Nanostructured copper thin films have been prepared using the pulsed laser deposition method. Optical absorption spectra of these films exhibit plasmonic absorption peaks around $619\\\\ {\\\\text{nm}}$, which suggests the formation of copper nanoparticles on their surfaces. Scanning electron micrographs of the films confirm the nanoparticle formation on the films surfaces. After laser beam passing through the thin films, the observed diffraction rings on a far-field screen have been recorded. Despite the smallness of the maximal axial phase shifts of the films, which have been obtained using the nonlocal $z$-scan theory, a series of low-intensity rings can be observed on the far field screen for some specific positions of the thin films from the focal point. It is shown that the best approach to determining the sign and magnitude of the nonlinear refractive index of thin samples is the application of the conventional close-aperture $z$-scan method.

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