| A procedure, for application in gravitational lensing using the geodesic deviation equation, is developed and used to determine the magnification of a source when the lens or deflector is modeled by a "thick" Weyl and "thick" Ricci tensor. This is referred to as the Thick Lens Model. These results are then compared with the, almost universally used, Thin Lens Model of the same deflector. We restrict ourselves to spherically symmetric lenses or, in the case of a thin lens, the projection of a spherically symmetric thin lens into the lens plane. Considering null rays that travel backward from the observer to the source, the null geodesic deviation equation is applied to neighboring rays as they pass through a region of space-time curvature in the vicinity of a lens. The thick lens model determines the magnification of a source for both transparent and opaque lenses. The null rays passing outside either the transparent or opaque lens are affected by the vacuum space-time curvature described by a Schwarzschild metric and transmitted via a component of the Weyl tensor with a finite extent. Rays passing through the transparent lens encounter the mass density of the lens, chosen to be uniform. Its influence on the null geodesics is determined by both the Weyl and Ricci tensor with the use of the Einstein equations. The curvature in the matter region is modeled by a constant Weyl and constant Ricci tensor. We apply the thick lens model to several theoretical cases. For most rays outside the matter region, the thick lens model shows no significant difference in magnification from that of the thin lens model; however large differences often appear for rays near the Einstein radius, both in the magnification and in the size of the Einstein radius. A small but potentially measurable discrepancy between the models arises in microlensing of a star. Larger discrepancies are found for rays traversing the interior of a transparent lens. This case could be used to model a galactic cluster. |
