A thick lens can be considered to be a block of glass with thin lenses attached to either side.It is equivalent to a pair of thin lenses separated by a distance in air proportional to the glass thickness. This “air equivalent ” thickness is determined by analysis to be equal to the glass thickness divided by the refractive index of the glass:
As illustrated below, the effect of inserting a glass block into the optical path is to increase the path length by an amount Δ equal to the difference between the ray intersection points with the optical axis before and after insertion of the block:
The increase in path length, Δ, is shown by analysis to be:
where t is the length of the space occupied by the block when it is inserted into the path and N is the refractive index of the glass block.
The ray diagrams below show image formation by a single thick lens and also by its thin lens counterpart which consists of two air spaced thin lenses. Mathematical expressions for the equivalent focal length of each is given. The source of the expression for the thick lens focal length can be found in basic texts on geometrical optics, e.g., Ref. 1; the expression for the thin lens pair focal length appears in the accompanying analysis of two thin lenses.
The focal lengths of the the thin lenses of zero thickness and infinite radius on one side are calculated from the radius of curvature and index data of the thick lens:
If the air separation of the two thin lenses is the air equivalent thickness of the thick lens thickness, i.e.,
then, equivalency between the thick lens and thin lens pair can be shown by substituting the above thin lens expressions into the thick lens equation. The path length increase Δ discussed above is illustrated in the lens diagrams below, also.