DOI: 10.1007/s00340-004-1501-y Appl. Phys. B 78, 775–780 (2004) Lasers and Optics Applied Physics B l. p ´ alfalvi 1, ✉ j. hebling 2 Z-scan study of the thermo-optical effect 1 Research Group for Nonlinear and Quantum Optics, Hungarian Academy of Sciences, 7624 P´ ecs, Ifj ´ us´ ag u. 6., Hungary 2 Department of Experimental Physics, University of P´ ecs, 7624 P´ ecs, Ifj ´ us´ ag u. 6., Hungary Received: 3 December 2003/ Final version: 19 February 2004 Published online: 16 April 2004 • © Springer-Verlag 2004 ABSTRACT In the case of cw Gaussian illumination, the thermo-optical n 2 cannot be characterised by a constant value. It is shown that, if absorption has both linear and nonlinear contributions, the thermo-optical n 2 consists of a position- and power-dependent term. Hence analytical formulae that assume a constant n 2 are no longer valid for the Z-scan fitting. In this paper a new Z-scan theory is introduced, which is applicable to the thermo-optical effect in the presence of both linear and non- linear absorption and an arbitrary extent of optical nonlinearity. The calculation technique can be used for large sample thick- ness too, by dividing it into thin slices. It was found that inside one slice, the distribution of the light-induced refraction is simi- lar to that for the case of graded index media. The propagation can be described through the transformation of the q-parameter. It is demonstrated that the Z-scan technique makes the sensi- tive measurement of the linear absorption coefficient possible. The linear and nonlinear absorption coefficients were experi- mentally determined for Mg doped LiNbO 3 to be α = 0.6m −1 and β = 2.9 × 10 −9 m/W, respectively. PACS 42.79.Ry; 78.20.Nv; 77.84.Dy 1 Introduction Z-scan is a well-established method for determin- ing nonlinear refraction and absorption. In this technique, the nonlinear sample is scanned along the propagation path (z-axis) of a focused Gaussian laser beam, around its focus. The intensity distribution of the beam induces a position- dependent ∆n(r, z ) = n 2 I(r, z ) change of refraction inside the sample, where n 2 is the coefficient of nonlinear refraction. This causes a divergence or a convergence of the laser beam, depending on the sample position and the sign of n 2 . This re- sults in the variation of the on-axis intensity in the far field as a function of the sample position. Monitoring the far field on-axis intensity versus the sample position makes it possible to determine the sign and magnitude of n 2 . Experimentally this is performed either by measuring the beam power pass- ing through a small size aperture [1–3], or by taking digital ✉ Fax: +36-72/501-571, E-mail: palfalvi@fizika.ttk.pte.hu recordings from the beam cross-section as a function of the sample position [4, 5]. The first Z-scan theory was proposed by M. Sheik-Bahae et al. [1]. In that paper, an analytical fit- ting formula for the Z-scan traces was deduced. The formula is valid only for a thin nonlinear medium, (where the sam- ple length is shorter than the z 0 Rayleigh range) and for weak nonlinearities (the maximal nonlinear phase shift is much less than π ). In 1991, M. Sheik-Bahae et al. extended the Z-scan theory to make it applicable for finite sample length and for large nonlinearities as well [6], although this method was based on numerical calculations. Later C.H. Kwak et al. de- duced an analytical formula which is valid for large optical nonlinearities, but only for short samples [3]. Beside nonlinear refraction, nonlinear absorption is also often present in materials. This means that absorption is de- scribed as d I(r, z ) =−I 2 (r, z )β dz , where β is the nonlinear absorption coefficient. J.A. Hermann and R.G. McDuff re- ported an analytical fitting formula, which takes the effect of nonlinear absorption into account as well. This formula con- tains n 2 and β as independent fitting parameters. The formula is valid for thick nonlinear optical media too, but only for weak nonlinearities. The n 2 coefficient usually means the nonlinear refraction related to the χ (3) cubic (Kerr) nonlinearity. In several materi- als, beside the Kerr type nonlinearity, thermo-optical nonlin- earity may also be present. In the case of using pulsed light for the examination, the material can be characterised by a con- stant n 2 value [1, 6, 7], but in the case of cw illumination it cannot [4]. A formula was deduced for the thermo-optical nonlinear refraction supposing linear absorption. But in [4], instead of this varying n 2 , an average n 2 was taken into ac- count in the calculation. Furthermore, the reported formula for n 2 contains some mistakes in the constants. In this paper we introduce a calculation method for the Z- scan curve, which is valid for finite sample length, and for large optical nonlinearities too. We will also show how can it be used for the general case, when the nonlinearity has a thermo-optical origin as well, and the absorption has both linear and nonlinear contributions. Recently we demonstrated that some Mg doped LiNbO 3 crystals show dominantly thermo-optical nonlinearity. It was also shown that light absorption is mainly nonlinear. In the present paper, we also report the results of an experiment