Abstract

Pure YAG single crystal was optically characterized in the wide spectral range (from far IR to vacuum UV) by applying the universal dispersion model. Data obtained from a broad range of characterization instruments and methods was simultaneously processed using least-square method and the result were compared with literature findings. The universal dispersion model describes individual elementary electron and phonon excitations in materials as separate contributions. For the first time recorded, an asymmetric Voigt peak approximation was used for modeling the contribution of one-phonon absorption in crystalline material. The optical constants are presented both graphically and in detailed dispersion parameters sets.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Single crystal yttrium aluminum garnet (YAG, Y3Al5O12) is the one of the most important host medium used for laser system, which together with the emerging solid-state pumping sources, made possible construction of high-power and high-efficiency laser systems. When doped with rare-earth ions (Nd, Er, Yb, Ho, Tm), it forms efficient active medium used in a wide range of systems, from low-power laser oscillators to multi-kW industrial or research systems [1]. Beside thin-disks, slabs or rods, the single crystal YAG fibers, both pure and doped, showed great potential in a variety of application [2]. The combination of its very low internal stress, high hardness and high heat and chemical resistance, makes the YAG single crystal an interesting alternative to sapphire for a variety of optics [3,4]. In high-power/high-energy laser system, the YAG can be even more attractive due to lack of birefringence, while its laser-induced damage threshold value scores high [58]. Due to its high non-linearity, the generation of supercontinuum in this solid-state material was also investigated [9].

Fabrication of the large size YAG substrates with low absorption for the specific laser wavelengths is still a technological challenge. The mechanism of the optical absorption is incomplete and depends on a number of manufacturing and post-processing. So understanding the optical properties of the YAG single crystal over a wide spectral range is crucial for using this material up to its very limits.

The broad range optical constants of crystalline materials can be precisely determined using different optical instruments combined with advanced modeling. Following the above mentioned, the universal dispersion model (UDM) [1012] is crucial for response function modeling, based on simultaneous experimental data processing, while a single physically realistic dispersion model is used. The UDM is a collection of elementary excitation models, fulfilling all tree fundamental conditions: time-reversal symmetry, Kramers-Kronig consistency and conformity with the sum rules.

2. Experimental data and data processing

The optical characterization presented in this paper is based on simultaneous processing of our experimental data (spectrophotometry and spectroscopic ellipsometry) and refractive index published by Zelmon et al. [13]. Our measurements were performed on four spectrophotometers: Bruker Vertex 70v, Bruker Vertex 80v, Perkin Elmer Lambda 1050 and McPherson VUVAS 1000 and two ellipsometers: Woollam IR-VASE and Horiba Jobin Yvon UVISEL (see Table 1). The experimental data were measured at room temperature (296–300 K) in the wide spectral range from far IR to vacuum UV. The experimental data enable us to precisely determine the spectral distribution of the phonon absorption in the IR and distribution of electronic excitations in UV regions. However, due to the systematic errors of these methods, it is not possible to determine the response function in the region of transparency with absolute accuracy as it is possible by minimal deviation method (MDM). Zelmon et al. measured a YAG prism by the MDM and tabulated the refractive index $n$ in his work. We assumed that YAG material of their prism is the same quality as our 2.555 mm thick (determined by a micrometer screw gauge) YAG plate – Czochralski grown in reducing atmosphere (Crytur [4]).

The data processing presented in this work is based on simultaneous fit of all measurements using equipments detailed in Table 1. For this purpose was used our internally developed optical characterization software newAD2 [15], enabling to perform heterogeneous data processing using least-square method (LSM) with Levenberg–Marquardt algorithm (LMA) [16] in the modified version, which automatically equalize the contributions of individual experimental data sets to the residual sum of squares [10,11]. This equalization allows the combination of individual data from different spectral ranges, obtained using different instruments, having distinct systematic errors at different spectral and angular densities of data points, while keeping sensitivity of the method to features in individual data sets. This modified LMA does not allow to define exactly the global quantity $\chi$ characterizing the quality of the fit for all the experimental data. For this reason why only the $\chi _i$ quantities defined for individual data sets are presented in Table 1. Note that the modified LMA provides all information about statistical uncertainties of the fit parameters and their correlations. For further details see Supplement 1 and out-final in Dataset 1 [17].

Tables Icon

Table 1. List of the experimental and tabulated data with spectral ranges (including resolution) and angle of incidence (AOI). The fitted experimental data $R$, $T$ and $I_\textrm {s},I_\textrm {c},I_\textrm {n}$ represent the reflectance, transmittance and associated ellipsometric quantities [14], respectively. The $\chi _i$ represents the quality of fits for corresponding data.

3. Results

3.1 Region of transparency

The refractive index of the undoped YAG single crystal, according to Zelmon et al. [13], was measured with the absolute accuracy better than $2 \times 10^{-4}$. The authors did not provide more precise estimation of errors, thus, we assumed for fitting a uniform accuracy $1 \times 10^{-4}$ for all the spectral points plotted in Fig. 1. Fit of YAG’s refractive index data by the UDM is also presented in the same figure. The root mean square (RMS) value of the difference between the UDM and the experimental data was found to be $1.26\times 10^{-4}$. It is known that the refractive index data in the region of transparency can be modeled by empirical Sellmeier formula [18]:

$$n(\lambda) = \left(1 + \sum_k \frac{A_k\lambda^{2}}{\lambda^{2} - \lambda_k^{2}} \right)^{1/2} ,$$
where summation is performed over individual terms. Zelmon et al. [13] showed that by using just two term Sellmeier formula, is sufficient for description of their experimental data (see also works of Sato and Taira [19] and Hrabovský et al. [20]). In Fig. 1 it is shown that Sellmeier model gives practically same fit as the UDM one. The RMS of the difference value between the Sellmeier model and experimental data was found to be $0.85\times 10^{-4}$, which is by 30% better than value achieved using the UDM fit. This is peculiar, because UDM should provide a more accurate dispersion dependence of the refractive index in the region of transparency, as it will be explained in the following paragraph.

 figure: Fig. 1.

Fig. 1. Spectral dependence of the refractive index $n$ of YAG crystal measured by Zelmon et al. [13] using the MDM. The curves represents fit of these data by universal dispersion model (UDM) and by harmonic oscillator model (HOM).

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The empirical Sellmeier formula is equivalent to real part of harmonic oscillator model (HOM) response function. Thus, the two-term Sellmeier formula can be understood as two independent harmonic oscillators with two resonant energies. The HOM is derived on the basis of classical equations of motion for bound charged particles. The complex response function can be written using generalized functions, as follows

$$\begin{aligned}\hat\varepsilon(E) = \hat n(E)^{2} = & \, 1 + \frac{2}{\pi} \frac{N_\textrm{el}}{E_\textrm{el}^{2} - E^{2}} + \frac{2}{\pi} \frac{N_\textrm{ph}}{E_\textrm{ph}^{2} - E^{2}}\\ & + {\textrm{i}} \frac{N_\textrm{el}}{E_\textrm{el}} \Big[\delta(E_\textrm{el} - E) - \delta(E_\textrm{el}+E) \Big] + {\textrm{i}} \frac{N_\textrm{ph}}{E_\textrm{ph}} \left[\delta(E_\textrm{ph} - E) - \delta(E_\textrm{ph}+E) \right] , \end{aligned}$$
where imaginary part of response function is written using four delta-functions at discrete resonant energies $\pm E_\textrm {el}$ and $\pm E_\textrm {ph}$. The resonant energies $E_\textrm {el}$ and $E_\textrm {ph}$ represents mean energies of electron and phonon excitations while the quantities $N_\textrm {el}$ and $N_\textrm {ph}$ are corresponding to transition strengths of these excitations. The transition strength quantities are related to the sum rule [21] as
$$\int_0^{\infty} E\, \varepsilon_{\mathrm{i}}(E) \, {\textrm{d}}E = \int_0^{\infty} F(E) \, {\textrm{d}}E = N_\textrm{el} + N_\textrm{ph} ,$$
where the function $F(E)=E \varepsilon _{\mathrm {i}}(E)$ is called transition strength function, i. e. distribution function of electron and phonon excitations. Transition strength quantities are proportional to density of the particles, square of their charge and inverse proportional to the mass.

 figure: Fig. 2.

Fig. 2. Spectral dependencies of the complex dielectric function $\hat \varepsilon$, calculated by universal dispersion model (UDM) and harmonic oscillator model (HOM) fitting Zelmon et al. [13] experimental data.

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Tables Icon

Table 2. Dispersion parameters of HOM by fitting Zelmon et al. [13] experimental data, compared with the transition strengths determined from UDM.

Four dispersion parameters of HOM are introduced in Table 2, while the corresponding spectral dependence is plotted in Fig. 2. Since HOM approximates the spectral distribution of electron and phonon excitations using only two discrete excitations, it is clear that the dielectric response in the region of transparency is a worse approximation than the one calculated by UDM.

In our opinion, it is a coincidence that systematic errors better correspond to the less precise model. Within UDM the distribution of electron and phonon excitations, i. e. transition strength function $F(E)$, are known relatively well (details are published in the following sections).

 figure: Fig. 3.

Fig. 3. Spectral dependencies of ellipsometric quantities displayed by the complex pseudodielectric function $\langle \hat \varepsilon \rangle$ and the degree of polarization $P$.

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The strength of all electron and phonon excitations calculated from UDM model as the sum of corresponding $N_x$ parameters can be compared to the transition strengths, predicted by HOM, from the fit of refractive index data (see Table 2). The differences between transition strengths are due to the asymmetric distribution of excitations described by $F(E)=E\varepsilon _{\mathrm {i}}(E)$ around the mean excitation energies $E_\textrm {el}$ and $E_\textrm {ph}$ (see Fig. 2). In addition, the actual electron density (transition strength) is much higher than determined from UDM, where the excitations of the core electrons have been neglected.

As mentioned earlier, the spectroscopic methods are not very sensitive to the absolute value of the refractive index or the extinction coefficient in the region of transparency. In the case of spectroscopic ellipsometry, this is mainly due to the uncertainty of the response from the sample surface. Figure 3 shows the ellipsometric data displayed using a pseudodielectric function $\langle \hat \varepsilon \rangle$ and the degree of polarization $P$. Because the measurement was performed on a relatively thick plate, the depolarization effects caused by the sample back side reflections had to be taken into account. A special algorithm truncating back side reflections implemented in newAD2 software was applied [14,15]. From Fig. 3 can be observed that the pseudodielectric function strongly depends on the angle of incidence and, in addition, contains jumps in the spectral points where the monochromators or photomultipliers are switched (for details see Supplement 1). Furthermore, the imaginary part of the pseudodielectric function, which in ideal case corresponds to the energy absorbed in the sample, is relatively high (0.2–0.3 at 6 eV). Later on, we will show that the real value of the imaginary part of the dielectric function is approximately $10^{-5}$. The high value of $\langle \varepsilon _{\mathrm {i}}\rangle$ can be explained by the non-ideal sample surface. Several surface effects can be accounted: surface roughness, adsorbed surface layer, polish-induced surface damage or naturally occurring surface effects associated with crystal termination (the surface dielectric response differs from bulk response, even for an ideal crystal surface, due to the broken symmetry).

Three approximations (theories) were tested for the description of surface and near-surface effects: Drude approximation (DA) [22,23], effective medium approximation (EMA) [24] and the Rayleigh-Rice theory (RRT) [2528]. The fits of experimental data using all models (including combinations) were practically equivalent (see Supplement 1). Although the choice of the surface model does not affect the fit of the experimental data, its choice may be important to minimize systematic errors in dispersion model parameter determination. The resulting model, combining RRT and EMA, is believed to realistically describe the surface. The RRT was used in this combination because this theory truthfully describes the bonding conditions at rough interfaces. For this model, the roughness is described using power spectral density function (PSDF). Then, if the Gaussian PSDF is assumed, the roughness is parameterized by the RMS of heights $\sigma$ and the autocorrelation length (lateral dimension of roughness) $\tau$. The EMA together with the Bruggeman formula [29] is a suitable choice for the description of the surface layer, because it simply allows to parameterize the optical response of this layer using thickness $d_\textrm {o}$ and packing density parameter $p_\textrm{m}$. The parameters obtained using this model were as follows: $\sigma =1.49(14)$ nm, $\tau =10.5(12)$ nm, $d_\textrm {o}=3.6(1.7)$ nm and $p_{\textrm{m}}=0.91(6)$.

From AFM data it is visible a very smooth surface, while the Fourier analysis reveals spatial frequencies corresponding to $\sigma =0.100(5)$ nm and $\tau =8.5(1.0)$ nm (see Supplement 1). Discrepancy between $\sigma$ parameter determined by optical method and by AFM can be explained by tip-surface convolution effects which underestimate $\sigma$ (tip radius is about 50 nm), therefore, the agreement can be considered as good as possible.

3.2 Absorption on localized states and Urbach tail

The existence of imperfections in the crystal network causes weak scattering (absorption). From the structure point of view, this scattering can be interpreted as excitations of the localized electrons with energies lying inside the forbidden energy band. Distribution of the localized states depends on purity and grow technology of the specific crystal. Response function can be effectively described as the sum of Gaussian broadened discrete transitions. Four discrete transitions were sufficient to describe the YAG sample. Thus, the contribution to the dielectric function can be written as follows [12,3033]

$$\begin{aligned} \hat\varepsilon_\textrm{loc}(E) & = \sum_{j=1}^{4} \frac{N_{\textrm{loc},j}}{\sqrt{2\pi} E_{\textrm{loc},j} B_{\textrm{loc},j}} \Bigg[ \frac{2}{\sqrt{\pi}} \mathrm{D}\!\left( \frac{E+E_{\textrm{loc},j}}{\sqrt{2} B_{\textrm{loc},j}} \right) - \frac{2}{\sqrt{\pi}} \mathrm{D}\!\left( \frac{E-E_{\textrm{loc},j}}{\sqrt{2} B_{\textrm{loc},j}} \right)\\ & + {\textrm{i}} \exp\!\left(-\frac{(E-E_{\textrm{loc},j})^{2}}{2 B_{\textrm{loc},j}^{2}}\right) - {\textrm{i}} \exp\!\left(-\frac{(E+E_{\textrm{loc},j})^{2}}{2 B_{\textrm{loc},j}^{2}}\right) \Bigg] , \end{aligned}$$
where $N_{\textrm {loc},j}$, $E_{\textrm {loc},j}$ and $B_{\textrm {loc},j}$ are transition strength of excitation of corresponding localized states, mean energy of the excitations and the Gaussian broadening parameter (RMS value). The symbol $\mathrm {D}(\cdot )$ denotes the Dawson function (integral) [34].

The proof data to demonstrate absorption at localized states is the UV-visible transmittance data shown in Fig. 4, where the Gaussian absorption bands are evident. The values of the dispersion parameters describing these absorption bands are introduced in Table 3.

 figure: Fig. 4.

Fig. 4. Spectral dependencies of the transmittance $T$ measured in the visible and UV spectral range and corresponding modeled imaginary part of dielectric function $\varepsilon _{\mathrm {i}}$. The separated UDM contributions of the absorption on localized states and Urbach tail are plotted by dash and dash–dot lines.

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 figure: Fig. 5.

Fig. 5. Spectral dependence of modeled imaginary part of dielectric function $\varepsilon _{\mathrm {i}}$ in the region of the electron excitations. The separated UDM contributions of the absorption on localized states and Urbach tail are plotted by dash and dash–dot lines.

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Figures 4 and 5 show that for energies higher than $6.3$ eV, the absorption in the YAG crystal increases exponentially. This exponential increase is associated with the well-known Urbach tail, which corresponds to the weak absorption below the band gap energy $E_\textrm {g}$. The Urbach tail can be explained as the transitions of the electrons from the localized valence states to the extended unoccupied conduction states and the transitions from the extended valence states to the localized unoccupied states [35]. In other worlds, the Urbach tail is the transition region between the absorption on localized states and the interband transitions regions. The Fermi energy is assumed to be in center of forbidden energy band. Thus, the minimum excitation energy is half of the band gap energy, i. e. excitation from the occupied Fermi level to the bottom of the unoccupied conduction band or from the occupied maximum of the valence band to the unoccupied Fermi level. In this case, at photon energies $E \in (0; E_\textrm {g}/2)$ no absorption is assumed. Then, up to the band gap energy, i. e. for photon energies $E \in (E_\textrm {g}/2; E_\textrm {g})$, the absorption corresponds to exponential Urbach tail. Finally, the absorption above the band energy is modeled by a rational function in which its behavior corresponds to the classical asymptotic behavior of bound states, i. e. $\varepsilon _\textrm {i,ut}(E) \approx 1/E^{3}$ for large energies. Thus, the exponential and rational part have the following forms

$$\varepsilon_\textrm{i,ut}(E) = \frac{A_\textrm{exp}}{E} \left[ \exp\!\left(\frac{E - E_\textrm{g}}{E_\textrm{u}}\right) - \exp\!\left(\frac{- E_\textrm{g}}{2E_\textrm{u}}\right) \right] \qquad \textrm{or} \qquad \varepsilon_\textrm{i,ut}(E) = \frac{A_\textrm{rat} (E-E_\textrm{x})^{2}}{E^{5}} ,$$
where $E_\textrm {u}$ is Urbach energy, dispersion parameter. The internal parameters $A_\textrm {exp}$, $A_\textrm {rat}$ and $E_\textrm {x}$ are chosen so the function $\varepsilon _\textrm {i,ut}(E)$ is smooth up to the first derivative in the point $E=E_\textrm {g}$ and simultaneously the sum rule is equal to transition strength of the Urbach tail excitations $N_\textrm {ut}$, third dispersion parameter of this contribution (see Table 3). The real part of the response function contribution is easily expressed by Kramers–Kronig integral (see Supplement 1). In Fig. 5 the imaginary part of the dielectric function is shown in a logarithmic scale, where exponential part of the Urbach tail is displayed as linear function below the band gap energy. For energies above the band gap energy the Urbach tail contribution overlap interband transitions.

Tables Icon

Table 3. Dispersion parameters describing electronic excitations.

3.3 Interband electron excitations

The interband transitions correspond to the excitations of the electrons from the extended valence states to the extended conduction states. The interband contribution is modeled as the sum of eight contributions. Each contribution represent an absorption structure above the band gap energy (see reflectance in Fig. 6). These structures (excitons) are modeled by Campi–Coriasso model [12,3638], where the imaginary part is zero for energies below the $E_\textrm {g}$, otherwise

$$\varepsilon_{\textrm{i,ex},k}(E) = \frac{2 N_{\textrm{ex},k} B_{\textrm{ex},k} \ (E-E_\textrm{g})^{2} \ \Theta(E-E_\textrm{g})}{ \pi E \Big[ \Big((E_{\textrm{ex},k}-E_\textrm{g})^{2} - (E-E_\textrm{g})^{2}\Big)^{2} + B_{\textrm{ex},k}^{2} \, (E-E_\textrm{g})^{2} \big] } ,$$
where $\Theta (\cdot )$ is the Heaviside step function. The parameter $E_{\textrm {ex},k}$ (central exciton energy) must fulfill the inequality $E_{\textrm {ex},k}>E_\textrm {g}$. The real part of the contribution is calculated by the Kramers–Kronig integral, where the analytic expressions can be found in [38]. The dielectric response of the interband electronic excitations is then calculated as the sum of excitonic contributions. Each exciton is described by three individual parameters, i. e. transition strength $N_{\textrm {ex},k}$, exciton energy $E_{\textrm {ex},k}$ and broadening parameter $B_{\textrm {ex},k}$, and one common parameter $E_\textrm {g}$ identical with band gap energy used in Urbach tail model. The fitted parameters are introduced in Table 3. It must be emphasized that not all eight excitons in the model are real absorption structures observed in experimental data. Only seven absorption structures can be observed in experimental reflectance data above the band gap energy (see Fig. 6). The absorption band corresponding to the eighth broad structure is used to estimate the dielectric response in the spectral range outside the measured data. The eighth absorption band uncertainty influences the overall uncertainty for transition strength parameters determination, but it does not influence the optical constants determination uncertainty in the experimental data range, especially for refractive index in the region of transparency. It should be emphasized that the optical constants predicted by UDM in this work for energies higher than 10 eV are only a very rough estimate, and that it is necessary to add synchrotron measurement data to determine the proper optical constants for higher energies. YAG crystal reflectance was measured on a synchrotron for energies up to 116 eV by Tomiki et al. [39,40]. A fast comparison to their measurements reveals a higher probability of systematic artifacts for energies higher than 9 eV in case of our optical constants data. Therefore, the existence of the excitons and credibility of optical constants in this region can be questioned.

 figure: Fig. 6.

Fig. 6. Spectral dependencies of the reflectance $R$ measured in the UV and VUV spectral range and corresponding modeled complex dielectric function $\hat \varepsilon$. The separated UDM Urbach tail contribution is plotted by dashed line. The exciton position are shown by arrows.

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The first-principles calculations [41] predict direct band gap for energy 4.71 eV while the presented UDM assumes an indirect band gap at energy 6.652(8) eV. The typical direct square-root band gap with the discrete excitons was not observed in the optical data, thus, a well known fact must be stated: today’s ab-initio calculations do not allow to correctly predict either the energy or the nature of the band gap [42] for such complicate system as the YAG crystal is.

3.4 One-phonon absorption

Primitive cell of YAG crystal contains 80 atoms, thus 240 normal modes of vibration exist in this crystal. However, due to the crystal symmetry only 17 modes are IR-active [4346]. The UDM incorporates these one-phonon absorption structures using Voigt peaks [12,47]. This model represents generalization of coupled harmonic oscillator model replacing Lorentzian distribution function by a more general Voigt profile. Moreover, asymmetric peak approximation is used for description of the coupling effects compared to the classical model, where the 17 vibration modes generates 136 coupling parameters (one parameter for each pair of peaks). Using this approximation, the coupling parameters are reduced to 16 independent parameters. As a result of this reduction, the asymmetric Voigt peak approximation (AVPA) yields better fit compared to the exact solution for classical equations of motion or by other approximations tested (for details see Supplement 1). The AVPA contribution to the dielectric function of one-phonon absorption peaks can be written using complex Faddeeva function $\hat {\mathrm {W}}(\cdot )$ in the following closed form

$$\begin{aligned} \hat\varepsilon_\textrm{1ph}(E) = & \frac{1}{\mathcal{C}_{\mathrm{N}}} \sum_{j=1}^{17} \left( N_{\textrm{ph},j} + {\textrm{i}}M_{\textrm{ph},j} \frac{E}{E_{\textrm{ph},j}} \right) \frac{{\textrm{i}}}{\sqrt{2\pi} E_{\textrm{ph},j} B_{\textrm{G},j}}\\ & \Bigg[ \hat{\mathrm{W}}\!\left(\frac{E-E_{\textrm{ph},j}+{\textrm{i}}B_{\textrm{L},j}/2}{\sqrt{2} B_{\textrm{G},j}}\right) - \hat{\mathrm{W}}\!\left(\frac{E+E_{\textrm{ph},j}+{\textrm{i}}B_{\textrm{L},j}/2}{\sqrt{2} B_{\textrm{G},j}}\right) \Bigg], \end{aligned}$$
where $B_{\textrm {G},j}$ and $B_{\textrm {L},j}$ are root mean square (RMS) of the Gaussian part and full with at half maximum (FWHM) of the Lorentzian part of the Voigt profile. These internal parameters are calculated from FWHM of the Voigt profile $B_{\textrm {ph},j}$ using following approximate equations
$$B_{\textrm{G},j} = \frac{B_{\textrm{ph},j}}{2\sqrt{2 \ln 2}} \sqrt{(1-a L_{\textrm{ph},j})^{2}-(1-a)^{2} L_{\textrm{ph},j}^{2}} , \quad a=0.5346 \quad B_{\textrm{L},j} = B_{\textrm{ph},j} L_{\textrm{ph},j} .$$

The parameter $L_{\textrm {ph},j}$ takes values between 0 (fully Gaussian model) and 1 (fully Lorentzian model). Finally, the spectral position and width of phonon excitations is usually introduced in wavenumber unit, thus, instead $E_{\textrm {ph},j}$ and $B_{\textrm {ph},j}$ in eV unit are used the parameters $\nu _{\textrm {ph},j}$ and $\beta _{\textrm {ph},j}$ in $\textrm{cm}^{-1}$ unit ($E_{\textrm {ph},j}= h \nu _{\textrm {ph},j}$ and $B_{\textrm {ph},j}= h \beta _{\textrm {ph},j}$, where $h$ is Planck’s constant).

For fully Lorentzian model the (7) must be calculated as an asymmetric Lorentzian peak approximation (ALPA)

$$\hat\varepsilon_\textrm{1ph}(E) = \frac{2}{\pi\mathcal{C}_{\mathrm{N}}} \sum_{j=1}^{17} \frac{N_{\textrm{ph},j} + {\textrm{i}}M_{\textrm{ph},j} E / E_{\textrm{ph},j}}{E_{\textrm{ph},j}^{2}+B_{\textrm{L},j}^{2}/4-E^{2}-{\textrm{i}}B_{\textrm{L},j} E} .$$

The previous ALPA form can be rewritten in the factorized form of damped harmonic oscillator model [12,48]. The parameter $M_{\textrm {ph},j}$ expresses the coupling between the $j$-th vibration mode and the rest of the system and it can have both positive and negative values. The factor $1/\mathcal {C}_{\mathrm {N}}$ is a normalization constant. Each absorption peak depends on five parameters: $N_{\textrm {ph},j}$ transition strength; $M_{\textrm {ph},j}$ asymmetry of the peak; $\nu _{\textrm {ph},j}$ wavenumber of vibration mode; $\beta _{\textrm {ph},j}$ FWHM of peak; $L_{\textrm {ph},j}$ strength of Lorentzian part. The condition for $M_{\textrm {ph},j}$ parameters together with the normalization constant

$$\sum_{j=1}^{17} \frac{M_{\textrm{ph},j}}{\nu_{\textrm{ph},j}} = 0 , \qquad \mathcal{C}_{\mathrm{N}} = \frac{1}{\sum_{j=1}^{17} N_{\textrm{ph},j}} \sum_{j=1}^{17} \left( N_{\textrm{ph},j} + \frac{M_{\textrm{ph},j} \beta_{\textrm{L},j}}{\nu_{\textrm{ph},j}} \right) ,$$
ensures convergence and validity of sum rule integral as the sum of $N_{\textrm {ph},j}$ parameters.

Tables Icon

Table 4. Dispersion parameters of the phonon excitations. $^{\dagger}$ Internal parameter $M_\textrm {ph17}$ is calculated according to condition (10).

All 84 dispersion parameters together with the internal parameter $M_\textrm {ph17}$ are introduced in Table 4. In the one-phonon absorption spectral range, the sample is non-transparent, so the response function can be measured by ellipsometry or by reflectivity in the FIR and MIR regions (see Figs. 7 and 8). From these figures it is seen that only 16 absorption peaks can be clearly identified. These 16 absorption peaks corresponds to vibration modes $j=1,\dots,11,13,\dots,17$. In the Hofmeister and Campbell work [45] a very weak 12-th vibration mode is assumed at $\textrm{460}\,cm^{-1}$. In the presented work, this vibration mode was found at similar frequency (see parameters in Table 4). The parameters of this peak was not possible to fit because due to the remaining systematic errors the fitting procedure is unstable and it have tendency to use the parameters for correction of the shape of peak 11 or 13. For details see Supplement 1. Assignment of all 17 vibration modes was done in Hofmeister and Campbell work [45] (see also [46]).

 figure: Fig. 7.

Fig. 7. Spectral dependencies of ellipsometric quantities displayed by the complex pseudodielectric function $\langle \hat \varepsilon \rangle$ in the one-phonon absorption spectral range.

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 figure: Fig. 8.

Fig. 8. Spectral dependencies of reflectance $R$ and transmittance $T$ measured in the spectral range of the one-phonon excitations and corresponding modeled complex dielectric function $\hat \varepsilon$. For the 12-th vibrational mode concern to Table 4 and details in text.

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 figure: Fig. 9.

Fig. 9. Spectral dependencies of reflectance $R$ and transmittance $T$ measured in the one- and multi-phonon regions and corresponding modeled imaginary part of dielectric function $\varepsilon _{\mathrm {i}}$.

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3.5 Multi-phonon absorption

In the case of multi-phonon absorption, all the phonons from all the branches and the entire Brillouin zone participate on absorption process. Thus, the two-phonon processes cover spectral range from zero frequency up to the double of maximum phonon frequency, three-phonon processes cover spectral range up to the triple of maximum phonon frequency and so on (see Figs. 9 and 10). The maximum phonon frequency ($818.2\,\textrm {cm}^{-1}$) was estimated as a longitudinal frequency of 17th vibrational mode. Multi-phonon absorption can be modeled by Gaussian broadened bands described by piecewise continuous functions, corresponding to the additive and subtractive joint density of states for phonon branches [12,49]. In this study, the modeling was simplified by model of Gaussian broadened discrete spectrum, where each excitation is described by 3 parameters (see Table 4). Because the probability of absorption processes decreases with increasing number of phonons participating on these processes, then, also the transition strength of phonon absorption processes decreases with the number of phonons. This is the reason why it is very difficult to separate them. For separation of individual absorption processes, it is necessary to perform multi-temperature measurements and to use the different temperature dependencies of individual absorption processes, as it was performed for crystalline silicon [50]. Therefore, it is impossible to say that wide structures 18 and 19 are multi-phonon excitations. These broad peaks can be interpreted as some corrections of the one-phonon excitations.

 figure: Fig. 10.

Fig. 10. Spectral dependencies of ellipsometric quantities displayed by pseudodielectric function $\langle \hat \varepsilon \rangle$ and the degree of polarization $P$ in the multi-phonon region.

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4. Conclusion

The goal of this work was to study the optical properties of YAG single crystal in wide spectral range by applying a universal dispersion model. The YAG single crystal, which is a popular laser host material, was studied from VUV to FIR range, using reflectance, transmittance and ellipsometric measurements, all performed at room temperature. The universal dispersion model described individual elementary electron and phonon excitations in material, presented in a unified formalism, which is parameterized using normalized transition strength functions, respecting conformity with the sum rules [1012]. Absorption phenomena (interband electronic excitations, absorption on localized states, Urbach tail and phonon absorptions) contributing to the dielectric response were discussed in detail. For the one-phonon contribution, the model used an asymmetric Voigt peak approximation (AVPA), which represented the best fit of the experimental data. The optical constants modeled in this study were determined with high absolute precision (in order of 10$^{-4}$). The results presented are an important starting point for a more complex research, where various functional dopants (Nd, Yb, Er, Ce, Sm, Tu, Ho, Cr, V, etc.) in the YAG’s matrix can be comprehensively studied. The work can be extended to a temperature-dependent dispersion model or it can be extrapolated to similar crystalline host material such as YAP and LuAG. Therefore, we showed that the UDM can be successfully used for determination of optical constants with absolute precision and for dissemination of optical constants data instead of tabulated values.

Funding

Ministerstvo Školství, Mládeže a Tělovýchovy (CZ.02.1.01/0.0/0.0/15_006/0000674, LM2018097, LM2018110); Horizon 2020 Framework Programme (NO 739573).

Acknowledgments

The authors thank Dr. Petr Klapetek for AFM measurement of YAG sample. The optical measurements were performed on the CEPLANT and CEITEC Nano Research Infrastructure.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available in Dataset 1 [17].

Supplemental document

See Supplement 1 for supporting content.

References

1. V. Lupei and A. Lupei, “Nd:YAG at its 50th anniversary: still to learn,” J. Lumin. 169, 426–439 (2016). [CrossRef]  

2. B. Liu and P. R. Ohodnicki, “Fabrication and application of single crystal fiber: review and prospective,” Adv. Mater. Technol. 6(9), 2100125 (2021). [CrossRef]  

3. Y. Nagata, R. Takeda, and M. Suginome, “High-pressure circular dichroism spectroscopy up to 400 MPa using polycrystalline yttrium aluminum garnet (YAG) as pressure-resistant optical windows,” RSC Adv. 6(111), 109726 (2016). [CrossRef]  

4. “Crytur web pages,” https://www.crytur.cz/materials/yag-undoped/.

5. D. E. Zelmon, K. L. Schepler, S. Guha, D. J. Rush, S. M. Hegde, L. P. Gonzalez, and J. Lee, “Optical properties of Nd-doped ceramic yttrium aluminum garnet,” in Laser-Induced Damage in Optical Materials, vol. 5647 of Proc. SPIEG. J. Exarhos, A. H. Guenther, N. Kaiser, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds. (2004), pp. 255–264.

6. B. T. Do and A. V. Smith, “Bulk optical damage thresholds for doped and undoped, crystalline and ceramic yttrium aluminum garnet,” Appl. Opt. 48(18), 3509–3514 (2009). [CrossRef]  

7. J. Vanda, M.-G. Mureșan, P. Čech, M. Hanuš, A. Lucianetti, D. Rostohar, T. Mocek, Š. Uxa, and V. Škoda, “Multiple pulse nanosecond laser-induced damage threshold on AR coated YAG crystals,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108050F.

8. Š. Uxa, V. Škoda, J. Vanda, and M.-G. Mureșan, “A comparison of LIDT behavior of AR-coated yttrium-aluminium-garnet substrates with respect to thin-film design and coating technology,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108051R.

9. N. Garejev, G. Tamošauskas, and A. Dubietis, “Comparative study of multioctave supercontinuum generation in fused silica, YAG, and LiF in the range of anomalous group velocity dispersion,” J. Opt. Soc. Am. B 34(1), 88–94 (2017). [CrossRef]  

10. D. Franta, D. Nečas, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to hafnia,” Appl. Opt. 54(31), 9108–9119 (2015). [CrossRef]  

11. D. Franta, D. Nečas, A. Giglia, P. Franta, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to magnesium fluoride,” Appl. Surf. Sci. 421, 424–429 (2017). [CrossRef]  

12. D. Franta, J. Vohánka, and M. Čermák, “Universal dispersion model for characterization of thin films over wide spectral range,” in Optical Characterization of Thin Solid Films, O. Stenzel and M. Ohlídal, eds. (Springer, 2018), Springer Series in Surface Sciences 64, pp. 31–82.

13. D. E. Zelmon, D. L. Small, and R. Page, “Refractive-index measurements of undoped yttrium aluminum garnet from 0.4 to 5.0 μm,” Appl. Opt. 37(21), 4933–4935 (1998). [CrossRef]  

14. I. Ohlídal, J. Vohánka, M. Čermák, and D. Franta, “Ellipsometry of layered systems,” in Optical Characterization of Thin Solid Films, vol. 64O. Stenzel and M. Ohlídal, eds. (Springer, Cham, 2018), pp. 233–267.

15. D. Franta, D. Nečas, and J. Vohánka, “Software for optical characterization newAD2,” http://newad.physics.muni.cz.

16. W. D. Marquardt, “An algorithm for least squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11(2), 431–441 (1963). [CrossRef]  

17. D. Franta, “Outpout files of newAD2 and Gwyddion software,” figshare, 2021, https://doi.org/10.6084/m9.figshare.16832731.

18. W. Sellmeier, “Ueber die durch die Aetherschwingungen erregten Mitschwingungen der Körpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien,” Ann. Phys. 223(11), 386–403 (1872). [CrossRef]  

19. Y. Sato and T. Taira, “Highly accurate interferometric evaluation of thermal expansion and dn/dT of optical materials,” Opt. Mater. Express 4(5), 876–888 (2014). [CrossRef]  

20. J. Hrabovský, M. Kučera, L. Paloušová, L. Bi, and M. Veis, “Optical characterization of Y3Al5O12 and Lu3Al5O12 single crystals,” Opt. Mater. Express 11(4), 1218–1223 (2021). [CrossRef]  

21. D. Franta, D. Nečas, and L. Zajíčková, “Application of Thomas–Reiche–Kuhn sum rule to construction of advanced dispersion models,” Thin Solid Films 534, 432–441 (2013). [CrossRef]  

22. P. Drude, “Ueber die Reflexion und Brechung ebener Lichtwellen beim Durchgang durch eine mit Oberflächenshichten behaftete planparallelr Platte,” Ann. Phys. 279(5), 126–157 (1891). [CrossRef]  

23. Z. Knittl, Optics of Thin Films (Wiley, London, 1976).

24. D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20(8), 3292–3302 (1979). [CrossRef]  

25. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Comm. Pure Appl. Math. 4(2-3), 351–378 (1951). [CrossRef]  

26. D. Franta and I. Ohlídal, “Ellipsometric parameters and reflectances of thin films with slightly rough boundaries,” J. Mod. Opt. 45(5), 903–934 (1998). [CrossRef]  

27. D. Franta and I. Ohlídal, “Comparison of effective medium approximation and Rayleigh–Rice theory concerning ellipsometric characterization of rough surfaces,” Opt. Commun. 248(4-6), 459–467 (2005). [CrossRef]  

28. J. Vohánka, M. Čermák, D. Franta, and I. Ohlídal, “Efficient method to calculate the optical quantities of multi-layer systems with randomly rough boundaries using the Rayleigh–Rice theory,” Phys. Scr. 94(4), 045502 (2019). [CrossRef]  

29. D. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. 416(7), 636–664 (1935). [CrossRef]  

30. M. E. Thomas, S. K. Andersson, R. M. Sova, and R. I. Joseph, “Frequency and temperature dependence of the refractive index of sapphire,” Infrared Phys. Technol. 39(4), 235–249 (1998). [CrossRef]  

31. D. De Sousa Meneses, J.-F. Brun, B. Rousseau, and P. Echegut, “Polar lattice dynamics of the MgAl2O4 spinel up to the liquid state,” J. Phys. Condens. Matter 18(24), 5669–5686 (2006). [CrossRef]  

32. D. De Sousa Meneses, M. Malki, and P. Echegut, “Structure and lattice dynamics of binary lead silicate glesses investigated by infrared spectroscopy,” J. Non-Cryst. Solids 352(8), 769–776 (2006). [CrossRef]  

33. R. Kitamura, L. Pilon, and M. Jonasz, “Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature,” Appl. Opt. 46(33), 8118–8133 (2007). [CrossRef]  

34. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series - 55 (National Bureau of Standards, 1964).

35. J. Tauc, “Optical properties of non-crystaline solids,” in Optical Properties of Solids, F. Abelès, ed. (North–Holland, 1972), pp. 277–313.

36. D. Campi and C. Coriasso, “Prediction of optical properties of amorphous tetrahedrally bounded materials,” J. Appl. Phys. 64(8), 4128–4134 (1988). [CrossRef]  

37. D. Campi and C. Coriasso, “Relationships between optical properties and band parameters in amorphous tetrahedrally bonded materials,” Mater. Lett. 7(4), 134–137 (1988). [CrossRef]  

38. D. Franta, M. Čermák, J. Vohánka, and I. Ohlídal, “Dispersion models describing interband electronic transitions combining Tauc’s law and Lorentz model,” Thin Solid Films 631, 12–22 (2017). [CrossRef]  

39. T. Tomiki, F. Fukudome, M. Kaminao, M. Fujisawa, Y. Tanahara, and T. Futemma, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region,” J. Phys. Soc. Japan 58(5), 1801–1810 (1989). [CrossRef]  

40. T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993). [CrossRef]  

41. Y.-N. Xu and W. Ching, “Electronic structure of yttrium aluminium garnet (Y3Al5O12),” Phys. Rev. B 59(16), 10530–10535 (1999). [CrossRef]  

42. P. Ondračka, D. Holec, and L. Zajíčková, “Predicting optical properties from ab initio calculations,” in Optical Characterization of Thin Solid Films, O. Stenzel and M. Ohlídal, eds. (Springer, Cham, 2018), Springer Series in Surface Sciences 64, pp. 83–104.

43. J. P. Hurrell, S. P. S. Porto, I. F. Chang, S. S. Mitra, and R. P. Bauman, “Optical phonons of yttrium aluminum garnet,” Phys. Rev. 173(3), 851–856 (1968). [CrossRef]  

44. G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10- to 55 000-cm−1 wave numbers,” Phys. Rev. 177(3), 1308–1314 (1969). [CrossRef]  

45. A. M. Hofmeister and K. R. Campbell, “Infrared spectroscopy of yttrium aluminum, yttrium gallium, and yttrium iron garnets,” J. Appl. Phys. 72(2), 638–646 (1992). [CrossRef]  

46. S. Kostić, Z. Lazarević, V. Radojević, A. Milutinović, M. Romčević, N. Romčević, and A. Valčić, “Study of structural and optical properties of YAG and Nd:YAG single crystals,” Mater. Res. Bull. 63, 80–87 (2015). [CrossRef]  

47. D. Franta, J. Vohánka, M. Bránecký, P. Franta, M. Čermák, I. Ohlídal, and V. Čech, “Optical properties of the crystalline silicon wafers described using the universal dispersion model,” J. Vac. Sci. Technol. B 37(6), 062907 (2019). [CrossRef]  

48. J. Humlíček, R. Henn, and M. Cardona, “Infrared vibrations in LaSrGaO4 an LaSrAlO4,” Phys. Rev. B 61(21), 14554–14563 (2000). [CrossRef]  

49. D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Dispersion model of two-phonon absorption: application to c-Si,” Opt. Mater. Express 4(8), 1641–1656 (2014). [CrossRef]  

50. D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, and I. Ohlídal, “Temperature-dependent dispersion model of float zone crystalline silicon,” Appl. Surf. Sci. 421, 405–419 (2017). [CrossRef]  

References

  • View by:

  1. V. Lupei and A. Lupei, “Nd:YAG at its 50th anniversary: still to learn,” J. Lumin. 169, 426–439 (2016).
    [Crossref]
  2. B. Liu and P. R. Ohodnicki, “Fabrication and application of single crystal fiber: review and prospective,” Adv. Mater. Technol. 6(9), 2100125 (2021).
    [Crossref]
  3. Y. Nagata, R. Takeda, and M. Suginome, “High-pressure circular dichroism spectroscopy up to 400 MPa using polycrystalline yttrium aluminum garnet (YAG) as pressure-resistant optical windows,” RSC Adv. 6(111), 109726 (2016).
    [Crossref]
  4. “Crytur web pages,” https://www.crytur.cz/materials/yag-undoped/ .
  5. D. E. Zelmon, K. L. Schepler, S. Guha, D. J. Rush, S. M. Hegde, L. P. Gonzalez, and J. Lee, “Optical properties of Nd-doped ceramic yttrium aluminum garnet,” in Laser-Induced Damage in Optical Materials, vol. 5647 of Proc. SPIEG. J. Exarhos, A. H. Guenther, N. Kaiser, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds. (2004), pp. 255–264.
  6. B. T. Do and A. V. Smith, “Bulk optical damage thresholds for doped and undoped, crystalline and ceramic yttrium aluminum garnet,” Appl. Opt. 48(18), 3509–3514 (2009).
    [Crossref]
  7. J. Vanda, M.-G. Mureșan, P. Čech, M. Hanuš, A. Lucianetti, D. Rostohar, T. Mocek, Š. Uxa, and V. Škoda, “Multiple pulse nanosecond laser-induced damage threshold on AR coated YAG crystals,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108050F.
  8. Š. Uxa, V. Škoda, J. Vanda, and M.-G. Mureșan, “A comparison of LIDT behavior of AR-coated yttrium-aluminium-garnet substrates with respect to thin-film design and coating technology,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108051R.
  9. N. Garejev, G. Tamošauskas, and A. Dubietis, “Comparative study of multioctave supercontinuum generation in fused silica, YAG, and LiF in the range of anomalous group velocity dispersion,” J. Opt. Soc. Am. B 34(1), 88–94 (2017).
    [Crossref]
  10. D. Franta, D. Nečas, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to hafnia,” Appl. Opt. 54(31), 9108–9119 (2015).
    [Crossref]
  11. D. Franta, D. Nečas, A. Giglia, P. Franta, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to magnesium fluoride,” Appl. Surf. Sci. 421, 424–429 (2017).
    [Crossref]
  12. D. Franta, J. Vohánka, and M. Čermák, “Universal dispersion model for characterization of thin films over wide spectral range,” in Optical Characterization of Thin Solid Films, O. Stenzel and M. Ohlídal, eds. (Springer, 2018), Springer Series in Surface Sciences 64, pp. 31–82.
  13. D. E. Zelmon, D. L. Small, and R. Page, “Refractive-index measurements of undoped yttrium aluminum garnet from 0.4 to 5.0 μm,” Appl. Opt. 37(21), 4933–4935 (1998).
    [Crossref]
  14. I. Ohlídal, J. Vohánka, M. Čermák, and D. Franta, “Ellipsometry of layered systems,” in Optical Characterization of Thin Solid Films, vol. 64O. Stenzel and M. Ohlídal, eds. (Springer, Cham, 2018), pp. 233–267.
  15. D. Franta, D. Nečas, and J. Vohánka, “Software for optical characterization newAD2,” http://newad.physics.muni.cz .
  16. W. D. Marquardt, “An algorithm for least squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11(2), 431–441 (1963).
    [Crossref]
  17. D. Franta, “Outpout files of newAD2 and Gwyddion software,” figshare, 2021, https://doi.org/10.6084/m9.figshare.16832731.
  18. W. Sellmeier, “Ueber die durch die Aetherschwingungen erregten Mitschwingungen der Körpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien,” Ann. Phys. 223(11), 386–403 (1872).
    [Crossref]
  19. Y. Sato and T. Taira, “Highly accurate interferometric evaluation of thermal expansion and dn/dT of optical materials,” Opt. Mater. Express 4(5), 876–888 (2014).
    [Crossref]
  20. J. Hrabovský, M. Kučera, L. Paloušová, L. Bi, and M. Veis, “Optical characterization of Y3Al5O12 and Lu3Al5O12 single crystals,” Opt. Mater. Express 11(4), 1218–1223 (2021).
    [Crossref]
  21. D. Franta, D. Nečas, and L. Zajíčková, “Application of Thomas–Reiche–Kuhn sum rule to construction of advanced dispersion models,” Thin Solid Films 534, 432–441 (2013).
    [Crossref]
  22. P. Drude, “Ueber die Reflexion und Brechung ebener Lichtwellen beim Durchgang durch eine mit Oberflächenshichten behaftete planparallelr Platte,” Ann. Phys. 279(5), 126–157 (1891).
    [Crossref]
  23. Z. Knittl, Optics of Thin Films (Wiley, London, 1976).
  24. D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20(8), 3292–3302 (1979).
    [Crossref]
  25. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Comm. Pure Appl. Math. 4(2-3), 351–378 (1951).
    [Crossref]
  26. D. Franta and I. Ohlídal, “Ellipsometric parameters and reflectances of thin films with slightly rough boundaries,” J. Mod. Opt. 45(5), 903–934 (1998).
    [Crossref]
  27. D. Franta and I. Ohlídal, “Comparison of effective medium approximation and Rayleigh–Rice theory concerning ellipsometric characterization of rough surfaces,” Opt. Commun. 248(4-6), 459–467 (2005).
    [Crossref]
  28. J. Vohánka, M. Čermák, D. Franta, and I. Ohlídal, “Efficient method to calculate the optical quantities of multi-layer systems with randomly rough boundaries using the Rayleigh–Rice theory,” Phys. Scr. 94(4), 045502 (2019).
    [Crossref]
  29. D. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. 416(7), 636–664 (1935).
    [Crossref]
  30. M. E. Thomas, S. K. Andersson, R. M. Sova, and R. I. Joseph, “Frequency and temperature dependence of the refractive index of sapphire,” Infrared Phys. Technol. 39(4), 235–249 (1998).
    [Crossref]
  31. D. De Sousa Meneses, J.-F. Brun, B. Rousseau, and P. Echegut, “Polar lattice dynamics of the MgAl2O4 spinel up to the liquid state,” J. Phys. Condens. Matter 18(24), 5669–5686 (2006).
    [Crossref]
  32. D. De Sousa Meneses, M. Malki, and P. Echegut, “Structure and lattice dynamics of binary lead silicate glesses investigated by infrared spectroscopy,” J. Non-Cryst. Solids 352(8), 769–776 (2006).
    [Crossref]
  33. R. Kitamura, L. Pilon, and M. Jonasz, “Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature,” Appl. Opt. 46(33), 8118–8133 (2007).
    [Crossref]
  34. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series - 55 (National Bureau of Standards, 1964).
  35. J. Tauc, “Optical properties of non-crystaline solids,” in Optical Properties of Solids, F. Abelès, ed. (North–Holland, 1972), pp. 277–313.
  36. D. Campi and C. Coriasso, “Prediction of optical properties of amorphous tetrahedrally bounded materials,” J. Appl. Phys. 64(8), 4128–4134 (1988).
    [Crossref]
  37. D. Campi and C. Coriasso, “Relationships between optical properties and band parameters in amorphous tetrahedrally bonded materials,” Mater. Lett. 7(4), 134–137 (1988).
    [Crossref]
  38. D. Franta, M. Čermák, J. Vohánka, and I. Ohlídal, “Dispersion models describing interband electronic transitions combining Tauc’s law and Lorentz model,” Thin Solid Films 631, 12–22 (2017).
    [Crossref]
  39. T. Tomiki, F. Fukudome, M. Kaminao, M. Fujisawa, Y. Tanahara, and T. Futemma, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region,” J. Phys. Soc. Japan 58(5), 1801–1810 (1989).
    [Crossref]
  40. T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993).
    [Crossref]
  41. Y.-N. Xu and W. Ching, “Electronic structure of yttrium aluminium garnet (Y3Al5O12),” Phys. Rev. B 59(16), 10530–10535 (1999).
    [Crossref]
  42. P. Ondračka, D. Holec, and L. Zajíčková, “Predicting optical properties from ab initio calculations,” in Optical Characterization of Thin Solid Films, O. Stenzel and M. Ohlídal, eds. (Springer, Cham, 2018), Springer Series in Surface Sciences 64, pp. 83–104.
  43. J. P. Hurrell, S. P. S. Porto, I. F. Chang, S. S. Mitra, and R. P. Bauman, “Optical phonons of yttrium aluminum garnet,” Phys. Rev. 173(3), 851–856 (1968).
    [Crossref]
  44. G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10- to 55 000-cm−1 wave numbers,” Phys. Rev. 177(3), 1308–1314 (1969).
    [Crossref]
  45. A. M. Hofmeister and K. R. Campbell, “Infrared spectroscopy of yttrium aluminum, yttrium gallium, and yttrium iron garnets,” J. Appl. Phys. 72(2), 638–646 (1992).
    [Crossref]
  46. S. Kostić, Z. Lazarević, V. Radojević, A. Milutinović, M. Romčević, N. Romčević, and A. Valčić, “Study of structural and optical properties of YAG and Nd:YAG single crystals,” Mater. Res. Bull. 63, 80–87 (2015).
    [Crossref]
  47. D. Franta, J. Vohánka, M. Bránecký, P. Franta, M. Čermák, I. Ohlídal, and V. Čech, “Optical properties of the crystalline silicon wafers described using the universal dispersion model,” J. Vac. Sci. Technol. B 37(6), 062907 (2019).
    [Crossref]
  48. J. Humlíček, R. Henn, and M. Cardona, “Infrared vibrations in LaSrGaO4 an LaSrAlO4,” Phys. Rev. B 61(21), 14554–14563 (2000).
    [Crossref]
  49. D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Dispersion model of two-phonon absorption: application to c-Si,” Opt. Mater. Express 4(8), 1641–1656 (2014).
    [Crossref]
  50. D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, and I. Ohlídal, “Temperature-dependent dispersion model of float zone crystalline silicon,” Appl. Surf. Sci. 421, 405–419 (2017).
    [Crossref]

2021 (2)

B. Liu and P. R. Ohodnicki, “Fabrication and application of single crystal fiber: review and prospective,” Adv. Mater. Technol. 6(9), 2100125 (2021).
[Crossref]

J. Hrabovský, M. Kučera, L. Paloušová, L. Bi, and M. Veis, “Optical characterization of Y3Al5O12 and Lu3Al5O12 single crystals,” Opt. Mater. Express 11(4), 1218–1223 (2021).
[Crossref]

2019 (2)

J. Vohánka, M. Čermák, D. Franta, and I. Ohlídal, “Efficient method to calculate the optical quantities of multi-layer systems with randomly rough boundaries using the Rayleigh–Rice theory,” Phys. Scr. 94(4), 045502 (2019).
[Crossref]

D. Franta, J. Vohánka, M. Bránecký, P. Franta, M. Čermák, I. Ohlídal, and V. Čech, “Optical properties of the crystalline silicon wafers described using the universal dispersion model,” J. Vac. Sci. Technol. B 37(6), 062907 (2019).
[Crossref]

2017 (4)

D. Franta, M. Čermák, J. Vohánka, and I. Ohlídal, “Dispersion models describing interband electronic transitions combining Tauc’s law and Lorentz model,” Thin Solid Films 631, 12–22 (2017).
[Crossref]

N. Garejev, G. Tamošauskas, and A. Dubietis, “Comparative study of multioctave supercontinuum generation in fused silica, YAG, and LiF in the range of anomalous group velocity dispersion,” J. Opt. Soc. Am. B 34(1), 88–94 (2017).
[Crossref]

D. Franta, D. Nečas, A. Giglia, P. Franta, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to magnesium fluoride,” Appl. Surf. Sci. 421, 424–429 (2017).
[Crossref]

D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, and I. Ohlídal, “Temperature-dependent dispersion model of float zone crystalline silicon,” Appl. Surf. Sci. 421, 405–419 (2017).
[Crossref]

2016 (2)

Y. Nagata, R. Takeda, and M. Suginome, “High-pressure circular dichroism spectroscopy up to 400 MPa using polycrystalline yttrium aluminum garnet (YAG) as pressure-resistant optical windows,” RSC Adv. 6(111), 109726 (2016).
[Crossref]

V. Lupei and A. Lupei, “Nd:YAG at its 50th anniversary: still to learn,” J. Lumin. 169, 426–439 (2016).
[Crossref]

2015 (2)

D. Franta, D. Nečas, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to hafnia,” Appl. Opt. 54(31), 9108–9119 (2015).
[Crossref]

S. Kostić, Z. Lazarević, V. Radojević, A. Milutinović, M. Romčević, N. Romčević, and A. Valčić, “Study of structural and optical properties of YAG and Nd:YAG single crystals,” Mater. Res. Bull. 63, 80–87 (2015).
[Crossref]

2014 (2)

2013 (1)

D. Franta, D. Nečas, and L. Zajíčková, “Application of Thomas–Reiche–Kuhn sum rule to construction of advanced dispersion models,” Thin Solid Films 534, 432–441 (2013).
[Crossref]

2009 (1)

2007 (1)

2006 (2)

D. De Sousa Meneses, J.-F. Brun, B. Rousseau, and P. Echegut, “Polar lattice dynamics of the MgAl2O4 spinel up to the liquid state,” J. Phys. Condens. Matter 18(24), 5669–5686 (2006).
[Crossref]

D. De Sousa Meneses, M. Malki, and P. Echegut, “Structure and lattice dynamics of binary lead silicate glesses investigated by infrared spectroscopy,” J. Non-Cryst. Solids 352(8), 769–776 (2006).
[Crossref]

2005 (1)

D. Franta and I. Ohlídal, “Comparison of effective medium approximation and Rayleigh–Rice theory concerning ellipsometric characterization of rough surfaces,” Opt. Commun. 248(4-6), 459–467 (2005).
[Crossref]

2000 (1)

J. Humlíček, R. Henn, and M. Cardona, “Infrared vibrations in LaSrGaO4 an LaSrAlO4,” Phys. Rev. B 61(21), 14554–14563 (2000).
[Crossref]

1999 (1)

Y.-N. Xu and W. Ching, “Electronic structure of yttrium aluminium garnet (Y3Al5O12),” Phys. Rev. B 59(16), 10530–10535 (1999).
[Crossref]

1998 (3)

M. E. Thomas, S. K. Andersson, R. M. Sova, and R. I. Joseph, “Frequency and temperature dependence of the refractive index of sapphire,” Infrared Phys. Technol. 39(4), 235–249 (1998).
[Crossref]

D. Franta and I. Ohlídal, “Ellipsometric parameters and reflectances of thin films with slightly rough boundaries,” J. Mod. Opt. 45(5), 903–934 (1998).
[Crossref]

D. E. Zelmon, D. L. Small, and R. Page, “Refractive-index measurements of undoped yttrium aluminum garnet from 0.4 to 5.0 μm,” Appl. Opt. 37(21), 4933–4935 (1998).
[Crossref]

1993 (1)

T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993).
[Crossref]

1992 (1)

A. M. Hofmeister and K. R. Campbell, “Infrared spectroscopy of yttrium aluminum, yttrium gallium, and yttrium iron garnets,” J. Appl. Phys. 72(2), 638–646 (1992).
[Crossref]

1989 (1)

T. Tomiki, F. Fukudome, M. Kaminao, M. Fujisawa, Y. Tanahara, and T. Futemma, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region,” J. Phys. Soc. Japan 58(5), 1801–1810 (1989).
[Crossref]

1988 (2)

D. Campi and C. Coriasso, “Prediction of optical properties of amorphous tetrahedrally bounded materials,” J. Appl. Phys. 64(8), 4128–4134 (1988).
[Crossref]

D. Campi and C. Coriasso, “Relationships between optical properties and band parameters in amorphous tetrahedrally bonded materials,” Mater. Lett. 7(4), 134–137 (1988).
[Crossref]

1979 (1)

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20(8), 3292–3302 (1979).
[Crossref]

1969 (1)

G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10- to 55 000-cm−1 wave numbers,” Phys. Rev. 177(3), 1308–1314 (1969).
[Crossref]

1968 (1)

J. P. Hurrell, S. P. S. Porto, I. F. Chang, S. S. Mitra, and R. P. Bauman, “Optical phonons of yttrium aluminum garnet,” Phys. Rev. 173(3), 851–856 (1968).
[Crossref]

1963 (1)

W. D. Marquardt, “An algorithm for least squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11(2), 431–441 (1963).
[Crossref]

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Comm. Pure Appl. Math. 4(2-3), 351–378 (1951).
[Crossref]

1935 (1)

D. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. 416(7), 636–664 (1935).
[Crossref]

1891 (1)

P. Drude, “Ueber die Reflexion und Brechung ebener Lichtwellen beim Durchgang durch eine mit Oberflächenshichten behaftete planparallelr Platte,” Ann. Phys. 279(5), 126–157 (1891).
[Crossref]

1872 (1)

W. Sellmeier, “Ueber die durch die Aetherschwingungen erregten Mitschwingungen der Körpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien,” Ann. Phys. 223(11), 386–403 (1872).
[Crossref]

Aiura, Y.

T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993).
[Crossref]

Andersson, S. K.

M. E. Thomas, S. K. Andersson, R. M. Sova, and R. I. Joseph, “Frequency and temperature dependence of the refractive index of sapphire,” Infrared Phys. Technol. 39(4), 235–249 (1998).
[Crossref]

Aspnes, D. E.

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20(8), 3292–3302 (1979).
[Crossref]

Bauman, R. P.

J. P. Hurrell, S. P. S. Porto, I. F. Chang, S. S. Mitra, and R. P. Bauman, “Optical phonons of yttrium aluminum garnet,” Phys. Rev. 173(3), 851–856 (1968).
[Crossref]

Bi, L.

Bránecký, M.

D. Franta, J. Vohánka, M. Bránecký, P. Franta, M. Čermák, I. Ohlídal, and V. Čech, “Optical properties of the crystalline silicon wafers described using the universal dispersion model,” J. Vac. Sci. Technol. B 37(6), 062907 (2019).
[Crossref]

Bruggeman, D.

D. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. 416(7), 636–664 (1935).
[Crossref]

Brun, J.-F.

D. De Sousa Meneses, J.-F. Brun, B. Rousseau, and P. Echegut, “Polar lattice dynamics of the MgAl2O4 spinel up to the liquid state,” J. Phys. Condens. Matter 18(24), 5669–5686 (2006).
[Crossref]

Campbell, K. R.

A. M. Hofmeister and K. R. Campbell, “Infrared spectroscopy of yttrium aluminum, yttrium gallium, and yttrium iron garnets,” J. Appl. Phys. 72(2), 638–646 (1992).
[Crossref]

Campi, D.

D. Campi and C. Coriasso, “Prediction of optical properties of amorphous tetrahedrally bounded materials,” J. Appl. Phys. 64(8), 4128–4134 (1988).
[Crossref]

D. Campi and C. Coriasso, “Relationships between optical properties and band parameters in amorphous tetrahedrally bonded materials,” Mater. Lett. 7(4), 134–137 (1988).
[Crossref]

Cardona, M.

J. Humlíček, R. Henn, and M. Cardona, “Infrared vibrations in LaSrGaO4 an LaSrAlO4,” Phys. Rev. B 61(21), 14554–14563 (2000).
[Crossref]

Cech, P.

J. Vanda, M.-G. Mureșan, P. Čech, M. Hanuš, A. Lucianetti, D. Rostohar, T. Mocek, Š. Uxa, and V. Škoda, “Multiple pulse nanosecond laser-induced damage threshold on AR coated YAG crystals,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108050F.

Cech, V.

D. Franta, J. Vohánka, M. Bránecký, P. Franta, M. Čermák, I. Ohlídal, and V. Čech, “Optical properties of the crystalline silicon wafers described using the universal dispersion model,” J. Vac. Sci. Technol. B 37(6), 062907 (2019).
[Crossref]

Cermák, M.

D. Franta, J. Vohánka, M. Bránecký, P. Franta, M. Čermák, I. Ohlídal, and V. Čech, “Optical properties of the crystalline silicon wafers described using the universal dispersion model,” J. Vac. Sci. Technol. B 37(6), 062907 (2019).
[Crossref]

J. Vohánka, M. Čermák, D. Franta, and I. Ohlídal, “Efficient method to calculate the optical quantities of multi-layer systems with randomly rough boundaries using the Rayleigh–Rice theory,” Phys. Scr. 94(4), 045502 (2019).
[Crossref]

D. Franta, M. Čermák, J. Vohánka, and I. Ohlídal, “Dispersion models describing interband electronic transitions combining Tauc’s law and Lorentz model,” Thin Solid Films 631, 12–22 (2017).
[Crossref]

D. Franta, J. Vohánka, and M. Čermák, “Universal dispersion model for characterization of thin films over wide spectral range,” in Optical Characterization of Thin Solid Films, O. Stenzel and M. Ohlídal, eds. (Springer, 2018), Springer Series in Surface Sciences 64, pp. 31–82.

I. Ohlídal, J. Vohánka, M. Čermák, and D. Franta, “Ellipsometry of layered systems,” in Optical Characterization of Thin Solid Films, vol. 64O. Stenzel and M. Ohlídal, eds. (Springer, Cham, 2018), pp. 233–267.

Chang, I. F.

J. P. Hurrell, S. P. S. Porto, I. F. Chang, S. S. Mitra, and R. P. Bauman, “Optical phonons of yttrium aluminum garnet,” Phys. Rev. 173(3), 851–856 (1968).
[Crossref]

Ching, W.

Y.-N. Xu and W. Ching, “Electronic structure of yttrium aluminium garnet (Y3Al5O12),” Phys. Rev. B 59(16), 10530–10535 (1999).
[Crossref]

Chrenko, R. M.

G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10- to 55 000-cm−1 wave numbers,” Phys. Rev. 177(3), 1308–1314 (1969).
[Crossref]

Coriasso, C.

D. Campi and C. Coriasso, “Relationships between optical properties and band parameters in amorphous tetrahedrally bonded materials,” Mater. Lett. 7(4), 134–137 (1988).
[Crossref]

D. Campi and C. Coriasso, “Prediction of optical properties of amorphous tetrahedrally bounded materials,” J. Appl. Phys. 64(8), 4128–4134 (1988).
[Crossref]

De Sousa Meneses, D.

D. De Sousa Meneses, J.-F. Brun, B. Rousseau, and P. Echegut, “Polar lattice dynamics of the MgAl2O4 spinel up to the liquid state,” J. Phys. Condens. Matter 18(24), 5669–5686 (2006).
[Crossref]

D. De Sousa Meneses, M. Malki, and P. Echegut, “Structure and lattice dynamics of binary lead silicate glesses investigated by infrared spectroscopy,” J. Non-Cryst. Solids 352(8), 769–776 (2006).
[Crossref]

Do, B. T.

Drude, P.

P. Drude, “Ueber die Reflexion und Brechung ebener Lichtwellen beim Durchgang durch eine mit Oberflächenshichten behaftete planparallelr Platte,” Ann. Phys. 279(5), 126–157 (1891).
[Crossref]

Dubietis, A.

Dubroka, A.

D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, and I. Ohlídal, “Temperature-dependent dispersion model of float zone crystalline silicon,” Appl. Surf. Sci. 421, 405–419 (2017).
[Crossref]

Echegut, P.

D. De Sousa Meneses, J.-F. Brun, B. Rousseau, and P. Echegut, “Polar lattice dynamics of the MgAl2O4 spinel up to the liquid state,” J. Phys. Condens. Matter 18(24), 5669–5686 (2006).
[Crossref]

D. De Sousa Meneses, M. Malki, and P. Echegut, “Structure and lattice dynamics of binary lead silicate glesses investigated by infrared spectroscopy,” J. Non-Cryst. Solids 352(8), 769–776 (2006).
[Crossref]

Franta, D.

J. Vohánka, M. Čermák, D. Franta, and I. Ohlídal, “Efficient method to calculate the optical quantities of multi-layer systems with randomly rough boundaries using the Rayleigh–Rice theory,” Phys. Scr. 94(4), 045502 (2019).
[Crossref]

D. Franta, J. Vohánka, M. Bránecký, P. Franta, M. Čermák, I. Ohlídal, and V. Čech, “Optical properties of the crystalline silicon wafers described using the universal dispersion model,” J. Vac. Sci. Technol. B 37(6), 062907 (2019).
[Crossref]

D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, and I. Ohlídal, “Temperature-dependent dispersion model of float zone crystalline silicon,” Appl. Surf. Sci. 421, 405–419 (2017).
[Crossref]

D. Franta, M. Čermák, J. Vohánka, and I. Ohlídal, “Dispersion models describing interband electronic transitions combining Tauc’s law and Lorentz model,” Thin Solid Films 631, 12–22 (2017).
[Crossref]

D. Franta, D. Nečas, A. Giglia, P. Franta, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to magnesium fluoride,” Appl. Surf. Sci. 421, 424–429 (2017).
[Crossref]

D. Franta, D. Nečas, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to hafnia,” Appl. Opt. 54(31), 9108–9119 (2015).
[Crossref]

D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Dispersion model of two-phonon absorption: application to c-Si,” Opt. Mater. Express 4(8), 1641–1656 (2014).
[Crossref]

D. Franta, D. Nečas, and L. Zajíčková, “Application of Thomas–Reiche–Kuhn sum rule to construction of advanced dispersion models,” Thin Solid Films 534, 432–441 (2013).
[Crossref]

D. Franta and I. Ohlídal, “Comparison of effective medium approximation and Rayleigh–Rice theory concerning ellipsometric characterization of rough surfaces,” Opt. Commun. 248(4-6), 459–467 (2005).
[Crossref]

D. Franta and I. Ohlídal, “Ellipsometric parameters and reflectances of thin films with slightly rough boundaries,” J. Mod. Opt. 45(5), 903–934 (1998).
[Crossref]

D. Franta, J. Vohánka, and M. Čermák, “Universal dispersion model for characterization of thin films over wide spectral range,” in Optical Characterization of Thin Solid Films, O. Stenzel and M. Ohlídal, eds. (Springer, 2018), Springer Series in Surface Sciences 64, pp. 31–82.

I. Ohlídal, J. Vohánka, M. Čermák, and D. Franta, “Ellipsometry of layered systems,” in Optical Characterization of Thin Solid Films, vol. 64O. Stenzel and M. Ohlídal, eds. (Springer, Cham, 2018), pp. 233–267.

D. Franta, D. Nečas, and J. Vohánka, “Software for optical characterization newAD2,” http://newad.physics.muni.cz .

Franta, P.

D. Franta, J. Vohánka, M. Bránecký, P. Franta, M. Čermák, I. Ohlídal, and V. Čech, “Optical properties of the crystalline silicon wafers described using the universal dispersion model,” J. Vac. Sci. Technol. B 37(6), 062907 (2019).
[Crossref]

D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, and I. Ohlídal, “Temperature-dependent dispersion model of float zone crystalline silicon,” Appl. Surf. Sci. 421, 405–419 (2017).
[Crossref]

D. Franta, D. Nečas, A. Giglia, P. Franta, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to magnesium fluoride,” Appl. Surf. Sci. 421, 424–429 (2017).
[Crossref]

Fujisawa, M.

T. Tomiki, F. Fukudome, M. Kaminao, M. Fujisawa, Y. Tanahara, and T. Futemma, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region,” J. Phys. Soc. Japan 58(5), 1801–1810 (1989).
[Crossref]

Fukudome, F.

T. Tomiki, F. Fukudome, M. Kaminao, M. Fujisawa, Y. Tanahara, and T. Futemma, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region,” J. Phys. Soc. Japan 58(5), 1801–1810 (1989).
[Crossref]

Fukutani, H.

T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993).
[Crossref]

Futemma, T.

T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993).
[Crossref]

T. Tomiki, F. Fukudome, M. Kaminao, M. Fujisawa, Y. Tanahara, and T. Futemma, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region,” J. Phys. Soc. Japan 58(5), 1801–1810 (1989).
[Crossref]

Ganaha, Y.

T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993).
[Crossref]

Garejev, N.

Giglia, A.

D. Franta, D. Nečas, A. Giglia, P. Franta, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to magnesium fluoride,” Appl. Surf. Sci. 421, 424–429 (2017).
[Crossref]

D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, and I. Ohlídal, “Temperature-dependent dispersion model of float zone crystalline silicon,” Appl. Surf. Sci. 421, 405–419 (2017).
[Crossref]

Gonzalez, L. P.

D. E. Zelmon, K. L. Schepler, S. Guha, D. J. Rush, S. M. Hegde, L. P. Gonzalez, and J. Lee, “Optical properties of Nd-doped ceramic yttrium aluminum garnet,” in Laser-Induced Damage in Optical Materials, vol. 5647 of Proc. SPIEG. J. Exarhos, A. H. Guenther, N. Kaiser, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds. (2004), pp. 255–264.

Guha, S.

D. E. Zelmon, K. L. Schepler, S. Guha, D. J. Rush, S. M. Hegde, L. P. Gonzalez, and J. Lee, “Optical properties of Nd-doped ceramic yttrium aluminum garnet,” in Laser-Induced Damage in Optical Materials, vol. 5647 of Proc. SPIEG. J. Exarhos, A. H. Guenther, N. Kaiser, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds. (2004), pp. 255–264.

Hanuš, M.

J. Vanda, M.-G. Mureșan, P. Čech, M. Hanuš, A. Lucianetti, D. Rostohar, T. Mocek, Š. Uxa, and V. Škoda, “Multiple pulse nanosecond laser-induced damage threshold on AR coated YAG crystals,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108050F.

Hegde, S. M.

D. E. Zelmon, K. L. Schepler, S. Guha, D. J. Rush, S. M. Hegde, L. P. Gonzalez, and J. Lee, “Optical properties of Nd-doped ceramic yttrium aluminum garnet,” in Laser-Induced Damage in Optical Materials, vol. 5647 of Proc. SPIEG. J. Exarhos, A. H. Guenther, N. Kaiser, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds. (2004), pp. 255–264.

Henn, R.

J. Humlíček, R. Henn, and M. Cardona, “Infrared vibrations in LaSrGaO4 an LaSrAlO4,” Phys. Rev. B 61(21), 14554–14563 (2000).
[Crossref]

Hofmeister, A. M.

A. M. Hofmeister and K. R. Campbell, “Infrared spectroscopy of yttrium aluminum, yttrium gallium, and yttrium iron garnets,” J. Appl. Phys. 72(2), 638–646 (1992).
[Crossref]

Holec, D.

P. Ondračka, D. Holec, and L. Zajíčková, “Predicting optical properties from ab initio calculations,” in Optical Characterization of Thin Solid Films, O. Stenzel and M. Ohlídal, eds. (Springer, Cham, 2018), Springer Series in Surface Sciences 64, pp. 83–104.

Hottier, F.

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20(8), 3292–3302 (1979).
[Crossref]

Hrabovský, J.

Humlícek, J.

J. Humlíček, R. Henn, and M. Cardona, “Infrared vibrations in LaSrGaO4 an LaSrAlO4,” Phys. Rev. B 61(21), 14554–14563 (2000).
[Crossref]

Hurrell, J. P.

J. P. Hurrell, S. P. S. Porto, I. F. Chang, S. S. Mitra, and R. P. Bauman, “Optical phonons of yttrium aluminum garnet,” Phys. Rev. 173(3), 851–856 (1968).
[Crossref]

Jonasz, M.

Joseph, R. I.

M. E. Thomas, S. K. Andersson, R. M. Sova, and R. I. Joseph, “Frequency and temperature dependence of the refractive index of sapphire,” Infrared Phys. Technol. 39(4), 235–249 (1998).
[Crossref]

Kaminao, M.

T. Tomiki, F. Fukudome, M. Kaminao, M. Fujisawa, Y. Tanahara, and T. Futemma, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region,” J. Phys. Soc. Japan 58(5), 1801–1810 (1989).
[Crossref]

Kato, H.

T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993).
[Crossref]

Kitamura, R.

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, London, 1976).

Kostic, S.

S. Kostić, Z. Lazarević, V. Radojević, A. Milutinović, M. Romčević, N. Romčević, and A. Valčić, “Study of structural and optical properties of YAG and Nd:YAG single crystals,” Mater. Res. Bull. 63, 80–87 (2015).
[Crossref]

Kucera, M.

Lazarevic, Z.

S. Kostić, Z. Lazarević, V. Radojević, A. Milutinović, M. Romčević, N. Romčević, and A. Valčić, “Study of structural and optical properties of YAG and Nd:YAG single crystals,” Mater. Res. Bull. 63, 80–87 (2015).
[Crossref]

Lee, J.

D. E. Zelmon, K. L. Schepler, S. Guha, D. J. Rush, S. M. Hegde, L. P. Gonzalez, and J. Lee, “Optical properties of Nd-doped ceramic yttrium aluminum garnet,” in Laser-Induced Damage in Optical Materials, vol. 5647 of Proc. SPIEG. J. Exarhos, A. H. Guenther, N. Kaiser, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds. (2004), pp. 255–264.

Liu, B.

B. Liu and P. R. Ohodnicki, “Fabrication and application of single crystal fiber: review and prospective,” Adv. Mater. Technol. 6(9), 2100125 (2021).
[Crossref]

Lucianetti, A.

J. Vanda, M.-G. Mureșan, P. Čech, M. Hanuš, A. Lucianetti, D. Rostohar, T. Mocek, Š. Uxa, and V. Škoda, “Multiple pulse nanosecond laser-induced damage threshold on AR coated YAG crystals,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108050F.

Lupei, A.

V. Lupei and A. Lupei, “Nd:YAG at its 50th anniversary: still to learn,” J. Lumin. 169, 426–439 (2016).
[Crossref]

Lupei, V.

V. Lupei and A. Lupei, “Nd:YAG at its 50th anniversary: still to learn,” J. Lumin. 169, 426–439 (2016).
[Crossref]

Malki, M.

D. De Sousa Meneses, M. Malki, and P. Echegut, “Structure and lattice dynamics of binary lead silicate glesses investigated by infrared spectroscopy,” J. Non-Cryst. Solids 352(8), 769–776 (2006).
[Crossref]

Marquardt, W. D.

W. D. Marquardt, “An algorithm for least squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11(2), 431–441 (1963).
[Crossref]

Milutinovic, A.

S. Kostić, Z. Lazarević, V. Radojević, A. Milutinović, M. Romčević, N. Romčević, and A. Valčić, “Study of structural and optical properties of YAG and Nd:YAG single crystals,” Mater. Res. Bull. 63, 80–87 (2015).
[Crossref]

Mitra, S. S.

J. P. Hurrell, S. P. S. Porto, I. F. Chang, S. S. Mitra, and R. P. Bauman, “Optical phonons of yttrium aluminum garnet,” Phys. Rev. 173(3), 851–856 (1968).
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T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993).
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J. Vanda, M.-G. Mureșan, P. Čech, M. Hanuš, A. Lucianetti, D. Rostohar, T. Mocek, Š. Uxa, and V. Škoda, “Multiple pulse nanosecond laser-induced damage threshold on AR coated YAG crystals,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108050F.

Mure?an, M.-G.

J. Vanda, M.-G. Mureșan, P. Čech, M. Hanuš, A. Lucianetti, D. Rostohar, T. Mocek, Š. Uxa, and V. Škoda, “Multiple pulse nanosecond laser-induced damage threshold on AR coated YAG crystals,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108050F.

Š. Uxa, V. Škoda, J. Vanda, and M.-G. Mureșan, “A comparison of LIDT behavior of AR-coated yttrium-aluminium-garnet substrates with respect to thin-film design and coating technology,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108051R.

Nagata, Y.

Y. Nagata, R. Takeda, and M. Suginome, “High-pressure circular dichroism spectroscopy up to 400 MPa using polycrystalline yttrium aluminum garnet (YAG) as pressure-resistant optical windows,” RSC Adv. 6(111), 109726 (2016).
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Necas, D.

D. Franta, D. Nečas, A. Giglia, P. Franta, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to magnesium fluoride,” Appl. Surf. Sci. 421, 424–429 (2017).
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D. Franta, D. Nečas, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to hafnia,” Appl. Opt. 54(31), 9108–9119 (2015).
[Crossref]

D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Dispersion model of two-phonon absorption: application to c-Si,” Opt. Mater. Express 4(8), 1641–1656 (2014).
[Crossref]

D. Franta, D. Nečas, and L. Zajíčková, “Application of Thomas–Reiche–Kuhn sum rule to construction of advanced dispersion models,” Thin Solid Films 534, 432–441 (2013).
[Crossref]

D. Franta, D. Nečas, and J. Vohánka, “Software for optical characterization newAD2,” http://newad.physics.muni.cz .

Ohlídal, I.

J. Vohánka, M. Čermák, D. Franta, and I. Ohlídal, “Efficient method to calculate the optical quantities of multi-layer systems with randomly rough boundaries using the Rayleigh–Rice theory,” Phys. Scr. 94(4), 045502 (2019).
[Crossref]

D. Franta, J. Vohánka, M. Bránecký, P. Franta, M. Čermák, I. Ohlídal, and V. Čech, “Optical properties of the crystalline silicon wafers described using the universal dispersion model,” J. Vac. Sci. Technol. B 37(6), 062907 (2019).
[Crossref]

D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, and I. Ohlídal, “Temperature-dependent dispersion model of float zone crystalline silicon,” Appl. Surf. Sci. 421, 405–419 (2017).
[Crossref]

D. Franta, M. Čermák, J. Vohánka, and I. Ohlídal, “Dispersion models describing interband electronic transitions combining Tauc’s law and Lorentz model,” Thin Solid Films 631, 12–22 (2017).
[Crossref]

D. Franta, D. Nečas, A. Giglia, P. Franta, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to magnesium fluoride,” Appl. Surf. Sci. 421, 424–429 (2017).
[Crossref]

D. Franta, D. Nečas, and I. Ohlídal, “Universal dispersion model for characterization of optical thin films over wide spectral range: Application to hafnia,” Appl. Opt. 54(31), 9108–9119 (2015).
[Crossref]

D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Dispersion model of two-phonon absorption: application to c-Si,” Opt. Mater. Express 4(8), 1641–1656 (2014).
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D. Franta and I. Ohlídal, “Comparison of effective medium approximation and Rayleigh–Rice theory concerning ellipsometric characterization of rough surfaces,” Opt. Commun. 248(4-6), 459–467 (2005).
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D. Franta and I. Ohlídal, “Ellipsometric parameters and reflectances of thin films with slightly rough boundaries,” J. Mod. Opt. 45(5), 903–934 (1998).
[Crossref]

I. Ohlídal, J. Vohánka, M. Čermák, and D. Franta, “Ellipsometry of layered systems,” in Optical Characterization of Thin Solid Films, vol. 64O. Stenzel and M. Ohlídal, eds. (Springer, Cham, 2018), pp. 233–267.

Ohodnicki, P. R.

B. Liu and P. R. Ohodnicki, “Fabrication and application of single crystal fiber: review and prospective,” Adv. Mater. Technol. 6(9), 2100125 (2021).
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Oliver, D. W.

G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10- to 55 000-cm−1 wave numbers,” Phys. Rev. 177(3), 1308–1314 (1969).
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Ondracka, P.

P. Ondračka, D. Holec, and L. Zajíčková, “Predicting optical properties from ab initio calculations,” in Optical Characterization of Thin Solid Films, O. Stenzel and M. Ohlídal, eds. (Springer, Cham, 2018), Springer Series in Surface Sciences 64, pp. 83–104.

Page, R.

Paloušová, L.

Pilon, L.

Porto, S. P. S.

J. P. Hurrell, S. P. S. Porto, I. F. Chang, S. S. Mitra, and R. P. Bauman, “Optical phonons of yttrium aluminum garnet,” Phys. Rev. 173(3), 851–856 (1968).
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Radojevic, V.

S. Kostić, Z. Lazarević, V. Radojević, A. Milutinović, M. Romčević, N. Romčević, and A. Valčić, “Study of structural and optical properties of YAG and Nd:YAG single crystals,” Mater. Res. Bull. 63, 80–87 (2015).
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S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Comm. Pure Appl. Math. 4(2-3), 351–378 (1951).
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G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10- to 55 000-cm−1 wave numbers,” Phys. Rev. 177(3), 1308–1314 (1969).
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S. Kostić, Z. Lazarević, V. Radojević, A. Milutinović, M. Romčević, N. Romčević, and A. Valčić, “Study of structural and optical properties of YAG and Nd:YAG single crystals,” Mater. Res. Bull. 63, 80–87 (2015).
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Romcevic, N.

S. Kostić, Z. Lazarević, V. Radojević, A. Milutinović, M. Romčević, N. Romčević, and A. Valčić, “Study of structural and optical properties of YAG and Nd:YAG single crystals,” Mater. Res. Bull. 63, 80–87 (2015).
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Rostohar, D.

J. Vanda, M.-G. Mureșan, P. Čech, M. Hanuš, A. Lucianetti, D. Rostohar, T. Mocek, Š. Uxa, and V. Škoda, “Multiple pulse nanosecond laser-induced damage threshold on AR coated YAG crystals,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108050F.

Rousseau, B.

D. De Sousa Meneses, J.-F. Brun, B. Rousseau, and P. Echegut, “Polar lattice dynamics of the MgAl2O4 spinel up to the liquid state,” J. Phys. Condens. Matter 18(24), 5669–5686 (2006).
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D. E. Zelmon, K. L. Schepler, S. Guha, D. J. Rush, S. M. Hegde, L. P. Gonzalez, and J. Lee, “Optical properties of Nd-doped ceramic yttrium aluminum garnet,” in Laser-Induced Damage in Optical Materials, vol. 5647 of Proc. SPIEG. J. Exarhos, A. H. Guenther, N. Kaiser, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds. (2004), pp. 255–264.

Sato, Y.

Schepler, K. L.

D. E. Zelmon, K. L. Schepler, S. Guha, D. J. Rush, S. M. Hegde, L. P. Gonzalez, and J. Lee, “Optical properties of Nd-doped ceramic yttrium aluminum garnet,” in Laser-Induced Damage in Optical Materials, vol. 5647 of Proc. SPIEG. J. Exarhos, A. H. Guenther, N. Kaiser, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds. (2004), pp. 255–264.

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W. Sellmeier, “Ueber die durch die Aetherschwingungen erregten Mitschwingungen der Körpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien,” Ann. Phys. 223(11), 386–403 (1872).
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T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993).
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J. Vanda, M.-G. Mureșan, P. Čech, M. Hanuš, A. Lucianetti, D. Rostohar, T. Mocek, Š. Uxa, and V. Škoda, “Multiple pulse nanosecond laser-induced damage threshold on AR coated YAG crystals,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108050F.

Š. Uxa, V. Škoda, J. Vanda, and M.-G. Mureșan, “A comparison of LIDT behavior of AR-coated yttrium-aluminium-garnet substrates with respect to thin-film design and coating technology,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108051R.

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G. A. Slack, D. W. Oliver, R. M. Chrenko, and S. Roberts, “Optical absorption of Y3Al5O12 from 10- to 55 000-cm−1 wave numbers,” Phys. Rev. 177(3), 1308–1314 (1969).
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Smith, A. V.

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M. E. Thomas, S. K. Andersson, R. M. Sova, and R. I. Joseph, “Frequency and temperature dependence of the refractive index of sapphire,” Infrared Phys. Technol. 39(4), 235–249 (1998).
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Suginome, M.

Y. Nagata, R. Takeda, and M. Suginome, “High-pressure circular dichroism spectroscopy up to 400 MPa using polycrystalline yttrium aluminum garnet (YAG) as pressure-resistant optical windows,” RSC Adv. 6(111), 109726 (2016).
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Taira, T.

Takeda, R.

Y. Nagata, R. Takeda, and M. Suginome, “High-pressure circular dichroism spectroscopy up to 400 MPa using polycrystalline yttrium aluminum garnet (YAG) as pressure-resistant optical windows,” RSC Adv. 6(111), 109726 (2016).
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T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993).
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Tanahara, Y.

T. Tomiki, F. Fukudome, M. Kaminao, M. Fujisawa, Y. Tanahara, and T. Futemma, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region,” J. Phys. Soc. Japan 58(5), 1801–1810 (1989).
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M. E. Thomas, S. K. Andersson, R. M. Sova, and R. I. Joseph, “Frequency and temperature dependence of the refractive index of sapphire,” Infrared Phys. Technol. 39(4), 235–249 (1998).
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T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993).
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T. Tomiki, F. Fukudome, M. Kaminao, M. Fujisawa, Y. Tanahara, and T. Futemma, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region,” J. Phys. Soc. Japan 58(5), 1801–1810 (1989).
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Š. Uxa, V. Škoda, J. Vanda, and M.-G. Mureșan, “A comparison of LIDT behavior of AR-coated yttrium-aluminium-garnet substrates with respect to thin-film design and coating technology,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108051R.

J. Vanda, M.-G. Mureșan, P. Čech, M. Hanuš, A. Lucianetti, D. Rostohar, T. Mocek, Š. Uxa, and V. Škoda, “Multiple pulse nanosecond laser-induced damage threshold on AR coated YAG crystals,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108050F.

Valcic, A.

S. Kostić, Z. Lazarević, V. Radojević, A. Milutinović, M. Romčević, N. Romčević, and A. Valčić, “Study of structural and optical properties of YAG and Nd:YAG single crystals,” Mater. Res. Bull. 63, 80–87 (2015).
[Crossref]

Vanda, J.

Š. Uxa, V. Škoda, J. Vanda, and M.-G. Mureșan, “A comparison of LIDT behavior of AR-coated yttrium-aluminium-garnet substrates with respect to thin-film design and coating technology,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108051R.

J. Vanda, M.-G. Mureșan, P. Čech, M. Hanuš, A. Lucianetti, D. Rostohar, T. Mocek, Š. Uxa, and V. Škoda, “Multiple pulse nanosecond laser-induced damage threshold on AR coated YAG crystals,” in Laser-Induced Damage in Optical Materials 2018: 50th Anniversary Conference, vol. 10805 of Proc. SPIEC. W. Carr, G. J. Exarhos, V. E. Gruzdev, D. Ristau, and M. Soileau, eds. (2018), p. 108050F.

Veis, M.

Vohánka, J.

J. Vohánka, M. Čermák, D. Franta, and I. Ohlídal, “Efficient method to calculate the optical quantities of multi-layer systems with randomly rough boundaries using the Rayleigh–Rice theory,” Phys. Scr. 94(4), 045502 (2019).
[Crossref]

D. Franta, J. Vohánka, M. Bránecký, P. Franta, M. Čermák, I. Ohlídal, and V. Čech, “Optical properties of the crystalline silicon wafers described using the universal dispersion model,” J. Vac. Sci. Technol. B 37(6), 062907 (2019).
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D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, and I. Ohlídal, “Temperature-dependent dispersion model of float zone crystalline silicon,” Appl. Surf. Sci. 421, 405–419 (2017).
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D. Franta, M. Čermák, J. Vohánka, and I. Ohlídal, “Dispersion models describing interband electronic transitions combining Tauc’s law and Lorentz model,” Thin Solid Films 631, 12–22 (2017).
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I. Ohlídal, J. Vohánka, M. Čermák, and D. Franta, “Ellipsometry of layered systems,” in Optical Characterization of Thin Solid Films, vol. 64O. Stenzel and M. Ohlídal, eds. (Springer, Cham, 2018), pp. 233–267.

D. Franta, J. Vohánka, and M. Čermák, “Universal dispersion model for characterization of thin films over wide spectral range,” in Optical Characterization of Thin Solid Films, O. Stenzel and M. Ohlídal, eds. (Springer, 2018), Springer Series in Surface Sciences 64, pp. 31–82.

D. Franta, D. Nečas, and J. Vohánka, “Software for optical characterization newAD2,” http://newad.physics.muni.cz .

Wang, C.

D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, and I. Ohlídal, “Temperature-dependent dispersion model of float zone crystalline silicon,” Appl. Surf. Sci. 421, 405–419 (2017).
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T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993).
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Yuri, M.

T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, “Optical spectra of Y3Al5O12 (YAG) single crystals in the vacuum ultraviolet region. II,” J. Phys. Soc. Japan 62(4), 1388–1400 (1993).
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Zajícková, L.

D. Franta, D. Nečas, L. Zajíčková, and I. Ohlídal, “Dispersion model of two-phonon absorption: application to c-Si,” Opt. Mater. Express 4(8), 1641–1656 (2014).
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D. Franta, D. Nečas, and L. Zajíčková, “Application of Thomas–Reiche–Kuhn sum rule to construction of advanced dispersion models,” Thin Solid Films 534, 432–441 (2013).
[Crossref]

P. Ondračka, D. Holec, and L. Zajíčková, “Predicting optical properties from ab initio calculations,” in Optical Characterization of Thin Solid Films, O. Stenzel and M. Ohlídal, eds. (Springer, Cham, 2018), Springer Series in Surface Sciences 64, pp. 83–104.

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D. E. Zelmon, D. L. Small, and R. Page, “Refractive-index measurements of undoped yttrium aluminum garnet from 0.4 to 5.0 μm,” Appl. Opt. 37(21), 4933–4935 (1998).
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D. E. Zelmon, K. L. Schepler, S. Guha, D. J. Rush, S. M. Hegde, L. P. Gonzalez, and J. Lee, “Optical properties of Nd-doped ceramic yttrium aluminum garnet,” in Laser-Induced Damage in Optical Materials, vol. 5647 of Proc. SPIEG. J. Exarhos, A. H. Guenther, N. Kaiser, K. L. Lewis, M. J. Soileau, and C. J. Stolz, eds. (2004), pp. 255–264.

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D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, and I. Ohlídal, “Temperature-dependent dispersion model of float zone crystalline silicon,” Appl. Surf. Sci. 421, 405–419 (2017).
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Supplementary Material (2)

NameDescription
Dataset 1       Outpout files of newAD2 and Gwyddion softwares
Supplement 1       Supplement 1

Data availability

Data underlying the results presented in this paper are available in Dataset 1 [17].

17. D. Franta, “Outpout files of newAD2 and Gwyddion software,” figshare, 2021, https://doi.org/10.6084/m9.figshare.16832731.

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Figures (10)

Fig. 1.
Fig. 1. Spectral dependence of the refractive index $n$ of YAG crystal measured by Zelmon et al. [13] using the MDM. The curves represents fit of these data by universal dispersion model (UDM) and by harmonic oscillator model (HOM).
Fig. 2.
Fig. 2. Spectral dependencies of the complex dielectric function $\hat \varepsilon$, calculated by universal dispersion model (UDM) and harmonic oscillator model (HOM) fitting Zelmon et al. [13] experimental data.
Fig. 3.
Fig. 3. Spectral dependencies of ellipsometric quantities displayed by the complex pseudodielectric function $\langle \hat \varepsilon \rangle$ and the degree of polarization $P$.
Fig. 4.
Fig. 4. Spectral dependencies of the transmittance $T$ measured in the visible and UV spectral range and corresponding modeled imaginary part of dielectric function $\varepsilon _{\mathrm {i}}$. The separated UDM contributions of the absorption on localized states and Urbach tail are plotted by dash and dash–dot lines.
Fig. 5.
Fig. 5. Spectral dependence of modeled imaginary part of dielectric function $\varepsilon _{\mathrm {i}}$ in the region of the electron excitations. The separated UDM contributions of the absorption on localized states and Urbach tail are plotted by dash and dash–dot lines.
Fig. 6.
Fig. 6. Spectral dependencies of the reflectance $R$ measured in the UV and VUV spectral range and corresponding modeled complex dielectric function $\hat \varepsilon$. The separated UDM Urbach tail contribution is plotted by dashed line. The exciton position are shown by arrows.
Fig. 7.
Fig. 7. Spectral dependencies of ellipsometric quantities displayed by the complex pseudodielectric function $\langle \hat \varepsilon \rangle$ in the one-phonon absorption spectral range.
Fig. 8.
Fig. 8. Spectral dependencies of reflectance $R$ and transmittance $T$ measured in the spectral range of the one-phonon excitations and corresponding modeled complex dielectric function $\hat \varepsilon$. For the 12-th vibrational mode concern to Table 4 and details in text.
Fig. 9.
Fig. 9. Spectral dependencies of reflectance $R$ and transmittance $T$ measured in the one- and multi-phonon regions and corresponding modeled imaginary part of dielectric function $\varepsilon _{\mathrm {i}}$.
Fig. 10.
Fig. 10. Spectral dependencies of ellipsometric quantities displayed by pseudodielectric function $\langle \hat \varepsilon \rangle$ and the degree of polarization $P$ in the multi-phonon region.

Tables (4)

Tables Icon

Table 1. List of the experimental and tabulated data with spectral ranges (including resolution) and angle of incidence (AOI). The fitted experimental data R , T and I s , I c , I n represent the reflectance, transmittance and associated ellipsometric quantities [14], respectively. The χ i represents the quality of fits for corresponding data.

Tables Icon

Table 2. Dispersion parameters of HOM by fitting Zelmon et al. [13] experimental data, compared with the transition strengths determined from UDM.

Tables Icon

Table 3. Dispersion parameters describing electronic excitations.

Tables Icon

Table 4. Dispersion parameters of the phonon excitations. Internal parameter M ph17 is calculated according to condition (10).

Equations (10)

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n ( λ ) = ( 1 + k A k λ 2 λ 2 λ k 2 ) 1 / 2 ,
ε ^ ( E ) = n ^ ( E ) 2 = 1 + 2 π N el E el 2 E 2 + 2 π N ph E ph 2 E 2 + i N el E el [ δ ( E el E ) δ ( E el + E ) ] + i N ph E ph [ δ ( E ph E ) δ ( E ph + E ) ] ,
0 E ε i ( E ) d E = 0 F ( E ) d E = N el + N ph ,
ε ^ loc ( E ) = j = 1 4 N loc , j 2 π E loc , j B loc , j [ 2 π D ( E + E loc , j 2 B loc , j ) 2 π D ( E E loc , j 2 B loc , j ) + i exp ( ( E E loc , j ) 2 2 B loc , j 2 ) i exp ( ( E + E loc , j ) 2 2 B loc , j 2 ) ] ,
ε i,ut ( E ) = A exp E [ exp ( E E g E u ) exp ( E g 2 E u ) ] or ε i,ut ( E ) = A rat ( E E x ) 2 E 5 ,
ε i,ex , k ( E ) = 2 N ex , k B ex , k   ( E E g ) 2   Θ ( E E g ) π E [ ( ( E ex , k E g ) 2 ( E E g ) 2 ) 2 + B ex , k 2 ( E E g ) 2 ] ,
ε ^ 1ph ( E ) = 1 C N j = 1 17 ( N ph , j + i M ph , j E E ph , j ) i 2 π E ph , j B G , j [ W ^ ( E E ph , j + i B L , j / 2 2 B G , j ) W ^ ( E + E ph , j + i B L , j / 2 2 B G , j ) ] ,
B G , j = B ph , j 2 2 ln 2 ( 1 a L ph , j ) 2 ( 1 a ) 2 L ph , j 2 , a = 0.5346 B L , j = B ph , j L ph , j .
ε ^ 1ph ( E ) = 2 π C N j = 1 17 N ph , j + i M ph , j E / E ph , j E ph , j 2 + B L , j 2 / 4 E 2 i B L , j E .
j = 1 17 M ph , j ν ph , j = 0 , C N = 1 j = 1 17 N ph , j j = 1 17 ( N ph , j + M ph , j β L , j ν ph , j ) ,

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