Effects of focusing on third-order nonlinear processes in isotropic media

Abstract

A theoretical analysis of third-order nonlinear interactions of focused laser beams is performed for the processes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{1} + \omega_{2} + \omega_{3} \rightarrow \omega_{4}, \omega_{1} + \omega_{2} - \omega_{3} \rightarrow \omega_{4}</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{1} - \omega_{2} - \omega_{3} \rightarrow \omega_{4}</tex> . The total power and far-field beam profile of the generated radiation is related to the total powers of the fundamental beams, to the tightness and location of the focus, and to the value of the difference between the wave vectors of the generated radiation and driving polarization. The optimum degree of wave-vector mismatch as a function of tightness and location of focus is determined for each of the three processes. The process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{1} + \omega_{2} - \omega_{3} \rightarrow \omega_{4}</tex> is found to be unique in that it is always optimized by focusing as tightly as possible. Experimental results, which verify the theory for the processes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{1} + \omega_{2} + \omega_{3} \rightarrow \omega_{4}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{1} + \omega_{2} - \omega_{3} \rightarrow \omega_{4}</tex> , are presented.