From the expression for optical power of a radial first-order graded-index (GRIN) lens with curved surfaces, we derive an expression for chromatic aberration. Our expressions for optical power and chromatic aberration are valid under the paraxial approximation. By applying a series of further simplifying assumptions, namely a thin lens and thin GRIN, we derive a set of equations with which one can design an achromatic GRIN lens. We also derive expressions for the dispersive property of a GRIN element. Our analysis enables us to derive the relationship between material pairs that indicate their suitability as a material pair for a GRIN achromat. We use this relationship to search a standard glass catalog for attractive GRIN material pairs for a particular achromat design. We compare the optical performance of our GRIN design to that of a conventional homogeneous doublet and demonstrate that our approach is capable of identifying material pairs that perform well for achromatic GRIN lenses which would not generally be considered for conventional achromatic design. We also demonstrate our approach is capable of designing GRIN achromats with superior performance.
© 2015 Optical Society of America
For over 100 years, researchers have recognized the potential offered by graded-index (GRIN) optical elements to improve the performance of optical systems . The technology received considerable attention in the 1970s [2, 3], when it found its primary application in fiber-based imaging systems. Fabrication of GRIN elements for such systems relied on diffusing salts into glass . However, more recently, researchers have fabricated GRIN elements by layering nanometer-thick polymer films of differing indices to produce a material with a desired effective index . Other techniques newly introduced for creating arbitrary three-dimensional distributions of refractive index rely upon rapid prototyping technology  and upon semiconductor fabrication [7, 8]. Spurred by these developments, interest in applying GRIN optics to conventional, i.e., non-fiber-based, imaging has been rekindled.
Although the ability to create an arbitrary index profile in space is the most alluring prospect of GRIN optics, a practical, near-term consideration is their utility as achromatic elements. Achromatic design using GRIN has been addressed in the literature [9–18]. But, except for a few cases , the dispersive and achromatic properties of a GRIN element have not been presented in a manner useful for optical designers. There exist few rules of thumb or intuition with regard to such design. Such rules would have benefited a recent survey of materials suitable for achromatic design .
The first attempt, known to us, to derive formulae for first-order chromatic aberrations of GRIN lenses was performed by Sands [11,13]. His starting point was the analysis by Buchdahl in his seminal work . Starting from the notion of the optical quasi-invariant and its change with wavelength, Sands derived formulae for the first-order longitudinal and lateral chromatic aberrations of gradient-index lenses  and applied them to selected examples of axial and radial index distributions . His formulae account for both surface and transfer chromatic aberration components of longitudinal and lateral color. Surface terms are not unlike those for a homogeneous lens, but transfer terms are unique. They do not exist in homogeneous lenses and represent a contribution from gradient dispersion. Others [15, 17] subsequently found that two formulae for transfer aberrations contained errors. These are correct in .
Unfortunately, the Buchdahl-Sands approach is not easily understood. It does not provide much intuition about aberrations and their physical meaning. Furthermore, the approach is not based on the materials that create the gradient index. Instead, the index function is described as a purely mathematical object.
The goal of our work is to address some of these deficiencies. To do so, we assume the paraxial approximation is valid. This allows us to use a ray optical model and its Gaussian coefficients (the ABCD-matrix elements) to analyze aberrations. For a GRIN lens, the Gaussian coefficients relate the optical properties of a GRIN singlet, in particular, its effective focal length, to its physical parameters, such as thickness, diameter, and refractive index.
The coefficients derived by Sands  and others  for a GRIN lens with a quadratic radial index distribution are the starting formulae for our analysis. From this, we derive the first-order chromatic aberration formulae, which allows us to analyze the change in optical power as a function of wavelength and provides us with a measure of longitudinal chromatic aberration. We then apply simplifying assumptions to elucidate the links between an element’s geometrical and material parameters and its optical performance. We leave an analysis of lateral color to a subsequent paper.
To the best of our knowledge, our approach to chromatic analysis of GRIN lenses is new. Our primary goal is to give optical designers a tool that provides them the intuition to generate a good initial design. We also wish to indicate how material properties affect design. Hopefully, this insight might lead to better GRIN designs, development of new GRIN materials, and may help novice designers.
We assume the GRIN is designed to operate nominally at the wavelength λ0. Unlike others who have examined chromatic behavior of GRIN elements , we choose to expand Eqs. (1) and (2) in terms of wavelength λ rather than the Buchdahl variable ω, a dimensionless quantity related to wavelength . Our analysis is valid regardless of the variable we use and we felt a presentation in λ would be less confusing to optical designers.
In Sec. 2 we introduce the radial gradient lens and, in Sec. 3, we derive expressions for its power and chromatic aberration. In Sec. 4 we examine these optical properties for the special case of a Wood lens. The full expressions derived in Sec. 3 are cumbersome in their length, if not their complexity. To uncover the underlying relationships between material and optical properties, we make a series of simplifying approximations in Sec. 5. A key contribution of our analysis is the discussion in Sec. 6 on the material relationship that allows achromatic designs to work over broad bandwidths. We provide concluding remarks in Sec. 7.
2. First-order radial gradients
Equation (1) is sufficiently general that it can represent axial (Γℓm = 0 for all m) and radial (Γℓm = 0 for all ℓ) gradients. In this work we are interested in focusing elements and, since axial gradients do not add optical power, we consider only radial gradients. More specifically, we consider only first-order radial gradients. Thus, we rewrite Eq. (1),
With reference to Fig. 1, we assume the front and back surfaces of the lens are spherical with radii Rf and Rb. The maximum element thickness is t and its diameter is D. The lens focal length is f. By definition, the indices at the center and at the edge of the GRIN element, nctr(λ) and nedg(λ), respectively, areEquation (9) introduces explicitly the notion that the material at the edge of a GRIN lens is different than the material at its center. The material at intermediate locations is a blended combination of these two boundary materials. Thus, if we know the dispersions for the edge and center materials, we can infer not just the index at all radial locations but also the dispersion. We note that for radial gradients higher than first-order, the representation of the gradient generator is not as compact.
If Γ(λ0) < 0, the lens center has greater refractive index than the edge. Since this corresponds to a lens with positive optical power, we refer to this as a positive GRIN. Similarly, Γ(λ0) > 0 indicates the lens center has less refractive index than the edge, which corresponds to a lens with negative power. We refer to this condition as a negative GRIN. However, rather than refering to Γ(λ0) to define the GRIN, we introduce the GRIN sign sG to increase clarity,
3. Optical properties of a GRIN lens
In this section, we use the expression for optical power to derive chromatic aberration. The form of our equations differs slightly from what is currently in the literature. However, we feel it simplifies analysis and lends itself to a more physical and intuitive presentation than previous work.
3.1. Optical power
We represent the chromatic behavior of focal length by f(λ), which is inversely related to optical power ϕ(λ),Eq. (16) of , the optical power for a GRIN lens is Eq. (12) reduces to the Lens Maker’s equation for conventional singlets .
We note that some treatments of GRIN elements define the gradient strength α(λ) such that Φ(λ) = α(λ)t. Although representing a gradient as a spatial frequency is useful when one wishes to understand the structure of the gradient in a material, we found that representing the gradient as a phase was more effective for optical design. This is evident in Sec. 5.
3.2. Chromatic aberrationAppendix. These terms are analogous to Abbe numbers in that they relate to dispersion, i.e., large values indicate small dispersion and small values, large dispersion. One can interpret the V -terms as the bandwidth Δλ over which a relative change in power Δϕ/ϕ takes place.
In the following, we use Eqs. (12) and (21) to consider optical and material achromatic lens design. To do so, we consider simplifying assumptions that reduce the complexity of the equations and provide us with physical insight into their meaning.
For our purposes, achromatization means dϕ(λ0)/dλ = 0, i.e., the lens power near the design wavelength λ0 does not change with λ. This is, of course, a property local to λ0. As we discuss below, the range over which one can expect this condition to hold within some tolerance is dictated by the choice of materials.
We define achromatic optical design when, given a gradient material, one achieves achromatization by specifying the lens’ geometric parameters. Conversely, we define achromatic material design when, given a range of geometric parameters, one determines material properties that are conducive to achromatization.
We begin our discussion of achromatic design by considering a Wood lens, a gradient lens that has no refractive power, due to its flat front and back surfaces. The chromatic properties of a Wood lens have been studied previously [14–16] and provide insight into more general achromatic design and GRIN lens behavior.
4. Wood lens
A Wood lens generates no power at its surfaces due to curvature. Thus, the variation in power with wavelength is determined solely by the dispersive properties of the gradient index. If ϕR(λ) = 0 and ϕR2(λ) = 0, the power and chromatic aberration (Eqs. (12) and (21)) reduce to,
If we set dϕ(λ0)/dλ = 0, use Eqs. (19) and (23), and use the expression for VG(λ) from the Appendix, we can express the edge material dispersion that insures achromatization in terms of the center material dispersion,
We compare Eq. (24) to results derived by Krishna and Sharma, who also considered achromatization of a Wood lens [15, 16]. We borrowed their definition of Ψ1(λ) (Eq. (14) in ), which we use in the Appendix and repeat here,Eqs. (6) and (8) in  are equivalent to our Eqs. (22) and (23), respectively. However, despite this agreement,  indicates that material achromatization occurs when Ψ1(λ0) = 0, which is in contrast to our Eq. (24). The two are equivalent only when nctr(λ) ≈ nedg(λ), which is implicit in Krishna and Sharma’s polynomial expansion of the gradient generator (Eq. (17) in ). Other than the paraxial approximation, we made no assumptions to derive Eq. (24).
For a thin lens such that Φ(λ0) << 1, Eq. (23) reduces to,Eqs. (19) and (20), if we assume C[ϕ(λ)] ≈ S[ϕ(λ)]/ϕ(λ) ≈ 1. For a Wood lens to have the same focal length for all wavelengths, the first derivatives of the dispersion curves for the center and edge materials must be equal to one another. In other words, their dispersion curves differ only by a constant index contrast Δn. (This also follows from Eq. (24) when C[ϕ(λ)] ≈ S[ϕ(λ)]/ϕ(λ) ≈ 1.)
Since real materials rarely meet this condition, we note the tradeoffs implied by Eq. (25) for a fixed diameter. For a fixed thickness, the smaller the difference in dispersions, the smaller the amount of chromatic aberration. If one wishes to hold chromatic aberration below some tolerance level, there exists a maximum thickness one can achieve and still meet that criterion. For a small difference in dispersion, the maximum thickness may hinder the design of a lens with sufficient power to meet the application.
If there is a target value for chromatic aberration, to balance the aberration of another element, for example, then the material relationships require a specific thickness to meet that goal; the greater the difference in the center and edge material dispersions, the thinner the lens can be.
This behavior is true also for variations in focal length due to other parameters, such as temperature. So long as the difference in index variation at the edge and at the center is small, the variation in focus is small. For a desired focal deviation, an element with a small variation from center to edge will have a greater thickness than an element with a large variation.
5. GRIN lens assumptions
In this section we consider approximations that simplify subsequent analysis and design. Our objective in doing so is to strip away extraneous details to elucidate core physical relationships that describe the behavior of an achromatic GRIN lens. These relationships enable the optical designer to think about the materials and geometries that work best for achromatic GRIN design. The utility of our approach is borne out in the next section on design, Sec. 6, wherein ray tracing validates the robustness of achromatic GRIN designs generated using paraxial and thin lens approximations.
5.1. Thin lens
In conventional optics, the thin lens approximation for the Lens Makers Equation holds when ϕR2(λ) << ϕR(λ). This is typically a valid approximation because, for all but weak meniscus lenses, ϕR2(λ) and dϕR2(λ)/dλ are a small fraction of ϕR(λ) and dϕR(λ)/dλ, given that the ratio of ϕR2(λ) to ϕR(λ) is on the order of t/(Rb − Rf).
We now consider the validity of this approximation when we add a GRIN element. For GRIN achromats to provide an advantage over conventional achromats, the achromats should be comparable in weight and volume. This restricts the class of lenses to conventional “pancake”-shaped lenses, as opposed to rod-like lenses, and maintains the above-mentioned condition t/(Rb − Rf) << 1.
Further, we recognize that rays bend more quickly at the abrupt interface between air and a curved high index material than they do through even a strong index gradient; the strong gradient index requires a propagation length on the order of the lens diameter to achieve ray deflections equal to those of homogeneous refraction . This suggests that the most efficient manner to design a GRIN achromat is to use surface refraction to generate optical power and GRIN dispersion to balance the color.
Depending on the materials used in the GRIN, the GRIN power ϕG(λ) may or may not be small compared to the refractive power ϕR(λ). We, therefore, need to keep the GRIN power term. But, because we want surface curvature to generate considerable optical power, i.e., the surface cannot be a weak meniscus, we require ϕR2(λ) << ϕR(λ). Finally, for the dispersion from the GRIN power to cancel the dispersion generated by the surface power, GRIN dispersion must dominate dϕR2(λ)/dλ just as much as dϕR(λ)/dλ does.
For these reasons, the conventional thin lens approximation, which ignores contributions from ϕR2(λ), remains a good approximation for designing the GRIN achromats. Under this approximation, the equations for power and chromatic aberration reduce toTable 1.
5.2. Thin lens-thin GRIN
Examination of Eq. (12) shows that the lens powers evolve with trigonometric functions of Φ(λ). From Eq. (20), because the term for the relative change in index is order one, the lens thickness t that corresponds to Φ(λ) = 2π is always much greater than the diameter D. For a GRIN lens to be an attractive replacement for conventional achromatic doublets, its thickness should be on the order of a standard doublet, which is generally less than the diameter. For GRIN lenses constrained to thicknesses less than their diameter, we must have Φ(λ) < 1. We refer to this as a thin GRIN, in which case, using a Taylor series expansion, the power reduces toTable 1.
5.3. Thin lens-very thin GRIN
For the simplest expressions, we consider Φ(λ) << 1. From Eq. (20), this can occur when the relative difference in refractive index between the center and edge is small and when the lens thickness is smaller than its diameter. Under this condition, the power and chromatic aberration areTable 1.
6. Achromatic design
Our analysis reveals that, assuming the paraxial and thin lens approximations are valid, we can model a GRIN lens as a doublet comprised of a homogeneous refractive lens and a Wood lens. See Fig. 2. All other approximations affect only the values that multiply the refractive and GRIN powers in the expressions for optical power and chromatic behavior. The equations that allow a designer to correct first-order chromatic aberration areTables 1 and 2.
6.1. Materials considerations for achromatic design
A central result of our analysis is that, once chromatic aberration is corrected at λ0, the lens is also nearly corrected at other wavelengths. The extent over which this is true depends upon the ratio ϕG(λ)/ϕR(λ) at nearby wavelengths. Since this ratio depends only upon the material dispersion curves,
In typical examples of a GRIN achromat, which allow for direct comparisons to conventional doublets, a negative GRIN power produces the dispersion required to balance that of a positive refractive power. To maximize the overall lens power while still achieving chromatic balance, the designer would prefer to build in as little negative GRIN power as possible. This suggests that attractive material pair candidates have a ratio VG(λ0)/VR(λ0) with a small magnitude. In the thin lens-very thin GRIN approximation, this reduces to:
Note that different material pairs optimize each expression. Materials suitable for a homogeneous doublet include one with high index and low dispersion, and a second material with low index and high dispersion. Materials well suited to an achromatic GRIN have indices that are large and similar but whose dispersions are quite different, one low and one high.
6.2. Thin lens-very thin GRIN design
We now provide examples of both analysis and design using the simplest forms for chromatic aberration. If we normalize the equations by the diameter D and assume we have selected materials for the center and edge, the geometric parameters are
Alternatively, if we fix the geometric parameters, the material parameters that insure achromatic performance are,
6.3.1. Material selection
We used Eqs. (37) and (38) to search the Schott catalog of optical glasses  for material pairs suitable to design homogeneous doublet and GRIN achromats. Because the equations are defined locally around λ0, to determine their maximum value within the visible spectrum, we evaluated them at several wavelengths between 0.4 and 0.7 μm. Given the difference between indices in the denominator of Eq. (37), the condition for a GRIN element, the sample rate must be sufficiently high to resolve the wavelength at which small differences occur.
We performed a similar search for material pairs that minimized Eq. (37). Note that this occurs when the edge and center materials have the same index at the operating wavelength but different dispersions. For the wavelength at which the materials share the same index, the GRIN element appears homogeneous despite the material inhomogeneity and Eq. (37) is zero. This implies that the GRIN element provides no optical power. Instead, it only corrects the chromatic dispersion of the refractive lens.
The Schott catalog contains multiple examples of glass pairs that have near identical indices of refraction in the visible spectrum yet distinct dispersion curves. We selected N-SK4 and NKzFS4 as a representative example from the candidate pairs available. Their indices and Abbe numbers are listed in Table 3.
6.3.2. Optical design
For both pairs of materials, we designed an achromatic doublet and radial GRIN lens. Our designs are summarized in Tables 4–6. Given the number of simplifying assumptions in our analysis, many of which are stressed by the applications we believe GRIN can positively impact, we chose to test our approach on a non-paraxial lens design with a 24-mm entrance pupil aperture, 25-mm clear aperture, and 72-mm focal length. The lens should be achromatic over the visible band defined conventionally by the wavelengths λF = 0.4861 μm, λd = 0.5876 μm, and λC = 0.6563 μm.
Except for thickness, the values in Tables 5 and 6 describe the front surface of the material indicated. The thickness value for air is the surface’s back focal length. We assumed the homogeneous doublets are cemented and designed with 1-mm minimum edge and center thicknesses. We constrained the thickness of the GRIN lens to be less than or equal to that of the doublet.
As documented previously , addressing chromatic aberration with GRIN exacerbates geometric aberrations, which one can compensate using surface asphere terms. Consequently, to provide a fair comparison to the aspheric GRIN lenses, we allowed the front surface of the doublet to contain aspheres as well. A conic constant sufficed to provide the maximum benefit. Finally, to accommodate the full f /3 aperture, we allowed the GRIN profile to deviate from a simple parabola and included high-order polynomial terms in the optimization process. With reference to Eq. (1) and Table 6, we included conic and even-order asphere surface terms up to order r18, and both even and odd index polynomial terms from r0 to r9. The values for the index terms are given for λd.
6.3.3. Optical analysis
The optical performance of the resulting four lenses is represented in Figs. 3 by their rayfans and in Table 7 by their calculated spot sizes. A rayfan represents the height of a ray in the image plane yi as a function of the ray’s height in the entrance aperture yl, which in our case is the lens front surface. We computed the spot sizes from the ray fans.
We produced the rayfans using a commercial raytrace package, ZEMAX®. However, because native ZEMAX cannot easily model high-order radial GRINs that contain aspheric terms in arbitrary material blends, we generated our own model to communicate surface shape and dispersive GRIN information to ZEMAX’s ray trace engine. To do so, we followed ZEMAX’s prescription for developing dynamically linked libraries.
We validated our model by comparing ZEMAX-computed wavefronts transmitted through GRIN test structures to those calculated from independently developed GRIN propagation code that is based on published algorithms [24, 25]. Wavefronts calculated using the two methods agreed to within picometers of optical path length.
Further, our model confirmed the utility of paraxial power as a valid model for analysis and design. Results predicted by Eq. (12) agreed with those generated by ray tracing to within a few percent.
The homogeneous doublet designs are tabulated in Table 4 and their optical performance is represented in Figs. 3(a) and 3(b) and Table 7. The performance represented in Figs. 3(a) and 3(b) reflects that of a typical homogeneous doublet. The limiting aberration for both designs is spherochromatism, the wavelength-dependent variation in spherical aberration. Note that spherochromatism of blue and red are opposite in sign. As reflected in the spot sizes presented in Table 7, glass pair 1 outperforms glass pair 2. The geometric and RMS spot sizes for glass pair 2 is seven times larger than that of glass pair 1, which we selected using the homogeneous glass criterion.
The GRIN designs are tabulated in Tables 5 and 6 and their optical performance is represented in Figs. 3(c) and 3(d) and Table 7. It is clear from these that the GRIN achromat we designed using a glass pair selected especially for this purpose yields the best optical performance. The nearly straight lines in Fig. 3(d) indicate that all high order aberrations are corrected for glass pair 2. The limiting aberration is now secondary color, which is measured by the distance between the focal plane for λd and the focal plane for the extremal wavelengths λF and λC.
It is interesting to note that glass pair 1 produces a GRIN achromat whose performance is comparable to, but still slightly better than, its homogeneous achromat. As we discuss below, the GRIN achromat from glass pair 1 was a difficult design to produce, especially when compared to the ease with which we found an achromatic GRIN using glass pair 2. Nonetheless, glass pair 2 produces a GRIN achromat whose spot size is approximately 70 percent that of glass pair 1.
Some comments are pertinent to the development of the GRIN lens designs. Although we use several polynomial terms to describe the index variation, as represented in Fig. 4, the indices are smoothly monotonic. Figure 4 represents the volume fraction γ(r) of the material blend, as opposed to its refractive index, whereEq. (42) is a consequence of effective medium theory .
From Fig. 4, it is evident that the optimal design has the most dispersive material at the lens edge. However, neither solution has the pure glass counterpart at the center. Instead, the center material is a blend, 30 percent for glass pair 1 and 15 percent for glass pair 2. Given that we selected the glass pairs from a list of existing materials, as opposed to designing them from scratch, they are not the optimal materials for either lens design. Consequently, the numerical optimizer identified a blend that performed better than a pure glass.
Note that each curve is qualitatively parabolic. It has a flat minimum in the center and rises smoothly at the edges. Its departure from a simple parabola indicates the degree to which computed light rays at relatively low f -numbers achieve good focus in contrast to those of paraxial ray theories.
As mentioned above, although we found a solution to the GRIN achromat design using glass pair 1, it was significantly more difficult to find than the solution using glass pair 2. The design for glass pair 2 converged quickly with minor effort. The design for glass pair 1 required long runs of non-local optimization algorithms before it settled into a practicable solution. From Table 6, it is evident that the solution requires many more index and asphere terms than glass pair 2 to achieve the performance depicted in Fig. 3(c), which is comparable to Fig. 3(d).
During the design process, we observed that we could achieve optical performance similar to that presented in Fig. 3(d) by thinning the glass pair 2 GRIN, but this required us to include more index and surface polynomial terms. However, there existed a minimum thickness beyond which we could no longer achieve the same optical performance. Therefore, we believe that the oscillations visible in the ray fan of Fig. 3(c) can be reduced if we used more index and surface terms, if we allowed the GRIN lens to be thicker, or both. This only highlights our point that design complexity increases when one uses materials for a conventional achromat in a GRIN achromat design, as opposed to materials selected for an achromatic GRIN.
From the expression for optical power of a radial first-order GRIN lens with curved surfaces, we derived an expression for chromatic aberration. By applying a set of simplifying assumptions, we combine chromatic behavior with optical power to derive a set of equations with which one can design a paraxial starting point for an achromatic GRIN lens.
We also used our analysis to identify relationships among suitable material pairs for a GRIN achromat. This enabled us to search within a standard glass catalog for attractive GRIN material pairs, along with a side-by-side comparison to a similar search for conventional doublet materials. We demonstrate that our analysis highlights material pairs which perform well for achromatic GRIN lenses which would not generally be considered in conventional achromatic design. We anticipate future progress in the selection of advantageous material pairs by refining the concepts introduced here.
With the advent of new technologies for producing gradient-index optics, these results should be instructive to those taking advantage of the opportunities afforded by GRIN optical design, as well as by those invested in the development of new materials which could open up new application areas for advanced GRIN optics.
To derive the chromatic behavior of a GRIN element, we differentiate the GRIN optical power. For a positive GRIN, the optical power isEq. (19), yields Eqs. (17)–(20), we determine Eq. (47) removes the need to include radicals in our equations. We note also that
We borrow the definition of Ψ1(λ) from Krishna and Sharma ,
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