We propose and demonstrate a method for generating broadband terahertz (THz) vortex beams. We convert a THz radially polarized beam into a THz vortex beam via achromatic polarization optical elements for THz waves and characterize the topological charge of the generated vortex beam by measuring the spatial distribution of the phase of the THz wave at its focal plane. For example, a uniform topological charge of is achieved over a wide frequency range. We also demonstrate that the sign of the topological charge can be easily controlled. By utilizing the orbital angular momentum of the beam, these results open new THz wave technologies for sensing, manipulation, and telecommunication.
© 2014 Optical Society of America
Vortex light beams, which are light beams with spiral wavefronts, have attracted extensive attention owing to their ability to carry orbital angular momentum [1,2]. In contrast to the spin angular momentum carried by a circularly polarized light beam, the orbital angular momentum of a vortex beam is attributed to the spatial phase distribution within the beam. The orbital angular momentum carried by a vortex beam is quantized and is per photon, where is an integer and indicates the sign and size of the orbital angular momentum. This quantized value is the “topological charge” of a vortex beam.
Vortex beams have numerous applications in optical technologies, such as optical tweezers , free-space optical communications , the fabrication of chiral nanostructures , and stimulated emission depletion microscopy . The extension of techniques for generating vortex beams into the terahertz (THz) spectral range has been awaited to extend the scope of THz technologies. For example, THz vortex beams are attractive for THz wireless communication because they can support an infinite number of orbital angular momentum eigenstates characterized by their topological charges [7,8]. In addition, the orbital angular momentum of vortex beams gives them potential applications in the optical manipulation of matter. For example, manipulation of the rotations of a quantum condensate is promising  because their elementary rotational excitations are found in the THz frequency region. In such applications, control of the topological charge of vortex beams over a broad spectral range is especially important. The broad spectral range characteristic is particularly important for fast THz wireless communication and for generation of intense few-cycle THz pulses . In the optical frequency range, broadband vortex beams are generated using axially-symmetric half-wave plate [11,12].
However, limited availability of achromatic THz polarization elements has made the generation of broadband THz vortex beams difficult. The conventional method for generating a vortex beam for use in optical-frequency-range applications such as fork-shaped holograms  and cylindrical lens converters  cannot be directly implemented in the THz frequency range. Recently, several methods for generating THz vortex beams have been reported, including the use of spiral phase plates  and arrays of V-shaped antennas . However, these methods are only applicable to narrow-bandwidth of the THz waves. They function appropriately for a targeted frequency but give unwanted modulation at other frequencies. In addition, devices used to generate THz vortex beams are designed for one specific topological charge, and changing its value by a single setup was difficult.
In this Letter, we propose and demonstrate a method for generating broadband and few-cycle THz vortex beams through conversion of a THz radially polarized beam into a THz vortex beam. With our method, one can easily control the sign of the topological charge by changing only the angle of the polarizer.
In our previous report, we demonstrated the generation of broadband THz radially polarized beams covering the frequency range of 1–2.5 THz . The direction of the electric field of a first-order radial mode varies in space, as depicted in the left side of Fig. 1. Namely, electric field vector of a radially polarized beam with th azimuthal order () in the frequency domain and in angular coordinate is described as
A quarter-wave plate and a polarizer convert the dependent electric field direction into the phase mapping. Specifically, the Jones matrix of a quarter-wave plate with its fast axis parallel to the axis is
Thus, the phase of the THz wave is proportional to the angular coordinate . A counterclockwise rotation around the center of the beam gives an increase in phase by : this is a characteristic of a vortex beam with a topological charge . This phase distribution is independent of the carrier frequency. Therefore, a broadband THz vortex beam is generated with this method. In this Letter, we focus on the case that initial THz beam is a radially polarized beam, i.e., .
Figure 2 shows the experimental setup. A regeneratively amplified Ti:sapphire laser system (Hurricane, Spectra-Physics Lasers, Inc.) with a repetition rate of 1 kHz, a center wavelength of 800 nm, a pulse duration of 100 fs, and a pulse energy of 0.8 mJ was used as the light source. The laser beam was then divided into the pump and probe beams. We generated a THz radially polarized beam from the pump beam by using a mode converter that consists of eight pieces of half-wave plates  and a nonlinear optical crystal. We controlled the polarization state of the generated THz beam through the polarization selection rule for optical rectification in nonlinear optical crystals that possess threefold rotational symmetry [17–19]. We used ZnTe with a (111) face as the nonlinear optical crystal. Not all the pump light was converted into the THz wave, and the remaining fundamental beam was blocked by Teflon and black polypropylene sheets. The generated THz radially polarized beam then passes through an achromatic quarter-wave plate (i.e., a high-resistance silicon Fresnel-rhomb prism  with an apex angle of 96.2°). Finally, the beam passes through a wire-grid polarizer, resulting in a THz vortex beam.
The spatial distribution of the electric field in the THz vortex beam was characterized via a two-dimensional scanning electro-optic (EO) sampling method using a ZnTe(110) crystal, which is an extension of the one-dimensional scanning method described in our previous paper . We scanned the probe-beam position by inserting an iris into the probe beam and scanned it two-dimensionally. The image of the iris was projected onto the EO crystal by a pair of lenses into a spot with diameter of , as shown in Fig. 2. Note that the second lens of the pair is a THz Tsurupica (PAX Co.) lens, which has the same focal length in both the visible and the THz frequency regions. So both the THz beam and the probe beam are on focus at the EO crystal.
Using this scanning EO sampling method, we measured the distribution of the generated THz waveforms to confirm that the generated THz beam is a vortex beam. Figure 3(a) shows the spatial distribution of the time-integrated electromagnetic power of the THz vortex beam. The temporal waveforms at the four points in Fig. 3(a) are presented in Fig. 3(b) to illustrate how the waveforms depend on the azimuthal angle . The intensity envelope of the waveforms were independent of , whereas the carrier-envelope offset phase increased monotonically as a function of . The snapshots of the electric field distribution are presented in Fig. 3(c) and indicate that the phase front rotates as time evolves. The generated THz waveforms were few-cycle pulses in the time domain, and were thus broadband in the frequency domain. The amplitude and the phase at each frequency were obtained by Fourier transformation of the measured THz waveforms. The distributions of the phase and amplitude at 0.75, 1.0, 1.25, 1.5, 1.75, and 2.0 THz are shown in Fig. 4(a). Phase singularities at the center of the beam are evident in all frequency components. The phase distribution around the singularity increases equally as the azimuthal angle increases. These features clearly indicate that the generated beam is a broadband vortex beam with a topological charge of .
Next, we changed the sign of the topological charge of a THz vortex beam, which was achieved by simply changing the transmission axis of the polarizer to . After passing the polarization elements, the electric field component along the axis, , is given by4). This is a vortex beam with a topological charge . Figure 4(b) shows the measured phase distributions of this counter-rotating vortex beam of . The phase of the beam decreases as the azimuth increases and changes by around the center. This feature indicates that the topological charge of this broadband vortex beam is .
To evaluate the mode purity of the generated vortex beam, we analyzed the charge distribution by calculating the correlation coefficients  between the measured electric fields and basis functions of Laguerre–Gaussian modes with topological charge . The decomposed power spectra are calculated by5. For the radial index, we rounded up to , which sufficed for converging in the present experiment. We confirmed that a vortex mode with is dominant in a wide frequency range. Because the segmented waveplate used to generate the THz radially polarized beam in these experiments was composed of eight pieces, we obtained a quasi-vortex beam. The power spectrum of modes with topological charge of in such a quasi-vortex beam was numerically estimated to be 95%. As seen in Fig. 5, the obtained ratio of the mode with at approximately 1.5 THz agreed well with this value.
One can extend the above radial-to-vortex mode conversion method to larger topological-charge numbers () by converting higher-azimuthal-order radially polarized beams () as described in the above equations. The azimuthal order of the incoming THz radially polarized beam can be controlled by tailoring the spatial distribution of the polarization state of the laser pulse that generates this THz radially polarized beam; this is easily achieved by employing optical elements such as spatial light modulators designed for the laser wavelength. Thus, the proposed method is readily applicable to generating vortex beams with arbitrary topological-charge numbers.
In conclusion, we demonstrated a method for generating broadband THz vortex beams via conversion of a THz radially polarized beam. We developed a simple method to control the sign of the topological charge by rotating the polarizer by 90°. We detected a spatial phase distribution using a scanning EO sampling method and confirmed that THz vortex beams with topological charges of and were generated. This method can be used to generate higher-order THz vortex beams via conversion of higher-order THz radially polarized beams. This method opens new horizons of measurement, manipulation, and communication technologies in the THz frequency range.
The authors would like to thank Z. Zheng for helpful discussion about our experiment and data analysis. This research was supported by the Photon Frontier Network Program, KAKENHI (20104002), Project for Developing Innovation Systems of the Ministry of Education, Culture, Sports, Science and Technology, Japan and by the Japan Society for the Promotion of Science through its FIRST Program and by the Center of Innovation Program from Japan Science and Technology Agency, JST. NK and TH acknowledge support by JSPS Research Fellowships.
1. A. M. Yao and M. J. Padgett, Adv. Opt. Photon. 3, 161 (2011). [CrossRef]
2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992). [CrossRef]
3. N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1943 (1998). [CrossRef]
4. P. Jia, Y. Yang, C. J. Min, H. Fang, and X.-C. Yuan, Opt. Lett. 38, 558 (2013). [CrossRef]
5. K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, Nano Lett. 12, 3645 (2012). [CrossRef]
6. K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, Nature 440, 935 (2006). [CrossRef]
7. R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, Science 338, 640 (2012). [CrossRef]
8. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004). [CrossRef]
9. D. Sanvitto, F. M. Marchetti, M. H. Szymańska, G. Tosi, M. Baudisch, F. P. Laussy, D. N. Krizhanovskii, M. S. Skolnick, L. Marrucci, A. Lemaître, J. Bloch, C. Tejedor, and L. Viña, Nat. Phys. 6, 527 (2010). [CrossRef]
10. T. Kampfrath, K. Tanaka, and K. A. Nelson, Nat. Photonics 7, 680 (2013). [CrossRef]
11. K. Yamane, Y. Toda, and R. Morita, Opt. Express 20, 18986 (2012). [CrossRef]
12. Y. Tokizane, K. Oka, and R. Morita, Opt. Express 17, 14517 (2009). [CrossRef]
13. N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992). [CrossRef]
14. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993). [CrossRef]
15. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996). [CrossRef]
16. J. He, X. Wang, D. Hu, J. Ye, S. Feng, Q. Kan, and Y. Zhang, Opt. Express 21, 20230 (2013). [CrossRef]
17. R. Imai, N. Kanda, T. Higuchi, Z. Zheng, K. Konishi, and M. Kuwata-Gonokami, Opt. Express 20, 21896 (2012). [CrossRef]
18. Z. Zheng, N. Kanda, K. Konishi, and M. Kuwata-Gonokami, Opt. Express 21, 10642 (2013). [CrossRef]
19. T. Higuchi, N. Kanda, H. Tamaru, and M. Kuwata-Gonokami, Phys. Rev. Lett. 106, 047401 (2011). [CrossRef]
20. Y. Hirota, R. Hattori, M. Tani, and M. Hangyo, Opt. Express 14, 4486 (2006). [CrossRef]