Analytical solution of Wassermann-Wolf differential equations for optical system aplanatism

Abstract

Optical systems for data storage and processing of information have diffraction limited image quality. This requires an exact fulfillment of aplanatic conditions on the whole system aperture and usually leads to the introduction of two more adjacent aspherical surfaces. For exact defmition of these aspheric surface shapes it is necessary to solve numerically a system of two first-order differential equations. For this purpose, one can use Runge-Kutta or Adams-Bashforth-Moulton algorithms or combination of them both. However, solutions often can not be found, particularly for systems with high and super high numerical aperture. If the solution is not found, it is not clear whether it exists or not and what is the reason for the lack of solution. We propose an analytical solution of Wassermann-Wolf differential equations for aplanatism that overcomes such disadvantages. We show that the solution of the system of two Wassermann-Wolf first-order differential equations is mathematically equivalent to the consecutive solution of a set of independent linear equations and the most important factor of the lack of solution is the critical angle of incidence of aperture rays at the two aspherical surfaces. The proposed algorithm allows reliable and effective design of aplanatic optical systems containing two neighboring aspherical surfaces with high and super high numerical aperture and diffraction limited image quality for an object at infinity. We illustrate the successful application of the algorithm to the design of blue DVD objective with super high (0.95) numerical aperture and diffraction limited image quality.