Explanation of domes from SPIE Press

Domes are mostly monocentric shells, which means that the second, or inside, radius, is shorter by the amount of the element's thickness than the first, or outside radius (see Fig. 1). Such an element has a focal length of

It is a negative element and contributes an overcorrected spherical aberration. Of course, one can take advantage of this fact in balancing the total system aberrations.1

Figure 1 Concentric dome is a negative element.

A dome with zero power can be derived by setting Eq. (3.2) to zero, i.e.,

Solving for R2 yields that R2 is equal to R1 less the axial optical displacement caused by the thickness of the dome, or

If this concept of a zero power lens is extended to a very thick element, a beam expander evolves (see Fig. 2). The magnification of a beam expander is the ratio between the exiting and entering beam diameters. With this definition, the thickness of a solid beam expander can be stated by

The second radius is simply

i.e., the magnification is the ratio of the two radii, independent of the lens material.

For example, a solid 4× germanium beam expander (N = 4) has a first radius of -10 mm. Therefore, its thickness according to Eq. (5.17) is 40 mm, and its second surface radius is -40 mm [Eq. (5.16) or (5.18)]. The spherical surfaces introduce spherical aberrations.

Figure 2 Solid beam expander with m = D/d.

Reference

1. H. Dubner, “Optical Design for Infrared Missile-Seekers,” Proceedings of the IRE (September 1959), pages 1537-1539.