Special phase matching of second-harmonic generation in helical ferroelectric liquid crystal cells

The helical structures in ferroelectric liquid crystals can be utilized to realize a special phase matching for second-harmonic generation (SHG) when two counter fundamental waves propagate along the helical axis and the wavelength of SHG is near the photonic (selective reflection) band edge. On the basis of the exact theory [Drevensek-Olenik and Copic, Phys. Rev. E 56, 581 (1997)], a simple analytical description is derived and some characteristic features of the special phase matching are shown. (1) Special phase matching is definitely achieved under particular combinations of polarization. (2) The SH spectrum is related to a subsidiary oscillating structure in the selective reflection spectrum. The maximum SH intensity is realized at the first dip of the oscillation near one of the edges in the selective reflection band. (3) The thickness (d) dependence of the maximum SH intensity is $d^4$ in thick cells, while it is $d^2$ for conventional phase matching. (4) The linewidth for the SH peak is $d^{-3}$ dependent, which is much narrower than in conventional phase matching.