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The excitation of a periodic structure composed of two- or three-dimensional metallic scatterers by a finite dipole source is studied through the application of the Array Scanning Method. The difficulties in the numerical implementation are discussed in details, providing general guidelines and suggesting a useful preprocessing procedure to determine the number of quadrature points needed in the involved spectral integrations to obtain a given accuracy in the field calculation. The problem of transmission and reflection from two typical periodic screens (arrays of apertures in a metallic plate and arrays of metallic spheres) is first addressed in order to show the differences between a finite-source and the conventional plane-wave excitation. Finally, results for the near field of a recently proposed two-dimensional periodic leaky-wave antenna (an array of metallic patches over a ground plane) are also presented examining for the first time its main characteristics.