Near-field THz micropolarimetry

: We introduce a method for rapid determination of anisotropic terahertz absorption with sub micron resolution and high spectral integrity in the terahertz range. The method is ideal for microscopic and environmentally sensitive materials such as 2-D materials and protein crystals where the anisotropic absorption is critical to understanding underlying physics. We introduce the idea of using an iso-response relationship between the THz polarization and electro optic probe polarization to enable stationary sample polarization measurements covering a full 2 π polarization dependence measurement.


Introduction
The terahertz (THz) spectroscopy is now common for solid-state and biological physics characterization [1,2]. Contactless THz conductivity measurements have become a standard for materials, especially when standard transport measurements are challenging [3,4]. While commercial THz spectroscopy systems are readily available for longitudinal transport studies, among underdeveloped tools is THz micropolarimetry which is critical to studying emerging solid state materials and biomacromolecules. Among the materials of particular interest for THz micropolarimetry are monolayer transition metal dichalcognides (TMD) such as MoS 2 and WSe 2 [5]. TMD's are excellent candidates as valleytronic devices, where the splitting of valence band provides low power state control. Using the magnetic proximity effect, exciton splitting as large as ∼ 0.6 THz/T has been observed [6]. Previously far-field THz conductivity measurements were performed on monolayer films produced using chemical vapor deposition [7]. However polarimetry measurements have not been reported, likely because typical grain/domain sizes are on the order of microns or less, requiring micropolarimetry [8]. THz microscopy demands near field optics in order to overcome the diffraction limited spot size ∼ 1 mm. Such near field spectroscopy is complicated by the sensitivity of TMD electronic properties to gas exposure, requiring measurements to be performed in vacuum [9]. Ultrahigh spatial resolution near field measurements could be performed in vacuum with THz nanoscopy [10], however polarimetry is not possible as the AFM tip-modified technique is limited to electric field polarization perpendicular to the sample. Other emergent materials system needing THz micropolarimetry include 2D ferromagnets [11]. Discovery of 2D ferromagnetism in Van der Waals materials has opened the possibility of integrating them as spin filters in spintronic devices which can be controlled via electric or magnetic fields. The characterization of magnons or spin waves in these materials is essential to evaluate the frequency cutoff for the devices. It was previously demonstrated using micro-Raman that the spin excitations in the 2D ferrmagnet material CrI 3 at 2.28 THz and 3.75 THz [12]. Lower energy excitations would be extremely challenging for micro-Raman, and THz micropolarimetery is needed to perform the necessary spectroscopic characterization.
Beyond inorganic materials science, THz micropolarimetry also has the potential to make enormous impact on molecular biology. Intramolecular collective vibrations in proteins and polynucleotides occur in the terahertz range [13], and there has been considerable speculation that these vibrations impact biological function via the efficiency of structural changes [14], electron transfer [15] and possibly coherent energy transfer [16]. While terahertz solution phase measurements of biomolecular samples have vastly improved [17][18][19], the spectra are fundamentally dominated by water absorption and a broad vibrational density of states [20][21][22]. Recently, it was shown that near-field, anisotropic measurements of aligned biomolecules, (realized using protein crystals) isolates vibrational bands based on their dipole transition direction [23]. This technique, anisotropic terahertz microscopy (ATM), uncovered previously unobserved narrow-band, frequency dependent features that switch with protein functional state [24]. The isolation of specific motions using ATM opens up unprecedented potential for protein engineering by identifying intramolecular vibrations that might enhance or inhibit function and possibly mapping of allosteric control targets [25].
ATM is capable of submicron resolution, full 360 control of the in-plane polarization, and was specifically built for environmental hydration control of protein samples. ATM could conceivably characterize 2D solid state materials as well. However there are some drawbacks for this near field polarimetry technique. Specifically the ATM method achieved near field spatial resolution by having the sample in direct contact with the detection electro optic crystal. To ensure uniform signal response as the relative crystal orientation and THz polarization change, the microscopic sample is rotated relative to both the electro optic detection crystal and the incident THz light polarization. Invariably such a rotation results in a microscopic displacement of the sample from the optic axis, and raster scans of the sample must be performed for each rotation to reposition the sample. This adds a significant amount of time to the measurement. The displacement of the sample also introduces potential variables, such as pixel-to-pixel variation due to defects in the EO detection crystal and signal drift. While stationary sample THz polarization spectroscopy techniques have been developed [26], these cannot readily be applied to microscopic samples requiring environmental control, such as protein crystals and TMD's. Recently a microscopic stationary sample method was introduced to address these concerns, polarization-varying ATM (PV-ATM) in which the THz polarization is rotated by rotating a wire-grid polarizer in the THz path [27]. This technique characterizes the anisotropic response a factor of six more rapidly than ATM, enabling reproducibility tests and identification of unique spectra for different materials. However the resulting spectra cannot be readily compared to computational results because of distortions from sample birefringence. PV-ATM can be used to establish a catalog of THz response, however ultimately to extract the conductivity tensor, or to achieve protein dynamical engineering a direct link to calculations is needed. The distortions arise from the variation of the THz intensity and the EO responsivity as the THz polarizer is rotated. While some distortions can be corrected by additional measurements, such as the THz intensity variation with polarization angle, the change in EO signal cannot be readily removed because it is not separable from the sample's own THz optical properties.
The electric field transmission is given by: Where ì E i(t) (θ, ν) is the incident (transmitted) electric field with polarization θ and frequency ν. The transmitted field is given by the sample's refractive index n(θ,ν) and polarization dependent absorption coefficient α(θ,ν). HereF is the Fresnel factor accounting for transmission changes for all the interfaces, and it is assumed changes in the Fresnel coefficients with angle are small. The ATM method isolates a small anisotropic response from a large isotropic background by self-referencing the polarization dependent transmission to calculate the change in absorbance with polarization angle, ∆abs(θ,ν): θ ref is the angle the THz polarization makes with respect to a defined reference direction for the sample. For example, a reference orientation associated with a molecular sample could be its static dipole direction. We aim to combine ATM's direct connection to the absorbance and constant THz intensity at the sample, with the stationary sample configuration of PV-ATM. Our proposed method, Ideal ATM (IATM), utilizes circularly polarized THz light, a wire grid polarizer to control the THz polarization and synchronous rotation of the near infrared (NIR) EO probe polarization to eliminate the polarization sensitivity of the EO detection. There are several advantages to this approach. First, the sample remains stationary. This removes reimaging of the sample center between rotations and avoids signal variation due to non uniformities in the EO detection crystal. Next, by using circularly polarized light before the polarizer, the light incident on the sample maintains a constant amplitude as the polarizer is rotated. Finally the variation in the EO responsivity is eliminated by synchronously rotating the EO probe polarization relative to the THz polarization in order to maintain a constant EO response. The overall technique of remaining on an iso-response curve for polarization dependent measurements can be used in the far field as well when environmental control and space limitations impede linear dichroism measurements using other techniques.

Experimental approach
The system proposed here is based on the near-field THz microscope designed by Planken [28] and utilized in the Acbas et al. and Niessen et al. studies [23,27]. An example schematic of the experimental setup can be seen in Fig. 1. The THz light may be generated by a variety of methods such as optical rectification or a photoconductive antenna [29] which lead to linearly polarized output. Conversion to circular polarization can be achieved by adding a prism, such as a silicon prism with appropriate geometry, to the THz path [30] . A wire-grid polarizer on a rotating stage is placed after the prism. With the addition of the prism, the polarization can be rotated without varying the THz intensity. An electro optic crystal is used for detection. For the specific parameters discussed here we assume the EO crystal is a (110) face ZnTe. In the near field configuration shown in Fig. 1 the sample is placed on a sample plate with a subwavelength diameter circular aperture, and this sample plate is placed directly on top of the EO detection crystal, which is mounted horizontally. The THz beam is focused onto the top EO crystal surface from above and the NIR is focused onto the top EO crystal surface from underneath. A high reflectivity coating on the top surface of the EO crystal reflects the NIR and the reflected beam is guided to a balanced detector. The THz light is transmitted through the sample to the EO crystal, inducing birefringence in the EO crystal which is sampled from underneath by the NIR beam. The change in the NIR polarization ∆I(τ), the EO response, is proportional to the THz electric field and is measured as a function of time delay τ . The Fourier transform gives the transmitted terahertz spectrum. In the frequency domain, the EO response of a (110) face ZnTe crystal relative to the THz and NIR polarizations has been determined by Planken [31]: θ THz and ϕ NIR are the THz and NIR polarization angles relative to the EO crystal [001] axis, E(ν,θ THz ,), is the amplitude of the THz field transmitted through the EO crystal, and R(θ THz ,φ NIR ) we will call the EO response function. The EO response dependence on the THz and NIR angles relative to the EO crystal [001] axis is shown in the surface plot of Fig. 2. The signal has been normalized to the maximum value. A typical THz time domain spectroscopy system using EO detection has θ THz and φ NIR set to values to optimize the signal, such as θ THz = φ NIR = 90°, shown by the blue dot on Fig. 2. If the polarization of the terahertz is rotated independently of the NIR, there will be variation in the responsivity, indicated by the white line on Fig. 2. To avoid this, we propose to have the terahertz and EO probe polarization change synchronously so as to retain a constant response. For example in Fig. 2 the black line and orange dashed lines show the pairs of angles which result in a constant THz signal when no sample is present. We refer to these paths as iso-response curves, φ NIR, iso (θ THz ). In the schematic in Fig. 1 a quarter-wave plate and linear polarizer on a rotating stage are placed in the NIR path before the EO detection crystal to realize this synchronous polarization rotation. Here we examine computationally the resulting spectra using this approach for samples with anisotropic resonances and compare these to the ATM approach. We find that in the case of continuous iso-response curves, samples with low birefringence have resulting spectra nearly identical to those measured by rotating the sample, however in the case of high birefringence or thick samples, the spectra are distorted relative to those where the sample alone is rotated. Alternatively we find that choosing iso-response curves which are piecewise continuous, but minimize the slope of φ NIR, iso (θ THz ) (shown by the dotted white lines in Fig. 2) removes the distortion, allowing for stationary sample anisotropic terahertz spectroscopy both in the near and far field configurations.

Simulations
The simulations of the Ideal ATM spectra are similar to the procedures for simulating the PV-ATM spectra of sucrose [27]. Here the THz polarization is modeled as circularly polarized instead of linearly polarized, thus as the wire grid polarizer before the sample is rotated, the THz amplitude remains constant. The polarization angles of the THz and NIR probe are referenced to the [001] ZnTe EO crystal axis, and the ZnTe [001] and [101] directions define the lab frame.
The transmitted THz electric field vector is calculated through Jones matrix calculus, with rotation into the THz polarizer frame, transmission through the polarizer, rotation back to the lab frame, rotation to the sample frame, transmission through the anisotropic sample, and rotation back to the lab frame. The sample's properties are modeled by the permittivity ε(ν) : where ν n , γ n , and A n are the frequency, linewidth, and amplitude of resonance n, ν is the frequency, and ε DC is the DC dielectric constant. The transmission through the sample is modeled by diagonal matrix components e i √ ε o (ν)2πνd and e i √ ε e (ν)2πνd , for the 0°and 90°absorption, respectively, where d is the sample thickness. We do not consider off diagonal components here, which would be appropriate for the transverse conductivity measurements of 2D ferromagnets and TMD's.
Combining Eq. 1 and Eq. 2 the measured ∆abs depends on the change in the relative θ THz and φ NIR and using 0°as the reference THz polarization direction we have: in the case where the THz polarization is unchanged by passing through the sample where E sample (ν,θ THz ,)is amplitude of the THz field transmitted through the sample reflecting the sample's anisotropic optical response and θ THz and ϕ NIR are the THz and NIR polarization angles relative to the EO crystal [001] axis. Ideally the only change in the signal arises from the sample's polarization dependent absorption: that is R(θ THz ,φ NIR )= constant. This can be accomplished by following an iso-response curve for the response surface shown in Fig. 2.
Supplementary Table 1 lists the θ THz and φ NIR values for iso-response curve 1 (black curve in Fig. 2) and iso-response curve 2 (orange curve in Fig. 2). However the innate birefringence that must accompany anisotropic absorbance leads to the rotation of THz polarization as it is transmitted through the sample. The polarization angle of the THz light exiting the sample being given by Θ sample (ν,θ THz ). This rotation effects the EO response: The simulations are performed on simple model systems. Two absorption peaks of equal magnitude and linewidth, A n = 100 and γ n = 1 cm −1 , are simulated to align along the 0°and 90°T Hz polarization directions at 50 cm −1 and 90 cm −1 , respectively, named Model Sample 1. In Model Sample 2, an off-axis peak is added oriented at 30°with frequency 70 cm −1 and the same amplitude and linewidth as used in model 1. Figure 3 shows surface plots of the ATM and IATM spectra for the model samples. The ATM method is the most direct method for measuring the near field anisotropic absorption, with the THz and EO probe polarizations remaining constant and the sample rotating. The method was previously validated using sucrose, where single crystal linear dichroism has been established with standard far field measurements [23]. For model sample 1, the calculated ATM spectrum in Fig. 3(a) is symmetric about 90°, and the resonant frequencies and orientations are readily read with the large positive ∆abs at 90 cm −1 indicating the resonance is oriented along this direction, whereas the large negative ∆abs at 50 cm −1 indicates the resonance is at 0°. Figure 4 shows the ATM angular dependence at 90 cm −1 and 50 cm −1 for model sample 1: and 90 cm −1 , 50 cm −1 and 70 cm −1 for model sample 2. One expects model sample 1 to act as a lossy narrowband polarizer with ∆abs = −2ln t(θ)

Results
where t is the transmission parallel to the initial polarization direction and t ⊥ is the transmission perpendicular to the initial polarization direction. At ν = 90 cm −1 t ∼ 1 and t ⊥ 1 due to the 90 cm −1 resonance at 90°, and at ν = 50 cm −1 t 1 and t ⊥ ∼ 1 due to the 50 cm −1 resonance at 0°. Fits to the angular dependencies of model sample 1 in Fig. 4(a) verify that ATM is consistent with this picture. Looking now at model sample 3 results in Fig. 3(d) we see the off axis resonance 30 slightly distorts the angular dependence at 50 and 90 cm −1 , removing the symmetry about 90°. Nevertheless the extremum of the angular dependences shown in Fig. 4(b) do coincide with the orientations of all three resonances. The determination of the resonant frequencies and the orientation of the dipole direction is straight forward in ATM.
The sample rotation requirements for ATM limits its utility. The rotational mount integrated into the near field sample position impedes sample loading, and the re imaging and repositioning with each angle leads to data taking times as long as 24 h. This is not practical for materials science, where high throughput is essential. The stationary sample approach of IATM could provide a substantial improvement. Figure 3 shows the IATM signals for following iso-response curve 1 (Fig. 3(b) and Fig. 3(e)) and iso-response curve 2 (Fig. 3(c) and Fig. 3(f)). We see in both cases the IATM surface plots are distorted from the symmetric resonances in ATM. For model sample 1 both techniques yield spectra which are 180°periodic, however the IATM spectra lack the symmetry about 90°.
In Fig. 5 we show the IATM angular dependence for 90 cm −1 , 50 cm −1 for model sample 1 (Fig. 5(a)) and 90 cm −1 , 50 cm −1 and 70 cm −1 for model sample 2 (Fig. 5(b)). First examining Fig. 5(a) the results for the iso-response curve 1, for ν = 90 cm −1 there are two orientations where ∆abs is maximum, at 90°and 150°, for ν = 50 cm −1 the minimum ∆abs is no longer at 90°but is shifted up to 150°. Now examining the results for iso-response curve 2 in Fig. 5(b), for ν = 90 cm −1 ∆abs is maximum, at 30°and 90°, and for ν = 50 cm −1 the minimum is shifted down to 30°. Figure 5(c) and Fig. 5(d) show the angular dependencies for the IATM result at 50 cm −1 , 70 cm −1 and 90 cm −1 for model sample 2 and iso response curves 1. For 90 cm −1 angular dependence no longer has any maximum at 90°, rather there are maxima at 107°and 163°. Similarly the orientation of the 50 cm −1 resonance and 70 cm −1 resonances are incorrectly determined by the extreme of the measured ∆abs. The results for iso-response curve 2 are similar in Fig. 5(d). The results suggest that while the IATM technique is faster than ATM, for a complex sample with many resonances, the orientation of the resonances will be incorrectly assigned. A closer examination of the THz polarization dependence of the EO signal reveals the cause of this lack of 90°symmetry, and the origin of the additional angle dependent peaks for IATM. Figure 6 shows the slope in the EO response function with respect to the incident terahertz polarization, dR(θ THz , φ NIR )/dθ THz for the different iso-response curves φ NIR, iso (θ THz ). This slope indicates the sensitivity of the EO response to changes in θ THz resulting from sample birefringence. We can see that for iso-response curve 1 this sensitivity is highest at 150, the same angle where an additional peak is seen for the 90 cm −1 resonance and the shifted minimum for the 50 cm −1 resonance in Fig. 5(a). Similarly we see that for iso-response curve 2 dR(θ THz , φ NIR )/dθ THz is peaked at 30°, the same orientation for the additional 90 cm −1 resonance and the shifted minimum for the 50 cm −1 resonance in Fig. 5(b). The secondary peaks and shifts in peak orientation arise from angles at which the EO response is most sensitive to changes in θ THz , which are inherent for anisotropic absorbance. To avoid these regions of high sensitivity to birefringence we consider a different iso-response function. In Fig. 2 we show a piecewise continuous iso-response curve which we label the low slope (ls) curve, φ NIR, iso−ls (θ THz ). In Fig. 6 we show the dR(θ THz , φ NIR )/dθ THz for this curve. While there are still regions where the value is higher, the slope remain less than half the value where the anomalous peaks occur. We calculate ∆abs using this set of angles for the θ THz and φ NIR, iso−ls (θ THz ) and the angular dependence of at 90 cm −1 and 50 cm −1 for model sample 1. The results are shown in Fig. 7. The results show that using φ NIR, iso−ls (θ THz ) the distortions are essentially eliminated and the identification of the resonant frequencies and their orientations is straight forward.

Discussion
Our modeling shows that using the configuration in Fig. 2 with synchronous control of θ THz and φ NIR according to the low slope iso-response curve will enable rapid near field anisotropic absorbance measurements with a stationary sample. The calculations also show that in general EO detection can be problematic for samples with high anisotropic absorbance and birefringence, and care must be taken to avoid artifacts. It is reasonable to wonder why ATM does not have these artifacts, as shown by sucrose calibration measurements [23]. This is due to the fact that the ATM θ THz -φ NIR configuration is less sensitive to small θ THz rotations caused by the sample. For ATM the EO crystal orientation and the polarizations of the THz and NIR are aligned to optimize the EO signal. In Fig. 2 we mark such a configuration with a blue dot, where θ THz = 90°and φ NIR = 90°. As the sample is rotated, the φ NIR and θ THz are held constant, however the transmitted THz polarization Θ sample (θ THz ) could lead to changes in the EO response R(Θ sample (θ THz ), φ NIR ) as indicated by the horizontal white line. In Fig. 6 we show the slope of white line, dR(θ THz , φ NIR )/dθ THz in green with a blue dot on this curve indicating a typical starting configuration. The slope at the optimized configuration point is zero, and the slight shifts in the transmitted THz polarization Θ sample (θ THz ) about the optimized value will have essentially no effect on the EO response.
The results also show why the fingerprinting technique, PV-ATM has increased structure relative to ATM [27]. In the PV-ATM configuration, only the THz polarization is rotated, following the EO response along the horizontal white line in Fig. 2. The PV-ATM biomolecular spectra show considerably more structure than ATM, particular near θ THz rotated 90°from the optimized configuration, that is at 180°on Fig. 2 and Fig. 6. We see that at this angle dR/dθ THz is a maximum, the point at which the EO signal is most sensitive to polarization rotation by the sample. This then suggests that PV-ATM is highly sensitive to birefringence or optical activity, and the additional narrow features seen using this technique may indeed be coming from resonances buried in the isotropic background. The relationship of observed resonant features in PV-ATM to the actual direction of the transition dipoles will be dependent on both the strength of the transition and the orientation of the dipole relative to the incident THz polarization, both unknown quantities. Thus, while PV-ATM shows promise for providing unique spectra for different biomacromolecules, it cannot be used for mapping specific vibrational directions. IATM using the optimized iso-response function locates the resonances correctly for both frequency and orientation.

Conclusion
We find that the Ideal ATM approach could provide an accurate and fast measurement of anisotropic terahertz response both in the near and far field. The polarization control needed can be realized with current technology, although considerable calibration is likely to be necessary. In the near field, the stationary sample technique removes the need for re imaging and positioning the sample or effects from spatial variation in the EO crystal. The sample space integrated with the EO crystal could readily be inserted in a cryostat or vacuum chamber for TMD and 2D magnet measurements. The speed of measurements would be sufficient for the essential survey data necessary to establish a field of biomolecular collective vibration spectroscopy. The proposed system should make THz micropolarimetry accessible to non-THz specialists for materials science and molecular biology applications.