1.

Motivation: DKIST Thermal Issues and $F$/ Numbers

In many astronomical spectropolarimeters, spectral fringes in intensity and polarization are the dominant source of error. These errors can either involve corrupting the measured signals or a skewing of the calibrations. Fringe amplitudes can be over 10% with strong changes in fringe characteristics over time, field angle, wavelength, and optical configuration. These fringes often have similar characteristics to the solar polarimetric signals. This similarity complicates the data analysis as fringe removal techniques can corrupt the measurement and skew the properties of the object derived from those measurements. Accurate tools to estimate fringe amplitudes and polarization characteristics are critical for assessing optical designs, evaluating the trade-offs in retarder location, and preparing techniques for fringe removal in postfacto processing of instrument data products. Fringes must be estimated in converging or diverging beams along with dependence on optical design properties such as cover windows, oil layers, and antireflection coatings. This must be coupled to thermal behavior as environmental and optical heat load control is critical for the instrument design and fabrication process. Particular challenges arise in modern solar instrumentation where beams are steeply converging and heat loads can be severe.

The Daniel K. Inouye Solar Telescope (DKIST) on Haleakalā, Maui, Hawai’i, is under construction and planning on science operations beginning in 2020. The off-axis altitude azimuth telescope has a 4.2-m diameter $F/2$ primary mirror (4.0 m illuminated). The secondary mirror creates an $F/13$ Gregorian focus. Five more mirrors then relay this beam to a suite of polarimetric instrumentation in the Coudé laboratory.^1–3 Modulating retarders are used in each of these instruments with beams with focal ratios varying from $F/8$ to $F/62$. Many of the proposed science cases rely on high spectral resolution polarimetry. We recently adopted the Berreman calculus to model many-crystal retarders along with antireflection coatings, oils, and bonding materials, and we refer to this work as H17 here.⁴ We use the Berreman calculus along with interferrometric calculations and thermal modeling to create fringe amplitude and Mueller matrix predictions for the DKIST instruments. We show how to predict fringe properties as well as to anticipate their amplitude in converging and diverging beams during the instrument design process. With our thermal modeling, we also can assess impacts from design choices on retarder performance and temporal instabilities limiting calibrations.

DKIST uses seven mirrors to feed the beam to the rotating Coudé platform.^1,5–9 Operations involve four polarimetric instruments spanning the 380-nm to 5000-nm wavelength range. At present design, three different retarders are in fabrication for use in calibration near the Gregorian focus.^8,10,11 These calibration retarders see a beam with 300 W of optical power, a focal ratio $F/13$ with an extremely large clear aperture of 105 mm. A train of dichroic beamsplitters in the collimated Coudé path allows for rapid changing of instrument configurations. Different wavelengths can be observed simultaneously by three polarimetric instruments covering 380 to 1800 nm all using the adaptive optics system.^8,9,12,13 Another instrument [cryogenic near-infrared spectropolarimeter (Cryo-NIRSP)] can receive all wavelengths using an all-reflective beam to 5000 nm wavelength but without adaptive optics.

Complex polarization modulation and calibration strategies are required for such a multi-instrument system.^{8–10,14–16} The planned 4-m European Solar Telescope, though on-axis, will also require similar calibration considerations.^17–20 Many solar and night-time telescopes have performed polarization calibration of complex optical pathways.^21–43 We refer the reader to recent papers outlining the various capabilities of the DKIST first-light instruments.^1,3,6,8,9

Berreman⁴⁴ formulated a $4\times 4\text{-}\text{matrix}$ method that describes electromagnetic wave propagation in birefringent media. The interference of forward and backward propagating electromagnetic waves inside arbitrarily oriented stacks of biaxial material is included in this very general theory. This Berreman calculus can be used to describe wave interference in multiple birefringent layers, crystals, chiral coatings, and other complex optical configurations with many birefringent layers of arbitrary optical axis orientation. A recent textbook by McCall, Hodgkinson, and Wu (MHW) has further developed and applied the Berreman calculus.⁴⁵ In this work, we assume basic familiarity with the MHW textbook⁴⁵ and the basic thin film calculations by Abeles and Heavens matrices.⁴⁶ This formalism is in common use in coating modeling software such as TFCalc or Zemax coating reports.

We adapted the Berreman formalism to the six-crystal achromatic retarders used in DKIST along with many-layer antireflection coatings, oil layers, and cover windows.⁴ In this paper, we use the Berreman calculus and add interference effects from converging and diverging beam variation across the aperture. We then show thermal models for our retarders under absorptive loads in the 300-W Gregorian beam. With associated spectral measurements of parts per million level absorption caused by antireflection coatings and crystal bulk material, we can accurately assess the spectral absorption though these retarder optics and predict thermal performance. The appendix details the thermal modeling. The fringe temporal instability caused by thermal loading is also measured in simple laboratory experiments to verify sensitivity. We predict fringe amplitudes and thermal timescales for DKIST retarders with application to typical solar telescope heat loads on similar calibration optics.

The Berreman calculus contains all polarization phenomena and is very general.⁴⁵ We can compute nonnormal incidence interference effects through multiple birefringent layers or thick crystals as required for converging beams. The main limitation of the Berreman formalism is in the assumption of complete beam overlap using plane waves of infinite spatial extent. In the Berreman formalism for a finite sized beam at nonnormal incidence, the multiple reflections inside a thick plate will, in practice, not overlap with the incoming beam. In the limit of no beam overlap, the Jones formalism is recovered. Berreman always assumes infinite coherence lengths and that all multiple reflections stay within the optical path. For most astronomical applications, this beam overlap assumption is reasonably valid as the crystals are thin compared to the beam diameter and the backreflected footprint is within a few percent of the diameter of the incoming beam. As we show in this paper, most optical systems with beams slower than $F/5$ and retarders placed not exactly in focal planes will have amplitudes and fringe characteristics well estimated by the Berreman formalism.

In this work, we follow standard notation for propagation of polarization through an optical system. The Stokes vector is denoted as $\mathbf{S}={[I,Q,U,V]}^T$. The Mueller matrix is the $4\times 4$ matrix that transfers Stokes vectors.^47–49 Each element of the Mueller matrix is denoted as the transfer coefficient.^49,50 For instance, the coefficient [0,1] in the first row transfers $Q$ to $I$ and is denoted $QI$. The first row terms are denoted $II$, $QI$, $UI$, $VI$. The first column of the Mueller matrix elements is $II$, $IQ$, $IU$, $IV$. In this paper, we will use the notation in Eq. (1)

We also will adopt a standard astronomical convention for displaying Mueller matrices. We normalize every element by the $II$ element to remove the influence of transmission on the other matrix elements as seen in Eq. (2). 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.

2.

Equal Inclination Fringes: Fringe Dependence on AOI and $F$/

Retarders are often used in converging and diverging beams. A range of incidence angles is present across the beam footprint for these optics. We compute the expected fringe amplitudes under some simple assumptions to compare with laboratory data.

We consider the limiting case of a thin window where we can neglect the incomplete overlap between the backreflected beam and the incoming beam. In this situation, we recover a simple division of amplitude-type interferometer for fringes of equal inclination sometimes called Haidingers fringes. Detailed descriptions are in several optical textbooks, including Born and Wolf Chapter 7⁵¹ and Hariharan Chapter 2.⁵² By tracing both the first-surface reflected ray and the ray that reflects off the back surface, a trigonometric relation between the two parallel but displaced reflected rays can be created. The optical thickness of the window is computed as $o=2dn/\lambda$. The phase difference between front-surface-reflected and back-surface-reflected rays is $2\pi \mathrm{o}\cos \theta$, where $\theta$ is the propagation angle in the medium. For small incidence angles, we can use the approximation that $\theta ={\theta }_{\mathrm{air}}/n$. We get bright fringes for constructive interference when $2dn\cos \theta$ plus the halfwave of phase upon reflection gives integer-waves of path. We get destructive interference at half-wave integer multiples.

For a beam of a given $F$/ number in air, the marginal ray represents the highest incidence angle in the beam at $\theta ={\tan }^{-1}(1/2F)$. The fastest beam seen by the DKIST and the Meadowlark high-resolution spectrograph we use here has an $F/8$ beam, which sees a maximum incidence angle of 3.67 deg. The DKIST Gregorian focus at $F/13$ would see a 2.20-deg incidence angle for the marginal ray. For the calculation of fringes, we must divide by the material refractive index to get the propagation angle in the medium.

We compute a simple example of the interference pattern across the clear aperture of a fused silica window. We use the Meadowlark Optics provided Heraeus Infrasil 302 sample, as measured in H17.⁴ The thickness is measured to be 1.1335 mm with the Heidenhain metrology system, and we compute a refractive index of $n=1.46$ at a measurement wavelength around 630 nm using the vendor provided equations. The optical thickness is 5253.7 waves for the on-axis beam. For a beam traveling through the part with a marginal ray incident at 3.67 deg for the Meadowlark Spex spectrograph $F/8$ beam, the refracted ray travels at 2.45 deg incidence inside the optic ($\theta /n$). The thickness for an inclined beam is $2dn/\lambda \cos \theta$. The marginal ray traverses a part thickness of 5258.5 waves. The difference is about 4.8 waves path between on-axis chief ray and the marginal ray for the $F/8$ beam. When computing the interference path difference, we use the equal inclination fringe equation $2\pi \mathrm{o}\cos \theta$ and we get the same 4.8 waves of path length between rays. The optical path difference (OPD) between chief and marginal rays can be computed as the factor $(1/\cos \theta -1)$. Once the optical path is known, the interference amplitudes can be calculated across the footprint as the ray incidence angle changes.

Figure 1 shows the OPD in waves across a rectangular aperture for this Infrasil sample. The beam is at $F/5$ on the extreme diagonal corners of the rectangle. We choose a nominal wavelength of 630 nm and the metrologized thickness to compute nearly destructive interference at the center of the optic oscillating over many waves of optical path across the rectangular aperture. We encode waves of OPD as the gray-scale color where white is integer multiples of one wave of path difference. Black is integer multiples of zero waves of path difference. The field angle was normalized from 0 to 1 along the $X$ and $Y$ axes of the image. The inner part of the beam footprint representing an $F/20$ or slower beam is within less than one wave of interference variation across the aperture. For a beam of $F/10$ illuminating more of the part, a few waves of interference would be seen.

Fig. 1

Waves OPD across a rectangular footprint for a beam at $F/5$. Field edges at $r=1$ correspond to an internal beam propagating at an angle of 3.91 deg refracted into an index $n=1.46$ medium. For a 1.1-mm-thick fused silica window, the backreflected chief ray traverses 5253.683 waves optical path. The backreflected $F/5$ marginal ray sees 11.6 waves of additional path in addition to double the incidence angle.

The standard equation for summing interfering waves of the same frequency is $A_1^2+A_2^2+2A_1{A}_2\cos \varphi$ where the wave amplitudes are denoted $A$ and the relative phase between waves is $\varphi$. Figure 2 shows the waves of phase path difference between chief and marginal rays from the center of the clear aperture. The in-air incidence angle runs from 0 deg at the center of the optic to 4 deg, near an $F/7$ beam following Fig. 1. As this phase represents the coherent interference term, the cosine of this OPD becomes $\varphi$ in the interference equation multiplying the two root-amplitude coefficients. We can see the blue curve of Fig. 2 sees seven peaks with constructive interference and six peaks with destructive interference as the OPD changes from zero to over six waves for a beam from collimated to $F/7$.

Fig. 2

The waves OPD between chief and marginal rays across the center of the rectangular aperture from Fig. 1 is shown using the left-hand $Y$ axis. The cosine of this term is shown in blue using the right-hand $Y$ axis. This determines the magnitude of interference and multiplies the square root field amplitudes.

We next translate the interference pattern across the clear aperture to a transmitted intensity at each incidence angle. We do this using the simple interference equation where the fringe amplitude is 4 $\sqrt{I_{\text{front}}}\sqrt{I_{\text{back}}}$. For Infrasil at 630 nm wavelength, the single-surface uncoated reflection is nominally 3.5%. The backreflection from the internal Infrasil-to-air interface would have an intensity of 96.5% of 3.5%, which is 3.4%. As electric fields add coherently, we take the square root of the intensity and add the fields. If the phase is 180 deg, destructive interference reduces the transmission of the optic to 86.25%. If the phase is 0 deg then coherent interference increases the transmission of the optic to 99.9%. As the effective angle of the incident ray is increased, the optic will have the thickness vary by several waves giving multiple constructive and destructive interference peaks.

Figure 3 shows an example of the electric field interference calculation across a simulated rectangular aperture for this Infrasil sample. Figure 3(a) shows four separate wavelengths solved to have 5253 waves plus 0.5, 0.75, 0, and 0.25 waves of path for the backreflected chief ray. Using these wavelengths corresponding to integer multiples of quarter-wave optical thickness, we can show the transmission for rays as functions of incidence angle across the footprint of the beam on the optic. The integer-wave multiple wavelength sees complete constructive interference, hence 99.9% transmission for the chief ray at zero incidence angle. As the incidence angle is increased, we see the first minimum transmission of 86.25% occur around 0.8-deg incidence angle corresponding to $F/36$ or only the inner 0.16 of the aperture radius. As the incidence angle increases and the ray encounters increasingly larger path lengths, we see oscillations between maximum and minimum transmission.

Fig. 3

(b) a grayscale image of transmission of the Infrasil using a wavelength of 630 nm computed by coherently summing electric fields. The grayscale goes linearly from 86.25% to 100% transmission for a fringe amplitude of 13.75% peak-to-peak. The optical path is 5253.683 waves for the backreflected chief ray at this wavelength. The $F/5$ reflected marginal ray sees an incidence angle of 3.91 deg in the index = 1.46 medium and 5.71 deg in air. This gives an optical path increase of 11.6 waves when reflected back to the first surface in the medium. (a) Transmission functions for wavelengths near 630 nm chosen to be exactly 0, 0.25, 0.5, and 0.75 waves optical interference path thickness for the chief ray. At part center, these rays see transmission of 99.9%, 93.1%, 86.3%, and 93.1% transmission, respectively. A wavelength of 629.95 nm gives perfect constructive interference and corresponds to the black curve. A wavelength of 630.01 nm is exactly half a fringe period later and gives perfect destructive interference at the center of the optic. The green and red curves show optical paths at 0.25 and 0.75 waves interference path which provide average transmission of 93.1% at the center of the optic. As the incidence angle increases toward the edge of the beam footprint, the ray sees increasing path length and oscillations of constructive and destructive interference. Note that a 1-deg incidence angle in air corresponds to $F/28.5$.

For the two curves of Fig. 3 at multiples of quarter-wave thickness, the chief ray sees a transmission of 93.1%, which is the noninterferometric transmission computed by independently considering the 3.5% loss from the first surface and 3.4% loss from the back surface. The first minimum and/or maximum occurs at incidence angles around 0.6 deg corresponding to a normalized radius of 0.12 or a beam of $F/47$.

Figure 3(b) shows the calculation of interference fringes at the single 630 nm wavelength across the full rectangular clear aperture out to the extreme edges of the $F/5$ beam. Given these fringes, an Infrasil window at 1.13 mm thickness and 630 nm wavelength would be expected to show high fringe amplitudes only for circular beams slower than roughly $F/40$ where the part is less than half-wave interference across the beam footprint. For beams faster than $F/20$, we are spatially averaging more than a full wave of optical path interference across the converging beam footprint. We should note that our Berreman calculus scripts were used to compute the curves in Fig. 3. To assess the fringe amplitude as a function of beam focal ratio, we can easily compute the average transmission over a footprint of a given $F$/ number by doing an intensity weighted aperture average.

We compute the dependence on beam $F$/ number and wavelength by running a large simulation over a full fringe spectral period. We selected 100 $F$/ numbers between $F/6$ and $F/120$. For each of these $F$/ numbers, we choose 100 wavelengths to cover at least a full fringe period. For the Infrasil window, we selected 0.15 nm of spectral bandpass to more than fully cover the 0.12-nm spectral fringe period.

For each of these simulations, we compute the spatial interference pattern across the aperture for transmission through the part as in Fig. 3. For each of these apertures, we can select transmission within a restricted $F$/ number to create the transmission function averaged over that aperture. We repeat this aperture average for all 100 $F$/ numbers and all 100 wavelengths. In Fig. 4, we show the typical transmission spectra for $F$/ numbers from 16 to 120 in(a). We show all 100 wavelengths in Fig. 4(b) as a function of $F$/ number.

Fig. 4

(a) The transmission spectrum at each simulated $F$/ number from $F$/ 120 to $F/16$. The fringes have the expected range from 86.25% to 99.99% when the beam is nearly collimated. As the $F$/ number approaches $F/20$, the fringe amplitude approaches nearly zero and the transmission is spectrally constant around 93%. (b) The transmission as a function of beam $F$/ number. In this panel, each wavelength is a different curve showing how the influence of $F$/ number creates alternating patterns of constructive and destructive interference at any individual wavelength that oscillates about the mean as the $F$/ number increases.

When averaging over the aperture, this simple geometric model predicts the fringe amplitude will go to zero at specific beam $F$/ numbers. This effect has a simple intuitive geometric explanation. When the marginal ray sees an additional half-wave of OPD from the chief ray, we will be averaging over a spatial pattern that has equal spatial areas of constructive and destructive interference. As we are averaging the beam spatially over an aperture, this would bring the transmission to a common average value. As these parts are typically several thousand waves thickness, all wavelengths in Fig. 4 share common null points.

Each ray sees an optical thickness of $[1-\cos (\theta /n)]2dn/\lambda$ waves. We can simply solve for integer multiples as $\theta ={\cos }^{-1}(1-m2dn/\lambda )$ where $m$ is an integer (0,1,2….). The $F$/ number for this fringe null is then computed as $f=1/2\tan \theta$. As an example, the 1.13-mm-thick Infrasil window at 630 nm wavelength sees half-wave multiples for beam $F$/ numbers of (17.6, 12.4, 10.1, 8.8, 7.8, and 7.1). To compute fringe maxima, simply calculate using half-wave multiples. For a 12-mm-thick quartz optic, the $F$/ numbers of the null points are $F/55$, 39, 32, 28, etc. An immediate conclusion is that the $F/13$ beam near DKIST Gregorian focus should be sufficiently fast that fringe amplitudes in 12-mm-thick crystal retarders will be averaged over many waves of OPD. Fringe amplitudes for the faster fringe periods will be reduced by factors of few to $>20$ for the DKIST super achromatic calibration retarders (SARs) and the polychromatic modulator (PCM) optics provided the beam $F$/ number is sufficiently fast. Given the DKIST retarders range from $F/13$ to $F/62$ and cover close to four octaves of wavelength, case-by-case consideration will be required.

The amplitude of the fringes decreases with $F$/ number as the aperture average drives the transmission toward the nominal average value. For the reflected beam, the equation is somewhat simpler to write in terms of the coherently summed electric field values. We note that the reflectivity ($R$) is the square of the $E$ field and use the standard equation for summing two waves of the same frequency but different phase offset. Equation (3) shows the circular area integral over the clear aperture. This equation considers an optic of normalized aperture radius $r$ where the incidence angle relates to the $F$/ number through $\tan \theta =r/2F$ as $r$ goes from 0 to 1. We can easily imagine integrating the area in a circular aperture weighted by the transmission functions of Fig. 3.

Figure 5 shows the deviation from the nominal average transmission of 93.1% as the beam $F$/ number is changed. The peak-to-peak fringe amplitude is roughly 14%. The Infrasil window at 630 nm wavelength shows a 2.8% fringe peak-to-peak at the first maximum near $F/21$. This maximum amplitude would occur when integrating over the aperture from the center out to an integer multiple of quarter-wave interference path corresponding to constructive interference on the outer annulus of the aperture. This 2.8% fringe is reduced by a factor of five from the collimated beam 14% fringe amplitude. The second maximum is near $F/16$ with 1.6% fringe, corresponding to an amplitude reduction factor of nine when averaging over more than two waves of aperture interference. The reduction factor is roughly 22 for five waves of aperture interference.

Fig. 5

The peak-to-peak fringe amplitude about the 93.1% average transmission as a function of $F$/ number for the 1.13-mm Infrasil window at 630 nm wavelength.

The dashed black line of Fig. 5 shows the $r^{-2}$ envelope expected for fringes as the integrated area increases with beam $F$/ number and fringes successively average over multiple fringe cycles. Given the relatively simple dependence of these equal inclination fringes on optic thickness, beam $F$/ number, and wavelength, we can construct amplitude reduction factors for the various spectral fringe components in the DKIST retarders for the wide range of operating wavelengths.

3.

Laboratory Measurements: Fringes with Beam $F$/ Number

Laboratory measurements are easily done with well-characterized samples and controlled environments. We use windows and crystal retarders of known thickness, low beam deflection, and small wavefront error in beams of controlled shape to verify the fringe behavior. In the Meadowlark facility, they have a SPEX 1401 double-grating 0.85-m Czerny–Turner spectrometer. The light source is an Energetiq broad band fiber-coupled plasma source using a $200\text{-}\mu \mathrm{m}$-diameter core fiber. The fiber output is nominally collimated to an $\sim 10\text{-}\mathrm{mm}$ diameter beam by a Thor labs 90-deg 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 and some spatial dependence on sampling the fiber exit illumination. Some mild nonuniformity is seen across the beam. The system is set up to 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 fills a 0.67-mm tall SPEX slit. Given the $200\text{-}\mu \mathrm{m}$-diameter fiber and the 50/15 magnification, the range of angles across the field is $\pm 0.38\mathrm{deg}$. At visible wavelengths, the measured resolving power is $\lambda /\delta \lambda$ in the 30,000 to 45,000 range. The slit is $35\mu \mathrm{m}$ in width and over 1 mm high to pass the full magnified $200\mu \mathrm{m}$ fiber core image. The $F/5$ beam entering the spectrograph is stopped to $F/8$ beam by a rectangular aperture on the collimating mirror inside the spectrograph.

The system uses photomultiplier tubes (PMT) to cover a range of wavelengths from the UV to NIR. For our nominal integration times, the standard PMT delivers a measured statistical signal-to-noise ratio around 1000. The system noise level is dominated by systematic errors for integration times longer than 0.1 s through drifts in the baseline count levels. The baseline count rate was measured to vary by roughly 10% in 200 min with a mostly linear trend; however, some erratic behavior of the bias offset was observed. We typically complete a measurement in a few minutes with a baseline scan measured before and after.

3.1.

Infrasil Window Fringes in Collimated and $F/8$ Beams

The first sample tested is a window of 1.1335 mm thickness of Heraeus Infrasil 302. The physical thickness was measured by a Heidenheim MT 60M metrology system with $\sim 0.5\mu \mathrm{m}$ thickness accuracy. Meadowlark measured the transmitted wavefront error (TWE) at 632.8 nm wavelength to be 0.021 waves peak-to-valley over an aperture of 12 mm diameter. The beam deviation through the Infrasil was measured to be 0.26 arcsec.

This window was illuminated 10 mm beam footprint when mounted in the collimated beam ahead of the 50-mm focal-length lens. As reported in H17,⁴ the nominal data sets recover the predicted fringe period at moderate spectral sampling of 0.080 nm step per measurement. In data sets presented here, we increased the spectral sampling to cover smaller bandpass at spectral steps of 0.002 nm giving an effective sampling at $\lambda /\delta \lambda$ of about 315,000. We thus sample the 16 pm full-width half-maximum (FWHM) instrument profile with about eight points giving us hundreds to thousands of measurements over a few fringe cycles.

We tested this Infrasil 302 window mounted in a converging beam before the slit, in the diverging beam after the slit and also in the collimated beam before the focusing lens. Figure 6 shows the collimated beam fringe amplitude is 11 times larger than the fringes detected in the converging and diverging beams. The Infrasil data sheet from the manufacturer (Heraeus) gives a refractive index of 1.457 at 632.8 nm wavelength. We compute the period as $\frac{{\lambda }^2}{2dn}=0.120\mathrm{nm}$ for both collimated and $F/8$ beams. A Fourier analysis of the fringe data shows that the fringe period does not significantly change, as expected. Given that the slit is a spatial filter at the focal plane and the beam is $F/5$ before hitting the internal spectrograph collimating mirror stop, comparing these measurements allows us to rule out significant impact of the slit spatial filtering. When mounted before the slit, the footprint was about 8 mm diameter as the optic was 10 mm downstream of the focusing lens. When mounted behind the slit, the footprint was of similar size.

Fig. 6

The transmission fringes for the Infrasil 302 window. The black curve shows fringes in the collimated beam with $\pm 5\%$ amplitude. The blue and green curves show the measured fringes with the window in a converging or diverging $F/8$ beam multiplied by a factor of 11 to match amplitudes. This factor shows the amplitude decreases strongly without a fringe period change in the diverging beam. The green curve shows the Infrasil mounted before the slit while blue shows the Infrasil mounted after the slit.

The theoretical calculation gives minimum and maximum transmissions of 86.25% and 99.99% for a fringe amplitude of about 13.75% at infinite spectral resolving power. In this data set, we achieve amplitudes of roughly 10%. We used our Berreman calculus Python code to compute fringes at a spectral sampling of $\lambda /\delta \lambda$ of 500,000. We then convolved the resulting Berreman fringes with Gaussian profiles of the appropriate FWHM to simulate reduced spectral resolving power. At reduced resolving power of $R=40,000$, we see a reduction of the fringe amplitude to 10% peak-to-peak, matching our measurements. For this window, the optical thickness seen by the nominal backreflected interfering wave is $2dn/\lambda \sim 5250$ waves of path. For an $F/8$ beam, the incoming ray in air sees an incidence angle of ${\tan }^{-1}(1/2F)=3.7\mathrm{deg}$. Refracted marginal ray propagates at an angle of 2.5 deg in the medium and would see an optical thickness difference of roughly 5.0 waves.

3.2.

Quartz Crystal Retarder: Measured Fringes in Collimated and $F/8$ Beams

A quartz crystal retarder sample was measured to have $575.4\mu \mathrm{m}$ physical thickness $\pm 0.5\mu \mathrm{m}$. The retarder was oriented with fast and slow axes at 45-deg orientation to the grating rulings and mirror fold orientations. The retarder was mounted in a collimated beam as well as in the $F/8$ beam mounted ahead of the slit. The baseline scans without the sample in the beam were also recorded. The TWE for the crystal is 0.034 waves peak-to-peak at 632.8 nm wavelength over a clear aperture of 12 mm diameter. Beam deviation was measured to be 0.21 arcsec.

A Fourier analysis of the collimated and $F/8$ data set gives nearly identical periods 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 similar to H17.⁴ The fringes were derived at spectral sampling of $\delta \lambda /\lambda =\mathrm{500,000}$. The theoretical fringe spectral period for this optic was $1.8\times$ longer than the Infrasil sample. As such, the resolving power of the spectrograph has less influence on the detected fringe amplitude. We use this quartz crystal retarder for the analysis of the $F$/ number dependence.

The higher refractive index quartz crystal has a transmission ranging from 99.99% to 81.8% for a fringe amplitude around 18%. When convolving this theoretical curve with a Gaussian profile at resolving power of $R=\mathrm{40,000}$, the amplitude only decreases to about 15%. 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. 7, with fringe amplitudes rising quickly to shorter and longer wavelengths. This is very similar to our Berreman calculation.

Fig. 7

The fringes measured in the quartz retarder. Black shows measurements in the collimated beam. Blue shows measurements of the $F/8$ beam multiplied by 5 to roughly match the collimated beam fringe amplitude. See text for details.

For this crystal data set, we measured with a 0.002-nm spectral step size from 614.0 to 614.5 nm to cover two fringes but keep the measurement time to less than 2 min. This is a significantly faster measurement time than the data sets on the Infrasil window in Fig. 6. The exact value of the baseline scan was somewhat more erratic for this data set even though the measurement time was significantly shorter. Baseline values changed at amplitudes up to several percent even for immediately repeated measurements without any perturbation of the system. The spectral shape of the baseline was much more stable. Given this uncertainty, we determined fringe amplitudes and phases by fitting sinusoidal functions allowing for a constant offset.

An example of a data set comparing collimated to diverging fringes on our crystal quartz sample is seen in Fig. 8. The curves have been fit by sinusoidal functions then normalized and centered using the fit parameters. The red curve shows the data with the crystal quartz retarder in the collimated beam. The fit fringe amplitude was 16.1% peak-to-peak. Blue shows the data with the crystal retarder in the diverging beam with a 1.9% peak-to-peak fringe amplitude. The ratio of fringe amplitudes computed using the sin-fit parameters is 8.4. Significantly more statistical noise is seen in the blue curve of Fig. 8 as the curves are normalized by the fit fringe amplitude. The periods are essentially identical again showing that converging beams do not impact the period calculation. We see excellent agreement in both curves in Fig. 8. We can conclude the SPEX setup is sufficient to measure fringe period, amplitude, and to assess impact of diverging beams on fringe properties. Some slight differences are seen between the Infrasil window fringe data set of Fig. 6 and the thinner quartz crystal data set of Fig. 8 after normalization by a sin-fit to data with a larger fringe amplitude. Differences arise from the increased spectral sampling, decreased measurement time, higher cadence of baseline scans, and the $1.8\times$ slower spectral fringe period of the crystal sample.

Fig. 8

The fringes measured in the quartz crystal retarder normalized and adjusted with sinusoidal fit parameters. Black shows the sine function fits. Blue shows the data with the crystal in the diverging $F/8$ beam after the slit. Red shows the data with the crystal in the collimated beam. As all curves overlap and are difficult to distinguish, we consider the sin function fits successful. See text for details.

Figure 9 shows the deviation from the nominal average transmission as the beam $F$/ number is changed. The theoretical peak-to-peak fringe amplitude is roughly 18%. The quartz retarder at 614 nm wavelength shows a $\pm 1.7\%$ fringe at the first maximum near $F/16$. This peak would occur when integrating over the aperture from the center out to an integer multiple of quarter-wave interference path corresponding to constructive interference on the outer annulus of the aperture. This 3.5% fringe is reduced by a factor of five from the collimated beam 18% fringe amplitude. The second maximum is near $F/12$ with $\pm 1\%$ fringe, corresponding to an amplitude reduction factor of nine when averaging over more than two waves of aperture interference.

Fig. 9

The peak-to-peak fringe amplitude about the average transmission as a function of $F$/ number for the 0.5745 mm crystal quartz retarder at 614 nm wavelength. The $r^{-2}$ amplitude envelope begins around half peak fringe amplitude.

We were suspicious that the residual angular divergence in the input light source may have been reducing spatial fringe amplitudes. As a further test of the SPEX system, we changed the fiber and feed optics to ensure that any small angular divergence from our light source did not impact measured fringe amplitudes or periods. We changed from a $200\text{-}\mu \mathrm{m}$-diameter fiber to a $50\text{-}\mu \mathrm{m}$-diameter fiber. With the 50-mm focal-length collimator and four times smaller fiber diameter, the field divergence in the beam decreased by a factor of 13.3. The field divergence in the beam is now $\pm 0.03\mathrm{deg}$ corresponding to the outer diameter of the fiber. When this $50\text{-}\mu \mathrm{m}$ core fiber is used, the crystal quartz measurement has a 16.5% fringe amplitude peak-to-peak. This represents a slight increase from the 16.1% amplitude found with the $200\text{-}\mu \mathrm{m}$ core fiber and shows that the impact of light source field divergence. In the diverging beam, the fringe amplitude significantly increased to 3.7% peak-to-peak. Changing the light source roughly doubled the amplitude of the fringes detected compared to using the $200\text{-}\mu \mathrm{m}$ core diameter fiber.

We performed a simple experiment to manually change the system $F$/ number by closing the iris in the collimated beam ahead of the lens focusing the beam on the spectrograph entrance slit. The nominal setting with the iris open gives a 100 mm round beam on the spectrograph collimating mirror. The full collimating mirror aperture is a 110-mm square that is fully illuminated without the iris. We would sequentially close the iris and manually measure the beam diameter on the collimating mirror mask.

Fringe measurements were performed with the crystal retarder mounted both in the collimated beam and again in the diverging beam. Occasional repeated measurements were performed with slight adjustments to the optical alignment to verify that the detected fringe amplitude was not sensitive to optical alignment or system stability.

Figure 10 shows the compiled results of this data set. Blue shows the fringe amplitudes detected with the crystal retarder mounted in the collimated beam. Fringe amplitudes remained near the nominal 16.5% fringe amplitude to within a small fraction of a percent as the iris was closed, and the beam diameter reduced from 100 to 40 mm. The 16.5% fringe amplitude is detected in all cases, showing the system resolution and optical alignment is stable upon changing the beam diameter with the iris. The black curve shows the fringe amplitudes detected when the retarder was mounted in the diverging beam. The bottom $X$ axis shows the manually measured beam diameter, and the top $X$ axis shows the corresponding spectrograph beam $F$/ number. Diameters ranged from 40 mm for the slowest beam of $F/20$ to 110 mm for the fastest beam of $F/8$. The collimating mirror aperture is a square at 110 mm per side. The iris was set to 100 mm at the widest, corresponding to roughly $F/9$.

Fig. 10

The fringe amplitudes measured in the quartz crystal retarder as a function of $F$/ number. Black shows the fringe amplitudes with the crystal quartz retarder mounted in the diverging beam. Dashed black shows an $r^{-2}$ law scaled to 16.25% fringe at 38 mm beam diameter. Blue shows the fringe amplitude in the same optical setup but with the crystal quartz retarder moved to the collimated beam before the slit. Blue shows the change in beam diameter did not degrade fringe amplitude in the collimated beam. The left-hand $Y$ axis shows the detected fringe amplitude in percent. The fringe amplitude reduction between blue and black curves as seen in the right-hand $Y$ axis. The bottom $X$ axis shows the manually measured beam diameter on the collimating mirror. The top $X$ axis shows a conversion of that beam diameter to $F$/ number within the spectrograph. See text for details.

The right-hand $Y$ axis of Fig. 10 shows how the fringes were reduced in amplitude at the corresponding $F$/ number and beam diameter. For instance, with the beam at 100 mm diameter and the system at $F/9$, the fringes were roughly 3.5% amplitude, which is only 20% of the nominal 16.5% fringe amplitude collimated. For the smallest beam diameter of 40 mm near $F/20$, the fringes detected with the crystal in both collimated and diverging beams are nearly the same with an amplitude of 80% that of the collimated case. We note that the $F/9$ limit here derived a fringe reduction factor of roughly five while the experiments above without an iris in the older setup derived a factor of 8.4. Given the potential issues with spectrograph alignment and manual optical positioning, this magnitude of uncertainty may be expected in our simple experiments.

At 614 nm wavelength, this $574.5\text{-}\mu \mathrm{m}$-thick quartz crystal has a refractive index of $n=1.552$ and the backreflected chief ray sees 2905 waves of optical path. At $F/10$, the marginal ray is traveling at an incidence angle of 2.9 deg and will see 1.5 waves of additional optical path compared to the chief ray. At $F/17.4$, the marginal ray would see exactly half a wave of path difference and destructive interference. This also corresponds to the $F$/ number where the measured fringe amplitude drops significantly in Fig. 10. Our system uses a multimode fiber-coupled plasma light source with imperfect coherence and mild filed divergence. We do not expect to match the fully coherent predictions for Haidingers fringes with multiple oscillations of constructive and destructive interference. However, we do recover the significant reduction by a factor of four in detected fringe amplitude when more than half a wave of path variation is seen across a beam footprint. These results were consistent and repeatable with low sensitivity to manual optical alignment. From the simple $r^{-2}$ behavior of Fig. 9, we do expect to see a break in the fringe amplitude curve around $F/30$. With our setup, this rapid reduction in fringe amplitude occurs closer to $F/20$. Given the alignment and potential issues with the double-grating spectrograph, we consider this agreement within the uncertainty of our simple manual experiments. We show additional details of the Meadowlark SPEX system in Appendix C.

4.

DKIST Retarders: Amplitude versus Beam $F$/ Number Predictions

With this simple $r^{-2}$ envelope for predicting fringe amplitudes as a function of beam $F$/ number, we can make simple predictions for the fringe amplitudes caused by the various DKIST retarder optics. We note that the various beamsplitters and dichroics part of the Adaptive Optics system (WFS-BS1) and the Facility Instrument Distribution Optics (FIDO) are both wedged. As such, there will be thousands of fringe periods spatially averaged across the clear aperture. We focus this paper on the crystal retarders, which are strictly plane parallel optics mounted in converging beams.

In Table 1, we show the optical thickness of the various crystal retarder interference paths. We also compute the associated physical thickness difference in waves [$2dn/\lambda \cos (\theta /n)$] seen by the marginal ray. For the visible spectropolarimeter (ViSP) instrument, the calibration retarder is crystal quartz working in an $F/13$ beam while the modulator is at $F/26$. The marginal ray is at an incidence angle of 2.20 deg in air for $F/13$ and at 1.10 deg for the $F/26$ beam. For the diffraction limited near-infrared spectropolarimeter (DL-NIRSP) at $F/8$, the marginal ray is at a 3.58-deg incidence angle while at 0.46 deg for the $F/62$ beam.

Table 1

Beam properties versus crystal thicknesses.

The number of waves path difference for the marginal ray is a proxy for the number of interference cycles across the clear aperture of the illuminated optic. The wavelength and $F$/ number dominate the behavior with some slight dependence on refractive index variation with wavelength. Using the standard formula from CVI, we get refractive indices of 1.568 at a wavelength of 393 nm falling to 1.546 at a wavelength of 854 nm and 1.536 at 1565 nm wavelength. For the ${\mathrm{MgF}}_2$ retarder, we derive a refractive index of 1.3596 at wavelength 3934 nm intended for Cryo-NIRSP observations of the Si IX spectral line. The modulating retarder for Cryo-NIRSP is in an $F/18$ converging beam mounted upstream of the spectrograph entrance slit. Table 1 shows the shortest period spectral fringe will see roughly two waves of interference over the converging beam aperture. The longest period fringe corresponding to a single crystal however will only see a small fraction of a wave. Thankfully, the individual crystals are at refractive index 1.35, and the oil layers between crystals have an index of 1.3, significantly reducing this spectral component of the fringe.

Simulations for the fringe amplitude at specific wavelengths can be simply computed using the Berreman scripts or the equal inclination fringe equations for isotropic materials. An aperture integral converts the predicted intensity for each incidence angle to the total transmission for a footprint. For this simple example, we do not compute the fully birefringent fringe spatially across a retarder crystal. However, this is straightforward in the Berreman formalism.

Figure 11 shows examples of how the fringe amplitude depends on wavelength, $F$/ number, and thickness. The deviation from the average unfringed calculation is shown in $\pm$ transmission. Colors show different wavelengths and crystal thicknesses. Solid lines show the spectral fringe. Dashed lines show the $r^{-2}$ fringe amplitude decrease behavior. The blue curves show a 2-mm thickness of an isotropic material of index 1.55 corresponding to crystal quartz at 396 nm wavelength corresponding to the shortest wavelength for the ViSP instrument. The ViSP modulator sees a diverging $F/26$ beam and should see fringe amplitudes less than $\pm 1\%$ as compared to the over $\pm 8\%$ collimated fringe. The air–crystal interface through the six-crystal optic produces the largest amplitude fringes in the Berreman calculus, but the aperture average would reduce this spectral component to $\pm 0.15\%$ per the dashed blue line.

Fig. 11

The $\pm$ transmission fringe amplitudes as functions of beam $F$/ number for uncoated crystals. Blue shows 2 mm thickness of crystal quartz at a wavelength of 396 nm. Black shows quartz at a wavelength of 1565 nm. Red shows the full 12-mm thick six-crystal stack at 630 nm wavelength. The $r^{-2}$ envelopes are scaled for each curve.

Black shows quartz but at a wavelength of 1565 nm appropriate for the science wavelength DL-NIRSP instrument camera arm. For the $F/62$ configuration, the beam is essentially collimated with minimal fringe amplitude reduction and peak amplitudes above $\pm 8\%$. The $F/24$ configuration would reduce the fringe magnitudes to below $\pm 3\%$.

Red shows the full 12-mm thickness of the six-crystal stack at a wavelength of 630 nm as intended for ViSP, visible tunable filter (VTF), and DL-NIRSP instruments. The full crystal stack is six times thicker. The fringes corresponding to this interface see significant reduction of amplitude. The calibration and modulation retarders all would see $\pm 0.3\%$ instead of $\pm 8\%$, a reduction of roughly 27 times. Antireflection coatings further reduce the fringe amplitude.

The results show that simple scaling relations apply. From beams of $F/28$ to $F/10$, the peak fringe amplitudes decrease by roughly a factor of three. As seen in Table 1, most optics in the converging beams see a few to several waves of optical path variation across the aperture. This gives fringe amplitude reductions that follow the linear trends of Fig. 11. In the six-crystal modulator, however, this amplitude prediction is modified by the oil at refractive index 1.3 reducing the magnitude of the back reflection as in H17.⁴

We compile a rough estimate of the fringe amplitude reduction using the $r^{-2}$ envelope in Table 2. The left column shows the OPD between chief and marginal rays. The right column shows the rough estimate of the fringe reduction factor for a single window or crystal. These rough estimates are simply rounded to factors of two where the circular clear aperture averages of Fig. 11 would have maxima. The table shows that a few waves of marginal ray OPD compared to the chief ray can reduce fringes by close to one order-of-magnitude. However, getting two orders of magnitude fringe reduction to levels below DKIST sensitivity would require at least a few tens of waves and unrealistically thick optics. We note that this rough estimate is not rigorously applicable to many-crystal optics as the beam overlap, incidence angles, and phase relationship between all the many internal reflections is not considered.

Table 2

OPD.

To synthesize these results for easy application to retarder design and DKIST optical configurations, we compute fringe properties as functions of beam $F$/ number and wavelength. Figure 12 shows the optical path as a function of wavelength for the DKIST retarders. Figure 12(b) shows the difference between chief and marginal ray optical paths for a few $F$/ numbers used for the DKIST retarders. For the six-crystal retarders at shorter DKIST wavelengths, the backreflected optical path is close to 20,000 waves for the 2 mm crystal and over 100,000 waves for the entire crystal stack. However, when assessing amplitudes of which fringe spectral components at which $F$/ number and wavelength, we need many waves of interference over the aperture to significantly reduce the measured fringe amplitude.

Fig. 12

(a) The optical path seen by the interfering beam ($2dn/\lambda$) through the DKIST retarder crystals as a function of wavelength. (b) The path difference between the chief and marginal rays when propagating through the DKIST retarders at varying $F$/ number. See text for details.

Figure 12(b) shows that significant amplitude reduction is expected only the shortest wavelengths and shortest period spectral components for beams faster than $F/30$. Thus, for DKIST, we expect to see a higher relative amplitude for the longer period spectral fringe components in steeper converging beams. This fringe period-dependent amplitude reduction factor can now be coupled with the Berreman predictions from H17⁴ using antireflection coatings to assess what fringe components will be present for the various DKIST instruments at specific wavelengths with a calibration retarder at $F/13$ and modulating retarders from $F/8$ to $F/62$.

5.

Fringe Thermal Stability: a Large Source of Error

Temperature sensitivity is a major concern for the stability of a retarder. Temporal instability often is the ultimate calibration limitation and DKIST expects the 300-W beam to impose strong usage constraints. With the Berreman calculus and simple analytical calculations, we can show how fringes and retarder optical properties depend on thermal effects. There are three main factors. First is physical expansion (through the coefficient of thermal expansion, CTE, $\alpha$). Second is the change in refractive index with temperature through the thermo-optic coefficient ($dn/dT$, TOC). Third is the dependence of crystal birefringence on temperature [$d(n1-n2)/dT$]. All three parameters cause the fringes and retarder properties to change during typical system operation.

The polarized fringes from the calibration retarders are strongly temperature dependent. As we have seen in H17,⁴ fringes in these many-crystal retarders cause variation in all three retardance properties: linear retardance magnitude, linear retardance fast axis orientation, and circular retardance (ellipticity). To assess the calibration limitations, we not only need the amplitude predictions for all spectral components but also the stability.

As a demonstration, we collected laboratory measurements as functions of temperature for quartz and ${\mathrm{MgF}}_2$ crystal as used in the DKIST retarders. In the Meadowlark Spex system, an enclosure with a heater was created for the crystal retarder sample. This system was set to heat by roughly 10°C over several hours. Small entrance and exit ports were cut in the enclosure to allow the 10-mm-diameter beam to pass unobstructed through the enclosure. In addition to a heater, a temperature sensor was coupled to the optic mount as a direct reading of the optic temperature. Given the slow heating rate and high crystal conductivity, we assume this temperature proxy is accurate enough for the purposes of demonstrating fringe drift with temperature.

Figure 13 shows the resulting high spectral resolution scans as functions of temperature for the crystals. In both cases, the fringe pattern moved very roughly about half a wave period during the 10 to 12°C of heating. Given that the samples are a fraction of a millimeter thick, the order-of-magnitude expected for the fringe drift is a fraction of a period per cm of optical path per °C of heating. In subsequent sections, we go through each physical effect.

Fig. 13

The transmission fringes through the sample optic as the temperature is raised by over 10°C. (a) The quartz retarder and (b) the ${\mathrm{MgF}}_2$ retarder.

We also note that we repeated this heating experiment for the Infrasil window sample. Instead of scanning fringes in wavelength, we monitored a single wavelength at cadences faster than 1 Hz. Similar behavior was recorded and no high-frequency errors were detected. Fringe drift temperature scales were similar. We focus on the crystal retarders here.

5.3.

Birefringence Variation with Temperature

The birefringence of a crystal optic is also a function of temperature.⁵⁸ The extraordinary and ordinary rays do not see the same refractive index change with temperature, creating a differential effect. In DKIST designs, this effect was incorporated to address concern for the temperature sensitivity of birefringence impacting the basic retarder design.^8,58 In addition, Sueoka found that the refractive index and birefringence models available in the literature did not adequately address the birefringence at longer wavelengths.¹¹ The uncertainty was significant enough to require DKIST to perform an independent assessment and to adapt our designs accordingly.

The CVI Melles-Griot Materials Handbook entry for crystal quartz gives both ordinary and extraordinary refractive indices in terms of a Laurent series equation following Eq. (4). In Sueoka,¹¹ the Handbook of Optics from the Optical Society of America (OSA) provides a five-term Sellmeier equation following the style of Eq. (5). Sueoka modified the ordinary refractive index following the equation plus the measured birefringence as a way to correct the equations for accurate birefringence predictions as required in the DKIST application.¹¹ Table 4 shows the coefficients for each refractive index equation.

In Fig. 14, we show the difference between CVI Handbook and OSA Handbook refractive indices as blue and black curves. The two equations diverge at $\sim 100\mathrm{ppm}$ amplitudes as well as diverge from each other by $\sim 100\mathrm{ppm}$ at wavelengths longer than 1000 nm. The modified birefringence difference is shown as the red curves using the red right-hand $Y$ axis. The birefringence only differs at levels of less than 15 parts per million, but this is enough to have impacted the modulation efficiency for retarders in the DKIST designs.¹¹ Included in the Sueoka¹¹ analysis is physical expansion and measurements of temperature perturbation from the TOC. The birefringence is predicted to change at amplitudes of a few parts per million when temperature is changed by 10°C as seen by the various red curves of Fig. 14.

Eq. (4)

$${\mathrm{n}}^2=A_0+A_1{\lambda }^2+\frac{A_2}{{\lambda }^2}+\frac{A_3}{{\lambda }^4}+\frac{A_4}{{\lambda }^6}+\frac{A_5}{{\lambda }^8},$$

Eq. (5)

$${\mathrm{n}}^2=1+\frac{B_1{\lambda }^2}{{\lambda }^2-C_1}+\frac{B_2{\lambda }^2}{{\lambda }^2-C_2}+\frac{B_3{\lambda }^2}{{\lambda }^2-C_3}+\frac{B_4{\lambda }^2}{{\lambda }^2-C_4}+\frac{B_5{\lambda }^2}{{\lambda }^2-C_5}.$$

Fig. 14

The refractive index difference between NSO and CVI catalog values for crystal quartz are shown in blue and black at amplitudes around 250 parts per million. The three red curves show the difference in birefringence between CVI and the thermally perturbed NSO equations. Values are shown in red on the right-hand $Y$ axis and are differences at the level of 10 parts per million.

Several have reported on the temperature coefficients of quartz and crystal ${\mathrm{MgF}}_2$ and athermalization of retarder designs.^56,59,60 Bicrystalline achromats can be constructed of positive and negative crystals to become thermally compensated at a single wavelength. By keeping crystal thickness ratios similar, athermal retarder designs can be created.^8,59 The various Pancharatnam style designs⁶¹ that have thin crystal components have lower thermal sensitivity than thicker many-order retarders. For multiwavelength designs, typically you cannot exactly solve both for a retardance at multiple wavelengths as well as athermal performance. However, you can balance thermal behavior against requirements on retardance, plate thickness, wavefront error, alignment tolerances, etc. to decrease the sensitivity to various effects.

Table 4

Refractive index coefficients for CVI Laurent and NSO Sellmeier.

CVI extraord.	2.38490e+00	−1.25900e−02	1.07900e−02	1.65180e−04	−1.94741e−06	9.36476e−08
CVI ordinary	$2.35728\mathrm{e}+00$	$-1.17000\mathrm{e}-02$	$1.05400\mathrm{e}-02$	$1.34143\mathrm{e}-04$	$-4.45368\mathrm{e}-07$	$5.92362\mathrm{e}-08$
NSO extraord. $B$	0.74700637	0.45865921	0.17833250	0.73225069	8.7421747
NSO extraord. $C$	0.063458831	0.11266214	0.11288341	9.1190338	54.983117
NSO ordinary $B$	0.663044	0.517852	0.175912	0.565380	1.675299
NSO ordinary $C$	0.060	0.106	0.119	8.844	20.742

The coefficients for the CVI six-term Laurent series of Eq. (4) for both extraordinary and ordinary beams are in the first two rows. The B and C coefficients of the five-term Sellmeier equations (10 coefficients each) for the ordinary and extraordinary beams of the NSO-modified fit are shown as the last four rows.

The DKIST retarder designs used coefficients for birefringence changes near ${10}^{-4}$ per °C for quartz and $5\times {10}^{-5}$ per °C for ${\mathrm{MgF}}_2$ crystals. These numbers agree with the Handbook of TOCs⁶² and are similar to other athermal designs.^56,59,60 Thus, the birefringence changes are the same order of magnitude as the refractive index changes.

5.4.

Fringe Thermal Sensitivity and Impact on Retarder Use Cases

Translating the fringe temperature dependence into a specific quantifiable impact on the calibration or observation process depends on many estimates of materials properties, heat loads, and calibration strategies. The order-of-magnitude estimates presented above show that the fringes are expected to change for the DKIST quartz and ${\mathrm{MgF}}_2$ retarders. We apply a simple linear thermal perturbation analysis using our Berreman calculus in H17⁴ to show the expected magnitude and character of thermal perturbations for the DKIST retarders.

We fit a sinusoidal function to the fringes of Fig. 13. For the single-crystal quartz at 0.5745 mm thickness and an observed wavelength of 625 nm, we found roughly a quarter-wave of fringe drift in 10.9 per °C of heating. The fringe period is computed as ${\lambda }^2/2dn$, which gives a spectral fringe period of 0.219 nm. The fit periods were at 99.3% of this value for the 24.5°C data set and 98.7% of the nominal period for the 35.4°C data set. The offset between fringes was computed at 83.7 deg phase or roughly 0.233 waves drift of the fringe.

Eq. (6)

$$\mathrm{OP}=\frac{d(1+\alpha \mathrm{\Delta }T)n(1+\mathrm{TOC}\mathrm{\Delta }T)}{\lambda }.$$

The Crystran Handbook of Optical Materials gives CTE values for crystal quartz as $\alpha =7.1\mathrm{e}\text{-}6\mathrm{per}°\mathrm{C}$ for the extraordinary beam and $\alpha =13.2\mathrm{e}\text{-}6\mathrm{per}°\mathrm{C}$ for the ordinary beam. The TOCs were much more similar in the Crystran Handbook with the $\mathrm{TOC}=-5.5\mathrm{e}\text{-}6\mathrm{per}°\mathrm{C}$ for the extraordinary beam and $6.5\mathrm{e}\text{-}6\mathrm{per}°\mathrm{C}$ for the ordinary beam. With these handbook values, we compute a simple perturbation of the thickness as $d(1+\alpha )$ and the refractive index changes to $n(1+\mathrm{TOC})$.

The perturbed optical path is computed as $dn/\lambda$ using the perturbations in thickness and refractive index linearly in the temperature change $\mathrm{\Delta }T$ shown in Eq. (6). In the Berreman calculus, we input the three-dimensional refractive index data, crystal orientation, and thickness. Each value is perturbed separately in the Berreman formalism. Note that for the 10°C change in the Meadowlark lab, the 0.5-mm quartz sample only expanded by 76 nm and the refractive index changed by 65 parts per million.

We use this simple linear thermal perturbation in our Berreman calculus fringe models to verify our calculations match laboratory data for thermal drifts of the crystals. Figure 15 shows such a calculation for this 0.57-mm-thick crystal around the 625-nm wavelength used in the Meadowlark Optics test setup.

Fig. 15

Our Berreman model using a thermal perturbation of 10°C following Eq. (6). (a) A narrow wavelength range of 400 pm matching the Meadowlark Spex bandpass of Fig. 13. We perform a sin function fit and find a phase offset of 0.21 waves or 74.5 deg when using the Crystran Handbook values. (b) A larger bandpass illustrating how the differential temperature sensitivity of extraordinary and ordinary beams causes a wavelength drift in the destructive interference.

Using this simple perturbation with Crystran Handbook values, our Berreman model gives a fringe phase shift of 74.5 deg or 0.207 waves of fringe thermal drift. This is very similar to the laboratory measured value of 83.7 deg and 0.233 waves drift using the Meadowlark Spex system. With this simple linear perturbation, we can easily compute the expected form and magnitude of thermal sensitivity in the DKIST and Keck Low Resolution Imaging Spectrograph with polarimetry (LRISp) six-crystal retarders.

6.

Application to the DKIST Retarder in DL-NIRSP AT F/62

We apply both the Berreman fringe amplitude estimates as well as the thermal perturbation analysis to a DKIST instrument and the six-crystal modulator. In the DL-NIRSP instrument, the DKIST project will install a six-crystal PCM, which includes antireflection coatings and oil layers between the crystals. The modulating retarder sees beams at either $F/24$ or $F/62$ depending on the configuration of the feed optics. We assess the worst-case $F/62$ beam for fringe amplitude and thermal stability. Our assessment above shows that the $F/62$ beam is essentially collimated and fringe amplitudes will not be reduced by the mild convergence of the beam. In addition, the DKIST Coudé laboratory is only stabilized to $\pm 1°\mathrm{C}$ plus possible thermal instability caused by imperfect temperature control on the rotation stage motors driving the crystal modulator.

We use the linear perturbation of crystal thickness and refractive index in Eq. (6) to modify the refractive indices and physical thickness for every layer in the six-crystal retarder design. The DL-NIRSP instrument mounts the modulator (PCM) just ahead of the focal plane formed on the fiber-bundle integral field unit input to the spectrograph. The DL-NIRSP has two infrared camera channels, which we will consider here. One channel is nominally used to observe two common solar lines at 1075 and 1083 nm wavelength. The second channel is optimized for two lines at 1430 and 1565 nm.

At these wavelengths, the nominal resolving power of the instrument is over 100,000. This instrument plans to spectrally sample the beam with a variety of user-selected modes. For this discussion, we assume sampling is in the range of several picometers as designed to fully sample the instrument profile at these wavelengths. This modulator includes the refractive index $\sim 1.3$ oil between crystal interfaces.

As discussed in H17, we removed the 10-mm-thick Infrasil cover windows from the optics. The design now only has six crystals, five oil layers, and antireflection coatings on each crystal surface. Each coating is modeled as an isotropic ${\mathrm{MgF}}_2$ layer designed as a quarter-wave of path at a central wavelength of 1300 nm. With a refractive index of about 1.38 at this wavelength, we compute 236-nm physical thickness for the coating. Table 5 shows the Berreman stack of birefringent materials used in the model. As described later, the modulators are designed as elliptical retarders that deliver efficient modulation over wide wavelength bandpasses. For the DL-NIRSP, we achieve sufficient efficiency from 500 to 2500 nm when using six-quartz crystals with the thicknesses and orientations specified in Table 5.

Table 5

DL PCM retarder.

In Fig. 16, we show the nominal Berreman model in black along with a 1°C thermally perturbed Berreman model in blue. We used a wavelength grid for the model at a constant spectral sampling of $\lambda /\delta \lambda$ of 500,000. The spectral resolving power was infinite and no simulation of instrument profile resolution degradation was applied. We also applied a thermal perturbation to the oil layer with an assumed CTE value of $\alpha ={10}^{-4}$ consistent with other oils. We do not have any data on the $dn/dT$ value for the oil.

Fig. 16

The Berreman model for the DL-NIRSP modulator with a bandpass of 1083.0 to 1083.4 nm covering a wavelength range of 400 pm. Computations were done at spectral sampling of 500,000 and no degradation of resolving power. We used a thermal perturbation of 1°C following the temperature stability specification for the DKIST Coudé laboratory and actively cooled rotation stages. Black shows the nominal model while blue shows an increase of 1°C and perturbations to the optical path following Eq. (6).

The dominant effect of the thermal perturbation is a shift of the entire pattern in wavelength by about 7.5 pm. This is roughly the spectral sampling for the DL-NIRSP. Given the temperature stability of the optic, it is possible that the fringe pattern could be stable at levels around a few resolution elements. We can easily fit simple optical models to the thermally perturbed Mueller matrix. In typical solar demodulation schemes, intensity fringes caused by transmission and/or diattenuation can be postfacto filtered in various ways. These techniques have various consequences for the fidelity of the derived solar signals when the fringes and real signals are similar.

We can use the Berreman calculus to highlight the impact of oil layers, bonding epoxies, coatings, and other materials between the crystals. As an example of the oil layer impact, we ran a grid of models where the oil layer thickness was either 7, 10, or $13\mu \mathrm{m}$ for each layer computed against each other layer. Given the five oil layers and three possible thicknesses, we computed 243 separate Berreman models. Diattenuation is dominated by Stokes $U$ at this particular DL-NIRSP wavelength with amplitudes up to 15% peak-to-peak. Figure 16 shows the $IQ$ and $IV$ terms are of order $\pm 1\%$. The elliptical retardance fringes vary spectrally by over 6 deg peak-to-peak.

The Fourier analysis shows that every layer gives rise to a fringe component at the appropriate spectral period. However, the interplay between the relatively thin layers and the relatively thick crystals creates amplitude variation and also changes a much lower period amplitude envelope for the fringes. Simply changing the isotropic oil layer thickness by a few microns can strongly vary peak fringe amplitudes. As an example of the relatively slow spectral variation caused by the oil layers in the DKIST designs, we show two Mueller matrices in Fig. 17. The black curve shows the nominal $10\text{-}\mu \mathrm{m}$ layer thickness while blue shows uniform $7\mu \mathrm{m}$ layer thickness. The Fourier analysis of any narrow spectral bandpass used by a DKIST instrument would be nearly the same, but there is a spectrally slow amplitude modulation that changes strongly with varying oil layer thickness.

Fig. 17

The Mueller matrix following Sec. 2 for the DL-NIRSP modulator around 1083 nm wavelength. The 1080 to 1090 nm bandpass at left, a narrow 1083.0 to 1083.5 nm bandpass at right.

Figure 17 only covers a 10-nm wide spectral bandpass but the transmission fringes change by over 8%, diattenuation can double and elliptical retardance change by degrees at specific narrow wavelengths used by solar spectropolarimeters. We have done several spectrophotometric tests to determine oil layer thickness between various materials. Values range from $5\mu \mathrm{m}$ to over $20\mu \mathrm{m}$. The detailed fringe spectra of each individual DKIST retarder will no doubt require testing at the highest spectral resolving powers for each specific wavelength planned.

7.

Summary: Predicting Retarder Fringe Amplitudes and Temporal Stability in Converging Beams with Thermal Loads

Polarization fringes are a major calibration limitation in astronomical spectropolarimeters. Designing systems with reduced fringe amplitudes and benign behavior is a challenge for modern large instrumentation. Calibration of DKIST instruments demands stringent temporal stability requirements as well as minimization of optical sensitivities to thermal changes. The temporal stability of optical components must be assured for DKIST in the presence of thermal loads from a 300-W beam and operations in the mountain summit environmental conditions. A systems-engineering level assessment of DKIST calibration processes requires these new tools for predicting polarization fringe amplitudes and their temporal behavior in converging and diverging beams. We showed simple calculations of Haidingers fringes (fringes of equal inclination) over a converging beam footprint to show fringe amplitude reduction dependence on beam $F$/ number. This combined with the Berreman formalism presents a tool to estimate full Mueller matrix and fringe behavior under design, thermal, and manufacturing perturbations. The fringe amplitude is subsequently reduced by the averaging over many waves of spatial fringes in converging or diverging beams, but the underlying fringe spectral periods remain unchanged. We verified the $r^{-2}$ fringe amplitude scaling relation with laboratory data on crystal retarder and window samples.

For the DKIST six-crystal retarders, the highest amplitude fringes from the air–crystal interfaces see the greatest reduction of amplitude in the converging beam as the marginal ray sees significantly more optical path upon backreflection through the entire crystal stack. The amplitude of polarization fringes can be significantly reduced by placing the retarder in a steeply converging beam in addition to using antireflection coatings as was done for the DKIST calibration retarders and certain instrument modulator configurations. This fringe amplitude reduction benefit in converging beams must be traded against effects of spatial nonuniformity, depolarization (as outlined in Sueoka¹⁰), and exacerbated thermal issues. The temporal stability of the fringes was assessed for DKIST by including physical expansion, the TOC, and the birefringence variation with temperature under heat load in Berreman models. These thermal sensitivities were also demonstrated for crystal retarders and windows in the lab with a high-resolution spectrograph.

These issues are common to any precision astronomical high-resolution spectropolarimeter. We included in Appendix D an on-sky demonstration. We showed fringe amplitude estimates and Berreman models for the six-crystal achromatic retarder used in the Keck 10 m diameter telescope and LRISp spectropolarimeter on Maunakea. This retarder is an excellent comparison case for DKIST and other astronomical systems as both use $F/13$ beams and a six-crystal achromatic retarder design. The fringes in LRISp are detected at amplitudes of a small fraction of a percent with thermal evolution over a night in outdoor conditions as reported in H15. This small amplitude is consistent with the Berreman predictions presented here after accounting for $F$/ number and low spectral resolving power. Berreman predicts large fringe amplitudes for a collimated beam and substantial dependencies on cement layer thickness and refractive index. We predict and detect significant reduction when convolving with low-resolution instrument profiles and averaging over the aperture in the $F/13$ converging beam. This on-sky demonstration of fringe properties validates the aperture-average in a converging beam as well as thermal perturbation when combined with the Berreman calculus.

With the design tools presented here, the DKIST team was able to assess fringe behavior for optics in varying $F$/ number beams. This formalism was also used to reassess the optical design with cover windows and to assess the temporal instabilities for the retardance as well as polarization fringe evolution. Considering the thermal protection provided by the calibration polarizer as well as an additional window mounted separately with the polarizer, predicted thermal loads are reduced by an order-of-magnitude and keep steady-state temperatures within 1°C of ambient. We show detailed thermal analysis of our retarders under various beam configurations in Appendices A and B.

This polarization fringe amplitude calculation was also used to predict the various fringe spectral component amplitudes for the DKIST modulating retarders, which work in beams from $F/8$ to $F/62$ and wavelengths from 380 to 5000 nm. We showed an example calculation for the DL-NIRSP instrument modulator in the $F/62$ configuration. The fringe amplitude $r^{-2}$ envelope calculation shows no significant fringe amplitude reduction for this configuration compared to a collimated beam. With spectral resolution up to 125,000, this DKIST instrument will see significant fringes at amplitudes over 10% for transmission, 15% for diattenuation, and several degrees for elliptical retardance. A simple thermal perturbation analysis was performed to show the likely drift of this modulator Mueller matrix using the 1°C temperature stability requirement for the DKIST Coudé laboratory. This modeling tool should be useful for future solar and night-time spectropolarimeters where fringes may be high amplitude, thermally unstable, and possibly mitigated using a range of techniques.

With this analysis we showed theoretical origins and laboratory verification of the $r^{-2}$ fringe amplitude envelope in converging beams. The Berreman calculus was used with thermal perturbations in refractive index through the TOC and the physical thickness through the CTE. These thermal perturbations were also experimentally verified in the laboratory. Predictions were made for DKIST instruments as well as for on-sky data from the Keck LRISp retarder. In Appendix D, by combining the Berreman calculus with thermal simulations and converging beam parameters, instrument designers now have tools to estimate likely fringe amplitudes for a wide variety of use cases and thermal conditions.