Electromagnetic waves in a Josephson junction in a thin film

We consider a one-dimensional Josephson junction in a superconducting film with a thickness that is much less than the London penetration depth. We treat an electromagnetic wave propagating along this tunnel contact. We show that the electrodynamics of a Josephson junction in a thin film is nonlocal if the wavelength is less than the Pearl penetration depth. We find the integrodifferential equation determining the phase difference between the two superconductors forming the tunnel contact. We use this equation to calculate the dispersion relation for an electromagnetic wave propagating along the Josephson junction. We find that the frequency of this wave is proportional to the square root of the wave vector if the wavelength is less than the Pearl penetration depth.