An optic to replace space and its application towards ultra-thin imaging systems

The study addresses the often-overlooked contribution of free-space propagation distances to the size of optical imaging systems. While metasurfaces and metalenses have enabled compact, flat optical elements, the millimeter-to-meter-scale spaces between lenses (and between lenses, objects, and image planes) still dominate overall system size and are critical to image formation. The authors propose a new optical element, the spaceplate, which of physical thickness d imparts to the transmitted field a phase equivalent to free-space (or background medium) propagation over an effective length d_eff = R d, where R is a compression factor. The research question is whether an optical component can replace physical propagation distance without introducing optical power (i.e., without changing magnification or NA), thereby enabling ultra-thin imaging systems and compact devices that rely on spatial propagation. The work develops the theory, proposes realizations, and demonstrates through simulation and experiments that spaceplates can advance focal planes and reproduce free-propagation effects within a much smaller physical thickness.

The paper situates the work within advances in flat optics and metasurfaces, highlighting applications such as achromatic metalenses, spatial mode generation, lasing, polarimetry, and holography. It contrasts the proposed spaceplate with prior metamaterial concepts that involve field compression (e.g., transformation optics devices like hyperlenses or concentrators), noting that distance compression per se has not been targeted to reduce imaging system length. Prior nonlocal (momentum-space) optical responses have mainly controlled transmittance magnitude (angular filtering, spatial differentiators) or used lens-based 4f systems to implement k-dependent transfer functions, which defeats the purpose of replacing free-space propagation. The authors emphasize that a spaceplate requires imparting a momentum-dependent phase φ_sp(kx, ky) matching background-medium propagation φ_BG(θ, d_eff), with unity transmittance and no change in k-vector direction, thereby providing no optical power. They also reference practical optical systems where space dominates (e.g., telescopes, spectrometers) and note constraints of telephoto lenses, motivating the need for an element that can break the trade-off between system length and magnification.

Theory: Using Fourier optics, the authors define the desired transfer function of a spaceplate as imparting the phase of propagation through a background medium over distance d_eff: H(k) = exp[i (2π n_BG / λ) d_eff cos θ]. The spaceplate must implement φ_sp(θ) = φ_BG(θ, d_eff) within thickness d, admitting a global phase offset and 2π wrapping in θ. The response is purely momentum-dependent (nonlocal in space), preserving k-vectors and thus not introducing optical power. Multilayer spaceplate design (simulation): They design a thin-film multilayer stack (alternating silicon and silica layers) exhibiting transverse translational invariance to achieve the required k-dependent phase. A genetic algorithm optimizes layer materials and thicknesses to fit φ_BG(θ) for NA ≈ 0.26 (θ ≤ 15°) at λ = 1550 nm, initially constraining total thickness to ~10 µm and up to 40 layers; two materials (Si, SiO2) and practical constraints (>10 nm per layer) are used. Fitness combines phase fit and derivative metrics over θ ∈ [0, 15°]. Transfer-matrix calculations compute complex transmission H = |H| exp(i φ_SP). The algorithm converged in ~4000 generations to a 25-layer, ~10 µm-thick design. Full-wave 2D FDTD simulations (PML boundaries) propagate a converging Gaussian beam through the structure to verify focus advancement and polarization performance. Homogeneous spaceplates (experiments): They analyze unstructured media that produce the required φ_BG via an angle-dependent refractive index n(θ), focusing on a uniaxial crystal implementation. A calcite prism (20.04 mm × 19.98 mm × 29.84 mm, optical axis along depth) immersed in background media (linseed oil for monochromatic beam tests, glycerol for color imaging) serves as a uniaxial spaceplate for e-polarized light, with expected R = n_o / n_BG ≈ 1.117 and Δ = (n_o − n_BG) d ≈ +3.494 mm for d ≈ 29.84 mm at 532 nm. They also implement a low-index spaceplate as an air-filled glass-faced cylindrical cell (d = 4.37 ± 0.06 mm). Experimental setup: A 532 nm diode laser (4.5 mW) is spatially filtered and focused (f1 = 100 mm) into a tank containing the background medium and spaceplate; polarization is controlled with waveplates and PBS. A field relay 4f lens system (f1 = 100 mm, f2 = 200 mm; separation 300 mm) images the transmitted field to a CCD. The camera is scanned along z; multiple frames are averaged and background-subtracted. Monochrome and color cameras are used for beam and imaging tests respectively. For broadband imaging, an incoherent white-light source illuminates a print; a lens system forms an image into a glycerol tank, with and without the calcite plate inserted before the image plane. Beam deflection (lateral displacement Δx) vs. incidence angle (to 35°) is recorded to compare with ideal free-propagation displacement over d_eff in oil.

- Multilayer metamaterial spaceplate (simulation): A 25-layer, ~10 µm-thick Si/SiO2 stack at λ = 1550 nm reproduces the target φ_BG(θ) for θ ≤ 15° (NA ≈ 0.26) with fitted compression factor R = 4.9; full-wave simulations of a focusing Gaussian beam show a focus advance Δz = −43.2 µm, corresponding to R ≈ 5.2, in agreement with design. The device shows polarization insensitivity and operation over a ~30 nm bandwidth in simulation. Although not fabricated, it demonstrates that R can exceed constituent refractive index ratios (n_Si = 3.48, n_SiO2 = 1.45, n_vac = 1), indicating no such fundamental bound on R. - Uniaxial homogeneous spaceplate (experiment): A calcite plate in a matched-index background medium reproduces the lateral displacement of a beam as if it had propagated through d_eff of the background, matching theory up to 35° incidence (NA ≈ 0.85 in oil). In broadband visible imaging within glycerol, inserting the calcite spaceplate advances the image focal plane by Δ = −3.4 mm (theory ≈ −3.5 mm) without changing magnification, NA, or field of view, demonstrating achromatic, broadband, high-NA operation for a modest compression factor R ≈ 1.12. - The spaceplate preserves k-vector direction (no optical power), advancing focus or image planes without magnification changes, validating the nonlocal k-space phase-control concept. - Practical outlook: Estimates suggest that R ≈ 10–40 (smartphone cameras, d_eff ~1–4 mm) and R ≈ 200–400 (VR headsets, d_eff ~2–4 cm) could be impactful with d ≈ 100 µm thin-film stacks, with relaxed polarization/bandwidth requirements in display or integrated photonics contexts.

The findings demonstrate the feasibility of replacing free-space (or background medium) propagation with a thin optical element that imparts the appropriate k-dependent phase, without introducing optical power. Simulations confirm that multilayer nonlocal metamaterials can achieve R ≈ 5 with polarization insensitivity and moderate bandwidth, while experiments with a homogeneous uniaxial medium validate physical realizability, achieving broadband, achromatic, high-NA operation and image advancement with preserved magnification. These results directly address the research question by showing that the dominant spatial extent in imaging systems—the propagation distance—can be substantially reduced without compromising imaging parameters. The work argues no obvious fundamental physical limit on R; however, trade-offs between R, numerical aperture, bandwidth, polarization, and transmission may exist and remain to be fully characterized. Potential applications in smartphones, VR headsets, and integrated photonics illustrate the relevance of spaceplates and delineate practical performance targets. The spaceplate concept complements lenses by operating in k-space; coupling nonlocal (k-space) control with local (x–y) metasurfaces could enable complete spatial control of light in monolithic devices.

This work introduces the spaceplate, an optical element that effectively replaces a length of space by imparting the phase associated with free propagation over a much longer effective distance. The authors develop the Fourier-optics framework, design and simulate a multilayer nonlocal metamaterial spaceplate with R ≈ 5, and experimentally demonstrate homogeneous spaceplates (uniaxial calcite and a low-index cell) that reproduce free propagation effects, including broadband visible image advancement with preserved magnification and high NA. The results establish feasibility and key performance attributes (polarization independence, broadband operation, high NA), though not yet in a single device with large R. Future research should aim to combine these attributes in practical designs, increase R via advanced multilayer optimization, engineered uniaxial media, or other homogeneous solutions, and map out trade-offs among R, bandwidth, NA, polarization, and efficiency to meet application-specific requirements.

- The multilayer metamaterial spaceplate was not fabricated and was evaluated via simulation only; it exhibits moderate R, small d_eff, and low transmittance in the presented design, making it presently impractical. - Demonstrations of broadband, achromatic, high-NA performance used a uniaxial calcite plate with a small compression factor (R ≈ 1.12) and polarization selectivity (effective for e-polarization as implemented). - Performance attributes (large R, broadband, high NA, polarization insensitivity, high transmission) were shown separately across different implementations; combining them in a single device remains unachieved. - The multilayer design targets a limited NA (θ ≤ 15°, NA ≈ 0.26) and narrow wavelength band (~30 nm) at 1550 nm. - Surface quality and index-matching imperfections (e.g., calcite surfaces, glycerol/oil matching) can introduce scattering or minor discrepancies in measurements. - Potential trade-offs between R and other parameters (bandwidth, NA, efficiency) are not fully established.