different material systems, especially those that are spatially inhomogeneous or cannot be prepared as large single crystals.

In recent years, research into mid-and far-infrared (3-100μm) polaritonics has rapidly expanded owing to the emergent physical phenomena in this wavelength range 1 . One growth area has been the study of van-der-Waals (vdW) materials 2,3 , with a layered structure that leads to effects including hyperbolicity [4][5][6][7] and ballistic transport 8 . They are prepared from bulk crystal samples through exfoliation, producing high-quality thin films on arbitrary substrates 9 . Many properties of vdW materials including phase transitions 10 , high mobility behavior 8,11 , carrier freeze-out 12 , and damping pathways 13 , cannot be studied at room temperature. These effects are essential for emerging quantum technologies 14 , in which polaritonics play a crucial role 15 . Therefore, developing techniques for studying infrared polaritonics in vdW crystallites at low temperatures is critical.

One approach to measuring polaritons in vdW materials is near-field microscopy, which exploits a sharp metal probe to get a precise spatial resolution (10nm) at a wide range of wavelengths from the infrared to THz 16 . However, the reliance on laser sources, the restricted sensitivity to polarization 17 , and the possibility of artifacts in the data 18 , make it less versatile for generalpurpose characterization or materials screening. Far-field techniques are diffraction-limited but make up for spatial resolution by offering excellent spectral bandwidth and polarization control, which can span the entire mid-infrared 1,19 . FTIR microscopy is the primary route to achieving this and can measure individual van-der-Waals flakes 20 . Several approaches to FTIR micro cryospectroscopy for small samples have been developed in past works 11,[21][22][23][24][25][26][27] . Parabolic focusing on sub-mm crystallites and subsequent gold overcoating for referencing 28 can measure relatively large crystals 22,29,30 . However, it is unsuited for single, exfoliated flake analysis with a size of 100μm or less. Transmission-based focusing setups leveraging light pipes are also effective on this length scale 26,27 . However, these cannot perform reflection-based spectroscopy, and are not suited to mapping. Finally, several groups have adapted a commercial He cryostat to a commercial FTIR microscope and have been used to study van-der Waals materials 11,21,24,31 .

However, commercial microscope systems often cannot be adapted for advanced infrared microscopy techniques. The fixed geometry limits the range of cryostats that can be used and the optical components that can be inserted into the beam path for advanced polarimetry or optical pumping experiments. Commercial systems also use extensive custom optics, which makes it challenging to reproduce their designs accurately in the laboratory and repairs require service visits. Finally, the costs associated with commercial microscope components can be significant. This paper presents the design for an infrared cryo-microscope that leverages off-the-shelf components capable of performing precision reflection and dielectric function measurements.

Our approach can be applied to any combination of FTIR and cryostat and can be optimized for polarimetry and other experiments. We study the temperature dependent optical properties of 10 B and 11 B enriched hexagonal boron nitride. These materials have record-breaking propagation lengths for hyperbolic phonon polaritons (HPhPs), owing to the reduced scattering in isotopically pure materials 13,18,[32][33][34] . Both h 10 BN and h 11 BN have been shown to have extended polariton lifetime at low temperatures 13,35 ; however, the full dielectric function is not available. While such information is available for hBN with naturally distributed boron isotopes 22 , the damping parameters and phonon tuning of isotopically pure materials is not available. First, we examine samples prepared on an Au mirror, which has strong background reflectance from the substrate that aids in referencing. We observe Fabry-Perot (FP) type absorption resonances 36,37 in the mid-IR, which is well suited to determining the temperature dependence of TO phonons in the infrared. We also use sapphire substrates, which are better suited for completing a full dielectric function analysis close to both TO and LO phonon energies. Our dielectric function analysis extends previous methods 7,20 implementing a method for backside reflection correction, and addressing systematic errors. We find that optical losses in isotopically pure hBN at room temperature were previously underestimated by a factor of 2. This parameter is critical for realistic simulations of polariton propagation in hBN materials and heterostructures. Our results are consistent with past measurements on Raman modes 13 , and hBN polaritons for which no model previously existed 35 . Our results for the dielectric function show comparable shifts in the TO phonon energy as the FP resonances, suggesting that tracking the properties of FP modes is a suitable mechanism for estimating the temperature dependence of phonon properties. Finally, we calculate the quality factors of polaritons as a function of temperature and show that our temperature-dependent dielectric function closely matches prior experiments by Ni 35 . This suggests our dielectric function improves on those previously published, even at room temperature, and can be used for predictive modeling for polaritonic devices.

Our microscope implements a conventional, finite conjugate length optical microscope using allreflective optical components, with a schematic shown in Fig. 1a. Indicative full beam power spectra are shown in Fig 1b , comparing the spectra from the microscope with a 3mm aperture (measured 60μm spot size) to a conventional configuration FTIR (SiC Glow Bar, KBr beamsplitter, and DLaTGS detector). For more details, please see the Supplemental Section 1.

Temperature-dependent micro-FTIR spectra of a flake of boron h 10 BN, which is 1220nm thick, exfoliated onto an Au-coated silicon substrates substrate at 300K and 5K are shown in Fig. 2a.

There are resonances are observed in the spectral ranges of approximately 840cm -1 and from 1200-1400cm -1 . We attribute these to a Fabry Perot-type resonance associated with the high index of the h 10 BN close to the TO phonon (located at 1393cm -1 ), which is reported to exceed 25 in prior works 32 . These have been reported previously in the context of FTIR-ATR measurements 36 , on bulk 38 , and on exfoliated flakes 3734 . The out-of-plane resonances can be measured due to the reflective objective's off-normal incidence angle. The narrow spikes in the range of 1400-1800cm -1 can be attributed to atmospheric water absorption. A detailed view of the resonance associated with the out-of-plane phonon and the lower Reststrahlen band of h 10 BN is shown in Fig. 2b. The linewidth is sharp, consistent with the long lifetime of the out-of-plane phonon 32 , and shows almost no temperature dependence. The resonances associated with the high energy phonon is also presented in Fig. 2c. A series of resonances in the spectra are present, with varying absorption strength and linewidth as they approach the TO phonon located at approximately 1393cm -1 . First, we consider the resonance frequency of these modes -they redshift with increasing temperature, with a tuning range of 1.8±0.1cm -1 , 2.7 cm -1 ±0.1and 2.5±0.1cm -1 , for modes FP1, FP2, and FP3 respectively. This is suggestive of temperature tuning of the TO phonon in hBN. The trend in linewidth between the modes is more subtle. The linewidth of FP resonances is related to a combination of material and radiative damping. As the refractive index increases close to the TO phonon, radiative damping decreases due to increased reflection at the interface between hBN and air. Meanwhile, light more readily gets absorbed by the TO phonon, so material absorption rises dramatically. This makes resonances sharper as they approach the TO phonon as the radiative losses decrease but then get significantly broader once the TO phonon losses increase. A more detailed discussion of modal lifetime for standing wave resonances can be found in prior work 39 . There is a slight change in the model linewidth with temperature -taking FP3 as an example, we can see that the full-width half maximum changes from 3.2 cm -1 to 3.7 cm -1 between 5K and 300K, suggesting a small but notable change in the mode lifetime. These results are compared with temperature dependent micro-FTIR spectra of a flake of boron hB 10 N with 700nm thickness exfoliated onto a sapphire substrate, as shown in Fig. 3a. We see several spectral features across the mid-infrared, including the phonon response of the sapphire substrate (900cm -1 and below), a sharp dip associated with the out of plane phonon of hBN at approximately 842cm -1 , and the in-plane Reststrahlen band from approximately 1395cm -1 to 1650cm -1 . The resonance associated with the out-of-plane phonon of h 10 Each mode redshifts with increasing temperature, with the mode labeled FP1 tuning 2.5±0.5cm -1 and FP2 tuning 2.6±0.1cm -1 . These can be directly contrasted with the results for FP3 on Au, which shows 2.5±0.1cm -1 of tuning. Similarly, the linewidth slightly broadens with temperature, going from 6.8±0.3cm -1 to 7.3±0.3cm -1 . This consistency between substrates suggests a phonon tuning of approximately 2.5cm -1 , and a reduction of the phonon lifetime. Similar results are seen for a flake of isotopically pure h 11 BN and a second thinner flake of h 10 BN, detailed in Supplemental Section 2.

To determine the optical constants of the hBN, we use a commercial piece of software WAVSE from J. A. Wollam, and homebuilt. a nonlinear least squares fitting algorithm based on a 4x4 transfer matrix formalism published in 40 The phonon energy continuously redshifts with increasing temperature for all flakes studied here, consistent with prior Raman studies by Cuscó. Through the deviations between the two parameters, we can estimate systematic errors at 0.9cm -1 , as discussed in more detail in SI section 6. We highlight that our results show a slightly lower room temperature phonon energy than that previously reported in the literature by Giles 32 , likely attributed to the lower resolution of prior measurements at 2cm -1 . The tuning of the extracted TO phonon is consistent with that of the FP modes, demonstrating that FP mode tuning can provide the tuning of the TO phonon. The trends observed for the phonon energy with temperature can be described by a combination of lattice contraction (increasing phonon energy) and phonon anharmonic interaction (decreasing phonon energy). For many semiconductors, this has led to an overall redshift with a temperature of a few wavenumbers. This contrasts with recent work on MoO3, with an overall phonon blueshift 31 . The differences between hBN and MoO3 are likely associated with weak anharmonicity in the MoO3 suppressing higher-order anharmonic perturbations to the phonons, giving a predominantly lattice contraction-induced blueshift of the phonon. Our results for damping 𝛤 are shown in Fig 4d-e are also consistent with recent Raman and s-

SNOM studies, showing that the phonon linewidths are reduced as the samples are cooled. Our results for damping are significantly larger than those reported in Giles 32 . We can attribute this to the harmonic model used in Giles 32 where an anharmonic model is more applicable 23 . As discussed in SI section 6 we estimate systematic errors during the fit of 0.47cm -1 . To better compare our results with those from Ni 35 , we use the temperature-dependent fitting function proposed in that work;

Where T is the temperature, 𝛾 is the low-temperature damping limit, and 𝑇 is the characteristic temperature. Fitting to Figure 4, we get a characteristic temperature of 1200±100K for the 700nm h 10 BN flake, 1110±60K for the 500nm thick h 10 BN flake and 1060±30K for the 750nm thick h 11 BN flake. These consistent values support the hypothesis that the damping is given mainly by acoustic phonon scattering. While this value is approximately twice the value presented by Ni 35 , this is likely due to uncertainties in extracted parameters. Finally, the change in the line widths of the FP modes in the reflectance spectrum on Au or Sapphire gives a comparable shift to that of the damping extracted from dielectric function analysis. As such, we believe that fitting the line shape of FP modes close to the TO phonon can approximate the temperature-dependent properties of ωTO and Γ. The framework for the dependence presented in equation 2 is a phenomenological model that considers the population of acoustic phonons.

This T 4 dependence will be general for any material with sufficiently high acoustic phonon energies to be the dominant scattering term, where isotopic impurities are insignificant. This dependence would, therefore, be anticipated to hold for other high phonon energy materials, such as silicon carbide, cBN, and AlN.

While our results for ωTO and 𝛤 are in general agreement with prior literature and extend the previously reported values , we find that both ωLO and 𝜀 parameters show some inconsistencies which prevents observation of temperature trends. (See Supplementary Section 5). While this is a limitation of our approach, we show in Supplementary Section 7 that our dielectric function can reproduce the FP mode tuning from the flake in Figure 2, suggesting that our model is accurate enough for predictive modeling. Finally, we assess the properties of the hyperbolic phonon polaritons in hexagonal boron nitride.

The expression gives the dispersion of hyperbolic polaritons in the high k limit 4 ;

Where d is the flake thickness, 𝜀 is the dielectric function above the hBN flake, 𝜀 is the dielectric function of the substrate, 𝜀 is the in-plane dielectric function, 𝜀 is the out-of-plane dielectric function, and l is an integer, set to 1 here. We examine the polariton dispersion (k'), and the mode Q factor (k'/k'') as shown in Fig. 5a/b. We only consider the extreme cases of 300K and 5K and calculate for both suspended hBN and on SiO2. The dispersion only changes slightly between room temperature and 5K. However, the mode quality factors show a dramatic change in behavior. The reduction in the losses at low-temperature results in a quality factor approximately 30% larger than at room temperature in the center of the Reststrahlen band for suspended flakes. This enhancement is reduced on SiO2 substrates, suggesting that the losses in SiO2 suppress the benefits of cryogenic cooling. We can compare our results against quality factor values taken from Ni 35 , and we show excellent agreement between our modeled values and prior experiments. These represent the closest agreement between s-SNOM and far-field studies reported for isotopically pure hBN, suggesting that our higher damping values than those of Giles 32 represent the actual loss in isotopically pure hBN.

To further explore the correlation between far-field and near-field measurements, we also calculate the polariton damping factor, which can be evaluated from 35

shows polariton dispersion for h 10 BN and h 11 BN on a SiO2 substrate and suspended in air. b)

shows the mode Q factor, defined as k'/k'', for h 10 BN and h 11 BN on both a SiO2 substrate and suspended, compared against data from Ni 35 . c) shows polariton damping rate and a temperature-dependent fit from our dielectric function, again compared against Ni 35 for hB 11 N.

In this paper, we have detailed a design of a cryogenic FTIR microscope constructed using offthe-shelf components capable of measuring the infrared optical properties of exfoliated van-der-Waals materials. We measure the temperature-dependent properties of 10 B and 11 B isotopically pure hBN, evaluated the full infrared dielectric function as a function of temperature. Our results are consistent with past Raman and polaritonic studies and suggest that the polariton losses at room temperature have been significantly underestimated in prior works. This allows us to explain experimentally measured properties of HPhPs at low temperatures and generalize prior results to the full Reststrahlen band. Our work and methods can be applied beyond hBN to a much more comprehensive set of van der Waals materials, including those with phase transitions, free electron plasmas, and other optical phenomena.

objective's back focal plane after the D-shaped half mirror to define the collection area analyzed by the detector. Using two pinholes reduces the presence of scattered light in the spectrum.

Motorized polarizers (ELL 14) are positioned alongside both pinholes to analyze the reflected polarization state and are equipped with KRS5 (Thorlabs WP25H-K) or ZnSe (Thorlabs WP25H-Z) polarizers. A magnetic mount mirror switches between a visible camera and an infrared detector. In the imaging position, an achromatic doublet lens focuses light onto a visible camera with a CCD area of ½''. We maximize the FOV to allow for easy sample identification by using a 4f configuration, enabling us to image the pinhole completely from 1mm through 12mm aperture sizes. Chromatic aberration is present between visible and IR ranges from the ZnSe window, which is compensated for in measurements by a focal shift of the cryostat between optical and infrared measurements. For infrared detection, we use a combination of mirrors and a 1'' 90-degree off-axis parabolic mirror with a focal length (fdet) of 1'', focusing on an MCT detector (IR Associates FTIR-22-0.25). The off-axis parabolic is located 30cm after the back focal length of the objective. The detector is mounted on a modular base plate so that it can be swapped out with other detectors for spectroscopy.

It is worth discussing the choice of components used in this microscope configuration and how they can be adapted for different experiments. The parameters to optimize for an FTIR microscope are light throughput, spectral resolution, and spatial resolution. To control these, one chooses the properties of the J-Stop, concave mirror, objective, detector, and collection optics.

The J-Stop should be selected to maintain sufficient spectral resolution at a given wavelength for the chosen applications. In our case, we decided to achieve a spectral resolution of 0.5cm -1 across the full spectrum of the MCT, which is adequate for most materials' science applications. A larger J-stop can be used if a high resolution is not required, providing additional light into the microscope. A concave mirror is then chosen to fill the pinhole (accounting for the demagnification factor) and better match the detector's acceptance angle, characterized via the fnumber of the components. The finite conjugate length objective lens f-number is typically low, and matching these can improve light throughput in the system (f=0.04 for the objective and f=0.1 for the concave mirror). The third choice is the objective selection, which should consider the sample sizes that need to be analyzed and the pinhole size range -here, 0-12mm. Here, we anticipate analyzing samples from 20μm-200μm, commensurate with the size of vdW flakes, choosing 50x (1mm-10mm pinhole). The final choice is the detector and collection optics. For HgCdTe detectors, typically smaller detectors offer dramatically higher responsivity due to reduced noise in small active area detectors 42 . However, smaller detectors also require a higher magnification factor to couple to the detector, which is challenging with off-the-shelf parabolic reflectors, which have a minimum focal length of around 25mm. We use a 25mm focal length off-axis parabolic, located approximately 300mm behind the pinhole, providing a magnification of roughly 11X between the iris and the detector. We choose to use a 250μm x 250μm HgCdTe detector with a 22μm cutoff, which offers a good compromise between detector sensitivity and bandwidth. Sensitivity can be improved by the choice of detectors -we have implemented Si, Ge, InSb, and pyroelectric detectors in addition to the MCT to cover the far-infrared and visible between different configurations.

Isotopically pure B 10 and B 11 enriched hBN samples were grown using the flux growth method described in the paper 34 . Crystals were then exfoliated onto c-plane sapphire substrates and Aucoated silicon, and sufficiently large (>50µm x 50µm) flakes were identified for spectroscopy using optical techniques and atomic force microscopy (see images A silicon diode temperature probe is mounted on the backside of the cryostat cold head to accurately monitor the sample's temperature. An Au mirror was placed next to the sample to provide an in-situ reference for all reflectance spectroscopy, and spectra were collected with a 0.5cm -1 spectral resolution.

Supplemental Section 2 -Reflectance Spectra for other Flakes studied.

In this section we show full infrared reflectance spectra for a 510nm flake of h 10 BN and a 750nm thick flake of h 11 BN. Supplemental Section 3 -Fitting Procedure Both vary the dielectric function parameters and construct transfer matrices that match the best fit to the reflectance data. The incident angle (20º), and polarization are set parameters, and the substrate uses room temperature literature values from 43 . The thickness of the flakes is determined by AFM, but is allowed to be fitted in a 5% range about the AFM measured values. The optimization first constructs the projected dielectric function for the material of interest using starting input parameters. The in-plane and out-of-plane dielectric function for hBN is modeled using the form: Figure S8, group velocity, propagation length and lifetime for polaritons in a 710nm thick flake of hBN of different isotopic content and on different substrates.