Abstract

The article presents a new method for rigorous simulation of the light diffraction on one-dimensional gratings. The method is capable to solve metal-dielectric structures in linear time and consumed memory with respect to structure complexity. Exceptional performance and convergence for metal gratings are achieved by implementing a curvilinear coordinate transformation into the generalized source method previously developed for dielectric gratings.

© 2013 Optical Society of America

1. Introduction

Diffraction gratings play an important role in numerous edge optical technologies. Most of applications require optimization procedures which often rely on direct simulation methods. When characteristic features of gratings scale to wavelengths of diffracting waves only rigorous numerical methods of the Maxwell’s equations solution can provide sufficiently accurate results. For large aperture diffractive structures the complexity of diffraction simulation methods defines whether a structure can be analyzed or not. This is why the search of computationally efficient ways to simulate various types of grating is currently of particular importance.

We proposed recently an efficient implementation of the generalized source method (GSM) [1] for grating diffraction simulation with low numerical complexity and memory consumption [2,3]. Similarly to the well-known Fourier modal method (FMM) [4] the GSM implementation implies operation in the Fourier space and slicing, nevertheless, the GSM overwhelms widely the FMM in terms of required computational resources and complexity of possibly solvable structures. For a large variety of dielectric gratings and diffractive optical elements the GSM is very probably the best choice of simulation method but for low-loss metal-dielectric structures it faces the same intrinsic difficulties as the FMM. The Fourier space poorly represents the field in metal-dielectric gratings where it is generally highly concentrated at corrugation surface, this leads to rapid increase of numerical errors.

Simulation of diffraction on metal-dielectric gratings is quite well performed by the C-method [4,5]. The C-method searches for the Fourier decomposition of modes in a curvilinear coordinates related with the grating profile. This approach demonstrates an exceptional convergence compared to the FMM, but exhibits high numerical complexity. Additionally, the C-method relies implicitly on the Rayleigh hypothesis [6], and cannot be applied, for example, to overhanging gratings or closely placed gratings of different shapes.

In this work we aim at extending the GSM applicability domain and to amend the method for simulation of metal-dielectric gratings. We reformulate the method in curvilinear coordinates, which transforms a corrugation profile to a plane interface similarly to the C-method. The new method will be referred to as GSMCC (Generalized Source Method in Curvilinear Coordinates). It will be shown that the GSMCC admits a wider class of allowable coordinate transforms in comparison with the C-method, and, since it does not use the Rayleigh hypothesis in any form, the GSMCC potentially has larger domain of applicability. The proposed method keeps the main advantage of the GSM: the linear time and memory consumption with respect to the problem complexity.

The paper is organized as follows. First, we define the diffraction problem to be solved, discuss properties of curvilinear coordinate systems needed for grating representation and review the coordinate transformation of the Maxwell’s equations. All the derivations in the article are made for the simplest case of collinear diffraction on a 1D grating. We show how to split the Maxwell’s equations in curvilinear coordinates into the “basis” and the “source” parts, and formulate the GSM in the form of an implicit self-consistent integral equation. This integral equation reduces to the system of linear algebraic equations. Numerical solution of such system is pretty the same as for the previously developed GSM [2,3], therefore, we review briefly the algorithm. Finally, we demonstrate the GSMCC numerical implementation and give an illustrative example of the enhanced transmission by a sinusoidally modulated thin metal layer.

2. Diffraction problem

To formulate the new method we start with the following diffraction problem. Consider a monochromatic plane wave with time dependence factor exp(iωt) incident on a periodically corrugated interface between two semi-infinite homogeneous media (Fig. 1). The interface is defined in Cartesian coordinates x1,x2,x3 by continuously differentiable function x3=ϑ(x1), with period Λ along axis X1, so that ϑ(x1+mΛ)=ϑ(x1), mZ. Diffraction is described by the Maxwell’s equations

×E=iωμ0H,×H=iωεE.
The permeability is supposed to be equal to that of the vacuum everywhere. Constant values of homogeneous permittivity below and above the grating interface x3=ζ(x1) will be referred to as substrate εs and cover permittivity εc respectively.

 figure: Fig. 1

Fig. 1 Problem under consideration: a plane wave diffraction on a smooth-profiled 1D metal- dielectric grating.

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3. Curvilinear coordinates

The GSMCC is formulated further in a simple case of the diffraction on a 1D grating. The GSM as it was presented in [2,3] solves the diffraction problem by treating it in one-dimensional Cartesian coordinate space and two-dimensional Fourier reciprocal space. In order to adapt the method to metal-dielectric structures and to get rid of Fourier representation of the grating volume we employ the cornerstone C-method idea and rewrite Eq. (1) in a curvilinear coordinate system z1,z2,z3 where the grating interface becomes a plane interface z3=0 [Fig. 2(a)2(b)]. For a 1D grating it is natural to take z2x2. Note that translation invariance of curvilinear coordinates (which is essential for the C-method) is not supposed here since we will not search for eigensolutions below and above plane z3=0. Moreover, we introduce curvilinear coordinates in some bounded semi-infinite region Ω containing the grating Ω={bx3b|bc} and not in the whole space (here maxf=c, and minf=c is assumed, without loss of generality). We assume also that new coordinates z1,z2,z3 continuously become the Cartesian coordinates x1,x2,x3 at Ω boundaries Ω={x3=±b} and coincide with them everywhere outside Ω.

 figure: Fig. 2

Fig. 2 Transformation of a grating corrugation profile. (a) Curvilinear coordinates introduced in a bounded region Ω = {–b ≤□x3 ≤□b | b ≥□c}. (b) Grating profile becomes a plane interface z3 = 0, and everywhere outside Ω the new coordinates coincide with the Cartesian ones.

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Generally, a curvilinear coordinate system allows for defining two bases. The first basis is composed of tangent vectors to curves xα=xα(z1,Z2,Z3), xα=xα(Z1,z2,Z3), and xα=xα(Z1,Z2,z3), α=1,2,3 with Zβ being some fixed constants. This is the contravariant basis eα=β(xβ/zα)iβ. Symbols iβ, β=1,2,3 denote the Cartesian orthonormal basis vectors. The second curvilinear basis is formed by normal vectors to surfaces zα=const. It is the covariant basis eα=β(zα/xβ)iβ. Scalar products gαβ=eαeβ and gαβ=eαeβ form the metric tensors. Covariant and contravariant components of any vector v are identified by lower and upper indices, as vα and vα, respectively. With this notation, Eq. (1) are written in the new coordinates (see, for example [4,7],)

ξαβγzβEγ=iωμ0ggαβHβ,ξαβγzβHγ=iωεggαβEβ,
where the summation over the repeating indices is assumed, g denotes determinant g=detgαβ, and ξαβγ is the Levi-Civita symbol (ξαβγ=1 for (α,β,γ) = (1,2,3), (3,1,2), (2,3,1), and ξαβγ=1 for (α,β,γ) = (1,3,2), (2,1,3), (3,2,1)). It is important to notice here that redefinition of tensors of electric permittivity and magnetic permeability as ε^αβ=εggαβ and μ^αβ=μ0ggαβ allows them to contain all the metric information, while Eq. (2) become mathematically equivalent to Eq. (1).

After having discussed the coordinate transformations and retrieved Maxwell’s equations in curvilinear coordinates we can proceed to the analysis of Eq. (2) by the GSM and to formulating the new method.

4. GSM implementation in curvilinear coordinates

To transform Eq. (2) to a self-consistent integral equation we exploit the formal similarity between Eqs. (1) and (2). The core of the GSM consists in constructing the solution in a basis structure defined by some functions εb(z) and μb(z) (the subscript means “basis”). In principle, such basis structure can be chosen arbitrarily. For example, one can simply take some scalar constants εb and μb. The solution of covariant Maxwell Eqs. (2) in such simple structure can be found for any given distribution of electric J(z) and magnetic M(z) sources:

ξαβγzβEγ=Mα+iωμbδαβHβ,ξαβγzβHγ=JαiωεbδαβEβ.
Note that these equations do not describe any real electromagnetic field as they do not follow from the Maxwell’s equations in the Cartesian coordinates. However, these equations are mathematically equivalent to the Maxwell’s equations in the Cartesian coordinates. This means that Eq. (3) can be formally obtained by replacing Cartesian coordinates and vector components in (1) by curvilinear ones leaving the differential operators unchanged. Thus, solutions to Eq. (3) are readily known. Presence of the magnetic current in Eq. (3) is new with respect to the former GSM implementations [2,3]. However, the analysis of magnetic sources M is very similar to that of electric sources J, and the fields emitted by magnetic currents are well known [8].

Because of the structure periodicity along z1axis the diffraction problem will be solved in two-dimensional Fourier space (reciprocal to axes z1 and z2) and one-dimensional coordinate space (axis z3). Therefore, the sources are determined in the form of plane currents

J=j(z3)exp(ik1z1+ik2z2),M=m(z3)exp(ik1z1+ik2z2).
where k1,2 denote in-plane wavevector components. The field response to these sources is found in the form of a sum of plane waves and the field proportional to the sources due to the singular term in the basis solution [8,9]:
(Eα(z)Hα(z))=(E0α(z)H0α(z))+exp(ik1z1+ik2z2)×{δα3(j3(z3)iωεbm3(z3)iωμb)+Yαβ+z3(jβ(ζ)mβ(ζ))exp[ik3(z3ζ)]dζ+Yαβz3(jβ(ζ)mβ(ζ))exp[ik3(z3ζ)]dζ}
with δαβ being the Kronecker symbol. Column (E0αH0α)T denotes the incident field. It is represented by plane waves exp(ik±z)=exp(ik1z1+ik2z2±ik3z3) propagating upwards and downwards relative to coordinate z3 (sign “±” behind the third component of wavevector k±=(k1,k2,±k3)) and having dispersion relation k3=(kb2k12k22)1/2, kb=ωεbμb. Components of matrix Y write
Yαβ±=(kα±kβ±δαβkb22ωεbk3ξαγβkγ2k3ξαγβkγ2k3kα±kβ±δαβkb22ωμbk3).
To avoid ambiguity we denote matrices and matrix elements by straight capital symbols, and scalar variables by italic type font throughout the text. The derivation of Eq. (5) is very similar to that of [2,3,9].

At the next step of the GSM Eq. (2) is rewritten in the form (3) introducing the generalized sources:

(JαMα)=iωVαβ(EβHβ),
where
Vαβ=(ε^αβεbδαβ00μ^αβμbδαβ).
By substituting these sources in Eq. (5) one can obtain an implicit integral equation for unknown fields. To simplify the analysis we move the δ-term in (5) from the right- to the left-hand side and introduce modified fields E˜and H˜ with the following z3-components:
E˜3(z)=E3(z)δ33J3(z)iωεb,H˜3(z)=H3(z)δ33M3(z)iωμb.
The other two components remain unaffected E˜1.2(z)=E1,2(z), H˜1,2(z)=H1,2(z). Combining Eqs. (7) and (9) we get
E˜3=E3+δ33(ε^3βεbδ3β)Eβεb=δ33ε^3βεbEβ,H˜3=H3+δ33(μ^3βμbδ3β)Hβμb=δ33μ^3βμbHβ.
It is clear that except for the factors 1/εb and 1/μb the modified field components E˜3 and H˜3 coincide with components of the displacement fields D3 and B3, respectively, and are continuous in Ω.

The last step consists in transition to the Fourier space conjugate to axis Z1. By definition of the discussed coordinate transformation, tensors ε^αβ and μ^αβ are periodic functions along axis z1. We assume that in new coordinates the period still equals Λ. Thus, the Fourier decomposition of fields and matrix V components is used:

f(z)=exp(ik2incz2)n=fn(z3)exp(ik1nz1)
where f denotes a decomposed function, k1n=k1inc+2πn/Λ, and k2inc is the wavevector projection along the grating grooves defined by the incident wave direction. As result of the Fourier transform all products of matrix V components by field amplitudes in Eq. (7) become convolution products and are expressed via Toeplitz matrix-by-vector multiplications.

Substituting generalized sources (7) and the modified field into (5) yields the integral equation on the unknown vector of the field Fourier components:

(Eαm(z3)Hαm(z3))=(E0α(z3)H0α(z3))δm0iω{z3exp[ik3(z3ζ)]Yαβ+Vβγ(Eγ(ζ)Hγ(ζ))dζ+z3exp[ik3(z3ζ)]YαβVβγ(Eγ(ζ)Hγ(ζ))dζ}.
Equation (12) is the self-consistent integral equation describing the grating diffraction in the curvilinear coordinate system. Information about particular coordinate transformation is fully contained in the Fourier image of matrix V components. The generalized sources generated by both spatial permittivity variations and curvilinear metric exist only in the introduced bounded region Ω. Thus, resolving Eq. (12) at the boundaries of region Ω provide the true amplitudes of outgoing diffraction waves in the Cartesian coordinate system. Moreover, since in the GSMCC the metric becomes a generalized source no additional effort is required to transform an incident field into the curvilinear coordinates as it is done in the C-method. Since the numerical method of resolving (12) is the same as that in the previous GSM implementations [2,3] and it is just briefly reviewed in the next section.

5. Numerical method

Reduction of Eq. (12) to a linear algebraic equation system is done in two steps. First the Fourier series are truncated to some finite maximum number NO (so thatNO/2nNO/2). It is important here to notice that in this work we consider continuous functions comprising matrix V (8), i.e. a continuously differentiable grating corrugation profile is assumed and two media below and above the corrugation interface are homogeneous. In this case no problem arises with the Fourier representation of discontinuous products (as in the FMM [1012] and GSM [2,3] implementations), and truncation of the Fourier series is mathematically justified.

Modified fields E˜ and H˜ can be represented as superposition of plane waves propagating in the basis homogeneous medium with εb and μb. Introducing two types of TE and TM diffraction plane waves with amplitudes ae± and ah±, respectively, the relation between the field components and plane wave amplitudes writes (see [3] for the details):

(E˜1mE˜2mE˜3mH˜1mH˜2mH˜3m)=Qm(aem+aemahm+ahm),
where
Qm=(k2γmk2γk1mk3mωεbγmk1mk3mωεbγmk1mγmk1mγmk2k3mωεbγmk2k3mωεbγm00γmωεbγmωεbk1mk3mωμbγmk1mk3mωμbγmk2γmk2γmk2k3mωμbγmk2k3mωμbγmk1mγmk1mγmγmωμbγmωμb00),
and γm=(k1m2+k22)1/2. Such substitution is important from the computational point of view since it naturally allows for reducing six mutually dependent field components to four independent amplitudes. Then, relation between source currents and emitted wave amplitudes takes the following form
(aem+aemahm+ahm)=Pm(J1mJ2mJ3mM1mM2mM3m)
with matrix Pm:

Pm=12(ωμbk2incγmk3mωμbk1mγmk3m0k1mγmk2incγγmk3mωμbk2incγk3mωμbk1mγmk3m0k1mγmk2incγmγmk3mk1mγmk2incγmγmkzωεbk2incγmk3mωεbk1mγmk3m0k1mγmk2incγmγmkzωεbk2incγmk3mωεbk1mγmk3m0).

The second discretization step implies replacement of the integral in Eq. (12) by a finite sum over slices along coordinate z3. Denoting slice thicknesses as Δh and the number of slices as NS=2b/Δh, slice center coordinates become zp3=Δh(2p+1NS)/2, p=0,1,,NS1. Exponential factors exp[±ik3m(z3ζ)] in Eq. (12) due to propagation of plane waves are combined in matrix Rmpq±:

Rmpq±=θpq±Δhexp[±ik3m(zp3zq3)],
where

θs±=12+{±12,s>00,s=012,s<0.

Thus, Eq. (12) is reduced to a set of linear algebraic equations on unknown amplitudes a=(aemp+aempahmp+ahmp)T of each harmonic (index n) in each slice (index p):

a=(IRPVQ)1ainc
Four matrices R, P, V, and Q are defined by (17), (16), (8), and (14), respectively. The size of vectors ainc and a equals to 4NSNO. Matrix Q is a block-diagonal matrix of size 6NSNO×4NSNO; matrix V is a block-Toeplitz matrix of size 6NSNO×4NSNO; elements of block-diagonal matrix P (matrix size is 4NSNO×6NSNO) are independent of slice index similarly to matrix Q. Matrix R is a block-Toeplitz matrix with respect to slice indices.

The curvilinear metric is treated in form of electromagnetic sources. Therefore, incident field amplitudes are written in non-transformed space as if the new coordinates coincide with the Cartesian ones. Thus, amplitudes of a plane incident wave are found in each slice by exponential propagation factors exp(ik3m|z3|) similarly to the standard Cartesian GSM formulation [2,3].

Equation (19) allows for calculating 4NSNO diffracted field amplitudes in each slice. In order to find 4NO amplitudes of the diffracted field at upper and lower boundaries of region Ω aout=(aenout+|z3=baenout|z3=bahnout+|z3=bahnout|z3=b)T we apply discretized Eq. (12) once again:

aout=ainc+TPVQ(IRPVQ)1ainc.
Matrix T of size 4NO×4NSNO contains exponential factors and “collects” the field diffracted in each slice at Ω boundaries:
Tmp+|z3=b=exp[ik3m(bzp3)],Tmp|z3=b=exp[ik3m(b+zp3)],Tmp+|z3=b=Tmp|z3=b=0.
For more complex basis media matrices R and T should be modified. For example, in [3] they are derived for a homogeneous plane layer placed between homogeneous substrate and cover of different materials.

Formally, Eqs. (19) and (20) is similar to the corresponding equations derived in [2,3] for diffraction in the Cartesian coordinates. Complexity of matrix inversion in (19) and (20) defines the complexity of the method. An exceptional performance of O(NONSlog(NONS)) for diffraction simulation relies on two powerful numerical techniques: the generalized minimal residual method (GMRES) [13] and the fast Fourier transform (FFT). The GMRES is used to iteratively invert matrices in (19), (20), and requires performing only matrix-vector multiplications. When performing these multiplications we use the fact that product RPVQ is a product of block-diagonal and block-Toeplitz matrices, while a product of Toeplitz matrix V by a vector can be rewritten as a convolution product by Toeplitz-extended-to-circulant matrix and calculated with the FFT. This allows performing matrix-vector multiplications required by each GMRES iteration in time which linearly grows relative to the matrix size. The memory consumption is also linear: O(NONS). A more detailed explanation of the numerical algorithm can be found in [2,3].

6. Numerical examples

The developed method was applied to calculate the diffraction on a sinusoidal grating x3=csin(Kx1). In this case, the solution can be compared with well recognized reference methods: either with the Rayleigh method [14] or with the C-method [5]. We define the following coordinate transform

x1z1,x2z2,x3={c(1|z3|/b)sin(Kx2)+z3,|z3|b,z3,|z3|>b.
Figure 3(a) demonstrates the analyzed sinusoidal corrugation example together with coordinate lines of the introduced transformation.

 figure: Fig. 3

Fig. 3 (a) Gold sinusoidal grating example and coordinate lines of the introduced curvilinear coordinate transformation. (b) Thin sinusoidal gold film placed in the air.

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To demonstrate the convergence consider a metallic sinusoidal grating with refractive index ng = 0.25 + 6.25i placed on a substrate of index ns = 1.5, and covered by air (nc = 1) [Fig. 3(a)]. Profile parameters are c = 0.1 μm, b = 0.11 μm, and Λ = 1 μm. We calculate the diffraction of a plane wave of wavelength λ = 0.6328 μm incident under 10° from the substrate side. Figure 4(a) shows the convergence of the calculated diffraction efficiency, and comparison with the Rayleigh method versus increasing number of slices NS with fixed number of diffraction orders NO = 64 (this number of harmonics guarantees the accuracy of the Rayleigh method solution to be better than the single floating point precision 108). Convergence is calculated as maximum absolute difference between corresponding diffraction amplitudes for two subsequent slice numbers. The GMRES breakpoint was chosen to be 1010. The graph demonstrates that for a given number of diffraction orders the GSMCC converges directly to the Rayleigh method (or alternatively to the C-method) solution. This is a quite important result since the conventional GSM exhibits poorer convergence even in the case of dielectric sinusoidal grating simulation (see the numerical examples in [2,3]). The next Fig. 4(b) shows the convergence of the method with the increasing number of diffraction orders for large number of slices NS = 2048. It is seen that the method quite rapidly and almost monotonically converges to better than a single floating point precision. Note also that for the both TE and TM polarizations the convergence is near the same.

 figure: Fig. 4

Fig. 4 Simulation results for the diffraction of a plane wave of wavelength λ = 0.6328 μm on a metal-dielectric sinusoidal grating with refractive index ng = 0.25 + 6.25i placed onto a substrate of index ns = 1.5, and covered by air (nc = 1) with c = 0.1 μm, b = 0.11 μm, and Λ = 1 μm. (a) Convergence of the GSMCC and comparison of the GSMCC with the Rayleigh method with better than 10−8 accuracy. (b) Convergence of the GSMCC with the increasing number of diffraction orders for large number of slices NS = 2048.

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Figure 4(a) demonstrates that solution obtained by the GSMCC and the Rayleigh method are pretty the same. To provide an additional demonstration of the method validity we compared the results for the diffraction on a sinusoidal dielectric grating (period 1 μm, depth 0.1 μm, refractive index 2.5; plane wave of wavelength 0.6328 μm is incident under angle θ=10) with the solution obtained by the FMM. Corresponding diffraction efficiencies are shown in Table 1. Both the GSMCC and the FMM solutions are calculated with NO = 63 and NS = 1024. It is seen that solutions coincide up to the 5th digit.

Tables Icon

Table 1. Comparison of diffraction efficiencies calculated by the GSMCC and the FMM for the diffraction of a plane wave on a sinusoidal corrugation interface separating air and a substrate of refractive index 2.5. Grating period is 1 μm, grating depth is 0.1 μm. A plane wave of wavelength 0.6328 μm is incident under 10° angle from the air side

In order to demonstrate the method abilities we calculate the enhanced transmission by a thin sinusoidal gold film (ng = 0.25 + 6.25i) [15] [Fig. 3(b)]. For the same corrugation depth of 0.1 μm and film thickness of 0.02 μm the reflection and transmission curves of a plane wave of wavelength λ = 0.6328 μm incident under 10° angle are shown in Fig. 5 versus the grating period. The enhanced transmission phenomenon is clearly observed for a range of grating periods around 0.53 μm and around 0.76 µm. The resonances are due to the coupling of the gold film surface plasmon in the ( ± 1)st grating orders. A detailed description of the phenomenon can be found, for example, in [16].

 figure: Fig. 5

Fig. 5 Reflection and transmission curves of a plane wave of wavelength λ = 0.6328 μm incident under 10° angle on a thin sinusoidal gold film (ng = 0.25 + 6.25i) of thickness 0.02 μm and corrugation depth 0.1 μm versus grating period.

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7. Conclusion

We present the integral curvilinear coordinate Fourier method for grating diffraction simulation. In this method, which we refer to as the GSMCC, advantages of the both GSM and C-method are combined together. The cornerstone idea of the C-method, namely coordinate transformation which converts a grating corrugation profile to a plane interface, was used to transform the corrugated interface into the continuous volume modification of the material properties. Such transformed diffraction problem is successfully solved by the GSM. Maxwell’s equations are split into the basis part with known analytical solution and the source part containing the curvilinear metric of permittivity and permeability modification. Contrary to the C-method, the curvilinear coordinate transformation is defined in a bounded region, and do not possess the translation invariance property which is essential in the C-method. Such transformation allows for considering closely placed gratings with profiles described by different functions. Additionally, the GSMCC does not rely on the Rayleigh hypothesis and potentially could be applicable to a larger variety of gratings in comparison with the C-method. The GSMCC is shown to have the linear numerical complexity relative to the number of discretization points as opposed to the cubic complexity of both the FMM and the C-method. To our knowledge this is the first work presenting a method with the described rationale, thus, all the derivation was made for a simple case of diffraction on 1D gratings with smooth corrugation profiles. Details of the GSMCC formulation for 2D gratings require a separate thorough explanation and this will be addressed in a future publication.

Acknowledgment

This work was supported in part by the Russian Ministry of Education and Science (Contract no. 14.A18.21.1946) and by the Russian Foundation for Basic Research (Grants 12-07-00592 and 12-07-00710).

References and links

1. A. V. Tishchenko, “Generalized source method: new possibilities for waveguide and grating problems,” Opt. Quantum Electron. 32(6/8), 971–980 (2000). [CrossRef]  

2. A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron. 40(6), 538–544 (2010). [CrossRef]  

3. A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf. 113(2), 158–171 (2012). [CrossRef]  

4. E. Popov, Gratings: Theory and Numerical Applications (Presses Universitaires de Provence, 2012).

5. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris) 11(4), 235–241 (1980). [CrossRef]  

6. K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013). [CrossRef]  

7. J. A. Schouten, Tensor Analysis for Physicists (Dover Publications, 1954).

8. J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939). [CrossRef]  

9. A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt. 7(6), 1425–1449 (1998). [CrossRef]  

10. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13(5), 1019–1023 (1996). [CrossRef]  

11. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13(4), 779–784 (1996). [CrossRef]  

12. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13(9), 1870–1876 (1996). [CrossRef]  

13. Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, 2003).

14. A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express 17(19), 17102–17117 (2009). [CrossRef]   [PubMed]  

15. I. Avrutsky, Y. Zhao, and V. Kochergin, “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated thin metal film,” Opt. Lett. 25(9), 595–597 (2000). [CrossRef]   [PubMed]  

16. Y. Jourlin, S. Tonchev, A. V. Tishchenko, C. Pedri, C. Veillas, O. Parriaux, A. Last, and Y. Lacroute, “Spatially and polarization resolved plasmon mediated transmission through continuous metal films,” Opt. Express 17(14), 12155–12166 (2009). [CrossRef]   [PubMed]  

References

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  1. A. V. Tishchenko, “Generalized source method: new possibilities for waveguide and grating problems,” Opt. Quantum Electron. 32(6/8), 971–980 (2000).
    [Crossref]
  2. A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron. 40(6), 538–544 (2010).
    [Crossref]
  3. A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf. 113(2), 158–171 (2012).
    [Crossref]
  4. E. Popov, Gratings: Theory and Numerical Applications (Presses Universitaires de Provence, 2012).
  5. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris) 11(4), 235–241 (1980).
    [Crossref]
  6. K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
    [Crossref]
  7. J. A. Schouten, Tensor Analysis for Physicists (Dover Publications, 1954).
  8. J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
    [Crossref]
  9. A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt. 7(6), 1425–1449 (1998).
    [Crossref]
  10. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13(5), 1019–1023 (1996).
    [Crossref]
  11. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13(4), 779–784 (1996).
    [Crossref]
  12. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13(9), 1870–1876 (1996).
    [Crossref]
  13. Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, 2003).
  14. A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express 17(19), 17102–17117 (2009).
    [Crossref] [PubMed]
  15. I. Avrutsky, Y. Zhao, and V. Kochergin, “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated thin metal film,” Opt. Lett. 25(9), 595–597 (2000).
    [Crossref] [PubMed]
  16. Y. Jourlin, S. Tonchev, A. V. Tishchenko, C. Pedri, C. Veillas, O. Parriaux, A. Last, and Y. Lacroute, “Spatially and polarization resolved plasmon mediated transmission through continuous metal films,” Opt. Express 17(14), 12155–12166 (2009).
    [Crossref] [PubMed]

2013 (1)

K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

2012 (1)

A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf. 113(2), 158–171 (2012).
[Crossref]

2010 (1)

A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron. 40(6), 538–544 (2010).
[Crossref]

2009 (2)

2000 (2)

I. Avrutsky, Y. Zhao, and V. Kochergin, “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated thin metal film,” Opt. Lett. 25(9), 595–597 (2000).
[Crossref] [PubMed]

A. V. Tishchenko, “Generalized source method: new possibilities for waveguide and grating problems,” Opt. Quantum Electron. 32(6/8), 971–980 (2000).
[Crossref]

1998 (1)

A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt. 7(6), 1425–1449 (1998).
[Crossref]

1996 (3)

1980 (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris) 11(4), 235–241 (1980).
[Crossref]

1939 (1)

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Avrutsky, I.

Chandezon, J.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris) 11(4), 235–241 (1980).
[Crossref]

Chu, L. J.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Edee, K.

K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

Granet, G.

K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13(5), 1019–1023 (1996).
[Crossref]

Guizal, B.

Jourlin, Y.

Kochergin, V.

Lacroute, Y.

Lalanne, P.

Last, A.

Li, L.

Maystre, D.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris) 11(4), 235–241 (1980).
[Crossref]

Morris, G. M.

Parriaux, O.

Pedri, C.

Plumey, J. P.

K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris) 11(4), 235–241 (1980).
[Crossref]

Shcherbakov, A. A.

A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf. 113(2), 158–171 (2012).
[Crossref]

A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron. 40(6), 538–544 (2010).
[Crossref]

Stratton, J. A.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Tishchenko, A. V.

A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf. 113(2), 158–171 (2012).
[Crossref]

A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron. 40(6), 538–544 (2010).
[Crossref]

Y. Jourlin, S. Tonchev, A. V. Tishchenko, C. Pedri, C. Veillas, O. Parriaux, A. Last, and Y. Lacroute, “Spatially and polarization resolved plasmon mediated transmission through continuous metal films,” Opt. Express 17(14), 12155–12166 (2009).
[Crossref] [PubMed]

A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express 17(19), 17102–17117 (2009).
[Crossref] [PubMed]

A. V. Tishchenko, “Generalized source method: new possibilities for waveguide and grating problems,” Opt. Quantum Electron. 32(6/8), 971–980 (2000).
[Crossref]

A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt. 7(6), 1425–1449 (1998).
[Crossref]

Tonchev, S.

Veillas, C.

Zhao, Y.

J. Opt. Soc. Am. A (3)

J. Optics (Paris) (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris) 11(4), 235–241 (1980).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (1)

A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf. 113(2), 158–171 (2012).
[Crossref]

Opt. Commun. (1)

K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

A. V. Tishchenko, “Generalized source method: new possibilities for waveguide and grating problems,” Opt. Quantum Electron. 32(6/8), 971–980 (2000).
[Crossref]

Phys. Rev. (1)

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Pure Appl. Opt. (1)

A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt. 7(6), 1425–1449 (1998).
[Crossref]

Quantum Electron. (1)

A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron. 40(6), 538–544 (2010).
[Crossref]

Other (3)

E. Popov, Gratings: Theory and Numerical Applications (Presses Universitaires de Provence, 2012).

J. A. Schouten, Tensor Analysis for Physicists (Dover Publications, 1954).

Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, 2003).

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Figures (5)

Fig. 1
Fig. 1 Problem under consideration: a plane wave diffraction on a smooth-profiled 1D metal- dielectric grating.
Fig. 2
Fig. 2 Transformation of a grating corrugation profile. (a) Curvilinear coordinates introduced in a bounded region Ω = {–b ≤□x3 ≤□b | b ≥□c}. (b) Grating profile becomes a plane interface z3 = 0, and everywhere outside Ω the new coordinates coincide with the Cartesian ones.
Fig. 3
Fig. 3 (a) Gold sinusoidal grating example and coordinate lines of the introduced curvilinear coordinate transformation. (b) Thin sinusoidal gold film placed in the air.
Fig. 4
Fig. 4 Simulation results for the diffraction of a plane wave of wavelength λ = 0.6328 μm on a metal-dielectric sinusoidal grating with refractive index ng = 0.25 + 6.25i placed onto a substrate of index ns = 1.5, and covered by air (nc = 1) with c = 0.1 μm, b = 0.11 μm, and Λ = 1 μm. (a) Convergence of the GSMCC and comparison of the GSMCC with the Rayleigh method with better than 10−8 accuracy. (b) Convergence of the GSMCC with the increasing number of diffraction orders for large number of slices NS = 2048.
Fig. 5
Fig. 5 Reflection and transmission curves of a plane wave of wavelength λ = 0.6328 μm incident under 10° angle on a thin sinusoidal gold film (ng = 0.25 + 6.25i) of thickness 0.02 μm and corrugation depth 0.1 μm versus grating period.

Tables (1)

Tables Icon

Table 1 Comparison of diffraction efficiencies calculated by the GSMCC and the FMM for the diffraction of a plane wave on a sinusoidal corrugation interface separating air and a substrate of refractive index 2.5. Grating period is 1 μm, grating depth is 0.1 μm. A plane wave of wavelength 0.6328 μm is incident under 10° angle from the air side

Equations (22)

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×E=iω μ 0 H, ×H=iωεE.
ξ αβγ z β E γ =iω μ 0 g g αβ H β , ξ αβγ z β H γ =iωε g g αβ E β ,
ξ αβγ z β E γ = M α +iω μ b δ αβ H β , ξ αβγ z β H γ = J α iω ε b δ αβ E β .
J=j( z 3 )exp( i k 1 z 1 +i k 2 z 2 ), M=m( z 3 )exp( i k 1 z 1 +i k 2 z 2 ).
( E α ( z ) H α ( z ) )=( E 0α ( z ) H 0α ( z ) )+exp( i k 1 z 1 +i k 2 z 2 )×{ δ α3 ( j 3 ( z 3 ) iω ε b m 3 ( z 3 ) iω μ b ) + Y αβ + z 3 ( j β ( ζ ) m β ( ζ ) )exp[ i k 3 ( z 3 ζ ) ]dζ + Y αβ z 3 ( j β ( ζ ) m β ( ζ ) )exp[ i k 3 ( z 3 ζ ) ]dζ }
Y αβ ± =( k α ± k β ± δ αβ k b 2 2ω ε b k 3 ξ αγβ k γ 2 k 3 ξ αγβ k γ 2 k 3 k α ± k β ± δ αβ k b 2 2ω μ b k 3 ).
( J α M α )=iω V αβ ( E β H β ),
V αβ =( ε ^ αβ ε b δ αβ 0 0 μ ^ αβ μ b δ αβ ).
E ˜ 3 ( z )= E 3 ( z ) δ 33 J 3 ( z ) iω ε b , H ˜ 3 ( z )= H 3 ( z ) δ 33 M 3 ( z ) iω μ b .
E ˜ 3 = E 3 + δ 33 ( ε ^ 3β ε b δ 3β ) E β ε b = δ 33 ε ^ 3β ε b E β , H ˜ 3 = H 3 + δ 33 ( μ ^ 3β μ b δ 3β ) H β μ b = δ 33 μ ^ 3β μ b H β .
f( z )=exp( i k 2 inc z 2 ) n= f n ( z 3 )exp( i k 1n z 1 )
( E αm ( z 3 ) H αm ( z 3 ) )=( E 0α ( z 3 ) H 0α ( z 3 ) ) δ m0 iω{ z 3 exp[ i k 3 ( z 3 ζ ) ] Y αβ + V βγ ( E γ ( ζ ) H γ ( ζ ) )dζ + z 3 exp[ i k 3 ( z 3 ζ ) ] Y αβ V βγ ( E γ ( ζ ) H γ ( ζ ) )dζ }.
( E ˜ 1m E ˜ 2m E ˜ 3m H ˜ 1m H ˜ 2m H ˜ 3m )= Q m ( a em + a em a hm + a hm ),
Q m =( k 2 γ m k 2 γ k 1m k 3m ω ε b γ m k 1m k 3m ω ε b γ m k 1m γ m k 1m γ m k 2 k 3m ω ε b γ m k 2 k 3m ω ε b γ m 0 0 γ m ω ε b γ m ω ε b k 1m k 3m ω μ b γ m k 1m k 3m ω μ b γ m k 2 γ m k 2 γ m k 2 k 3m ω μ b γ m k 2 k 3m ω μ b γ m k 1m γ m k 1m γ m γ m ω μ b γ m ω μ b 0 0 ),
( a em + a em a hm + a hm )= P m ( J 1m J 2m J 3m M 1m M 2m M 3m )
P m = 1 2 ( ω μ b k 2 inc γ m k 3m ω μ b k 1m γ m k 3m 0 k 1m γ m k 2 inc γ γ m k 3m ω μ b k 2 inc γ k 3m ω μ b k 1m γ m k 3m 0 k 1m γ m k 2 inc γ m γ m k 3m k 1m γ m k 2 inc γ m γ m k z ω ε b k 2 inc γ m k 3m ω ε b k 1m γ m k 3m 0 k 1m γ m k 2 inc γ m γ m k z ω ε b k 2 inc γ m k 3m ω ε b k 1m γ m k 3m 0 ).
R mpq ± = θ pq ± Δhexp[ ±i k 3m ( z p 3 z q 3 ) ],
θ s ± = 1 2 +{ ± 1 2 ,s>0 0,s=0 1 2 ,s<0 .
a= ( IRPVQ ) 1 a inc
a out = a inc +TPVQ ( IRPVQ ) 1 a inc .
T mp + | z 3 =b =exp[ i k 3m ( b z p 3 ) ], T mp | z 3 =b =exp[ i k 3m ( b+ z p 3 ) ], T mp + | z 3 =b = T mp | z 3 =b =0.
x 1 z 1 , x 2 z 2 , x 3 ={ c( 1 | z 3 | /b )sin( K x 2 )+ z 3 , | z 3 |b, z 3 , | z 3 |>b.

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