2.1.

Comparison with Zemax Coating Calculations

Many readers will be familiar with the Abeles and Heavens calculus from thin film coating calculations. These are standard calculations included in common software such as TFCalc, Zemax, or other industry standard modeling packages. We used Zemax version 15.5 to verify that our Berreman calculation gives expected results in the limit of isotropic materials. In Zemax, we set up a coating that used similar refractive index values for isotropic amorphous ${\mathrm{MgF}}_2$ coatings, a fused silica window and identical material thicknesses to compare our calculations. For those readers familiar with Zemax, the software has a coating analysis tool that uses the isotropic Abeles and Heavens matrices to compute the reflectivity, diattenuation, and retardance for a specified layering of isotropic coating materials. The Abeles and Heavens matrix formula can be considered the isotropic limit of the Berreman formalism where no birefringent materials are used in the calculation.

In Fig. 3, we show how our software reproduces the familiar results from Zemax. We show the diattenuation and retardance for the thick ${\mathrm{MgF}}_2$ coating. Two waves thickness at 2000 nm central wavelength gives 2924 nm physical thickness of isotropic ${\mathrm{MgF}}_2$ coated on top of fused silica at an incidence angle of 45 deg. The Zemax result is shown in blue while our calculations are shown in black. The curves overlap completely with differences of less than 0.01 retardance and ${10}^{-5}$ diattenuation. For completeness, we also tested several other limiting cases. Bare windows do indeed produce diattenuation but not retardance in both software packages as expected. We tested several other material interfaces such as a coatings among various glass substrates. We find good agreement between Zemax and Berreman predictions in all cases. We also find similar levels of agreement between the thin film calculator program TFCalc and our calculations here. Likely these computational disagreements arise from how the Zemax specified coating refractive indices are interpolated to the wavelength of interest. Both Zemax and our Berreman code adopt the same equation for refractive index of the fused silica substrate.

Fig. 3

The comparison of Zemax version 15.5 coating analysis tools and our Berreman calculations for a single thick layer of isotropic ${\mathrm{MgF}}_2$ coated on a fused silica window illuminated at a 45-deg incidence angle. The blue curves show the Zemax result for coating thickness of 2.0 waves at 2000 nm primary wavelength. The black curve shows our derived values from the Berreman calculus code using similar values for the refractive index of the coating and window materials. The curves overlap completely showing good agreement between our code and the industry standard Zemax calculations. (a) The diattenuation oscillating about an average value of $\sim 93\%$. (b) Retardance oscillating about 0 deg.

2.2.

Laboratory Verification of Intensity Fringes in an Isotropic Substrate

We verified these fringe amplitude and period predictions in the laboratory using a high-resolution spectrometer and a window. Meadowlark optics is fabricating the DKIST retarders and has several samples of the various window and crystal materials used in DKIST optics. To experimentally show the fringe periods and amplitudes, we derive here the fringe period and amplitude based on some simple physical assumptions outlined below.

The fringes are coherent interference between the incoming wave and the backreflected wave. The backreflected wave traverses the material twice. As such, the reflected beam sees an optical thickness of twice the physical thickness ($d$) times the refractive index ($n$). We expect constructive interference when the window is integer multiples of half-wave thickness. We expect destructive interference when the window is integer multiples of quarter-wave thickness (just like antireflection coatings). To compute the fringe period, we solve for the wavelength interval required to change the optical path traversed by the backreflected exactly one wave in

For optics that are much thicker than a wavelength, we can approximate ${\lambda }_1$ and ${\lambda }_2$ as about the same. For typical windows and crystals at visible wavelengths, the optic is at least millimeters thick and the fringe period is less than a part per thousand of the wavelength. With this approximation, we can simplify Eq. (3) to get a simple formula for the fringe period in wavelengths ($\delta \lambda$) expressed as the wavelength squared (${\lambda }^2$) divided by the optical path for the backreflected beam ($2dn$).

With this formula, it is straightforward to compute the expected fringe period as functions of the optical properties. As an example, a 1-mm-thick window with a refractive index of 1.5 observed at 500 nm wavelength would see a fringe period of 0.083 nm. Since the difference between wavelengths ${\lambda }_1$ and ${\lambda }_2$ is one part in 6000, the assumption of ${\lambda }_1\sim {\lambda }_2$ is quite reasonable.

In the Meadowlark facility, a SPEX 1401 double grating 0.85-m Czerny–Turner spectrometer is available with several sensor options. The light source is an Energetiq broadband fiber coupled plasma source using a $200\text{-}\mu \mathrm{m}$-diameter core fiber. The fiber output is collimated to a $\sim 10\text{-}\mathrm{mm}$ beam by a Thor labs 90-fold angle silver-coated off-axis parabola mirror with an effective focal length of 15 mm. The fiber light source and the OAP collimating mirror will produce some polarization expected to be at amplitudes less than a few percent at visible wavelengths. For this mounted OAP, the beam diameter is set by the exit of the housing after the mirror at an 11 mm diameter. The mirror is oversized and mounted before this aperture, giving rise to a small field dependence of less than $\pm 0.4\mathrm{deg}$. Some spatial dependence on sampling the fiber core is introduced by having the system beam stop be a rectangular mask on the collimating mirror after the slit.

At visible wavelengths, the measured resolution is $\frac{\lambda }{\delta \lambda }$ in the 25,000 to 45,000 range depending primarily on how fully the grating is illuminated. The instrument profile of 0.016-nm full-width half-maximum was measured with a neon discharge lamp at 653 nm and the profile has Gaussian shape giving a resolving power of 40,800. Other lines at 585, 609, 633, and 725 nm with slightly different beam diameters gave resolving power in the range of 25,000 to 49,000 depending on grating illumination.

The system is set up to nominally have a 10-mm-diameter collimated beam that is focused on the spectrograph entrance slit via a 50-mm focal-length singlet. The fiber core is magnified by the 15 to 50 mm ratio and would theoretically fill a 0.67-mm tall slit. The slit is 35 μm in width and is over 1 mm high to pass the full fiber core image. The $f/5$ beam entering the spectrograph is stopped to $f/7.8$ beam by a rectangular aperture on the collimating mirror inside the spectrograph. The system uses photomultiplier tubes to cover a range of wavelengths and delivers a typical measured signal-to-noise ratio around 1000. This noise level is dominated by systematic errors for integration times longer than 0.1 s. The baseline count rate was measured to vary by roughly 10% in 200 min with a mostly linear trend.

Meadowlark tested a window over the 625- to 635-nm wavelength range with the SPEX 1401 system. The window was 1.1335 mm thickness of Heraeus Infrasil 302. Thickness was measured by a Haidenhain MT 60M metrology system with $\sim 0.5\text{-}\mu \mathrm{m}$ thickness accuracy. The measured transmitted wavefront error (TWE) at 632.8 nm wavelength is 0.021 waves peak-to-valley (PV) over an aperture of 12 mm. The beam deviation through the window is measured to be 0.26 arc sec over the same footprint.

A spectral sampling of 0.079 nm was used to sample the beam at roughly two points per FWHM at the highest measured resolving power of $R=\mathrm{45,000}$. We use 1 s per point integration for high signal-to-noise ratio at reasonable speed. Over the 10-nm bandpass at 0.079-nm sampling, we collected 1260 points. Scans were made with and without the sample to derive a transmission spectrum. The beam was collimated and had a 10-mm-diameter footprint on the window. The transmission spectrum was dominated by a simple sinusoidal intensity fringe with a period of $\sim 0.12\mathrm{nm}$ with transmission values oscillating between 87% and 99%. This measurement agrees very well with Eq. (4). The Infrasil data sheet from the manufacturer (Heraeus) gives a refractive index of 1.457 at 632.8 nm. We compute the period as $\frac{{\lambda }^2}{2dn}=0.121\mathrm{nm}$.

The 10-nm bandpass records roughly 84 fringe periods allowing us to do a simple Fourier analysis. Figure 4 shows the power spectrum of the measurements. The blue curve shows the power in the raw detected intensity data without calibration by a baseline. The black curve shows the power in the transmission spectrum computed after calibration of the data using the no-sample scan. The vertical dashed red line shows the predicted fringe frequency from Eq. (4) which agrees with the measured peak in both raw counts and calibrated transmission. The agreement between blue and black curves as well as the absence of significant power at other periods shows that the detector window internal to the spectrograph does not contribute to the fringes through the calibration process.

Fig. 4

(a) The data and (b) power spectrum for spectrophotometric data set. The intensity is recorded for 1260 transmission measurements covering 625 to 635 nm wavelength using the Meadowlark SPEX 1401 spectrograph and a 1.1335-mm-thick Heraeus Infrasil 302 window. The power spectrum is computed as (${\mathrm{FFT}}^2$). For (a) intensity data, strong oscillations of detected intensity are seen in blue with the baseline no-sample measurement in black. There is some oscillation in the no-sample baseline data attributed to the PMT sensor window glass. However, the amplitude is much lower and the period of oscillation in this baseline is much longer than detected in the Infrasil window data set. (b) Shows two power spectra. The blue curve shows the power in the raw intensity data uncalibrated for the baseline. The black curve shows the power in the transmission spectrum of the sample scan calibrated by the no-sample scan. The vertical dashed red line shows the predicted fringe period from Eq. (4). The black dashed line shows the period we find in the no-sample baseline data. The measured power is dominated by the single fringe period created by the Infrasil window which does not change through the calibration process. The fringe period agrees with the theory. The blue and black curves show minimal differences between raw data and baseline-calibrated transmission. Other optics inside the spectrograph (e.g., detector window) do not contribute significantly to calculating window fringe periods.

To compute the likely fringe amplitude, the expected electric field amplitudes must be added or subtracted coherently. As a first-order approximation for highly transmissive optics, we can simply consider the reflection from the back surface interfering coherently with the front surface. The initial reflection has an intensity of 3.5% at the interface between air and Infrasil. The transmitted beam reflects off the back surface and then transmits again through the front surface with an intensity that scales with the back surface reflectivity twice multiplied by the transmission of air to Infrasil. This is roughly 96.5% ($3.5\%\times 96.5\%$) $\sim 3.38\%$.

Electric field amplitudes go as the square root of the intensity. With this assumption of a single reflected beam interfering, we can predict the total intensity as constructive and destructive interference at each wavelength. We can compute the intensity interference amplitude using the predicted reflection and transmission coefficients after converting to field amplitudes as in Eq. (5). For our nominal 3.5% front surface reflection interfering with a 3.3% back surface contribution, we expect to see the transmission oscillate from 86% to 99.9% for complete destructive and constructive interference. We detected transmission (intensity) fringes from 87% to 99%, which we consider to be a good experimental match. Detected fringe amplitude can be decreased by reduced spectral resolution, optic wavefront flatness, optical wedge, and several instrumental degradations. Simple single-reflection theory predicts 14% intensity variation and detect 12% variation.

3.1.

Multiorder Quartz Retarder Data and Models

We show a Berreman model for a multiorder quartz crystal retarder compared to high spectral resolution scans with the Meadowlark Spex instrument. The retarder thickness was measured to have $575.4\pm 0.5\mu \mathrm{m}$ thickness and was mounted for testing with retarder fast and slow axes at 45 deg to the grating rulings. This thickness corresponds to about 16.5 waves of retardance at 630 nm wavelength. The TWE for the crystal is 0.034 waves PV at 632.8 nm wavelength over a clear aperture of 12 mm. Beam deviation was measured to be 0.21 arc sec.

The retarder was mounted in a collimated beam and baseline scans without the sample were also recorded. A slight blemish in the beam attributed to the fiber was identified and the beam was stopped to roughly $f/10$ inside the spectrograph to ensure this blemish did not impact the data. This beam underfilled the grating and likely had reduced spectral resolving power. In the data sets presented in this section, we increased the spectral sampling to cover 30-nm bandpass at steps of 2 pm giving an effective sampling at $\frac{\lambda }{\delta \lambda }$ of about 315,000. We sample the 16-pm FWHM beam with about eight points with a total of roughly 15,000 points over the bandpass.

A Fourier analysis of the data set gives a fringe period of 0.22 nm. The power spectrum is dominated by a single somewhat broad peak at 0.22 nm without any other significant features in the 0.05- to 0.5-nm period range. The theoretical period of ${\lambda }^2/2dn$ is 0.2226 nm for the extraordinary beam of refractive index 1.551 and 0.2239 nm for the ordinary beam at a refractive index of 1.543.

We created fringe predictions with our Berreman code as shown in Fig. 8. The fringes were derived at spectral sampling of $\delta \lambda /\lambda =\mathrm{500,000}$ and infinite spectral resolving power. To show the impact of instrument resolving power on fringe amplitude, we convolve these Berreman predictions with Gaussian profiles of varying widths corresponding to the Spex range of $R=\mathrm{20,000}$ to $R=\mathrm{40,000}$.

Fig. 8

(a) Transmission fringes for the quartz retarder computed using our Berreman code over the bandpass 610 to 650 nm. Black shows the Berreman calculation at spectral sampling of 500,000. Blue and red show the fringes convolved with a Gaussian instrument profile of resolving power 40,000 and 20,000, respectively. (b) The Meadowlark Spex 1401 data set. Transmission ranges from 89% to 100% for a fringe amplitude of 11%, consistent with the Berreman calculations at $R=\mathrm{20,000}$ resolving power.

The higher refractive index quartz crystal has transmission ranging from 99.99% to 81.8% for a fringe amplitude around 18% using the simple single-reflection analytic equation. When convolving this theoretical curve with a Gaussian profile at resolving power of $R=\mathrm{20,000}$, the amplitude decreases to about 12%. In addition, the interference between the extraordinary and ordinary beams gives rise to a much slower amplitude modulation at a period of roughly 35 nm at 630 nm wavelength. The measurements show the minimum fringe amplitude clearly around 631 nm wavelength in Fig. 8 with fringe amplitudes rising quickly to shorter and longer wavelengths. This is very similar to our Berreman calculation of Fig. 8.

Fig. 9

The linear retardance predicted for the three SPINOR bicrystalline achromats. SPINOR retarder designs at 633 nm wavelength are to $1/4$-wave, $3/8$-wave, and $1/2$-wave retardance. The $1/4$ wave is shown in green with crystals at (0.960209 and 1.098837 mm) thickness. The $3/8$-wave retarder is shown in blue with crystals of (1.438883 and 1.648256 mm) thickness. The $1/2$-wave retarder is shown in red with crystals at (1.920417 and 2.197675 mm) thickness.

Fig. 10

The half-wave retarder mounted with the SPINOR polarizing beam splitter near the slit. Courtesy Doug Gilliam, DST staff.

4.1.

Air Gap and Coating Impact on Fringe Amplitude and Spectral Frequency

For the analysis we present, we model the air gap as $20\mu \mathrm{m}$. The gap is set by spacer spheres placed on the outer edge of the aperture. On one side of the aperture, the spacers are slightly thicker. This results in a slight tilt to the second optic. This does reduce ghosting for the modulator and half-wave retarders near focal planes. However, the small angle has minimal impact on the polarization fringes reported here as the beam overlap for coherent interference is not substantially changed.

The antireflection coating is almost certainly a many-layer broadband antireflection (BBAR) coating of unknown formulation. The company ZC&R had developed a coating to nominally cover the 450- to 1600-nm wavelength range. According to the DST staff, the coating should have roughly 1% reflectivity from 450 to 650 nm wavelength and 0.5% out to 1600 nm wavelength with some slow spectral oscillations. We have seen no records of the measured coating performance and will consider the impact of coating variation in later sections.

For simplicity, we use a single-layer quarter-wave of isotropic ${\mathrm{MgF}}_2$ with a central wavelength near 600 nm to illustrate how the fringe amplitude is reduced when using AR coatings on this part. Although even isotropic materials can produce retardance and diattenuation when used at nonzero incidence, we are not considering incidence angle variation in this section. We proceed with an analysis of the on-axis beam only using this coating to illustrate how the SPINOR fringes are dominated by the internal reflections and birefringence of the two crystals.

For our analysis of thick crystals, the concept of a fringe spectral period is reasonably well defined. As shown in Table 1, our crystals are typically several thousand waves of optical path thickness. The coherent interference with wavelength through multiples of quarter- and half-waves is dominated by the change in wavelength. Typical crystals of $>1\mathrm{mm}$ thickness are thousands of waves optical path thickness at visible wavelengths. This optical thickness is computed as the physical thickness times the refractive index divided by the wavelength ($d*n/\lambda$). With an essentially constant refractive index over a narrow bandpass, the wavelength changes by one wavelength divided by the optical path and the interference will change from constructive to destructive. This allows us to apply simple Fourier analysis to the fringes for thick crystals and to speak of the spectral period of the fringe as in Eq. (4).

As an example, the SPINOR modulator at 400 nm has quartz crystal thicknesses of (5637.9, 5603.5) waves for the $e$- and $o$-beams and sapphire crystal thicknesses of (7326.7, 7361.6) waves. The quartz crystal has 34.46 waves retardance while the sapphire crystal has $-34.95$ waves retardance. The crystals mounted with axes aligned add retardance to become a $-0.492$-wave retarder.

This bicrystalline achromat has $e$- and $o$-beam optical path lengths of (12964.6 and 12965.1) waves. The internal interface between quartz and sapphire as well as the exit surface reflection will both contribute to the fringes. The fringe period from the combined optical thickness is 0.03 nm and wavelength of 400 nm. There will be additional contributions from the reflections internal to the crystals. As examples, the $o$-beams contribute fringes with periods of $400\mathrm{nm}/7360=0.054\mathrm{nm}$ and $400\mathrm{nm}/5603=0.07\mathrm{nm}$. In the air-spaced design, there will be an additional contribution coming from multiple internal reflections inside the air gap as well as multiple internal reflections from internal faces to other internal faces.

The Berreman calculus solves the eigenvalue problem that simultaneously satisfies all boundary conditions for the electric field on the entrance and exit of the optic. It is useful for thick crystal retarders to think of the primary contributions to the fringes as the internal reflections with fringe periods scaling as the optical thickness of the crystal. As an example, Fig. 11 shows the predicted fringe period for the three different SPINOR modulators using ${\lambda }^2/2dn$. We have computed the fast Fourier transforms of various Mueller matrix elements and retardance fits on a wide variety of data sets. For thick crystals, there is excellent agreement between the predicted and calculated fringe spectral period. We will apply this technique in later sections to SPINOR data as well as to many-crystal retarders at other telescopes.

Fig. 11

The fringe period predicted using the optical thickness of the quarter-wave retarder and Eq. (4) for period (${\lambda }^2/2dn$). The o- and e-beam fringe periods are both plotted, but are within the thickness of the plotted line. Blue shows the fringe through the quartz crystal, green shows the sapphire crystal, and black shows both crystals together as a bicrystalline achromat. See text for details.

We have created fringe models for the three SPINOR retarders considering several design choices. Crystals are either optically contacted or air-gapped as well as coated and uncoated. For each model, we can fit a linear retarder Mueller matrix to the Berreman calculus results. For the case of zero incidence angle with exact alignment of the crystals, a linear retarder fit is exact for this optic. In later sections, we discuss nonzero incidence angle and errors in manufacture (crystal orientation and polish) that produces elliptical retardance (linear + circular retardance). For the simple models of SPINOR bicrystalline achromats, the fit to the linear retardance fast axis orientation is within numerical precision ($<1e-11$) of zero. The theoretical prediction computed as a series of linear retarders agrees perfectly with the Berreman calculus.

To estimate fringe amplitudes, we apply a simple spectral analysis technique. We have the fringe calculations at a spectral sampling of $1/\mathrm{2,000,000}$ but the fringes vary on a wavelength scale of a few parts per thousand. We first subtract the theoretical linear retardance from the fringe model prediction to give linear retardance variation about the design. We then compute the maximum deviation in spectral bins of 50 samples to produce a curve showing the maximum fringe amplitude in narrow bandpasses sampled at $1/\mathrm{40,000}$.

Figure 12 shows examples of these fringe amplitude estimates for the SPINOR $3/8$-wave modulator. The green curve on the bottom shows an optically contacted, uncoated optic. The linear retardance varies by roughly $\pm 0.5\mathrm{deg}$ to $\pm 1.4\mathrm{deg}$ across the 400- to 1200-nm wavelength range. The blue curve shows the same bicrystalline achromat but with a single layer of isotropic ${\mathrm{MgF}}_2$ as an antireflection coating with a central wavelength of 550 nm on every surface and a $20\text{-}\mu$ air gap between crystals. The variation in linear retardance for this coated, gapped model is now less than $\pm 0.5\mathrm{deg}$ near the AR-coated wavelengths but the air gap causes amplitudes to rise over $\pm 1.5\mathrm{deg}$ to $\pm 3.0\mathrm{deg}$ at long wavelengths. The Mueller matrix of the air-gapped optic now combines several fringe periods with mixing between the relatively large period air-gap fringe and relatively small period crystal fringes. Figure 13 shows the Mueller of the air-gapped, AR-coated SPINOR $3/8$-wave modulator in a narrow spectral bandpass from 425.7 to 426.0 nm. As seen later, this is a bandpass where we have SPINOR observation and is representative of the complexity in analyzing the diattenuation and retardance variation caused by these retarders. The diattenuation and linear retardance are predicted to contain a mix of several spectral periods with amplitudes that depend strongly on the antireflection coating performance and the air-gap.

Fig. 12

The amplitude of the linear retardance fringes in narrow spectral bandpasses of $R=\mathrm{10,000}$ spectral sampling for the SPINOR $3/8$-wave modulator. Black is the linear retardance fringe magnitude using a $20\text{-}\mu \mathrm{m}$ air gap and a single-layer antireflection coating. Blue shows an uncoated model with crystals optically contacted. See text for details.

Fig. 13

The Mueller matrix of the SPINOR third-wave bicrystalline achromatic retarder used as the modulator at a wavelength of 425.7 nm. The [0,0] Mueller matrix element has been used to normalize all other matrix elements as in Eq. (2). The retarder has roughly 172 retardance, close to half-wave. Spectral sampling is $\lambda /\delta \lambda =\mathrm{2,000,000}$. This simulation used a single-layer quarter-wave thickness of isotropic ${\mathrm{MgF}}_2$ coating to reduce reflections at 550-nm central wavelength. The crystals were air-spaced with a $20\text{-}\mu \mathrm{m}$ gap.

4.2.

SPINOR Observations and Fringe Properties

We collected observations of sunspots in March of 2016 at the DST using several configurations of the SPINOR spectrograph. We collected a minimum of several hours observing at wavelengths of 426, 431, 458, 514, 526, and 614 nm to test six separate wavelengths.

The $3/8$-wave retarder used as a rotating modulator was placed in the converging $f/36$ beam located just ahead of the SPINOR spectrograph slit. The data were reduced and processed using the standard data analysis software available at the National Solar Observatory website.⁶¹ Data reduction was performed by Christian Beck. Example demodulated observations of Stokes $Q$ are shown in Fig. 14 for all six wavelengths. In each panel of Fig. 14, there is a sunspot and visible changes in the polarized spectra. Often in the sunspot, the intensity drops substantially and the fringe character changes. There are clear spectral features corresponding to real solar magnetic signatures, but fringes cover and often dominate the entire $Q$ image for most wavelengths.

Fig. 14

Examples of reduced and demodulated Stokes $Q$ measured by SPINOR at the Dunn Solar Telescope while observing a sunspot. The $3/8$-wave modulator as well as the $1/2$-wave retarder on the polarizing BS were in the optical path during observations. In each image, there are gray regions corresponding to the center of the sunspot (horizontal, spectral direction). There are real Stokes $Q$ signals in several spectral lines seen as spatially localized, saturated white/black regions. Each image is otherwise covered with large amplitude, quasiperiodic polarization fringes caused by the bicrystalline retarder. The wavelengths are (a) 425.8 nm, (b) 526.2, (c) 514.2 nm, and (d) 614.9 nm. The lower two images use a different sensor than the others and have a $256\times 512\text{pixel}$ format. The shorter-wavelengths are observed with a $512\times 512\text{pixel}$ format camera.

We analyze the fringes by masking out regions of the image corresponding to the sunspot and real signals. The SPINOR data were then Fourier transformed to create a power spectrum for the six wavelengths observed.

At each wavelength, the effective optical thickness of the retarder is different. Table 2 shows the optical thickness for the $3/8$-wave retarder (modulator). We adopt the $3/8$-wave modulator crystal thickness scaled from the nominal design values specified by the DST staff. The numerical solution for the exact retardance specification using our refractive index formulas solved for a crystal quartz thickness of 1.438883026 mm and a sapphire crystal thickness of 1.648256206 mm. Thickness must be specified to subnanometer precision or solved exactly in the Berreman calculus using refractive index formulas as a few microns of crystal typically corresponds to a quarter-wave of retardance. The quartz crystal has a net retardance (bias) of about 21.2 waves at 615 nm while the sapphire crystal has roughly $-21.7$ waves retardance bias for the SPINOR modulator.

Table 2

SPINOR 3/8-wave retarder optical properties.

The first column of Table 2 shows the six wavelengths we observed at the DST. The second column shows optical path for the quartz crystal extraordinary beam ($dn/\lambda$). The third column shows the sapphire crystal extraordinary beam optical thickness. The fourth column shows the summed part thickness for the bicrystalline achromat (ignoring the air gap of a few waves thickness). The next two columns show the net retardance for the quartz and sapphire crystals. The predicted fringe period in picometers (${\lambda }^2/2dn$) for the sapphire crystal e-beam is then shown to reference Fig. 15. The final column shows the single pixel wavelength sampling for the SPINOR sensor as observed during our campaign. At short wavelengths, we sample a 31-pm fringe period with 3-pm sampling giving 10 pixels per fringe. At long wavelengths, we sample at 65-pm fringe at 4 pm giving 16 pixels per fringe.

Fig. 15

The spectral power of the measured Stokes $Q$ fringes in regions without noticeable signals. All six wavelengths are plotted with a different color: 425.8, 431.5, 458.8, 514.2, 526.2, and 614.9 nm. Each spectral fringe power spectra had the spectral period normalized by the expected fringe period based on the thickness of the sapphire crystal e-beam (${\lambda }^2/2dn$). The dominant fringe power is around 94.5% of the predicted period for the sapphire crystal as seen by the vertical red dashed line. As the quartz piece is thinner and lower index, the fringe caused by the quartz crystal is 1.30 times larger than for the sapphire crystal. The combined crystals (quartz + sapphire) create fringes with periods roughly 0.565 times smaller than the sapphire crystal alone. See text and Table 2 for details.

To show all power spectra on the same spectral period array, we normalized the $x$ axis by the individual periods predicted for the sapphire crystal fringe at each wavelength. Both crystals in the optic are a few thousand waves thick with a birefringence of 20 to 30 waves. There is little difference between ordinary and extraordinary beam fringe periods.

These six fringe spectral periods are listed in Table 2. As the quartz piece is thinner (1.44 versus 1.65 mm) and the refractive index is lower (1.56 versus 1.78), the fringe caused by the quartz crystal is 1.30 times larger than for the sapphire crystal. Figure 15 shows the fringe power clearly in several measurements. The combined crystals (quartz + sapphire) have periods roughly 0.565 times smaller than the sapphire crystal alone. As an example from Table 2, the quartz crystal optical thickness is $\sim 5290$ waves at 425.8 nm while the sapphire crystal is $\sim 6870$ waves. Both crystals together account for $\sim \mathrm{12,150}$ waves of optical path. The fringe period would be 31 pm for the sapphire crystal, roughly 40 pm for the quartz crystal and 15 pm for both crystals combined. At this wavelength, SPINOR samples the spectrum with 2.78 pm per pixel. SPINOR resolves the shortest period fringe with roughly 5.5 samples per fringe period. The dominant power caused by the sapphire crystal is sampled at 11.1 pixels per fringe period.

5.1.

DKIST Calibration Retarder

In this section, we show models for the three DKIST calibration retarders designed to be 0.25 wave linear retarders in specific bandpasses. Each calibration retarder contains six individual retarder crystal plates. This increases the complexity of the retarder fringe model and shows the interference effects from multiple internal interfaces in birefringent media. DKIST will produce three additional modulating retarders that are designed to be elliptical. We will focus on the three linear retarders for simplicity.

A common retarder design tool was introduced by Pancharatnam⁶³ to make a superachromatic retarder (SAR) as a combination of three retarders. By using three retarders together, the wavelength range for achromatic linear retardance is greatly increased. For our particular application, we make each of these three retarders as subtraction-pairs of crystal plates. This way there are six total crystals, following the Pancharatnam design where each of the three linear retarders is now comprised of two crystals used in subtraction.

There are many degrees of freedom if one chooses different materials, retardance values, and orientations for all six crystals. The Pancharatnam designs can be simplified by choosing just one or two materials and also by making the outer two retarders identical. This simple design uses an $A\text{-}B\text{-}A$ type alignment where the two outer crystal pairs are mounted with their fast axes aligned. Provided the bicrystalline pairs are treated as perfect linear retarders, there is a simple theoretical formula for the linear retardance of such an $A\text{-}B\text{-}A$ design.

If we take the retardance of the A crystals as ${\delta }_A$ and the B crystals as ${\delta }_B$, and the relative orientation between the $A$ and $B$ crystal pairs as $\theta$, we can write the formula for the resulting superachromatic optic retardance ($\mathrm{\Delta }$) and fast axis orientation ($\mathrm{\Theta }$) as in Eqs. (6) and (7).⁶³

Often, a further constraint is to make all three pairs identical. There is still an orientation offset between the inner $B$ pair and the outer $A$ pairs. This way, a simple Pancharatnam design would only use two materials (such as quartz and ${\mathrm{MgF}}_2$ crystal) and a manufacturer would only polish each material to one specific thickness. This way, the retarder has three identical subtraction pairs achromats with an orientation of [0, X, 0] and only two thicknesses to vary for a three-dimensional optimization problem. Each DKIST calibration retarder was designed using six crystals grouped into three pairs using either all quartz or all ${\mathrm{MgF}}_2$ crystals. Each two-crystal group is a subtraction pair.

Crystals were mounted with their fast and slow axes crossed in order to perform the subtraction. The quartz plates are all designed to be 30 waves of retardance thick while the ${\mathrm{MgF}}_2$ crystals are 40-waves retardance. Both the quartz and ${\mathrm{MgF}}_2$ crystal thicknesses end up being a little over 2 mm physical thickness. As the Berreman calculus includes all interference from all interfaces, we show here the impact of adding antireflection coatings, air gaps, refractive index-matching oils, optical contacting, and/or the inclusion of cover windows often used to achieve specific flatness and polish specifications. Figure 18 shows the Berreman-derived Mueller matrix including all optics, coatings and oil from Table 3.

For our large retarders, the 120-mm diameter and 2-mm thickness prevented optical contact as a viable method for bonding the crystals together. Our calibration retarders must work in a 300-W beam of 105 mm diameter with substantial flux at short and long wavelengths. This heat load, UV flux, and high aspect ratio places stringent requirements on any materials including coatings and oils placed between the crystals.

DKIST performed tests on several types of refractive index-matching oil to ensure survivability and performance when observing wavelengths from 380 to 5000 nm while being exposed to nonzero flux for wavelengths as short as 300 nm. Other epoxies and cements were also considered and ruled out, as they would risk catastrophic failure of the part should the material ever degrade over the 40-year lifetime of the observatory.

There were stringent restrictions on TWE and also beam deflection for these optics. Beam deflection in particular causes image motion as the optic rotates. For rotating retarders used as modulators, this is a major source of error in processing polarimetric image sequences. In response to all these concerns, the DKIST team decided to require relatively thick (10 mm) cover windows on both exit and entrance interfaces to the crystal stacks. These windows are thought to assist with wavefront error and beam deflection, but we show here there are also strong consequences for polarization fringe amplitudes and spectral periods. There are several other strong impacts to beam wobble and heating from including cover windows.

The calibration retarders were designed as SARs with a traditional quarter-wave linear retarder design goal. Of course, calibration does not require only linear retardance and does not require particularly chromatic behavior.^{15,64,65–74} Elliptical, chromatic retarders are suitable, but the DKIST project decided to follow this common quarter-wave linear strategy at the time procurement decisions were made.

There are three subtraction crystal pairs designed to work in the combination $A\text{-}B\text{-}A$ when used as a calibration (linear) retarder. This design strategy and the crystal thicknesses were also influenced by other manufacturing reasons. The elliptical retarders used as a rotating modulator in each DKIST instrument covering the same wavelength range could be made using $B\text{-}A\text{-}B$ crystals with a change of orientation. In this way, we only need to manufacture crystals of two different thicknesses $(A,B)$ but can make linear retarders and elliptical (efficient) rotating modulators using relatively few design parameters.

5.2.

Quartz Visible Wavelength Retarder

The first retarder we consider for DKIST is nominally designed to cover 380 to 1100 nm wavelength range. The visible spectropolarimeter (ViSP) instrument spans roughly this wavelength range. The DKIST project often refers to this retarder as the ViSP SAR. However, this optic is useful over a wide wavelength range and can be used by any instrument. It will be used by other instruments in the same wavelength range such as the visible tunable filter or the Diffraction-Limited Near-Infrared Spectropolarimeter. As an example, all DKIST instruments should be using the 854-nm bandpass for some activities and all instruments can use this retarder for calibration at this wavelength.

For this quartz visible-range calibration retarder, the crystal plates were designed to be 30 waves retardance thick with the $A$-bicrystalline achromat to be polished to 0.476 waves linear retardance at 633.443 nm wavelength. The $B$-bicrystalline achromat was designed to be polished to 0.328 waves linear retardance at the same wavelength. By orienting these three bicrystalline achromats as $A\text{-}B\text{-}A$ with orientations of (0, 70.25, 0), a retarder that is close to a quarter-wave linear retardance is created. The two cover windows chosen for this optic are 10 mm thickness of Heraeus Infrasil 302.

When performing a Berreman calculation, we also need to specify all other materials in the stack. The properties of the ViSP six-crystal SAR for calibration are shown in Table 3. Each of the crystals is about 2 mm thick.

Table 3

ViSP SAR retarder.

The refractive index-matching oil has been measured on several substrates and is typically in the range of 5 to $15\mu \mathrm{m}$ thick. We use $10\mu \mathrm{m}$ physical thickness and a constant refractive index of 1.3 for the present modeling. The Cauchy equation on the data sheet for the oil does show an almost constant $n=1.3$ across the full wavelength range. We use a 94.4-nm physical thickness of isotropic ${\mathrm{MgF}}_2$ for the antireflection coating to simulate the central wavelength 525 nm specified for the AR coatings.

From the Berreman calculus Mueller matrix model, we can derive the retardance and diattenuation. Figure 19 shows fits to the linear retardance components of the Mueller matrix. The linear retardance follows the predicted design and illustrates how the interference fringes change strongly as the many layers from Table 3 contribute backward reflected radiation.

Fig. 19

The best fit linear retardance and fast axis to the Berreman calculus model of the visible-wavelength quartz retarder design with all optical elements listed in Table 3. The wavelength range goes from 380 to 1600 nm. The linear retardance amplitude is close to a quarter-wave (90 deg magnitude) throughout the visible wavelength range and the fast axis orientation changes by about 20 deg as in the theoretical design. The amplitude of the linear retardance fringes is 10 deg to 20 deg in magnitude. Note that the fringes are elliptical and a similar magnitude of circular retardance is not captured in this fit.

As expected for a many-layer part, there is a substantial circular retardance in the fringes. The residual Mueller matrix values in the $QUV$ to $QUV$ terms are up to 0.05 when comparing best-fit linear retarder models to the fully elliptical Berreman fringe model. An elliptical retarder fit to this model completely reproduces these terms, showing that the elliptical retardance of the fringe is easily detected and fit.

5.3.

MgF₂ Calibration Retarder

A six-crystal ${\mathrm{MgF}}_2$ retarder was designed as a linear quarter-wave retarder the 2500 to 4000 nm wavelength range. The DKIST instrument called CryoNIRSP presently plans to have filters covering at least 1000 to 4600 nm. This infrared wavelength range required the use of an IR-transparent material. Obvious choices included sapphire and ${\mathrm{MgF}}_2$ crystals. Given the manufacturing feasibility some years ago, the DKIST team decided to proceed with a design that included only ${\mathrm{MgF}}_2$ crystals and used ${\mathrm{CaF}}_2$ windows to achieve the required flatness and polish specifications. The two ${\mathrm{CaF}}_2$ cover windows are 10 mm thick each.

This retarder uses the same $A\text{-}B\text{-}A$ style of three bicrystalline achromats but with ${\mathrm{MgF}}_2$ crystals instead of quartz. The thickness are (2273.14 and $2153.11\mu \mathrm{m}$) for $A$ and (2333.21 and $2153.11\mu \mathrm{m}$) for $B$. These correspond to net retardance of 40 waves per plate. The $A\text{-}B\text{-}A$ pairs are polished to net retardances of 2.230, 3.346, and 2.230 waves retardance at 633.443 nm. The orientations for the bicrystalline pairs are (0, 107.75, 0). We used the revised birefringence measurements from Sueoka as directly measured over a wide wavelength range.¹⁰

The six crystals combine to a thickness of about 13 mm and we use the same $10\mu \mathrm{m}$ thickness for the refractive index-matching oil between the crystals. The antireflection coating modeled here used a central wavelength of 1500 nm and we computed a physical thickness of 271.7 nm for the isotropic ${\mathrm{MgF}}_2$ coating applied on all just the four window surfaces. We show in later sections the impact of thicker, thinner, and no coating. The actual design coating has a central wavelength of 2500 nm and a corresponding thickness of 452.9 mm but it is unclear yet if this thick of a coating is feasible. As the ${\mathrm{MgF}}_2$ crystals have low refractive index and our oil has a close matching refractive index of around 1.3, antireflection coatings are not considered for the ${\mathrm{MgF}}_2$ crystals.

We computed the Mueller matrix using the Berreman calculus and fit a linear retarder model. In contrast to the quartz retarders, this retarder has a strong change in retardance at visible wavelengths as it was optimized for achromatic performance in the 1500- to 4000-nm wavelength range. Figure 20 shows the linear retardance fits. The linear retardance magnitude is near an integer multiple of quarter-wave retardance in the 2500- to 4000-nm wavelength range as expected for the relatively thick $A\text{-}B\text{-}A$ design at (2.230, 3.346, and 2.230) waves retardance bias per plate pair. At shorter wavelengths, the retardance of the optic drops in magnitude to zero near 705 nm. The sign of the retardance changes and the magnitude then continues to change.

Fig. 20

The fit linear retardance and fast axis to the CryoNIRSP SAR design which includes six crystals, two AR-coated cover windows, and oil between all interfaces. (a) Linear retardance and (b) the fast axis orientation. The design wavelength range was 2500 to 4000 nm. The linear retardance drops through zero and changes sign at short wavelengths. Near zero, the definition of the fast axis is problematic and noise amplification leads to a vertical line. For clarity, we have wrapped the linear retardance around 360 deg.

5.4.

Design Choices: Trade-Offs for Removing Retarder Windows

When designing retarders, there are several decisions that can benefit from a quantitative trade off of polarization fringe properties against wavefront error, beam deflection, cost, coating thickness, and other system level choices.

Often, there are restrictions on the transmitted wavefront quality, polish, and beam deflection that impose challenges for relatively high aspect ratio crystals. The DKIST retarders are 2 mm thick by 120 mm across for a 60:1 aspect ratio. With thin crystal plates, there is difficulty manufacturing a part with low wavefront distortion, retardance uniformity of $<0.01$ waves, environmental insensitivity, and benign response to heating through optical absorption. Often times, a solution is to put cover windows on the part. While windows provide many benefits, the impact on polarization fringes must be considered as a trade against potential benefits.

Fringes may be spectrally resolved, spatially variable, and can be minimized with coatings. Having fringe predictions to include with other performance metrics allows more realistic systems engineering. For the specific case of the DKIST calibration retarders, the high flux $f/13$ beam at the Gregorian focus presents special challenges. The set of DKIST polarimetric optics must work with a diffraction limited large aperture optical system in a 300 watt beam with a 105 mm clear aperture. The optics must be mounted and rotate near focal planes with substantial flux to wavelengths shorter than 320 nm. The set of optics must be used from 380 nm to as long as 5000 nm with good efficiency and often simultaneous operation of many instruments. This imposes constraints on wavefront error, polishing uniformity, wedge and beam deflection, surface polish, and thermal response to name a few. In this section, we outline a few typical choices involving inclusion or removal of cover windows that may be encountered in design of retarders for large aperture telescopes.

As an example of Berreman fringe predictions, Fig. 21 shows the computed Mueller matrix for the DKIST ${\mathrm{MgF}}_2$ calibration retarder with and without the 10-mm thick ${\mathrm{CaF}}_2$ cover windows. The black curve shows the window-covered optic with transmission fringes ranging up to 0.4. Diattenuation terms have amplitudes over 0.2 and the $QUV$ to $QUV$ retardance elements of the Mueller matrix are fringing over 0.2. The blue curve shows the part without these coated ${\mathrm{CaF}}_2$ cover windows. The transmittance, diattenuation, and retardance fringes all decrease by factors of 2 to $>3$ when removing the ${\mathrm{CaF}}_2$ cover windows with intensity fringes always at or below 10%.

Fig. 21

The polarization fringes for window-covered (black) and no-window (blue) designs for the ${\mathrm{MgF}}_2$ retarder. The nominal design used ${\mathrm{CaF}}_2$ cover windows (black) on top of the six ${\mathrm{MgF}}_2$ crystal retarders. The ${\mathrm{CaF}}_2$ window to air interfaces is antireflection coated with isotropic ${\mathrm{MgF}}_2$ at a central wavelength of 1500 nm (nominal design). When the cover windows are removed (blue), the fringes see much less optical path, the diattenuation amplitudes are substantially reduced in the $I$ to $QUV$ and $QUV$ to $I$ matrix elements. The retardance fringes are also substantially reduced as seen in the $QUV$ to $QUV$ elements. The theoretical Mueller matrix was subtracted from all Berreman-computed elements to isolate the fringes. The [0,0] element ($II$) was used to normalize all elements and separate transmission effects from polarization effects as in Eq. (2).

The spectral frequency and complexity of the fringe pattern also increase strongly when cover windows are added. The increase in optical path from cover windows also increases the sensitivity of the part to thermal issues. Figure 22 shows the DKIST calibration retarder with and without the 10-mm-thick ${\mathrm{CaF}}_2$ cover windows in a narrow wavelength region starting at 854 nm covering 0.1 nm. In Fig. 22, the blue curve again shows the optic with cover windows. Intensity fringes are seen in the $II$ matrix element with amplitudes of 0.2. The blue curve shows the part without ${\mathrm{CaF}}_2$ cover windows. The transmission fringes are nearly zero, the diattenuation decreases amplitude by over $2\times$ in $IQ$ and $QI$. The circular polarizance $VI$ and $IV$ go to nearly zero without windows. The complexity and spectral content of the fringes greatly decreases without windows. These amplitude and frequency decreases are all strong motivations to not use cover windows.

Fig. 22

The residual Mueller matrix differences between an ideal retarder $A\text{-}B\text{-}A$ stack Mueller matrix model and the Berreman fringed Mueller matrix model. The [0,0] matrix element $II$ has been used to normalize all subsequent Mueller matrix elements to separate transmission effects from polarization. The nominal window-covered design is shown in black and the no-window uncovered crystal model is shown in blue for the ${\mathrm{MgF}}_2$ calibration retarder. This figure shows a narrow wavelength band from 854 to 854.1 nm covering a 100-pm bandpass where many DKIST instruments intend to observe. When the cover windows are removed (blue), the fringes see much less optical path and the fringe amplitudes are substantially reduced.

The amplitude and spectral frequency of the fringes depend strongly on the refractive index mismatch among air, the windows, and the two indices in the crystals. Antireflection coatings can be useful to reduce fringe amplitudes, but they must be considered carefully for other risks. The index mismatch between the ${\mathrm{CaF}}_2$ and the crystal ${\mathrm{MgF}}_2$ $e$- and $o$-beams is roughly (0.045 and 0.055) at 854 nm wavelength.

Two of the DKIST calibration retarders are made of crystal quartz with Heraeus Infrasil 302 fused silica cover windows.^7,9,10 For the fused silica to crystal quartz e- and o-beam interfaces, this refractive index mismatch is (0.085 and 0.107) at 854 nm wavelength. If the quartz crystals were left uncoated, the fringes in these retarders would be substantially worse. The quartz crystals are antireflection coated with isotropic ${\mathrm{MgF}}_2$, substantially reducing the internal fringe amplitude around the antireflection coating design wavelength.

Antireflection coatings come with benefits as well as risks. Each crystal must be polished to a specific retardance in subtraction. If a coating run fails, as some did for DKIST, the damaged crystals must be repolished. For such thin crystals, this can result in loss of the crystal at significant schedule and cost impact. Coatings also absorb heat and add complication for the DKIST calibration retarders in the 300 W beam. We can use the Berreman calculus to predict fringe behavior for a variety of antireflection coatings. In Fig. 23, we show the fringe amplitude for the [0,0] Mueller matrix element ($II$, transmission) for the visible-wavelength optimized quartz achromatic calibration retarder.

Fig. 23

The polarization fringe amplitudes for the optic transmittance using various scenarios for our visible wavelength optimized quartz-based retarder. The Berreman-predicted Mueller matrix was computed at spectral sampling of $\lambda /\delta \lambda$ of 2,000,000 and subtracted from the ideal perfect retarder design. Fringe amplitudes were computed in wavelength bins of roughly 100 nm. Blue shows no coating on the window exterior interfaces. Green shows a single-layer isotropic ${\mathrm{MgF}}_2$ coating. Dashed black shows the same coating but with both Infrasil windows removed. Red shows an 18-layer BBAR coating from Infinite Optics on the window exterior interfaces. See text for details.

In all models, there is 75.5-nm physical thickness of isotropic ${\mathrm{MgF}}_2$ as an antireflection coating on internal crystal interfaces. This corresponds to a quarter-wave of optical path at 525 nm central wavelength, the common single-layer AR coating. The oil thickness is set to $10\mu \mathrm{m}$ among all six crystals and both window internal interfaces.

To easily show the fringe amplitudes, we computed the Berreman models at spectral sampling of $\lambda /\delta \lambda$ of 2,000,000. These models were then subtracted from the theoretical Mueller matrix to compute residual fringe amplitudes. These residual fringe amplitudes were then binned spectrally by a factor of 400,000 to create smooth, low-resolution wavelength dependence that represents the envelope for polarization fringes. Fringe frequencies have periods $\ll \mathrm{nm}$ but the amplitudes have envelopes that effectively follow the antireflection coating behavior as seen in Fig. 23.

The blue curve shows fringe amplitudes for a Berreman model with an uncoated Infrasil 302 fused silica windows. The green curve shows a quarter-wave isotropic ${\mathrm{MgF}}_2$ antireflection coating on the Infrasil 302 window external surfaces (as well as internal) with a central coating wavelength of 525 nm (75.5-nm physical thickness). The red curve shows fringe amplitudes when an 18-layer BBAR formula is coated on the exterior cover window surfaces interfaced with air. The BBAR coating was $\sim 1\%$ reflectivity at all wavelengths in the 380- to 1100-nm wavelength range.

The black curve shows the fringe amplitude for a model without any Infrasil cover windows at all. In this no cover window Berreman model, there are single-layer AR coatings on all quartz crystal internal interfaces as seen in Fig. 24. The exterior-facing quartz crystals have the nominal 525-nm central wavelength isotropic ${\mathrm{MgF}}_2$ antireflection coating. The internal interfaces have this same coating as well as the index-matching oil. Without cover windows, the fringe period would be roughly three times larger as the optic is roughly three times thinner optically. The amplitude in Fig. 23 is higher than the red curve because the 18-layer BBAR coating is a much better antireflection coating than a single layer of isotropic ${\mathrm{MgF}}_2$.

Fig. 24

The reflectivity for coatings used in the Berreman fringe calculations. Green curves show a simple single layer of isotropic ${\mathrm{MgF}}_2$ at quarter-wave thickness for a central wavelength of 525 nm. The dashed green line shows this coating interfaced to the oil at $n\sim 1.3$. The solid blue curve shows a BBAR coating with 14 layers optimized for the 380- to 900-nm bandpass on quartz crystal interfaced to air. The dashed blue curve shows 14 layers of the same material but with slightly different layer thicknesses optimized for the crystal quartz to index-matching oil interface. See text for details.

5.5.

Antireflection Coatings Impact Fringes

As an example of the usefulness of this tool, we computed several fringe models using various types of antireflection coatings on the various interfaces. Fringe amplitudes are directly impacted by the electric field amplitude in the backward traveling waves. In many-crystal stack retarders, the antireflection coating properties become very critical in achieving low fringe amplitudes.

Figure 24 shows example coatings used in our fringe calculations. For the visible wavelength quartz-based retarders used in the 380- to 900-nm wavelength range, we applied a traditional antireflection coating as a simple quarter-wave thickness of (isotropic) ${\mathrm{MgF}}_2$ with a central wavelength of 525 nm. This corresponds to a physical thickness of 94 nm and is shown as the solid green line for the air to quartz crystal interface in Fig. 24. As we have a refractive index-matching oil in between the interior crystal interfaces, we also show this same single-layer ${\mathrm{MgF}}_2$ coating as the dashed-green line in Fig. 24 when the interface is oil to quartz crystal. This simple quarter-wave ${\mathrm{MgF}}_2$ coating gives a minimum reflectivity at the central 525-nm wavelength of about 1% for the air interface and 0.1% for the oil interface. For reference, the crystal quartz to oil interface has a reflectivity of about 0.9% at 380 nm falling to 0.7% at 1500 nm wavelength without coatings. Uncoated crystal quartz has a $\sim 5\%$ reflection to air at 380 nm wavelength falling to about 4.5% reflection at 1500 nm wavelength.

BBAR coatings offer substantial reduction of fringe amplitude over a wide wavelength range. This translates to improved optical performance when these retarders are used for modulation or calibration with high spectral resolution astronomical instruments. As an example, we show in Fig. 24 a 14-layer coating formula that has been optimized for the air–quartz interface in the 380- to 900-nm wavelength range as the solid black line. Another 14-layer BBAR coating formula using the same materials has been optimized for the oil–quartz interface as the dashed blue line with slightly different layer thicknesses. As expected, these BBAR coatings reduce the reflectivity substantially. The improvement compared to the simple single-layer isotropic ${\mathrm{MgF}}_2$ coating is a factor of $2\times$ at the central wavelength of 525 nm but is up to $5\times$ improvement across the specified bandpass range. Wide wavelength ranges are typically required of astronomical spectropolarimeters as instruments usually cover the visible and near-infrared bandpasses.

These coatings cause a strong decrease in polarization fringe amplitudes and they are easily included in the Berreman formalism. Figure 25 shows the difference between the theoretical Mueller matrix and the Berreman-computed Mueller matrix including fringes. The theoretical Mueller matrix is computed using ideal linear retarder Mueller matrices following the crystal orientations in Table 3. There are no fringes in this calculation. We then compute Mueller matrix models with the Berreman formalism for the same stack of crystals but now we include various AR coating layers and refractive index-matching oil layers as in Table 3. The Berreman formalism computes a Mueller matrix-matching theory, but now including all polarization and intensity fringes.

Fig. 25

The difference between the ideal Mueller matrix (no fringes) and the Berreman-computed Mueller matrix (with fringes). The ideal Mueller matrix is computed as a stack of ideal linear retarders following Table 3. This ideal Mueller matrix is subtracted from the Berreman-calculated Mueller matrix, leaving only residual fringes. This calculation does not use the Infrasil fused silica windows nor the coatings and oil layers required for such a window listed in Table 3. Black shows fringes when a single-layer ${\mathrm{MgF}}_2$ coating is used only on the exterior air–quartz interface and no coatings are applied at the oil–quartz interfaces. Red shows single-layer ${\mathrm{MgF}}_2$ coatings on all surfaces, oil and air interfaces. The blue curve shows the 14-layer BBAR coatings from Fig. 24 applied to all interfaces.

The black curve in Fig. 25 shows fringes when a single-layer ${\mathrm{MgF}}_2$ coating is used only on the exterior air–quartz interface and no coatings are applied at the oil–quartz interfaces. Transmission fringes are up to 40% with the $I$ to $QUV$ elements up to 20% and retardance elements at roughly 0.2. The red curve shows the nominal design with single-layer ${\mathrm{MgF}}_2$ coatings on all interfaces, oil and air. These additional internal coatings reduce the fringes most significantly near the 525-nm central wavelength. The fringe reduction between black and red curves shows that coating the internal interfaces is critical even though we do have an oil at refractive index 1.3. We note that a refractive index 1.5 oil would have been preferable. However, searches for oils that could meet all requirements including $30+$ year lifetime without service, no degradation with UV flux exposure, no absorption bands from 380 to 5000 nm wavelength, and additional constraints were unsuccessful.

We then simulate fringes amplitudes with BBAR coatings on all interfaces as the blue curves of Fig. 25. The BBAR coatings do produce a significant improvement at all wavelengths but the improvement is strongest at wavelengths away from the 525-nm central wavelength of the single-layer coating. The BBAR coatings further reduce the fringes by another factor of several, showing strong benefit.

5.6.

Expected Fringe Periods for AO-Assisted DKIST Instruments

We use the Berreman framework to predict the expected fringe periods for the major transmissive optics in the DKIST optical path.

In the Coudé Lab, the AO-corrected beam is then distributed by the dichroic BSs of the FIDO for simultaneous use of several instruments. These FIDO dichroics are reconfigurable and will have special coatings. Most instruments observe transmitted through one to several beam splitters. However, the optics and coatings have yet to be procured as of mid 2017 so exact prediction of fringe amplitude and wavelength range will be performed later.

There are two 10-mm-thick Infrasil 302 windows originally included in the design of the DKIST retarders.^7,9,10 During polarization calibration of an instrument, there would be four of these 10-mm thick windows in the beam. There are two retarders, one at Gregorian focus for calibration and one acting as a modulator each with two windows. We have recently removed the windows from all retarders to mitigate fringes following successful testing for other performance parameters (TWE, beam deflection, and uniformity). Each of the DKIST retarders consists of six crystals of roughly 2 mm thickness each and there will be a fringe at all thickness combinations. During polarization calibration, there is also a window included in one of the wire grid polarizer assemblies we call calibration polarizer 2 (CP2). This window is presently designed as 25 mm thickness of Heraeus Infrasil 302. This window will absorb long wavelengths to provide thermal mitigation for the retarders.

We compile the predicted fringe period as a function of wavelength in Fig. 26. Fringes for BS1, the FIDO dichroics, the CP2 window, retarder crystals, and the now-removed retarder windows (Ret) all produce resolvable fringes. Symbols in Fig. 26 mark several common observing wavelengths. The thicker the optic, the smaller the fringe period. With the thicker optics, the required spectral resolution required to resolve the fringe also increases. As an example, at 396.8 nm wavelength, the Infrasil substrates for the FIDO beam splitters produce a fringe with a period of 1.25 pm. The required spectral sampling to measure this fringe at two points per wave would be $R=\mathrm{638,000}=\frac{\lambda }{\delta \lambda }$. For the same 43-mm-thick FIDO substrate measured at a wavelength of 1565 nm, the fringe period is 19.7 nm and two-point sampling would require spectral sampling at $R=\mathrm{159,000}$. The retarder crystals produce fringes that are easily resolved by all DKIST high spectral resolution instruments. The various windows also produce fringes that are marginally to fully resolved.

Fig. 26

The predicted fringe periods created by the DKIST transmissive optics for instruments using the AO-assisted beam. The dashed orange curves show spectral sampling at typical DKIST rates ($\lambda /\delta \lambda$) of 50,000 to 200,000. A typical calibration observation would use all listed optics including one to four dichroic BSs, the WFS-BS1, six quartz crystals, and a 25-mm-thick window in the calibration polarizer (CP2). Beam $f/\text{numbers}$ range from $f/13$ to collimated so fringe amplitude predictions are not listed. See text for details.