Analytical frequency-doubled HG modes in astigmatic mode converters for composite vortex beam generation

We present a powerful analytical framework that fully characterizes the propagation and astigmatic mode conversion (AMC) of frequency-doubled Hermite-Gaussian (HG) modes, enabling the deterministic generation of composite vortex beams (CVBs). Our approach is based on a rigorous modal decomposition, in which any frequency-doubled HG mode is expanded as a linear superposition of HG basis modes with analytically derived, fixed coefficients. The propagation of a frequency-doubled HG mode is described as the linear superposition of its constituent HG modes, each accumulating a distinct Gouy phase while maintaining its modal coefficient. Fundamentally, AMC acts as a unitary transformation, mapping the complete HG basis onto the full Laguerre-Gaussian (LG) basis in Hilbert space. This framework naturally expresses the resulting CVB as a linear superposition of LG modes, each inheriting the coefficient of its corresponding HG mode. The validity of our model is rigorously confirmed by excellent agreement between theoretical analysis, numerical simulations, and experimental measurements, which accurately reproduce the transverse intensity and phase profiles. This work establishes a groundbreaking paradigm for on-demand generation and control of complex structured light, offering significant advancements for optical manipulation, quantum communication, and high-dimensional information processing.