Analytical Grounds for Modern Theory of Two-Dimensionally Periodic Gratings

Rigorous models of one-dimensionally periodic diffraction gratings made their appearance in the 1970s, when the corresponding theoretical problems had been considered in the con‐ text of classical mathematical disciplines such as mathematical physics, computational mathematics, and the theory of differential and integral equations. Periodic structures are currently the objects of undiminishing attention. They are among the most called-for disper‐ sive elements providing efficient polarization, frequency and spatial signal selection. Fresh insights into the physics of wave processes in diffraction gratings are being implemented in‐ to radically new devices operating in gigahertz, terahertz, and optical ranges, into new ma‐ terials with inclusions ranging in size from microto nanometers, and into novel circuits for in-situ man-made and natural material measurements.


Introduction
Rigorous models of one-dimensionally periodic diffraction gratings made their appearance in the 1970s, when the corresponding theoretical problems had been considered in the context of classical mathematical disciplines such as mathematical physics, computational mathematics, and the theory of differential and integral equations. Periodic structures are currently the objects of undiminishing attention. They are among the most called-for dispersive elements providing efficient polarization, frequency and spatial signal selection. Fresh insights into the physics of wave processes in diffraction gratings are being implemented into radically new devices operating in gigahertz, terahertz, and optical ranges, into new materials with inclusions ranging in size from micro-to nanometers, and into novel circuits for in-situ man-made and natural material measurements.
However, the potentialities of classical two-dimensional models [1][2][3][4][5][6][7] are limited. Both theory and applications invite further investigation of three-dimensional, vector models of periodic structures in increasing frequency. In our opinion these models should be based on time-domain (TD) representations and implemented numerically by the mesh methods [8,9]. It follows from the well-known facts: (i) TD-approaches are free from the idealizations inherent in the frequency domain; (ii) they are universal owing to minimal restrictions imposed on geometrical and material parameters of the objects under study; (iii) they allow explicit computational schemes, which do not require inversion of any operators and call for an adequate run time when implementing on present-day computers; (iv) they result in data easy convertible into a standard set of frequency-domain characteristics. To this must be added that in recent years the local and nonlocal exact absorbing conditions (EAC) have been derived and tested [6,7]. They allow one to replace an open initial boundary value problem occurring in the electrodynamic theory of gratings with a closed problem. In addi-tion, the efficient fast Fourier transform accelerated finite-difference schemes with EAC for characterizing different resonant structures have been constructed and implemented [10].
It is evident that the computational scheme solving a grating problem must be stable and convergent, computational error must be predictable, while the numerical results are bound to be unambiguously treated in physical terms. To comply with these requirements, it is important to carry out theoretical analysis at each stage of the modeling (formulation of boundary value and initial boundary value problems, determination of the correctness classes for them, study of qualitative characteristics of singularities of analytical continuation for solutions of model boundary value problems into a domain of complex-valued frequencies, etc.).
In the present work, we present a series of analytical results providing the necessary theoretical background to the numerical solution of initial boundary value problems as applied to two-dimensionally periodic structures. Section 1 is an Introduction. In Section 2 we give general information required to formulate a model problem in electrodynamic theory of gratings. Sections 3 and 4 are devoted to correct and efficient truncation of the computational space in the problems describing spatial-temporal electromagnetic wave transformation in two-dimensionally periodic structures. Some important characteristics and properties of transient and steady-state fields in regular parts of the rectangular Floquet channel are discussed in Sections 5 and 7. In Section 6, the method of transformation operators (the TD-analog of the generalized scattering matrix method) is described; by applying this method the computational resources can be optimized when calculating a multi-layered periodic structure or a structure on a thick substrate. In Section 8, elements of spectral theory for two-dimensionally periodic gratings are given in view of its importance to physical analysis of resonant scattering of pulsed and monochromatic waves by open periodic resonators. g = {x, y, z} is the point in a three-dimensional spaceR 3 ; x, y, and z are the Cartesian coordinates; E → (g, t) = {E x , E y , E z } and H → (g, t) = {H x , H y , H z } are the electric and magnetic field vectors; η 0 = (μ 0 / ε 0 ) 1/2 is the intrinsic impedance of free space; ε 0 and μ 0 are permittivity and permeability of free space; j → (g, t) is the extraneous current density vector; χ ε (g, t), χ μ (g, t), and χ σ (g, t) are the electric, magnetic, and specific conductivity susceptibilities; f 1 (t) * f 2 (t) = ∫ f 1 (t − τ) f 2 (τ)dτstands for the convolution operation.
We use the SI system of units. From here on we shall use the term "time" for the parametert, which is measured in meters, to mean the product of the natural time and the velocity of light in vacuum.
With no frequency dispersion in the domainG ⊂ R 3 , for the points g ∈ G we have where δ(t) is the Dirac delta-function;ε(g), μ(g), and σ(g) are the relative permittivity, relative permeability, and specific conductivity of a locally inhomogeneous medium, respectively. Then equations (1) and (2) take the form: In vacuum, where ε(g) = μ(g) = 1 andσ(g) = 0, they can be rewritten in the form of the following vector problems [6]: , By Δ we denote the Laplace operator. As shown in [6], the operator grad divE → can be omitted in (5) from the following reasons. By denoting the volume density of induced and external electric charge through ρ 1 (g, t) and ρ 2 (g, t), we can write grad divE → = ε 0 −1 grad(ρ 1 + ρ 2 ). In homogeneous medium, where ε and σ are positive and non-negative constants, we have ρ 1 (g, t) = ρ 1 (g,0)exp( − tσ / ε), and if ρ 1 (g,0) = 0, then ρ 1 (g, t) = 0 for anyt > 0. The remaining term ε 0 −1 gradρ 2 can be moved to the right-hand side of (5) as a part of the function defining current sources of the electric field.
To formulate the initial boundary value problem for hyperbolic equations (1)-(6) [12], initial conditions at t = 0 and boundary conditions on the external and internal boundaries of the domain of analysis Q should be added. In 3-D vector or scalar problems, the domain Q is a part of the R 3 -space bounded by the surfaces S that are the boundaries of the domains intS , filled with a perfect conductor:Q = R 3 \ intS . In the so-called open problems, the domain of analysis may extend to infinity along one or more spatial coordinates.
The system of boundary conditions for initial boundary value problems is formulated in the following way [11]: • on the perfectly conducting surface S the tangential component of the electric field vector is zero at all times t Ε tg (g, t) | g ∈S = 0 for t ≥ 0; the normal component of the magnetic field vector on S is equal to zero (H nr (g, t) | g ∈S = 0), and the function H tg (g, t) | g ∈S defines the so-called surface currents generated on S by the external electromagnetic field; • on the surfacesS ε,μ,σ , where material properties of the medium have discontinuities, as well as all over the domainQ, the tangential components E tg (g, t) and H tg (g, t) of the electric and magnetic field vectors must be continuous; • in the vicinity of singular points of the boundaries ofQ, i.e. the points where the tangents and normals are undetermined, the field energy density must be spatially integrable; • if the domain Q is unbounded and the field {E → (g, t), H → (g, t)} is generated by the sources having bounded supports in Q then for any finite time interval (0,T ) one can construct a closed virtual boundary M ⊂ Q sufficiently removed from the sources such that The initial state of the system is determined by the initial conditions att = 0. The reference states E → (g,0) and H → (g,0) in the system (1), (2) or the system (3), (4) are the same as E → (g,0) and ∂ E → (g, t) / ∂ t | t =0 (H → (g,0)and ∂ H → (g, t) / ∂ t | t =0 ) in the differential forms of the second order (in the terms oft), to which (1), (2) or (3), (4) are transformed if the vector H  → (vectorE  → ) is eliminated (see, for example, system (5), (6)). Thus, (5) should be complemented with the initial conditions The functionsφ → (g), ψ → (g), and F → (g, t) (called the instantaneous and current source functions) usually have limited support in the closure of the domainQ. It is the practice to divide current sources into hard and soft [9]: soft sources do not have material supports and thus they are not able to scatter electromagnetic waves. Instantaneous sources are obtained from the pulsed wave U → i (g, t) exciting an electrodynamic structure: φ → (g) = U → i (g,0)and itself should satisfy the corresponding wave equation and the causality principle. It is also important to demand that the pulsed signal has not yet reached the scattering boundaries by the momentt = 0.
The latter is obviously impossible if infinite structures (for example, gratings) are illuminated by plane pulsed waves that propagate in the direction other than the normal to certain infinite boundary. Such waves are able to run through a part of the scatterer's surface by any moment of time. As a result a mathematically correct modeling of the process becomes impossible: the input data required for the initial boundary value problem to be set are defined, as a matter of fact, by the solution of this problem.

Time Domain: Initial Boundary Value Problems
The vector problem describing the transient states of the field nearby the gratings whose geometry is presented in Figure 1 can be written in the form Here, Q is the closure ofQ, χ ε,μ,σ (g, t)are piecewise continuous functions and the surfaces S are assumed to be sufficiently smooth. From this point on it will be also assumed that the continuity conditions for tangential components of the field vectors are satisfied, if required.
The domain of analysis Q = R 3 \ intS occupies a great deal of the R 3 -space. The problem formulated for that domain can be resolved analytically or numerically only in two following cases. • The problem (10) degenerates into a conventional Cauchy problem (intS = ∅ , the medium is homogeneous and nondispersive, while the supports of the functionsF → (g, t), φ → (g), and ψ → (g) are bounded). With some inessential restrictions for the source functions, the classical and generalized solution of the Cauchy problem does exist; it is unique and is described by the well-known Poisson formula [12].
• The functionsF → (g, t), φ → (g), and ψ → (g) have the same displacement symmetry as the periodic structure. In this case, the domain of analysis can be reduced to Q N = {g ∈ Q : 0 < x < l x ; 0 < y < l y }, by adding to problem (10) periodicity conditions [7] on lateral surfaces of the rectangular Floquet channelR = {g ∈ R 3 : 0 < x < l x ; 0 < y < l y }.
The domain of analysis can also be reduced to Q N in a more general case. The objects of analysis are in this case not quite physical (complex-valued sources and waves). However, by simple mathematical transformations, all the results can be presented in the customary, physically correct form. There are several reasons (to one of them we have referred at the end of Section 3) why the modeling of physically realizable processes in the electromagnetic theory of gratings should start with the initial boundary value problems for the images describing the actual sources: From (11) it follows that or, in other symbols, The use of the foregoing conditions truncates the domain of analysis to the domainQ N , which is a part of the Floquet channelR, and allows us to rewrite problem (10) in the form and It is known [6][7][8] that initial boundary value problems for the above discussed equations can be formulated such that they are uniquely solvable in the Sobolev spaceW 2 On this basis we suppose in the subsequent discussion that the problem (13) for all t ∈ 0,T has also a generalized solution from the space

Exact Absorbing Conditions for the Rectangular Floquet Channel
In this section, we present analytical results relative to the truncation of the computational space in open 3-D initial boundary value problems of the electromagnetic theory of gratings. In Section 3, by passing on to some special transforms of the functions describing physically realizable sources, the problem for infinite gratings have been reduced to that formulated in the rectangular Floquet channel R or, in other words, in the rectangular waveguide with quasi-periodic boundary conditions. Now we perform further reduction of the domain Q N to the region Q L N = { g ∈ Q N : | z | < L } (all the sources and inhomogeneities of the Floquet channel R are supposedly located in this domain). For this purpose the exact absorbing conditions [6,7,10,13,14] for the artificial boundaries L ± (z = ± L ) of the domain Q L N will be constructed such that their inclusion into (13) does not change the correctness class of the problem and its solutionE .
From here on we omit the superscripts N in (13). By applying the technique similar to that described in [13,14], represent the solution E → (g, t) of (13) in the closure of the domains where the superscript '+ ' corresponds to z ≥ L and '− ' to z ≤ − L and the following notation is used: {μ nm (x, y)} (n, m = 0, ± 1, ± 2,...) is the complete in L 2 (R z ) orthonormal system of the functions μ nm (x, y) = (l x l y ) −1/2 exp(iα n x)exp(iβ m y); The space-time amplitudes u → nm ± (z, t) satisfy the equations Equations (14) and (15) are obtained by separating variables in the homogeneous boundary value problems for the equation Δ − ∂ 2 / ∂ t 2 E → (g, t) = 0 (see formula (5)) and taking into account that in the domains A and B we have grad div E → (g, t) = 0 andF → E (g, t) = 0. It is also assumed that the field generated by the current and instantaneous sources located in Q L has not yet reached the boundaries L ± by the moment of timet = 0.
The solutions u → nm ± (z, t) of the vector problems (15), as well as in the case of scalar problems [13,14], can be written as The above formula represents nonlocal EAC for the space-time amplitudes of the field E → (g, t) in the cross-sections z = ± L of the Floquet channelR. The exact nonlocal and local absorbing conditions for the field E → (g, t) on the artificial boundaries L ± follow immediately from (16) and (14): and It is obvious that the magnetic field vector H → (g, t) of the pulsed waves }outgoing towards the domains A and B satisfies similar boundary conditions onL ± . The boundary conditions for E → (g, t) and H → (g, t) (nonlocal or local) taken together reduce the computational space for the problem (13) to the domain Q L (a part of the Floquet channelR) that contains all the sources and obstacles. Now suppose that in addition to the sources j } being incident on the boundary L + at timest > 0. The field is assumed to be nonzero only in the domainA. Since the boundary conditions (17), (18) remain valid for any pulsed wave outgoing through L ± towards z = ± ∞ [13,14], then the total field {E → (g, t), H → (g, t)}is the solution of the initial boundary value problem (13) in the domain Q L with the boundary conditions (17) or (18) on L − and the following conditions on the artificial boundaryL + : Here U is the pulsed wave outgoing towardsz = + ∞. It is generated by the incident wave U → i (g, t) ('reflection' from the virtual boundaryL + ) and the sources j

Some Important Characteristics of Transient Fields in the Rectangular Floquet Channel
For numerical implementation of the computational schemes involving boundary conditions like (19) or (20), the function U → i (g, t) for t ∈ 0,T and its normal derivative with respect to the boundary L + are to be known. To obtain the required data for the wave U generated by a given set of sources j , the following initial boundary value problem for a regular hollow Floquet channelR are to be solved: The function ρ 2 A (g, t) here determines the volume density of foreign electric charge.
First we determine the longitudinal components E z i and H z gof the domain R for all timest > 0. Let us consider the scalar initial boundary value problems following from (21): By separating of the transverse variables x and y in (22) represent the solution of the problem as To determine the scalar functions v nm(z,E ) (z, t) andv nm(z,H ) (z, t), we have to invert the following Cauchy problems for the one-dimensional Klein-Gordon equations: A are the amplitudes of the Fourier Let us continue analytically the functionsv A by zero on the semi-axis t0 and pass on to the generalized formulation of the Cauchy problem (24) [12]: where δ(t) and δ  (25) can be written as Relations (23) and (26) completely determine the longitudinal components of the field Outside the bounded domain enclosing all the sources, in the domainG ⊂ R, where the waves generated by these sources propagate freely, the following relations [6,14] are valid: in which are the scalar Borgnis functions such that and (see representation (23)) Hence the functions U E ,H (g, t) as well as the transverse components of the field { E The foregoing suggests the following important conclusion: the fields generated in the reflection zone (the domainA) and transmission zone (the domainB) of a periodic structure are uniquely determined by their longitudinal (directed along z-axis) components and can be represented in the following form (see also formulas (14) and (23)). For the incident wave we have for the reflected wave U → s (g, t)(which coincides with the total field U → (g, t) ifU and for the transmitted wave (coinciding in the domain B with the total fieldU → (g, t)) we can write In applied problems, the most widespread are situations where a periodic structure is excited by one of the partial components of TE-wave (withE z i (g, t) = 0) or TM -wave (with H z i (g, t) = 0) [7]. Consider, for example, a partial wave of orderpq. Then we have The excitation of this kind is implemented in our models in the following way. The time function v pq(z,H ) (L , t) or v pq(z,E ) (L , t)is defined on the boundaryL + . This function determines the width of the pulseU → i (g, t), namely, the frequency range K 1 , K 2 such that for all frequencies k from this range (k = 2π / λ, λis the wavelength in free space) the value where ṽ pq(z,H orE ) (L , k ) is the spectral amplitude of the pulsev pq(z,H orE ) (L , t), exceeds some given valueγ = γ 0 . All spectral characteristics f (k ) are obtainable from the temporal characteristics f (t) by applying the Laplace transform For numerical implementation of the boundary conditions (19) and (20) and for calculating space-time amplitudes of the transverse components of the wave U → i (g, t) in the cross-section z = L of the Floquet channel (formulas (27) and (29)), the function (v pq(z,H orE ) ) ′ (L , t) are to be determined. To do this, we apply the following relation [7,14]: which is valid for all the amplitudes of the pulsed wave U → i (g, t) outgoing towards z = − ∞ and does not violate the causality principle.

Evolutionary basis of a signal and transformation operators
Let us place an arbitrary periodic structure of finite thickness between two homogeneous dielectric half-spaces z 1 = z − L > 0 (withε = ε 1 ) and z 2 = − z − L > 0 (withε = ε 2 ). Let also a local coordinate system g j = {x j , y j , z j } be associated with each of these half-spaces (Figure 2).
Assume that the distant sources located in the domain A of the upper half-space generate a primary wave U →  t)). In Section 5, we have shown that the fields under consideration are uniquely determined by their longitudinal components, which can be given, for example, as: (see also formulas (30)-(32)). Here, as before, {μ nm (x, y)} n,m=−∞ ∞ is the complete (inL 2 (R z )) orthonormal system of transverse eigenfunctions of the Floquet channel R (see Section 4), while the space-time amplitudes u nm( j,E ) (z j , t) and u nm( j,H ) (z j , t) are determined by the solutions of the following problems (see also problem (15)) for the one-dimensional Klein-Gordon equations: Compose from the functionsv nm(1,E ) (z 1 , t), v nm(1,H ) (z 1 , t), u nm( j,E ) (z j , t), u nm( j,H ) (z j , t)and the eigenvalues λ nm (n, m = 0, ± 1, ± 2,...) the such that their members are defined according to the rules depicted in Figure 3. The sets v (1) (z 1 , t) and u ( j) (z j , t) are said to be evolutionary bases of signals U →  Let us introduce by the relations The structure of the operators given by (40) can be detailed by the formula which reflects general properties of solutions of homogeneous problems (37), i.e. the solutions that satisfy zero initial conditions and are free from the components propagating in the direction of decreasingz j . The derivation technique for (41) is discussed at length in [6,13,14].
Let us construct the algorithm for calculating scattering characteristics of a multilayer structure consisting of two-dimensionally periodic gratings, for which the operatorsS AA , S BA , S pq AB , and S pq BB are known. Consider a double-layer structure, whose geometry is given in Figure 4. Two semi-transparent periodic gratings I and II are separated by a dielectric layer of finite thickness M (hereε = ε 2 (I) = ε 1 (II)) and placed between the upper and the bottom dielectric half-spaces with the permittivity ε 1 (I) andε 2 (II), respectively. Let also a pulsed wave like (35) be incident onto the boundary z 1 (I) = 0from the Floquet channelA.
Retaining previously accepted notation (the evident changes are conditioned by the presence of two different gratings I and II), represent the solution of the corresponding initial boundary value problem in the regular domainsA, B, and C in a symbolic form The first terms in the square brackets correspond to the waves propagating towards the do-mainC, while the second ones correspond to the waves propagating towards the domain A

By denoting
Analytical Grounds for Modern Theory of Two-Dimensionally Periodic Gratings http://dx.doi.org/10.5772/51007 according to formulas (38)-(42), we construct the following system of operator equations:  I  I  I  I Z  II   I  I  I  I Z  II   II  II Z  I   II  II Z  I . (formulas (40) and (41)), we relate a pair 'field → directional derivative with respect to the propagation direction' to increase numerical efficiency of the corresponding computational algorithms.
The amplitudes ũ nm(z,E orH ) ± (k) form the system of the so-called scattering coefficients of the grating, namely, the reflection coefficients They follow from the complex power theorem (Poynting theorem) in the integral form [11] ∮ S L When deriving (50), (51) we have also used the equations relating z-components of the eigenmode of the Floquet channel U → (g, k ) : Ẽ z (g, k ) = Ae ±iΓz μ(x, y) and H z (g, k) = Be ±iΓz μ(x, y) (subscripts nm are omitted) with its longitudinal components: Here, According to the Lorentz lemma [11], the fields { E (2) , H → (2) } resulting from the interaction of a grating with two plane TM -waves satisfy the following equation From (57), using (54) and (56), we obtain , p, q, r, s = 0, ± 1, ± 2,...
-the reciprocity relations, which are of considerable importance in the physical analysis of wave scattering by periodic structures as well as when testing numerical algorithms for boundary problems (53), (54). k ) : g, k, A) be incident on the grating from the domainA, as in the case considered above, while another wave U → −r ,−s(E ) i (2) (g, k ) : Ẽ z i (2) (g, k , B) fromB. Both of these waves satisfy equation (57), whence we have and , p, q, r, s = 0, ± 1, ± 2,...

General properties of the grating's secondary field
Let now k be a real positive frequency parameter, and let an arbitrary semi-transparent grating ( Figure 1) be excited from the domain A by a homogeneous TM -or TE -wave The terms of infinite series in (54) and (61) are z-components of nm-th harmonics of the scattered field for the domains A andB. The complex amplitudes R pq(E orH ) nm(E orH ) and T pq(E orH ) nm(E orH ) are the functions ofk, Φ x , Φ y , as well as of the geometry and material parameters of the grating.
It is obvious that the propagation directions k → nm of homogeneous harmonics of the secondary field depends on their ordernm, on the values of k and on the directing vector of the incident wavek → pq i :k x i = α p , k y i = β q ,k z i = − Γ pq . According to (50) and (62), we can write the following formulas for the values, which determine the 'energy content' of harmonics, or in other words, the relative part of the energy directed by the structure into the relevant spatial radiation channel: (for TM -case) and Generation of the nonspecularly reflected mode of this kind is termed the auto-collimation.
The amplitudes R pq(E orH ) nm(E orH ) or T pq(E orH ) nm(E orH ) are not all of significance for the physical analysis. In the far-field zone, the secondary field is formed only by the propagating harmonics of the orders nm such thatReΓ nm ≥ 0. However, the radiation field in the immediate proximity of the grating requires a consideration of the contribution of damped harmonics (nm : ImΓ nm > 0).
Moreover, in some situations (resonance mode) this contribution is the dominating one [6].
• The upper lines in (50) and (62) Analytical Grounds for Modern Theory of Two-Dimensionally Periodic Gratings http://dx.doi.org/10.5772/51007 The first equation in (70) proves that the efficiency of transformation of the TM -or TE -wave into the specular reflected wave of the same polarization remains unchanged if the grating is rotated in the plane x0y about z-axis through180°. The efficiency of transformation into the principal transmitted wave of the same polarizations does not also vary with the grating rotation about the axis lying in the plane x0y and being normal to the vector k → 00 ( Figure 5).
• When r = s = p = q = 0 we derive from (58), (59), (63), and (64) that That means that even if a semi-transparent or reflecting grating is non symmetric with respect to the any planes, the reflection and transmission coefficients entering (71) do not depend on the proper changes in the angles of incidence of the primary wave.
• Relations (50), (58) allow the following regularities to be formulated for ideal (σ(g, k) ≡ 0) asymmetrical reflecting gratings. Let the parametersk, Φ x , and Φ y be such that ReΓ 00 (Φ x , Φ y ) > 0 and ReΓ nm (Φ x , Φ y ) = 0 forn, m ≠ 0. If the incident wave is an inhomogene- SinceR pq(E ) It is easy to realize a physical meaning of the equation (73) and of similar relation for TEcase, which may be of interest for diffraction electronics. If a grating is excited by a damped harmonic, the efficiency of transformation into the unique propagating harmonic of spatial spectrum is unaffected by the structure rotation in the plane x0y about z-axis through180°. The above-stated corollaries have considerable utility in testing numerical results and making easier their physical interpretation. The use of these corollaries may considerably reduce amount of calculations.

Elements of Spectral Theory for Two-Dimensionally Periodic Gratings
The spectral theory of gratings studies singularities of analytical continuation of solutions of boundary value problems formulated in the frequency domain (see, for example, problems (53), (54) and (60), (61)) into the domain of complex-valued (nonphysical) values of real parameters (like frequency, propagation constants, etc.) and the role of these singularities in resonant and anomalous modes in monochromatic and pulsed wave scattering. The fundamental results of this theory for one-dimensionally periodic gratings are presented in [4,6,7]. We present some elements of the spectral theory for two-dimensionally periodic structures, which follow immediately form the results obtained in the previous sections. The frequency k acts as a spectral parameter; a two-dimensionally periodic grating is considered as an open periodic resonator.

Canonical Green function
Let a solution G 0 (g, p, k) of the scalar problem is named the canonical Green function for 2-D periodic gratings. In the case of the elementary periodic structure with the absence of any material scatterers, the problems of this kind but with arbitrary right-hand parts of the Helmholtz equation are formulated for the monochromatic waves generated by quasi-periodic current sources located in the region|z| < L .
In the domainπ < argk ≤ 3π / 2, the situation is similar only the signs of ReΓ nm are opposite.
On the subsequent sheets (each of them with its own pair{k; Γ nm (k) }), the signs (root branches) of Γ nm (k) are opposite to those they have on the first sheet for a finite number of n andm.
The cuts (solid lines in Fig. 6) originate from the real algebraic branch pointsk nm ± = ± | λ nm | . In the vicinity of some fixed point K ∈ K the function G 0 (g, p, k) can be expanded into a Loran series in terms of the local variable [17] Therefore, this function is meromorphic on the surfaceK . Calculating the residuals Res k=k G 0 (g, p, k ) at the simple polesk ∈ {k nm ± }, we obtain nontrivial solutions of homogeneous (U → i (g, k ) ≡ 0) canonical (ε(g, k) ≡ 1, μ(g, k ) ≡ 1,intS = ∅ ) problems (53), (54) and (60), (61): E → (g, k nm ± )={Ẽ x , Ẽ y , Ẽ z } ; Ẽ x, y or z = a x, y or z exp i(α n x + β m y) and H → (g, k nm ± )=(ik nm ± η 0 ) −1 rotE → (g, k nm ± ) the uniqueness allows one to estimate roughly a domain where elements of the set Ω k are localized and simplify substantially the subsequent numerical solution of spectral problems owing to reduction of a search zone of the eigen frequencies. The uniqueness theorems serve also as a basis for application of the 'meromorphic' Fredholm theorem [20] when constructing well grounded algorithms for solving diffraction problems as well as when studying qualitative characteristics of gratings' spectra [4,7].
• There are no free oscillations whose eigen frequencies k j are located on the upper halfplane (Imk0) of the first sheet of the surfaceK . This can be verified by taking into account the upper relation in (82), the function Γ nm (k) onC k , and the inequalitiesV 1 ≥ 0, V 2 > 0,V 3 > 0.
• If σ(g) ≡ 0 (the grating is non-absorptive), no free oscillations exist whose eigen frequencies k j are located on the bottom half-plane (Imk < 0) of the sheet C k between the cuts corresponding to the least absolute values ofk nm ± . In Figure 6, this region of the first sheet of K and the above-mentioned domain are shaded by horizontal lines.
By inverting homogeneous operator equation (85), we can construct a numerical solution of the spectral problem given by (79), (80) [4,6], in other words, calculate the complex-valued eigen frequencies k j and associated eigen waves U → (g, k j ) = {E → (g, k j ), H → (g, k j )} or free oscil-

Conclusion
The analytical results presented in the chapter are of much interest in the development of rigorous theory of two-dimensionally periodic gratings as well as in numerical solution of the associated initial boundary value problems. We derived exact absorbing boundary conditions truncating the unbounded computational space of the initial boundary value problem for two-dimensionally periodic structures to a bounded part of the Floquet channel.
Some important features of transient and steady-state fields in rectangular parts of the Floquet channel were discussed. The technique for calculating electrodynamic characteristics of multi-layered structure consisting of two-dimensionally periodic gratings was developed by introducing the transformation operators similar to generalized scattering matrices in the frequency domain. In the last section, the elements of spectral theory for two-dimensionally periodic gratings were discussed.