How can I simulate a super-Gaussian laser pulse that propagates in free space?

[Hatisa]

In a 2-D PIC code I currently launch a pure Gaussian beam with:

begin:laser
    boundary = x_min
    intensity  = intensity0
    lambda     = lambda0
    pol_angle  = 0
    phase      = -2.0*pi/lambda0 * y**2/(2.0*RC) + gouy
    profile    = gauss(y,0,w_bnd)
    t_profile  = gauss(time,time_delay,t_duration)
end:laser

Here, the spatial intensity distribution of the beam is also Gaussian. The phase variation -2.0 * pi/lambda0 * y^2/(2.0 * RC) + gouy is consistent with the phase evolution of a Gaussian beam during propagation. However, for a super-Gaussian beam, besides changing the profile to supergauss, the phase term must also be modified. How should I set this phase parameter?
I need to simulate the propagation of a super-Gaussian beam in free space, where the temporal profile is Gaussian with a full width at half maximum (FWHM) of 30 fs, and the spatial profile is super-Gaussian. Any guidance from experts and scholars would be greatly appreciated! Thank you!