Vectorial, non-paraxial generalization of metasurface optics for on-demand polarization transformations along the optical path

Dorrah, Rubin, Zaidi, Tamagnone and Capasso (Nature Photonics 15, 287-296; 2021) demonstrated single form-birefringent metasurfaces that realize an arbitrarily prescribed, propagation-distance-dependent 2x2 Jones matrix $\tilde{F}(z)$, so that a beam's polarization is transformed on demand as it propagates, for any incident polarization. The synthesis is a matrix-valued Fourier superposition of Bessel harmonics that is exact only for the scalar (paraxial) wave equation, and the encoding (dual-matrix holography) reproduces the target only after spatial filtering of replicated diffraction orders. The open problem is to generalize this framework to a full vectorial electromagnetic treatment -- a synthesis that carries all three field components, remains valid at large cone angles and subwavelength structure, and enables genuine three-dimensional (longitudinal-component) polarization control along the optical path -- while removing the linked practical constraints: the freedom-versus-efficiency trade-off of the replicated-order encoding, the restriction to unitary symmetric single-layer Jones matrices, the absence of in-situ reprogrammable implementations, and the unquantified scaling, tolerance and robustness limits. The most direct successor, the 2026 extension to volumetric 3D vectorial holography, continues to operate in the transverse two-component (paraxial) framework with the same replicated-order encoding, and the vectorial non-paraxial core remains open.

Background

Polarization -- the direction in which the electric field of a light wave oscillates -- is described, for fully coherent paraxial beams, by a two-component complex Jones vector, and a linear polarization optical element (polarizer, wave plate) by a $2\times 2$ Jones matrix acting at a single transverse plane. Metasurfaces are planar arrays of subwavelength dielectric or metallic nanostructures ('meta-atoms') that shape a transmitted wavefront locally; rotated anisotropic nanopillars act as local wave plates whose retardance (dynamic phase) and orientation (geometric, or Pancharatnam-Berry, phase) define a local Jones matrix. Matrix Fourier optics (Zaidi et al., Optics Express 29, 39065; 2021) treats the metasurface as a spatially varying Jones-matrix distribution and analyzes it with polarization-resolved diffraction. A limitation of all conventional polarization optics, inherited by such planar elements, is that the specified operation applies at one plane: the polarization state downstream is then fixed by free-space propagation.

Dorrah et al. (Nature Photonics 15, 287-296; 2021) removed this limitation for paraxial beams. They synthesize a transverse distribution of Jones matrices $\tilde{U}(r,z)=\mathrm{e}^{-i\omega t}\sum_{m=-N}^{N}\tilde{A}^{(m)}\,J_0\!\big(k_r^{(m)}r\big)\,\mathrm{e}^{i k_z^{(m)} z},\qquad \big[k_r^{(m)}\big]^2+\big[k_z^{(m)}\big]^2=k_0^2,$ where $J_0$ is the zeroth-order Bessel function, $r$ is the radial coordinate, $k_0=\omega/c$, and the $2\times 2$ matrix coefficients $\tilde{A}^{(m)}$ are obtained from the designer target $\tilde{F}(z)$ by a matrix-valued Fourier projection over the implementation range $L$. Bessel harmonics are chosen because each is an exact scalar-Helmholtz solution and non-diffracting, so distinct polarization operations can be encoded at different distances. Multiplying the synthesized field $\tilde{U}(r,z=0)$ by any incident Jones vector yields a propagating field whose on-axis polarization follows $\tilde{F}(z)$ regardless of the incident polarization -- behaviour that would otherwise need several polarization-sensitive holograms on separate spatial light modulators (SLMs). The device level uses 'dual-matrix holography': each target matrix is decomposed (via singular-value decomposition) into a sum of two unitary matrices that are spatially interlaced on one lossless phase-only metasurface of TiO$_2$ nanopillars; the price is replicated diffraction-order spectra that must be Fourier-filtered in $k$-space, plus the intrinsic fact that rotated-birefringent single-layer cells realize only symmetric (and unitary) local Jones matrices. The demonstrated devices (a $z$-dependent analyser, half-wave plate and quarter-wave plate at $\lambda=532\,$nm, $462\,\mu$m aperture, $L=10\,$mm) were analyzed under the scalar paraxial approximation, in which the field has only two transverse components and no longitudinal ($z$-directed) electric-field component.

The paraxial approximation is accurate only for small angles between wave vectors and the optical axis (small cone angles). At large cone angles or subwavelength structure sizes, the full vectorial Maxwell theory must be used: the field acquires a longitudinal component, polarization is no longer strictly transverse, and the local-Jones description of a subwavelength-featured mask breaks down. 'Genuine three-dimensional polarization control' therefore means prescribing the full polarization ellipse, including its longitudinal component, as a function of position along the path -- beyond what any two-component Jones description can express.

Problem Statement

Generalize the metasurface synthesis of on-demand polarization transformations along the optical path beyond the scalar paraxial regime, and complete the framework it implies. Concretely, construct and demonstrate a synthesis that maps an arbitrarily prescribed, propagation-distance-dependent polarization operation -- now including the longitudinal (non-transverse) component of the electric field, i.e. genuine three-dimensional polarization control -- onto a single physical element or compact cascade, such that: (1) the construction is a full vectorial electromagnetic solution of Maxwell's equations valid at large cone angles and subwavelength structure sizes, reducing to the existing matrix Fourier/Bessel synthesis $\tilde{U}(r,z)=\sum_m \tilde{A}^{(m)} J_0(k_r^{(m)}r)\mathrm{e}^{i k_z^{(m)} z}$ in the paraxial limit (where $J_0$ is the zeroth-order Bessel function, $k_r^{(m)}$ and $k_z^{(m)}$ are the transverse and longitudinal wave numbers of harmonic $m$, and $\tilde{A}^{(m)}$ are $2\times 2$ matrix coefficients fixed by the target), and respecting conservation of energy and of total angular momentum (the spin angular momentum carried by the polarization together with the orbital angular momentum of the field, any change in the polarization-carried spin being compensated by the orbital part) for unknown incident polarization; (2) the encoding removes, or quantitatively bounds, the freedom-versus-efficiency trade-off of dual-matrix holography, in which replicated diffraction-order spectra must be discarded by $k$-space (transverse spatial-frequency) filtering; (3) the realization is not restricted to unitary, symmetric local Jones matrices of a single birefringent layer, extending to general (and, desirably, non-reciprocal or strongly chiral) responses in compact form; (4) a practical route exists to in-situ reprogramming of the target longitudinal transfer function $\tilde{F}(z)$ (active or reconfigurable implementation) at high efficiency; and (5) the scaling laws relating aperture, achievable axial range and resolution, fabrication tolerances, spectral bandwidth, and robustness to fabrication error, partial coherence or partial polarization of the input, and propagation through scattering or anisotropic media are quantified. These are facets of one objective -- turning the demonstrated paraxial, two-component, fixed-function device class into a general and practical vectorial framework. The core objective is item (1): a full vectorial electromagnetic solution of Maxwell's equations, valid at large cone angles and subwavelength structure sizes, reducing to the existing matrix Fourier/Bessel synthesis in the paraxial limit and respecting conservation of energy and of total angular momentum for unknown incident polarization, which realizes a prescribed longitudinal polarization transformation for arbitrary incident polarization on a single physical element or compact cascade.

References

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