Terahertz and Microwave Optical Properties of Single-Crystal Quartz and Vitreous Silica and the Behavior of the Boson Peak

applied sciences

Article Terahertz and Microwave Optical Properties of Single-Crystal Quartz and Vitreous Silica and the Behavior of the Boson Peak

Mira Naftaly * and Andrew Gregory

National Physical Laboratory, Teddington TW11 0LW, UK; [email protected] * Correspondence: [email protected]

Featured Application: Z-cut quartz is conﬁrmed as a suitable reference material for measurements of complex permittivity at terahertz and microwave frequencies, and data between 0.2 and 6 THz are provided. Silica glass has also been evaluated.

Abstract: Z-cut single-crystal quartz and vitreous silica (silica glass or fused silica) were evaluated for use as reference materials for terahertz and microwave measurements of complex permittivity, with Z-cut quartz confirmed as being suitable. Measurements of refractive indices and absorption coefficients for o-ray and e-ray in quartz and for vitreous silica are reported at frequencies between 0.2 and 6 THz and at 36 and 144 GHz, and compared with data reported in the literature. A previously unreported broad band was seen in the extraordinary absorption of quartz. The Boson peak in silica glass absorption was examined, and for the first time, two negative relationships have been observed: between the refractive index and the Boson peak frequency, and between the Boson peak height and its frequency. Keywords: single-crystal quartz; silica glass; complex permittivity; terahertz; microwave; Boson peak Citation: Naftaly, M.; Gregory, A. Terahertz and Microwave Optical Properties of Single-Crystal Quartz and Vitreous Silica and the Behavior 1. Introduction of the Boson Peak. Appl. Sci. 2021, 11, The aim of this work was twofold. First, to evaluate the suitability of Z-cut single- 6733. https://doi.org/10.3390/ crystal quartz and glassy silica as reference materials for terahertz (THz) and microwave app11156733 (MW) measurements. The current lack of reference materials suitable for calibrating and testing measurements of complex permittivity at THz and MW frequencies is an acknowl- Academic Editor: Petrarca Massimo edged and pressing issue that requires being addressed [1,2]. Quartz and silica appear to be good candidate materials due to their wide availability and affordability, standard Received: 10 June 2021 fabrication techniques, and high chemical purity. A reference material for spectroscopy Accepted: 19 July 2021 Published: 22 July 2021 should have highly reproducible properties that vary less than typical measurement uncertainty, and that are constant among all reputable manufacturers and production batches.

Publisher’s Note: MDPI stays neutral To verify this, we examined five samples of Z-cut quartz and seven samples of vitreous with regard to jurisdictional claims in silica. We report our findings in detail, confirming that Z-cut quartz is a good reference published maps and institutional affil- material. However, silica glass of nominally the same grade exhibits significant variations iations. in both refractive index and absorption at THz frequencies, although the refractive indices at 36 GHz show negligible variation. The second aim of this work, which is directly related to establishing reference materials, was to obtain accurate data on the complex permittivity of quartz and silica. Although there is extensive literature on both quartz [3–14] and vitreous silica [15–26], the reported Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. data vary significantly. If quartz and/or silica are to serve as reference materials, their This article is an open access article complex permittivities must be accurately known. We report values of the refractive index distributed under the terms and and absorption coefficient for the o-ray in quartz between 0.2 and 6 THz, for the e-ray in conditions of the Creative Commons quartz between 0.2 and 5 THz, and for vitreous silica between 0.2 and 4.5 THz. We also Attribution (CC BY) license (https:// report measurements at 36 GHz for the o-ray in quartz and for all silica glass samples. In creativecommons.org/licenses/by/ addition, we have examined variations in the Boson peak in silica glass, and report for the 4.0/). first time a negative relationship between refractive index and Boson peak frequency.

Appl. Sci. 2021, 11, 6733. https://doi.org/10.3390/app11156733 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 6733 2 of 13

2. Materials and Methods 2.1. Material Samples To study single-crystal quartz, ﬁve samples of Z-cut quartz and one sample of Y-cut quartz were used. α-Quartz is a uniaxial crystal and, therefore, birefringent. In a Z-cut crystal the transmitted beam travels parallel to the crystal axis; therefore, measurements on Z-cut quartz yield optical parameters for the ordinary ray. In a Y-cut crystal, the beam travels normal to the crystal axis; therefore, if the beam is polarized, both ordinary and extraordinary optical parameters may be measured. All Z-cut quartz crystals were approximately 4 mm in thickness; the Y-cut crystal was 16.9 mm thick. Seven samples of fused silica (Heraeus Spectrosil 2000) were obtained. All were approximately 2 mm thick. All samples had their thickness measured with a micrometer that had 1 µm resolution.

2.2. Terahertz Time-Domain Spectroscopy Measurements THz time-domain spectroscopy (TDS) measurements were carried out using a Ter- aFlash pro spectrometer from Toptica Photonics set up in a standard optical configuration with four F/2 parabolic mirrors [27,28]. The THz beam path was purged with dry air to eliminate absorption from atmospheric water vapor. Samples were placed in the collimated section of the beam, and laser alignment was used to ensure that they were positioned normal to the THz beam. The frequency resolution was 5 GHz for quartz samples, and 10 GHz for silica glass samples. The THz beam was 98% polarized, making possible o-ray and e-ray measurements on the Y-cut quartz sample. The frequency-dependent (ν) refractive index (n(ν)) and absorption coefficient (α(ν)) of each sample were calculated by the standard method of applying Fourier transform to the time-domain data to obtain the frequency-dependent field amplitude (E(ν)) and phase (φ(ν)), and using the equations [27]: φs(ν) − φre f (ν) c n(ν) = 1 + (1) 2πνd " # 2 (n + 1)2 E (ν) α(ν) = − ln s (2) d 4n Ere f (ν) where c is the speed of light, d is the sample thickness, and the subscripts s and ref refer to the sample and reference data, respectively. THz TDS has an important advantage in measurements of THz optical properties of materials, in that it enables unambiguous, straightforward determination of refractive index and absorption coefficient [27,28]. The measurement uncertainty had two main components: uncertainty in sample thickness; and measurement repeatability. The thickness of each sample was measured using a precision micrometer (Mitutoyo 340–711) in 10 different locations on the sample area, and the standard deviation was taken as the thickness uncertainty. The contribution of thickness uncertainty was frequency-independent. Repeatability uncertainty was evaluated by multiple measurements of a sample, removing and re-positioning it in the beam four times, and calculating the standard deviation of refractive index and absorption coefficient. Repeatability uncertainty decreases with the signal-to-noise ratio (SNR), and is larger at low frequencies (<0.5 THz) and at high frequencies (>5 THz); however, in the region of high SNR, it is lower than thickness uncertainty. The dominant contribution was used as an estimate of the uncertainty in the obtained values of refractive index and absorption coefficient. Appl. Sci. 2021, 11, x FOR PROOFREADING 3 of 14

Appl.2.3. Sci. Open2021, 11 Resonator, 6733 Measurements 3 of 13 MW measurements were performed using the open resonator technique, which uti- lizes the fact that the resonance frequency and the Q-factor of a resonator are affected in a 2.3. Open Resonator Measurements known way by the presence of a dielectric material in the resonant cavity. The open resonator employedMW measurements was of the were plano performed-concave using type, the openshown resonator in Figure technique, 1, which uti- lizes the fact that the resonance frequency and the Q-factor of a resonator are affected in a which uses fundamentalknown-mode way Gaussian by the presence beam ofresonances a dielectric materialfor measurement. in the resonant Sample cavity. s were positioned on the planeThe mirror, open and resonator a Vector employed Network was Analyzer of the plano-concave (VNA) (Rohde type, shownand in Figure1, Schwarz ZVA50) was usedwhich to record uses fundamental-mode the complex transmission Gaussian beam coefficient. resonances The for Q measurement.-factors Samples and resonant frequencieswere were positioned then obtained on the plane by mirror, fitting anda resonance a Vector Network model Analyzerto the data (VNA). (Rohde and Measurements of permittivitySchwarz and ZVA50) loss angle was used were to recordobtained the complexby the fixed transmission-frequency coefficient. tuna- The Q-factors ble length method [29].and For resonant each sample, frequencies measurements were then obtained were made by fitting at three a resonance resonant model to the data. Measurements of permittivity and loss angle were obtained by the fixed-frequency tunable modes close to 36 GHz, setlength by methodchanging [29 ].the For VNA each frequency, sample, measurements to check that were measurements made at three resonant modes are not affected by coincidentclose to higher 36 GHz,-order set by modes. changing The the Q- VNAfactor frequency, of the empty to check cavity that was measurements are approximately 130,000. Thenot affectedtechnique by requires coincident that higher-order sample thickness modes. The be approximately Q-factor of the empty an cavity was integer number of half wavelengthsapproximately in 130,000. the medium The technique of the requires material. that Uncertainties sample thickness we bere approximately evaluated from contributionsan integer associated number of with half wavelengths dimensions, in the Q- mediumfactors, ofand the material.theory. UncertaintiesThe were evaluated from contributions associated with dimensions, Q-factors, and theory. The Gaussian beam is highly polarized, which allows the two axes of uniaxial Y-cut quartz Gaussian beam is highly polarized, which allows the two axes of uniaxial Y-cut quartz crystals to be measured crystalsat different to be measuredspecimen at rotations. different specimen For Z-cut rotations. quartz, For measurem Z-cut quartz,ents measurements are expected to be independentare expected of the to bespecimen independent rotation. of the specimen rotation.

Figure 1. Fabry-Perot open Figureresonator 1. Fabry-Perot used for openMW resonatormeasurements. used for MW measurements.

0 Note that MW measurementsNote that typically MW measurements yield values typically of real yield permittivity values of real ε′ and permittivity loss ε and loss angle δ, whereas THz TDS measures refractive index n and absorption coefﬁcient α. The angle δ, whereas THz TDStwo measures sets of parameters refractive can index be inter-converted n and absorption by using coefficient the relations: α. The two sets of parameters can be inter-converted by using the relations: ε0 = n2 − k2 (3) 휀′ = (푛2 − 푘2) (3)

00 휀″ = 2푛푘 ε = 2nk (4) (4) tan δ = ε00 /ε0 (5) tan 훿 = 휀″/휀′ (5) k = αc/4πν (6) 푘 = 훼푐/4휋휈 (6) where k is extinction and ε” is the imaginary part of complex permittivity. where k is extinction and ε″ isFor the low-loss imaginary materials, part such of complex as those studied permittivity. here, permittivity ε0 at MW frequencies For low-loss materials,is expected such as to those have negligible studied here, dispersion. permittivity This is because ε′ at MW the Kramers-Kronig frequencies relationship n k n is expected to have negligiblebetween dispersion. the real and This imaginary is because parts ofthe the Kramers refractive-Kronig index ( andrelationship) requires that d /dν ∝ k (except in the vicinity of an absorption peak) [30]. When n>>k, n has negligible dispersion, between the real and imaginaryand so does partsε0. of the refractive index (n and k) requires that d푛/d휈 ∝ 푘 (except in the vicinity of an absorption peak) [30]. When n>>k, n has negligible dispersion, and so does ε′.

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3. Results and Discussion 3.1. Single-Crystal Quartz Five quartz samples cut normal to the crystal axis (Z-cut) were measured, yielding THz optical parameters for the ordinary ray (o-ray, traveling parallel to the crystal axis) transmission. Excellent reproducibility among tested samples was seen: variation in obtained optical parameters was found to be comparable with the measurement uncertainty. In addition, one Y-cut crystal was also measured (cut parallel to the crystal axis) in order to obtain the optical parameters for the extraordinary ray (e-ray, traveling normal to the crystal axis) transmission. The refractive indices for ordinary and extraordinary rays in quartz crystal are pre- sented in Figure 2, and the absorption coefficients in Figure 3. Quartz is a trigonal uniaxial crystal, and therefore birefringent and dichroic. Appl. Sci. 2021, 11, 6733 4 of 13 Birefringence is defined as the difference between extraordinary and ordinary refractive indices, and it is evident in Figure 2 that it is positive and increasing with frequency. The values of birefringence at a few selected frequencies are given in Figure 4a, where it is3. seen Results to rise and from Discussion 0.048 at 1 THz to 0.054 at 5 THz (the values at 3.5 THz and 4 THz are omitted3.1. Single-Crystal due to anomalous Quartz dispersion in the o-ray data). In contrast, in the visible range, the birefringenceFive quartz of samples quartz i cuts 0.009, normal i.e., toa factor the crystal of five axis smaller (Z-cut) than were at THz measured, frequencies. yielding THzThe optical well- parametersknown optical for thephonon ordinary resonance ray (o-ray, [3,8,11] traveling in the ordinary parallel to ray the spectrum crystal axis) is seentransmission. to be at 3.855 Excellent ± 0.005 THz. reproducibility The absorption among maximum tested samples is 10.7 ± was0.1 cm seen:−1, and variation the peak in FWHMobtained (full optical width parameters at half-maximum) was found is to 145 be comparable± 5 GHz. Figure with the4b shows measurement a Voigt uncertainty. fit to the absorptionIn addition, peak. one The Y-cut Voigt crystal function was also accounts measured for the (cut broadening parallel to that the may crystal arise axis) due in to order the presenceto obtain of the crystal optical defects. parameters It is seen for thethat extraordinary the peak is asymmetric: ray (e-ray, traveling whereas normalon the low to the- frequencycrystal axis) side transmission. the fit is good, on the high-frequency side there is an additional broad wing, causedThe refractive by the rising indices edge for of ordinary the strong and optical extraordinary phonon raysat 11.8 in THz. quartz crystal are pre- sented in Figure2, and the absorption coefﬁcients in Figure3. Quartz is a trigonal uniaxial crystal, and therefore birefringent and dichroic.

2.26

2.24

2.22

2.20

2.18

2.16 Refractiveindex 2.14 no

2.12 ne

2.10 0 1 2 3 4 5 6 Appl. Sci. 2021, 11, x FOR PROOFREADING 5 of 14 Frequency (THz)

Figure 2. The ordinary (no) and extraordinary (ne) refractive indices of quartz crystal. The uncertain- Figure 2. The ordinary (no) and extraordinary (ne) refractive indices of quartz crystal. The uncertainties are shown by light blue shading. ties are shown by light blue shading.

12 )

-1 a 10 o ae

Absorption(cm coefficient 2

0 0 1 2 3 4 5 6 Frequency (THz)

Figure 3. The ordinary (αo) and extraordinary (αe) absorption coefﬁcients of quartz crystal. The Figure 3. The ordinary (αo) and extraordinary (αe) absorption coefficients of quartz crystal. The un- uncertainties are shown by light blue shading. The optical phonon resonance in the ordinary spectrum certainties are shown by light blue shading. The optical phonon resonance in the ordinary spectrum is at is3.855 at 3.855 ± 0.005± 0.005 THz. THz.

12 a b ) data 0.054 -1 10 Voigt fit

8 0.052

Birefringence 0.050 4

0.048 Absorptioncoefficient (cm 2

0 1 2 3 4 5 3.4 3.6 3.8 4.0 4.2 Frequency (THz) Frequency (THz)

Figure 4. (a) Birefringence of quartz, defined as ne—no. (b) Voigt fit to the phonon resonance peak at 3.855 ± 0.005 THz.

An important motivation for this work was to establish accurate values of THz and MW optical parameters for quartz and silica, in the light of significant variation found in the published data. This variation is especially large in the case of absorption coefficients, as seen in the figures below. Figures 5 and 6 present the results of this work together with the data reported in the literature [3–12]. Different types of instrumentation were used for these measurements. Early workers employed Fourier Transform spectrometers (FTS) [3– 6,11], and a grating spectrometer [7,8]; in recent years, THz TDS has dominated the field [7,10–12]. Our refractive index data and the values of birefringence show excellent agreement with the majority of published results. In the case of ordinary absorption, our data are in good agreement with the literature at frequencies up to about 4 THz. However, there is considerable variability in the reported intensity of the phonon resonance at 3.855 THz, which varies from 5.8 cm−1 [8] to 13.1 cm−1 [11], whereas our value is 10.7 cm−1. Some of the variation may be attributable to differences in the quality of the quartz crystals studied: crystals with low concentrations of defects and impurities exhibit stronger and

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12 )

-1 a 10 o ae

Absorption(cm coefficient 2 Appl. Sci. 2021, 11, 6733 5 of 13

0 0 1 2 3 4 5 6 Birefringence is deﬁned asFrequency the difference (THz) between extraordinary and ordinary refractive indices, and it is evident in Figure2 that it is positive and increasing with frequency. TheFigure values 3. of The birefringence ordinary (αo) and at a extraordinary few selected (α frequenciese) absorption arecoefficients given inof quartz Figure crystal.4a, where The unit is seencertainties to rise are from shown 0.048 by atlight 1 THzblue shading. to 0.054 The at 5optical THz phono (the valuesn resonance at 3.5 in THz the ordinary and 4 THz spectrum are omittedis at 3.855 due to± 0.005 anomalous THz. dispersion in the o-ray data). In contrast, in the visible range, the birefringence of quartz is 0.009, i.e., a factor of ﬁve smaller than at THz frequencies.

12 a b ) data 0.054 -1 10 Voigt fit

8 0.052

Birefringence 0.050 4

0.048 Absorptioncoefficient (cm 2

0 1 2 3 4 5 3.4 3.6 3.8 4.0 4.2 Frequency (THz) Frequency (THz)

Figure 4. (a) Birefringence of quartz, deﬁned as n —n .(b) Voigt ﬁt to the phonon resonance peak at 3.855 ± 0.005 THz. Figure 4. (a) Birefringence of quartz, defined eas neo—no. (b) Voigt fit to the phonon resonance peak at 3.855 ± 0.005 THz.

TheAn well-known important opticalmotivation phonon for this resonance work was [3 ,to8,11 establish] in the accurate ordinary values ray spectrum of THz and is −1 seenMW to be optical at 3.855 parameters± 0.005 for THz. quartz The and absorption silica, in maximumthe light of issignificant 10.7 ± 0.1 variation cm , andfound the in peakthe FWHM published (full data. width This at half-maximum)variation is especially is 145 large± 5 GHz. in theFigure case of4 absorptionb shows a Voigtcoefficients, fit to the absorption peak. The Voigt function accounts for the broadening that may arise due Appl. Sci. 2021, 11, x FOR PROOFREADING as seen in the figures below. Figures 5 and 6 present the results of this work6 togetherof 14 with to thethe presence data reported of crystal in the defects. literature It is[3– seen12]. Different that the peaktypes isof asymmetric:instrumentation whereas were used on the for low-frequencythese measurements. side the fit Early is good, workers on the employed high-frequency Fourier Transf side thereorm isspectrometers an additional (FTS) broad [3– wing,6,11 caused], and bya grating the rising spectrometer edge of the [7,8]; strong in recent optical years, phonon THz at TDS 11.8 has THz. dominated the field narrower[7,10An phonon important–12]. resonan motivationces [31]. At for frequencies this work above was to 4 establishTHz, there accurate are large values inconsist- of THz and encies in the published data, precluding comparison. Impurities may be responsible for MW opticalOur parameters refractive index for quartz data and and the silica, values in theof birefringence light of significant show excellent variation agreement found in some of the variations in the published data at both THz and MW frequencies (see below). the publishedwith the majority data. This of published variation results. is especially In the large case of in ordinary the case ofabsorption absorption, our coefficients, data are in In contrast, our data for extraordinary absorption disagrees with the majority of lit- as seengood in agreement the figures with below. the literature Figures5 at and frequencies6 present up the to resultsabout 4 of THz. this However, work together there is erature. It is, however, consistent with the work of [7]. Our e-ray data indicates that there withconsiderable the data reported variability in the in the literature reported [3 –intensity12]. Different of the typesphonon of resonance instrumentation at 3.855 were THz, is a weak and broad absorption band centered at 3.2 THz, previously unreported at room usedwhich for these varies measurements. from 5.8 cm−1 Early[8] to workers13.1 cm−1employed [11], whereas Fourier our value Transform is 10.7 spectrometers cm−1. Some of temperature. Notably, [9] observed a weak resonance at 3.9 THz at 10 K. (FTS)the [3 variation–6,11], and may a grating be attributable spectrometer to differences [7,8]; in in recent the quality years, of THz the TDS quartz has crystals dominated stud- the fieldied: crystals[7,10–12 ]. with low concentrations of defects and impurities exhibit stronger and

a 2.22 [3] 2.20 [4] [5] 2.18 [7] [8] 2.16 [11] NPL 2.14

Refractive index 2.12

2.10

2.08 0 1 2 3 4 5 6 Frequency (THz)

Figure 5. Cont.

b 14

) [3] -1 12 [4] [5] 10 [7] [8] 8 [11] NPL 6

Absorption coefficient(cm 2

0 0 1 2 3 4 5 6 Frequency (THz)

Figure 5. THz optical parameters for the ordinary ray transmission in quartz crystal. Data reported in the literature are shown together with the results of this work. (a) refractive index; (b) absorption coefficient.

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narrower phonon resonances [31]. At frequencies above 4 THz, there are large inconsistencies in the published data, precluding comparison. Impurities may be responsible for some of the variations in the published data at both THz and MW frequencies (see below). In contrast, our data for extraordinary absorption disagrees with the majority of literature. It is, however, consistent with the work of [7]. Our e-ray data indicates that there is a weak and broad absorption band centered at 3.2 THz, previously unreported at room temperature. Notably, [9] observed a weak resonance at 3.9 THz at 10 K.

a 2.22 [3] 2.20 [4] [5] 2.18 [7] [8] 2.16 [11] NPL 2.14

Refractive index 2.12

2.10

2.08 0 1 2 3 4 5 6 Appl. Sci. 2021, 11, 6733 6 of 13 Frequency (THz)

b 14

) [3] -1 12 [4] [5] 10 [7] [8] 8 [11] NPL 6

Absorption coefficient(cm 2

0 0 1 2 3 4 5 6 Frequency (THz) Appl. Sci. 2021, 11, x FOR PROOFREADING 7 of 14 Figure 5. THz optical parameters for the ordinary ray transmission in quartz crystal. Data reported Figure 5. THz optical parameters for the ordinary ray transmission in quartz crystal. Data reported in thein literature the literature are shown are together shown with together the results with of the this results work. of(a) this refractive work. index; (a) refractive (b) absorption index; (b) absorp- coefficient.tion coefﬁcient.

a 2.26 [3] 2.24 [4] [7] [8] 2.22 [9] NPL 2.20

2.18 Refractiveindex

2.16

2.14 0 1 2 3 4 5 6 Frequency (THz)

b 3.0

) [3] -1 2.5 [4] [7] [8] 2.0 [9] NPL 1.5

1.0

Absorption(cm coefficient 0.5

0.0 0 1 2 3 4 5 6 Frequency (THz)

FigureFigure 6. THz 6. opticalTHz optical parameters parameters for the extraordinary for the extraordinary ray transmission ray transmission in quartz crystal. in quartzData re- crystal. Data portedreported in the inliterature the literature are shown are together shown with together the results with of the this results work. of(a) thisrefractive work. index; (a) refractive (b) index; absorption(b) absorption coefficient. coefﬁcient.

A Z-cut quartz crystal (designated Q1) with thickness 4.118 ± 0.003 mm was measured in the open resonator at 36 GHz. The results were ε′ = 4.435 ± 0.007 and tanδ = (37 ± 6) × 10−6. The measured ε′ is consistent with the results published by other workers [13,29], although the tanδ is higher than the value given in [13]. Our measurements confirm that Z-cut single-crystal quartz is an excellent reference material for THz and MW measurements. However, care must be taken in selecting samples of suitable thickness for THz TDS measurements [32]: between 2 and 5 mm is preferred.

3.2. Vitreous Silica It is known that different grades of silica glass have slightly different THz optical properties. In this work, we aimed to test variability among different batches of the same nominal grade and among different samples fabricated from the same batch. Glasses from four different batches of Spectrosil 2000 were examined, designated A, B, C, D. Batch A

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Our refractive index data and the values of birefringence show excellent agreement with the majority of published results. In the case of ordinary absorption, our data are in good agreement with the literature at frequencies up to about 4 THz. However, there is considerable variability in the reported intensity of the phonon resonance at 3.855 THz, which varies from 5.8 cm−1 [8] to 13.1 cm−1 [11], whereas our value is 10.7 cm−1. Some of the variation may be attributable to differences in the quality of the quartz crystals studied: crystals with low concentrations of defects and impurities exhibit stronger and narrower Appl. Sci. 2021, 11, x FOR PROOFREADINGphonon resonances [31]. At frequencies above 4 THz, there are large inconsistencies in the 8 of 14 published data, precluding comparison. Impurities may be responsible for some of the variations in the published data at both THz and MW frequencies (see below). In contrast, our data for extraordinary absorption disagrees with the majority of hadliterature. two samples; It is, however, batch consistent D had three with samples; the work of batches [7]. Our C e-ray and data D had indicates one sample that there each. Their THzis a weakoptical and properties broad absorption are shown band centered in Figures at 3.2 7 THz,and previously8; their MW unreported properties at room are listed in temperature. Notably, [9] observed a weak resonance at 3.9 THz at 10 K. Table A1.Z-cut quartz crystal (designated Q1) with thickness 4.118 ± 0.003 mm was measured in the openSignificant resonator differences at 36 GHz. The are results evident were amongε0 = 4.435 the± 0.007THz and refractive tanδ = (37 indices± 6) × 10 of− 6the. glasses examinedThe measured (Figureε0 is consistent7). There with is a the high results consistency published bybetween other workers samples [13, 29A1], althoughand A2; whereas theythe tandifferδ is higherfrom samples than the value B and given C. Samples in [13]. B and C are closer to each other than they are to samplesOur measurements A1 and A2. conﬁrmSamples that D1, Z-cut D2 single-crystal, and D3 are quartz very issimilar an excellent to samples reference A1 and A2. Samplesmaterial D2 for THzand and D3 MWare measurements.identical within However, measurement care must beuncertainty taken in selecting to samples samples A1 and A2. of suitable thickness for THz TDS measurements [32]: between 2 and 5 mm is preferred. However, sample D1 differs slightly from D2 and D3. These results, based on the limited number3.2. Vitreous of samples Silica examined, indicate that there is a good consistency between glass samples fromIt is knownthe same that batch, different but grades that inter of silica-batch glass variability have slightly may different be significant THz optical in some cases. Theseproperties. should In be this taken work, into we aimed consideration to test variability in employing among different silica glas batchess as ofa reference the same material fornominal THz measurements. grade and among The different THz samples absorption fabricated coefficients from the (Figure same batch. 8) of Glasses all glasses from are simi- four different batches of Spectrosil 2000 were examined, designated A, B, C, D. Batch A had lar up to about 2 THz, with increasingly significant differences appearing above 3 THz. two samples; batch D had three samples; batches C and D had one sample each. Their THz optical properties are shown in Figures7 and8; their MW properties are listed in Table1.

1.99

1.98

1.97 A1 A2

B Refractive index 1.96 C D1 D2 D3 1.95 0 1 2 3 4 5 Frequency (THz)

Figure 7. Refractive indices of Spectrosil 2000 silica glass samples. Figure 7. Refractive indices of Spectrosil 2000 silica glass samples.

40 )

-1 35 A1 A2 30 B C 25 D1 D2 20 D3 15

10 Absorptioncoefficient (cm 5

0 0 1 2 3 4 5 Frequency (THz)

Figure 8. Absorption coefficients of Spectrosil 2000 silica glass samples.

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had two samples; batch D had three samples; batches C and D had one sample each. Their THz optical properties are shown in Figures 7 and 8; their MW properties are listed in Table 1. Significant differences are evident among the THz refractive indices of the glasses examined (Figure 7). There is a high consistency between samples A1 and A2; whereas they differ from samples B and C. Samples B and C are closer to each other than they are to samples A1 and A2. Samples D1, D2, and D3 are very similar to samples A1 and A2. Samples D2 and D3 are identical within measurement uncertainty to samples A1 and A2. However, sample D1 differs slightly from D2 and D3. These results, based on the limited number of samples examined, indicate that there is a good consistency between glass samples from the same batch, but that inter-batch variability may be significant in some cases. These should be taken into consideration in employing silica glass as a reference material for THz measurements. The THz absorption coefficients (Figure 8) of all glasses are similar up to about 2 THz, with increasingly significant differences appearing above 3 THz.

1.99

1.98

1.97 A1 A2

B Refractive index 1.96 C D1 D2 D3 1.95 0 1 2 3 4 5 Frequency (THz)

Appl. Sci. 2021, 11, 6733 Figure 7. Refractive indices of Spectrosil 2000 silica glass samples. 8 of 13

40 )

-1 35 A1 A2 30 B C 25 D1 D2 20 D3 15

10 Absorptioncoefficient (cm 5

0 0 1 2 3 4 5 Frequency (THz)

Figure 8. Absorption coefﬁcients of Spectrosil 2000 silica glass samples. Figure 8. Absorption coefficients of Spectrosil 2000 silica glass samples. Table 1. Permittivity, loss, refractive index, and absorption coefﬁcient of quartz and silica samples were measured using the open resonator at 36 GHz and 144 GHz, and using time-domain spectroscopy at 1 THz.

Material 36 GHz 144 GHz 1 THz ε0 δ RI Abs (cm−1) ε0 δ RI Abs (cm−1) RI Abs (cm−1) ±0.012 (µrad) ±0.003 × 10−3 ±0.012 (µrad) ±0.003 × 10−3 ±0.001 Q1 (o-ray) 4.435 37 ± 12 2.105 0.6 ± 0.1 2.108 0.15 ± 0.08 A1 3.830 510 ± 40 1.957 7.6 ± 0.6 3.837 1370 ± 110 1.959 78 ± 6 1.953 2.5 ± 0.2 A2 3.830 520 ± 40 1.957 7.7 ± 0.6 3.830 1400 ± 120 1.957 80 ± 6 1.953 2.7 ± 0.2 B 3.834 426 ± 30 1.958 6.3 ± 0.4 1.957 2.7 ± 0.2 C 3.834 435 ± 30 1.958 6.4 ± 0.4 1.958 2.9 ± 0.2 D1 3.817 255 ± 20 1.954 3.8 ± 0.3 1.954 2.3 ± 0.2 D2 3.821 258 ± 20 1.955 3.8 ± 0.3 1.953 2.5 ± 0.2 D3 3.826 356 ± 20 1.956 5.3 ± 0.3 1.953 2.6 ± 0.2

Significant differences are evident among the THz refractive indices of the glasses examined (Figure7). There is a high consistency between samples A1 and A2; whereas they differ from samples B and C. Samples B and C are closer to each other than they are to samples A1 and A2. Samples D1, D2, and D3 are very similar to samples A1 and A2. Samples D2 and D3 are identical within measurement uncertainty to samples A1 and A2. However, sample D1 differs slightly from D2 and D3. These results, based on the limited number of samples examined, indicate that there is a good consistency between glass samples from the same batch, but that inter-batch variability may be significant in some cases. These should be taken into consideration in employing silica glass as a reference material for THz measurements. The THz absorption coefficients (Figure8) of all glasses are similar up to about 2 THz, with increasingly significant differences appearing above 3 THz. At THz frequencies absorption in amorphous materials approximates the Debye vibrational density of states (VDOS), described by the relationship α ∝ ν2. In terms of MW loss, this translates into δ ∝ ν (see Equations (3)–(6)), which is a well-known behavior in MW measurements [33]. However, it is well known that glasses exhibit low-frequency vibrational modes in excess of the Debye VDOS, termed the Boson peak [34,35]. The Boson peak can be revealed by plotting α/ν2 as a function of frequency, where it appears as a broad band. It has been shown that in silica glass the Boson peak occurs at around Appl. Sci. 2021, 11, 6733 9 of 13

1 THz [34–38] and that it is very sensitive to small variations in the glass properties, and in particular to the presence of OH groups [35–39]. Figure9 depicts the Boson peak in the glasses studied, revealing variations in glass absorption properties that cannot be easily distinguished in Figure8. As in the case of the Appl. Sci. 2021, 11, x FOR PROOFREADING 10 of 14 refractive indices in Figure7, the glass pair A1 and A2 are very similar; while B and D are close to each other but differ from A1 and A2. However, batch D differs signiﬁcantly from the other glasses; moreover, unlike the refractive indices, glasses D1 and D2 are very similar, while D3 differs from them.

3.5

A1 A2

) 3.0 B -2 C

THz D1

-1 D2

cm 2.5 D3

(

2 a / n a / 2.0

1.5 0 1 2 3 4 Frequency (THz)

Figure 9. The Boson peak in Spectrosil 2000 silica glass samples. Figure 9. The Boson peak in Spectrosil 2000 silica glass samples. It appears in Figure9 that the Boson peak height is correlated with its frequency: in glasses where the peaks are at a higher frequency, their heights are reduced. This is 1.962 a consistent with data reported inb the literature [35–39]. Moreover, by considering Figure9 together with the refractive indices1.0 in Figure7, it appears that glasses with lower refractive

1.960 indices have Boson peaks shifted to higher frequencies. To examine these correlations, Figure 10a plots the relationship between the refractive index at 2 THz and the Boson peak 0.8 frequency (the value at 2 THz was chosen because it is the region with a low slope and high 1.958 signal-to-noise ratio). Figure 10b likewise plots the relationship between the Boson peak height and its frequency, where the0.6 peak height is calculated as the difference between

1.956 peak maximum and the baseline at 3 THz. In addition to the seven samples studied in Boson peak height (a.u.)

Refractive index2 THz @ detail above, data from another seven samples are included in Figure 10. The provenance and grade of these samples were not0.4 known; therefore, they were not included in the main 1.954 0.8 0.9 1.0study. However,1.1 1.2 the relationships in0.8 Figure 100.9 are made1.0 apparent1.1 due1.2 to the fact that Boson peak frequencydata from (THz) a relatively large number of samplesBoson (14 peak in total)frequency are (THz) included, unlike previous work where only two to three samples were examined. Figure8 confirms the existence of Figure 10. (a) Relationship betweenstrong the negative refractive correlations index of silica between glasses (a) at refractive 2 THz and index their and Boson Boson peak peak frequency. frequency, (b) and (b) Relationship between the BosonBoson peak height peak heightand its andfrequency its frequency. in silica glasses. To our The knowledge, peak height this is measured is the first as explicitthe differ- report of ence between peak maximum andthese its relationshipsbase at 3 THz (see using Figure multiple 7). samples of silica glass. Although it is outside the scope of this work to attempt to explain this behavior, it may be speculated that it arises from polarizability,As in the case which of single is affected-crystal by quartz, material it is microstructure important to compare and impurities, our data and on inTHz turn, opticaldetermines parameters the refractive of silica with index those and influencesreported in absorption. the literature. Figure 11 presents the data reportedAs in thein the case literature of single-crystal [15–24] together quartz, with it is importantthe results toof comparethis work, our shown data as on the THz meanoptical and standard parameters deviation of silica of with the thosesamples reported studied. in The the reported literature. data Figure that 11 falls presents outside the thedata shown reported range are in the excluded. literature Unlike [15–24 the] together case of quartz, with the the results great majority of this work, of these shown results as the weremean obtained and standard using FTS deviation [13–18,22 of], the whil samplese two used studied. TDS The[19,21]; reported a review data of that all published falls outside datathe was shown reported range in are [20]. excluded. Unlike the case of quartz, the great majority of these results wereOur obtained results are using seen FTS to [be13 –in18 good,22], whileagreement two used with TDS the majority [19,21]; a of review the reported of all published data. However,data was it reportedis evident in that [20]. large variations exist in the reported values of both refractive index and absorption coefficient, confirming the necessity for accurate measurements.

Appl. Sci. 2021, 11, x FOR PROOFREADING 10 of 14

3.5

A1 A2

) 3.0 B -2 C

THz D1

-1 D2

cm 2.5 D3

(

2 a / n a / 2.0

1.5 0 1 2 3 4 Frequency (THz)

Appl. Sci. 2021, 11, 6733 10 of 13 Figure 9. The Boson peak in Spectrosil 2000 silica glass samples.

1.962 a b 1.0

1.960

0.8

1.958

0.6

1.956 Boson peak height (a.u.) Refractive index2 THz @ 0.4

1.954 0.8 0.9 1.0 1.1 1.2 0.8 0.9 1.0 1.1 1.2 Boson peak frequency (THz) Boson peak frequency (THz) Appl. Sci. 2021, 11, x FOR PROOFREADING 11 of 14

FigureFigure 10. 10.(a) Relationship(a) Relationship between between the refractivethe refractive index index of silica of silica glasses glasses at 2 THz at 2 and THz their and Boson their peakBoson frequency. peak frequency. (b) Rela- (b) tionshipRelationship between between the Boson the peakBoson height peak andheight its and frequency its frequency in silica in glasses. silica glasses. The peak The height peak height is measured is measured as the differenceas the differ- betweenence between peak maximum peak maximum and itsbase and atits 3base THz at (see 3 THz Figure (see7). Figure 7).

a 1.99As in the case of single-crystal quartz, it is important to compare our data on THz optical parameters of silica with those reported in the literature. Figure 11 presents the data1.98 reported in the literature [15–24] together with the results of this work, shown as the mean and standard deviation of the samples studied. The reported data that falls outside the shown range are excluded. Unlike the case of quartz, the great majority of these results were1.97 obtained using FTS [13–18,22], while two used TDS [19,21]; a review of all published data was reported in [20].

1.96Our results are seen to be in good agreement [15] with the majority of the reported data.

Refractive index However, it is evident that large variations [18]exist in the reported values of both refractive index and absorption coefficient, confirming [21] the necessity for accurate measurements. 1.95 [23] NPL

1.94 0 1 2 3 4 5 Frequency (THz)

b 40

) [15] -1 35 [16] [17] 30 [18] [19] 25 [20] [21] 20 [23] NPL 15

10 Absorption coefficient(cm 5

0 0 1 2 3 4 5 Frequency (THz)

Figure 11. THz optical parameters of Spectrosil 2000 silica glass, shown as the mean (heavy blue line) Figure 11. THz optical parameters of Spectrosil 2000 silica glass, shown as the mean (heavy blue line)and and standard standard deviation deviation (light (light blue blue shading) shading) of of the the seven seven samples samples examined. examined. Data Data reported reported in thein liter- the literatureature are shownare shown together together with with theresults the results of this of work.this work. (a) refractive (a) refractive index; index; (b) absorption (b) absorption coefficient. coefficient.

The open resonator measurements on the seven silica glass samples gave values of refractive index that were consistent within uncertainty. The measured loss of Spectrosil 2000 is noted as being particularly low in comparison with other grades of fused silica [26]. The absorption coefficient, however, is seen to vary by a factor of two between batches. It must, therefore, be concluded that this material is not suitable as a reference material for dielectric loss at microwave and millimeter-wave frequencies. At MHz frequencies, Spectrosil 2000 can be useful as a reference material for dielectric loss because, despite the observed variability, its loss is far below the resolution of typical experiments.

Table 1. Permittivity, loss, refractive index, and absorption coefficient of quartz and silica samples were measured using the open resonator at 36 GHz and 144 GHz, and using time-domain spectroscopy at 1 THz.

Material 36 GHz 144 GHz 1 THz ε′ δ RI Abs (cm−1) ε′ δ RI Abs (cm−1) RI Abs (cm−1) ±0.012 (μrad) ±0.003 × 10−3 ±0.012 (μrad) ±0.003 × 10−3 ±0.001

Appl. Sci. 2021, 11, 6733 11 of 13

Our results are seen to be in good agreement with the majority of the reported data. However, it is evident that large variations exist in the reported values of both refractive index and absorption coefficient, confirming the necessity for accurate measurements. The open resonator measurements on the seven silica glass samples gave values of refractive index that were consistent within uncertainty. The measured loss of Spectrosil 2000 is noted as being particularly low in comparison with other grades of fused silica [26]. The absorption coefficient, however, is seen to vary by a factor of two between batches. It must, therefore, be concluded that this material is not suitable as a reference material for dielectric loss at microwave and millimeter-wave frequencies. At MHz frequencies, Spectrosil 2000 can be useful as a reference material for dielectric loss because, despite the observed variability, its loss is far below the resolution of typical experiments.

4. Conclusions Single-crystal quartz and vitreous silica were evaluated for use as reference materials for MW and THz measurements of complex permittivity. A secondary aim of this work was to produce a set of accurate data in order to clarify the large variations in the data reported in the literature. Single-crystal Z-cut quartz was confirmed as a highly reproducible material suitable for use as a reference. Ordinary and extraordinary refractive indices and absorption coefficients were measured between 0.2 and 6 THz. The optical phonon resonance in the ordinary ray spectrum was determined to be at 3.855 ± 0.005 THz, with the absorption maximum at 10.7 ± 0.1 cm−1 and the FWHM of 145 ± 5 GHz. The results were compared with the published literature. The ordinary refractive index at 36 GHz was found to be equal within measurement uncertainty to that at 1 THz, indicating a flattening of dispersion at frequencies below 1 THz. Silica glass samples from four different batches were examined and had their refractive indices and absorption coefficients measured between 0.2 and 4.5 THz. Variability among samples was found to be sufficiently significant to render silica glass unsuitable as a reference material. The results were compared with the published literature. As in quartz, the refractive index at 36 and 144 GHz was found to be equal within measurement uncertainty to that at 1 THz, indicating a flattening of dispersion at frequencies below 1 THz. In order to examine in more detail variations in absorption behavior, the value of α(ν)/ν2 was plotted vs. frequency, revealing large differences in the Boson peak profiles. By studying multiple glass samples, it was possible for the first time to observe the negative relationships between refractive index and Boson peak frequency, and between Boson peak height and its frequency.

Author Contributions: Conceptualization, M.N. and A.G.; methodology, M.N. and A.G.; formal analysis, M.N. and A.G.; investigation, M.N. and A.G.; writing—original draft preparation, M.N.; writing—review and editing, M.N. and A.G. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the project TEMMT, grant number 18SIB09, under the EMPIR H2020 program of the European Union, and by the National Measurement System Programs Unit of the UK Department for Business, Innovation, and Skills. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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