Carrier envelope phase of few cycle pulsed Hermite–Gaussian beam

The complex q parameter method is used to analyze the propagation of the few cycle pulsed Hermite–Gaussian beam in the free space within the paraxial condition, and an approximate formula for the carrier envelope phase (CEP) is deduced by using the zero-order approximation in the amplitude and first-order approximation in the phase. The validity of the approximate formula is verified by the numerical simulation methods, which shows that they fit very well with each other on condition that the pulse duration is more than 5 fs or the propagation distance is longer than about 3 Rayleigh length. The order of the Hermite function, the beam waist and the position of the axis play important roles in the CEP of the few cycle pulsed Hermite–Gaussian beam; the conclusion is as follows: the CEP trends to −(m+n+1)π/2 in the far field and their variety are in inverse proportion to the beam waist on the axis. The beam waist is larger, the CEP is smaller and its variation changes slowly along the propagation distance. The variation of the CEP is in direct proportion to r2 on any z  -plane, and the maximal values all occur at the position z=ZR(ω0)/3.