volume | PIER Journals

Abstract

This study concerns the 2D inverse problem of the retrieval, using external field data, of either one of the two physical parameters, constituted by the real and imaginary parts of the permittivity, of a z-independent cylindrical dielectric specimen subjected to an external, z-independent, quasistatic electric field. Six other parameters enter into the inverse problem. They are termed nuisance parameters because: 1) they are not retrieved during the inversion and 2) uncertainty as to their actual values can adversely affect the accuracy of the retrieval of the permittivity. This inverse problem is shown to have an exact, mathematically-explicit, solution, both for continuous and discrete input data, whose properties, with respect to the various nuisance parameter uncertainties, are analyzed, first in a mathematical, and subsequently in a numerical manner for noiseless data. It is found that: a) optimal inversion requires data registered at only a small number of sensors, b) the inverse solution, satisfying pre-existing physical constraints, exists and is unique. Moreover, the inverse solution is shown to be unstable with respect to three nuisance parameter uncertainties, the consequence of which is large retrieval inaccuracy for small nuisance parameter uncertainties, acting either individually or in combination.

1. Banks, H. T. and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Birkhauser, Boston, 1989.
doi:10.1007/978-1-4612-3700-6

2. Bauer, N., K. Fajans, and S. Z. Lewin, "Refractometry," Physical Methods of Organic Chemistry, A. Weissberger (ed.), Interscience, New York, 1960. Google Scholar

3. Bohren, C. F. and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983.

4. Chen, L., Z.-Y. Lei, R. Yang, X.-W. Shi, and J. Zhang, "Determining the effective electromagnetic parameters of bianisotropic metamaterials with periodic structures," Progress In Electromagnetics Research M, Vol. 29, 79-93, 2013.
doi:10.2528/PIERM13010204 Google Scholar

5. Chen, X., T. M. Grzegorczyk, B.-I. Wu, J. Pacheco Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E, Vol. 70, 016608, 2004.
doi:10.1103/PhysRevE.70.016608 Google Scholar

6. Chylek, P., V. Ramaswamy, A. Ashkin, and J. M. Dziedzic, "Simultaneous determination of refractive index and size of spherical dielectric particles from light scattering data," Appl. Opt., Vol. 22, 2302-2307, 1983.
doi:10.1364/AO.22.002302 Google Scholar

7. Emery, A. F., "The effect of correlations and uncertain parameters on the efficiency of estimating and the precision of estimated parameters," Inverse Engineering Handbook, K. A. Woodbury (ed.), CRC Press, Boca Raton, 2003. Google Scholar

8. Hadamard, J. S., Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Oxford University Press, Oxford, 1923.

9. Hasar, U. C., J. J. Barroso, C. Sabah, Y. Kaya, and M. Ertugrul, "Differential uncertainty analysis for evaluating the accuracy of S-parameter retrieval methods for electromagnetic properties of metamaterial slabs," Opt. Express, Vol. 20, 29002-29022, 2012.
doi:10.1364/OE.20.029002 Google Scholar

10. Hasar, U. C., J. J. Barroso, M. Ertugrul, C. Sabah, and B. Cavusoglu, "Application of a useful uncertainty analysis as a metric tool for assessing the performance of electromagnetic properties retrieval methods of bianisotropic metamaterials," Progress In Electromagnetics Research, Vol. 128, 365-380, 2014.
doi:10.2528/PIER12040802 Google Scholar

11. Lefeuve-Mesgouez, G., A. Mesgouez, E. Ogam, T. Scotti, and A. Wirgin, "Retrieval of the physical properties of an anelastic solid half space from seismic data," J. Appl. Geophys., Vol. 88, 70-82, 2013.
doi:10.1016/j.jappgeo.2012.09.010 Google Scholar

12. Morse, P. M. and H. Feshbach, Methods of Theoretical Phyics, Mc Graw-Hill, New York, 1953.

13. Neittaanmaki, P., M. Rudnicki, and A. Savini, Inverse Problems and Optimal Design in Electricity and Magnetism, Clarendon Press, Oxford, 1996.

14. Pluchino, A. B., S. S. Goldberg, J. N. Dowling, and C. M. Randall, "Refractive-index measurements of single micron-sized carbon particles," Appl. Opt., Vol. 19, 3370-3372, 1980.
doi:10.1364/AO.19.003370 Google Scholar

15. Sambuelli, L., "Uncertainty propagation using some common mixing rules for the modelling and interpretation of electromagnetic data," Near Surf. Geophys., Vol. 7, 285-296, 2009. Google Scholar

16. Scotti, T. and A. Wirgin, "Multiparameter identification of a lossy fluid-like object from its transient acoustic response," Inverse Prob. Sci. Engrg., Vol. 22, 1228-1258, 2014.
doi:10.1080/17415977.2013.867485 Google Scholar

17. Seitz, F., The Modern Theory of Solids, Dover, New York, 1987.

18. Smith, D. R., D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E, Vol. 71, 036617, 2005.
doi:10.1103/PhysRevE.71.036617 Google Scholar

19. Von Hippel, A. R., Dielectrics and Waves, Chapman & Hall, London, 1954.

20. Vuye, G. and T. Lopez-Rios, "Precision in the ellipsometric determination of the optical constants of very thin films," Appl. Opt., Vol. 21, 2968-2971, 1982.
doi:10.1364/AO.21.002968 Google Scholar

21. Yilmaz, T., R. Foster, and Y. Hao, "Detecting vital signs with wearable wireless sensors," Sensors, Vol. 10, 10837-10862, 2010.
doi:10.3390/s101210837 Google Scholar

22. Young, K. F. and H. P. R. Frederikse, "Compilation of the static dielectric constant of inorganic solids," J. Chem. Phys. Ref. Dat, Vol. 2, 313-408, 1973.
doi:10.1063/1.3253121 Google Scholar

Dr. Armand Wirgin