Basic paraxial characteristics of gradient-index optical elements

Greisukh, G.I., Levin, I.A., Ezhov, E.G.

For Russian citation (Opticheskii Zhurnal):

Грейсух Г.И., Левин И.А., Ежов Е.Г. Основные параксиальные характеристики градиентных оптических элементов // Оптический журнал. 2026. Т. 93. № 3. С. 71–78. http://doi.org/10.17586/1023-5086-2026-93-03-71-78

Greisukh G.I., Levin I.A., Ezhov E.G. Basic paraxial characteristics of gradientindex optical elements [in Russian] // Opticheskii Zhurnal. 2026. V. 93. № 3. P. 71–78. http://doi.org/10.17586/1023-5086-2026-93-03-71-78

For citation (Journal of Optical Technology):

Abstract:

Subject of the study. Paraxial characteristics of a radial-gradient-index plane-parallel plate and lens. Purpose of the study. Demonstration of the possibility of obtaining convenient for practical use and at the same time reliable expressions for determining the paraxial characteristics of radial-gradient-index optical elements. The research method. Theoretical and numerical analysis. The main results. Formulas are presented for calculating the optical power, the front and back focal lengths, and the distances to the front and back principal planes of a radial-gradient-index plane-parallel plate and lens. It is shown that a significant simplification of these formulas is achieved only when the parameter K, which includes the base refractive index and the element thickness along with the first radial gradient-index coefficient, is much less than unity. It is shown that the relative errors in calculating the paraxial characteristics do not exceed 1.12% at K < 0.25 for a radial-gradient-index plate and at K < 0.15 for a radial-gradient-index lens. It is concluded that relatively simple algebraic expressions allow obtaining the values of paraxial characteristics with high accuracy even in the case of optically strong radial-gradient-index elements. Algebraic expressions for the dispersion characteristics of a radial-gradient-index lens, including only the optical power calculated at three wavelengths are also presented. The relative errors of their calculation are estimated and it is shown that the chromatic properties of such a lens are described by relatively simple algebraic expressions with very high accuracy. Practical significance. The presented results will provide an opportunity to obtain reliable values of the paraxial and dispersion characteristics of optical systems with gradient-index elements only by supplementing the traditional calculation procedure with simple algebraic expressions.

Keywords:

radial-gradient-index optical elements, paraxial characteristics, algebraic formulas for paraxial and dispersion characteristics, relative calculation error

OCIS codes: 080.2740, 080.3630, 160.4670

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Greisukh G.I., Levin I.A., Ezhov E.G. Methodology for the design of infrared objectives with homogeneous, gradient-index, and diffractive elements // J. Opt. Technol. 2025. V. 92. № 3. Р. 183‒188. https://doi.org/10.1364/JOT.92.000183