1. Introduction

Recent developments in terahertz (THz) technology have made THz electromagnetic waves commonly used tools in laboratories. In particular, the generation and detection of broadband THz pulses have been developed as a powerful tool for spectroscopical investigation of the optical properties of matter over a wide range of frequencies from the sub-THz to the tens-of-THz range [1, 2]. More recently, control of materials’ properties (including electronic, phononic, and magnetic properties) was achieved using such THz pulses [3, 4].

THz electromagnetic waves are transverse, and thus have polarization degrees of freedom. Control and detection of THz polarization states are expected to further broaden the scope of THz technology. For example, polarization-dependent THz spectroscopy can be used to discriminate enantiomers of biomolecules and macromolecules [5], image anisotropic materials [6–8], and perform noncontact measurements of the electronic properties of semiconductors such as the density and mobility of their carriers [9, 10].

Polarization-dependent THz measurements require precise determination of the magnitude and direction of the electric field. A frequently employed method is to detect the change in the THz polarization state using two polarizers in the crossed-Nicol configuration [11–13]. Changes in the THz polarization states at the sub-mrad level can be detected by this method. Hence, it has been employed for various applications such as observations of the quantized Hall resistance in graphene [14].

However, in this method, the sensitivity of the polarization angles is restricted by the extinction ratio of the polarizers. In the THz frequency range, wire-grid polarizers (WGPs) are most commonly used for this purpose. Efforts to improve the extinction ratio of these polarizers have been reported [15]; however, the extinction ratio of commercially available WGPs is of the order of 10⁻³ [16]. This value is inferior to that of the polarizers for visible frequencies by more than two orders of magnitude, and this poor extinction ratio has limited the sensitivity in THz polarization measurements to 0.5 mrad in the data acquisition time of 300 ms [17].

In the crossed-Nicol configuration, a small change in the polarization state of a linearly polarized THz wave can be detected with high sensitivity. Therefore, when a large polarization change is to be measured with high accuracy, the THz beam should be converted into a linearly polarized beam via additional optics, such as waveplates, for polarization control. However, the lack of waveplates with sufficient accuracy in the broadband THz range has hindered the precise determination of arbitrary polarization states. Instead, measurements with different polarizer angles are compared to determine the polarization states over a wide range of polarization angles [9]. However, the sensitivity of this method is restricted by the statistical error, which arises from the mechanical instability of the measurement system during the time required to change the polarizer angles. A WGP-free method for determining THz electric field vectors was developed by van der Valk et al. [18]. They employed a zincblende crystal with a (111) surface for THz electro-optic (EO) sampling. This configuration was equally sensitive to two orthogonal THz electric fields. After passing through the THz detection crystal, the probe pulse was separated into two orthogonal polarization components, which were detected by two different detectors. Diverting the components through two paths limits the precision of polarization measurement. In addition, misalignment of these components introduces unavoidable systematic errors.

Several methods have been proposed to overcome this limitation and thereby improve the sensitivity of THz polarization measurements, which include the use of polarization-sensitive detection elements and the acquisition of fast signals that depend on the THz polarization states [19–24]. Yet, the statistical error has not been sufficiently suppressed and dominates the sensitivity, which is comparable with that of the measurements performed using crossed-Nicol configurations.

Because of the low sensitivity of conventional THz polarization measurements, application is limited to cases in which relatively large polarization rotations are expected. For example, non-contact THz Hall mobility measurements are applicable only to semiconductors with high carrier mobility, such as graphene, GaAs, and Si [10, 14, 17], and are unsuitable for investigating important functional materials with low carrier mobility, such as compound semiconductors and organic semiconductors, used in solar cells and flexible organic devices, respectively. For example, under a magnetic field of 1 T, the Hall effect yields an approximate polarization rotation of 0.1 mrad at 1 THz in InGaN thin films for solar cells with a thickness of 100 nm, carrier mobility of 100 cm²/Vs, and carrier density of 10¹⁸ cm⁻³. Thus, to characterize a wider range of materials, we must drastically improve the sensitivity of THz polarization measurement. Achievement of sub-mrad polarization sensitivity is expected to revolutionize the characterization of non-contact materials, which is crucial for various electronic devices.

In this study, we developed an EO sampling method with probe-polarization modulation; this method allowed us to determine the THz electric-field vectors with arbitrary directions with sub-mrad angular accuracy and precision. Detection and generation of the THz waves were optimized, and fast recording of data significantly suppressed the statistical error: a statistical error of 0.1 mrad was achieved after signals were recorded over a period of 660 ms. This precision per square root of the accumulation time is superior to that reported in any previous study related to measurements of THz polarization states. The performance of the method was confirmed by measuring a sample WGP.

We provide an overview of the three main features of the presented method. First, we measured the THz electric field using an EO crystal with a threefold rotational axis. In addition, we modulated the polarization azimuth of the probe laser pulse and detected the change of its polarization state. This configuration does not require any mechanical motion of optical elements such as the polarizer and EO crystal. Therefore, it improves the stability of the measurement. The effect of misalignment of the optical setup, such as the angle of the EO crystal, is also minimized in this configuration, which suppresses the systematic error. Furthermore, fast modulation of the probe beam suppresses the common-mode noise related to the slow fluctuation in the setup.

Second, we employed intense THz pulses to increase the signal to a value far exceeding the noise level. Therefore, we generated THz pulses by optical rectification of a laser pulse with a tilted pulse front in a LiNbO₃ (LN) crystal, which is known as a method for generating THz pulses with high conversion efficiency [25, 26]. In addition, a regeneratively amplified laser system is preferable for increasing the intensity of the THz wave because an increase in the pulse energy of the fundamental laser improves the conversion efficiency. Although the regeneratively amplified systems have been considered to be less stable than laser oscillators, recent developments have resulted in the realization of regenerative amplification with sufficient stability. In particular, an amplifier with direct laser-diode pumping results in high stability similar to that of the oscillators. We employed this state-of-art technology in pulsed laser generation to achieve highly efficient and stable generation of THz waves.

Third, we optimized the data recording method. The pulse repetition rate of the regeneratively amplified system is relatively low (14.0 kHz). Therefore, the balance-detection signal created by the probe pulse, which is proportional to the THz electric field, can be recorded pulse by pulse. We also optimized the temporal window in the analog-to-digital conversion to improve the signal-to-noise ratio (SNR). The optimization of the whole experimental setup resulted in a detection method of THz electric fields with an angular sensitivity at the sub-mrad level.

2. Electro-optic sampling method with probe-polarization modulation

The proper orientation of the nonlinear crystal employed for the THz detection by the EO-sampling method is important because this crystalline orientation determines the components of the THz electric-field vector that can be detected. Nonlinear optical crystals with zincblende structures, such as ZnTe and GaP, are widely used for the EO-sampling method. Crystals with (110) surfaces are most commonly used to maximize the detection efficiency for in-plane THz electric fields. Another choice is a crystal with a (100) surface. In this case, the THz-electric-field component perpendicular to the surface, i.e., the longitudinal component, can be detected [27, 28].

In this study, we chose a different crystalline orientation, specifically, a zincblende crystal with a (111) surface that has threefold rotational symmetry. In previous reports, we developed methods for controlling the polarization properties of the radiation generated through second-order nonlinear optical processes in threefold rotational symmetric systems [27, 29–33]. For example, we controlled the polarization states of the THz waves generated through optical rectification of femtosecond laser pulses propagating along a threefold axis (i.e., the [111]-axis) of a cubic nonlinear optical crystal [27, 29–32]. In this experimental geometry, tailoring the polarization state of the fundamental laser achieved generation of a THz transverse electric field with an arbitrary direction, i.e., any angle in the transverse plane. The EO sampling is the inverse process of the optical rectification; thus, the same crystalline orientation is expected to provide an opportunity to determine the direction of the THz electric field via the EO-sampling method. As mentioned above, conventional methods employing two different detection units [18] introduce unavoidable statistical and systematic errors. To eliminate these errors, we subject the probe pulses to temporal polarization modulation and obtain signals by a single detection unit, as described below. This method drastically decreases the statistical and systematic errors and realizes highly accurate and precise THz polarization measurements.

A schematic of the detection of the THz electric field using a nonlinear optical crystal with a (111) surface is presented in Fig. 1. A linearly polarized probe laser pulse and THz pulse are incident normal to the crystal surface. After the probe pulse interacts with the THz electric field in the crystal, the polarization state of the probe pulse is converted by a quarter waveplate with its slow axis oriented 45° away from the X-axis. Note that we define the $[1\overline{1}0]$-axis of the EO crystal as the X-axis and the $[11\overline{2}]$-axis as the Y-axis. The probe pulse is then separated into two orthogonal polarization components by a Wollaston prism, and the difference signal S between the intensities of the two components is detected by a balanced photodetector. We define the angle between the polarization direction of the probe pulse and the X-axis as θ and the angle between the direction of the THz electric-field vector and the X-axis as ϕ, as shown in Fig. 1. The obtained balanced signal S is described as [34]

 

Fig. 1 Schematic of the EO-sampling method. A zincblende crystal with a (111) surface is used for EO sampling. Green and red arrows in the balloon indicate the directions of the electric field vectors of the THz and probe pulses, respectively.

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Modulation of θ in time domain is convenient for measuring S as a function of θ. One method for temporal modulation of θ is to use a photoelastic modulator (PEM) and a quarter waveplate. A PEM is an optical device that modulates the polarization state of light. A piezoelectric transducer introduces acoustic vibrations to the transparent optical element in the PEM, and thus modulates its birefringence at the modulation frequency. Consider a horizontally (// X) polarized probe pulse that passes through the PEM rotated in the XY plane by 45°. The polarization state of the probe pulse becomes elliptical, and ellipticity is the sine function of the phase retardation caused by the acoustic vibration in the PEM. This elliptically polarized probe pulse then passes through a quarter waveplate whose slow axis is horizontally aligned. The probe pulse then reverses to be linearly polarized, but its azimuthal angle is modulated from the original angle (θ = 0) as

When the angles of the PEM and quarter waveplate are deviated from the aforementioned ideal values, θ is shifted by a constant value. These systematic errors limit the accuracy of the measurement. In measuring the THz wave response of a sample to an incoming THz wave, we compare the transmitted THz waveforms with and without the sample. Because the systematic errors are the same in both geometries, they can be eliminated.

3. Experiment

Confirmation of the polarization state of the probe pulse and maximization of the SNR of the measurements are important to obtain the direction and magnitude of the THz electric-field vector with high accuracy. Figure 2 shows the experimental setup with the detection method proposed in this study.

 

Fig. 2 Schematic of the experimental setup. P1, polarizer; PEM, photoelastic modulator; LN crystal, LiNbO₃ crystal; PM1 and PM2, parabolic mirrors; EO crystal, (111)-oriented GaP crystal; WP, Wollaston prism; BD, balanced photodetector.

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Intense THz pulses are generated by an optical rectification process in a LN crystal using the tilted pulse front excitation technique. As the fundamental light source, a laser-diode-pumped Yb:KGW regenerative amplifier (PHAROS-SP1.5 mJ, Light Conversion, Ltd.) with a center wavelength of 1028 nm, repetition rate of 14.0 kHz, average power of 5.5 W, and pulse duration of 200 fs was employed. The output beam of the laser was divided into two beams: a pump beam for THz wave generation and a probe beam for detection. The pulse front of the pump beam was tilted using a diffraction grating (1,200 lines/mm), and the image of the grating was created in a 1.3 mol% Mg-doped LN prism using a pair of lenses (focal lengths of 80 mm and 50 mm). The conversion efficiency from the laser to the THz radiation was approximately 10⁻⁴. The temporal waveform of the THz pulse was recorded by scanning the temporal delay introduced to the probe beam by a translational stage.

For polarization modulation of the probe beam, a PEM (II/FS50, HINDS Instruments) and a quarter waveplate were used. The peak retardation and modulation frequency of the PEM were δ₀ = 0.8π and F = 49.8 kHz, respectively. The polarization-modulated probe beam overlapped with the THz wave in an EO crystal, a 450-μm-thick GaP (111) crystal along its [111]-axis. Then, the probe pulse passed through a quarter waveplate whose slow axis was rotated 45° from the X-axis. Finally, the X- and Y-polarization components of the probe beam were separated with a Wollaston prism, and the difference between their intensities was detected by a balanced photodetector.

We recorded the detected signals at the balance detector pulse by pulse. Because of the pulsed nature of the probe pulse, signals were recorded at time t_m = m/f_r + t₀, where m is an integer, f_r is the repetition rate of the laser pulses, and t₀ is an offset between the PEM modulation and the laser repetition. To set the offset t₀ to be zero for any PEM modulation period, the repetition rate f_r of the laser was synchronized to a value commensurate to the PEM modulation frequency F = 49.8 kHz. For example, in the case where the repetition rate f_r is selected to be 9F, as shown in Fig. 3(a), the phase of the PEM modulation at a probe pulse at t_m is 2π × m/9. The same principle is also valid if the ratio f_r/F is a fraction, which is helpful when a reduction of f_r is desired to satisfy the requirement set by the upper limit of the recording speed of the data acquisition device. In our experiment, we chose f_r = 9F/32 = 14.0 kHz. As shown in Fig. 3(b), the same data set obtained for f_r = 9F was also obtained in this case. Notably, the order of the PEM phase was shuffled in the data set as shown in Fig. 3(b); however, the correct order was easily retrieved. The effective modulation frequency of the polarization azimuthal angle of the probe beam was F/32 = 14.0 kHz/9 = 1.56 kHz.

 

Fig. 3 The probe-pulse polarization modulation scheme using the PEM. (a) High repetition rate case. The repetition frequency of the laser pulse is 9 times of the PEM modulation frequency. (b) Low repetition rate case. The repetition frequency of the laser pulse is 9/32 times of the PEM modulation frequency. The blue solid lines indicate the phase shift introduced by the PEM. δ₀ is defined as the amplitude of the PEM modulation. Red triangles indicate the timings when probe pulses are passed through the PEM. Orange arrows show the probe-pulse polarization directions incident to the EO crystal.

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In the EO-sampling method, elimination of the effect of birefringence caused by residual strain inside the EO crystal is important. It is also necessary in our polarization-modulated EO sampling. A commonly employed method of extracting the THz-field-induced signals is to detect probe pulses under two conditions—with and without the THz field—and subsequently take the difference between the two signals. Therefore, we blocked half of the pulse trains for the THz generation by an optical chopper rotating at a frequency of 1.56 kHz/4 = 389 Hz. This frequency was synchronized to one-fourth of the effective modulation frequency.

We used a scope coder (DL850E, Yokogawa Meters & Instruments Corporation) to record the signals from the balanced detector pulse by pulse. The scope coder was triggered by an external clock that was synchronized with the laser pulse repetition. Thus, the sampling rate was 14.0 kS/s. Note that the sampling rate should be sufficiently smaller than the bandwidth of the scope coder (300 kHz) to separate signals between neighboring probe pulses. This limitation for the sampling rate gives the upper limit for the laser repetition rate, and we chose f_r = 9F/32 = 14.0 kHz. The details of the data analysis are described in Appendix B.

4. Results

First, we measured a THz pulse in the absence of a sample. The obtained THz waveform was almost linearly polarized, as shown in Fig. 4(a). This result is reasonable because we employed the tilted pulse front excitation technique by using an LN crystal in this experiment. We subsequently measured THz waveforms with a WGP placed at the sample position shown in Fig. 2. We define the in-plane angle of WGP with the maximum transmittance as the origin (0°) and performed the measurements by setting the angles to 0°, 45°, 90°, and −45°. The measured THz waveforms are shown in Figs. 4(b)-4(e). The directions of linear polarizations are consistent with the angles of the WGP, which indicates that THz electric fields were measured correctly by this method. The investigation of the performance of this measurement method is described in the following sections.

 

Fig. 4 THz waveform acquired by the EO-sampling method with probe-polarization modulation. (a) THz waveform without a sample. (b)-(e) THz waveforms transmitted through the WGP. In-plane angle of the WGP: (b) 0°, (c) 90°, (d) 45°, and (e) −45°.

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4.1 Accuracy of THz polarization measurements

We investigated the accuracy of the direction of the THz electric-field vector obtained using the presented method. We placed a WGP at the sample position, as shown in Fig. 2, and measured the direction of the THz electric-field vector by setting the WGP to several different angles in the XY plane. We performed these measurements by setting the delay stage at the position where the THz electric field intensity was maximal. The measured relation between the direction of the THz electric-field vector and that of the WGP is shown in Fig. 5. In Figs. 5(a) and 5(b), the data are plotted in the range between ± 6° (approximately ± 100 mrad) and ± 0.25° (approximately ± 5 mrad), respectively. The black lines in these figures correspond to the case where the direction of the THz electric-field vector and that of the WGP are identical. The experimentally obtained data are plotted on these lines in Fig. 5. These results indicate that the THz electric field was measured appropriately.

 

Fig. 5 Relationship between the orientation of the WGP and the direction of the measured THz electric field vector. The black solid lines indicate that the orientation of the WGP and the direction of the measured THz electric-field vector are the same. (a) The data in the range between ± 100 mrad. (b) The data in the range between ± 5 mrad.

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4.2 Precision of THz polarization measurements

We examined the statistical error Δϕ of the direction of the measured THz electric-field vector by following the previous research [22]. These measurements were also performed with the delay stage set at the position where the maximum THz electric field intensity was obtained. A histogram of the results obtained from one hundred measurements is presented in Fig. 6(a). The accumulation time used for each measurement was 660 ms. The standard deviation of the direction of the THz electric-field vector obtained by fitting this histogram with Gaussian function was approximately 0.1 mrad, which corresponds to the statistical error Δϕ.

 

Fig. 6 (a) Histogram of the direction of the measured THz electric-field vector obtained from one hundred measurements. The accumulation time of each measurement was 660 ms. (b) Accumulation time dependence of the statistical error of the direction of the measured THz electric field. The blue dotted line indicates the statistical error determined by the shot-noise limit of the probe pulse.

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The dependence of the statistical error on the accumulation time of the measurements is shown in Fig. 6(b). As is evident in the figure, the statistical error is inversely proportional to the one-half power of the accumulation time, i.e., the precision of the measurement is improved with increasing accumulation time. The blue dotted line in Fig. 6(b) is a theoretical line corresponding to the shot-noise limit of the probe pulse. We discuss the difference between the experimental data and the shot-noise limit in the discussion section.

In the experiments, the SNR was 9.8 × 10³, as calculated from the ratio between the peak intensity of the THz electric field and the standard deviation of the background noise after an accumulation time of 660 ms. When the SNR is sufficiently greater than 1, the statistical error of the direction of the measured THz electric-field vector is inversely proportional to the SNR. Therefore, the statistical error estimated from the previously described SNR is Δϕ ~1/SNR ~1/9.8 × 10³ = 0.10 mrad, which is consistent with the experimentally obtained value shown in Fig. 6(a). The SNR increases in proportion to the one-half power of the accumulation time; hence, the statistical error is inversely proportional to the square root of the accumulation time, as shown in Fig. 6(b). Therefore, the statistical error becomes smaller when the accumulation time is increased.

4.3 Example of a measurement using WGP

The Jones matrix $T=\left(\begin{array}{cc}{t}_{XX}& t_{XY}\\ t_{YX}& t_{YY}\end{array}\right)$ of samples can be determined from the electric-field vector of a THz wave transmitted through a sample and that of the reference THz wave [13]. THz waveforms must be measured twice with the sample positioned at two different angles in the XY plane. We experimentally obtained the Jones matrix of a WGP as $T_{WG}=\left(\begin{array}{cc}{t}_{\perp }& 0\\ 0& t_{//}\end{array}\right)$, where $t_{\perp }$and $t_{//}$ are matrix elements of the transmission and reflection axes, respectively, i.e., $t_{\perp }>>t_{//}$. Details of the calculation are described in Appendix C. The energy transmittances of the transmission and reflection axes are $T_{\perp }=|t_{\perp }{|}^2$ and $T_{//}=|t_{//}{|}^2$, respectively, and $T_{\perp }/T_{//}$ corresponds to the extinction ratio of the WGP.

The experimentally obtained $T_{\perp }$ and $T_{//}$ are shown in Fig. 7. The data represented by solid (dotted) lines in Fig. 7(a) were obtained by the THz wave through the WGP with the in-plane angles set to 0° and 90° ( ± 45°), as shown in Figs. 4(a) and 4(b) (Figs. 4(c) and 4(d)), respectively. The extinction ratios calculated from $T_{\perp }$ and $T_{//}$ are shown in Fig. 7(b). Measurement of an extinction ratio greater than 60 dB, as shown in Fig. 7(b), requires a more accurate measurement of the THz electric-field vector. Therefore, the experimental result that the extinction ratios obtained by a combination of WGP angles (0°, 90°) and ± 45° are identical within the error margin indicates that a THz electric-field vector with an arbitrary direction can be measured with high accuracy via this method.

 

Fig. 7 (a) Transmittance of the WGP ($T_{\perp }$,$T_{//}$). (b) Extinction ratio of the WGP. Solid (dotted) lines were obtained for a THz wave through the WGP when the in-plane angles were set to 0° and 90° ( ± 45°).

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5. Discussion

Here, we consider the origins of statistical error Δϕ and systematic error δϕ in our experiments. First, we estimate the value of statistical error, which can be reduced by increasing the accumulation time. The SNR in our setup was limited by imperfect synchronization between the PEM and the laser pulses and by fluctuations in the intensity of the laser pulses. The imperfection in the synchronization results from the limitation of the laser-pulse output timing of the system because the operation of a regeneratively amplified system with an external clock results in timing jitter of 13 ns, which corresponds to the repetition time of the oscillator that supplies the seed pulse to the regeneratively amplified system. Therefore, a timing jitter smaller than the oscillator period of 13 ns may have been induced between the PEM and the laser pulses. Given the modulation frequency of 49.8 kHz in the PEM and the recording time of 660 ms, the statistical error in the angle of the THz electric-field vector is calculated to be Δϕ ~0.063 mrad.

The fluctuation of laser intensity is another cause of statistical error. The intensity fluctuation of the laser pulse was 0.07% pulse-by-pulse when the signal acquisition time was 100 s. If the intensity of the probe pulse is assumed to be the same, the statistical error that originates from the laser fluctuation can be estimated as Δϕ ~0.006 mrad.

The measured statistical error of 0.1 mrad is slightly greater than the aforementioned statistical errors, which might result from the modulation frequency and the peak retardation of the PEM having slightly fluctuated. To decrease these statistical errors, an electro-optic modulator (EOM) could be used instead of a PEM because the modulation frequency of a PEM is difficult to change, whereas that of an EOM is easily changed by controlling the frequency of the external AC voltage. The effect of fluctuation of the pulse intensity also can be analytically compensated for by measuring the pulse intensity pulse-by-pulse. These improvements are required to measure the shot-noise limit, where the statistical error is less than 10 μrad with an accumulation time of 1 s.

Next, we consider the systematic error. The systematic error resulting from misalignment of an optical element was removed in our experimental setup because we employed the polarization modulation technique. Therefore, only the incident angle of the probe pulse and the THz wave to the EO crystal can cause systematic error. Crystals with a (111) surface orientation are known to be only sensitive to in-plane electric fields and not to longitudinal ones [34]. Hence, because the THz electric-field vector is projected onto the crystal plane, the sensitivity to the field parallel to the incident plane (P-polarized component) is cosβ times larger than that of the field perpendicular to the P-polarized component (S-polarized component), where β is the angle of incidence of the THz wave to the crystal. If β << 1, the systematic error can be described as

We can estimate this systematic error by rotating the THz electric-field vector with the WGP and examining the difference between the rotating angle of the WGP and that of the measured THz electric-field vector. Although we have already examined this difference, as shown in Fig. 5, the difference between the rotating angle of the WGP and that of the measured THz electric-field vector is within the error determined by the extinction ratio of the WGP. However, if a WGP with a higher extinction ratio is used, we can estimate the systematic error and correct the incident angle of the probe pulse and THz wave.

6. Conclusion

In this study, we developed an EO-sampling method with probe-polarization modulation. This method enables the THz electric field of arbitrary polarization states to be determined with high accuracy. We used the method to measure the electric-field vectors of THz waves passed through a WGP. The measured polarization angle showed good agreement with the angle expected from the rotation angle of the WGP. The accuracy of the polarization angle determined by this method was 0.1 mrad when the data acquisition time was 660 ms. The accuracy was limited by the loss of synchronization between the PEM and the laser pulse and by fluctuations in the intensity of the laser pulse.

If these limitations can be overcome, the measurement accuracy of the polarization state would be less than 10 μrad with an accumulation time of 1 s. Such high-precision THz polarization measurements could be used to measure the density and mobility of carriers in organic semiconductors without contact and to discriminate enantiomers of biomolecules.

Because the data acquisition speed is limited only by the modulation frequency of the PEM and optical chopper in this method, we achieved faster data acquisition by increasing the modulation frequency. In fact, during our preparation of this manuscript, the application of high-speed THz polarization measurements using a similar method has been proposed [35].

Appendix A

Here, we explain in detail EO sampling using a zincblende crystal with a (111) surface orientation.

We define the $[1\overline{1}0]$- and $[11\overline{2}]$- axes of the crystal as the X- and Y-axes, respectively, as shown in Fig. 1. When a THz wave propagates along the [111]-axis, the unit vector of the slow axis e₊ and that of the fast axis e₋ and the difference in the refractive indexes Δn along them, which originates from birefringence induced by the EO effect, are represented by [34]

(5)$$\Delta n=n_{+}-n_{-}=\frac{2}{\sqrt{6}}r{n}^3{E}^{\text{THz}},$$

(6)$$e{}_{\pm }=\left(\begin{array}{c}\cos \left(-\frac{\varphi }{2}\pm \frac{\pi }{4}\right)\\ \sin \left(-\frac{\varphi }{2}\pm \frac{\pi }{4}\right)\end{array}\right),$$

where $E^{\text{THz}}=\sqrt{{(E_X^{\text{THz}})}^2+{(E_Y^{\text{THz}})}^2}$ is the magnitude of the in-plane THz electric-field vector, $\varphi =\mathrm{arc}\tan (E_Y^{\text{THz}}/E_X^{\text{THz}})$ is the angle between the THz electric field and the X-axis of the crystal, and, n₊ and n₋ are refractive indices of slow and fast axes, respectively.

Notably, in these equations, Δn depends only on E^THz and does not depend on ϕ. In contrast, e_± depends only on ϕ and does not depend on E^THz. Therefore, we can obtain E^THz and ϕ by determining Δn and e_±, respectively. Here we can determine Δn and e_± with high accuracy using a common polarization modulation method in the visible and near-infrared regions.

The Joens vector of a linearly polarized probe beam with an azimuth angle θ relative to the X-axis is described as

(7)$$E^{\text{probe,i}}=E^{\text{probe}}\left(\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right).$$

After the probe beam passes through the (111)-oriented EO crystal and quarter waveplate rotated 45° from the X-axis, the polarization becomes

(8)$$\begin{array}{l}{E}^{\text{probe,f}}=\left(\begin{array}{c}{E}_X^{\text{probe,f}}\\ E_Y^{\text{probe,f}}\end{array}\right)\\ =E^{\text{probe}}R\left(\frac{\pi }{4}\right)J\left(\frac{\pi }{2}\right)R\left(-\frac{\pi }{4}\right)R\left(-\frac{\varphi }{2}+\frac{\pi }{4}\right)J\left(\delta \right)R\left(\frac{\varphi }{2}-\frac{\pi }{4}\right)\left(\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right).\end{array}$$

Here, we define the phase retardation δ, the Jones matrix of rotation R, and the Jones matrix of phase retardation plate J as

(9)$$\delta =\frac{2\pi \Delta nL}{\lambda },$$

(10)$$R(\alpha )=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right),$$

(11)$$J(\delta )=\left(\begin{array}{cc}{e}^{i\delta /2}& 0\\ 0& e^{-i\delta /2}\end{array}\right),$$

where L is the thickness of the EO crystal and λ is the center wavelength of the probe beam. The X- and Y-components of the probe beam are separated by a Wollaston prism and are detected by a balanced photodetector. The obtained balanced signal is

(12)$$\begin{array}{l}S=\frac{\Delta I}{I}=\frac{|E_X^{\text{probe,f}}{|}^2-|E_Y^{\text{probe,f}}{|}^2}{|E_X^{\text{probe,f}}{|}^2+|E_Y^{\text{probe,f}}{|}^2}\\ =\sin \left(\frac{2\pi L\Delta n}{\lambda }\right)\sin (2\theta +\varphi )\cong \frac{4\pi Lr{n}^3{E}^{\text{THz}}}{\sqrt{6}\lambda }\sin (2\theta +\varphi ),\end{array}$$

where ΔI is difference between the intensity of the X- and Y-components of the probe beam and I is the total intensity of the probe beam. We assume that the phase-matching condition is satisfied.

Appendix B

In Fig. 8, we show a schematic of the data analysis after signal acquisition. Figure 8(a) shows the all signals collected in 60 s. The amplitude of signals is shown to depend on the position of the delay stage. Figure 8(b) is an enlarged version of Fig. 8(a) in the range of 29.0–29.5 s (this region corresponds to the gray-highlighted area in Fig. 8(a)). Figure 8(b) clearly shows that the intensities of signals change every 150 ms due to changes in the position of the delay stage. Figure 8(c) is an expanded version of Fig. 8(b) in the range of 29.29–29.31 s (this region corresponds to the gray-highlighted area in Fig. 8(b)). As is evident in the figure, the signals and THz waves are modulated by 389 Hz, which corresponds to the frequency of the optical chopper. Figure 8(d) is an expanded version of Fig. 8(c) in the range of 29.2995–29.3025 s. When THz waves enter the EO crystal, the acquired signals oscillate because the azimuth angles of the polarizations of the probe pulses are modulated by the PEM.

 

Fig. 8 Plots of the data analysis. (a) All signals acquired during a measurement. (b) Enlarged figure of Fig. 8(a). (c) Magnified view of Fig. 8(b). THz waves are on (off) in the red (blue) areas. (d) Magnified view of Fig. 8(c). (e) Results after the subtraction of the value of the EO signals acquired in the absence of a THz wave from that of the signals acquired in the presence of a THz wave. (f) Fitting results after reordering of the data in Fig. 8(e). The black line represents the fitting curve.

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The on–off transition of the modulation by the optical chopper appears every 36 signals, which corresponds to the ratio between the repetition rate of the laser pulses (14.0 kHz) and the modulation frequency of the optical chopper (389 Hz). Therefore, 18 signals were acquired in the each region with and without a THz wave incident to the EO crystal. Because the modulation frequency of the PEM is effectively 1.56 kHz, probe pulses with the same polarization state enter the EO crystal every 9 signals. Therefore, by subtracting the value of the EO signals without THz waves from that of the signals with THz waves taken before 18 pulses, we can remove the effect of the residual strain birefringence. However, some signals acquired when THz waves are partially blocked by the optical chopper should be removed to avoid lowering the quality of the results. Therefore, we only used signals taken when THz waves were completely blocked by or entirely passed through the optical chopper. In Fig. 8(d), the signals used for the analysis are enclosed by solid lines. Figure 8(e) shows the results after the subtraction.

As previously mentioned, the repetition rate of the regeneratively amplified laser system was synchronized to 9/32 times the modulation frequency of the PEM; consequently, the sequence of the polarization state of the probe pulses are complicated, as shown in Fig. 3(b). The numbers shown in the top of Fig. 8(e) correspond to the pulse numbers in Fig. 3(b). Specifically, the pulses with the same numbers represent the same polarization state. According to these numbers, we can reorder the signals. Thus, we fit the data according to the function given in Eq. (3), as shown in Fig. 8(f) and obtained the values of E^THz and ϕ. The THz waveform was obtained by conducting the aforementioned processes at every delay time τ.

Appendix C

Here, we show how to determine the transmittance and extinction ratio of a WGP experimentally. We define the Jones matrix of a WGP as T. An electric-field vector of a THz wave that passes through a WGP rotated α in plane from the X-axis can be written as

(13)$$E_{\text{s}}^1=R(\alpha )TR(-\alpha )E_{\text{r}},$$

where E_r is THz wave without samples, and R is the Jones matrix of rotation described in Eq. (10). In the same manner, an electric-field vector of a THz wave that passes through a WGP rotated α + ξ in the plane from the X-axis can be written as

(14)$$E_{\text{s}}^2=R(\alpha +\xi )TR(-\alpha -\xi )E_{\text{r}}.$$

Therefore, T can be calculated by

(15)$$R(\alpha )TR(-\alpha )=\left[E_{\text{s}}^1{E}_{\text{s}}^{2+}\right]{\left[E_{\text{r}}^{}{E}_{\text{r}}^{+}\right]}^{-1},$$

where E_r⁺,E_s²⁺ are defined as

(16)$$\begin{array}{l}{E}_{\text{r}}^{+}=R(-\xi )E_{\text{r}}\\ E_{\text{s}}^{2+}=R(-\xi )E_{\text{s}}^2.\end{array}$$

Because the WGP is achiral, the transmittance of transmission axis $t_{\perp }$ and that of reflection axis $t_{//}$ can be obtained via diagonalization of the left side of (15) by rotating matrix R(α).

Acknowledgments

We appreciate K. Yoshioka for helpful discussions and important suggestion of experimental laser technique. This research was supported by the Photon Frontier Network Program, KAKENHI (20104002), and Project for Developing Innovation Systems of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, by JSPS through its FIRST Program, and by the Center of Innovation Program from Japan Science and Technology Agency, JST. NN acknowledge supports by JSPS Research Fellowship.

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