Abstract
The direct problem in reflection is the calculation of the reflection amplitudes (and thence the reflectivities and the ellipsometric ratio), given the characteristics of the reflecting profile. Inverse or inversion problems consist in the estimation of the profile characteristics, given some experimental reflection or transmission data (or both).
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Further Readings
Introductory and review articles on inverse problems
Dyson FJ (1976) Old and new approaches to the inverse scattering problem. In: Lieb EH, Simon B, Wightman AS (eds) Studies in mathematical physics, essays in Honor of Valentine Bargmann. Princeton University Press, Princeton, pp 151–167
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Keller JB (1976) Inverse problems. Am Math Monthly 83:107–118
Newton RG (1970) Inverse problems in physics. SIAM Rev 12:346–355
Papers and books on the inverse problem of scattering and diffraction theory
Agranovich ZS, Marchenko VA (1963) The inverse problem of scattering theory. Gordon and Breach, New York
Bargmann V (1949a) Remarks on the determination of a central field of force from the elastic scattering phase shifts. Phys Rev 75:301–303
Bargmann V (1949b) On the connection between phase shifts and scattering potential. Rev Mod Phys 21:488–493
Chadan K, Sabatier PC (1977) Inverse problems in quantum scattering theory. Springer, New York
Faddeev LD (1963) The inverse problem in the quantum theory of scattering. J Math Phys 4:72–104
Jost R, Kohn W (1952) Equivalent potentials. Phys Rev 88:382–385
Marchenko VA (1955) The construction of the potential energy from the phases of the scattered waves. Dokl Akad Nauk SSSR 104:695–698 Math. Rev. 17:740 (1956))
Nieto-Vesperinas M (2006) Scattering and diffraction in physical optics, 2nd edn. World Scientific, Singapore
Collections of papers on electromagnetic and optical inverse problems
Baltes HP (ed) (1978) Inverse source problems in optics. Springer, Berlin
Baltes HP (ed) (1980) Inverse scattering problems in optics. Springer, Berlin
Boemer WM, Jordan AK, Kay IW (eds) (1981) IEEE transactions on antennas and propagation. Inverse Methods Electromagnet AP-29(2):185–417
Devaney AJ (ed) (1985) Inverse problems in propagation and scattering. J Opt Soc Am A2:1901–2061 (In relation to Section 11-5, see especially the papers by Landouceur HD and Jordan AK, and by Jaggard DL and Kim Y.)
Another important inversion problem arises in the extraction of the electron density as a function of height from the measured times of travel of nearly monochromatic radio pulses which are reflected from the ionosphere. The problem reduces to that of solving Abel’s integral equation, and is related to several inverse problems in mechanics (Keller 1976, quoted above). The ionospheric case is considered in detail by
Budden KG (1961) Radio waves in the ionosphere. Cambridge University Press, Cambridge (Chapter 10)
Budden KG (1985) The propagation of radio waves. Cambridge University Press, Cambridge (Chapter 12)
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Lekner, J. (2016). Inverse Problems. In: Theory of Reflection. Springer Series on Atomic, Optical, and Plasma Physics, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-319-23627-8_11
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