Diffraction of the gaussian beam by the logarithmical axicon: overcoming the diffraction limit

We have obtained explicit analytical expressions for complex amplitude of light, describing Fresnel diffraction of the Gaussian beam by a logarithmical axicon (LA). Also, equation has been obtained for on-axis intensity of the Gaussian beam diffracted by LA. Estimating equation for effective radius of the beam has been derived. This equation shows inverse relation between the beam radius and "strength" parameter of the axicon. By FDTD simulation we show that LA can be used to overcome the diffraction limit: near the LA full with of the beam at half-maximum of intensity can be fifth part of the free space wavelength.