Nonlinear absorption coefficient

For the application of QDs to three-dimensional biological imaging, a large two-photon absorption cross section is required to avoid cell damage by light irradiation. For application to optoelectronics, QDs should have a large nonlinear refractive index as well as fast response. Two-photon absorption and the optical Kerr effect of QDs are third-order nonlinear optical effects, which can be evaluated from the third-order nonlinear susceptibility, or the nonlinear refractive index, y, and the nonlinear absorption coefficient, p. Experimentally, third-order nonlinear optical parameters have been examined by four-wave mixing and Z-scan experiments. 

The shape of the Z-scan curve can be modified if a nonlinear absorption or nonlinear transmission (absorption bleaching) takes place in the sample, e.g., due to the presence of an imaginary part of 3) of the material. The curves then become asymmetrical due to increased absorption or transmission when the sample is close to the focal plane. By analyzing the shape of such a modified Z-scan curve, one can determine the nonlinear absorption coefficient f32 or the related imaginary part of 3). Alternatively, to determine the nonlinear absorption properties of a sample, the total transmission through the sample can be monitored, i.e., the total intensity of the transmitted beam can be measured without an aperture, as a function of the sample position with respect to the focal plane. Such an experiment is usually referred to as an open aperture Z-scan. It is often used for the investiga-... 

Information about nonlinear absorptive properties of a sample can be derived simply by measuring the sample transmission as a function of the incident light intensity. We note that Eq. (5) can therefore be inverted to read T-1 = 1 + fi2LI a linear dependence of the inverse transmission on the incident intensity can be used to determine the j82 value. Because of the ease of determining the value of fi2 with open-aperture Z-scan, Z-scan is often the preferred method for quickly determining the nonlinear absorption coefficient. Point-by-point transmission measurements can be undertaken to verify the applicability of Eq. (5). 

In particular, the third-order nonlinear susceptibility is a complex quantity, (ty) = Re[ f (real part of X s related to the to the nonlinear refraction index, [20], and its imaginary component — which is the component of interest to our work - is associated to the material nonlinear absorption coefficient, 02 [20], through... 

In a composite material, as described here, the effective third-order nonlinear susceptibility should depend linearly with the concentration of the inclusions in a low filling fraction regime. In that way, the nonlinear absorption coefficient of the medium, associated to the Im[ (ru)] and consequently to the two-photon absorption processes, should also be a function of the inclusions concentration. 

Experimental Techniques in Nonlinear Absorption Measurements of TPA yield nonlinear absorption coefficient/TPA cross section (compare Ref. [54] for further details). The nonlinear absorption coefficient [j is related to 8 as [i = 8NAcch x 10 3 (fi in cm/GW, /VA Avogadro s number with 6.02 x 1023 moP1, cch = chromophore concentration in mol/L).11 In general, the following techniques have been applied to quantify the TPA cross section[53, 54], which are complementary methods for determination of either [j or 8. 

Mo and Qq e the linear refractive index and absorption coefficient, respectively ( 2) y is the nonlinear refraction coefficient, while (3 is the nonlinear absorption coefficient. By developing the relation between the electric displacement and the electric field and neglecting the terms proportional to ff, one easily obtains the link between these coefficients and the complex third-order nonlinear susceptibility,... 

We now focus on the main subject of this contribution, namely the optical Kerr effect. Depending on the material characteristics and experimental conditions - that is, on laser wavelength and power as well as on metal and matrix kinds and relative amounts - the nonlinear absorption coefficient (3 is found to be either negative or positive. The influence of each of these parameters on the nonlinear response will be examined in details in forthcoming sections. 

Change in the refractive index can be induced by either a resonant or a nonresonant process. For a resonant process, the frequency of the incident light overlaps with an electronic absorption band, by either a one-photon or a multiphoton process. The energy is absorbed by the sample and an excited state population is generated. This induces a transient change in the absorption spectrum of the material due to the bleaching of the ground state absorption and/or the appearance of the excited state absorption. A nonlinear absorption coefficient a2 can be defined similarly to Eq. (15) ... 

In this equation, A is the polarizability, B is the hyperpolarizability, and Xo1, %jui3) are the 1st, 2nd, and 3rd order susceptibilities, respectively. These are tensors, where i,j, k, l correspond to the space coordinates (x, y, z) and crystal axes. The refractivity is related to Xij"-The occurrence of birefringence in anisotropic media is a direct consequence of the fact that X,fu is a tensor. Susceptibilities of order higher than one are called nonlinear susceptibilities. The nonlinear refractive index, n2, and the nonlinear absorption coefficient, a2, both depend on the intensity of light, /. They are defined by Eqs. (5.10) and (5.11),... 

Note that ri2 is a differently dehned nonlinear index that relates the refractive index change to the square of the electric held amplitude]. The nonlinear absorption coefficient is related to the imaginary part of through [13] ... 

While self-focusing and self-defocusing are manifestations of the refractive part of the degenerate cubic optical nonlinearity, the absorptive part results in variation of the total power of the beam transmitted through the sample as a function of z. This can be monitored with a detector that integrates the power in the whole beam, and the changes of such power as a function of z (in the so-called open-aperture scan) can be directly related to the nonlinear absorption coefficient of the sample, a2. 

The exposure time (t) needed to induce a fixed birefringence (i.e. fixed pabs) for a given P(X) is inversely proportional to the square of the exposure intensity for a two photon process. Similarly, for fixed pabs tmd intensity, the exposure dose is inversely proportional to 3(A,), the nonlinear absorption coefficient. Relative values of 3 calculated from the exposure needed to induce a fixed birefringence can therefore be normalized by comparing the absolute change in UV absorption (also proportional to the number of bonds broken) for both UV and two-photon exposure to infer the number of excitations. This treatment assumes similar scission quantum yields for both two-photon and UV excitation. 

Doping with active species (e.g., Au or Ag NPs, and CdSe or PbS QDs) can result in new NLO materials, such as those with significant third-order NLO susceptibility. The Z-scan method is used to characterize NLO effect in nanocomposite sol-gel films. Z-scan measures the nonlinear refiactive index and nonlinear absorption coefficient of a material [22]. The measurement setup is shown in Figure 22.3. A standard open-aperture Z-scan apparatus is used to measure the nonlinear extinction coefficients of materials, using a laser with a pulse duration of fs, ps, or at least ns as an excitation source, to avoid thermal effects. All the Z-scan measurements described in this chapter were carried out at room temperature. Sol-gel films doped with noble metal NPs (e.g., Au, Ag, and Cu) or QDs (e.g., CdS and PbS) have been investigated by Z-scan measurements [23,24]. 

Nanosized noble metal gold nanoparticles (AuNPs 10 nm)-doped dielectric composite titania films [323] with large third-order nonlinear susceptibility were reported to be applied to optical information processing devices, such as optical switch or all optical logical gates. The third-order nonlinear refractive index and nonlinear absorption coefficient of the films were on the order of 10 cm and 10 cm with nonlinear susceptibility = 6.88 x 10 ° esu. 

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