Two-Scale Approach to an Asymptotic Solution of Maxwell Equations in Layered Periodic Media
aa r X i v : . [ m a t h - ph ] M a y Two-Scale Approach to an Asymptotic Solution of Maxwell Equations in LayeredPeriodic Medium
M. V. Perel a) and M. S. Sidorenko b) St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034,Russia. (Dated: 27 March 2018)
An asymptotic investigation of monochromatic electromagnetic fields in a layeredperiodic medium is carried out in the assumption that the wave frequency is closeto the frequency of a stationary point of a dispersion surface. We find solutions ofMaxwell equations by the method of two-scale asymptotic expansions. We establishthat the principal order of the expansion of a solution dependent on three spatialcoordinates is the sum of two differently polarized Floquet-Bloch solutions, each ofwhich is multiplied by a slowly varying envelope function. We derive that the envelopefunctions satisfy a system of differential equations with constant coefficients. In newvariables, it is reduced to a system of two independent equations, both of them areeither hyperbolic or elliptic, depending on the type of the stationary point. Theenvelope functions are independent only in the planar case. Some consequences arediscussed. a) Electronic mail: [email protected] b) Electronic mail: [email protected] . INTRODUCTION The propagation of electromagnetic waves in media with periodic changes of dielectricpermittivity and magnetic permeability, in the so-called photonic crystals, is a subject ofmany investigations (see, for example, ). The popularity of such studies is caused, on theone hand, by unusual properties of such media, which are yet not completely known andwhich promise new applications. On the other hand, there is a technological progress in thecreation of artificial structures with prescribed properties.The problem of wave propagation in a quickly oscillating medium can be usually re-duced to a problem of wave propagation in an effective homogeneous medium. The mathe-matic methods applied in finding the effective medium are methods of homogenization , , ,i.e., asymptotic methods in the long wavelength approximation: kb → ∞ , where k is thewavenumber, b is the period of oscillations of the medium. One of the methods of thederivation of asymptotic formulas is the method of two-scale expansions. The solution isassumed to depend on fast and slow distance variables. It turns out that the principal termis a function of only a slow variable and satisfies homogenized equations, which may beinterpreted as equations in an effective medium. Another approach is based on the spectralpoint of view: the field can be represented as a superposition of Floquet-Bloch solutionscorresponding to the lower part of the spectrum.A lot of physical works use the concept of effective medium. Peculiarities of a dispersivesurface in an effective medium are responsible for unusual phenomena of wave propagation insuch structures, which are often not observed in nature. For example, if one of the principalcomponents of the effective electric or magnetic tensor has an opposite sign with respect toother two principal components, the dispersive surface in such a medium has a hyperbolicpoint. Such media are named hyperbolic ones and are studied intensively; see, for example, .Another example is a composite material consisting of layers of metals and dielectrics. Theisofrequency dispersive surface for such a structure may have a plane part, i.e., one of thecomponents of the wavevector may be almost independent of the other two. The waves maypropagate without distortion in such a medium, see .The present work was motivated by papers of Longhi , , in which the possibilities of theexistence of localized waves in 2D and 3D periodic structures were studied. The idea wasto take the frequency of a monochromatic electromagnetic field equal to the frequency of2he stationary point of the dispersive (band) surface in the periodic structure. The field in2D and 3D photonic crystals was represented as a superposition of Wannier functions withan envelope, which satisfies the equation with constant coefficients. For hyperbolic (saddle)point the envelope satisfies the wave equation, where one of the spatial coordinates standsfor time. By choosing for the envelope one of the known solutions of the wave equation withfinite energy, see, for example, , one may obtain localized waves in the periodic structure.The papers , were mainly concentrated on physical aspects of the problem. We are moreinterested in careful mathematical analysis and obtain new results in the 3D case.We noted in our paper that the local hyperbolic behavior of a dispersive surface occurseven for the simplest dielectric layered periodic structures, for example, for a structure withalternating layers of dielectric. In this case we studied a field dependent only on two spatialcoordinates in and found and numerically confirmed the following physical phenomenon:undistorted beams can propagate in such a structure and there are only two permitteddirections of a beam in the medium, which differs by a sign from the angle with the normalto layers. A similar phenomenon was found before by numerical simulation in 2D crystalsin ; but it was not investigated analytically.In the present paper, we study monochromatic electromagnetic fields with a frequencyclose to that of a stationary point of the dispersive surface, which may be a hyperbolic or anelliptic one. The field is studied in a layered structure with periodically varying dielectricpermittivity and the magnetic permeability. We deal with fields dependent on all the threespatial coordinates. Our aim is to elaborate a mathematical asymptotic approach for thedescription of solutions of Maxwell equations.The two-scale method in the homogenization theory is applied usually to equations writ-ten in divergent form. Our asymptotic scheme is applied to Maxwell equations in matrixform, which was first suggested in the book by Felsen and Marcuvitz . This form was alsoused in the turning points problem for the Maxwell equations in , for other particular equa-tions in , , , and for the operators in the general form . Therefore we believe that resultsmay have generalizations to other problems. The matrix form of the Maxwell equations isgiven in Section III.In Section IV we discuss difficulties in studying electromagnetic fields dependent on threespatial variables caused by the fact that the stationary points are common for waves of boththe TM and the TE polarizations, and that the concepts of TM and TE polarizations depend3n the direction of propagation and at the stationary point itself they loose their meaning.In Section V, we obtain some integral formulas for second derivatives of the dispersionfunctions, which are applied in the next section. In Section VI, we develop the two-scaleexpansions method for our problem. Apart from the period b in z direction we introducethe second scale of length, which is a scale of field variation in the ( x, y ) plane denoted by L .We regard the ratio between b and L as a small parameter χ . We consider the entire formalasymptotic series and show that the recurrent system for subsequent approximations can besolved step by step. In Section VI, we found that the principal term of the two-scaled field isobtained as a sum of two differently polarized Floquet-Bloch solutions at the stationary pointand each of these solutions has a slowly varying envelope function. The envelope functionssatisfy a system of two partial differential equations with constant coefficients. In SectionVII, we show that these equations are independent only if the envelopes depend on twospatial coordinates. A qualitative consequence of equations is that undistorted beams canpropagate in the medium. In the general 3D case, each envelope function can be expressedin terms of two functions. These functions are solutions of two equations, which are of theHelmholtz type or the Klein-Gordon-Fock type for elliptic or hyperbolic stationary points,respectively. The fact that the wave field is described by a system of two equations wasnot predicted in the papers , . Our paper is concluded with two Appendices. Appendix1 contains known results important for the present paper from the Floquet-Bloch theory,which are given in our notation. Appendix 2 contains proofs of two lemmas.Our case may be regarded as homogenization near a stationary point of the dispersivesurface in contrast to a more usual situation, where the frequency corresponds to the lowerpart of the spectrum and obtained in the assumption that kb →
0. Our case demands anassumption kb ∼ / √ ε av µ av , where ε av , µ av are typical permittivity and permeability. Theequations for envelopes are analogous to equations in an effective medium and can be usedfor a qualitative description of the wavefield. II. STATEMENT OF THE PROBLEM
A monochromatic electromagnetic field satisfies the Maxwell equationsrot E = ikµ H , rot H = − ikε E , (1)4here ε ( z + b ) = ε ( z ), µ ( z + b ) = µ ( z ); ε and µ are piecewise continuous.The medium with alternating dielectric layers is a practically important special case.We seek solutions under two assumptions. The first assumption is about the parametersof the problem. The Maxwell equations contain two parameters of length dimension: thewavelength λ = 2 π/k and the period of the medium b . We introduce the third parameterof length dimension L , which is the scale of variation of the field in the ( x, y ) plane. Weassume that the parameter χ ≡ b/L is small: χ = b/L ≪ . (2)The second assumption is related to the frequency of the monochromatic field ω . To statethis assumption, we need some concepts: the quasimomentum p , the dispersive surface ω = ω ( p ), and Floquet-Bloch solutions. We determine and discuss in detail all these conceptsin Appendix 1. This assumption means that the frequency ω is close to that of the stationarypoint of the dispersive surface ω ∗ : ω = ω ∗ + χ δω, δω ∼ , (3)where ω ∗ satisfies the condition ∇ ω | p ∗ = 0 , ω ∗ = ω ( p ∗ ) . (4)We shall see that these stationary points are minima and saddle points. There is also anadditional condition, which we shall introduce later. III. MATRIX FORM OF MAXWELL EQUATIONS
We represent the Maxwell equations in matrix form for the sake of brevity and generalityof subsequent asymptotic considerations: kP Ψ = − i b Γ · ∇ Ψ , Ψ = EH , (5) b Γ · ∇ ≡ Γ ∂∂x + Γ ∂∂y + Γ ∂∂z , (6)where P = εI µI , Γ = γ − γ , Γ = − γ − γ , Γ = γ − γ , = − , γ = − , γ = − , (7)and k/ √ ε av µ av = ω/c , where c is the speed of light in vacuum.We introduce two types of inner products. Let v ( z ) and w ( z ) be 6-component complex-valued vector functions; then < v , w > = X j =1 v j w j , j = 1 , . . . , (8)where the bar over the symbol stands for complex conjugation. If v and w are periodic withperiod b and piecewise continuous, we define ( v , w ) as follows:( v , w ) = b Z < v ( z ) , w ( z ) > dz. (9)The Umov-Poynting vector, averaged over time, for a monochromatic field of frequency ω ,i.e., the energy flux density of this field, averaged over T = 2 π/ω , is determined as s = 12 Re E × H . (10)It is easy to check that < Ψ , Γ j Ψ > = 4 s j , j = 1 , ,
3; ( Ψ , Γ j Ψ ) = 2 b Z Re (cid:2) E × H (cid:3) j dz. (11)The density of electromagnetic energy, averaged over time, in the case of real ε and µ reads u = ε | E | + µ | H | . (12)We note that < Ψ , PΨ > = 4 u, ( Ψ , PΨ ) = b Z ( ε | E | + µ | H | ) dz. (13) IV. THE FLOQUET-BLOCH SOLUTIONS AND THE DISPERSIONRELATION
Since the properties of the medium do not depend on x, y , we shall seek particular solu-tions in the form Ψ B ( x, y, z ; p ) = e i ( p x x + p y y ) Φ ( z ; p ) , Φ = EH , (14)6here the parameters p x and p y are lateral components of the wave vector. The componentsof the vector-valued function Φ ( z ; p ) satisfy a system of ordinary differential equations withperiodic coefficients: k PΦ + i Γ ∂ Φ ∂z = p x Γ Φ + p y Γ Φ . (15)We are going to obtain Floquet-Bloch solutions of this system. However there are difficultiesowing to the vector nature of the problem. It is well known (see, for example, ) thatan appropriate choice of the coordinate system allows one to split the system (15) intotwo independent subsystems. These subsystems describe waves of two polarizations: thetransverse electric wave and the transverse magnetic waves, which are named the TE andTM waves, respectively. We call the coordinates, in which such a splitting occurs, the naturalones. A. Two types of Floquet-Bloch solutions in the natural coordinates
To determine the type of a solution, we should clarify, which component of the waveis transverse to the propagation plane, the electric or the magnetic one. The propagationplane passes through the lateral wave vector ( p x , p y ) and the z axis. To find the TM andTE modes, it is convenient to rotate the axes in the ( x, y ) plane through an angle γ , sothat in the new coordinates, e p y = 0, e p x ≡ p k = p p x + p y . We denote fields in the rotatedcoordinate system by a tilde. The waves of TE and TM types are as follows: e Φ E = (cid:16) , E ⊥ , , H k , , p k kµ E ⊥ (cid:17) t , e Φ H = (cid:16) E k , , − p k kε H ⊥ , , H ⊥ , (cid:17) t . (16)The system of equations (15) in such coordinates splits into two subsystems: i ∂H ⊥ ∂z = − kεE k ,i ∂E k ∂z = − (cid:18) k εµ − p k kε (cid:19) H ⊥ . , i ∂H k ∂z = (cid:18) k εµ − p k kµ (cid:19) E ⊥ ,i ∂E ⊥ ∂z = kµH k . (17)In the special case p k = 0, the splitting into TM and TE waves has no meaning: bothsystems can be reduced to the following one: i dE dz = − kµH ; i dH dz = − kεE , (18)7here E k | p k =0 = E , H ⊥ | p k =0 = H , E ⊥ | p k =0 = − E , H k | p k =0 = H . (19)We introduce the new notation for vector-functions obtained by means of the passage to thelimit p k → e Φ E , e Φ H : e Φ E | p k → → Φ X , e Φ H | p k → → Φ Y , (20)where Φ X = ( E , , , , H , t , Φ Y = (0 , − E , , H , , t . (21)The superscript X (or Y ) indicates that the vector-function has the nonzero first (or second)component.The obtained systems of ordinary linear differential equations (17), (18) for E k , H ⊥ and E ⊥ , H k and for E and H , respectively, are systems with piecewise continuous periodiccoefficients, because ε and µ are piecewise continuous. Each system has two Floquet-Blochsolutions; for details, see Appendix 1. We denote the components of the second solution bythe subscript 2. For example, the two solutions of the first subsystem of (17) for TM wavesread E k H ⊥ = e ip z z U H + ( z ; p z , p k , ω ) , E k H ⊥ = e − ip z z U H − ( z ; p z , p k , ω ) . (22)The solutions depend on the parameters of the equations p k and ω , and also on the real-valued parameter p z , which is called the quasimomentum and which is related to p k and ω by the formula p z = p z ( p k , ω ) . (23)This relation ensures that the functions U H ± are periodic, U H ± ( z + b ; p z , p k , ω ) = U H ± ( z ; p z , p k , ω ) , (24)and continuous as functions of the z variable. We call these functions Floquet-Bloch ampli-tudes . The relation (23) can be reduced to the following one: ω = ω H ( p ) . (25)We call this equation the dispersion relation . The function ω H ( p ) is called the dispersionfunction . It is a multisheeted function on [ − π/b, π/b ) × R , and its derivation is discussed in8he Appendix 1. Substituting (25) into (22), we obtain E k and H ⊥ as functions of z and p .The Floquet-Bloch solutions and the dispersion function for the waves of TE type ( E ⊥ , H k )and for the solutions E and H of the system (18) are obtained analogously. The solutions(22) are linearly independent if p z = 0 , ± π/b ; see Appendix 1. However, this particular case p z = 0 , ± π/b is important in the present paper.We are interested here in effects that arise if the frequency of the problem is close tothe frequency of one of the stationary points of some sheet of the dispersive surface. InAppendix 1, we show that the multisheeted dispersive surfaces ω = ω H ( p ) and ω = ω E ( p )have stationary points on each sheet, at the points p ∗ , p k∗ = 0, p z ∗ = 0 , ± π/b , where ∇ ω E p ∗ = 0. In these points the sheets of the dispersive surfaces for TM and TE polarizationstouch each other, and therefore ∇ ω H p ∗ = ∇ ω E p ∗ = 0 and ω E ( p ∗ ) = ω H ( p ∗ ) ≡ ω ∗ . At thepoint p ∗ , formulas (22) may give the same solution, which is bounded at infinity and whichis periodic for p z ∗ = 0 and anti-periodic for p z ∗ = ± π/b. Then the second solution, which islinearly independent with the first one, grows at infinity ( for details, see Appendix 1). It isthis case that is of interest to us in the present paper. The solutions read E k H ⊥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = p ∗ = E H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p z = p z ∗ = e ip z ∗ z U H + ( z ; p z ∗ , , ω ∗ ) , E k H ⊥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = p ∗ = E H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p z = p z ∗ = e ip z ∗ z h e − ip z ∗ b zb U H + ( z ; p z ∗ , , ω ∗ ) + Q H ( z ; p z ∗ , ω ∗ ) i , (26)where the vector-valued function Q H is a periodic and continuous function of the variable z . The function Q H does not depend on the argument p k∗ , because p k∗ = 0 , and we are notgoing to use the second solution (26) for p k = 0.The solutions of the second subsystem (17) are expressed in the following way: E ⊥ H k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = p ∗ = − E H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p z = p z ∗ , E ⊥ H k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = p ∗ = − E H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p z = p z ∗ . (27) B. Floquet-Bloch solutions in an arbitrary coordinate system
We have found solutions E k , H ⊥ and E ⊥ , H k of the systems (17). These solutions werefound in the coordinate system rotated by an angle γ , which characterizes the direction of9ave propagation. In the initial coordinate system, these solutions read: Φ H = E k cos γE k sin γ − kε p k H ⊥ − H ⊥ sin γH ⊥ cos γ , Φ E = − E ⊥ sin γE ⊥ cos γ H k cos γH k sin γ kµ p k E ⊥ . (28)The solutions (28) do not have a limit for p k → p x , p y . Thislimit exists in any direction determined by γ and depends on the direction γ : Φ H | p k → → cos γ Φ X − sin γ Φ Y , Φ E | p k → → sin γ Φ X + cos γ Φ Y . (29)Here we take into account (19) and the definitions of Φ X and Φ Y from (21).Below we will use the derivatives ∂ Φ H /∂p j , ∂ Φ E /∂p j , j = x, y, z , which are linear combi-nations of the derivatives of the functions E k , H ⊥ and E ⊥ , H k . These functions are analyticfunctions of the parameter p k , since the coefficients of the system (17) are analytic functionsof p k . It is easy to obtain ∂E k ∂p x = 2 p k ∂E k ∂p k cos γ. (30)This yields ∂E k /∂p x | p k =0 = 0 . Similarly, we derive that the derivatives of H ⊥ , E ⊥ and H k with respect to the variables p x and p y are zero for p k = 0. Finally, we get ∂ Φ H ∂p x (cid:12)(cid:12) p x = p y =0 = ∂ Φ H ∂p y (cid:12)(cid:12) p x = p y =0 = (cid:18) , , − H kε , , , (cid:19) t , (31) ∂ Φ E ∂p x (cid:12)(cid:12) p x = p y =0 = ∂ Φ E ∂p y (cid:12)(cid:12) p x = p y =0 = (cid:18) , , , , , − E kµ (cid:19) t . (32)These derivatives do not depend on the direction γ .We introduce the unified notation Φ f for the Floquet-Bloch solutions of different types: Φ f ( z ; p ) = e ip z z ϕ f ( z ; p ) , f = H, E, X, or Y, (33)where ϕ f ( z + b ; p ) = ϕ f ( z ; p ). The six-component vector-functions ϕ f ( z, p ) are calledthe Floquet-Bloch amplitudes. Let us discuss formula (33) in more detail. The Floquet-Bloch solutions Φ f ( z ; p ) for f = H, E are determined by formula (28), where E k , H ⊥ and E ⊥ , H k are the first solutions (22) of the systems (17). We assume that the frequency ω
10n the formulas (22) is expressed in terms of p on one of the sheets of the multisheetedfunction ω = ω f ( p ). We also take into account the fact that p k = p x + p y , cos γ = p x /p k , sin γ = p y /p k . The number of the sheet is omitted for brevity. The Floquet-Bloch solutions Φ f ( z ; p ), f = X, Y , are defined only for p k = 0, i.e., for p = p ≡ (0 , , p z ); their polarizationis not defined.The limit of the Floquet-Bloch amplitudes in the direction determined by γ follows from(29): ϕ H | p k → → cos γ ϕ X − sin γ ϕ Y , ϕ E | p k → → sin γ ϕ X + cos γ ϕ Y . (34)In particular, if γ = 0, then p k = p x . If γ = π/
2, then p k = p y . From (34) we obtain ϕ H ( z ; p ) | p x → ,p y =0 → ϕ X ( z ; p ) , ϕ E ( z ; p ) | p x → ,p y =0 → ϕ Y ( z ; p ) , (35) ϕ H ( z ; p ) | p x =0 ,p y → → − ϕ Y ( z ; p ) , ϕ E ( z ; p ) | p x =0 ,p y → → ϕ X ( z ; p ) . (36)Next, we need to obtain the derivatives of the solutions ϕ H and ϕ E with respect to theparameters p x and p y at the point p k = 0. Owing to (31) and(32) and the definition (33),we get ∂ ϕ H ∂p x | p ∗ = ∂ ϕ H ∂p y | p ∗ = (cid:18) , , − H kε , , , (cid:19) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p z ∗ e − ip z ∗ z , (37) ∂ ϕ E ∂p x | p ∗ = ∂ ϕ E ∂p y | p ∗ = (cid:18) , , , , , − E kµ (cid:19) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p z ∗ e − ip z ∗ z . (38)These derivatives do not depend on the direction γ . These functions are periodic in z ,because ϕ H , ϕ E are periodic for all p . V. SOME AUXILIARY RELATIONS
Our aim is to obtain relations for the derivatives of the dispersion functions by means ofthe Floquet-Bloch amplitudes. To do this, we give equations for these amplitudes and theirderivatives with respect to the parameters p j , j = x, y, z . A. Equations for the Floquet-Bloch amplitudes and their derivatives
We deal with six-component solutions of the Maxwell equations: Ψ fB ( x, y, z ; p ) = e i ( p x x + p y y ) Φ f ( z ; p ) , f = E, H, (39)11here Φ f is of the form (33). Here the superscript f stands for the type of the field: f = E corresponds to the TE polarization and f = H corresponds to the TM one. Inserting (39)into Maxwell equations, we rewrite them in the form A f ( p ) ϕ f ( z ; p ) = 0 , (40)where A f ( p ) ≡ ω f ( p ) c P + i Γ ∂∂z − p · b Γ , p · b Γ ≡ p x Γ + p y Γ + p z Γ . (41)We note that the equations for both types differ only by the dispersion function ω f ( p ).These equations are valid for any p , and thus we can take the derivatives of these equationswith respect to p . By taking the derivatives with respect to parameters p j , j = x, y , weobtain A f ( p ) ∂ ϕ f ∂p j = − ∂ A f ∂p j ϕ f , (42) A f ( p ) ∂ ϕ f ∂p j = − c ∂ ω f ∂p j P ϕ f − ∂ A f ( p ) ∂p j ∂ ϕ f ∂p j , (43)where ∂ A f ∂p j = 1 c ∂ω f ∂p j P − Γ j . (44)The derivatives with respect to p z are discussed below at the end of the Section. Now weproceed with specifying formulas (42), (44) at the stationary points p ∗ . Such points for eachbranch of the multisheeted dispersion functions ω f ( p ), f = E, H , differ only by p z ∗ , and allof them have p k∗ = 0. Solutions ϕ f ( z ; p ) are not continuous as functions of two variables p x and p y near the point p x = p y = 0. However, for any fixed angle γ , they can be calculatedby means of the passage to the limit, see (34). The derivatives at p k = 0 do not depend on γ , see (37), (38). We indicate these derivatives at p = p ∗ by the asterisk subscript. Uponsubstitution p = p ∗ , the operator A is denoted as A f ( p ∗ ) ≡ A ∗ ( p ∗ ) , f = E, H, (45)and no more depends on the wave type TM or TE.Let M be a class of six-component vector-valued functions of z , which are periodic witha period b , piecewise smooth on the period and their components with numbers 1 , , , A ∗ ( p ∗ ) is symmetric on the functions from M .12inally, passing to the limit in (42), (44), in view of (35), (36) we obtain A ∗ ∂ ϕ H ∗ ∂p x = Γ ϕ X ∗ , A ∗ ∂ ϕ E ∗ ∂p x = Γ ϕ Y ∗ , (46) A ∗ ∂ ϕ H ∗ ∂p y = − Γ ϕ Y ∗ , A ∗ ∂ ϕ E ∗ ∂p y = Γ ϕ X ∗ . (47)Here we have introduced the notation ϕ X ( z ; p ∗ ) = ϕ X ∗ , ϕ Y ( z ; p ∗ ) = ϕ Y ∗ . (48)We note that the terms containing the derivative ∂ω f /∂p j vanish at the stationary point,since this derivative vanishes.For the second derivatives of the Floquet-Bloch amplitudes, in view of (43), (44), and(35), (36) we derive A ∗ ∂ ϕ H ∗ ∂p x = − c ∂ ω H ∗ ∂p x P ϕ X ∗ + 2 Γ ∂ ϕ H ∗ ∂p x , (49) A ∗ ∂ ϕ E ∗ ∂p x = − c ∂ ω E ∗ ∂p x P ϕ Y ∗ + 2 Γ ∂ ϕ E ∗ ∂p x , (50) A ∗ ∂ ϕ H ∗ ∂p y = 1 c ∂ ω H ∗ ∂p y P ϕ Y ∗ + 2 Γ ∂ ϕ H ∗ ∂p y , (51) A ∗ ∂ ϕ E ∗ ∂p y = − c ∂ ω E ∗ ∂p y P ϕ X ∗ + 2 Γ ∂ ϕ E ∗ ∂p y . (52)We emphasize that the directional limit of ϕ H and ϕ E at the point p ∗ can be expressed by ϕ X ∗ , as well as by ϕ Y ∗ depending on the direction γ .Now we proceed to the calculation of the derivatives with respect to p z at the point p ∗ ,i.e., p k∗ = 0, p z = p z ∗ . If p k = p k∗ = 0, i.e., p = p ≡ (0 , , p z ), we have ω H ( p ) = ω E ( p ) ≡ ω ( p ) (53)and A H ( p ) = A E ( p ) ≡ A ( p ). The Floquet-Bloch amplitudes ϕ X , ϕ Y (see (33), (21) )satisfy the same equation A ( p ) ϕ f ( z ; p ) = 0 , f = X, Y. (54)By differentiating (54) with respect to p z and then by passing to the limit p z → p z ∗ , we get A ∗ ∂ ϕ X ∗ ∂p z = Γ ϕ X ∗ , A ∗ ∂ ϕ Y ∗ ∂p z = Γ ϕ Y ∗ . (55)13y taking the second derivatives of (54) and then by passing to the limit p z → p z ∗ , we obtain A ∗ ∂ ϕ X ∗ ∂p z = − c ∂ ω ∗ ∂p z P ϕ X ∗ + 2 Γ ∂ ϕ X ∗ ∂p z , (56) A ∗ ∂ ϕ Y ∗ ∂p z = − c ∂ ω ∗ ∂p z P ϕ Y ∗ + 2 Γ ∂ ϕ Y ∗ ∂p z . (57) B. Derivatives of dispersion functions
Now we get integral relations containing derivatives of the Floquet-Bloch amplitudes andderivatives of dispersion functions. We take the inner product of (46), (47), and (55) with ϕ X ∗ or ϕ Y ∗ and, taking into account the fact that A ∗ ϕ X ∗ = 0 , A ∗ ϕ Y ∗ = 0 , (58)by Lemma 1 (see Appendix 2) we find (cid:0) ϕ f ∗ , Γ j ϕ f ∗ (cid:1) = 0 for any j = 1 , , f = X, Y ; f = X, Y. (59)Henceforth, we use the fact that ϕ f and its derivatives with respect to parameters p j , j = x, y, z belong to M .We are going to apply the same operations to formulas (49–52) and (56–57). We introducethe notation u f f ∗ ≡ (cid:0) ϕ f ∗ , P ϕ f ∗ (cid:1) , f = X or Y, f = X or Y, (60)where u ff has the meaning of the density of energy averaged over time (see Section III), and¨ ω f ∗ ≡ ∂ ω f ∂p x (cid:12)(cid:12)(cid:12)(cid:12) p = p ∗ , ¨ ω f ∗ ≡ ∂ ω f ∂p y (cid:12)(cid:12)(cid:12)(cid:12) p = p ∗ , f = H or E ; ¨ ω ∗ ≡ ∂ ω ∂p z (cid:12)(cid:12)(cid:12)(cid:12) p = p ∗ , (61)where ¨ ω ∗ = ¨ ω H ∗ = ¨ ω E ∗ , because (53) is valid for any p z . We arrive at the relations¨ ω H ∗ c u XX ∗ = 2 (cid:18) ϕ X ∗ , Γ ∂ ϕ H ∗ ∂p x (cid:19) , ¨ ω E ∗ c u Y Y ∗ = 2 (cid:18) ϕ Y ∗ , Γ ∂ ϕ E ∗ ∂p x (cid:19) , (62)¨ ω H ∗ c u Y Y ∗ = − (cid:18) ϕ Y ∗ , Γ ∂ ϕ H ∗ ∂p y (cid:19) , ¨ ω E ∗ c u XX ∗ = 2 (cid:18) ϕ X ∗ , Γ ∂ ϕ E ∗ ∂p y (cid:19) , (63)¨ ω ∗ c u XX ∗ = 2 (cid:18) ϕ X ∗ , Γ ∂ ϕ X ∗ ∂p z (cid:19) , ¨ ω ∗ c u Y Y ∗ = 2 (cid:18) ϕ Y ∗ , Γ ∂ ϕ Y ∗ ∂p z (cid:19) . (64)If f = f , then u f f ≡
0. Moreover, u XX = u Y Y by the definitions of ϕ X ∗ , ϕ Y ∗ , and P .14 . Additional relations Multiplying equations (46), (47) and (55) by ϕ X ∗ and ϕ Y ∗ in such a way that on theleft-hand side we obtain ( ϕ X ∗ , P ϕ Y ∗ ), which vanishes, we find the following relations: (cid:18) ϕ Y ∗ , Γ ∂ ϕ H ∗ ∂p x (cid:19) = (cid:18) ϕ X ∗ , Γ ∂ ϕ E ∗ ∂p x (cid:19) = (cid:18) ϕ Y ∗ , Γ ∂ ϕ H ∗ ∂p y (cid:19) = 0 , (65) (cid:18) ϕ X ∗ , Γ ∂ ϕ E ∗ ∂p y (cid:19) = (cid:18) ϕ Y ∗ , Γ ∂ ϕ X ∗ ∂p z (cid:19) = (cid:18) ϕ X ∗ , Γ ∂ ϕ Y ∗ ∂p z (cid:19) = 0 . (66)Now we mention some other useful relations with the derivatives of ϕ f . The derivativesof ϕ f with respect to p x and p y coincide; see (31) and (32). This fact yields Γ ϕ X ∗ = − Γ ϕ Y ∗ , Γ ϕ Y ∗ = Γ ϕ X ∗ . (67)The same relations can be obtained by direct computations by (21), (7) not only at thepoint p ∗ but at the point p = (0 , , p z ) for any p z . The direct calculations with the help of(31), (32) and (7) show that Γ ∂ ϕ H ∗ ∂p j = 0 , Γ ∂ ϕ E ∗ ∂p j = 0 , j = x, y. (68) VI. THE TWO-SCALED ASYMPTOTIC DECOMPOSITION
We give an asymptotic representation of some special solutions of Maxwell equations inthe entire space under several assumptions:1. the vertical period b of the medium is small as compared with the horizontal scale ofthe field, and the relation between the scales is characterized by the small parameter χ ,2. the frequency ω is close to the frequency ω ∗ of the stationary point p ∗ of one ofthe sheets of the dispersion function ω = ω f ( p ) , f = H, E , i.e., the frequency ω ∗ isdetermined by the relations ω ∗ = ω E ( p ∗ ) = ω H ( p ∗ ) , ∇ ω f ( p ∗ ) = 0 , f = H, E. (69)We assume that ω = ω ∗ + χ δω, δω ∼ . (70)15. We assume that there is one bounded and one unbounded Floquet-Bloch solution ofthe periodic problem (18) at the point p ∗ .Our aim is to find the asymptotics of solutions of the Maxwell equations in the followingform: Ψ = Ψ ( z, ρ ) , ξ ≡ χx, η ≡ χy, ζ ≡ χz, ρ = ( ξ, η, ζ ) (71)where χ ≪ . In the direction transverse to the layers, the field has two scales, one of themis determined by the slow variable ζ = χz , and the other is given by the variable z . In theplane of the layers in the directions x and y , the field depends only on the slow variables ξ = χx , η = χy .We seek a solution in the form of a two-scaled asymptotic series Ψ ( z, ρ ) = Φ ( z, ρ ) e i ( p x ∗ ξ + p y ∗ η ) /χ , Φ ( z, ρ ) = φ ( z, ρ ) e ip z ∗ z , (72) φ ( z, ρ ) = X n ≥ χ n φ ( n ) ( z, ρ ) , φ ( n ) ( z + b, ρ ) = φ ( n ) ( z, ρ ) , ρ = ( ξ, η, ζ ) . (73)In the case under consideration, the stationary point is p x ∗ = p y ∗ = 0.We assume that φ ( n ) ∈ M for every n as functions of z , see the definition after formula(45). Also these functions are infinitely differentiable with respect to slow variables.The Maxwell equations in new variables read k ∗ PΨ + i Γ ∂ Ψ ∂z = − iχ b Γ · ∇ ρ Ψ − χ δωc PΨ , k ∗ = ω ∗ c , (74)where b Γ · ∇ ρ ≡ Γ ∂∂ξ + Γ ∂∂η + Γ ∂∂ζ . Substituting the asymptotic series (72) in (74), we obtain a set of equations A ∗ φ (0) = 0 , A ∗ φ ( n ) = F ( n ) , (75)where A ∗ Ψ = k ∗ PΨ + i Γ ∂ Ψ ∂z − p ∗ Γ Ψ , (76) F (1) = − i b Γ · ∇ ρ φ (0) , (77) F ( n ) = − i b Γ · ∇ ρ φ ( n − − δωc P φ ( n − , n ≥ . (78)Prior to solving the set of equations, we are going to find the relations between theparameters of the problem, which ensures that all nonzero terms on the right-hand side of1676) are of the same order. We assume that the variables E and H and the parameters ε and µ in formulas (1) are already normalized as follows E = r ε av µ av e E , H = r ε av µ av e H , where ε = e εε av , µ = e µµ av , k ∗ = √ ε av µ av e k ∗ , where ε av , µ av are typical dielectric permittivity and magnetic permeability, these param-eters may be large, ε and µ are of order unity, and e ε, e µ are the original parameters of theequation. The variables e k ∗ and k ∗ mean the wave number in vacuum and in the mediumwith parameters ε av and µ av , respectively, k ∗ = √ ε av µ av ω/c , where c is the speed of light invacuum. The second and the third (if nonzero) terms in the right-hand side of (76) are oforder of 1 /b . The first and the second terms are of the same order if √ ε av µ av ω/c ∼ /b. (79)This means that the case under consideration differs from the well-known case ωb/c → A. The principal order
The equation of principal order term is the equation for the Floquet-Bloch amplitudes atthe stationary point p ∗ . It does not contain the derivatives with respect to slow variables ρ = ( ξ, η, ζ ) and its coefficients do not depend on ρ . Its solutions may depend on ρ as onparameters. We seek the principal term in the form of φ (0) ( z, ρ ) = α ( ρ ) ϕ X ∗ ( z ) + α ( ρ ) ϕ Y ∗ ( z ) , φ (0) ∈ M , (80)where ϕ f ∗ ( z ) = ϕ f ( z ; p ∗ ) = e − ip z ∗ z Φ f ( z ; p ∗ ) , (81) f = X, Y , and Φ f ( z ; p ) for p k = 0 are defined in (21) by means of the functions E and H , which satisfy a system (18). The functions α , α are arbitrary scalar functions of slowvariables ρ . Additional restrictions on these arbitrary functions will arise later. B. First-order approximation
The equation for the first-order term φ (1) of the expansion has the form A ∗ φ (1) = − i b Γ · ∇ ρ φ (0) , φ (1) ∈ M . (82)17n order to get the solution of the system belonging to the class M , we must impose addi-tional conditions. Lemma 2 . A solution from the class M of the equation A ∗ φ = F exists if and only ifthe following conditions are satisfied: (cid:0) ϕ X ∗ , F (cid:1) = 0 , (cid:0) ϕ Y ∗ , F (cid:1) = 0 . (83)The proof of the Lemma 2 is given in the Appendix 2.Now we check the solvability conditions for the first-order approximation (82), i.e., wemust check that (cid:16) ϕ X ∗ , b Γ · ∇ ρ φ (0) (cid:17) = 0 , (cid:16) ϕ Y ∗ , b Γ · ∇ ρ φ (0) (cid:17) = 0 , (84)which are reduced to the following conditions (cid:16) ϕ X ∗ , b Γ ϕ X ∗ (cid:17) · ∇ ρ α + (cid:16) ϕ X ∗ , b Γ ϕ Y ∗ (cid:17) · ∇ ρ α = 0 , (85) (cid:16) ϕ Y ∗ , b Γ ϕ X ∗ (cid:17) · ∇ ρ α + (cid:16) ϕ Y ∗ , b Γ ϕ Y ∗ (cid:17) · ∇ ρ α = 0 , (86)where, for example, (cid:16) ϕ X ∗ , b Γ ϕ X ∗ (cid:17) · ∇ ρ α ≡ (cid:0) ϕ X ∗ , Γ ϕ X ∗ (cid:1) ∂α ∂ξ + (cid:0) ϕ X ∗ , Γ ϕ X ∗ (cid:1) ∂α ∂η + (cid:0) ϕ X ∗ , Γ ϕ X ∗ (cid:1) ∂α ∂ζ . (87)These conditions are satisfied at the stationary point owing to (59).Now let us find the exact formula for the solution φ (1) . The right-hand side of the equation(82) can be written as follows: F (1) = − i b Γ · ( ∇ ρ α ) ϕ X ∗ − i b Γ · ( ∇ ρ α ) ϕ Y ∗ . (88)Taking into account (67), we replace the terms containing Γ by the terms containing Γ .Collecting the resulting terms, we obtain F (1) = − i (cid:18) ∂α ∂ξ − ∂α ∂η (cid:19) Γ ϕ X ∗ − i (cid:18) ∂α ∂η + ∂α ∂ξ (cid:19) Γ ϕ Y ∗ − i ∂α ∂ζ Γ ϕ X ∗ − i ∂α ∂ζ Γ ϕ Y ∗ . (89)Now the right-hand side contains four terms. Instead of solving (82), we solve four indepen-dent vector equations with right-hand sides containing each of the terms and then take thesum of their solutions. We add also a solution of the homogeneous equation. The particu-lar solutions of four equations coincide with solutions of (46), (47), and (55). We find thefollowing solution: φ (1) = − i (cid:18) ∂α ∂ξ − ∂α ∂η (cid:19) ∂ ϕ H ∗ ∂p x − i (cid:18) ∂α ∂η + ∂α ∂ξ (cid:19) ∂ ϕ E ∗ ∂p x − i ∂α ∂ζ ∂ ϕ X ∗ ∂p z − i ∂α ∂ζ ∂ ϕ Y ∗ ∂p z + α (1)1 ϕ X ∗ ( z ) + α (1)2 ϕ Y ∗ ( z ) , (90)18here α (1)1 , are new arbitrary functions of the slow variables ρ = ( ξ, η, ζ ). The subscript ∗ stands to show that all the derivatives with the respect to p j , j = x, y, z are taken at p = p ∗ . C. The second-order approximation
Now let us consider the equation on the second-order approximation A ∗ φ (2) = − i b Γ · ∇ ρ φ (1) , φ (2) ∈ M . (91)The solution φ (1) depends on four unknown functions α j , j = 1 , α (1) j , j = 1 , . Thesolvability conditions of (91) yield equations for two of them: i (cid:16) ϕ X ∗ , b Γ · ∇ ρ φ (1) (cid:17) + (cid:18) ϕ X ∗ , δωc P φ (0) (cid:19) = 0 , (92) i (cid:16) ϕ Y ∗ , b Γ · ∇ ρ φ (1) (cid:17) + (cid:18) ϕ Y ∗ , δωc P φ (0) (cid:19) = 0 . (93)For brevity, we introduce the notation τ ≡ (cid:18) ∂α ∂ξ − ∂α ∂η (cid:19) , τ ≡ (cid:18) ∂α ∂η + ∂α ∂ξ (cid:19) . (94)Let us calculate every term separately i (cid:16) ϕ X ∗ , b Γ · ∇ ρ φ (1) (cid:17) = (cid:18) ϕ X ∗ , b Γ ∂ ϕ H ∗ ∂p x (cid:19) · ∇ ρ τ + (cid:18) ϕ X , b Γ ∂ ϕ E ∗ ∂p x (cid:19) · ∇ ρ τ + (cid:18) ϕ X ∗ , b Γ ∂ ϕ X ∗ ∂p z (cid:19) · ∇ ρ ∂α ∂ζ + (cid:18) ϕ X ∗ , b Γ ∂ ϕ Y ∗ ∂p z (cid:19) · ∇ ρ ∂α ∂ζ . (95)Here we take into account that the coefficients of α (1)2 and α (1)1 vanish owing to (59).For example, the first term in the right-hand side of (95) reads (cid:18) ϕ X ∗ , b Γ ∂ ϕ H ∗ ∂p x (cid:19) · ∇ ρ τ ≡ (cid:18) ϕ X ∗ , Γ ∂ ϕ H ∗ ∂p x (cid:19) ∂τ ∂ξ + (cid:18) ϕ X ∗ , Γ ∂ ϕ H ∗ ∂p x (cid:19) ∂τ ∂η + (cid:18) ϕ X ∗ , Γ ∂ ϕ H ∗ ∂p x (cid:19) ∂τ ∂ζ . (96)According to (62), the first term is proportional to ¨ ω H ∗ . The second term vanishes owingto (31) and the last relation of (65), the third term vanishes by (68). Analogously, (cid:18) ϕ X ∗ , b Γ ∂ ϕ E ∗ ∂p x (cid:19) · ∇ ρ τ ≡ (cid:18) ϕ X ∗ , Γ ∂ ϕ E ∗ ∂p x (cid:19) ∂τ ∂ξ + (cid:18) ϕ X ∗ , Γ ∂ ϕ E ∗ ∂p x (cid:19) ∂τ ∂η + (cid:18) ϕ X ∗ , Γ ∂ ϕ E ∗ ∂p x (cid:19) ∂τ ∂ζ . (97)19ccording to (63), the second term is proportional to ¨ ω E ∗ and two other terms vanish:the first one owing to (65), and the third one because of (68).Now we proceed to the last two terms in (95). By (64)-(66), we obtain (cid:18) ϕ X ∗ , b Γ ∂ ϕ X ∗ ∂p z (cid:19) · ∇ ρ ∂α ∂ζ = (cid:18) ϕ X ∗ , Γ ∂ ϕ X ∗ ∂p z (cid:19) ∂ α ∂ζ = ¨ ω ∗ c u XX ∗ ∂ α ∂ζ , (98) (cid:18) ϕ X ∗ , b Γ ∂ ϕ Y ∗ ∂p z (cid:19) · ∇ ρ ∂α ∂ζ = (cid:18) ϕ X ∗ , Γ ∂ ϕ Y ∗ ∂p z (cid:19) ∂ α ∂ζ = 0 . (99)We note that since u XY ∗ = 0, we get, with account of (80) and (60), (cid:18) ϕ X ∗ , δωc P φ (0) (cid:19) = δωc u XX ∗ α . (100)Finally, the condition (92) yields ∂τ ∂ξ ¨ ω H ∗ + ∂τ ∂η ¨ ω E ∗ + ∂ α ∂ζ ¨ ω ∗ + 2 δωc α = 0 . (101)We omit here the nonzero common factor u XX ∗ / u Y Y ∗ / i (cid:16) ϕ Y ∗ , b Γ · ∇ ρ φ (1) (cid:17) = (cid:18) ϕ Y ∗ , b Γ ∂ ϕ H ∗ ∂p x (cid:19) · ∇ ρ τ + (cid:18) ϕ Y , b Γ ∂ ϕ E ∗ ∂p x (cid:19) · ∇ ρ τ + (cid:18) ϕ Y ∗ , b Γ ∂ ϕ X ∗ ∂p z (cid:19) · ∇ ρ ∂α ∂ζ + (cid:18) ϕ Y ∗ , b Γ ∂ ϕ Y ∗ ∂p z (cid:19) · ∇ ρ ∂α ∂ζ . (102)Here, for example, (cid:18) ϕ Y ∗ , b Γ ∂ ϕ H ∗ ∂p x (cid:19) · ∇ ρ τ ≡ (cid:18) ϕ Y ∗ , Γ ∂ ϕ H ∗ ∂p x (cid:19) ∂τ ∂ξ + (cid:18) ϕ Y ∗ , Γ ∂ ϕ H ∗ ∂p x (cid:19) ∂τ ∂η + (cid:18) ϕ Y ∗ , Γ ∂ ϕ H ∗ ∂p x (cid:19) ∂τ ∂ζ . (103)The second term here is proportional to ¨ ω H ∗ according to (63). The first and the third termsvanish by (65) and (68), respectively. The term (cid:16) ϕ Y ∗ , b Γ ∂ ϕ E ∗ (cid:14) ∂p x (cid:17) · ∇ ρ τ is treated analo-gously to (97) by using the second relation of (62), (66) with (47), and (68). Analogouslyto (98) and (99), we obtain (cid:18) ϕ Y ∗ , b Γ ∂ ϕ X ∗ ∂p z (cid:19) · ∇ ρ ∂α ∂ζ = 0 , (cid:18) ϕ Y ∗ , b Γ ∂ ϕ Y ∗ ∂p z (cid:19) · ∇ ρ ∂α ∂ζ = ¨ ω ∗ c u XX ∗ ∂ α ∂ζ . (104) (cid:18) ϕ Y ∗ , b Γ ∂ ϕ H ∗ ∂p x (cid:19) · ∇ ρ τ = − ¨ ω H ∗ c u XX ∗ ∂τ ∂η . (105)20gain, by (65) and (66), we get − ∂τ ∂η ¨ ω H ∗ + ∂τ ∂ξ ¨ ω E ∗ + ∂ α ∂ζ ¨ ω ∗ + 2 δωc α = 0 . (106)Taking into account the definition of τ and τ (94) and the fact that ¨ ω H ∗ = ¨ ω H ∗ and¨ ω E ∗ = ¨ ω E ∗ , we rewrite the equations for α and α as follows: ∂ α ∂ξ ¨ ω H ∗ + ∂ α ∂η ¨ ω E ∗ + ∂ α ∂ζ ¨ ω ∗ + 2 δωc α − ∂ α ∂ξ∂η (¨ ω H − ¨ ω E ) = 0 ,∂ α ∂ξ ¨ ω E ∗ + ∂ α ∂η ¨ ω H ∗ + ∂ α ∂ζ ¨ ω ∗ + 2 δωc α − ∂ α ∂ξ∂η (¨ ω H − ¨ ω E ) = 0 . (107) D. Higher-order approximations
Now we turn to the set of equations (75-78). By considering several recurrent equations,we conclude that the approximation of n th order has the form φ ( n ) = α ( n )1 ϕ X ∗ ( z ) + α ( n )2 ϕ Y ∗ ( z ) + G ( n ) (cid:16) α ( n − , α ( n − , . . . α , α (cid:17) , (108)where G ( n ) is the linear combination of the derivatives of the functions α ( k ) j , j = 1 , , k =1 . . . n −
1, with respect to the variables ξ, η, ζ with known coefficients. For example, in theprincipal order, owing to the formula (80), G (0) ≡
0. In the first order approximation, by(90), G (1) contains the derivatives of α (0) j ≡ α j , j = 1 , , up to the first order: G (1) ≡ − i (cid:18) ∂α ∂ξ − ∂α ∂η (cid:19) ∂ ϕ H ∗ ∂p x − i (cid:18) ∂α ∂η + ∂α ∂ξ (cid:19) ∂ ϕ E ∗ ∂p x − i ∂α ∂ζ ∂ ϕ X ∗ ∂p z − i ∂α ∂ζ ∂ ϕ Y ∗ ∂p z . (109)To find the approximation φ (2) we consider the following inhomogeneous equations: A ∗ Υ Hj = Γ j ∂ ϕ H ∗ ∂p x , A ∗ Υ Ej = Γ j ∂ ϕ E ∗ ∂p x , (110) A ∗ Υ Xj = Γ j ∂ ϕ X ∗ ∂p z , A ∗ Υ Yj = Γ j ∂ ϕ Y ∗ ∂p z , j = 1 , , . (111)Each of these equations is a system of the form (75) with nonzero right-hand sides. It isnecessary to check the solvability of these equations in the class M . To do this, we checkthe conditions imposed in Lemma 2. This means that the conditions (83) must be satisfied: (cid:0) ϕ X ∗ , F (cid:1) = 0 , (cid:0) ϕ Y ∗ , F (cid:1) = 0 , (112)21here F stands for each expression on the right-hand sides of the equations (110), (111).By taking all the possible combinations of the form (cid:18) ϕ f ∗ , Γ j ∂ ϕ H ∗ ∂p x (cid:19) = 0 , (cid:18) ϕ f ∗ , Γ j ∂ ϕ E ∗ ∂p x (cid:19) = 0 , (cid:18) ϕ f ∗ , Γ j ∂ ϕ X ∗ ∂p z (cid:19) = 0 , (cid:18) ϕ f ∗ , Γ j ∂ ϕ Y ∗ ∂p z (cid:19) = 0 , j = 1 , , , f = X, Y, (113)we obtain 24 conditions, 18 of them are satisfied (which follows from (65), (66), (68)), and 6are not satisfied (which follows from (62), (63) and (64)). Nonhomogeneous equations fromthe set (110), (111), for which the solvability conditions are satisfied, have solutions in theclass M . The other equations also have solutions, which do not belong to the class M andare nonperiodic.Next we take the sum of the equations for Υ Hj , Υ Ej , Υ Xj , Υ Yj , j = 1 , , , with coeffi-cients such that the sum of the right-hand side expressions coincides with F (2) ≡ − i b Γ ·∇ ρ φ (1) .The solution of the obtained sum of equations belongs to the class M , because the conditionsof solvability, which are reduced to (107), are satisfied. Thus, we get G (2) ≡ − i ∂α (1)1 ∂ξ − ∂α (1)2 ∂η ! ∂ ϕ H ∗ ∂p x − i ∂α (1)1 ∂η + ∂α (1)2 ∂ξ ! ∂ ϕ E ∗ ∂p x −− i ∂α (1)1 ∂ζ ∂ ϕ X ∗ ∂p z − i ∂α (1)2 ∂ζ ∂ ϕ Y ∗ ∂p z −− X j =1 ∇ j Γ j (cid:20)(cid:18) ∂α ∂ξ − ∂α ∂η (cid:19) Υ Hj ( z ) + (cid:18) ∂α ∂η + ∂α ∂ξ (cid:19) Υ Hj ( z ) ++ ∂α ∂ζ Υ Xj + ∂α ∂ζ Υ Yj (cid:21) , ∇ = ∂∂ξ , ∇ = ∂∂η , ∇ = ∂∂ζ . (114)In an analogous manner, we can derive that the term G (3) contains the first derivativesof α (2)1 and α (2)2 , the second derivatives of α (1)1 and α (1)2 , and the third derivatives of α and α . The approximation φ ( k ) contains yet unknown functions α ( k ) j , j = 1 ,
2, and G ( k ) , whichdepends on the first derivatives of α ( k − and α ( k − , the second derivatives of α ( k − and α ( k − , and the derivatives of the k th order of α and α . The solvability conditions of φ ( k +1) , (cid:16) ϕ X ∗ , b Γ · ∇ ρ φ ( k ) (cid:17) = 0 , (cid:16) ϕ Y ∗ , b Γ · ∇ ρ φ ( k ) (cid:17) = 0 , (115)provide equations for α ( k − and α ( k − . By analogy with the section VI C, we obtain a22ystem of partial differential equations: ∂ α ( k − ∂ξ ¨ ω H ∗ + ∂ α ( k − ∂η ¨ ω E ∗ + ∂ α ( k − ∂ζ ¨ ω ∗ + 2 δωc α ( k − −− ∂ α ( k − ∂ξ∂η (¨ ω H ∗ − ¨ ω E ∗ ) = A ( k − (cid:16) α ( k − , α ( k − , . . . , α , α (cid:17) , (116) ∂ α ( k − ∂ξ ¨ ω E ∗ + ∂ α ( k − ∂η ¨ ω H ∗ + ∂ α ( k − ∂ζ ¨ ω ∗ + 2 δωc α ( k − −− ∂ α ( k − ∂ξ∂η (¨ ω H ∗ − ¨ ω E ∗ ) = A ( k − (cid:16) α ( k − , α ( k − , . . . , α , α (cid:17) , (117)where A (0)1 = A (0)2 = 0, A ( k − , A ( k − for k > α ( l ) j , j = 1 , , l = 1 , . . . , k − , with respect to the variables ξ, η, and ζ . This means that equations for approximations of all orders (75-78) can be solved step bystep. VII. SOLUTION OF THE EQUATIONS FOR α We have obtained the equations (107) with constant coefficients, which describe thebehavior of the envelopes of the field. Now we are going to discuss the methods of solvingthem. The simplest case arises if the field does not depend on one of the lateral coordinates,for example, on η . In this case, the equations for α and α are separated: ∂ α ∂ξ ¨ ω H ∗ + ∂ α ∂ζ ¨ ω ∗ + 2 δωc α = 0 ,∂ α ∂ξ ¨ ω E ∗ + ∂ α ∂ζ ¨ ω ∗ + 2 δωc α = 0 , (118)where the coefficients are defined in (61) and are the derivatives of the dispersion functionsfor TM and TE type waves, calculated at stationary points. Equations (118) can be ellipticor hyperbolic ones depending on the type of a stationary point. According to the Appendix1, we have ¨ ω f ∗ > f = E, H . Equations (118) are elliptic if ¨ ω ∗ >
0, and hyperbolicif ¨ ω ∗ <
0. We solve these equations by means of the Fourier method performing theFourier transform with respect to the variable ξ . By dividing the obtained equations by thecoefficient ¨ ω ∗ , we derive the following equations for the functions b α j ( p ξ , ζ ) : ∂ b α j ∂ζ − ¨ ω j ∗ ¨ ω ∗ p ξ b α j + 2 δωc ¨ ω ∗ b α j = 0 , j = 1 , , (119)23 ω ∗ ≡ ¨ ω H ∗ , ¨ ω ∗ ≡ ¨ ω E ∗ . (120)Solutions of (119) are determined by the integral α j ( ρ ) = 12 π Z R dp ξ e ip ξ ξ (cid:2)b α − j ( p ξ ) e ip jζ ζ + b α + j ( p ξ ) e − ip jζ ζ (cid:3) , (121)where ρ = ( ξ, , ζ ) and p jζ = s − p ξ ¨ ω j ∗ ¨ ω ∗ + 2 δωc ¨ ω ∗ . (122)If ¨ ω ∗ > < p ξ ¨ ω j ∗ / ¨ ω ∗ < δω/c ¨ ω ∗ , then the integral describes the propagatingwaves governed by the elliptic equation. The components with such values of p ξ that p ξ ¨ ω j ∗ > δω/c do not propagate. If ¨ ω ∗ <
0, then introducing the notation σ j , q , we obtain p jζ = q p ξ σ j + q , σ j = ¨ ω j ∗ | ¨ ω ∗ | , q = − δωc | ¨ ω ∗ | . (123)If δω >
0, only the components with large p ξ propagate. The equation (119) turns to aKlein-Gordon-Fock type equation.If δω = 0, the equations (118) are the one-dimensional wave equations, where the variable ζ plays the role of time, and σ plays the role of the speed of wave propagation. The solutionscan be found by the D’Alambert method and read α j = F j ( ξ − σζ ) + G j ( ξ + σζ ) , (124)where F j and G j are some functions. If the function F j is localized for ζ = 0 near ξ = 0,it is localized for any ζ near the line ξ − σζ = 0 and propagates undistorted. This meansthat in the medium under consideration there is the possibility of existence of non-distortingbeams, all of them having the the same angle with z axis equal to ϕ f = ± arctg s ¨ ω f ∗ | ¨ ω ∗ | , f = H, E. (125)We have discussed and studied this effect numerically in .Now we proceed to the general case, where the field depends on both lateral coordinates ξ, η . We obtained equations (101) and (106) for the functions τ , τ defined by (94). Let ustake the derivative of the equation (101) with respect to ξ and of the equation (106) withrespect to η and then calculate the difference. We obtain an equation for τ . To get the24quation for τ , we differentiate (101) with respect to η and equation (106) with respect to ξ and take the sum of the results. Thus, we find the equations for the functions τ , τ ¨ ω H ∗ (cid:18) ∂ τ ∂ξ + ∂ τ ∂η (cid:19) + ¨ ω ∗ ∂ τ ∂ζ + 2 δωc τ = 0 , ¨ ω E ∗ (cid:18) ∂ τ ∂ξ + ∂ τ ∂η (cid:19) + ¨ ω ∗ ∂ τ ∂ζ + 2 δωc τ = 0 . (126)These equations are either hyperbolic or elliptic ones depending on the sign of ¨ ω ∗ . Theirsolutions obtained by the Fourier transform read τ j ( ρ ) = 1(2 π ) Z R dp ξ dp η e ip ξ ξ + ip η η (cid:2)b τ + j ( p ξ , p η ) e − ip jζ ζ + b τ − j ( p ξ , p η ) e ip jζ ζ (cid:3) , (127)where p ζ = s − ¨ ω H ∗ ¨ ω ∗ ( p ξ + p η ) + 2 δωc ¨ ω ∗ , p ζ = s − ¨ ω E ∗ ¨ ω ∗ ( p ξ + p η ) + 2 δωc ¨ ω ∗ , (128)and b τ ± j ( p ξ , p η ) are such functions that the integrals (127) and the integrals, obtained bytaking the derivative of (127), converge. For ¨ ω ∗ < δω > p ξ + p η propagate.Now we are going to derive the equations for the original functions α , α . We take thederivatives of two relations (94) with respect to η and ξ , respectively, and take their sumand difference. We obtain △ α = ∂τ ∂ξ + ∂τ ∂η , △ α = ∂τ ∂ξ − ∂τ ∂η , △ ≡ ∂ ∂ξ + ∂ ∂η . (129)By the Fourier integral we get the following expressions for the functions α , α : α ( ρ ) = − i (2 π ) Z R dp ξ dp η e i ( p ξ ξ + p η η ) p ξ b τ + p η b τ p ξ + p η ,α ( ρ ) = − i (2 π ) Z R dp ξ dp η e i ( p ξ ξ + p η η ) p ξ b τ − p η b τ p ξ + p η , (130)where b τ j ≡ b τ + j ( p ξ , p η ) e − ip jζ ζ + b τ − j ( p ξ , p η ) e ip jζ ζ . (131)We also require the functions b τ j p ξ / ( p ξ + p η ) and b τ j p η / ( p ξ + p η ) to be continuous and integrable.25he expressions for α , α can also be rewritten in the polar coordinate system ( p ρ , γ ) as α ( ρ ) = − i (2 π ) ∞ Z dp ρ π Z dγ e i ( p ρ ξ cos γ + p ρ η sin γ ) (cos γ b τ + sin γ b τ ) ,α ( ρ ) = − i (2 π ) ∞ Z dp ρ π Z dγ e i ( p ρ ξ cos γ + p ρ η sin γ ) (cos γ b τ − sin γ b τ ) , (132)where p ρ = q p ξ + p η , cos γ = p ξ /p ρ , sin γ = p η /p ρ .We express also the principal approximation of the asymptotic solutions in terms of TMand TE solutions at the stationary point determined by the passage to the limit in everydirection (29). Substituting solution of (107) given by (132) in formula (80), we obtain theintegral representation EH ( z, ρ ) ≃ − i (2 π ) ∞ Z dp ρ π Z dγ e i ( p ρ ξ cos γ + p ρ η sin γ ) ( b τ Φ H ∗ ( z ) + b τ Φ E ∗ ( z )) , (133)where Φ H ∗ ( z ) and Φ E ∗ ( z ) depend on the direction of propagation according to (29). VIII. CONCLUSIONS
We have elaborated a formal asymptotic approach for monochromatic electromagneticfields in a layered periodic structure. The frequency of the field is close to that of a stationarypoint p ∗ of one of the sheets of the dispersive surface ω = ω f ( p ). Here f stands for thetype of the polarization, which may be H for TM- or E for TE- polarization. The dispersivesurfaces for waves of different polarizations are distinct, but the stationary points of themcoincide. For the conditions listed in Section VI, we found asymptotic series for the solutionsof Maxwell equations (5) by the two-scale expansions method. The field is assumed to bea function of a fast variable z and slow variables ρ ; see (71). The field in the principalapproximation is represented as a linear combination of Floquet-Bloch solutions of differentpolarizations Φ f ∗ ( z ) , f = X or Y with slowly varying envelopes α j ( ρ ) , j = 1 ,
2. It reads Ψ = EH ( z, ρ ) ≃ α ( ρ ) Φ X ∗ ( z ) + α ( ρ ) Φ Y ∗ ( z ) , (134)where Φ f ∗ ( z ) ≡ Φ f ( z, p ∗ ) = e ip z ∗ z ϕ f ∗ , f = X, Y, (135)26here p ∗ = (0 , , p z ∗ ), p z ∗ = 0 , ± π/b. The functions Φ f ( z, p ) are Floquet-Bloch solutions ofthe system (15). For p = p ∗ , they are expressed in terms of ( E , H ) (see, (21)), which areFloquet-Bloch solutions of the system (18) with the quasimomentum p z ∗ .The envelope functions α j , j = 1 ,
2, are defined by the equations with constant coeffi-cients (107). These coefficients are the second derivatives of the dispersion functions, i.e., thecoefficients of the Tailor expansion of ω = ω f ( p ) near the stationary points p ∗ = (0 , , p z ∗ ) ω = ω ∗ + 12 ¨ ω f ∗ ( p x + p y ) + 12 ¨ ω ∗ ( p z − p z ∗ ) + . . . , (136)where ¨ ω f ∗ , ¨ ω ∗ are the second derivatives of the dispersion function ω with respect to p x and p z calculated at the stationary point. Since the problem is axially symmetric, thederivatives of the dispersion functions with respect to p x and p y are equal and depend onthe type of the polarization, while the derivatives with respect to p z do not depend on thetype of the polarization.The system (107) can be split by introducing new functions τ j , j = 1 , , by means of (94).As was obtained in Section VII, the functions τ and τ satisfy the equations (126). Thecoefficients α j , j = 1 , , are expressed in terms of both τ j , j = 1 , , in (132). The system canbe split into two separated equations only if the field does not depend on one of the spatialcoordinates. Such analysis was absent in the papers of Longhi , .An interesting particular case arises if ¨ ω ∗ <
0. Then the equations for τ and τ arehyperbolic of the Klein-Gordon-Fock type. The coordinate ζ stands for time. If additionally δω = 0, there are wave equations. The effects that arise if the field depends only on onelateral coordinate were studied both analytically and numerically in our paper .The investigation of qualitative consequences of the obtained results in the case of twolateral coordinates are out of scope of the present paper and will be discussed in the nextpublications. We expect that, by choosing the localized solutions of envelopes, we canconstruct beam-like solutions of the Maxwell equations. The obtained formulas enable usalso to find a change of polarization of the field in passing through the layered periodicstructure.The results may be generalized to another equations, which can be written in matrixform (5). 27 PPENDIX 1The dispersion relation
To make the paper self-contained, we obtain the dispersion relation and find stationarypoints of the dispersive surfaces. To do this, we consider the Floquet-Bloch solutions of thefirst subsystems (17): E k H ⊥ = e ipz U U , (137)where U , U are the periodic functions of z with period b .We accomplish the first of subsystems (17) with the initial data E k = 1 , H ⊥ = 0 anddenote the solution of such a Cauchy problem by e , h . We introduce another solution e , h , which satisfies (17) and the initial data E k = 0 , H ⊥ = 1. Both solutions are smoothin the intervals, where ε, µ are continuous, and continuous at the points of discontinuity ofthe parameters ε, µ , but, generally speaking, they are not periodic. These solutions dependon p k , ω , because the coefficients of (17) depend on these parameters. These solutions arelinear independent and form a basis in the space of the solutions of the first subsystem of(17). We introduce the matrix M of the solutions ( e , h ) t and ( e , h ) t as follows: M ( z ; p k , ω ) = e e h h ( z ; p k , ω ) . (138)The matrix M ( b ; p k , ω ) is then a monodromy matrix: M ( z + b ; p k , ω ) = M ( z ; p k , ω ) M ( b ; p k , ω ) . (139)We seek the Floquet-Bloch solutions in the following form: E k H ⊥ = M ( z ; p k , ω ) β β , (140)where ( β , β ) t is the eigenvector corresponding to the eigenvalue λ of the problem: (cid:0) M ( b ; p k , ω ) − λ I (cid:1) β β = , (141)here I is the identity matrix. The equation for λ is expressed in terms of the determinantand the trace of the matrix M . The determinant of the matrix M ( b ; p k , ω ) is the Wronskian28f the solutions ( e , h ) t and ( e , h ) t of the system (17) at z = b . It is constant by theOstrogradskiy-Liuville theorem and, thus, can be calculated for any z . For z = 0 it is equalto 1, so det M ( b ; p k , ω ) = 1 and λ satisfies the quadratic equation λ + Sp M ( b ; p k , ω ) λ + 1 = 0 . (142)In order to obtain the Floquet-Bloch solutions, we require | λ | = 1 and we may assume λ = e ip z b , where p z is real-valued. Then λ = e − ip z b and M ( b ; p k , ω ) + M ( b ; p k , ω ) = 2 cos ( p z b ) . (143)This equation establishes the relation between p z , p k and ω , so we consider below only two ofthese three variables as free parameters. We denote F ≡ M ( b ; p k , ω ) + M ( b ; p k , ω ). Thefunction F ≡ F ( p k , ω ) depends on the problem parameters ω, p k analytically, because thecoefficients of the system (17) depend on these parameters analytically if ω = 0. Thereforethe dispersion relation (143) can be written in the form of F ( p k , ω ) − cos ( p z b ) = 0 . (144)The function F is an oscillating real-valued function of ω , its minima and maxima are largerthan or equal 1 (see, for example, ). The corresponding p z changes from − π/b to π/b . If |F | is greater than 1 for some interval of ω , then the bounded solutions do not exist andsuch interval is called the forbidden zone. For each interval of ω (the allowed zone), where |F | ≤ F of ω is monotone, there exists an inverse function ω = ω Hj ( p ) . (145)If |F | = 1 and ∂ F /∂ω = 0, two intervals of monotone behavior of F touch each other atthis point. Further, we assume that ∂ F /∂ω = 0 . This follows from the assumption that thesystem has one bounded and one unbounded solutions on the boundary of the interval ofthe monotone behavior of F ; see, for example, .Analogous considerations for the second subsystem of (17) yield the dispersion relation ω = ω Ej ( p ) for the wave of the TE polarization.In other words, the problem under consideration can be treated as a spectral problem forthe operator (cid:0) − i P − Γ · ∇ + P − ( p · Γ ) (cid:1) ϕ = ωc ϕ , ϕ ∈ M , (146)29here ω plays the role of the spectral parameter and p is an external parameter of theproblem. For fixed p , the operator has a discrete set of eigenvalues ω = ω j . For p ∈ R × ( − π/b, π/b ), ω j ( p ) forms a sheet of number j . The stationary points of the dispersive surface
We are going to find at least one stationary point of a single-valued function ω Hj , where j is the number of the sheet of the dispersive surface. Below we omit j for the sake of brevity.We consider only such zones, where ∂ F /∂ω = 0 on the entire zone. We take the derivativeof the implicitly defined function ∂ω H ∂p k = − p k ∂ F ∂ ( p k ) , ∂ F ∂ω , ∂ω H ∂p z = − b sin p z b (cid:30) ∂ F ∂ω . (147)We note that ∂ω H /∂p z vanishes for p z = 0 , ± π/b . The function ω H ( p ) is a periodic functionof p z with period 2 π/b . The derivative ∂ω H /∂p k vanishes at least at the point p k = 0 since F depends on p k . We denote the stationary points with asterisk subscript: ω ∗ = ω H ( p ∗ ), p x ∗ = p y ∗ = 0, p z ∗ = 0 , ± π/b . We note that such stationary points are the bounds of theforbidden zones since cos p z ∗ b = ±
1, and hence F = ± ω . First, we find the second derivative withrespect to p z from the formula (147). At a stationary point it reads ∂ ω H ∂p z = − b cos p z ∗ b (cid:30) ∂ F ∂ω − b sin p z ∗ b ∂∂p z (cid:18) ∂ F ∂ω (cid:19) . (148)The second term is equal to zero owing to (144), and the first term has a constant nonzeronumerator and a denominator with the sign changing on each sheet of the function ω . Nowwe consider the second derivative of ω with respect to p x (or p y ). By (62), we get¨ ω H ∗ c u XX ∗ = 2 (cid:18) ϕ X ∗ , Γ ∂ ϕ H ∗ ∂p x (cid:19) , (149)where ∂ ω H ∂p x (cid:12)(cid:12)(cid:12)(cid:12) p ∗ ≡ ¨ ω H ∗ . (150)By the formula (31) and the definition of the matrix (7), we conclude that the scalar producton the right-hand side of the formula (149) is equal to the integral of | H | / ( kε ) with respectto z and hence it is positive if ε >
0. Since u XX > ε > , µ > ω H ∗ >
0. This means that the stationary points of ω are of two different types: hyperbolicand elliptic, depending on the sign of the second derivative with respect to p z .30 he Floquet-Bloch amplitudes We seek the Floquet-Bloch solutions in the form (140), where ( β , β ) t is the eigenvectorof the monodromy matrix (141), which corresponds to the eigenvalue λ . Up to an arbitraryconstant factor, β = M , β = M − λ . We denote the variable U corresponding tothe choice of the sign ± in the exponent e ± ip z b with a subscript ± . We consider TM-typesolution and denote it by the superscript H . Thus, U H + = e − ip z z M e h + ( M − λ ) e h . (151)Here ( e j , h j ) t , j = 1 , z and on the parameters p k , ω . The mon-odromy matrix also depends on p k , ω . We assume that the Floquet-Bloch solutions areexpressed in terms of p . We note that p k = p x + p y and take into account the dispersionrelation (145). The formula (137) takes a form: E k H ⊥ ( z ; p ) = e ip z z U H + ( z ; p z , p k , ω H ( p )) , U H + = U U . (152)The second Floquet-Bloch solution is obtained by replacing p z by − p z , and U H + by U H − .An exception is the case p z = p z ∗ , p z ∗ = 0 , ± π/b. Each point p z ∗ corresponds to theboundary of the forbidden zone, where |F ( p k , ω ) | = 1. Then the double root of the dispersionequation (142) is λ = λ = ±
1. We assume that M = 0. Formulas (152), (151) determinethe unique, up to the constant factor, solution for p = p ∗ (i.e., p k = p k∗ = 0, p z = p z ∗ ). Weseek the second linearly independent solution in the form of E k H ⊥ ( z ; p ∗ ) = M ( z ; 0 , ω ∗ ) γ γ , (153)where the coefficients are found from the equation( M ( b ; 0 , ω ∗ ) − λ I ) γ γ = β β . (154)Then the solution (153) satisfies the relation E k H ⊥ ( z + b ; p ∗ ) = λ E k H ⊥ ( z ; p ∗ ) + E k H ⊥ ( z ; p ∗ ) , (155)31nd can be written as E k ( z ; p ∗ ) H ⊥ ( z ; p ∗ ) = e ip z ∗ z h zλb U H + ( z ; p z ∗ , , ω ∗ ) + Q H ( z ; p z ∗ , ω ∗ ) i , (156)where U H + , Q H are periodic functions of the variable z , the function U H + is defined by (151).We note that U H + and U H − are proportional at the point p ∗ .For waves of the TE polarization, the Floquet-Bloch amplitudes are obtained analogously. APPENDIX 2
Let M be a class of six-component vector-valued functions of z , which are periodic withperiod b , piecewise smooth on the period, and their components with numbers 1 , , , A piecewise function is a function that can be broken into a finite number of distinct pieces andon each piece both the function and its derivative are continuous, even though the whole functionmay have a jump discontinuity at points between the pieces. ) Lemma 1 .The operator A ∗ defined by (41) and (45) on the class of functions M is symmetric, i.e.,for any w , v ∈ M the following relation is valid:( v , A ∗ w ) = ( A ∗ v , w ) . (157)This fact follows from the definition of the operator A ∗ . Since P is a real-valued matrix and v , w ∈ M , we derive by integration by parts that( v , A ∗ w ) = (cid:18) v , k ∗ Pw + i Γ ∂ w ∂z − p z ∗ Γ w (cid:19) = ( A ∗ v , w ) + i M X m =0 [ < v , Γ w > ] m , (158)where < ., . > is a scalar product dependent on z ; see (8). By the definition of Γ , see (7),we get < v ( z ) , Γ w ( z ) > = v ( z ) w ( z ) − v w ( z ) − v ( z ) w ( z ) + v ( z ) w ( z ) . (159)We denote a jump of any scalar function h ( z ) at the point z m by[ h ] m = h ( z m + 0) − h ( z m − , m = 0; [ h ] = h ( b ) − h (0) . (160)32he summation over all discontinuities of v and w is implied in (158). Since v , w ∈ M , thefunction < v ( z ) , Γ w ( z ) > is continuous, the terms outside the integral in (158) vanish, andthe symmetry of A ∗ is proved.Now we proceed to the proof of Lemma 2 , which states that a solution from the class M of the equation A ∗ φ = F (161)exists if and only if the following conditions are satisfied: (cid:0) ϕ X ∗ , F (cid:1) = 0 , (cid:0) ϕ Y ∗ , F (cid:1) = 0 , (162)where the operator A ∗ is defined by the formula (76), the functions ϕ X ∗ ( z ) , ϕ Y ∗ ( z ) are definedby (48), (33), and (21) and belong to M .In order to prove the lemma, we take the scalar product of (161) and ϕ X ∗ : (cid:0) ϕ X ∗ , A ∗ φ (cid:1) = (cid:0) ϕ X ∗ , F (cid:1) . (163)First, if a solution of (161) φ ∈ M , then the scalar product, by the symmetry of A ∗ , canbe rewritten as (cid:0) ϕ X ∗ , A ∗ φ (cid:1) = (cid:0) A ∗ ϕ X ∗ , φ (cid:1) , and since A ∗ ϕ X ∗ = 0, it follows that the scalarproduct on the right-hand side of the equation (163) is also equal to zero.Now let (cid:0) ϕ X ∗ , F (cid:1) = 0. The equation (161) for the vector-valued function φ = ( φ , φ , φ ,φ , φ , φ ) T has the form kεφ + i ∂φ ∂z − p z ∗ φ = F kµφ + i ∂φ ∂z − p z ∗ φ = F kεφ = F kµφ − i ∂φ ∂z + p z ∗ φ = F kεφ − i ∂φ ∂z + p z ∗ φ = F kµφ = F . (164)It splits into the pair of nonhomogeneous systems by the components with numbers 1,5 and2,4, respectively. The components with numbers 3 and 6 are found explicitly. We considerone of the subsystems for the components 1,5, and the second system is considered similarly.The solution of the subsystem can always be represented as a sum φ φ = A U H + ( z ; p z ∗ , , ω ∗ ) + B h zbλ U H + ( z ; p z ∗ , , ω ∗ ) + Q H ( z ; p z ∗ , ω ∗ ) i + e φ e φ , A, B are some coefficients dependent on slow variables as on parameters, and ( e φ , e φ ) t indicates the particular vector solution of the nonhomogeneous equation, which in generaldoes not belong to the class M . We show now that this solution can be chosen continuousat all the points inside (0 , b ). To get the continuity inside (0 , b ), we note that the solution( e φ , e φ ) t is always smooth on the intervals, where the parameters ε, µ are continuous. It canhave jumps at the boundary points of these intervals. At boundary points inside the period(0 , b ), the jumps can be compensated by addition of the solutions of homogeneous equationwith appropriate coefficients, which depend on intervals. However, at the end of the period,the solution ( e φ , e φ ) t may differ from the value at the beginning of the period with jump([ e φ ] , [ e φ ] ) t ; see (160) for explanation of the notation. Integrating by parts, we obtain thefollowing expression: (cid:0) ϕ X ∗ A ∗ φ (cid:1) = (cid:0) A ∗ ϕ X ∗ , φ (cid:1) + E (0; p ∗ ) [ φ ] + H (0; p ∗ ) [ φ ] = E (0; p ∗ ) [ φ ] + H (0; p ∗ ) [ φ ] = (cid:0) ϕ X ∗ , F (cid:1) = 0 , (165)where [ φ ] [ φ ] = 1 λ B U H + ( b ; p z ∗ , , ω ∗ ) + [ e φ ] [ e φ ] . (166)The jump of one of the components e φ or e φ of the particular solution at the end of the periodcan always be compensated by an appropriate choice of the coefficient B of the nonperiodicsolution of the homogeneous equation. The jump of the other component vanishes by (165)if H (0; p ∗ ) = 0, E (0; p ∗ ) = 0. Then the constructed solution to the problem A ∗ φ = F belongs to the class M and the lemma is proved. If E (0; p ∗ ) or H (0; p ∗ ) is equal to zero,we take the beginning of the period at another point. ACKNOWLEDGEMENTS
This work was supported by RFBR grant 140200624 (M.V.P. and M.S.S.) and by SPbGUgrant 11.38.263.2014 (M.V.P). 34
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