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 ⁡ ( k 2 a ) + i ( 1 + γ 2 ) α 1 k 1 cos ⁡ ( k 2 a ) ] , E = i γ 2 α 1 2 α 2 2 k 1 2 k 2 2 sin ⁡ ( k 2 a ) , α j k j 2 = Ω j 2 + ρ j 2 / 4 − β j − ρ j / 2 − α j k j ⊥ 2 − H ˜ 0 j . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeabbaaaaeaacaWLjaGaem4qam0aaSbaaSqaaiabgglaXcqabaGccqGH9aqpdaqadaqaaiabigdaXiabgglaXIGaciab=n7aNnaaCaaaleqabaGaeGinaqdaaaGccaGLOaGaayzkaaGae8xSde2aaSbaaSqaaiabigdaXaqabaGccqWFXoqydaWgaaWcbaGaeGOmaidabeaakiabdUgaRnaaBaaaleaacqaIXaqmaeqaaOGaem4AaS2aaSbaaSqaaiabikdaYaqabaGccyGGJbWycqGGVbWBcqGGZbWCdaqadaqaaiabdUgaRnaaBaaaleaacqaIYaGmaeqaaOGaemyyaegacaGLOaGaayzkaaGaeyOeI0IaemyAaKMae83SdC2aaWbaaSqabeaacqaIYaGmaaGcdaqadaqaaiab=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f7aHnaaDaaaleaacqaIXaqmaeaacqaIYaGmaaGccqWFXoqydaqhaaWcbaGaeGOmaidabaGaeGOmaidaaOGaem4AaS2aa0baaSqaaiabigdaXaqaaiabikdaYaaakiabdUgaRnaaDaaaleaacqaIYaGmaeaacqaIYaGmaaGccyGGZbWCcqGGPbqAcqGGUbGBdaqadaqaaiabdUgaRnaaBaaaleaacqaIYaGmaeqaaOGaemyyaegacaGLOaGaayzkaaGaeiilaWcabaGae8xSde2aaSbaaSqaaiabdQgaQbqabaGccqWGRbWAdaqhaaWcbaGaemOAaOgabaGaeGOmaidaaOGaeyypa0ZaaOaaaeaacqqHPoWvdaqhaaWcbaGaemOAaOgabaGaeGOmaidaaOGaey4kaSYaaSGbaeaacqWFbpGCdaqhaaWcbaGaemOAaOgabaGaeGOmaidaaaGcbaGaeGinaqdaaaWcbeaakiabgkHiTiab=j7aInaaBaaaleaacqWGQbGAaeqaaOGaeyOeI0YaaSGbaeaacqWFbpGCdaWgaaWcbaGaemOAaOgabeaaaOqaaiabikdaYaaacqGHsislcqWFXoqydaWgaaWcbaGaemOAaOgabeaakiabdUgaRnaaDaaaleaacqWGQbGAcqGHLkIxaeaacqaIYaGmaaGccqGHsislcuWGibasgaacamaaBaaaleaacqaIWaamcqWGQbGAaeqaaOGaeiOla4caaaaa@F792@

Using expression (13), it is possible to rewrite the equation (9) as

R = F + F − − G + G − F + F − + G + G − , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGucqGH9aqpjuaGdaWcaaqaamaakaaabaGaemOray0aaSbaaeaacqGHRaWkaeqaaiabdAeagnaaBaaabaGaeyOeI0cabeaaaeqaaiabgkHiTmaakaaabaGaem4raC0aaSbaaeaacqGHRaWkaeqaaiabdEeahnaaBaaabaGaeyOeI0cabeaaaeqaaaqaamaakaaabaGaemOray0aaSbaaeaacqGHRaWkaeqaaiabdAeagnaaBaaabaGaeyOeI0cabeaaaeqaaiabgUcaRmaakaaabaGaem4raC0aaSbaaeaacqGHRaWkaeqaaiabdEeahnaaBaaabaGaeyOeI0cabeaaaeqaaaaacqGGSaalaaa@4108@

where

F ± = A 2 { [ cos ⁡ ( k 1 b / 2 ) − i γ 4 sin ⁡ ( k 1 b / 2 ) ± 1 ] cos ⁡ ( k 2 a ) − − [ ( γ 2 ζ ± 1 ) sin ⁡ ( k 1 b / 2 ) + i ( γ 2 ζ − 1 ± 1 ) cos ⁡ ( k 1 b / 2 ) ] sin ⁡ ( k 2 a ) } + + A γ α 1 k 1 { ζ [ cos ⁡ ( k 1 b / 2 ) − i γ 2 sin ⁡ ( k 1 b / 2 ) ] sin ⁡ ( k 2 a ) + i ( 1 + γ 2 ) cos ⁡ ( k 1 b / 2 ) cos ⁡ ( k 2 a ) } + + i γ 2 α 1 2 k 1 2 ζ cos ⁡ ( k 1 b / 2 ) sin ⁡ ( k 2 a ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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z7a6jabgglaXkabigdaXaGaayjkaiaawMcaaiGbcohaZjabcMgaPjabc6gaUnaabmaabaWaaSGbaeaacqWGRbWAdaWgaaWcbaGaeGymaedabeaakiabdkgaIbqaaiabikdaYaaaaiaawIcacaGLPaaacqGHRaWkcqWGPbqAdaqadaqaaiab=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n7aNnaaCaaaleqabaGaeGOmaidaaOGae8xSde2aa0baaSqaaiabigdaXaqaaiabikdaYaaakiabdUgaRnaaDaaaleaacqaIXaqmaeaacqaIYaGmaaGccqWF2oGEcyGGJbWycqGGVbWBcqGGZbWCdaqadaqaamaalyaabaGaem4AaS2aaSbaaSqaaiabigdaXaqabaGccqWGIbGyaeaacqaIYaGmaaaacaGLOaGaayzkaaGagi4CamNaeiyAaKMaeiOBa42aaeWaaeaacqWGRbWAdaWgaaWcbaGaeGOmaidabeaakiabdggaHbGaayjkaiaawMcaaiabcYcaSaaaaaa@0795@

G ± = A 2 { [ i sin ⁡ ( k 1 b / 2 ) − γ 4 cos ⁡ ( k 1 b / 2 ) ∓ 1 ] cos ⁡ ( k 2 a ) + + [ ( γ 2 ζ − 1 ± 1 ) sin ⁡ ( k 1 b / 2 ) + i ( γ 2 ζ ± 1 ) cos ⁡ ( k 1 b / 2 ) ] sin ⁡ ( k 2 a ) } + + A γ α 1 k 1 { ζ [ i sin ⁡ ( k 1 b / 2 ) − γ 2 cos ⁡ ( k 1 b / 2 ) ] sin ⁡ ( k 2 a ) − ( 1 + γ 2 ) sin ⁡ ( k 1 b / 2 ) cos ⁡ ( k 2 a ) } − − γ 2 α 1 2 k 1 2 ζ sin ⁡ ( k 1 b / 2 ) sin ⁡ ( k 2 a ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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n7aNnaaCaaaleqabaGaeGOmaidaaOGae8xSde2aa0baaSqaaiabigdaXaqaaiabikdaYaaakiabdUgaRnaaDaaaleaacqaIXaqmaeaacqaIYaGmaaGccqWF2oGEcyGGZbWCcqGGPbqAcqGGUbGBdaqadaqaamaalyaabaGaem4AaS2aaSbaaSqaaiabigdaXaqabaGccqWGIbGyaeaacqaIYaGmaaaacaGLOaGaayzkaaGagi4CamNaeiyAaKMaeiOBa42aaeWaaeaacqWGRbWAdaWgaaWcbaGaeGOmaidabeaakiabdggaHbGaayjkaiaawMcaaiabcYcaSaaaaaa@0445@

ζ = α₂k₂/α₁k₁.

The reflection amplitude for a multilayer structure consisting of N layers, is defined by expression (8).

Note that as calculating the reflection amplitude (14) requires extracting the square root of complex expressions, there will be branching points, whenever the following conditions are met,

F₊F_- = 0,

G₊G_- = 0.