Fields and propagation characteristics in vacuum of an ultrashort tightly focused radially polarized laser pulse

Analytic expressions for the electric and magnetic fields of a radially polarized ultrashort and tightly focused laser pulse, propagating in vacuum, are derived from scalar and vector potentials satisfying simple initial conditions. It is shown that for a pulse of axial length comparable to a wavelength, only the zeroth (lowest-order) term in a power-series expansion of the vector potential is needed. A procedure is outlined which may be used to obtain the fields analytically, to any desired order. Most of the needed analytic work is done that would lead to the vector potential from which the fields may be derived and the main expressions are given.

Figure 1

Variation of the initial (at $t=0$) normalized intensity with distance along the propagation direction, at points on the axis of propagation $\left(x=0=y\right)$ of pulses for which (a) $w_0=L=2{\lambda }_0$, (b) $w_0=2{\lambda }_0$ and $L=0.6{\lambda }_0$, (c) $w_0=0.6{\lambda }_0$ and $L=2{\lambda }_0$, and (d) $w_0=L=0.6{\lambda }_0$. Note that $w_0=2{\lambda }_0$ and $0.6{\lambda }_0$ correspond to the Rayleigh lengths $z_r\sim 12.57{\lambda }_0$ and $1.13{\lambda }_0$, respectively. To produce each panel of figures, fields to orders $n=0\text{–}3$ have been employed.

Figure 2

Snapshots of the normalized intensity along the propagation direction, at points on the axis of propagation $\left(x=0=y\right)$ of pulses for which $w_0=L=0.6{\lambda }_0$, taken at (a) $t=1$ fs and (b) $t=1$ ps. To produce each panel of figures, fields to orders $n=0\text{–}3$ have been employed.

Figure 3

Density plots in the focal plane $\left(z=0\right)$ perpendicular to the propagation direction, of the normalized intensities. The panel on the left is for $t=0$ and that on the right is for $t=1$ fs. The plots are for a pulse with $w_0=0.6{\lambda }_0=L$.

Figure 4

Variations in the focal plane $\left(z=0\right)$ of the normalized intensity distributions. Panel on the left is for $t=0$ and that on the right is for $t=1$ fs. All parameters are the same as in Fig. 3.

Figure 5

Variation of the normalized axial intensity with distance along the propagation direction, at points on the axis of propagation $\left(x=0=y\right)$ of a pulse that is both ultrashort and tightly focused ($w_0=L=0.6{\lambda }_0$ and $z_r\sim 1.13{\lambda }_0$). Shown are snapshots at times $t=0,3\mathrm{fs}$ and $1\mathrm{ps}$.