Detection of photonic orbital angular momentum with micro- and nano-optical structures

Abstract

Light with an optical orbital angular momentum (OAM) has attracted an increasing amount of interest and has found its way into many disciplines ranging from optical trapping, edge-enhanced microscopy, high-speed optical communication, and secure quantum teleportation to spin-orbital coupling. In a variety of OAM-involved applications, it is crucial to discern different OAM states with high fidelity. In the current paper, we review the latest research progress on OAM detection with micro- and nano-optical structures that are based on plasmonics, photonic integrated circuits (PICs), and liquid crystal devices. These innovative OAM sorters are promising to ultimately achieve the miniaturization and integration of high-fidelity OAM detectors and inspire numerous applications that harness the intriguing properties of the twisted light.

Keywords

orbital angular momentum (OAM) / optical vortices / singular optics / spatial light modulator / surface plasmon polariton (SPP) / holography / photonic integrated circuit (PIC)

1 Introduction

It is well known that circularly polarized light carries spin angular momentum (SAM) [1]. Allen et al. recognized that, contrary to SAM which can only take one of the two values±ħ, light with an azimuthal phase dependence of exp(ilϕ) carries an orbital angular momentum (OAM) that is characterized by lħ, where l is the topoloical charge (TC) that can be any integer [2]. Light with OAM has been intensively explored and applied to various applications including high-speed optical communications [3,4], optical spanners [5], edge-enhanced microscopy [6], and quantum optics [7,8]. To take full advantage of the intriguing property of dynamic rotation and the unbounded state space of OAM beams, it is crucial to develop highly efficient, compact, and high-resolution OAM sorters with parallel detection capabilities. Several methods based on conventional optical elements have been demonstrated including fork holograms [8,9] and cascaded Mach–Zehnder interferometers [10]. However, these methods are either low in efficiency or unsuitable for miniaturization and integration to realize on-chip OAM detection. In the current paper, we review the latest progress on OAM detection with the use of a plasmonic lens and grating, meta-hologram, micro-ring resonator, hybrid conformal mapper, etc. These techniques have been proven to be effective, efficient, and capable of being integrated into a compact platform.

2 Plasmonics

Surface plasmon polaritons (SPPs) are the collective oscillations of free electrons, and they propagate along the metal-dielectric interface [11]. The interaction between SPPs and optical OAM states has attracted great interest recently [12,13]. Spiral phase information can be imparted onto the excited SPPs when they are excited by a vortex beam [14–16]. Light incident on a plasmonic lens made of open slots perforated in a thin metallic film either couples to SPPs propagating along the surface or directly transmits through the lens [17,18]. Evanescent SPP waves and directly transmitted propagating waves can interfere near the plasmonic lens surface and form a unique intensity distribution that is dependent on the OAM state of the incident beam. To demonstrate this OAM detection scheme, a ring-shaped slot with a 200 nm width and a 70 nm depth is etched into the gold film by focused ion beam (FIB) lithography, forming a plasmonic lens with an inner diameter of 6 mm, as illustrated in Fig. 1 [19]. A near-field scanning optical microscope (NSOM) probe on the collection mode scans over the plasmonic lens, and near-field images of the intensity distribution of the electric field above the plasmonic lens are captured by the NSOM.

Fig.1 (a) Schematic of light with spiral phase. The insets are the transverse intensity profiles of Laguerre-Gaussian (LG) modes with different |l|. (b) Schematic of experimental setup. The plasmonic lens is excited by LG modes from the SiO2 substrate side and imaged by a NSOM probe working on collection mode. Inset 1 is the diagram of a single ring plasmonic lens and the coordinates used in analytical derivation. The illumination is along the z-direction. Inset 2 is the scanning electronic microscope micrograph of the plasmonic lens fabricated in gold film on SiO2 substrate. Adapted from Ref. [19]

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The intensity distributions are shown in Fig. 2. Because the SPPs are only excited by the locally TM polarized wave, concentric bright arcs are generated by an incident linearly polarized LG beam. For l = 0, corresponding to a fundamental Gaussian illumination, a solid spot appears in the center. For |l| = 1, a central dark spot shows up due to destructive interference. For |l| = 2, a bright spot reemerges. The intensity rotation direction is dependent on the sign of the topological charge: clockwise for negative charges and anticlockwise for positive charges. Light with a topological charge of |l|>2 always results in a dark spot. However, the size of the central dark area can be exploited to determine the OAM mode index.

Fig.2 Intensity distributions of the optical field near the plasmonic lens surface excited by photons with different OAMs. (a)−(e) are the NSOM images for l = 0, +1, −1, +2 and −2, respectively. (f)−(j) are the corresponding numerical simulation results. The excitation polarization is shown by the white arrow in (a). The scale bar in (f) and the color bar in (j) are also applicable to the other images of intensity distributions. Adapted from Ref. [19]

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One disadvantage in plasmonic lens-based detection is the requirement of a near-field scanning mechanism to collect surface plasmons (SPs), which leads to a slow detection speed. To overcome this limit, a multi-sector meta-hologram structure with various geometrical TCs has been reported that separates the incident vortex beam into SP waves with a propagation direction depending on the OAM mode [20]. The design of the multi-sector meta-hologram is illustrated in Fig. 3. A holographic pattern with a depth of 75 nm and an area of 10 mm × 10 mm is fabricated on a gold film with a thickness of 200 nm. The holographic pattern is obtained by interfering a converging SP wave with a free-space vortex beam. The hologram is divided into four sectors with equal areas but different geometrical TCs. Assuming that a vortex beam with a TC of l_i passes through a metallic structure with a geometrical TC of l_g, the TC of the SP wave is given by l_s = l_i + l_g. Consequently, different OAM modes can be differentiated when identifying the propagating direction of the SP wave because a larger l_s leads to a larger steering angle.

Fig.3 (a) Schematic of the metahologram. The holographic pattern has four sectors, which are designed by considering the interference between a converging SP wave with a vortex beam carrying different OAMs. The inset corresponds to the designed TC for each sector. (b) Binary version of the interferogram. Adapted from Ref. [20]

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The number of distinguishable OAM states by the four-sector meta-hologram is estimated to be

(1)$N={\displaystyle \frac{\mathrm{1.22}\lambda /(W/\mathrm{2}f)}{\lambda f/(\mathrm{4}W)}}\approx \mathrm{10},$

where f is the focal length of the lens and W is the width of the hologram. The OAM detectable range of the meta-hologram increases linearly with the number of sectors and is independent of the incident wavelength and the focal length of the hologram. Figure 4 renders the simulated near-field intensity distributions when a circular polarized vortex beam with a TC of 1 is normally incident on the meta-hologram. As expected, the illumination is coupled into four bunches of focused SP waves with steering angles depending on the different TCs carried by the SPs, which are (0, 2, 3, -1) for the (A, B, C, D) sectors of the meta-hologram, respectively.

Fig.4 Numerical simulations of the SP intensity distribution of the metahologram for circularly polarized illumination with a TC of 1. Adapted from Ref. [20]

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A detector can be integrated into the rear side of the substrate to spatially sample the SP waves that are collected by an array of subwavelength holes with a period of 200 nm in the focal region of each channel of the meta-hologram. The data measured by the detectors are normalized to the maximum value of the signals and discretized into five levels from 0.2 to 1 with a step size of 0.2. Each level is indicated by a different color as shown in Fig. 5. Clearly, the OAM states ranging from −6 to 6 can be differentiated by the sequence of discrete signals. A higher number of OAM states can be discerned by the application of more discretization levels, as long as the signal-to-noise ratio is satisfied in practice.

Fig.5 Lookup table for identifying incident OAM by encoding the OAM mode with discretized signal levels. Adapted from Ref. [20]

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Approaches based on the principle of holography have been reported to be integrated into plasmonic photodiodes and can selectively detect the orbital angular momentum of light [21]. Another intriguing alternative is to exploit plasmonic nanoantennas and convert the information on the OAM of light into spectral information using bright and dark modes [22]. All of these solutions are based on the interaction of OAM beams with plasmonics, and the sign and magnitude of topological charges are unveiled accordingly.

3 Photonic integrated circuits

With recent developments in nanotechnology, photonic integrated circuits (PICs) have been employed in a wide range of applications and have achieved multiple functionalities that were previously accomplished by bulky optical components such as ring-based Mach–Zehnder modulators [23–25]. PIC-integrated OAM detectors can be made to be very compact and efficient for OAM-based optical communication systems that have great potential in boosting the data rate.

Figure 6 shows the schematic diagram of an OAM detector that is composed of a circular resonator, angular gratings, and an access waveguide [26]. Periodic grating elements are patterned on the inner wall of the circular resonator. The width of the gratings is about 1/10 of the grating period. The periodic variation of the dielectric constant provided by the angular gratings helps couple the eigenmodes of the resonator into other modes with well-controlled OAM states [27].

Fig.6 Diagram of the proposed OAM receiver. The circular resonator with angular gratings patterned along the inner wall couples the normally incident azimuthally polarized vortex beam to an access waveguide. Adapted from Ref. [26]

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When the mode matching condition is satisfied, the receiving efficiency of the device can be maximized, which is defined as the ratio of the power coupled with the access waveguide to the incident power. The coupling between the guided whispering gallery modes (WGMs) and the illumination mode can be characterized by the topological charge l of the received vortex beam:

(2)$l=p-g\times q,$

where p and q are the number of optical periods and grating elements around the circular resonator, respectively, and g is any integer. The value of p can be calculated as

(3)$p=\mathrm{2}\mathrm{\pi }R{n}_{\mathrm{e}\mathrm{f}\mathrm{f}}/\lambda ,$

where R and n_eff are the inner radius and the effective index of the WGM of the circular resonator, respectively. The wavelength of illumination is denoted by l. The diffraction order g is determined by the material system [27]:

(4)$(n_{\mathrm{e}\mathrm{f}\mathrm{f}}-\mathrm{1}){\displaystyle \frac{\mathrm{\Lambda }}{\lambda }}<g<(n_{\mathrm{e}\mathrm{f}\mathrm{f}}+\mathrm{1}){\displaystyle \frac{\mathrm{\Lambda }}{\lambda }}.$

When l is around 1550 nm, R equals 3.9 mm, L is 598 nm, and n_eff is 2.6, so the value of g can only be 1. Then Eq. (2) is reduced to

(5)$l=p-q.$

Numerical simulations are performed with a three-dimensional finite element method model (COMSOL). Figure 7 shows the receiving spectrum of the device for different OAM states from -2 to 2. The receiving efficiencies at various resonances are measured between 6% and 28%, which can be enhanced by an increase in the size of the circular resonator or an optimization of the gap distance between the resonator and access waveguide.

Fig.7 Receiving spectrum of the OAM receiver. Adapted from Ref. [26]

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Considering a circularly polarized vortex beam normally illuminating the circular resonator, Eq. (5) can be rewritten according to the conservation of total angular momentum (AM) in a closed physical system:

(6)$j=p-q,$

where j = s + l is the total AM and s is the spin angular momentum. As illustrated in Fig. 8, the corresponding OAM of each resonant wavelength strongly depends on the spin of the light, making this device suitable for the identification of vortex beams with various polarization states.

Fig.8 Resonant wavelengths for vortex beam with different SAMs. Adapted from Ref. [26]

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GeSe is reported to support high-speed reversible transitions between phases and thus enables the reconfiguration possibility of the proposed OAM receiver [28]. As shown in Fig. 9(a), a very thin, annular photonic cap on a silicon resonator is formed by cladding a GeSe film with a thickness of 60 nm. Through doping the GeSe layer and the lower half of the silicon resonator to provide electrical contact for both surfaces of the cladding layer, GeSe film can be pulsed with voltage for a binary phase transition. A linear and continuous phase change can be accomplished by patterning dense electrons on the GeSe film. The duty cycle of the gratings can be controlled by turning on a different number of electrodes. As an example, a 60 nm thick GeSe grating with a duty cycle of 0.5 is deposited on top of the circular resonator, and the total number of GeSe grating is chosen to be 41, as shown in Fig. 9(b).

Fig.9 Diagram of the composite OAM receiver that has a (a) GeSe annular film and (b) GeSe gratings consist of alternate states on top of the resonator as the cladding layer. The duty cycle of the gratings is 0.5. Adapted from Ref. [26]

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Figure 10 shows the receiving spectrum tuned by the binary phase change of GeSe, enabling the switch of detected OAM between two different states at nearly the same wavelength. In addition, the upper cladding layer offers the function of cancelling the vertical leakage through destructively interfering the leakage propagation mode in the circular resonator, resulting in a significantly increased quality factor [29].

Fig.10 Receiving spectrum of the composited device shown in Fig. 5(a) when the GeSe is in (a) amorphous state and (b) crystal state, respectively. Adapted from Ref. [26]

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Integrated nanophotonic circuits have been demonstrated to be interfaced with single-mode fibers [30] to interfere with on-chip generated Gaussian beams [31] and to covert the free space vortex beam and the in-plane guiding wave to each other by introducinga holographic grating on top of a dielectric waveguide [32]. PICs have also been experimentally demonstrated for free-space spatial division-multiplexing optical transmission with multiplexed OAM states over a topological charge range of -2 to+2 [33]. These techniques introduce vast opportunities of integrating conventional optical systems and functionalities onto a silicon chip.

4 Spatial light modulators

Spatial light modulators (SLMs) empower researchers to have access to structured light with phase control on a pixel by pixel level. With the ingenious combination of additional optical elements, SLMs render the full control of all four degrees of freedom of a structured beam [34]. SLMs have been intensively utilized for the proof-of-principle demonstration of novel OAM sorters that are promising in the integration onto a chip. The hybrid conformal mapper is one of the candidates.

Conformal mapping offers a highly efficient solution to differentiate different OAM states based on a log-polar transform [35]. The circular intensity distribution of an OAM mode is unwrapped to a straight line, and the helical phase gradient is transformed to the transverse phase gradient. However, the overlap between adjacent modes undermines the sorting resolution. To solve this problem, a fan-out element and the corresponding phase corrector can be added to the optical system to compress the width of the main lobe of each mode. Therefore, four pieces of custom diffractive and refractive optical elements with three lenses in between are implemented in the previous experimental demonstrations, as illustrated in Fig. 11(a) [36–38].

Fig.11 Comparison of two high-resolution OAM sorters. (a) Previous demonstrations with four custom refractive/diffractive elements (log-polar mapper, mapper corrector, fan-out element, fan-out corrector) and three lenses in between. (b) The novel scheme with only two custom phase elements (quadratic fan-out mapper, dual-phase corrector) and no lens in between. Adapted from Ref. [39]

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A significantly simplified high-resolution OAM sorter that comprises a quadratic fan-out mapper and a dual-phase corrector shown in Fig. 11(b), has recently been demonstrated [39]. The quadratic fan-out mapper simultaneously performs the log-polar transformation, fan-out beam copying, and beam focusing. The dual-phase corrector corrects both the distorted phase from the log-polar mapping and the phase jumps generated from the beam copying process.

Fig.12 Numerical simulation and experimental results for the three-copy fan-out case of OAM sorting. (a) Simulation results show that different OAM modes (l = −2, −1, 0, 1 and 2) are sorted into a set of parallel lines with various vertical positions. (b) Experimental results verify the simulation results. All experimental images use the same scale bar and coordinates. Adapted from Ref. [39]

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The OAM sorter is demonstrated using a single SLM with the two-phase elements implemented on the two halves of the widescreen SLM. The simulation and experimental results for the three-copy fan-out case are presented in Figs. 12(a) and 12(b), respectively. It is clearly shown that different OAM modes (l = -2, -1, 0, 1, and 2) are sorted into a set of parallel lines with various vertical positions. The sorting resolution is proportional to the number of copies of the unwrapped beam. Because this high-resolution OAM sorter is composed of only two phase elements, it is very promising to be integrated in a compact device. In fact, a similar OAM sorter based on conformal mapping without this fan-out function has been demonstrated recently on one single piece of a refractive element [40].

With the use of SLMs, gradually changing-period gratings are demonstrated for the measurement of the OAM states from Hermite–Gaussian-like diffraction patterns [41]. Annular gratings are also reported to be capable of determining the sign and magnitude of the topological charge from the orientation and number of dark fringes [42]. The inhomogeneous and anisotropic q-plates made of liquid crystals have been exploited for the encoding and decoding in alignment-free quantum communication [43].

5 Conclusions

In conclusion, micro- and nano-structures have shown great potential in the design of compact and integrated OAM detectors. By the interaction of twisted light with plasmonic structures, PICs and novel diffractive elements unfold unprecedented properties that can be exploited to create on-chip devices capable of discerning OAM states with high fidelity. These new devices may find important applications in optical communications and quantum information processing with OAM states.

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