Strong coupling of quantized spin waves in ferromagnetic bilayers

We formulate a strong-coupling theory for perpendicular standing spin waves (PSSWs) in ferromagnetic bilayers with the interlayer exchange coupling (IEC). Employing the Hoffmann boundary condition and the energy-flow continuity across the interface, we show that the PSSWs are still quantized but with nonintegral quantum numbers, similar to the case with a surface pinning but in sharp contrast to that with a free surface. The magnon-magnon coupling is characterized by the spectrum splitting which is linear with the IEC in the weak-coupling region but getting saturated in the strong-coupling limit. Analytical predictions are verified by full micromagnetic simulations with good agreement.