The paper is devoted to application of geometrical optics formalism [1] to the description of behavior of spin waves propagating in a ferromagnetic medium with non-uniform distribution of magnetic parameters. Use of this approach enables to obtain a necessary veering of propagation of spin waves (in particular, a focusing) with the help of artificial inhomogeneities of medium’s magnetic parameters of the given configuration, and also by change of value of an external magnetic field.

In the papers [2, 3] refractive index of a spin beam has been defined, and its behavior was explored on a boundary of two homogeneous ferromagnetic media with different parameters of exchange interaction and biaxial magnetic anisotropy. In the present paper, bifocal spin-wave lens is proposed to be constructed on the base of inclusion of biaxial ferromagnetic part inside a uniaxial one. We have calculated the dependencies of the refraction indexes and corresponding focal distances on the wave frequency, value of external permanent magnetic field and magnetic properties of medium. It is shown that the opportunity to change these “optical” parameters in a wide region of values by only change in the external magnetic field while keeping constant the frequency and magnetic parameters of structure.

Let's consider an unbounded ferromagnetic medium consisting of two half-infinite homogeneous parts. The first part is ferromagnet having uniaxial anisotropy which has a value of saturation magnetization M₀₁, value of parameter of exchange interaction α₁, value of uniaxial magnetic anisotropy β₁ and value of spin fixing parameter L₁. The second part is biaxial ferromagnet having values of corresponding parameters M₀₂, α₂, β_2, L₂ and rhombic magnetic anisotropy ρ₂. These parts contact along a plane xOz. The easy axis of such structure is directed along axis Oz. The hard axis of biaxial ferromagnet and external magnetic field is directed along axis Oy. Also, plain z = 0 separates the given structure from vacuum.

The energy density of such magnetic structure in exchange mode looks like [4]:

Θ(x) is the step function; M_j = M_0jm_j, m_j are unit vectors in the direction of magnetization, j=1,2; A is the parameter that characterizes exchange interaction between half-spaces at y=0. Note that the case A=0 is equivalent to the absence of a coupling between layers through an interface, and A→∞ corresponds to an ideal (in a coupling sense) boundary [3].

Using methods similar to those that were used in papers [2, 5] one can find expression for spin wave spectrum in the uniaxial medium:

Analogically, expression for spin wave spectrum in the second medium is given by

Following from Eq. (4) and Eq. (5),

If a spin wave wavelength λ satisfies the condition of geometrical optics λ << a, where a is the characteristic size of an inhomogeneity, then an analogy of classic Hamilton-Jacoby equation can be used [2, 5]:

Critical angle of complete reflection is given by

The complex reflection amplitude in interface is given by [2]

Estimate material’s parameters when a lens is thin and incident angle is small. Obviously, we have to provide a necessary lens transparency. An intensity of reflected wave is defined by the square of reflection amplitude module and, according to Eq. (9), is given by |R|² ≈ [(α₁ - α₂γ²n)/(α₁ + α₂-γ²n)]² (for small incident angles and A→∞). Demanding a conformity to the condition |R|² < η, where η is a necessary smallness of reflection coefficient, we obtain a limitation on n and, therefore, on α, β, ρ, ω, L, M₀ and H₀:

In particular, at α₁ = α₂, M₀₁ = M_02, L₁ = L₂, reflection coefficient is less than 10%, if 0.52<n<.92.

To satisfy the condition of geometrical optics λ << a, a thickness of lens or mirror is restricted by

As it is seen from (9), parameters for lens can be easily provided for wide spectrum of magnetic materials [6]. In particular, the condition (11) for thin lens gives permissible values a > 10^-4 ÷ 10^-6 cm.

Focal length of thin lens is given by:

As it can be seen from Eq. (7) it is possible to observe effect of birefringence of surface spin wave. Effect of birefringence can also be observed on boundary of two biaxial ferromagnets in magnetic field directed along the easy axis (as it is shown in paper [2]) but only if the value of α₂L₂² is relatively large. In this case, effect of birefringence can be observed even for (relatively) small values of α₂L₂², but only if the values of magnetic anisotropies are large enough.

The Fig. (1) shows the dependencies of refraction indexes of both branches of spin wave n⁺ and n^- on value of a spin wave frequency. We see different frequency dependencies for n⁺ and n^-, so we can obtain a wide diapason of their relative values. It is worth noting that “negative” branch of spin wave exists only for particular range of values of spin wave frequency. As for characteristics of media that are used for Fig. (1), point ω = 6.13×10¹¹ s^-1 represents beginning of forbidden zone for “negative” branch.

Fig. (1) Dependencies of refraction indexes of both branches of spin wave n+ and n- on value of a spin wave frequency ω. α1 = 5.44×10-11 cm2, α2 = 6×10-11 cm2, β1 = 10, β2 = 15, ρ2 = 2, L1 = 25×105 cm-1, L2 = 30×105 cm-1, M01 = 105 G, M02 = 95 G, H0 = 1500 Oe.

The Fig. (2) shows the dependencies of refraction indexes of both branches of spin wave n⁺ and n^- on value of external homogeneous magnetic field. We see different frequency dependencies for n⁺ and n^-, so we can obtain a wide diapason of this relative values.

Fig. (2) Dependencies of refraction indexes of both branches of spin wave n+ and n- on value of external homogeneous magnetic field H0. α1 = 5.44×10-11 cm2, α2 = 6×10-11 cm2, β1 = 10, β2 = 15, ρ2 = 2, L1 = 25×105 cm-1, L2 = 30×105 cm-1, M01 = 105 G, M02 = 95 G, ω = 2×1011 s-1.

The Fig. (3) shows dependencies of focal distances f ⁺ and f ^– on a spin wave frequency.

Fig. (3) Dependencies of focal distances f + and f – on value of a spin wave frequency ω. α1 = 5.44×10-11 cm2, α2 = 6×10-11 cm2, β1 = 10, β2 = 15, ρ2 = 2, L1 = 25×105 cm-1, L2 = 30×105 cm-1, M01 = 105 G, M02 = 95 G, H0 = 1500 Oe.

The Fig. (4) shows the dependencies of focal distances of both branches of spin wave on value of external homogeneous magnetic field.

Fig. (4) Dependencies of focal distances f + and f – on value of external homogeneous magnetic field H0. α1 = 5.44×10-11 cm2, α2 = 6×10-11 cm2, β1 = 10, β2 = 15, ρ2 = 2, L1 = 25×105 cm-1, L2 = 30×105 cm-1, M01 = 105 G, m02 = 95 G, ω = 2×1011 s-1.

The Fig. (5) shows the dependencies of refraction indexes of both branches of spin wave n⁺ and n^- on value of a parameter of exchange interaction (.

Fig. (5) Dependencies of refraction indexes of both branches of spin wave n+ and n- on value of parameter of exchange interaction α 1. α2 = 6×10-11 cm2, β1 = 10, β2 = 15, ρ2 = 2, L1 = 25×105 cm-1, L2 = 30×105 cm-1, M01 = 105 G, M02 = 95 G, ω = 2×1011 s-1, H0 = 1500 Oe.

The Fig. (6) shows dependencies of focal distances f ⁺ and f ^– on value of a parameter of exchange interaction (.

Fig. (6) Dependencies of focal distances f + and f – on value of parameter of exchange interaction α1. α2 = 6×10-11 cm2, β1 = 10, β2 = 15, ρ2 = 2, L1 = 25×105 cm-1, L2 = 30×105 cm-1, M01 = 105 G, M02 = 95 G, ω = 2×1011 s-1, H0 = 1500 Oe.

Thus, we can see several ways to change the “optical” parameters of spin-wave lens, in particular, the opportunity to change them in a wide range of values by only changing the external magnetic field while keeping constant the frequency and magnetic parameters of structure. This fact allows one to use the results of this research in applications of spin-wave electronics.