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The inter-conversion between the Hermite-Gaussian (HG) modes and the Laguerre-Gaussian (LG) modes is discussed. The HG beams carrying a cross phase can evolve into the LG modes, and vice versa, a LG mode with the cross phase can also transform to the HG mode. This conversion process is accompanied by the intensity rotations of optical beams, and their angular velocities and acceleration are both radially dependent. Initially, the outer intensity peak and the inner intensity hollow rotate in the opposite directions. After that they tend to rotate in the same direction with different velocities. Different patterns can be generated in a controllable way by adjusting the cross phase coefficients. The theoretical results provide a controllable approach for modes generation by engineering the phase structure.
Liang et al. (Tue,) studied this question.