FIGURE CAPTIONS Fig. 1. Cross-sections of the coplanar (a) and the microstrip (b) broadband FMR trans- ducers with a sample on top. Fig. 2. Solution of the exchange-free equations. (a): Red solid line: real part of the out-of- plane wavenumber Q . Blue dashed line: its imaginary part. Green dotted line: assymptotics for Q for large wavenumbers ( Q = k ). (b): Thin red solid line: Amplitude of the microwave field h$_{xk}$ at the rear film surface y = L . Thin blue dashed line: amplitude of h$_{xk}$ at the front film surface y = 0 on which the microwave flux is incident. Green dotted line: assymptotics for both for large wavenumbers k . Thick black solid line in both panels: Dispersion of the Damon-Eshbach wave in an insulating film, coincides with graphical accuracy with the result obtained including the electric conductivity. Parameters of calculation: film thickness: 40 nm, film saturation magnetisation 4 πM$_{s}$ =10000 G, applied field is 2000 Oe, film conductivity is 4 . 6 · 10 6 Sm/m. Fig. 3. Microwave broadband FMR absorption data for single-layer Permalloy films. Microstrip transducer is 1.5 mm in width. Left-hand panels (a-d): driving frequency is 7.5 GHz. Right-hand panels (e-h): 18 GHz. Film thicknesses: (a) and (e): 30 nm; (b) and (f): 40 nm; (c) and (g): 60 nm; (d) and (h): 80 nm. Fig. 4. Cavity FMR (Panels (a-c), dashed line) and Brillouin light scattering (BLS) data (Panel (d)) in comparison with broadband FMR data (Panels (a-c), solid lines) for the single layer films with different thicknesses. (a): 40 nm; (b): 60 nm; and (c): 80nm. Driving frequency is 9.47 GHz. Broadband transducer: the same 1.5mm-wide microstrip. Cavity: Varian-4 ESR spectrometer cavity. BLS data are taken for the 60nm-thick film at an incidence angle of 5 angle degree and in an applied field of 500 Oe. ”F” in Panel (d) indicates the fundamental (”Damon-Eschbach”) BLS peak, ”SSWM1” is the first (odd- symmetry) standing spin wave mode, and ”SSWM2” is the second (even-symmetry) standing spin wave mode. Fig. 5. (a) Resonant mode frequencies as a function of applied bias field. Inset: calculated resonant fields as a function of saturation magnetization 4 πM$_{s}$ (given in Gauss) for cobalt. Driving frequency for the inset is 13.47 GHz. Solid lines in the inset: calculation; dashed lines: measured values for the bi-layer sample. (b): Relative mode intensities. Coplanar transducer with a central conductor 0.3 mm in width is used. Black triangles: single layer 74nm-thick film. Red dots: bi-layer film (Py:74nm, Co:10nm). Black dashed 23