Effective radius of curvature of Hermite–Gaussian array beams

Analytical expressions for the effective radius of curvature, R, of Hermite–Gaussian (H–G) array beams propagating in free space for both coherent and incoherent combinations are derived. It is shown that for the two types of beam combination a minimum of the effective radius of curvature, Rmin, appears as the propagation distance z increases. For the coherent combination, R is larger than that for the incoherent combination. The position zmin where the effective radius of curvature reaches its minimum is further away from the source plane for the coherent combination than that for the incoherent combination. For the two types of beam combination, R and zmin increase with increasing beam number, increasing beam separation distance, increasing waist width, and decreasing beam order and wavelength. In particular, the R of single H–G beams is always smaller than that of H–G array beams; the R of Gaussian array beams is always larger than that of H–G array beams.