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Non-Local Aspects of Second Harmonic Generation at a Metal Surface

A general non-local expression for the second harmonic generated current in a metal is derived within an explicit spatially dependent quadratic response formalism. In the limit of an instantaneous response earlier hydrodynamical results are corroborated. Surface sensitivity and spatial extension of the induced current are discussed. Gauge invariance is used to express the second harmonic current only in terms of the electric and magnetic fields instead of the gauge-dependent potentials A and ϕ.