## Accurate one-dimensional inverse scattering using a nonlinear renormalization technique

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

http://dx.doi.org/10.1364/JOSAA.2.001922

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### Abstract

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

**History**

Original Manuscript: May 30, 1985

Manuscript Accepted: July 25, 1985

Published: November 1, 1985

**Citation**

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)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-2-11-1922

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### References

- J. W. Strutt, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).
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- 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.
- 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]
- 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.
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- See, e.g., I. Kay, “The inverse scattering problem,” NYU Res. Rep. No. EM-74 (New York University, New York, 1955).
- 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]
- 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).
- 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]

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