Perfect self-similar beams

We present a new type of beams which are exact solutions to the paraxial Helmholtz equation. Such beams feature strict self-similar transverse profile during the propagation, and the peak intensity remains unchanged. These beams constitute a unique beam mode in free space, which can be regarded as an intermediate mode between non-diffracting beams and traditional self-similar beams. Thus, we name these beams the “perfect self-similar beams”. We show that the transverse profile of these beams scales parabolically during propagation. As such, our work reveals a previously unnoticed beam mode which holds both main characteristics of self-similar beams and non-diffracting beams, and expands conceptually the special beam modes.