# Light-Induced Ring Pattern in a Dye-Doped Nematic Liquid Crystal
## 1 Introduction
This phenomenon is understood as a result of the different scales in the transport processes of the concentration of the _cis_ state and the orientational order of the liquid crystal. Thin films of liquid crystals without dye  
Figure 1: Ring patterns induced by light in a dye-doped liquid crystal cell (DDLCC). (**a**) Schematic representation of the experimental system. The blue and red bars, respectively, account for the molecules of the liquid crystal and azo-dye. The cell is illuminated by a blue and green beam. The snapshot accounts for the observed ring patterns. A transversal plane in the DDLCC is schematically represented. The areas under higher blue laser irradiation are more disordered, while the zones less illuminated preserve the nematic order. (**b**) Isomers of the molecule methyl red methyl ester. (**c**) Two snapshots showing the observed pattern (upper panel) and snapshot with the beam that induces photo-isomerization superimposed (bottom panel). dopants subjected to strong electromagnetic fields in their cross section exhibit diffraction rings [14; 23]. The above phenomenon is associated with phase modulation or autofocusing of light when is diffracted in the NLC. This medium is a good approximation to a Kerr medium [2; 3; 4; 5; 6; 24], that is, the envelope of the light is under the effect of phase modulation, a cubic term for the envelope and diffraction. Hence, the conjunction of diffraction and phase modulation produces the emergence of diffraction rings. Note that the thickness of the observed rings decreases with the square distance from the center of the light beam. A similar phenomenon is observed in dye-doped liquid crystals when subjected to a coherent light with a moderate and low light intensity [25; 26; 27]. Because of the presence of dye-dopants, the nonlinear response can be achieved for light power of a few milliwatts [28]. The emergence of these rings has been associated with phase modulation. However, the morphology and dynamics of the ring patterns are not consistent with that expected for self-modulation rings observed for large intensities of light [14; 23]. Furthermore, in this type of description, the dynamical behavior of the _cis_ state concentration is passive. In other words, this concentration is enslaved to the system dynamics. The article aims to elucidate and characterize the origin of the ring patterns observed when illuminating a dye-doped nematic liquid crystal cell with a light beam and planar anchoring. Based on an experimental setup with two parallel coherent beams, one of which induces photo-isomerization and the other the illumination, the formation of ring patterns is studied. Figure 1 illustrates the typical observed ring patterns. This type of setup allows us to separate the induction of the photo-isomerization and the observation of the self-organized patterns. To use these two coherent beams, we have synthesized methyl red methyl ester as dye-dopant, photo sensitive only to one of the light beams, and a commercial E7 (Instec Inc., Boulder, CO, USA) as a liquid crystal matrix. The methylred methyl ester was used as a dye-dopant (cf. Figure 1). Unlike methylred, the methylred methyl ester is more soluble and less viscous in E7 due to the absence of hydrogen bonds from the carboxylic acid group of the methylred. In addition, this structural modification prevents an intramolecular hydrogen bonding interaction with one of the nitrogens of the azo group, causing a faster _cis-trans_ isomerization [29]. Theoretically, based on a mathematical model that accounts for the coupling between the concentration of the _cis_ state and the order parameter, we elucidate the emergence of the rings as forming patterns in an inhomogeneous medium. The bifurcation diagram is analytically characterized. The emergence, propagation, and establishment of the ring patterns are in fair agreement with the experimental observations.