Open access peer-reviewed chapter

Analytical Grounds for Modern Theory of Two- Dimensionally Periodic Gratings

By L. G. Velychko, Yu. K. Sirenko and E. D. Vinogradova

Submitted: February 24th 2012Reviewed: June 24th 2012Published: November 14th 2012

DOI: 10.5772/51007

Downloaded: 886

1. 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 [1-7] 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 addition, 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.

2. Fundamental Equations, Domain of Analysis, Initial and Boundary Conditions

Space-time and space-frequency transformations of electromagnetic waves in diffraction gratings, waveguide units, open resonators, radiators, etc. are described by the solutions of initial boundary value problems and boundary value problems for Maxwell’s equations. In this chapter, we consider the problems of electromagnetic theory of gratings resulting from the following system of Maxwell’s equations for waves propagating in stationary, locally inhomogeneous, isotropic, and frequency dispersive media [9,11]:

rotH(g,t)=η01[E(g,t)+χε(g,t)E(g,t)]t+χσ(g,t)E(g,t)+j(g,t),E1
rotE(g,t)=η0[H(g,t)+χμ(g,t)H(g,t)]t,E2

where

g={x,y,z}is the point in a three-dimensional spaceR3;

x, y, and zare the Cartesian coordinates;

E(g,t)={Ex,Ey,Ez}and H(g,t)={Hx,Hy,Hz}are the electric and magnetic field vectors;

η0=(μ0/ε0)1/2is the intrinsic impedance of free space;

ε0and μ0are permittivity and permeability of free space;

j(g,t)is the extraneous current density vector;

χε(g,t), χμ(g,t), and χσ(g,t)are the electric, magnetic, and specific conductivity susceptibilities; f1(t)f2(t)=f1(tτ)f2(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 domainGR3, for the points gGwe have

χε(g,t)=δ(t)[ε(g)1],χμ(g,t)=δ(t)[μ(g)1],χσ(g,t)=δ(t)σ(g),

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:

rotH(g,t)=η01ε(g)E(g,t)t+σ(g)E(g,t)+j(g,t),E3
rotE(g,t)=η0μ(g)H(g,t)t.E4

In vacuum, where ε(g)=μ(g)=1andσ(g)=0, they can be rewritten in the form of the following vector problems [6]:

[Δgraddiv2t2]E(g,t)=FE(g,t),tH(g,t)=η01rotE(g,t),FE(g,t)=η0tj(g,t)E5

or

[Δ2t2]H(g,t)=FH(g,t),η01tE(g,t)=rotH(g,t)j(g,t),FH(g,t)=rotj(g,t).E6

By Δwe denote the Laplace operator. As shown in [6], the operator graddivEcan be omitted in (5) from the following reasons. By denoting the volume density of induced and external electric charge through ρ1(g,t)and ρ2(g,t), we can write graddivE=ε01grad(ρ1+ρ2). In homogeneous medium, where εand σare positive and non-negative constants, we have ρ1(g,t)=ρ1(g,0)exp(tσ/ε), and if ρ1(g,0)=0, then ρ1(g,t)=0for anyt>0. The remaining term ε01gradρ2can 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=0and boundary conditions on the external and internal boundaries of the domain of analysis Qshould be added. In 3-D vector or scalar problems, the domain Qis a part of the R3-space bounded by the surfaces Sthat are the boundaries of the domainsintS, filled with a perfect conductor:Q=R3\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 Sthe tangential component of the electric field vector is zero at all times t

Εtg(g,t)|gS=0fort0;E7

the normal component of the magnetic field vector on Sis equal to zero (Hnr(g,t)|gS=0), and the function Htg(g,t)|gSdefines the so-called surface currents generated on Sby the external electromagnetic field;

  • on the surfacesSε,μ,σ, where material properties of the medium have discontinuities, as well as all over the domainQ, the tangential components Etg(g,t)and Htg(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 Qis unbounded and the field {E(g,t),H(g,t)}is generated by the sources having bounded supports in Qthen for any finite time interval (0,T)one can construct a closed virtual boundary MQsufficiently removed from the sources such that

{E(g,t),H(g,t)}|gM,t(0,T)=0.E8

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

Ε(g,0)=φ(g),tΕ(g,t)|t=0=ψ(g),gQ¯.E9

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 Ui(g,t)exciting an electrodynamic structure: φ(g)=Ui(g,0)andψ(g)=[Ui(g,t)/t]|t=0. The pulsed signal Ui(g,t)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.

3. 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

{rotH(g,t)=η01[E(g,t)+χε(g,t)E(g,t)]t+χσ(g,t)E(g,t)+j(g,t),rotE(g,t)=η0[H(g,t)+χμ(g,t)H(g,t)]t,g={x,y,z}Q,t>0Ε(g,0)=φE(g),H(g,0)=φH(g),gQ¯Εtg(g,t)|gS=0,Hnr(g,t)|gS=0,t0.E10

Here, Q¯is the closure ofQ, χε,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=R3\intS¯occupies a great deal of the R3-space. The problem formulated for that domain can be resolved analytically or numerically only in two following cases.

Figure 1.

Geometry of a two-dimensionally periodic grating.

  • 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 toQN={gQ:0<x<lx;0<y<ly}, by adding to problem (10) periodicity conditions [7] on lateral surfaces of the rectangular Floquet channelR={gR3:0<x<lx;0<y<ly}.

The domain of analysis can also be reduced to QNin 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 fN(g,t,Φx,Φy)of the functions f(g,t)describing the actual sources:

f(g,t)=f˜(z,t,Φx,Φy)exp(2πiΦxxlx)exp(2πiΦyyly)dΦxdΦy=fN(g,t,Φx,Φy)dΦxdΦyE11

From (11) it follows that

fN{fNx}(x+lx,y,z,t,Φx,Φy)=e2πiΦxfN{fNx}(x,y,z,t,Φx,Φy),

fN{fNy}(x,y+ly,z,t,Φx,Φy)=e2πiΦyfN{fNy}(x,y,z,t,Φx,Φy)

or, in other symbols,

D[fN](x+lx,y)=e2πiΦxD[fN](x,y),D[fN](x,y+ly)=e2πiΦyD[fN](x,y).

The use of the foregoing conditions truncates the domain of analysis to the domainQN, which is a part of the Floquet channelR, and allows us to rewrite problem (10) in the form

E(g,t)=EN(g,t,Φx,Φy)dΦxdΦy,H(g,t)=HN(g,t,Φx,Φy)dΦxdΦyE12

and

{rotHN(g,t)=η01[EN(g,t)+χε(g,t)EN(g,t)]t+χσ(g,t)EN(g,t)+jN(g,t),rotEN(g,t)=η0[HN(g,t)+χμ(g,t)HN(g,t)]t,gQN,t>0ΕN(g,0)=φEN(g),HN(g,0)=φHN(g),gQ¯ND[EN(HN)](lx,y)=e2πiΦxD[EN(HN)](0,y),0ylyD[EN(HN)](x,ly)=e2πiΦyD[EN(HN)](x,0),0xlxΕtgN(g,t)|gS=0,HnrN(g,t)|gS=0,t0.E13

It is known [6-8] that initial boundary value problems for the above discussed equations can be formulated such that they are uniquely solvable in the Sobolev spaceW21(QT), where QT=Q×(0,T)and0tT. 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 W21(QN,T)and that the uniqueness theorem is true in this space. Here symbol ×stands for the operation of direct product of two sets, (0,T)and [0,T]are open and closed intervals, Wmn(G)is the set of all elements f(g)from the space Lm(G)whose generalized derivatives up to the order ninclusive also belong toLm(G). Lm(G)is the space of the functions f(g)={fx,fy,fz}(forgG) such that the functions |f...(g)|mare integrable on the domainG.

4. 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 Ror, in other words, in the rectangular waveguide with quasi-periodic boundary conditions. Now we perform further reduction of the domain QNto the region QLN={gQN:|z|<L}(all the sources and inhomogeneities of the Floquet channel Rare 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 QLNwill be constructed such that their inclusion into (13) does not change the correctness class of the problem and its solutionEN(g,t),HN(g,t).

From here on we omit the superscripts Nin (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 A={gR:z>L}and B={gR:z<L}in the following form:

E(g,t)=n,m=unm±(z,t)μnm(x,y),{x,y}R¯z,t0,E14

where the superscript ‘+’ corresponds to zLand ‘’ to zLand the following notation is used:

Rz=(0<x<lx)×(0<y<ly);

{μnm(x,y)}(n,m=0,±1,±2,...) is the complete in L2(Rz)orthonormal system of the functionsμnm(x,y)=(lxly)1/2exp(iαnx)exp(iβmy);

αn=2π(Φx+n)/lx, βm=2π(Φy+m)/ly, andλnm2=αn2+βm2.

The space-time amplitudes unm±(z,t)satisfy the equations

{[2t2+2z2λnm2]unm±(z,t)=0,t>0unm±(z,0)=0,tunm±(z,t)|t=0=0,{zLzL}.E15

Equations (14) and (15) are obtained by separating variables in the homogeneous boundary value problems for the equation [Δ2/t2]E(g,t)=0(see formula (5)) and taking into account that in the domains Aand Bwe have graddivE(g,t)=0andFE(g,t)=0. It is also assumed that the field generated by the current and instantaneous sources located in QLhas not yet reached the boundaries L±by the moment of timet=0.

The solutions unm±(z,t)of the vector problems (15), as well as in the case of scalar problems [13,14], can be written as

unm±(±L,t)=0tJ0[λnm(tτ)]unm±(±L,τ)dτ,t0.E16

The above formula represents nonlocal EAC for the space-time amplitudes of the field E(g,t)in the cross-sections z=±Lof 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):

E(x,y,±L,t)=n,m={0tJ0[λnm(tτ)][0lx0lyE(x˜,y˜,z,τ)z|z=±Lμnm(x˜,y˜)dx˜dy˜]dτ}μnm(x,y),{x,y}R¯z,t0E17

and

E(x,y,±L,t)=2π0π/2WE(x,y,t,φ)tdφ,{x,y}R¯z,t0{[2t2sin2φ(2x2+2y2)]WE(x,y,t,φ)=E(g,t)z|z=±L,{x,y}Rz,t>0WE(x,y,t,φ)|t=0=WE(x,y,t,φ)t|t=0=0,{x,y}R¯zD[WE](lx,y)=e2πiΦxD[WE](0,y),0ylyD[WE](x,ly)=e2πiΦyD[WE](x,0),0xlx,t0.E18

Here, unm±(±L,τ)=unm±(z,τ)/z|z=±L, J0(t)is the zero-order Bessel function, the superscript ‘’ stands for the complex conjugation operation, WE(x,y,t,φ)is some auxiliary function, where the numerical parameter φlies in the range0φπ/2.

It is obvious that the magnetic field vector H(g,t)of the pulsed waves U(g,t)={E(g,t),H(g,t)}outgoing towards the domains Aand Bsatisfies 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 QL(a part of the Floquet channelR) that contains all the sources and obstacles.

Now suppose that in addition to the sourcesj(g,t), φE(g), andφH(g), there exist sourcesjA(g,t), φEA(g), and φHA(g)located in Aand generating some pulsed wave Ui(g,t)={Ei(g,t),Hi(g,t)}being incident on the boundary L+at timest>0. The field Ui(g,t)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 QLwith the boundary conditions (17) or (18) on Land the following conditions on the artificial boundaryL+:

Es(x,y,L,t)=n,m={0tJ0[λnm(tτ)][0lx0lyEs(x˜,y˜,z,τ)z|z=Lμnm(x˜,y˜)dx˜dy˜]dτ}μnm(x,y),{x,y}R¯z,t0E19

or

Es(x,y,L,t)=2π0π/2WE(x,y,t,φ)tdφ,{x,y}R¯z,t0{[2t2sin2φ(2x2+2y2)]WE(x,y,t,φ)=Es(g,t)z|z=L,{x,y}Rz,t>0WE(x,y,t,φ)|t=0=WE(x,y,t,φ)t|t=0=0,{x,y}R¯zD[WE](lx,y)=e2πiΦxD[WE](0,y),0ylyD[WE](x,ly)=e2πiΦyD[WE](x,0),0xlx,t0.E20

Here Us(g,t)={Es(g,t),Hs(g,t)}=U(g,t)Ui(g,t)(gA,t>0) is the pulsed wave outgoing towardsz=+. It is generated by the incident wave Ui(g,t)(‘reflection’ from the virtual boundaryL+) and the sourcesj(g,t), φE(g), andφH(g).

5. 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 Ui(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 Ui(g,t)generated by a given set of sourcesjA(g,t), φEA(g), andφHA(g), the following initial boundary value problem for a regular hollow Floquet channelRare to be solved:

{[2t2+Δ]{EiHi}={η0jA/t+ε01gradρ2ArotjA}={FEAFHA},g={x,y,z}R,t>0{Ei(g,0)Hi(g,0)}={φEAφHA},{Ei(g,t)/t|t=0=η0rotHi(g,0)Hi(g,t)/t|t=0=η01rotEi(g,0)}={ψEAψHA},gR¯D[Ei(Hi)](lx,y)=e2πiΦxD[Ei(Hi)](0,y),0ylyD[Ei(Hi)](x,ly)=e2πiΦyD[Ei(Hi)](x,0),0xlx,t0.E21

The function ρ2A(g,t)here determines the volume density of foreign electric charge.

First we determine the longitudinal components Eziand Hziof the field {Ei,Hi}at all points gof the domain Rfor all timest>0. Let us consider the scalar initial boundary value problems following from (21):

{[2t2+Δ]{EziHzi}={Fz,EAFz,HA},gR,t>0{Ezi(g,0)Hzi(g,0)}={φz,EAφz,HA},{Ezi(g,t)/t|t=0Hzi(g,t)/t|t=0}={ψz,EAψz,HA},gR¯D[Ezi(Hzi)](lx,y)=e2πiΦxD[Ezi(Hzi)](0,y),0ylyD[Ezi(Hzi)](x,ly)=e2πiΦyD[Ezi(Hzi)](x,0),0xlx,t0.E22

By separating of the transverse variables xand yin (22) represent the solution of the problem as

{Ezi(g,t)Hzi(g,t)}=n,m={vnm(z,E)(z,t)vnm(z,H)(z,t)}μnm(x,y)E23

To determine the scalar functions vnm(z,E)(z,t)andvnm(z,H)(z,t), we have to invert the following Cauchy problems for the one-dimensional Klein-Gordon equations:

{[2t2+2z2λnm]{vnm(z,E)(z,t)vnm(z,H)(z,t)}={Fnm(z,E)AFnm(z,H)A},t>0,<z<{vnm(z,E)(z,0)vnm(z,H)(z,0)}={φnm(z,E)Aφnm(z,H)A},t{vnm(z,E)(z,t)vnm(z,H)(z,t)}|t=0={ψnm(z,E)Aψnm(z,E)A},<z<,n,m=0,±1,±2,....E24

HereFnm(z,E)A, φnm(z,E)A, ψnm(z,E)AandFnm(z,H)A, φnm(z,H)A, ψnm(z,H)Aare the amplitudes of the Fourier transforms of the functionsFz,EA, φz,EA, ψz,EAandFz,HA, φz,HA, ψz,HAin the basic set{μnm(x,y)}.

Let us continue analytically the functionsvnm(z,E)(z,t), vnm(z,H)(z,t)andFnm(z,E)A, Fnm(z,H)Aby zero on the semi-axis t0and pass on to the generalized formulation of the Cauchy problem (24) [12]:

B(λnm)[vnm(z,E)(z,t)vnm(z,H)(z,t)][2t2+2z2λnm2]{vnm(z,E)(z,t)vnm(z,H)(z,t)}={Fnm(z,E)AFnm(z,H)A}δ(1)(t){φnm(z,E)Aφnm(z,H)A}δ(t){ψnm(z,E)Aψnm(z,H)A}={fnm(z,E)fnm(z,H)},<z<,<t<,n,m=1,±2,±3,...,E25

where δ(t)and δ(m)(t)are the Dirac delta-function and its derivative of the orderm. Taking into account the properties of the fundamental solution G(z,t,λ)=(1/2)χ(t|z|)J0(λt2z2)of the operator B(λ)[6,13,14] (χ(t)is the Heaviside step function), the solutions vnm(z,E)(z,t)and vnm(z,H)(z,t)of equations (25) can be written as

{vnm(z,E)(z,t)vnm(z,H)(z,t)}=G(z,t,λnm){fnm(z,E)fnm(z,H)}=12t|zω|J0(λnm(tτ)2(zω)2)({Fnm(z,E)AFnm(z,H)A}δ(1)(t){φnm(z,E)Aφnm(z,H)A}δ(t){ψnm(z,E)Aψnm(z,H)A})dωdτ,<z<,t0,n,m=1,±2,±3,....E26

Relations (23) and (26) completely determine the longitudinal components of the field{Ei,Hi}.

Outside the bounded domain enclosing all the sources, in the domainGR, where the waves generated by these sources propagate freely, the following relations [6,14] are valid:

{Ei=(2UExz2UHyt)x+(2UEyz+2UHxt)y+(2UEz22UEt2)zη0Hi=(2UEyt+2UHxz)x+(2UExt+2UHyz)y+(2UHz22UHt2)zE27

in which

UE,H(g,t)=n,m=unmE,H(z,t)μnm(x,y)E28

are the scalar Borgnis functions such that[Δ2/t2][UE,H(g,t)/t]=0. Equations (23), (26)-(28) determine the field {Ei,Hi}at all points gof the domain Gfor all timest>0. Really, since at the time point t=0the domain Gis undisturbed, then we have [Δ2/t2]UE,H=0(gG,t>0). Hence, in view of (27), (28), it follows:

Ez=2UEz22UEt2=(2UEx2+2UEy2)=n,m=λnm2unmEμnm,

η0Hz=2UHz22UHt2=(2UHx2+2UHy2)=n,m=λnm2unmHμnm

and (see representation (23))

unmE(z,t)=(λnm)2vnm(z,E)(z,t),unH(z,t)=η0(λnm)2vnm(z,H)(z,t).E29

Hence the functions UE,H(g,t)as well as the transverse components of the field {Ei,Hi}are determined.

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

{Ezi(g,t)Hzi(g,t)}=n,m={vnm(z,E)(z,t)vnm(z,H)(z,t)}μnm(x,y),gA¯,t0,E30

for the reflected wave Us(g,t)(which coincides with the total field U(g,t)ifUi(g,t)0) we have

{Ezs(g,t)orEz(g,t)Hzs(g,t)orHz(g,t)}=n,m={unm(z,E)+(z,t)unm(z,H)+(z,t)}μnm(x,y),gA¯,t0E31

and for the transmitted wave (coinciding in the domain Bwith the total fieldU(g,t)) we can write

{Ez(g,t)Hz(g,t)}=n,m={unm(z,E)(z,t)unm(z,H)(z,t)}μnm(x,y),gB¯,t0.E32

In applied problems, the most widespread are situations where a periodic structure is excited by one of the partial components of TE-wave (withEzi(g,t)=0) or TM-wave (withHzi(g,t)=0) [7]. Consider, for example, a partial wave of orderpq. Then we have

Ui(g,t)=Upq(H)i(g,t):Hzi(g,t)=vpq(z,H)(z,t)μpq(x,y)

or

Ui(g,t)=Upq(E)i(g,t):Ezi(g,t)=vpq(z,E)(z,t)μpq(x,y).

The excitation of this kind is implemented in our models in the following way. The time function vpq(z,H)(L,t)or vpq(z,E)(L,t)is defined on the boundaryL+. This function determines the width of the pulseUi(g,t), namely, the frequency range [K1,K2]such that for all frequencies kfrom this range (k=2π/λ, λis the wavelength in free space) the value

γ=|v˜pq(z,HorE)(L,k)|maxk[K1;K2]|v˜pq(z,HorE)(L,k)|

where v˜pq(z,HorE)(L,k)is the spectral amplitude of the pulsevpq(z,HorE)(L,t), exceeds some given valueγ=γ0. All spectral characteristics f˜(k)are obtainable from the temporal characteristics f(t)by applying the Laplace transform

f˜(k)=0f(t)eiktdtf(t)=12πiαiα+f˜(k)eiktdk,0αImk.E33

For numerical implementation of the boundary conditions (19) and (20) and for calculating space-time amplitudes of the transverse components of the wave Ui(g,t)in the cross-section z=Lof the Floquet channel (formulas (27) and (29)), the function (vpq(z,HorE))(L,t)are to be determined. To do this, we apply the following relation [7,14]:

vpq(HorE)(L,t)=0tJ0[λpq(tτ)](vpq(HorE))(L,t)dτ,t0.E34

which is valid for all the amplitudes of the pulsed wave Ui(g,t)outgoing towards z=and does not violate the causality principle.

6. Transformation Operator Method

6.1. Evolutionary basis of a signal and transformation operators

Let us place an arbitrary periodic structure of finite thickness between two homogeneous dielectric half-spaces z1=zL>0(withε=ε1) and z2=zL>0(withε=ε2). Let also a local coordinate system gj={xj,yj,zj}be associated with each of these half-spaces (Figure 2). Assume that the distant sources located in the domain Aof the upper half-space generate a primary wave U1i(g,t)={E1i(g,t),H1i(g,t)}being incident on the artificial boundary L+(on the planez1=0) as viewed fromz1=.

Denote by Ujs(g,t)={Ejs(g,t),Hjs(g,t)}the fields resulting from scattering of the primary wave U1i(g,t)in the domains A(where the total field isU(g,t)={E(g,t),H(g,t)}=U1s(g,t)+U1i(g,t)) and B(whereU(g,t)=U2s(g,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:

{Ezi(g,t)Hzi(g,t)}=n,m={vnm(1,E)(z1,t)vnm(1,H)(z1,t)}μnm(x,y),z10,t0E35
{Ezs(g,t)Hzs(g,t)}=n,m={unm(j,E)(zj,t)unm(j,H)(zj,t)}μnm(x,y),zj0,t0,j=1,2E36

(see also formulas (30)-(32)). Here, as before, {μnm(x,y)}n,m=is the complete (inL2(Rz)) orthonormal system of transverse eigenfunctions of the Floquet channel R(see Section 4), while the space-time amplitudes unm(j,E)(zj,t)and unm(j,H)(zj,t)are determined by the solutions of the following problems (see also problem (15)) for the one-dimensional Klein-Gordon equations:

{[εj2t2+2zj2λnm2]unm(j,EorH)(zj,t)=0,t>0unm(j,EorH)(zj,0)=0,tunm(j,EorH)(zj,t)|t=0=0,j=1,2,n,m=0,±1,±2,....E37

Compose from the functionsvnm(1,E)(z1,t), vnm(1,H)(z1,t), unm(j,E)(zj,t), unm(j,H)(zj,t)and the eigenvalues λnm(n,m=0,±1,±2,...) the setsv(1)(z1,t)={vp(1)(z1,t)}p=, u(j)(zj,t)={up(j)(zj,t)}p=, and {λp}p=such that their members are defined according to the rules depicted in Figure 3. The sets v(1)(z1,t)and u(j)(zj,t)are said to be evolutionary bases of signals U1i(g,t)andUjs(g,t). They describe completely and unambiguously transformation of the corresponding nonsine waves in the regular Floquet channels Aand Bfilled with dielectric.

Figure 2.

A two-dimensionally periodic grating between two dielectric half-spaces as element of a multi-layered structure.

Let us introduce by the relations

up(j)(0,t)zjup(j)(zj,t)|zj=0=0tq=[SpqAA(tτ)δj1+SpqBA(tτ)δj2]vq(1)(0,τ)dτ,t0,p=0,±1,±2,...,j=1,2E38
u(j)(0,t)={up(j)(0,t)}p=[SAAδj1+SBAδj2][v(1)(0,τ)],t0,j=1,2E39

the boundary (on the boundarieszj=0) transformation operators SAAand SBAof the evolutionary basis v(1)(z1,t)of the wave U1i(g,t)incoming from the domainA. Here δmnstands for the Kronecker delta, the operators’ elements SnmXYspecify the space-time energy transformation from the domain Yinto the domain Xand from the mode of order minto the mode of ordern.

It is evident that the operators SAAand SBAworking in the space of evolutionary bases are intrinsic characteristics of the periodic structure placed between two dielectric half-spaces. They totalize an impact of the structure on elementary excitations composing any incident signalU1i(g,t). Thus forvq(1)(0,t)=δqrδ(tη), where ris an integer andη>0, we have up(1)(0,t)=SprAA(tη)andup(2)(0,t)=SprBA(tη). We use this example with an abstract nonphysical signal by methodological reasons in order to associate the transformation operators’ components SprAA(tτ)and SprBA(tτ)with an ‘elementary excitation’.

Figure 3.

Construction of sets of the values v p ( 1 ) , u p ( j ) , and λ p ( p = 0, ± 1, ± 2,... ) from sets of the values v n m ( 1, E ) , u n m ( j , E ) , v n m ( 1, H ) , u n m ( j , H ) , and λ n m ( m , n = 0, ± 1, ± 2,... ): (a) p = 0,1,2,... ; (b) p = − 1, − 2, − 3,... .

The operators SAAand SBAdetermine all the features of transient states on the upper and bottom boundaries of the layer enclosing the periodic structure. Secondary waves outgoing from these boundaries propagate freely in the regular Floquet channels Aand Btherewith undergoing deformations (see, for example, [6]). The space-time amplitudes up(j)(zj,t)of the partial components of these waves (the elements of the evolutionary bases of the signalsUjs(g,t)) vary differently for different values of pandj. These variations on any finite sections of the Floquet channels Aи Bare described by the diagonal transporting operators Z0z1Aand Z0z2Bacting according the rule:

u(j)(zj,t)={up(j)(zj,t)}=[Z0z1Aδj1+Z0z2Bδj2][u(j)(0,τ)],j=1,2.E40

The structure of the operators given by (40) can be detailed by the formula

up(j)(zj,t)=1εj0J0[λp(tτ)2εjzj2]χ(tτεjzj)up(j)(0,τ)dτ,t0,zj0,p=0,±1,±2,...,j=1,2,E41

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 decreasingzj. The derivation technique for (41) is discussed at length in [6,13,14].

6.2. Equations of the operator method in the problems for multilayer periodic structures

The operators SAAand SBAcompletely define properties of the periodic structure excited from the channelA. By analogy with (38) we can determine transformation operators SBBand SABfor evolutionary basis v(2)(z2,t)={vp(2)(z2,t)}p=of the wave U2i(g,t)={E2i(g,t),H2i(g,t)}incident onto the boundary z2=0from the channelB:

up(j)(0,t)=0tm=[SpqAB(tτ)δj1+SpqBB(tτ)δj2]vq(2)(0,τ)dτ,t0,p=0,±1,±2,...,j=1,2.E42

Let us construct the algorithm for calculating scattering characteristics of a multilayer structure consisting of two-dimensionally periodic gratings, for which the operatorsSAA, SBA, SpqAB, and SpqBBare 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=ε2(I)=ε1(II)) and placed between the upper and the bottom dielectric half-spaces with the permittivity ε1(I)andε2(II), respectively. Let also a pulsed wave like (35) be incident onto the boundary z1(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 Cin a symbolic form

U(A)=p=[vp(1)(z1(I),t)+up(1)(z1(I),t)]μp(x,y),U(B)=p=[up(2)(z2(I),t)+up(1)(z1(II),t)]μp(x,y),U(C)=p=up(2)(z2(II),t)μp(x,y).

The first terms in the square brackets correspond to the waves propagating towards the domainC, while the second ones correspond to the waves propagating towards the domain A(Figure 4). The set {μp(x,y)}p=is formed from the functionsμnm(x,y), (n,m=0,±1,±2,...), while the set {λp}p=is composed from the valuesλnm, (n,m=0,±1,±2,...) (Figure 3).

Figure 4.

Schematic drawing of a double-layered structure.

By denoting

u(j)(I)zj(I)u(j)(zj(I),t)|zj(I)=0,u(j)(I)={up(j)(zj(I),t)}|zj(I)=0,

according to formulas (38)-(42), we construct the following system of operator equations:

{u(1)(I)=SAA(I)[v(1)(I)]+SAB(I)Zz1(II)=0MB[u(1)(II)]u(2)(I)=SBA(I)[v(1)(I)]+SBB(I)Zz1(II)=0MB[u(1)(II)]u(1)(II)=SBB(II)Zz2(I)=0MB[u(2)(I)]u(2)(II)=SCB(II)Zz2(I)=0MB[u(2)(I)].E43

Equations (43) clearly represent step-by-step response of the complex structure on the excitation by the signal U1i(g,t)with the evolutionary basis v(1)(z1(I),t)={vp(1)(z1(I),t)}p=(or simplyv(1)(I)). Тhus, for example, the first equation can be interpreted as follows. A signal u(1)(I)(the secondary field inA) is a sum of two signals, where the first signal is a result of the reflection of the incident signal v(1)(I)by the gratingI, while another one is determined by the signal u(1)(II)being deformed during propagation in the channel Band interaction with the gratingI.

By method of elimination the system (43) is reduced to the operator equation of the second kind

u(2)(I)=SBA(I)[v(1)(I)]+SBB(I)Zz1(II)=0MBSBB(II)Zz2(I)=0MB[u(2)(I)]E44

and some formulas for calculating the electromagnetic field components in all regions of the two-layered structure. The observation time tfor the unknown function u(2)(I)from the left-hand side of equation (44) strictly greater of any moment of time τfor the function u(2)(I)in the right-hand side of the equation (owing to finiteness of wave velocity). Therefore equation (44) can be inverted explicitly in the framework of standard algorithm of step-by-step progression through time layers. Upon realization of this scheme and calculation of the boundary operators by (38), (42), the two-layered structure can be used as ‘elementary’ unit of more complex structures.

Turning back to (38)-(42), we see that the operators entering these equations act differently that their analogues in the frequency domain, where the boundary operators relate a pair ‘field field’. Reasoning from the structure of the transport operators Z0z1Aand Z0z2B(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.

7. Some Important Properties of Steady-State Fields in the Rectangular Floquet Channel

7.1. Excitation by a TM-wave

Let a grating (Figure 1) be excited form the domain Aby a pulsed TM-wave Ui(g,t)=Upq(E)i(g,t):Ezi(g,t)=vpq(z,E)(z,t)μpq(x,y)and the region QLis free from the sourcesj(g,t), φE(g), andφH(g). The field generated in the domains Aand Bis determined completely by their longitudinal components. They can be represented in the form of (31), (32). Define steady-state fields {E˜(g,k),H˜(g,k)}(see formula (33) withImk=0) corresponding to the pulsed fields{Ei,Hi}, {Es,Hs}in Aand the pulsed field {E,H}inB, by their z-components:

{E˜zi(g,k)H˜zi(g,k)}={v˜pq(z,E)(k)0}eiΓpq(zL)μpq(x,y),gA¯E45
{E˜zs(g,k)H˜zs(g,k)}=n,m={u˜nm(z,E)+(k)u˜nm(z,H)+(k)}eiΓnm(zL)μnm(x,y),gA¯E46
{E˜z(g,k)H˜z(g,k)}=n,m={u˜nm(z,E)(k)u˜nm(z,H)(k)}eiΓnm(z+L)μnm(x,y),gB¯E47

where the following notation is used:v˜pq(z,E)(k)vpq(z,E)(L,t), u˜nm(z,EorH)±(k)unm(z,EorH)±(±L,t), Γnm=(k2λnm2)1/2, ReΓnmRek0, ImΓnm0[7].

The amplitudes u˜nm(z,EorH)±(k)form the system of the so-called scattering coefficients of the grating, namely, the reflection coefficients

Rpq(E)nm(H)=u˜nm(z,H)+(k)v˜pq(z,E)(k),Rpq(E)nm(E)=u˜nm(z,E)+(k)v˜pq(z,E)(k),n,m=0,±1,±2,...E48

specifying efficiency of transformation of pq-th harmonic of a monochromatic TM-wave into of order nm-th harmonics of the scattered field {E˜s,H˜s}in the reflection zone, and the transmission coefficients

Tpq(E)nm(H)=u˜nm(z,H)(k)v˜pq(z,E)(k),Tpq(E)nm(E)=u˜nm(z,E)(k)v˜pq(z,E)(k),n,m=0,±1,±2,...E49

determining the efficiency of excitation of the transmitted harmonics in the domainB.

These coefficients are related by the energy balance equations

n,m=1λnm2[(|Rpq(E)nm(E)|2+|Tpq(E)nm(E)|2)±η02(|Rpq(E)nm(H)|2+|Tpq(E)nm(H)|2)]{ReΓnmImΓnm}=1λpq2{ReΓpq+2ImΓpqImRpq(E)pq(E)ImΓpq2ReΓpqImRpq(E)pq(E)}1ε0{W1W2},p,q=0,±1,±2,,E50
W1=ε0η0kQLσ(g,k)|E˜(g,k)|2dg,W2=QL[μ0μ(g,k)|H˜(g,k)|2ε0ε(g,k)|E˜(g,k)|2]dgE51

They follow from the complex power theorem (Poynting theorem) in the integral form [11]

SL([E˜×H˜]ds)=QLdiv[E˜×H˜]dg=ikη0QLμ|H˜|2dgikη0QLε|E˜|2dgQLσ|E˜|2dgE52

whereε(g,k)1=χ˜ε(g,k)χε(g,t), μ(g,k)1=χ˜μ(g,k)χμ(g,t), σ(g,k)=χ˜σ(g,k)χσ(g,t), dsis the vector element of the surface SLbounding the domainQL. Equations (50)-(52) have been derived starting from the following boundary value problem for a diffraction grating illuminated by a plane TM-waveU˜pq(E)i(g,k):E˜zi(g,k)=exp[iΓpq(zL)]μpq(x,y):

{η0rotH˜(g,k)=ikε¯(g,k)E˜(g,k),rotE˜(g,k)=ikη0μ(g,k)H˜(g,k),gQLD[E˜(H˜)](lx,y)=e2πiΦxD[E˜(H˜)](0,y),0yly,|z|<LD[E˜(H˜)](x,ly)=e2πiΦyD[E˜(H˜)](x,0),0xlx,|z|<LΕ˜tg(g,k)|gS=0,H˜nr(g,k)|gS=0E53
{E˜z(g,k)H˜z(g,k)}={10}eiΓpq(zL)μpq(x,y)+n,m={Rpq(E)nm(E)(k)Rpq(E)nm(H)(k)}eiΓnm(zL)μnm(x,y),gA¯,{E˜z(g,k)H˜z(g,k)}=n,m={Tpq(E)nm(E)(k)Tpq(E)nm(H)(k)}eiΓnm(z+L)μnm(x,y),gB¯.E54

When deriving (50), (51) we have also used the equations relating z-components of the eigenmode of the Floquet channel

U˜(g,k):E˜z(g,k)=Ae±iΓzμ(x,y)andH˜z(g,k)=Be±iΓzμ(x,y)E55

(subscripts nmare omitted) with its longitudinal components:

E˜x=βkη0λ2H˜zαΓλ2E˜z,E˜y=αkη0λ2H˜zβΓλ2E˜z,H˜x=αΓλ2H˜z+βkη0λ2E˜z,H˜y=βΓλ2H˜zαkη0λ2E˜z.E56

Here, ε¯(g,k)=ε(g,k)+iη0σ(g,k)/k, μ(x,y)=(lxly)1/2exp(iαx)exp(iβy), Γ=k2λ2,λ2=α2+β2.

According to the Lorentz lemma [11], the fields {E˜(1),H˜(1)}and {E˜(2),H˜(2)}resulting from the interaction of a grating with two plane TM-waves

U˜pq(E)i(1)(g,k):E˜zi(1)(g,k)=exp[iΓpq(Φx,Φy)(zL)]μpq(x,y,Φx,Φy)and

U˜r,s(E)i(2)(g,k):E˜zi(2)(g,k)=exp[iΓr,s(Φx,Φy)(zL)]μr,s(x,y,Φx,Φy),

satisfy the following equation

SL(([E˜(1)×H˜(2)][E˜(2)×H˜(1)])ds)=0.E57

From (57), using (54) and (56), we obtain

Rpq(E)rs(E)(Φx,Φy)λp,q2(Φx,Φy)Γpq(Φx,Φy)=Rr,s(E)p,q(E)(Φx,Φy)λr,s2(Φx,Φy)Γr,s(Φx,Φy),p,q,r,s=0,±1,±2,...E58

– 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).

Assume now that the first wave U˜pq(E)i(1)(g,k)::E˜zi(1)(g,k)=exp[iΓpq(Φx,Φy)(zL)]μpq(x,y,Φx,Φy)=U˜pq(E)i(1)(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):E˜zi(2)(g,k,B)=exp[iΓr,s(Φx,Φy)(z+L)]μr,s(x,y,Φx,Φy)is incident fromB. Both of these waves satisfy equation (57), whence we have

Tpq(E)rs(E)(Φx,Φy,A)λp,q2(Φx,Φy)Γpq(Φx,Φy)=Tr,s(E)p,q(E)(Φx,Φy,B)λr,s2(Φx,Φy)Γr,s(Φx,Φy),p,q,r,s=0,±1,±2,...E59

7.2. Excitation by a TE-wave

Let a grating be excited form the domain Aby a pulsed TE-wave Ui(g,t)=Upq(H)i(g,t):Hzi(g,t)=vpq(z,H)(z,t)μpq(x,y)and the region QLis free from the sourcesj(g,t), φE(g), andφH(g). The field generated in the domains Aand Bis determined completely by their longitudinal components. They can be represented in the form of (31), (32). Define steady-state fields {E˜(g,k),H˜(g,k)}corresponding to the pulsed fields{Ei,Hi}, {Es,Hs}in Aand the pulsed field {E,H}inB, by their z-components as was done for the TM-case (see equations (45)-(47)). Introduce the scattering coefficientsRpq(H)nm(E), Rpq(H)nm(H), Tpq(H)nm(E), and Tpq(H)nm(H)by the relations like (48). These coefficients can be determined from the problems

{η0rotH˜(g,k)=ikε¯(g,k)E˜(g,k),rotE˜(g,k)=ikη0μ(g,k)H˜(g,k),gQLD[E˜(H˜)](lx,y)=e2πiΦxD[E˜(H˜)](0,y),0yly,|z|<LD[E˜(H˜)](x,ly)=e2πiΦyD[E˜(H˜)](x,0),0xlx,|z|<LΕ˜tg(g,k)|gS=0,H˜nr(g,k)|gS=0,E60
{E˜z(g,k)H˜z(g,k)}={01}eiΓpq(zL)μpq(x,y)+n,m={Rpq(H)nm(E)(k)Rpq(H)nm(H)(k)}eiΓnm(zL)μnm(x,y),gA¯,{E˜z(g,k)H˜z(g,k)}=n,m={Tpq(H)nm(E)(k)Tpq(H)nm(H)(k)}eiΓnm(z+L)μnm(x,y),gB¯E61

and satisfy the following relations, which are corollaries from the Poynting theorem and the Lorentz lemma:

n,m=1λnm2[(|Rpq(H)nm(H)|2+|Tpq(H)nm(H)|2)±1η02(|Rpq(H)nm(E)|2+|Tpq(H)nm(E)|2)]{ReΓnmImΓnm}==1λpq2{ReΓpq+2ImΓpqImRpq(H)pq(H)ImΓpq2ReΓpqImRpq(H)pq(H)}1μ0{W1W2},p,q=0,±1,±2,E62

and

Rpq(H)rs(H)(Φx,Φy)λp,q2(Φx,Φy)Γpq(Φx,Φy)=Rr,s(H)p,q(H)(Φx,Φy)λr,s2(Φx,Φy)Γr,s(Φx,Φy),p,q,r,s=0,±1,±2,...E63
Tpq(H)rs(H)(Φx,Φy,A)λp,q2(Φx,Φy)Γpq(Φx,Φy)=Tr,s(H)p,q(H)(Φx,Φy,B)λr,s2(Φx,Φy)Γr,s(Φx,Φy),p,q,r,s=0,±1,±2,...E64

7.3. General properties of the grating’s secondary field

Let now kbe a real positive frequency parameter, and let an arbitrary semi-transparent grating (Figure 1) be excited from the domain Aby a homogeneous TM- or TE-wave

U˜pq(EorH)i(g,k):{E˜zi(g,k)orH˜zi(g,k)}=eiΓpq(zL)μpq(x,y),p,q:ImΓpq=0.E65

The terms of infinite series in (54) and (61) are z-components of nm-th harmonics of the scattered field for the domains AandB. The complex amplitudes Rpq(EorH)nm(EorH)and Tpq(EorH)nm(EorH)are the functions ofk, Φx, Φy, as well as of the geometry and material parameters of the grating. Every harmonic for which ImΓnm=0and ReΓnm>0is a homogeneous plane wave propagating away from the grating along the vectorknm:kx=αn, ky=βm, kz=Γnm(inA; Figure 5) or kz=Γnm(inB). The frequencies ksuch that Γnm(k)=0(k=knm±=±|λnm|) are known as threshold frequency or sliding points [1-6]. At those points, a spatial harmonic of order nmwith ImΓnm>0are transformed into a propagating homogeneous pane wave.

It is obvious that the propagation directions knmof homogeneous harmonics of the secondary field depends on their ordernm, on the values of kand on the directing vector of the incident wavekpqi:kxi=αp, kyi=βq,kzi=Γpq. 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:

(WR)pqnm=(|Rpq(E)nm(E)|2+η02|Rpq(E)nm(H)|2)ReΓnmλnm2λpq2Γpq=(WR)pq(E)nm(E)+(WR)pq(E)nm(H),(WT)pqnp=(|Tpq(E)nm(E)|2+η02|Tpq(E)nm(H)|2)ReΓnmλnm2λpq2Γpq=(WT)pq(E)np(E)+(WT)pq(E)np(H)E66

(for TM-case) and

Figure 5.

On determination of propagation directions for spatial harmonics of the field formed by a two-dimensionally periodic structure.

(WR)pqnm=(|Rpq(H)nm(H)|2+1η02|Rpq(H)nm(E)|2)ReΓnmλnm2λpq2Γpq=(WR)pq(H)nm(H)+(WR)pq(H)nm(E),(WT)pqnp=(|Tpq(H)nm(H)|2+1η02|Tpq(H)nm(E)|2)ReΓnmλnm2λpq2Γpq=(WT)pq(H)nm(H)+(WT)pq(H)nm(E)E67

(for TE-case). The channel corresponding to the nm-th harmonic will be named ‘open’ ifImΓnm=0. The regime with a single open channel (nm=pq) will be called the single-mode regime.

Since|kpqi|=|knm|=k, the nm-th harmonic of the secondary field in the reflection zone propagates in opposition to the incident wave only if αn=αpand βm=βqor, in other notation, if

n=2Φxpandm=2ΦyqE68

Generation of the nonspecularly reflected mode of this kind is termed the auto-collimation.

The amplitudes Rpq(EorH)nm(EorH)or Tpq(EorH)nm(EorH)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 nmsuch thatReΓnm0. However, the radiation field in the immediate proximity of the grating requires a consideration of the contribution of damped harmonics (nm:ImΓnm>0). Moreover, in some situations (resonance mode) this contribution is the dominating one [6].

7.4. The simplest corollaries of the reciprocity relations and the energy conservation law

Let us formulate several corollaries of the relations (50), (58), (59), and (62)-(64) basing on the results presented in [3] and [7] for one-dimensionally periodic gratings and assuming thatε(g,k)0, μ(g,k)0, andσ(g,k)0.

  • The upper lines in (50) and (62) represent the energy conservation law for propagating waves. IfImΓpq=0, the energy of the scattered field is clearly related with the energy of the incident wave. The energy of the wave U˜pq(EorH)i(g,k)is partially absorbed by the grating (only ifW10), and the remaining part is distributed between spatial TM- and TE-harmonics propagating in the domains Aand B(the wave is reradiating into the directionsz=±). If a plane inhomogeneous wave be incident on a grating (ImΓpq>0), the total energy is defined by the imaginary part of reflection coefficientRpq(EorH)pq(EorH), which in this case is nonnegative.

  • The relations in the bottom lines in (50), (62) limit the values ofn,m=|Rpq(E)nm(E)|2λnm2ImΓnm, n,m=|Tpq(E)nm(E)|2λnm2ImΓnm, etc. and determine thereby the class of infinite sequences

l¯2={a={anm}nm=:nm=|anm|2n2+m2}E69

or energetic space, to which amplitudes of the scattered harmonicsRpq(E)nm(E), Tpq(E)nm(E), etc. belong.

  • It follows from (58), (59), (63), and (64) that for all semi-transparent and reflecting gratings we can write

(WR)00(EorH)00(EorH)(Φx,Φy)=(WR)00(EorH)00(EorH)(Φx,Φy),(WT)00(EorH)00(EorH)(Φx,Φy,A)=(WT)00(EorH)00(EorH)(Φx,Φy,B).E70

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 x0yabout 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 x0yand being normal to the vector k00(Figure 5).

  • When r=s=p=q=0we derive from (58), (59), (63), and (64) that

R00(EorH)00(EorH)(Φx,Φy)=R00(EorH)00(EorH)(Φx,Φy),T00(EorH)00(EorH)(Φx,Φy,A)=T00(EorH)00(EorH)(Φx,Φy,B).E71

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, Φx, and Φybe such that ReΓ00(Φx,Φy)>0and ReΓnm(Φx,Φy)=0forn,m0. If the incident wave is an inhomogeneous plane waveU˜±p,±q(E)i(g,k,±Φx,±Φy), then

(|R±p,±q(E)00(E)(±Φx,±Φy)|2+η02|R±p,±q(E)00(H)(±Φx,±Φy)|2)ReΓ00(±Φx,±Φy)λ002(±Φx,±Φy)==2ImR±p,±q(E)±p,±q(E)(±Φx,±Φy)ImΓ±p,±q(±Φx,±Φy)λ±p,±q2(±Φx,±Φy).E72

SinceRpq(E)pq(E)(Φx,Φy)=Rp,q(E)p,q(E)(Φx,Φy), we derive from (72)

|Rp,q(E)00(E)(Φx,Φy)|2+η02|Rp,q(E)00(H)(Φx,Φy)|2==|Rp,q(E)00(E)(Φx,Φy)|2+η02|Rp,q(E)00(H)(Φx,Φy)|2.E73

It is easy to realize a physical meaning of the equation (73) and of similar relation for TE-case, 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 x0yabout 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.

8. 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 kacts as a spectral parameter; a two-dimensionally periodic grating is considered as an open periodic resonator.

8.1. Canonical Green function

Let a solution G˜0(g,p,k)of the scalar problem

{[Δg+k2][G˜0(g,p,k)]=δ(gp),g={xg,yg,zg}R,p={xp,yp,zp}QLD[G˜0](lx,yg)=e2πiΦxD[G˜0](0,yg),0ygly,|zg|LD[G˜0](xg,ly)=e2πiΦyD[G˜0](xg,0),0xglx,|zg|LG˜0(g,p,k)=n,m={Anm(p,k)Bnm(p,k)}e±iΓnm(zgL)μnm(xg,yg),g{A¯B¯}E74

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.

Let us construct G˜0(g,p,k)as a superposition of free-space Green functions:

G˜0(g,p,k)=14πn,m=exp[ik|gpnm|]|gpnm|e2πinΦxe2πimΦy,pnm={xp+nlx,yp+mly,zp}.E75

By using in (75) the Poisson summation formula [15] and the tabulated integrals [16]

exp(ipx2+a2)x2+a2eibxdx=πiH0(1)(a|p2b2|)andH0(1)(px2+a2)eibxdx=2exp(iap2b2)p2b2,

where H0(1)(x)is the Hankel function of the first kind, we obtain

G˜0(g,p,k)=i2lxlyn,m=ei[αn(xgxp)+βm(ygyp)]exp[i|zgzp|Γnm]ΓnmE76

The surface Kof analytic continuation of the canonical Green function (76) into the domain of complex-valued kis an infinite-sheeted Riemann surface consisting of the complex planes kCwith cuts along the lines(Rek)2(Imk)2λnm2=0, n,m=0,±1,±2,..., Imk0(Figure 6). The first (physical) sheet Ckof the surface Kis uniquely determined by the radiation conditions for G˜0(g,p,k)in the domains AandB, i.e. by the choice of ReΓnmRek0and ImΓnm0on the axisImk=0. On this sheet, in the domain0argk<π, we haveImΓnm>0, while ReΓnm0for 0argkπ/2and ReΓnm0forπ/2argkπ. In the domain 3π/2argk<2πfor finite number of functions Γnm(k)(with nand msuch that(Rek)2(Imk)2λnm2>0), the inequalities ImΓnm<0and ReΓnm>0hold; for the rest of these functions we have ImΓnm>0andReΓnm0. In the domainπ<argk3π/2, the situation is similar only the signs of ReΓnmare opposite. On the subsequent sheets (each of them with its own pair{k;Γnm(k)}), the signs (root branches) of Γnm(k)are opposite to those they have on the first sheet for a finite number of nandm. The cuts (solid lines in Fig. 6) originate from the real algebraic branch pointsknm±=±|λnm|.

Figure 6.

Natural domain of variation of the spectral parameter k : the first sheet of the surface K .

In the vicinity of some fixed point KKthe function G˜0(g,p,k)can be expanded into a Loran series in terms of the local variable [17]

κ={kK;K{knm±}kK;K{knm±}.

Therefore, this function is meromorphic on the surfaceK. Calculating the residuals Resk=k¯G˜0(g,p,k)at the simple polesk¯{knm±}, 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,knm±)={E˜x,E˜y,E˜z};E˜x,yorz=ax,yorzexp[i(αnx+βmy)]andH˜(g,knm±)=(iknm±η0)1rotE˜(g,knm±)E77

where ax,yorzare the arbitrary constants. These solutions determine free oscillations in the space stratified by the following conditions:

D[E˜(H˜)](x+lx,y)=e2πiΦxD[E˜(H˜)](x,y),D[E˜(H˜)](x,y+ly)=e2πiΦyD[E˜(H˜)](x,y).E78

8.2. Spectrum qualitative characteristics

Let a set Ωkof the points {k¯j}jKsuch that for all k{k¯j}jthe homogeneous (spectral) problem

{η0rotH˜(g,k)=ikε¯(g,k)E˜(g,k),rotE˜(g,k)=ikη0μ(g,k)H˜(g,k),gQLD[E˜(H˜)](lx,y)=e2πiΦxD[E˜(H˜)](0,y),0yly,|z|<LD[E˜(H˜)](x,ly)=e2πiΦyD[E˜(H˜)](x,0),0xlx,|z|<LΕ˜tg(g,k)|gS=0,H˜nr(g,k)|gS=0,E79
{E˜z(g,k)H˜z(g,k)}=n,m={Anm(E)(k)Anm(H)(k)}eiΓnm(zL)μnm(x,y),gA¯{E˜z(g,k)H˜z(g,k)}=n,m={Bnm(E)(k)Bnm(H)(k)}eiΓnm(z+L)μnm(x,y),gB¯E80

has a nontrivial (not necessarily unique) solution U˜(g,k¯j)={E˜(g,k¯j),H˜(g,k¯j)}be called the point spectrum of the grating. It is obvious that these solutions characterize the so-called free oscillations, whose field pattern, structure of their spatial harmonics and behavior of these harmonics for large |z|and tare determined by the value of k¯j=Rek¯j+iImk¯jand by a position of the point k¯j(the eigen frequency associated with a free oscillationU˜(g,k¯j)) on the surface K[4,6,7]. By continuing analytically the problems (53), (54) and (60), (61) together with their solutions U˜(g,k)={E˜(g,k),H˜(g,k)}into the domain Kof the complex-valuedk, we detect poles of the function U˜(g,k)at the pointsk=k¯j. In the vicinity of these poles, the desired solutions can be represented by the Loran series in terms of the local on Kvariable κ[17]. The analytical findings of this kind may form the basis for detailed study of physical features of resonant wave scattering by one-dimensionally and two-dimensionally periodic structures [4,6,7,18,19].

Derive now the conditions that constrain existence of nontrivial solutions of the problem (79), (80). These conditions can be considered as uniqueness theorems for the problems (53), (54) and (60), (61) formulated for different domains of the surfaceK. Notice that the study of the uniqueness allows one to estimate roughly a domain where elements of the set Ωkare 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].

Assume that grating scattering elements are nondispersive (ε(g,k)=ε(g), μ(g,k)=μ(g), andσ(g,k)=σ(g)). In this case, the analytical continuation of the spectral problem (79), (80) into the domain of complex-valued kare simplified considerably. From the complex power theorem in the integral form formulated for the nontrivial solutions U˜(g,k¯j)like

SL([E˜×H˜]ds)=QLdiv[E˜×H˜]dg=ikη0QLμ|H˜|2dgikη0QLε|E˜|2dgQLσ|E˜|2dgE81

the following relations result:

n,m=1λnm2{(ReΓnmRek+ImΓnmImk)(ImΓnmRekReΓnmImk)}[(|Anm(E)|2+|Bnm(E)|2)±η02(|Anm(H)|2+|Bnm(H)|2)]=1ε0{Imk(V3+V2)V1Rek(V3V2)}E82

Notation:k=k¯j, E˜=E˜(g,k¯j), Γnm=Γnm(k¯j), Anm(E)=Anm(E)(k¯j), etc., and

V1=ε0η0QLσ|E˜|2dg, V2=QLε0ε|E˜|2dg,V3=QLμ0μ|H˜|2dg.

No free oscillations exist whose amplitudes do not satisfy equations (82). From this general statement, several important consequences follow. Below some of them are formulated for gratings withε(g)>0, μ(g)>0, andσ(g)0.

  • There are no free oscillations whose eigen frequencies k¯jare located on the upper half-plane (Imk0) of the first sheet of the surfaceK. This can be verified by taking into account the upper relation in (82), the function Γnm(k)onCk, and the inequalitiesV10, V2>0,V3>0.

  • If σ(g)0(the grating is non-absorptive), no free oscillations exist whose eigen frequencies k¯jare located on the bottom half-plane (Imk<0) of the sheet Ckbetween the cuts corresponding to the least absolute values ofknm±. In Figure 6, this region of the first sheet of Kand the above-mentioned domain are shaded by horizontal lines.

  • If σ(g)>0on some set of zero-measure pointsgQL, then there are no elements k¯jof grating’s point spectrum Ωkthat are located on the real axis of the planeCk.

Investigation of the entire spectrum of a grating, i.e. a set of the pointskK, for which the diffraction problems given by (53), (54) and (60), (61) are not uniquely solvable, is a complicated challenge. Therefore below we do no more than indicate basic stages for obtaining well grounded results. The first stage is associated with regularization of the boundary value problem describing excitation of a metal-dielectric grating by the currents J˜(g,k)J(g,t)located in the domainQL:

{η0rotH˜(g,k)=ikε¯(g,k)E˜(g,k)+J˜(g,k),rotE˜(g,k)=ikη0μ(g,k)H˜(g,k),gQLD[E˜(H˜)](lx,y)=e2πiΦxD[E˜(H˜)](0,y),0yly,|z|<LD[E˜(H˜)](x,ly)=e2πiΦyD[E˜(H˜)](x,0),0xlx,|z|<LΕ˜tg(g,k)|gS=0,H˜nr(g,k)|gS=0,E83
{E˜z(g,k)H˜z(g,k)}=n,m={Anm(E)(k)Anm(H)(k)}eiΓnm(zL)μnm(x,y),gA¯,{E˜z(g,k)H˜z(g,k)}=n,m={Bnm(E)(k)Bnm(H)(k)}eiΓnm(z+L)μnm(x,y),gB¯.E84

By regularization is meant (see, for example, [7]) a reduction of the boundary value electrodynamic problem to the equivalent operator equation of the second kind

[E+B(G˜0,S,ε¯,μ,k)]X=Y,EX=XE85

with a compact (in some space Wof vector fields) finite-meromorphic (in local on Kvariablesκ) operator-function B(G˜0,S,ε¯,μ,k)[20,21]. If the problem given by (83), (84) is considered separately for metal gratings (intS¯and Sare sufficiently smooth surfaces;ε¯(g,k)=μ(g,k)1) and dielectric gratings (intS¯=, ε¯(g,k)=ε(g)and μ(g,k)=μ(g)are sufficiently smooth functions), then its regularization can be performed by applying the potential theory methods [4,7,22].

In the second stage, the following statements should be proved: (i) the resolvent [E+B(k)]1(kK) of the problem in (85) is a finite-meromorphic operator-function; (ii) its poles are located at the points k=k¯j(j=1,2,3,...); (iii) the entire spectrum coincides with its point spectrumΩk; (iv) Ωkis nothing more than a countable set without finite accumulation points. All these statements are corollaries of the ‘meromorphic’ Fredholm theorem [4,20,21] and the uniqueness theorem proved previously.

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¯jand associated eigen waves U˜(g,k¯j)={E˜(g,k¯j),H˜(g,k¯j)}or free oscillations of an open two-dimensionnaly periodic resonator. Commonly, this operation is reduced to an approximate solution of the characteristic equation like:

det[C(k)]=0.E86

Here C(k)is some infinite matrix-function; the compactness of the operator B(k)ensures (i) existence of the determinant det[C(k)]and (ii) the possibility to approximate the solutions k¯of equation (86) by the solutions k¯(N)of the equation det[C(k,N)]with the matrix C(k,N)reduced to dimensionN×N.

Let k¯be a root of characteristic equation (86) that do not coincide with any pole of the operator-functionB(k). The multiplicity of this root determines the multiplicity of the eigen value k¯of homogeneous operator equation (85), i.e. the value M=M(1)+M(2)+...+M(Q)[21]. Here, Qis the number of linearly-independent eigen functionsU˜(q)(g,k¯); q=1,2,...,Q(the number of free oscillations) corresponding to the eigen value (eigenfrequency)k¯, while M(q)1is the number of the associated functionsU˜(m)(q)(g,k¯);m=1,2,...,M(q)1. The order of pole of the resolvent [E+B(k)]1(and of the Green function G˜(g,p,k)of the problem in (83), (84)) for k=k¯is determined by a maximal value ofM(q).

9. 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.

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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L. G. Velychko, Yu. K. Sirenko and E. D. Vinogradova (November 14th 2012). Analytical Grounds for Modern Theory of Two- Dimensionally Periodic Gratings, Solutions and Applications of Scattering, Propagation, Radiation and Emission of Electromagnetic Waves, Ahmed Kishk, IntechOpen, DOI: 10.5772/51007. Available from:

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