Eikonal Aberrations in Gradient Cylindrical Lenses
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NNA AND FEEDER SYSTEMS
Eikonal Aberrations in Gradient Cylindrical Lenses A. S. Venetskiya, * and V. A. Kaloshina aKotelnikov
Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow, 125009 Russia *e-mail: [email protected] Received October 11, 2019; revised November 15, 2019; accepted November 25, 2019
Abstract—Approximate formulas are derived for eikonal on the output surface of a cylindrical dielectric lens with a gradient of refractive index along the Cartesian coordinate for a source that is displaced from the focal point on the symmetry axis of the lens. The first formula is obtained for the paraxial region and represents an expansion in powers of the longitudinal and transverse displacement of the source including terms of the first and second orders of smallness, and the second (integral) formula has no limitations on the numerical aperture. The first formula is used to obtain an equation of the focal curve. Errors of the formulas in the calculations of the eikonal and its aberrations on the output surface of the lens are analyzed. The second formula is used to find optimal parameters of the lens with parabolic variation in permittivity that provides minimum aberrations. DOI: 10.1134/S1064226920080124
INTRODUCTION Cylindrical lenses made of smoothly inhomogeneous (gradient) dielectric material are employed in optical and radio-frequency ranges of electromagnetic waves. The first cylindrical lenses with plane surfaces and quadratic variation in the refractive index have been made of gelatin and studied in the optical range [1]. In optics, such lenses are known as Wood lenses [2]. A specific gradient lens (plane lens with the first focus on the lens surface and the second focus at infinity) has been synthesized in the framework of geometrical optics, and the exact solution to the problem has been represented as a dependence of the refractive index on the Cartesian coordinate or the radial coordinate in cylindrical coordinates [3]. Such a lens is known as the Mikaelian lens [4]. The solution of [3] has been generalized in [5] to a lens made of an anisotropic dielectric material. The theory of geometrical optical synthesis of gradient lenses with plane input and output surfaces and an external focus of [2] employs the classical optical theory of aberrations. The theory of synthesis that involves expansion of the refractive index in even powers of the distance from the lens axis has been developed in [6], and the theory for curvilinear surface has been proposed in [7, 8]. A recurrence procedure for the synthesis of such lenses based on the model of a layered lens with constant refractive indices in the layer has been developed in [9]. Curvilinear rays lead to complications in the analysis of eikonal (optical path) and its aberrations in gradient lenses for a source that is displaced from the focal point. The classical optical theory of aberrations for such lenses describes the ray aberration as a 2D series
(expansion in powers of the angle of vision and lens aperture) [2]. An abe
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