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Abstract
We experimentally demonstrated focusing of light with orbital angular momentum (OAM) using an 8-element circular array of linear antennas. A spiral phase plate was used to generate a vortex beam with an OAM of ħ in the terahertz (THz) frequency region. We used THz near-field microscope to directly measure the phase vortex. A beam profile with a center dark spot and 2π phase rotation was observed in the small center gap region of the circular array antenna after the vortex beam excitation. The beam size is reduced by a factor of 3.4 ± 0.2. Half-wave resonance of the antenna element is responsible for the focusing function, indicating the scalability of this method to other frequency regions. This method will enable deep subwavelength focusing of light with OAM and eliminate the obstacle for the observation of the dipole forbidden transition with finite OAM of the vortex beam.
© 2017 Optical Society of America
1. Introduction
Since the recognition of the orbital angular momentum (OAM) of light associated with a helical phasefront [1], a new class of phenomena which cannot be realized with conventional Gaussian beam has been discovered [2–6]. Theory predicts many more intriguing phenomena, such as a modification of optical selection rules in the quantum mechanical vortex-light—matter interaction [7–10]. However, to the best of our knowledge, these phenomena involving the OAM transfer between light and quantum mechanical systems (such as the observation of the dipole forbidden transition) have not been demonstrated, especially in solids. The main obstacle lies in the size mismatch between the spot size w0 of the vortex beam and the coherent length lc of a material system (typically w0 >> lc). Since the OAM of light is defined by the spatial phase structure over the beam spot, the coherent length lc of a material system has to be comparable to w0 for the effective OAM transfer. Otherwise (i.e. w0 >> lc), a material system only experiences a local electric field at some point of a vortex, which is indistinguishable from a tilted plane wave without an OAM. For this reason, the transition probability involving the OAM transfer depends on the following factor; (lc/w0) [11]. In the visible region, this factor is extremely small (~10−3) due to the short coherent length (or wavefunction in the case of atomic systems) of the material system (lc ~10−10 m) and the diffraction-limited spot size (w0 ~10−7 m). A further difficulty arises in the longer wavelength region, for example in the terahertz (THz) frequency region, where physically important resonances exist in solid state materials.
To explore the nature of the quantum mechanical interaction between vortex-light and matter, we need to establish a way to focus a vortex beam beyond the diffraction limit. Heeres et al. proposed a scheme based on a resonant optical antenna array [12]. Using the finite difference time domain (FDTD) simulation, they numerically demonstrated the vortex phase structure of the incident beam can be focused in the subwavelength region at the center of circularly arranged stripes of gold (circular array antenna). However, experimental verification of this method in the optical region is challenging. In this paper, we used THz near-field imaging technique to experimentally demonstrate vortex beam focusing by circular array antenna. Our time-domain measurement technique allows us to directly detect electric field including phase information. We reduced the beam size by a factor of 3.4 while keeping the original OAM. Near-field THz imaging also revealed that half-wave resonance of the antenna element is responsible for the focusing function, which indicates the scalability of this method to other frequency regions simply by adjusting the antenna length.
2. Experimental
We used a THz microscope [13–15], which enables us to perform time-resolved near-field imaging [Fig. 1(a)]. Linearly polarized coherent THz pulses with a Gaussian beam profile are generated via optical rectification in a LiNbO3 (LN) crystal by tilted-pulse-front excitation scheme [16,17]. In order to convert the Gaussian THz beam into vortex beam, we used a spiral phase plate (SPP) [18–21]. We used ZEONEX (refractive index 1.52) to make a 16-level SPP nominally designed for 0.45 THz with a step height of 1.29 mm. This SPP generates a vortex beam carrying OAM of ħ with reasonable efficiency and quality from 0.3 to 0.6 THz. When performing vortex beam excitation, the SPP was inserted in the collimated part of the THz beam [Fig. 1(a)]. To reduce and match the spot size of the incident beam to the size of the array antenna, we put a 2-mm-radius silicon bullet lens (3 mm thick) in contact with the antenna structure.
Fig. 1 (a) Schematic of the experimental setup. The magnified view in the upper right is not drawn to scale. THz pulses are focused to the circular antenna by two lenses (Lens1: Tsurupica lens, f = 50 mm. Lens2: Silicon bullet lens). The top and bottom surfaces of the EO crystal substrate are coated with high-reflection (HR) and anti-reflection film (3 μm thick) for the probe pulse at 780 nm, respectively. The circular array antennas are fabricated directly on the top surface of the EO substrate. An x-cut LiNbO3 with the thickness of 20 μm is used to compensate for the birefringence of the EO crystal. BS: beam splitter, SP: short pass filter, LP: long pass filter, QWP: quarter wave plate, PBS: polarizing beam splitter. (b) Structure of the circular array antenna. The vector e0 shows the polarization direction of the incident THz pulse. (c) Dimensions of the antenna element. The blue sinusoidal wave represents half-wave resonance.
In order to perform two-dimensional electro-optic (EO) imaging, we used a probe beam with a large spot size for EO sampling in an x-cut LN crystal (10 μm thick) stacked on a glass substrate (500 μm thick). The image of the probe beam at the EO crystal is relayed to a 16 bits CMOS camera with a polarization analyzer unit. The short pass (SP) and long pass (LP) filters are used to enhance the detection sensitivity using the probe spectrum filtering technique [15].
The circular array antenna structures are fabricated directly on the top surface of the EO substrate using photolithographic and vacuum deposition technique. This enables us to observe the near-field distribution around the antenna structure with a spatial resolution of ~10 μm. The circular array antenna [Fig. 1(b)] consists of 8 gold bars [100 nm thick, Fig. 1(c)] with the length L = 115 μm and width w = 20 μm, circularly arranged to make an inner gap space with radius r. We made two antenna arrays with different inner radiuses of 30 μm (for Gaussian beam) and 150 μm (for vortex beam). As shown in Fig. 1(b), none of the gold bars are aligned perpendicular to the polarization direction (e0), ensuring that all the antenna elements are excited by the electric field component projected along their long sides.
3. Results and discussion
3.1 Vortex beam generation using a spiral phase plate
Before discussing the focusing effect of the array antenna, let us characterize the incident THz beam, which is measured without the antenna structure. Figure 2(a) shows the temporal snapshots of the THz pulse passing through the EO crystal taken without the SPP (see Visualization 1 for movie). A uniform phase distribution around the center (white cross in Fig. 2(a)) suggests the THz pulse generated by LN is very close to the ideal Gaussian beam. The electric field waveform and its frequency spectrum at the center is shown in Figs. 2(c) and 2(d), respectively, which shows a single cycle, broadband nature. Figure 2(b) shows the temporal snapshots of the THz pulse passing through the EO crystal measured with the SPP (see Visualization 2 for movie). One can see a vortex phase structure, which indicates the existence of the OAM.
Fig. 2 (a) Time-domain EO images of the Gaussian beam (see Visualization 1) and (b) vortex beam (see Visualization 2). The black arrow represents the direction of the polarization. (c) Electric field waveform at the center of the Gaussian beam. (d) Frequency spectrum of the waveform shown in (c).
To characterize the vortex beam at the resonance frequency of the antenna (see section 3.2 below), we extracted and analyzed the intensity and phase information in the frequency-domain. Figure 3 shows intensity and phase images at 0.6 THz calculated from the data shown in Fig. 2. While the Gaussian beam shows a single intensity peak at the center [Fig. 3(a)] and almost flat phase distribution [Fig. 3(b)], the vortex beam has an intensity null at the center [Fig. 3(d)] and 2π phase rotation with a center singular point [Fig. 3(e)]. The black open circles in Figs. 3(c) and 3(f) show the phase values as a function of the azimuthal angle ϕ, taken along the dotted circles in Figs. 3(b) and 3(e), respectively. These data clearly show that the THz beam after the SPP has an OAM of ħ at 0.6 THz. The phase gradient in the radial direction [Fig. 3(b)] indicates that the imaging plane (position of the EO crystal) is not precisely at the beam waist, and the phase curvature of the Gaussian beam is detected [22]. The twisted phase structure prominent in the outer edge of the vortex beam [Fig. 3(e)] is also due to the same displacement effect [23]. The intensity profile of the vortex does not look like a perfect ring, but has two peaks presumably due to the deviation from the design frequency of the SPP and also some imperfection of our SPP [18].
Fig. 3 (a) Frequency-domain intensity and (b) phase images of the Gaussian beam at 0.6 THz. Each pixel is plotted by taking the 0.6 THz Fourier component of the waveform at each pixel of the microscope field (such as the one shown in Fig. 2(c)). The intensity images are normalized by the maximum value. (c) The azimuthal angle dependence of the phase of the Gaussian beam. The data points (black open circle) are taken along the dotted circle with the radius of 25 μm shown in (b). (d) Frequency-domain intensity and (e) phase images of the vortex beam at 0.6 THz. (f) The azimuthal angle dependence of the phase of the vortex beam. The data points (black open circle) are taken along the dotted circle with the radius of 25 μm shown in (e). The origin of the horizontal axis is chosen so that the phase starts from -π. The red open circles in (c) and (f) are taken from Fig. 4(e) and Fig. 5(c), along the white circle with the same radius of 25 μm, respectively.
For the later calculation of the reduction factor [(beam size without antenna)/(beam size with antenna)], we defined the sizes of the Gaussian beam and vortex beam as follows. We used the full width at half maximum (FWHM) for the Gaussian beam. The FWHM at 0.6 THz is 190 μm. For the vortex beam, we used the ring diameter (distance between two intensity peaks), which is 310 μm at 0.6 THz.
3.2 Focusing of Gaussian beam
Figure 4(a) shows the temporal snapshots of the near-field distribution around the circular array antenna (r = 30 μm) after the illumination by the Gaussian beam (see Visualization 3 for movie). In the inner space of the array antenna, one can see long-lived oscillation (blue and red) with the period of 1.65 ps. The electric field waveform at the center (white cross in (a)) is shown in Fig. 4(b). This clearly shows that damped sinusoidal oscillation is present even after 3.54 ps, where the incident THz pulse has gone through the antenna (time region enclosed by the dashed line). This indicates the presence of the antenna resonance. The power spectrum of the oscillating part enclosed by the dashed line shows that the resonance frequency is 0.6 THz [Fig. 4(c)].
Fig. 4 (a) Time-domain EO images of the near-field distribution around the circular array antenna illuminated by the Gaussian beam (see Visualization 3). The circular array antenna with the inner radius of 30 μm is used and shown by the dotted lines. The color scales are optimized at each frame for the sake of clarity. (b) The electric field waveform at the center of the array antenna (white cross in (a)), showing the damped sinusoidal oscillation. (c) The power spectrum of the waveform enclosed by the dashed line in (b) (after 3.54 ps). A clear peak at 0.6 THz is seen. (d) Intensity and (e) phase image at 0.6 THz. The radius of the white circle in (e) is 25 μm. The Fourier transformation is performed using the time-domain data after 3.54 ps.
To see the field characteristics at the resonance frequency, we calculated the intensity and phase image at 0.6 THz [Figs. 4(d) and 4(e)]. Note that in generating these images, only the time-domain data after 3.54 ps are used for the Fourier transform to eliminate the influence of the incident beam. The intensity distribution shows the resonant component is concentrated in the gap region of the array antenna. The intensity pattern does not look circular probably because all the gold bars are not equally excited by the linearly polarized wave and therotation symmetry is lowered. The FWHM is about 50 μm, which means the beam size reduction factor of 3.8 ± 0.4. Although the precise determination of the enhancement factor is technically difficult, we did not see any notable enhancement (enhancement factor ~1). The phase image shows a uniform distribution in the gap region. This means that the phase distribution is not distorted by the uneven excitation efficiency of the gold bars. The phase values taken along the white circle shown in Fig. 4(e) are plotted in Fig. 3(c) by red open circles. One can see a flat phase distribution similar to the incident Gaussian beam. These results unambiguously show that the circular array antenna focuses the incident Gaussian beam inside the gap region.
The antenna resonance frequency depends on the refractive index [ng in Fig. 1(b)] of the medium surrounding it. In our case, the antenna structure is in air and sandwiched by a silicon lens and substrate (with high-reflection coating and LN) as shown in Fig. 1(a). This makes it difficult to estimate the effective refractive index that determines the resonance frequency. It might be expressed by a combination of the refractive index of air (n = 1.00), silicon (n = 3.41), glass (n = 1.96), high-reflection coating materials (n ~2) and LN (n = 5.11) with appropriate weightings depending on the spatial extent of the electric field. Since this is too complex to calculate, we adopted a three-layer model that consists of air, silicon and glass as the first approximation and simply used an average of their indexes to estimate the effective index. As to the glass layer, we neglected the contribution from high-reflection coating material (3 μm) and LN (10 μm) because the spatial extent of the electric field (a few hundred μm) is much larger and the contribution from the glass would be dominant. Although the air layer (nominally 100 nm) is also very thin, we didn’t neglect it as the electric field would be strong in the antenna gap. The effective refractive index estimated this way is ng = (1 + 3.41 + 1.96)/3 = 2.12, which means that the resonance frequency of 0.6 THz corresponds to the wavelength of 225 μm. Considering the length of the gold bars (L = 115 μm), this suggests each gold bar acts as a half-wave dipole antenna [Fig. 1(c)], in agreement with the FDTD simulation [12].
Looking at the phases around the other edges of the gold bars in Fig. 4(e), one can notice these are mostly green, i.e. shifted by π from those around the center of the antenna array. This also supports the half-wave dipole antenna resonance described above. This observation directly confirms the following focusing mechanism; in-phase excitation of all the dipole antennas induce oscillating currents with a common phase and produce electric field with a uniform phase distribution in the inner space. Since the antenna resonance is scalable, this focusing method should be applicable in other frequency regions as long as the metal behaves like perfect electric conductor. This concept is expected to work even in the near-infrared and optical region by taking into account an antenna geometry and material properties [24].
3.3 Focusing of vortex beam
The same mechanism should be exploited to realize focusing of vortex beam. In this case, the antennas are excited with different phases and the vortex phase distribution is expected in the center gap region. Figure 5(a) shows the temporal snapshots of the near-field around the circular array antenna (r = 150 μm) after the illumination by the vortex beam (see Visualization 4 for movie). In the gap region, we can see an out-of-phase electric field distribution (red and blue) across the center point (black cross in (a)). This pattern appears to be rotating with the period of ~1.7 ps (0.6 THz) as shown in the second and third frames in Fig. 5(a). This pattern is recognizable up to 8.34 ps, where the incident vortex beam has already gone through the antenna. This suggests that a similar antenna resonance as we saw in the Gaussian beam excitation is happening. The same resonance frequency of 0.6 THz also indicates that this is the antenna resonance since the length of the gold bar is the same.
Fig. 5 (a) Snapshots of the near-field distribution around the circular array antenna (dotted lines) illuminated by the vortex beam (see Visualization 4). The color scales are optimized at each frame for the sake of clarity. (b) Intensity and (c) phase image at 0.6 THz, showing an intensity null at the center (white cross) and the 2π phase rotation. The figures in the upper right of (b) and (c) show enlarged images of the region enclosed by the corresponding dashed square. Unfortunately, we could not observe the phases around the outer edges of the gold bars due to the limited microscope view field. The radius of the white circle in (c) is 25 μm. The Fourier transformation is performed using the time-domain data after the incident THz pulse has gone through the sample. The black and white crosses in the figures indicate the same center point.
To characterize the beam profile inside the gap region, we calculated the intensity and phase image at 0.6 THz [Figs. 5(b) and 5(c)]. The figures in the upper right of Figs. 5(b) and 5(c) show enlarged images of the region enclosed by the corresponding dashed square. In generating these images, only the time-domain data after 5.34 ps are used for the Fourier transform to eliminate the influence of the incident beam. Although the intensity profile in the gap region is complicated partly due to the non-uniform excitation of the gold bars, it shows ring-like structure with two lobes and intensity null at the center. The ring diameter is 90 μm, which means the beam size reduction factor of 3.4 ± 0.2. Note that two intensity peaks exist in the region of the intensity null of the original beam. Most importantly, the phase image shows 2π phase rotation with a singular point at the center. The phase values taken along the white circle shown in Fig. 5(c) are plotted in Fig. 3(f) by red open circles. One can see a similar 2π phase rotation as the incident vortex beam. These results clearly demonstrate that the circular array antenna reduces the spot size of the incident vortex beam.
Much tighter focusing is expected by the circular array antenna with the inner radius of 30 μm. As the inner radius becomes smaller, accurate spatial matching between the centers of the incident vortex beam and the circular array antenna becomes much more crucial to transfer the vortex phase structure. In the current experimental situation, the best achievable accuracy is about 30 μm. This should be the reason why the focused vortex center in Fig. 5 is a little bit off the center of the circular array antenna. This also significantly alters the intensity distribution in the gap from the ideal one. For this reason, we have not succeeded in focusing vortex beam using the smaller circular array antenna at this point.
4. Conclusion
We experimentally demonstrated focusing of the Gaussian beam and vortex beam with an OAM of ħ in the THz frequency region using 8-element circular array antennas. We reduced the beam size of the Gaussian and vortex beam by a factor of 3.8 ± 0.4 and 3.4 ± 0.2, respectively. Near-field THz microscopy revealed that half-wave resonance of the antenna element is responsible for the focusing function. This indicates the scalability of this method to other frequency regions simply by adjusting the antenna length, thus opening a way to explore many kinds of quantum mechanical interaction between vortex-light and matter.
Funding
JSPS KAKENHI (16K17529 and 26247052).
Acknowledgments
We thank Professor François Blanchard for his technical assistance. We also thank Professor Keiji Sasaki for valuable discussions. The SPP used in this study was made with the assistance of the technical division of graduate school of science, Kyoto university. The Institute for Integrated Cell-Material Sciences is supported by the World Premier International Research Center Initiative at the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan. This work was supported in part by Kyoto University Nano Technology Hub in “Nanotechnology Platform Project” sponsored by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.
References and links
1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]
2. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]
3. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003). [CrossRef] [PubMed]
4. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002). [CrossRef] [PubMed]
5. K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013). [CrossRef] [PubMed]
6. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006). [CrossRef] [PubMed]
7. R. Grinter, “Photon angular momentum: selection rules and multipolar transition moments,” J. Phys. B 41(9), 095001 (2008). [CrossRef]
8. G. F. Quinteiro and P. I. Tamborenea, “Electronic transitions in disk-shaped quantum dots induced by twisted light,” Phys. Rev. B 79(15), 155450 (2009). [CrossRef]
9. K. Sakai, K. Nomura, T. Yamamoto, and K. Sasaki, “Excitation of multipole plasmons by optical vortex beams,” Sci. Rep. 5(1), 8431 (2015). [CrossRef] [PubMed]
10. H. Fujita and M. Sato, “Ultrafast generation of skyrmionic defects with vortex beams: printing laser profiles on magnets,” Phys. Rev. B 95(5), 054421 (2017). [CrossRef]
11. A. Alexandrescu, D. Cojoc, and E. Di Fabrizio, “Mechanism of angular momentum exchange between molecules and Laguerre-Gaussian beams,” Phys. Rev. Lett. 96(24), 243001 (2006). [CrossRef] [PubMed]
12. R. W. Heeres and V. Zwiller, “Subwavelength focusing of light with orbital angular momentum,” Nano Lett. 14(8), 4598–4601 (2014). [CrossRef] [PubMed]
13. F. Blanchard, A. Doi, T. Tanaka, H. Hirori, H. Tanaka, Y. Kadoya, and K. Tanaka, “Real-time terahertz near-field microscope,” Opt. Express 19(9), 8277–8284 (2011). [CrossRef] [PubMed]
14. F. Blanchard, A. Doi, T. Tanaka, and K. Tanaka, “Real-time, subwavelength terahertz imaging,” Annu. Rev. Mater. Res. 43(1), 237–259 (2013). [CrossRef]
15. F. Blanchard and K. Tanaka, “Improving time and space resolution in electro-optic sampling for near-field terahertz imaging,” Opt. Lett. 41(20), 4645–4648 (2016). [CrossRef] [PubMed]
16. J. Hebling, G. Almási, I. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large area THz-pulse generation,” Opt. Express 10(21), 1161–1166 (2002). [CrossRef] [PubMed]
17. H. Hirori, A. Doi, F. Blanchard, and K. Tanaka, “Single-cycle terahertz pulses with amplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO3,” Appl. Phys. Lett. 98(9), 091106 (2011). [CrossRef]
18. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5), 321–327 (1994). [CrossRef]
19. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127(4), 183–188 (1996). [CrossRef]
20. K. Miyamoto, K. Suizu, T. Akiba, and T. Omatsu, “Direct observation of the topological charge of a terahertz vortex beam generated by a Tsurupica spiral phase plate,” Appl. Phys. Lett. 104(26), 261104 (2014). [CrossRef]
21. K. Miyamoto, B. J. Kang, W. T. Kim, Y. Sasaki, H. Niinomi, K. Suizu, F. Rotermund, and T. Omatsu, “Highly intense monocycle terahertz vortex generation by utilizing a Tsurupica spiral phase plate,” Sci. Rep. 6(1), 38880 (2016). [CrossRef] [PubMed]
22. A. Yariv, Quantum Electronics, 3rd ed. (John Wiley and Sons, 1988).
23. J. He, X. Wang, D. Hu, J. Ye, S. Feng, Q. Kan, and Y. Zhang, “Generation and evolution of the terahertz vortex beam,” Opt. Express 21(17), 20230–20239 (2013). [CrossRef] [PubMed]
24. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98(26), 266802 (2007). [CrossRef] [PubMed]
