Optics

Optics

Sep 1, 2025

Generative Radial Geometry for Optics: From Aspheres to Metasurfaces

Generative Radial Geometry (GRG) treats optical elements as radially generated structures optimized from the center outward. This viewpoint enables targeted aberration correction, compact/lightweight systems, and fast iteration via simulation—spanning refractive aspheres, GRIN profiles, diffractive zone plates, and radially programmed metasurfaces.

What GRG means in optics

In GRG, a surface or refractive-index profile is parameterized radially and then optimized against a merit function (e.g., wavefront error, Strehl, or MTF) over the full pupil. Rather than restricting to a low-order conic/asphere, GRG explores richer radial degrees of freedom that better align the phase over wide apertures and fields.

GRG Modes & Selection for Optics. We use three base modes: Δx Trapezoidal (fast area-like radial sums, lower estimate), Projection Corridor (paired lower/upper area bounds via right/midpoint sampling), and Chord-Sum at Δu (circumference/loops on the full turn, lower; pair with tangent-sum for an upper bound). In optics they map to pupil integrals (energy/WFE), certified bounds on merit terms, and phase/loop controls on rings/zones.

Δx Trapezoidal (Cartesian)

Target: area-weighted radial sums (pupil cross-sections) • Estimate: lower

  • Encircled energy, WFE/OPD RMS, throughput vs radius.
  • Baseline fast integration in design loops.

Projection Corridor

Target: same as above • Estimate: lower & upper (certified)

  • Bounds for energy, WFE, Strehl/MTF contributions.
  • Use to stop/refine when gap ≤ tolerance.

Chord-Sum at Δu (Radial)

Target: loops on rings (full turn) • Estimate: lower (pair with tangent-sum ↑)

  • Zone edges, phase wraps (2π), vortex/Bessel rings.
  • Perimeter/phase uniformity & ring tiling quality.

Picker by task. Quick merit updates → Trapezoid; need guarantees → Corridor; ring/zone phase & perimeter → Chord-Sum (± tangent-sum); 3D volume/weight/thermal → shell/frusta extensions.

Why GRG improves optical design

  • Targeted aberration correction: spherical aberration, coma, and astigmatism can be minimized across large apertures by flattening phase error over the beam.
  • Compact, lightweight systems: high-performance single elements can replace multi-lens groups; GRG also supports radial GRIN to combine focus and correction in one body.
  • Advanced diffractive elements: optimized radial zone partitioning for resolution, efficiency, and robustness—supporting multi-order/multi-focal designs.
  • Metasurfaces & LIDAR: radial placement/shaping of nano-scatterers for phase/amplitude control (beam shaping, Bessel/vortex), wide-FOV collimation, compact 360° sensing.
  • Rapid simulation-to-prototype: GRG loops tightly with FEM/FDTD engines to explore candidate radial profiles quickly.

Design workflow (radial parametrization → merit function → solver)

A typical GRG workflow parameterizes the sag or index as a radial series; samples the pupil; defines a merit function over wavefront or image metrics; and solves for the radial coefficients (often via linearization around a base conic).

Radial surface model

We model the sag z(r) as a base conic plus a radial polynomial tail:

$$ z(r) \;=\; \frac{c\,r^{2}}{\,1+\sqrt{\,1-(1+k)c^{2}r^{2}\,}\,} \;+\; a_{4}r^{4} \;+\; a_{6}r^{6} \;+\; a_{8}r^{8}. $$
Axis r vs z(r); blue curve shows base conic; accent curve shows GRG-corrected sag. r z base conic GRG tail

Sampling and merit function

Sample radii across the clear aperture and minimize wavefront residual at wavelength λ for the target focal length:

$$ \min_{a_4,a_6,a_8}\;\sum_{i} \bigl[\Phi\bigl(z(r_i)\bigr)\bigr]^2, $$
Circle aperture with 5 concentric rings and sample points.

Linearized solve

Linearizing in the coefficients leads to a system

$$ M \begin{pmatrix} a_{4}\\[2pt] a_{6}\\[2pt] a_{8} \end{pmatrix} \;=\; b. $$

GRG modes: Use Trapezoid for fast radial contributions to WFE/energy; switch to Corridor when you need a guaranteed bound on residuals; use Chord-Sum on rings to enforce phase consistency (2π wraps) and to check zone/perimeter uniformity; add tangent-sum to bracket loop metrics.

Worked example: aspheric–radial singlet at λ = 550 nm

For a collimated beam at λ=550 nm focused to f=100 mm with D=25 mm, take a base paraboloid (k=–1) with curvature c=0.010 mm⁻¹ (R=100 mm). Solving the linearized system yields:

$$ \begin{aligned} a_{4} &= 2.50\times 10^{-11}\,\mathrm{mm}^{-3},\\ a_{6} &= -1.20\times 10^{-16}\,\mathrm{mm}^{-5},\\ a_{8} &= 3.00\times 10^{-22}\,\mathrm{mm}^{-7}. \end{aligned} $$

With these coefficients, the residual wavefront error across the pupil falls below λ/20 and the MTF at 50 lp/mm improves by ~30% relative to the spherical counterpart:

$$ \begin{aligned} \mathrm{WFE}_{\mathrm{rms}} &\le \frac{\lambda}{20},\\ \mathrm{MTF}\!\left(50\,\mathrm{lp/mm}\right) &\approx 1.30\,\times\,\mathrm{MTF}_{\text{spherical}}. \end{aligned} $$
MTF curves vs spatial frequency; GRG shows higher contrast near 50 lp/mm. spatial freq (lp/mm) MTF 50 ~30% gain @ 50 lp/mm (GRG) spherical GRG Two small fan plots suggesting reduced aberrations after correction. Ray fan (tangential) Ray fan (sagittal)

GRG modes: Trapezoid for quick MTF/Strehl-related radial sums; Corridor to certify WFE/energy targets before freezing coefficients; Chord-Sum (+ tangent-sum) to monitor ring-wise phase/perimeter and edge artifacts.

Diffractive optics and zone plates

GRG optimizes the radial subdivision and phase steps of diffractive optical elements (DOEs) to jointly maximize resolution, first-order efficiency, and fabrication tolerance, and can be extended to multi-order/multi-focal targets.

Concentric rings with varying widths representing an optimized radial phase profile.

GRG modes: Chord-Sum to place ring edges by phase advance along the circumference (2π steps, vortex charge checking); Corridor to bound first-order efficiency/EE across radii; Trapezoid for throughput and fill-factor roll-ups.

Metasurfaces and 360° sensing (LIDAR)

For metasurfaces, GRG prescribes the radial distribution and geometry of nano-scatterers (meta-atoms) to deliver the desired phase/amp profile—enabling beam shaping (Bessel, vortex), wide-FOV collimation, or compact 360° sensing modules.

Dots arranged in concentric rings with varying densities and azimuthal offsets.

GRG modes: Chord-Sum for ringwise phase conformity and azimuthal uniformity (Bessel/vortex); Corridor to bracket efficiency across pupil zones under scatterer quantization; Trapezoid for fast aggregate metrics during layout sweeps.

GRIN materials and fabrication

Beyond pure surface shaping, GRG covers radial gradient-index (GRIN) profiles co-optimized with geometry, compatible with precision 3D printing, advanced lithography, and layered deposition processes.

Circle with radial gradient indicating higher index in the center. n(r)

GRG modes: Trapezoid/Corridor for OPD and energy integrals through shells (robustness to dispersion/tolerances); Chord-Sum to verify iso-phase circles after co-optimizing n(r) + sag.

Simulation loop and time-to-prototype

A GRG toolchain linked to wave/EM solvers rapidly screens candidate radial profiles, reducing the number of physical iterations needed before fabrication.

Blocks: Parametrization → Solver → EM Simulation → Tolerance → Prototype. Radial parametrization Solver EM simulation Tolerance Prototype

GRG modes: Loop fast with Trapezoid; certify with Corridor (stop when the bound gap is within tolerance); validate ring/zone behavior with Chord-Sum (+ tangent-sum) before tape-out.

Practical notes: tolerances, costs, and when GRG is worth it

  • Tolerancing: include manufacturing/assembly error budgets in the merit function so the solved profile remains robust after polishing, litho rounding, or index dispersion.
  • Cost trade-offs: in commodity systems, classic low-order aspheres may stay cheaper; GRG shines when performance/weight constraints justify advanced profiles.
  • System impact: a single GRG element can replace a small lens group—simplifying alignment and improving thermal stability—especially with GRIN or metasurface layers.

Conclusion

GRG augments—not replaces—classical optical design. By combining Δx Trapezoidal, Projection Corridor, and Chord-Sum at Δu (with tangent-sum for upper loop bounds), you get fast merit updates, certified guarantees, and precise ring/zone control across refractive, diffractive, GRIN, and metasurface optics.

Encircled Energy (EE) via GRG Corridor — bounds on Rτ

Definition. For a pupil-normalized PSF intensity \(I\), the encircled energy at radius \(R\) is

$$ E(R)=\frac{\iint_{x^2+y^2\le R^2}\! I(x,y)\,dA}{\iint_{x^2+y^2\le R_{\max}^2}\! I(x,y)\,dA}\,. $$

Axisymmetric shortcut. If \(I(x,y)=I(r)\) then

$$ E(R)=\frac{2\pi\int_0^{R} r\,I(r)\,dr}{2\pi\int_0^{R_{\max}} r\,I(r)\,dr}\,. $$

GRG corridor (discrete). Choose radii \(0=r_0

Lower/upper cumulative EE (up to the smallest index with \(r_{k+1}\le R\)):

$$ E^{\downarrow}(R)=\frac{\sum_{k\,:\,r_{k+1}\le R} A_k\,I_{k}^{\min}}{\sum_{k=0}^{K-1} A_k\,I_{k}^{\mathrm{mid}}},\qquad E^{\uparrow}(R)=\frac{\sum_{k\,:\,r_{k+1}\le R} A_k\,I_{k}^{\max}}{\sum_{k=0}^{K-1} A_k\,I_{k}^{\mathrm{mid}}}, $$
where \(I_k^{\min}, I_k^{\max}\) can be chosen as \(\{I_k^{\mathrm{out}}, I_k^{\mathrm{in}}\}\) for monotone-decreasing \(I(r)\), or via angular min/max for general PSFs. Always \(E^{\downarrow}(R)\le E(R)\le E^{\uparrow}(R)\).

Bracketing the EE radius. For a target \(\tau\) (e.g., 0.80):

  1. \(R_{\tau}^{-}=\min\{R:\,E^{\uparrow}(R)\ge \tau\}\) is a lower bound on the true \(R_{\tau}\).
  2. \(R_{\tau}^{+}=\min\{R:\,E^{\downarrow}(R)\ge \tau\}\) is an upper bound on the true \(R_{\tau}\).
  3. Refine by increasing annuli (or bisect within the bracket). Stop when \(R_{\tau}^{+}-R_{\tau}^{-}\le\) tolerance.

Method picker. Axisymmetric/on-axis: use annular corridor above (fast). General PSF or field points: use GRG Projection Corridor on a quarter (area bounds), or polar cells with per-cell min/max. Use Chord-Sum (+ tangent-sum) to validate ring perimeter/phase uniformity when EE is tied to zone edges.

E_up above true E, E_down below, horizontal tau line; bracket on R. R E τ Rτ− Rτ+