Demonstration of a terahertz pure vector beam by tailoring geometric phase

We demonstrate the creation of a vector beam by tailoring geometric phase of left- and right- circularly polarized beams. Such a vector beam with a uniform phase has not been demonstrated before because a vortex phase remains in the beam. We focus on vortex phase cancellation to generate vector beams in terahertz regions, and measure the geometric phase of the beam and its spatial distribution of polarization. We conduct proof-of-principle experiments for producing a vector beam with radial polarization and uniform phase at 0.36 THz. We determine the vortex phase of the vector beam to be below 4%, thus highlighting the extendibility and availability of the proposed concept to the super broadband spectral region from ultraviolet to terahertz. The extended range of our proposed techniques could lead to breakthroughs in the fields of microscopy, chiral nano-materials, and quantum information science.

group also illustrated an achromatic axially-symmetric wave plate based on internal reflections and a generation of vector beams in the visible 22,28 , middle infrared 18 , and terahertz 29 spectral regions. To verify details of the pure vector beam generated by use of those radial polarization converters, one must verify that the generated beam removes the vortex phase that can result from geometric phase. Although the vortex phase must be reduced in applications of pure vector beams, few studies have rigorously determined the spatial distribution of phase and its polarization of generated vector beams 2,3,7,8,15,17,23 , having observed only intensity distributions transmitted through a polarizer. The radial polarization converters used in these studies, moreover, are limited to narrow wavelength bands because broadband vortex phase cancellation has not been previously demonstrated.
In this paper, we specifically show a pure vector beam that eliminates the vortex phase by tailoring geometric phase at 0.36 THz. Our proposed concept of vortex phase cancellation considers symmetry of the polarization of the light and asymmetry of the polarization converter. The asymmetry of the radial polarization converter enables one to remove vortex phase in generated vector beam. Geometric phase, which is induced by the spatial distribution of polarization in the vector beam, determines whether the generated vector beam has vortex phase. For this reason, we also propose single shot determination of polarization distribution in the generated vector beam. Our work provides insights that could lead to new frontiers in highly important scientific fields involving singular optics 30 and quantum communications 31,32 using pure vector beams, which are expected to represent a breakthrough for surpassing the diffraction limit, high-density transmission of signals, and generating longitudinal electric fields by light. We focus on the proof-of-principle demonstration of a terahertz (THz) pure vector beam by tailoring the geometric phase. In our demonstration, we also show that our concept is not limited to the THz region only and enables us to extend this approach to different spectral regions.

Results
Pure vector beams with radial polarization. We theoretically investigate the spatial distribution of polarization and vortex phase of vector beams. The key concept for reduction of vortex phase in vector beams is the cancellation of vortex phase by tailoring left-and right-hand circularly polarized optical vortices. This cancellation is achieved from the relationship between symmetry of polarized light and a rotational asymmetry of a polarization converter (see Fig. 1). In the figure, the incident beam, which has linear polarization described at "A", is converted into output beam "B" via radial polarization converters as a non-axially-symmetric half-wave plate (Fig. 1a). The incident linear polarization consists of a pair of left and right circular polarizations with uniform phase (Fig. 1b). Considering polarization conversion of the incident circular polarizations by the non-axially-symmetric half-wave plate, left and right circular input polarizations with uniform phase are reversed to give right and left circular output polarization. The phase distributions of the circularly polarized beams are spirally distributed with vortex phase because of the non-axially-symmetric half-wave plate (Fig. 1c). After the circularly polarized vortex beams are combined, the output beam illustrated at "B" becomes a radially polarized beam with uniform phase because the vortex phase is eliminated.
To verify the abovementioned concept, we estimate the phase of the generated vector beam by calculating the geometric phase induced by cyclic changes of the polarization 33 . The changes of the generated beam's polarization state are sketched on a Poincaré sphere (Fig. 1d). When the incident linear polarization (red circle) is converted into radial polarization, changes of the polarization state on the Poincaré sphere is described on the equator (blue dashed line) of the Poincaré sphere because the radial polarization has linear polarizations corresponding with the angle θ. The area closed by cyclic changes of the polarization on the Poincaré sphere is zero. As a result, the generated beam becomes a pure vector beam having radial polarization and uniform phase. For linear polarization, which consists of left and right circular polarization, the non-axially-symmetric half-wave plate enables cancellation of the vortex phase.
We calculate the phase retardance generated by internal reflections within the non-axially-symmetric halfwave plate. We investigate the phase retardance along the slope angle of the non-axially-symmetric half-wave plate (Fig. 2a) as well as THz frequency dependence of the phase retardance (Fig. 2b). Phase retardance is related to the slope angles and refractive indices n = 1.52 (solid, blue) of high-density polyethylene (HDPE) and n = 1.44 (dashed, red) polytetrafluoroethylene (PTFE). (see the subsection "Internal Fresnel Reflections" in the Methods section) We estimate that total internal reflection produces achromatic phase retardance in the frequency region from 0.1 to 1.6 THz (for more information, refer the subsection non-axially-symmetric half-wave plate, "Jones calculus" in the Method sections, and Supplementary Note 1 and 2). HDPE generates a retardance of 180° due to four internal reflections at the angle of 55° due to create the radially polarized beam with uniform phase.
Beam profile of the vector beam. In our proof-of-principle experiments, we constructed an experimental setup consisting of a vector beam polarization converter and polarization analyzer (see the subsection "Experimental setup" in the Methods section and Supplementary Note 3). We obtain an image (Fig. 3a) using a pyroelectric camera after light beam passes through an axially-symmetric wave plate and a THz polarizer. In general, such images likely indicate a radially polarized beam because the transparency angle of the THz polarizer was set at 0°. Details of the spatial distribution of polarization of the generated vector beam, however, are difficult to observe in the image of intensity distribution. To resolve this issue, we develop a full analytical theory to determine the spatial distribution of polarization on the vector beam. For effective analysis of the vector beam, the image was transformed from Cartesian x-y coordinates to polar coordinates as shown in Fig. 3b. From Fig. 3a and b, we can evaluate the polarization distribution of the vector beam because the intensity distribution is modulated along the angle θ. The intensity distribution enables single shot determination of the polarization on the vector beam 29 .  Fig. 3b, we determine the two-dimensional distribution of the polarization, such as ellipticity (Fig. 4a) and azimuth angle (Fig. 4b). The measured ellipticity and azimuth are consistent with the theory of pure vector beams with radial polarization because the ellipticity is nearly zero and the azimuth varies linearly. For this case, the vector beam is radially polarized. However, we cannot determine whether the vector beam successfully removes the vortex phase because its phase distribution was not known.
To elucidate whether the generated beam reduces the vortex phase, we investigate the phase as well as the polarization of the vector beam. Simultaneously determining both properties of the vector beam is difficult because an identical setup is required for a wavefront sensor and an interferometer. Stokes parameters determined by single-shot determination of the spatial distribution of the polarization on the vector beam are normalized to obtain the geometric phase. The Stokes parameters are distributed on the Poincaré sphere (Fig. 5a), a s′ 1 -s′ 2 plane projection (Fig. 5b), a s′ 1 -s′ 3 plane projection (Fig. 5c), and a s′ 2 -s′ 3 plane projection (Fig. 5d). Evaluating the Poincaré representation, we confirm the changes of the polarization of the vector beam corresponded to the polarization distribution on the Poincaré sphere shown in Fig. 1d (for more information, see Supplementary Note 4).

Geometric phase of the vector beam with radial polarization.
To demonstrate the reduction of the vortex phase, we evaluate the geometric phase of the experimentally generated beam. The two-dimensional distribution (Fig. 6a) and the radius dependence ( Fig. 6b)  smaller than ±6.28 rad, which is that of the radially polarized optical vortex. As a result, the phase of the vector beam was uniformly distributed because the vortex phase of the vector beam eliminated 96%. To the best of our knowledge, this work provides the first demonstration of the tailoring of the geometric phase due to a pure vector beam. This approach, however, is limited by the pulse width of the beam because of the influence of the group velocity dispersion.

Discussion
We showed a pure vector beam by tailoring geometric phase in 0.36 THz. As shown in Fig. 2, the non-axially-symmetric half-wave plate is also calculated achromatic properties of phase retardance. Such achromatic phase retardance is resulted by frequency dependence of refractive index as shown in Fig. 2b. Our   experimental results containing not only the polarization distribution but also the geometric phase agreed well with calculated results. In our previous study, we have also shown achromatic properties of vectorial vortex beams using an axially-symmetric wave plate at the frequencies of 0.16 and 0.36 THz 29 . For this reason, our proposed device is possible to generate achromatic pure vector beam in THz regions.
The range of our proposed approach extends to all spectral regions from ultraviolet to terahertz. Table 1 shows the possible materials for the use in the non-axially-symmetric half-wave plate for the different spectral regions. SiO 2 , BK7, ZnSe, PTFE, HDPE, and Tsurupica enable fabrication of the segmented rhomb in each spectral region. However, for the segmented non-axially-symmetric half-wave plate shown in Fig. 1a, practical wave plate materials are limited for ultraviolet, visible, and infrared regions because of issues associated with the complex assembly and high processing accuracy. To overcome this drawback, we propose constructing a continuous non-axially-symmetric half-wave plate body as shown in Fig. 7a. In previous study, the continuous body has realized from form birefringence fabricated by three-dimensional printer 26 . However, the device is limited by the frequency of the THz light source because of frequency dependence of form birefringence. On the other hand, our proposed continuous body provides the generation of achromatic pure vector beams by the internal Fresnel reflection. Our continuous non-axially-symmetric half-wave plate body consists of two rotational bodies "A" and "B" (Fig. 7b). These optical elements are formed by as the volume generated by rotating a Fresnel rhomb through an angle of 90° along the axes z A and z B , respectively. After passing through the continuous non-axially-symmetric half-wave plate, four internal reflections generate the vector beam. We numerically calculate the spatial distribu-  Figure 6. Details of the measured geometric phase. A requirement for a pure vector beam with radial polarization is a uniform phase and not a vortex phase around the beam. To demonstrate that the generated beam becomes the pure vector beam, the geometric phase is calculated from area enclosed by Stokes parameters on a Poincaré sphere shown in Fig. 5. The geometric phase is uniformly distributed around the beam (a). The geometric phase is plotted as the function of the angle θ, which reaches the maximum of −0.23 rad at r = 20 pixels and θ = 2π rad (b). These results provide evidence that the vortex phase has been removed because the determined geometric phase is low compared with a radially polarized optical vortex with a geometric phase of 2π rad.  Table 1. Possible configurations for a continuous non-axially-symmetric half-wave plate. D and λ indicate the aperture diameter of optical element and emerging wavelength, respectively. SiO 2 in the ultraviolet region provides a retardance of 180° due to four internal reflections at a the slope angle of 50°. SiO 2 and BK7 on visible region give a retardance of 180° due to eight and four internal reflections at a slope angle of 52° and 55°, respectively. ZnSe yeilds a retardance of 180° due to four internal reflections at a slope angle of 65°. PTFE gives a retardance of 162° due to four internal reflections at a slope angle of 55°. HDPE, and Tsurupica also generate a retardance of 180° due to four internal reflections at a the slope angle of 55°, respectively. As compared with other spectral regions, D/λ in the THz region is small. However, the aperture size of optical element is large enough to employ a THz light source.  Table 1. This helps to achieve pure vector beam with radial polarization. However, precisely fabricating an optical element as shown in Fig. 1a in the ultraviolet, visible, and infrared regions is difficult because the optical configuration is complicated. (a) To address this drawback, we propose a non-axially-symmetric half-wave plate with a continuous body. (b) The non-axially-symmetric half-wave plate consists of two rotational bodies "A" and "B" of Fresnel rhomb. We numerically simulated Jones vectors of [E x (θ), E y (θ)] T of the vector beam along the angle θ after passing through the continuous non-axially-symmetric half-wave plate. confirms that it should be possible to generate a pure vector beam with radial polarization via the proposed continuous non-axially-symmetric half-wave plate of Fig. 7a. In addition to radial polarization, azimuthally polarized beam, which is another of pure vector beams, also enables beam generation by changing the polarization angle at 0° of the incident linear polarization. The continuous body of this non-axially-symmetric half-wave plate is suitable for use in the superbroadband spectral region from ultraviolet to THz as shown in Table 1. We expect to create the continuous body because three-dimensional printing of transparent fused silica glass enables the construction of complex shapes on optical elements 34 .

Spectral regions Wavelength Material Slope angle (degrees) D/λ
In conclusion, we have proposed and demonstrated a pure vector beam without vortex phase by tailoring the geometric phase of the beam. This paper reports the first demonstration of a pure vector beam with radial polarization achieved by the concept for the reduction of vortex phase. We also proposed a continuous body extended to the segmented non-axially-symmetric half-wave plate. Our technique enables the vector beam to be manipulated in singular optics. Our proposed concept can also be used in conjunction with generic technologies in other different fields, such as super resolution imaging, material science, multiplexing quantum information science, high energy physics, and even astronomy. Non-axially-symmetric half-wave plate. A non-axially-symmetric half-wave plate, which possesses phase retardance of φ = π, is required for a generation of a pure vector beam with radial polarization and a uniform phase. A fast axis of the non-axially-symmetric half-wave plate is rotated with azimuth of θ/2 as a function of angle θ. In this experiment, we fabricated the non-axially-symmetric half-wave plate with four Fresnel rhomb segments. (see the subsections "Jones calculus" in the Methods section and Supplementary Note 1).

Internal
Jones calculus. When a uniform beam with linear polarization is incident onto a non-axially-symmetric half-wave plate that is acting as a radial polarization convertor (see Fig. 8a), the polarization of the vector beam can be calculated via Jones calculus [36][37][38] to evaluate the spatial distribution of polarization and its phase in the vector beam. The Jones vector of the linear polarization at π/2 is described by E in = (0, 1) T as shown in Fig. 8b. For calculation between the Jones vector and Jones matrix, the polarization of the vector beam, which is converted Figure 8. Conversion of vector beam by use of a non-axially-symmetric half-wave plate. (a) A non-axiallysymmetric half-wave plate consists of four segments oriented at azimuth angles of 0, π/4, π/2, and 3π/4 radians. A segment is formed similar to a pair of Fresnel rhombs, resulting in internal reflections at slope angles. Based on four internal reflections, the relative relationship between orthogonal polarizations produces a phase retardance of π. (b,c) When an incident beam with Jones vector (0, 1) T (i.e. linearly polarized at π/2 rad), the polarization after passing through segments oriented at 0, π/4, π/2, and 3π/4 rad can be determined as Jones vectors of (0, −1) T , (1, 0) T , (0, 1) T , and (−1, 0) T via the use of Jones calculus. The vector beam is radially polarized with uniform phase. from the non-axially-symmetric half-wave plate fabricated in four parts, can be obtained in Fig. 8c (for more information, refer Supplementary Note 1). Figure 9a and b show a schematic drawing and a picture of the experimental setup for polarization conversion and polarization analysis of vector beams. The polarization conversion for the vector beams uses a THz light source, a THz lens (lens diameter d = 30 mm and focus length f = 100 mm), and the non-axially-symmetric half-wave plate with four segments. Figure 9c shows a picture of a non-axially-symmetric half-wave plate. The polarization analysis for the vector beams consists of an axially-symmetric wave plate, a wire-grid polarizer, a THz lens, and a pyroelectric array camera (  A vertically polarized beam with Stokes parameters of (1, −1, 0, 0) T is converted into a vector beam with spatial distribution of Stokes parameters (s 0 (θ), s 1 (θ), s 2 (θ), s 3 (θ)) T . The polarization of the vector beam generated by the polarization conversion is determined by polarization analysis through the use of an axially-symmetric wave plate, a wire-grid polarizer set at 0, and a pyroelectric array camera.

Experimental setup.
ScIentIfIc REPORTS | (2018) 8:8690 | DOI:10.1038/s41598-018-26964-7 method, which is commonly used for fringe pattern analysis in optical metrology 39,40 . Using Fourier transform method, Fourier spectrum of P(k) is expressed as where k denotes angular frequency;  −1 represents an inverse Fourier transform operator; and C 0 (k), C 2 (k), and C 4 (k) represent Fourier spectra; a superscript asterisk " * " represents a complex conjugate. By individually extracting the C 0 (k), C 2 (k), and C 4 (k) components in Fourier space, the spatial distribution of Stokes parameters s A0 (θ) − s A3 (θ) are determined as where  , R, and I are the Fourier transform operator and the operators to extract the real and imaginary parts of their argument, respectively. For the spatial distribution of Stokes parameters, we determine the normalized Stokes parameters, ellipticity, and azimuth angle, respectively 38 (for more information, refer to Supplementary Note 4).

Determination of the geometric phase.
To evaluate the phase distribution of the generated vector beam, we introduce geometric phase using the spatial distribution of Stokes parameters. The geometric phase was first suggested by Berry and Pancharatnam 33,41 , and it is related to cyclic changes of polarization on a Poincaré sphere with azimuth order of  = 0. Here, we can estimate the geometric phase using the changes of the polarization represented by the Stokes parameters of (1, −1, 0, 0) T , (1, cos2θ cos2ε, sin2θ cos2ε, sin2ε) T , and (1, cos2θ cos2ε, sin2θ cos2ε, 0) T , where ε is ellipticity and θ is azimuth, respectively. The surface area Ω of a spherical triangle is expressed as