## Inverse diffraction grating of Maxwell’s equations in biperiodic structures |

Optics Express, Vol. 22, Issue 4, pp. 4799-4816 (2014)

http://dx.doi.org/10.1364/OE.22.004799

Acrobat PDF (1727 KB)

### Abstract

Consider a time-harmonic electromagnetic plane wave incident on a perfectly conducting biperiodic surface (crossed grating). The diffraction is modeled as a boundary value problem for the three-dimensional Maxwell equation. The surface is assumed to be a small and smooth deformation of a planar surface. In this paper, a novel approach is developed to solve the inverse diffraction grating problem in the near-field regime, which is to reconstruct the surface with resolution beyond Rayleigh’s criterion. The method requires only a single incident field with one polarization, one frequency, and one incident direction, and is realized by using the fast Fourier transform. Numerical results show that the method is simple, efficient, and stable to reconstruct biperiodic surfaces with subwavelength resolution.

© 2014 Optical Society of America

## 1. Introduction

1. R. Petit, ed., *Electromagnetic Theory of Gratings* (Springer, 1980). [CrossRef]

2. G. Bao, D. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A **12**, 1029–1042 (1995). [CrossRef]

3. Z. Chen and H. Wu, “An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures,” SIAM J. Numer. Anal. **41**, 799–826 (2003). [CrossRef]

5. Y. Wu and Y. Y. Lu, “Analyzing diffraction gratings by a boundary integral equation Neumann-to-Dirichlet map method,” J. Opt. Soc. Am. A **26**, 2444–2451 (2009). [CrossRef]

6. G. Bao, L. Cowsar, and W. Masters, *Mathematical Modeling in Optical Science*, Vol. 22 of Frontiers in Applied Mathematics (SIAM, 2001). [CrossRef]

## 2. Model problem

### 2.1. Maxwell’s equations

*ρ*,

*z*) ∈ ℝ

^{3}, where

*ρ*= (

*x*,

*y*) ∈ ℝ

^{2}. As seen in Fig. 1, the problem may be restricted to a single period of Λ = (Λ

_{1}, Λ

_{2}) in

*ρ*due to the periodicity of the structure. Let the surface in one period be described by

*S*= {(

*ρ*,

*z*) ∈ ℝ

^{3}:

*z*=

*ϕ*(

*ρ*), 0 <

*x*< Λ

_{1}, 0 <

*y*< Λ

_{2}}, where

*ϕ*∈

*C*

^{2}(ℝ

^{2}) is a biperiodic function satisfying

*ϕ*(

*ρ*+ Λ,

*z*) =

*ϕ*(

*ρ*,

*z*). The grating surface function

*ϕ*is assumed to be in the form where

*ψ*∈

*C*

^{2}(ℝ

^{2}) is a biperiodic function with period Λ and is called the grating profile, and

*ε*is sufficiently small and is called the surface deformation parameter.

*= {(*

_{S}*ρ*,

*z*) ∈ ℝ

^{3}:

*z*>

*ϕ*(

*ρ*), 0 <

*x*< Λ

_{1}, 0 <

*y*< Λ

_{2}} the space above

*S*, which is filled with some homogeneous medium characterized by a positive constant wavenumber

*κ*. Denote Ω = {(

*ρ*,

*z*) ∈ ℝ

^{3}:

*ϕ*(

*ρ*) <

*z*<

*h*, 0 <

*x*< Λ

_{1}, 0 <

*y*< Λ

_{2}} be the domain bounded below by

*S*and bounded above by the plane surface Γ = {(

*ρ*,

*z*) ∈ ℝ

^{3}:

*z*=

*h*, 0 <

*x*< Λ

_{1}, 0 <

*y*< Λ

_{2}}, where

*h*> max

_{0 < x < Λ1,0 < y < Λ2}

*ϕ*(

*ρ*).

**E**

^{inc},

**H**

^{inc}) be the incoming plane waves that are incident upon the grating surface from above, where Here

*α*= (

*α*

_{1},

*α*

_{2}),

*α*

_{1}= sin

*θ*

_{1}cos

*θ*

_{2},

*α*

_{2}= sin

*θ*

_{1}sin

*θ*

_{2}, and

*β*= cos

*θ*

_{1}, where

*θ*

_{1}and

*θ*

_{2}are the latitudinal and longitudinal incident angles, which satisfy 0 ≤

*θ*

_{1}<

*π*/2, 0 ≤

*θ*

_{2}< 2

*π*. Denote by

**d**= (

*α*

_{1},

*α*

_{2}, −

*β*) the unit propagation direction vector. The unit polarization vectors

**t**= (

*t*

_{1},

*t*

_{2},

*t*

_{3}),

**s**= (

*s*

_{1},

*s*

_{2},

*s*

_{3}) satisfy

**t**·

**d**= 0 and

**s**=

**d**×

**t**, which gives explicitly For normal incident, i.e.,

*θ*

_{1}= 0, we have

*α*

_{1}= 0,

*α*

_{2}= 0,

*β*= 1, and

*s*

_{1}=

*t*

_{2},

*s*

_{2}= −

*t*

_{1},

*s*

_{3}= 0. Hence we get from |

**t**| = |

**s**| = 1 that

*t*

_{3}= 0.

**E**is the total electric field and

**H**is the total magnetic field. Due to the homogeneous medium, the electromagnetic fields satisfy the divergence free condition: We consider the perfect electric conductor condition: where

*ν*= (

_{S}*ν*

_{1},

*ν*

_{2},

*ν*

_{3}) ∈ ℝ

^{3}is the unit normal vector on

*S*, given explicitly as Here

*ϕ*=

_{x}*∂*(

_{x}ϕ*x*,

*y*) and

*ϕ*=

_{y}*∂*(

_{y}ϕ*x*,

*y*) are the partial derivatives.

### 2.2. Transparent boundary condition

*into a bounded domain Ω, a transparent boundary condition needs to be imposed on Γ.*

_{S}*n*= (

*n*

_{1},

*n*

_{2}) ∈

^{2}and denote

*α*= (

_{n}*α*

_{1}

*,*

_{n}*α*

_{2}

*), where*

_{n}*α*

_{1}

*= 2*

_{n}*πn*

_{1}/Λ

_{1}and

*α*

_{2}

*= 2*

_{n}*πn*

_{2}/Λ

_{2}. For a biperiodic function

*u*(

*ρ*) with period Λ in

*ρ*, it has the Fourier series expansion For any vector field

**u**= (

*u*

_{1},

*u*

_{2},

*u*

_{3}), denote its tangential component on Γ by where

*ν*

_{Γ}= (0, 0, 1) is the unit normal vector on Γ.

**u**(

*ρ*,

*h*) = (

*u*

_{1}(

*ρ*,

*h*),

*u*

_{2}(

*ρ*,

*h*), 0) on Γ, where

*u*is a biperiodic function in

_{j}*ρ*with period Λ, we define a boundary operator

*T*: where

*v*is also a biperiodic function in

_{j}*ρ*with the same period Λ. Here

*u*and

_{j}*v*have the following Fourier expansions and the Fourier coefficients

_{j}*u*and

_{jn}*v*satisfy

_{jn}29. G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. **79**, 1–34 (2009). [CrossRef]

### 2.3. Reduced model problem

**E**= (

*E*

_{1},

*E*

_{2},

*E*

_{3}). Noting the divergence condition (6), we may reduce Maxwell’s equations to the Helmholtz equation for

*E*: The divergence free condition (6) can be explicitly written as Substituting (8) into (7) yields The transparent boundary condition (10) becomes where the Fourier coefficients of

_{j}*H*

_{1}and

*H*

_{2}are given by Here

*E*

_{1n}(

*h*) and

*E*

_{2}

*(*

_{n}*h*) are the Fourier coefficients of

*E*

_{1}(

*ρ*,

*h*) and

*E*

_{2}(

*ρ*,

*h*), respectively.

*ϕ*(

*ρ*). The inverse problem is to reconstruct the function

*ϕ*(

*ρ*) from the tangential trace of the total field measured at Γ, i.e.,

*δ*is the noise level. In particular, we are interested in the inverse problem in the near-field regime where the measurement distance

*h*is much smaller than the wavelength

*λ*= 2

*π/κ*.

## 3. Transformed field expansion

### 3.1. Change of variables

**Ẽ**= (

*Ẽ*

_{1},

*Ẽ*

_{2},

*Ẽ*

_{3}) and let

*Ẽ*(

_{j}*x̃*,

*ỹ*,

*z̃*) =

*E*(

_{j}*x*,

*y*,

*z*) under the transformation. After tedious but straightforward calculations, it can be verified from (12) that the total electric field, upon dropping the tilde, satisfies the equation where

*z*=

*h*reduces to

### 3.2. Power series solution

*E*in a power series of

_{j}*ε*: Substituting

*ϕ*=

*εψ*into

*c*and inserting (20) into (16), we may derive where

_{j}*ψ*=

_{x}*∂*(

_{x}ψ*x*,

*y*) and

*ψ*=

_{y}*∂*(

_{y}ψ*x*,

*y*) are the partial derivatives.

*z*= 0, we have where

*z*=

*h*reduces to where

### 3.3. Zeroth order term

*k*= 0, we have The divergence free condition (22) reduces to The perfect electric conductor boundary condition (23) is Using (28) and (29), we have

*f*are periodic functions of

_{j}*ρ*, they have the Fourier expansions where

*f*

_{j}_{0}= −2i

*κt*exp(−i

_{j}*κh*) and

*f*= 0 for

_{jn}*n*≠ 0.

*z*= 0: and the boundary conditions at

*z*=

*h*:

### 3.4. First order term

*k*= 1, we have where

*z*= 0 is Evaluating (39) at

*z*= 0 gives The transparent boundary condition (25) becomes where The divergence free condition (26) gives one more boundary condition on

*z*=

*h*: Consider the Fourier expansions for periodic functions of

*ψ*(

*ρ*): where

*z*= 0: and boundary conditions at

*z*=

*h*:

## 4. Reconstruction formula

*E*(

_{j}*ρ*,

*h*),

*j*= 1, 2 is the exact data and

*δ*is the noise level.

*z*=

*h*and replacing

*E*(

_{j}*ρ*,

*h*) with

*𝒪*(

*ε*

^{2}) and

*𝒪*(

*δ*) yield which is the linearization of the nonlinear inverse problem and enables us to find an explicit reconstruction formula for the linearized inverse problem.

*ϕ*=

*εψ*and thus

*ϕ*=

_{n}*εψ*, where

_{n}*ϕ*is the Fourier coefficient of

_{n}*ϕ*. Substituting (44) into (46), we deduce that where

*δ*

_{0}

*the Kronecker’s delta function.*

_{n}*β*and (47) that it is well-posed to reconstruct those Fourier coefficients

_{n}*ϕ*with |

_{n}*α*| <

_{n}*κ*, since the small variations of the measured data will not be amplified and lead to large errors in the reconstruction, but the resolution of the reconstructed function

*ϕ*is restricted by the given wavenumber

*κ*. In contrast, it is severely ill-posed to reconstruct those Fourier coefficients

*ϕ*with |

_{n}*α*| >

_{n}*κ*, since the small variations in the data will be exponentially enlarged and lead to huge errors in the reconstruction, but they contribute to the super resolution of the reconstructed function

*ϕ*. To obtain a stable and super-resolved reconstruction, we may adopt a regularization to suppress the exponential growth of the reconstruction errors.

51. G. Bao and J. Lin, “Near-field imaging of the surface displacement on an infinite ground plane,” Inverse Probl. Imag. **7**, 377–396 (2013). [CrossRef]

*h*, the cut-off wavenumber

*κ*

_{c}is chosen in such a way that which implies that the spatial frequency will be cut-off for those below the noise level. More explicitly, we have which indicates

*κ*

_{c}>

*κ*as long as SNR > 0 and super resolution may be achieved.

*ϕ*are computed, the grating surface function can be approximated by

_{n}*ϕ*.

## 5. Numerical experiment

*ε*,

*h*, and

*δ*on the reconstruction results. As seen in Fig. 2, we consider two types of grating profiles: one is a smooth function with finitely many Fourier modes and another is a non-smooth function with infinitely many Fourier modes. Although the method requires that the grating profile function

*ψ*(

*ρ*) is

*C*

^{2}(ℝ

^{2}), it is still applicable to non-smooth functions numerically.

*z*direction so that no artificial wave reflection occurs to ruin the wave field inside the domain. Adaptive refinement technique [29

29. G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. **79**, 1–34 (2009). [CrossRef]

56. PHG (Parallel Hierarchical Grid), http://lsec.cc.ac.cn/phg/.

*H*

^{1}and

*H*(curl) element. In order to generate the tetrahedral mesh with a biperiodic structure, we generate a non-uniform hexahedral mesh firstly and divide each hexahedron into six tetrahedrons. The linear system resulted from finite element discretization is solve by the multifrontal massively parallel sparse direct solver [57

57. P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent, “A fully asynchronous multifrontal solver using distributed dynamic scheduling,” SIAM J. Matrix Anal. Appl. **23**, 15–41 (2001). [CrossRef]

58. P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent, and S. Pralet, “Hybrid scheduling for the parallel solution of linear systems,” Parallel Comput. **32**, 136–156 (2006). [CrossRef]

**E**

^{inc}= (1, 0, 0)exp(−i

*κz*), i.e.,

*t*

_{1}= 1 and

*t*

_{2}=

*t*

_{3}= 0, and only the first component of the electric field,

*E*

_{1}(

*ρ*,

*h*), needs to be measured. The wavenumber is

*κ*= 2

*π*, which corresponds to the wavenlength

*λ*= 1. Define by

*R*the unit rectangular domain, i.e.,

*R*= [0, 1.0

*λ*] × [0, 1.0

*λ*]. The computational domain is

*R*× [

*ϕ*, 1.0

*λ*] with the PML region

*R*× [0.5

*λ*, 1.0

*λ*]. The scattering data

*E*

_{1}(

*ρ*,

*h*) is obtained by interpolation into the uniform 256 × 256 grid points on the measurment plane

*z*=

*h*. In all the figures, the plots are rescaled with respect to the wavelength

*λ*to clearly show the relative size, and the meshes are done in 32 × 32 instead of 256 × 256 grid points in order to reduce the display sizes. To test the stability of the method, some relative random noise is added to the scattering data, i.e., the scattering data takes the form where rand stands for uniformly distributed random numbers in [−1, 1]. The relative

*L*

^{2}(

*R*) error is defined by where

*ϕ*is the exact surface function and

*ϕ*is the reconstructed surface function.

_{δ,ε}**Example 1**. This example illustrates the reconstruction results of a smooth grating profile with finitely many Fourier modes. The exact grating surface function is given by

*ϕ*(

*ρ*) =

*εψ*(

*ρ*), where the grating profile function

*ε*. The measurement is taken at

*h*= 0.4

*λ*and no additional random noise is added to the scattering data, i.e.,

*δ*= 0. This test is to investigate the influence of surface deviation parameter on the reconstructions. In (46), higher order terms of

*ε*are dropped in the power series to linearize the inverse problem and to obtain the explicit reconstruction formula. As expected, the smaller the surface deviation

*ε*is, the more accurate is the approximation of the linearized model to the original nonlinear model problem. Table 1 shows the relative

*L*

^{2}(

*R*) error of the reconstructions with four different surface deformation parameter

*ε*= 0.2

*λ*, 0.1

*λ*, 0.05

*λ*, 0.025

*λ*for fixed measurement distance

*h*= 0.4

*λ*. It is clear to note that the error decreases from 85.0% to 9.86% as

*ε*decreases from 0.2

*λ*to 0.025

*λ*.

*δ*and the measurement distance

*h*. In practice, the scattering data always contains certain level of noise. To test the stability and super resolving capability of the method, we add an amount of 5% random noise to the scattering data. Table 2 reports the relative

*L*

^{2}(

*R*) error of the reconstructions with four different measurement distance

*h*= 0.4

*λ*, 0.3

*λ*, 0.2

*λ*, 0.1

*λ*for fixed

*ε*= 0.025

*λ*. Comparing the results for the same

*ε*= 0.025

*λ*and

*h*= 0.4

*λ*in Tables 1 and 2, we can see that the relative error increases dramatically from 9.86% by using noise free data to 86.3% by using 5% noise data. The reason is that a smaller cut-off should be chosen to suppress the expotentially increasing noise in the data and thus the Fourier modes of the exact grating surface function can not be recovered for those higher than the cutoff frequency, which leads to a large error and poor resolution in the reconstruction. A smaller measurement distance is desirable in order to have a large cut-off frequency, which enhances the resolution and reduces the error. As can be seen in Table 2, the reconstruction error decreases from 86.3% by using

*h*= 0.4

*λ*to as low as 12.0% by using

*h*= 0.1

*λ*even for 5% noise data. Figure 3 plots the reconstructed surfaces by using

*h*= 0.4

*λ*, 0.3

*λ*, 0.2

*λ*, 0.1

*λ*. Comparing the exact surface profile in Fig. 2(a) and the reconstructed surface in Fig. 3(d), we can see that the reconstruction is nearly perfect and the difference is really minor by carefully checking the contour plots.

**Example 2**. This example illustrates the reconstruction results of a non-smooth grating profile with infinitely many Fourier modes, as seen in Fig. 2(b). The exact grating surface function is given by

*ϕ*(

*ρ*) =

*εψ*(

*ρ*), where the grating profile function Clearly, the profile function (51) is nondifferentiable and its Fourier coefficients decay slowly. Comparing with the grating profile (50), it is more challenging to obtain as good reconstructions as those in Example 1 since a much higher cutoff frequency is desirable to recover as many Fourier modes as possible for (51).

*ε*by using noise-free data. The measurement is taken at

*h*= 0.2

*λ*. Table 3 presents the relative

*L*

^{2}(

*R*) error of the reconstructions with four different surface deformation parameter

*ε*= 0.1

*λ*, 0.05

*λ*, 0.025

*λ*, 0.0125

*λ*. The error decreases from 72.9% to 15.2% as

*ε*decreases from 0.1

*λ*to 0.0125

*λ*. Based on these results, the following observation can be made: a smaller deformation parameter

*ε*yields a better reconstruction; smaller

*ε*and

*h*are required in order to obtain comparable error with that in Table 2 for Example 1 due to the non-smooth nature of the grating surface function of Example 2.

*δ*and the measurement distance

*h*. An amount of 5% random noise is added to the scattering data. Table 4 reports the relative

*L*

^{2}(

*R*) error of the reconstructions with four different measurement distance

*h*= 0.2

*λ*, 0.1

*λ*, 0.05

*λ*, 0.025

*λ*for fixed

*ε*= 0.0125

*λ*. Comparing the results for the same

*ε*= 0.0125

*λ*and

*h*= 0.2

*λ*in Tables 3 and 4, we can see that the relative error is more than doubled from 15.2% by using noise-free data to 39.9% by using 5% noise data. Again, the reason is that a smaller cut-off is chosen to suppress the expotentially increasing noise in the data and thus higher Fourier modes of the exact grating surface function can not be recovered. A smaller measurement distance helps to enhance the resolution and reduce the error. In Table 4, the reconstruction error decreases from 39.9% by using

*h*= 0.2

*λ*to as low as 11.9% by using

*h*= 0.025

*λ*. Figure 4 shows the reconstructed surfaces by using

*h*= 0.2

*λ*, 0.1

*λ*, 0.05

*λ*, 0.025

*λ*. Comparing the exact surface profile in Fig. 2(b) and the reconstructed surface in Fig. 4(d), we can see that a good reconstruction can still be possible when using a small measurement distance.

## 6. Conclusion

## Acknowledgments

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52. | S. Carney and J. Schotland, “Inverse scattering for near-field microscopy,” App. Phys. Lett. |

53. | S. Carney and J. Schotland, “Near-field tomography,” MSRI Ser. Math. Appl. |

54. | H. Ammari, J. Garnier, and K. Solna, “Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging,” Proc. Am. Math. Soc. |

55. | H. Ammari, J. Garnier, and K. Solna, “Partial data resolving power of conductivity imaging from boundary measurements,” SIAM J. Math. Anal. |

56. | PHG (Parallel Hierarchical Grid), http://lsec.cc.ac.cn/phg/. |

57. | P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent, “A fully asynchronous multifrontal solver using distributed dynamic scheduling,” SIAM J. Matrix Anal. Appl. |

58. | P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent, and S. Pralet, “Hybrid scheduling for the parallel solution of linear systems,” Parallel Comput. |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(290.3200) Scattering : Inverse scattering

(180.4243) Microscopy : Near-field microscopy

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: January 3, 2014

Revised Manuscript: February 5, 2014

Manuscript Accepted: February 6, 2014

Published: February 21, 2014

**Citation**

Gang Bao, Tao Cui, and Peijun Li, "Inverse diffraction grating of Maxwell’s equations in biperiodic structures," Opt. Express **22**, 4799-4816 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4799

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