Optical transmission and spectroscopic ellipsometry data of fused Quartz: Estimating dielectric function with Krönig-Kramers analysis

Abstract

The manuscript discusses the procedure of estimating refractive index from optical transmission data of a solid material. Fused Quartz is used as the standard sample in this manuscript. First the refractive index data is measured directly using the spectroscopic ellipsometry technique. Subsequently, the transmission data is collected with a transmission spectrometer. The transmission data is then converted into extinction coefficient and subsequently transformed in to refractive index, by direct Krönig-Kramers (KK) analysis code. Further using singly and doubly subtractive KK analysis, the refractive index is estimated. The derived set of refractive indices from KK analysis, are then compared to the measured spectroscopic ellipsometry data. The refractive index and extinction coefficient are then used to estimate the relative dielectric permittivity function of the material.

Specifications Table

Subject	Engineering & Materials science
Specific subject area	Evaluation of optical constants of materials derived from Spectroscopy
Type of data	Graph, Table, Raw Data, Analysed Data.
Data collection	The raw transmission spectrum of Fused Silica substrate is measured in an Agilent Cary 5000 series UV–VIS transmission system. The same substrate’s refractive index is measured with a J.A. Woolam Spectroscopic Ellipsometry system. The transmission spectrum is used for Krönig-Kramers analysis and compared with the ellipsometry data
Data source location	School of Sciences, Woxsen University, Telangana, India, 502,345
Data accessibility	Repository name: Mendeley Data Data identification number: 10.17632/jz2wc5rzyx.4 Direct URL to data: https://data.mendeley.com/datasets/jz2wc5rzyx/4
Related research article	none

1. Value of the Data

• •The raw data discussed in this manuscript is collected for fused Quartz sample, which is transparent in visible to near infra-red region of light and is widely used as substrate for film growth and optical windows, due to its excellent optical transmission properties. The fused quartz spectral data is a valuable initial point for understanding the dielectric function of grown films.
• •Due to the isotropic nature of fused Quartz, the transmission data and ellipsometry data of Quartz is used as a calibration standard for building a library of transmission spectra and dielectric function of optical materials. The fused Quartz dataset would be useful to Geologists and Gemologists employed by mining and jewellery industries, who often work with the optical properties of transparent crystals.
• •Fused Quartz is used as a test-case for Krönig-Kramers (KK) analysis, in order to extract refractive index from extinction coefficient. The raw data and the code in this manuscript can be used by other researchers for validating their KK analysis code of a transparent sample, which is weakly absorbing, isotropic, non-scattering dielectric, in a flat substrate form of known thickness.
• •The KK analysis programs show the application of subtractive KK analysis in estimation the refractive index from extinction spectrum of a material for a broad wavelength region, with as few as two known refractive data points.

2. Background

Estimating refractive index of an unknown transparent solid is commonly done by using either spectroscopic ellipsometry [1] or laser interferometry techniques [2]. Spectroscopic ellipsometry (SE) needs prior information about the composition of the sample and samples with rough surface morphology needs significant modelling [3]. Laser Interferometry can measure refractive index at the used Laser’s wavelength, but it cannot measure refractive index over a broad wavelength region. Both SE and Interferometry need advanced instrumental facilities. On the other hand, measuring transmission/extinction spectrum is relatively easier. Due to causality condition, [4,5] the extinction coefficient and refractive index are linked by the KK relations.

KK analysis can provide an estimate of refractive index over a broad region of spectrum, by only using sample’s extinction spectrum [6,7]. Subtractive KK analysis used along with either SE data or interferometry data, gives a better estimate of the sample’s refractive index over a broad range [7]. The Refractive index ($\mathit{\eta }$) is related to the extinction co-efficient ($\mathit{\kappa }$) using KK relation as [[4], [5], [6], [7]]

$$\mathit{\eta }\left(\mathit{\lambda }\right)=1+\frac{2}{\mathit{\pi }}\mathit{P}\underset{0}{\overset{\infty }{\int }}\left(\frac{\mathit{\kappa }\left({\mathit{\lambda }}^{\prime }\right)}{{\mathit{\lambda }}^{\prime }}\right)·\left(\frac{{\mathit{\lambda }}^2}{{\mathit{\lambda }}^2-{{\mathit{\lambda }}^{\prime }}^2}\right)\mathbf{d}{\mathbf{\lambda }}^{\prime }$$ (1)

If the sample’s refractive index is known at p different wavelengths and the refractive index of sample at i^th wavelength is $\eta$(λ_i) = $\eta$_i. Then the p-fold subtractive Krönig-Kramers relation is given by [[6], [7], [8], [9]]

$$\begin{array}{ccc}\hfill \mathit{\eta }\left(\mathit{\lambda }\right)& =& 1+\sum _{\mathbf{i}=1}^{\mathbf{p}}\left\{\left(\frac{{\mathit{\lambda }}_{\mathbf{i}}^2}{{\mathit{\lambda }}^2}\right)·\left(\frac{{\prod }_{\mathbf{j}\ne \mathbf{i}}^{\mathbf{p}}\left({\mathit{\lambda }}_{\mathbf{j}}^2-{\mathit{\lambda }}^2\right)}{{\prod }_{\mathbf{j}\ne \mathbf{i}}^{\mathbf{p}}\left({\mathit{\lambda }}_{\mathbf{j}}^2-{\mathit{\lambda }}_{\mathbf{i}}^2\right)}\right)·\left[{\mathbf{n}}_{\mathbf{i}}-1\right]\right\}+\frac{2}{\mathit{\pi }}\left\{\prod _{\mathbf{i}=1}^{\mathbf{p}}({\mathit{\lambda }}^2-{\mathit{\lambda }}_{\mathbf{i}}^2)\right\}·\mathit{P}\underset{0}{\overset{\infty }{\int }}\left(\frac{{{\mathit{\lambda }}^{\prime }}^{\left(2\mathbf{p}-1\right)}}{{\mathit{\lambda }}^{\left(2\mathbf{p}-2\right)}}\right)\hfill \\ & & ·\left[\frac{\mathit{\kappa }\left({\mathit{\lambda }}^{\prime }\right)}{({{\mathit{\lambda }}^{\prime }}^2-{\mathit{\lambda }}^2)·{\prod }_{\mathbf{i}=1}^{\mathbf{p}}\left({{\mathit{\lambda }}^{\prime }}^2-{\mathit{\lambda }}_{\mathbf{i}}^2\right)}\right]\mathbf{d}{\mathbf{\lambda }}^{\prime }\hfill \end{array}$$ (2)

3. Data Description

The raw data shown in the Fig. 1(a) and (b) is available at Mendeley Data repository [10], as two ascii text files: “FQ_JGS2_Transmission_stepsize.dat” and “Spectroscopic_Ellipsometry_FQ_JGS2.dat”. These files are found in “Grouped Raw Data” folder.

Fig. 1.

(a) shows the spectroscopic ellipsometry data for the fused Quartz sample. (b) shows the transmission spectrum of fused Quartz substrate collected at 0.5 nm, 1 nm, 2 nm, 5 nm and 10 nm spectral resolution.

FQ_JGS2_Transmission_stepsize.dat file has a data matrix with 10 columns representing 5 sets of transmission spectrum of fused Quartz at 0.5 nm, 1 nm, 2 nm, 5 nm and 10 nm sequentially. The wavelength data is measured in nanometers (nm) and the transmittance data is in percentage values. The wavelength values range from 200 nm to 800 nm, whereas the transmittance values range from 80% to approximately 95%.

Spectroscopic_Ellipsometry_FQ_JGS2.dat file has a data matrix with 7 columns representing measurement of refractive index and extinction coefficients at three angles of incidence set at three values 65°, 70° and 75° respectively. The wavelength resolution is approximately 0.6 nm for SE data The corresponding refractive index and extinction coefficients are labelled (n65, k65); (n70, k70) and (n75, k75).

The measured transmission data is organized into five files according to spectral resolution, in the folder “Individual Raw Spectral Files” for using as code inputs with “FQTS_Znm.dat”, where Z = 0.5 nm, 1 nm, 2 nm, 5 nm and 10 nm. The SE refractive index measured at 65° is placed in text file “FQ_SE_n65.dat”

The transmission data collected at 0.5 nm resolution and 200 nm to 800 nm range is extrapolated and archived in folder “Extrapolated Data” with name “FQTS_0.5nm_extrapolation.dat”. Similarly, the transmission data collected at 0.5 nm resolution and longer range (LR) 200 nm to 1800 nm range is extrapolated in UV, IR and both UV+IR regions and the corresponding data files are stored in the folder “Extrapolated Data” with file names “FQTS_0.5nm_LR_extrp_uv.dat”, “FQTS_0.5nm_LR_extrp_ir.dat” and “FQTS_0.5nm_LR_extrp_uv_ir.dat”. The data text file and code file nomenclature is described in following Table 1.

Table 1.

Shows the file nomenclature of data and code files used in this manuscript [[10], [11], [12]].

File name	Description	Input	Output
FQ_JGS2_Transmission_stepsize.dat	Grouped Transmittance Raw data	Not Applicable	Not Applicable
Spectroscopic_Ellipsometry_FQ_JGS2.dat	Grouped Spectroscopic ellipsometry Raw data	Not Applicable	Not Applicable
FQTS_0.5 nm.dat	Fused Quartz transmission spectral raw data at 0.5 nm resolution over range of 200 nm to 800 nm	• Code_2SKKanalysis_FQ.m • Code_SSKKanalysis_FQ.m • Code_direct_kkanalysis_FQ.m • 2SKK_FQ.py • SSKK_FQ.py • direct_KK_FQ.py The set of these six files is denoted by SET A for brevity	Refractive index (n) & extinction coefficient (k) plot
FQTS_0.5nm_LR.dat	Fused Quartz transmission spectral raw data at 0.5 nm resolution over range of 200 nm to 1800 nm	• Code_direct_kkanalysis_FQ.m • direct_KK_FQ.py The set of these two files is denoted by SET B for brevity	Refractive index (n) & extinction coefficient (k) plot
FQTS_1nm.dat	Fused Quartz transmission spectral raw data at 1 nm resolution over range of 200 nm to 800 nm	SET A	Refractive index (n) & extinction coefficient (k) plot
FQTS_2nm.dat	Fused Quartz transmission spectral raw data at 2 nm resolution over range of 200 nm to 800 nm	SET A	Refractive index (n) & extinction coefficient (k) plot
FQTS_5nm.dat	Fused Quartz transmission spectral raw data at 5 nm resolution over range of 200 nm to 800 nm	SET A	Refractive index (n) & extinction coefficient (k) plot
FQTS_10nm.dat	Fused Quartz transmission raw data at 10 nm resolution overrange of 200 nm to 800 nm	SET A	Refractive index (n) & extinction coefficient (k) plot
FQ_SE_n65.dat	Spectroscopic ellipsometry Raw data at angle of incidence 65o	Not Applicable	Not Applicable
FQTS_0.5nm_extrapolation.dat	Fused Quartz transmission spectral data at 0.5 nm resolution over the range 1 nm to 800 nm (extrapolated from 1 nm to 200 nm)	SET B	Refractive index (n) & extinction coefficient (k) plot
FQTS_0.5nm_LR_extrp_uv.dat	Fused Quartz transmission spectral data at 0.5 nm resolution over the range 200 nm to 1800 nm (extrapolated from 170 nm to 200 nm)	SET B	Refractive index (n) & extinction coefficient (k) plot
FQTS_0.5nm_LR_extrp_ir.dat	Fused Quartz transmission spectral data at 0.5 nm resolution over the range 200 nm to 2700 nm (extrapolated from 1800 nm to 2700 nm)	SET B	Refractive index (n) & extinction coefficient (k) plot
FQTS_0.5nm_LR_extrp_uv_ir.dat	Fused Quartz transmission spectral data at 0.5 nm resolution over the range 170 nm to 2700 nm (extrapolated from 170 nm to 200 nm and from 1800 nm to 2700 nm)	SET B	Refractive index (n) & extinction coefficient (k) plot

4. Experimental Design, Materials and Methods

Flat fused Quartz also known as fused Silica (JGS2-grade: 150 ppm Hydroxyl content) substrate of dimensions 20 mm x 20 mm x 1 mm is procured from Vritra Technologies, Delhi, India. The sample is then placed on stage of a Spectroscopic ellipsometer system (J.B. Woolam) and the corresponding Raw data for three angles of incidence at (65°,70° and 75°) is displayed in Fig. 1(a).

The sample transmittance was measured in a Cary 5000 UV–VIS spectrometer (Agilent) over the range of 200 nm to 800 nm, with a combination of Deuterium arc lamp and Tungsten Halogen lamp. This combination of lamps emitting white light is used for transmittance measurement at five different resolutions 0.5 nm, 1 nm, 2 nm, 5 nm and 10 nm respectively. The corresponding data is shown in Fig 1(b). The collected transmission spectral data variation at various resolutions is within 2%. The transmission data is then converted to extinction coefficient $\left(\mathit{\kappa }\right)$, by using the following relation: [13]

$$\mathit{\kappa }=\frac{-\mathit{2}.\mathit{303}\mathit{\lambda }·\log \left(\%\mathbf{T}/\mathit{100}\right)}{\mathit{4}\mathit{\pi }\mathit{t}}$$ (3)

Here, t is thickness of the sample (t = 1 mm), %T is the % transmittance and $\lambda$ is the wavelength. Once the extinction coefficient is obtained then by using KK analysis, the corresponding refractive index can be obtained by using Eq. (1).

The main experimental constraint for this approach is that the measurement of transmission spectrum/extinction co-efficient over infinite rage is not possible. Experimental constraints would truncate the region of measurement to a finite range of wavelengths (a = 200 nm, b = 800 nm is chosen here). Here the upper wavelength limit is b and lower wavelength limit is a of the measured transmittance spectrum.

As fused Quartz is an isotropic non-absorbing dielectric material, [14,15] we make two assumptions for analyzing, the transmission spectrum in Fig. 1(b). The first assumption is that at wavelength higher than b, the transmittance is unchanged and has saturated. Fused Quartz absorbs strongly in infrared (IR) region [16]. The second assumption is that at wavelengths shorter than wavelengths shorter than a, the transmittance linearly decreases till zero. The experimental data is linearly extrapolated till zero wavelength to obtain the values of transmittance as shown in Fig. 2(a).

Fig. 2.

shows the step-by-step procedure followed for retrieving refractive index and relative permittivity in this manuscript.

To understand the effect of extrapolation of measured transmission data in both ultraviolet (UV) and IR regions, transmission data is recorded in a longer range 200 nm to 1800 nm for 0.5 nm resolution. The data is extrapolated as shown in Fig. 2(c) to align with reports of broadband transmission data of fused Quartz reported at [16].

Under these two assumptions, the Eq. (1) reduces to the following Eq. (4).

$$\mathit{\eta }{\left(\mathit{\lambda }\right)}_{\mathbf{extrp}}=1+\frac{2}{\mathit{\pi }}P\underset{0}{\overset{\mathbf{b}}{\int }}\left(\frac{\mathit{\kappa }\left({\mathit{\lambda }}^{\prime }\right)}{{\mathit{\lambda }}^{\prime }}\right)·\left(\frac{{\mathit{\lambda }}^2}{{\mathit{\lambda }}^2-{{\mathit{\lambda }}^{\prime }}^2}\right)\mathbf{d}{\mathbf{\lambda }}^{\prime }+\frac{2}{\mathit{\pi }}P\underset{\mathbf{b}}{\overset{\infty }{\int }}\left(\frac{\mathit{\kappa }\left({\mathit{\lambda }}^{\prime }\right)}{{\mathit{\lambda }}^{\prime }}\right)·\left(\frac{{\mathit{\lambda }}^2}{{\mathit{\lambda }}^2-{{\mathit{\lambda }}^{\prime }}^2}\right)\mathbf{d}{\mathbf{\lambda }}^{\prime }$$ (4)

The second integral from (b, ∞) can be simplified considering $\mathbf{k}\left({\mathit{\lambda }}^{\prime }\right)$ is constant over the range.

$$\mathit{\eta }{\left(\mathit{\lambda }\right)}_{\mathbf{extrp}}=1+\frac{\mathit{2}}{\mathit{\pi }}\mathit{P}\underset{0}{\overset{\mathbf{b}}{\int }}\left(\frac{\mathit{\kappa }\left({\mathit{\lambda }}^{\prime }\right)}{{\mathit{\lambda }}^{\prime }}\right)·\left(\frac{{\mathit{\lambda }}^2}{{\mathit{\lambda }}^2-{{\mathit{\lambda }}^{\prime }}^2}\right)\mathbf{d}{\lambda }^{\prime }+\left[\frac{\mathit{\lambda }·\mathit{\kappa }\left(\mathbf{b}\right)}{\mathit{\pi }}\right]·\ln \left\{\left|\frac{{\mathit{b}}^{\mathbf{2}}}{{\mathit{b}}^{\mathbf{2}}-{\mathit{\lambda }}^{\mathbf{2}}}\right|\right\}$$ (5)

The extinction spectrum is plugged in Eq. (5) and numerically computed with GNU Octave and Python codes archived at GitHub repository links [11,12]. The corresponding refractive index is shown in Fig. 2(b) along with measured refractive index from SE technique. Similarly, the refractive index from direct KK analysis is shown in Fig 2(d) for extrapolated data in UV region and measured data. Despite the extrapolating the data in both UV and IR region the change in the estimated value of refractive index using the direct KK analysis is <0.002 as observed from Fig. 2(d). Also, it can be seen that the extrapolation in IR region contributes to the estimate more than the extrapolated data in UV region.

The larger estimate of refractive index data in Fig. 2(d) is reportedly attributed to the OH group absorption in IR region [16]. The discrepancy in refractive index in Fig. 2(d) from the measured refractive index from SE in 2(b), is the result of strong transmittance dips due to absorption of hydroxyl ions in IR region beyond 2700 nm, whose KK transform response, extends into visible and UV regions. This cannot be accurately accounted from measurement of transmittance in a truncated spectral region in visible and near IR regions.

To get an accurate estimate of refractive index from direct KK analysis, the experimental range of spectrum collection should be infinitely large for exact numerical integration of Eq. (1). All transmittance dips over the infinite spectral region will contribute to refractive index estimate. So, ideally measurement of large range of transmittance values, typically over a range of several microns of wavelength will give a better estimate of refractive index. This is not possible in real time measurements due to detector range limitations. In such case of finite spectral range, direct numerical evaluation of Eq. (1), over a truncated region will give erroneous estimates of refractive index.

Alternatively, the refractive index can be evaluated by using subtractive KK relations in Eq. (2). The estimates from subtractive KK relations have rapid convergence [7,8]. Singly subtractive Krönig Kramers (SSKK) and Doubly subtractive Krönig Kramers (DSKK) relations are often employed when the refractive index of the sample is known only at certain wavelengths.

In this manuscript, we consider refractive indices measured with SE, at two wavelengths λ₁= 208.0 nm and λ₂= 800.99 nm corresponding to $\mathit{\eta }$₁= 1.5423 and $\mathit{\eta }$₂=1.4433 for subtractive KK analysis. Extrapolated transmittance data is not considered for subtractive KK analysis in this manuscript. We have used the raw data for subtractive KK analysis, which is truncated from 200 nm to 800 nm.

Using the code from repositories, [11,12] the refractive index is estimated with direct evaluation of KK transform integral using extrapolated transmission data in Fig. 2(a). The estimated refractive index is shown in Fig. 2(b) along with SE data. SSKK analysis is run with the initial condition as $\mathit{\eta }$₂=1.4433 and λ₂= 800.99 nm. DSKK analysis is run with initial conditions as ($\mathit{\eta }$₁= 1.5423; λ₁= 208.0 nm) and ($\mathit{\eta }$₂=1.4433; λ₂= 800.99 nm). The refractive index of estimated from direct and subtractive KK analysis is shown in Fig. 3(a).

Fig. 3.

(a) shows the truncated raw data and the extrapolated data for the fused Quartz sample. (b) shows the estimated refractive index of fused Quartz substrate collected at 0.5 nm spectral resolution against the measured SE data at 65°. (c) shows the longer-range transmittance measurement from 200 nm to 1800 nm, which is extrapolated in both UV and IR regions. (d) shows the refractive index estimated using direct KK analysis code for measured and extrapolated data in UV, IR and UV + measured + IR regions.

The real part (${\mathit{\eta }}^2-{\mathit{\kappa }}^2$) and imaginary part ($2\mathit{\eta }\mathit{\kappa }$) of relative permittivity of fused Quartz are extracted from the refractive index and extinction co-efficient, are shown in Fig. 3(b). To visualize the spectral resolution dependence of estimated refractive index with doubly subtractive KK relations, the corresponding transformed refractive index from transmission data is shown in Fig. 4(a) and (b).

Fig. 5.

(a) shows the estimated refractive index data from doubly subtractive KK analysis code at 5 nm and 10 nm resolution plotted against the measured SE data at 65° for the sample. Fig. 5(b) shows the estimated refractive index data from doubly subtractive KK analysis at 0.5 nm, 1 nm and 2 nm resolution plotted against the measured SE data at 65°.

Fig. 4.

(a) shows the estimated refractive index data from direct, singly and doubly subtractive KK analysis code at 0.5 nm resolution plotted against the measured SE data at 65° for the sample. (b) shows the calculated dielectric function of fused Quartz sample using estimated data from doubly subtractive KK analysis.

To quantify the deviation in estimation of the refractive index, the absolute mean deviation (AMD) and root mean squared deviation (RMSD) of the index estimated from direct, SSKK and DSKK methods, the deviation in the estimated refractive index data over wavelength range 200 nm to 800 nm is compared with the refractive index estimate from the SE data in the following Table 2.

Table 2.

Shows the deviation estimation of Refractive index relative to the SE data of fused Quartz.

Spectral Resolution	Direct KK		SSKK		DSKK
of fused Quartz	AMD	RMSD	AMD	RMSD	AMD	RMSD
0.5 nm	0.46474	0.46533	0.02279	0.03286	0.00329	0.00419
1 nm	0.46487	0.46546	0.00108	0.14927	0.0033	0.00419
2 nm	0.46486	0.46547	0.00103	0.14811	0.00319	0.00409
5 nm	0.46552	0.46615	0.00102	0.14747	0.00301	0.00362
10 nm	0.45871	0.46320	0.00101	0.14705	0.00276	0.00334

Limitations

• 1.The transmission spectrum in this manuscript is limited in the wavelength range, due to instrumental factors at both deep ultraviolet and far infrared regions of the light spectrum. The data does not capture any strong light absorption or scattering processes that might occur in those wavelength regions. The data represents a weakly absorbing isotropic, non-scattering dielectric, in a flat substrate of known thickness (1 mm).
• 2.If the sample is strongly absorbing or has a complicated shape, which does not allow transmission of light through it, then recording transmittance will be experimentally challenging. In that case retrieving Refractive index from Reflectance of the sample would be more convenient. We are working on a manuscript, with such a scenario.
• 3.Also, if the transparent sample is in thin film form on an opaque substrate, transmittance measurement would not be possible. In such case the reflectance due to Fabry-Perot oscillations, the reflectance/transmittance can oscillate rapidly. This has to be carefully accounted for using appropriate retrieval procedures [[17], [18], [19]].
• 4.The spectroscopic ellipsometry technique, the beam interaction volume depth does not exceed beyond 10 µm from the surface. So, the extinction coefficient data from spectroscopic ellipsometry is not representative of the bulk volume extinction of sample.
• 5.On the other hand, the transmittance is related to the volume averaged loss in the intensity of light beam, so extinction coefficient evaluated from the transmission spectrum is a better estimate of the sample’s overall extinction.

Ethics Statement

The authors confirm that they have read and followed the ethical requirements for publication in Data in Brief and confirming that the current work does not involve human subjects, animal experiments, or any data collected from social media platforms.

CRediT authorship contribution statement

Sanat Kumar Gogoi: Writing – review & editing, Data curation, Software. Harshavardhan Reddy Kalluru: Writing – review & editing, Conceptualization, Methodology, Software.

Data Availability

• Mendeley DataTransmission spectrum and Spectroscopic Ellipsometry data of JGS2-fused Silica (Original data).