We develop a systematic approach for simultaneous extraction of the dispersion relations and profiles of multiple modes in periodic waveguides though a special global optimization procedure applied to near-field electric field measurements in the waveguide plane. We apply this method to perform in-depth analysis of experimental data on wave propagation close to an interface between waveguide sections with different dispersion characteristics, and we successfully identify several modes contributing to the experimentally measured fields. We find clear evidence that when the group velocity is reduced across the interface, evanescent modes that facilitate the excitation of propagating slow-light waves appear, confirming previous theoretical predictions.