Three photon absorption in ZnO and ZnS crystals

We report a systematic investigation of both three-photon absorption (3PA)spectra and wavelength dispersions of Kerr-type nonlinear refraction in wide-gap semiconductors. The Z-scan measurements are recorded for both ZnO and ZnS with femtosecond laser pulses. While the wavelength dispersions of the Kerr nonlinearity are in agreement with a two-band model, the wavelength dependences of the 3PA are found to be given by (3Ephoton/Eg-1)5/2(3Ephoton/Eg)-9. We also evaluate higher-order nonlinear optical effects including the fifth-order instantaneous nonlinear refraction associated with virtual three-photon transitions, and effectively seventh-order nonlinear processes induced by three-photon-excited free charge carriers. These higher-order nonlinear effects are insignificant with laser excitation irradiances up to 40 GW/cm2. Both pump-probe measurements and three-photon figures of merits demonstrate that ZnO and ZnS should be a promising candidate for optical switching applications at telecommunication wavelengths.Comment: 13 pages, 7 figure

Evaluation functions and composition operators on Banach spaces of holomorphic functions

Let $B(\Omega)$ be the Banach space of holomorphic functions on a bounded connected domain $\Omega$ in $\mathbb C^n$, which contains the ring of polynomials on $\Omega$. In this paper we investigate the properties of evaluation functions on $B(\Omega)$ to show that the evaluation functions span the dual spaces $(B(\Omega ))^{*}$ of $B(\Omega )$ when $B(\Omega )$ is reflexive. We give the necessary and sufficient conditions for the space $B(\Omega )$ to be reflexive via the evaluation functions. Moreover, under suitable assumptions on $\Omega$ and $B(\Omega)$, we establish a characterization of holomorphic self-map $\varphi:\Omega\to\Omega$ such that the composition operator $C_\varphi$ is a Fredholm operator on $B(\Omega)$ by using the symbols of composition operators to construct a linearly independent function sequence, which extends the recent result by Guangfu Cao, Li He and Kehe Zhu. Our proof does not depend on the boundary behaviour of the evaluation functions