Study Of The Two-photon Absorption Equation Under The Ultrafast Laser

At present, the methods measuring the two-photon absorption(TPA) cross section of medium molecules(or atoms) domestic and overseas, can be summarized into two kinds, i.e. physical and chemical. Physical methods include Z-scan technique and nonlinear transmittance(NLT) method, while chemical methods mainly include the two-photon induced fluorescence(TPIF) method and the two-photon transient spectrum method, etc. After comparing these two kinds of methods, researchers from university of California, professor P. M. Rentzepis et al, found that the values of molecular TPA cross section of Rhodamine( standard fluorescent material) determined by these two kinds of methods, have a big difference. And they are more inclined to the results of chemical method. In general, physical method for measuring this is based on the traditional nonlinear absorption equation. Owing to lack of the term of light intensity change over time, the nonlinear transmittance obtained by this equation cannot give expression to the temporal variation characteristics of high-peak power and ultrafast laser pulses. It deserves further study. In consideration of the rapid development and wide application of ultrafast laser technology in recent years, it will be conductive to further reveal the TPA character of medium molecules by studying the influences of ultra-fast laser pulses in the TPA equation and the corresponding nonlinear transmittance, especially to non fluorescent materials.Based on Maxwell’s equation and considered with the transient characteristics of ultrafast laser pulse, we deduced and obtained a TPA nonlinear partial differential equation under ultrafast laser acting for third order nonlinear homogeneous medium system of non magnetic, non conductive and no free charge, and we neglected the influence of the linear absorption and dispersion. Take Gaussian pulse laser for instance. Considering the characteristics of Gaussian function, based on the approximation that the quadratic component of first order time derivation of light intensity and the second order time derivation of light intensity are in direct proportion to the square of light intensity, we arrived at an approximate solution. After analyzed the pulse width of the approximate solution, we found that the pulse width is broaden with propagation of laser. According to the approximate solution, we arrived at them relevant expression of nonlinear transmittance. The results show that, the related nonlinear transmittance based on the approximate solution depends not only on the peak intensity of the pulse laser, sample thickness and TPA coefficient, but also the two main physical quantities describing the characteristics of laser pulse, the frequency and the full width at half maximum(FWHM). Using numerical fitting and take Rhodamine etc.(The fitting data is gotten from professor P. M. Rentzepis’ s paper.). Our simulations results show that, the TPA coefficient obtained based on nonlinear transmittance of of transient characteristics of the ultra-short pulse is larger than the traditional one. This result is qualitatively consistent with the corresponding value that was determined by two-photon induced fluorescence method under the same conditions.The above results show that, the transient and pulse characteristics of ultra-fast laser pulse played an important role on determining the TPA coefficient using nonlinear transmittance physical method.