# Geometrical trade-offs

A common material pairing for achromatic singlets in the IR regime, Si and Ge are ideal candidates for analysis due to their desirable properties [5]. When looking over the mid-IR (MIR) regime (3-5 gm), Si and Ge possess refractive index values of 3.4253 and 4.0249 at a wavelength of 4 gm and Abbe numbers of 236.5 and 107.5, respectively, thus facilitating a range of potential designs. The index distribution of the mixture is governed by volume-filling-fraction mixing rules of the form n_{Mix} = *fn _{H}* + (1 -

*f)n*where

_{L},*n*and

_{H}*n*refer to the high- and low-index materials, respectively. With the analytical expressions developed in Section 6.5.1.3, one can analyze the system for potential solutions. A summary of this is given in Fig. 6.60, where the sum of the absolute values of the surface radii (Fig. 6.60a) and inverse sum of the surface curvatures (Fig. 6.60b) relative to the lens diameter are plotted as a function of the relative thickness and ratio of radial/axial contribution to the GRIN and shown on a log

_{L}_{10 }scale to highlight the dynamic range of values. Ratios of 1,0 mean the GRIN is purely radial or purely axial, respectively, while positive and negative ratios indicate that the GRIN is converging

*(f*0) or diverging

_{G}>*(f*0) in accordance with (1.55). All cases utilize the full

_{G}<*An*range of the material system and are assigned a focal length corresponding to

*f/2.5*to stress the validity of these expressions.

**Figure 6.60 **Lens surface curvature as a function of the GRIN ratio and lens thickness for a //2.5 Si-Ge achromatic radial-axial singlet: (top) total surface radii and (bottom) total surface curvature.

As can be seen in Fig. 6.60, the trade-offs between thickness, surface curvature, and GRIN type (*i.e*., radial, axial, or radial-axial) are easy to distinguish. Furthermore, the majority of potential solutions exist in the negative ratio half-space. In Fig. 6.60a, values of 2 (dark red) and 0.3 (dark blue), on the log_{10} scale, correspond to solutions with total surface radii that are 100 and 2 times the lens diameter, respectively. These solutions exist in the negative halfspace due to the necessity to balance out the positive dispersion of the Si-Ge mixture (v_{G} = 33.5), while a GRIN system with a negative Abbe number (v_{G} < 0) would have solutions existing in the positive half-space. The two dark red “branches” correspond to half-plano geometries where the surfaces transit from positive to negative curvatures (or vice versa) with the upper and lower branches belonging to the front and back surfaces, respectively. Meanwhile, Fig. 6.60b shows the effective “flatness” of the lens where the flattest solutions exist in-between the two branches seen in Fig. 6.60a. An interesting feature in the solution-space is the purely radial crossing point at *T/D ~* 0.02 (*i.e*., a 50:1 aspect ratio), which corresponds to the requisite thickness (1.50) of radial GRIN lenses. In general, moving away from the purely radial solution and into the radial-axial *(i.e., *spherical) regime results in an increase in lens thickness in order to maintain a solution without radical curvature. While this potentially limits the range of geometries achievable, not all lens thicknesses nor GRIN profiles may be realizable from a manufacturing standpoint. The purely axial solution is also accessible for all lens thicknesses and is realizable due to the large background indices of Si and Ge.

While the geometrical-optics analysis of achromatic GRIN singlets was simple for the radial, axial, and radial-axial (i.e., ~ spherical) regimes, analysis for higher-order profiles such as those generated by qTO would be much more involved. However, the takeaway is clear: the dispersive properties of the surfaces and GRIN profile need to balance out in order to yield the desired achromatic performance. Moreover, this outcome can also be seen when examining the problem from a qTO-inspired perspective.