This report focuses on a design method for gradient index (GRIN) lens antennas with controllable aperture field distributions. First, we derive differential equations representing optical paths in a gradient index medium with two optical surfaces by using geometrical optics, and then we formulate a novel design method for GRIN lens antennas based on these equations. The Levenberg-Marquardt algorithm is applied as a nonlinear least squares method to satisfy two conditions-focusing and shaping the aperture field distribution-thus realizing a prescribed radiation pattern. The conditions can be fulfilled by optimizing only the index (or permittivity) distribution, whereas the shapes of the optical surfaces remain as free parameters that can be utilized for other purposes, such as reducing reflection losses that occur on the surfaces, as illustrated in this report. A plano-concave GRIN lens is designed as an example, applying the proposed method, to realize a sidelobe level of -30 dB pseudo Taylor distribution, and a maximum sidelobe level of -29.1 dB was observed, indicating it is sufficiently accurate for practical use. In addition, we discuss the convergence of this method considering the relationship between the number of the initial conditions and the differential order of the design equations, factoring in scale invariance of the design equations.

In this paper, two techniques are proposed for accelerating and stabilizing the Levenberg-Marquardt (LM) method where its conventional stabilizer matrix (identity matrix) is superseded by (1) a diagonal matrix whose elements are column norms of Jacobian matrix J, or (2) a non-diagonal square root matrix of J TJ. Geometrically, these techniques make constraint conditions of the LM method fitted better to relevant cost function than conventional one. Results of numerical simulations show that proposed techniques are effective when both column norm ratio of J and mutual interactions between arguments of the cost function are large. Especially, the technique (2) introduces a new LM method of damped Gauss-Newton (GN) type which satisfies both properties of global convergence and quadratic convergence by controlling Marquardt factor and can stabilize convergence numerically. Performance of the LMM techniques are compared also with a damped GN method with line search procedure.

This paper presents a fast inversion method for electromagnetic imaging of cylindrical dielectric objects with the optimal regularization parameter used in the Levenberg-Marquardt method. A novel procedure for choosing the optimal regularization parameter is proposed. The method of moments with pulse-basis functions and point matching is applied to discretize the equations for the scattered electric field and the total electric field inside the object. Then the inverse scattering problem is reduced to solving the matrix equation for the unknown expansion coefficients of a contrast function, which is represented as a function of the relative permittivity of the object. The matrix equation may be solved in the least-squares sense with the Levenberg-Marquardt method. Thus the contrast function can be reconstructed by the minimization of a functional, which is expressed as the sum of a standard error term on the scattered electric field and an additional regularization term. While a regularization parameter is usually chosen according to the generalized cross-validation (GCV) method, the optimal one is now determined by minimizing the absolute value of the radius of curvature of the GCV function. This scheme is quite different from the GCV method. Numerical results are presented for a circular cylinder and a stratified circular cylinder consisting of two concentric homogeneous layers. The convergence behaviors of the proposed method and the GCV method are compared with each other. It is confirmed from the numerical results that the proposed method provides successful reconstructions with the property of much faster convergence than the conventional GCV method.

This paper provides a normalized Iterative Feedback Tuning (IFT) method that assures the boundedness of the gradient vector estimate (ρ) and the Hessian matrix estimate without the assumption that the internal signals are bounded. The proposed method uses the unbiased Gauss-Newton direction by the addition of the 4-th experiment. We also present blended control criteria and a PID-like controller as new design choices. In examples, the normalized IFT method results in a good convergence although the internal signal or the measurement noise variance is large.