Fig. 2: Numerical investigation of the high-Q resonance of a nanophotonic resonator.

a Nanoresonator on a three-layer substrate. The substrate is infinitely extended in x and y direction. The geometrical parameters p₁, p₂, …, p₅ are the reference values from Koshelev et al.³². b Calculated eigenfrequency \(\tilde{\omega }=(1.17309-0.00296i)\times 1{0}^{15}\,{{{{{{{{\rm{s}}}}}}}}}^{-1}\) corresponding to the high-Q resonance. The other red crosses shown are the two eigenfrequencies which are closest to \(\tilde{\omega }\). The circular integration contour \(\tilde{C}\) with the center ω₀ = 2πc/(1600 nm) and the radius r₀ = ω₀ × 10⁻² is used for computing Riesz projections. c Electric field intensity \(|{\tilde{{{{{{{{\bf{E}}}}}}}}}}|^{2}\) corresponding to the high-Q resonance. d Convergence of the eigenfrequency sensitivities \(\partial \tilde{\omega }/\partial {p}_{i}\) with respect to the polynomial degree d of the FEM ansatz functions. The sensitivities are computed at the parameter reference values given in (a). Relative errors \({{{{{\rm{err}}}}}}_{{{{{{{{\rm{real}}}}}}}},i}=| {{{{{{{\rm{Re}}}}}}}}(\frac{\partial \tilde{\omega }}{\partial {p}_{i}}(d)-\frac{\partial \tilde{\omega }}{\partial {p}_{i}}({d}_{{{{{{{{\rm{ref}}}}}}}}}))/{{{{{{{\rm{Re}}}}}}}}(\frac{\partial \tilde{\omega }}{\partial {p}_{i}}({d}_{{{{{{{{\rm{ref}}}}}}}}}))|\), where d_ref = 6. e Relative errors err_imag,i for the imaginary parts of the sensitivities; cf. (d).