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Abstract
We report on a polarizing interferometer-ellipsometer arrangement that overcomes the need for additional measurements with a retarder for the unambiguous determination of the ellipsometric parameters in the far infrared spectral range. It consists of a Martin–Puplett interferometer and a wire-grid polarizer as an analyzer. The application of such interferometer-ellipsometer is experimentally demonstrated on a polyethylene sample deriving the refractive index and the thickness in the spectral range between 15 and ${35}\;{{\rm cm}^{- 1}}$. Based on these results, a similar solution without a retarder for the mid-infrared spectral region is additionally proposed.
Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. INTRODUCTION
Ellipsometry allows one to derive the so-called ellipsometric parameters ${ \tan}\psi$ and $\Delta$ upon reflection or transmission under an oblique angle from a sample and moreover to derive the degree of polarization of the reflected light. An ordinary infrared ellipsometer is built up of a Fourier transform (FT) spectrometer modulating the light, which then passes a polarizer and hits the sample at an oblique angle. The light experiences a change in amplitude and phase by interacting with the sample and passes further through an analyzer to the detector. A complete ellipsometric experiment requires two sets of measurements to obtain the experimental data for ${\cos}\Delta$ and ${\sin}\Delta$ [1,2] in a precise and unambiguous manner, namely, one set of data taken with a retarder and the other without a retarder. If one exchanges the beam splitter of the FT spectrometer by a polarizing beam splitter, the set of measurements with a retarder become obsolete. Here, the interferograms already contain the cos- and sin-Fourier transformations, which are connected with ${\cos}\Delta$ and ${\sin}\Delta$, respectively.
Dignam and Baker [3] reported such possibility to measure dichroism and ellipsometric data employing a Martin–Puplett interferometer (MPI) [4]. This kind of interferometer already incorporates a polarizer as a beam-splitting optical element. However, the setup reported is rather complex and uses polarized radiation as a source.
Ishida et al. [5] have introduced a polarizing Michelson interferometer with flat mirrors. Bomem Inc. produced such a Michelson interferometer (MI) with a polarizing beam splitter for measurement of optically active samples and ellipsometry measurements, but the data evaluation only considered the cos-Fourier transformation [6]. The input and output beam of this interferometer go both in the same direction, so the output beam has to be tilted for detection. The advantage of the Bomem Inc. design is the application of a common interferometer where only the standard beam splitter has to be exchanged by a polarizing beam splitter. In addition, no polarizer has to be placed at the input of the MI for ellipsometric data acquisition.
Both experimental arrangements have the same characteristic, namely, the electrical fields from the two arms of the interferometer (IF) are perpendicular to each other. An interference signal arises only if an analyzer combines the fields of both IF arms in front of the detector.
Here we report an MPI-ellipsometer configuration without any retarder for a complete ellipsometric measurement. This concept was reported in short by us in a workshop [7] and will now be presented in more detail in this paper. Our evaluation of the experimental data differs considerably from the evaluation of Dignam and Baker. In contrast, we consider the complex Fourier transformation for four analyzer angles, providing us with the sin- and cos-transformations for each angle. Therefore, eight inverse transformations can now be combined to obtain the optical data of the sample. The evaluation is exemplarily shown for a polyethylene foil as the sample and terahertz radiation from the electron storage ring BESSY II as the source.
Fig. 1. Scheme of the Martin–Puplett-ellipsometer configuration for the unambiguous determination of the ellipsometric parameters.
2. FOURIER TRANSFORMATION GENERATED BY THE MPI-ELLIPSOMETER CONFIGURATION
The MPI-ellipsometer configuration proposed here is shown in Fig. 1. Normally, the azimuthal angle α of the polarizing beam splitter of an MPI is 45° with respect to the optical plane of the interferometer, ensuring the entire intensity measured at the exit port but with the consequence of not having unambiguous access to ${\cos}\Delta$. Our approach differs from this by choosing an angle unlike 45°. The two perpendicular components of the electrical fields generated by the MPI at the exit are oriented according to the azimuthal angle of the polarizing beam splitter of the MPI in relation to the plane spanned by the two interferometer arms:
with $E_x^{(1)}$ is the electrical field from the moving mirror in the $x$ direction, $E_y^{(1)}$ is the electrical field from the moving mirror in the $y$ direction, $E_x^{(2)}$ is the electrical field from the fixed mirror in the $x$ direction, $E_y^{(2)}$ is the electrical field from the fixed mirror in the $y$ direction, $\tilde{v}$ is the wavenumber, the optical path difference ${ z} = 2\pi \tilde{v}D$ of the traveling waves from the moving arm, and $D$ is the mirror displacement.The perpendicular components of the electrical fields after the reflection at the ellipsometric sample are
with ${\rho _x},\;{\rho _y}$ the amplitudes and ${\delta _x},\;{\delta _y}$ the phases of the Fresnel coefficients ${r_x},\;{r_y}$ of the two perpendicular directions.Then the intensity measured behind an analyzer as a function of the azimuthal angle $\beta$ in respect to the reflection plane of the sample is given by
After subtraction of the constant term not depending on $D$ from the latter, we can write the Fourier integral, which is measured with the instrument
Obviously, the corresponding coefficients can be obtained by the inverse Fourier cosine and sine transformation, respectively. Its experimental evaluation makes ellipsometric measurements with a retarder obsolete for deriving ${\sin}\Delta$ as will be shown in the experimental part of the paper.
The chosen value of $\alpha$ is important in order to obtain all the relevant information to derive the ellipsometric parameters from the measurement. Equation (12) contains the product ${\cos\!2}\alpha \;\cos\!\Delta$, which means that for $\alpha = {45}^\circ$ the cos-transformation becomes zero with the consequence that ${\cos}\Delta$ cannot be evaluated. The best compromise for the azimuthal angle of the beam splitter is for $\alpha = {22.5}^\circ$. Further, it is convenient to measure at analyzer angles of $\beta = {0}^\circ$, 45°, 90°, 135° which results in four double-sided interferograms.
The above formalism derived here for an MPI is also applicable to common MIs when a polarizing beam splitter is inserted as it is introduced for the Bomem configuration [6].
3. EXPERIMENT
The MPI-ellipsometer concept as described above was realized with a homebuilt and compact MPI in conjunction with a wire-grid polarizer as an analyzer. The polarizing beam splitter of the MPI could not be changed and had a fixed azimuthal angle of 45°. In order to get the desired angle of 22.5° to also obtain ${\cos}\Delta$ from the measurements, the plane of reflection or transmission of the sample had to be tilted by 22.5°.
The experiments were performed with terahertz (THz) synchrotron radiation as a source. The radiation was delivered by the infrared beamline [8] of the electron storage ring BESSY II at ring energy of 1.7 GeV and 250 mA current stored. The detector was an InSb hot-electron bolometer, and the signal was recovered by a lock-in amplifier using the revolution frequency of the storage ring of 1.25 MHz as reference frequency. Double-sided interferograms of 500 equidistant points were taken at azimuthal angles of the analyzer of 0°, 45°, 90°, and 135°. A polyethylene (PE) sheet with a nominal thickness of 0.5 mm was used as a sample. Since the reflection of such sample is rather low compared to the transmitted intensity, the measurements were performed for experimental convenience in transmittance geometry under oblique angle of incidence without losing the generality of the concept.
4. DATA EVALUATION AND EXPERIMENTAL RESULTS
The PE foil was measured in transmittance under an angle of incidence of 65°. Figure 2 shows the interferograms measured for the four azimuthal angles of the analyzer. In order to derive ${\tan}{\psi}$, ${\cos}\Delta$, and ${\sin}\Delta$, the inverse Fourier cosine and sine transformation of the experimental interferograms shown in Fig. 3 have to be calculated numerically. The analytical expressions for the inverse Fourier cosine and sine transformation can be derived for selected values of $\alpha$ and $\beta$ from Eq. (2). In particular, one obtains from the inverse Fourier cosine transformation
and which directly allows for the calculation ofFig. 2. Interferograms for the transmittance of a polyethylene film at four azimuthal angles of the analyzer as described in the text.
Fig. 3. Spectral coefficients obtained by the inverse Fourier cosine and sine transformation of the interferograms for four different azimuthal angles of the analyzer shown in the Fig. 2.
The $\Delta$ values may not be correct for the sample measured, since they can be influenced by the instrument itself and by the data sampling. As an example, a deviation of the numerical zero-point of the interferogram from the exact position can add an additional phase. A second set of measurements with no sample in the beam path provides us with the corrective value ${\Delta_{\rm reference}}$. Finally, the corrected $\Delta$ values of the sample are
From the ellipsometric parameters, the total degree of polarization from the sample, ${P}$, can be calculated using the Stokes parameter ${{ s}_{1 - 3}}$ [1,2]:
Finally, the degree of polarization of the phase is given by
which can reach values ${\lt}{1}$, for example, caused by possible scattering of infrared radiation at the sample surface or films, or for ellipsometric schemes in attenuated total reflection (ATR) geometry by possible imperfection of the bulk crystal as is observed for Ge, Si, ZnSe, and KRS5.The optical constants and the thickness of the foil are obtained by fitting the experimental ellipsometric data to a film model using an optimization procedure from the MATLAB toolbox. The best fits for ${\tan}\Psi$ and $\Delta$ in comparison to the data measured at an angle of incidence of $\varphi = {65}^\circ$ are shown in Fig. 4 and result in ${ n} = {1.46},\;{k} = {0}$, and a film thickness of ${ d} = {0.054}\;{\rm cm}$. These data are very close to the nominal data of our sample. Refractive indices of low-density PE of 1.51 [9] measured with a dispersive FT spectrometer and of about 1.5 [10] measured with a time-domain spectrometer are reported for the 1 THz region. Our value for the refractive index is slightly lower, possibly due to imperfections in the setup. A mathematical treatment and a dedicated discussion on the influence of such imperfections are out of the scope of this proof-of-principle experiment and will be conducted in a separate paper.
Fig. 4. Experimentally obtained $\tan {\psi}$ and $\Delta$ and the corresponding fits for the PE sample film.
Fig. 5. Interferometer ellipsometer setup with polarizing mirrors in the interferometer arms for measurements without a retarder in the mid-infrared spectral range.
5. DISCUSSION
Interferometers with a polarizing beam splitter are reported by Martin and Puplett [4], Dignam and Baker [3], Ishida et al. [5], and they are also commercialized by Bomem Inc. [6]. Including our new configuration for a polarizing-ellipsometer, all constructions have the same characteristic that the electrical vectors from the different interferometer arms are perpendicular to each other. The interference signal arises only behind a second polarizer, that is, an analyzer in the optical path before the detector is necessary to implement.
Dignam and Baker [3] report the general relation for the intensity of such scheme, but in another form than that reported here. Even though they discussed the application for ellipsometric measurements, the full potential of the relation for the intensity was not evaluated. Only the Fourier cosine transformation was calculated for a fixed analyzer in a laborious ellipsometry arrangement.
The radiation output for the Bomem configuration [6] is in the direction of the source, which demands a deviation in the ray directions for the input and output paths. No radiation is leaving the other output for this ellipsometer-interferometer configuration.
The MI with a polarizing beam splitter proposed by Ishida et al. that is commercialized by Bomem Inc. also benefits from Eq. (12) derived for our MPI-ellipsometer configuration. The expression becomes applicable for those concepts when the factor ${{\sin}^2}{2}\alpha$ is omitted to account for the same direction to the source of the input and output paths. This gives an advantage in the signal-to-noise ratio due to a higher intensity measured, since the first polarizer placed before the polarizing beam slitter is unnecessary for ellipsometric measurements. A fixed azimuthal angle of 45° of the polarizing beam splitter is insufficient to get a set of complete ellipsometric parameters with the Bomem arrangement. The measurements have to be extended to four measurements with four azimuthal angles of the analyzer, and a data evaluation as described above becomes applicable to obtain the ellipsometric parameters from the sample.
We also like to discuss a new and third variant of an interferometer-ellipsometer, which opens the possibility for complete ellipsometric measurement without a retarder, even in the mid-infrared spectral range. This newly proposed ellipsometer arrangement consists of an interferometer with an amplitude-dividing beam splitter, and it utilizes polarizing flat mirrors for the fixed and the moving mirror. The azimuthal orientations of the polarizing mirrors are perpendicular to each another and are tilted by $\alpha$. The polarizing flat mirrors can be manufactured by metal stripes on an appropriate dielectric surface. The beam splitter could also be realized as a polarizer instead of an amplitude-dividing beam splitter without altering the concept and when specific requirements, e.g., from the spectral range, have to be met. Figure 5 shows the arrangement for such kind of polarizing interferometer-ellipsometer. The polarizing mirrors of the two interferometer arms meet the same requirements as in the Dignam and Baker arrangement. Again, the evaluation of the measurements is the same as described above.
6. SUMMARY
We present a concept for a Martin–Puplett interferometer-ellipsometer and the concept of the data evaluation, where no additional measurements with a retarder are necessary in order to obtain the ellipsometric parameters and with it the optical constants of a sample in an unambiguous way. This concept differs essentially from previously proposed and realized interferometer-ellipsometer schemes, which are unable to provide ${\cos}\Delta$ and ${\sin}\Delta$ unambiguously and independently without employing an additional retarder.
The concept was experimentally proven in the THz spectral range on a polyethylene foil and with synchrotron radiation as source.
Originating from the Martin–Puplett interferometer-ellipsometer, a new interferometer-ellipsometer scheme with polarizing interferometer mirrors is proposed, which allows an unambiguous ellipsometric evaluation of a sample, even with an amplitude-dividing beam splitter. This concept may be of advantage in the mid-infrared spectral range where polarizing beam splitters are virtually uncommon.
Disclosures
The authors declare no conflicts of interest.
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