In this paper, we use a powerful mathematical concept based on Fisher information (FI) to estimate the cross-sensitivity of multi-parameter sensors. FI has been used in various branches of natural and social sciences [1], in both classical and quantum systems [2]. Lately, it has been used for estimation of interferometer sensitivity in quantum optics [3], as well as in classical optical interferometers [4]. Here, we expand its application to fibre grating sensors that are sensitive to several environmental (temperature, humidity) and mechanical (strain, pressure, curvature) parameters simultaneously [5]. FI is a local probability measure of obtaining an unknown parameter θ from the data measured at the sensor output, X. In the theory of measurement, FI directly enters the relation which determines the lower bound of the measurement uncertainty known as Cramer-Rao bound (CRB = 1/ FI (θ ) ). The sensor sensitivity is optimized by maximization of the corresponding FI. In the case of a multi-parameter sensor, FI assumes a matrix form defined via the probability distribution function f(X| θ) of the measured variable X conditioned by θ, where both X and θ are vector quantities. ( ) ( ) i n j n f X FI E i j i j ; 1,2,... ; 1,2,..., 2 ln , = = ∂ ∂ ∂ = − θ θ θ θ . Here, we develop a comprehensive model of the spectral response of fibre gratings to changes in multiple parameters and apply Fisher's formalism to optimize their sensitivity in the realistic parameter space. We start from a 1-parameter model of a long-period grating sensor of curvature and corroborate our approach by comparison with the conventional sensitivity models and the experimental data. We then expand the model to describe a 2-parameter long-period grating sensor of curvature and temperature. Based on this analysis, we recommend the multi-parameter interrogation scheme (estimator) and optimize grating parameters to render the highest sensitivity.V International School and Conference on Photonics and COST actions: MP1204, BM1205 and MP1205 and the Second international workshop "Control of light and matter waves propagation and localization in photonic lattices" : PHOTONICA2015 : book of abstracts; August 24-28, 2015; Belgrad
