We present a novel and straightforward Fibonacci lens, which is created by binarizing and adding a spiral phase to its radial phase function. It is shown that this new element yields a pair of petal-like modes due to several segments of the Fibonacci lens, where each segment is phase shifted by pi compared to its adjacent segment. Also, the generated petals have highly controllable rotation, and their locations are adjustable, in which their focal distance ratio is equal to a mathematical parameter of the so-called golden ratio. Also, by changing the topological charge, an arbitrary even number of petals is obtained. (C) 2020 Optical Society of America