Single-Oscillation Two-Dimensional Solitons of Magnetic Polaritons

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Single-Oscillation Two-Dimensional Solitons of Magnetic Polaritons

H. Leblond¹and M. Manna²

1Laboratoire POMA, CNRS-UMR 6136, Universite´ d’Angers, 2 Bd Lavoisier 49045 Angers Cedex 1, France

2Physique Mathe´matique et The´orique, CNRS-UMR 5825, Universite´ Montpellier II, 34095 Montpellier, France (Received 18 December 2006; published 8 August 2007)

The propagation of bulk polaritons in ferromagnetic slab is considered through a short-wavelength approximation. Neither the damping nor the demagnetizing field do affect essentially the propagation and stability of the line soliton. The stable line soliton may be destroyed by background instability: the latter is suppressed in a narrow strip. The unstable line soliton decays into lumps, which can be described both numerically and through a variational approach. Lump interactions are mentioned.

DOI:10.1103/PhysRevLett.99.064102 PACS numbers: 05.45.Yv, 41.20.Jb, 75.30.Ds

Waves in ferromagnetic media are highly nonlinear, as well in the exchange, magnetostatic, electromagnetic (EM), or optical domains [1,2]. A wide range of soliton- type propagation phenomena has been predicted theoretically [3–6], and some of these predictions have been confirmed experimentally [7], leading, e.g., to the recent observation of envelope magnetostatic spin wave two- dimensional (2D) solitons [8]. Analogous behaviors are predicted in the so-called EM or polariton regime, but the experiments are much more difficult in this regime due to the wave velocity close to the speed of light. The linear theory of the polariton modes has been developed [2]

(chap. 4), [9,10], and validated experimentally [11,12], it involves many complications due to finite-size effects. On the other hand, the study of solitary waves, or single- oscillation waves, which is fundamental for soliton theory from both the water wave and the mathematical viewpoints [13], is now a fast-growing field in nonlinear optics.

Our aim is to describe the nonlinear propagation of solitary EM waves in a ferromagnetic slab. These waves belong to the bulk polariton modes. It must be noticed that, owing to the great complexity of the nonlinear study, it is necessary to simplify considerably the linear description of the wave and material.

We consider a ferromagnetic slab lying in thexyplane,x being the propagation direction. The slab is magnetized to saturation by an in-plane external fieldH¹₀ directed along y. The evolution of the magnetic fieldHis governed by the Maxwell equations, which reduce to

rrH Hc²@²_tHM; (1)

wherec1=

₀"~

p is the speed of light with"~the scalar permittivity of the medium. The magnetization densityM obeys the Landau-Lifschitz equation, which reads as

@_tM ₀M^H_eff

M_sM^ M^H_eff; (2) where is the gyromagnetic ratio, ₀ the magnetic per- meability of the vacuum, <0the damping constant, and M_s the saturation magnetization. The effective magnetic

field isH_eff HNM, whereN is the demagnetizing factor tensor. In the configuration considered,Nis diagonal with N_x; N_y; N_z 0;0;1. In contrast with the case of magnetostatic spin waves, the bulk polariton wavelengthes are extremely large with regard to the exchange length, and hence inhomogeneous exchange can be neglected. We also assume that the crystalline and surface anisotropy of the sample can be neglected. The quantities M,H, and tare rescaled into₀M=c,₀H=c, and ct, so that the con- stants₀=candcin Eqs. (1) and (2) are replaced with 1, M_s by m₀M_s=c and by ~=₀, which is dimensionless.

To derive the nonlinear propagation equation we follow strictly the short wave approximation scheme, as in [14].

We introduce the variables

"¹xt; yy; "t: (3) The variable allows us to describe the shape of the wave propagating with speed c, it assumes a short wavelength about1=". The slow timeaccounts for the propagation on distances very large with regard to the wavelength. The transverse variable y has an intermediate scale, as in Kadomtsev-Petviashvili (KP) type expansions [15]. The fieldMis expanded as

M M₀"M₁"²M₂. . .;

where theM_jare functions of (,y,), and analogously for H. This approach neglecting the continuity and pinning conditions at the film boundary is valid in a thick enough slab, see the discussion of the effect of the film thickness on EM spin wave spectrum in Ref. [16].

We further assume a weak damping. In yttrium iron garnet (YIG) films, envelope solitons have been observed [7]. The observations can be accounted for using a nonlinear Schro¨dinger-type model including the damping term. It can be derived for the Landau-Lifschitz and Maxwell Eqs. (1) and (2), assuming that the dimensionless damping constant ~ is small, of order "², where " is a perturbative parameter measuring the amplitude of the magnetic pulse [17]. In YIG films, ~ can be as small as PRL99,064102 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending

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10⁴[18], which would correspond to the assumption~

"², with a perturbative parameter "’0:01. In a first stage, we consider the less restrictive assumption ~

", which allows a stronger damping.

Running the perturbation procedure, we obtain the asymptotic system

C_XT BB_XC_YYB_Y; (4) B_XT BC_XB_YYC_YB _X=2; (5) where the subscripts denote partial derivatives (i.e. C_Y

@_YC, and so on). We have set C_X 1H₀^y=m, B M^x₁=2m, X m=2, Ymy, Tm. Only Eq. (5) contains a damping term. Apart from this term, system (4) and (5) exactly coincides with the system derived in [14], in which the demagnetizing field was non considered.

Hence, within the considered approximation, and for the geometry considered, the demagnetizing field has no effect on the wave propagation.

The numerical scheme used for solving the model with- out damping [14] is straightforwardly generalized to take it into account. Figure1shows the effect of damping on a 1D pulse. Since it is far from the soliton, the pulse undergoes strong oscillations, and its velocity isc.

In the case of a YIG slab, we can assume a smaller damping~"², instead of ~"as above. Then is merely replaced by zero in Eqs. (4) and (5) and the damping completely disappears from the system, which proves that it is negligible within the considered approximation, in contrast with the case of envelope solitons [17]. The results obtained in a bulk medium with zero damping apply: the line soliton is stable ifv < ²=81’0:24[14].vis the soliton parameter, and coincides with both the background level and the soliton velocity, so thatv >1corresponds to a magnetization having the same direction as the applied field.

The stable line soliton involvesv <1, which yields an instability of the transverse modulation of the background, confirmed by numerical resolution of system (4) and (5).

We will see that it can be inhibited if the propagation occurs in a narrow stripe. The background is defined by a

uniform value of the components

H₀^y m1C_X; H₀^z mB_X (6) of the magnetic field. It corresponds thus to solutions of the form CXf, BXg, where f andg are constants. In order to analyze their modulation instability, we introduce a scaled time XT, and assume that CXfY;

XT,BXgY; XT. Using a small time approxima- tion (the terms proportional toXTare neglected), we get

2f g²g_Yf_YY; (7) 2gfgg_YYf_Yg=2X:^ (8) Seeking for solutions of the form

f ve^iqY. . .; (9) ge^iqY. . .; (10) we get the relation

2q²2q²v ~ q²; (11) where v~v=2X. It allows to retrieve the stability condition v >~ 1. Whenv <~ 1, the instability is due to the small wave numbers q, i.e., long wavelengths. If some filtering of the long wavelengths is introduced, the stability can be recovered. Since the allowed wave numbers are at most 2=l, the background can be stabilized by limiting the widthlof the sample, according tol <2=

1v~ p . A numerical example and confirmation of this property is given in Fig. 2. The initial data is a line soliton [14]

Bx; y;0 2v=coshxx₁, Cx; y;0 v2 tanhx x₁ xx₁, where x₁ is some Gaussian perturbation, withv 0:4. The maximal width which is expected to eliminate the transverse instability of the background is

l2=

p1:4

’5:31, according to the above formula. The computation has been performed using l5:30, and pe- riodic boundary conditions in the transverse direction. The instability is suppressed, which confirms numerically the above analysis. In physical units, the stabilization threshold widthlis

0 5

10 15

20 15 25

16 17

18 19

−1

−0.5 0

H 0.5

z

x

t

FIG. 1 (color online). The time evolution of a quasi-1D pulse, with a strong damping. Initial data: BC tanhxx₀. Normalized damping constant: 0:4.

FIG. 2 (color online). An initially perturbed line soliton after t8stable propagation. The background is stabilized using the narrowing of the stripe where propagation occurs. The line soliton velocity isv 0:4, and damping is neglected ( 0).

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l 2c

₀

H₀^yM_s

q : (12)

Considering typical values as H₀2 kOe, M_s or4M_s 1800 Oe,₀ 1:759 10⁷ rads¹Oe¹, c8:66 10⁷ ms¹, we getl’1:6 cm.

This result presents some analogy with the transverse stabilization of a spin wave envelope solitons in a narrow YIG film [19]. The authors observed stable propagation of a line soliton in a narrow stripe of width 1.5 mm, while the transverse self-focusing occurred in a wide film of 18 mm width, in accordance with our result. However, the width threshold depends on the exact transverse modes and pinning conditions at the edges of the stripe [20]. Since our analysis considers pure Fourier modes, it can only give an order of magnitude for it.

The velocity in the laboratory frame is V c1"²v, hence the stable line soliton of Fig. 2 is

‘‘supraluminous’’, in the sense that it travels faster than light in the medium (but slower than light in vacuum).

Numerical resolution of the system (4) and (5), starting from an unstable line soliton transversely perturbed as an initial data, shows the breaking of the line soliton into several parts, which continue to evolve as stable two- dimensionally localized entities. A characteristic example is given on Fig.3. The stability of the fragments depend on the parametervof the line soliton. They begin to stabilize forv1:1and become stable abovev1:5. The shape of the formed structure is shown on Fig.4; it is comparable to that of a KP lump [21]. However, a characteristic of the lump is that it decays algebraically in all directions, while a 2D soliton is expected to be exponentially localized. The present structures decay algebraically in the x direction, and exponentially in theyone, precisely

B^x!1 x³⁼²; B^y!1 e^y³⁼² (13) as can be determined from the numerical data.

The breaking up of a line soliton into lumps has already been described in the case of the KP I equation [22]. We observe here the same phenomenon, except that (i) the lumps have a transverse velocity component, and (ii) the various lumps have different parameters. Indeed, in the case of KP I, all emitted lumps are identical, and travel

along the x axis. The asymmetry of the present model, which is induced by the application of an external field to the ferromagnetic slab, is evidently at the origin of this feature.

System (4) and (5) derives from the Lagrangian density L C_XC_TB_XB_TC²_YB²_Y2CB_YC_XB²=2;

(14) through L= C0, L= B0. We seek for traveling solutions of (4) and (5), including a background fieldH~₀ 0; am;0. Therefore B and C must have the form B BXvT; YwT, C aXC⁰XvT; YwT, which yields the effective Lagrangian density (dropping the primes)

Leff ¹₂C_XC_TvC_XwC_Y B_XB_TvB_XwB_Y C_Y² B_Y²2CB_Y C_XB²aB²: (15) We make use of the variational approximation method with the ansatz

BpexpX²=f²Y²=g²; (16) C XYexpX²=f²Y²=g²: (17) The Gaussian shape in (17) does not exactly match the numerical results, it is used for reasons of tractability.

Treating p,, ,f², g² as the dynamical variables, and a, v, and w as parameters, we obtain a three parameter

−10 0 10

0 20 40 60 80

−10 0 10

0 20 40 60 80

−10 0 10

0 20 40 60 80

−10 0 10

0 20 40 60 80

−10 0 10

0 20 40 60 80

−10 0 10

0 20 40 60 80

y

0

t = 4.2 5.1 7.2 15.9 24.3

y x

H z

FIG. 3 (color online). Emission of three localized pulses from an unstable line soliton withv2(damping is neglected).

−2 0

2 0

5 10

−4

−2 0 2

−2 0

2 0

5 10

−4

−2 0 2

H y H z

x

y y

x

FIG. 4 (color online). The lump emitted by the line soliton.

−5 05 10 0

30 20 40

−6

−4

−2 0 2 4

−50 5 10 0

30 20 40

−6

−4

−2 0 2 4

−50 5 0 20 10 40 30

−5 0 5

−505 10 0

30 20 40

−5 0 5

t = 1.8 t = 0

H

H y

z x

y

x

y

x

y x

y H y

H z

FIG. 5 (color online). Example of evolution from initial data given by the variational approximation.

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family of stationary lumps, explicitly in terms of q g²=f²,v, ands=.

The stability of the lumps can be discussed numerically as follows: The evolution of the pulse is computed, taking the variational solution as initial data. Then the stable lumps are expected to adjust to their exact value, whereas the unstable ones would decay to some different state, eventually some stable lump. We observe that most of the energy is always propagated as a lump, even if its speed mays differ from the input. For an adequate choice of the parameters, the agreement can be very good, corresponding to a stable lump; see Fig. 5. All parameters of the variational solution can be fitted from numerical data.

Setting s, v, and a to the values obtained, we get as a typical example p; ; f²; g²; w 17:05;7:81;8:67;

1:57;1:87 from the fitting, and (19.75, 8:82, 8.58, 1.45,3:64) from the variational formulas. The agreement is good (notice that the parameters of the variational solution vary very quickly withs,v, andq).

The space variables are normalized with respect to 1=mc=₀M_s, while the field amplitudes are normalized with respect to the saturation magnetization M_s. In physical units, the width and length of the lump express as y_lgc=₀M_s and x_l2"fc=₀M_s, while its transverse velocity isW "cw. Assuming values typical for YIG, as ₀ 1:759 10⁷ rads¹Oe¹, M_s 1800 Oe,"_r12, and taking"10², the lump charac- teristics corresponding to our simulations arex_l’0:14to 0.21 mm, y_l’3:3 to 3.7 mm, V ’8:66 10⁷1–2 10⁴v 8:633 10⁷ to 8:646 10⁷ ms¹, and W ’ 4:1 10⁶ to1:6 10⁶ ms¹.

Some observations about lump interactions can be drawn from the simulations. Two lumps can merge together, or one be absorbed by another. After that, three situations have been observed: (i) the absorbed lump is reemitted. In this case the reemitted lump is shifted for- wards for an appreciable distance (Fig.6). (ii) the absorbed lump is not reemitted. It is properly speaking a merging of the two lumps. (iii) The two lumps annihilate together, and their energy is dispersed and diffracted.

In conclusion, we have described theoretically 1D and 2D stable solitary waves belonging to the bulk polariton

propagation mode. The prolongments of this study are twofold: on one hand, some experimental validation is required, and the theory should be improved to take into account several important properties of the magnetic material and wave which have been neglected here. On the other hand, system (4) and (5) is a 2D generalization of the sine-Gordon equation, which can be studied from a mathematical point of view. Especially, the question of its inte- grability and universality raises.

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2 4 6 8 101214 15

20 25 30 35 40

2 4 6 8 1012 20 25 30 35 40 45

0 5 10

25 30 35 40 45 50

0 5 10

35 40 45 50 55

−2 0 2 4 6 8 40 45 50 55 60 65

t = 0 2.4 4.8 8.4 10.8

H

x

y z

FIG. 6 (color online). Interaction of two lumps: one lump is absorbed by the other, and then reemitted (backwards in the lab frame). Notice that the picture frame moves with the lumps.

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