Therefore, the filters can simultaneously exhibit constant passband group delay and high rejection asymmetrical slopes out of band. A systematic extraction procedure and the normalization expressions to accomplish the synthesis have Ž been presented. To validate it, two different filters fourth . and fifth orders have been synthesized, calculating their circuital component values as well as their coupling matrices, and showing their theoretical responses. Likewise, a second- order filter with two transmission zeros has been designed to illustrate the applicability of these kinds of responses. The Ž filter has only one type of electromagnetic coupling capaci- . tive or inductive, depending on the structure , showing that an elliptic response can be obtained without changing the sign of the elements of the coupling matrix. Three different rectangular waveguide geometries have been designed using the mode-matching technique. The comparison with the the- oretical response shows an excellent agreement. REFERENCES 1. J.D. Rhodes, Theory of electrical filters, Wiley, New York, 1976, p. 6. 2. J.D. Rhodes, A low-pass prototype network for microwave linear phase filters, IEEE Trans Microwave Theory Tech MTT-18 Ž . 1970 , 290301. 3. R.J. Cameron, General prototype network-synthesis methods for Ž . microwave filters, ESA J 6 1982 , 193206. 4. W. Gross, Synthese von Hochfrequenzbandfiltern mit Unwegkop- Ž . plungen, Hochfreq Elektroakust 75 1966 , 6775. 5. B. Easter and K.J. Powell, Direct-coupled resonator filters with Ž . improved selectivity, Electron Lett 4 1968 , 415416. 6. U. Rosenberg and W. Hagele, Advanced multimode cavity filter ¨ design using sourceload-resonance circuit cross coupling, IEEE Ž . Microwave Guided Wave Lett 2 1992 , 508510. 7. J. Liang and W.D. Blair, High-Q TE01 mode DR filters for PCS wireless base stations, IEEE Trans Microwave Theory Tech 46 Ž . 1998 , 24932500. 8. U. Rosenberg and W. Hagele, Consideration of parasitic bypass couplings in overmoded cavity filter design, IEEE Trans Mi- Ž . crowave Theory Tech 42 1994 , 13011306. 9. H. Bell, Canonical lowpass prototype network for symmetric Ž . coupled-resonator bandpass filters, Electron Lett 10 1974 , 265266. 10. H. Bell, Canonical asymmetric coupled-resonator filters, IEEE Ž . Trans Microwave Theory Tech MTT-30 1982 , 13351340. 11. K.A. Zaki, C. Chen, and A. Atia, Modeling of coupling by probes in dual mode cavities, IEEE MTT-S Int Symp Dig, New York, NY, 1988, pp. 515518. 12. J.M. Reiter and F. Arndt, Rigorous analysis of arbitrarily shaped H- and E-plane discontinuities in rectangular waveguides by a full-wave boundary contour mode-matching method, IEEE Trans Ž . Microwave Theory Tech 43 1995 , 796801. 13. J.R. Montejo-Garai, Synthesis of N-even order symmetric filters with N transmission zeros by means of source-load cross cou- Ž . pling, Electron Lett 36 2000 , 232233. 14. R.J. Cameron, Fast generation of Chebyshev filter prototypes with asymmetrically-prescribed transmission zeros, ESA J 6 Ž . 1982 , 8395. 15. R.J. Cameron, General coupling matrix synthesis methods for Chebyshev filtering functions, IEEE Trans Microwave Theory Ž . Tech 47 1999 , 433442.  2001 John Wiley & Sons, Inc. REFRACTIVE-INDEX PROFILE OF FIBER OPTICS Moataza A. Hindy 1 1 Department of Electrical Engineering University of Bahrain Tsa-Town, State of Bahrain Recei  ed 4 December 2000 ABSTRACT: An optimized approach for obtaining the complex refrac- ti  e-index profile of graded-index fiber optics is presented. The measured optical reflections from the sample are used to minimize the norm of the difference between measured and calculated data. An iterati  e technique based on a one-dimensional search in the descent direction is applied for obtaining a new estimate of the refracti  e-index profile. Numerical experiments are used to apply the gradient technique in sol  ing optical in erse problems.  2001 John Wiley & Sons, Inc. Microwave Opt Technol Lett 29: 252256, 2001. Key words: optical fiber; refracti  e index; index profile I. INTRODUCTION Graded-index fiber exhibits far less intermodal dispersion due to its refractive-index profile. The different group velocities of the modes tend to be normalized by index grading. Obtain- ing the actual refractive-index profile will lead to accurate modal analysis, determination of the acceptance angle, the Ž . numerical aperture and normalized frequency  -number , the number of modes propagating within the fiber core, the impulse response, and consequently, the information-carrying  capacity of the fiber. In 1 , direct measurement of the refractive indexes of substrates and guiding layers in slab waveguides is presented. The method is based on the excita- tion of leaky waves in substrates and guided waves in guiding layers, owing to the etching of the grating coupler of the top of the substrates. Based on observations of diffracted or guided beams, the optical characteristics of materials are  obtained. In the work of 2 , the authors present a nonde- structive experimental method to measure the monotonically varying refractive-index profiles of planar waveguides using Lloyd’s setup, taking into account the multiple reflections  inside the sample. In 3 , the authors obtained a functional form of the refractive-index profile of a planar microlens  from total shearing interferometric measurements. In 4 and  5 , the authors presented mathematical experimental meth- ods to reconstruct the refractive-index profile of a planar optical waveguide. They used sets of measured effective re- fractive indexes, measured with TE and TM polarization, at different wavelengths. A simple interferometric technique for mapping the refractive-index profile of an optical fiber is  presented in 6 . The WKB inverse method and reflectivity  calculations are presented in 7 to obtain the refractive-index profile in a planar waveguide. Electromagnetic probing of an   inhomogeneous stratified medium is presented in 8  9 . The inverse scattering problem of reconstructing the permittivity and conductivity profiles in an inhomogeneous isotropic lossy dielectric slab has been studied in the time and frequency   domains in the literature 10  17 . These approaches need much smoothing, and the initially assumed profile must be close to the actual one. The electrical parameters cannot be evaluated at large depths, and the reconstruction process   diverges. In 18 , the authors used the costate method in the Ž . time domain, through the knowledge measuring of incident, MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 29, No. 4, May 20 2001 252