Modeling Broadband Cloaking Using Nano-Assembled Plasmonic Meta-Structures

Interior field

To compute the interior field, we make use of the Method of Fundamental Solutions (MFS) which is also known as the Discrete Source Method. This method relies on knowing Green's function, denoted by \(G_{1}\), which satisfies \begin{equation} \left( \nabla^{2} + k_{1}^{2} \right) G_{1} = -\delta(r - r'), \end{equation} and the Sommerfeld radiation condition. It is given explicitly by \begin{equation} G_{1}(r - r') = \frac{e^{\mathrm{i} k_{1} | r - r '|}}{4 \pi | r - r' |}, \end{equation} which is a spherical wave centered at \(r'\) propagating in the whole space with wavenumber \(k_{1}\).

The key observation is that when \(r' \not\in D\), \(G_{1}\) exactly satisfies the differential equation for \(\psi^{\text{int}}\) since \(\delta(r - r') = 0\) for \(r \in D\) and \(r' \not\in D\). Because of this, we consider a set of \(M\) points \(r_{j}^{\text{int}} \in E\) for \(j = 1, \cdots, M\) and form the following approximation, $$\psi^{\text{int}}(r) \sum_{j =1}^{M} c^{\text{int}}_{j} G_{1}(r - \rho^{\text{int}}_{j}),r \in D. $$ This approximation is an exact solution of the differential equation for \(\psi^{\text{int}}\) for any set of points \(\rho_{j}^{\text{int}} \in E\) for \(j = 1, \cdots, M\). It is an approximation of the solution of this scattering problem because it does not exactly satisfy the boundary conditions. Rather, we will require that this approximation satisfies the boundary conditions only at a set of \(M\) points, which is called a collocation method.

In practice, we would like to have these points uniformly sample the interface \(B\) to make the approximation of the interface conditions as accurate as possible. Suppose the set of \(M\) points \(\rho^{\text{bdy}}_{j} \in B\) for \(j = 1, \cdots, B\) uniformly sample (or close to uniformly sample) the sphere of radius \(a_{0}\). Then, we set $$\rho^{\text{int}}_{j} = \rho^{\text{bdy}}_{j} + \ell \nu_{j}, \quad j = 1, \cdots, M, $$ with \(\nu_{j}\) denoting the unit outward normal at \(r_{j}\), and \(\ell > 0\) denoting a user-defined length parameter which we typically set to be smaller than a wavelength.

Exterior field

To approximate \(\psi^{\text{ext}}\) we make use of Green's identity which essentially states that the exterior field is given by a superposition of surface and volume sources. Here, surface sources correspond to scattering by the dielectric sphere and volume sources correspond to the \(N\) metal nanoparticles. In what follows, we approximate the surface sources using the Method of Fundamental Solutions and the volume sources using the self-consistent Foldy-Lax scattering theory, which is a scalar wave theory closely related to what is used in the Discrete Dipole Approximation.

In what follows, we represent \(\psi^{\text{ext}}\) as the following sum, \[\psi^{\text{ext}} = \psi^{\text{inc}} + \underbrace{\psi^{B} + \sum_{n = 1}^{N} \Psi_{n}}_{\psi^{\text{s}}}, \] with \(\psi^{B}\) denoting the "surface source" field scattered by the dielectric sphere and \(\Psi_{n}\) denoting the "volume source" field scattered by the \(n\)th metal nanoparticle. We describe our approximations for each of these scattered fields below.

Field scattered by the dielectric sphere

We use the Method of Fundamental solutions to compute the field scattered by the dielectric sphere. Here, we use the Green's function \(G_{0}\) satisfying $$ \left( \nabla^{2} + k_{0}^{2} \right) G_{0} = -\delta(r - r'), $$ and the Sommerfeld radiation condition, which is given explicitly by \[ G_{0}(r - r') = \frac{e^{\mathrm{i} k_{0} | r - r' |}}{4 \pi | r - r' |}.\] Just as we have done for the interior field, we introduce the set of M points,

\[\rho_{j}^{\text{ext}} = \rho^{\text{bdy}}_{j} - \ell \nu_{j},\quad j = 1, \cdots, M. \]

Field scattered by the metal nanoparticles

Let us denote the center of the \(n\)th metal nanoparticle by \(R_{n}\). In what follows, we assume that each of the \(N\) metal nanoparticles is small enough that we can account for its scattering properties by a scattering coefficient, \(\alpha_{n}\), which may depend on \(\lambda\). It follows that the field scattered by the \(n\)th metal nanoparticle is given by \[ \Psi_{n}(r) = \alpha_{n} G_{0}(r - R_{n}) \Psi_{E}(R_{n}),\] with \(G_{0}\) denoting the Green's function defined above and \(\Psi_{E}(R_{n})\) denoting the exciting field evaluated at \(R_{n}\). Supposing that \(R_{n}\) and \(\alpha_{n}\) for \(n = 1, \cdots, N\) are known, all that remains to be determined is \(\Psi_{E}(R_{n})\) for \(n = 1, \cdots, N\).

To determine this exciting field, we use Foldy-Lax theory which we describe below. Foldy-Lax theory is a self-consistent multiple scattering theory. It is not an exact theory, but it does account for infinitely many interactions between the metal nanoparticles, so it provides a useful multiple scattering model. This theory is limited to settings in which the metal nanoparticles are not too densely distributed over the dielectric sphere so that these scattering elements are statistically independent.

We model the field exciting the \(n\)th metal nanoparticle as the following sum, \[ \Psi_{E}(R_{n}) = \psi^{\text{inc}}(R_{n}) + \psi^{B}(R_{n}) + \sum_{\substack{n' = 1\\ n' \neq n}}^{N} \sigma_{n'} G_{0}(R_{n} - R_{n'}) \Psi_{E}(R_{n'}). \] This equation gives the exciting field at \(r_{n}\) as the sum of the incident field \(\psi^{\text{inc}}\), the field scattered by the dielectric sphere \(\psi^{B}\), and the fields scattered by all of the other \(N-1\) metal nanoparticles evaluated at \(r_{n}\). By rearranging this equation and considering it for each and all \(N\) metal nanoparticles, we obtain the \(N \times N\) linear system, \[ \Psi_{E}(R_{n}) - \sum_{\substack{n' = 1\\ n' \neq n}}^{N} \sigma_{n'} G_{0}(R_{n} - R_{n'}) \Psi_{E}(R_{n'}) = \psi^{\text{inc}}(R_{n}) + \psi^{B}(R_{n}), \quad n = 1, \cdots, N. \] Upon solution of this linear system, we obtain \(\Psi_{E}(r_{n})\) for \(n = 1, \cdots, N\). This result can then be used to evaluate \[ \Psi(r) = \sum_{n = 1}^{N} \alpha_{n} G_{0}(r - R_{n}) \Psi_{E}(R_{n}), \quad r \in E. \] This field gives the field scattered by the entire set of \(N\) metal nanoparticles. Because this field is given as a superposition of Green's functions, each of which satisfies the Sommerfeld radiation condition, it follows that \(\Psi\) satisfies the Sommerfeld radiation condition.

Interface conditions

At this point we have introduced approximations for the interior and exterior fields. Both are exact solutions of the associated differential equations and the exterior field satisfies the Sommerfeld radiation condition. All that remains is to require that these fields satisfy interface conditions given in Section 1.

Let us review what we have so far. The approximation of the interior field is given by \[ \psi^{\text{int}}(r) \cong \sum_{j = 0}^{M} c_{j}^{\text{int}} G_{1}(r - \rho_{j}^{\text{int}}) \quad r \in D. \] The approximation of the exterior field is given by \[ \psi^{\text{ext}}(r) \cong \psi^{\text{inc}}(r) + \underbrace{\sum_{j = 1}^{M} c_{j}^{\text{ext}} G_{0}(r - \rho_{j}^{\text{ext}})}_{\psi^{B}} + \sum_{n = 1}^{N} \alpha_{n} G_{0}(r - R_{n}) \Psi_{E}(R_{n}), \quad r \in E,\] with \(\Psi_{E}(R_{n})\) satisfying \[ \Psi_{E}(R_{n}) - \sum_{\substack{n' = 1\\ n' \neq n}}^{N} \sigma_{n'} G_{0}(R_{n} - R_{n'}) \Psi_{E}(R_{n'}) \] \[ = \psi^{\text{inc}}(R_{n}) + \sum_{j = 1}^{M} c_{j}^{\text{ext}} G_{0}(R_{n} - \rho_{j}^{\text{ext}}), \quad n = 1, \cdots, N. \] The quantities \(c_{j}^{\text{int}}\) for \(j = 1, \cdots, N\), \(c_{j}^{\text{ext}}\) for \(j = 1, \cdots, N\), and \(\Psi_{E}(R_{n})\) for \(n = 1, \cdots, N\) are to be determined.

By requiring that the interface condition for the fields is satisfied exactly on the \(M\) boundary points, \(\rho^{\text{bdy}}_{i} \in B\) for \(i = 1, \cdots, M\), we find that

\[ \sum_{j = 0}^{M} c_{j}^{\text{int}} G_{1}(\rho^{\text{bdy}}_{i} - \rho_{j}^{\text{int}}) - \sum_{j = 1}^{M} c_{j}^{\text{ext}} G_{0}(\rho^{\text{bdy}}_{i} - \rho_{j}^{\text{ext}}) - \sum_{n = 1}^{N} \alpha_{n} \] \[ G_{0}(\rho^{\text{bdy}}_{i} - R_{n}) \Psi_{E}(R_{n}) = \psi^{\text{inc}}(\rho^{\text{bdy}}_{i}),\quad i = 1,\cdots, M. \]

\[\sum_{j = 0}^{M} c_{j}^{\text{int}} \partial_{\nu} \\ G_{1}(\rho^{\text{bdy}}_{i} - \rho_{j}^{\text{int}}) - \sum_{j = 1}^{M} c_{j}^{\text{ext}} \partial_{\nu} G_{0}(\rho^{\text{bdy}}_{i} - \rho_{j}^{\text{ext}})\] \[-\sum_{n = 1}^{N} \alpha_{n} \partial_{\nu} G_{0}(\rho^{\text{bdy}}_{i} - R_{n}) \Psi_{E}(R_{n})=\partial_{\nu} \psi^{\text{inc}}(\rho^{\text{bdy}}_{i}), \quad i = 1, \cdots, M.\]

We now present the codes that computes the solution of this linear system and uses that result to compute the field scattered by this cloaking structure.

Computing the total scattering cross-section

After solving the system of linear equation described in the last section (avobe), thescattered filed is given by,

\[\psi^{s}(r) = \psi^{B} + \sum_{n=1}^{N} \Psi_{n}\approx \sum_{j=1}^{M}c_{j}^{ext}G_{0}(r-r_{j}^{ext}) + \sum_{n=1}^{N}\alpha_{n}G_{0} (r-r^{NP}_{n})\Psi_{E}(r_{n}^{NP}), r \in E.\]

The scattered field evaluated at \(\textbf{r}= R\hat{o}\) with \(R=|r|\) and \(\hat{o} = \textbf{r} / |\textbf{r}|\) in the far-field corresponding to \(R>>1\) behaves like a spherical wave and is given by,

\[\psi^{s} \sim f(\hat{o},\hat{i}) \frac{e^{ik_{0}}R}{R} \]

Here, \(f(\hat{o},\hat{i})\) is the scattering amplitude for the scattered field in the far field in the direction \(\hat{o}\) when the particle is illuminated by a plane wave propagating in the direction \(\hat{i}\) with unit amplitude.

Suppose we have set \(\psi^{inc}\) to be a plane wave of unit amplitude propagating in the direction \(\hat{i}\) and we have used that compute the the solution of the last three equation described in the section above. Using \(|R\hat{o}-r^{'}|\sim R- \hat{o}.r^{'}\) for \(R>>1\), in the far-field Green's function is ,

\[G_{0}(R\hat{o}-r^{'}) \sim e^{ik_{0}\hat{o}.r^{'}}\frac{e^{ik_{0}R}}{4\pi R}, R>>1\]

Replacing \(G_{0}\) by the first two equations from this section we get,

\[\psi^{s}(r) \sim \left[ \frac{1}{4\pi} \sum_{j=1}^{M} c_{j}^{ext} e^{ik_{0}\hat{o}.r^{ext}_{j}} + \sum_{n=1}^{N} \alpha_n e^{ik_{0}\hat{o}.r^{NP}_{n}} \Psi_{E}(r_{n}^{NP}) \right]\frac{e^{ik_{0}R}}{R}, R>>1\]

By comparing this last equation with the second equation from beginning of this section we find the scattering amplitude using our method given as,

\[f(\hat{o}, \hat{i}) = \frac{1}{4\pi} \sum_{j=1}^{M} c_{j}^{ext} e^{ik_{0}\hat{o}.r^{ext}_{j}} + \frac{1}{4\pi} \sum_{n=1}^{N} \alpha_n e^{ik_{0}\hat{o}.r^{NP}_{n}} \Psi_{E}(r_{n}^{NP}) \]

According to the Optical Theorem of the forward scattering theorem, we can compute the total cross-section of the scattering structure through evaluation of ,

\[\sigma_{t} = \frac{4\pi}{k_{0}}Im[f(\hat{i},\hat{i})]\]

Now we can compute the total cross-section by substituiting equation above the last equation into this formula. Remarkably, the last equation of this section allows for the determination of \(\sigma_{t}\) by only measuring scattered power in the forward direction.