The phason dispersion relation has been measured in the smectic-C* phase of the ferroelectric liquid crystal p-decyloxybenzylidene-/?'-amino-2-methylbutyl cinnamate (DOBAMBC), together with the temperature dependence of the phason and amplitudon excitations, by optical-mixing spectroscopy. The overdamped phason was found to be nearly gapless in contrast to crystalline incommensurate systems where the modulation wave is pinned by frozen defects. A theory of light scattering by phasons and amplitudons in ferroelectric liquid crystals has been developed.PACS numbers: 61.30. Gd, Whereas the presence of acousticlike phason modes 1,2 in structurally incommensurate (IC) crystals has been by now clearly demonstrated by inelastic neutron scattering, 3,4 NMR, 5 EPR, 6 and a variety of other techniques, no truly gapless phason seems to have been observed so far, and the question of the possible existence of such a mode is still open. The phason represents the sliding of the IC modulation wave recovering the broken translational high-temperature periodicity of the phase. Unlike true acoustic modes it becomes overdamped in the longwavelength limit. 7 The theory of IC systems 1 ' 2 predicts the existence of a gapless phason mode in the IC phase in addition to the opticlike amplitudon branch. Impurities 8 and higher-order commensurability effects 9 may produce a locking of the modulation wave to the underlying lattice and introduce a gap A 0 into the phason spectrum:(o$=At + K 2 (q-q 0 y (1) NMR, 5,10 EPR, 6 and neutron-scattering data 3,4 have indeed shown that A^ is of the order of 10 10 Hz in solid IC dielectrics.To resolve the question of the existence of a gapless phason we decided to look for a system with no frozen impurities and with negligible higher-order commensurability effects. 9 Ferroelectric smectic-C*-type (Sm-C*) liquid crystals, 11 " 13 where the periodicity of the helicoidal orientational ordering (~10 3 interlayer distances) is incommensurate to the periodicity of the onedimensional density modulation, fulfill both the above conditions. Here we report on what we believe to be the first observation of the dispersion relation for a nearly gapless phason. The system investigated is the ferroelectric liquid crystal p-decyloxybenzylidene-p '-amino-2-methylbutyl cinnamate (DOBAMBC), 11 where the periodicity of the IC helicoidal modulation wave is of the order of the wavelength of light (i.e., a few micrometers) and not in the x-ray region as in most IC solids. We determined by optical-mixing spectroscopy the phason dispersion as well as the temperature dependence of the phason and amplitudon excitations in the Sm-C* phase in the vicinity of the first Bragg peak together with the temperature dependence of the soft mode of the smectric-^ (Sm-A) phase.In the Sm-^4 phase, the long axes of the molecules are oriented perpendicular to the smectic layers so that the molecular director is n = (0,0,rt z ). Below T C in the Sm-C* phase the molecules tilt away from the normal to the smectic layers. The order parameter of the ...