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The Inverse Scattering Problem in Structural Determinations

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Inverse Scattering Problems in Optics

Part of the book series: Topics in Current Physics ((TCPHY,volume 20))

Abstract

The aim of this contribution is to formulate, and to attempt to construct a solution to, the inverse scattering problem. In Sect.2.1, the fundamental epistemological foundations of this task are outlined. Section 2.2 presents the direct scattering problem, and the mathematical equipment required for treating the inverse problem is given in Sect.2.3.

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References

  1. Plato: The Collected Dialogues ( Princeton University Press, Princeton 1973 )

    Google Scholar 

  2. K.R. Popper: The Logic of Scientific Discovery ( Hutchinson, London 1972 )

    Google Scholar 

  3. J.W. Yolton: The Philosophy of Science of A.S. Eddington ( Martinus Nijhoff, The Hague 1960 )

    Google Scholar 

  4. H.P. Baltes: “Introduction”, in Inverse Source Problems in Optics, ed. by H.P. Baltes, Topics in Current Physics, Vol. 9 ( Springer, Berlin Heidelberg, New York 1978 )

    Chapter  Google Scholar 

  5. I. Kant: Critique of Pure Reason, translated by N. Kemp Smith ( Macmillan, London 1964 )

    Google Scholar 

  6. I. Berlin: The Purpose of Philosophy“, in Concepts and Categories, ed. by I. Berlin ( Hogarth Press, London 1978 ) pp. 1–11

    Google Scholar 

  7. A. Eddington: The Philosophy of Physical Science ( Cambridge University Press, Cambridge 1939 )

    MATH  Google Scholar 

  8. J. Agassi: Science in Flux ( Reidel, Dordrecht 1975 )

    Book  Google Scholar 

  9. A. Eddington: Relativity Theory of Protons and Electrons ( Cambridge University Press, Cambridge 1936 )

    Google Scholar 

  10. K.R. Popper: Conjectures and Refutations ( Routledge and Kegan Paul, London 1963 )

    Google Scholar 

  11. J.P. Eckermann: Conversations with Goethe ( Dent and Sons, London 1951 )

    Google Scholar 

  12. Marcus Annaeus Lucanus: The Civil War (The Pharsalia) ( Heinemann, London 1928 )

    Google Scholar 

  13. M. Born, E. Wolf: Principles of Optics ( Pergamon Press, Oxford 1975 )

    Google Scholar 

  14. M.J. Beran, G.B. Parrent, Jr.: Theory of Partial Coherence (Prentice-Hall,Englewood Cliffs 1964 )

    Google Scholar 

  15. P.H. Morse, H. Feschbach: Methods of Theoretical Physics ( McGraw-Hill, New York 1953 )

    MATH  Google Scholar 

  16. G. Ross: Opt. Acta 25, 57 (1978)

    Article  ADS  Google Scholar 

  17. J.W. Goodman: Introduction to Fourier Optics ( McGraw-Hill, New York 1968 )

    Google Scholar 

  18. U. Frisch: Ann. d’Astrophys. 29, 645 (1966); 30, 565 (1967)

    Google Scholar 

  19. D.A. Jarvis: The Characterization of Orientation in Anisotropie Materials by Means of Elastic Light Scattering; Ph.D. Thesis, Queen Elizabeth College, University of London (1977)

    Google Scholar 

  20. R.E.A.C. Paley, N. Wiener: Fourier Transforms in the Complex Domain (American Mathematical Society, Ann Arbor, Michigan 1934 )

    Google Scholar 

  21. R.P. Boas: Entire Functions ( Academic Press, New York 1954 )

    MATH  Google Scholar 

  22. K. Chadan, P.C. Sabatier: Inverse Problems in Quantum Scattering Theory (Springer, Berlin, Heidelberg, New York 1977 )

    Google Scholar 

  23. B.Ja. Levin: Distribution of Zeros of Entire Functions (American Mathematical Society, Providence, Rhode Island 1964 )

    Google Scholar 

  24. T. Kawata: Fourier Analysis in Probability Theory ( Academic Press, New York 1972 )

    MATH  Google Scholar 

  25. E.C. Titchmarsh: Proc. Lond. Math. Soc. (2) 25, 283 (1925)

    Article  MathSciNet  Google Scholar 

  26. G. Ross, M.A. Fiddy, M. Nieto-Vesperinas, M.W.L. Wheeler: Proc. Roy. Soc. (London) A360, 25 (1973)

    ADS  MathSciNet  Google Scholar 

  27. H.B. Voelcker, A.A.G. Requicha: IEEE Comm. 21, 933 (1973)

    Article  Google Scholar 

  28. B.A. Fuks: Introduction to the Theory of Analytic Functions of Several Complex Variables (American Mathematical Society, Providence, Rhode Island 1963 )

    Google Scholar 

  29. M. Plancherel, G. Polya: Commentari Mathematici Helvetici 9,224 (1936–7)

    Google Scholar 

  30. G. Ross, M.A. Fiddy, M. Nieto-Vesperinas, I. Manolitsakis: Opt. Acta 26, 229 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  31. D. Gabor: “Light and Information”, in Proc. Symposium on Astronomical Optics and Related Subjects, ed. by Z. Kopal ( North-Holland, Amsterdam 1956 ) pp. 17–30

    Google Scholar 

  32. G. Ross: Phil. Trans. Roy. Soc. London A268, 177 (1970)

    Article  ADS  Google Scholar 

  33. J.T. Winthrop: J. Opt. Soc. Am. 67, 833 (1977)

    Article  Google Scholar 

  34. C.E. Shannon: Bell Systems Techn. J. 27, 623 (1948)

    MathSciNet  Google Scholar 

  35. B.J. Hoenders, H.A. Ferwerda: Optik 37, 542 (1973)

    Google Scholar 

  36. A.S.B. Holland: Introduction to the Theory of Entire Functions ( Academic Press, London 1973 )

    MATH  Google Scholar 

  37. G. Ross, M.A. Fiddy: Opt. Acta 25, 205 (1978)

    Article  ADS  Google Scholar 

  38. H.M. Nussenzveig: Causality and Dispersion Relations ( Academic Press, London 1972 )

    Google Scholar 

  39. J.S. Toll: Phys. Rev. 104, 1760 (1956)

    Article  ADS  MathSciNet  Google Scholar 

  40. D.L. Misell, R.E. Burge, A.H. Greenaway: J. Phys. D 7, L27 (1974)

    Article  ADS  Google Scholar 

  41. R.E. Burge, M.A. Fiddy, A.H. Greenaway, G. Ross: Proc. Roy. Soc. London A350, 191 (1976)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  42. A.H. Greenaway: Opt. Lett. 1, 10 (1977)

    Article  ADS  Google Scholar 

  43. H.A. Ferwerda: The Phase Reconstruction Problem for Wave Amplitudes and Coherence Functions“, in Inverse Source Problems in Optics,ed. by H.P. Baltes, Topics in Current Physics, Vol.9 (Springer, Berlin, Heidelberg, New York 1978) Chap.2

    Google Scholar 

  44. D.L. Misell: J. Phys. D 6, L6 and 2200 (1973)

    Google Scholar 

  45. R.W. Gerchberg, W.O. Saxton: Optik 35, 237 (1972)

    Google Scholar 

  46. D.L. Misell: Adv. in Optical and Electron Microscopy 7, 185 (1978)

    Google Scholar 

  47. G. Ross, G.J. Brownsey: Plastics and Rubber: Materials and Applications 1, 109 (1976)

    Google Scholar 

  48. E. Wolf: J. Opt. Soc. Am. 60, 18 (1970)

    Article  ADS  Google Scholar 

  49. P.D. Rowley: J. Opt. Soc. Am. 59, 1946 (1969)

    Article  Google Scholar 

  50. R. Dändliker, K. Weiss: Opt. Commun. 1, 323 (1970)

    Article  ADS  Google Scholar 

  51. W. Carter: J. Opt. Soc. Am. 60, 306 (1970)

    Article  ADS  Google Scholar 

  52. A. Klug, R.A. Crowther: Nature 238, 435 (1972)

    Article  ADS  Google Scholar 

  53. M.V. Berry, D.F. Gibbs: Proc. Roy. Soc. Lond. A314, 143 (1970)

    Article  ADS  Google Scholar 

  54. D.W. Sweeney, C.M. Vest: Appl. Opt. 12, 2649 (1973)

    Article  ADS  Google Scholar 

  55. H.H. Lipson, W.A. Wooster: Interpretation of X-ray Diffraction Photographs ( Macmillan, London 1960 )

    Google Scholar 

  56. H.G. Schmidt-Weinmar: J. Opt. Soc. Am. 65, 1059 (1975)

    Article  ADS  Google Scholar 

  57. A.F. Fercher, H. Bartelt, H. Becker, E. Wiltschko: Appl. Opt. 18, 2427 (1979)

    Article  ADS  Google Scholar 

  58. B.J. Hoenders: The Uniqueness of Inverse Problems“, in Inverse Source Problems in Optics,ed. by H.P. Baltes, Topics in Current Physics, Vol.9 (Springer, Berlin, Heidelberg, New York 1978) Chap.3

    Google Scholar 

  59. N. Wiener: The Fourier Integral and Certain of its Applications ( Dover, New York 1933 )

    Google Scholar 

  60. A. Papoulis: Probability, Random Variables and Stochastic Processes ( McGraw-Hill, New York 1965 )

    MATH  Google Scholar 

  61. J.L. Doob: Stochastic Processes ( Wiley, New York 1967 )

    Google Scholar 

  62. G. Ross: Opt. Acta 15, 451 (1968)

    Article  ADS  Google Scholar 

  63. E. Lukacs: Characteristic Functions ( Griffin, London 1960 )

    MATH  Google Scholar 

  64. R.C. Bourret: Can. J. Phys. 40, 782 (1962)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  65. R.C. Bourret: Nuovo Cimento 26, 1 (1962)

    MATH  MathSciNet  Google Scholar 

  66. J.B. Keller: Proc. Symp. Appl. Math. 13, 227 (1962)

    Google Scholar 

  67. Yu.V. Linnik: Decomposition of Probability Distributions ( Oliver and Boyd, London 1964 )

    MATH  Google Scholar 

  68. E.C. Titchmarsh: Introduction to the Theory of the Fourier Integral (Oxford University Press, Oxford 1948 )

    Google Scholar 

  69. G.N. Watson: A Treatise on the Theory of Bessel Functions ( Cambridge University Press, Cambridge 1958 )

    Google Scholar 

  70. G. Ross: Opt. Acta 16, 95 (1969)

    Article  ADS  Google Scholar 

  71. G. Ross, R.L. Addleman: J. Phys. D 6, 1537 (1973)

    Article  ADS  Google Scholar 

  72. L.H. Koopmans: The Spectral Analysis of Time Series ( Academic Press, New York 1974 )

    MATH  Google Scholar 

  73. G.M. Jenkins, D.G. Watts: Spectral Analysis and its Applications ( Holden-Day, San Francisco 1968 )

    MATH  Google Scholar 

  74. R.B. Blackman, J.W. Tukey: The Measurement of Power Spectra ( Dover, New York 1958 )

    Google Scholar 

  75. S. Bochner: Lectures on Fourier Integrals ( Princeton University Press, Princeton 1959 )

    MATH  Google Scholar 

  76. E. Parzen: Technometrics 3, 167 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  77. M. Nieto-Vesperinas: “Statistics of Amorphous Absorbing Media and Object Wave Determination Using the Theory of Entire Functions”; Ph.D. Thesis, Queen Elizabeth College, University of London (1978)

    Google Scholar 

  78. G.F. Chew: The Analytic S Matrix (Benjamin, New York 1966 )

    Google Scholar 

  79. N.J. Bershad: J. Opt. Soc. Am. 59, 157 (1969)

    Article  ADS  Google Scholar 

  80. J.W. von Goethe: Faust, translated by W.H. Bruford ( Dent and Sons, London 1954 )

    Google Scholar 

  81. Lakatos, I.: Mathematics, Science and Epistemology, ed. by J. Worral, G. Currie (Cambridge University Press 1978 )

    Chapter  Google Scholar 

  82. Grene, M.: The Knower and the Known ( Faber and Faber, London 1966 )

    Google Scholar 

  83. Ronkin, L.I.: An analogy of the canonical product for entire functions of several complex variables. Trans. Moscow Math. Soc. 18 (1968)

    Google Scholar 

  84. Bochner, S., Martin, W.: Several Complex Variables (Princeton University Press 1948) Nussenzveig, H.M.: Phase problem in coherence theory. J. Math. Phys. 8, 561 (1967)

    Google Scholar 

  85. Hoenders, B.J.: On the solution of the phase retrieval problem. J. Math. Phys. 16, 1719 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  86. Wolf, E.: Is a complete determination of the energy spectrum of light possible from measurements of the complex degree of coherence? Proc. Phys. Soc. 80, 1269 (1962)

    Article  ADS  Google Scholar 

  87. Helstrom, C.W.: Resolvable degrees of freedom in observation of a coherent object. J. Opt. Soc. Am. 67, 833 (1977)

    Article  ADS  Google Scholar 

  88. Mitsui, T.: “X-ray diffraction studies of membranes”, in Advances in Biophysics, ed. by Masao Kotani, Vol.10 (University Park Press 1978) p. 115

    Google Scholar 

  89. O’Neill, E.L., Walther, A.: The question of phase in image formation. Opt. Acta 10, 33 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  90. Saxton, W.O.: Computer Techniques for Image Processing in Electron Microscopy ( Academic Press, New York 1978 )

    Google Scholar 

  91. de Bruijn, N.G.: The roots of trigonometric integrals. Duke. Math. J. 17, 197 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  92. de Brange, L.: Hilbert Spaces of Entire Functions (Prentice-Hall, New York 1968 )

    Google Scholar 

  93. Bond, F.E., Cahn, C.R.: On sampling the zeros of bandwidth limited signals. I.R.E. Trans. Information Theory, September 110 (1958)

    Google Scholar 

  94. Levinson, N.: Gap and density theorems. Amer. Math. Soc. Colloq. Publ. 26 (1940)

    Google Scholar 

  95. Cartwright, M.L.: On the zeros of certain integral functions I and II. Quart. J. Math. (Oxford Series) (1) 1, 38 (1930) and 2, 113 (1931)

    Article  Google Scholar 

  96. Cartwright, M.L.: Integral Functions, Cambridge tracts in mathematics and mathematical physics, No.44 (Cambridge University Press 1955 )

    Google Scholar 

  97. Stein, R.S., Wilson, P.R.: Scattering of light by polymer films possessing correlated orientation fluctuations. J. Appl. Phys. 33, 1914 (1962)

    Article  ADS  Google Scholar 

  98. Stein, R.S., Hotta, T.: Light scattering from oriented polymer films. J. Appl. Phys. 35, 2237 (1964)

    Article  ADS  Google Scholar 

  99. Stein, R.S., Erhardt, P.F., Clough, S.B., Adams, G.: Scattering of light by films having non-random orientation fluctuations. J. Appl. Phys. 37, 3980 (1966)

    Article  ADS  Google Scholar 

  100. Roe, R.J., Krigbaum, W.R.: Description of crystalline orientation in polycrystalline materials having fibre texture. J. Chem. Phys. 40, 2608 (1964)

    Article  ADS  Google Scholar 

  101. Krigbaum, W.R., Roe, R.J.: Crystalline orientation in materials having fibre texture II. A study of strained samples of cross linked polyethylene. J. Chem. Phys. 41, 737 (1964)

    Article  ADS  Google Scholar 

  102. Roe, R.J.: Description of crystalline orientation in polycrystalline materials III. General solution to pole figure inversion. J. Appl. Phys. 36, 2024 (1965)

    Article  ADS  Google Scholar 

  103. Roe, R.J. Methods of description of orientation in polymers. J. Polymer Sci., Pt. A-2, 8, 1187 (1970)

    Google Scholar 

  104. Nomura, S., Kawai, H., Kumira, I., Kagiyama, M.: General description of orientation factors in terms of expansion of orientation distribution function in a series of spherical harmonics. J. Polymer Sci. Pt. A-2, 8, 383 (1970)

    Article  Google Scholar 

  105. Goldstein, M., Michalik, E.R.: Theory of scattering by an inhomogeneous solid possessing fluctuations in density and anisotropy. J. Appl. Phys. 26, 1450 (1955)

    Article  MATH  ADS  Google Scholar 

  106. Debye, P., Bueche, A.M.: Scattering by an inhomogeneous solid. J. Appl. Phys. 20, 518 (1949)

    Article  ADS  Google Scholar 

  107. Ward, I.M.: Optical and mechanical anisotropy of crystalline polymers. Proc. Phys. Soc. 80, 1176 (1962)

    Article  ADS  MathSciNet  Google Scholar 

  108. Nieto-Vesperinas, M., Ross, G., Fiddy, M.A.: “The theory of light scattering in absorbing media”, in: Optica Hoy y Mariana, ed. by J. Bescos, A. Hidalgo, L. Plaza, J. Santamaria ( Instituto de Optica, Madrid 1978 ) pp. 747–750

    Google Scholar 

  109. Ross, G., Fiddy, M.A., Nieto-Vesperinas, M., Manolitsakis, I.: “The nature of the analytically-of scattered fields in the Fresnel and Fraunhofer region”, in: Optica Hoy y Mariana, ed. by J. Bescos, A. Hidalgo, L. Plaza, J. Santamaria ( Instituto de Optica, Madrid 1978 ) pp. 739–742

    Google Scholar 

  110. Fiddy, M.A., Ross, G., Nieto-Vesperinas, M.: “The encoding of structural information, in scattering and image formation, by the zeros of entire functions”, in: Optica Hoy y Mariana, ed. by J. Bescos, A. Hidalgo, L. Plaza, J. Santamaria ( Instituto de Optica, Madrid 1978 ) pp. 743–746

    Google Scholar 

  111. Ross, G., Fiddy, M.A., Moezzi, H., Nieto-Vesperinas, M: “The inverse problem in light scattering”, in: Antennas and Propagation, Vol.I., International Symposium Digest (IEEE 1979) pp. 232–235

    Google Scholar 

  112. Fiddy, M.A., Ross G.: Analytic Fourier optics: the encoding of information by complex zeros. Opt. Acta 26, 1139–1146 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  113. Ross, G., Jarvis, D.A.: The characterization of orientation in anisotropic media by means of elastic light scattering. I. The inhomogeneity of anisotropic media and its description. Opt. Acta 27, 359–370 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  114. Ross, G., Jarvis, D.A.: The characterization of orientation in anisotropic media by means of elastic light scattering. II. The elastic scattering of light by an inhomogeneous medium. Opt. Acta 27, 371–384 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  115. Ross, G., Fiddy, M.A., Moezzi, H.: The solution to the inverse scattering problem, based on zero location from two measurements. Opt. Acta (in press)

    Google Scholar 

  116. Ross, G., Nieto-Vesperinas, M.: Theory of light scattering by amorphous absorbing media. Opt. Acta (in press)

    Google Scholar 

  117. Fienup, J.R.: Reconstruction of an object from the modulus of its Fourier transform. Opt. Lett. 3, 27–29 (1978)

    Article  ADS  Google Scholar 

  118. Bruck, Yu. M., Sodin, L.G.: On the ambiguity of the image reconstruction problem. Opt. Commun. 30, 304–308 (1979)

    Article  ADS  Google Scholar 

  119. Boucher, R.H.: “Convergence of algorithms for phase retrieval from two intensity distributions”, in: Proc. I.O.C.C. Meeting, Washington April 8, 1980, to be published by S.P.I.E.

    Google Scholar 

  120. Misell, D.L.: “The phase problem in electron microscopy”, in: Advances in Optical and Electron Microscopy, Vol. 7, ed. by V.E. Cosslett, R. Barer ( Academic Press, London 1978 ) pp. 185–219

    Google Scholar 

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Ross, G., Fiddy, M.A., Nieto-Vesperinas, M. (1980). The Inverse Scattering Problem in Structural Determinations. In: Baltes, H.P. (eds) Inverse Scattering Problems in Optics. Topics in Current Physics, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-81472-3_2

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  • DOI: https://doi.org/10.1007/978-3-642-81472-3_2

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