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Abstract
We demonstrate sub-terahertz vortex beam generation using a spiral metal reflector that can be used for both polarizations. A vortex beam is a ring-shaped beam that possesses sub-wavelength null in the center formed by angular phase variation. While the sub-terahertz vortex beams have gained increasing attention for a wide range of applications in sensing and communications, techniques for generating them are still accompanied by challenges. For example, the use of a phase plate, which is common in the optical regime, suffers from intrinsic losses of dielectric materials in the sub-terahertz regime. Moreover, holographic diffraction gratings, which could replace transmissive components, are inefficient and sensitive to the polarization. To reconcile these challenges, here we design a reflector type metal component with a spiral surface shape. We firstly derive a direct equation to design its shape. We then experimentally validate the design by mapping the radiation pattern of a vortex beam for the WR10 frequency band (75 to 110 GHz) in both of the orthogonal polarizations. The result confirms an inexpensive and versatile approach to generate a vortex beam in the sub-terahertz regime.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Vortex beams are ring-shaped beams that possess sub-wavelength null in the center formed by angular phase variation. The annular intensity profile along with the orbital angular momentum has found a wide range of applications such as super-resolution [1,2], high-capacity communication [3], and photophoretic force generation [4–7]. While those applications have been discussed mainly in the optical regime, much attention has recently been paid to the millimeter-wave and terahertz regimes [8–13]. Terahertz vortex beams can also be excited and propagated using THz fibers [14,15]. To generate vortex beams at those frequencies, several approaches have been considered. For example, spiral phase plates that impose azimuthally varying phase delay based on path lengths have been implemented with spiral thickness [16,17]. The use of fork-shaped diffraction gratings [18–22] or computer-generated holograms [23,24] have also been considered for flat implementation, which could also be in reflection configuration [25,26]. Simultaneous generation of two vortex beams with opposite angular momenta in the opposite directions have also been demonstrated [19]. Despite these efforts, there still exist two fundamental challenges. Firstly, the use of common dielectric materials suffers from intrinsic losses in the sub-terahertz regime. Secondly, diffraction gratings and holograms are sensitive to polarization. To reconcile these challenges, here we design a spiral metal reflector to generate a vortex beam. We derive a direct equation to design its spiral shape. We then experimentally validate the design by mapping the radiation pattern of the vortex beam for the WR10 frequency band (75 to 110 GHz) in both of the orthogonal polarizations. The result confirms an inexpensive and versatile approach to generate a vortex beam in the sub-terahertz regime.
2. Method
2.1 Reflector design
A vortex beam is characterized by its annular intensity and topological charge [27]. The annular intensity is distinct from that of a Gaussian beam, which has a Gaussian field distribution with the highest value at the center. In a vortex beam, on the other hand, a null point is formed at the center by the spiral phase distribution. The topological charge is an integer that represents the number of twists in the orbital motion of the vortex beam. We consider the topological charge of 1 in this paper. In order to convert a Gaussian beam into a vortex beam, we design a reflector with a spiral surface that generates the spiral phase as illustrated in Fig. 1. Figure 1(a) shows a process in which an incident Gaussian beam is converted into a vortex beam. Figure 1(b) shows a photo of the prototyped reflector. It also illustrates that rays reflected from different parts of the surface acquire different phases owing to the different path lengths. In our design, we take inward tilt of the wavefront defined by an angle $\theta$ into account, which contributes to control the beam divergence (Fig. 1(a)). The inward tilt is indeed a well-known characteristic of a Bessel beam [28], which is quasi-diffraction-free and has conventionally been implemented with an axicon lens [29]. Hence, the vortex beam in this study could also be referred to as a Bessel vortex beam [30,31].
Fig. 1. (a) Conversion of a Gaussian beam into a vortex beam by the reflector. (b) Photo of the prototyped reflector. A pair of the incident points in the axial symmetry generates the opposite phases.
Here we analytically derive the required shape of the spiral reflector. We consider a situation illustrated in Fig. 2. The incident rays are oriented in the $x$-direction and distributed uniformly on the $yz$-plane. After reflecting off the surface, the group of the rays form a vortex beam in the $z$-direction. Let $\rm {P_0}$ denote a point of incidence on the surface when its azimuth angle $\phi$ measured from the $x$-axis becomes $\phi =0\,^{\circ }$ (i.e. on the $xz$-plane). Let $\rm {Q}$ denote the point on the $z$-axis which is uniquely defined by the inward tilt angle of $\theta$ with respect to $\rm {P_0}$. The angle $\theta$ is the sole design parameter of our reflector, which is fixed to $\theta =5\,^{\circ }$ in this paper. We express the distance between $\rm {P_0}$ and $\rm {Q}$ as $r_0$. Now we consider another point of incidence $\rm {P}$ with $\phi \neq 0\,^{\circ }$ (i.e. off the $xz$-plane) from which the reflected ray reaches $\rm {Q}$, keeping the same angle $\theta$. We can then specify the required surface as a locus of $\rm {P}$ when sweeping $r_0 \left (r_0\ge 0\right )$ and $\phi \left (0\le \phi \le 2\pi \right )$. To satisfy that the phase of the reflected ray varies linearly with $\phi$, the path length $r$ between $\rm {P}$ and $\rm {Q}$ must satisfy the following equation,
where $\lambda$ is the wavelength of interest. In this paper, we set $\lambda$ = 3.3 mm (91 GHz). Considering the geometry illustrated in Fig. 2, the location of $\rm {P}$ is described asFig. 2. Coordinate system to describe the spiral surface. The locus of $\rm {P}$ when sweeping $r_0$ and $\phi$ defines the surface to be designed. The inset shows the variation of $r$ as a function of $\phi$ based on Eq. (1).
2.2 Experimental setup
To confirm the generation of a vortex beam around the designed frequency, we measure the radiation pattern for the WR10 band from 75 GHz to 110 GHz. Figure 3 shows the experimental setup. The WR10 band signal is generated by 6 times frequency multiplication of microwaves from a signal generator using an Amplifier-Multiplier Chain (AMC-I, Virginia Diodes, hereinafter AMC). The signal is modulated with a modulator and delivered to the reflector via a horn antenna. The polarization of the incident waves to the reflector is set so that the electric field is oriented in either the $z$-direction or $y$-direction. We refer to the former horizontal polarization as polarization 1 and the latter vertical polarization as polarization 2. We use a Zero-biased Schottky Barrier Diodes (WR10ZBD-F06, Virginia Diodes, hereinafter ZBD) as a detector. To map out the spatial distribution of the radiation, we translate the ZBD by using a 3-axes automated stage (OSMS26-100, OptoSigma). To acquire the voltage output from the ZBD, we use a lock-in amplifier (LI5650, NF Corporation). The modulation frequency is 130 Hz and the duty ratio is 1 %. As mentioned in Section 2, we set $\theta =5\,^{\circ }$ in this study. Therefore, the vortex beam is formed around $z=d/4\tan {5}\,^{\circ }$ = 71 mm, where $d$ = 25 mm in Fig. 3 indicates a typical beam diameter determined by the area of the reflector irradiated with the horn antenna. The automated stage is therefore moved to cover this area.
Fig. 3. Experimental setup for sub-terahertz vortex beam pattern measurement. Definition of the $xyz$ axes is also shown.
3. Results
3.1 Observation of the vortex beam at the center frequency
Figure 4 shows the measured radiation pattern at 91 GHz in polarization 1. Alongside the result of (a) 3D scan, the cross sections in (b) $xy$-, (c) $yz$-, and (d) $xz$-planes are shown. The cross-section in the $xy$-plane is plotted when the distance between the end edge of the reflector and the ZBD is 71 mm. It shows ring-shaped intensity around the null at the center, confirming the generation of a vortex beam. We need to mention that our current experimental setup can measure the spatial distribution of the intensity but not the phase. Therefore, we cannot directly characterize the topological charge of the vortex beam by observing the spiral phase evolving along the propagation axis. Yet, as we discussed in the Sec. 2.1, a topological charge of 2 is obtained for 182 GHz based on Eq. (1), and thus we conclude the annular intensity at 91 GHz manifests a topological charge of 1. Fig. 4(e) and (f) show a close-up view around the null and its 1D profile cut out at $y=25$ mm. From these results, the inner diameter of the ring defined by the 1/$e$ width [33,34] is about 14.3 mm. The field intensity of the null reaches $-63$ dB as compared to the highest intensity of the ring part. The cross-sections in the $xz$- and $yz$-planes show stripe patterns appearing periodically in the $z$-directions. This can be attributed to standing waves caused by the interference between the propagating and reflecting waves to and from the ZBD. We provide a more detail discussion on this in Section 3.3.
Fig. 4. Experimentally observed vortex beam in polarization 1. (a) 3-dimensional view. (b) $xy$-plane. (c) $yz$-plane. (d) $xz$-plane. (e) Close-up view of the $xy$-plane in the range of 10 mm $\times$ 10 mm with a scanning resolution of 0.25 mm. (f) 1D profile of the beam pattern along the $x$-direction when $y=25$ mm.
As mentioned in Section 1, it is expected that the spiral reflector generates vortex beams irrespective of the incident polarization. To confirm this, we also measure the radiation pattern in polarization 2. Similarly to polarization 1, the generation of the vortex beam is observed as shown in Fig. 5(a). The cross-section in the $xy$-plane is plotted when the distance between the reflector and the ZBD is 69 mm. It should be noted that we have slightly adjusted the detector position, from 71 mm to 69 mm, in order to observe the null in the polarization 2 at a signal level comparable to the polarization 1. To characterize the polarization 2, we need to reassemble the configuration by rotating both the horn antenna and the ZBD for 90$^\circ$. There, in addition to the finite error of the manual alignment, the field distribution impinging on the reflector converted from the TE$_{10}$ mode of the rectangular waveguide makes the measurement for the polarization 2 not perfectly identical to that for the polarization 1, involving a slight discrepancy in the appearance the standing waves between the polarization 1 and 2. The 1D profile in the $x$-direction shown in Fig. 5(b) estimates the inner diameter of the ring to be 12.9 mm. The null line is observed along the $z$-axis in Fig. 5(c) and (d). We thus confirm that the vortex beam can be generated both in polarization 1 and 2.
Fig. 5. Experimentally observed vortex beam in polarization 2. (a) $xy$-plane (b) 1D profile of the beam pattern along the $x$-direction when $y=25$ mm. (c) $yz$-plane. (d) $xz$-plane.
3.2 Bandwidth characteristics
While we have designed our spiral reflector for the center frequency of 91 GHz, here we evaluate its bandwidth characteristics by measuring the radiation patterns at different frequencies. The AMC generates 75 to 110 GHz by sweeping the original microwave signal from 12.5 to 18.3 GHz. Figure 6(a) compares the 1D profiles of the null for different frequencies in polarization 1 when the distance between the reflector and the ZBD is 71 mm. For all the frequencies, we observe the ring shaped intensity distribution with the null intensity of around $-55$ dB. We also observe that the location of the null slightly shifts for different frequencies. We think this is because the incident wavefront is not perfectly uniform in both terms of amplitude and phase and fluctuates with the frequency, due to the finite accuracy of the fabrication and alignment of the horn antenna. We evaluate the variation of the inner diameter of the ring defined by the 1/$e$ width as a function of the frequency. Figure 6(b) summarizes the result, in which the averages of the 1/$e$ widths measured in the $x$- and $y$-directions are plotted. We observe that the inner diameter of the ring is about 15 mm in the vicinity of 91 GHz and deviates at off-center frequencies. The variation of the the inner diameter relative to that at 91 GHz is less than 30% from 81 to 103 GHz. It should be mentioned that the result at 75 GHz is excluded from Fig. 6(b) because the 1/$e$ width can be determined only in the $x$-direction due to the distortion of the beam shape. Next, we consider the spatial uniformity of the ring shape. For that purpose, we evaluate the field intensity around the null along a circle with a radius of 13 mm, which is about the half of the outer diameter of the ring. The result is plotted in Fig. 6(c), in which the horizontal axis indicates the arc length $l$ from 0 to 26$\pi$ mm and the vertical axis is the measured intensity. We find that the intensity becomes less uniform at off-center frequencies. The ratio of the highest and lowest intensity along the circumference is less than $10$ dB from 83 to 102 GHz. The bandwidth characteristics with respect to polarization 2 can be characterized in the same way. The 1D profiles in the $x$-direction are shown in Fig. 6(d). The variation of the inner diameter of the ring as a function of the frequency is summarized in Fig. 6(e). The variation of the inner diameter relative to that at 91 GHz is less than 30% from from 82 to 110 GHz. The field distribution along the circumference around the null is also shown in Fig. 6(f). The ratio of the highest and lowest intensity along the circumference is less than $10$ dB from 79 to 109 GHz. In the end of this section, we show the measured field distributions propagating in the $z$-direction at 75, 93, and 109 GHz in Fig. 7(a)-(c) for polarization 1 and (d)-(f) for polarization 2. From these results, we see that a null line can be formed along the propagation axis even at off-center frequencies although the minimum intensity becomes larger due to imperfect destructive interference.
Fig. 6. (a) 1D profiles of the beam pattern along the $x$-direction when the frequency is swept from 75 to 110 GHz. (b) Comparison of the inner diameter at each frequency. (c) Comparison of the field intensity evaluated along the circumference with a radius of 13 mm around the null. While (a)-(c) are for polarization 1, the corresponding results for polarization 2 are shown in (d)-(f).
Fig. 7. Radiation pattern in the $xz$-plane at (a)(d) 75 GHz, (b)(e) 93 GHz, (c)(f) 109 GHz. (a)-(c) are polarization 1 and (d)-(f) are polarization 2.
3.3 Stripe artifact
Here we discuss the stripe artifact appearing in Figs. 4, 5, and 7. We attribute the artifact to the standing waves between the reflector and the detector. As explained in Section 2, the wavefront of our vortex beam tilted by an angle $\theta =5^\circ$ like a Bessel beam. Since the wavenumber of a Bessel beam is calculated by $k_z=k_0/\cos {\theta }$, where $k_0$ is the wavenumber of a plane-wave in vacuum, the wavenumber of a standing wave is given as $k_z/2$, if it exists. Figure 8(a) shows the field distribution measured in the $z$-direction at $y=16$ mm on the $yz$-plane and Fig. 8(b) shows its Fourier transform. By searching the peaks in Fig. 8(b), we can extract the wavenumber of the stripe pattern. Table 1 compares the theoretical and measured values, confirming that the wavenumber of the stripe pattern corresponds to that of the standing waves. We thus conclude that the observed stripe patterns are because of the standing waves between the reflector and the detector.
Fig. 8. (a) 1D profile of the beam pattern along the $z$-direction when the frequency is swept from 75 to 110 GHz ($y=16$ mm). (b) Fourier transform of (a).
Table 1. Comparison of theoretical and actual values of the wavenumber.
4. Conclusion
In this study, we have designed and experimentally demonstrated the generation of sub-terahertz vortex beams using a spiral metal reflector that can be used for both polarizations. We have derived an analytical formula to design the spiral reflector that converts an incident plane wave into a vortex beam with a wavefront tilted inward like a Bessel beam. We have prototyped the reflector for a center frequency of 91 GHz by mechanically processing an aluminium block and measured the radiation pattern. The results have confirmed the generation of a vortex beam for the entire WR10 band for incident plane waves in both of the orthogonal polarizations although the diameter and the intensity of the null shows slight dependence on the frequency. This work offers an inexpensive and versatile approach to generate a vortex beam in the sub-terahertz regime. We believe that it will be valuable for a variety of sub-terahertz applications including high-capacity communication and high-resolution imaging.
Funding
Japan Society for the Promotion of Science (20H02154); Precursory Research for Embryonic Science and Technology (JPMJPR18J9).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Dataset 1 [32].
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