Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media

The formation and the evolution of the far-field patterns of the Gaussian beams passing through a thin self-focusing medium and a thin self-defocusing medium are studied using the Fresnel-Kirchhoff diffraction theory, and the effects of the different Kerr media and the different Gaussian beams on the far-field pattern formation and evolution are analysed by taking into consideration both the change in the additional phase shift induced by the refractive index and the change in the sign of the radius of wavefront curvature of the laser beam passing through the nonlinear medium. Our results show that, when either the divergent Gaussian beam passes through the self-defocusing medium or the convergent Gaussian beam passes through the self-focusing medium, the far-field intensity distribution pattern is a series of thick diffraction rings with a central dark spot; while, when either the divergent Gaussian beam traverses the self-focusing medium or the convergent Gaussian beam traverses the self-defocusing medium, the far-field intensity distribution pattern is a series of thin diffraction rings with a central bright spot. Our results also show that whether the central dark spot or the central bright spot occurs depends mainly on the sign of the product of the additional phase shift and the radius of the wavefront curvature. When the sign is negative, that is, when the divergent Gaussian beam passes through the self-defocusing media, or the convergent Gaussian beam passes through the self-focusing media, the central dark spot surrounded by the thick diffraction rings will emerge in the far field; while, when the sign is positive, that is, when the divergent Gaussian beam traverses the self-focusing media, or the convergent Gaussian beam passes through the self-defocusing media, the central bright spot with the thin diffraction rings will occur in the far field. The difference between the diffraction patterns is attributed to the interplay of the strong spatial self-phase modulation caused by the refractive index change of the medium and the changes in sign of the radius of the wavefront curvature of the incident Gaussian beam. The results obtained in this paper are of significance to the design of practical nonlinear optical limiters for the eye or sensor protection and many other applications.