Z-scan analytical theory for material with saturable absorption and two-photon absorption
Introduction
Nonlinear absorption materials have attracted extensive attentions. Because these materials with different nonlinear absorption processes (such as saturable absorption, reverse saturable absorption, two-photon absorption and multi-photon absorption), are promising in the different applications of science and technology [1], [2], [3], [4]. For example, the saturable absorption (SA) materials (the transmittance increases with the increase of optical intensity) have been used in the lasers as Q-switching elements. Two-photon absorption (TPA), multi-photon absorption (MPA) and reverse saturable absorption (RSA) materials (their transmittances reduce with the increase of optical intensity) have been used in two-photon microscopy and optical limiters. Therefore, it is necessary to identify their nonlinear absorption effects, and to determine their nonlinear absorption parameters, such as the saturable intensity for saturable absorber, the MPA coefficient for multi-photon absorbing material. It is well known that the open-aperture (OA) Z-scan technique, which was present by Sheik-Bahae et al. [5], has been extensively used as an effective and convenient tool for exploring the nonlinear absorption properties of various materials [5], [6], [7], [8], [9]. However, some reports have revealed that many materials exhibit more than one nonlinear absorption process simultaneously under the excitation of intense laser pulses, which make the change of OA Z-scan curves dependent on the intensity or wavelength of the used laser [10], [11], [12], [13], [14], [15], [16]. Rao et al. have found that the nonlinear absorption effect in Rhodamine B changed from SA to RSA with the increase of the concentration or intensity at 600 nm [10]. Rojo et al. have reported that the nonlinear absorption properties of a novel organic material originated from the effect of SA and induced absorption (namely the transmission decreased with the increase of the on-axes peak intensity) simultaneously, as the input intensity is intense [14]. Gao et al. have explained the transformation from SA to RSA using the phenomenological model of the combination of SA and TPA, and estimated the saturable intensity and the nonlinear absorption coefficient by simulating the experimental curves [12]. Cassano et al. have investigated a novel composition of Ru under different input intensity using Z-scan technique, and observed the transformation of OA Z-scan curves from RSA to SA [16]. It is of great importance to distinguish the nonlinear absorption effects and to evaluate the nonlinear absorption coefficients for the materials possessing more than one nonlinear absorption effect.
Adomian decomposition method (ADM) has been present and developed during the 1980s by George Adomian, and has attracted more and more attentions as an effective tool for solving linear and nonlinear differential, integral equations, and partial differential equations [17], [18]. By the use of ADM, the solutions of differential equations can be written in a series form, as . With a recurrence formula, all components un can be calculated quickly. Ultimately, the sum of un should approach to the approximate or exact solution of differential equations. In the year of 2000, Sancez et al. introduced Adomian decomposition method into the domain of nonlinear optics, firstly, and acquired the analytical expression of Gaussian beam passing through the saturable absorber [19]. And then, Gu et al. investigated the Z-scan theories of saturable absorption [20] and two-photon absorption saturation [21] by using the Adomian decomposition method, respectively.
In this work, we develop the OA Z-scan theory for the simultaneous appearance of both SA and TPA by the use of the Adomian decomposition method. We obtain the analytical expressions of the OA Z-scan traces for the phenomenological model of the combination of SA and TPA [12], [13]. On the basis of the analytical expressions, we explore the interplay between SA and TPA in the Z-scan traces under different laser intensity conditions. Finally, both the saturable intensity and TPA coefficient for nonlinear materials could be obtained quickly and efficiently from the experimental Z-scan traces.
Section snippets
Theory
Here, we apply the model of Ref. [12], [13] for the OA Z-scan theory of the materials with the concurrence of SA and TPA processes. This model, which has successfully explained the nonlinear absorption effects dependence on the input intensity for the aqueous solution of Pt nanospheres [12], consists of two terms as follows, the first term stands for the saturable absorption (Is is the saturable intensity), and the second represents the two-photon absorption (β is the two-photon absorption
Discussion
For the most cases, the optical nonlinearity is detected under the excitation of laser pulses. In this analysis, we discuss Z-scans for a pulsed laser by Eq. (23) at different levels of laser intensities. As a test, taking the parameters in Ref. [12] for the nonlinear material with the appearance of both SA and TPA processes, we compare our analytical solutions with numerical simulations. When we fit the Z-scan experimental results with Eq. (23), however, the sum upper limit of Eq. (23) can
Conclusions
Based on the Adomian decomposition method, we investigated the OA Z-scan theory for the material with the simultaneous appearance of both SA and TPA effects, and acquired the analytical expressions of the OA Z-scan curves. We also explored the change of the OA Z-scan curves with the analytical solutions under different input intensity, and compared with the numerical simulations, when the SA and TPA compete against each other. The results showed that our analytical solutions agree well with the
Acknowledgments
This work is supported by the National Science Foundation of China under Grant Nos. 60778006 and 60878023.
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