Complex frequency technique for linear and second harmonic optical properties of metallic surfaces

We describe a complex frequency technique for evaluating the linear and quadratic dielectric responses of metal surfaces, illustrated by application to the surface of jellium. The electric susceptibilities are shorter-range functions of the spatial coordinates at complex frequency, whereas their general behaviour is complicated, long-range and highly oscillatory at real frequency. As a result the linear and the second harmonic electric charges induced by an optical perturbation are then numerically easier to calculate at complex frequency. As the functions which characterize the optical behaviour of the metal surface are analytic in the upper complex frequency half-plane, the dielectric response at real frequency can be deduced by analytic continuation from the results at complex frequency. We illustrate and discuss this approach, which should be useful for studying more realistic models of a surface in which the crystal potential is included, and where a direct calculation of the dielectric response is difficult to obtain at real frequency.