^{1}, D. V. Kadygrob

^{1}, Z. A. Mayselis

^{1}, T. M. Slipchenko

^{1}, S. E. Savel’ev

^{2}and V. A. Yampol’skii

^{3,a)}

### Abstract

The nonlinear response of a layered superconductor to symmetric (in magnetic field)electromagnetic excitation has been theoretically investigated. An ambiguous dependence of the phase of the reflected signal on the amplitude of the irradiating wave is predicted. This causes hysteresis jumps in the dependence of the surface reactance of the superconductor on . If the frequency of the irradiating field is close to the Josephson plasma frequency, this unusual nonlinear effect can be observed when the amplitudes of the ac field are not very strong. The conditions for the appearance of hysteresis are obtained. Expressions for the phase shift of the reflected wave are derived, using the coupled sine-Gordon equations. Moreover, a class of solutions of these equations that are discontinuous in the coordinate are studied that correspond to a continuous spatial distribution of the magnetic field in the superconductor. Such solutions result in the appearance of additional branches in the dependence of the phase shift of the reflected wave on the incident wave amplitude.

I. INTRODUCTION

II. FORMULATION OF THE PROBLEM

A. Geometry of the problem

B. The equations for the electromagnetic field in vacuum and in a layered superconductor

III. THE PHASE SHIFT OF THE REFLECTED WAVE

A. Thin samples,

B. Thick samples,

IV. CONCLUSION

### Key Topics

- Superconductors
- 23.0
- Plasma waves
- 18.0
- Magnetic fields
- 12.0
- Superconductivity
- 8.0
- Electric fields
- 6.0

## Figures

Geometry of the problem. A layered superconducting plate illuminated by -polarized electromagnetic waves with the magnetic field symmetrical relative to the center of the sample.

Geometry of the problem. A layered superconducting plate illuminated by -polarized electromagnetic waves with the magnetic field symmetrical relative to the center of the sample.

Phase portrait . The value of is the amplitude of the wave solution of Eq. (5) for phase , and is its derivative with respect to the dimensionless coordinate . The phase portrait shows only those trajectories that correspond to a magnetic field distribution symmetric relative to the middle of the sample. Motion along the heavy solid curves corresponds in phase trajectories with the variation of coordinate inside the sample. The points denote the boundaries of the sample at , and the direction of motion is indicated by the arrows.

Phase portrait . The value of is the amplitude of the wave solution of Eq. (5) for phase , and is its derivative with respect to the dimensionless coordinate . The phase portrait shows only those trajectories that correspond to a magnetic field distribution symmetric relative to the middle of the sample. Motion along the heavy solid curves corresponds in phase trajectories with the variation of coordinate inside the sample. The points denote the boundaries of the sample at , and the direction of motion is indicated by the arrows.

Normalized total magnetic field at the boundary of the sample vs amplitude , described by the cubic parabola of Eq. (7). The arrows denote motion along the curve as the amplitude of the incident wave varies.

Normalized total magnetic field at the boundary of the sample vs amplitude , described by the cubic parabola of Eq. (7). The arrows denote motion along the curve as the amplitude of the incident wave varies.

Principal panel: phase shift of the reflected wave vs dimensionless amplitude of the incident wave (numerical simulation). The values of the parameters are as follows: , , , and . The arrows along the curves show how the phase shift varies as the amplitude periodically varies. The first jump in the dependence at is shown by a downward arrow. The value of the jump equals . The second (reverse) jump at is shown by an upward arrow. The value of the jump is . Lower inset: The dependence on an altered scale. Upper inset: A comparison of the numerical and analytic results for the dependence. The points correspond to a numerical simulation, and the solid curve to the dependence implicitly given by Eqs. (25).

Principal panel: phase shift of the reflected wave vs dimensionless amplitude of the incident wave (numerical simulation). The values of the parameters are as follows: , , , and . The arrows along the curves show how the phase shift varies as the amplitude periodically varies. The first jump in the dependence at is shown by a downward arrow. The value of the jump equals . The second (reverse) jump at is shown by an upward arrow. The value of the jump is . Lower inset: The dependence on an altered scale. Upper inset: A comparison of the numerical and analytic results for the dependence. The points correspond to a numerical simulation, and the solid curve to the dependence implicitly given by Eqs. (25).

Phase shift of the reflected wave vs dimensionless amplitude of the incident wave, taking into account the discontinuous solutions of Eq. (11) (numerical simulation). The values of the parameters are as follows: , , , and . The arrows along the curves show how the phase shift changes as the amplitude varies periodically. The jumps in the dependence as increases at points *2* and *5* are shown by downward arrows. The jumps as decreases at points *1* and *4* (reverse jumps) are shown by upward arrows. Inset: phase shift of the reflected wave vs dimensionless amplitude of the incident wave, obtained using only the continuous solutions of Eq. (11) (numerical simulation). The shaded region corresponds to amplitudes for which none of the three constructed branches is determined.

Phase shift of the reflected wave vs dimensionless amplitude of the incident wave, taking into account the discontinuous solutions of Eq. (11) (numerical simulation). The values of the parameters are as follows: , , , and . The arrows along the curves show how the phase shift changes as the amplitude varies periodically. The jumps in the dependence as increases at points *2* and *5* are shown by downward arrows. The jumps as decreases at points *1* and *4* (reverse jumps) are shown by upward arrows. Inset: phase shift of the reflected wave vs dimensionless amplitude of the incident wave, obtained using only the continuous solutions of Eq. (11) (numerical simulation). The shaded region corresponds to amplitudes for which none of the three constructed branches is determined.

Phase portrait . Motion along the thick solid curves corresponds in phase trajectory with the variation of coordinate inside the sample. The points indicate the boundaries of the sample at , and the exes denote the points of discontinuity. The direction of motion is indicated by the solid arrows, and the points of discontinuity are connected by arrow-tipped dashed curves. The values of the parameters for the given trajectory with discontinuities are as follows: ,

Phase portrait . Motion along the thick solid curves corresponds in phase trajectory with the variation of coordinate inside the sample. The points indicate the boundaries of the sample at , and the exes denote the points of discontinuity. The direction of motion is indicated by the solid arrows, and the points of discontinuity are connected by arrow-tipped dashed curves. The values of the parameters for the given trajectory with discontinuities are as follows: ,

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