# Attenuated Total Reflection at THz Wavelengths: Prospective Use of Total Internal Reflection and Polariscopy

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## Abstract

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## 1. Introduction

## 2. Peculiarities of ATR Measurements

#### 2.1. Attenuated Total Reflection

#### 2.2. A Useful Condition When Atr Becomes Invalid

#### Example: Water in Butter around Water Freezing Conditions

#### 2.3. Polarisation and Field Enhancement at the Interface of ATR Prism

#### 2.4. Polarisation Analysis with Atr

#### 2.5. Feasibility Test

#### 2.6. Combined Spectral Filters and Polarisers

## 3. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Phase and Amplitude Changes in Atr

**Figure A1.**Visualisation of Equations (A1)–(A4). Reflection coefficients ${R}_{TE,TM}$ vs. angle of incidence ${\theta}_{i}$ for the ATR prism, e.g., ${n}_{1}=2.12$ at THz spectral range. Absorption in air is added by the imaginary part of refractive index ${k}_{2}={10}^{-2}$ and ${10}^{-3}$. The inset shows ATR geometry and conventions. (

**b**) The phase change upon reflection for the TE and TM modes governed by the real part of refractive indices $n={n}_{2}/{n}_{1}$. The polarising Brewster angle is given by ${\theta}_{B}={tan}^{-1}\left(n\right)$ and the critical angle ${\theta}_{c}={sin}^{-1}\left(n\right)$. Thinner lines correspond to the case when the sample has a refractive index of ${n}_{2}=1.5$. The right inset in (

**b**) shows the polarisation ellipse after the analyser.

## Appendix B. Experimental: Thz Beamline at Australian Synchrotoron

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**Figure 1.**(

**a**) Photo and schematics of ATR setup used for temperature dependent absorbance spectroscopy at the THz/Far-IR beamline [15,16]. (

**b**) Visualisation of frustrated TIR (Equation (2)) for different angles of incidence and refractive indices n of the slabs. The index $n=1.6$ also corresponds to the case when two $n=2.4$ (diamond) slabs are separated by ${n}_{gap}=1.5$ (bio-sample), i.e., $\frac{n}{{n}_{gap}}=\frac{2.4}{1.5}=1.6$.

**Figure 2.**(

**a**,

**b**) Photo and schematics of the beam path in ATR unit. Three reflections are encountered from the IN-port to the sample (mirrors m1–3) and extra six for the path sample-to-OUT port (m5–10). (Image: courtesy Dr. Jeff Kuehl) (

**c**) Schematics of measurements with top reflecting mirror when TIR conditions are not fulfilled due to the high refractive index ${n}_{2}$ of the sample. Plot shows the evanescent field ratio for the p-pol. (TM) and s-pol. (TE) ${d}_{p}/{d}_{s}$ (Equation (3)) vs. the angle of incidence ${\theta}_{i}$. For the non-polarised illumination, the penetration depth of evanescent field is ${d}_{ev}=\frac{{d}_{s}+{d}_{p}}{2}$. Critical angle is at ${\theta}_{c}=arcsin\phantom{\rule{3.33333pt}{0ex}}\left(\frac{{n}_{2}}{{n}_{1}}\right)$.

**Figure 3.**Temperature dependence of butter sample (2-mm-thick) at 1 and 2 THz measured with Au top-mirror [15]; wavelength of 1 THz is 0.3 mm and 2 THz is 0.15 mm. The insets show the reflectance intensity vs. the wavenumber; air (no Au top-mirror) reflectance is taken as the reference. Dotted-lines (black) are eye guide splines of the experimental data.

**Figure 4.**Maxwell’s scaling (change nm to $\mathsf{\mu}$m for the same refractive index): numerical finite difference time domain (FDTD; lumerical solutions) calculations for optical wavelengths of $\lambda =1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m – 0.65 $\mathsf{\mu}$m to simulate evanescent field at the ATR geometry used (at THz radiation) and at the angle of incidence of ${\theta}_{i}={45}^{\xb0}$ (ATR interface is 45${}^{\xb0}$-tilted). Top row shows E-field enhancement when a ∼$\lambda /20$-diameter sphere of refractive index $n=1.3$ (water or ice at visible), 2.2 (water or ice at THz) and gold (Au) touches the ATR prism of $n=2.4$ (diamond); all intensity maps are calculated for the same wavelength $\lambda =1050$ nm. Bottom row shows the evanescent field at the interface of $\lambda =1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m – 0.65 $\mathsf{\mu}$m; the interference maxima are recognisable in the $n=2.4$ media (in the wavelength, there is $\lambda /n$). Calculations are carried out for the p-pol. E-field with strength ${E}_{0}=1$; hence, $E/{E}_{0}$ scale represents enhancement.

**Figure 5.**(

**a**) Geometry and conventions of ATR; photo of the Bruker Alpha unit setup on the THz-beamline compartment. ATR conventions: incident beam (IN) has ${E}_{s,p}$ polarisations selected by a wire grid polariser incident on the ATR diamond (${n}_{1}=2.42$ [29]) prism; plane of incidence is xz. The reflected beam is elliptically polarised due to phase and amplitude changes incurred upon reflection (e.g., air case ${n}_{2}=1$). The sample is contributing with absorption and phase change for the output beam (OUT). At the interface (sample) the evanescent field ${E}_{TM}$ (or ${E}_{z}$) and ${E}_{TE}$ or (${E}_{y}$) defines the light–matter interaction. (

**b**) The reflected beam can be expressed via ${E}_{s,p}$ projections measured analyser and given by Equation (4). The angle $\phi $ is the orientation of the wire grid polariser in the incidence side (IN); $\phi ={0}^{\xb0}$ corresponds to pure ${E}_{s}$ component incident onto the ATR prism. The amplitude span in the polar plot is [0–1]. The two components of the ${E}_{p}$ and ${E}_{s}$ are shown separately by the dipole-like figures (each term in Equation (4)).

**Figure 6.**Numerical example of polarisation analysis at the ATR output (OUT) based on Equation (4). The angular dependence of synchrotron THz radiation (at THz beamline) comprised of 20% linear polarisation and 80% isotropic (circular) [28] is shown. (

**b**) Sample holder for four-polarisation (4-Pol) ATR measurements: instead of changing the linear polarisation of light beam (THz), the sample orientation can be changed. Sample shown is isotropic THz absorber—black paper.

**Figure 7.**(

**a**) Schematic of the ATR unit (Pike) used in this study. From the port IN to OUT, there are ten reflections (mirrors m1–10). (

**b**) Cross-polarised performance of the ATR unit. The polarisation is set horizontal ($\phi ={0}^{\xb0}$) at the IN port. This corresponds to the maximum of linear polarisation of synchrotron radiation extracted by the first mirror (slotted mirror at the bending magnet of the synchrotron). THz spectrum over 30–630 cm${}^{-1}$ (0.9–18.9 THz) is integrated (spectral range is defined by the mylar beamsplitter, and the detector is a Si bolometer). The water vapour absorption coefficient at 1 atm over the same spectral window [36] is shown in the inset. (

**c**) Co-rotation of the mesh-grid polariser and analyser (at the IN and OUT ports): dots are experimental data and fits by ${E}_{s}^{2}\propto {cos}^{8}(\phi +\Delta {\psi}_{s})$ and ${E}_{p}^{2}\propto {sin}^{8}(\phi +\Delta {\psi}_{p})$ intensity components (see text for details). Four angles separated by 45${}^{\xb0}$ degrees are selected, where the local maxima or minima are observed (Angles 1–4). (

**d**) Polarisation analysis of signal at port OUT for Angles 1–4 (at the IN port): experimental data by markers 1–4 and fit by Equation (4). The legend shows the fit functions explicitly.

**Figure 8.**(

**a**) SEM images of THz filters defined by photolithography: ∼6 $\mathsf{\mu}$m-thick AZ4562 resist (applied on surface) and SU8 laminate 20–500 $\mathsf{\mu}$m thickness (SUEX from DJ Microlaminates Ltd.; applied or free-standing.) The period, length and width $P,L,W$ define the central wavelength ${\lambda}_{0}$ and the bandwidth $\Delta \lambda $ of the filter [38]; gap $G=P-L$. (

**b**) Superimposed five THz bands filter based on the cross apertures (

**a**). The binary mask (false colour) and the $(0-2\pi )$ phase map of the filter (see details in Figure 9). Spectrum of Australian synchrotron radiation at the THz beamline [40] is shown with selected five frequencies (star-markers).

**Figure 9.**(

**a**) Image of super-pixel mosaic design with $5\times 5$ elements. (

**b**) A block scheme of the Gerchberg–Saxton algorithm with scattering degree $\sigma =d/D$. Images of spatially randomly multiplexed cross-filters with for (

**c**) $\sigma =0.1$, (

**d**) $\sigma =0.15$ and (

**e**) $\sigma =0.2$. SEM images of the fabricated (

**f**) mosaic type and (

**g**) randomly-multiplexed cross filters on gold-coated SU8 films on silicon substrate.

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**MDPI and ACS Style**

Ryu, M.; Ng, S.H.; Anand, V.; Lundgaard, S.; Hu, J.; Katkus, T.; Appadoo, D.; Vilagosh, Z.; Wood, A.W.; Juodkazis, S.;
et al. Attenuated Total Reflection at THz Wavelengths: Prospective Use of Total Internal Reflection and Polariscopy. *Appl. Sci.* **2021**, *11*, 7632.
https://doi.org/10.3390/app11167632

**AMA Style**

Ryu M, Ng SH, Anand V, Lundgaard S, Hu J, Katkus T, Appadoo D, Vilagosh Z, Wood AW, Juodkazis S,
et al. Attenuated Total Reflection at THz Wavelengths: Prospective Use of Total Internal Reflection and Polariscopy. *Applied Sciences*. 2021; 11(16):7632.
https://doi.org/10.3390/app11167632

**Chicago/Turabian Style**

Ryu, Meguya, Soon Hock Ng, Vijayakumar Anand, Stefan Lundgaard, Jingwen Hu, Tomas Katkus, Dominique Appadoo, Zoltan Vilagosh, Andrew W. Wood, Saulius Juodkazis,
and et al. 2021. "Attenuated Total Reflection at THz Wavelengths: Prospective Use of Total Internal Reflection and Polariscopy" *Applied Sciences* 11, no. 16: 7632.
https://doi.org/10.3390/app11167632