OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 2, Iss. 11 — Nov. 1, 1985
  • pp: 1922–1930

Accurate one-dimensional inverse scattering using a nonlinear renormalization technique

D. L. Jaggard and Y. Kim  »View Author Affiliations

JOSA A, Vol. 2, Issue 11, pp. 1922-1930 (1985)

View Full Text Article

Enhanced HTML    Acrobat PDF (934 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



A nonlinear method based on the inversion of the Riccati equation is presented here for the one-dimensional nondispersive inverse-scattering problem. This method avoids the significant errors in both amplitude and phase that plague most linearized (e.g., Born or its varients) inversion schemes. Instead, a nonlinear approximation to the Riccati equation is used for the accurate determination of the refractive-index amplitude from reflection data. This information is subsequently used to stretch the coordinates so as to remove the phase-accumulation error. The resulting refractive-index reconstructions are therefore accurate both in amplitude and in longitudinal placement as evidenced by the excellent comparison with exact theory. The method is applicable to both continuous and discontinuous refractive profiles and is supported by experimental measurements.

© 1985 Optical Society of America

Original Manuscript: May 30, 1985
Manuscript Accepted: July 25, 1985
Published: November 1, 1985

D. L. Jaggard and Y. Kim, "Accurate one-dimensional inverse scattering using a nonlinear renormalization technique," J. Opt. Soc. Am. A 2, 1922-1930 (1985)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. W. Strutt, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).
  2. M. Born, “Quantenmechanik der Stossvorgänge,” Z. Phys. XXXVII, 803–827 (1926).
  3. B. Ambarzumian, “On the problem of the diffuse reflection of light,” J. Phys. (Moscow) VIII, 65–75 (1944).(Here the Riccati equation is derived using an invarient embedding method. The result is expressed as an integral equation for the reflection coefficient rather than the usual differential equation.)
  4. J. R. Pierce, “A note on the transmission line equation in terms of impedance,” Bell Syst. Tech. J. 22, 263–265 (1943).(In this short note, the Riccati differential equation is derived using circuit concepts and physical insight. It is noted that the same equation describes the electron-optics equation for paraxial trajectories.) [CrossRef]
  5. L. R. Walker, N. Wax, “Non-uniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946). [CrossRef]
  6. S. A. Schelkunoff, “Remarks concerning wave propagation in stratified media,” in The Theory of Electromagnetic Waves, M. Kline, ed. (Interscience, New York, 1951), pp. 181–192.
  7. For a modern derivation of the Riccati equation, see, e.g., F. T. Ulaby, R. K. Moore, A. K. Fung, Fundamentals and Radiometry, Vol. 1 of Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1981), pp. 82–84.
  8. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).
  9. C. H. Greenewalt, W. Brandt, D. D. Friel, “Iridescent colors of hummingbird feathers,” J. Opt. Soc. Am. 50, 1005–1013 (1960). [CrossRef]
  10. I. A. Kozlov, “The limit of applicability of the first approximation for determining the reflection coefficient in the theory of nonuniform lines,” Radio Eng. Electron. Phys. 14, 132–133 (1969).
  11. I. S. Gaydabura, “A method for linearization of the equation of an inhomogeneous line,” Radio Eng. Electron. Phys. 16, 1625–1627 (1971).
  12. M. Lahlou, J. K. Cohen, N. Bleistein, “Highly accurate inversion methods for three-dimensional stratified media,” Soc. Ind. Appl. Math. 43, 726—758 (1983). [CrossRef]
  13. The techniques include invariant embedding, use of the impedance concept and a direct manipulation of Maxwell’s equations or the Helmholtz equation. See Refs. 5–8 for examples.
  14. The dispersive case for a cold plasma is treated by D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbational and high frequency profile inversion,” IEEE Trans. Antennas Propag. AP-31, 804–803 (1983),while the nondispersive case is discussed in Ref. 8. [CrossRef]
  15. Note that r(p) must satisfy energy conservation and causality. However, r̂(p) is a mathematical transformation that does not necessarily satisfy the restriction |r̂(p)|<1.
  16. J. W. Strutt, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917). [CrossRef]
  17. S. M. Rytov, “Diffraction of light by ultrasonic wave,” Izv. Akad. Nauk SSSR Ser. Fiz. 2, 223–259 (1937).
  18. See, e.g., I. Kay, “The inverse scattering problem,” NYU Res. Rep. No. EM-74 (New York University, New York, 1955).
  19. See, e.g., D. L. Jaggard, K. E. Olson, “Numerical reconstruction of dispersionless refractive profiles,” J. Opt. Soc. Am. A 2, 1931–1936 (1985). [CrossRef]
  20. D. L. Jaggard, P. V. Frangos, “Profile reconstruction for dispersionless layered dielectrics with imprecise and band-limited data,” IEEE Trans. Antennas Propag. (to be published).
  21. H. D. Ladouceur, A. K. Jordan, “Renormalization of an inverse-scattering theory for inhomogeneous dielectrics,” J. Opt. Soc. Am. A 2, 1916–1921 (1985). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited