Alternative Models of the Rotating Beam

Controlling the intensity behavior of a wavefront along its axis of propagation can assist, for example, with optical manipulation, trapping, and vortex generation. A particularly interesting beam with applications in three-dimensional imaging is the rotating beam [1, 2]. A certain class of diffracting elements placed in the aperture plane of a digital camera can yield a point-spread function that rotates with defocus, providing a direct indication of object depth. Piestun et al. have found numerous applications for this element, for example, in a camera and a microscope [3, 4]. Typically, a rotating beam is generated from considering a Gauss-Laguerre (GL) modal beam decomposition, which offers stable solutions to the scalar Helmholtz equation with rotational symmetry about the propagation axis. A superposition of two or more GL modes offers one solution to a propagation-intensity eigenfunction equation, generating intensities that repeat along the propagation trajectory [5]. The mathematical theory behind these coherent modal superpositions is well understood. We approach the problem of generating and modeling the rotating beam from several new perspectives. First, we show how approximate forms of rotating beams can be created through the use of iterative phase retrieval procedures, with two different algorithms. Using a few desired intensities as input, these algorithms converge to a solution for an amplitude and phase distribution to approximate a three-dimensional intensity distribution that rotates along the propagation axis. New forms of rotating beams, like partially coherent beams, can be explored using a non-linear optimization approach. Second, the propagation of rotating beams within a GRIN medium is examined. Rotation rate and beam size oscillate as a function of propagation distance, offering a unique space to examine beam angular momentum.