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Inverse Design

from ikarus.inverse import MetaAtom, free, pixels, Target, optimize

Declare what you want; let evolution do the drafting. Three steps: define a parameterized metaatom, state one or more targets, call optimize. Binary pixels evolve by bit-flip, continuous parameters by SBX/PM, under a mixed-variable GA (one objective) or NSGA-III (several).

Optional dependency

Needs pymoo: pip install "ikarus-rcwa[inverse]".

flowchart LR
    A["MetaAtom<br/>free + pixels DOF"] --> O["optimize"]
    T["Target(s)<br/>minimize / maximize / match"] --> O
    O --> Rres["OptimizeResult"]
    Rres --> RC[".metaatom → ready RCWA"]

Degrees of freedom

free(low, high) -> Free

Mark a continuous parameter (height or period) as a free DOF bounded to [low, high] (SI units).

pixels(nx, ny, symmetry=None) -> Pixels

Mark the patterned-layer topology as a free binary pixel map. symmetry shrinks the search space and enforces the physical symmetry:

symmetry Meaning Constraint
None all nx*ny pixels free —
"mirror_x", "mirror_y", "mirror_xy" reflection symmetry —
"c2" 180° rotation —
"c4" 90° rotation square grid
"c4v" 90° rotation + mirrors square grid

Pixels.n_free is the independent bit count (an 8×8 c4v grid → just 10 bits); Pixels.expand(bits) rebuilds the full (nx, ny) 0/1 grid.

Parametric shapes

A Shape (Cross, SplitRing, Ellipse, …) used as a topology turns each of its free(...) parameters into a real DOF named shape__<param> (e.g. shape__arm_length, shape__angle). This optimizes a physically interpretable meta-atom — arm widths, radii, rotation — instead of a pixel grid, over far fewer variables. See Lesson 7.

from ikarus.shapes import Cross
from ikarus.inverse import free

topology = Cross(arm_length=free(0.3, 0.95), arm_width=free(0.1, 0.45),
                 angle=free(0, 90))   # 3 free DOF + a clean, manufacturable shape

MetaAtom

MetaAtom(period, cover, substrate, polarization="linear", pol_angle=0.0)

A parameterized 3-region metaatom: cover / patterned layer / substrate. period and the pattern height may be fixed floats or free(...) ranges; the topology may be a fixed array, a pixels(...) map, or a parametric Shape with free parameters.

add_pattern(topology, materials, height) -> MetaAtom

Add the single patterned layer (0 -> materials[0], etc.).

variables() -> dict

{name: ('real', (lo, hi)) | ('binary',)} for every free DOF (period, height, px0, px1, …) — also your search space when bringing your own optimizer.

n_dof -> int

Number of free degrees of freedom.

build(params, n_orders) -> RCWA

The concrete RCWA for one parameter assignment (no source set).

atom = MetaAtom(period=180e-9, cover="Air", substrate="SiO2")
atom.add_pattern(topology=pixels(8, 8, symmetry="c4v"),
                 materials=["Air", "Si3N4"],
                 height=free(40e-9, 200e-9))
print(atom.variables())
# {'height': ('real', (4e-08, 2e-07)), 'px0': ('binary',), ...}

Target

One figure of merit. Build with a classmethod:

Target.maximize(metric, at=None, band=None, order=(0, 0), **kw)
Target.minimize(metric, at=None, band=None, order=(0, 0), **kw)
Target.match(metric, value, at=None, band=None, order=(0, 0), **kw)

Metrics

Metric Meaning
"R", "T" Diffraction efficiency into order (default specular (0,0); order=None → total).
"r_co", "t_co" Complex zero-order coefficient (co-pol).
"r_cross", "t_cross" Cross-pol coefficient (0 for linear polarization).
"r_phase", "t_phase" Phase (rad), matched modulo \(2\pi\).

Wavelengths (pick one)

Argument Meaning
at=1550e-9 one wavelength
at=[1064e-9, 1550e-9] a discrete set
band=(lo, hi) or band=(lo, hi, n) a sampled range (n defaults to 8)

Options

Keyword Default Meaning
order (0, 0) diffraction order; None/"total" for the sum
weight 1.0 scales this target's contribution
worst_case False aggregate wavelengths by the worst point, not the mean
name auto label used in report() 
# AR coating: minimize reflection across a band, robustly.
ar = Target.minimize("R", band=(300e-9, 600e-9, 6), worst_case=True)

# Beam steering: shove power into the +1 reflected order.
steer = Target.maximize("R", order=(1, 0), at=1550e-9)

# A metalens pixel: pin the transmission phase.
phase = Target.match("t_phase", value=1.57, at=1550e-9)

optimize

optimize(atom, targets, n_orders=8, algorithm="auto",
         pop=100, n_gen=60, seed=0, verbose=True) -> OptimizeResult
Argument Default Description
atom — a MetaAtom.
targets — a Target or list (≥ 2 → multi-objective Pareto).
n_orders 8 harmonic truncation for every forward solve.
algorithm "auto" GA if one objective, NSGA-III if several; or "ga", "nsga2", "nsga3".
pop, n_gen 100, 60 population size and generations.
seed 0 RNG seed — runs are reproducible.
verbose True print the per-generation pymoo table.
progress False show one progress bar over the generations (sets verbose=False).

OptimizeResult

Member Description
params Best parameter dict (first Pareto point if multi-objective).
metaatom The optimized structure as a ready-to-simulate RCWA.
report() -> str Human-readable summary (objective + parameters, or the Pareto front).
X, F Raw best parameters and objective value(s).
multi True for multi-objective runs.

Complete example — broadband AR coating

import numpy as np
from ikarus.inverse import MetaAtom, free, pixels, Target, optimize

atom = MetaAtom(period=180e-9, cover="Air", substrate="SiO2")
atom.add_pattern(topology=pixels(8, 8, symmetry="c4v"),
                 materials=["Air", "Si3N4"], height=free(40e-9, 200e-9))

target = Target.minimize("R", band=(300e-9, 600e-9, 6), worst_case=True)
best = optimize(atom, target, n_orders=6, pop=16, n_gen=10, seed=0)
print(best.report())

coating = best.metaatom                       # a ready RCWA
coating.set_source(wavelength=450e-9, theta=0, polarization="linear")
print("R @ 450 nm:", coating.simulate()[2].R_total)

Best practices

  • Pin BLAS to one thread for these tight loops (why it's worth ~10×).
  • Keep the metaatom subwavelength for effective-medium behavior — no parasitic diffraction lanes during evolution.
  • worst_case=True for broadband robustness; a discrete at=[...] list when only specific lines matter.
  • Exploit symmetry: 8×8 c4v is a 10-bit search, 8×8 free is 64 bits — a difference of eighteen orders of magnitude in search-space size.
  • Start with small pop/n_gen to gauge runtime, then scale.
  • Competing goals (high T and a phase)? Pass a list of targets and read the Pareto front from report().