Spin-decoupled metasurface for simultaneous detection of spin and orbital angular momenta via momentum transformation

With inherent orthogonality, both the spin angular momentum (SAM) and orbital angular momentum (OAM) of photons have been utilized to expand the dimensions of quantum information, optical communications, and information processing, wherein simultaneous detection of SAMs and OAMs with a single element and a single-shot measurement is highly anticipated. Here, a single azimuthal-quadratic phase metasurface-based photonic momentum transformation (PMT) is illustrated and utilized for vortex recognition. Since different vortices are converted into focusing patterns with distinct azimuthal coordinates on a transverse plane through PMT, OAMs within a large mode space can be determined through a single-shot measurement. Moreover, spin-controlled dual-functional PMTs are proposed for simultaneous SAM and OAM sorting, which is implemented by a single spin-decoupled metasurface that merges both the geometric phase and dynamic phase. Interestingly, our proposed method can detect vectorial vortices with both phase and polarization singularities, as well as superimposed vortices with a certain interval step. Experimental results obtained at several wavelengths in the visible band exhibit good agreement with the numerical modeling. With the merits of ultracompact device size, simple optical configuration, and prominent vortex recognition ability, our approach may underpin the development of integrated and high-dimensional optical and quantum systems.


Section 1: A mathematical deduction of the PMT
The PMT that transforms vortex beams with different OAM modes into in-plane rotation of focusing patterns can be mathematically expressed as: where φ is the azimuthal angle, l is a topological charge of an OAM mode, Φ(φ) represents the phase profile of a mask, and Δφ is the rotation angle. After simple mathematical transformation, the above equation can be rewritten as: When the Δφ is infinitely small, the difference operation can be approximately taken as differential operation, i.e., Since the rotation angle Δφ is topological dependent, here we assume:

Section 2: A phenomenological explanation of the PMT
A typical azimuthal-quadratic phase profile is shown in Fig. S1(A). It is symmetric about φ l = 0 and changes rapidly with the azimuthal coordinates especially approaching ±π, where the equivalent wavevector becomes evanescent. By flattening the azimuthal-quadratic phase profile into a cylindrical-quadratic phase profile, one can easily grasp that such a phase profile will generate a focusing pattern at the azimuthal angle of φ l = 0 for a normal plane wave (i.e., l = 0). As indicated in Fig.   S1(B) and (C), when a light beam carrying an arbitrary OAM mode is normally projected on the phase mask, the symmetric direction of the total phase profile will rotate clockwise or anti-clockwise depending on the incident OAM modes. As a consequence, the propagation wave region and evanescent wave region will interchange, causing a rotation of the focusing pattern.

Section 4: Discussions on radius choice of the azimuthal-quadratic phase mask
To obtain fine focusing patterns through PMT, the radius of the phase mask should be optimized. Here we take a left-handed circularly polarized (LCP) vortex beam with a topological charge of l = 10 as an example. By exploiting the vectorial angular spectra theory, the focusing patterns through an azimuthal-quadratic phase mask with the same coefficient l 0 = 40 but different radius are presented in Fig. S3.
We can see that when the inner radius is 0 μm (i.e., a common circular shape phase mask) the intensity spreads along the azimuth and forms a sweat-heart shaped pattern.
Therefore, the most intensity is not well focused into a spot. By increasing the inner radius to 50 μm, the intensity spreading is gradually suppressed. With a further increment of the inner radius, the intensity spreading phenomenon emerges again.
Therefore, for obtaining a fine focusing pattern and improving the intensity utilization, the inner radius is selected as 50 μm when the outer radius is 80 μm. Note that, when the inner radius and outer radius are simultaneously scaled, the focusing pattern just scales with a ratio.

Measurement procedure:
The light from the He-Ne laser is modulated by a reflective SLM, which can generate anticipated vortex beams via computer generated holograms. We use an iris to pick up only one diffraction order and the combination of a linear polarizer (LP) and a quarter-wave plate (QWP) to generate LCP incidence. The generated LCP OAM is then focused by a lens so that its lateral size matches with the size of the metasurface based phase mask. When light passes through the metasurface, a 50× objective lens mounted on a 2D translation platform is utilized to magnify the optical patterns on the camera. The 2D translation platform is firstly tuned to make the metasurface located at the imaging plane of the objective lens so that we can check whether good alignment between the metasurface and OAM modes generated by SLM is achieved.
Subsequently, the imaging plane of the objective lens is changed to the focusing plane of the phase mask and another pair of QWP and LP is inserted before the camera to eliminate the co-polarization components.

Section 6: Diffraction efficiency of the fabricated geometric metasurface
To characterize the diffraction efficiency of the metasurface, a geometric phase metasurface-based beam deflector with a deflection angle of 15° is designed and fabricated. As indicated in Fig. S5(A), although the TiO 2 nanopillars are separated from each other, there are also some fabrication errors, which causes a degraded diffraction efficiency compared with the design. The measured diffraction efficiency around 633 nm is about 30%, as shown in Fig. S5(B). To quantitatively characterize the extent of mode overlap, we define the detection efficiency as the intensity fraction in the weak cross-talk region, indicated by the shadowed areas in Fig. S6. When Δl = 1, there is about 77.8% intensity overlap between them and only one peak appears at the combined intensity curve, which cannot be distinguished. With the expansion of mode interval from Δl = 1 to Δl = 6, the detection efficiency increases from 22.2% to 82.2%. Fig. 2(D) Fig. 2(D). The differences between the measured and theoretical azimuthal coordinate are no more than ±0.01 radian. In this situation, the peak intensity is still located within the weak cross-talk area of each mode (For a coefficient l 0 = 40, the azimuthal separation between the intensity peaks of two superimposed adjacent modes is 1/40 = 0.025 radian), as indicated by the shadow area. The correct ratio of detection within ±20 OAM modes with different position offsets. When the centre-misalignment is beyond ± 3 μm, the correct ratio drops below 90%.

Section 10: Simulated results of LCP vortices up to 100 orders at 633 nm
The intensity patterns for arbitrary OAM modes up to 100 orders with an interval of 1 are displayed in Fig. S9~S14. We can see that all the focusing patterns locate on a fixed circle without obvious pattern variation and there is only an azimuthal rotation for different OAM modes.