# First-order perturbation theory of eigenmodes for systems with interfaces: data

In the related research, an exact first-order perturbation theory is presented for eigenmodes in systems with interfaces causing material discontinuities. It is shown when interfaces deform, higher-order terms of the perturbation series can contribute to the eigenmode frequencies in first order in the deformation depth. In such cases, the first-order approximation is different from the usual diagonal approximation and its single-mode result. Additional first-order corrections from all higher-order terms are extracted, enabling to recover the diagonal formalism in a modified form. A general formula for the single-mode first-order correction to electromagnetic eigenmodes in systems with interfaces is derived, capable of treating dispersive, magnetic, and chiral materials of arbitrary shape.

Data is given separately for each figure in the paper.

Figure 1:

(a) Schematic illustration of a Boundary Perturbation as a permittivity perturbation of the sphere, with its radius R changing by h.

(b) Effect of the BP on the wave number k of first RS with orbital number l = 1 in a sphere of permittivity ε = 4 surrounded by vacuum.

(c) Relative error of RS wave numbers calculated without static modes (squares), with all static modes (circles), and with 1000 static modes included (stars), for two different BPs as given.

fig1.opju: Sheet 'sizePert' contains the data, tab 'permittivity change plot' relates to (a), tab 'error vs h/R' relates to (b), and tabs 'error, h/R = 0.01' and 'error, h/R = 0.001' relate to (c).

Figure 3:

(a) Real and (b) imaginary parts of the wave number of the dipolar surface plasmon mode of a silver sphere perturbed to an ellipsoid, as sketched in the inset of (b). The mode degeneracy is shown in brackets, and m is its magnetic quantum number.

fig3.opju: Sheet 'shapeDeform' contains the data, tab '1st order' contains the first-order solutions obtained via the two different approaches (Eq.5 and Eq.18), tab 'COMSOL' contains the numerical data from COMSOL, the tab 'ellipse and circle plot' contains the data for the inset.

Figure 4:

Left:

(a) Field amplitude of a mode of a cylinder with kd = 2.9766 − 0.2014i in a plane containing the cylinder axis.

(c) Real and (e) imaginary parts of the mode wave number versus height change δh for first-order perturbations (lines) and COMSOL (dots), with the unperturbed mode highlighted in red and a sketch of the cylinder and its height perturbation in (c).

Right:

As left but for a mode with kd = 1.2496 − 0.0808i.

All modes have a magnetic quantum number of |m| = 1 and are twice degenerate (2) as noted in (c).

fig4.opju: Sheet 'cylinder' contains the data, with the tabs (labeled according to kd of the modes) containing the data for (c)-(f). (a) and (b) are screenshots from COMSOL.

Figure 5:

(a) Wave numbers of TM modes with angular momentum l = 1 of a dielectric sphere with permittivity ε = 4 and radius R, surrounded by vacuum.

(b) Illustration of the homogeneous permittivity perturbation across the sphere.

(c) Perturbed wave number of the fundamental RS with the permittivity change given by the color code.

(d) Magnitude of the matrix elements |Vnn| and |Snn| of the perturbation.

(e) Relative error of the perturbed RS wave numbers calculated with (crosses) and without (dots) inclusion of static modes.

fig5.opju: Sheet 'epsilonPert' contains the data, tab 'Unperturbed modes' relates to (a), tab 'permittivity change plot' relates to (b), tab 'error vs delta epsilon' relates to (c), tabs 'error, delta epsilon = 0.04' and 'error, delta epsilon = 0.004' relate to (d) and (e).

Figure 6:

(a) Illustration of the permittivity perturbation corresponding to a radius reduction of the sphere by −h.

(b) Perturbed wave number of the fundamental RS with the sphere radius change given by the color code.

(c) Magnitude of the matrix elements |Vnn| and |Snn| of the perturbation.

(d) Relative error of the perturbed RS wave numbers calculated with (crosses) and without (dots) inclusion of static modes.

(e) Absolute value of the error of the perturbed RS wave numbers relative to the wave number change due to the perturbation, as functions of Re(k_nh), for three selected modes, and k_ex denote the exact perturbed wavenumber. The dashed line indicates a

linear scale.

fig6.opju: Sheet 'sizePert' contains the data. Tab 'permittivity change plot' relates to (a), tab 'error vs h/R' relates to (b), tabs 'error, h/R = 0.01' and 'error, h/R = 0.001' relates to (c) and (d), and all remaining tabs relate to (e).

Figure 7:

(a) Wave numbers of TM modes with angular momentum l = 50 of a dielectric sphere with radius R and permittivity ε = 4, surrounded by vacuum. Inset shows a zoom of the high-quality whispering-gallery modes in a logarithmic scale.

(b) Magnitude of the matrix elements |Vnn| and |Snn| of the perturbation.

(c) Relative error of the perturbed RS wave numbers calculated with (crosses) and without (dots) inclusion of static modes.

(d) Absolute error of the perturbed RS wave numbers relative to the wave number change due to the perturbation, as functions of Re(k_nh), for three selected modes. Green line is the fundamental whispering-gallery mode.

fig7.opju: Sheet 'sizePert' contains the data. Tab 'Unperturbed modes' relates to (a), tabs 'error, h/R = 0.01' and 'error, h/R = 0.001' relates to (b) and (c), and all remaining tabs relate to (d).

Research results based upon these data are published at https://doi.org/10.1103/PhysRevResearch.5.013209