2000 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

IEEE Journal of Lightwave Technology
Volume 18 Number 5, May 2000

An Iterative Method for Optical Reconstruction of Graded Index Profiles in Planar Dielectric Waveguides

Page 729.

Abstract:

A nondestructive technique for the reconstruction of refractive index profiles in planar waveguides is presented and analyzed. The approach is based on the integral scattering equations, which permit to relate the refractive index of an inhomogeneous layer to the reflected field intensity at different incidence angles. From this formulation, an iterative algorithm is developed, such as at each iteration step the problem is formulated as the minimization of a functional representing the error between the measurements and the model data. The recovered profile is then used to improve the validity of the approximation in performing the next step. In this approach, the unknown index profile is represented as the sum of a finite series of basis functions avoiding to select a priori the particular functional form (e.g., Gaussian function, complementary error function, etc). The practical effectiveness of this approach is demonstrated by numerically simulating the measurements for different planar waveguides. The influence of measurement uncertainty and noise on the stability of the technique is also evaluated.

References

1. D. Brooks and S. Ruschin, "Improved near-field method for refractive index measurements of optical waveguides", J. Lightwave Technol., vol. 8, pp.  254-256, 1996.
2. M. L. von Bibra and A. Roberts, "Refractive index reconstruction of graded-index buried channel waveguides from their mode intensities", J. Lightwave Technol. , vol. 15, pp.  1695-1699, 1997.
3. F. Caccavale, F. Segato, I. Mansour and M. Gianesin, "A finite differences methods for the reconstruction of refractive index profile from near-filed measurements", J. Lightwave Technol., vol. 16, pp.  1348-1353, 1998.
4. K. S. Chiang, "Construction of refractive-index profiles of planar dielectric waveguide from distribution of effective indexes", J. Lightwave Technol., vol. 14, pp.  385-391, 1985.
5. K. S. Chiang, C. L. Wong, H. P. Chan and Y. T. Chow, "Refractive index profiling of graded-index planar waveguide from effective indexes measured for both mode types and at different wavelengths", J. Lightwave Technol., vol. 14, pp.  827-832, 1996.
6. F. Caccavale, F. Gonella, G. Caneva and I. Mansour, "Iterative simplex-finite difference methods for characterization of optical waveguide", J. Lightwave Technol., vol. 14, pp.  1825-1831, 1996.
7. P. Mathey, P. Jullien and J. L. Bolzinger, "Refractive-index profile reconstructions in planar waveguides by the WKB inverse method and reflectivity calculations", J. Opt. Soc. Am. B., vol. 12, pp.  1663-1670, 1995.
8. W. A. Ramadan, E. Fazio and M. Bertolotti, "Measurements of the refractive-index profile of planar waveguides by the use of a double Lloyd's interferometer", Appl. Opt., vol. 35, pp.  6173-6178, 1996.
9. D. Tonova, A. Paneva and B. Pantchev, "Determination of refractive index profiles of gradient optical waveguides by ellipsometry", Opt. Commun., vol. 150, pp.  121-125, 1998.
10. J. M. Ortega and W. C. Rheiboldt, Iterative Solution of Nonlinear Equations in Several Variables , San Diego, CA: Academic Press, 1970, ch. 7.
11. W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method", IEEE Trans. Med. Imag. , vol. 9, pp.  218-225, 1990.
12. O. S. Haddin and E. S. Ebbini, "Imaging strongly scattering media using a multiple frequency distorted Born iterative method", IEEE Trans. Ultrason. Ferroelect. Freq. Contr., vol. 45, pp.  1485-1496, 1998.
13. R. V. Ramaswamy and R. Srivastava, "Ion-exchanged glass waveguides: A review", J. Lightwave Technol., vol. 6, pp.  984-1001, 1988.
14. A. G. Tijhuis, Electromagnetic Inverse Profiling: Theory and Numerical implementation , Utrecht: The Netherlands: VNU Science, 1987, pp.  108-114. 
15. M. Slaney, A. C. Kak and L. E. Larsen, "Limitations of imaging with first order diffraction tomography", IEEE Trans. Microwaves Theory Tech., vol. 32, pp.  860-874, 1984.
16. A. Franchois and C. Pichot, "Microwave imaging complex permittivity reconstruction with a Leveberg-Marquardt method", IEEE Trans. Antennas Propagat., vol. 45, pp.  203-215, 1997.
17. W. Singer, B. Dobler, H. Schreiber, K. Brenner and B. Messerdchmidt, "Refractive index measurements of gradient-index microlenses by diffraction tomography", Appl. Opt., vol. 35, pp.  2167-2171,  1996.
18. M. Kildemo, R. Ossikovski and M. Stchakovsky, "Measurements of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near IR ellipsometry", Thin Solid Films, vol. 313-314, pp.  108-113, 1998.
19. A. H. Cook, Interference of Electromagnetic Waves, Oxford: U.K.: Clarendon, 1971.
20. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, New York: Academic, 1994.
21. L. Zeni, R. Bernini and R. Pierri, "Reconstruction of doping profiles in semiconductor materials using optical tomography", Solid-State Electron., vol. 43, pp.  761-769, 1999.
22. R. Pierri, R. Persico and R. Bernini, "Information content of Born field scattered by an embedded slab: Multifrequency, multiview and multifrequency-multiview cases", J. Opt. Soc. Amer. A., vol. 16, pp.  2392-2399, 1999.
23. T. C. Wedberg and J. J. Stamnes, "Experimental examination of the quantitative imaging properties of optical diffraction tomography", J. Opt. Soc. Amer. A., vol. 12, pp.  493-500, 1995.
24. M. H. Maleki, A. J. Devaney and A. Schatzberg, "Tomographic reconstruction from optical scattered intensities", J. Opt. Soc. Amer. A., vol. 9, pp.  1356-1363, 1992.