---
title: Meta-Aspheric Lens (MAL)
url: https://www.emergentmind.com/topics/meta-aspheric-lens-mal
type: topic
---

# Meta-Aspheric Lens (MAL)

to=arxiv_search.search  สามสิบเอ็ดjson
{"query":"\"meta-aspheric lens\" metasurface aspheric lens arXiv","max_results":10}
to=arxiv_search.search  天天中彩票怎么json
{"query":"\"Planar Lenses at Visible Wavelengths\" OR \"Designing large, high-efficiency, high-numerical-aperture, transmissive meta-lenses for visible light\" OR \"Field of View Based Optimization of Aspheric Designed Geometric-Phase Doublet Lenses\"","max_results":10}
A Meta-Aspheric Lens (MAL) is a metasurface-based optical element that implements an aspheric focusing phase profile in a flat or near-flat form factor; in one usage, it is a planar metasurface that reproduces the phase delay of an aspheric lens, and in another, it is a single optical element that monolithically integrates an aspheric refractive surface with a patterned metasurface on one substrate [1605.02248], [2509.23599]. Across the cited literature, MALs are used to realize high numerical aperture focusing, off-axis aberration correction, compact imaging, tunability, and, in one early plasmonic implementation, sub-wavelength hot-spots generated by superoscillation [1211.1496]. The common principle is wavefront synthesis by sub-wavelength meta-atoms so that the transmitted field approximates a target aspheric phase profile.

## 1. Conceptual scope and historical development

The foundational planar-lens formulation in the visible was established by Khorasaninejad et al. in "Planar Lenses at Visible Wavelengths" [1605.02248]. In that work, high-aspect-ratio titanium dioxide metasurfaces were fabricated and designed as meta-lenses with \( \mathrm{NA} = 0.8 \), and diffraction-limited focusing was demonstrated at wavelengths of \(405\ \mathrm{nm}\), \(532\ \mathrm{nm}\), and \(660\ \mathrm{nm}\). The same work states that the metasurface can mimic the phase of a conventional refractive asphere through geometric phase, so that the MAL’s “equivalent” surface is the aspheric profile
\[
z(r) = \sqrt{r^2 + f^2} - f.
\]

An earlier precursor is the plasmonic meta-lens of Roy, Rogers, and Zheludev, which used ring-shaped nano-grooves milled in a \(50\ \mathrm{nm}\) gold film on glass and produced foci of \(160\ \mathrm{nm}\) \((0.2\lambda)\) in diameter when illuminated by a wavelength of \(800\ \mathrm{nm}\) [1211.1496]. That work does not define MAL in the monolithic refractive–metasurface sense, but it explicitly implements the ideal aspheric focusing phase
\[
\phi_{\mathrm{ideal}}(r) = -k\,[\sqrt{r^2 + f^2} - f]\ \mathrm{mod}\ 2\pi,
\]
and therefore constitutes an early meta-aspheric focusing surface.

Large-area high-NA transmissive meta-lenses were systematized by Byrnes et al., who described a design method for large-area meta-lenses with computational cost almost independent of lens size and reported three \( \mathrm{NA}=0.94 \) designs with minimum feature size of \(100\ \mathrm{nm}\) [1511.04781]. Hornburg et al. later studied aspheric phase pattern geometric-phase doublet lenses, optimized monochromatically and polychromatically, and fabricated a liquid-crystal implementation of an aspheric doublet [2205.11286]. Tunable MAL architectures were then realized through dielectric elastomer actuators and MEMS-actuated Alvarez metasurfaces, extending MAL functionality from static focusing to electrically controlled defocus, astigmatism correction, and image shift [1708.01972], [2001.07800].

A later near-infrared formulation makes the definition explicit: the MAL is “a single optical element that monolithically integrates an aspheric refractive surface with a patterned metasurface on one substrate,” achieving both wavefront shaping and high-order aberration correction in a volume of only \(0.02\ \mathrm{cm}^3\) [2509.23599]. This suggests that the term now spans both pure planar meta-aspheres and integrated refractive–metasurface stacks.

## 2. Phase engineering and equivalence to refractive aspheres

For a normally incident plane wave of wavelength \( \lambda \) in air focused to focal length \( f \), the visible-wavelength TiO\(_2\) MAL uses the phase delay
\[
\phi(r) = -\frac{2\pi}{\lambda}\left[\sqrt{r^2 + f^2} - f\right],
\]
with \( \phi(0)=0 \) and \( r \equiv \sqrt{x^2+y^2} \) [1605.02248]. The same source gives the correspondence to a refractive asphere: a conventional aspheric lens in glass with index \( n>1 \) has surface sag \( z(r) \) chosen so that the optical-path-length from any point to the focus is constant, yielding
\[
z(r) = \sqrt{r^2 + f^2} - f,
\]
and transmitted phase relative to the center
\[
\phi(r)= -\frac{2\pi}{\lambda}(n-1)\,z(r).
\]
For small \( r/f \), this reduces to \( z \approx r^2/(2f) \), the familiar parabolic approximation.

Byrnes et al. generalized this relation to an aspheric surface with conic constant \( K \) and higher-order terms [1511.04781]. In that treatment, the sag is
\[
z(r) = \frac{c r^2}{1 + \sqrt{1 - (1+K)c^2 r^2}} + \sum_{i=2}^{M} A_{2i} r^{2i},
\]
where \( c=1/R \), \( K \) is the conic constant, and \( A_4,A_6,\ldots \) are higher-order aspheric coefficients. The required optical phase is then
\[
\phi_{\mathrm{asph}}(r) = \frac{2\pi}{\lambda}(n-1)\,z(r),
\]
or, in the pure metasurface interpretation in air, \( n-1 \rightarrow 1 \). They further write
\[
\phi_{\mathrm{asph}}(r) \simeq \frac{2\pi}{\lambda}\left[\sqrt{r^2+f^2}-f\right] + \frac{2\pi}{\lambda}\sum_{i=2}^{M}\alpha_{2i} r^{2i},
\]
with \( \alpha_{2i}\equiv(n-1)A_{2i} \), so that higher-order coefficients can be tuned to correct residual aberrations.

The near-infrared monolithic MAL adopts a different parameterization: the radial phase profile is expanded in even polynomials of normalized radius \( p=r/R \),
\[
\phi(r,\lambda)=\sum_{n=0}^{N} a_n(\lambda)\cdot (r/R)^{2n},
\]
with \( N=10 \) in that design [2509.23599]. The same work models true dispersion and off-axis incidence through
\[
\phi_{\mathrm{meas}}(r,\lambda,\theta)=f[d(r),\lambda,\theta],
\]
where \( d(r) \) is the meta-atom diameter and \( f \) is calibrated by FDTD plus measured a-Si refractive index \( n(\lambda) \). Hornburg et al. used two related aspheric representations for geometric-phase doublets: a Zernike-fringe surface with only three terms, \( Z_3 \), \( Z_8 \), and \( Z_{15} \), and a polynomial expansion
\[
\Phi(r)=2\pi\sum_{i=1}^{5} C_i r^{2i}
\]
[2205.11286]. These formulations differ in basis, but all of them encode the same objective: phase synthesis beyond a simple quadratic lens.

## 3. Numerical aperture, field correction, and aberration structure

For a lens of radius \( R=D/2 \) and focal length \( f \), numerical aperture is defined by
\[
\mathrm{NA}=n_{\mathrm{medium}}\sin\theta_{\max},
\]
and, in air,
\[
\mathrm{NA}=\frac{R}{\sqrt{R^2+f^2}}=\frac{D/2}{\sqrt{(D/2)^2+f^2}}
\]
[1605.02248]. In the visible TiO\(_2\) MALs, \( D=240\ \mu\mathrm{m} \) and \( f=90\ \mu\mathrm{m} \) give \( \mathrm{NA}\approx 0.8 \), while larger-diameter devices such as \( D=2\ \mathrm{mm} \), \( f=0.725\ \mathrm{mm} \) likewise yield \( \mathrm{NA}=0.8 \) for imaging applications.

The same source states that because \( \phi(r) \) exactly follows the non-paraxial profile, all spherical and higher orders are intrinsically corrected for on-axis, monochromatic focus, and no residual spherical aberration remains [1605.02248]. Off-design wavelengths show chromatic focal shift and spot broadening due to metasurface dispersion; narrowband laser operation renders this negligible for microscopy and spectroscopy.

Wide-field correction requires a different optimization target. The monolithic NIR MAL reaches a half-angle \( \theta_{\max}=50.75^\circ \), hence full field of view \( 2\theta_{\max}\simeq 101.5^\circ \), by ray-tracing marginal rays through the asphere plus metasurface and optimizing \( \phi(r) \) and the aspheric sag \( z(r) \) to correct spherical, coma, and astigmatism up to \( \theta_{\max} \) [2509.23599]. Its merit function minimizes RMS wavefront error \( W \) across field,
\[
W(\theta,r)=\Sigma_{m,n} c_{mn}\cdot Z_{mn}(\theta,r)+\mathrm{residual\ terms},
\]
with optimization conditions \( \partial W/\partial a_n = 0 \) and \( \partial W/\partial a_{2n}(\mathrm{asphere})=0 \) under the total track length constraint.

Hornburg et al. treated the field-of-view problem explicitly in doublet geometric-phase lenses [2205.11286]. Their spherical singlet is diffraction-limited on-axis but has severe coma by \( \pm 7^\circ \), whereas the MAL doublet retains \(<120\ \mu\mathrm{m}\) RMS spot up to \( \pm 7^\circ \), reducing off-axis aberration by \(\sim 50\%\). This establishes a distinct MAL design regime: rather than maximizing on-axis NA alone, the lens is optimized for field-angle robustness through distributed aspheric phase terms and a spacer-dependent power balance.

A different notion of focusing appears in the superoscillatory plasmonic meta-lens [1211.1496]. There, sub-wavelength spots smaller than the conventional diffraction limit occur in low-intensity regions characteristic of superoscillatory focusing. The source explicitly notes that defining an effective numerical aperture from the spot size would formally give a “superoscillatory NA” \(>1\), but such a definition is not meaningful for superoscillations. This addresses a common misconception: sub-\(\lambda\) hot-spots in a metasurface do not necessarily imply a conventional high-NA imaging lens.

## 4. Meta-atom platforms and optical architectures

Several distinct meta-atom implementations appear in the literature. In the visible TiO\(_2\) MAL, each “pixel” is a high-aspect-ratio TiO\(_2\) nanofin acting as a half-wave plate, with rotation
\[
\theta(x,y)=\phi(r)/2,
\]
so that geometric Pancharatnam–Berry phase imposes the local phase shift \( \phi(r) \) [1605.02248]. The nanofin dimensions are tuned for the design wavelength. For \(660\ \mathrm{nm}\): \(H=600\ \mathrm{nm}\), \(W=85\ \mathrm{nm}\), \(L=410\ \mathrm{nm}\), \(S=430\ \mathrm{nm}\). For \(532\ \mathrm{nm}\): \(H=600\ \mathrm{nm}\), \(W=95\ \mathrm{nm}\), \(L=250\ \mathrm{nm}\), \(S=325\ \mathrm{nm}\). For \(405\ \mathrm{nm}\): \(H=600\ \mathrm{nm}\), \(W=40\ \mathrm{nm}\), \(L=150\ \mathrm{nm}\), \(S=200\ \mathrm{nm}\). The corresponding look-up table covers \(0\rightarrow 2\pi\) with nearly uniform conversion efficiency \((\ge 90\%)\) over the design wavelength.

Byrnes et al. departed from isolated-pillar lookup tables and treated each local cell as a tiny 2D periodic “beam-deflector” grating [1511.04781]. In each rectangular cell, one places \(N_{\mathrm{pillars}}\) pillars, each with \(5N_{\mathrm{pillars}}\) degrees of freedom \((x_j,y_j,a_j,b_j,\theta_j)\). The transmitted field in the \(+1\) diffraction order has amplitude \(A(p_j)\) and phase \( \phi_{\mathrm{trans}}(p_j) \), and the geometry is adjusted so that \( \phi_{\mathrm{trans}}(p_j)\approx \phi_{\mathrm{target\ cell}}=\pi/2 \) while \( |A(p_j)| \) is maximized. Their figure of merit is
\[
\mathrm{FOM}=\langle |A_{+1}|\cdot \sin[\phi_{+1}-\phi_{\mathrm{ref}}]\rangle_{\lambda,\mathrm{pol}}.
\]
This architecture is aimed at macroscopic \( \mathrm{NA}=0.94 \) lenses with weak computational scaling in diameter.

The monolithic NIR MAL uses nanocylinders in amorphous silicon and models the phase operator through the complex transmission coefficient \( t_x \) for x-polarized light [2509.23599]:
\[
\phi(d,\lambda,\theta)=\arg\{t_x(d,\lambda,\theta)\},\qquad
T(d,\lambda,\theta)=|t_x|^2.
\]
This is coupled to a refractive asphere in a single stack, rather than treated as a standalone planar metalens.

Geometric-phase doublet MALs built in liquid-crystal films implement phase through the local optical axis of a half-wave retardation layer [2205.11286]:
\[
\Phi_{\mathrm{GP}}(x,y)=\pm 2\alpha(x,y).
\]
Two such plates, separated by a variable spacer, supply distributed aberration correction in a doublet geometry. Tunable MALs introduce further architectures. In the dielectric elastomer implementation, the starting hyperboloidal phase is
\[
\phi_0(x,y)=\frac{2\pi}{\lambda}\,[\sqrt{x^2+y^2+f_0^2}-f_0],
\]
and uniform stretch transforms the coordinates so that the focal length scales as \( f(s)=s^2f_0 \) to first order [1708.01972]. In the MEMS-actuated Alvarez lens, two complementary cubic phase plates,
\[
\phi_1(x,y)=+A\cdot(x^3+xy^2),\qquad
\phi_2(x,y)=-A\cdot(x^3+xy^2),
\]
generate a quadratic lens phase when laterally shifted by \( \pm \Delta \), giving
\[
f(\Delta)=\frac{\pi}{2A\lambda\Delta}
\]
[2001.07800]. This is not a hyperboloidal asphere in the singlet sense, but it is a metasurface realization of a freeform aspheric tuning principle.

## 5. Fabrication pathways and manufacturability

The visible TiO\(_2\) MAL is fabricated by a single-layer lithography plus ALD process [1605.02248]. The stated sequence is: spin and bake a ZEP 520A electron-beam resist to thickness \(H=600\ \mathrm{nm}\); pattern resist by e-beam lithography with the rotation map \( \theta(x,y)=\frac{1}{2}\phi(r) \); develop resist to yield vertical resist posts; conformal atomic-layer deposition of amorphous TiO\(_2\) \((n\approx 2.4)\); blanket reactive-ion etch to remove the top TiO\(_2\) film; strip resist, leaving freestanding, high-aspect-ratio TiO\(_2\) nanofins with sidewalls of approximately \(90^\circ\). The summary states that the resulting process produces vertical, atomically smooth nanofins compatible with foundry deep-UV steppers for high-volume manufacture.

The monolithic NIR MAL emphasizes wafer-level integration [2509.23599]. The metalens is fabricated on a D263T glass wafer by depositing \(10\ \mathrm{nm}\) SiO\(_2\) by PECVD, then \(550\ \mathrm{nm}\) a-Si, followed by a \(30\ \mathrm{nm}\) Cr hard mask and \(30\ \mathrm{nm}\) SiO\(_2\) protection layer; electron-beam lithography in maN-2403 resist and ICP etch define the a-Si nanostructures. The asphere is fabricated on a separate wafer by spin coating UV-curable resist, UV exposure by laser direct writing \((95\ \mathrm{kHz})\) using a nanoimprint mold, and development to achieve the aspheric sag profile \( z(r) \). Wafer-level bonding aligns the two wafers with an active-alignment mask to \(<1\ \mu\mathrm{m}\) error while leaving a controlled air gap above the metasurface. The final device requires only one dicing step and no additional mechanical fixtures.

Byrnes et al. frame manufacturability as a design constraint rather than only a process constraint [1511.04781]. Their optimization enforces minimum separation \(\ge 100\ \mathrm{nm}\), minimum pillar diameter \(\ge 100\ \mathrm{nm}\), maximum aspect ratio \(\sim 5{:}1\), and fixed pillar height \(=550\ \mathrm{nm}\). They also note that the central \(1\%\) of the area is handled with a constant-period hexagonal lattice method, while the outer \(99\%\) uses beam-deflector tiling, which is a practical response to the large local phase gradients near the periphery.

Other fabrication routes extend the MAL concept to different materials and actuation platforms. The superoscillatory plasmonic meta-lens is patterned by focused-ion-beam lithography in a \(50\ \mathrm{nm}\) gold film with groove width \(34\ \mathrm{nm}\) and \(\pm 5\ \mathrm{nm}\) tolerance [1211.1496]. The tunable elastomeric MAL uses a water-based transfer process in which a-Si posts fabricated on a Ge/GeO\(_2\) sacrificial layer are transferred onto pre-stretched VHB and integrated with single-walled carbon nanotube transparent electrodes [1708.01972]. The MEMS Alvarez MAL uses PECVD Si\(_3\)N\(_4\), stepper lithography, ICP etch, DRIE, HF-vapor release, and face-to-face alignment with \(50\ \mu\mathrm{m}\) Kapton spacers [2001.07800]. Hornburg et al. use photo-aligned liquid-crystal films written by direct-write laser scanning and assembled with NOA-61 and a \(1\ \mathrm{mm}\) N-BK7 spacer [2205.11286]. Taken together, these reports show that MAL fabrication spans e-beam-defined dielectric nanostructures, plasmonic FIB patterning, wafer-bonded hybrid stacks, soft-matter geometric-phase films, and semiconductor-compatible MEMS processes.

## 6. Performance regimes, applications, and limitations

The visible TiO\(_2\) MAL with \( \mathrm{NA}=0.8 \) yields focusing efficiencies of \(86\%\) at \(405\ \mathrm{nm}\), \(73\%\) at \(532\ \mathrm{nm}\), and \(66\%\) at \(660\ \mathrm{nm}\), with diffraction-limited spots given by \( \mathrm{FWHM}\approx 0.51\lambda/\mathrm{NA} \): approximately \(280\ \mathrm{nm}\), \(375\ \mathrm{nm}\), and \(450\ \mathrm{nm}\), respectively [1605.02248]. Measured wave-aberration RMS \( \lesssim 0.06\lambda \) and Strehl ratio \( \approx 0.8 \) confirm near-ideal, aberration-free performance at the design wavelength. The same work reports that the meta-lenses can resolve nanoscale features separated by sub-wavelength distances and provide magnification as high as \(170\times\) with image qualities comparable to a state-of-the-art commercial objective.

The monolithic NIR MAL targets a different regime: a \(101.5^\circ\) field of view, \(3.39\ \mathrm{mm}\) total track length, F/1.64 aperture, and volume \(0.02\ \mathrm{cm}^3\) for imaging near \(940\ \mathrm{nm}\) [2509.23599]. Simulated and measured sagittal/tangential MTF at \(50\ \mathrm{lp/mm}\) are \(>0.65\) on-axis and \(\ge 0.31\) at \( \theta_{\max}=50.75^\circ \), with agreement within \(0.03\). The device resolves USAF 1951 Group \(-1\) and \(0\) \((\approx 50\ \mathrm{lp/mm})\) with Michelson contrast \(>0.89\) and CNR \(>3.1\), maintains field curvature below \(\pm 5\ \mu\mathrm{m}\) sag across the FOV, and geometric distortion below \(3\%\). Relative illumination exceeds \(90\%\) centrally and remains at least \(75\%\) at the edge. Imaging demonstrations include dorsal hand vein imaging under a \(940\ \mathrm{nm}\) LED, eye tracking at \(0^\circ\), \(25^\circ\), and \(50^\circ\) field angles, and computational pixel super-resolution using a MambaIR network.

Byrnes et al. reported predicted efficiencies of \(79\%\) for a single-wavelength \(580\ \mathrm{nm}\) \( \mathrm{NA}=0.94 \) collimator, \(68\%\) for a dichroic \(580\ \mathrm{nm}\) focus plus \(450\ \mathrm{nm}\) pass-through design, and broadband efficiency peaking at approximately \(75\%\) at \(580\ \mathrm{nm}\) for a \(500\text{–}650\ \mathrm{nm}\) design [1511.04781]. Their chromatic focal shift of approximately \(\pm 10\ \mu\mathrm{m}\) across the broadband case is stated to be as expected for diffractive optics. Hornburg et al. reported a measured MAL average RMS spot of \(104\ \mu\mathrm{m} \pm 5.5\ \mu\mathrm{m}\) over \(\{-7^\circ,-3^\circ,0^\circ,3^\circ,7^\circ\}\), compared with a measured singlet reference average of \(74.5\ \mu\mathrm{m} \pm 30.0\ \mu\mathrm{m}\), reflecting the trade-off between slightly larger on-axis spot and improved field uniformity [2205.11286].

Tunable MALs emphasize dynamic range and control. The dielectric elastomer device achieves focal length tuning \(>100\%\), specifically \(f_0=50\ \mathrm{mm}\rightarrow f_{\max}\approx 103\ \mathrm{mm}\) \((107\%\ \Delta f)\) at \(3\ \mathrm{kV}\) for the single-layer device, with focusing efficiency \(91\%\) before transfer and approximately \(62.5\% \pm 2\%\) after transfer over the full tuning range [1708.01972]. It also performs dynamic corrections, including astigmatism and image shift. The MEMS Alvarez MAL achieves total uniaxial displacement of \(6.3\ \mu\mathrm{m}\) for DC voltage up to \(20\ \mathrm{V}\), focal position tuning over \(68\ \mu\mathrm{m}\), and \(1460\) diopters change in optical power, with power consumption below \(100\ \mathrm{nW}\) DC and resonant frequency around \(3.4\ \mathrm{kHz}\) [2001.07800].

The principal limitations are also explicit in the cited works. Visible meta-aspheres show chromatic focal shift and spot broadening off design wavelength [1605.02248]. The NIR MAL is currently limited to a bandwidth of approximately \(30\ \mathrm{nm}\) around \(940\ \mathrm{nm}\), and extending to multi-band NIR requires more complex dispersion engineering; electron-beam lithography is also described as slow for mass production, motivating step-and-repeat nanoimprint or deep-UV stepper lithography [2509.23599]. The superoscillatory plasmonic lens produces its smallest hot-spots in low-intensity tails and is limited by absorption in gold and finite cluster-size interactions [1211.1496]. The MEMS Alvarez prototype reports small side lobes and elongated spot due to a \(50\ \mu\mathrm{m}\) axial gap and slight lateral misalignment [2001.07800]. These constraints indicate that MALs are not a single-performance class but a family of architectures trading off bandwidth, efficiency, field angle, tunability, manufacturability, and system compactness.

A plausible implication is that MAL research has bifurcated into two main trajectories. One trajectory pursues high-NA or superoscillatory planar focusing through increasingly precise phase control on a single surface [1605.02248], [1211.1496], [1511.04781]. The other integrates aspheric refractive power, metasurface correction, and, in some cases, actuation into wafer-level or semiconductor-compatible optical systems aimed at compact cameras, eye tracking, and adaptive imaging [2509.23599], [1708.01972], [2001.07800].

Source: https://www.emergentmind.com/topics/meta-aspheric-lens-mal