We prove a subconvexity bound for the central value $L(\frac{1}{2},{\it\chi})$ of a Dirichlet $L$-function of a character ${\it\chi}$ to a prime power modulus $q=p^{n}$ of the form $L(\frac{1}{2},{\it\chi})\ll p^{r}q^{{\it\theta}+{\it\epsilon}}$ with a fixed $r$ and ${\it\theta}\approx 0.1645<\frac{1}{6}$, breaking the long-standing Weyl exponent barrier. In fact, we develop a general new theory of estimation of short exponential sums involving $p$-adically analytic phases, which can be naturally seen as a $p$-adic analogue of the method of exponent pairs. This new method is presented in a ready-to-use form and applies to a wide class of well-behaved phases including many that arise from a stationary phase analysis of hyper-Kloosterman and other complete exponential sums.

Analytical far-field expressions for the transverse electric mode and transverse electric magnetic mode terms, and the energy flux distributions of vortex Airy beams are derived based on the vector angular spectrum of the beam and the stationary phase method. The physical pictures of vortex Airy beams from the vectorial structure are illustrated and the energy flux distributions are demonstrated in far-field. The influences of the beam parameters, especially the exponential factor, on the energy flux distributions of vortex Airy beams and its transverse electric mode and transverse electric magnetic mode terms are discussed. This work provides a new understanding of the propagation behaviors and applications of a vortex Airy beam.