Fluxon coupling in dual thin films

The coupling force for an isolated fluxon which extends through two thin films is calculated in the thin-film approximation. When the distance between centers of the fluxon in the two films is R, the force of attraction is $F_c=\left(\frac{{\phi }_0}{{\mu }_0}\right)\left[\frac{d_1{d}_2}{(d_1{\lambda }_2^2+d_2{\lambda }_1^2)}\right]A_{\frac{d}{{\lambda }^2}}\left(R\right)$; $A_{\frac{d}{{\lambda }^2}}\left(R\right)$ is the vector potential describing the magnetic field distribution of a fluxon in a film for which $\frac{d}{{\lambda }^2}$ has the effective value $\frac{(d_1{\lambda }_2^2+d_2{\lambda }_1^2)}{{\lambda }_1^2{\lambda }_2^2}$. The film thicknesses are d₁ and d₂, the penetration depths are λ₁ and λ₂, and φ₀ is the flux quantum. The result is obtained by using a simple superposition principle, and is valid when d₁≪λ₁, d₂≪λ₂, the fluxon core radii are much less than $\frac{2{\lambda }^2}{d}$, and the separation between the films is small compared to λ₁ or λ₂.