## Abstract

Gradient refractive index (GRIN) materials are attractive candidates for improved optical design, especially in compact systems. For GRIN lenses cut from spherically symmetric GRIN material, we derive an analogue of the “lens maker’s” equation. Using this equation, we predict and demonstrate via ray tracing that an achromatic singlet lens can be designed, where the chromatic properties of the GRIN counterbalance those of the lens shape. Modeling the lens with realistic materials and realistic fabrication geometries, we predict we can make an achromatic singlet with a 19 mm focal length using a matrix of known polymers.

©2013 Optical Society of America

## 1. Introduction

The potential benefits of GRIN optics have been well documented [1–3] but only more recent advances in materials manufacturing techniques, such as ion exchange and polymer composite systems, allow for well-behaved, useful GRIN optics [4, 5]. GRIN materials exhibit inhomogeneous surface refraction and continuous bulk focusing, a significantly more complex optical behavior than standard lenses. Because of this complexity, development in GRIN optical modeling and design has typically followed known manufacturing capabilities, which includes both man-made advances and the multitude of natural GRIN lenses found in biological eyes.

A recent advance in manufacturing nanolayered polymer composite systems creates a new class of optical material: moldable GRIN sheets with a highly arbitrary index profile and a relatively high material index range, δn (the value 0.08 has already been demonstrated) [5–7]. The manufacturing technology, shown schematically in Fig. 1 , reliably produces the designed index profile in a clear, moldable, and polishable lens. Using this new material system, researchers already demonstrated a GRIN lens with corrected on-axis spherical aberrations where the layers were molded into a spherical GRIN plano-convex lens [5]. Geometric aberration correction primarily occurred at the plano-interface where the index profile varied across the aperture, much like the correction from a curved surface intersecting an axial gradient. Chromatic compensation remained to be considered.

Prior analysis of chromatic aberration compensation in GRIN lenses has emphasized refractive index distributions that are either radial [8, 9] or general frameworks using families of polynomials [10, 11]. It is well established that axial GRIN distributions can perform spherical aberration compensation when limited to monochromatic systems [12].

The current work explores chromatic aberration compensation within a GRIN singlet lens, where spherical GRIN distributions are polished into plano-convex elements. While the GRIN distributions are spherical, the lens surfaces are not concentric with the GRIN distributions. Since the lens surface shape “cuts across the grain” of the GRIN distribution, GRIN both alters refraction at the lens surfaces and curves the light path inside the lens (referred to in prior work as the surface and transfer contributions respectively [10]). We apply a scalar wave-based analysis which demonstrates the principle of GRIN chromatic aberration compensation. We derive a focal length function similar to the “lens maker’s” equation, which allows analytical matching of focal length at two distinct wavelengths. The analysis accounts for surface and transfer contributions, within the paraxial approximation to ray propagation. This new analysis is called for because the spherically symmetric GRIN distributions used here do not fit readily into the general framework cited above [10, 11]. That prior analysis assumed a certain degree of separation of variables between the axial and radial GRIN variations (Eq. (1) of reference 10). In the present case, axial and radial index variations are connected through a square root of sum of squares relationship. This relationship is not supported in the cited framework. For further validation, we use commercial ray trace software [13] to analyze axial chromatic aberration and to explore achromat GRIN design space. The analysis below considers the GRIN to be continuous. Any effects of discrete layering due to the fabrication method described above are considered outside the scope of this paper. For those interested, an initial estimation of discrete effects has been calculated [14]. That treatment shows the effect to be minimal for current fabrication techniques. Let us return to the analysis at hand. As compared to a homogeneous lens of similar shape, the focal point shift over visible wavelengths is reduced by 4 orders of magnitude, from millimeters to sub-micron, in an optimized achromat GRIN singlet.

## 2. Theory

Our aim is to develop analytic expressions helpful for achromatizing spherical GRIN singlets. To that end, we assumed a specific GRIN geometry and utilized the thin lens approximation. The lens geometry is depicted in Fig. 2 . The lens is cut from a mold with a spherical GRIN index distribution which varies linearly with distance from the spherical origin. A plano-convex lens is shaped and polished out of this mold, in such a way that the center of GRIN curvature lies on the optical axis but well outside the lens, either in front of or behind it.

To determine the focusing properties of the lens, we compute the wavefront transmitted through the optic when illuminated by plane parallel light. In accordance with the thin lens approximation, the phase-front is calculated as if light remains parallel to the propagation axis throughout the lens element. Because of the lens and GRIN symmetry the wavefront is cylindrically symmetric, so it is sufficient to compute the wavefront *ϕ*(*y*) along one axis *y* perpendicular to the propagation direction *z*:

*k*is the vacuum wavevector of the light, Δ

_{o}*z*(

*y*) is the distance in air from the vertex plane to the lens surface at height

*y*,

*t*is the center thickness of the lens,

_{c}*r*(

*y,z*) is proportional to the distance from the GRIN center of curvature at point (

*y,z*) within the lens, and

*n*(

*r*) is the index of refraction at

*r*. The first term is the accumulated phase in air before striking the curved front surface, while the second represents the phase from traversing the GRIN.

We find *r*(*y,z*) from

*R*is the radius of curvature of the GRIN contour at the lens vertex, with the usual sign convention

_{G}*R*>0 if the center lies to the right of the lens and

_{G}*R*<0 otherwise. Note that for

_{G}*R*<0,

_{G}*r*takes on negative values within the lens. As mentioned above, we model a simple linear relationship for the index of refraction:where

*n*is the index of refraction of the lens at its vertex and

_{0}*a*is the rate of index change with distance. (Because it will come up later in the text, we point out that for all values of

*R*the index of refraction

_{G}*n*at the back vertex of our lenses is given by

_{1}*n*=

_{1}*n*–

_{0}*a t*.) Assuming that

_{c}*R*>>(

_{G}*y,z*), which implies paraxial (y) and thin lens (z) assumptions, and inserting Eq. (2) into Eq. (3) gives an approximate GRIN profile along the integral

*z*depends on the radius of curvature of the front surface. Denoting the radius of curvature of the lens

*R*and assuming that

_{L}*R*>>

_{L}*y*, again a paraxial assumption:

*f*is written as:

*f*(

*λ*) of the plano-convex GRIN lens:

*a*, n

_{0}and

*f*on wavelength, λ.

The first term of Eq. (8) is simply what the lens maker’s equation predicts for a plano-convex lens:

*f*= (1/

*f*

_{1}+ 1/

*f*

_{2})

^{−1}. We can see how the geometric optical power, the first term, compares with the GRIN optical power, the second term. Critically, the power of the GRIN component depends not on a single index value, but rather the

*rate of change*in index via the slope parameter,

*a*. The GRIN power also grows as the GRIN curvature, 1/R

_{G}, becomes stronger. (In the limit of R

_{G}→ 0, Eq. (8) is invalid, as small R

_{G}violates the approximation R

_{G}>> (y,z).) Note that the GRIN power can be both positive and negative. If

*a*and

*R*have opposite signs then the GRIN adds focal power to the lens. On the other hand, if they have the same sign then they serve to weaken the lens power.

_{G}Equation (8) lays the foundation for achromatizing the GRIN singlet. For that reason we highlight the wavelength dependence of the parameters for the first time, which arises from the fact that real materials exhibit dispersion. The wavelength dependence of *n _{0}* is easy enough to understand – whatever material is at the vertex of the lens will exhibit some level of optical dispersion which is described by

*n*(

_{0}*λ*). The wavelength dependence of

*a*(

*λ*) is similar. At the back vertex of the lens (

*z*=

*t*and

_{c}*y*= 0) the index at some reference wavelength

*λ*is defined by [

_{ref}*n*(

_{0}*λ*) –

_{ref}*a*(

*λ*)

_{ref}*t*]. Just as for the front vertex, this index function is associated with a

_{c}*material*composition at the back vertex. This different material composition will exhibit its own dispersion characteristic, likely of different shape than that of the front vertex. Suppose the back material has both a higher index of refraction and steeper dispersion than the front vertex. Then

*a*(

*λ*) would be negative-valued, creating the higher index. Also,

_{ref}*a*(

*λ*) would have greater magnitude than

_{blue}*a*(

*λ*), leading to higher dispersion at the back vertex.

_{red}## 3. Achromat design

Due to the number of parameters involved in Eq. (8), we can choose lens parameters which allow for identical focal lengths at two different wavelengths. To see how the parameters are chosen, we consider the difference in optical powers 1/*f* for the same lens at two different wavelengths *λ _{blue}* and

*λ*:

_{red}*a t*) is equal to the difference in index between the front and back vertices (

_{c}*n*–

_{0}*n*). If we restrict ourselves to materials with normal dispersion and assume a plano-convex lens, then the first term in Eq. (10) will always be positive. To obtain a chromatically balanced lens, therefore, the second term must also be positive, and equal to the first term.

_{1}To obtain an idea of the lens geometries required for a balanced lens, assume that the material at the front vertex has a higher index, and correspondingly higher dispersion, than the material at the back vertex. In this case the dispersion at the front vertex (*n _{0}^{blue}* –

*n*) would be greater than that at the back vertex (

_{0}^{red}*n*–

_{1}^{blue}*n*) and the GRIN radius of curvature

_{1}^{red}*R*would have to be greater than zero to get a balanced lens. Taking the same materials but reversing their order, so that the high-index material is located at the back vertex, requires the sign of

_{G}*R*to change in order to maintain a chromatic balance.

_{G}In designing these lenses, we consider the GRIN radius *R _{G}* and material properties fixed and solve for the radius of curvature

*R*which sets the focal length difference in Eq. (10) equal to zero:

_{L}*n*= [

_{j}*n*(

_{j}*λ*) -

_{blue}*n*(

_{j}*λ*)] at surface j. We express the balanced focal length, where f

_{red}*= f*

_{red}*, by combining Eq. (8) at λ*

_{blue}*with Eq. (11):*

_{red}## 4. Achromat simulations

To validate the idea of an achromatic GRIN lens, we simulated several lenses in the optical design program ZEMAX^{®}. While ZEMAX has the capability to trace rays through gradient index materials, and furthermore has a built in capability to model a spherical GRIN profile, it does not possess the native ability to model dispersion in a spherical GRIN lens. Therefore we were required to develop our own software model which could communicate dispersive GRIN information to ZEMAX’s ray trace engine. Each base material’s dispersion is modeled by Cauchy’s formula, as described in [15] on p. 17, Eq. (1).37-1.39, with coefficients calculated solely from index and abbe number. This integrated model was validated by comparing ZEMAX-computed wavefronts transmitted through GRIN test cases to those calculated from independently-developed GRIN propagation code, based on published algorithms [16, 17]. Wavefronts calculated using the two methods agreed to within picometers of optical path length.

The approach to ZEMAX modeling used below is meant to be consistent with the theoretical approach above. Each lens blends two ideal materials described by refractive index, n, and Abbe number, V, to create a linear variation in index versus R_{G} consistent with Eq. (3). The two materials' index, Abbe values (n, V) represent a fairly typical polymer (1.48, 60) and a high index dispersive polymer (1.70, 20) unless noted otherwise. An appropriate value of index slope, a, is chosen so that the index never exceeds the limits of the base polymers throughout the lens volume. Each lens is 1 mm thick; however to first order the lens thickness value is immaterial as long as Δn_{0}-Δn_{1} remains unchanged. Lens shape is plano-convex, as illustrated in Fig. 2. We optimize each lens by fixing the GRIN profile and allowing the software's optimization routine to select a surface power that shows least chromatic effect.

With the ability to model a dispersive GRIN in ZEMAX, we first set out to validate the balanced focal length equation, Eq. (12). Choosing the Fraunhofer F and C lines as the blue and red wavelengths, respectively, used to balance the focal length via Eqs. (10) and (11) provides a natural way to use the definition of the Abbe number *V _{j}* to compute

*Δn*=

_{j}*V*/ (

_{j}*n*– 1), with

_{j}*n*as the d-line wavelength of material

_{j}*j*. There’s no unambiguous way to compute

*n*(

*λ*) from

_{red}*n, V*, however, so rather than using Eq. (12) to report the balanced focal length we report the focal length at the d-line as calculated from Eq. (8) and the curvature computed in Eq. (11). Included in Table 1 are focal lengths calculated from Eq. (8) and those determined by a ray trace through the ZEMAX model. Achromatic focal length match can be achieved both for standard material pairs displaying increasing dispersion with index and for more exotic materials pairs with decreasing dispersion with index, and in each case both for negative and positive GRIN curvature. Table 1 enumerates these four different combinations, where GRIN curvature is held constant at 20 mm but can be positive or negative. The exotic case of decreasing dispersion with index is included for academic interest; such materials are not currently in use. Agreement between ray traced and theoretically predicted focal lengths is within ± 2% of the paraxial theory.

Higher power lenses result when the GRIN contours curve in opposite direction to the surface radius, e.g. when R_{G} is negative and R_{L} positive. This higher power results from rays encountering a more rapidly varying radial GRIN profile.

Figure 3
further explores axial chromatic aberration by showing how the image plane location shifts, relative to the d-line focal point, versus wavelength over the visible spectrum. The GRIN radius R_{G} is fixed at 20 mm. The left plot shows the focal shift for both a non-GRIN lens with R_{L} = 27 mm (solid line), and two less than fully optimized GRIN singlets with R_{L} = 26 and 28 mm (2 dashed lines). As R_{L} is varied, the GRIN can both over-correct (R_{L} = 28 mm) and under-correct (R_{L} = 26 mm) the chromatic focal shift. We find the best overall focal shift at R_{L} = 27.095 mm, depicted in the right plot in Fig. 3 (note the expanded vertical scale).

The use of n_{0} and n_{1}, index at the front and back vertex of the lens, in Table 1 obscures an important consideration in a real spherically-varying GRIN lens. While one extreme index appears at a surface vertex, the other typically appears at a surface edge. Consider a lens spanning a maximum material index range of 1.48 to 1.70, with R_{G} = −8 mm and a = −0.18. As Fig. 4
shows, the minimum index occurs at the vertex of the front surface (Z = 0, left plot), but the maximum index occurs at the *edges* of the back surface (X = ± 2 mm, right plot). This boundary issue is incorporated into our ZEMAX model.

We move on to explore the parameter space of an optimized achromat GRIN singlet. Specifically, how does the choice of the second material's index and Abbe number and the lens' GRIN radius impact the lens' optical power? To evaluate this we probed design points in this parameter space while optimizing the on-axis point spread function response of the system at the F, d, C wavelengths simultaneously. We included a conic term on the convex surface in order to minimize spherical aberration, allowing the GRIN effects to counter chromatic aberration. Figure 5
shows a graph of the optimized focal length as the GRIN radius (R_{G}) and second blended material (represented by its index, n_{2}, and Abbe number, V) are varied. The graph clearly shows a near-linear dependence of *f* on R_{G}, further validating the prediction in Eq. (12). Since the Abbe number and base index of one polymer are held fixed (n = 1.48, V = 60), this search focuses on the δn and δV of the system. Clear trends emerge: increased optical power results from decreasing R_{G}, increasing δn, and increasing δV. Increasing n_{2}, V from 1.57, 35 to a higher and more dispersive 1.7, 20 more than doubles optical power.

## 5. Conclusion

Nano-layered polymer composites offer an exciting new optical material for manufacturing high-performance GRIN lenses. A significant theoretical contribution herein is the derivation of equations that provide a starting point for ray-based optimization of achromatic GRIN singlets. These equations relate optical power to lens surface radius, gradient iso-index contour radius, refractive index range and dispersion. Three points stand out in GRIN singlet design. First, a strong dependence of δn on λ is at least as important as the magnitude of δn. Increasing the relative dispersion between the two polymer constituents substantially increases the power of the resultant achromat. Second, more aggressive molding of the GRIN contours to achieve a small R_{G} is desirable. Chromatic effects in the GRIN are stronger for small R_{G}, resulting in a higher power achromatized system. Higher power lenses result when R_{G} is opposite in sign to the surface radius, since rays encounter a more rapidly varying radial GRIN profile. Third, placing the lower dispersion material at the highest power lens surface allows the achromat to achieve the highest overall optical power.

This work shows the principle of an achromat GRIN singlet lens through a first order theoretical treatment verified by ray trace analysis. Future work should explore GRIN singlet lenses of more sophisticated shape than plano-convex and provide a fuller treatment of residual aberration. Further, the analysis of material properties indicates the need for further research on layering polymers with a broader variety of dispersion curves.

## Acknowledgments

The authors gratefully acknowledge support from the Office of Naval Research.

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