Optical characterization linear contribution

Measurement of non-linear optical properties [580] also provides a means for characterizing size-quantized semiconductor particles. Third-order optical non-linearity of size-quantized semiconductor particles has been discussed in terms of resonant and non-resonant contributions [11]. Resonant non-linearity is expected to increase with decreasing particle size and increasing absorption coefficients. 

The polarization P is given in tenns of E by the constitutive relation of the material. For the present discussion, we assume that the polarization P r) depends only on the field E evaluated at the same position r. This is the so-called dipole approximation. In later discussions, however, we will consider, in some specific cases, the contribution of a polarization that has a non-local spatial dependence on the optical field. Once we have augmented the system of equation B 1.5.16. equation B 1.5.17. equation B 1.5.18. equation B 1.5.19 and equation B 1.5.20 with the constitutive relation for the dependence of Pon E, we may solve for the radiation fields. This relation is generally characterized tlirough the use of linear and nonlinear susceptibility tensors, the subject to which we now turn. 

Optical properties are usually related to the interaction of a material with electromagnetic radiation in the frequency range from IR to UV. As far as the linear optical response is concerned, the electronic and vibrational structure is included in the real and imaginary parts of the dielectric function i(uj) or refractive index n(oj). However, these only provide information about states that can be reached from the ground state via one-photon transitions. Two-photon states, dark and spin forbidden states (e.g., triplet) do not contribute to n(u>). In addition little knowledge is obtained about relaxation processes in the material. A full characterization requires us to go beyond the linear approximation, considering higher terms in the expansion of h us) as a function of the electric field, since these terms contain the excited state contribution. 

Bruckner and Kondratenko (2006) used a similar approach to characterize VOx/Ti02 catalysts. In a separate TPR experiment carried out with a quartz reactor equipped with a UV-vis fiber optical probe, the relationship between the "absorbance" at 800 nm and the degree of reduction as determined from H2 consumption via mass spectrometry was established. The absorbance at 800 nm increased with increasing reduction of the vanadium, but not linearly. During the catalytic reaction experiment, the absorbance at 800 nm was then used to determine the average valence of vanadium. Because contributions of reduced titanium species in the analyzed spectral range could not be excluded, only a lower limit of the vanadium oxidation state could be determined, which was 4.86 at 523 K and C3H8/02 = 1 1. 

---