Multi-Foci Metalens

• Multi-Foci Metalens is a metasurface device that uses engineered subwavelength phase profiles to create multiple independent focal spots in one or more planes.
• Optimization algorithms, including inverse-design methods, minimize sidelobes and ensure uniform spot intensities with variations below 5%.
• Implementations use diverse platforms such as dielectric nanopillars, metallic nanostructures, and phase-change materials for advanced imaging and communication.

A multi-foci metalens is a metasurface optical device engineered to focus incident electromagnetic waves onto multiple, spatially distinct, and simultaneously addressable focal spots. Unlike conventional monofocal lenses, these ultrathin diffractive elements leverage subwavelength-scale phase modulation to achieve arbitrary control of both the number, position, and weighting (i.e., intensity) of focal points in one or more planes. The multi-foci paradigm encompasses pointwise arrays, longitudinal focal stacks, user-defined curves, and polarization- or wavelength-multiplexed spot sets, and is central to advanced imaging, multi-channel communications, trapping arrays, and information-processing applications across the electromagnetic spectrum.

1. Underlying Phase Profile Construction

The core principle of multi-foci metalens design is the synthesis of a spatial phase distribution $\phi(x,y)$ on the device surface so that the propagating field at the desired focal plane(s) reconstructs the specified set of focal spots. For $N$ distinct target foci at positions $(x_i, y_i, z_i)$, the ideal phase at each aperture position $(x, y)$ can be expressed as

$$\phi_{\text{req}}(x, y) = \mathrm{Arg} \left\{ \sum_{i=1}^N w_i \exp\left[ -j k_0 \left( \sqrt{(x - x_i)^2 + (y - y_i)^2 + z_i^2} - z_i \right) \right] \right\}$$

where $w_i$ controls the relative amplitude at focus $i$, $k_0=2\pi/\lambda$ is the free-space wavenumber, and $\lambda$ is the operational wavelength (Mao et al., 12 Mar 2025, Abdolali et al., 2020, Andreoli et al., 29 Oct 2024). This “superposed phase” prescription—sometimes referred to as the “direct argument method”—has been employed for both single-layer and multilayer metasurface realizations.

In application-specific variants:

• The direct argument is applied for spatial arrays in the focal plane, often with additional amplitude weights to normalize spot intensities (Mao et al., 12 Mar 2025, Abdolali et al., 2020).
• For longitudinal multi-foci, a composite of hyperbolic phase terms across focal lengths is used (e.g., tri-focal metalens) (Sun et al., 18 Feb 2024).
• In full-Stokes polarimetric systems, independent Jones-matrix profiles are optimized so that each of $M$ target polarization states is sent to its own designated focus, with the full phase vector designed via adjoint-based inverse optimization (Wang et al., 2023).

2. Physical Platforms and Meta-Atom Architectures

Multi-foci metalenses have been demonstrated using a diverse range of material stacks and meta-atom geometries, dictated by operational wavelength and application context.

• Dielectric nanopillar arrays: High-index (e.g., Ge, TiO$_2$, Si$_3$N$_4$) pillars with tailored height and lateral geometry are widely used for the visible through mid-IR (Wang et al., 2023, Sun et al., 18 Feb 2024, Mao et al., 12 Mar 2025, Hou et al., 27 Jul 2024). Si$_3$N$_4$ platforms support near-diffraction-limited, longitudinally multiplexed foci via nanofins with propagation phase and geometric phase tuning (Sun et al., 18 Feb 2024).
• Metallic nanoslits and nanorods: Gold nanoslits structured along confocal conics realize geometric-phase multi-foci operation in the visible and near-IR (Bao et al., 2016).
• Phase-change and reconfigurable materials: VO$_2$ nanofilms functioning as thermally/electrically/optically triggered phase shifters enable real-time, digitally reconfigurable multi-foci operation at THz frequencies, exploiting their insulator-to-metal phase transitions for high-speed switching (Abdolali et al., 2020).
• Atomic emitter arrays: Theoretical proposals show lattices of cold atoms can be engineered, via spatially varying interatomic spacings across stacked 2D arrays, to yield phase profiles that realize arbitrary multi-foci lensing with robust, low-loss characteristics (Andreoli et al., 29 Oct 2024).
• Hybrid meta-atoms: Polymer resist pillars with conformal atomic-layer-deposited TiO$_2$ coatings achieve full $0$–$2\pi$ phase coverage, polarization insensitivity, and low-loss uniform focal arrays suitable for scalable nanoimprint fabrication (Mao et al., 12 Mar 2025).

3. Optimization Algorithms and Uniformity Control

Direct phase superposition,

$$\phi_{\rm super}(x,y) = \mathrm{Arg} \left[ \sum_{j=1}^N \exp(i \phi_j(x, y)) \right],$$

yields nonideal performance for large $N$ due to strong sidelobe formation, non-uniform spot intensities, and cross-talk (Mao et al., 12 Mar 2025).

To address this, inverse-design strategies based on gradient-descent optimization, angular-spectrum propagation, and adjoint-field methods are employed. The loss is constructed to minimize the squared field error at each focus, enforce uniformity among peak intensities, and suppress off-target light:

$$\mathcal{L} = W_{\text{foc}} \frac{1}{N}\sum_{j=1}^{N} \left| |U(x_{j},y_{j},f)| - U_{\text{target}}(x_{j},y_{j}) \right|^2 + W_{\text{non}} \frac{1}{P} \sum_{(x,y)\notin\{x_j,y_j\}} \left| U(x,y,f) \right|^2$$

where $U(x, y, f)$ is the propagated field at focus, and optimization is performed over the phase profile $\phi(x, y)$. This approach yields focal arrays with standard deviation $\sigma < 5\%$ in intensity and sidelobes $< 10\%$ of the main peak, outperforming phase superposition in both contrast and fidelity (Mao et al., 12 Mar 2025, Wang et al., 2023).

4. Reconfigurable, Electrically Tunable, and Polarization-Multiplexed Multi-Foci

Reconfigurability and multi-channel selection are achieved via several mechanisms:

• Electrically tunable layering: Cascading $N$ polarization-multiplexed bi-focal metalenses interleaved with voltage-controlled nematic liquid crystal (LC) waveplates yields $2^N$ independently selectable focal channels. The parabolic phase of each layer adds reciprocally, setting the effective focal length:

$1/f_e = \sum_{j=1}^{N} 1/f_j^{(u_j)}$

where $u_j\in\{|D\rangle,|A\rangle\}$ is the polarization state at layer $j$. Efficiency up to $10\%$ and FWHM down to $19\,\mu$m are achieved with eight-channel switches in a 6-mm-thick multilayer stack (Ma et al., 16 May 2025).

• Polarization control: Dual- or multi-foci can be partitioned across orthogonal linear or circular polarization channels, such that, for instance, each co- and cross-polarized transmission carries a different focal phase: $$\phi_{\text{co}}(x,y;\lambda)\approx\phi_1(x, y; \lambda)$$ and $$\phi_{\text{cross}}(x,y;\lambda)\approx\phi_2(x, y; \lambda)$$. This enables simultaneous and independent point formation for multiple orthogonal input states, with theoretical efficiency per channel up to $80.5\%$ (Hou et al., 27 Jul 2024), or four-foci full-Stokes polarimetric routing with $54.6\%$ total efficiency (Wang et al., 2023).
• Phase-change materials: VO$_2$-based reconfigurable metalenses permit arbitrary and real-time adjustment of the number, position, intensity, and width of focal spots, with sub-ns to sub-s switching speeds depending on bias modality (Abdolali et al., 2020).

5. Fabrication Technologies and Scalability

Large-area, wafer-scale, and scalable multi-foci metalens manufacturing can be realized by

• Nanoimprint lithography (NIL): Combined with atomic layer deposition (ALD) for conformal high-index coatings, NIL enables batch fabrication of polarization-insensitive, highly uniform focal arrays ($\sigma < 3.5\%$) across arbitrary geometries, with measured spot FWHM within 1.1$\times$ the diffraction limit and $70\%$ transmission (Mao et al., 12 Mar 2025).
• Electron-beam lithography and reactive-ion etching: Provide sub-20 nm fidelity in high-index dielectrics (e.g., Si$_3$N$_4$, Ge), critical for phase accuracy and spot Strehl ratio (Sun et al., 18 Feb 2024, Wang et al., 2023).
• Focused ion beam (FIB) milling: Used for fabricating metal nanoslit arrays in Au, achieving geometric-phase encoding for conic-shaped multifocal lenses (Bao et al., 2016).
• Layered atomic arrays: Theoretical designs recommend stacking three sub-wavelength 2D atomic lattices, with precise spatial control over lattice constant and interlayer spacing, to realize the target phase and maximize cooperative transmission (Andreoli et al., 29 Oct 2024).

Critical process parameters include aspect-ratio control, resist viscosity, anti-adhesion measures, and ALD uniformity for pillar-based meta-atoms, as well as subwavelength alignment and retardance uniformity for polarization-multiplexed and electrically tunable devices (Mao et al., 12 Mar 2025, Ma et al., 16 May 2025).

6. Performance Metrics and Application Domains

Multi-foci metalenses are characterized by focusing efficiency, spot uniformity, crosstalk, speed, chromatic performance, and fabrication scalability. Representative metrics include:

• Efficiency: Up to $80.5\%$ per channel in dual-focal, broadband, polarization-insensitive metalenses (Hou et al., 27 Jul 2024); $\sim$30–40% for THz VO$_2$-based reflection metalenses (Abdolali et al., 2020); $10\%$ in electrically tunable 8-channel stacks (Ma et al., 16 May 2025).
• Uniformity: Intensity variation $\sigma < 5\%$ across multiple foci (Mao et al., 12 Mar 2025, Wang et al., 2023).
• Spot size and Strehl ratio: FWHM within 10–15% of the diffraction limit; Strehl ratios $\gtrsim$0.8 for all foci in Si$_3$N$_4$ tri-focal devices (Sun et al., 18 Feb 2024).
• Crosstalk: Inter-focus isolation $\geq10$ dB for foci separated by more than twice the FWHM (Abdolali et al., 2020).
• Switching speed: Picosecond to sub-second range for phase-change devices; milliseconds for liquid-crystal switching (Abdolali et al., 2020, Ma et al., 16 May 2025).
• Scalability: Large-area wafer-imprinted arrays for trapping, imaging, and optical tweezers; compact, monolithic integration in infinity-corrected, multi-magnification microscopes (Sun et al., 18 Feb 2024, Mao et al., 12 Mar 2025).

Core application areas include multifunctional optical trapping arrays, AR/VR depth scanning, LiDAR and 3D sensing, parallel nanolithography, multi-channel terahertz or wireless communication, full-vector polarimetric imaging, and compact, microscopy-grade objective lenses with tunable or multi-plane imaging capability.

7. Extensions, Limitations, and Outlook

• Extension to arbitrary focal patterns: Inverse-Fourier and propagation-phase techniques permit the design of continuous focus curves or 2D patterns (e.g., lines, rings, alphanumeric shapes) subject to the constraints of sampling pitch, NA, and propagation-induced amplitude variation (Ye et al., 2019).
• Longitudinal multi-foci: Multiple focal planes with engineered relative magnifications and longitudinal spacing can be realized by harmonic-phase superpositions or polarization-multiplexing (Sun et al., 18 Feb 2024).
• Achromatic and broadband design: Dispersion-engineered meta-atom libraries and global optimization (e.g., particle swarm) enable achromatic, polarization-independent multi-foci metalenses with sub-4% focal length deviation across broad visible bands (Hou et al., 27 Jul 2024).
• Limitations: Current challenges include chromatic and polarization crosstalk in complex multiplexing, efficiency loss in multi-layer stacks (glass/alignment losses), spot uniformity for $N \gg 10$ without inverse design, fabrication tolerances ($<$10-20 nm) for high-NA, and limitations in simultaneous bandwidth, efficiency, and focus count (Ma et al., 16 May 2025, Mao et al., 12 Mar 2025).
• Atomic array approaches: Offer the prospect of ultra-low-loss, robust multifocal metasurfaces with efficiency $\sim$80–90%, provided the subwavelength lattice engineering is experimentally achievable (Andreoli et al., 29 Oct 2024).

In concluding, the multi-foci metalens framework unifies a broad class of metasurface-based focusing devices under a phase-engineered, subwavelength-thick paradigm, with demonstrated scalability, reconfigurability, and application reach spanning classical and quantum photonics (Mao et al., 12 Mar 2025, Abdolali et al., 2020, Sun et al., 18 Feb 2024, Andreoli et al., 29 Oct 2024, Ma et al., 16 May 2025).