Axicon Gaussian Laser Beams

If a laser beam is to have radial transverse polarization, the transverse electric must vanish on the symmetry axis, which is charge free. However, we can expect a nonzero longitudinal electric field on the axis, considering that the rays of the beam that converge on its focus each has polarization transverse to the ray, and hence the projections of their electric fields onto the axis all have the same sign. This contrasts with the case of linearly polarized Gaussian laser beams [2, 3, 4, 5] for which rays at 0 and 180 azimuth to the polarization direction have axial electric field components of opposite sign. The longitudinal electric field of axicon laser beams may be able to transfer net energy to charged particles that propagate along the optical axis, providing a form of laser acceleration [6, 7, 8]. Although two of the earliest papers on Gaussian laser beams [9, 10] discuss axicon modes (without using that term, and without deducing the simplest axicon mode), most subsequent literature has emphasized linearly polarized Gaussian beams. We demonstrate that a calculation that begins with the vector potential (sec. 2.1) leads to both the lowest order linearly polarized and axicon modes. We include a discussion of Gaussian laser pulses as well as continuous beams, and find condition (8) for the temporal pulse shape in sec. 2.2. The paraxial wave equation and its lowest-order, linearly polarized solutions are reviewed in secs. 2.3-4. Readers familiar with the paraxial wave equation for linearly polarized Gaussian beams may wish to skip directly to sec. 2.5 in which the axicon mode is displayed. In sec. 2.6 we find an expression for a guided axicon beam, i.e., one that requires a conductor along the optical axis.