Carrier–envelope phase of ultrashort pulsed Hermite–Gaussian beams on propagation

An analytical expression of the carrier–envelope (CE) phase of ultrashort pulsed Hermite–Gaussian beams on propagation has been deduced in detail. Numerical calculations have been carried out to illustrate the evolution of the CE phase of ultrashort pulsed Hermite–Gaussian beams on propagation with different parameters. The results show that the order of the Hermite function greatly affects the evolution of the CE phase. The higher the order of the Hermite function, the larger the CE phase is. The beam waist affects the CE phase in the near field, especially in a few Rayleigh distances. In addition, the variation in the CE phase on the axis is larger than that on the beam periphery on propagation. (letter)