Interaction of the fluxon with the finite size dipole impurity in the long Josephson junction is investigated. The impurity has polarity and will be referred to as a dipole impurity because it also has a direction and, consequently, changes its sign under the space inversion transform x → −x. Such a model is used to describe the inductively coupled to the Josephson transmission line qubit and the misaligned Abrikosov vortex that penetrates into the long Josephson junction. We derive the approximate equations of motion for the fluxon center of mass and its velocity. With the help of these equations we demonstrate that pinning and scattering of the fluxon on the impurity differs significantly from the case of the point impurity which is modelled by the derivative of the Dirac's δ-function.

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Fluxon transmission through impurities of different shape in a quasi-onedimensional long Josephson junction is investigated. The junction width is significantly less than its length but, at the same time, is of the order of the Josephson penetration length λ J or exceeds it. The retrapping current on the impurities of the point, line and rectangular shape is computed as a function of the junction width both numerically and analytically. Good agreement between the analytic formulae and the numerical simulation results for the intermediate (several λ J ) junction width is observed.