectral features that reach up to 7% transmission. In the right-hand graphic of Fig. 28, we collected several spectral features from both Dichroics A and B. We have centered them on a relative scale by the central wavelength seen in the legend. We then normalized them by the peak transmission. The wavelength-dependent profile of each feature is complex, but the widths are roughly 1 to 2 nm FWHM.

Fig. 28

Transmission measurements of Infinite Optics dichroics in the reflection bandpass. (a) Shows transmission for the run 8-5652 Dichroic A. Three narrow features at amplitudes of 1% to 7% transmission are seen in this bandpass and are predicted in the TFCalc model. (b) Shows a compilation of several narrow features in both run 7-2802 Dichroic B and run 8-5652 Dichroic A. Some features are Gaussian in spectral profile while others are complex. Amplitudes range from a fraction of a percentage to 7%.

5.5.

Summary of Dichroic Coatings: Spectral Features and Optimization

In Sec. 5, we have shown the nominal DKIST designs for the FIDO. The FIDO dichroic coating designs are shown using 25 to 96 layers and thicknesses from 1.5 to $8.8\mu \mathrm{m}$. These designs are compared with as-built samples of many-layered coatings. We have shown NLSP spectropolarimetric data, the nominal thin film models and fits of those models to our as-built data. The first example dichroic (A, run 8-5652), shown in Sec. 5.2, has 61 layers and shows spectrally smooth behavior in both reflection and transmission with high reflectivity and a sharp transition. An alternate dichroic (B, run 7-2802) example, shown in Sec. 5.3, has very high reflectivity but spectrally narrow bands of high diattenuation along with retardance oscillations of over five waves in the reflection bandpass of 420 to 680 nm. We have then shown these very narrow features are quite real, measurable, and expected in various designs, in Sec. 5.4. We have presented spectrophotometric measurements with a resolving power of 20,000 on the Meadowlark SPEX system and example TFCalc design files showing how different dichroic designs can be quite sensitive to manufacturing errors and create these narrow but quite strong spectral features.

Given these measurements of several dichroic styles, DKIST optimized our designs not only for high reflectivity but also for low diattenuation and spectrally smooth retardance while also being constrained by the minimum stress and wavefront error requirements. The C-BS-950 design presents challenges, given the nominal 96 layers, but the sensitivity was minimized and the retardance should maintain spectral gradients of $<10\mathrm{deg}$ per nanometer of bandpass for almost all of the required bandpass with particular emphasis on key DKIST spectral wavelengths. We are presently performing a thorough coating design verification and repeatability study to ensure that DKIST achieves the required optical performance on the FIDO optics.

We have shown how the thin film design tools such as TFCalc can be used to verify that dichroic coatings fall within tolerances and also to predict the presence and magnitude of narrow spectral features. Many astronomical instruments are now observing simultaneously with other instruments often covering narrow bandpass at high spectral resolving power. To ensure success in designing and calibrating these instruments, the properties of all the many-layered coatings in the system must be known. By controlling the design, benign performance can be assured. For DKIST, we have shown mirrors, broadband antireflection coatings, and dichroics in the above sections. We now combine the optics into their appropriate groups and create a system-level model for DKIST with all coatings on all optics.

6.1.

Instrument Feed Optics: ViSP Reflectivity and Polarization with Many Layers

The ViSP has three enhanced protected silver mirrors between the last FIDO optic reflection and the spectrograph entrance slit. The original specification had very high reflectivity mirrors using 29 dielectric layers and a physical thicknesses of 3 microns. Thick, many-layer designs can have very narrow spectral features that impact polarization calibration and heightened sensitivity to various manufacturing and environmental conditions. Though the ViSP team has stripped and recoated their optics, we consider what would have been the polarization properties of these feed optics.

For the baseline ViSP feed mirrors coatings, we were given a coating design file for the coating with roughly 29 layers of dielectric material over silver. Figure 33 shows the NLSP measurements of a witness sample retardance measurement at 45-deg incidence along with the TFCalc model fit to that data set in the left-hand graphic. With this TFCalc model fit to retardance at 45 deg over wavelengths 380 to 800 nm, we can then predict performance for the baseline ViSP mirrors before stripping at appropriate incidence angles outlined in Table 9. ViSP FM1 at 28-deg incidence is shown in blue. The ViSP-powered optic F1 is shown in black at 2.2-deg incidence. The ViSP-powered optic F2 is shown in green at 12.3-deg incidence.

Fig. 33

(a) Shows retardance measured with NLSP for the ViSP feed mirror coating witness sample at 45-deg incidence from 380 to 900 nm as the oscillating black curve. The TFCalc model fit to that NLSP retardance is shown as the dashed red line. With this successful TFCalc fit, we then predict the retardance for the ViSP mirrors FM1, F1, and F2 at the appropriate incidence angles. (b) Shows vendor-measured reflected diattenuation at the appropriate ViSP incidence angles in the 380- to 520-nm bandpass where the mirrors show very strong changes.

Table 9

ViSP feed.

The polarization introduced by these three mirrors combine with the diattenuation and retardance introduced by the tilted broadband antireflection-coated glass substrate of the slit mask to create static elements of the ViSP system modulation matrix. The modulator is mounted behind the slit mask. Table 9 shows the three ViSP feed optics between the FIDO and the ViSP modulator. The ViSP slit masks are deposited on a glass substrate mounted at 5.4-deg incidence angle to the beam. Both sides of this substrate are coated with a broadband antireflection coating with reflectivity averaging 1.0% over the 380- to 900-nm bandpass. Each side has slightly different coating behavior with both coatings having spectral oscillations ranging from 0.2% to 1.5% across the bandpass from shot-to-shot variation. Given the low incidence angle and high transmission, the retardance and diattenuation contributions are very small from this optic.

Figure 34 shows the parallel (P) and perpendicular (S) polarization state reflectivity of the ViSP fold mirrors measured at the appropriate incidence angles for the ViSP fold mirrors in the left-hand graphic, measured several months after applying the original (now stripped) coating. The ViSP FM1 is at 28-deg incidence plotted as the solid lines. The ViSP F2 mirror is at 12.3-deg incidence and is shown as the dashed lines. As expected, when the incidence angle increases, the complex spectral pattern generally shifts to the blue and changes morphology. The DKIST-enhanced silver mirror coating is shown as a comparison using black lines. Solid line shows a model while a dashed line shows reflectivity data for one of the DKIST mirrors (M10). Two dashed vertical lines show the Ca II spectral lines of interest for ViSP at 393.4 and 396.9 nm, which drove the coating design.

Fig. 34

(a) Shows the $S$ and $P$ reflectivity measured by the vendor for a witness sample from the single coating shot covering all three ViSP fold mirrors. The solid lines show incidence angle 28 deg for ViSP FM1 and dashed lines show 12 deg for ViSP F2. The black lines show both data (dashed) and models (solid) of the DKIST silver for comparison. (b) Shows the reflectivity of the combined ViSP mirrors with each mirror at the appropriate incidence angle. Black shows the highly enhanced coating with the measured reflectivity issue for the three combined optics. Blue shows the nominal DKIST silver formula (D.EAg) modeled for each optic at the appropriate incidence angle. Green shows the stripped and recoated mirrors as measured using the formula EAg1-420, the same as for the FIDO mirrors.

We note that the band of reflectivity at 80% to 90% was not present in the original test coating metrology. A manufacturing error gave rise to the actual mirror coatings behaving differently than the preliminary test coatings. The TFCalc models fit to vendor data on the ViSP-enhanced silver coating failed to reproduce the relatively low reflectivity curves measured by the vendor in the 450- to 490-nm bandpasses. However, the model does show that the wavelength shift from incidence angles of 12.3 deg to 2.2 deg are very small. We do not have vendor reflectivity data at 2.2-deg incidence, but we can approximate the F1 mirror as the average of S and P reflectivity curves at 12.3 deg with zero diattenuation and zero retardance. Using this assumption, we can make a reflectivity model of the three combined ViSP feed mirrors as the black curve shows in the right-hand graphic of Fig. 34. The system throughput would have generally been around 96%. We show common atomic spectral lines in Table 10 intended for ViSP observations that are noted as vertical dashed lines in some figures of this section. However, there is a drop to roughly 70% to 75% throughput that will impact observations of spectral lines, such as Ba II at 455.4 nm, Sr I at 460.7 nm, and ${\mathrm{H}}_{\beta }$ at 486.1 nm shown as vertical dashed lines. In addition, there is a drop to 90% throughput for the Ca II NIR line at 854.2 nm. Given this metrology result, these mirrors had been stripped and recoated in October 2018.

Table 10

Ex. lines.

The new system throughput after stripping and recoating the three ViSP feed mirrors are shown as the green curves in the right-hand graphic of Fig. 34. These curves are generated using spectrophotometry on witness samples for the two appropriate coating shots. The blue curve shows the model prediction for the nominally specified DKIST-enhanced protected silver. As shown above, the as-coated performance for this coating can vary substantially so we only show the nominal modeled performance. Each coating shows a few percent throughput loss at various wavelengths, but the polarization curves are very smooth, following the two-layer coating models shown in previous sections.

Figure 35 shows the polarization model for the three ViSP feed optics combined at the appropriate incidence angles for the older (baselined) 29 layer coatings. The Mueller matrices are created for each optic and multiplied in sequence to create the model. The reflectivity and diattenuation from the vendor data are used while the retardance is predicted from the TFCalc model fit to our NLSP retardance measurements. The left-hand graphic of Fig. 35 shows wavelengths between 380 and 550 nm a different vertical scale than the right graphic covering 500 to 950 nm. Around 450 nm there is a change of retardance of about one full wave over roughly 15-nm bandpass as the retardance goes from zero to over half wave and back. The spectral change in retardance in this bandpass goes over 60-deg retardance per nanometer of bandpass. This instrument would have to be calibrated at high spectral resolving power to account for the rapid spectral changes in this bandpass. Figure 35 also includes vertical lines that denote ViSP spectral lines of interest. Fortunately, the 434.1 nm ${\mathrm{H}}_{\gamma }$ line and 453.1 nm Ti I line avoid this coating spectral retardance feature. There is a band of relatively high diattenuation at slightly longer wavelengths of 445 to 490 nm with magnitudes of over $\pm 4\%$.

Fig. 35

Retardance and diattenuation from the combination of ViSP feed optics FM1, F1 and F2 before stripping of this 29 layer coating. (a) Shows the 380- to 550-nm wavelength region. Retardance varies by almost 200 deg around 445-nm wavelength as seen by the black curve on the left-hand $Y$ axis. The diattenuation is shown in blue using the right-hand blue $Y$ axis, varying by $\pm 4\%$ across adjacent wavelengths. (b) Shows longer wavelengths on reduced $Y$ scales.

The right-hand graphic of Fig. 35 shows the three mirrors at longer wavelengths. The retardance oscillates between 0 deg and 45 deg with an oscillation period much larger than the nanometer scale ViSP bandpass. There is a small diattenuation feature around the 854.2 nm Ca II spectral line. Thus, we could have expected some mild continuum polarization gradients in calibrations. We show, in Sec. 10.4, further confirmation of spectrally narrow reflectivity and polarization artifacts using our own NLSP metrology in an image-rotator (K-cell)-type configuration even at incidence angles of 11 deg. Though these coatings may have offered high reflectivity, when high spectral resolving power and polarization are considered, these coatings could have caused substantial calibration complexity.

6.2.

DKIST and FIDO Configured for DL-NIRSP On-Disk Observations

In this section we show example calculations of the DL-NIRSP system modulation matrix and group model terms, given the uncertainties in coatings. We use the $F/24$ configuration and describe the impact of incomplete knowledge of coating properties in computing expected Mueller matrices for the optics. In Table 11, we show which beam splitters and mirrors would be installed at which stations. We choose a setup with maximum transmission in the wavelength range from 465 to 1800 nm, allowing DL-NIRSP to cover the full wavelength range within its capabilities. The DL-NIRSP sees high transmission through the windows and the dichroic beam splitter C-BS-465. In the last two stations, CL3 and CL4, we put in a window with antireflection coatings on both sides.

Table 11

FIDO Config 1.

As a demonstration, we chose to install the FIDO window with an uncoated front surface C-W2 in the optical station CL2a. This window would send the $\sim 4\%$ Fresnel surface reflection to VBI-blue so that instrument could perform its function as a context imager at wavelengths from 380 to 465 nm for the ViSP spectrograph slit. The beam transmitted through C-W2 would contain significant flux for wavelengths shorter than 465 nm due to the dichroic reflection off C-BS-465. ViSP can be configured to observe wavelengths shorter than this, subject to the constraint on physical spacing between the camera arms. With the antireflection-coated window C-W1 in station CL3, the beam going to VTF and the VBI-red systems would have very little flux and render these systems undesirable for use in this configuration. In Fig. 36, we show a simplified cartoon of the FIDO setup and the wavelength ranges for the various instruments.

Fig. 36

The FIDO layout for configuration 1.

In the lower half of Table 11 we also show an example of some camera configurations possible with this setup. For this particular observing setup, the ViSP and the blue arm of VBI receiving 4% of the flux would both be able to simultaneously observe in a few relevant channels.

We choose two example wavelengths for the Ca II UV line at 393 nm and the Ba II line at 455 nm as a convenient second wavelength. The ViSP is flexibly configured so several alternate choices are possible. We list the three cameras in DL-NIRSP with three example wavelengths. We had chosen the shortest wavelength presently planned for the visible channel at 530 nm. We note that alternates include 587, 630, 789, and 854 nm, which can be easily observed in this FIDO configuration. Similarly, the two NIR cameras could also observe 1075, 1080, or 1430 nm.

In Fig. 37, we show the transmission functions of all mirror groups and also all coudé optics. We use the silver refractive indices that are closer to the measured mirror reflection values for this calculation. The WFS-BS1 optic is assumed to have a bare fused silica front surface reflection computed directly from the Fresnel equations. The back surface is coated with WBBAR1, as described in Sec. 4. We perform the same calculation for the FIDO windows C-W1 and C-W3 where both surfaces are coated with WBBAR1. The FIDO dichroic C-BS-465 utilizes the transmission from our best-fit TFCalc model on the front surface and the WBBAR1 coating on the back surface.

Fig. 37

The DL-NIRSP system transmission in this configuration for all mirror groups beyond M1 and M2: DKIST M3 to M10, BS1, FIDO optics, and instrument feed mirrors.

We show in Fig. 38 the retardance and diattenuation for propagation through the FIDO optics in configuration 1 as well as the WFS-BS1. The diattenuation is a relatively small 1% to 2% across the wavelength range transmitted to DL-NIRSP. The retardance is 10 deg at 530-nm wavelength caused almost entirely by the front surface dichroic coating of C-BS-465.

Fig. 38

The retardance in black and diattenuation in blue caused by the transmissive optics WFS-BS1, C-BS-465, C-W1, and C-W3 combined as installed for FIDO configuration 1.

The diattenuation in the DL-NIRSP modulation matrix is a roughly equal combination of the transmissive optics in FIDO and the WFS from Fig. 38, along with the three high incidence angle fold mirrors within the instrument relay optics. For the retardance, the three high incidence mirrors in the DL-NIRSP optics dominate with magnitudes up to 100 deg varying strongly across the visible and NIR wavelength region. The FIDO optics is $<10\mathrm{deg}$ at most wavelengths and are a very small portion of the system retardance. The DKIST telescope mirror retardance from Fig. 31 is also significant at roughly one-third this magnitude and are also time-dependent. Diattenuation of the DKIST mirror groups has a magnitude similar to the combined FIDO optics and DL-NIRSP feed optics before the modulator.

In summary, the telescope, FIDO optics, and DL-NIRSP feed mirrors give roughly 70% throughput from 525 to 1600 nm wavelength. The DL-NIRSP feed optics is the dominant source of retardance at magnitudes up to 100 deg. The diattenuation is roughly equal combinations of feed mirrors and FIDO optics in this configuration but with combined magnitudes less than several percent.

6.3.

DKIST and FIDO Configured for Multi-Instrument Spectropolarimetry

We consider in this section a configuration for the FIDO optics that operates all cameras in all three AO-assisted DKIST polarimetric instruments.

In Fig. 39, we show a schematic for which FIDO dichroic beam splitters and windows will be installed in which stations. The WFS-BS1 sends the 4% Fresnel reflection from the uncoated fused silica window to the AO system. In station CL2, we mount dichroic C-BS-555 where we reflect wavelengths shorter than 555 nm toward ViSP and VBI-blue. In station CL3, we mount dichroic C-BS-680 to reflect wavelengths between 555 and 680 nm toward VTF and VBI-red. The final station CL4 has the window C-W3 with broadband antireflection coefficients on all surfaces for high transmission. We show the beam splitters and stations in Table 12.

Fig. 39

The FIDO layout for a configuration where all polarimetric cameras are operated simultaneously.

Table 12

FIDO Config 2.

In station CL2a, we install the window C-W2 which sends the 4% Fresnel reflection from the uncoated front surface toward VBI-blue. This camera operates as a context imager for ViSP with limited flux. ViSP can be configured to operate three cameras over a diverse range of spectral lines. In Table 13, we arbitrarily choose the three ViSP camera wavelengths to cover the Ca II $H$ and $K$ lines at 393 nm, the $H_{\beta }$ line at 486 nm, and the photospheric Fe I line at 525nm. In station CL3a, we install the dichroic C-BS-643 which would transmit 643 to 680 nm toward the VBI-red camera. The reflected beam would allow VTF to observe in the 555 to 643 nm wavelength range. It has one filter for the $H_{\alpha }$ line at 656 nm in this wavelength range.

Table 13

Cameras: 2.

In Table 13, we list the VTF as observing this 630-nm line. We note that VTF technically operates three separate cameras. Two cameras are synchronized and are operated as a dual-beam spectropolarimeter as the etalons scan the spectral line in steps of a few picometers. These two cameras can be compared to the spectrograph instrument calibrations as these cameras scan in wavelength to sample the spectral line. The third camera is also synchronized with the two polarimetric cameras. This third camera is set to observe continuum wavelengths adjacent to the spectral line. These three cameras all operate at wavelengths very close to 630 nm and are treated as one wavelength range, similar to spectrograph calibrations for our purposes here.

A highly transparent FIDO window C-W3 is installed in station CL4. The DL-NIRSP would thus receive all wavelengths from 680 nm to the cutoff of the Infrasil transmission near 3000-nm wavelength. The antireflection coatings and dichroics all become fairly reflective for wavelengths longer than 1800 nm. We configure DL-NIRSP in Table 13 for observation at 854 nm using the visible wavelengths camera. We set the two NIR cameras to 1083 and 1565 nm.

Figure 40 shows the BS1 and FIDO optics throughput to the first instrument optic. We show throughput in the UV bandpass as this is important for calculating the optical degradation in polymer optics and the oils used in the crystal modulators. The ViSP feed is shown in blue, receiving roughly 90% of the flux from 390- to 540-nm wavelength. The VBI blue channel is shown in yellow, receiving only a few percent of the flux over a similar bandpass. The VTF feed is shown in magenta with roughly 90% of the flux in the 565- to 630-nm wavelength. The VBI red system is shown in red and receives 90% of the flux at 656 nm wavelength to observe the $H_{\alpha }$ line. The DL-NIRSP is shown in black receiving light at wavelengths longer than 690 nm. From Fig. 40, we can also see that no instrument sees more than a few percent of the flux at longer wavelengths outside the intended bandpass. The VTF, DL-NIRSP, and VBI-red channels do receive some UV flux at varying wavelengths depending on the imperfect UV reflectivity of the combined optics.

Fig. 40

The flux through combined optics from BS1 the last FIDO element as combined and installed for FIDO configuration 3. The optical throughput delivered to the first feed optic of each subsystem is shown in different colors.

Figure 41 shows the retardance and diattenuation in the beam propagated through BS1 and the FIDO optics as configured for this use case. The left graphic shows the appropriate wavelength range for the beam to the first lens of the VTF. In this configuration, the VTF would only be observing the 630-nm spectral line. The right-hand graphic shows the beam sent to the DL-NIRSP first-fold mirror DL-FM1. We listed three wavelengths of 854, 1083, and 1565 nm. However, this configuration could easily support other DL-NIRSP configurations for any wavelength longer than 690 nm.

Fig. 41

The retardance in black and diattenuation in blue for the beam through WFS-BS1 and FIDO optics C-BS-555, C-BS-643, C-BS-680, and windows C-W2 and C-W3 as combined for FIDO configuration 3. (a) Shows the combined polarization response of the beam to VTF in the band for observation at 630 nm. (b) Shows the full wavelength range available to DL-NIRSP in this configuration from 700 to 1600 nm.

The VTF mirrors have not yet been coated but the nominal specified coating is the same as the GREGOR telescope mirrors. We show our NLSP measurements and two-layer model fits to this GREGOR coating in Sec. 10.1. The VTF feed mirror polarization response will essentially come from the two combined fold mirrors at 45-deg incidence. The $\sim 20\mathrm{deg}$ retardance and zero diattenuation from FIDO in Fig. 41 will likely be only a small contribution to the modulation matrix properties. For DL-NIRSP, we draw the same conclusions as in Sec. 6.2.

6.4.

Summary: Mueller Matrices of DKIST Instruments: FIDO and Coudé Optics

In Sec. 6, we have presented models of the DKIST mirrors grouped together between rotation axes for the on-axis beam. We have shown how typical manufacturing tolerance errors can impact reflectivity, retardance, and diattenuation. Variation directly follows the layer thickness variation derived from two-layer model fits to our NLSP measurements. We have then shown examples of the additional polarization effects from mirror combinations in two DKIST instruments: ViSP and DL-NIRSP. In the case of DL-NIRSP, the nominal DKIST silver formula coated on several high incidence angle mirrors creates significant variability in the expected polarization properties in response to manufacturing variability even though all mirrors have been coated in the same run. In addition, this instrument did not choose the nominal DKIST silver coating for some of their mirrors, complicating the modeling of the entire system. However, the DL-NIRSP mirrors measured in Sec. 10.7 showed that their alternate coating is broadly similar to the DKIST silver in magnitude and spectral smoothness of polarization properties. For the ViSP instrument optics of Sec. 6.1, the original specification of a highly enhanced 29-layer coating on the feed mirrors would have significantly complicated the spectral behavior. We have estimated the reflectivity, diattenuation, and retardance using a combination of vendor reflectivity and diattenuation, vendor modeling, and NLSP retardance measurements. We were able to predict the Mueller matrix of these combined optics had they not been stripped and recoated with one of our nominal FIDO-enhanced protected silver coatings. We note that the Cryo-NIRSP uses an all-reflective feed by inserting a mirror (M9a) in front of the AO deformable mirror (DM) at a low incidence angle of 9 deg and with this same FIDO EAg coating. The two other Cryo-NIRSP feed mirrors are at very low incidence angles of 4 deg and 1 deg. Thus, the polarization properties of the Cryo-NIRSP feed optics are substantially simplified compared to DL-NIRSP. In addition, the VTF feed mirrors are essentially twofold mirrors around 45 deg. They are clocked with respect to the coudé floor, introducing both $QV$ and $UV$ retardance in the frame of their analyzer. But this is similar to DL-NIRSP, predictable with witness sample data, and easily calibrated. With this information, and a few more coating samples delivered in the near future, we have the Mueller matrix of every metal-coated optic in the telescope and on the coudé floor.

We then showed examples of how the dichroics of the FIDO could be configured for two use cases. The first configuration of Sec. 6.2 shows maximum transmission to DL-NIRSP with relatively simple and spectrally smooth Mueller matrices. In Sec. 6.3, we add the complexity of operating polarimeters in all AO-assisted instruments simultaneously. FIDO is configured for ViSP from 380 to 555 nm, the VTF at 630 nm and DL-NIRSP from 680 to 1800 nm. The predicted throughput from the combined reflections and transmissions are shown along with the retardance and diattenuation expected for each system. This process can be used for all DKIST instruments and updated in the future to reflect the as-built coating performance once the FIDO dichroics are procured and we have measured those as-coated witness samples in the near future.

10.1.

DKIST VTF- and GREGOR-Enhanced Protected Silver Commercial Mirrors

The Kiepenheuer Institute for Solar Physics (KIS) provided us GREGOR telescope witness samples for test in summer 2017. These witness samples were also the same enhanced protected silver coating formula nominally proposed for the VTF instrument that KIS was constructing for installation on DKIST. The KIS staff was informed that the coating could be modeled as a high-index dielectric layer over a low-index dielectric layer on top of a silver metal layer similar to the models presented here and in our prior work.^11,67

Four repeated data sets are collected at 45-deg incidence angle. Figure 51 shows the retardance and diattenuation properties of the Mueller matrix derived from measurements. The $IQ/II$ diattenuation Mueller matrix element is shown in the right-hand graphic. The inverse tangent of the $VV$ and $UV$ elements is used as a proxy for the linear retardance component. Agreement between visible and NIR spectrographs is quite good as seen by the overlap in the 950- to 1050-nm wavelength range. The extreme edges of each data set are removed for this analysis by stitching together the data sets at 1020-nm wavelength. We ignore longer wavelengths measured by the visible spectrograph and we ignore shorter wavelength data measured by the NIR spectrograph.

Fig. 51

NLSP measurements and corresponding Berreman model fits. (a) Shows retardance fits to an enhanced protected silver mirror used at the GREGOR solar telescope along with some model fits using common materials. (b) Shows diattenuation along with predictions for the retardance-fit Berreman models. Three different material combinations are shown that are more successful at fitting retardance than several other material combinations.

We have attempted to fit Berreman models of many material choices in two-layer configurations to the retardance measurements. The three best fits are selected for Fig. 51. All three models represent relatively high refractive index materials on top of a relatively moderate refractive index material. The retardance behavior is similar to the DKIST-enhanced protected silver in that there are two wavelengths around blue and NIR where the retardance is zero. If only the visible-wavelength data set is used, the best-fit model layer thickness changes slightly. As an example, the best-fit layer thicknesses are (62 and 60 nm) when using only VIS spectrograph data as opposed to (65 and 56nm) when using the full VIS+NIR range for a ${\mathrm{HfO}}_2$ over ${\mathrm{Al}}_2{\mathrm{O}}_3$ formula model.

10.2.

Commercial Mirror Samples: Big Bear Solar Observatory and Newport PAg

We also aim to help create a telescope model comparison for the BBSO. Many observatories use commercial off-the-shelf (COTS) optics. It is straightforward to create polarization performance models. We were given witness samples from the BBSO aluminum-coated mirrors as well as a COTS-protected metal-coated mirror from Newport used at BBSO. This Newport coating has been used on several mirrors as part of the BBSO optical train. The BBSO uses several other commercial mirrors in their optical train and knowledge of each coating is required to create a detailed polarization model.

We measure the Mueller matrix of the Newport mirror at 45-deg incidence angle with NLSP. The linear retardance in the $UV$ plane as well as the $IQ$ diattenuation is derived using the Mueller matrix. The resulting stitched data sets and Berreman models are shown in Fig. 52. There are two models that fit the Newport coating equally well. The first fit is 137-nm thickness of the ${\mathrm{SiO}}_2$ model coated over 14-nm thickness of the ${\mathrm{Al}}_2{\mathrm{O}}_3$ model using the Boidin et al.⁶⁸ refractive index formula using index 1.67 at 850-nm wavelength. A similarly good model fit is seen for 77-nm thickness of the ${\mathrm{SiO}}_2$ model coated over 71-nm thickness of the ${\mathrm{Al}}_2{\mathrm{O}}_3$ model with refractive index of 1.55 at 850-nm wavelength. Both ${\mathrm{SiO}}_2$ models have refractive index of 1.55 at 850-nm wavelength. The 0.5-deg retardance step barely visible at 1020 nm is the stitching wavelength where VIS spectrograph data are concatenated with NIR spectrograph data.

Fig. 52

Commercial Newport-enhanced protected silver. (a) Shows retardance fits and some model fits using common materials. (b) Shows measured diattenuation along with the Berreman models for diattenuation (not fit).

For each vendor the BBSO used, we requested information on the coating prescription. As expected, only limited and incomplete information could be obtained, if any was even available. This is clearly insufficient for creating polarization models of reasonable fidelity at observatories with powered optics at varying incidence angles.

10.3.

Commercial Mirror Samples: DKIST and A Thorlabs-Protected Silver K-Cell

We tested three Thorlabs-protected silver mirrors we use in our DKIST laboratory. These Thorlabs mirrors have a different wavelength dependence of retardance and diattenuation than several of the previous samples reported in the main paper. This difference is likely caused by substantially thicker coating layers as suggested below by our fitting process outlined above. In Fig. 53, we show the retardance in black and diattenuation in blue using the right-hand $Y$ axis measured with NLSP in the reflective configuration for each of the three samples. The retardance curves cross 180 deg at four wavelengths over the DKIST instrument wavelength range. The diattenuation for this sample has a somewhat strong peak near 600 nm. One mirror was measured in July 2017, while the other two mirrors was measured in February 2018.

Fig. 53

NLSP measurements of three Thorlabs-protected silver mirrors. Black shows retardance using the left-hand $Y$ axis. Blue shows diattenuation on the right-hand $Y$ axis. The first sample is measured 7 months before the second two, and the diattenuation is shifted by 0.25% for the curves to match.

The three mirror samples were all procured at the same time. While it is possible that they were all coated in the same shot, there is no guarantee of their coating heritage. We measure significant differences in both the diattenuation and the retardance. The diattenuation measured in the first mirror is spatially offset by 0.25% in Fig. 53 to match values at NIR wavelengths. As we have shown above for the DKSIT silver mirrors in Fig. 46, there are significant impacts to diattenuation for even small variations in the layer thicknesses of the dielectrics in the model. Our 0.25% offset is a likely a combination of real systematic difference between measurements as outlined for NLSP in prior sections and also real variations between these commercial mirrors.

The retardance curves for all three samples match to within 3 deg, but there are easily detectable difference between nominally identical samples. We show in Fig. 54 the difference between mirror retardance for Thorlabs mirrors number 3 and 2 in black. Both mirrors were measured on the same day with NLSP at 45-deg incidence angle. We also show the difference between sample 3 and sample 1 in blue as well as sample 2 and sample 1 in red.

Fig. 54

The difference in retardance between Thorlabs-protected silver mirrors. Sample numbers 2 and 3 measured with NLSP on the same day. Sample 1 had been measured in July 2017. Differences are more than an order of magnitude larger than systematic errors.

We note that repeatability tests shown above in Figs. 43 and 44 showed that after remounting and aligning a mirror, we can reproduce a measurement to better than 0.05 deg. Estimates of retardance computed by using different combinations of the $UV$ Mueller matrix elements agree to better than 0.3 deg. Both of these systematic errors are an order of magnitude smaller than the difference measured here between Thorlabs mirrors.

With these data sets, we can attempt to fit various coating models to the retardance measurements. Figure 55 shows the retardance and diattenuation measurements as the dashed black lines. The prediction from the best fit to retardance from our two-layer Berreman coating models are shown as solid lines of varying color. Retardance is shown on the left with the diattenuation on the right. The dielectric layer thicknesses found in the fit are shown in the legend of the retardance graph.

Fig. 55

Commercial Thorlabs-protected silver mirror and corresponding fits. (a) Shows retardance fits and some model fits using common materials. (b) Shows measured diattenuation along with the Berreman models for diattenuation (not fit). For this mirror the dielectric layers are much thicker and the fits to retardance data are significantly worse.

We have attempted to fit a large range of two-layer coating models but found that only a few reasonably reproduced the retardance curves. These better-fitting models generally use relatively high refractive index material curves such as ZnS, ${\mathrm{TiO}}_2$ from TFCalc or vendor references, and ${\mathrm{Al}}_2{\mathrm{O}}_3$ from either Zemax or Boidin et al.⁶⁸ references. The red curve shows a coating model with a single layer of ${\mathrm{TiO}}_2$ at 185 nm physical thickness using the Boidin et al.⁶⁸ refractive index curves from refractiveindex.info. The blue curve shows another model with layer of amorphous ${\mathrm{SiO}}_2$ at 199 nm physical thickness coated on top of 104 nm of ${\mathrm{Al}}_2{\mathrm{O}}_3$. In all models fit, the layers end up with thicknesses of significantly $>150\mathrm{nm}$. The two-layer coating models described in the main text had total thickness around or $<100\mathrm{nm}$. All models diverge at longer wavelength where it is possible that both the metal and the refractive index curves for the dielectrics have higher error.

The right-hand graphic of Fig. 55 shows the diattenuation. All models consistently underestimate the diattenuation for blue and NIR wavelengths. The spectral feature around 600 nm shows a diversity of results between the models. We do not expect diattenuation to be modeled well as this parameter is not included in the fit. We showed above in Sec. 3.1 that the metal complex refractive index can strongly influence the fit and can be highly variable between coating process and vendors.

We combined the three Thorlabs mirrors to create a sample where the beam enters and exits in translation and tilt exactly following the unperturbed incoming beam. This setup is often called an image rotator, derotator, K-mirror, or K-cell. Common designs use relatively high incidence angles near 60 deg for the outer two mirrors and a smaller angle for the inner mirror. In our setup, we are able to place these round 1-in. diameter mirrors into kinematic mounts with incidence angles near 50 deg to 60 deg for the outer mirrors and 10 deg to 17 deg for the inner mirror. Table 16 shows the incidence angles estimated for the two outer mirrors (${\theta }_1$ and ${\theta }_3$) as well as the lower incidence angle inner mirror.

Table 16

K-cells.

Using a HeNe laser with a $\sim 3\text{-}\mathrm{mm}$ beam, we aligned the optical footprint location on each mirror as well as the individual mirror tilts. We ensured that the exit beam roughly matched the unperturbed beam in height and spot location over a $\sim 1\text{-}\mathrm{m}$ path with and without the K-cell translated into the beam. As a further alignment step, our NLSP software provides tools to show the symmetry of detected flux upon 180-deg rotation of the NLSP retarders as the various optics are rotated in 10-deg steps through 360 deg. This diagnostic shows photometric errors over 0.5% if the beam is not well centered on the spectrograph fibers. We perform additional tilt adjustment of each mirror after aligning for translation and height in the NLSP beam to maximize the flux through the system (minimize vignetting on the fiber) as well as to ensure the $\sim 4\mathrm{mm}$ NLSP beam is centered on the optics.

In Fig. 56, we show the NLSP-measured retardance and diattenuation as solid colored lines. The narrower $K$-cell is shown as blue. The moderate $K$-cell configuration is shown in green. A wider $K$-cell is shown in red. Dashed lines show a two-layer Berreman coating model for the three mirrors operating in series at the appropriate incidence angles. We only show one of our better fit two-layer models as 200 nm of ${\mathrm{SiO}}_2$ over 100 nm of ${\mathrm{Al}}_2{\mathrm{O}}_3$ coated over silver modeled using the default TFCalc values. There is a sharp change in the predictions for wavelengths short of 400 nm. This is caused by a strong change in the silver refractive index complex component in the default TFCalc values interpolated here. The value of $k$ changes from 1.93 at 400 nm to 5.85 at 375-nm wavelength. We use linear interpolation, which accounts for the sharp change in behavior at 400 nm.

Fig. 56

Transmission measurements from a three-mirror K-cell using our three commercial Thorlabs-protected silver mirrors. (a) Shows retardance measurements and a Berreman prediction using our best-fit two-layer coating model. (b) Shows measured diattenuation along with the Berreman models for diattenuation (not fit). Note that the sharp change in behavior for wavelengths short of 400 nm is caused by a strong change in the silver refractive index complex component in the default TFCalc values used here.

This issue with TFCalc default metal refractive index and all other metal refractive index values is drastically seen by the failure of the Berreman models to match the measured reflectivity through the $K$-cell. We show the three $K$-cell reflectivity measurement of the wider configuration as solid red in Fig. 57. The Berreman model using the default TFCalc values is shown in Fig. 57 as a red dashed line. There is reasonably good agreement in spectral shape and throughput for wavelengths longer than 420 nm. At short wavelengths, however, the high complex index of $k=5.85$ in the TFCalc silver creates reflectivity values over 50% higher than observed. We additionally show two different sets of vendor values as black and blue. These values, respectively, overestimate and underestimate the throughput of this three-mirror system at longer wavelengths. Both silver refractive index values overestimate reflectivity at the shortest wavelengths.

Fig. 57

The combined three-mirror reflectivity of the wider K-cell with Thorlabs silver mirrors. Dashed lines show two-layer coating models with 200 nm of ${\mathrm{SiO}}_2$ over 100 nm of ${\mathrm{Al}}_2{\mathrm{O}}_3$ protecting silver. The complex refractive index of the silver varies between models, giving rise to strong changes in predicted reflectivity.

These models are not expected to match in detail as this COTS mirror has an unknown coating in an unknown process. Additional work is required to adjust the refractive index of the dielectric and the metal to match this particular optic. The two outer mirrors are at relatively high incidence with the inner mirror at small incidence angles. With all folds roughly in the same plane, we expect the retardance to be somewhat larger than double the magnitude about 180 deg than that of a single mirror at 45 deg. This is easily seen by a comparison between the single mirror of Fig. 55 and the three mirrors in Fig. 56.

One major benefit of testing $K$-mirror-type setups is that a much larger range of incidence angles can be tested. The polarization predictions need to be valid from near normal incidence to 45 deg to account for the full range of DKIST reflections. By building this type of setup, we can further constrain our simple coating models to ensure that accurate predictions are derived from these coatings as we have applied to telescopes such as DKIST and AEOS.^11,67

10.4.

Image Rotator K-Cell: A NLSP Sample with Both ViSP and Thor Silver Mirrors

The $K$-cell-type sample can be useful in deriving reflectivity of individual mirrors using the well-calibrated transmission arm of NLSP. Given the various measurement techniques applied to the ViSP many-layered silver mirrors, we confirm the reduced transmission bands as well as the non-negligible polarization impacts at low incidence angles. We had taken the Thorlabs K-cell in the narrow configuration and had replaced the low-incidence middle mirror with the ViSP witness samples. This allows extraction of the polarization and reflectance properties at an incidence angle around 11 deg, similar to the ViSP F2 mirror at 12.3-deg incidence.

We show in the left-hand graphic of Fig. 58 the retardance and diattenuation of the combined three mirrors in the narrow configuration. The smooth, large magnitude structures of Fig. 56 are reproduced by the two high-incidence Thorlabs silver mirrors at 50.4-deg incidence. However, we now see significant narrow spectral features in both retardance and diattenuation around 450- to 500-nm wavelength. Additional small amplitude ripples are also seen in the retardance curve caused by the many-layered enhanced silver coating design. We attribute these narrow spikes and broader ripples to the ViSP many-layered mirrors.

Fig. 58

(a) Shows retardance and diattenuation for a K-cell made of two Thorlabs-protected silver mirrors at high incidence angles and a ViSP witness sample at low incidence angles. (b) Shows the reflectivity of the ViSP sample after compensating for the two Thorlabs mirror reflections. The ViSP sample adds narrow spectral features seen around 450-nm wavelength and substantial drops in reflectivity in two separate bandpasses.

In the right-hand graphic of Fig. 58, we extract the reflectivity of the ViSP samples by dividing the reflectivity of the two Thorlabs mirrors derived in the last section. We use the cube-root of the Thorlabs K-cell of Fig. 57 as an approximation of an individual Thorlabs mirror. The various incidence angles tested did not show an appreciable difference in reflectivity. Dividing the measured K-cell reflectivity by this Thorlabs reflectivity curve twice results in a reflectivity curve for the ViSP witness sample that very well correlates with the vendor data presented in the main paper. The black and blue reflectivity curves show two separate witness samples that each had different storage histories over roughly 9 months between coating and measurement. Though their storage conditions varied, the reflectivity and polarization properties between the two samples are remarkably consistent. In testing at the vendor site immediately after coating, the low reflectivity band between 450 and 500 nm is not present.

10.5.

Protected Silver Mirrors: DKIST Feed Optics with Zygo Silver

DKIST has used an enhanced protected silver from Zygo for the feed mirror M9a as well as one instrument feed mirror (DL-FM1) inside the DL-NIRSP instrument relay optics. Figure 59 shows the retardance and diattenuation measured in a $K$-cell with witness samples from Zygo. Samples are roughly 8 years old and are stored in laboratory conditions. These coatings use an ion-assisted deposition process and are described as having a few layers over silver. The samples are stored in a laboratory environment that is roughly 50% humid in the summer and a bit drier in the winter. The $K$-cell sample is measured at the same 50 deg to 11 deg to 50 deg configuration. The two different curves of Fig. 59 show the difference when we replaced the first fold at 50-deg incidence with a separate sample from a separate coating shot. There is only a small difference in retardance with a fraction of a percent change in diattenuation.

Fig. 59

An enhanced protected silver from Zygo in a $K$-cell configuration. Black shows retardance, while blue shows diattenuation using the right-hand $Y$ axis. A model is a thin dash-dotted line.

In the fall of 2018, the M9a and DL-FM1 optics had been coated. Figure 60 shows the reflectivity and diattenuation. The test run is shown in red. The M9a coating run is shown in green. The DL-FM1 coating run is shown in blue. These coatings are not blue-enhanced as they feed infrared optimized instruments working at wavelengths longer than 450 nm. The diattenuation is less than half percent for wavelengths longer than 500 nm. There is a fairly prominent absorption feature around $3\text{-}\mu \mathrm{m}$ wavelength and a corresponding change in diattenuation. Both curves show discontinuities around 1000 nm wavelength from the change in spectrophotometric equipment, also illustrating the level of systematic error present in the data. The reflectivity curves show that there are differences of roughly 1% between coating runs.

Fig. 60

Diattenuation of $S$- and $P$-polarization states measured by Zygo at 8 deg and 45 deg incidence in (a). Reflectivity of $S$- and $P$-polarization states measured by Zygo at 8 deg and 45 deg incidence in (b). Colors show different coating runs.

10.6.

Protected Silver Mirrors: DKIST VBI and BBSO with Edmund Optics

We show additional examples of polarization properties derived from NLSP Mueller matrix measurements for commercial protected silver mirrors relevant to DKIST and BBSO. The VBI instrument in DKIST uses an Edmund Optics off-the-shelf protected silver mirror. We took the actual 6-inch diameter mirror from the VBI optical path and measured the Mueller matrix at 45-deg incidence in the center of the optic. The BBSO optical path also contains a few mirrors that are a commercial Edmund Optics protected silver coating with polarization properties possibly similar to this mirror. Figure 61 shows the retardance and diattenuation derived from the measurements. Diattenuation is rapidly increasing in magnitude for wavelengths shorter than 420 nm. For visible and NIR wavelengths, the diattenuation stays below 1% magnitude with a few sign changes. The retardance is broadly similar to the two-layer simple models we have presented in this paper with retardance crossing the theoretical 180-deg magnitude at wavelengths around 525 and 1100 nm.

Fig. 61

An Edmund Optics-protected silver used in VBI-blue (and possibly similar to BBSO mirrors). Black shows retardance and blue shows diattenuation on the right-hand $Y$ axis.

10.7.

Protected Silver Mirrors: DKIST DL-NIRSP and EMF

The DL-NIRSP instrument contracted a vendor who ultimately procured some coatings that were not the specified DKIST silver. Once we discovered this alternate coating was present on several mirrors in the DL-NIRSP optics, we directly measured one of the flat DL-NIRSP mirrors that had used this alternate coating. Figure 62 shows measurement of a coating on a DL-NIRSP spectrograph mirror. Diattenuation values for the DL-NIRSP mirror are slightly higher, but still below 2% magnitude. This coating shows much stronger oscillations of diattenuation at short wavelengths. However, the DL-NIRSP only observes wavelengths $>525\mathrm{nm}$ and this coating has $<1\%$ diattenuation for wavelengths $>700\mathrm{nm}$. The retardance also crosses the theoretical 180-deg magnitude at effectively three wavelengths: 380, 700, and 1080 nm. The reflectivity, retardance, and diattenuation are acceptable for DKIST purposes, especially since only one DL-NIRSP feed mirror uses this coating and only at 3.8-deg incidence angle.

Fig. 62

A coating by EMF used on two of the DL-NIRSP spectrograph mirrors as well as the spherical steering mirror feed optic. Black shows retardance and blue shows diattenuation on the right-hand $Y$ axis.

10.8.

Protected Silver Mirrors: DKIST M8 and EMF Blue-Optimized AG99

DKIST also utilized EMF for one of the coudé feed mirrors: M8. We used a blue-modified flavor of the AG99 formula to ensure that we did not compromise the 393-nm wavelength throughput. Metrology was recorded at 8-deg incidence while the M8 off-axis parabola was mounted at roughly 5-deg incidence for the chief ray.

Figure 63 shows the spectrophotometry from EMF at 8-deg incidence for two coating shots. The test coating shot number 2094 is shown as dashed lines while the coating shot deposited on the actual DKIST M8 is shown in solid lines. As is typical, the test data show spectral reflectivity variation of up to 3% at some wavelengths. The test shot run 2094 had three samples distributed within the clear aperture shown as black, blue, and green dashed lines. There is a fourth sample located outside the clear aperture of the M8 optic that shows somewhat similar spectral behavior but with significant deviation at short wavelengths. The two solid lines of the actual coating shot show tests on a glass slide and also a standard witness sample coupon. As the M8 optic is coated, these samples are by definition outside the clear aperture of the optic and are only representative of the actual M8 coating within the limits of the coating spatial uniformity as deposited in this chamber. The test run data show a similar drop in blue reflectivity though with some differences in the spectral oscillations.

Fig. 63

The blue enhanced AG99 coating by EMF used on DKIST M8. Test run data are shown as dashed lines. Solid lines show witness samples adjacent to the M8 optic during the coating run.

10.9.

Protected Silver Mirrors: DKIST C-M1 and M9, ViSP Mirrors F1, F2, and FM1

We had several mirrors coated with an Infinite Optics-enhanced protected silver with formulas EAg1-420 and -450. We list in Table 17 the coating run number associated with each of the optics. The FIDO mirror C-M1 was in the same shot as the first ViSP feed mirror (FM1). The second two ViSP mirrors were in a second shot. A separate coating shot was done for DKIST M9.

Table 17

IOI EAg runs.

We obtained three witness samples from each coating shot containing DKIST optics. With these, we were able to create a $K$-cell image rotator-type setup to use as a sample in the transmissive arm of NLSP. Figure 64 shows the retardance on the left and diattenuation on the right for this sample.

Fig. 64

(a) Shows retardance and (b) shows diattenuation. The chamber 6 test shot 7759 is shown in green, shot 7766 in blue, shot 7767 in magenta, and the chamber 9 shot for DKIST M9 in red. There are actually 12 red curves and 9 curves of all other colors as we repeated each optical alignment three or four times, and repeated each test at each alignment three times. Some enhanced statistical noise is seen in the diattenuation graph in (b).

We used the narrow $K$-cell setup with approximate incidence angles of 49 deg on the outer two samples and 8.6 deg on the inner sample. We only had two samples for the chamber six test shot 7759 so we substituted a DKIST sample from M10 spatial position U. At the low incidence angle of 8.6 deg, we expected retardance impact of less than a few degrees when modeling this system. The four curves represented the same materials but with different thicknesses of each material. All of these coatings were slightly thinner than the DKIST-specified coating with zero retardance values at correspondingly shorter wavelengths.

Figure 65 shows the reflectivity measured at IOI from 300 to 2200 nm. We follow the same color scheme as Fig. 64. One sample has been measured to 5000-nm wavelength with reflectivity slowly increasing above 99% as predicted in the coating model. All coatings have $>96\%$ reflectivity at the 393-nm Ca K line.

Fig. 65

The reflectivity of each coating by Infinite Optics measured at 8-deg incidence. Vertical dashed lines show spectral lines of interest.

10.10.

Protected Silver Mirrors: ViSP Fold Mirrors FM2, FM3, FM4, and RMI EAg

The ViSP feed optics include a fold mirror (FM2) coated with a Rocky Mountain Instruments-enhanced protected silver. The optic is mounted in the $F/32$ diverging beam after the slit reflecting at an incidence angle of 47.7 deg toward the modulator. This optic sees footprints of only a few millimeters and is included in the system modulation matrix as it is ahead of the modulator. In Fig. 66, we show the reflectivity left and diattenuation right for 45-deg incidence. The reflectivity is mostly over 96% with an exception around 420-nm wavelength. The vertical dashed lines show spectral channels of interest. The diattenuation oscillates spectrally with six zero crossings in the nominal ViSP bandpass. The diattenuation is never $>1\%$ though the sign and magnitude changes along with the spectral oscillations.

Fig. 66

Spectrophotometry for S (black) and P (blue) polarization states at 45-deg incidence shown in (a). Diattenuation derived from S&P reflectivity is shown in (b). Vertical dashed lines show typical solar-observing wavelengths.

10.11.

Bare Aluminum Mirrors: DKIST Primary and Test Mirrors Coated at AFRL

The DKIST primary mirror (M1) is coated at the AFRL coating chamber located adjacent to DKIST on the summit of Haleakala, Maui. We also have a full-sized commissioning mirror as a replica of M1. Both are coated by AFRL with bare aluminum. Reflectivity of multiple samples in each coating run measured by a Varian Cary 5000 at Gemini in Hilo is shown in Fig. 67.

Fig. 67

The reflectivity measured at 7-deg incidence angle for three samples coated with the actual M1 science grade mirror and another six samples measured during coating of the full-sized M1 commissioning mirror.

We tested two witness samples from the first coating on the science mirror. Figure 68 shows the retardance on the left and the diattenuation on the right. We ran our two-layer fitting routines using the Berreman calculus in our Python scripts. However, for this run we let the thickness of the aluminum metal layer be the second thickness variable on a grid from 40- to 200-nm thickness in steps of 10 nm. The top layer was the aluminum oxide using the Boidin refractive index curves running from 0 to 10 nm in steps of 0.5 nm. We found a coating model of 2.5 nm ${\mathrm{Al}}_2{\mathrm{O}}_3$ over 50 nm aluminum as the best fit when using the TFCalc aluminum metal refractive indices. We also noted that we have corrected the diattenuation in Fig. 68 for a linear offset to highlight the often crude spectral sampling of typical coating models. We found the same 2.5 nm of oxide but a thinner 40-nm metal layer when using an internal NSO aluminum metal interpolation.

Fig. 68

(a) Shows retardance and (b) shows diattenuation of the DKIST M1 science mirror witness sample from spatial position 3 measured in reflection at 45-deg incidence angle. Blue shows the NLSP data. Green shows spatial position 5. The solid black line shows the Berreman code fit when using the TFCalc refractive indices for aluminum. Dashed black shows the fit with an internal NSO aluminum refractive index interpolation.

Coatings on large mirrors are expected to be spatially variable. We note that we have interferometric testing of larger samples distributed throughout the chamber during testing showing that the coating is physically between 90- and 150-nm thick across the 4-m aperture so neither of these metal thickness fits are close to a direct thickness measurement. A recent study on polarization aberrations and the impact on the Habex system by Breckenridge et al.⁸¹ shows spatial variation in the retardance in reflection for a 3.75-m diameter mirror coated at the University of Arizona in Fig. 19.⁸¹ This form birefringence measurement at magnitudes of 0.002 radians retardance required a special setup developed, built, and measured by. B Daugherty.⁸¹ As we have detailed above, getting correct values for the optical constants of the coated aluminum is critical for matching the data with high spectral accuracy. The Breckenridge et al.⁸¹ work shows that spatial variation is present and measurable across large area mirrors. Given the factor of $\sim 2$ thickness variation in the DKIST aluminum coating, we certainly anticipate spatial variations in the mirror at some undetermined magnitude due to the varying properties of the aluminum across the mirror.