Fundamental Gaussian wave

Abstract

The secondary source for the approximate paraxial beams and the exact full waves is a current sheet that is situated on the plane z = 0. The beams and the waves generated by the secondary source propagate out in the +z direction in the space 0 < z < ∞ and in the -z direction in the space -∞ < z < 0. The response of the electric current source given by Eq. (1.24) obtained in the paraxial approximation is the fundamental Gaussian beam. The same current source for the full Helmholtz wave equation yields the fundamental Gaussian wave. For the electric current source given by Eq. (1.24), the Helmholtz wave equation is solved to obtain the exact vector potential. The electromagnetic fields are derived, the radiation intensity distribution is determined, and its characteristics are analyzed. The time-averaged power transported by the fundamental Gaussian wave in the ±z direction is obtained. The complex power is evaluated and from there the reactive power is found. The time-averaged power carried by the fundamental Gaussian wave in the ±z direction increases, reaches a maximum greater than 1, decreases, and approaches the value of 1 corresponding to the fundamental Gaussian beam as the parameter kw0 is increased. The reactive power does not vanish for the fundamental Gaussian wave. The reactive power decreases, reaches zero, decreases further, reaches a minimum, then increases and approaches zero (that is, the limiting value for the corresponding fundamental Gaussian beam) as kw0 is increased.