General analytical solution for the electromagnetic grating diffraction problem

Implementing the modal method in the electromagnetic grating diffraction problem delivered by the curvilinear coordinate transformation yields a general analytical solution to the 1D grating diffraction problem in a form of a T-matrix. Simultaneously it is shown that the validity of the Rayleigh expansion is defined by the validity of the modal expansion in a transformed medium delivered by the coordinate transformation.


Introduction
The Rayleigh hypothesis (RH) was first formulated in [1] and then applied to the theory of diffraction gratings in [2]. If one considers the reflection of a plane wave from the plane interface between two homogeneous media, there exist only three waves: the incident plane wave, the reflected outgoing plane wave, and the transmitted (refracted) plane wave. Considering reflection from a sinusoidal interface, Rayleigh looked for a solution in a similar form, assuming that the field above and under the grating interface only consists of outgoing waves with constant amplitudes. Whereas such assumption is proved to be true for the regions outside the grating, it is considered as doubtful within the grating region by many researches, and any method based on the RH is still regarded as approximate (e.g., [3,4]).
Despite a number of works rationalizing a limited applicability of the RH [5][6][7][8][9][10], there was evidence calling into question the established theoretical limits [11,12]. Furthermore, numerical validity of the RH for deep sinusoidal gratings, even for the correct near field simulation, which contradicted the admitted belief, was demonstrated in [13] (see also [15]). This article presents a theoretical analysis based on a concept formulated in [14]. The Chandezon Method (CM) [16,17], and the True Modal Method (TMM) [18][19][20] reputed to be rigorous in the diffraction theory are used here to shed light on the problem. Both of these methods are well established and yield stable and correct results when applied to deep gratings. We show that the association of a basic CM idea (the coordinate transformation, which does not depend on any hypothesis) and the TMM technique (a construction of the modal basis of the true permittivity and permeability profile) leads to the demonstration of the validity of the RH providing that the modal expansion is complete, and to an analytical solution to the grating diffraction problem.
In the article we, first, show that the CM known to rigorously solve grating diffraction problems by means of a coordinate transformation is actually identical to the Rayleigh hypothesis provided the fields are represented in the basis of true modes of the transformed structure instead of a basis of the diffraction orders calculated by means of the Fourier decomposition of the transformed structure. Second, the coordinate transformation approach implemented on the basis of the modes of the transformed structure will be demonstrated to lead to an exact analytical solution to a wide range of grating problems. Moreover, such analytical solutions appear to be identical to those obtained directly on the basis of the RH. To our knowledge, this is the first time when the solution to a general diffraction problem is found in a closed analytical form. Two grating examples being important for practical applications (sinusoidal and saw-tooth profiles) are chosen to illustrate these two steps. At the end of the paper we discuss consequences of these results for electromagnetic simulation.

Problem formulation and notations
This work refers to the 1D plane grating linear diffraction problem which requires one to solve Maxwell's equations for a given incident field together with boundary conditions (continuity of tangential field components) at a periodically corrugated interface between two homogeneous isotropic media described by dielectric permittivities ε a,b and magnetic permeabilities µ a,b (see illustration in Fig. 1). Due to linearity of the problem the electromagnetic fields will be implicitly assumed to be harmonic with exp(− jωt) time dependence factor.
Consider a Cartesian coordinate system (x 1 , x 2 , x 3 ), whose axis X 3 is perpendicular to the grating plane, and axis X 1 indicates periodicity direction. Corrugation profile is supposed to be defined by a continuous and piecewise twice differentiable function f ( According to the principles of the CM we will implement a transformation from Cartesian coordinates (x 1 , x 2 , x 3 ) with unit orts i α , α = 1, 2, 3, to curvilinear coordinates (z 1 , . Contravariant and covariant basis vector sets of the new coordinate system are e α = (∂x β /∂z α )i β and e α = (∂z α /∂x β )i β respectively. The two bases are mutually orthogonal: e α · e β = δ β α . Here and further summation over the repeating index is implied. Scalar products of basis vectors yield metric tensor components g αβ = e α · e β and g αβ = e α · e β with g = det{g αβ }. Conventionally, covariant and contravariant components of any vectorF are identified by lower and upper indices asF α andF α , where the tilde is used to distinguish curvilinear vector components from components in the Cartesian coordinates F α . Corresponding relations read For more details on the tensor notations we refer readers to [21]. Source-free Maxwell's equations in the Cartesian system read Curls in left-hand sides of Eqs. (2) are written via Levi-Civita symbols ξ αβγ , and Kronecker delta symbol δ αβ is kept here for consistency with representation in curvilinear coordinates, where vector components with upper and lower indices differ. In curvilinear coordinates Maxwell's equations include the metric tensor components: Eqs. (3) are quite similar to Eqs. (2). The only difference is in the permittivity and the permeability tensors, which can be redefined as with χ standing either for ε or µ. Such similarity leads to an important conclusion: a solution to the electromagnetic problem in the curvilinear coordinate system is equivalent to a solution in Cartesian coordinates (x 1 ,x 2 ,x 3 ) (bars are used here to avoid confusion with the initial coordinates) supposing that properties of the medium are determined in accordance with Eq. (4). We refer to this problem and the corresponding solution as reciprocal. This means that there exists a transformed medium with permittivityε and permeabilityμ, in which the fields written in the Cartesian system (x 1 ,x 2 ,x 3 ) have the same coordinate dependences as the unknown fieldsẼ andH in the curvilinear coordinate system (z 1 , z 2 , z 3 ). Every solution in the transformed medium corresponds to a solution of the initial problem providing that all the initial conditions and fields are correctly translated. The inverse transformation defined by Eqs. (1) yields the required solution in the initial Cartesian coordinates ( and back is not a coordinate transformation within a given diffraction problem, but a formal replacement of one diffraction problem with another, which is possible due to the similarity of Maxwell's equations (2) and (3). Thus, we will rely on the following claim. Given a boundary electromagnetic problem and a curvilinear coordinate system in which the boundary coincides with a coordinate plane, there exists a volume electromagnetic problem such as any solution in Cartesian coordinate system (x 1 ,x 2 ,x 3 ) has a corresponding solution to the initial problem in the transformed coordinate system (z 1 , z 2 , z 3 ) expressed by the same coordinate functions.

Coordinate transformation of the grating region
The CM substitutes the electromagnetic grating diffraction problem by another one which deals with plane boundary but with changed permeability and permittivity tensors. The introduced tensorsε andμ determine an electromagnetic response of the medium. Therefore, a choice of the curvilinear coordinate system is a very important step. If one chooses new coordinates so that tensor g αβ depends only on one coordinate, the reciprocal problem will be one-dimensional. This dramatically simplifies the resolution of the problem. In the considered case we can take In accordance with Eqs. (4) and (5), the permittivity and permeability tensors of the transformed structure areχ αβ = χM αβ with Consider here two illustrative examples. The first one is a sinusoidal corrugation between two isotropic media with profile function (the upper part of Fig. 2a): It follows from Eq. (7) that The permeability and the permittivity of the reciprocal medium appear to be smoothly modulated along the grating period, as the lower part of Fig. 2a illustrates. The other practically important example is a saw-tooth corrugation (Fig. 2b): where d 2 = Λ − d 1 . The coordinate transformation yields a reciprocal stratified structure described by two tensors per period

Modal solution to the diffraction problem
The next step in the analysis is the resolution of the diffraction problem in the transformed medium with material tensors given by Eqs. (6) and (7) (with coordinates (z 1 , z 2 , z 3 ) being substituted by ( x 1 ,x 2 ,x 3 )). The transformed structure is composed of periodically stratified media being translational invariant alongx 3 , in which an exact field solution (which also will be denoted with a bar asF) can be expressed in terms of modes propagating up and down along coordinatex 3 . Once the modal basis of such volume grating is defined, any field solution in the grating region is represented by a superposition of grating modes [18]. Since no dependence assumed alongx 2 direction, all grating modes split into TE and TM ones. The electric field of each TE mode is directed along theX 2 axis, and so does the magnetic field of all TM modes.
Suppose that each mode propagates up or down the grating with the propagation constant β q , q ∈ Z. Since the grating interface corresponds to planex 3 = 0 in the reciprocal problem, the modal spectrum should be retrieved for tensorsε =ε a andμ =μ a in the regionx 3 > 0, and forε =ε b andμ =μ b in the regionx 3 < 0. In other words the modal propagation constants and amplitudes will be different below and above the plane interface. Denote the propagation constant and the mode amplitude as β a q and a ± q , respectively, in the regionx 3 > 0. Here "plus" sign corresponds to the upward propagation. Analogously, the symbols β b q and b ± q are used further to describe a mode of order q in regionx 3 < 0.
Without loss of generality let us consider region thex 3 < 0 (derivation forx 3 > 0 is absolutely the same). Thex 2 component of the electric field of the q-th TE mode is where function ψ b q (x 1 ) describes modal field distribution along the grating period. FieldẼ T E 2q satisfies a quasi-periodicity condition: imposed by an incident field with wavevector projection onx 1 direction equal to k inc 1 . Substitution of Eq. (12) into the first set of Maxwell's equations yields the magnetic field of the TE modē Further substitution of this field components into the second set of Maxwell's equations provides a differential equation on function ψ b q (x 1 ): which can be reduced to the Riccati type equation. Analogous considerations and derivations for the q-th TM mode yield the field components and absolutely the same differential equation on function φ b q (x 1 ) as Eq. (15). This means that all TE and TM modes of the same index have similar modal field distribution defined by solution ψ b q (x 1 ) ≡ φ b q (x 1 ) of Eq. (15). General solution of Eq. (15) can be searched in form [22] with an unknown function G(x 1 ). This function satisfies the homogeneous Helmholtz equation Therefore, in any region R i = {x 1 :x 1,i−1 <x 1 <x 1,i }, withx 1,i being some constants, a of continuous derivative f (x 1 ) a general solution of Eq. (15) writes: where constants C i and D i should to be related by vertical boundary conditions. Consider the modal fields at the vertical interfacex 1 =x 1,i between some two adjacent domains R i and R i+1 . Continuity of the tangent field component gives a remarkable result: This means that each mode is either a pure right (D i = 0) or a pure left (C i = 0) propagating wave. According to Eqs. (12), (13), and (19), dispersion equations for the right and left propagating modes are Thus, the propagation constant of the q-th order grating mode is where k 1q = k inc 1 + 2πq/Λ, index q runs from −∞ to ∞, and 0 ≤ arg(β b q ) < π. Additionally, substitution of the last relation into Eqs. (12), (14), and (16) demonstrates that the absence of reflections at the vertical interfacesx 1 =x 1,i is due to equality of mode impedances in all domains where F and G stand for fields E and H for the TE polarization and vice versa for the TM polarization.

Grating T-matrix
In accordance with Eq. (22) the transverse modal field components relative tox 3 direction are proportional toF where notations are the same as for Eq. (23), and χ denotes either µ or ε in the TE or the TM case respectively, as before. Similar relations hold for the upper medium. Continuity of the tangent field components at the interfacex 3 = 0 leads to the equations relating modal amplitudes below and above the interface: 1 It is known in the modal method theory that normalizing modal fields makes available analytical expressions for T-matrix components. To this end, we will use here the following integrals 1 Λ Λ ∫ 0 (β a,b p − β a,b q ) ± (k 1p + k 1q ) f (x 1 ) exp j(k 1q − k 1p )x 1 ± (β a,b q + β a,b p ) f (x 1 ) dx 1