Open Access
Add to BibSonomy
Abstract
Recent advancements in additive manufacturing have enabled new methods of fabricating gradient-index (GRIN) optics by blending multiple materials in the deposition process. A design study highlighting the advantages of multi-material GRIN optics is presented. It is shown that additional materials in the GRIN allow for higher orders of color correction. A new multi-material refractive index representation, which constrains the GRIN to real materials, is also presented.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Achromatizing a lens is a task with which every designer is intimately familiar. Without proper care, a refractive design form will always have some amount of chromatic aberration [1]. The singlet is the simplest refractive design starting point, and it will often be improved by cementing it with another element, forming a doublet. This can correct the primary chromatic aberrations, but unless the material pair is carefully chosen, secondary color will remain [2]. Triplets, often seen in microscope objectives, can then be employed to correct some higher orders of chromatic aberrations, such as secondary color [3].
An advanced technique for improving a lens system is the use of gradient-index (GRIN) optics. Various studies have shown that, all other variables being equivalent, a system with GRIN outperforms a homogeneous system across numerous categories [4–6]. One such category is chromatic performance. Due to its unique dispersion properties, GRIN optics afford additional levels of color correction not achievable by homogeneous lenses.
A GRIN singlet can be used to correct primary chromatic aberrations, analogous to a homogeneous cemented doublet [7]. In a cemented doublet, the dispersion of the two materials and powers of the two elements balance one another,
where $\Phi $ and V are the power and Abbe numbers of the respective elements. The Abbe number is defined as where n is the refractive index; the Abbe number describes how dispersive a material is.As stated earlier, a homogeneous triplet could be used to improve the homogeneous doublet performance. Analogous to this process is the addition of a third material to the typically binary GRIN; previous research has shown the benefit of using a ternary GRIN [8]. To further improve a ternary GRIN, a fourth material could be added. This fourth material, introducing additional degrees of freedom, should allow for higher levels of color correction not previously seen in GRIN designs. The design study outlined in this paper is the first example of modeling a four-material GRIN.
2. Multi-material index representation
Before beginning the design study, a new refractive index representation needs to be constructed that accurately models the multi-material GRIN and constrains the index profile to realizable mixtures of the materials. Previously, Lippman et al. [9] modeled a multi-material GRIN by varying the GRIN coefficients with respect to wavelength. And while this significantly improved their design, it did not guarantee that the GRIN could be made with available materials.
In this work, the freeform-GRIN (F-GRIN) representation introduced by Yang et al. [10] is extended to include any number of materials. A linear composition model is assumed, which, although not exact, approximately matches the behavior of optical plastics [5,11]; if deemed necessary by the designer, several transformation techniques exist to model nonlinear compositions [4,12]. The index is described as follows:
where ${n_{GRIN}}$ is the refractive index of the GRIN at each point spatially and spectrally, ${n_i}$ is the refractive index of the ${i^{th}}$ material, and N is the total number of materials used. ${\gamma _i}$ is the composition ratio, defined as the volume fraction of a material in a unit volume.${\gamma _i}$ is therefore bounded on the interval of [0,1] and constrained by Maintaining the bounded interval and Eq. (4) while optimizing is required to ensure the manufacturability of a GRIN profile. A ${\gamma _i}$ value of less than 0 requires a negative amount of that material, which is unphysical, and a value of greater than 1 requires more than 100% contribution of that material, which is also unphysical.The remainder of the derivation will be carried out for $N = 3$, but the steps can be repeated for any number of materials. Spatial and spectral dependence is implied throughout for brevity.
This first step eliminates ${\gamma _3}$, reducing the dimensionality of the problem. It also ensures that Eq. (4) is true everywhere.The task of describing the F-GRIN at every point spatially then becomes a task of defining ${\gamma ^{\prime}}$. Identical to Yang et al. [10], the Fringe Zernike polynomials define the lateral dimension $(x,y)$ and the Legendre polynomials define the axial dimension $(z)$. This choice in polynomials provides flexibility for use in systems that are rotationally or non-rationally symmetric. However, a designer is free to choose any functional basis to suit their design needs.
3. Visualizing the three-dimensional material space
Designing achromatic doublets and apochromatic triplets often involves heavy use of glass charts. These 2D charts plot a material’s refractive index versus Abbe number and a material’s Abbe number versus partial dispersion, which for the purposes of this paper is defined as
where ${n_F}$, ${n_d}$, and ${n_C}$ are the refractive indices at the F, d, and C wavelengths, respectively. The Abbe number describes the slope of the refractive index curve and the partial dispersion describes the departure from a linear behavior. Knowing where your materials sit in this space is crucial to achieving high levels of color correction [13].A set of tools has been developed to visualize where multi-material F-GRINs exist in this same three-dimensional space. These tools have been instrumental in picking three and four-material combinations, for visualizing whether a design is manufacturable (i.e., is $\gamma $ negative or greater than unity), and for intuition as to why using more materials often yields higher levels of achromatization.
Fig. 1. Achievable index, Abbe number, and partial dispersion values for a binary (left), ternary (center), and quaternary (right) GRIN. The labeled endpoints correspond to the materials in Table 2.
The plots shown in Fig. 1 are examples of a binary, ternary, and quaternary GRIN, plotted in ${n_d}$ vs ${V_d}$ vs ${P_{F,d}}$ space. The black lines span the boundaries of possible index and dispersion values achievable by each combination, while the endpoints of the line segments are the parent materials. Note that a homogeneous material is a single point in the 3D space, a binary GRIN forms a curve, a ternary GRIN forms a surface, and a quaternary GRIN forms a volume. Using a binary GRIN severely limits the dispersion values a GRIN can achieve, whereas a quaternary GRIN has the ability to freely explore the 3D dispersion space; a ternary GRIN exists somewhere in-between – more room to explore the space, but still ultimately bounded in some dimensions.
4. Design examples
A design study is proposed to validate the model and support the discussions outlined earlier in the paper. The F-GRIN representations in Eqs. (5) and (6) were encoded in a dynamic-link library (DLL) as a user-defined GRIN in CODE V. The goal of the design study is to optimize an f/2.8, 1° full field-of-view (FFOV) GRIN singlet in the visible, with special attention paid to the chromatic effects of adding additional materials to the GRIN. A Qcon asphere [14], varying up to 10th order, is used on the front surface to correct higher orders of spherical aberration and isolate the GRIN impact to be solely chromatic in nature. The design study specifications are summarized in Table 1.
Materials for all the GRIN designs were chosen from a catalog provided by Nanovox LLC. Although the material compositions are proprietary, Nanovox LLC has publicly discussed their performance in optical applications [15,16]. Motivation for the material selection is as follows: material 1 had the lowest Abbe number, material 2 had the highest partial dispersion, material 3 had one of the lowest refractive indices and lowest partial dispersions, and material 4 had one of the lowest refractive indices and highest partial dispersions. All of these decisions were made in order to provide a diversity of values so that the optimizer could find the best GRIN solution.
Table 1. Specification table for the singlet design study. Rotationally symmetric and axially constant F-GRIN terms are used [10].
First, a homogeneous NBK7 singlet is shown for comparison.
Fig. 2. Homogeneous singlet (left). Transverse ray error plot (right), scale of 40 μm. The line colors, ordered from longest to shortest wavelength, are red, green, cyan, blue, and magenta. Average RMS wavefront performance is 1.31 wvs at 587.6 nm.
The transverse ray error plot in Fig. 2 shows all five wavelengths coming to separate longitudinal focuses. The largest separation is between the two ends of the spectrum, signifying that the design is limited by primary color.
Table 2. Material information for the GRIN design study. These four dispersive materials were chosen to provide a diversity of refractive index, Abbe numbers, and partial dispersion values.
Fig. 3. Binary GRIN singlet (left).$\Delta {n_d}$= 0.123. Transverse ray error plot (right), scale of 25 μm. Average RMS wavefront performance is 0.744 wvs.
As seen in Fig. 3, the addition of the binary GRIN (materials 1 and 3 from Table 2) has significantly improved the performance, reducing the spot size and the RMS wavefront performance by a factor of ∼2. The transverse ray error plot in Fig. 3 shows the main reason why: the longest and shortest wavelengths now focus to near the same longitudinal position, signifying that the primary color has been nearly corrected. The lens is now limited by secondary color, i.e., the two ends of the spectrum (red and magenta) do not focus to the same longitudinal position as the center of the spectrum (cyan). It is worth noting that a large amount of optical power is being introduced by the GRIN (large $\Delta {n_d}$) to try and correct the residual chromatic aberrations.
Fig. 4. Dispersion curves for the binary GRIN. Left is a 3D perspective view. Center is a view in ${n_d}$vs ${V_d}$space. Right is a view in ${n_d}$vs ${P_{F,d}}$space.
Figure 4 overlays the dispersion properties of the binary GRIN with the material bounds. The black curve spans the bounds, while the blue curve is the actual range used by the design in Fig. 3. The binary GRIN is limited to the refractive index, Abbe, and partial values spanned by the two parent materials. The system has very limited freedom to move towards the optimal position in the dispersion space, which results in the system being limited by secondary color.
Fig. 5. Ternary GRIN singlet (left). $\Delta {n_d}$= 0.0260. Transverse ray error plot (right), scale of 1 μm. Average RMS wavefront performance is 0.026 wvs, which is diffraction limited by the Marechal criterion.
Fig. 6. Composition ratios of the three materials used in the ternary GRIN. All plots share the same colorbar, which represents$\gamma $.${\rho _x}$and${\rho _x}$are the normalized pupil coordinate of the part.
The ternary GRIN (shown in Fig. 5), which uses materials 1-3 from Table 2, achieves levels of color correction not afforded by the traditional binary GRIN. The spot size and RMS wavefront performance are both reduced by a factor of ∼25. The transverse ray error plots show the correction of secondary color, as now our five wavelengths come to two separate longitudinal focuses (where in Fig. 3 they came to three separate longitudinal focuses). Note that the superior color correcting capabilities of the ternary GRIN were achieved with a $\Delta {n_d}$ of only 0.026 (compared to the binary value of 0.123). Figure 6 is a plot of the composition ratios of the three individual materials for the design in Fig. 5. These plots can be used to ensure manufacturability (i.e., are any values outside the range [0,1]). They also are needed for fabrication via additive manufacturing as they show the relative amount of each ink needed at each point in the lens.
Fig. 7. Dispersion curves for the ternary GRIN. Left is a 3D perspective view. Center is a view in ${n_d}$vs ${V_d}$space. Right is a view in ${n_d}$vs ${P_{F,d}}$space.
Figure 7 overlays the dispersion properties of the ternary GRIN with the material bounds. The black curves span the bounds, while the blue curve is the actual range used by the design in Fig. 5. The addition of a third material allows the system to expand along another dimension compared to the binary system; recall that a binary GRIN is bound to a curve while a ternary GRIN is bound to a surface. This additional degree of freedom provides the ability to correct the secondary color. However, the surface bounded by the three materials is tilted at a certain angle in ${n_d}$ vs ${V_d}$ vs ${P_{F,d}}$ space, and there is no reason to expect that this is the optimum orientation. The ideal scenario would be if the surface were allowed to rotate freely in space, which is what is effectively achieved by a quaternary GRIN.
Fig. 8. Quaternary GRIN singlet (left).$\Delta {n_d}$= 0.0447. Transverse ray error plot (right), scale of 1 μm. Average RMS wavefront performance is 0.014 wvs, which is diffraction limited.
Fig. 9. Composition ratios of the four materials used in the quaternary GRIN. All plots share the same colorbar. Note the very slight change in${\gamma _2}$across the part diameter. Minimum value of ${\gamma _4}$ is 0.051.
Now, using materials 1-4 from Table 2, all five wavelengths come to the same longitudinal focus (seen in Fig. 8). Compared to the ternary GRIN, the quaternary GRIN improves the RMS wavefront performance by nearly a factor of 2 (see Table 3 for a performance summary). In fact, the chromatic performance has improved to the point that the system as modeled is now limited by monochromatic aberrations (namely, astigmatism and Petzval). These monochromatic, field-dependent aberrations can then be corrected with the addition of axially dependent F-GRIN terms [10], but this is beyond the scope of the design study. Figure 9 is a plot of the composition ratios of the four individual materials for the design in Fig. 8. It is interesting to note that there is an almost constant amount of material 2 across the lens. The significance of this is not yet known. When used in a multi-element system, this quaternary GRIN provides the useful design tool of being able to contribute optical power without changing the chromatic aberration balance of the system. Also, as the material can effectively be made dispersionless, this has immediate applications in waveguides and photonic integrated circuits (PICs).
Table 3. Performance summary of the design study.
Figure 10 overlays the dispersion properties of the quaternary GRIN with the material bounds. The black curves span the bounds, while the blue curve is the actual range used by the design in Fig. 8. The addition of a fourth material allows the system total freedom to explore the ${n_d}$ vs ${V_d}$ vs ${P_{F,d}}$ space and find the optimal chromatic solution. It is worth noting that the GRIN designs in Figs. 3, 5, and 8 all form 1D curves in the 3D dispersion space. For a binary GRIN, the blue curve is only permitted to translate along the curve spanning the two parent materials (a single dimension of movement). For a ternary GRIN, the blue curve is free to rotate and translate within the plane of the surface bounded by the three parent materials (two dimensions of movement). For the quaternary GRIN, the blue curve can rotate and translate anywhere within the volume bounded by the four parent materials (three dimensions of movement).
Fig. 10. Dispersion curves for the quaternary GRIN. Left is a 3D perspective view. Center is a view in ${n_d}$vs ${V_d}$space. Right is a view in ${n_d}$vs ${P_{F,d}}$space.
It is this additional dimension of movement that allows for complete achromatization at five wavelengths. When a microscope objective is corrected at four wavelengths, it is called a superapochromat [3]; a taxonomy search reveals that the prefix hyper is greater than super, so a lens which is corrected at five or more wavelengths should be called a hyperapochromat. This paper is the first example of a hyperapochromatic singlet.
Using a quaternary GRIN is also a useful way to survey a material design space, similar to a glass catalog global optimizer. If, after optimizing the four-material solution, it is found that a pair of two materials spans the same curve traced out by the four-material solution, that two-material solution will give equivalent performance. This is analogous to how a homogeneous doublet and triplet can produce roughly equivalent chromatic performance, but only if the right two materials are chosen for the doublet [2,3,13]. If no two materials exist that match the dispersion properties of the quaternary solution, this information could be used to motivate the creation of new, exotic materials that have the requisite dispersion characteristics.
5. Conclusion
Achromatizing lenses can often be a challenging design task. However, the process becomes almost trivial when introducing multi-material GRIN elements (as long as the solution sits within the material boundaries). In this paper, a new refractive index representation for ternary and quaternary GRIN were introduced. The derivation shown can easily be repeated for any number of materials. Finally, a polychromatic singlet design study was conducted to verify the model and highlight the advantages of using multi-material GRIN singlets, showing the first example of a hyperapochromatic singlet. And while the hyperapochromat is a physically realizable design, it cannot yet be manufactured reliably. New manufacturing methods that have better control over the spatial refractive index change and non-destructive metrology techniques that reconstruct the 3D refractive index are currently being investigated.
Acknowledgments
The authors thank Synopsys, Inc. for the student license of CODE V.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors of Ref. [16] upon reasonable request.
References
1. W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008).
2. J. Bentley and C. Olson, Field Guide to Lens Design, (SPIE, 2012).
3. Y. Zhang and H. Gross, “Systematic design of microscope objectives. Part I: System review and analysis,” Adv. Opt. Technol. 8(5), 313–347 (2019). [CrossRef]
4. A. Visconti, “Design and Fabrication of Large Diameter Gradient-Index Lenses for Dual-Band Visible to Short-Wave Infrared Imaging Applications,” Ph.D. thesis (University of Rochester, 2015), http://hdl.handle.net/1802/29662
5. P. McCarthy, “Gradient-Index Materials, Design, and Metrology for Broadband Imaging Systems,” Ph.D. thesis (University of Rochester, 2015), http://hdl.handle.net/1802/29303
6. J. Corsetti, “Design of ZnS/ZnSe Gradient-Index Lenses in the Mid-Wave Infrared and Design, Fabrication, and Thermal Metrology of Polymer Radial Gradient-Index Lenses” Ph.D. thesis (University of Rochester, 2017), http://hdl.handle.net/1802/33048
7. J. P. Bowen, J. B. Caldwell, L. R. Gardner, N. Haun, M. T. Houk, D. S. Kindred, D. T. Moore, M. Shiba, and D. Y. H. Wang, “Radial gradient-index eyepiece design,” Appl. Opt. 27(15), 3170–3176 (1988). [CrossRef]
8. J. N. Mait, G. Beadie, R. A. Flynn, and P. Milojkovic, “Dispersion design in gradient index elements using ternary blends,” Opt. Express 24(25), 29295–29301 (2016). [CrossRef]
9. D. H. Lippman, R. Chou, A. X. Desai, N. S. Kochan, T. Yang, G. R. Schmidt, J. L. Bentley, and D. T. Moore, “Polychromatic annular folded lenses using freeform gradient-index optics,” Appl. Opt. 61(3), A1–A9 (2022). [CrossRef]
10. T. Yang, N. Takaki, J. Bentley, G. Schmidt, and D. T. Moore, “Efficient representation of freeform gradient-index profiles for non-rotationally symmetric optical design,” Opt. Express 28(10), 14788–14806 (2020). [CrossRef]
11. P. McCarthy and D. T. Moore, “Optical design with gradient-index elements constrained to real material properties,” in Optical Fabrication and Testing (OFT) (2012), paper OTu4D.2.
12. S. P. Wu, E. Nihei, and Y. Koike, “Large radial graded-index polymer,” Appl. Opt. 35(1), 28–32 (1996). [CrossRef]
13. D. Ren and J. R. Allington-Smith, “Apochromatic lenses for near-infrared astronomical instruments,” Opt. Eng. 38(3), 586 (1999). [CrossRef]
14. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007). [CrossRef]
15. 15. S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injketaided additive manufacturing techniques,” in IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting (IEEE) (2015), pp. 605–606.
16. G. Williams Jr., “Additive manufacturing of freeform gradient index optics,” in SPIE Optifab (2021), paper 11889-25.
