ISSN 1062-8738, Bulletin of the Russian Academy of Sciences: Physics, 2007, Vol. 71, No. 11, pp. 1604–1606. © Allerton Press, Inc., 2007. Original Russian Text © A.V. Vashkovskii, E.G. Lokk, 2007, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2007, Vol. 71, No. 11, pp. 1645–1647.
Properties of Noncollinear Magnetostatic Waves in Magnetic Films A. V. Vashkovskii and E. G. Lokk Institute of Radio Engineering and Electronics (Fryazino Branch), Russian Academy of Sciences, sq. Vvedenskogo 1, Fryazino, Moscow oblast, 141190 Russia e-mail:
[email protected] Abstract—Some effects and phenomena that can be implemented in ferrite films using dipole spin or magnetostatic waves are described. DOI: 10.3103/S106287380711041X
Let us consider a ferrite film or plate located in free space. We will introduce the Cartesian coordinate system in such a way that the x axis is perpendicular to the film plane, the z axis is directed along the vector of the tangent uniform magnetic field H 0 , and the y axis is perpendicular to the vector H 0 . Since the vector H 0 forms a preferential direction in the film, the analogy with a uniaxial crystal is useful in the investigation of propagation and properties of magnetostatic waves. Before starting the consideration, we will refine some definitions for an anisotropic magnetic film. The
does not lead to a change in the distribution of the magnetic potential Ψ(x) at a symmetric point). The z axis has antisymmetric properties with respect to a surface magnetostatic wave (since mapping of each point of the constant-frequency curve for a surface magnetostatic wave with respect to the kz axis leads to a change in the distribution of the magnetic potential Ψ(x) at the symmetric point: the surface magnetostatic wave corresponding to the symmetric point is localized at the opposite film surface). For a backward bulk magnetostatic wave, the z axis is optical (Fig. 1), and the sym-
direction in which the wave vector k and the vector of group velocity V are collinear for any wave polarization will be referred to as the optical axis. This definition is based on the symmetry of the properties of the medium with respect to the wave. We will also generalize the definition of the forward and backward waves to the case where the vectors V and k are noncollinear. In an isotropic medium, a wave whose vectors V and k are codirectional is referred to as the forward wave and a wave whose vectors V and k are oppositely directed is referred to as the backward wave. Generally, the waves for which the scalar product (Vk ) > 0 and (Vk ) < 0 should be considered as forward and backward waves, respectively. These formulations are applicable to determination of the wave character in any medium. Many properties of magnetostatic waves can be revealed using the constant-energy or constant-frequency curves. For example, for a surface magnetostatic wave, the y axis is both the optical axis (upon propagation along this axis, the vectors V and k are collinear) and the symmetry axis (Fig. 1) (since mapping of each point of the constant-frequency curve for a surface magnetostatic wave with respect to the ky axis 1604
kz, cm–1 1500 1000 3
V
500
k
2
1
0 k
–500
υ
4 –1000 –1500 –1500 –1000 –500
0
500
1000 1500 ky, cm–1
Fig. 1. Constant-frequency curves for magnetostatic waves in a ferrite film with the thickness s = 10 µm and the saturation magnetization 4πM0 = 1750 G in the external field H0 = 300 Oe: (1, 2) a surface magnetostatic wave with the frequency f = 2800 MHz and (3, 4) a backward bulk magnetostatic wave with the frequency f = 2120 MHz. The orientations for two arbitrary pairs of vectors V and k are shown.
PROPERTIES OF NONCOLLINEAR MAGNETOSTATIC WAVES IN MAGNETIC FILMS
metry properties of the z and y axes with respect to this wave are the same as with respect to the surface magnetostatic wave (it was found that noncollinear backward bulk magnetostatic waves are also (as surface magnetostatic waves) nonreciprocal [1]). Thus, a tangentially magnetized ferrite film has an antisymmetric optical axis with respect to a backward bulk magnetostatic wave (z axis) and a symmetric optical axis with respect to a surface magnetostatic wave (y axis). In the case of a noncollinear magnetostatic wave, it is important to know the orientations of the group velocity vector (the angle ψ between the vector V and the y axis) and the wave vector (the angle ϕ between the vector k and the y axis). For a surface magnetostatic wave, the dependence ψ(ϕ) is monotonic and one-toone. For a backward bulk magnetostatic wave, this dependence is nonmonotonic and may have two extrema. Therefore, upon propagation of a backward bulk magnetostatic wave, a situation may arise where two rays with different directions and magnitudes of the vector k propagate in the same direction (Fig. 2; for example, the wave propagating in the direction ψ = −60° may have ϕ = 54° or 85°). Let us now consider the characteristic features of the reflection of magnetostatic waves. Let a magnetostatic wave be incident on a plane mirror (straight-line edge of a ferrite film). Depending on the angle between the optical axis of the medium and the normal to the plane mirror, we can select some characteristic types of reflection differing in the mutual position of the incident and reflected waves. When the optical axis coincides with the normal to the plane mirror, the properties of the medium on the left and on the right from the normal are symmetric and reflection occurs according to Euclid’s law: the angle of incidence is equal to the reflection angle. When the optical axis deviates from the normal, the wave properties of the medium are asymmetric with respect to the normal. As a result, the reflection angle differs from the angle of incidence [2]. With further deviation of the optical axis, reverse (or negative) reflection arises; the most unusual situation occurs when, in the case of oblique incidence on the mirror, the reflected ray emerges in the direction opposite to the incident ray. At a strong deviation of the optical axis from the normal, the rays of the forward noncollinear wave, incident on the mirror in a wide sector (40°–45°), are reflected into a narrow sector (2°–3°); i.e., the plane mirror collects the reflected rays into a narrow beam. With a further deviation of the optical axis, a situation arises where the normal to the mirror plane coincides with the asymptote to the constant-frequency curve for a surface magnetostatic wave; in this case, any ray incident on the plane mirror under any angle does not give a reflection. The reason for this effect is that the second branch of the constant-frequency curve lies on the other side of the asymptote and
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ψ, deg 90 60 30 1
0 –30 –60
2
–90 –120 –150 –180 –90 –60 –30 0
30 60 90 120 150 180 ϕ, deg
Fig. 2. Dependences of the group velocity ψ on the orientation of the wave vector ϕ in a ferrite film with s = 10 µm, 4πM0 = 1750 G, and H0 = 300 Oe: (1) a surface magnetostatic wave with f = 2800 MHz (corresponds to curve 1 in Fig. 1) and (2) a backward bulk magnetostatic wave with f = 2120 MHz (corresponds to curve 3 in Fig. 1).
there are no reflected rays satisfying the boundary conditions on the mirror surface. Note that when a backward bulk magnetostatic wave is reflected from the plane mirror, two reflected rays may arise [1]. All the effects described above have been experimentally observed. In investigation of the wave reflection, one should also bear in mind that, in contrast to isotropic media, where the method of changing the angle of incidence of a wave on the plane mirror (by rotating the mirror or the excitation antenna) is unimportant, in the case of an anisotropic ferrite film, these two methods will lead to absolutely different results, because the orientations of the mirror and antenna will also change with respect to the optical axis. Obviously, upon rotation of the mirror, the parameters of the incident wave (wavelength λi , wave vector k i , group velocity V i , and the corresponding angles ϕi and ψi) remain constant, whereas upon rotation of the antenna the parameters of the incident wave will be different for each new value of the incidence angle (because the orientations of the vectors k i and V i will change with respect to the optical axis). Analysis of the ray reflection from an arbitrarily oriented plane mirror makes it possible to understand how a plane wave will be reflected from a round inhomogeneity (in an experiment, it may be a hole in a film). If the edges of an inhomogeneity are smooth and its size greatly exceeds the wavelength, the reflection from it will be similar to that from a mirror. When a collinear
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wave is incident on a round mirror, the near arc of the circle gives a backward reflection, converging into a kind of a node on the y axis, while the lateral part of the circle gives a reflection in the direction of the incident wave [3]. It turns out that some part of the backward reflected rays can be focused into a point. To this end, the boundary should be shaped not as an arc of a circle but in a more complicated way. The calculation and experiment demonstrated that the focal spot diameter can be several tenths of a millimeter. A stripe exciter of a specified shape can also focus rays. It is noteworthy that focusing of magnetostatic wave rays is performed by a convex rather than a concave surface, as in electrodynamics of isotropic media.
ACKNOWLEDGMENTS This study was supported in part by the Russian Foundation for Basic Research, project no. 07-0200233, and the Program “Development of the Scientific Potential of Higher School,” project no. 2.1.1.4639. REFERENCES 1. Vashkovskii, A.V. and Lokk, E.G., Usp. Fiz. Nauk, 2006, vol. 176, no. 4, p. 403. 2. Vashkovskii, A.V. and Zubkov, V.I., Radiotekh. Elektron. (Moscow), 2003, vol. 48, no. 2, p. 149. 3. Vashkovskii, A.V. and Zubkov, V.I., Radiotekh. Elektron. (Moscow), 2005, vol. 50, no. 6, p. 670.
BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES: PHYSICS
Vol. 71
No. 11
2007
