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A solution of the inverse scattering problem for dielectric circular cylindrical scatterers has been presented on the basis of the exact electromagnetic theory of scattering. Two new contributions have been made. First, a new concept of characteristic functions has been introduced. It has been shown that the extremum properties of these functions determine the radii uniquely from the coefficients of the cylindrical wave expansion of the scattered field amplitude. Second, it has been shown that the complex amplitude coefficients can be uniquely retrieved from the measurement of the far-field irradiance alone and that no phase information is necessary. This has been achieved by giving a Fourier representation for the scattered far-field irradiance; the Fourier coefficients are obtained in terms of the amplitude coefficients. Determination of the amplitude coefficients from the Fourier coefficients is facilitated by a special relation that has been found to exist between the real and the imaginary parts of each amplitude coefficient. Some results of our numerical investigation have also been included.