Crescent waves are localized modes pinged to a circular boundary with a shape similar to a crescent moon . To form crescent waves, the original proposal is to extend the concept of surface modes for electronic Tamm states localized at the interface separating periodic crystal line lattice and homogeneous vacuum space. With the introduction of concentric rings, nonlinear surface waves form at the edge of guiding structures, where an abrupt termination of the periodic potential happens. Direct observations of optical surface states have been demonstrated in photonic lattice edges and periodic waveguide arrays. In this work, we introduce symmetry-breaking in geometry by analyzing crescent waves in an elliptical ring, which is also fabricated experimentally in a microcavity based on vertical cavity surface emission lasers. It is the symmetry-breaking that makes the formation power for crescent waves along the semi-major axis of an elliptical ring to be lower than that along the semi-minor axis, resulting in a thresholdless operation condition .
 C. P. Jisha, Y. Y. Lin, T. D. Lee, and R.-K. Lee, “Crescent waves in optical cavities,” Phys. Rev. Lett. 107, 183902 (2011).
 K. H. Kuo, Y. Y. Lin, and R.-K. Lee, “Thresholdless crescent waves in an elliptical ring,” Opt. Lett. 32, 1077 (2013).