Practical Implementation of Accurate Finite-Element Calculations for Electromagnetic Scattering by Nanoparticles

Abstract

The finite-element method (FEM) is increasingly used as a numerical tool to support experimental and theoretical studies of the optical properties of nanoparticles, in contexts such as surface-enhanced spectroscopy, molecular plasmonics, metamaterials, and optical trapping. Here, we investigate the validity of such calculations, focusing in particular on numerically challenging cases involving strong optical (plasmon) resonances and solutions with large electric field gradients and intensities. These are exemplified by elongated metallic nanoparticles and two closely spaced metallic spheres (dimer), where highly localized regions of intense electric field enhancements occur at the tip or in the gap, so-called electromagnetic hot-spots. We assess the accuracy of the FEM solutions by comparing the result to exact analytic solutions based on the T-matrix method for an elongated particle and generalized Mie theory for a dimer. Particular attention is given to the electromagnetic properties that have seldom been studied in this context, notably near-field properties such as surface-field enhancement factors and far-field radiation profiles. We also demonstrate explicitly how the accuracy of the FEM predictions can be inferred from the solution of two problems with different mesh and bounding box parameters. Such a numerical check is crucial in practice as no exact solutions are in general available to compare with. While we chose the commercial software COMSOL to illustrate our results, the methods and conclusions are equally applicable to other FEM implementations. We provide for convenience full details of how to set up these calculations in COMSOL, which we hope will allow readers to easily reproduce them and seamlessly adapt them to their modeling needs. We expect this work will cement the FEM as a reliable method for routine calculation of electromagnetic scattering by nanoparticles.

Appendix A: Additional Details for the Practical Implementation in COMSOL

One example of COMSOL report file is provided as a Supplementary Material for the 60-nm diameter gold sphere embedded in a medium with 𝜖₁ = 4 simulation. It contains the full details of the simulations set up and should allow other COMSOL user to rerun the very same simulations.

A.1 General Comments

In the following, we assume the user is already familiar with COMSOL and only highlight a number of important comments/tips/clarifications for setting up this simulation, along with common traps/mistakes:

• All simulations were set up using the “Electromagnetic Waves, Frequency Domain (emw)” node of COMSOL 5.3 (which is part for example of the wave optics or radio frequency modules). This mode solves for the complex electric field, which is a vector quantity denoted Ex, Ey, Ez. Most other predefined quantities in the model are called as emw.variablename and can be seen in the emw node provided that the option “equation view” has been selected in the “Model Builder” window. Additional variables can be defined by the user prior to the computation.

• For convenience, we define a new parameter in the “Global Definitions” of the model builder called lambda corresponding to the wavelength in nm. Wavelength dependence is then carried out as a parametric sweep on lambda.

• It should be noted that COMSOL uses the engineer convention for the definition of the complex fields, i.e., with a factor $\exp (+i\omega t)$ for the time dependence; hence, the imaginary part of the dielectric function is negative for passive materials. In practice, 𝜖 is simply the complex-conjugate of what it would be with the $\exp (-i\omega t)$ convention more commonly used in physics. For example, the relative dielectric function of gold is obtained from the analytic expression of Ref. [32] as:

conj(1.54*(1-1/(177.5^2*((1/lambda)^2 +i/(14500*lambda)))) +1.27/470*(exp(i*(-pi/4))/(1/470-1/lambda -i/1900)+exp(-i*(-pi/4))/(1/470+1/lambda +i/1900))+1.1/325*(exp(i*(-pi/4))/(1/325 -1/lambda-i/1060)+exp(-i*(-pi/4))/(1/325 +1/lambda+i/1060)))

• The variable epsilon1 is defined in the global definitions and is the relative dielectric constant, 𝜖₁, of the external/surrounding medium.

• In order to implement PMLs on the bounding box, it is important to solve the problem for “scattered field”, i.e., for E_sca as opposed to solving directly for the total field E = E_sca + E_inc. This option is specified in the “physics” (emw) node. We will only consider plane waves for the incident field E_inc (called “background electric field” in the emw node). This is set as $E_{0} \exp (-ik_{1} z) \mathbf {e}_{x}$ (i.e., x-polarized and propagating along + z) with $k_{1} =k_{0} \sqrt {\epsilon _{1}}$.

A.2 Computation of Electromagnetic Properties in COMSOL

The electromagnetic (EM) properties considered in this work are defined in the main text. In COMSOL, they can be computed by defining new “variables” as described below.

• For convenience, we first define several commonly used integrals as “components couplings” in the “definitions” node of COMSOL.

  int_surf: surface (double) integral on the NP surface.

  int_vol: volume (triple) integral inside the NP.

  int_OT: this is defined as an integral over a point located in the forward scattering direction and therefore is not per se an integral, but just a convenient way of evaluating the forward scattering amplitude to calculate the extinction cross section with the optical theorem (see below).

• The scattering cross section is calculated by integrating the flux of the complex Poynting vector of the scattered field S_sca across the nanoparticle surface:

  $$ \sigma_{\text{sca}}= \frac{1}{S_{0}}\iint\limits_{\text{NP}} \text{Re}\left( \mathbf{S}_{\text{sca}} \cdot \mathbf{n}\right) dS, $$

  where ${S_{0}=\frac {{E_{0}^{2}}}{2 Z_{1}}}$ is the power density of the incident field and $Z_{1} = Z_{0} / \sqrt {\epsilon _{1}}$ with Z₀ = μ₀c =≈ 376.73 Ω the characteristic impedance of vacuum. E₀ is chosen arbitrarily as 1 V/m in all the simulation, but by linearity the results do not depend on this choice. For this we therefore define the following parameters:

E0 = 1 [V/m] Z1 = Z0_const / sqrt(epsilon1) S0 = E0^2 / (2*Z1)

And the variables:

nrelPoav=nx*up(emw.relPoavx) +ny*up(emw.relPoavy) +nz*up(emw.relPoavz) sigma_sc = int_surf(nrelPoav)/S0

• The absorption cross section is calculated from the volume integral of the total power dissipation density inside the NP:

  $$ \sigma_{\text{abs}}= \frac{1}{S_{0}}\iiint\limits_{\text{NP}} Q_{h} dV. $$

  In COMSOL, this gives:

  sigma_abs = int_vol(emw.Qh)/S0

• The extinction cross section is calculated in two ways, either from energy conservation or from the optical theorem (OT):

  
  

  k1 = emw.k0 * sqrt(epsilon1) sigma_ext = sigma_sc+sigma_abs sigma_extOT = -4*pi/k1 *int_OT(imag(emw.Efarx*1[m]))/E0

• The surface-averaged local field intensity enhancement factor is obtained from

  $$ \langle M \rangle=\frac{1}{A}\iint\limits_{\text{NP}}\left| \frac{\mathbf{E}(\mathbf{r})}{E_{0}} \right|^{2} dS $$

  where A is the surface area of the particle and E is the surface field (outside the NP). Explicitly:

  
  

  sigma_geom = int_surf(1) M_ave = int_surf((up(emw.normE))^2) /sigma_geom

It is important to point out that for surface integrals of non-continuous quantities (like the electric field across the NP surface),COMSOL will by default take the average value of the outside and inside integrand. This is clearly not what we want, so we need to specify that we choose the norm of the electric field on only one side of the surface. Every surface in COMSOL has two opposite intrinsic normal unit vectors defined at every point, called up and down. Their respective coordinates are denoted (unx,uny,unz) and (dnx,dny,dnz) and by plotting those vectors in an arrow plot, it is possible to check which side is up and which is down. In our case, the up side is the one outside the NP, hence the use of up(emw.normE) to specify the field norm outside.

A.3 Setting up the Perfectly Matched Layers

As the computational domain has to be meshed, a bounding box yielding appropriate physical conditions needs to be defined. The nature of the bounding box then depends on the problem at stake. Periodic boundary conditions will for example be chosen to simulate an array of particles. In the present case of an isolated particle, the bounding box has to be designed so as to prevent any reflection of the incident or scattered waves. One of the most efficient approach to achieve this is to set the bounding box as perfectly matched layers (PML). COMSOL provides ready-to-use PML settings, leaving the user to simply choose the most appropriate PML geometry for his problem, which is usually straightforward, though it remains the user’s responsibility to ensure an adequate meshing of this region, meaning that a sufficient number of mesh elements is present across the thickness of the PML. Failing to do so will result in an inefficient PML and lead to an inaccurate solution. For typical PML thicknesses of a fraction of the incident wavelength, the default meshing typically results in mesh nodes present only on the surfaces enclosing the PML. In other words, the PML region would only consist of a single mesh layer. To avoid this and refine the meshing, it is necessary to use a swept mesh across the PML domain, where the number of elements can be specified explicitly.

Note that for a cubical (“cartesian”) PML region, one should make sure to mesh the cube faces using “mapped surfaces.” All the internal surfaces of the cube have then to be converted into triangles in order to match the tetrahedral meshing of the volume.

The rest of the geometry can be meshed using standard methods, where triangles and tetrahedrons are used to mesh the surfaces and the volumes. The size of the meshing can be adjusted either with predefined COMSOL settings (...coarse, fine, finer,...) or with custom settings. Different elements in the geometry can be meshed with different precision by adding a “size” node to the considered element. See the provided example file for further details.

A.4 Far-Field Computations

We are interested in retrieving the optical far-field properties, such as the radiation profile or scattering matrix, of the particle under study. The size of the bounding box and our computing resources (and time) being finite, we have to resort to other solutions than just evaluating the fields “far” from the NP. A common approach for such problems is the use of near-field to far-field transformations, where the far-field amplitudes are obtained typically from integrals of near-field quantities. For scattering by a particle, the most common approach is based on the Stratton-Chu formula [27], where the far-field is obtained as an integral over the particle surface. In COMSOL 5.3, a “far-field” module has been implemented based on this approach. We summarize this method below to clarify the notations and conventions of far-field calculations in COMSOL.

The basic idea is that the knowledge of the EM field at the surface of the NP is able to provide the EM field anywhere else, explicitly (see [28] or Eq. 5.168 in [11]):

$$ \begin{array}{@{}rcl@{}} \mathbf{E}_{\text{sca}}(\mathbf{r}) &=& \iint\limits_{\text{NP}} dS^{\prime} \left\{ - i \omega \mu_{0} [\mathbf{n} \times \mathbf{H}(\mathbf{r}^{\prime})] \cdot G\&\#x20E1; (\mathbf{r}, \mathbf{r}^{\prime})\right. \\ &&\quad\quad\quad\quad\left. + [\mathbf{n} \times \mathbf{E}(\mathbf{r}^{\prime})] \cdot \left[\boldsymbol \nabla \times G\&\#x20E1;(\mathbf{r}, \mathbf{r}^{\prime}) \right] \right\} \end{array} $$

(1)

where $G\&\#x20E1;(\mathbf {r}, \mathbf {r}^{\prime })$ is dyadic Green’s function in the surrounding medium:

$$ \begin{array}{@{}rcl@{}} G\&\#x20E1;(\mathbf{r},\mathbf{r}^{\prime}) = \left( I\&\#x20E1; +\frac{1}{{k_{1}^{2}}} \boldsymbol \nabla \otimes \boldsymbol \nabla \right)g(\mathbf{r}, \mathbf{r}^{\prime}) \end{array} $$

(2)

with

$$ g(\mathbf{r}, \mathbf{r}^{\prime})= \frac{e^{-ik_{1} \vert \mathbf{r} - \mathbf{r}^{\prime} \vert}}{4 \pi \vert \mathbf{r} - \mathbf{r}^{\prime} \vert} $$

the scalar Green’s function, I⃡ the identity dyadic, and $k_{1}=k_{0}\sqrt {\epsilon _{1}}$ the wavenumber in the medium. Note that all those expressions are written here using the $\exp (+i\omega t)$ convention used by COMSOL.

Following [28], we define the induced electric p ≡n ×H and magnetic m ≡n ×E dipole moments.

As we are interested in the far-field (FF) ($r \longrightarrow \infty$, E(r) and H(r) perpendicular to r), the following approximations can be made [28]:

$$ \begin{array}{@{}rcl@{}} \mathbf{p} \cdot G\&\#x20E1;(\mathbf{r}, \mathbf{r}^{\prime}) &\overset{\textsf{FF}}{\backsimeq} &\frac{e^{-ik_{1}R}}{4 \pi R}\left[\mathbf{p} - (\mathbf{p}\cdot\mathbf{e}_{r}) \mathbf{e}_{r}\right], \end{array} $$

(3)

$$ \begin{array}{@{}rcl@{}} \mathbf{m} \cdot \left[\boldsymbol \nabla \times G\&\#x20E1;(\mathbf{r}, \mathbf{r}^{\prime})\right] & \overset{\textsf{FF}}{\backsimeq}& -ik_{1} \frac{e^{-ik_{1}R}}{4 \pi R} [\mathbf{e_{r}} \times \mathbf{m}], \end{array} $$

(4)

$$ \begin{array}{@{}rcl@{}} \frac{e^{-ik_{1}R}}{R} &\overset{\textsf{FF}}{\backsimeq}& \frac{e^{-ik_{1}r}}{r}e^{ik_{1} \mathbf{e_{r}}\cdot\mathbf{r}^{\prime}},\quad \mathbf{e_{r}}=\frac{\mathbf{r}}{r}. \end{array} $$

(5)

where $R=|\mathbf {r} - \mathbf {r}^{\prime }|$.

These approximations justify writing the scattered field in the far-field region as:

$$ \begin{array}{@{}rcl@{}} \mathbf{E}_{\text{sca}}(\mathbf{r}) \overset{\textsf{FF}}{\backsimeq} \frac{e^{-ik_{1}r}}{r} \left( E_{\text{far}}^{\theta}(\theta,\varphi) \mathbf{e}_{\theta}+E_{\text{far}}^{\varphi}(\theta,\varphi) \mathbf{e}_{\varphi} \right). \end{array} $$

(6)

This expression defines the far-field amplitudes $E^{\theta }_{\text {far}}$ and $E^{\varphi }_{\text {far}}$ as they are defined in COMSOL, but we note that different conventions may be used for this definition. We also note that within this definition, the far-field amplitudes have the dimension of electric field times length. We can also write the far-field components as:

$$ \begin{array}{@{}rcl@{}} E_{\text{far}}^{\theta} = \frac{-i}{4\pi} \iint\limits_{\text{NP}} dS^{\prime} \left[ \omega \mu_{0} p_{\theta}-k_{1} m_{\varphi} \right]e^{ik_{1}\mathbf{e}_{r}\cdot \mathbf{r}^{\prime}} \\ E_{\text{far}}^{\varphi} = \frac{-i}{4\pi} \iint\limits_{\text{NP}} dS^{\prime} \left[ \omega \mu_{0} p_{\varphi}+ k_{1} m_{\theta} \right]e^{ik_{1}\mathbf{e}_{r}\cdot \mathbf{r}^{\prime}} \end{array} $$

or equivalently as

$$ \begin{array}{@{}rcl@{}} E_{\text{far}}^{\theta} &=& \frac{-i k_{1}}{4\pi} \iint\limits_{\text{NP}} dS^{\prime} \left[ Z_{1} p_{\theta}- m_{\varphi} \right]e^{ik_{1}\mathbf{e}_{r}\cdot \mathbf{r}^{\prime}} \\ E_{\text{far}}^{\varphi} &=& \frac{-i k_{1}}{4\pi} \iint\limits_{\text{NP}} dS^{\prime} \left[ Z_{1} p_{\varphi}+m_{\theta} \right]e^{ik_{1}\mathbf{e}_{r}\cdot \mathbf{r}^{\prime}} \end{array} $$

(7)

where $Z_{1} = Z_{0} / \sqrt {\epsilon _{1}}$ has been defined earlier.

The far-field module in COMSOL provides a ready-made implementation of those formulae. It is necessary to add a “far-field domain” node in the “emw” node, which automatically adds a far-field option in the “post-processing” node of COMSOL. This simplifies the output of variables that are defined on the unit sphere (i.e., only depend on 𝜃 and φ). Variables for the far-field amplitudes defined above can then be computed and plotted.

The SPlaC implementation [8, 33] of Mie theory provides the elements S₁(𝜃) and S₂(𝜃) of the scattering amplitude matrix with the same definitions and conventions as in Ref. [43]:

$$ \begin{array}{@{}rcl@{}} \mathbf{E}_{\text{sca}}(\mathbf{r}) \overset{\textsf{FF}}{\simeq} \frac{e^{ik_{1}r}}{-ik_{1} r} E_{0} \left[ \cos\varphi S_{2}(\theta) \mathbf{e}_{\theta} + \sin\varphi S_{1}(\theta)\mathbf{e}_{\varphi} \right]. \end{array} $$

(8)

Taking into account the different convention for time-dependence ($\exp (-i\omega t)$), we therefore have the following equivalence between this formulation and the one of COMSOL summarized above:

$$ \begin{array}{@{}rcl@{}} E_{\text{far}}^{\theta} (\theta,\phi=0) &\equiv& S_{2}(\theta)^{*}/(ik_{1}) \end{array} $$

(9)

$$ \begin{array}{@{}rcl@{}} E_{\text{far}}^{\phi} (\theta,\phi=\pi/2) &\equiv& -S_{1}(\theta)^{*}/(ik_{1}) \end{array} $$

(10)

These formulae were used to test the results of FEM far-field computations against Mie theory.