A method for solving Riemann–Hilbert boundary value problems in nonreciprocal wave propagation

Abstract

A new method for solving periodic boundary value problems in nonreciprocal wave propagation is described. A characteristic function 0 and its complementary function 0 are introduced after the particular problem has been reduced to a Riemann-Hilbert boundary value problem based on complex variable theory. The Taylor coefficients n and n of 0 and 0 , respectively, compose a system of linear equations to deduce a characteristic equation. A numerical example is presented for magnetostatic surface waves propagating in a ferrite film on which metallic stripes are deposited periodically.

Diffraction of plane waves from a strip grating on a ferrite substrate

“…(12) and (13) are rewritten as where Employing the unknowns x n s, we define the function of a complex variable Z as Thus, Eqs. (18) and (19) are reduced to the nonhomogeneous RiemannHilbert problem, which is a boundary value problem of the analytic function FZ [13]. When K !…”

“…4, we next show that the scattering resonances are caused by MSSW excitation in YIG. Figure 4 exist between 1.99 and 3.17 gHz [19]. In a periodic structure, the spatial harmonics of a wave propagating along the y-axis appear with a period of 2S/l along the k y -axis.…”

“…We know that if the MSSWs have a positive (negative) group velocity dZ /dk y , their energies are confined in the vicinity of the top (bottom) surface of the YIG [20]. This confinement becomes marked as the frequency becomes high [19]. Thus, we understand that the MSSWs excited on the top surface of the YIG give rise to the dull resonances and that the MSSWs excited on the bottom surface cause the sharp resonances.…”

“…The intensity of J z is normalized to J z 0 2k x / ZP 0 2 CCCCC H 0 / P 0 , which is the value of J z when the whole surface of the YIG is metallized. The asymptotic expression for J z near the edge is deduced from the characteristic function F 0 Z [19]. If r is the distance from the edge, we have Thus, the amplitude of J z is proportional to r 1 / 2 and the argument of J z varies with ln r [23].…”

“…Furthermore, it has been applied to the mode analysis of waves propagating parallel to the strips [17] and traversing the strips [18]. Although the conventional RiemannHilbert method is applicable only to the reciprocal case, the authors have recently developed a nonreciprocal RiemannHilbert method in order to reveal the propagating modes and the field behavior near the edges in a nonreciprocal medium [19].…”