Research and method

Key equations behind the solver.

The model studies small, high-contrast multilayer scatterers through a normalized limiting mode. That single radial field profile controls the leading resonance and the layer-weighted effective response.

Limiting mode and normalization

The solver discretizes the radial integral eigenproblem on a trapezoidal grid. The mode is normalized in the three-dimensional unit-ball convention.

4π ∫₀¹ u₀(r)²r²dr=1

U_kModal loading: 4π∫_Bku₀(r)r²dr

E_kModal energy: 4π∫_Bk|u₀(r)|²r²dr

U_β,kNonlinear loading: 4π∫_Bku₀(r)³r²dr

F_kQuartic weighting: 4π∫_Bk|u₀(r)|⁴r²dr

First-order resonance correctors

The forward calculator reports both supplied forms so the distinction is visible rather than hidden.

λ_h≈λ₀+hλ₁

2018 reference form

λ_1,A=−i λ₀^5/2W₀U₀/(4π)

Updated narrative form

λ_1,B=−i λ₀^5/2W₀²/(4πD₀)

The updated RI-EMT narrative introduces the Fredholm energy denominator. In the comparison code, W₀=η₁U_in+η₂U_out+β₁U_β,in and D₀=η₁E_in+η₂E_out.

Inverse-design weighting

Loading definition

η_eff=(η₁U₁+η₂U₂)/(U₁+U₂)

β_eff=(β₁U_β,1+β₂U_β,2)/(U_β,1+U_β,2)

Energy definition

η_eff,en=(η₁E₁+η₂E₂)/(E₁+E₂)

β_eff,en=(β₁F₁+β₂F₂)/(F₁+F₂)

The energy option is the default used in the updated inverse-design comparison. The quartic Kerr weighting is shown as the provisional nonlinear extension described in that program.

Numerical defaults

Forward gridN=1000,r₂=1,u guess=2,λ guess=2.2

Forward tolerancefsolve xtol=10⁻¹² with residual validation

Inverse gridN=300,maximum 50 design iterations

Inverse updateRelaxation=0.5,design tolerance=10⁻⁶