Introduction

Multitasking shared-aperture systems have initially emerged as phased array radar antennas1, 2, 3. Recently, the shared-aperture concept has been suggested as a platform for multifunctional optical phased array antennas, accomplished by a reflective metasurface4. Metasurfaces consist of metallic or dielectric subwavelength nanoantennas, capable of manipulating light by controlling the local amplitude and phase of an incident electromagnetic wave5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17. Shared-aperture interleaved phased arrays are formed by the random interspersing of sub-arrays, thus resulting in a device with high flexibility in multifunctional wavefront generation and the angular resolution of the shared aperture. Each sub-array is associated with a specific phase function, sparsely sampled at randomly chosen lattice points. The random sampling as well as the antennas’ size affect the dominant mechanism responsible for creating an undesired background noise—for subwavelength nanoantennas, speckle noise appears, whereas for larger antennas, most of the noise is attributed to sidelobes2, 4. The noise imposes a fundamental limitation to the system’s information capacity. Therefore, knowledge of such limitations can guide the design of optimal devices based on multifunctional metasurface. Here the interleaved geometric-phase metasurface (GPM) served as a platform to study those limitations.

The shared-aperture GPM is obtained by the incorporation of the geometric-phase mechanism into interleaved structures, which enables spin-control capabilities. GPMs are composed of anisotropic nanoantennas that generate a local phase delay, corresponding to an orientation function φg (x, y)=2σθ (x, y)6, 7, 12, 18, 19. Here θ(x, y) is the in-plane nanoantenna orientation and σ=±1 denotes the polarization helicity, that is, right (σ+) or left (σ−) circular polarization, respectively. Due to the interaction with the nanoantennas of varying orientations, the light’s polarization state attained at different points traverses various paths on the Poincaré sphere. This results in a Pancharatnam–Berry phase pick-up between any pair of nanoantennas, which is equal to half the area of the corresponding geodesic triangle Ω on the Poincaré sphere7, 18 (Figure 1b).

Figure 1
figure 1

Interleaved GPM concept. (a) Sparse geometric-phase profiles and the resultant interleaved phase (right panel). Left panel depicts the corresponding SEM images of the interleaved GPM. (b) Illustration of the geometric phase pick-up principle via the Poincaré sphere. The phase pick-up φg is equal to half the area of the geodesic triangle Ω. (c) Measured diffraction efficiency at various wavelengths for a linear phase GPM (red circles), and the corresponding FDTD simulation (green) and effective medium theory calculation (purple). Right inset illustrates the designed Si-BB oriented at an angle θ, where h=260 nm, L=600 nm, t=70 nm and Λ=200 nm. Left inset depicts the measured far-field intensity distributions of the linear phase GPM for σ+ and σ− illumination at a wavelength of 630 nm. (d, e) Measured spin-flip momentum deviation of different optical angular momentum wavefronts, where σ denotes the incident spin and k0 is the incident wave number. Insets show the interference patterns of a plane-wave with the vortex beams for σ− and l=1,2,3, verifying the desired topological charges. (f) Calculated mutual information (blue line, left axis) and measured total number of bits (red dots, right axis) for a varying number of channels (N); dashed black line depicts the distinguishability limit for the generated channels. Inset shows the calculated relation between the number of nanoantennas in each of the lattice’s dimensions (n) and the channel capacity (C). FDTD, finite-difference time-domain; SNR, signal-to-noise ratio.

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Materials and methods

Fabrication of Si-GPM

The poly-Si thin film with a thickness of 260 nm was grown at a temperature of 600 °C on a SiO2 (fused silica) substrate. The photoresist CSAR 6200.09 (Allresist GmbH, Strausberg, Germany) with a thickness of 190 nm was deposited on the poly-Si film and baked at 150 °C for 1 min. For the pattern transfer onto the poly-Si film, the photoresist mask was made using Raith e-LiNE E-beam lithography system (Raith, Dortmund, Germany), then baked at 130 °C for 1 min and etched using deep reactive-ion etching by F-ICP Plasma Therm system (Plasma-therm, Saint Petersburg, FL, USA) for 30 s.

Metasurface design parameters

Figure 2 shows the scanning electron microscopy (SEM) image of a resultant interleaved GPM with the designed Si-building blocks (BBs). Each Si-BB is composed of several nanorods, where their number is determined according to the orientation angle θ so as to fill an area of 600 × 600 nm2. The nanorods are of 70 nm width and 260 nm depth, arranged 200 nm apart from each other (center to center, see also Figure 1e). The metasurface diameter was of 50 μm. These parameters were maintained for all the fabricated metasurfaces.

Figure 2
figure 2

SEM image of the interleaved Si-based GPM.

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Experimental set-up

Figure 3 depicts the experimental set-up used in all of our measurements. The input beam, generated by a supercontinuum laser source (Fianium Supercontinuum SC450, Fianium, Southampton, UK) was temporally modulated by an acousto-optic modulator (Fianium AOTF V1, Fianium, Southampton, UK) facilitating the use of multi-wavelength laser light. The beam was then spatially filtered and collimated. A non-polarizing beam splitter followed by a reflection mode phase only spatial light modulator (Hamamatsu PPM X8267, Hamamatsu, Shizuoka Pref. Hamamatsu City, Japan) were used in order to generate the helical phase profiles needed for the optical angular momentum (OAM) spectropolarimeter metasurface (OSPM) experiments (Figure 6). The beam’s polarization state was set using a polarization generator (linear polarizer followed by a quarter wave plate), and the beam was subsequently focused on the sample. The diffracted light was collected using an objective lens (Olympus SLMPlan 50 × 0.45, Olympus Corporation, Shinjuku, Tokyo, Japan) and the obtained magnified image was then Fourier transformed using a second lens. The output beam’s polarization state was analyzed using a polarizer–analyzer (Figure 3a) or two linear polarizers placed parallel and set at a relative angle of 45° (Figure 3b; for the OSPM experiments). The obtained images were taken using a RGB CMOS camera (Ximea MQ022CG-CM, Ximea Corp. Münster, Germany).

Figure 3
figure 3

Experimental set-up. (a) A single polarizer used in the optical rotatory dispersion and spectropolarimetry experiments. (b) Two linear polarizers placed parallel and set at a relative angle of 45° for the optical angular momentum spectropolarimeter metasurface experiments. The focal lengths for the two lenses are 80 and 50 mm, respectively. BS, beam splitter; Pol., polarizer; QWP, quarter wave plate; SLM, spatial light modulator.

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Stokes parameters

In order to determine the Stokes parameters, the detected pattern upon a CMOS camera should be provided with the intensities and IL45, IL0. The spin-dependent beams determine the intensities , whereas the spin-independent beams, projected on a linear polarizer, determine the linearly polarized intensities IL0 and IL45, respectively. The Stokes parameters are then calculated by the following expressions: ; ; and S1=2IL0/η−S0; S2=2IL45/η−S0, where η is determined by a calibration experiment20.

Results and discussion

We designed an efficient dielectric GPM, using a Si-based BB, constructed of several nanorods that operate as an anisotropic π retarder. The BBs were arranged in a square array with a lattice constant of 600 nm (see Materials and Methods). The realized GPM provides a maximal diffraction efficiency of ~85%, which was measured for the transmission mode of a single channel, in agreement with an effective medium theory calculation and FDTD simulation (Figure 1c). Note, the imaginary part of the refractive index of the Si was taken into account by the FDTD calculations12.

In order to demonstrate the interleaving concept, we designed a shared-aperture Si-GPM (Figure 1a) generating multiple off-axis wavefronts carrying OAM (scalar vortices). Therefore, the BB’s orientation was arranged according to the relation 2θj(x, y)= (k·r+lϕ)j, to obtain a linear momentum redirection of σkj and scalar vortices of topological charge σlj, where ϕ is the azimuthal angle and rj=(x, y)j is the position of each BB at the jth sub-array. We observed spin-dependent OAM carrying wavefronts with the desired topological charges of ±1, ±2 and ±3 at a wavelength of 630 nm (Figure 1d and 1e). Moreover, diffraction-limited spot-sizes corresponding to the GPM’s full aperture were confirmed (Supplementary Figs. S1 and S2).

The limitation of the number of modes (channels) that can be generated by an interleaved GPM is analyzed in terms of information capacity by means of the Wigner position-momentum (x, k) distribution in phase-space21. The spin-dependent Wigner function of a shared-aperture GPM is represented by , where is the phase function corresponding to N interleaved channels for σ=±1 illumination. We estimate the GPM’s channel capacity (C), that is, the number of distinct channels that can be opened, utilizing the spin-dependent mutual information (MI), , where and are the marginal density functions in position and momentum space (Supplementary Fig. S3), respectively22, 23. Figure 1f depicts the numerically calculated MI for an interleaved GPM size of 40 × 40 μm2, where the total number n × n of nanoantennas is 80 × 80. The number of bits of information carried by all channels N log2 (1+SNR) was obtained via measurements of the signal-to-noise ratio(SNR) for each channel as a function of the number of channels, N (Figure 1f; Supplementary Fig. S1). These results are in agreement with the relation MI≈N log2 (1+SNR) as predicted by information theory24. The GPM’s channel capacity is bounded by the external condition where each channel is allocated with a single bit (SNR=1). Thus, a value of can be estimated from Figure 1f, as well as from the relation C=sup{MI}. Our simulations show that the channel capacity is also affected by the total number of nanoantennas and was found to increase according to (Figure 1f, inset), wherein the fourth power evidently equals the phase-space dimensionality. Note, the intensity per channel decreases according to 1/N2, as a result of speckle noise4 (Supplementary Fig. S1). The introduced information capacity analysis enables to choose a certain shared-aperture metasurface dimension and amount of nanoantennas to achieve a high SNR in demultiplexing light in different frequencies, polarization states and OAM by packing several functions into a single area. For a metasurface demultiplexing four channels and 5 bits dynamic range imaging system, we estimated the GPM parameters to be n≈80, and D≈50 μm for 600 nm size of BB, providing the required 20 bits of information capacity (Figure 1f, inset). Consequently, we implemented GPMs with such parameters that operate within the fundamental limits.

GPMs enable space-variant polarization manipulation, producing a broad class of wavefronts such as vectorial vortices. This class of complex beams, signatured with a polarization singularity, is of special interest for optical fiber communication systems, optical tweezers, tractor beams and laser beam shaping25, 26. Interleaved GPMs facilitate the generation of multiple vectorial vortices by coherent superposition of multiplexed scalar vortices with opposite helicities (Figure 4a). This superposition is attained by illuminating the GPM with linearly polarized light. The resultant beams are characterized by axially symmetric polarization states. Therefore, projecting these vectorial vortices on a linear polarizer reveals a petal-shaped diffraction pattern. A change in the incident linear polarization angle is manifested by a constant phase difference between the right and left circular polarizations. Hence, the fringes of the pattern rotate when the input linear polarization angle is altered. This phenomenon can be utilized to measure the specific rotation of a chiral molecule, which enables, for example, the distinction between two enantiomers at a certain wavelength. Moreover, knowledge of the specific rotation’s dispersion provides more detailed characterization of optically active materials, which has numerous applications in medicine, as well as the pharmaceutical and food industries.

Figure 4
figure 4

ORD measurement by interleaved GPM. (a) Schematic of vectorial vortex beams with winding numbers l=1 and l=2 emerging from a GPM (SEM image) illuminated with linearly polarized light; red and blue helices represent scalar vortices of opposite helicities and OAMs. (b–d) Observed diffraction patterns of interleaved vectorial vortices generated by using linearly polarized light passing through (S)-limonene (b), air (c) and (R)-limonene (d) at wavelengths of 550, 630 and 725 nm, with a fixed angle polarizer–analyzer. (e) Enlarged spots (corresponding to b–d, top down) illustrating the wavelength-dependent rotation of the vectorial vortices for (S)- and (R)-limonene. (f) ORD measurements of (S)- and (R)-limonene, represented by red circles and black squares, respectively. Insets show the corresponding chemical structure. ORD, optical rotatory dispersion.

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Interleaved GPM under polychromatic illumination, which generates multiple off-axis vectorial vortex beams, enables real-time measurement of optical rotatory dispersion (ORD; see Supplementary Information). To this end, we realized a GPM consisting of four interleaved phase profiles and . The GPM of 50 μm diameter was placed between two orthogonal linear polarizers (see Materials and Methods for the experimental set-up) and vectorial vortex beams with winding numbers of 1 and 2 were obtained. When (S)- and (R)-limonene solutions were placed after the first polarizer, wavelength-dependent rotation of the fringes was observed (Figure 4b–4d). Moreover, the ORD of (S)-limonene manifested opposite behavior compared to its enantiomer, (R)-limonene (Figure 4e and 4f), in agreement with the literature27.

Interleaved dielectric metasurface enables the implementation of a multitasking device for spectropolarimetric measurements, to instantaneously determine the polarization state of light at different wavelengths. Recently, metasurface-based polarimetry20, 28, chiroptical spectroscopy29 and reflective spectropolarimetry based on gap-plasmon nanoantennas4 have been presented. Here we present a simple and direct spectropolarimetry approach based on a transmissive interleaved Si-based GPM. The spectropolarimeter metasurface (SPM) is composed of three interleaved phase profiles; first, a chiroptical channel of phase function , enabling circular dichroism spectroscopy by performing a projection of the incident wave’s polarization state |ψ〉 on the circular polarization component 〈σ|ψ〉. The projection on the linear polarization components, 〈L0|ψ〉 and 〈L45|ψ〉, is obtained by two additional binary phases and , each providing two spatially separated diffraction orders, where Uπ(ξ)={π, for ξ>0; 0 otherwise}. Note, the additional phase shift between left- and right-circular polarization (σπ/8) rotates an incident linear polarization by 45°, thus facilitating the use of a single polarizer–analyzer (Figure 5a). The SPM of 50 μm diameter was normally illuminated by a supercontinuum light source that was passed through an acousto-optic modulator and a polarization state generator. In this manner, predetermined polarization states were measured at wavelengths of 550, 590, 630, 660 and 725 nm, spanning the Stokes parameters basis (s0, s1, s2 and s3) as depicted on the Poincaré sphere (Figure 5b–5f; see Materials and Methods). Moreover, a spectral resolving power of was measured by taking the far-field profile (Figure 5g), in agreement with the Rayleigh criterion of the shared-aperture.

Figure 5
figure 5

Si-based SPM. (a) Schematic set-up of the SPM illuminated by an elliptically polarized polychromatic light source; the projection on different polarization components is then captured by a charge-coupled device camera. (b–f) Measured far-field intensities for different input polarizations: σ+ (b), σ− (f), linear at 0° (c) and linear at 45° (e), for wavelengths of 550, 590, 630, 660 and 720 nm, also illustrated on the Poincaré sphere (d). (g) The corresponding measured spectrum (black) and the spectral resolving power (red) obtained with an SPM of 50 μm diameter. SPM, spectropolarimeter metasurface.

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Information-carrying OAM beams may leverage optical communications and quantum information processing30, as they introduce an additional degree of freedom for increasing the capacity of free-space communications. For data transmission to capitalize on this additional degree of freedom requires the detection, characterization and sorting of the OAM optical modes. Several techniques have been suggested to measure the topological charge of a structured optical wavefront31, 32. We incorporate the spectropolarimetry and OAM sensing into a GPM device that would ultimately facilitate a simultaneous detection of the frequency, polarization and OAM of light. The determination of the last parameter is achieved by the projection of an incident mode |lin〉 on a set of orthogonal OAM states |lj〉, which indicates the location of the incident OAM in the set. For this purpose, we designed a shared-aperture GPM based on the harmonic response (HR) approach4. Here the HR phase function is expanded according to , resulting in a finite number of dominant multiplexed OAM harmonic orders with identical intensities Al, where lk is the momentum redirection of the lth order. The phase function was optimized to achieve maximal diffraction efficiency for −2≤l≤2 (Ref 33). By modifying the HR function, the geometric-phase profile was obtained, generating spin-dependent dispersive OAM wavefronts possessed with circularly polarization states, ranging from l=−2 to 2 (Figure 6a and 6b; see also Supplementary Fig. S4). We utilized the HR structure to achieve a multifunctional OSPM. This metasurface modifies the phase of an incident wavefront by interleaved HR phase profiles and . Illuminating the OSPM with a plane-wave of elliptic polarization results in four symmetric sets of annular spots with a bright spot at the center, whereas those generated by are transferred through linear polarizers at 0° and 45° to determine the linearly polarized components (Figure 6c; see Materials and Methods for the experimental set-up). Given an incident beam possessing an arbitrary helical phase exp (ilinϕ), the winding numbers of the diffracted orbital harmonics are modified by accumulating the value of the beam’s topological charge lin. Particularly, this modification shifts the bright spot from its original location l=0 to l=−lin, enabling the determination of the incident OAM value. We demonstrate the OSPM capabilities by illuminating it with various OAM modes, which were generated by external phase modulation at different polarization states, and wavelengths (Figure 6d–6i).

Figure 6
figure 6

Si-based OSPM. (a, b) Measured spin-flip momentum deviation of OAM harmonic wavefronts, demonstrating five off-axis spin-dependent scalar vortices at wavelengths of 550 and 670 nm; σ± denotes the incident spin. (c) Schematic set-up of the OSPM of 50 μm diameter illuminated by an arbitrarily polarized polychromatic light source. The image captured by the charge-coupled device camera was obtained for an elliptically polarized beam at λ1,2,3=550, 590 and 633 nm with l=0. (d–i) Measured far-field intensities for different wavelengths, polarizations and OAMs. Note, |λ, P, l〉 stands for the incident beam state, representing the wavelength, the polarization and the OAM, respectively.

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Conclusions

The shared-aperture interleaved GPM paves the way for multimodal real-time detection devices, integrated-on-chip, for multi-parameter light characterization and manipulation of light–matter interactions. The ability to encode, decode and measure the fundamental properties of light may find applications in quantum information processing, communication and computation systems. The introduced analysis of the fundamental limitations for multiplexed phased antenna arrays can also have an impact on the design of radio frequency antenna systems.