Consider a system consisting of three parts, whose planes of contact are parallel to the yz plane. The first and third parts (in the direction of x-axis) are homogeneous, biaxial, semi-infinite ferromagnetic media, with an N-layer ferromagnetic structure between them. The modulated parameters are exchange interaction, α, uniaxial, β, and rhombic, ρ, magnetic anisotropy, and saturation magnetization, M₀. As shown in figure 1, each layer of the N-layer ferromagnetic structure consists of two layers with thicknesses a and b. Parameters α, β, ρ and M₀ have values α₁, β₁, ρ₁, M₀₁ and α₂, β₂, ρ₂, M₀₂ in corresponding layers. The easy axis is parallel to an external homogeneous permanent magnetic field, H₀, itself parallel to z-axis.

Using the formalism of spin density 8, the magnetization can be written as

M j ( r , t ) = M 0 j Ψ j + ( r , t ) σ Ψ j ( r , t ) , j = 1 , 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaacbeGae8xta00aaSbaaSqaaiabdQgaQbqabaGccqGGOaakcqWFYbGCcqGGSaalcqWG0baDcqGGPaqkcqGH9aqpcqWFnbqtdaWgaaWcbaGaeGimaaJaemOAaOgabeaakiabfI6aznaaDaaaleaacqWGQbGAaeaacqGHRaWkaaGccqGGOaakcqWFYbGCcqGGSaalcqWG0baDcqGGPaqkiiqacqGFdpWCcqqHOoqwdaWgaaWcbaGaemOAaOgabeaakiabcIcaOiab=jhaYjabcYcaSiabdsha0jabcMcaPiabcYcaSaqaaiabdQgaQjabg2da9iabigdaXiabcYcaSiabikdaYiabcYcaSaaaaaa@51DB@

where Ψ_j's are quasiclassical wave functions playing the role of a spin density order parameter, r is a radius-vector of the Cartesian coordinate system, t is time, and σ are Pauli matrices.

Write Lagrange's equations for Ψ_j as

i ℏ ∂ Ψ j ( r , t ) ∂ t = −   μ 0 H e j ( r , t ) σ Ψ j ( r , t ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGPbqAcqWIpecAjuaGdaWcaaqaaiabekGi2kabfI6aznaaBaaabaGaemOAaOgabeaadaqadaqaaiabhkhaYjabcYcaSiabdsha0bGaayjkaiaawMcaaaqaaiabekGi2kabdsha0baakiabg2da9iabgkHiTiaaykW7iiGacqWF8oqBdaWgaaWcbaGaeGimaadabeaakiabhIeainaaBaaaleaacqWGLbqzcqWGQbGAaeqaaOWaaeWaaeaacqWHYbGCcqGGSaalcqWG0baDaiaawIcacaGLPaaaiiqacqGFdpWCcqqHOoqwdaWgaaWcbaGaemOAaOgabeaakmaabmaabaGaeCOCaiNaeiilaWIaemiDaqhacaGLOaGaayzkaaGaeiilaWcaaa@54D3@

where μ₀ is a Bohr magneton, Hej=−∂wj∂Mj+∂∂xk∂wj∂(∂Mj/∂xk) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabhIeainaaBaaaleaacqWGLbqzcqWGQbGAaeqaaOGaeyypa0JaeyOeI0scfa4aaSaaaeaacqGHciITcqWG3bWDdaWgaaqaaiabdQgaQbqabaaabaGaeyOaIyRaeCyta00aaSbaaeaacqWGQbGAaeqaaaaakiabgUcaRKqbaoaalaaabaGaeyOaIylabaGaeyOaIyRaemiEaG3aaSbaaeaacqWGRbWAaeqaaaaadaWcaaqaaiabgkGi2kabdEha3naaBaaabaGaemOAaOgabeaaaeaacqGHciITdaqadaqaamaalyaabaGaeyOaIyRaeCyta00aaSbaaeaacqWGQbGAaeqaaaqaaiabgkGi2kabdIha4naaBaaabaGaem4AaSgabeaaaaaacaGLOaGaayzkaaaaaaaa@5118@, and w_j is energy density. Note that, in an exchange mode when the condition L ≫ l = a + b is satisfied (here L is material's characteristic length), energy density has the following form in each of homogeneous parts,

w j = α 2 ( ∂ m j ∂ x k ) 2 + β 2 ( m j x 2 + m j y 2 ) + ρ 2 m j x 2 − H 0 M j z , j = 1 , 2. MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaem4DaC3aaSbaaSqaaiabdQgaQbqabaGccqGH9aqpjuaGdaWcaaqaaGGaciab=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@66D9@

Here it is taken into account, that the material is magnetized parallel to e_z in the ground state, Mj2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabd2eannaaDaaaleaacqWGQbGAaeaacqaIYaGmaaaaaa@2E94@(r, t) = const and M_j(r, t) = M_0je_z + m_j(r, t), where m_j(r, t) is a small deviation of the magnetization from the ground state.

Then, using the linear perturbation theory, it is possible to write down the solution of (2) as

Ψ j ( r , t ) = exp ⁡ ( i μ 0 H 0 t / ℏ ) ⋅ ( 1 χ j ( r , t ) ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHOoqwdaWgaaWcbaGaemOAaOgabeaakiabcIcaOGqabiab=jhaYjabcYcaSiabdsha0jabcMcaPiabg2da9iGbcwgaLjabcIha4jabcchaWnaabmaabaGaemyAaKgcciGae4hVd02aaSbaaSqaaiabicdaWaqabaGccqWGibasdaWgaaWcbaGaeGimaadabeaakiabdsha0jabc+caViabl+qiObGaayjkaiaawMcaaiabgwSixpaabmaabaqbaeqabiqaaaqaaiabigdaXaqaaiab+D8aJnaaBaaaleaacqWGQbGAaeqaaOGaeiikaGIae8NCaiNaeiilaWIaemiDaqNaeiykaKcaaaGaayjkaiaawMcaaiabcYcaSaaa@5207@

where χ_j(r, t) is a small function characterizing the deviation of a magnetization from the ground state. Linearizing equation (2) while taking into account equation (4), we obtain the following equation,

− i ℏ 2 μ 0 M 0 j ∂ χ j ( r , t ) ∂ t = ( α j Δ − β j − ρ j 2 − H ˜ 0 j ) χ j ( r , t ) − ρ j 2 χ j ∗ ( r , t ) , i ℏ 2 μ 0 M 0 j ∂ χ j ∗ ( r , t ) ∂ t = ( α j Δ − β j − ρ j 2 − H ˜ 0 j ) χ j ∗ ( r , t ) − ρ j 2 χ j ( r , t ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeGabaaabaGaeyOeI0scfa4aaSaaaeaacqWGPbqAcqWIpecAaeaacqaIYaGmiiGacqWF8oqBdaWgaaqaaiabicdaWaqabaGaemyta00aaSbaaeaacqaIWaamcqWGQbGAaeqaaaaadaWcaaqaaiabgkGi2kab=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D8aJnaaDaaabaGaemOAaOgabaGaey4fIOcaamaabmaabaGaeCOCaiNaeiilaWIaemiDaqhacaGLOaGaayzkaaaabaGaeyOaIyRaemiDaqhaaOGaeyypa0ZaaeWaaeaacqWFXoqydaWgaaWcbaGaemOAaOgabeaakiabfs5aejabgkHiTiab=j7aInaaBaaaleaacqWGQbGAaeqaaOGaeyOeI0scfa4aaSaaaeaacqWFbpGCdaWgaaqaaiabdQgaQbqabaaabaGaeGOmaidaaOGaeyOeI0IafmisaGKbaGaadaWgaaWcbaGaeGimaaJaemOAaOgabeaaaOGaayjkaiaawMcaaiab=D8aJnaaDaaaleaacqWGQbGAaeaacqGHxiIkaaGcdaqadaqaaiabhkhaYjabcYcaSiabdsha0bGaayjkaiaawMcaaiabgkHiTKqbaoaalaaabaGae8xWdi3aaSbaaeaacqWGQbGAaeqaaaqaaiabikdaYaaakiab=D8aJnaaBaaaleaacqWGQbGAaeqaaOWaaeWaaeaacqWHYbGCcqGGSaalcqWG0baDaiaawIcacaGLPaaacqGGSaalaaaaaa@B73C@

where H˜0j MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdIeaizaaiaWaaSbaaSqaaiabicdaWiabdQgaQbqabaaaaa@2E94@ = H₀/M_0j.

Expressing χ*(r, t) from the first of these equations and substituting it into the second, we obtain the following equation for magnetization dynamics:

− ℏ 2 ( 2 μ 0 M 0 j ) 2 ∂ 2 χ j ( r , t ) ∂ t 2 = = [ α j 2 Δ 2 − 2 α j ( β j + ρ j 2 + H ˜ 0 j ) Δ + ( β j + H ˜ 0 j ) ( β j + ρ j + H ˜ 0 j ) ] χ j ( r , t ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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j7aInaaBaaaleaacqWGQbGAaeqaaOGaey4kaSscfa4aaSaaaeaacqWFbpGCdaWgaaqaaiabdQgaQbqabaaabaGaeGOmaidaaOGaey4kaSIafmisaGKbaGaadaWgaaWcbaGaeGimaaJaemOAaOgabeaaaOGaayjkaiaawMcaaiabfs5aejabgUcaRmaabmaabaGae8NSdi2aaSbaaSqaaiabdQgaQbqabaGccqGHRaWkcuWGibasgaacamaaBaaaleaacqaIWaamcqWGQbGAaeqaaaGccaGLOaGaayzkaaWaaeWaaeaacqWFYoGydaWgaaWcbaGaemOAaOgabeaakiabgUcaRiab=f8aYnaaBaaaleaacqWGQbGAaeqaaOGaey4kaSIafmisaGKbaGaadaWgaaWcbaGaeGimaaJaemOAaOgabeaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaiab=D8aJnaaBaaaleaacqWGQbGAaeqaaOWaaeWaaeaacqWHYbGCcqGGSaalcqWG0baDaiaawIcacaGLPaaacqGGUaGlaaaaaa@882F@

Performing the Fourier transformations on y and z coordinates and time, t, we can write

[ α j 2 ∂ 4 ∂ x 4 − 2 α j ( β j + ρ j 2 + H ˜ 0 j ) ∂ 2 ∂ x 2 + α j 2 k j ⊥ 4 + 2 α j k j ⊥ 2 ( β j + ρ j 2 + H ˜ 0 j ) + + ( β j + H ˜ 0 j ) ( β j + ρ j + H ˜ 0 j ) − Ω j 2 ] χ j ω , k ⊥ ( x ) = 0. MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGabaaabaWaamqaaeaaiiGacqWFXoqydaqhaaWcbaGaemOAaOgabaGaeGOmaidaaKqbaoaalaaabaGaeyOaIy7aaWbaaeqabaGaeGinaqdaaaqaaiabgkGi2kabdIha4naaCaaabeqaaiabisda0aaaaaGccqGHsislcqaIYaGmcqWFXoqydaWgaaWcbaGaemOAaOgabeaakmaabmaabaGae8NSdi2aaSbaaSqaaiabdQgaQbqabaGccqGHRaWkjuaGdaWcaaqaaiab=f8aYnaaBaaabaGaemOAaOgabeaaaeaacqaIYaGmaaGccqGHRaWkcuWGibasgaacamaaBaaaleaacqaIWaamcqWGQbGAaeqaaaGccaGLOaGaayzkaaqcfa4aaSaaaeaacqGHciITdaahaaqabeaacqaIYaGmaaaabaGaeyOaIyRaemiEaG3aaWbaaeqabaGaeGOmaidaaaaakiabgUcaRiab=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j7aInaaBaaaleaacqWGQbGAaeqaaOGaey4kaSIae8xWdi3aaSbaaSqaaiabdQgaQbqabaGccqGHRaWkcuWGibasgaacamaaBaaaleaacqaIWaamcqWGQbGAaeqaaaGccaGLOaGaayzkaaGaeyOeI0IaeuyQdC1aa0baaSqaaiabdQgaQbqaaiabikdaYaaaaOGaayzxaaGae83Xdm2aa0baaSqaaiabdQgaQbqaaiab=L8a3jabcYcaSiabhUgaRnaaBaaameaacqGHLkIxaeqaaaaakmaabmaabaGaemiEaGhacaGLOaGaayzkaaGaeyypa0JaeGimaaJaeiOla4caaaaa@A360@

Here, Ω_j = ωħ/2μ₀M_0j, ω is frequency, and k_⊥ = (0, k_y, k_z).

The spin wave reflection amplitude from an N-layer structure can be represented as 9,

R N = R 1 − exp ⁡ ( 2 i q l N ) 1 − R 2 exp ⁡ ( 2 i q l N ) , MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGudaWgaaWcbaGaemOta4eabeaakiabg2da9iabdkfasLqbaoaalaaabaGaeGymaeJaeyOeI0IagiyzauMaeiiEaGNaeiiCaaNaeiikaGIaeGOmaiJaemyAaKMaemyCaeNaemiBaWMaemOta4KaeiykaKcabaGaeGymaeJaeyOeI0IaemOuai1aaWbaaeqabaGaeGOmaidaaiGbcwgaLjabcIha4jabcchaWjabcIcaOiabikdaYiabdMgaPjabdghaXjabdYgaSjabd6eaojabcMcaPaaacqGGSaalaaa@4ED1@

where R is an amplitude of reflection from semi-infinite multilayer structure (N = ∞),

R = ( r + 1 ) 2 − τ 2 − ( r − 1 ) 2 − τ 2 ( r + 1 ) 2 − τ 2 + ( r − 1 ) 2 − τ 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGucqGH9aqpjuaGdaWcaaqaamaakaaabaWaaeWaaeaacqWGYbGCcqGHRaWkcqaIXaqmaiaawIcacaGLPaaadaahaaqabeaacqaIYaGmaaGaeyOeI0ccciGae8hXdq3aaWbaaeqabaGaeGOmaidaaaqabaGaeyOeI0YaaOaaaeaadaqadaqaaiabdkhaYjabgkHiTiabigdaXaGaayjkaiaawMcaamaaCaaabeqaaiabikdaYaaacqGHsislcqWFepaDdaahaaqabeaacqaIYaGmaaaabeaaaeaadaGcaaqaamaabmaabaGaemOCaiNaey4kaSIaeGymaedacaGLOaGaayzkaaWaaWbaaeqabaGaeGOmaidaaiabgkHiTiab=r8a0naaCaaabeqaaiabikdaYaaaaeqaaiabgUcaRmaakaaabaWaaeWaaeaacqWGYbGCcqGHsislcqaIXaqmaiaawIcacaGLPaaadaahaaqabeaacqaIYaGmaaGaeyOeI0Iae8hXdq3aaWbaaeqabaGaeGOmaidaaaqabaaaaiabcYcaSaaa@56AA@

q is a Bloch wave vector defined by,

e x p ( i q l N ) = ( τ + 1 ) 2 − r 2 + ( τ − 1 ) 2 − r 2 ( τ + 1 ) 2 − r 2 − ( τ − 1 ) 2 − r 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@60A9@

l = a + b is the structure period, r and τ are the complex amplitudes of reflection and transmission accordingly for a single symmetric (with regard to its center) period.

Since the equations (5) have the form similar to Schrödinger equation, amplitudes of reflection and transmission for a single period can be found using the corresponding method of quantum mechanics.

For a material consisting of two homogeneous parts in contact along the yz plane, it is possible to write down the energy density as,

w = ∑ j = 1 2 θ [ ( − 1 ) j x ] w j + A δ ( x ) M 1 M 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG3bWDcqGH9aqpdaaeWbqaaGGaciab=H7aXnaadmaabaWaaeWaaeaacqGHsislcqaIXaqmaiaawIcacaGLPaaadaahaaWcbeqaaiabdQgaQbaakiabdIha4bGaay5waiaaw2faaiabdEha3naaBaaaleaacqWGQbGAaeqaaaqaaiabdQgaQjabg2da9iabigdaXaqaaiabikdaYaqdcqGHris5aOGaey4kaSIaemyqaeKae8hTdq2aaeWaaeaacqWG4baEaiaawIcacaGLPaaacqWHnbqtdaWgaaWcbaGaeGymaedabeaakiabh2eannaaBaaaleaacqaIYaGmaeqaaOGaeiilaWcaaa@4C84@

where A is the constant describing a coupling along the interface, θ(x) is a step function, and w_j's are defined by (3). After integrating the equations of movement of magnetic moment in the vicinity of an interface, we obtain the following boundary conditions (indexes ω, k_⊥ are omitted),

A γ ( χ 2 − χ 1 ) + α 1 χ ′ 1 = 0 , A ( χ 2 − χ 1 ) + γ α 2 χ ′ 2 = 0 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGabaaabaGaemyqaeecciGae83SdC2aaeWaaeaacqWFhpWydaWgaaWcbaGaeGOmaidabeaakiabgkHiTiab=D8aJnaaBaaaleaacqaIXaqmaeqaaaGccaGLOaGaayzkaaGaey4kaSIae8xSde2aaSbaaSqaaiabigdaXaqabaGccuWFhpWygaqbamaaBaaaleaacqaIXaqmaeqaaOGaeyypa0JaeGimaaJaeiilaWcabaGaemyqae0aaeWaaeaacqWFhpWydaWgaaWcbaGaeGOmaidabeaakiabgkHiTiab=D8aJnaaBaaaleaacqaIXaqmaeqaaaGccaGLOaGaayzkaaGaey4kaSIae83SdCMae8xSde2aaSbaaSqaaiabikdaYaqabaGccuWFhpWygaqbamaaBaaaleaacqaIYaGmaeqaaOGaeyypa0JaeGimaaJaeiilaWcaaaaa@52CE@

where γ = M₀₂/M₀₁, and prime means the derivative with x. These boundary conditions will be applied at each interface of the multilayer structure.

Represent an incident wave with function χ_I = exp(ik₁x), a reflected wave with function χ_r = r exp(ik˜1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdUgaRzaaiaWaaSbaaSqaaiabigdaXaqabaaaaa@2D7F@x), and a wave transmitted through a single layer with function χ_τ = τ exp(ik₁x). Here k₁ and k˜1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdUgaRzaaiaWaaSbaaSqaaiabigdaXaqabaaaaa@2D7F@ are the wave vectors of incident and reflected waves and accordingly k˜1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdUgaRzaaiaWaaSbaaSqaaiabigdaXaqabaaaaa@2D7F@ = -k₁. Substituting these expressions into (12) together with the expression χ_layer = C₁ exp(ik₂x) + C₂ exp(-ik₂x) describing the wave within the intermediate layer, for each of two borders of a single period we come to the expressions for spin wave amplitudes of reflection and transmission,

r = exp ⁡ ( i k 1 b ) ⋅ A 2 C − + A D − + E A 2 C + + A D + + E , τ = exp ⁡ ( − i k 1 a ) ⋅ 2 A 2 α 1 α 2 k 1 k 2 A 2 C + + A D + + E , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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f7aHnaaBaaabaGaeGymaedabeaacqWFXoqydaWgaaqaaiabikdaYaqabaGaem4AaS2aaSbaaeaacqaIXaqmaeqaaiabdUgaRnaaBaaabaGaeGOmaidabeaaaeaacqWGbbqqdaahaaqabeaacqaIYaGmaaGaem4qam0aaSbaaeaacqGHRaWkaeqaaiabgUcaRiabdgeabjabdseaenaaBaaabaGaey4kaScabeaacqGHRaWkcqWGfbqraaGaeiilaWcaaaaa@79D4@

where

C ± = ( 1 ± γ 4 ) α 1 α 2 k 1 k 2 cos ⁡ ( k 2 a ) − i γ 2 ( α 1 2 k 1 2 ± α 2 2 k 2 2 ) sin ⁡ ( k 2 a ) , D ± = γ α 1 α 2 k 1 k 2 [ ( 1 ± γ 2 ) α 2 k 2 sin ⁡ (