Reconstruction and modeling of the complex refractive index of nonlinear glasses from terahertz to optical frequencies

Adam Pacewicz; Jaroslaw Cimek; Bartlomiej Salski; Michal Walczakowski; Ryszard Buczynski; Ryszard Buczynski

doi:10.1364/OE.431430

1. Introduction

Despite decades of research, a relative lack of portable high-power terahertz (THz) radiation sources, known as the THz-gap, persists. One of the recently proposed schemes for its closure is degenerate four wave mixing (DFWM) of two optical pulses of carrier frequencies $f_1$ and $f_2$ chosen in such a way that one of the mixing products, e.g. $2f_1 - f_2$, falls into the THz range. A nonlinear photonic crystal fiber seems to be a natural choice for the mixing structure, opening the possibility for the realization of an all-fiber THz source. Optical sources of such kind have immense potential for mass adoption in science and industry [1]. While several numerical studies of the THz-DFWM generation scheme exist [2,3], they seem not to take into account real-world THz-range absorption values, which will undoubtedly be one of the main limiting factors of the efficiency of the source. Within the context of optical to THz conversion, it is therefore of interest to study THz range absorption properties of glasses possessing considerable nonlinearities in the optical/infrared range. Establishing the complex refractive index of typical nonlinear glasses at THz frequencies will pave the way for more accurate numerical studies geared towards a practical realization of the fiber.

We report on the refractive index investigation of a set of 12 selected in-house developed borosilicate and tellurite as well as heavy metal oxide silicate, germanate and fluoride soft glasses possessing good thermomechanical properties for fiber drawing [4]. Although chalcogenide glass families are known to have transmission windows in the terahertz gap [5], they are considerably more expensive and brittle [6].

The linear complex refractive index was extracted over the frequency range of ca. 0.15 THz to 200 THz via Kramers-Kronig transform (KKT) analysis of the combined data from FTIR and THz-TD spectrometers. In frequency ranges at both ends of the spectrum, there is relatively good agreement between the results obtained via the KKT and other methods. Broadband quantitative investigations of this kind (THz to optical) into the complex refractive index of vitreous materials are extremely rare [7,8]. One of the reasons for this are the complexities of the data-processing involved in both measurement techniques, which arise especially when multiple reflections from the sample faces occur, as explained later in the paper. A sub-goal of this work is therefore to provide a concise reference on how to synergize the measurement data from two different types of spectrometers.

There have been several investigations, both theoretical and experimental, into the far-infrared absorption of amorphous materials [9–11], which shows a universal power-law dependence [12]. The parameters of the power-law model, described later in the text, exhibit strong correlations with multiple mechanical and optical glass properties and thus constitute a complementary means of characterization [13,14]. However, there has been very little literature on how to unequivocally extract those parameters. We, therefore, propose a new systematic procedure and apply it to study the relationships between the far-infrared absorption parameters and other glass properties.

Table 1. Compositions of in-house developed glasses expressed in % mol. Nonlinear indices of refraction ($n_2$) at infrared (IR) frequencies after [6,15]. It should be noted that the composition of glass TCG-75921 is different from the one used in a previous work [4, p. 137]. *Full name TZN-732005/2La/Cl. **Synthesized from dehydrated materials and with the employment of the bubbling process.

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2. Spectral features of vitreous materials

Table 2. Compositions (expressed in % mol) of glasses whose literature absorption data were used for far-infrared modeling described in section 7. Nonlinear indices of refraction ($n_2$) at infrared (IR) frequencies after [4,6,16]. The data for fused silica (FS) were provided through private communication with M. Naftaly.

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Vitreous (also known as glassy or supercooled) dielectric materials exhibit so-called $\alpha$- (for $f <$1 kHz), $\beta$- (for $f <$ 10 GHz) and $\gamma$-relaxation processes ($f <$ 10 THz) [19]. The $\gamma$ process [20], in the bulk of the contemporary literature referred to as the vibrational density of states (VDOS) peak [7], is determined mainly by vibrations and librations of molecules, which give rise to a broad spectral feature located at terahertz frequencies. Anomalies in the vibrational spectra in excess of the classical Debye density of states [21] ($\propto f^{2}$) are known as the boson peak, owing to its connection with the Bose distribution function [21]. The temperature dependence of the VDOS peak is usually negligible in inorganic glasses [10], so measurements in this work were limited to room temperature only. The characteristics of the boson peak are closely related to many material physical properties such as specific heat, heat conduction or superconductivity, which is one of the main reasons why this phenomenon is intensely studied [22]. The boson peak is not a unique property of disordered materials, which has recently been explained theoretically as the result of competition between elastic mode propagation and diffusive damping [22]. For more information about relaxation processes in vitreous materials the reader is referred to the broad literature [7,19,23].

3. Experimental details

3.1 Sample preparation

All the glasses were manufactured in-house using the conventional melt and quench technique [6] and originally formed into ca. 2 mm and 5 mm thick double-side-polished slabs. The samples were thinned and repolished to a high degree as necessary for transmission-mode measurements. The minimum thickness amounted to ca. 180 µm, which is thick enough for the chemical composition to be considered unchanged. For glass manufacturing details the reader is referred to [6]. The compositions and nonlinear refractive indices of the manufactured glasses are provided in Table 1. In the far-infrared range modeling study described in section 7, in addition to the glasses listed in Table 1, glass data available in the literature was included. The compositions of the additional glasses are given in Table 2.

3.2 THz-TDS spectroscopy

To cover the frequency range up to a few hundred GHz transmission terahertz time domain spectroscopy (THz-TDS) measurements were performed on the 2 mm samples. Time-domain single-pulse traces with and without the sample were recorded. Three methods for solving the inverse problem of computing the complex permittivity were compared: an iterative [24] and explicit [13] approximate approach, and a rigorous approach employing a Nelder-Mead minimization algorithm of the complex transfer function [25]. It was assessed that the differences between the results produced by the algorithms, which become more evident as the material’s loss tangent increases, were lower than the experimental uncertainties. For the thinned slabs, multiple pulses in the sample trace were recorded, which necessitated the use of the rigorous method of minimizing the complex transfer function. Although the Nelder-Mead minimization algorithm generally yielded correct results, in certain situations jumps in the frequency dependence of the permittivity occurred. A gradient descent algorithm [26] used instead of Nelder-Mead was found to be more reliable.

For most samples, additional measurements in reflection mode were necessary to cover the frequency range up to the lowest usable frequency of the FŢIR measurement. It is well known that reflection-mode THz-TDS measurements are poorly conditioned. However, such an approach was admissible given the fact that we did not expect any sharp spectral features in the missing frequency ranges. To obtain the complex refractive index $\hat {n} = n + ik$, explicit formulas for the transverse magnetic (TM)-polarized beam were applied [27]:

(1)$$\hat{n} = \frac{\sin\theta_i}{\sin \hat{\theta}_t},$$

where $\hat {\theta }_t$ is the complex refraction angle:

(2)$$\hat{\theta}_t ={-}\frac{1}{2}\arcsin\left[\frac{(S_{11}-1)\sin 2\theta_i}{S_{11}+1}\right],$$

where $S_{11}$ is the complex amplitude reflection coefficient and $\theta _i =$ 45° is the THz-TD spectrometer angle of incidence provided in the datasheet. The results were highly sensitive to the value of the incidence angle $\theta _i$ and the phase of $S_{11}$. In order to bring the permittivity obtained via Eq. (1) into agreement with the results of transmission mode measurements, it was necessary to introduce a small incidence angle correction, $\Delta \theta _i = \theta _i - 45^{\circ }$, and a small frequency-dependent phase shift justified by the imperfect overlap of the sample face and reference (gold plate) planes, $\Delta L$:

(3)$$\angle S_{11} = \frac{2 \pi f \Delta L}{c}$$

The $\Delta L$ shifts did not exceed a few micrometers and the $\Delta \theta _i$ corrections a few degrees. Both are listed in Table S1 in Supplement 1. The absorption per unit length, $\alpha$, used later in the paper, is defined as [28]

(4)$$\alpha = \frac{1}{L}\ln\left(\frac{P_0}{P_T}\right),$$

and is twice the value of the real part of the propagation constant, $L$ is the sample thickness, $P_T$ ($P_0$) is the transmitted (incident) power, and $\ln$ denotes the natural logarithm.

3.3 FTIR spectroscopy

To extend the THz-TDS measurements, FTIR measurements, both in the reflection and transmission mode, were undertaken with the use of a Bruker VERTEX 80v bench top spectrometer [29] in near-vacuum conditions (1 mbar to 3 mbar pressure). Reflection measurements were performed in three bands: far-infrared (FIR) (ca. 2 THz to 20 THz), mid-infrared (MIR) (ca. 20 THz to 150 THz) and near-infrared (NIR) (ca. 50 THz to 240 THz), each covered by a different thermal radiation source: a FIR mercury lamp, a silicon carbide MIR rod (commercial name: Globar), and a tungsten-filament NIR lamp, with aperture 3.5 mm each. The beam was guided by parabolic mirrors onto a room temperature - deuterated triglycine sulfate (RT-DTGS) crystal detector in the FIR range, and a room temperature - deuterated L-alanine triglycine sulfate (RT-DLaTGS) detector in the MIR and NIR ranges. A KBr beamsplitter for the MIR and NIR ranges was chosen, and a 6 µm mylar beamsplitter was selected for the FIR range. The samples were illuminated at an angle of $\theta =$ 15°. All measurements were performed with a resolution of ca. 0.06 cm⁻¹. The MIR transmission measurements were performed in a similar configuration, but with a source aperture of 8 mm and the beam focused perpendicularly onto the sample by a 150 mm focal length mirror. The transmission-mode measurement data were used to remove reflections from the sample back surface visible in the reflectance spectrum. Depending on the sample, these multiple reflections began reaching the detector starting from frequencies between 37 THz and 90 THz. In all the transmission measurements, the sample thickness was much larger than the wavelength (12 times or more). Under such conditions, transmitted power readings were averaged out and thus no Fabry-Perot oscillations were visible in the reflectance spectra. This resulted from the fact that the sample surface irregularities were appreciable with respect to the wavelength and that the frequency resolution step was much larger than the period of oscillations [30].

4. Kramers-Kronig reflectance analysis

Assuming perpendicular incidence, the complex refractive index $\hat {n} = n + ik$ at frequency $f_0$ can be obtained from the complex reflection coefficient $r(f_0) = \sqrt {R(f_0)}\exp [i\phi (f_0)]$ via [31]

(5)$$\hat{n} = \frac{r(f_0) + 1}{r(f_0) - 1} ,$$

where $R(f_0)$ is the reflectance and $\phi (f_0)$ is the spectral phase, which can be recovered via the KKT transform of the whole $R(f)$ spectrum. Equation (5) is applicable to illumination with a non-polarized beam at small angles since the incident beam can be decomposed equally into parallel and perpendicular polarizations, where the associated reflectance deviations vs. incidence angle are approximately of opposite signs and compensate each other [32], as confirmed by our numerical calculations for the experimental conditions. We have found the most practical KKT variant to be [31,33,34]

(6)$$\phi(f_0) = \frac{f_0}{\pi} \int_{0}^{\infty} \frac{\log[R(f)/R(f_0)]}{f_0^{2} - f^{2}} df.$$

To obtain $R(f)$, the FTIR reflectances from multiple bands were combined, together with the reflectance calculated from the THz-TDS refractive index via [35]

(7)$$R = \frac{(n - 1)^{2} + k^{2}}{(n + 1)^{2} + k^{2}}.$$

The THz-TDS reflectance calculated with Eq. (7) was extrapolated to zero frequency at a constant level with the use of 100 linearly spaced data points before the application of the KKT phase reconstruction with Eq. (6). No extrapolations into the ultra-violet (UV) ranges [33] have been taken into account in the final results since they negligibly affected the refractive index in the FIR range, which is the main focus of this study.

Admittedly, for the majority of samples there were noticeable discontinuities between the reflectance $R(f)$ in each of the bands. Therefore, a data "merging strategy" was applied (see Figs. S1 and S2 in Supplement 1 for examples). Typically the FIR measurement results were the most repeatable ones out of the three FTIR bands and they were in good agreement with the THz-TDS reflectance. In most such cases, the continuity of $R(f)$ could be obtained by scaling only the NIR reflectance to match the MIR reflectance. Upon scaling, $R(f)$ was continuous and consistent with THz-TDS data as well as data measured or modeled at optical frequencies in other studies, e.g., using interferometric techniques [4].

Secondary reflections reaching the detector above a certain frequency were removed by computing the refractive index analytically [36,37] in the range where MIR transmission data were available and using it to compute the reflectance off the front side of the sample only. This method of extracting $\hat {n}$ will be referred to as the transmission-reflection (TR) method later in the paper. The removal of secondary reflections before Kramers-Kronig analysis enabled us to obtain smooth refractive index curves at MIR frequencies and above (see Fig. 1). Otherwise, the refractive index showed a characteristic dip, followed by an upshot [17]. It can be remarked that an incremental improvement of our results could be achieved by combining $R(f)$ with data obtained using the THz-TDS and TR methods in multiply-subtractive KKT algorithms [34].

Fig. 1. Summary of the real [(a), (c)] and imaginary [(b), (d)] parts of permittivity obtained for the investigated glasses by Kramers-Kronig transform (KKT) analysis of the reflectance. The absorption coefficients up to 10 THz are shown in (e), (f).

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5. Results

A summary of the complex permittivity results for all the investigated glasses ($\hat {\epsilon } = \hat {n}^{2} = \epsilon ' + i \epsilon ''$) obtained by KKT analysis of the reflectance is shown in Fig. 1. For all the samples it exhibits a similar textbook frequency dependence [38]: $\epsilon '$ peaks at around 1 THz, decreases sharply while $\epsilon ''$ passes through a peak, after which multiple absorption peaks occur until $\epsilon '$ plateaus at optical frequencies. Three samples (UV-710, ZBLAN, TWPN/I/6) show negative permittivity in narrow frequency bands, which is expected in the vicinity of absorption bands [39]. As it can be seen, the borosilicate composition UV-710 shows by far the lowest level of losses among all the samples, making it the most interesting one in the context of terahertz generation. The frequency dependence of $\epsilon '$ and $\epsilon ''$ of the UV-710 composition obtained via KKT reflectance analysis is compared in Fig. 2 against the permittivity extracted using three other methods: transmission- and reflection-mode THz-TDS, transmission-reflection (TR) analysis of FTIR data and interferometric methods using a Michelson interferometer (MI) [4]. Similar plots showing relatively good agreement between all the methods were obtained for all the other samples. The largest discrepancies are between the dielectric loss spectra below 1 THz, likely owing to the constant extrapolation of the reflectance to zero frequency for the purpose of KKT.

Fig. 2. Real (a) and imaginary (b) part of permittivity measured for the UV-710 glass using different methods: Kramers-Kronig transform (KKT) analysis of the combined reflectance, transmission- and reflection-mode terahertz time domain spectroscopy (THz-TDS), transmission-reflection (TR) analysis of Fourier transform infrared (FTIR) data. Results obtained using a Michelson interferometer (MI) method have been added after [4].

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6. Broadband permittivity modeling

In order to better understand the measured spectra, in the following section different material-dependent models applied to the UV-710 borosilicate glass composition results will be discussed. Another possible approach to the modeling problem, which has not been considered here, is the use of purely material-independent permittivity models [40].

6.1 Debye-like relaxation models

To the best of our knowledge, there have been no reported attempts to apply permittivity models to the terahertz VDOS or boson peak in vitreous materials. The classical Debye model [23] describes the orientational relaxation of dipolar non-interacting molecule. In many materials, this assumption is not valid. To account for the experimentally observed discrepancies, multiple phenomenological relaxation models (called "anomalous") having linear, passive and causal properties have been introduced [23,41]. To describe the VDOS peak at ca. 1 THz, we have applied the Havriliak-Negami (HN) model given by [42]

(8)$$\hat{\epsilon}(f) = \epsilon_{\infty} + \frac{\Delta \epsilon}{\left[1 + ({{i\mkern 1mu}}2 \pi f \tau)^{\alpha}\right]^{\gamma}},$$

where $\epsilon _{\infty }$ is the relative permittivity resulting from instantaneous dielectric polarization mechanisms, $\Delta \epsilon = \epsilon _s - \epsilon _{\infty }$, $\epsilon _s$ is the static relative permittivity, $\tau$ is the characteristic time constant (relaxation time), and parameters $\alpha$, $\gamma$ obey $\alpha > 0$, $\alpha \gamma \le 1$ [41]. In contrast to the original Debye permittivity model, Eq. (8) can reproduce the slight increase of the permittivity at frequencies on the rising edge of the VDOS peak (in the frequency domain). Unfortunately, neither the HN model given by Eq. (8) nor its recently introduced generalization [43]

(9)$$\hat{\epsilon}(f) = \epsilon_{\infty} + \Delta \epsilon\frac{({{i\mkern 1mu}}2 \pi f \tau)^{\mu-1}}{\left[1 + ({{i\mkern 1mu}}2 \pi f \tau)^{\alpha}\right]^{\gamma}}$$

can satisfactorily fit the experimental data since the $\epsilon '$ frequency-domain roll-off is much slower than Eq. (8) or Eq. (9) predict. Simultaneously, the $\epsilon ''$ level is grossly overestimated by those models. For this reason, we have applied Eq. (8) only up to a certain frequency, beyond which $\epsilon '$ was fitted using shape language modeling (SLM), i.e. least squares spline modeling up to 19 THz with the use of shape primitives [44]. The used primitives include convexness, monotonicity, a negative third derivative, and fixed values of slopes at the endpoints, and the number of knots (breakpoints between spline segments). Spline fitting of $\epsilon '$ yields no information on $\epsilon ''$, so the $\epsilon ''$ fit was obtained by the KKT transform of the $\epsilon '(f)$ model in the whole available frequency range (including the Lorentz-type peak models to be described in the following section):

(10)$$\epsilon^{\prime\prime}(f_0) = \frac{2f_0}{\pi} \int_{0}^{\infty} \frac{\epsilon'(f_0) - \epsilon'(f)}{f_0^{2} - f^{2}} df,$$

at each frequency $f_0$. In the case of the UV-710 sample, we were also able to obtain a promising fit using the alternative Jurlewicz-Weron-Stanislavsky (JWS) model [41], but in this paper we will limit the discussion to a description of the "engineering" SLM approach due to its generality. The parameters of Eq. (8) fitted to UV-710 are provided in Table 3.

Table 3. Parameters of the Havriliak-Negami (HN) model applied to the UV-710 glass permittivity. The upper model frequency is denoted by $f_{max}$. The SLM model with 35 knots was applied from $f_{max}$ up to 19 THz.

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6.2 Lorentz-type oscillator models

In crystalline solids, higher-energy vibrational modes leading to absorption peaks in the IR range can usually be described by Lorentz oscillator model [45, p.8]. In glassy materials, however, the absorption bands are broadened due to structural disorder and are usually well described by Gaussian distributions of Lorentz oscillators [46]. Brendel and Bormann [47] provided a useful analytical approximation for such a peak shape:

(11)$$\hat{\epsilon}(f) = \epsilon_{\infty} + \frac{{{i\mkern 1mu}}\sqrt\pi A}{2\sqrt 2 a \sigma} \left[ \exp\left(\frac{1}{2} a_m^{2}\right) \textrm{erfcx}(a_m) + \exp\left(\frac{1}{2} a_p^{2}\right) \textrm{erfcx}(a_p) \right],$$

where $\textrm {erfcx}$ is the scaled complementary error function [48], $a_m = \frac {a - f_p}{{{i\mkern 1mu}}\sigma }$, $a_p = \frac {a + f_p}{{{i\mkern 1mu}}\sigma }$, $a = -\sqrt {f^{2} - {{i\mkern 1mu}}f_c f}$, $f_p$ is the oscillator frequency, $f_c$ is the damping constant, $\sigma$ is the standard deviation of the Gaussian distribution, $A$ is the oscillator strength. A high (low) $\sigma /f_c$ ratio indicates a strong Gaussian (Lorentzian) peak character. It should be noted that Eq. (11) is not, as is frequently misidentified, a convolution of a Lorentz oscillator and Gaussian function, but rather a weighted sum over the peak frequencies. A convolution would require a reversal and shift of one of the two spectral distribution functions, which is not the case [45,47].

Two drawbacks of Eq. (11) are that, for low resonance frequencies of single THz and less, its imaginary part tends to infinity, and that it cannot describe asymmetric peaks. Both limitations are offset if a model assuming a beta distribution of Lorentz oscillators [49] is introduced:

(12)$$\hat{\epsilon}(f) = \epsilon_{\infty} + \frac{Aw\Gamma(u + v)}{\Gamma(u)\Gamma(v)} \int_{0}^{w^{{-}1}} \frac{\left(wx\right)^{u-1}\left(1 - wx\right)^{v-1}}{x^{2} - f^{2} + {{i\mkern 1mu}}f f_{c}} dx$$

where $w = \frac {1}{4f_p}\left (\frac {v - 1}{u - 1} + 1\right )$, $u = \frac {\beta }{2}\left [\left (\frac {f_p}{\sigma }\right )^{2} \beta (2 - \beta ) - 1\right ]$, $v = (1-\frac {\beta }{2})\left [\left (\frac {f_p}{\sigma }\right )^{2} \beta (2 - \beta ) - 1\right ]$, $\beta \in (0,2)$ is the asymmetry parameter and $\Gamma$ is the gamma distribution. For the symmetric case ($\beta = 1$), the shapes of the Gaussian and beta peaks are essentially the same [49].

At optical frequencies, the permittivity is frequently modeled using a 3-pole Lorentz oscillator model:

(13)$$\hat{\epsilon}(f) = \epsilon_{\infty} + \Delta \epsilon \sum_{n=1}^{3} \frac{A_n}{f_{p,n}^{2} - f^{2} + {{i\mkern 1mu}}f f_{c,n}}$$

The Lorentz oscillator model was chosen instead of the perhaps more frequently used Sellmeier model [28] since the former does not account for losses. The Lorentz-type model parameters are provided in Tables 4 to 6. To summarize, the beta distribution model of Eq. (12) was applied to the first IR absorption peak at ca. 16 THz. Another six peaks were modeled using Brendel-Bormann’s Eq. (11), and the plateau of $\epsilon '$ at optical frequencies was fitted with the triple-pole Lorentz model given by Eq. (13). The full UV-710 model plotted against the permittivity is provided in Fig. 3. This section could not be complete without mentioning Ref. [45]. It summarizes extensive broadband experimental results of IR spectra of glasses together with high-quality model fits, which, contrary to our piecewise models, have the advantage of being continuous. However, it seems that none of the fits presented in Ref. [45] assumed a Debye-like relaxation model. It can be presumed that this is a consequence of the limit of the measurement frequency on the lower end, which was ca. 600 GHz at best, and which, as it seems, did not make the necessity of using such a model evident.

Fig. 3. Real (a) and imaginary (b) part of permittivity extracted for the UV-710 glass through the Kramers-Kronig transform (KKT) of the combined reflectance (blue dots). The solid line represents the material-dependent permittivity model (a) and its KKT via Eq. (10)(b).

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Table 4. Beta distribution parameters for UV-710.

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Table 5. Gaussian weighted Lorentz oscillator model parameters for UV-710. The lower and upper frequencies of the model are denoted by $f_{min}$ and $f_{max}$.

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Table 6. Triple-pole Lorentz parameters for UV-710: $\epsilon _{\infty } =$ 1.035, $\Delta \epsilon =$ 0.191, minimum frequency $f_{min}$ = 32.28 THz.

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7. Far-infrared modeling

7.1 Introduction

One of the well-known models of far-infrared absorption is Schlömann and Strom’s power-law formula [13]

(14)$$n\alpha = K(hf)^{\beta} g(f) ,$$

where $h$ is Planck’s constant and

(15)$$g(f) = 1 - \left[1 + \left( \frac{2\pi}{V_D}\right)^{2} f^{2} l^{2} \right]^{{-}2},$$

where $V_D$ is the material’s Debye velocity of sound and $l$ is the charge correlation length. It should be noticed that the $g(f)$ function factor in Eq. (14) predicts a reduction in the absorption at low frequencies, which is indeed the case for fused silica [13] and some chalcogenide glasses [50], but not for all the other glasses analyzed in this work. The $K$ and $\beta$ parameters in Eq. (14) are of interest since they allow one to assess multiple optical and mechanical glass properties, such as the refractive index, melting temperature or thermal expansion coefficient [13]. A breakdown of Eq. (14) is expected at frequencies approaching the Ioffe-Regel crossover frequency [5], which is typically very close to the boson peak frequency. The parameter $K$ is given by [11,13]

(16)$$K = \frac{|e^{*}|^{2} N \left[\frac{1}{3}(n^{2} + 2)\right]^{2}}{4\pi\epsilon_0 \hbar^{2} \rho c V_D^{3} },$$

where $N$ is the density of charge fluctuations of amplitude $|e^{*}|$, $c$ is the speed of light, $\epsilon _0$ is the vacuum permittivity, $\hbar$ is the reduced Planck constant, and $\rho$ is the mass density.

Fitting experimental data with Eq. (14) has the disadvantage that the value of the $K$ parameter is strongly correlated with the $\beta$ parameter and very sensitive to its changes. Namely, changes of $\beta$ of tens of percent correspond to changes in $K$ by tens of orders of magnitude. To make $K$ independent of the value of $\beta$, the parameter $Kh^{2}$ is often evaluated [11,13] upon setting $\beta = 2$. Obviously, a consequence of such an evaluation is the frequency dependence of $Kh^{2}$ when $\beta \ne 2$. We have chosen to evaluate $Kh^{2}$ at 650 GHz since it is within or close to the fitting range chosen (see section 7.2 later in the paper) for all the 21 samples treated in this paper.

A slightly different version of Eq. (14) was used in Ref. [18]:

(17)$$n\alpha = Kf^{\beta}$$

The value of the parameter $K$ evaluated using Eq. (17) varies by only a few orders of magnitude between samples if $f$ is expressed in units of THz. To facilitate comparisons with other work, both $Kh^{2}$ (in units of s² cm⁻¹) with Eq. (14) and $K$ (in units of cm⁻¹) with Eq. (17) have been determined.

Comparison of the glass permittivity at optical and terahertz frequencies yields information about its polarizability, and therefore about the covalent/ionic character of its chemical bonds [17,51]. The permittivity at THz frequencies is larger due to the ionic contribution, $P_i$, to the electron polarizability, $P_{e}$, which together constitute the total polarizability $P_{tot} = P_e + P_i$. The relation between permittivity and polarizability can be approximated by the Clausius-Mossotti equation [52]

(18)$$P_{e} = \frac{\epsilon_{opt} - 1}{\epsilon_{opt} + 2}\frac{M}{\rho}\frac{3}{4\pi N_A},$$

where $M$ is the molar mass, $\rho$ is the mass density, $N_A$ is Avogadro’s number, $\epsilon _{opt}$ is the permittivity at optical frequencies. The total polarizability can be evaluated using the THz permittivity ($\epsilon _{THz}$):

(19)$$P_{tot} = \frac{\epsilon_{THz} - 1}{\epsilon_{THz} + 2}\frac{M}{\rho}\frac{3}{4\pi N_A}.$$

The character of the bonds can be assessed using the polarizability ratio [51]

(20)$$a = \frac{P_i}{P_e}.$$

7.2 Choice of fitting range

The most straightforward way to obtain the $K$ and $\beta$ parameters from Eqs. (14) and (17) is to linearize them by taking the logarithm of both sides and perform least-squares linear regression. We have observed though that the fitted parameter values can vary significantly depending on the chosen frequency fitting range. To the best of our knowledge, this fact has not been addressed in the literature thus far.

The choice of the range should require that it be as wide as possible and be located in the middle frequency range, between low frequencies and the frequency of the first boson spectral feature (not including). The approximate location of the boson peaks can be determined by plotting the quantity $\alpha /f^{2}$ [7]. Additionally, it should be chosen so as to maximize $\beta$ given that the $n\alpha$ product follows the power law only in the middle frequency range [see Fig. 4(a)], and its rate of change with increasing frequency is lower outside of that range.

Fig. 4. Parts (a) and (c) depict fits to the experimental data for the ZBLAN and SPB-51 samples (see Table 1). Limiting the frequency range on both ends becomes necessary since the model underestimates (overestimates) the absorption parameter $n\alpha$ at lower (higher) frequencies. The dashed lines serve as guides to the eye. Parts (c) and (d), corresponding to ZBLAN and SPB-51, respectively, show the influence of the choice of the minimum length of the fitting frequency range on the $\beta$ parameter. There exists a local RMSE minimum, and the RMSE quickly increases as the fitted frequency range widens.

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Since there is a slight difference between the complex refractive index data obtained from transmission- and reflection-mode THz-TDS data and by Kramers-Kronig analysis of the reflectance from both spectrometers (see Fig. 2), only the THz-TDS data were used. For samples CS-1030 and TZN-732005 (see Table 1) only transmission-mode data were included since the reflection-mode measurement results showed too much noise.

The proposed systematic procedure consists in determining the fitting frequency range that maximizes $\beta$ multiple times for an increasing minimum range length. Selection of the absorption parameters is made on the basis of the resultant curve of RMSE vs. the minimum number of frequency points. For different samples, the curve took different shapes, two exemplary ones being shown in Fig. 4, with a common feature: the stabilization of the $\beta$ and $K$ parameters around a local minimum of RMSE. Whenever multiple local minima were present, the one corresponding to the largest frequency range was chosen. A simple way to use the proposed algorithm is to compute a vector of differences of the RMSE values and take the $\beta$ and $K$ values corresponding to last change of sign in the RMSE differences from negative to positive. Remarkably, such an approach was applicable to all the 21 investigated samples.

7.3 Results and discussion

The THz data for all the glasses listed in Table 1 have been fitted with Eqs. (14) and (17) using the systematic procedure outlined in the preceding section. The estimated parameters are provided in Table 7. In each case the upper frequency of the fitting range lies below the first boson peak frequency ($f_{BP}$), which ranges from 600 GHz for the SF6 glass (which has the highest PbO content) to 1.67 THz for UV-710 (which is the highest among all borosilicate samples).

Table 7. Absorption parameters $\beta$ and $K$ obtained using the procedure proposed in section 7.2 (compare Fig. 4).

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Some comments should be made on the results for the 9 glasses taken from the literature. Our value of $\beta$ for fused silica of 2.01 is in agreement with Refs. [11,13,14]; however, it was obtained for a different updated data set provided by M. Naftaly through private communication. Similarly, in BK-7 $\beta$ agrees with the value (2.6 ± 0.1) from [14], but not with the later reported value (2.28 ± 0.05) in [13]. However, for Pyrex, $\beta =$ 2.131 is higher than the highest literature value of 2.0 [14], but the higher value is expected according to the hypothesis [14,18] that $\beta$ increases with increasing disorder in the glass network, caused in this case by the addition of 6 % mol of glass network modifiers, Na₂O and Al₂O₃ [53]. The highest value among all the glasses, $\beta =$ 3.336, belongs to SK10, which is much higher than the previously reported [13] range of (2.8 ± 0.2). The higher value is, however, consistent with the fact that SK10 also possesses the greatest share of alkali network modifiers of 53.3%. The fitted value of $\beta$ for the in-house developed composition UV-710 is consistent with the described results. The correlation between the fraction of network modifiers and the value of the $\beta$ and $K$ parameters is summarized in Fig. 5.

Fig. 5. Far-infrared absorption parameters $\beta$ and $K$ as a function of the percentage of contained alkali network modifiers for the investigated borosilicate glasses and fused silica.

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With regard to the two lead silicate glasses SF6 and SF10, we have obtained $\beta$ values larger than in Ref. [13], i.e. 2.86 vs. 2.23 and 2.94 vs. 2.38, respectively. However, the $Kh^{2}$ values are very similar to the reported ranges [13], 7.56×10^-22 s² cm⁻¹ vs. (7.00±0.20)×10^-22 s² cm⁻¹ and 3.930×10^-22 s² cm⁻¹ vs. (4.30±0.10)×10^-22 s² cm⁻¹. Finally, our $K$ and $\beta$ results for the chalcogenide glasses S1 and S2 [18] agree within a few percent or less but for the $K$ parameter of the S3 glass, 3.73×10² cm⁻¹ vs. the reported range (2.80±0.01)×10² cm⁻¹.

To summarize, a part of the modeling results ($K$ and $\beta$ parameters) obtained using the proposed systematic approach to fitting the absorption data is in excellent agreement with other work. The new approach has revealed a clear correlation between $\beta$ and alkali network modifier content, which did not follow from previously published data. These two facts give strong credibility to the new method.

Some of the glasses investigated in the course of this work show extremely high levels of loss, namely, 4 of the 12 glass compositions posses values of $Kh^{2}$ in excess of 1×10^-24 s² cm⁻¹, which is significantly higher than for any previously reported glasses [13]. These compositions are (in descending order of $Kh^{2}$) SPB-51, TWPN/I/6, TCG-75921, SPB-54. Although these high values of $Kh^{2}$ are not correlated with high $\beta$ values, the latter are generally closer to the value of 3 than to the typically assumed $\beta \approx 2$. One can therefore inquire if $Kh^{2}$ is a good measure of the far-infrared absorption. The dependence of the $K$ and $Kh^{2}$ parameters on the refractive index at 0.5 THz is depicted in Fig. 6. In accordance with Eq. (16), both measures show an approximately $\propto n^{4}$ dependence, and only $K$ will be analyzed in the following sections.

Fig. 6. Absorption parameters $K$ and $Kh^{2}$ as a function of the refractive index both show approximately a linear relationship on the double logarithmic plot. The slopes of the dashed lines are equal to 4.09 and 4.05, respectively.

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Table 8 lists the total and electronic polarizabilities computed with Eqs. (18) and (19) and the $a = P_i / P_e$ ratio. The covalency/ionicity of the glass bonds can be assessed by comparing the value of $a$ with that of fused silica [51]. In this work this "reference" value is 0.93, computed at frequencies 0.5 THz ($P_{tot}$) and 100 THz ($P_e$). In other similar work [17] this value amounted to 0.81 due to a different choice of frequencies at which $P_{tot}$ and $P_e$ were evaluated. All glasses except ZBLAN and UV-710 exhibit a covalent character as $a < 0.93$. No direct correlation between the value of $a$ and $\beta$ can be observed. With the exception of the ZBLAN glass, $P_{tot}$ values are comparable with the total molecular polarizability computed using the recommended values after [54]. Despite the fact that the differences in the refractive index between THz and optical frequencies are quite large, they translate via Eq. (19) to moderate values of $P_{tot}$, which are comparable with the ones in Ref. [51].

Table 8. A summary of the parameters of the in-house developed glasses listed in Table 1. $M$ - molar mass, $\rho$ - mass density, $\epsilon _{THz}$ - real part of permittivity at 0.5 THz, $\epsilon _{opt}$ - real part of permittivity at 100 THz, $P_m$ - mean molecular polarizability after [54], $P_{tot}$ - total molecular polarizability from Eq. (19) at 0.5 THz, $a$ - ratio of the ionic to the electronic polarizability from Eqs. (18) to (20). The value of $a$ for fused silica for the chosen frequencies is 0.93 (0.81 in [17]), which can be considered as a "reference" value enabling the assessment of the covalent/ionic character of the chemical bonds. Some of the values of $P_{tot}$ are missing since respective sample densities were not available at the time of writing.

View Table | View all tables in this article

Relatively little attention in the literature is devoted to the dependence of $\beta$ on other glass properties. The $\beta$ vs. $K$ and $\beta$ vs. $n$ at 0.5 THz are depicted in Fig. 7. The highest values of $K$ belong to the TCG-75921 and SPB-51 glasses. Since the substitution of SiO₂ in PBG-08 with GeO₂ in PBG-08/40Ge did not increase the $K$ parameter significantly, we can conclude that Tl and Bi₂O₃ likely cause the increase in the polarizability of those glass compositions. This is justifiable by noting that Tl(I) shows quasi-random coordination numbers (numbers of nearest neighbors for a given atom) [55] and the coordination number of Bi(III) can exceed 9 [56], whereas for Pb(II) it is only up to 6 [53], leading to larger charge disorder (dangling bonds) with a greater number of possible spatial arrangements of the atoms in the glassy network [18]. Figure 7 shows that the $\beta$ parameter is highly correlated with $K$. The character of the correlation is highly dependent on the type of glass family. For borosilicate glasses $\beta$ increases much faster with $K$ than for other glass families. For chalcogenide glasses, a reverse trend is observed, namely the linear decrease of $\beta$ with increasing $K$. Calculations assuming a constant $V_D$ show that the charge density - amplitude quotient $N|e^{*}|^{2}$ also increases with increasing $K$ for these glasses. This suggests the presence of additional factors influencing the $\beta$ parameter other than polarizability or charge disorder. Despite having comparable or higher $K$-values, the SPB-51 and SPB-54 glasses show a relatively low $\beta$ value. This could be attributed to the fewer number of different components, which may contribute to a more ordered glass network. It is not surprising that Fig. 7(b) bears close resemblance to Fig. 7(a). Differences can be owed to the fact that $K$ depends on multiple glass parameters besides $n$ [Eq. (16)].

Fig. 7. (a) Relationship between the $\beta$ parameter and $K$ from Eq. (17). The dashed lines serve as a guide to the eye. (b) The $\beta$ vs. $n$ dependence.

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The relationship between the absorption parameter $K$ and the glass transition temperature $T_g$ is addressed in [57]. While there is a rough tendency for lower glass transition temperature materials to have a higher $K$ parameter Fig. 8, this is not the rule. $K$ increases with increasing $T_g$ for the 3 borosilicate compositions UV-710, BK7, SK10 with the greatest alkali network modifier content, and decreases with increasing $T_g$ for the 4 lead-bismuth-silicate glasses PBG-08, CS-1030, SPB-54, and SPB-N12.

Fig. 8. Relationship between the $K$ parameter from Eq. (17) and the glass transition temperature $T_g$. The dashed lines serve as a guide to the eye.

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8. Conclusion

A method of obtaining the complex refractive index of amorphous materials in an ultrabroadband range of 0.15 THz to 200 THz requiring two relatively easily available spectroscopic tools, a FTIR and a THz-TDS spectrometer, has been described in detail. The method was applied to the characterization of a set of 12 in-house developed glasses belonging to several types. The paper will highly likely be useful for the characterization of other amorphous materials, e.g., for applications in broadband modeling problems and experiments.

The most attractive glass in the context of wavelength translation experiments is the borosilicate composition UV-710; however, all the materials exhibit quite high THz absorption. Material-dependent permittivity models, aided by shape language modeling (SLM), have been reviewed and applied to modeling the UV-710 glass, most notably in the vicinity of the boson peak.

An unequivocal method of extracting Schlömann and Strom’s far-infrared absorption model parameters has been proposed. The new systematic approach revealed a strong correlation between $\beta$ and alkali network modifier content, which did not follow from previously published data, as well as correlations between the $\beta$ and $K$ parameters for different glass families.

Funding

Ministerstwo Nauki i Szkolnictwa Wyższego (DI 2016 021146); Fundacja na rzecz Nauki Polskiej (TEAM-TECH 2016-1/3).

Acknowledgments

Scientific work financed from budget funds for science in 2017-2020 as a research project under the "Diamond Grant" program. This work was also partially supported by the TEAM-TECH 2016-1/3 Project entitled "High-precision techniques of millimeter and sub-THz band characterization of materials for microelectronics" operated within the Foundation for Polish Science TEAM TECH Program co-financed by the European Regional Development Fund, Operational Program Smart Growth 2014-2020.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available in Ref. [58].

Supplemental document

See Supplement 1 for supporting content.

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Symbol	Composition (% mol)	$n_{2}$ (m² W⁻¹)
ZBLAN	53% ZrF₄, 20% BaF₂, 20% NaF, 4% LaF₃, 3% AlF₃	2.10×10^-20
TWPN/I/6	55% TeO₂, 38% WO₃, 5% Na₂O, 2% Nb₂O₅	5.11×10^-19
TZN-732005*	73% TeO₂ 20% ZnO 5% Na₂O; 2% La₂O₃	-
UV-710	54.2% SiO₂, 24.7% B₂O₃, 10.3% Li₂O,	9.30×10^-20
	5% Na₂O, 4.9% K₂O, 0.9% Al₂O₃
PBG-08	40% SiO₂, 30% PbO, 13% Ga₂O₃, 10% Bi₂O₃, 7% CdO	4.13×10^-19
PBG-08/40Ge	40% GeO₂; 30%PbO; 13% Ga₂O₃; 10% Bi₂O₃; 7% CdO	-
PBG-81**	40% SiO₂, 30% PbO, 13% Ga₂O₃, 10% Bi₂O₃, 7% CdO	1.95×10^-19
CS-1030	30% SiO₂, 30%PbO, 16% Ga₂O₃, 14% Bi₂O₃, 10% CdO	1.14×10^-18
TCG-75921	21% GeO₂; 30.5% PbO; 16% Bi₂O₃; 16% Ga₂O₃;	-
	9% CdO; 7.5% Tl₂O
SPB-51	50% SiO₂; 10% PbO; 40% Bi₂O₃	-
SPB-54	50% SiO₂; 40% PbO; 10% Bi₂O₃	-
SPB-N12	50% SiO₂; 20% PbO; 10% Bi₂O₃; 15% ZnO; 5% BaO	-

Symbol	Composition (% mol)	$n_{2}$ (m² W⁻¹)
Fused silica (FS)	100% SiO₂	4.3×10^-20
BK7	70% SiO₂; 11.5% B₂O₃; 9.5% Na₂O; 7.5% K₂O;	3.5×10^-20
	1.5% BaO [14]
Pyrex	81% SiO₂; 13% B₂O₃; 4% Na₂O; 2% Al₂O₃ [14]	-
SK10	48.2% BaO; 30.6% SiO₂; 11.7% B₂O₃; 5% Al₂O₃;	-
	2.0% ZnO; 0.9%As₂O₃; 0.8%Sb₂O₃; 0.7%PbO;
	0.1% Na₂O [17]
SF6	70.5% PbO; 27.7% SiO₂; 1% K₂O; 0.5% Na₂O;	1.61×10^-19
	0.3% As₂O₃ [17]
SF10	55.7% PbO; 35.3% SiO₂; 4% TiO₂; 2.5% K₂O;	> 1×10^-19
	2% Na₂O; 0.5% As₂O₃ [17]
S1	20% La; 20% Ga; 60% S (La₂₀Ga₂₀S₆₀) [18]	-
S2	16% La; 24% Ga; 60% S (La₁₆Ga₂₄S₆₀) [18]	-
S3	12% La; 28% Ga; 60% S (La₁₂Ga₂₈S₆₀) [18]	-

Equation (8) parameters
$ϵ_{\infty}$	$ϵ_{s}$	$τ$ (ps)	$α$	$γ$	$f_{m a x}$ (THz)
2.275	3.427	0.105	1.411	0.396	2.568

Equation (12) parameters
$f_{p}$ (THz)	$f_{c}$ (THz)	$σ$ (THz)	$A$	$β$	$f_{m a x}$ (THz)
16.141	15.625	4.421	2.996×10²⁷	0.924	40

Equation (11) parameters
$f_{p}$ (THz)	$σ$ (THz)	$f_{c}$ (THz)	A	$f_{m i n}$ (THz)	$f_{m a x}$ (THz)
16.0	3.250	0.10	5.980×10¹	0.03	32.0
21.263	2.433	0.10	1.057×10²	0	-
24.034	2.361	0.23	1.862×10²	0	-
26.952	2.427	1.0	6.118×10²	0	-
30.335	2.805	0.87	1.474×10³	19.0	32.3
41.586	3.869	1.0	2.712×10²	20.0	57.5

Reconstruction and modeling of the complex refractive index of nonlinear glasses from terahertz to optical frequencies

Abstract

1. Introduction

2. Spectral features of vitreous materials

3. Experimental details

3.1 Sample preparation

3.2 THz-TDS spectroscopy

3.3 FTIR spectroscopy

4. Kramers-Kronig reflectance analysis

5. Results

6. Broadband permittivity modeling

6.1 Debye-like relaxation models

6.2 Lorentz-type oscillator models

7. Far-infrared modeling

7.1 Introduction

7.2 Choice of fitting range

7.3 Results and discussion

8. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (8)

Tables (8)

Equations (20)

Optics Express

Equation (13) parameters
$f_{p}$ (THz)	$f_{c}$ (THz)	A
2.809×10¹	6.269	7.213
1.942×10³	5.081×10^-7	0.551
3.573×10³	4.340×10^-4	6.269

sample	$β$	$K$	$K h^{2}$	$f_{m i n}$	$f_{m a x}$	$f_{B P}$
		(cm⁻¹)	(cm⁻¹ s²)	(THz)	(THz)	(THz)
PBG-08	2.915	1.318×10³	8.887×10^-22	0.605	0.694	1.06
ZBLAN	2.393	4.502×10^-22	3.800×10^-22	0.439	1.074	1.38
TCG-75921	2.999	2.091×10³	1.360×10^-21	0.368	0.652	0.98
SPB-51	2.767	2.082×10³	1.496×10^-21	0.403	0.733	1.00
SPB-54	2.690	1.625×10³	1.207×10^-21	0.368	0.721	0.89
PBG-08/40Ge	2.963	1.501×10³	9.912×10^-22	0.409	0.700	1.11
PBG-81	2.877	1.180×10³	8.087×10^-22	0.403	0.830	1.11
SPB-N12	2.868	1.093×10³	7.521×10^-22	0.409	0.851	1.11
CS-1030	2.892	1.434×10³	9.761×10^-22	0.439	0.682	1.12
TWPN/I/6	2.641	1.894×10³	1.437×10^-21	0.368	0.943	1.22
UV-710	2.873	1.658×10²	1.139×10^-22	0.700	1.252	1.67
TZN-732005	3.051	1.433×10³	9.112×10^-22	0.486	0.724	1.27
BK-7	2.662	1.940×10²	1.459×10^-22	0.598	1.039	1.07
S1	3.082	3.676×10²	2.307×10^-22	0.688	0.950	1.00
S2	3.231	3.176×10²	1.869×10^-22	0.713	0.842	$> 1$
S3	3.042	3.730×10²	2.381×10^-22	0.659	0.918	1.00
SK10	3.336	2.652×10²	1.491×10^-22	0.601	0.864	0.94
Pyrex	2.131	6.591×10¹	6.230×10^-23	0.612	0.962	1.00
Fused silica (FS)	2.010	4.646	4.626×10^-24	0.635	1.058	1.06
SF6	2.862	1.096×10³	7.558×10^-22	0.428	0.604	0.60
SF10	2.941	5.892×10²	3.929×10^-22	0.518	0.676	0.71

sample	$M$	$ρ$	$ϵ_{T H z}$	$ϵ_{o p t}$	$ϵ_{T H z}$	$P_{m}$	$P_{t o t}$	$a$
	(g mol⁻¹)	(g cm⁻³)	@0.5 THz	@100 THz	$- ϵ_{o p t}$	(Å³)	(Å³)
TCG-75921	238	7.410	21.43	4.301	17.132	11.19	11.103	0.664
TZN-732005	142	-	20.15	4.101	16.049	10.33	-	0.701
TWPN/I/6	184	5.881	27.64	4.252	23.388	11.80	11.166	0.728
PBG-08/40Ge	189	-	18.00	3.765	14.234	9.75	-	0.772
CS-1030	193	6.580	18.31	3.739	14.567	10.28	9.913	0.786
SPB-51	239	6.190	21.31	3.812	17.501	10.96	13.323	0.801
PBG-08	171	5.798	16.60	3.526	13.070	9.59	9.801	0.835
PBG-81	171	-	15.95	3.483	12.465	9.59	-	0.839
SPB-54	166	6.550	17.37	3.441	13.927	8.31	8.486	0.884
SPB-N12	141	5.640	15.04	3.276	11.766	9.79	8.176	0.910
UV-710	61	-	6.236	1.821	4.060	17.87	-	1.257
ZBLAN	140	4.500	10.72	2.150	8.572	24.53	9.419	1.758