The universal approach to calculation of the linear microwave response of thin superconducting films in perpendicular magnetic field with the trapped magnetic flux

The microwave response of thin superconducting films placed in the perpendicular magnetic field brings us the variety of unexpected phenomena in the dynamics of the mixed state of superconductors. The variety of dependences in the high-frequency response in nonlinear regime as well as in linear are measured in thin films of superconductors in a magnetic field. One of the features to be explained is the nonmonotonous dependence of the surface resistance (or quality factor or energy losses) on the static (linear response) or on the amplitude of high-frequency (nonlinear response) magnetic fields. The variety of experimental results and theoretical approaches to explain them are still controversial and not any generally accepted mechanism of such behavior still exists. The general task of analysis of the experimental data is to distinguish between real physics of the vortex matter and some artificial experimental factors like the geometry of sample measurement methods etc. Laying aside the nonlinear response of superconductors the main attention in this work is paid to the linear response of the thin films in a perpendicular magnetic field. The general method of calculation of energy losses caused by vortex motion in thin superconducting films placed in perpendicular magnetic field is proposed. As an example the responses of coplanar and strip-line resonators are calculated. The dependences calculated are compared with the available experimental data.