IDENTIFICATION OF A SOURCE TERM AND A COEFFICIENT IN A PARABOLIC DEGENERATE PROBLEM

The globally in time existence and uniqueness of solutions to inverse problems is one of the most difficult questions to be answered. Even though the direct problems are well-posed in the sense of Hadamard (i.e. existence, uniqueness and stability results hold true), the inverse ones generally are not. The situation gets more complicated if the equation contains more than one unknown coefficient, and even more if the unknown functions depend on different variables. We consider the following identification abstract problem in a general Banach space $X$: find a function $u:[0,T] \to X,$ a coefficient $a_1:[0,T] \to \mathbb R$ and a vector $z \in X$ such that the initial-value problem \begin{align} &\frac{1}{a_0(t)}\ u'(t)-Au(t)-a_1(t)u(t)\!=\!f(t)z+g(t), \qq u(0)=u_0 \label{zi2} \end{align} is fulfilled, where $a_0(t)>0$ and $a_0(t)=0$ only in some negligible set, while $A:D(A)\subset X \to X$ is a closed linear operator, $f$ is scalar functions and $g$ is a $X$-valued source term. The occurrence of two unknowns require to introduce two additional conditions. We choose the first as nonlocal one in the integral form \imi \!\!\varphi(t)u(t)d\mu(t)\!=\!h, where $\mu$ is a Borel measure on the interval $[0,T].$ The latter is of the following form: $\Phi[u(t)]=k(t), \: t\!\in\! [0,T],$ where $\Phi$ is a prescribed linear continuous functional. Here the functions $h$, $k, \varphi$ are scalar. So, we investigate the problem (\ref{zi2}) along with these additional conditions. We study explicitly the case of the \textit{Dirac measure} concentrated at $t=T_1, 0<T_1\leq T$ and the one of an \textit{absolutely continuous measure $\mu.$} This thesis is devoted to investigation of inverse problems for degenerate parabolic equations aiming at the determination of one time-dependent coefficient $a_1$ and a spatial source term $z.$ So, the goal of this work is to find sufficient conditions on our data and operator $A$ under which the problem turns out to be well-posed. By means of Semigroup Theory and the Banach fixed-point theorem, we can find out sufficient conditions on the data $(f, g, u_0, h, k)$ ensuring \textit{global-in-time} existence and uniqueness for the solution $(a_1,u,z) \in L^1(0,T; {\mathbb R})\times \bigl[W^{1,1}(0,T;X) \cap L^\infty(0,T; D(A))\bigr]\times X.$ Moreover, a continuous dependence of Lipschitz type of the solution on the data is provided. We stress that we are obliged to introduce an unusual distance involving the data accounting for the degeneracy of function $a_0$. Finally, using a suitable metric for the data, we apply such results to a concrete parabolic problem

Optimization of Z-scan technique inside a 4f system

International audienc

Linear optical characterization of transparent thin films by the Z-scan technique

We report experimental characterization of a very small rectangular phase shift (&lt;0.3 rad) obtained from the far-field diffraction patterns using a closed aperture Z-scan technique. The numerical simulations as well as the experimental results reveal a peak-valley configuration in the far-field normalized transmittance, allowing us to determine the sign of the dephasing. The conditions necessary to obtain useful Z-scan traces are discussed. We provide simple linear expressions relating the measured signal to the phase shift. A very good agreement between calculated and experimental Z-scan profiles validates our approach. We show that a very well known nonlinear characterization technique can be extended for linear optical parameter estimation (as refractive index or thickness)

Nonlinear Characterization Techniques inside a 4f System

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Absolute measurement of the nonlinear refractive indices of reference materials

We report absolute measurements of the nonlinear refractive index on carbon disulfide ( CS 2 ) and fused silica. These materials are commonly used as standard references in nonlinear optical experiments. To obtain more accurate values than those usually used, we have combined the z-scan method inside a 4-f imaging system (in order to analyze the spatial distortion of the diffracted pump beam) with the “Kerr shutter” experiment (to evaluate the temporal pulse width durations for three different wavelengths such as 1064, 532, and 355 nm). We obtained surprisingly n 2 values one order of magnitude less than the one usually taken into account in the picosecond regime and a more significant dispersion of the nonlinear refraction index. Experimental and simulated Z-scan transmittance profiles as well as acquired autocorrelation functions in the Kerr-gating experiments are presented here in order to validate our measurements

Determination of photo-induced changes in linear optical coefficients by the Z-scan technique

We introduce a Z-scan technique as a tool to characterize small phase shift (&lt;1 rad) and photodarkening, both effects induced inside photosensitive materials by light illumination. Theoretical analysis supported by experiments is presented for permanent refraction and absorption Gaussian profiles. Simple relations are derived in order to estimate the changes in the linear coefficients. Particularly, we investigate quantitatively the photo-induced modifications in the linear optical constants of As2S3 caused by subbandgap irradiation (17 ps, 1064 nm)

Kerr-induced nonlinear focal shift in presence of nonlinear absorption

We report a theoretical formulation for the nonlinear Kerr-induced focal shift of converging lenses in a high intensity regime. A numerical relation expressing the on-axis intensity of a focused Gaussian beam is derived in the case of a nonlinearly absorbing and diffracting lens induced in a medium. The concept of an effective Fresnel number is used to provide a simple linear relationship between the focal displacement and the nonlinear phase distortions. The influence of nonlinear absorption on the sensitivity of the focal shift measurements is also discussed