Optics of chiral smectic liquid crystals near their Lifshitz point

The reflectance spectra from chiral smectic-C (Sm-${\mathit{C}}^{\mathrm{*}}$) liquid-crystalline structure distorted by a magnetic field were calculated and analyzed near their Lifshitz point where coexistence of the three phases Sm-A–Sm-C–Sm-${\mathit{C}}^{\mathrm{*}}$ occurs. Using the 4×4 matrix formulation of Maxwell’s equations we found that an infinite series of reflection peaks that correspond to even harmonics of the helicoidal pitch appear in the case of normal incidence due to the modulation of the tilt angle. Additional infinite odd sequences of peaks appear only at oblique incidence. The peaks’ heights decay strongly with the order and the harmonic. For those that exist in the undistorted structure, they decay also with the perturbation exhibiting critical behavior. For those that appear due to the perturbation, they exhibit a maximum at a universal value of the field at H/${\mathit{H}}_{\mathit{L}}$=1/(2${)}^{1/4}$. We give full explanations to the polarization characteristics by calculating the form factor for scattering that provides selection rules for the scattering events. Analytic expressions were obtained within the framework of the kinematical theory of scattering. It is shown that the theory is valid in the range very close to the transition, suggesting a way of investigating this type of phase transition by reflection spectra techniques.