Abstract
1. INTRODUCTION
The spin of a photon is the quantum mechanical counterpart of the polarization of light, and each photon of left-/right-handed circular polarized (LHCP/RHCP) light carries a spin angular momentum of , where is the reduced Planck constant [1]. In addition to spin angular momentum (SAM), a photon of circular polarization can also have orbital angular momentum (OAM) which is dependent on the spatial degrees of freedom of light [2], including the intrinsic orbital angular momentum due to optical vortex within the beam and external orbital angular momentum due to the beam propagation in the transverse direction [3]. The spin-orbit interactions (SOIs) between these three types of angular momentums underpin a variety of interesting phenomena [4], including optical spin Hall effect [5–7], spin-directional coupling [8–12], and spin-dependent beam shaping [13–16], which have potential applications in quantum communication and spin-based optical elements. The underlying physical mechanism of SOI can be explained by the geometric Berry phase originating from the coupling between the SAM and the local coordinates of the media system [17], that is, rotating the transverse coordinates of the inhomogeneous media induces opposite phase shifts for LHCP and RHCP incidences. Conventionally, SOIs have been studied in helical fibers [18], gratings [13], and anisotropic crystals [15]. These systems are either of low efficiency, application-limited, or bulky. Recently, metasurfaces comprising artificial subwavelength planar structures have presented unprecedented abilities to manipulate the polarization, amplitude, and phase of light [19], offering a competitive artificial platform for flexible and considerably enhanced SOI effects [4,20,21].
The building element of SOI metasurfaces is generally called a Pancharatnam–Berry optical element (PBOE) [13,22]. A perfect PBOE will completely flip the input circular polarization to its orthogonal state with an extra geometric phase shift that is double the element’s orientation angle. Single layer plasmonic metasurfaces were first utilized to study SOI [16,23]. However, their power efficiency was low due to the compromise between anisotropy and transmittance [23]. To increase the SOI efficiency, multilayer plasmonic, coherently controlled, and dielectric metasurfaces have been adopted [22,24,25]. Specifically, dielectric metasurfaces consisting of high contrast dielectric elements have recently attracted increasing research interest [26,27]. For such a dielectric element, the anisotropy is flexibly tailorable over a large range, and the material loss can be made negligible by properly selecting dielectric materials, endowing it with a significant advantage in the search for a perfect PBOE.
Besides the power efficiency, another challenge of SOI is that the inherent geometric phase shifts of a PBOE over different circularly polarized inputs are equal in magnitude with opposite signs [16]. Therefore, the achieved functionalities are limited only through the geometric phase. To make the imposed phase shifts for RHCP and LHCP inputs independent and thus the spin-control more flexible, further transport phases have been introduced by utilizing PBOEs of various geometries [28]. Through the combination of the transport phase and the geometric phase, independent and complete phase and wavefront control over orthogonal circular polarizations has been demonstrated. Such a method has greatly expanded the applications of PBOEs in polarization optics [29–31], including polarization detection [32], splitting [33], polarization contrast imaging [34], multiplexing, and encryption [35–37].
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In this context, we present a design method of spin-dependent all-silicon dielectric metasurfaces by simultaneously involving the geometric phase and transport phase in the constituting PBOEs. The PBOEs are anisotropic rectangular posts distributed in a tetragonal lattice on silicon substrate. Two functional metasurface devices are designed and experimentally characterized in the terahertz regime, including a spin-selective Bessel/half-wave metasurface device and a spin-dependent Bessel/vortex beam generator. The Bessel/half-wave metasurface generates a cross-polarized Bessel beam under the LHCP incidence, whereas it only flips the polarization under the RHCP incidence. The Bessel/vortex beam generator produces Bessel and vortex beams separately under the LHCP and RHCP incidences. Such metasurface devices have the potential to help realize miniature THz systems, with key applications in spin switching, contrast imaging, coding, and encryption.
2. DESIGN STRATEGY
A perfect PBOE functions as a half-wave plate, i.e., there is a 180° phase difference between the phase shifts along the local fast and slow axes of the PBOE. The corresponding rotated transmission matrix in the Cartesian coordinate system can be expressed as [33]
Figure 1.(a) Schematic of the spin-dependent metasurface, which consists of PBOEs and can generate two arbitrarily designable beam functions,
To further investigate the phase control and cross-polarization conversion performances of the 8 fundamental PBOEs, the electric field distributions at 1.0 THz are simulated, as shown in Fig. 2(a). It can be seen that the output phases of and are out of phase in each PBOE, and both of them gradually change by 180° from No. 1 to No. 8. As for the last 7 PBOEs (No. 9 to No. 15), the corresponding simulated electric field distributions under - and -polarized incidences are the same as those of the first 7 PBOEs under - and -polarized incidences. Figures 2(b) and 2(c) illustrate the simulated transmission amplitude spectra and of the 8 fundamental PBOEs, whereas Fig. 2(d) illustrates the corresponding polarization conversion rates (PCRs) calculated by . It can be seen that the -polarized (cross-polarized) transmissions are obviously larger than those of the -polarized (co-polarized) transmissions at around 1.0 THz for all PBOEs, where the PCRs all nearly approach 1. To show the cross-polarization conversion performances more clearly, the corresponding , and PCR values at 1.0 THz marked by the dashed red lines are extracted in Fig. 2(e).
Figure 2.(a) Simulated electric field distributions of
3. RESULTS AND DISCUSSION
To experimentally verify that the selected PBOEs meet the two criteria, i.e., perfect cross-polarization conversion and complete phase coverage, a polarization conversion deflector (PCD) is first constructed by patterning the PBOEs into a linear phase distribution along the -axis. To enable the complete phase coverage of 360° with a phase increment of 22.5°, the super unit cell comprises 16 PBOEs of which the corresponding phase shifts range from 0° to 337.5° in sequence. Therefore, the periods are along the -direction and along the -direction. The orientation angles of all the PBOEs are 45°. Deep reactive ion etching process is employed for the fabrication; see Appendix B. The total size of the fabricated PCD sample is , which is the same for all devices in this work. Figure 3(a) shows the optical microscope image of the fractional PCD sample.
Figure 3.(a) Microscopic image of the fabricated PCD. Scale bar: 300 μm. (b) and (c) The measured normalized intensity distributions as a function of the diffraction angle and the frequency of the
The PCD was then characterized using our customized all-fiber terahertz time-domain spectrometer (TDS) system [38]. For bandwidth characterization and simplicity, a linearly polarized input is used here. Figures 3(b) and 3(c) show the measured - and -polarized outputs under -polarized input. It can be seen that almost all the -polarized input is converted to -polarized light and deflected to the st diffraction order, which confirms both the ability to achieve a full phase coverage and the 180° phase difference between and . Notably, the PCD also supports broadband operation. To illustrate the bandwidth, the power efficiency as a function of frequencies is shown in Fig. 3(d). Here, the power efficiency is defined as the ratio of the output power at the desired polarization and diffraction order to the overall output power. Figure 3(d) highlights that the maximum power efficiency is about 91% and appears at 1.1 THz. The deviation of this operating frequency from the design is likely due to fabrication accuracy. Thus, we focus on the following device performance at 1.1 THz. Figure 3(e) shows the measured normalized intensity curves of the - and -polarized outputs at 1.1 THz. The measured deflection angle is , which coincides with the general Snell's law [39].
From the measured results of the PCD, it can be concluded that the selected basic PBOEs can well satisfy our requirements in achieving efficient and independent control over the complete phases of the orthogonal circular polarizations at 1.1 THz. Following the above experimental confirmation, we further fabricate two functional devices.
One is a Bessel/half-wave metasurface, which serves as a Bessel generator under the LHCP incidence but as a half-wave plate under the RHCP incidence. To generate the Bessel beam under the LHCP incidence, a linear gradient of along the polar coordinate is required, assembling the conventional axicon phase distribution. Therefore, [40]. Here, is the polar coordinate, and indicates the deflection angle of the output light with regards to the optical axis and is dependent on the radial span of the 360° phase coverage along the polar direction by .
Meanwhile, attributing to the inherent cross-polarization conversion function of the PBOE metasurface, does not need any localized alteration but keeps a constant . Then, the metasurface can be constructed with the selected PBOEs of the transport phase and the rotation angle . Figure 4(a) shows the optical microscope image of the fabricated Bessel/half-wave metasurface for which is set as 1.24 mm. The metasurface was characterized by a fiber-based near-field scanning THz microscopy (FNSTM) [41]. Figure 4(b) illustrates the measured normalized RHCP intensity profile in the longitudinal plane under the LHCP incidence from the substrate side. It is clearly seen that there is a typical diffraction-free feature in a distance longer than along the propagation direction. The transverse normalized intensity and phase profiles at 8 mm above the metasurface are also measured, as shown in Figs. 4(c) and 4(d). Both profiles show typical Bessel features: an intense central spot is surrounded by weak concentric rings due to the limited size of the metasurface, and an almost linear phase gradient along the radial direction. Under the RHCP incidence, the measured longitudinal normalized LHCP intensity, and the transverse normalized intensity and phase profiles are shown in Figs. 4(e)–4(g). It can be observed that the beam keeps propagating as a Gaussian-type wave. Thus, under the RHCP incidence, the metasurface only performs the cross-polarization conversion function as a half-wave plate.
Figure 4.(a) Microscopic image of a fraction of the fabricated Bessel/half-wave metasurface. Scale bar: 300 μm. (b)–(d) The measured normalized RHCP intensity distributions at the longitudinal propagation plane at
The second device is a Bessel/vortex metasurface, which is capable of producing Bessel and vortex beams separately by switching between the LHCP and RHCP incidences. The Bessel beam for the LHCP input is designed to the same specifications as the former device, while the first-order vortex beam for the RHCP input is designed with a topological charge by [42], where is the azimuthal angle. Using the same design principle, we can obtain the and required to be patterned on the metasurface. Figure 5(a) shows the optical microscope image of the fabricated sample. Under the LHCP incidence, the measured normalized RHCP longitudinal normalized intensity profile at is illustrated in Fig. 5(b). Figures 5(c) and 5(d) illustrate the transverse intensity and phase profiles at . Similar Bessel features are observed as those seen in Figs. 4(b), 4(c), and 4(d). Figure 5(e) illustrates the measured normalized LHCP longitudinal intensity profile at , and Figs. 5(f) and 5(g) illustrate the transverse intensity and phase profiles at , under the RHCP incidence. Two distinct divergent lobes can be seen from the longitudinal propagation plane, in which the intensity along their central line is almost 0. Meanwhile, the transverse intensity profile in Fig. 5(f) presents a doughnut spot, and the phase profile presents a helical phase of 360° encircling the phase singularity at the center. These features coincide with the st vortex beam as expected.
Figure 5.(a) Microscopic image of a fraction of the fabricated Bessel/vortex metasurface. Scale bar: 300 μm. (b)–(d) The measured normalized RHCP intensity distributions at the longitudinal propagation plane at
4. CONCLUSION
Terahertz all-silicon dielectric metasurfaces for spin-dependent and complete wavefront control are realized using PBOEs through introducing extra geometry inhomogeneity besides the rotation inhomogeneity. In this work, a simplified design strategy is utilized to facilitate the metasurface design attributing to the structure symmetry. Based on the simplified design, several exemplar metasurfaces are accomplished using the minimized fundamental constituting elements. One is used to confirm the performance of the selected PBOEs, while the other two metasurfaces demonstrate functional devices—a Bessel/half-wave metasurface and a Bessel/vortex metasurface. The measured results agree well with our design. Such dielectric metasurfaces are easy to fabricate, compatible with semiconductor integration, and have promising applications in ultra-compact and spin-involved optical terahertz imaging system and communication systems for polarization contrast imaging, coding, and encryption.
APPENDIX A: SIMULATION METHODS
To select the proper PBOEs, CST Microwave Studio is used to simulate the transmissions (including amplitude and shift) of the silicon posts, in which we set and raster scan the lateral dimensions and . Periodic boundaries are applied along the - and -directions. Plane waves of and polarizations separately illuminate the structure layer from the substrate side. The corresponding outputs are detected using probes positioned at the far-field on the air side. After getting all the simulated transmissions, we selected the proper 15 PBOEs required for an 8-level phase control at 1.0 THz; see Table
APPENDIX B: SAMPLE FABRICATION
The silicon dielectric metasurfaces were fabricated using conventional photolithography and deep reactive ion etching (DRIE) [
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