Analytic expression for the critical-current interference pattern

Derive an analytic expression for the critical-current interference pattern produced by a Beenakker–Furusaki current-phase relation in a wide Josephson junction under an out-of-plane magnetic field.

Background

The paper analyzes how a spatially varying superconducting phase across a wide Josephson junction modifies the current-phase relation, critical current, and Josephson inductance in an out-of-plane magnetic field. For a Beenakker–Furusaki current-phase relation, the authors evaluate the critical-current interference pattern numerically and report that it shows no visible deviation from a sinc shape, independently of the channel transparency.

Although the Josephson-inductance interference pattern can be related analytically to the underlying current-phase relation, the corresponding critical-current pattern lacks an analytic formula in the treatment presented. Obtaining such a formula would replace the numerical evaluation used in the paper and provide a closed-form characterization of the transparency-independent critical-current interference pattern.