5G and IEEE 802.11ay introduce the use of the millimeter band as one promising solution to provide broadband wireless communication at multi-Gb/s user data rate. Due to the severe path-loss at such frequencies, it is generally assumed that large antenna arrays are used at the base station to steer narrow beams and build highly directionnal communication links towards the terminal points. However, broader and less directional beams are also of high interest in some of the steps involved in the establishment or the maintenance of the communication links. Indeed, search of a large area by narrow beams becomes too time consuming and link outage becomes more critical, thus affecting the latency and the robustness of the communications. A method enabling an adaptation of the beam widths is then worthwhile to consider. In this article, we investigate how narrow beams naturally produced by large antenna arrays can be broadened to adapt the beam width to a desired angular sector. We consider that the multi-antenna processing is performed by phase shifters on the radio-frequency stage since its digital counterpart is hardly feasible in practice at such high frequencies. The main idea of our systematic phase-only beam broadening technique relies on the determination of a quadratic phase excitation law from a desired beam width and steering angle. We first lead a thorough analysis of the radiation behavior regarding the coefficients of such quadratic excitation. We then propose a calculation method for determining the polynomial coefficients as a function of the desired beam width and steering angle. This non-iterative beam broadening method is described for boresight and non-boresight directions and is intended for discrete antenna arrays.