As discussed in Section 6.4, qTO cannot explicitly be used to develop a color-corrected lens in the optical regime. However, it can be used as a thought experiment to inspire a methodology for generating a color-correcting GRIN lens. When coupled with the geometrical- optics solutions developed above, a powerful classically inspired set of design rules can be developed.

Consider a homogeneous lens that introduces zero chromatic aberration to an incident ray bundle. In order for the lens to behave like an achromat, it would have to either be made of a non-dispersive material (v = •) or have a wavelength-dependent geometry. The former condition is the trivial solution, while the latter would require an adaptive wavelength-dependent geometry. Obviously, such a design is physically intractable when the lens needs to simultaneously operate over a wide wavelength range. However, as has been discussed, qTO provides the framework for getting one geometry to “behave” like another, at least in an electromagnetic sense. With qTO, an achromatic lens can be achieved then by transforming the requisite lens geometry at each wavelength to a single desired shape, which is then realized through a resulting wavelength-dependent inhomogeneous (i.e., GRIN) refractive index distribution.

The required curvatures for such a homogeneous lens are shown in Fig. 6.61a at three different wavelengths: short, center, and long. If these proposed geometries are considered as existing in the “virtual space,” qTO can be applied to transform them to a single geometry that exists in the “physical space.” In practice, the geometries at the short and long wavelengths are transformed back to the geometry at the center wavelength and the desired behaviors at these wavelengths are embedded in a spatially varying refractive index distribution as seen in Fig. 6.61b. If we assume a target geometry of a homogeneous biconvex at the center wavelength and a background material with normal dispersion (i.e., dn/dA < 0), then the GRIN needs to be slightly diverging at the short wavelength to counteract the excess power of the surface attributed to the higher base material index of refraction. Similarly, a slightly converging profile at the long wavelength is necessary to increase the optical surface power to maintain the desired total power over all three wavelengths. This correction is achieved with a finite refractive material index range over which there is also a finite amount of material dispersion. The dispersion curves for the min, mid, and max refractive index values in the lens are shown in Fig. 6.62 and compared to the ideal non-dispersive case. These dispersion curves have Abbe numbers of v = 30, v = 50, v = 100, and v = •, respectively. Note that this is only a hypothetical material system that has its min and max index dispersion lines all pass through the same value at the center wavelength, a phenomenon not possible with conventional materials. However, this thought experiment captures the physics required for color correction with GRIN media: A spatially varying Abbe number profile is required to compensate for the underlying dispersions of the background material and lens surfaces.

To illustrate the potential of a qTO-inspired achromatic GRIN singlet, consider the flattened biconvex lens shown in Fig. 6.63. For this example, a fictitious GRIN material is considered with various assumptions for its underlying dispersion properties. The two cases considered here are that of a homogeneous Abbe number distribution and that of a material with parallel dispersion lines (i.e Sn = constant). The former case is the condition for a homogeneous lens and the latter guarantees that the magnitude of the gradient An is unchanging with wavelength. Depictions of these two cases are given in Fig. 6.64 and ray trace results are summarized in Table 6.1.

Figure 6.61 Depiction of (a) geometric transformation via qTO and (b) resulting color-correcting GRIN profiles.

Table 6.1 Chromatic performance of two dispersion cases

Dispersion case

RMS spot diameter

f/Sf

Constant Abbe

85 pm

50

Constant Sn

1 pm

10,000

Figure 6.62 Dispersion curves for materials of different Abbe numbers.

Figure 6.63 Index distribution of flattened biconvex lens.

Figure 6.64 Chromatic performance of the flattened biconvex lens with dispersion assumptions of (a, b) constant Abbe number and (c, d) parallel dispersion lines.

The constant Abbe number case had a value of v = 50, while the constant Sn case possesses a spatially varying Abbe number with a median value v = 50. The lens was chosen to have f/5 performance and illuminated with the typical F, d, and C (486.13 nm, 587.56 nm, 656.27 nm) spectral lines. Figure 6.64b shows that the constant Abbe number case is diffraction limited at the center wavelength (the green dots in the spot diagram) but shows significant chromatic aberration at the other two wavelengths leading to focal drift of f/Sf = 50, which is what the equivalent homogeneous lens would possess. Meanwhile, the spot diagram of the constant Sn case (see Fig. 6.64d) shows that all three wavelengths are diffraction limited, and thus chromatic aberration has been nearly eliminated in the lens. In fact, the lens achieves a f/Sfvalue of 10,000 due to its ideal dispersion behavior. Because Дп(А) is constant, the GRIN power is unchanging with wavelength and there will be no chromatic focal drift in the flat lens. What is interesting is that this condition exists regardless of the Abbe number of the base material. Figure 6.65 shows the focal drift behavior of the two cases studied here as a function of the base materials Abbe number. The constant Abbe number case behaves exactly as a homogeneous lens does, while the Sn = constant case achieves a level of color correction more than two orders of magnitude higher even as the base material’s Abbe number approaches zero (i.e., infinitely dispersive).

Figure 6.65 Dispersion curves for materials of different Abbe numbers.

While the color-correcting condition for a flat GRIN lens is to have parallel dispersion lines and thus a constant GRIN power with wavelength, this behavior will change as power is introduced in the lens surfaces. The total system power should be conserved, while the dispersion of the gradient will have to oppose that introduced by the surfaces. While the mathematics governing color correction was laid out in Eq. 6.41, it is not immediately clear what impact this has on the material system or geometry. To illustrate this effect, the ideal dispersion behaviors necessary to correct chromatic aberration were found for bisymmetric lenses of varying surface curvatures using CMA-ES [15, 18]. The results for lenses with front surface radii curvature of ±50 mm are summarized in Fig. 6.66.

Figure 6.66 GRIN and effective material dispersion slope profiles for bisymmetric f/5 lenses with radii of curvature (a,c) R = -50 mm and (b, d) R = 50 mm.

For the biconcave lens, a converging GRIN profile is needed to maintain the desired f/5 focal length and radially varying Sn value that decreases with radial position from the constant value of the parallel slopes case. Meanwhile, the highly curved biconvex lens required a diverging profile to counteract the positive surface powers and a Sn that is spatially varying, but is increasing with radial position. Of course the flat lens condition exists between these two extremes; therefore, there will be a trend as the curvature goes from positive to negative and vice versa. This trend is captured in Fig. 6.67.

Figure 6.67 The optimal Abbe number and effective material dispersion slope profiles for several bisymmetricf/5 radial GRIN lenses.

As previously discussed, the 8n trend does indeed sweep from positive to negative while passing through the constant case for an infinitely flat lens geometry. Moreover, as Sn(x, y) and the index value n(x, y) are functions of the radial position, so is the Abbe number v(x, y). What is interesting is that each lens geometry was found to require the same distribution v(x, y) to maintain their color- correction behavior. The changes in index distribution are balanced by the variations in the dispersion slope to yield the unvarying Abbe number profiles. Similar behaviors are seen for lenses of differentf/#: all geometries will require the same profile. While only focusing lenses were investigated in this section, these results can be extended to other qTO-enabled devices outside the optical regime. Since chromatic aberration can be corrected with materials that are highly dispersive in nature, this provides a potential path forward for extending the bandwidth of devices that incorporate resonant metamaterials.

This TO-inspired discussion gives insight into the dispersive properties of these optical systems. The previous section gave approximate analytical design equations for color correction, while this section examined the phenomenon through the perspective of qTO, showcasing the potential of using qTO to better understand the physics of a given problem.