Parabasal theory is a technique in geometrical optics, which describes the behavior of light rays located near some defined base ray rather than the optical axis. In this work, we are concerned with parabasal rays, which lie in a sufficiently small neighborhood of a chief ray and develop some formulas for the parabasal quantities of the chief ray. The parabasal quantities of a chief ray are shown to be intimately related to the coefficients of the first-order differential equations of the chief ray. Using the relations, we derive parabasal formulas containing parabasal refractive indices and parabasal powers from the first-order differential equations. These parabasal formulas turn out to be decoupled differential equations of the first-order differential equations so that highly coupled differential equations for a chief ray can be solved systematically. In addition, we apply parabasal formulas to the paraxial region by taking the limits of the formulas. These limits give necessary conditions expressed in terms of Gaussian brackets for various initial design requirements of optical lens systems. Those necessary conditions do not seem to be derivable by using only paraxial theory without the parabasal approaches developed in this work.
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