Rayleigh Range of Hermite-Gaussian Array Beams

An analytical expression for the Rayleigh range of Hermite-Gaussian (H-G) array beams is derived. It is shown that under the non-phase-locked case the Rayleigh range z_R increases monotonously with the increasing beam number M, the beam separation distance x_d and the beam waist widthw₀, and with decreasing the beam order m. However, under the phase-locked case there exists oscillatory behavior of z_R versus m and x_d. For Gaussian array beams, under the phase-locked case, z_R is always larger than that under the non-phase-locked case. However, it holds true only when x_d is small enough or w₀ is large enough for H-G array beams. In addition, z_R of Gaussian array beams is always larger than that of H-G array beams.