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A novel broad-band telecom laser source is used to make a lateral-shear scanning-wavelength interferometer for measuring the thickness of thin plates. We show that the wide tunability range allows us to detect samples down to tens of microns with a relative uncertainty of less than 0.5%. A comparable accuracy in the thickness characterization of double-layer structures is also demonstrated. In turn, the wide tunability range needs the dispersion law of the materials to be taken into account in the model for correct thickness evaluation, although simultaneous measurement of dispersion and thickness are in principle possible with this technique.