Spherical Aberration Correction Using Refractive-Diffractive Lenses with an Analytic-Numerical Method

We propose an alternative method to design diffractive lenses free of spherical aberration for monochromatic light. Our method allows us to design diffractive lenses with the diffraction structure recorded on the last surface; this surface can be flat or curved with rotation symmetry. The equations that we propose calculate the diffraction profiles for any substratum, for any f-number, and for any position of the object. We use the lens phase coefficients to compensate the spherical aberration. To calculate these coefficients, we use an analytic-numerical method. The calculations are exact, and the optimization process is not required. 1. Introduction Spherical aberration is, in many cases, the most important of all primary aberrations, because it affects the whole field of the lens, including the vicinity of the optical axis. It is due to different focus positions for a marginal ray, meridional ray, and paraxial rays. An alternative to minimize the spherical aberration is to use diffractive optical elements (DOE). Diffractive lenses are essentially gratings with a variable spacing groove which introduces a chromatic aberration that is worse than conventional refractive/reflective optical elements. In some applications, an optical component may require a diffractive surface combined with a classic lens element. By using the diffractive properties, it is possible to design hybrid elements to obtain an achromatically corrected element [1]. In other cases, the requirements can be satisfied by just using a diffractive element. In general, iterative methods are used to design these lenses [2]. Also, some people have used analytical third-order and numerical integrator methods to design diffractive lenses [1, 3]. The diffractive lenses we describe in this paper are limited to monochromatic applications; however, our proposed method is valid for a wide range of wavelengths. We use lens phase coefficients to compensate spherical aberration. To calculate these coefficients, we use an analytic-numerical method. The calculations are exact, simple and quick. A process of optimization is not required. The manufacturing problem of diffractive lenses is not considered here; to solve this problem you can read Castro-Ramos et al. [4]. First, we describe the diffractive lenses theory. Also, we give a brief derivation of the general grating equation to trace a couple of light rays through a rotationally symmetrical surface. Then, we establish the analytic-numerical method to minimize spherical aberration. We propose some heights to correct the spherical aberration.