Objective  In view of the wide application of nonlinear optical materials in the field of modern optics and optoelectronic information, the search for high-performance nonlinear optical materials is a common concern. In particular, an important content of nonlinear optical effect and its application is partly to develop characterization techniques and measurements of material nonlinear coefficients. In the past few decades, many new materials have been synthesized to get larger and larger molecular multi-photon-absorption cross-section through new characterization techniques. The common methods of the nonlinear optical measurement include degenerate four-wave mixing, nonlinear transmittance, Z-scanning technique, etc. In a multi-photon-absorption experiment, many factors such as the stability of excitation pulse laser with high light intensity (MW·cm^-2, GW·cm^-2, or higher), high accuracy of measuring equipment, and the suitable sample would lead to the increase of the experimental cost, difficulty and complexity. So, it may be a good effort to find a predictable method to estimate two- and three-photon-absorption behaviors according to the linear absorption spectrum. As far as we know, the quantum impedance Lorentz oscillator (QILO) model just has, to some extent, the predictive ability upon the linear absorptive behavior of medium.

  Methods  QILO model was recently established and proposed, in which the classical Lorentz oscillator had been quantized via Bohr-Sommerfeld quantum theory and 1- and 2-photon-absorption selection rules of quantum mechanics. QILO's parameters including the linear or nonlinear param, the damping coefficient, and the oscillator strength have been expressed in terms of the typical quantum physical quantity, such as effective quantum number, Bohr radius, and the ground state energy of hydrogen atom. On the basis of QILO model, the reference formulae for calculating the fourth- and fifth-order nonlinear effect parameters of the oscillator are further derived theoretically and expressed in terms of effective quantum number, electronic charge and mass, and Bohr radius. Then, the single-, two-, and three-photon-absorption properties of the ferrocene derivative containing fluorene are investigated in detail. By fitting the linear absorption spectrum of the studied material, the effective quantum number before and after the electronic transition near the linear absorption peak of 400 nm wavelength is calculated by use of QILO model. As a prediction, the molecular two- and three-photon-absorption cross-sections of the same material are numerically calculated. The prediction results are compared with the experimental data in the literature.

  Results and Discussions   The 1-, 2-, and 3-photon-absorption properties of ferrocene derivative containing fluorene with R=NO₂ substituent are investigated using QILO model. The obtained major results are indicated in the fitting diagram of the linear absorption spectrum of the molecule (Fig.2(b)), the fitting diagram of the two-photon-absorption (2PA) cross-section (Fig.2(c)), and the curve of the three-photon-absorption (3PA) cross-section with the wavelength change (Fig.2(d)). The results of the theoretical numerical curves show that the 2PA cross-section of the compound molecule near 793 nm are about 0.49\times 10^-20\;\mathrmc\mathrmm^4\cdot\mathrmG\mathrmW^-1, and the 3PA cross-sections near 1 260 nm and 1 314 nm are 2.01\times 10^-25\;\mathrmc\mathrmm^6\cdot\mathrmG\mathrmW^-2 and 1.00\times 10^-25\;\mathrmc\mathrmm^6\cdot\mathrmG\mathrmW^-2, respectively. These values are in good agreement with the experimental ones. Additionally, the 2PA and 3PA processes of ferrocene derivatives containing fluorene with R=NO₂ substituent, based on QILO model, are taken as an example to discuss how to separate the 3PA process and ignore the 2PA effect under high light intensity in detail.

  Conclusions  QILO model can describe well the single-, two-, and three-photon-absorption properties of the ferrocene derivatives containing NO₂ as substituent. In the light of the QILO's characteristic that multi-photon-absorption cross-section can be estimated according to the linear absorption spectrum of the medium, QILO model may provide us a theoretical analysis method for finding the materials with large two- and three-photon-absorption cross-sections so as to reduce the comprehensive experimental cost in studying multi-photon processes. The model can also be extended to other nonlinear optical processes. The QILO model exhibits itself an advantage of its great reduction of the calculation complexity and high cost confronting the first principle in dealing with both linear and nonlinear properties of optoelectronic materials as well.