## 18 Aug Gradient Index Lens Design Key Concepts

FiveFocal presented the results of our work for DARPA’s Manufacturable Gradient Index Optics Program (M-GRIN), where we demonstrated efficient optimization with GRIN materials in novel lens designs. In this post, we’d like to share a couple of practical considerations for GRIN lens design.

## Background

The application of GRIN material in lens design has been presented by P.J. Sands, Duncan Moore and his group at the University of Rochester, Paul Manhart, Florian Bociort and others with recent contributions by Guy Beadie, Joe Mait, Richard Flynn, and Predrag Milojkovic.1,2,3,4,5,6 Due to limitations in manufacturing and metrology, GRIN lenses haven’t seen enough use in practice for designers to be familiar with them. In anticipation of the broader use of GRIN based on the work of Voxtel, Rochester Precision Optics, and PolymerPlus, we highlight some of the key concepts for designing GRIN lenses formulated as a linear combination of two homogenous materials.

## Certain Degrees of Freedom are Critical to Reach Material Limits

In any lens design, it is often difficult to know if you’ve reached a global optimum; this is especially true if there is limited application of design theory to apply to the solution space.

Radial (aka ‘cylindrical’) GRIN lenses perform a similar function to an aspheric achromatic doublet, and like their achromatic doublet counterpart, they can often be limited by sphero-chromatism. However, given the right degrees of freedom in the radial GRIN profile, the solution can be improved so the element reaches a global minimum determined by the two materials selected for the GRIN. This effect is shown in the following with two f/4 radial GRIN aspheric singlets designed with identical materials. The GRIN on the left has radial GRIN terms to the fourth order, while the design on the right has terms to the sixth order. The additional GRIN term enables the element to reach the limit of secondary spectrum (shown by a dominant defocus of the green wavelength with the red and blue wavelengths in focus).

## Constraints can (Easily) Restrict GRIN Elements from Material Limits

Since GRIN materials are challenging to manufacture consistently at large physical volumes, elements are often constrained in diameter or thickness. In the following we demonstrate two f/2.8 aspheric radial GRIN singlets designed with identical materials. When constrained to 8 mm thickness, the solution on the left under-corrects the Fraunhofer C line. By extending to 9 mm, the element is able to achieve the secondary spectrum limit established by the two materials in the GRIN. This same limit can be achieved with the 8mm thickness by increasing the f/# to 3.4.

## Low GRIN Partial Dispersion Slope Leads to Better Secondary Spectrum

To this point, we’ve demonstrated that a global minimum can be identified with a material set if the element is limited by secondary spectrum. As stated before, this is the condition where the element is in focus at two wavelengths (indicated by flat blue and red lines in the ray fan) with the median wavelength out of focus (indicated by the sloped green line in the ray fan). The secondary spectrum is a limit imposed by the properties of the two materials; for a single element lens, no additional GRIN terms, aspheric terms, or thickness will change this limit.

Since it is a limit imposed by the material properties, the secondary spectrum can be improved through material selection. In a uniform material achromatic doublet, we know that the change to the power due to secondary spectrum is given by

where p_{i} is the partial dispersion between the d and C lines and V_{i} is the Abbe number of the i’th lens in the doublet. For a GRIN lens we have the same equation with a slightly different notation,

where *L* indicates the thin lens component of the element and *G* indicates the GRIN component of the element. We are explicitly ignoring any impact due to the lens thickness.

In the approximation of a thin lens with a very thin GRIN, the partial dispersion and Abbe number for the thin lens is as typically defined. For the GRIN, we have

The Δ indicates the difference between the extreme indices of refraction in the element. By reducing the slope given by , one can improve the secondary spectrum limit.

The following figure demonstrates two f/4 radial GRIN aspheric singlets optimized to reach their secondary spectrum limit. The design on the right has a d-line transverse ray error that is one-fourth the ray error in the design on the left due to fact that a) all conditions are met for it to reach its secondary spectrum limit and b) the slope of the partial dispersion is 0.00035 compared to the slope of the partial dispersion on the left of 0.0014 (a factor of 4 difference).

## Conclusions

DARPA’s M-GRIN program improved the state-of-the-art for manufacturable GRIN elements, their metrology, and their design. In anticipation of their more common use in lens design, we shared some of the fundamental lessons we learned through some simple singlet examples including:

- The necessary degrees of freedom to reach the material limit
- The concept of the minimum element thickness to reach the performance limit in light of the lens’s F/#
- How to choose the best material combinations to minimize secondary spectrum and maximize overall performance

By applying these principles with the right materials, lenses designed with GRIN elements can replace aspheres, aspheric achromatic doublets, and do considerable work for achromatization, weight reduction, and athermalization.

### References

1. P.J. Sands, “Inhomogeneus Lenses, V. Chromatic Paraxial Aberrations of Lenses with Axial or Cylindrical Index Distributions”, JOSA. 61, 1495-1500 (1971).

2. Danette P. Ryan-Howard and Duncan T. Moore, “Model for the chromatic properties of gradient-index glass”, Appl. Optics. 24, 4356-4365 (1985).

3. Paul K. Manhart and Richard Blankenbecler, “Fundamentals of macro axial gradient index optical design and engineering,” Opt. Eng. 36 (6), 1607-1621 (1997).

4. Florian Bociort, “Chromatic paraxial aberration coefficients for radial gradient-index lenses,” J. Opt. Soc. Am. 13, 1277-1284 (1996).

5. Joseph N. Mait, Guy Beadie, Predrag Milojkovic, and Richard A. Flynn, “Chromatic analysis and design of a first-order radial GRIN lens,” Opt. Express 23, 22069-22086 (2015).

6. Guy Beadie, Joseph N. Mait, Richard A. Flynn, and Predrag Milojkovic, “Materials Figure of Merit for Achromatic Gradient Index (GRIN) Optics,” Proc. SPIE 9822, 98220Q (2016).

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