Lie algebraic treatment of optical systems in higher aberration orders

Using the technique of Lie operator algebra, we present a recursive formulation for calculating the third and fifth order aberrations of a general optical system and express the third and fifth order aberration coefficients in the 7×7 and 12×12 matrix forms, respectively. One advantage of our formulae is their explicit algebraic expressions suitable for practical application and numerical calculations, another is that the formulae provide a formulation of the matrix method for general nonlinear transformation and the generalized matrix method employing Lie algebraic tools proposes a new view for aberration optics computation. With the method it should be possible to evaluate all the aberration terms for any optical system. Applications of the matrix method are illustrated with thick lens and some well known imaging systems.