Calculation of Spectral Optical Constants Using Combined Ellipsometric and Reflectance Methods for Smooth and Rough Bulk Samples

Abstract

Spectra of the optical constants n and k of a substance are often deduced from spectroscopic measurements, performed on a thick and homogeneous sample, and from a model used to simulate these measurements. Spectra obtained for n and k using the ellipsometric method generally produce polarized reflectance simulations in strong agreement with the experimental measurements, but they sometimes introduce significant discrepancies over limited spectral ranges, whereas spectra of n and k obtained with the single-angle reflectance method require a perfectly smooth sample surface to be viable. This paper presents an alternative method to calculate n and k. The method exploits both ellipsometric measurements and s-polarized specular reflectance measurements, and compensates for potential surface scattering effects with the introduction of a specularity factor. It is applicable to bulk samples having either a smooth or a rough surface. It provides spectral optical constants that are consistent with s-polarized reflectance measurements. Demonstrations are performed in the infrared region using a glass slide (smooth surface) and a pellet of compressed ammonium sulfate powder (rough surface).

Keywords

Optical constants complex refractive index ellipsometry polarized reflectance specularity factor infrared

Introduction

The optical constants of a substance are key elements when describing and modeling physical phenomenology involving reflection, transmission, absorption, and emission. They are the real part, n, and the imaginary part, k, of the complex refractive index of the substance. Generally, spectral optical constants are obtained indirectly by comparing experimental measurements, acquired with an optical spectroscopic instrument, with a mathematical model simulating these measurements. The values of n and k as a function of the wavelength are deduced using either analytical equations or fitting processes.

A common method used to determine the spectra of n and k of a given substance is the ellipsometric method, in which an ellipsometer acquires spectroscopic ellipsometry measurements of a sample.^1–3 A typical ellipsometer uses two polarization modules, each containing a linear polarizer. The first module, or “polarizer” module, is placed before the sample and controls the polarization state of the source beam that illuminates the sample. The second module, or “analyzer” module, is placed after the sample and independently selects the polarization component measured by the detector from the beam that is specularly reflected by the sample. In some designs, a compensator is used to overcome potential polarization dependencies of the optical components. The ellipsometer acquires a set of measurements, with predefined fixed configurations of its polarization modules, and determines the polarization change induced by the sample reflection. The two spectral parameters typically retrieved (and combined with a model to deduce n and k) are the amplitude ratio, $tan Ψ_{p / s}$ , and the phase shift, $Δ_{p / s}$ , between the p- and s-polarized reflection coefficients. The technique only requires relative measurements and is considered highly accurate, reproducible, and robust against source power fluctuations.² Typically, in addition to the ellipsometric parameters, an ellipsometer can also measure separately the s- and p-polarized reflectances R_s and R_p of the sample.

The values obtained for n and k from the ellipsometric method should be applicable for subsequent simulation purposes. However, unexpected discrepancies are sometimes observed between polarized reflectance simulations, performed with effective optical constants deduced from ellipsometric measurements, and direct polarized reflectance measurements, performed with the exact same combination of sample, ellipsometer, and experimental configuration. Examples of such discrepancies will be discussed later in the paper.

Another method used to determine the spectra of n and k for a given substance is the fixed angle reflectance method. In this method, specular reflectance measurements are acquired from the sample with either a polarized spectrometer (e.g., an ellipsometer) or an unpolarized spectrometer, depending on whether polarization is accounted for or not. Methodologies have been developed previously to exploit s-polarized measurements acquired at a high angle of observation,⁴ or unpolarized measurements acquired near normal incidence.^5–7 The reflectance method uses the Kramers-Kronig transformation (KKT)⁸ to rebuild the essential phase information not acquired by the reflectance measurements. It requires a sample with a flat and perfectly smooth surface to give unbiased values of the optical constants.⁷

This paper presents and demonstrates an alternative method to calculate the spectral optical constants of a substance. The method combines both ellipsometric and specular reflectance measurements and applies to bulk samples with either smooth or rough flat surfaces.

Details of the samples investigated for this analysis, the physical parameters measured, and the ellipsometer used to perform the measurements are given in the Measurement Details section.

The alternative approach can be subdivided into three main steps, as illustrated in the diagram of Fig. 1 and detailed in the “Optical constants calculation” section. First, the basic effective optical constants n and k are obtained by the ellipsometric method, exploiting the measured ellipsometric parameters, $Ψ_{p / s meas}$ and $Δ_{p / s meas}$ , and the angle of observation, $θ_{o}$ . Second, the s-polarized specular reflectance $R_{s meas}$ , measured experimentally at $θ_{o}$ , is compensated for potential scattering from the surface roughness to estimate a rescaled s-polarized specular reflectance, $R_{s}^{'}$ . This step is centered on the determination of a specularity factor, Γ _s , which is the ratio of $R_{s meas}$ over the corresponding specular reflectance $R_{s sim}$ simulated using n, k, and $θ_{o}$ in the Fresnel equation. Third, the adjusted optical constants $n'$ and $k'$ are obtained by the reflectance method, exploiting $R_{s}^{'}$ instead of $R_{s meas}$ . The Kramers–Kronig consistency between $n'$ and $k'$ is ensured by the use of the KKT in the calculation process.

Discrepancies observed with the ellipsometric method and improvements obtained with the proposed alternative method are illustrated for two different samples (one smooth, one rough), at four angles of observation. The improvement that occurs when the alternative method is applied at any of these angles is discussed in the Results and Discussion section. An additional investigation has been performed in the p-polarization plane to assess the likelihood that the isotropy assumption made when deriving the optical constants holds.

The proposed alternative method is intended to be a simple and practical way to calculate optical constants. To keep the demonstration simple, only a high-level analysis is performed throughout this paper. Consistent results obtained with a different ellipsometer and depolarization curves are presented online as Supplemental Material.

Measurement Details

For this analysis, two samples were investigated. The first substance was glass, in the form of a 1 mm thick microscope slide. Prior to the measurements on this sample, black electrical tape was attached to its back side to cancel the corresponding reflection. The second substance was ammonium sulfate (AS), in the form of a pellet of compressed powder. To form a rigid pellet with a flat surface, the powder (product A5132 from Sigma-Aldrich) was put as delivered into a 35 mm diameter die and subjected to 50 tons of pressure for a period of 30 min.

The optical instrument used to perform the measurements was an IR-VASE Mark II ellipsometer (J.A. Woollam Company). It is a rotating-compensator Fourier-transform infrared (FT-IR) spectroscopic ellipsometer^3,9,10 controlled by proprietary software. It uses a silicon carbide glow bar and a Michelson-style FT-IR spectrometer with cesium iodine optics as the intensity-modulated light source, and a deuterated triglycine sulfate (DTGS) detector. The measurement components are aligned in the following order, which is called the PSCA configuration: fixed polarizer (P), vertical sample (S), step-scan rotating compensator (C), and fixed analyzer (A). The step-scan rotating compensator adjusts to a new position between each measurement. The instrument is optimized to acquire polarized signals between 333 and 5900 cm^–1 (1.7–30 µm), with a spectral grid expressed in wavenumber (in cm^–1).

The four parameters measured were the ellipsometric amplitude angle of the specular reflection ( $Ψ_{p / s meas}$ ), the ellipsometric phase angle of the specular reflection ( $Δ_{p / s meas}$ ), the s-polarized specular reflectance ( $R_{s meas}$ ), and the p-polarized specular reflectance ( $R_{p meas}$ ). These four parameters depend on both the wavenumber and the angle of observation, $θ_{o}$ . For each sample, each of the four parameters was measured at four different angles of observation, $θ_{o}$ ∈ ${45^{\circ}, 55^{\circ}, 65^{\circ}, 75^{\circ}}$ . Each measurement was acquired with a spectral resolution set to 4 cm^–1 (i.e., a data point every 2 cm^–1) and with 500 scans, averaged to increase the signal-to-noise ratio (SNR).

For each sample, the reflectance and ellipsometric measurements were performed consecutively, maintaining the exact same experimental conditions between the measurements. Reference measurements were acquired first without a sample present. Subsequently, the sample was placed in the path of the beam and the detector was moved automatically to acquire the polarized signals reflected by the sample at each $θ_{o}$ investigated, one angle at a time. Each spectrum of $R_{s meas}$ (or $R_{p meas}$ ) was measured with both polarization modules aligned to let the vertical (or horizontal) linear polarization pass through. Finally, at each $θ_{o}$ investigated, the spectra of $Ψ_{p / s meas}$ and $Δ_{p / s meas}$ were measured simultaneously, with a polarizer module fixed at $45^{\circ}$ and with a compensator rotating automatically in the analyzer module to take successively predefined acquisition configurations.

The four parameters measured with the ellipsometer can be expressed in terms of the polarized specular reflection coefficients, r_s and r_p, each being a complex value with an amplitude, $| r |$ , and a phase, Δ, such that

r_{s} = | r_{s} | e^{j Δ_{s}}

(1a)

r_{p} = | r_{p} | e^{j Δ_{p}}

(1b)

These specular reflection coefficients depend on both the wavenumber and $θ_{o}$ .

By definition, the complete measured ellipsometric information is linked to the ratio $r_{p} / r_{s}$ by the relation

ρ = tan Ψ_{p / s meas} e^{j Δ_{p / s meas}} = r_{p} / r_{s}

(2)

Taken separately, the measured ellipsometric angles are then expressed by

tan Ψ_{p / s meas} = | ρ | = | r_{p} | / | r_{s} |

(3a)

Δ_{p / s meas} = ∠ ρ = Δ_{p} - Δ_{s}

(3b)

The measured polarized specular reflectances can be expressed by

{R_{s meas} = Γ | r_{s} |}^{2}

(4a)

{R_{p meas} = Γ | r_{p} |}^{2}

(4b)

where the parameter Γ is a specularity factor. This factor represents the relative portion of the theoretical value, ${| r |}^{2}$ , effectively measured in the direction of the specular reflection, taking into account the non-specular scattering due to surface roughness. Its value ranges between 0, for an extremely rough surface, and 1, for a perfectly smooth surface.

Exploiting Eqs. 3 a, 4 a, and 4 b, $Ψ_{p / s meas}$ can be related to $R_{s meas}$ and $R_{p meas}$ by

Ψ_{p / s meas} = \arctan \sqrt{R_{p meas} / R_{s meas}}

(5)

Optical Constants Calculation

The proposed alternative method for calculating the optical constants combines both ellipsometric and specular reflectance measurements and applies to bulk samples with flat surfaces, either smooth or rough. This method can be subdivided into the three following main steps.

Application of the Ellipsometric Method

The spectra of the ellipsometric parameters, $Ψ_{p / s meas}$ and $Δ_{p / s meas}$ , and the value of the angle of observation, $θ_{o}$ , are used to calculate the basic effective spectra of the optical constants, n and k, as follows. The model used is a standard bulk, homogeneous, and isotropic (BHI) sample model.

The detected signal is modeled exclusively as the specular reflection occurring at a single interface, perfectly smooth and flat, between the incident medium (air) of isotropic refractive index n_i and the sample being analyzed, assumed to be homogeneous and to have the isotropic complex refractive index

\tilde{n} \equiv n + jk

(6)

At each wavenumber, the electric field incident on the sample is assumed to be a uniform plane wave with a propagation direction (or a wave vector) making an angle $θ_{i}$ to the normal of the interface. The reflected signal is measured in the observation direction making an angle $θ_{o}$ on the opposite side of the interface normal. Since specular reflection is observed,

θ_{o} = θ_{i}

(7)

With these assumptions, the modeled specular reflection is characterized by the well-known Fresnel polarized reflection coefficients

r_{s} = \frac{n_{i} cos θ_{i} - \tilde{n} cos \tilde{θ}}{n_{i} cos θ_{i} + \tilde{n} cos \tilde{θ}}

(8a)

r_{p} = \frac{\tilde{n} cos θ_{i} - n_{i} cos \tilde{θ}}{\tilde{n} cos θ_{i} + n_{i} cos \tilde{θ}}

(8b)

\tilde{θ} = \arcsin (\frac{n_{i} sin θ_{i}}{\tilde{n}})

(8c)

Equation 8 c defines the angle of refraction, $\tilde{θ}$ , in the sample using Snell’s law. For the typical case involving a lossless incident medium (i.e., n_i is purely real), an oblique angle of incidence on the sample ( $θ_{i}$ is not null) and a lossy sample substance ( $\tilde{n}$ is complex), the value of $\tilde{θ}$ is complex. A complex angle of refraction describes a refracted wave for which the wave vector and the gradient of the amplitude are not parallel. Except for $θ_{i}$ , all the other parameters in Eqs. 8 a through 8 c depend on the wavenumber.

As stated by Eq. 2, the ratio $r_{p} / r_{s}$ is linked to the complete ellipsometric information measured, represented by $ρ$ $=$ $tan Ψ_{p / s meas} e^{j Δ_{p / s meas}}$ . Equations 2 and 8 a through 8 c are combined and the ratio $\tilde{n} / n_{i}$ is isolated, giving

\frac{\tilde{n}}{n_{i}} = sin θ_{i} \sqrt{{1 + tan}^{2} θ_{i} (\frac{1 - ρ}{1 + ρ})^{2}}

(9)

When n_i is purely real (e.g., with $n_{i} \approx 1$ for air), the basic effective spectra of the optical constants n and k are calculated analytically with Eq. S1 (Supplemental Material).

Compensation for Potential Surface Scattering

The measured specular reflectances $R_{s meas}$ and $R_{p meas}$ are compensated for potential scattering from surface roughness as follows.

The spectra of the polarized specular reflectances are simulated for direct comparison with the measurements. For a perfectly flat and smooth BHI sample’s surface (r_s and r_p described by Eqs. 6 and 8 a through 8 c), the simulations are performed with

{R_{s sim} = | r_{s} |}^{2}

(10a)

{R_{p sim} = | r_{p} |}^{2}

(10b)

When n_i is purely real (e.g., with $n_{i} \approx 1$ for air), $R_{s sim}$ and $R_{p sim}$ are calculated with Eqs. S.2 (Supplemental Material).

To compensate for potential losses due to surface scattering, the smoothness of the sample surface is estimated. This is done by comparing the measured and simulated specular reflectances, using Eqs. 4 a, 4 b, 10 a and 10 b rewritten as

Γ_{s} = R_{s meas} / R_{s sim}

(11a)

Γ_{p} = R_{p meas} / R_{p sim}

(11b)

The specularity factor Γ represents the relative portion of the simulated specular reflectance, $R_{sim}$ , that is effectively measured by the detector or, in other words, that is effectively traveling in the direction of the specularly reflected ray during the experimental measurement.

In theory, Γ is expected to be polarization independent and have a spectral curve showing a relatively slow and monotonic decrease as the wavenumber increases (or as the wavelength decreases).^11–14 The monotonic decrease is due to the fact that a given surface modulation has a fixed size, which may appear smooth at a longer wavelength and rough at a shorter one. The value of Γ tends towards 1 for a perfectly flat and smooth surface. Conversely, for rough surfaces, the value of Γ is lower than 1 (the rougher the surface, the lower the value, with the minimum value being 0) and it depends on $θ_{o}$ and on the wavenumber. At $θ_{o}$ $=$ $45^{\circ}$ , Γ becomes the specularity index.¹³

For this analysis, Γ is calculated independently in each polarization plane. Since $R_{s meas}$ generally has a higher SNR than $R_{p meas}$ , Γ _s is generally less noisy and is, therefore, chosen to compensate for the scattering from surface roughness.

The spectrum of Γ _s obtained with Eq. 11 a highlights the discrepancies between $R_{s meas}$ and $R_{s sim}$ , whether they come from surface roughness or from any other source. In order to compensate only for scattering from surface roughness, features different from the expected behavior must be removed from the spectrum of Γ _s , which results in an adjusted version, $Γ_{s}^{'}$ .

More than one technique could be considered to produce $Γ_{s}^{'}$ . For the examples provided in the Results and Discussion section, each curve of $Γ_{s}^{'}$ was obtained by filtering Γ _s strongly with a Savitzky–Golay filter¹⁵ and by performing a polynomial fit on arbitrary anchor points of the filtered curve, selected at wavenumber values for which Γ _s behaves as expected.

Considering $Γ_{s}^{'}$ to be valid in both polarization planes (i.e., homogeneous surface reflectance), this adjusted parameter can be used to compensate for surface roughness scattering in each measured polarized reflectance, giving

R_{s}^{'} = R_{s meas} / Γ_{s}^{'}

(12a)

R_{p}^{'} = R_{p meas} / Γ_{s}^{'}

(12b)

Application of the Reflectance Method

The adjusted s-polarized reflectance spectrum $R_{s}^{'}$ and the corresponding value of $θ_{o}$ are used to calculate the adjusted spectra of the optical constants $n'$ and $k'$ as follows. The method used is that already developed for high-angle s-polarized reflectance measurements⁴ but employs $R_{s}^{'}$ instead of the original specular reflectance measurement. In order for the analysis to be valid (and the Fresnel equations operant), the considered reflection must correspond to that of a perfectly specularly reflecting surface.⁷

Even when it is deduced from a sample with a rough surface, the value of $\sqrt{R_{s}^{'}}$ can be viewed as the amplitude $| r_{s} |$ that would be expected from a BHI sample of the same substance if it had a perfectly flat and smooth surface. Mathematically,

r_{s} = \sqrt{R_{s}^{'}} e^{j Δ_{s}}

(13)

The corresponding phase, Δ _s , is obtained by applying the KKT to $\ln \sqrt{R_{s}^{'}}$ . This is done with a matrix-vector multiplication algorithm.¹⁶

Once both the adjusted amplitude $\sqrt{R_{s}^{'}}$ and the corresponding phase Δ _s are determined, the adjusted optical constants are calculated as follows. Equations 8 a, 8 c, and 13 are combined, $\tilde{n}$ is replaced by the adjusted version

\tilde{n}' \equiv n' + jk'

(14)

and the ratio

\tilde{n}' / n_{i}

is isolated, which results in the alternative equation

\frac{\tilde{n}'}{n_{i}} = \sqrt{{sin}^{2} θ_{i} + {cos}^{2} θ_{i} (\frac{1 - \sqrt{R_{s}^{'}} e^{j Δ_{s}}}{1 + \sqrt{R_{s}^{'}} e^{j Δ_{s}}})^{2}}

(15)

When n_i is purely real (e.g., with $n_{i} \approx 1$ for air), the adjusted spectra of the optical constants $n'$ and $k'$ can be calculated analytically with Eq. S3 (Supplemental Material).

Results and Discussion

In the figures presented in this section, the measured and calculated spectra are only displayed for limited spectral ranges, relevant to the discussed observations. Results for glass are displayed between 1145 and 1355 cm^–1, on the left-hand side of the figures. Results for AS are displayed between 2750 and 3550 cm^–1 for most parameters and between 1950 and 4550 cm^–1 for the specularity factor, on the right-hand side of the figures. In some cases where curves are superimposed, only regularly spaced subsets of the available data are displayed to help distinguish between curves.

Ellipsometric Method

For each sample investigated, the basic effective spectra of the optical constants (n and k) are obtained from the ellipsometric measurements ( $Ψ_{p / s meas}$ and $Δ_{p / s meas}$ ).

Photos of the samples are displayed in the top part of Fig. 2. The glass slide appears dark due to its transparency in the visible spectral range, its very smooth surfaces (hence, the reflection of some items in the room), and the black electrical tape fastened on its back surface. The pellet of compressed AS powder appears white due to its rougher front surface and its strong scattering in the visible spectral range.

Selected portions of the measured spectra of $Ψ_{p / s}$ and $Δ_{p / s}$ are shown as the “measured” curves in the middle and bottom parts of Fig. 2, respectively, for each sample at the four angles of observation investigated. For the glass sample, the SNR is good at spatial frequencies $< 1250$ cm^–1, but it drops significantly between 1250 and 1310 cm^–1, particularly for $θ_{o} = 45^{\circ}$ and $55^{\circ}$ . This behavior will be explained with the next results.

Figure 3 shows the basic effective optical constants spectra, n and k, obtained from the ellipsometric method. These spectra were calculated with Eq. S1 (Supplemental Material), using the measured values of $Ψ_{p / s}$ and $Δ_{p / s}$ displayed in Fig. 2 and the corresponding value of $θ_{o}$ ( $θ_{i} = θ_{o}$ for specular reflection). Even though the spectra of n (or k) are expected to be the same at every $θ_{o}$ for a BHI sample, they were computed independently at each of the four angles investigated, for comparison purposes. The spectral curves of their standard deviations (1 $σ$ ) are reported at the bottom of Fig. 3.

For the glass sample, the curves of n (or k) agree well with each other below 1250 cm^–1, whereas a significant discrepancy occurs between 1250 and 1310 cm^–1. This difference is related to the low SNR observed over the same spectral range in the measured values of $Ψ_{p / s}$ and $Δ_{p / s}$ at $θ_{o} = 45^{\circ}$ and $55^{\circ}$ (see Fig. 2). From Fig. 3, this behavior occurs when n → 1 and k → 0 for glass. The lower SNR is a consequence of the amplitude of the reflected light signal diminishing as the optical constant values approach those of the surrounding medium ( $n_{i} \approx 1$ for air).

For the AS sample, the curves of n agree well with each other under 3285 cm^–1. The same is observed for k, except for the curve at $75^{\circ}$ , which deviates from the others under 3000 cm^–1. A plausible explanation for this behavior is presented later.

Compensation for Surface Scattering and Discrepancies Observed

The s-polarized specular reflectance spectra acquired experimentally ( $R_{s meas}$ ) and modeled using n and k ( $R_{s sim}$ ) are compared to generate an adjusted version of the specularity factor ( $Γ'_{s}$ ) and to compensate the measurements for the potential scattering from surface roughness. Some discrepancies observed with the ellipsometric method are highlighted in the process, in both polarization planes, with the help of Figs. 4 and 5.

Selected portions of the spectral curves of $R_{s meas}$ (measured with the ellipsometer) and $R_{s sim}$ (simulated with Eq. 10 a using n and k from Fig. 3) are shown in the top part of Fig. 4, for both samples and at each value of $θ_{o}$ investigated. Corresponding spectra of $R_{p meas}$ and $R_{p sim}$ (simulated with Eq. 10 b) are shown in the middle part of the same figure. These reflectance spectra are used in Eqs. 11 a and 11 b to give the raw specularity factors Γ _s and Γ _p displayed in Fig. 5 (curves obtained using n, k). Γ _s is discussed here and Γ _p is discussed later.

At each angle $θ_{o}$ , the spectrum of Γ _s highlights all the discrepancies between $R_{s meas}$ and $R_{s sim}$ . In regions where no discrepancies exist, Γ _s behaves as expected when only scattering from the sample surface is considered. It stays close to 1 for the smooth glass sample and it decreases monotonically with wavenumber for the rougher AS sample. In regions where discrepancies exist, Γ _s is characterized by unexpected significant spectral features, at 1270 cm^–1 for the glass sample (with Γ _s ≫ 1 meaning $R_{s meas}$ ≫ $R_{s sim}$ ) and at 3285 cm^–1 for the AS sample. Using the technique mentioned earlier, the anomalous behavior is removed in the adjusted versions of the s-polarized specularity factor, $Γ_{s}^{'}$ , shown in the top part of Fig. 5.

Rescaled versions of the reflectance measurements are added in Fig. 4, for direct comparison with the simulations. They correspond to the curves of $R_{s}^{'}$ and $R_{p}^{'}$ calculated with Eqs. 12 a and 12 b using the $Γ_{s}^{'}$ curves that are displayed in the top part of Fig. 5. The rescaling brings $R_{meas}$ closer to $R_{sim}$ and compensates for surface scattering without altering the characteristic spectral features of the measurement. The original and rescaled measurements are practically identical for the glass sample, while they differ for the AS sample. In the latter case, the rescaling facilitates the visual comparison and clearly highlights the observed discrepancies discussed here.

The bottom part of Fig. 4 reports the curves of the discrepancy between

R'

and

R_{sim}

for both substances, at each angle, and in each polarization plane. Each of these discrepancy curves tends towards 0 outside the discrepancy region and contains a spectral feature within the discrepancy region. The peak-to-peak amplitudes of the discrepancies are reported in Table 1 under PP

[R_{sim} (n, k) - R']

Table 1.

Peak-to-peak (PP) amplitude of the discrepancy observed between the rescaled measured reflectance $R'$ and the reflectance $R_{sim}$ simulated using optical constants n and k (ellipsometric method) or $n'$ and $k'$ (alternative method), and relative improvement obtained with the alternative method, for two substances at four angles in the polarization planes s and p.

Substance	Spectral range	$θ_{o}$	PP $[R_{sim} (n, k) - R']$		PP $[R_{sim} (n', k') - R']$		Improvement
Substance	Spectral range	$θ_{o}$	s	p	s	p	s	p
Glass	1200–1300 cm^–1	$45^{\circ}$	0.014	0.004	0.000	0.001	100%	72%
		$55^{\circ}$	0.019	0.008	0.000	0.002	100%	70%
		$65^{\circ}$	0.033	0.020	0.000	0.005	100%	76%
		$75^{\circ}$	0.093	0.077	0.000	0.008	100%	89%
AS	2900–3400 cm^–1	$45^{\circ}$	0.014	0.002	0.000	0.003	100%	–25%
		$55^{\circ}$	0.020	0.002	0.000	0.002	100%	–21%
		$65^{\circ}$	0.026	0.004	0.000	0.005	100%	–14%
		$75^{\circ}$	0.034	0.013	0.000	0.012	100%	10%

For the glass sample analyzed, a discrepancy is observed between 1200 and 1300 cm^–1. On the one hand, all the curves of $R_{s}^{'}$ and $R_{p}^{'}$ present a dip reaching very low reflectance values between 1267 and 1273 cm^–1. The minimum value observed in these curves goes from pure noise (less than 0.1% absolute reflectance in the p-polarization plane at $θ_{o} = 45^{\circ}$ ) to 0.024 (2.4% absolute reflectance in the s-polarization plane at $θ_{o} = 75^{\circ}$ ). Here, the global low reflectance at 1270 cm^–1 is due to a strong similarity between the refractive indices of the sample (n → 1, k simultaneously small) and those of the surrounding medium (air, $n_{i} \approx 1$ ). This behavior is analogous to the Christiansen effect, observed when the refractive indices of a powdered substance match those of its surrounding medium at a specific wavenumber.¹⁷ On the other hand, a dip is also present for all the corresponding simulations, but it is sharper, deeper, and systematically shifted towards 1275 cm^–1. The simulations are not as noisy as the curves of n and k from which they are calculated, even at $θ_{o} = 45^{\circ}$ and $55^{\circ}$ . As shown for $θ_{o} = 75^{\circ}$ in the bottom panel of Fig. 4, the discrepancy between $R_{s sim}$ and $R_{s}^{'}$ behaves as follows. As the wavenumber increases, the discrepancy first reaches a maximum value of 0.074 (absolute 7.4 % reflectance difference in one direction) and then decreases to a minimum value of –0.019 (absolute 1.9 % reflectance difference in the other direction) for a total peak-to-peak amplitude change of 0.093 (absolute 9.3 % shift in the reflectance difference). All the other discrepancy curves displayed for glass show a similar behavior.

With the AS sample, a discrepancy is observable particularly in the s-polarization plane between 2900 and 3400 cm^–1. In this region, $R_{s sim}$ has a spectral feature at 3285 cm^–1 which is not significantly present in $R_{s}^{'}$ (or $R_{s meas}$ ), at the four angles investigated. In this case, $R_{s}^{'}$ and $R_{s sim}$ remain significantly above zero inside the discrepancy region. At $θ_{o} = 75^{\circ}$ , the discrepancy between $R_{s sim}$ and $R_{s}^{'}$ behaves as follows. As the wavenumber increases, the discrepancy first reaches a minimum value of –0.016 and then increases to a maximum value of 0.018 for a total peak-to-peak amplitude of 0.034 (absolute 3.4 % shift in the reflectance difference). The peak-to-peak amplitudes of the other discrepancy curves displayed for AS are reported in Table 1.

The consistency between the reflectance measurements and the ellipsometric measurements can be investigated by comparing both sides of Eq. 5. The left-hand side of the equation ( $Ψ_{p / s meas}$ ) was presented previously in the middle part of Fig. 2. The right-hand side of the equation is added in the same graphs, represented by the curves labeled from $R_{meas}$ . A very good agreement is observed, except when the curves become noisy. The curves calculated from $R_{p meas} / R_{s meas}$ are noisier than the curves of $Ψ_{p / s meas}$ that are measured directly.

Reflectance Method Using a Rescaled Spectrum

The adjusted spectra of the optical constants ( $n'$ and $k'$ ) are obtained from the rescaled s-polarized reflectance spectrum ( $R_{s}^{'}$ ).

Considering $\sqrt{R_{s}^{'}}$ to be the amplitude of r_s as written in Eq. 13, the corresponding phase Δ _s is obtained by applying the KKT to $\ln \sqrt{R_{s}^{'}}$ . The results are shown in Fig. 6 as the curves labeled from $R_{s}^{'}$ , for both samples at the four angles investigated.

For comparison purposes, corresponding simulations using n, k, and $θ_{o}$ (as $θ_{i}$ ) are shown in Fig. 6 as the curves labeled from n, k. They were calculated analytically with Eq. S4 (Supplemental Material). The greatest differences observed between the curves of Δ _s from $R_{s}^{'}$ and from n, k are found around 1270 cm^–1 for glass and around 3285 cm^–1 for AS.

Figure 7 shows the adjusted spectral optical constants, $n'$ and $k'$ , obtained with the reflectance method. These spectra were calculated with Eqs. S3 (Supplemental Material) using the rescaled values of $R_{s}^{'}$ displayed in Fig. 4 and the corresponding values of Δ _s from $R_{s}^{'}$ displayed in Fig. 6 at each value investigated for $θ_{o}$ ( $θ_{i} = θ_{o}$ in specular reflection).

For both samples, the adjusted curves of $n'$ (or $k'$ ) are in close agreement, as demonstrated by the small standard deviation. For the AS sample, the curve of $k'$ at $θ_{o} = 75^{\circ}$ now agrees well with the other three. For the glass sample, the curves of $n'$ are much less noisy, and therefore more reliable, than those of n displayed in Fig. 3. It is now clear that the glass sample has optical constants with a real part approaching exactly one near 1270 cm^–1 and with an imaginary part simultaneously tending to small values. This result explains the low reflectance analogous to a Christiansen minimum observed for both $R_{s meas}$ and $R_{p meas}$ at all angles (see Fig. 4), the low SNR observed for $Ψ_{p / s meas}$ and $Δ_{p / s meas}$ (see Fig. 2), and the noisy artifacts observed for n and k (see Fig. 3) near 1270 cm^–1.

When the sample does not have a perfectly smooth surface, as is the case for the AS sample, the alternative method presented still produces optical constants that do not vary with the angle of observation (except for the effect of noise), which is the expected behavior. In comparison, using the original measurements $R_{s meas}$ directly in the reflectance method produces the curves shown in the insets of Fig. 7. These spectra vary with $θ_{o}$ , which is undesirable. They all differ from the spectra obtained from $R_{s}^{'}$ , and those obtained from $R_{s meas}$ at $θ_{o} = 75^{\circ}$ differ the most in spite of the higher SNR. This comparison illustrates the benefit and the necessity of rescaling $R_{s meas}$ prior to using it in the reflectance method for samples not having a perfectly smooth surface.

Figure 8 presents the results from the analysis performed in Fig. 4, using $n'$ and $k'$ instead of n and k in Eqs. 10 a and 10 b to calculate $R_{s sim}$ and $R_{p sim}$ . The new peak-to-peak amplitudes of the discrepancies observed between the simulations and the rescaled measurements are reported in Table 1 under PP $[R_{sim} (n', k') - R']$ for each polarization plane. An estimation of the relative improvement obtained with the adjusted optical constants is reported at the end of the table, for each polarization plane. This improvement was calculated as a percentage using

\begin{array}{l} Improvement (%) \\ = \frac{PP [R_{sim} (n', k') - R'] - PP [R_{sim} (n, k) - R']}{PP [R_{sim} (n, k) - R']} \times 100 \end{array}

(16)

From Fig. 8 and Table 1, the polarized reflectance simulations were improved by the proposed alternative method as follows. When $n'$ and $k'$ are used, $R_{s sim}$ no longer differs from $R_{s}^{'}$ and the improvement is 100% for both samples at the four angles investigated. This perfect fit is due to the optical constants being adjusted precisely to better fit the rescaled s-polarized measurements, at each angle of observation considered independently. For the AS sample, the extraneous spectral feature previously observed at 3285 cm^–1 no longer appears in the curves of $R_{s sim}$ . The improvement is also very good in $R_{p sim}$ for the glass sample, with a relative decrease of 70–89 % in the peak-to-peak amplitude of the discrepancies. However, no real improvement is observed in $R_{p sim}$ for the AS sample.

In practice, the optical constants $n'$ and $k'$ are affected less by noise, and are therefore more valid, when deduced at higher observation angles, as can be observed in Fig. 7. This is attributable to an increasing SNR in the s-polarized measurement when $θ_{o}$ varies from $45^{\circ}$ to $75^{\circ}$ .

Since they are constant with $θ_{o}$ , the spectra of $n'$ and $k'$ obtained at $75^{\circ}$ can be used for further simulations at different angles. For example, assuming a purely real value for n_i, adjusted simulations of $Ψ_{p / s}$ and $Δ_{p / s}$ can be calculated analytically with Eq. S5 (Supplemental Material). Simulations using Eq. S5 and the spectra of $n'$ and $k^{'}$ obtained for glass at $75^{\circ}$ are displayed in the left-hand side of Fig. 2, at each angle investigated, as curves labeled from $n_{75}^{'}, k_{75}^{'}$ . These adjusted simulations are much smoother than the original ellipsometric measurements, $Ψ_{p / s meas}$ and $Δ_{p / s meas}$ , even though they come from measurements taken with the exact same combination of sample, acquisition instrument, experimental conditions, and number of averaged scans. In comparison, simulations using n and k obtained at each corresponding angle $θ_{o}$ with the ellipsometric method regenerate the original values of $Ψ_{p / s meas}$ and $Δ_{p / s meas}$ (verified, not shown).

Likelihood of the Isotropy Assumption

The proposed alternative method focuses on fitting reflectance measurements in the s-polarization plane. The effect of that fitting process in the p-polarization plane is illustrated by the p-polarized specularity factor (Γ _p , Eq. 11 b) curves displayed for each sample in the bottom part of Fig. 5. These are shown for either n and k (ellipsometric method) or $n'$ and $k'$ (alternative method presented). Due to a very low SNR in $R_{p meas}$ at the lowest values of $θ_{o}$ , some curves of Γ _p are omitted in the figure for clarity.

The adjusted optical constants $n'$ and $k'$ were obtained by converting Γ _s to $Γ_{s}^{'}$ , free of unwanted artifacts, and by using the BHI sample model, which assumes isotropic optical constants. Since Γ is expected to be the same in both polarization planes, the improvement of Γ _s (i.e., its replacement by $Γ_{s}^{'}$ ) should also be reflected in Γ _p , provided the optical constants of the sample are really isotropic.

From Fig. 5, when n and k from the ellipsometric method are used, Γ _p contains as many strong unexpected spectral features as Γ _s does. When $n'$ and $k'$ (computed using $Γ_{s}^{'}$ ) are used instead, almost all the unexpected features are removed from Γ _p for the glass sample investigated. This tends to indicate that the optical constants of this sample are really isotropic. For the AS sample, the unexpected feature observed in Γ _p is modified with the adjustment, but its amplitude did not decrease. This suggests that the optical constants of this sample are not perfectly isotropic.

AS is known to crystallize with an orthorhombic structure and to show very little anisotropy in crystal form.¹⁸ One would expect the weak anisotropy to be eliminated once the product is reduced to powder, since all the fragments would be randomly oriented. However, the pressing method used to form the pellet may have favored a common orientation of the lattice of the different fragments, partially restoring some anisotropy in the compressed sample. For weak anisotropy, the method proposed is expected to give a good approximation of the effective s-polarized optical constants and therefore of the effective optical constants at near normal incidence.

For the AS sample, anisotropy could explain why the curve of k obtained from an isotropic model deviated significantly at $θ_{o} = 75^{\circ}$ , and why simulations using these n and k produced an extraneous spectral feature not observed in the reflectance measurements.

Conclusion

This paper presented an alternative method to calculate the optical constants of a substance. The method combines the information from both ellipsometric measurements and polarized reflectance measurements acquired at a common angle of observation. It also makes use of the Kramers-Kronig transformation. The alternative method produces optical constants that perfectly fit s-polarized experimental reflectance measurements. It applies to an oblique and preferably high angle of observation, and to smooth as well as rough sample surfaces. The method relies on an isotropic model and can assess the likelihood of this assumption by performing an additional analysis in the p-polarization plane.

A demonstration of the method was performed independently at four angles of observation and with two different samples. When the optical constants calculated with the ellipsometric method were replaced by those determined with the alternative method for the simulations, the results obtained for a glass slide showed an important improvement around 1270 cm^–1 in both polarization planes, at the four angles investigated. Strong artifacts were removed in the simulations and the slide seemed smooth and isotropic. A similar analysis performed with a pellet of compressed ammonium sulfate powder showed an important improvement at 3285 cm^–1 only in the s-polarization plane, at the four angles investigated. A spectral feature was corrected in the simulations and the pellet seemed rougher than the glass slide, and also slightly anisotropic. The presented trends are not attributable to the specific instrument or samples used for the demonstration, as supported by additional graphs provided in the Supplemental Material.

The alternative method proposed is more robust than the ellipsometric method in spectral regions where the optical constants of the substance being analyzed approach those of the surrounding medium, which produces very low reflectance in both polarization planes. As an explanation, the alternative method is based on the s-polarized signal, which generally has a higher SNR than the p-polarized signal for a bulk sample observed at high angle. By comparison, the ellipsometric method exploits both polarization planes and is affected more by the lower SNR in the p-polarization plane.

The method estimates a specularity factor in the s-polarization plane and assumes its value to be the same in both polarization planes. The values obtained for the optical constants are expected to remain the same for different samples of the exact same isotropic substance showing different degrees of surface roughness. Further investigations would have to be performed to estimate the degree of roughness for which this expectation remains valid.

To keep the analysis simple and practical, high-level discussions were presented, since a full description of the underlying phenomenology would involve many additional considerations. To name only a few, oscillator models could be used to derive purer intrinsic values for the optical constants (instead of the effective values discussed), a more complex simulation process including an intermediary layer of mixed air and substance could be employed to model surface roughness effects (the many modeling parameters affording more degrees of freedom than the single specularity factor used as a “fudge factor”), and the potential imperfections of each optical component involved in the measurement device could be investigated. Each of these considerations is an individual topic of in-depth study. Most of them could still be explored when the alternative method is used. For example, the method could be used first to provide alternative effective values of the optical constants and adjusted versions of the ellipsometric parameters, and then the intermediary layers and oscillator models could be used to derive purer intrinsic values from the adjusted effective spectra.

Supplemental Material

sj-pdf-1-asp-10.1177_00037028211047898 - Supplemental material for Calculation of Spectral Optical Constants Using Combined Ellipsometric and Reflectance Methods for Smooth and Rough Bulk Samples

Supplemental material, sj-pdf-1-asp-10.1177_00037028211047898 for Calculation of Spectral Optical Constants Using Combined Ellipsometric and Reflectance Methods for Smooth and Rough Bulk Samples by Gilles Fortin in Applied Spectroscopy

Supplemental Material

sj-pdf-2-asp-10.1177_00037028211047898 - Supplemental material for Calculation of Spectral Optical Constants Using Combined Ellipsometric and Reflectance Methods for Smooth and Rough Bulk Samples

Supplemental material, sj-pdf-2-asp-10.1177_00037028211047898 for Calculation of Spectral Optical Constants Using Combined Ellipsometric and Reflectance Methods for Smooth and Rough Bulk Samples by Gilles Fortin in Applied Spectroscopy

Footnotes

Acknowledgments

The author would like to thank Annie Martinet for providing the pellet used for this work, and also Dr Jean-Marc Thériault and Dr Thomas E. Tiwald for helpful discussions.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Gilles Fortin

Supplemental Material

All supplemental material mentioned in the text is available in the online version of the journal.

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Supplementary Material

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