Construction of Bi-Pearcey beams and their mathematical mechanism

We present a theoretical expression in the form of the Pearcey function by deducing the Fresnel diffraction distribution of an elliptic line. Then, we numerically simulate and experimentally generate this kind of new Pearcey beams by using the Fresnel diffraction of optical ellipse line. This kind of beams can be referred to as Bi-Pearcey beams because their appearance of the topological structure is very similar to the combination of two face-to-face classical Pearcey beams. It is no doubt that so-called Bi-Pearcey beams are the new member of a family of form-invariant Pearcey beams. Subsequently, we also provide the theoretical mechanism of generating Bi-Pearcey beams based on the Zeeman catastrophe machine of catastrophic theory. By solving the critical equation of potential function of Bi-Pearcey beams generated by an ellipse line, we find that the optical morphogenesis of Bi-Pearcey beams is determined by the number of roots of the critical equation. The critical equation of potential function of Bi-Pearcey beams is a classical Cartan equation, which has at most three real roots. For the Fresnel diffraction of ellipse line, three real roots of the critical equation are corresponding to three stable points and represent three diffraction lines, hence they can be used to examine the optical topological structure of Bi-Pearcey beams. By choosing the appropriate control variable of Bi-Pearcey beams, two diffraction lines of an ellipse line overlap, and the strong caustic line of Bi-Pearcey beams is correspondingly generated when the two of the three real roots of the critical equation are equal. If the three real roots of the critical equation are all equal, the strongest cusps of Bi-Pearcey beams are generated, accordingly. Moreover, the equation of the caustic line and their positions of four cusps of Bi-Pearcey beams are given by solving the control variable equation of Bi-Pearcey beams. In conclusion, we elucidate the mathematical mechanism of topical morphogenesis of Bi-Pearcey beams based on catastrophic theory.