Optics

Sep 1, 2025

Scientific correction — 13 October 2026

The application intervals and charts on this historical page are proposals or illustrations, not certified bounds for arbitrary fields or validated performance results. Certification requires assumptions on each integrand, a proved quadrature residual, rounding control, and propagation through subsequent transformations. Regular circular-sector bounds do not automatically bound optical, chemical, or anisotropic quantities.

Current reference: Circle and Area from Binary Radial Modules.

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), proposed (not independently 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 (proposed (not independently 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, proposed (not independently 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+