### 物理代写|光学代写Optics代考|PHYSICS3540

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## 物理代写|光学代写Optics代考|Self-consistency of Adjoint Analysis for Topology

In frequency domain, the field variables of the optical waves are complex, comprising the amplitude and phase of the field. Cost functions of the topology optimization problems are usually included the conjugate of the field variables, e.g., the energy functionals, which are the product of the field variables and their conjugates. The energy functionals are popularly used, because of the well-posedness of their least square forms. For the sensitivity analysis of a topology optimization problem, adjoint method is popularly used, where the first-order variational is implemented for the corresponding augmented Lagrangian [10].

During the adjoint analysis procedure, it is necessary to implement the variational of the conjugate operation to the field variables. Mathematically, the conjugate operator is Gâteaux differential instead of Fréchet differential, to the field variables [130]. This was ignored in several previous researches, such as the literatures in the Refs. $[77,87]$. The Gâteaux differentiability of the conjugate operator can cause the incompleteness of the adjoint sensitivity, i.e., the adjoint sensitivity of the real-valued cost

functions is complex instead of real for the design variable, which is real-valued distribution defined on the computational domain. If this incomplete sensitivity is used directly, the design variable will be evolved to be complex during the iterative procedure, where the initial of the design variable is set to be real. This adjoint sensitivity is then self-inconsistent from the pointview of keeping the real-value property of the design variable. Therefore, the real part extraction operator is used to extract the real part of the derived adjoint sensitivity, and to artificially enforce the self-consistency of the adjoint sensitivity.

The consequence of such enforced self-inconsistency is that the derived structural topology has dependence on the phase of the incident wave. This phase-dependence is unreasonable, because the incident waves can not be inherently distinguished by only altering their phases. To solve the problem on the self-inconsistency of the adjoint sensitivity, Fréchet differentiability should be ensured for the cost function. The conjugate operator in the cost function can be removed by splitting the complex field variables into the corresponding real and imaginary parts respectively defined on real functional spaces instead of complex functional spaces. Then the Gâteaux differentiability induced by the conjugate operator is avoided. The splitting of the complex variables brings about the splitting of the wave equations, which are complex partial differential equations. The method of splitting complex partial differential equations or variational problems into their corresponding coupled systems for the real and imaginary parts of the field variables has been systematically discussed in the Refs. $[3,65]$.

## 物理代写|光学代写Optics代考|Dielectric Material Based Topology Optimization

It has been mentioned that the control of optical waves is realized by structures with complex spacial configurations using pre-selected materials and the incident waves can have complicated polarizations. Most of those situations cannot be reduced into two dimensional, except for a minority of cases involving linear polarized waves. Most of the reports on topology optimization in optics have focused on applications, including beamsplitters $[80,86]$, photonic crystals $[37,89]$, cloaks $[7,8,38]$, sensors and resonators $[104,105]$, metamaterials $[28,77,137]$, excitation of surface plasmons [10], and electromagnetic and optical antennas [35, 36, 48, 134], without presenting the systemical topology optimization methodology for optical waves propagating in three-dimensional space. Therefore, it is necessary to develop a unified and systematic topology optimization approach that sufficiently considers the physical complexity of three-dimensional optics.

It is not straightforward to develop the finite element-based topology optimization method for optical waves in three-dimensional space, because the divergence-free condition needs to be enforced. In the two-dimensional transverse electric or magnetic wave cases, the divergence-free conditions are automatically satisfied during the reducing procedure of the Maxwell’s equations with deriving the Helmholtz equations, and the node element-based Galerkin finite element method can be naturally used to directly discretize the Helmholtz equations $[53,69]$. Being different from the two-dimensional cases, the divergence-free conditions can not be automatically satisfied in solving the three-dimensional optical waves with the node element-based Galerkin finite element method, and this results in the spurious solutions.

For this problem, two dominant approaches have been developed to enforce the divergence-free conditions and eliminate the spurious solutions. The first approach is to add a penalty term with the least square form of the divergence-free condition to the weak form of the wave equation, and then discretize the weak form with node elements. However, the use of penalty term can not eliminate the divergence of the solution completely and it affects the solution accuracy. Therefore, the divergencefree condition can not be satisfied accurately by the penalty approach [53]. The second approach is the use of edge elements that assign degrees of freedom to the edges rather than to the nodes of the elements, where the vector basis with inherent satisfaction of divergence-free condition is used to implement interpolation $[53,69,72,116]$. The edge elements have also solved the problems on the inconvenience of imposing boundary conditions at material interfaces and the difficulty in treating conducting and dielectric edges and corners due to field singularities $[53,69]$. Therefore, the edge element-based finite element method is the more reasonable choice for discretizing the three-dimensional wave equations and developing the topology optimization method for three-dimensional optical waves.

## 物理代写|光学代写Optics代考|Metal Material-Based Topology Optimization

A metal surface with a negative real part of the permittivity can trap optical waves with achieving surface plasmon polaritons $[31,81]$. The metals used for surface plasmon polaritons are usually noble metals, e.g., silver (Ag), gold (Au) and Aluminum (Al). At optical frequencies, the metal’s free electrons can sustain, under certain conditions, oscillations with distinct resonance frequencies $[66,127,140]$. The existence of surface plasmons is characteristic for the interaction between metal and light, where the Kretschmann-Raether and Otto configurations are commonly used for plasmon excitation.

Many innovative concepts of surface plasmon polaritons have been developed over the past few years, e.g., localized surface plasmon resonances [100], extraordinary optical transmission [30] and transformational plasmon optics $[52,60]$. Correspondingly, many related applications have also been proposed for surface plasmon polaritons, e.g., biomolecular manipulation and labeling [23], surface enhanced Raman spectroscopy [70], chemical and biological sensors [33], photo-voltaics [22], nearfield lithography and imaging [84], optical trapping [67, 76], nano optic circuits [1], opto-electronic devices, wavelength-tunable filters, optical modulators [16,34,41, $42]$, plasmonic Luneburg lens and surface plasmonic cloaking $[52,60]$.

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## MATLAB代写

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