Lie Algebraic Treatment Of Optical Systems In Higher Aberration Orders

Lie algebraic theory is a technique that is applied in studying geometrical optics from a new point of view and it is a complementary method compared with those traditional methods of studying optical systems. It has the feature that it gives an explicit form between the initial and final quantities and simplifies the calculation of high-order aberrations, when characterizing an optical system. In this paper, we have studied Lie algebraic theory applied in the calculation of higher-order aberrations in the fiber and in the interface of two different fibers, and have also studied a recursive formulation for calculating the fifth order aberrations of a general optical system. The main concerns are as follows:First, higher-order aberration formulation in GRIN fiber is derived. The fifth order aberrations formulation is calculated, when a light ray is transforming through the continuous refractive-index media. Optical fiber is a widely used media and optical element composed of fiber provides more variables for optimizing an optical system.Second, higher-order aberration formulation in the interface between two different refraction index medium is obtained. The higher order aberration formulation is calculated when a ray is refracted in the interface between two fiber indexes. This can be concluded that Lie algebraic theory is applied in calculating the higher-order aberrations of this particular optical element. All of these calculations can be solved using Matlab program. It can be considered as an extension from the homogeneous media.Third, a recursive formulation for calculating the fifth order aberrations of a general optical system is derived. The matrix method of calculating the fifth order aberrations is derived from the recursive formulation for calculating the third order aberrations. This algebraic theory provides a more explicit and convenient tool for modulating and designing an optical system. Compared with traditional methods like ray-tracing method, it can more effectively characterize an optical system. As an application example, the explicit expressions for the fifth order aberrations of thick lens imaging systems are given.