Topology Optimization-Based Computational Design Methodology for Surface Plasmon Polaritons

Abstract

This paper presents the topology optimization-based computational design methodology for nanostructures in surface plasmon polaritons. Using the proposed method, nanostructures can be designed solely based on the user’s desired performance specification for the surface plasmon polaritons. This topology optimization-based computational design methodology is implemented based on the material interpolation with hybrid formulation of logarithmic and power law approaches, to mimic the metal surface with exponential decay of the electromagnetic field. The constructed computational design problem is analyzed using the continuous adjoint method, and the filter and projection techniques are utilized to ensure the minimum length scale in the obtained nanostructures. The outlined design methodology is used to investigate the nanostructures for localized surface plasmonic resonances, extraordinary optical transmission, and surface plasmonic cloaking, respectively. For localized surface plasmonic resonances and extraordinary optical transmission, the metallic nanostructures are designed with spectra peaks at the prescribed wavelengths and the shift of the spectra peak is controlled by solving the computational design problem corresponding to a different incident wavelength; for surface plasmonic cloaking, the cloak covered at a curved metal-dielectric interface is designed to bound the surface plasmon polariton at the interface and remove the radiation, where the conventional simple isotropic dielectric readily available in nature is used instead of the material possessing gradient electromagnetic properties with challenges on realization for optical frequencies.

Appendix

The details on deriving the adjoint equations and derivative of the computational design problem in Eq. 9 are presented as follows. According to the Lagrangian multiplier method, the augmented Lagrangian functional corresponding to the variational problem in Eq. 9 can be introduced as

$$ \begin{array}{ll} \hat{J} = & \displaystyle{\int}_{\Omega} A\left(H_{zs},\nabla H_{zs}, \rho_{fp}; \rho\right)\,\mathrm{d}{\Omega} + {\int}_{\partial{\Omega}} B\left(H_{zs}\right)\,\mathrm{d}{\Gamma} \\ & + {\Phi}\left(H_{zs},\nabla H_{zs}, \rho_{fp}; \rho\right) + {\Psi}\left(\rho_{f},\nabla \rho_{f}; \rho\right) \end{array} $$

(23)

where

$$ \begin{array}{llllll} &{} {\Phi}\left(H_{zs},\nabla H_{zs}, \rho_{fp}; \rho\right) = \displaystyle {\int}_{\Omega} - \epsilon_{r}^{-1} \nabla \left(H_{zs}+H_{zi}\right) \cdot \nabla \tilde{H}_{zs}^{*}\\ & \qquad+ {k_{0}^{2}} \mu_{r} \left(H_{zs}+H_{zi}\right) \tilde{H}_{zs}^{*} + \displaystyle{\int}_{{\Gamma}_{ab}} \bigg(- j k_{0} \sqrt{\epsilon_{r}^{-1} \mu_{r}} H_{zs} \\ & \qquad+ \epsilon_{r}^{-1} \nabla H_{zi} \cdot \mathbf{n} \bigg) \tilde{H}_{zs}^{*} \,\mathrm{d}{\Gamma} + \displaystyle{\int}_{{\Gamma}_{pd}} \epsilon_{r}^{-1} \nabla \left(H_{zs}+H_{zi}\right) \cdot \\ & \qquad\mathbf{n} \tilde{H}_{zs}^{*} \,\mathrm{d}{\Gamma} + \displaystyle{\int}_{{\Gamma}_{ps}} \epsilon_{r}^{-1} \nabla \left(H_{zs}+H_{zi}\right) \cdot \mathbf{n} \tilde{H}_{zs}^{*} \,\mathrm{d}{\Gamma} \\ & \qquad+ \displaystyle{\int}_{{\Gamma}_{sm}} \epsilon_{r}^{-1} \nabla H_{zi} \cdot \mathbf{n} \tilde{H}_{zs}^{*} \,\mathrm{d}{\Gamma} \end{array} $$

(24)

$$\begin{array}{@{}rcl@{}} {\Psi}\left(\rho_{f},\nabla \rho_{f}; \rho\right)&=& {\int}_{\Omega} r^{2} \nabla \rho_{f} \cdot \nabla \tilde{\rho}_{f}^{*} \,\mathrm{d}{\Omega} \\ &&+ {\int}_{\Omega} \rho_{f} \tilde{\rho}_{f}^{*} \,\mathrm{d}{\Omega} - {\int}_{\Omega} \rho \tilde{\rho}_{f}^{*} \,\mathrm{d}{\Omega} \end{array} $$

(25)

$\tilde {H}_{zs}$ and $\tilde {\rho }_{f}$ are the adjoint variables of H _{z s} and ρ _f , respectively; $\tilde {H}_{zs}$ satisfies the same periodicity as H _{z s} . H _{z s} and ρ _f are distributions in $\mathcal {H}^{1}\left ({\Omega }\right )$, the first-order Sobolev space defined on Ω; $\tilde {H}_{zs}$ and $\tilde {\rho }_{f}$ are distributions in $\mathcal {H}^{1*}\left ({\Omega }\right )$, the dual space of $\mathcal {H}^{1}\left ({\Omega }\right )$; ρ is the distribution in $\mathcal {L}^{2}\left ({\Omega }\right )$, the second-order Lebesgue integrable functional space; for functional space, ∗ represents the dual space; for complex, ∗ represents the conjugate operation. Because of the periodicity of the field expressed by the periodic boundary condition, it is satisfied that

$$ \begin{array}{ll} & \displaystyle{\int}_{{\Gamma}_{pd}} \epsilon_{r}^{-1} \nabla \left(H_{zs}+H_{zi}\right) \cdot \mathbf{n} \tilde{H}_{zs}^{*} \,\mathrm{d}{\Gamma} \\ & \qquad + \displaystyle {\int}_{{\Gamma}_{ps}} \epsilon_{r}^{-1} \nabla \left(H_{zs}+H_{zi}\right) \cdot \mathbf{n} \tilde{H}_{zs}^{*} \,\mathrm{d}{\Gamma} = 0 \end{array} $$

(26)

Then Eq. 24 can be reduced to be

$$\begin{array}{@{}rcl@{}} & {} {\Phi}\left(H_{zs},\nabla H_{zs}, \rho_{fp}; \rho\right) = \displaystyle{\int}_{\Omega} - \epsilon_{r}^{-1} \nabla \left(H_{zs}+H_{zi}\right) \cdot \nabla \tilde{H}_{zs}^{*}\\ &{\kern-6.2pc} + {k_{0}^{2}} {\upmu}_{r} \left(H_{zs}+H_{zi}\right) \tilde{H}_{zs}^{*} \,\mathrm{d}{\Omega} \\ & \qquad + \displaystyle{\int}_{{\Gamma}_{ab}} \left(- j k_{0} \sqrt{\epsilon_{r}^{-1} {\upmu}_{r}} H_{zs} + \epsilon_{r}^{-1} \nabla H_{zi} \cdot \mathbf{n} \right) \tilde{H}_{zs}^{*} \,\mathrm{d}{\Gamma} \\ & {\kern-2.9pc} + \displaystyle {\int}_{{\Gamma}_{pd}\bigcup{\Gamma}_{ps}\bigcup{\Gamma}_{sm}} \epsilon_{r}^{-1} \nabla H_{zi} \cdot \mathbf{n} \tilde{H}_{zs}^{*} \,\mathrm{d}{\Gamma} \end{array} $$

(27)

From the first-order variational of $\hat {J}$, one can obtain

$$\begin{array}{@{}rcl@{}} &{}\left<{{\delta\hat{J}}\over{\delta\rho}},\delta\rho\right>_{\mathcal{L}^{2*},\mathcal{L}^{2}}\,=\,\left<J_{H_{zs}},\delta H_{zs}\right>_{\mathcal{H}^{1*},\mathcal{H}^{1}} + \left<{\Phi}_{H_{zs}},\delta H_{zs}\right>_{\mathcal{H}^{1*},\mathcal{H}^{1}}\\ & {\kern-1.7pc}+ \left<J_{\nabla H_{zs}},\nabla \delta H_{zs}\right>_{\boldsymbol{\mathcal{L}}^{2*},\boldsymbol{\mathcal{L}}^{2}} + \left<{\Phi}_{\nabla H_{zs}},\nabla \delta H_{zs}\right>_{\boldsymbol{\mathcal{L}}^{2*},\boldsymbol{\mathcal{L}}^{2}} \\ & + \left<J_{\rho_{fp}}{\rho_{fp}}_{\rho_{f}},\delta \rho_{f}\right>_{\mathcal{H}^{1*},\mathcal{H}^{1}} + \left<{\Phi}_{\rho_{fp}}{\rho_{fp}}_{\rho_{f}},\delta \rho_{f}\right>_{\mathcal{H}^{1*},\mathcal{H}^{1}} \\ & {\kern-3.8pc}+ \left<{\Psi}_{\rho_{f}},\delta \rho_{f}\right>_{\mathcal{H}^{1*},\mathcal{H}^{1}} + \left<{\Psi}_{\nabla\rho_{f}},\nabla \delta \rho_{f}\right>_{\boldsymbol{\mathcal{L}}^{2*},\boldsymbol{\mathcal{L}}^{2}} \\ & {\kern-7.3pc}+ \left<J_{\rho},\delta \rho\right>_{\mathcal{L}^{2*},\mathcal{L}^{2}} + \left<{\Psi}_{\rho},\delta \rho\right>_{\mathcal{L}^{2*},\mathcal{L}^{2}} \end{array} $$

(28)

where 〈⋅,⋅〉 represents the dual pairing between two dual spaces; $\delta H_{zs}\in \mathcal {H}^{1}\left ({\Omega }\right )$, $\delta \rho _{f}\in \mathcal {H}^{1}\left ({\Omega }\right )$ and $\delta \rho \in \mathcal {L}^{2}\left ({\Omega }\right )$ are, respectively, the first-order variational of H _{z s} , ρ _f , and ρ; $\boldsymbol {\mathcal {L}}^{2}\left ({\Omega }\right )$ is the vector valued second-order Lebesgue integrable functional space. According to the Kurash-Kuhn-Tucker condition [52] and setting

$$\begin{array}{@{}rcl@{}} &&{} \left<J_{H_{zs}},\delta H_{zs}\right>_{\mathcal{H}^{1*},\mathcal{H}^{1}} + \left<{\Phi}_{H_{zs}},\delta H_{zs}\right>_{\mathcal{H}^{1*},\mathcal{H}^{1}} \\ &&+ \left<J_{\nabla H_{zs}},\nabla \delta H_{zs}\right>_{\boldsymbol{\mathcal{L}}^{2*},\boldsymbol{\mathcal{L}}^{2}} + \left<{\Phi}_{\nabla H_{zs}},\nabla \delta H_{zs}\right>_{\boldsymbol{\mathcal{L}}^{2*},\boldsymbol{\mathcal{L}}^{2}} = 0 \\ &&\left<J_{\rho_{fp}}{\rho_{fp}}_{\rho_{f}},\delta \rho_{f}\right>_{\mathcal{H}^{1*},\mathcal{H}^{1}} + \left<{\Phi}_{\rho_{fp}}{\rho_{fp}}_{\rho_{f}},\delta \rho_{f}\right>_{\mathcal{H}^{1*},\mathcal{H}^{1}} \\ &&+ \left<{\Psi}_{\rho_{f}},\delta \rho_{f}\right>_{\mathcal{H}^{1*},\mathcal{H}^{1}} + \left<{\Psi}_{\nabla\rho_{f}},\nabla \delta \rho_{f}\right>_{\boldsymbol{\mathcal{L}}^{2*},\boldsymbol{\mathcal{L}}^{2}} = 0 \end{array} $$

(29)

the adjoint equations of Eqs. 1 and 6 in weak forms can be obtained as follows:

$$\begin{array}{@{}rcl@{}} &&{} \displaystyle {\int}_{\Omega} - \epsilon_{r}^{-1} \nabla \delta H_{zs} \cdot \nabla \tilde{H}_{zs}^{*} \\ && + \bigg({k_{0}^{2}} {\upmu}_{r} \tilde{H}_{zs}^{*} + {\partial A \over \partial H_{zs}} - \nabla \cdot {\partial A \over\partial \nabla H_{zs}}\bigg) \delta H_{zs} \,\mathrm{d}{\Omega}\\ &&+ {\int}_{{\Gamma}_{ab}} {}\bigg(\,-\, j k_{0} \sqrt{\epsilon_{r}^{-1} {\upmu}_{r}} \tilde{H}_{zs}^{*} \,+\, {\partial A \over \partial {\nabla H_{zs}}}\cdot \mathbf{n} \,+\, {\partial B \over \partial H_{zs}}\bigg) \delta H_{zs} \,\mathrm{d}{\Gamma} \\ &&+\displaystyle{\int}_{{\Gamma}_{pd}\bigcup{\Gamma}_{ps}\bigcup{\Gamma}_{sm}} \left( {\partial A \over \partial {\nabla H_{zs}}}\cdot \mathbf{n} + {\partial B \over \partial H_{zs}}\right) \delta H_{zs} \,\mathrm{d}{\Gamma} = 0\\ && \Longleftrightarrow \\ &&{} {\int}_{\Omega} - \epsilon_{r}^{-1} \nabla \tilde{H}_{zs}^{*} \cdot \nabla \hat{\tilde{H}}_{zs} \\ &&+ \bigg({k_{0}^{2}} {\upmu}_{r} \tilde{H}_{zs}^{*} + {\partial A \over \partial H_{zs}} - \nabla \cdot {\partial A \over\partial \nabla H_{zs}}\bigg) \hat{\tilde{H}}_{zs} \,\mathrm{d}{\Omega} \\ && + \displaystyle{\int}_{{\Gamma}_{ab}} \bigg({}-{} j k_{0} \sqrt{\epsilon_{r}^{-1} {\upmu}_{r}} \tilde{H}_{zs}^{*} {}+{} {\partial A \over \partial {\nabla H_{zs}}}{}\cdot \mathbf{n} {}+{} {\partial B \over \partial H_{zs}}\bigg) \hat{\tilde{H}}_{zs} \,\mathrm{d}{\Gamma} \\ && + \displaystyle{\int}_{{\Gamma}_{pd}\bigcup{\Gamma}_{ps}\bigcup{\Gamma}_{sm}} \left( {\partial A \over \partial {\nabla H_{zs}}}\cdot \mathbf{n} + {\partial B \over \partial H_{zs}}\right) \hat{\tilde{H}}_{zs} \,\mathrm{d}{\Gamma} = 0,\\ && \text{for}~\forall \hat{\tilde{H}}_{zs} \in \mathcal{H}^{1}\left({\Omega}\right) \end{array} $$

(30)

and

$$\begin{array}{@{}rcl@{}} && {}{\int}_{\Omega} r^{2} \nabla \delta \rho_{f} \cdot \nabla \tilde{\rho}_{f}^{*} \\ &&{}+ \bigg[ \tilde{\rho}_{f}^{*} + {{\partial A}\over{\partial \rho_{fp}}} {{\partial \rho_{fp}}\over{\partial \rho_{f}}} - {{\partial \epsilon_{r}^{-1}}\over{\partial \rho_{fp}}} {{\partial \rho_{fp}}\over{\partial \rho_{f}}} \nabla \left(H_{zs}+H_{zi}\right) \cdot \nabla \tilde{H}_{zs}^{*} \bigg] \delta \rho_{f} \,\mathrm{d}{\Omega}\\ && {}+ {\int}_{{\Gamma}_{ab}} \left(- j k_{0} {{\partial \sqrt{\epsilon_{r}^{-1}}}\over{\partial \rho_{fp}}} \sqrt{{\upmu}_{r}} H_{zs} \,+\, {{\partial \epsilon_{r}^{-1}}\over{\partial \rho_{fp}}} \nabla H_{zi} \cdot \mathbf{n} \right)\tilde{H}_{zs}^{*} {{\partial \rho_{fp}}\over{\partial \rho_{f}}} \delta \rho_{f} \,\mathrm{d}{\Gamma} \\ && {}+ {\int}_{{\Gamma}_{pd}\bigcup{\Gamma}_{ps}\bigcup{\Gamma}_{sm}} {{\partial \epsilon_{r}^{-1}}\over{\partial \rho_{fp}}} {{\partial \rho_{fp}}\over{\partial \rho_{f}}} \nabla H_{zi} \cdot \mathbf{n} \tilde{H}_{zs}^{*} \delta \rho_{f} \,\mathrm{d}{\Gamma} = 0 \\ && \Longleftrightarrow \\ && {\int}_{\Omega} r^{2} \nabla \tilde{\rho}_{f}^{*} \cdot \nabla \hat{\tilde{\rho}}_{f} \\ &&{}+ \bigg[ \tilde{\rho}_{f}^{*} + {{\partial A}\over{\partial \rho_{fp}}} {{\partial \rho_{fp}}\over{\partial \rho_{f}}} - {{\partial \epsilon_{r}^{-1}}\over{\partial \rho_{fp}}} {{\partial \rho_{fp}}\over{\partial \rho_{f}}} \nabla \left(H_{zs}+H_{zi}\right) \cdot \nabla \tilde{H}_{zs}^{*} \bigg] \hat{\tilde{\rho}}_{f} \,\mathrm{d}{\Omega}\\ && {}+ {\int}_{{\Gamma}_{ab}} \left(- j k_{0} {{\partial \sqrt{\epsilon_{r}^{-1}}}\over{\partial \rho_{fp}}} \sqrt{{\upmu}_{r}} H_{zs} + {{\partial \epsilon_{r}^{-1}}\over{\partial \rho_{fp}}} \nabla H_{zi} \cdot \mathbf{n} \right) \tilde{H}_{zs}^{*} {{\partial \rho_{fp}}\over{\partial \rho_{f}}} \hat{\tilde{\rho}}_{f} \,\mathrm{d}{\Gamma} \\ && {}+ {\int}_{{\Gamma}_{pd}\bigcup{\Gamma}_{ps}\bigcup{\Gamma}_{sm}} {{\partial \epsilon_{r}^{-1}}\over{\partial \rho_{fp}}} {{\partial \rho_{fp}}\over{\partial \rho_{f}}} \nabla H_{zi} \cdot \mathbf{n} \tilde{H}_{zs}^{*} \hat{\tilde{\rho}}_{f} \,\mathrm{d}{\Gamma} = 0,\\ && {}\text{for}~\forall \hat{\tilde{\rho}}_{f} \in \mathcal{H}^{1}\left({\Omega}\right) \end{array} $$

(31)

Based on the arbitrariness of $\hat {\tilde {H}}_{zs}$ and $\hat {\tilde {\rho }}_{f}$ and Gauss theory [52], the weak forms in Eqs. 30 and 31 can be transformed into the strong forms of the adjoint Eqs. 11 and 12. After the derivation of the adjoint equations, the adjoint derivative of the computational design problem can be further obtained as

$$ \begin{array}{lll} & \left<{{\delta\hat{J}}\over{\delta\rho}},\delta\rho\right>_{\mathcal{L}^{2*},\mathcal{L}^{2}} \\ = & \left<J_{\rho},\delta \rho\right>_{\mathcal{L}^{2*},\mathcal{L}^{2}} + \left<{\Psi}_{\rho},\delta \rho\right>_{\mathcal{L}^{2*},\mathcal{L}^{2}} \\ = & {\int}_{\Omega} \left({{\partial A}\over{\partial\rho}}- \tilde{\rho}_{f}^{*} \right) \delta\rho \,\mathrm{d}{\Omega} \end{array} $$

(32)

which can be transformed into the strong form

$$ \begin{array}{l} {{\delta\hat{J}}\over{\delta\rho}} = {{\partial A}\over{\partial\rho}}- \tilde{\rho}_{f}^{*},~\text{in}~{\Omega} \end{array} $$

(33)

In the adjoint derivative, only the real part is utilized because the design variable ρ is the distribution defined on real space.