# Galerkin method for the scattering problem of strip gratings

## Abstract

In this paper, the diffraction problem of periodic strip gratings is considered. The previous study of this problem usually concentrated on the numerical method; however, we try to analyze this problem and the convergence of the numerical solution from the mathematical point of view in this work. By use of the Dirichlet to Neumann operator on the slit between two strips, we reformulate the problem to an operator equation. The well-posedness of the solution to the operator equation is proved. The Galerkin method is applied to solve this operator equation and the convergence result of the numerical solution is also derived. Finally, some numerical experiments are presented to show the effectiveness of our method and verify the theoretical convergence result.

## Keywords

Strip grating Dirichlet to Neumann map Helmholtz equation Galerkin method- MoM
method of moments

- FMM
Fourier modal method

- CBCM
combined boundary conditions method

- FDM
finite difference method

- FEM
finite element method

- DtN
Dirichlet to Neumann

- dofs
degree of freedoms

## 1 Introduction

The scattering theory of periodic structures has a wide variety of applications, for example, the micro-optics and the antenna engineering. In optics, the periodic structure is also called diffraction gratings. Introduction to the problem of electromagnetic diffraction through periodic structures and the corresponding numerical methods can be found in [1]. The reviews on the diffractive optics technology and the mathematical analysis of diffraction gratings are presented in [2] and [3], respectively.

In this paper, we focus on the diffraction problem of periodic perfectly conducting strip gratings. This is a classic model which has been investigated by many researchers. In [4] the method of moments (MoM) is employed to analyze the diffraction problem of strip gratings located in free space. In papers [5, 6] the same numerical method is used to solve the strip grating problem where the strips are printed on a dielectric substrate. As to the improved MoM for scattering problem of periodic strip gratings, we refer to paper [7] and the references therein. In addition to the MoM, there are also many other numerical methods for the strip gratings problem. For example, the singular integral equation approach is proposed to deal with the plane wave diffraction by an infinite strip grating at oblique incidence in [8]. Fourier modal method (FMM), also called combined boundary conditions method (CBCM), is introduced in [9, 10, 11]. CBCM is applied for solving finite strip grating problem in [11]. Because of the slow convergence and Gibbs phenomenon at the tips of the strips, CBCM has been substantially improved in [12] with the aid of adaptive spatial resolution in [11]. The parametric formulation of CBCM in [12] improves the convergence rate, the computational efficiency, and the numerical accuracy. In [13] the authors employ this improved method in multilayered structures of strip gratings. More references about CBCM can be found in [14]. In mathematics, because the grating diffraction problem is governed by differential equation, the finite difference method (FDM) and the finite element method (FEM) can also be applied in solving the problem. In the numerical experiment part of this paper, we compare our method with the FDM.

The studies above concentrate on the mathematical model and the numerical method of the strip gratings. In this paper, we will reformulate the diffraction problem of strip grating into an operator equation by use of the Dirichlet to Neumann (DtN) operator. Then we will give a rigorous mathematical analysis of the solution and the Galerkin method which is used to solve the operator equation. The convergence of the Galerkin method has also been verified by the numerical experiments.

The paper is organized as follows. In Sect. 2, we give some notations for describing the strip gratings problem and reformulate this problem into an operator equation by use of DtN operator on the slit between two strips. In Sect. 3, the well-posedness of the solution to the operator equation derived in Sect. 2 is proved. The well-posedness contains the existence, the uniqueness, and the stability of the solution. In Sect. 4, the Galerkin method is employed to solve the operator equation. The uniqueness and the convergence of the numerical solution are proved. We also obtain the error estimate of the numerical solution in Theorem 5. The concrete computing process is given at the end of this section. In Sect. 5, some numerical experiments are presented to show the effectiveness of the Galerkin method and convergence order proved in Sect. 4 is also verified by numerical Example 1 in this section.

## 2 Formulation of the problem

In this section, some notations are firstly presented to help us describe the diffraction problem of periodic strip gratings with period *d*. Then the definition of quasi-periodic, the famous Rayleigh expansion of diffraction field, and the DtN operator are introduced. Finally, we reformulate the diffraction problem into an operator equation through simple calculation and the knowledge prepared above.

*Γ̃*, see Figure 1. Denote the slit and the perfect conductive strip in one period by \(\varGamma_{0}\) and \(\varGamma_{1}\), respectively:

*Γ*as follows:

*d*. Moreover, we require the diffracted field \(u_{d}\) to satisfy the bounded outgoing wave condition in \(S_{+}\) and \(S_{-}\).

## 3 Well-posedness analysis

In this section, we will prove the well-posedness of the solution to (17). In order to give the uniqueness and existence of the solution to (17), we introduce the following definitions of spaces and norms.

*s*, define Sobolev spaces

In the following, we will present two lemmas about the properties of spaces \(H^{\frac{1}{2}}_{*}(\varGamma_{0})\), \(H^{-\frac{1}{2}}_{*}(\varGamma_{1})\) and the operator *T*. With the aid of these two lemmas, we can obtain the well-posedness of the solution to equation (17).

### Lemma 1

*The operator*\(T: H^{\frac{1}{2}}_{\alpha ,*}(\varGamma_{0})\rightarrow H^{-\frac{1}{2}}_{\alpha}(\varGamma_{0})\)*is a linear bounded operator*.

### Proof

### Lemma 2

*The embedding operator*\(I_{*}: H^{\frac {1}{2}}_{\alpha,*}(\varGamma_{0})\rightarrow H^{-\frac{1}{2}}_{\alpha}(\varGamma_{0})\)*is a compact operator*.

### Proof

The operator \(I_{*}\) can be decomposed into \(I_{*}=R \circ I \circ E_{0} \), where \(E_{0}\) is the zero extension operator from \(H^{\frac {1}{2}}_{\alpha,*}(\varGamma_{0})\) to \(H^{\frac{1}{2}}_{\alpha}(\varGamma)\), *I* is the embedding operator from \(H^{\frac{1}{2}}_{\alpha}(\varGamma)\) to \(H^{-\frac{1}{2}}_{\alpha }(\varGamma)\), and *R* is the restriction operator from \(H^{-\frac {1}{2}}_{\alpha}(\varGamma)\) to \(H^{-\frac{1}{2}}_{\alpha}(\varGamma_{0})\). Since \(E_{0}\) and *R* are bounded and *I* is compact, we obtain that \(I_{*}\) is compact. □

### Theorem 1

*Assume that*\(\beta_{n}\neq0\)

*for all*\(n\in \mathbb{Z}\),

*then the homogeneous equation*

*has only one solution*\(u=0\).

### Proof

### Theorem 2

*Assume that*\(\beta_{n}\neq0\)

*for all*\(n\in \mathbb{Z}\).

*Then*,

*for any*\(g\in H^{-\frac{1}{2}}_{\alpha}(\varGamma_{0})\),

*the operator equation*\(Tu=g\)

*has a unique solution*\(u\in H^{\frac{1}{2}}_{\alpha,*}(\varGamma_{0})\),

*and*

*where*\(C>0\)

*is a constant independent of*

*g*.

### Proof

*B*: \(H^{\frac{1}{2}}_{\alpha,*}(\varGamma _{0})\to H^{-\frac{1}{2}}_{\alpha}(\varGamma_{0})\) and the corresponding bilinear form \(b(\cdot,\cdot)\) as follows:

*T*and \(I_{*}\) are bounded, \(b(\cdot,\cdot)\) is a bounded bilinear form on \(H^{\frac{1}{2}}_{\alpha,*}(\varGamma_{0})\times H^{\frac{1}{2}}_{\alpha,*}(\varGamma_{0})\). Next, we will show that \(b(\cdot ,\cdot)\) has a lower bound. By simple calculation, we can derive the following two inequalities:

*B*has a bounded inverse \(B^{-1}: H^{-\frac{1}{2}}_{\alpha }(\varGamma_{0})\to H^{\frac{1}{2}}_{\alpha,*}(\varGamma_{0})\). Then the operator equation \(Tu=g\) can be rewritten as

*B*. Because \(I_{*}\) is a compact operator, by use of Fredholm alternative theorem, the operator equation \(Tu=g\) has a unique solution \(u\in H^{\frac{1}{2}}_{\alpha,*}(\varGamma _{0})\) and

*g*. □

## 4 Galerkin method

### Theorem 3

*The Galerkin equation* (18) *has a unique solution*\(u_{N} \in V_{N}\).

### Proof

The next theorem follows from standard estimates for Galerkin method associated with compact operator equation, so we omit the proof here and refer to [15] for details. □

### Theorem 4

*Assume that*\(\beta_{n}\neq0\)

*for all*\(n\in \mathbb{Z}\),

*u*

*is the solution of*(17),

*and*\(u_{N}\)

*is the solution of*(18).

*When*

*N*

*is large enough*,

*where*

*C*

*is a positive constant independent of*

*N*.

### Theorem 5

*Assume that*\(\beta_{n}\neq0\)

*for all*\(n\in \mathbb{Z}\), \(u\in H^{s}_{p}(\varGamma_{0})\ (s>\frac{1}{2})\)

*is the solution of*(17),

*and*\(u_{N}\)

*is the solution of*(18),

*then when*

*N*

*is large enough*,

*or*

*where*

*C*

*is a positive constant independent of*

*N*.

### Proof

## 5 Numerical results

In this section we demonstrate the numerical results of our method. All computations are performed using MATLAB. In all the following examples, we set the period \(d=4\), the length of the slit \(L=2\), and \(\varGamma_{0}=\{(x_{1},0); 0< x_{1}<2\}\).

### Example 1

*N*, and the corresponding convergence orders are coincident with Theorem 5.

\(\Vert u-u_{N} \Vert _{\frac {1}{2},p,\varGamma_{0}}\) and \(\Vert u-u_{N} \Vert _{\frac{1}{2},*,\varGamma _{0}}\) with respect to *N*

| 2 | 4 | 8 | 16 | 32 | 64 |
---|---|---|---|---|---|---|

\(\Vert u-u_{N} \Vert _{\frac{1}{2},p,\varGamma_{0}}\) | 9.0460E–02 | 2.6825E–02 | 7.1011E–03 | 1.8020E–03 | 4.5188E–04 | 1.1302E–04 |

order | 1.7537 | 1.9175 | 1.9785 | 1.9956 | 1.9995 | |

\(\Vert u-u_{N} \Vert _{\frac{1}{2},*,\varGamma_{0}}\) | 8.4367E–02 | 2.5019E–02 | 6.6220E–03 | 1.6799E–03 | 4.2110E–04 | 1.0522E–04 |

order | 1.7537 | 1.9177 | 1.9788 | 1.9959 | 1.9997 |

### Example 2

*k*is small. Further, in the left one of Fig. 3, we also show the results given by FDM (finite difference method). We demonstrate two results given by FDM with 32 and 64 nodes in one period. Comparing with the FDM, we can see the results of our Galerkin method with 20 dofs (degree of freedoms) are coincident with the results of FDM with 64 dofs, i.e., our Galerkin method needs fewer dofs than the FDM.

### Example 3

### Example 4

## 6 Conclusion

In this paper, we study the scattering problem of strip gratings. By use of the continuity of the total field across the slit in one period and the Dirichlet to Neumann map, this problem is reformulated to an operator equation on the slit. The well-posedness of the solution to the operator equation is proved and Galerkin method is employed to solve this operator equation. We also derive the error estimate for the Galerkin method and numerical examples show that our method is effective.

## Notes

### Acknowledgements

The authors would like to thank the referees for their valuable suggestions which helped to improve this work.

### Availability of data and materials

Not applicable.

### Authors’ contributions

The main idea of this paper was proposed by EZ, the theoretical analysis was carried out by the two authors together, and the numerical experiments were conducted by EZ. Both authors read and approved the final manuscript.

### Funding

This work is supported by the Fundamental Research Funds for the Central Universities (Grant no. 3132017053, 3132018226), the Doctoral Scientific Research Foundation of Liaoning Province (Grant no. 20170520289), TianYuan Special Funds of the National Natural Science Foundation of China (Grant no. 11626054), and the National Natural Science Foundation of China (Grant no. 11601056).

### Competing interests

The authors declare that they have no competing interests.

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