Nonlinear electromagnetic wave propagation in isotropic and anisotropic antiferromagnetic media

Transcription

1 Nonlinear electromagnetic wae propagation in isotropic anisotropic antiferromagnetic media V. Veerakumar Department of Physics, Manipal Institute of Technology, MAHE Deemed Uniersity, Manipal India. N. Mastorakis Hellenic Naal Academy Hatzikyriakou 8539, Piraeus, Greece We study the nonlinear propagation of electromagnetic wae EMW] in an isotropic charge-free antiferromagnetic medium by considering the two-sublattice model at low energy configurations. It is found that in the case of isotropic medium the dynamics is goerned by a perturbed MKdV equation. It is also found that the magnetic field component of the EMW decelerates the shape is distorted. Further the magnitude of the magnetic field is getting damped as time progressed. The aboe results are due to the interlocking of the adjacent antiparallel spins in the antiferromagnetic medium. In the case of an anisotropic medium the EM wae propagation is goerned by the completely integrable deriatie NLS equation which possess soliton solutions. I. Introduction There has been increased interest in the recent times in the study of nonlinear systems which are integrable show regular behaior in the form of solitons. In this context ferromagnet with different magnetic interactions are inherently nonlinear in nature. It is found that the effect of nonlinearity in the ferromagnet with bilinear exchange interaction leads to localized structures called magnetic solitons ]. The reason for the magnetic solitons in ferromagnetic medium is the integrable nature of the Lau-Lifshitz (LL) equation goerning the spin dynamics in ferromagnet. The other class of interesting integrable nonlinear system is the electromagnetic wae (EMW) propagation in optical fibers. Hasegawa Tappert ] showed theoretically that an optical pulse in a dielectric fiber medium during propagation form an enelope soliton due to Kerr effect in the medium which was experimentally demonstrated by Mollenauer etal 3]. Another class of nonlinear system is the antiferromagnetic media. Though nonlinear spin dynamics of ferromagnetic systems hae been studied extensiely, 4 ], the study of antiferromagnetic spin systems is still in the infant stage, because, unlike in ferromagnets, in the case of antiferromagnets the adjacent spins are aligned antiparallel to each other hence the system is treated as a two sublattice model thus the dynamics is goerned by highly nontriial coupled nonlinear partial differential equations. Very recently some progress has been made in understing the spin dynamics of the one dimensional classical continuum isotropic antiferromagnetic systems 5]. Interesting finite energy solutions including multi-solitons hae been obtained in these systems only in certain limiting cases ]. Also, when the adjacent antiparallel spins are locked together its dynamics is identified interms of twists other classes of spin configurations 5]. In a ery different context the interaction of EMW in ferromagnetic medium has become ery important especially in relation to ferrite deices at microwae frequencies such as ferrite loaded wae guides 6]. This problem was extensiely studied from the linear iew point by linearizing the equation describing the spin dynamics in ferromagnets 7]. Howeer, this study was not able to explain the rich the interesting nonlinear dynamics obsered in the ferromagnetic resonance experiments such as saturation of the main ferromagnetic resonances (FMR), absorption of energy in the FMR etc. at high power leels. This was explained by seeral authors by studying the exact nonlinear equation (for details refer 8]. Also, this problem of propagation of EMW in a ferromagnetic medium in the context of dispersion less propagation was studied it was found that the magnetic field component of the EMW is modulated in the form of solitons as it propagates in a ferromagnetic medium the magnetization excitations in the medium are goerned by soliton modes 9 3]. In this context antiferromagnetic materials are equally important as they also exhibit long range order, with the exchange interaction leading to an antiparallel ordering of spins in the minimum energy configuration. Hence, the study of propagation of EMW in antiferromagnetic medium is also equally interesting the same can be studied along the lines of ferromagnetic medium. This topic of problem is found to hae potential applications in the areas of magneto optical recording, switching etc. 4]. In the present paper, we study EMW propagation in an antiferromagnetic medium, by soling the spin

2 equation coupled with Maxwell s equations using a reductie perturbation method. In section II, we present the mathematical model for the antiferromagnetic system construct the spin equation for the problem the Maxwell s equations for EMW propagation. We study EMW propagation in isotropic as well as anisotropic antiferromagnetic medium using a reductie perturbation method in section III conclude the results in section IV. II. Antiferromagnetic model EWW equation The Heisenberg model of the effectie Hamiltonian for an anisotropic antiferromagnet with N-spins in an inhomogeneous time dependent external magnetic field H(r, t) can be written as H = i JSi S i+ + A(S x i ) ˆγS i H ]. () Here S i represents the spin angular momentum operator which in the classical limit can be replaced by three component ectors S i = (Si x, Sy i, Sz i ) the external magnetic field H(r, t) = (H x, H y, H z ) in our problem is infact the magnetic field component of the propagating EMW. In Eq.(), J represents the exchange integral which take only alues less than zero, A is the anisotropy parameter ˆγ = gµ B where g is the gyromagnetic ratio µ B is the Bohr magneton. The dynamics of spins can be considered from a classical point of iew by treating spin as a dynamical ariable suitable canonical equations can be obtained in analogy with the spinless nonrelatiistic particle. Therefore, for the gien spin Hamiltonian, we can write down the classical equations of motion for the spin ectors S a,i S b,i in the continuum limit as S a (r, t) = S a J(S b + λ S b AS x a n + ˆγH], () S b (r λ], t) = S b J(S a λ S a + ASb x n + ˆγH], (3) S a = (Sa x, Sa, y Sa), z S b = (Sb x, S y b, Sz b ), S a = = S b. While writing the equations of motion (II) for the two sublattices, we hae considered the positie negatie x directions as the easy axes of magnetization for the sublattices a b respectiely. Eqs.(II) in the classical continuum limit describe the spin dynamics in the a b sublattices of the anisotropic antiferromagnet in the presence of an external inhomogeneous time dependent magnetic field which in our problem is the magnetic field component of the propagating EMW. Eqs.(II) are analogous to the Lau-Lifshitz equation in ferromagnets 5]. Now adding subtracting Eqs.() (3) defining two unit ectors M M which we call as the staggered total magnetization ectors in the form 4] M = ( ρ ) S (S a S b ), M = Sρ (S a + S b ), where M = (M x, M y, M z ) M = (M x, M y, M z ), M M = ρ = ( + S a S b S ), we obtain M M = 4JSρ(M M ) + JλSρ ( ρ ) (M M x ) JSλ ρ (M M ) + ˆγ(M H) x AS ( ρ )M x M + ρ M x M ] n,(4) ρ { = JSλ ρ } x (M M ) + ˆγ(M H) AS ρ (M x M + M x M) n. (5) On substituting M M in Eqs.(II) assuming that the low energy configurations correspond to S a S b S S a + S b, when ρ, after suitable rescaling redefinition of parameters, the dynamics is dominated by the equation M = M {ˆγH Jλ M AM x n)}, (6) The aboe equation represents the dynamics of magnetization in a classical continuum Heisenberg anisotropic antiferromagnetic medium in the presence of an inhomogeneous time dependent external magnetic field when the adjacent antiparallel spins are locked together. The isotropic limit of Eqs.(II) in the absence of any external field has been soled twist-like excitations hae been found 3 5] The dynamics of electromagnetic field in a material medium is described by Maxwell s equations which in the absence of stationary moing charges can be written as 6] E =, B =, (7) E = B, H = ɛ E. (8) Here the fields H(r, t) = (H x, H y, H z ), B(r, t) = (B x, B y, B z ) E(r, t) = (E x, E y, E z ) hae the usual meaning of the magnetic field, magnetic induction electric field respectiely ɛ is the dielectric constant of the medium. The aboe three field in antiferromagnetic medium is connected by the relation 6] B = µ H + M], where µ is the permeability of the medium. Now, taking curl on both sides of the second of Eq.(8) using the first of (8) the relation connecting H, B M after little algebra we obtain H + M] = c H (.H) ]. (9) Here c = µɛ is the elocity of propagation of the EMW in the antiferromagnetic medium. Eq.(9) describes

3 3 the propagation of EMW in antiferromagnetic medium. Thus, the set of coupled equations (6) (9) completely describe the propagation of EMW in an anisotropic antiferromagnetic medium when the adjacent antiparallel spins are locked under low energy configurations. III. EMW in antiferromagnetic medium In order to find the nature of propagation of EMW in the antiferromagnetic medium, we now hae to sole the set of coupled equations (6) (9). Howeer, the nonlinear character of Eq.(6) makes the problem of soling them difficult therefore we restrict ourseles to one dimension. We are interested in finding soliton solutions to these equations so that one can hae lossless propagation of EMW in the medium in the form of electromagnetic solitons. Infact, we are going to study the nonlinear modulation of the slowly arying EM plane wae of small but finite amplitude in the antiferromagnetic medium into soliton using a reductie perturbation method deeloped by Tanuiti Yajima 7]. Very recently, the reductie perturbation method has been successfully used to study the electromagnetic soliton propagation in different ferromagnetic media by the present authors also by few others 9 3]. Experience suggests that the magnetization the magnetic fields hae to be exped uniformly nonuniformly respectiely in the case of isotropic anisotropic media. Also an examination of the dispersion relation for the plane EMW propagation suggests us to introduce different coordinate stretchings in the case of isotropic anisotropic media. This has made us to treat the propagation of EMW in isotropic anisotropic media separately. A. Isotropic medium We first consider the propagation of EMW in an isotropic antiferromagnetic medium therefore try to sole the set of coupled equations (6) (9) in one dimension in the isotropic limit by setting A =. We exp the magnetization the magnetic field about an undisturbed uniform state specified by M H as M = M + εm + ε M..., () H = H + εh + ε H... () Without loss of generality we assume that the one dimensional plane waes propagate along the x-direction therefore assume that H M can be treated as functions of (x t) alone where is the speed of the wae. We also introduce slow ariables to separate the system into rapidly arying slowly arying parts by stretching the time the newly introduced wae ariable as τ = ε 3 t ξ = ε(x t), where ε is a small parameter. Also we make the lattice spacing to be ery small by rescaling λ = ελ λ x = λ y = λ z = λ. We now substitute Eqs.(III A) use the newly introduced slow ariables in the one dimensional (say x) component equations of (6) (9) collect the coefficients of different powers of ε, sole the resultant equations. On soling the equations at O(ε ), we obtain H x = M x =, H y = km y, Hz = km z, where k = c. Similarly on soling the equations at O(ε ), after using the solutions at O(ε ) we obtain H x = M x, H y = km y, Hz = km z, M y M z = σm z M x, () = σm y M x. (3) Here σ = (+k). Finally, at O(ε ), we obtain M x M y M z H x = M x, (4) Hy km y y ] = M, (5) Hz km z ] = M z (6) = Jλ M z M y M y M z + M y (Hz km z ) M z (H y km y )], (7) = σ M z M x + M z M x ], (8) = σ M y M x + M y M x ]. (9) While writing equations (III A), we hae used the results of the preious orders of perturbation. To proceed further we assume that the EMW is propagating obliquely at an angle θ with reference to the uniform fields in the medium. Hence we represent the unperturbed uniform magnetization M in terms of polar coordinates in a unit sphere. As the magnetization M is restricted to (y-z)-plane in the lowest order of perturbation (i.e.m x = ), we can choose the azimuthal angle φ as π so that M is represented by M = (, sin θ(ξ), cos θ(ξ)). () In iew of this, Eq.() can be rewritten as M x = σ which on using in Eq.(7) gies θ = β θ + cos θ sin θ ] θ. () sin θdξ cos θdξ. ()

4 component of the EMW on interacting with the magnetization of the medium, generates excitation of magnetization in the form of soliton(em spin soliton) in the medium also the plane EMW is modulated in the form of soliton (EM soliton). In the general case when J (β ), then we hae the full PMKDV equation (3) which on soling will bring out the effect of perturbation due to spin-spin exchange interaction on the soliton during eolution. This can be done using a soliton perturbation theory, 9] based on the IST theory. The results show that a small structural difference will lead to slow ariation of soliton parameters distortion of the soliton shape. The one soliton solution of Eq.(3) under perturbation can be written in the form ] f = g (τ) sech(u) W (u, τ)], (6) FIG. : Eolution of M x for η =., =. σ =. Here = σ β = Jλ. While writing Eq.() we hae rescaled τ σ k τ. After differentiating Eq.() twice using the results of the preious steps integrating with respect to ξ after many lengthy calculations, we obtain the following perturbed modified Kortewegde-Vries (PMKDV) equation ( f +3 f f f = β f + )] ξ f f dξ, (3) where f = θ. It may be noted that when J =,(then β = ) the right h side of Eq.(3) anishes thus reducing to the well known completely integrable modified Korteweg-de-Vries (MKDV) equation for which the N- soliton solutions hae been found using the Inerse Scattering Transform (IST) method 8]. For instance, the one-soliton solution of the MKDV equation is written as 8]. ] η η f = sech (ξ ητ), (4) where η is a real constant. Knowing f, θ can be straight away calculated further the components of magnetization M can be obtained from the Eqs.() (). Knowing M the magnetic field H can be ealuated hence the magnetic induction B can be obtained from the linear relation connecting H, B M. For instance, the x-component of magnetization at the O(ε) is gien by M x = ] η η σ sech (ξ ητ). (5) Fig.() shows the eolution of M x for σ =, η =. =. when J = β =. This implies that when the Zeeman energy dominates oer the exchange energy (i.e. when J of the antiferromagnetic medium that normally happens at microwae frequency the magnetic field where u = g (τ) ξ φ (τ)] the parameters g (τ) φ (τ) are found from the relations dg dτ = R du coshu, (7) dφ dτ = 4g + 4g R udu coshu. (8) Here R sts for the right h side of Eq.(3). The correction to the soliton namely W (u, τ) is determined from a cumbersome expression which has the asymptotic forms W = 3g 4 u exp ( u) R du, u, (9) coshu W = 3g 4 σ u Rdu, u,. (3) where σ = 8 gdτ. On ealuating the integrals in Eqs.(7) (8) the elocity (τ) = dφ dτ of the soliton is found to eole as (τ) = () + 4 ()] 3 βτ, (3) where () is the initial elocity of the soliton. From Eq.(3) we obsere that the soliton is getting decelerated, as an illustration we hae shown the deceleration of M x for () = β = 3 4 in Fig(3). Eqs.(III A) gie the ariation of soliton shape by the asymptotic relations W = ( β 6g )u exp ( u) as u W = βπ g exp (σ u) as u. Again, knowing f from Eq.(6), the alue of θ can be found using the relation f = θ then the magnetization of the medium consequently the magnetic induction the magnetic field component of the EMW can also be computed as done in the unperturbed case. For example the alue M x when (J ) is gien by M x = σ g (τ) sech(u) W (u, τ)], (3) Here u, g W takes the form as gien below Eq.().

5 the resultant equations at O(ε ), we obtain the relation H = M, H = km, where k (H /M ) = c. At O(ε ), after using the results at O(ε ) we finally obtain H x = M x also ( ) H H km ] =, (37) M x = Jλ M z M y M y M z ] + M y M z ] ξ dξ M z M y dξ,(38) FIG. : Deceleration of M x for () = β = 3 4 B. Anisotropic Medium Now, we study the propagation of EMW in the anisotropic antiferromagnetic medium by soling the full (A ) one dimensional ersion of the coupled equations (6) (9). The anisotropic character of the medium suggests us to make a nonuniform expansion of the magnetization M magnetic field H. Since, the easy axis of magnetization of the anisotropic medium lies parallel to the direction of propagation (x-direction), we assume that at the lowest order of expansion the magnetization of the medium the magnetic field lie parallel to the anisotropic axis turn around to the y z plane at higher orders. M x = M + εm x + ε M x +..., (33) M = ε M + εm +...], (34) H x = H + εh x + ε H x +..., (35) H = ε H + εh +...],. (36) where = y, z ɛ is the same perturbation parameter used in the isotropic limit. Also the magnetic field is exped in the same way. It may be noted that along the direction of propagation of the EMW (i.e.) along x- direction the magnetization the magnetic field hae been exped about uniform alues M H respectiely. Before carrying out the analysis, for considering the slowly arying part of the EMW we introduce the slow ariables ξ = (x t) τ = ε t, based on the nonuniform expansion which are different from the isotropic case. We now substitute the expansions of M H as gien in Eqs.(III B) in the component form of the one dimensional ersion of Eqs.(6) (9) collect the terms proportional to different powers of ε. On soling M y M z = JλM + AM M z M z M + σm x M z = JλM M y σm x M y AM + M M z dξ, (39) M y M y dξ. (4) Here we hae rescaled τ τ k. In order to identify the aboe set of equations with more stard nonlinear partial differential equations, we define ψ = (M y im z ), ψ = M x. (4) The aboe definitions suggest that M x = M y + M z, which is in accordance with the nonuniform expansion we hae made for the magnetization of the medium. After a single differentiation of Eqs.(39) (4) using the transformation X = ξ + Aτ, the resultant equations can be combined together after some simple algebra we finally obtain iψ τ + ψ XX + i ψ ψ] X =, (4) M +ija X. where σ = M X is rescaled as M +ijλ Here the suffices X τ represent partial deriaties. It may be erified that on using the definitions (4), Eq.(38) can also be rewritten in the form of Eq.(4). From Eq.(4), it can be obsered that een if the lattice parameter is made smaller by writing λ ελ, the magnetization shows the same dynamics except for a change in the coefficient of the nonlinear term. Thus, Eq.(4) represents the dynamics of magnetization in an anisotropic antiferromagnetic medium at the first order of expansion when the EMW propagates through it. Eq.(4) is the well known completely integrable deriatie nonlinear Schrödinger (DNLS) equation. The DNLS equation has been soled by Kaup Newell 3] using IST also soled by Liu etal using Hirota s bilinearisation procedure 3] N-soliton solutions were obtained. For instance the one soliton solution can be written as ψ = P sech(ζ + A )tanh(ζ + A ). (43)

6 magnetization excitations due to the interaction of the magnetic field component of the EMW is found hae the similar excitations. Though the spin dynamics of this isotropic antiferromagnetic system is found to hae soliton excitations as its isotropic ferromagnetic counterpart, the results of the EMW propagation in this medium is not similar to that in isotropic ferromagnetic medium, where the magnetic field component of the EMW the magnetization of the medium are modulated in the form of solitons, This change in the dynamics of the electromagnetic field the magnetization of the medium is attributed to the interlocking of the adjacent antiparallel spins in the two-sublattice of the antiferromagnet. Acknowledgments FIG. 3: Eolution of M x for Ω R =.8, Ω I =.3, η R =. =. V.V wishes to thank MAHE for proiding the research support. where ζ = Ω R (Ω R X + τ) + η () R, Ω = Ω R + iω I, is a complex constant where Ω R is the real part Ω I is the imaginary part of Ω respectiely, η () R is the real part of the complex constant η associated with the soliton. P = expiω I (Ω I τ + X) + A] A = η () R + A, A = ln( i Ω ). Using Eq.(43) in Eqs.(4) we 8Ω R can find the components of magnetization of the medium at O(ε ). For example the magnetization component M x is gien as M x = sech (ζ + A )tanh (ζ + A ). (44) Here ζ takes the form as gien below Eq.(43). Fig.(6) represents the eolution of M x for the alues of Ω R =.8, Ω I =.3, η R =. =.. Knowing the magnetization, the magnetic field components of the EMW can be written down straight away. IV. Conclusions In this paper, we studied the propagation of EMW in an isotropic charge-free antiferromagnetic medium by treating the system as a two-sublattice model. This problem is analyzed using an uniform perturbation analysis by stretching the space time ariables perturbing the fields. It is found that the dynamics of the magnetic field component of the EMW the magnetization excitations are goerned by perturbed modified KdV equation. This shows that as the magnetic field component of the EMW interacts with the magnetization of the antiferromagnetic medium, it is modulated in the form of soliton as it further propagates in the medium the amplitude of the soliton decreases gets damped. Further, the elocity of the soliton also decreases. It is also found that the shape of the soliton is also distorted. The ] M. Lakshmanan, Phys. Lett. A6, 53 (977). ] A. Hasegawa, Y. Kodama, Solitons in Optical Communications, Oxford Uniersity Press, Oxford, ] L. F. Mollenauer R. H. Stolen, J. P. Gordon, Phys. Lett. 45, 95 (98). 4] K. Nakamura T. Sasada, Phys. Lett. A 48, 3 (974) 5] H. J. Mikeska M. Steiner, Ad. Phys. 4, 9 (99). 6] A. M. Koseich, B. A. Iano, A. S.Koalo, Phys. Rep. 95, 7 (99). 7] M. Daniel, M. D. Kruskal, M. Lakshmanan K. Nakamura, J. Math. Phys. 33, 77 (99). 8] K. Porsezian, M. Daniel M. Lakshmanan, J. Math. Phys. 33, 87 (99). 9] M. Daniel, K. Porsezian M. Lakshmanan, J. Math. Phys. 35, 6498 (994). ] M. Daniel R. Amuda, Phys. Re. B 53, R93 (996). ] B. V. Costa, M. E. Gourea A. S. T. Pires, Phys. Re. B 47, 559 (993). ] R. Balakrishnan, A. R. Bishop, R. Doloff, Phys. Re. Lett. 64, 7 (99); Phys. Re. B 47, 38 (99). 3] M. Daniel A. R. Bishop, Phys. Lett. A 69, 6 (99) 4] M. Daniel R. Amuda, Phys. Lett. A 9, 46 (994) 5] R. Balakrishnan R. Blumenfeld, Phys. Lett. A 37, 69 (997); Phys. Letts. A 37, 73 (997). 6] R. F. Soohoo, Theory Applications of Ferrites, Prentice-Hall, London, 96. 7] A. D. M. Walker J. F. Mckenzie, Proc. R. Soc. A, 339, 7 (985). 8] R. W. Dawon, Magnetism: A Treatise on Modern Theory Materials, (Eds.) G. T. Rado H. Suhl, Academic Press, New York, ] I. Nakata, J. Phys. Soc. Jpn. 6, 79 (99). ] H. Leblond, J. Phys. A: Math. Gen. 34, 9687 (). ] M. Daniel, V. Veerakumar R. Amuda, Phys. Re. E 55, 369 (997).

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