Estimation of local error by a neural model in an inverse scattering problem

¹ Laboratoire Dispositifs et Instrumentation pour l'Opto-électronique et Micro-ondes, EA 3526, 23 rue du Docteur Paul Michelon, 42023 Saint-Etienne Cedex 2, France
² Laboratoire Traitement du Signal et Instrumentation, UMR CNRS 5516, 10 rue Barrouin, 42000 Saint-Etienne, France
³ Laboratoire d'Océanographie Dynamique et de Climatologie, UMR CNRS 7617, 4 place Jussieu, 75005 Paris, France
⁴ Centre de Recherche en Informatique du CNAN, 292 rue Saint Martin, 75141 Paris Cedex 03, France

Corresponding author: stephane.robert@univ-st-etienne.fr

Received: 20 October 2004
Revised: 15 March 2005
Accepted: 12 April 2005
Published online: 21 June 2005

Abstract

Characterization of optical gratings by resolution of inverse scattering problem has become a widely used tool. Indeed, it is known as a non-destructive, rapid and non-invasive method in opposition with microscopic characterizations. Use of a neural model is generally implemented and has shown better results by comparison with other regression methods. The neural network learns the relationship between the optical signature and the corresponding profile shape. The performance of such a non-linear regression method is usually estimated by the root mean square error calculated on a data set not involved in the training process. However, this error estimation is not very significant and tends to flatten the error in the different areas of variable space. We introduce, in this paper, the calculation of local error for each geometrical parameter representing the profile shape. For this purpose a second neural network is implemented to learn the variance of results obtained by the first one. A comparison with the root mean square error confirms a gain of local precision. Finally, the method is applied in the optical characterization of a semi-conductor grating with a 1 m period.