Frequency-Tunable Multiplexing Lens

• The paper demonstrates the use of advanced metamaterials and MEMS to achieve frequency-dependent focusing by tailoring phase profiles for each channel.
• It employs multiplexing modalities such as frequency-division, polarization, and material-state switching to enable simultaneous, low-crosstalk channel routing.
• Numerical optimization and integrated photonic circuits underpin the scalable design, with applications in plasma diagnostics, imaging, communications, and quantum information processing.

A frequency-tunable lens with multiplexing capability is an optical or electromagnetic device whose focal length and spatial focusing behavior depend functionally on the operating frequency, and which is engineered to precisely route, focus, or process multiple frequency channels in parallel. Modern research draws on metamaterials, metasurfaces, MEMS, phase-change materials, magneto-optics, and advanced beam steering platforms to realize such lenses across microwave, THz, infrared, and visible regimes. Multiplexing capability typically refers to the simultaneous or reconfigurable focusing or routing of several frequency or polarization channels, enabling parallel sensing, communication, or information processing applications.

1. Principles of Frequency-Dependent Focusing

Frequency-tunable focusing in lens systems relies on imparting a frequency-dependent phase profile on incident wavefronts, so the waist (focus) of the resulting Gaussian beam or EM mode is located at a prescribed distance for each frequency channel. In metamaterial implementations, the lens is constructed from concentric annular zones, each composed of miniaturized phase-shifting unit cells designed to provide a phase shift

$$\varphi(n, f) = \frac{\pi f}{c\, R_\text{adj}(f)}\, r_n^2 + \varphi_0(f)$$

where $r_n$ is the radial coordinate of the nth zone, $f$ is the frequency, $R_\text{adj}(f)$ is the adjusted radius of curvature calibrated for diffraction/aperture effects, and $\varphi_0(f)$ is an arbitrary offset. The overall lens creates a frequency-dependent focal length ($\ell(f)$), yielding reverse chromatic aberration (RCA)—the focal length increases with frequency. For example, in (Hammond et al., 2014), a metamaterial lens for DIII-D tokamak diagnostics achieves focal lengths from 1.37 m to 1.97 m across frequencies 83–130 GHz, directly mapping spectral channels to their emission layer locations.

Metasurface approaches similarly engineer subwavelength patterns (e.g., silicon or phase-change nanodisks) whose diameter or material state selects the local phase shift (spanning up to $2\pi$), with overall phase profiles matching classic lens equations for targeted frequency-dependent focusing at micro- and nano-scale (Afridi et al., 2019, Arbabi et al., 2017).

2. Multiplexing Modalities

Multiplexing is realized via spatial, temporal, frequency-division, polarization, or material-state degrees of freedom:

• Frequency-Division Multiplexing (FDM): Simultaneous routing or focusing of multiple frequencies, each to a spatially distinct region (e.g., emission layers in a tokamak). The lens phase profile is reverse-engineered such that each frequency channel has a different focal waist position (Hammond et al., 2014, Wang et al., 2017, Smith et al., 2023).
• Polarization Multiplexing: Metasurfaces patterned with polarization-sensitive elements (e.g., rods and crosses) support bifocal or multifocal operation, where orthogonal input polarizations see different effective lens phase profiles, producing distinct foci (Markovich et al., 2017, Shamuilov et al., 2020).
• Material-State Multiplexing: In metasurfaces based on phase-change materials (e.g., GST, VO₂), writing/erasing patterns with light or temperature dynamically reconfigures the device's function, enabling switching between multi-focus, wavelength-dispersive, or chromatically corrected lens states (Wang et al., 2015, Kargar et al., 2019).
• Spatial Beam Steering Arrays: Lens-assisted beam-steering platforms (LABS) integrate photonic switching and collimation to route frequency-stable light to N spatially separated nodes, as seen in free-space optical networks with up to 16 simultaneous outputs (Hu et al., 2022).
• Microwave-to-Optical Conversion: Atomic vapor cells leverage Doppler-broadened spectra to achieve highly tunable (>500 MHz) conversion between microwave and optical frequency channels, with simultaneous handling of multiple input channels and amplitude-phase correlated control (frequency-domain beam splitting) (Smith et al., 2023).

3. Device Architectures and Optimization Methodologies

Designing frequency-tunable multiplexing lenses requires:

• Unit Cell Database and Target Functions: For metamaterial lenses, a database of simulated unit cells (with tunable geometric parameters) is created. For each annular zone, a goal function measures deviation from a target phase shift/transmittance over benchmark frequencies, e.g.,

$$G(g, w, \varphi_t) = \sum_{i=1}^{6} \frac{[\delta\varphi(g, w, f_i) - \varphi_{t1}(f_i)]^2}{90} + \frac{[T(g, w, f_i) - T_t]^2}{T_t}$$

where ($g, w$) are dimension parameters, $f_i$ frequencies, $\delta\varphi$ simulated phase, $T$ transmittance (Hammond et al., 2014).

• Numerical Optimization: Full-wave EM simulations and iterative refinement are deployed to minimize aberrations and meet design targets. Deviations of 5–7% in focal position are typical when fine-tuning unit cell arrangements for multi-frequency alignment (Hammond et al., 2014).
• Integrated Multiplexing Circuits: Photonic switch chips (e.g., silica 1×16 switches) connect to fiber arrays and lenses for multiple output beams, leveraging geometric beam steering and monolithic transceiver arrangement to minimize phase noise (Hu et al., 2022).
• MEMS Actuation: For ultrafast varifocal metasurfaces, electrostatic actuation of submicron-separated dies enables >60 diopters focal length tuning for micron-scale displacement, with kHz modulation bandwidth and (potentially) chip-level integration of multiplexed arrays (Arbabi et al., 2017).

4. Demonstrated Performance and Physical Constraints

Empirical results indicate:

• Resolution and Alignment: In DIII-D, the spatial accuracy of ECE layer mapping is within a fraction of a Rayleigh length ($\ll \lambda$), critical for high-resolution plasma diagnostics (Hammond et al., 2014).
• Transmission and Aberration Control: Uniform, high transmittance is prioritized across all zones; finite-aperture diffraction and pass-band shifts remain challenges for wideband performance, with lower-frequency zones showing transmittance drops under certain pass-band conditions (Hammond et al., 2014, Wang et al., 2015).
• Multiplexing Isolation: In RL-SD spectrum decomposers, measured isolation between output channels is typically ≳10 dB, with minimal crosstalk enabled by calibration arrays and port sampling (Wang et al., 2017).
• Dynamic Range and Speed: MEMS metasurface lenses achieve 15–23% tuning of focal length (exceeding Rayleigh length) within response times of 100 ms (thermal, (Afridi et al., 2019)) to 2.6–5.6 kHz resonance frequencies (mechanical, (Arbabi et al., 2017)).
• Frequency Conversion Bandwidth: Atomic vapor solutions offer up to 550 MHz tunability in microwave-to-optical conversion, compatible with quantum information multiplexing applications and frequency-bin encoding (Smith et al., 2023).
• Fractional Frequency Instability: Free-space LABS achieves fractional frequency instability values of $4.5 \times 10^{-17}$ (1 s averaging) and $7.7 \times 10^{-20}$ (20,000 s averaging) for optical time-frequency transfer across 50 m links, maintaining metrological stability for clock networks (Hu et al., 2022).

5. Applications in Sensing, Imaging, Communications, and Quantum Domains

Frequency-tunable multiplexing lenses find integration into diverse domains:

• Plasma Diagnostics: ECE radiometry in tokamaks benefits from frequency-resolved spatial mapping for monitoring thermal emission layers without mechanical adjustment (Hammond et al., 2014).
• Millimeter Wave Communications: Path division multiplexing (OPDM) exploits lens arrays to focus each propagation path to dedicated RF chains, reducing hardware complexity and achieving near-eigenmode MIMO capacity (Zeng et al., 2015).
• Adaptive Imaging: MEMS-integrated metasurface lenses with varifocal tuning allow chip-scale 3D imaging, fiber-tip microscopes, and beam scanning, bypassing the inertia and slow speed of traditional varifocal optics (Arbabi et al., 2017, Afridi et al., 2019).
• THz Imaging and Holography: VO₂ coding metasurfaces and bifocal polarization-sensitive metasurfaces provide dynamic beam steering, multiplexed focusing, and support encrypted or reconfigurable optical coding across GHz–THz–visible ranges (Kargar et al., 2019, Markovich et al., 2017, Wang et al., 2015).
• Optical Time-Frequency Networks: Integrated lens-antenna beam steering enables multi-user, chip-scale, high-stability frequency transfer for clock distribution and metrology (Hu et al., 2022).
• Quantum Information Processing: Frequency-division multiplexed atomic vapor interfaces support coherent conversion and manipulation of frequency-bin qubits, amplitude-phase control of multi-channel quantum information, and serve as quantum transducers bridging microwave and optical domains (Smith et al., 2023, Hiemstra et al., 2019).

6. Limitations, Scalability, and Prospective Developments

Key challenges and future directions include:

• Aperture/Finite Size Effects: Diffraction from finite apertures or cell arrays leads to displacement of beam waists and undesired sidelobes, necessitating optimization of meta-atom density and aperture control (Hammond et al., 2014, Kargar et al., 2019).
• Material Response Constraints: Phase-change materials (VO₂, GST) introduce hysteresis and uniformity concerns for large-scale multiplexed arrays; scaling up requires precise temperature or light management (Kargar et al., 2019, Wang et al., 2015).
• Speed and Power Trade-Offs: MEMS and thermo-optical designs are limited by air/vacuum damping and thermal diffusion, with millisecond to microsecond response times, and require efficient actuation for multiplexing at high bandwidth (Arbabi et al., 2017, Afridi et al., 2019).
• Cross-Talk and Channel Isolation: Polarization and frequency multiplexed metasurfaces must minimize inter-channel crosstalk (meta-atom resonance separation, material anisotropy engineering) for robust performance (Markovich et al., 2017, Wang et al., 2017).
• Ultrafast Tunability: Magneto-optic OMLs present picosecond focal length tuning, expandable to multi-functional lens-grating-hologram operations via tailored magnetic field profiles and advanced 2D materials (Shamuilov et al., 2020).
• Generalization: The numerical optimization, calibration, and port-sampling methods are applicable to other lens architectures, including optical frequencies via dispersion engineering and lithographic metasurface integration (Wang et al., 2017).

In summary, frequency-tunable lenses with multiplexing capability represent a convergence of advanced materials science, electromagnetic design, and integrated photonics. Their ability to map, route, and process multiple frequency, polarization, and spatial channels in parallel—while providing dynamic, remote, or ultra-fast reconfigurability—underpins an expanding array of fundamental research and applied technologies in modern optics, sensing, communication, and quantum information science.