# An auxiliary field approach for computing optical resonances in dispersive media

**Part of the following topical collections:**

## Abstract

We report on an auxiliary field approach for solving nonlinear eigenvalue problems occurring in nano-optical systems with material dispersion. The material dispersion can be described by a rational function for the frequency-dependent permittivity, e.g., by a Drude-Lorentz model or a rational function fit to measured material data. The approach is applied to compute plasmonic resonances of a metallic grating.

## Keywords

Maxwell’s equations Material dispersion Nonlinear eigenvalue problems Auxiliary field approach## Introduction

Detailed knowledge on the resonant states of nano-optical systems is essential for understanding the physical properties of the systems and for designing related photonic devices [1, 2, 3]. With numerical approaches it is possible to compute the resonant states, which are typically solutions to nonlinear eigenvalue problems (NLEVPs) arising from Maxwell’s equations. The material dispersion described by the permittivity causes the nonlinearity of the eigenproblems. A multitude of numerical solution techniques are used for solving the NLEVPs, such as linearization, iterative projection methods and contour integral methods [4, 5, 6].

In nano-optics, linearization with physically derived auxiliary fields is a common approach [7, 8, 9, 10, 11, 12]. In this work, we report on an auxiliary field approach based on modeling the permittivity with rational functions. We implement the approach using an iterative projection method. Motivated by scatterometry applications, the numerical realization is applied to compute resonant states of a metallic line grating.

## Auxiliary field approach for dispersive nano-optical systems

where **E**(**r**,*ω*) is the electric field. The permittivity tensor *ε*(**r**,*ω*), depending on the complex angular frequency *ω* and the position **r**, describes the material dispersion and the spatial distribution of materials. For optical frequencies, the permeability tensor *μ*(**r**,*ω*) typically equals the vacuum permeability *μ*_{0}. Equation 1 becomes a non-Hermitian problem in the presence of open boundary conditions or lossy materials.

*B*(

*ω*), which is based on

*ε*(

**r**,

*ω*). If the permittivity model

*ε*(

**r**,

*ω*) is a rational function of the frequency with poles of order one, e.g., a Drude model [15] or a rational fit of measured material data, the matrix

*B*(

*ω*) has the form

where \(B_{0},\dots,B_{N} \in \mathbb {C}^{n \times n}\) are matrices resulting from the partial fraction decomposition and \(\omega _{1},\dots,\omega _{N} \in \mathbb {C}\) are the poles of the rational function. Note that physical dispersion models have to satisfy Kramers-Kronig relations to ensure causality.

*ω*and corresponding eigenvectors

*u*, an implementation of the shift-and-invert Arnoldi method is applied [16]. For this, the shifted eigenvalue \(\tilde {\omega } = \omega - \sigma \), the shifted poles \(\tilde {\omega _{i}} = \omega _{i} - \sigma, i=1,\dots,N\), and the auxiliary fields

*σ*is the chosen shift. As the matrices

*B*

_{1},…,

*B*

_{N}have only non-zero entries for degrees of freedom of the discretization corresponding to the dispersive object, the auxiliary fields

*u*

_{1},…,

*u*

_{N}can be restricted to this subset. However, for the sake of a simpler notation, we define them on the entire domain. Using the auxiliary fields with Eq. (3) to reformulate Eq. (2) yields

*u*and the auxiliary fields

*u*

_{0},…,

*u*

_{N}. The linear eigenvalue problem in Eq. (5) is solved by applying the Arnoldi method to

assuming that \(\tilde {u}\) is suitably scaled. The Arnoldi iteration typically converges to the largest eigenvalue, i.e., to the smallest shifted eigenvalue \(\tilde {\omega } = \omega - \sigma \). Thus, the eigenvalue *ω* of the NLEVP in Eq. (2) which is closest to the shift *σ* is obtained. Note that the auxiliary field approach increases the dimension of the eigenvalue problem with the number of poles of the rational function.

**Remark**For the computation of the Krylov subspace

*n*rows in Eq. (4) with the initial vector \(\tilde {v}\) for the right-hand side lead to

Instead of solving the linear system \(\tilde {\!A}\tilde {u} = \tilde {\!B}\tilde {v}\) to generate the Krylov subspace \(\mathcal {K}_{m}\), the system \(\hat {\!A}u = \tilde {\omega }\hat {f}\) is solved yielding *u* and Eq. (7) is used to achieve *u*_{0},…,*u*_{N}.

where *s*(*ω*) is a source term. Setting *ω*=*σ* yields \(\hat {\!A} = \left [A-\sigma ^{2}B(\sigma)\right ]\). Thus, the implementation of a scattering solver can also be used in the framework of solving eigenproblems.

## Application to metallic grating

*λ*=650 nm [18]. For the relative permittivity of the gold grating, a one-pole Drude model

*ω*

_{p}=1.26e+16 s

^{−1}is the plasma frequency and

*γ*=1.41e+14 s

^{−1}is the damping coefficient. The permittivity is then given by

*ε*(

*ω*)=

*ε*

_{0}

*ε*

_{r}(

*ω*), where

*ε*

_{0}is the vacuum permittivity. The chosen shift is

*σ*=2

*π*

*c*/(650 nm), where

*c*is the speed of light. Different finite element degrees

*p*=1,…,6 and a fixed mesh containing about 1e+03 triangles are applied. Corners are a known issue considering systems containing metals. To deal with the occurring field singularities at the corners, refinements with a minimum edge length of about 0.016 nm are used. Bloch boundary conditions with a Bloch vector of [2

*π*/(5

*a*),0,0] enforce the periodicity in

*x*direction. To realize the open boundary conditions in

*z*direction, perfectly matched layers (PMLs) are used. Convergence of the PML method is ensured by applying an adaptive numerical realization of the PML method [13]. The relative error of the eigenvalue

*ω*is shown in Fig. 2, where the reference solution

*ω*

_{ref}is the eigenvalue computed with

*p*=6. Convergence to the reference solution is observed. For the finite element degree

*p*=5, the eigenvalue

*ω*=2

*π*

*c*/(649.1397576+11.0601049

*i*nm±(6.2e−06+1.5e−06

*i*nm)) is obtained.

*σ*=2

*π*

*c*/(650 nm) as before is used to initialize the mass matrix

*B*(

*σ*). Equation 2 becomes linear and is solved with the shift-and-invert Arnoldi method. The resulting eigenvalue

*ω*

_{iter}is then used to update

*B*(

*ω*

_{iter}) and to repeat the procedure until

*ω*

_{iter}does not change up to a chosen tolerance. In Table 1, the relative difference between the results from the auxiliary field approach, denoted by

*ω*, and the results from the fixed-point iteration, denoted by

*ω*

_{iter}, are shown. An abort condition for the fixed-point iteration with a tolerance of 1e−08 is chosen for the real and imaginary parts of

*ω*

_{iter}. This leads to about 10 iterations. For all finite element degrees

*p*=1,…,5, matching results for the two approaches with a relative difference smaller than 4.1e−10 are obtained.

Comparison of eigenvalues computed with the auxiliary field approach and with the fixed-point iteration, denoted by *ω* and *ω*_{iter}, respectively

| \(\left | \frac {\text {Re}(\omega -\omega _{\text {iter}})} {\text {Re}(\omega _{\text {iter}})} \right |\) | \( \left | \frac {\text {Im}(\omega -\omega _{\text {iter}})} {\text {Im}(\omega _{\text {iter}})} \right |\) |
---|---|---|

1 | 6.6e−13 | 1.2e−10 |

2 | 1.0e−12 | 4.1e−10 |

3 | 9.9e−13 | 4.0e−10 |

4 | 9.4e−13 | 4.1e−10 |

5 | 9.3e−13 | 4.0e−10 |

## Conclusions

We have reported on an approach for computing eigensolutions to Maxwell’s equations in dispersive media. Auxiliary fields are used to linearize the corresponding NLEVP. The resulting linear eigenvalue problem is then solved with the shift-and-invert Arnoldi method. The approach has been applied to a metallic line grating and the results for the eigenvalues have been validated by an implementation of a fixed-point iteration.

## Notes

### Acknowledgements

We acknowledge Martin Hammerschmidt for helpful discussions.

### Funding

This work is carried out in the framework of MATHEON supported by Einstein Foundation Berlin (ECMath project OT9). This work is partially funded through the project 17FUN01 “BeCOMe” within the Programme EMPIR. The EMPIR initiative is co-founded by the European Union’s Horizon 2020 research and innovation program and the EMPIR Participating Countries.

### Availability of data and materials

Not applicable.

### Authors’ contributions

LZ implemented the numerical method. FB and SB performed the numerical experiments. FB wrote the manuscript with input from all authors. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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