We present a theoretical method for analyzing the first-order optics of stabilized zoom lenses with two focal-length-variable elements. The zoom equations are established through the use of the Gaussian brackets method. This is done because the optical power of the focal-length-variable elements varies during the zooming process. The first and second derivatives and the Hessian matrix of the zoom equations with respect to the Gaussian parameters are determined using the equations. These parameters could represent the sensitivity of the zoom ratio of the system to changes in the corresponding system variables. We select the initial values of these system variables, i.e. the magnification of the focal-length-variable element and the structure parameters of the fixed lens group, to be close to the steepest gradient direction. Here the sensitivity of the system focal length is high with respect to variations in the zoom variables. This process leads to an increase in the zoom ratio of the zoom system. The results show successful four-group stabilized zoom lens designs with 2:1 and 5:1 zoom ratios, using two deformable mirrors as focal-length-variable elements. This system, with the inherent characteristics of a steepest gradient, could miniaturize zoom systems.
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