Determination of the polarization states of an arbitrary polarized terahertz beam: Vectorial vortex analysis

Vectorial vortex analysis is used to determine the polarization states of an arbitrarily polarized terahertz (0.1–1.6 THz) beam using THz achromatic axially symmetric wave (TAS) plates, which have a phase retardance of Δ = 163° and are made of polytetrafluorethylene. Polarized THz beams are converted into THz vectorial vortex beams with no spatial or wavelength dispersion, and the unknown polarization states of the incident THz beams are reconstructed. The polarization determination is also demonstrated at frequencies of 0.16 and 0.36 THz. The results obtained by solving the inverse source problem agree with the values used in the experiments. This vectorial vortex analysis enables a determination of the polarization states of the incident THz beam from the THz image. The polarization states of the beams are estimated after they pass through the TAS plates. The results validate this new approach to polarization detection for intense THz sources. It could find application in such cutting edge areas of physics as nonlinear THz photonics and plasmon excitation, because TAS plates not only instantaneously elucidate the polarization of an enclosed THz beam but can also passively control THz vectorial vortex beams.

Vectorial vortex analysis is used to determine the polarization states of an arbitrarily polarized terahertz (0.1-1.6 THz) beam using THz achromatic axially symmetric wave (TAS) plates, which have a phase retardance of D 5 1636 and are made of polytetrafluorethylene. Polarized THz beams are converted into THz vectorial vortex beams with no spatial or wavelength dispersion, and the unknown polarization states of the incident THz beams are reconstructed. The polarization determination is also demonstrated at frequencies of 0.16 and 0.36 THz. The results obtained by solving the inverse source problem agree with the values used in the experiments. This vectorial vortex analysis enables a determination of the polarization states of the incident THz beam from the THz image. The polarization states of the beams are estimated after they pass through the TAS plates. The results validate this new approach to polarization detection for intense THz sources. It could find application in such cutting edge areas of physics as nonlinear THz photonics and plasmon excitation, because TAS plates not only instantaneously elucidate the polarization of an enclosed THz beam but can also passively control THz vectorial vortex beams.
T he polarization states of a beam can be varied by the optical anisotropy of materials using the electro-and magnetic-optical effects as external triggers. In addition, optical scattering and reflection processes can create changes in polarization. Furthermore, variations in vibrational and rotational spectral structure can be induced by light-matter interactions, due to the external electrical field of a polarized laser beam. For these reasons, polarization determination is important in materials science 1 , pharmacology 2 , and even astronomy 3 . The frequency response of a polarized beam permits elucidation of atomic and molecular dynamics 4 , stereo and chiral structures of a protein 5 , and charge density distributions 6 . Interestingly, the use of THz-wave polarization pulse shaping with arbitrary vector control to examine molecular dynamics has been reported in the last few years 7,8 . The vector control of electric fields has been realized, enabling the use of nonlinear optical crystals and femtosecond lasers. Terahertz time-domain spectroscopy (THz-TDS) enables sequential electrical fields to be obtained with a high time resolution.
Vectorial vortex (VV) beam technology is able to control a beam's spatially variable polarization and phase, and can generate longitudinal electrical fields 9 , optical vortices 10-13 , and higher order Poincaré beams 14 carrying orbital angular momentum. This technology has led to exciting discoveries and applications such as electron acceleration 9,15 , optical trapping 16 , super-resolution microscopy 17,18 , and even the creation of new functional materials including metamaterials 19 with photo-alignment 20,21 and chiral nanostructures 22 . The information describing the light-matter interactions is contained in the VV beam. By decoding this information, it should be possible to understand the light-matter interaction in more detail. Intense THz beams generated using gyrotrons, free electron lasers (FELs), and coherent synchrotron radiation (CSR) in particular have attracted interest for their use as next-generation beams in cutting-edge areas of physics such as nuclear fusion, isotope separation, and electron acceleration. To examine light-matter interactions using intense THz beams, it is important to determine the polarization characteristics [23][24][25] . However, conventional methods cannot easily obtain the polarization states of a beam from an intense THz source, because it is necessary to modulate the polarization both mechanically and electrically 26 . Even THz-TDS is not suitable for characterizing the electrical fields from such sources. Instead it is necessary to instantaneously elucidate the polarization states of intense THz sources. In this paper, VV analysis is used to determine the polarization states of arbitrarily polarized 0.1-1.6 THz beams using achromatic axially symmetric wave plates, which have a phase retardance of D 5 163u and are made of polytetrafluorethylene (PTFE). Linearly polarized beams are converted into THz vectorial vortex (TVV) beams having no spatial and wavelength dispersion. The polarization states are reconstructed for incident THz beams of unknown but uniform polarization. The polarization determination is also demonstrated at frequencies of 0.16 and 0.36 THz. Furthermore, the polarization states of TVV beams are estimated after they pass through THz achromatic axially-symmetric wave (TAS) plates. The results obtained by solving the inverse source problem agree with the values used in experiments.

Results
Determination of the polarization states of an arbitrarily polarized THz beam. As shown in Fig. 1(a), TAS plates, a THz polarizer, and a THz camera are used to determine the polarization state of an arbitrary but uniformly polarized THz beam. Consider an incident THz beam linearly polarized at 90u, represented by a dot on the Poincaré sphere in Fig. 1(b). After passing through a pair of TAS plates with an integrated retardance of 90u, the polarization states are axisymmetrically modulated by a function of the azimuthal angle h of the beam. The spatially varying polarizations are represented as a figure of eight on the Poincaré sphere shown in Fig. 1(c). The output passes through a linear polarizer oriented at angle h 0. The resulting Stokes vector S 1 (r, h) is given by S 1 (r, h) 5 P(h 0 )?T(h)?S 0 , where S 0 5 (s 00 , s 01 , s 02 , s 03 ) T is the Stokes vector of an incident THz beam of unknown polarization state, T(h) is the Mueller matrix for the TAS plates, r is the radial direction, P(h 0 ) is the Mueller matrix for the polarizer 27,28 , and h~tan {1 y x .
The output beam is analyzed by a linear polarizer oriented at h 0 5 0u. The resulting intensity distribution I(r, h) varies periodically with a frequency of 2h and 4h, i.e.,   The standard method for analyzing the polarization state of an arbitrary beam is to rotate a retarder in front of a linear polarization analyzer and take several intensity measurements. The TAS plates allow this modulation to occur angularly, and the polarization properties of the beam can be estimated from a single image, that is, a snapshot. The TAS plates effectively provide both the basis of the polarization determination and the TVV beam control. Proof-ofprinciple experiments are conducted as follows. Figure 2(a) illustrates the conversion of a linearly polarized beam into a TVV beam. An experiment is performed to demonstrate that the system can be used in reverse to determine the polarization of a uniformly polarized THz beam. A linearly polarized THz beam passes through the TAS plates as shown in Fig. 1. The polarization states of the arbitrarily polarized THz beams are modulated by the plates. The intensity distribution is captured by a pyroelectric camera and plotted in Fig. 2(b) after extracting values from the 2D image. The intensity varies periodically with a frequency 2h and 4h. After calculating the discrete Fourier transform, the Fourier coefficients a n and b n are evaluated as shown in Fig. 2(c). Figure 2(d) lists the Stokes parameters for the THz beam evaluated from the captured image.
Uniform arbitrary polarization states of the incident THz beams are also analyzed. A mechanical rotation stage changes the polarization of the beams by rotating the THz source. The incident beams are polarized at typical values of 245u, 0u, 45u, and 90u. Figure 3 shows theoretical results compared to measured THz images. The THz beams are converted into TVV beams and modulated according to the incident polarization states. The experimental images agree with the theoretical results. Using the polarization analysis shown in Fig. 2, the Stokes parameters, the degree of polarization (DOP), the ellipticity, and its azimuth are analyzed. The theoretical values and experimental results are given in Table 1, and the ellipticities and azimuths are plotted in Fig. 4. The ellipticities of the incident beams are almost zero. The azimuths of the incident THz beams are found to be 246.3u, 2.2u, 49.3u, and 89.7u, respectively. The standard deviations of the ellipticities and azimuths are 0.06 and 0.25u, respectively. Note that s 02 /s 00 5 0.06 6 0.00 in the third line of Table 1 because s 02 /s 00 5 0.06 6 0.004(6) and the s 00 components in Table 1 are s 00 (245u) 5 711, s 00 (0u) 5 1121, s 00 (45u) 5 780, and s 00 (90u) 5 1086.
A proof-of-principle experiment is conducted for the inverse source problem. An elliptically polarized beam is used for this purpose, because it is not easy to generate circularly polarized THz beams. The experimental results are presented in Fig. 5. Firstly, compare linearly and elliptically polarized incident beams on the TAS plates. When the incident beam is linearly polarized, the THz image is captured by the camera after passing through the TAS plates and the THz polarizer, as shown in Fig. 5(a1). Stokes parameters of (790, 2248, 248, 54) T are computed from the image. By substituting these parameters into Eq. (1), the intensity distributions are reconstructed theoretically in Fig. 5(a2). The experimental and theoretical intensity distributions about angle h are determined from the intensity values on the circles respectively shown in Figs. 5(a1) and 5(a2). The experimental and theoretical distributions plotted in Fig. 5(a3) agree with each other. Next, elliptical incident polarization is used to confirm   the theory for arbitrarily polarized THz beams. The incident beam is elliptically polarized using a Fresnel rhomb made of high-density polyethylene (HDPE) positioned before the TAS plate. The polarization state is then determined by the TAS plates as shown in Fig. 5(b1). Calculated Stokes parameters of (4953, 21283, 632, 1376) T are found for the incident beam. As shown in Fig. 5(b2), the intensity distributions are again reconstructed, as in the linear polarization case. A comparison between experiment and theory is graphed in Fig. 5(b3).

Discussion
Polarization determination can solve the inverse source problem associated with the conversion from arbitrary polarized THz beams to TVV beams. The simulations indicate that a phase retardance of D 5 163u by fourth-order Fresnel reflections occurs at a slope angle of b 5 55u for TAS plates made of PTFE. The ellipticity and the azimuth distributions for a TVV beam can be determined using the rotating polarizer method. Arbitrarily polarized THz beams can be analyzed. The measured values agree with theory, although imperfections in the optical components lead to some depolarization that is not accounted for in the inversion algorithms.
The method can determine not only the arbitrary polarization state of an incident beam, but also the polarization state of a VV beam after transmitting through the TAS plates. Polarization can be determined using PTFE across the spectral range from 0.1 to 1.6 THz. It is important, however, to account for the absorption coefficient of the THz materials. The accuracy of the determination of the Stokes parameters is estimated to be about 60.1. The precision and resolution of the measured parameters could be improved by stabilizing the THz source and camera, and by refining the THz alignment.
It is possible to extend the investigations to beams produced by intense THz sources such as gyrotrons, FELs, and CSR. The present method enables not only the examination of light-matter interactions through polarization determination, but also the clarification of additional polarization properties of intense THz beams such as   TVV beam control. That could find application to nonlinear plasmon excitation, isotope separation, and electron acceleration. The method also suggests a new universal concept for polarization detection for all wavelength regions from UV to THz.
The deterioration of the DOP is influenced by the extinction in the THz polarizer, by the sensitivity of the pyroelectric camera, and by the accuracy of the THz alignment. Also, crosstalk of the polarization occurs when the TVV beam is focused into the pyroelectric camera,  as well as polarization scrambling (depolarization) by the thickness of the PTFE at the center of the TAS plates, which is comparable to the wavelength of the THz beams at l 5 1.875 mm. The theoretical analysis assumes that the entire system is composed of ideal retarders and polarizers. In practice, depolarizing effects should be accounted for in the system calibration to improve the accuracy of the polarization estimates. If the fabrication and performance of the TAS plates and other elements could be improved, the Mueller matrices of the system components would more closely match those for an ideal setup and the overall system accuracy would improve. A similar effect has been noted for infrared polarimeters using polarizers with low extinction ratios 29 . In spite of these issues, the method convincingly demonstrates TVV beam control, and the components can be used as the basis of an angularly modulated polarization detector. With proper calibration, even partially polarized THz beams can be measured.

Method
THz achromatic axially symmetric wave plate (TAS plate). Based on the Fresnel reflection coefficients, the total internal reflection of a THz beam introduces a phase retardance d(n,b) between the p and s orthogonal polarization states given by Here n and b are the refractive index of the material and the incident angle, respectively 30 . Whereas b is a constant, n depends on the wavelength l. The output beam is totally internally reflected and produces a phase retardance of d (n,b) between the p and s states in the THz region 27,30 . Figure 7 sketches the experiments. When an arbitrarily polarized THz beam is incident on a TAS plate with a concave conical surface (similar to an element rotated about the optical axis), the reflected beam is transformed into a conical beam after reflecting off the sloping surface. The beam then becomes ring-shaped, because the reflection is omnidirectionally generated along the optical axis shown in Fig. 7(a). The polarization states of the beam are axially symmetrically modulated as a function of the azimuthal angle h of the beam, converting it into a VV beam.
Rotating analyzer method. The relationship between the input S 2 and output S 3 Stokes vectors after the VV conversion is Through the action of the angularly varying Mueller matrix in Eq. (3), the polarization state of the TVV can be passively controlled. The Stokes parameters are determined to be The spatially variable intensity distribution in a 2D image is    Simulation of phase retardance in the THz region. To manipulate the phase retardance using internal Fresnel reflections, the necessary slope angles b for a TAS plate are calculated using the material properties of PTFE. The long-dashed line in Fig. 7(b) indicates the refractive index n 5 1.44 in the 0.1-1.6 THz region 32 . Three phase retardances are determined by the refractive indices of n 5 1.40, 1.44, and 1.55. The relationship between the slope angle and the phase retardance is graphed as the three curves in Fig. 7(b). Assuming the refractive index to be n 5 1.44, the slope angle is estimated assuming a small angle dependence. The phase retardance is d 5 40.75u due to a single reflection in the case of b 5 55u, which implies the total phase retardance is D 5 163.0u after four reflections. Figure 7(c) is a schematic of a TAS plate which has dimensions of D 1 5 50 mm, D 2 5 100 mm, L 5 35.7 mm, and t 5 5 mm. The plate was manufactured on a precision lathe.
Evaluation of TVV beams by the rotating analyzer method. To demonstrate that a TAS plate can be used for the passive control of TVV beams, the rotating polarizer method is employed. Figure 8 Since the polarization states of the VV beam are spatially scrambled, the extinction ratio cannot be eliminated, as observed at the center of the VV beams in Fig. 8(a).
Check for the inverse source problem. To determine the inverse source solution using linear system theory, the THz images are recalculated by rotating the THz polarizer through the coordinate transformation of the system sketched in Fig. 1(a). Figure 9 graphs both the measured results and the theoretical values of the ellipticity and its azimuth. The ellipticity is nearly zero, while the azimuth varies linearly. The experimental results agree with the theory, demonstrating the proof-of-principle for polarization determination using the inverse source conversion from a linearly polarized beam to a VV beam. Figure 9 | Check for the inverse source solution using linear system theory. To determine the inverse problem, the ellipsometric parameters are recalculated from the THz images captured by rotating the THz polarizer through the coordinate transformation of the system shown in Fig. 1(a). The theoretical parameters are plotted as the red line for the azimuth and the blue line for the ellipticity. The experimental values are given by the red circles for the azimuth and by the blue diamonds for the ellipticity. The ellipticities are almost zero, whereas the azimuth varies linearly.