When a pair of coils is positioned on opposite sides of a superconducting thin film, measurement of their mutual inductance may in principle be used to infer the penetration depth &lgr; in the superconductor. We have studied how to optimize this measurement with respect to coil design, and have found that the approach that has been generally used is far from the optimum. Useful simplifications to the expression relating mutual inductance to penetration depth are derived. An analysis of the sources of uncertainty in determining &lgr; is presented. For an optimized coil set, the major source of uncertainty often is uncertainty of the thickness of the film. The sensitivity to changes in &lgr; is also studied; it is shown that this can approach 1 pm for a typical high temperature superconductor sample. Finally, it is shown that the analysis may be extended to normal metal films, with the skin depth playing a role similar to that of the penetration depth in superconductors. Measurement of a high conductivity normal metal foil can be used to check calculated calibration factors for a coil set or to determine the skin depth. ©1997 American Institute of Physics.