Abstract

We use first-order optical principles to examine the ability of gradient index (GRIN) lenses to correct chromatic aberrations. We consider radial GRIN lenses with flat surfaces, with a flat diffractive surface, and with curved surfaces. We model the GRIN material system as a locally varying, subwavelength blend of three materials. In this model, we demonstrate that the color-correcting properties of each lens type can be expressed solely in terms of the dispersion properties of the base materials. We find, at this level of approximation, that the material condition for a two-material GRIN achromat with curved surfaces is identical to that for a homogeneous doublet achromat comprised of the same two materials. For the more general case of three-material, ternary GRIN elements, we use the theory to develop a figure-of-merit-based optimization approach. This allows us to identify promising material combinations without first fabricating a GRIN element. The optimization approach can be applied to alternate GRIN geometries and arbitrary glass catalogs. We use our model to search a large, commercial glass catalog to identify the best achromatic glass combinations for the three different GRIN lenses described above. Significant numerical effort was required to identify which glass combinations performed best. Ternary glass combinations are necessary to achieve good achromatic performance for flat geometries. Diffraction combined with a graded-index enables improved color correction for the same optical power or nearly a factor of two increase in power for the same level of color correction. Glass pairs that perform well as an achromatic doublet also perform well chromatically when blended in a GRIN singlet.

1. Introduction

Since 1733, when Hall first used two materials with disparate dispersive properties to reduce chromatic aberrations [1], the first-order design of achromatic doublets has been fundamental to optical design. The development of holographic and diffractive technology in the 1980s provided an alternative for correcting chromatic aberrations in imaging [2]. That the chromatic dispersion for a diffractive lens is opposite in sign to that of a refractive element and is, to first order, independent of its material properties allows one to fabricate an achromatic refractive-diffractive lens from a single material. Recent advances in the fabrication of graded-index (GRIN) optical elements prompted renewed interest in the technology for imaging applications and, subsequently, their ability to correct chromatic aberrations [3–5].

In this work, we examine further the ability of GRIN-based lenses to correct chromatic aberrations. Our primary focus here is to establish criteria, extended from our previous results [6], that identify advantageous material combinations for GRIN lenses. To do so, we use first-order optical principles to analyze the performance of several lens types. In our model, we describe the GRIN material at any point as an effective medium mixture of three independent materials. The variation in index is defined by a variation of material volume fractions within the lens; the refractive index and dispersion follow naturally from the weighted combination of base material optical properties.

Using this model, we define an achromatic lens performance figure of merit (FOM) that depends solely on the optical properties of the base materials. Our search over available glass blends reveals material combinations with high figures of merit.

We begin in Section 2 with the definition and description of the chromatic behavior for three singlet lenses: refractive, diffractive, and GRIN. In Section 3, we consider the optical and chromatic properties of a GRIN lens assuming the index is created using nanolayers of different materials. We follow this in Section 4 with a general treatment of achromatic doublet design and consider the achromatic design of the lenses represented in Fig. 1, a Wood lens, which is a GRIN lens with flat surfaces, a hybrid diffractive-Wood lens, and a GRIN lens, which we show is a doublet of Wood and refractive lenses. We introduce our FOM in Section 5 and, in Section 6, discuss our search of a glass catalog to identify GRIN material combinations that optimize the FOM. Section 7 summarizes our results and provides some concluding remarks.

 figure: Fig. 1

Fig. 1 Achromatic lenses. (a) Wood. (b) Hybrid diffractive-Wood. (c) GRIN (refractive-Wood).

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2. Singlets

2.1. Refractive lens

The optical power ϕR(λ) of a thin refractive lens fabricated from a material with refractive index n(λ) and curvatures Rf and Rb on its front and back surfaces is [7]

ϕR(λ)=(1Rf1Rb)[n(λ)1],
where λ represents wavelength. By definition, the change in optical power as a function of wavelength is
dϕR(λ)dλ=ϕ˙R(λ),=ϕR(λ)VR(λ),
where V(λ) is a generalization of the Abbe representation of dispersion [8],
V(λ)=ϕ(λ)ϕ˙(λ).
Throughout our treatment, we use the dot-notation, e.g., ϕ̇(λ), to indicate differentiation with respect to wavelength λ. Thus,
VR(λ)=n(λ)1n˙(λ).
Although the quantity (λ) is related to dispersion, i.e., index as a function of wavelength, to ensure clarity in this work, we refer to it as the index slope.

2.2. Diffractive lens

The optical power ϕD(λ) of a diffractive lens with diameter D and number of zones M is [9]

ϕD(λ)=8MλD2.
Application of Eq. (3) to Eq. (5) yields the continuous dispersion
VD(λ)=λ.

2.3. GRIN lens

From the Appendix, the paraxial power of a radial GRIN lens with refractive indices nctr(λ) and nedg(λ) at its center and edge, respectively, diameter D, thickness t, and curvatures Rf and Rb on its front and back surfaces is

ϕG(λ)=ϕR(λ)C[Φ(λ)]+[ϕW(λ)+ϕR2(λ)]Snc[Φ(λ)].
The functions C[Φ(λ)] and Snc[Φ(λ)] are defined in the Appendix. The function Φ(λ) is the gradient strength,
Φ(λ)=(8t2D2)[|nedg(λ)nctr(λ)|nctr(λ)].
We note C(0) = Snc(0) = 1. The index profile n(r, λ) that generates this power is
n(r,λ)=nctr(λ)+[nedg(λ)nctr(λ)](rD)2.

As discussed in [4], a good approximation to ϕG(λ) keeps just the refractive and Wood lens terms,

ϕG(λ)=ϕR(λ)+ϕW(λ),
ϕ˙G(λ)=ϕR(λ)VR(λ)+ϕW(λ)VW(λ),
where the individual powers are
ϕR(λ)=(1Rf1Rb)[nctr(λ)1],
ϕW(λ)=(8tD2)[nctr(λ)nedg(λ)].
The dispersion of a Wood lens VW(λ) is given by
VW(λ)=nctr(λ)nedg(λ)n˙ctr(λ)n˙edg(λ).
Note that a Wood lens in which the change in refractive index from center to edge is small relative to the change in index slopes is highly dispersive. A lens with a small change in index slopes is almost nondispersive.

One interpretation of Eqs. (10) and (11) is that a GRIN lens can be modeled as a doublet consisting of a refractive lens and a Wood lens [4]. A second interpretation is possible if one recognizes the equivalence between Eq. (10) and

ϕG(λ)=ϕctr(λ)+ϕedg(λ),
where
ϕctr(λ)=[1Rf(1Rb8tD2)][nctr(λ)1],
ϕedg(λ)=[(1Rb8tD2)1Rb][nedg(λ)1].
Equations (15)(17) represent a standard, homogeneous doublet comprised of pure materials of nctr(λ) followed by nedg(λ). The doublet has the same outer surface curvatures of the GRIN lens with a common interface like a cemented doublet.

This equivalence indicates that, for thin lenses which satisfy the paraxial approximation, a radial GRIN singlet performs like a homogeneous doublet. A positive consequence of this is that it provides familiar ground for optical engineers faced with a GRIN lens design. Material choices from conventional optics provide good intuition for materials to consider in GRIN lens solutions.

However, this equivalence can also explain the difficulties we encountered attempting to design GRIN achromats that outperform conventional achromatic doublets [10]. Given that our derivation of Eq. (10) adopts a tightly-specified GRIN profile, it is not surprising that it forces the GRIN element to adhere so closely to the properties of a standard lens that it behaves like one.

This equivalence is negated for thick lenses. When lens thickness is unconstrained, there exist material pairs that perform well as GRIN achromats, but poorly as homogeneous doublets [10].

3. GRIN blends

The fabrication process to create a GRIN profile, as well as the constituent materials used, influence the chromatic properties of the final element. A common method to create GRIN profiles is salt diffusion into glass [11]. However, other methods have recently been introduced [12–15]. Many of these rely on mixing independent materials at length scales smaller than the wavelength of light, to create a medium whose effective refractive index is a combination of the independent material indices. By varying the volume ratio of the materials spatially one can realize an index gradient.

Models for effective medium systems vary in their complexity and provide a rich research history of their own. If the constituents interact as they are blended (e.g., due to some chemical reaction) then no simple linear relationship exists between the optical response of the blend and its base materials. Even for noninteracting blends, the optical response depends on the details of the microscopic geometry and is not always predictable [16]. While one may be unable to predict the proper model, the possible permutations of noninteracting effective media are bounded and the bounds are well understood [16].

The model we choose is based on the description of light propagating through a nanolayered composite, normal to the layer interfaces. In this geometry, the dielectric constant of the effective medium is given by the volume average of the constituent dielectric constants [17]. Recently, we explored the advantages of using three materials to realize a gradient [5] and showed that a third material gives designers the freedom to specify independently index and index slope.

The diagram in Fig. 2 helps us visualize this. In this plot, each point on the graph represents an optical material, parameterized by its refractive index and its index slope at a specified wavelength λ0. The plot is inspired by the standard glass map of index versus Abbe number. Both representations display sets of individual glasses parameterized by their index and dispersion characteristics. In the remainder, because we use an actual glass catalog in our investigation, we refer to glass and optical material interchangeably.

 figure: Fig. 2

Fig. 2 Dispersion map representation of binary and ternary material blends. Binary blends exist on the periphery of the region indicated. Ternary blends exist within its interior. The red vertical line indicates a graded ternary blend whose index slope is constant at the design wavelength.

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If one considers a material blend of two glasses, the set of available compositions is a curve whose endpoints are represented by the base glasses. Fabrication of a GRIN element allows one to pick a glass from the continuum of blends that exist on the curve, in contrast to typical glass maps which consist of isolated points. For a binary blend, however, the relationship between index and index slope on the curve is fixed, i.e., picking a glass with a specified index slope fixes its index and vice versa.

If one considers a blend of three glasses, the set of available compositions is represented by the approximately triangular region whose vertices are the values for the base glasses. Consider the materials represented by the vertical, dashed line in Fig. 2. This line represents glass compositions whose index slope at the design wavelength are equal but whose index value differs. That one can blend materials with these properties illustrates how the third material breaks the dependency between index and index slope in a binary blend. In later sections, we use plots like Fig. 2 to illustrate material combinations that yield optimal achromatic lenses.

We now derive the optical properties of a ternary blend of three materials a, b, and c with refractive properties na(λ), nb(λ), and nc(λ), such that nc(λ0) > nb(λ0) > na(λ0) [5]. Under these conditions and assuming TE-polarized illumination, the square of the refractive index is [17]

n2(r,λ)=na2(λ)+γb(r)[nb2(λ)na2(λ)]+γc(r)[nc2(λ)na2(λ)],
where the functions γa(r), γb(r), and γc(r) indicate the volume fraction of material in a unit volume. They are, therefore, bounded 1 ≥ γ(r) ≥ 0 and constrained,
γa(r)+γb(r)+γc(r)=1.
We used Eq. (19) to arrive at the final form of Eq. (18). Note in Eq. (18) that the spatial structure and the chromatic properties of the optical materials are separable. The index slope of this blended material is
n˙(r,λ)=[1n(r,λ)]{na(λ)n˙a(λ)+γb(r)[nb(λ)n˙b(λ)na(λ)n˙a(λ)]+γc(r)[nc(λ)n˙c(λ)na(λ)n˙a(λ)]}.
Thus, the refractive indices and their index slopes at the lens center and edge are
nctr(λ)={na2(λ)+γb(0)[nb2(λ)na2(λ)]+γc(0)[nc2(λ)na2(λ)]}1/2,
nedg(λ)={na2(λ)+γb(D2)[nb2(λ)na2(λ)]+γc(D2)[nc2(λ)na2(λ)]}1/2,
n˙ctr(λ)=[1nctr(λ)]{na(λ)n˙a(λ)+γb(0)[nb(λ)n˙b(λ)na(λ)n˙a(λ)]+γc(0)[nc(λ)n˙c(λ)na(λ)n˙a(λ)]}.
n˙edg(λ)=[1nedg(λ)]{na(λ)n˙a(λ)+γb(D2)[nb(λ)n˙b(λ)na(λ)n˙a(λ)]+γc(D2)[nc(λ)n˙c(λ)na(λ)n˙a(λ)]}.

4. Achromatic doublet design

Given our analysis in Sections2 and 3, we are now ready to consider achromatic design of doublet lenses. The optical power for a pair of lenses with individual powers ϕ1(λ) and ϕ2(λ) is

ϕ(λ)=ϕ1(λ)+ϕ2(λ),
and the continuous change in optical power for this doublet is
ϕ˙(λ)=ϕ˙1(λ)+ϕ˙2(λ),=ϕ1(λ)V1(λ)+ϕ2(λ)V2(λ).
This section discusses how to correct primary color by combining two lenses among the three we consider: refractive [Eq. (1)], diffractive [Eq. (5)], and radial Wood [Eq. (13)].

4.1. Generalized achromatic design

For the purpose of this work, an ideal achromat satisfies

ϕ˙1(λ)=ϕ˙2(λ),
for all λ. If this is true, the optical power in Eq. (25) is constant for all wavelengths. We acknowledge that holding the paraxial power sum constant with wavelength is not the same as holding the back focal length constant, which forces each color to focus on a common image plane. As we will show, our approach allows us to isolate chromatic properties of the material pair from geometrical factors of the lenses.

Our approach to generalized achromatic doublet design separates materials-based, wavelength-dependent functions h(λ) from constant C geometric-based terms. This allows us to represent the power of a doublet as

ϕ(λ)=C1h1(λ)+C2h2(λ).
As formulated here, the h functions are unitless quantities, while the constants C have units of optical power. To correct first-order chromatic aberrations, we constrain the doublet so that it is achromatic only at the design wavelength λ0,
C2h˙2(λ0)=C1h˙1(λ0).
The geometric factors are, therefore,
C1=h˙2(λ0)ϕ(λ0)h1(λ0)h˙2(λ0)h2(λ0)h˙1(λ0),
C2=h˙1(λ0)ϕ(λ0)h1(λ0)h˙2(λ0)h2(λ0)h˙1(λ0).

We can now express the ratio of ϕ(λ) to ϕ(λ0) as a function only of the material, wavelength-dependent functions:

ϕ(λ)ϕ(λ0)=h1(λ)h˙2(λ0)h2(λ)h˙1(λ0)h1(λ0)h˙2(λ0)h2(λ0)h˙1(λ0).
This is an important relationship. By examining only the dispersion of each material, we can deduce the relative wavelength dependence of a doublet, regardless of that doublet’s geometry or power. This allows us to examine the chromatic compatibility of different materials for a given doublet type. This is a necessary, but not sufficient, condition for determining a good material pair. Solving Eqs. (28) and (29), together, provides information on the total lens power and element power ratio, which allows us to assess more fully the suitability of the materials for a given doublet.

As a notional example, consider two materials whose material properties are related via

h1(λ)=h2(λ)+Δn.
A doublet from such materials can be made perfectly achromatic from the perspective of Eq. (29) if C2 = −C1. Given these conditions, from Eq. (28), the lens power is
ϕ(λ)=C1Δn.
Note that, for a fixed optical power, the index difference Δn and element geometry (e.g., thickness or surface curvature as represented by C1) are inversely proportional. In other words, if Δn is small, a reasonable doublet power requires C1 (and, by extension, C2 = −C1) to be large. Thus, despite the apparent attractiveness of the chromatic properties of the material pair, the lens thickness or surface curvatures may be extreme for a reasonable lens. Indeed, in the limit as Δn approaches 0, this is like trying to make a doublet using two elements cut from the same glass. Clearly, this is not possible, which illustrates the need to examine the geometric C factors as well as the wavelength-dependence of Eq. (32).

4.2. Wood lens design

From Eq. (13), the optical power of a Wood lens is

ϕ(λ)=8tD2nctr(λ)8tD2nedg(λ).
In the generalized achromat form,
C1=8tD2,
C2=8tD2,
h1(λ)=nctr(λ),
h2(λ)=nedg(λ).

A Wood lens is achromatic if there exist materials that satisfy

nctr(λ)=nedg(λ)+Δn.
Under these conditions, the lens power is
ϕ(λ)=8tΔn/D2.
If the optical power and lens diameter are fixed, the index difference and lens thickness are inversely proportional. The larger the Δn, the thinner the lens to achieve a given power. Conversely, if Δn is small, the lens must be thick to achieve reasonable power.

4.3. Diffractive Wood lens design

The addition of a diffractive surface to a Wood lens produces a lens with optical power given by

ϕ(λ)=8MλD2+8tD2[nctr(λ)nedg(λ)].
In our generalized achromat form,
C1=8Mλ0D2,
C2=8tD2,
h1(λ)=λλ0,
h2(λ)=nctr(λ)nedg(λ).
We inserted a constant λ0 in Eqs. (39) and (41) to ensure consistent units.

This doublet combines two elements with greatly differing dispersion properties. The derivative of h1(λ) is 1/λ0. The derivative of a typical index curve n(λ) is much less than 1/λ0 and, since the derivative of h2(λ) is the difference between two such values, its value, in general, yields an even smaller number. To achieve an achromatic balance for typical material choices, this suggests, from Eq. (29), that the magnitude of C2 must be much greater than the magnitude of C1. In the absence of material choices that allow for a more equitable power balance, an achromatic combination of these elements will be substantively a Wood lens, with only a small diffractive-power perturbation.

4.4. Radial GRIN singlet

The paraxial power of a curved, radial GRIN singlet is given by

ϕ(λ)=(1Rf1Rb)[nctr(λ)1]+8tD2[nctr(λ)nedg(λ)].
As we noted in Eq. (15), rearranging terms allows us to identify the dispersive and geometric factors,
C1=1Rf(1Rb8tD2),
C2=(1Rb8tD2)1Rb,
h1(λ)=nctr(λ)1,
h2(λ)=nedg(λ)1.
As suggested by Eqs. (44)(47), material considerations for achromatizing a curved GRIN singlet are identical to those for achromatizing a conventional, homogeneous doublet.

5. Figure of merit

Section 4 codifies the design of three GRIN-based lenses in a way that highlights how material choices affect their chromatic and lens power properties. Realistic materials do not, of course, allow for perfectly achromatic doublets that have high power and small geometric factors. It is beneficial, therefore, to consider a comparative approach to material selection that allows us to find optimal material combinations.

To do so, we define a FOM based on the dispersion of two materials. The FOM is our attempt to use a single number to quantify how well the materials might perform in a given doublet. There are many ways to define such a FOM and, in our previous work [6,10], we explored several of them. In this work, we use

m12=[h1(λ0)h2(λ0)h˙1(λ0)h˙2(λ0)]2σ[h1(λ)h2(λ)h˙1(λ0)h˙2(λ0)]+δσ,ϕ(λ0)σ[ϕ(λ)ϕ(λ0)],
where σ[g(λ)] is the standard deviation with respect to wavelength,
σ[g(λ)]={(1Nλ)i=1Nλ[g(λi)g¯]2}1/2,
g¯=(1Nλ)i=1Nλg(λi),
and Nλ is the number of spectral samples. Unlike in some of our previous work [6], we do not consider geometric factors in the FOM. We are interested here in minimizing chromatic variations based solely on material properties.

The approximation in the second line of Eq. (48), clearly inconsistent from a units standpoint, illustrates our intent: the FOM is approximately the lens power at the design wavelength divided by the standard deviation of the relative power over the operating bandwidth. The equivalence would be exact if the numerator in the second equation were ϕ(λ0)/C1 and δσ = 0, with the understanding that ϕ(λ) satisfies ϕ̇(λ0) = 0.

We use the parameter δσ to provide a threshold below which the variation in ϕ(λ)/ϕ(λ0) over wavelength does not impact the FOM. When comparing different material combinations whose chromatic variations are smaller than δσ, optical power becomes the differentiating feature.

As we show later, δσ can be used to tradeoff optical power and chromatic correction. If a designer wishes to emphasize optical power over color correction, they can use the results we present in the next section as a guide to determine values of δσ to achieve this. The higher the value of δσ, the less important color correction is to the FOM.

To avoid singularities in calculating the FOM, it is important to identify identical, or nearly identical, glass pairs beforehand. This is not entirely an academic concern. Consider that many modern glasses have been developed without certain materials, such as lead, and some popular glasses have been re-formulated to perform across increased temperature ranges. By design, the dispersions for the reformulated glass and legacy glass are almost identical, yet both glasses remain in the catalog. For example, the Schott Glass catalog lists both N-BK7 and BK7, where the former is the lead-free version of the latter. This leads to singularities when one calculates the FOM using these two materials.

6. Figure of merit search

We use the FOM defined in Section 5 to rank the chromatic compatibility of materials contained in the Zemax version of the Schott Glass catalog [18]. We did not choose this catalog because we expect to fabricate GRIN elements from the materials it lists. Indeed, we suspect it is not possible to blend most of the glasses from this catalog into GRIN elements. Rather, we chose the catalog for its collection of glasses that broadly covers the glass map for visible optics. Further, its materials are well-known to optical designers. Thus, designers can draw insight from glasses with which they are familiar, but are applied to the unfamiliar territory of GRIN optical design.

Our FOM-based search considers combinations of three glasses to form a GRIN element. From a single set of three glasses (a triplet), we consider one blend of volume ratios (γb(0), γc(0)) the center material nctr(λ) and a second blend (γb(D/2), γc(D/2)) from the same triplet the edge material nedg(λ). We systematically searched pairs of blends until we found center and edge formulations that generated the highest FOM for that triplet. We performed this search for every possible glass triplet from the catalog and ranked the results in order of FOM.

We reiterate that our search is over different blends within a single triplet, not over different triplets at the center and edge. Nonetheless, such a search is complex. Although we describe our search in detail below, we acknowledge upfront that, given what we learned about the topology of the search space, we would structure it differently in the future. We encourage others to consider how to improve upon our approach.

The first stage of our search process separated in a coarse manner low-FOM triplets from high-FOM ones. For every triplet, volume ratios were discretized in 10% increments, as shown in Fig. 3. This generated 66 discrete blends and 2145 blend pairs. The best FOM calculated from the 2145 blend pairs was saved for each triplet.

 figure: Fig. 3

Fig. 3 Dispersion map representation of ternary material blends for quantized volume ratios.

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To calculate our FOM, we used a design wavelength of λd = 587.5618 nm and computed the standard deviation using Nλ = 25 wavelengths between 400 nm and 700 nm, inclusive. Our algorithm assigns materials to the center or edge automatically to produce an achromat with positive, rather than negative, power.

The complete Schott Glass catalog contains 160 glasses. As described in Section 5, we needed to remove near-identical glasses from the catalog before performing the search. To identify near-identical glasses, we compared their dispersion functions by computing the refractive index and its first two derivatives with respect to wavelength, and . We computed these values for each glass at N = 30 evenly spaced wavelengths between 400 nm and 700 nm. First and second derivatives were calculated in μm−1 and μm−2, respectively. For every material pair a and b, we computed

n^=(1N)i=1N[|na(λi)nb(λi)|+|n˙a(λi)n˙b(λi)|+|n¨a(λi)n¨b(λi)|].
We considered materials a and b identical if < 0.035. After we identified the near-identical glasses and kept only one representative per group, our final catalog contained 130 glasses.

For each of the three lenses we considered, we explored FOM-based searches using four different values of δσ ranging from 0 to 1 × 10−6. In other words, for each lens, we executed our search process four times, once for each value of δσ. After examining our results, we concluded that the materials found using δσ = 0 provided a good compromise between achromatic performance and lens power. These are the results presented in Tables 1–6.

Tables Icon

Table 1. Material Triplet Rankings for an Achromatic Wood Lens.

In summary, we performed a coarse search of (130 × 129 × 128)/6 = 357, 760 glass triplets by calculating the FOM for each of the 2145 discrete blend pairs in a single triplet. That is, we calculated (357760 × 2145) ≈ 7.7 × 108 FOMs. We repeated this process for four different values of δσ for each of the three lens types discussed in Section 4. In total, this represents calculating over 9 × 109 FOMs to complete our initial, coarse search. As illustrated in the histograms we present below, the vast majority of triplets exhibited very small FOM values.

For each lens, we subjected the top 100 material triplets identified in our coarse search to a refined search. The goal of the refined search was to identify more accurately the center and edge blends of a single triplet that generated the highest FOM. That is, we assumed volume fractions were continuous and no longer quantized in 10% increments.

Our initial refined search was a least-squares optimization of the FOM function for a given triplet. As such, we treated the FOM as a scalar function of four continuous variables: the two independent volume fraction ratios for the center and edge material blends, i.e., γb(0), γc(0), γb(D/2), and γc(D/2) in Eqs. (21)(24).

When searching over a continuum of glass combinations, it is important to prevent the edge and center glasses from becoming equal to one another. As mentioned in Section 5, we wish to avoid FOM calculations for identical glasses. Therefore, to search over a continuum of blends within a fixed triplet of glasses, we modified the FOM,

mcontinuum=m12(γ1+γ),
where
γ={[γb(0)γb(D/2)]2+[γc(0)γc(D/2)]20.05}5.
The numerator inside the parentheses of Eq. (53) is the Euclidean distance between the volume fractions of the center and edge material compositions. When the compositions are more than 5% apart, by this metric, the modification factor leaves m12 nearly unchanged. If the glass compositions are closer than 5%, however, the modification factor rapidly approaches zero, which penalizes the selection of near-identical glasses.

We initially thought the least-squares approach would be a straightforward way to refine the optimum volume fraction values. The reality turned out to be more difficult. Due to the topology of the search space, convergence to globally optimum solutions was both challenging and time consuming. We found that regions surrounding good solutions occupy only a small volume of the search space. The regions are long and narrow, and the FOM varies minimally within them. Further, many of the optimal solutions were located on the boundaries of the available space. Thus, it was difficult to find good regions within which to search and least-squares optimization was ill-suited to convergence. Search algorithms better-suited to the topology of our search space would have saved us a significant computational time.

Instead, without our current appreciation for the search space topology, we applied an ad hoc, iterative algorithm. For a single material triplet, we used a Monte Carlo approach to randomly select a collection of seed values out of the entire space of possible blends, and performed a least-squared optimization for each. We used the best-performing pairs as anchor points for an iterative, progressively finer-mesh solution search. While one glass composition was held fixed, the other glass composition was changed on a discrete grid of up to 41 × 41 compositions over the two-dimensional glass-map space, as per Fig. 3, with even grid spacing, centered on the original composition. FOM values were computed (not least-squared optimized) for the discrete pairs from the grid. The best glass combination from this grid was held fixed while the first glass composition was altered in a similar way. The new solution served as a starting point for the same type of search, but with the grid spacing reduced by 30%. With this combination of grid size and grid reduction, there is significant overlap of area from one grid to the next, but not a degenerate set of points. The initial grid spacing was in 5% volume fraction increments. Naturally, nonphysical grid points (such as negative volume fraction values, or values greater than 1) were ignored. Nine iterations were performed, resulting in a final volume fraction grid spacing of about 3 × 10−6. This final solution was subjected to a least-squared optimization.

As expected by such a complicated search space topology, the refined search resulted in final FOM values that dramatically altered our original, coarse-search material ranking. Despite the significant effort we expended to optimize the FOM values reported here, we note that different optimization algorithms may yield different material rankings. Additionally, we point out that the FOM may be sensitive to small fluctuations in material volume fraction near the optimal solution. A careful examination of “how small” is beyond the scope of this work. Isolated cases have shown sensitivity down to volume fraction fluctuations in the fourth decimal place. Readers who wish to implement this approach for real materials should perform their own sensitivity analysis, or incorporate sensitivity constraints in the FOM definition, for solutions of manufacturable interest.

6.1. Wood lens

The achromatic condition [Eq. (29)] for a Wood lens (see Eqs. (34) and (35)) implies 1(λ0)/2(λ0) = 1 in Eq. (48). Thus, the FOM for a Wood lens reduces to

mWL=[nctr(λ0)nedg(λ0)]2σ[nctr(λ)nedg(λ)].

Although we calculated the FOM for all ∼ 0.8 billion Wood-lens blends, Fig. 4 is a histogram of only the maximum FOM for each of the 357,760 material triplets. As one can infer from the logarithmic scale of the vertical axis, the percentage of well-performing triplets is extremely small. The top 100 triplets were selected for the refined search process. The top 10 triplets from our refined search are presented in Tables 1 and 2.

 figure: Fig. 4

Fig. 4 Histogram of triplet combinations for achromatic Wood lenses from the coarse search, plotted on a semi-log10 scale. Vertical dashed lines indicate boundaries for the top 10 and top 100 FOM values, out of the 357,760 glass combinations.

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Tables Icon

Table 2. Parameters for Achromatic Wood Lenses.

As mentioned above, the refined search perturbed our original rankings. The best material triplet from our initial search (mWL = 1373) is second in our final rankings and our best final triplet was initially ranked 9th with mWL = 1012. It is reassuring that 5 of the top 10 final material triplets were in our initial top 10. We note, however, that the 10th best final triplet was initially ranked 79th with mWL = 603, which increased to 1946.

Examination of the FOMs and their associated materials reveals that the triplets are often clumped in groups. This is readily apparent in Table 2. Note the number of triplet blends whose index slopes are near −0.0771 or −0.0710. It is clear that N-KZFS11 and KZFSN4 are common to the best-performing triplets for a Wood lens achromat. When blended with members of the LAK family of glasses, these triplets provide the largest values of Δn with the least dispersion over the visible waveband.

A graphical representation of several of the best performing material triplets is presented in Fig. 5. As one might expect, material blends with the highest FOM appear on this plot as near-vertical lines. These blends provide the largest possible index difference Δn with the smallest possible difference in index slope Δ. The best performing triplet, representative of triplets that contain N-KZFS11, has a solid black perimeter and is labeled “1.” The second best performing triplet, representative of those containing KZFS12, has a dotted green perimeter labelled “2.” The triangle with the dotted blue perimeter labelled “4” represents the outlier of the top 10 triplets, with N-BAF3 as the edge glass composition.

 figure: Fig. 5

Fig. 5 Index versus index slope representation of selected Wood lenses. The base glasses form vertices of triangles, with the lens solution forming a line that transects the triangle. The best-performing material triplet is indicated by its solid black perimeter. Labels indicate the triplet’s rank in Table 1.

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The shape of the triangles and the optimal blends appears to suggest an alternate, simpler search than ours. Note that, for any triplet, one can identify the glass whose index slope lies between the other two and draw a straight line to the edge formed by the blend of the two other glasses. One might think that searching among all such bisections of triplet triangles for the largest Δn ought to provide a set of FOM solutions similar to ours. Recall, however, that Fig. 5 is a single-wavelength representation and does not account for variations in optical properties across the desired waveband.

Such a (rapid) search of triangles drawn for the d-line wavelength identifies SF66:N-LASF31:LITHOSIL-Q as the triplet with the largest bisector, Δn = 0.247. However, the dispersion of this blend is such that σn) = 0.001, which yields a FOM of only 59. A modified search that attempts to account for the dispersion at only the wavelengths used to calculate the Abbe number does not fare much better. It identified the triplet LAK9G15:SF5:N-SF57, which has Δn = 0.073 and near-perfect agreement at those wavelengths, but a full-bandwidth FOM of just 98.

If the FOM search is modified to maximize Δn/σn), rather than Δn2/σn), the glass combinations that rise to the top are those with excellent dispersion control, but small differences in index, Δn ≈ 0.01. In other words, it would take a very thick GRIN lens to elicit any power from these material blends. In a realistic imaging situation that contains any off-axis light rays, such a thick lens would produce large geometric aberrations.

Figure 6 compares the optical performance of lenses designed with two different material systems. The upper half of Fig. 6 presents a design using the SF66:N-LASF31:LITHOSIL-Q material system identified in our simple triangle search described above, while the lower half presents a design using N-LAK9:N-KZFS11:N-LASF9, the top-ranked material blend in Table 1.

 figure: Fig. 6

Fig. 6 Wood lens designs based on two different material systems: (top) Material system from simple triangle search, (bottom) Top-ranked solution from Table 1. Lens design details discussed in the text.

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Each lens is f/5 with a 10-mm diameter and a 50-mm focal length lens. The refractive index distribution is parabolic at λd = 0.58756 μm and ranges from the high-index material at the center to the low-index material at the edge. The material properties of the bottom design are detailed in Table 2. In the top lens, the center material is pure N-LASF31. The edge material is a 0.3442:0.6558 mixture of SF66:LITHOSIL-Q, which has the same index slope (λd) as N-LASF31. The lenses were designed in Zemax OpticStudio, using GRIN design tools available publicly at GitHub [19].

The SF66:N-LASF31:LITHOSIL-Q design, whose 0.247 index difference is larger than the 0.0801 difference of N-LAK9:N-KZFS11:N-LASF9, achieves the same focal length with a thinner element. However, the transverse ray aberration plots (ray height in the focal plane versus ray height incident on the lens) on the right show clearly the improved color control of N-LAK9:N-KZFS11:N-LASF9. The aberration plot indicates the LITHOSIL-Q design generates a 20-μm diameter geometric spot for blue light. Further, since blue light incident on the bottom of the lens lands below the optic axis in the focal plane, the best focal plane for blue light lies behind the focal plane. The best focus for red and green light lies in front of the focal plane. In contrast, geometric rays from the N-LASF9 lens all lie within 1.5 μm of the optic axis in the focal plane. Since the diffraction-limited Airy radius for this lens is about 3.6 μm, these materials are capable of diffraction-limited performance.

Lenses with perfectly flat surfaces at this f/# exhibit significant spherical aberration. Since the purpose of these designs is to illustrate material compatibility, not to optimize a specific lens problem, the spherical aberration was removed using an aspheric back surface, with terms from the 4th, 6th, and 8th order radial polynomial optimized. Because the total sag of each aspheric surface is less than 1.5 μm, the back surfaces appear flat in Fig. 6. Without the aspheric correction, the smallest geometric spot diameters for the LITHOSIL-Q and N-LASF9 lenses in Fig. 6 are 29.6 μm and 21.6 μm, respectively.

6.2. Diffractive Wood lens

The FOM for a diffractive Wood lens is

mDWL=[1nctr(λ0)nedg(λ0)λ0[n˙ctr(λ0)n˙edg(λ0)]]2σ[λλ0nctr(λ)nedg(λ)λ0[n˙ctr(λ0)n˙edg(λ0)]].

Similar to our discussion of Fig. 4, Fig. 7 is a histogram of the best coarse-search FOM value for each of the 357,760 possible triplet combinations. As before, the percentage of well-performing triplets is extremely small. The top 100 triplets were selected for the refined search process. The top 10 triplets from our refined search are presented in Tables 3 and 4.

 figure: Fig. 7

Fig. 7 Histogram of triplet combinations for achromatic diffractive Wood lenses from the coarse search, plotted on a semi-log10 scale. Vertical dashed lines indicate boundaries for the top 10 and top 100 FOM values, out of the 357,760 glass combinations.

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Tables Icon

Table 3. Material Triplet Rankings for an Achromatic Diffractive Wood Lens.

Tables Icon

Table 4. Parameters for Achromatic Diffractive Wood Lenses.

Again, as we saw for the Wood lens, the rankings after the refined FOM search differed from those of the coarse search. Only three of the top ten triplets found in the coarse search are in the final ten triplets listed in Tables 3 and 4. The top-ranked triplet, with mDWL = 29770, was originally ranked 22nd, with mDWL = 5308. The top-ranked triplet from the coarse search is the 4th-ranked final triplet in Table 3. Its initial FOM was mDWL = 13411 and final FOM, mDWL = 17202. The 3rd ranked triplet with mDWL = 22194 exhibited the largest change in ranking. It started in 69th place with mDWL = 3778.

A comparison between the third column in Tables 2 and 4 reveals that the power of a diffractive-Wood lens designed using chromatically-matched materials is only marginally better than that for a chromatically-matched Wood lens. However, a comparison between the variances σ reveals that the wavelength variation of a diffractive-Wood lens is an order of magnitude better than that of a Wood lens.

Clearly, the diffractive surface provides a degree of freedom that improves wavelength correction of a Wood lens. This freedom can be exploited in one of two ways. First, as shown here, one can generate the same optical power as a Wood lens with an order of magnitude reduction in chromatic variation.

Alternatively, if a designer is content with the degree of chromatic control offered by a Wood lens, they can design diffractive versions with material combinations that have Wood-lens-like values of σ while providing increased optical power. To identify such solutions, the designer should use a non-zero value for δσ as we discussed at the end of Section 5.

Given that a typical value of σ in Table 2 is 10−6, we examined the material combinations we generated for the diffractive-Wood lens search using δσ = 10−6. The reduced emphasis on chromatic correction yielded material combinations that generate optical powers greater than the lenses in Tables 3 and 4. In fact, the material combination of a 84:16 N-LASF44:P-SF67 center composition and an edge glass of pure SF5 generated ϕ(λo)/C1 = 0.150 and σ = 2.7 × 10−6. The index difference between center and edge is Δn = 0.1483. Thus, diffraction generated a chromatic variance σ comparable to the Wood lenses in Table 2 yet with twice the optical power. In particular, compare the performance of this diffractive-Wood lens to the 5th ranked Wood lens.

Returning to Table 4, it is clear that the F2, SF2, and SF5 glasses perform the same role here as N-KZFS11 and N-KZFS4 did for the Wood lens: they serve as common denominators for the majority of the best-performing triplet solutions in this search space. By this measure, the 6th and 9th ranked triplets stand out by virtue of their different edge glass composition. The 5th solution is also notable for being a true blend: neither the center nor edge material is close to being a pure glass.

A graphical representation of several of the solutions is presented in Fig. 8. The best-performing triplet, indicated by its solid, black perimeter and labeled “1,” is representative of most of the top-10 triplets. Two of the outliers, the 5th and 9th ranked triplets, are also represented. The 9th solution is interesting, in that among all the solutions in Tables 14, it is the only one in which the middle vertex lies above the line joining the most and least dispersive glasses.

 figure: Fig. 8

Fig. 8 Index versus index slope representation of selected diffractive Wood lens FOM solutions. The base glasses form vertices of triangles, with the lens solution forming a line which transects the triangle. The best-performing solution is outlined by the triangle with solid lines. Each triangle is labeled with its solution’s corresponding rank in Table 3.

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Note the qualitative similarity to the solutions depicted in Fig. 5; the best material blends appear on this plot as near-vertical lines. This is a result of the behavior discussed at the end of Section 4.3. Achieving achromatic balance between a highly dispersive diffractive lens and a normally dispersive Wood lens material pair necessarily shifts the balance of power to the Wood lens. Thus, these blends are similar to those well suited to Wood lenses in Section 6.1. The slight slope indicated in the blends represents the degree to which some, slight dispersion differences between the center and edge materials balance the dispersion of the diffractive component.

6.3. GRIN singlet

The FOM for a GRIN singlet lens is

mGSL=[(nctr(λ0)1)n˙ctr(λ0)n˙edg(λ0)(nedg(λ0)1)]2σ[(nctr(λ)1)n˙ctr(λ0)n˙edg(λ0)(nedg(λ)1)].
As noted in the discussion of Sec. 2.3, the paraxial analysis for a radial GRIN singlet which varies from the center material defined by nctr(λ) to an edge material nedg(λ) is identical to the paraxial analysis of a thin, homogeneous doublet comprised of a crown glass with nctr(λ) and flint glass with nedg(λ).

Figure 9 is the histogram of the best coarse-search FOM value for each of the 357,760 possible triplet combinations, per our discussion of Fig. 4. As before, the percentage of well-performing triplets is extremely small. The top 100 triplets were selected for the refined search process. The top 10 triplets from our refined search are presented in Tables 5 and 6.

 figure: Fig. 9

Fig. 9 Histogram of triplet combinations for achromatic GRIN singlet lenses from the coarse search, plotted on a semi-log10 scale. Vertical dashed lines indicate boundaries for the top 10 and top 100 FOM values, out of the 357,760 glass combinations.

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Tables Icon

Table 5. Material Triplet Rankings for an Achromatic GRIN Singlet Lens.

As opposed to the solutions found for Wood lenses and diffractive Wood lenses, however, there is little to distinguish the 10 different solutions found here. All 10 blends have at their centers near-pure N-FK58. N-LAK34 and N-LAK8 have almost identical (λ) curves and, although the dispersion curve for N-LAK33A is not quite as identical, it is close. N-LAK33A has the additional benefit of an index 0.025 higher, which yields a higher FOM. Consequently, all the glasses in column c are nearly superfluous. It is clear that the very best match is a small perturbation of the material pair N-FK58:N-LAK34. Because it takes only a small perturbation, we speculate any glass can be used to produce similar perturbations. Of the 100 top triplets identified by the coarse search, 78 of them involved N-FK58 as the majority component of center material, including the top 40. Glass N-FK51, a similar glass, is the first non-(N-FK58) glass in that list.

Figure 10 represents the set of solutions presented in Tables 5 and 6. N-FK58 is the lower-right vertex, while N-LAK34 is the top vertex in the glass-space triangle. As indicated by the thick, red line, the ternary blends are almost identical to a N-FK58:N-LAK34 binary blend. Moreover, as we discussed in Section 2.3, this indicates that the performance of a lens fabricated from this ternary blend is marginally better than that of a homogeneous doublet fabricated from N-FK58 and N-LAK34.

 figure: Fig. 10

Fig. 10 Index versus index slope representation of the best-performing blend for an achromatic GRIN singlet. The vertices of the triangle are defined by the base glasses. The optimal blend for the lens is defined by a line that transects the triangle. The blend is labeled with the triplet’s rank in Table 5.

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In hindsight, this should perhaps come as little surprise. The commercial glass catalog represents the culmination of over a century of optical glass development. Since the most basic extension of a lens singlet in optical design is the achromatic doublet, one should expect that glass scientists have been able to formulate glass pairs that are extremely well-suited to this application.

Tables Icon

Table 6. Parameters for Achromatic GRIN Singlet Lenses.

7. Summary and concluding remarks

We used first-order geometric optics and effective medium theory to analyze the wavelength dependence of focal power for three GRIN-based lenses, including a Wood lens, a diffractive-Wood doublet, and a GRIN lens (a refractive-Wood doublet). Further, we used our analysis to define a figure-of-merit that allowed us to identify high-performing glass blends for the three lenses we considered.

Among the insights provided by our analysis, one of the most important is that, within the paraxial approximation, the design of an achromatic homogeneous doublet and that of an achromatic GRIN lens are equivalent. This equivalence allows designers to leverage their knowledge and experience in traditional achromatic optical design to the selection of materials suitable for achromatic GRIN design.

Based on the material combinations our approach identified for each lens, we make several specific observations. We note that the best visible-bandwidth Wood lens material combinations all have an index contrast Δn ≈ 0.07. Further, the relative focal power of lenses made from these materials varies by less than 4 parts in 106 over the waveband of interest. Clearly, material combinations exist that yield higher index contrasts and focus light faster for the same volume of material. But, as we showed, these combinations are less capable of controlling color.

Material combinations found after the addition of a diffractive surface to a Wood lens reduced chromatic variations of the combined element by almost an order of magnitude, without loss of element power. Alternatively, the results suggest that power can be improved by at least a factor of two without increasing the chromatic variation of even the best Wood lenses. The combination of GRIN and diffractive technologies to produce a thin, flat achromatic lens is an intriguing result. However, predictions of improved performance must be weighed carefully against the possible complexity of fabrication.

Finally, our exhaustive search of all possible glass triplets from the Schott catalog indicates that the optimal combination is, effectively, a glass pair. In hindsight, given the equivalence between a homogeneous doublet and a GRIN lens, this is not surprising. For nearly 140 years, optical glass chemists have striven to develop glasses that provide the best achromatic doublet performance possible. Our results suggest that they succeeded. Given the multiplicity of glasses available to tweak dispersion curves, our best solution is little changed from a pair they formulated.

We caution that this is not a general conclusion about material pairs versus triplets. For applications where there exist few materials, e.g., infra-red glasses and plastics, the “glass-map” is less densely populated than visible glasses. In these applications, one must search the available material catalogs as we have here to determine whether triplets offer an advantage over pairs.

We close with a rather obvious statement: the material combinations we identified depend on our definition of the FOM. No combination of real materials exists that can focus simultaneously all wavelengths of light arbitrarily rapidly within a negligible volume of optical material, so there is no solution that will always rise to the top of every search. Tradeoffs in performance always exist (e.g., optical power vs. lens volume vs. chromatic variations in power) so different combinations will be identified by different definitions of the FOM. Based on our previous experience, we believe our FOM identifies material combinations that provide a good, global advantage over multiple all-optic metrics of a lens element.

It is possible to modify the FOM to include other design factors, such as thermal compatibility, mechanical durability, chemical resistance, cost, and even availability. If one chooses to include these factors, or adjust the weight given to each, a re-defined FOM will identify material combinations different from the ones we presented in Tables 16. We hope the details presented here provide readers with sufficient guidance to run an effective material search using FOMs relevant to their applications.

Appendix: GRIN lens optical power and dispersion

This treatment of GRIN lens optical power is based on [4]. The lens has refractive indices nctr(λ) and nedg(λ) at its center and edge, respectively, such that the index at the design wavelength λ0 varies parabolically as a function of radius. Note from our GRIN model [Eq. (18)], a parabolic index distribution implies a non-parabolic distribution of volume fraction. Note also that, at wavelengths other than λ0, the index distribution is also not parabolic. In general, though, this departure is small for most materials.

The lens has diameter D, front surface curvature Rf, back surface curvature Rb, and thickness t. From Sands [20], the paraxial optical power of the lens is

ϕG(λ)=ϕR(λ)C[Φ(λ)]+[ϕW(λ)+ϕR2(λ)]Snc[Φ(λ)],
where
Snc[Φ(λ)]=S[Φ(λ)]Φ(λ).
We define the sign sG of the gradient strength,
sG=sgn[nctr(λ)nedg(λ)],
where sgn is the signum function, such that sG > 0 corresponds to a positive lens, in which case
C[Φ(λ)]=cosΦ(λ),
S[Φ(λ)]=sinΦ(λ),
and sG < 0 corresponds to a negative lens and
C[Φ(λ)]=coshΦ(λ),
S[Φ(λ)]=sinhΦ(λ).

The individual optical powers are

ϕR(λ)=(1Rf1Rb)[nctr(λ)1].
ϕR2(λ)=[nctr(λ)1]2tRfRbnctr(λ),
ϕW(λ)=(8tD2)[nctr(λ)nedg(λ)].
Further, the gradient strength is characterized by Φ(λ),
Φ(λ)=(8t2D2)[|nedg(λ)nctr(λ)|nctr(λ)],

To examine the chromatic behavior of the GRIN lens, we differentiate Eq. (57),

ϕ˙G(λ)={C[Φ(λ)]ϕ˙R(λ)+ϕR(λ)C˙[Φ(λ)]}+{Snc[Φ(λ)][ϕ˙W(λ)+ϕ˙R2(λ)]+[ϕW(λ)+ϕR2(λ)]S˙nc[Φ(λ)]}=ϕR(λ)[C[Φ(λ)]VR(λ)+C˙[Φ(λ)]]+ϕR2(λ)[Snc[Φ(λ)]VR2(λ)+S˙nc[Φ(λ)]]+ϕW(λ)[Snc[Φ(λ)]VW(λ)+S˙nc[Φ(λ)]],=ϕR(λ)VRth(λ)+ϕR2(λ)VR2th(λ)+ϕW(λ)VWth(λ),
where
VRth(λ)=VR(λ)C[Φ(λ)]sGVR(λ)Φ2(λ)Snc[Φ(λ)][Φ˙(λ)/Φ(λ)],
VR2th(λ)=VR2(λ)Snc[Φ(λ)]+VR2(λ){C[Φ(λ)Snc[Φ(λ)]}[Φ˙(λ)/Φ(λ)],
VWth(λ)=VW(λ)Snc[Φ(λ)]+VW(λ){C[Φ(λ)]Snc[Φ(λ)]}[Φ˙(λ)/Φ(λ)],
and
VR(λ)=nctr(λ)1n˙ctr(λ),
VR2(λ)={nctr(λ)[nctr(λ)1]nctr(λ)+1}[1n˙ctr(λ)],
VW(λ)=nctr(λ)nedg(λ)n˙ctr(λ)n˙edg(λ).
Finally, from Eq. (67),
Φ˙(λ)Φ(λ)=(12)[nedg(λ)nctr(λ)nedg(λ)][n˙ctr(λ)nctr(λ)n˙edg(λ)nedg(λ)].

References

1. J. A. Bennett, “Peter Dolland answers Jesse Ramsden,” Sphaera 8, 5 (1998).

2. T. Stone and N. George, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt. 27, 2960–2971 (1988). [CrossRef]   [PubMed]  

3. R. A. Flynn, E. F. Fleet, G. Beadie, and J. S. Shirk, “Achromatic GRIN singlet lens design,” Opt. Express 21, 4970–4978 (2013). [CrossRef]   [PubMed]  

4. J. N. Mait, G. Beadie, P. Milojkovic, and R. A. Flynn, “Chromatic analysis and design of a first-order radial GRIN lens,” Opt. Express 23, 22069–22086 (2015). [CrossRef]   [PubMed]  

5. J. N. Mait, G. Beadie, R. A. Flynn, and P. Milojkovic, “Dispersion design in gradient index elements using ternary blends,” Opt. Express 24, 29295–29301 (2016). [CrossRef]   [PubMed]  

6. G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” Proc. SPIE 9822, 98220Q (2016). [CrossRef]  

7. J. W. Goodman, Introduction to Fourier Optics, 3rd Ed. (Roberts & Company Publishers, 1988).

8. A. E. Conrady, Applied Optics and Optical Design, Part One (Dover Publications, Inc., 1985).

9. D. A. Buralli and G. M. Morris, “Design of diffractive singlets for monochromatic imaging,” Appl. Opt. 30, 2151–2158 (1991). [CrossRef]   [PubMed]  

10. G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Ternary versus binary material systems for gradient index optics,” Proc. SPIE 10181, 1018108 (2017). [CrossRef]  

11. S. N. Houde-Walter and D. T. Moore, “Gradient-index profile control by field-assisted ion exchange in glass,” Appl. Opt. 24, 4326–4333 (1985). [CrossRef]   [PubMed]  

12. S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606. [CrossRef]  

13. S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013). [CrossRef]  

14. L. Sisken, C. Smith, A. Buff, M. Kang, K. Chamma, P. Wachtel, J. D. Musgraves, C. Rivero-Baleine, A. Kirk, M. Kalinowski, M. Melvin, T. S. Mayer, and K. Richardson, “Evidence of spatially selective refractive index modification in 15GeSe(2)-45As(2)Se(3)-40PbSe glass ceramic through correlation of structure and optical property measurements for GRIN applications,” Opt. Mater. Express 7, 3077–3092 (2017). [CrossRef]  

15. A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013). [CrossRef]  

16. V. A. Markel, “Introduction to the Maxwell Garnett approximation: tutorial,” J. Opt. Soc. Am. A 33, 1244–1256 (2016). [CrossRef]  

17. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, Inc., 1988).

18. “Optical glass - overview,” https://www.us.schott.com/advanced_optics/english/download/index.html.

19. “Mgrinx64_radial_v102.dll,” https://github.com/GrinLens/Zemax-Radial-and-Ball-GRIN/releases.

20. P. J. Sands, “Inhomogeneous lenses, III. Paraxial optics,” J. Opt. Soc. Am. 61, 879–885 (1971). [CrossRef]  

References

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  1. J. A. Bennett, “Peter Dolland answers Jesse Ramsden,” Sphaera 8, 5 (1998).
  2. T. Stone and N. George, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt. 27, 2960–2971 (1988).
    [Crossref] [PubMed]
  3. R. A. Flynn, E. F. Fleet, G. Beadie, and J. S. Shirk, “Achromatic GRIN singlet lens design,” Opt. Express 21, 4970–4978 (2013).
    [Crossref] [PubMed]
  4. J. N. Mait, G. Beadie, P. Milojkovic, and R. A. Flynn, “Chromatic analysis and design of a first-order radial GRIN lens,” Opt. Express 23, 22069–22086 (2015).
    [Crossref] [PubMed]
  5. J. N. Mait, G. Beadie, R. A. Flynn, and P. Milojkovic, “Dispersion design in gradient index elements using ternary blends,” Opt. Express 24, 29295–29301 (2016).
    [Crossref] [PubMed]
  6. G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” Proc. SPIE 9822, 98220Q (2016).
    [Crossref]
  7. J. W. Goodman, Introduction to Fourier Optics, 3rd Ed. (Roberts & Company Publishers, 1988).
  8. A. E. Conrady, Applied Optics and Optical Design, Part One (Dover Publications, Inc., 1985).
  9. D. A. Buralli and G. M. Morris, “Design of diffractive singlets for monochromatic imaging,” Appl. Opt. 30, 2151–2158 (1991).
    [Crossref] [PubMed]
  10. G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Ternary versus binary material systems for gradient index optics,” Proc. SPIE 10181, 1018108 (2017).
    [Crossref]
  11. S. N. Houde-Walter and D. T. Moore, “Gradient-index profile control by field-assisted ion exchange in glass,” Appl. Opt. 24, 4326–4333 (1985).
    [Crossref] [PubMed]
  12. S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
    [Crossref]
  13. S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
    [Crossref]
  14. L. Sisken, C. Smith, A. Buff, M. Kang, K. Chamma, P. Wachtel, J. D. Musgraves, C. Rivero-Baleine, A. Kirk, M. Kalinowski, M. Melvin, T. S. Mayer, and K. Richardson, “Evidence of spatially selective refractive index modification in 15GeSe(2)-45As(2)Se(3)-40PbSe glass ceramic through correlation of structure and optical property measurements for GRIN applications,” Opt. Mater. Express 7, 3077–3092 (2017).
    [Crossref]
  15. A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
    [Crossref]
  16. V. A. Markel, “Introduction to the Maxwell Garnett approximation: tutorial,” J. Opt. Soc. Am. A 33, 1244–1256 (2016).
    [Crossref]
  17. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, Inc., 1988).
  18. “Optical glass - overview,” https://www.us.schott.com/advanced_optics/english/download/index.html .
  19. “Mgrinx64_radial_v102.dll,” https://github.com/GrinLens/Zemax-Radial-and-Ball-GRIN/releases .
  20. P. J. Sands, “Inhomogeneous lenses, III. Paraxial optics,” J. Opt. Soc. Am. 61, 879–885 (1971).
    [Crossref]

2017 (2)

2016 (3)

2015 (1)

2013 (3)

R. A. Flynn, E. F. Fleet, G. Beadie, and J. S. Shirk, “Achromatic GRIN singlet lens design,” Opt. Express 21, 4970–4978 (2013).
[Crossref] [PubMed]

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
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1998 (1)

J. A. Bennett, “Peter Dolland answers Jesse Ramsden,” Sphaera 8, 5 (1998).

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1971 (1)

Baer, E.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

Beadie, G.

Bennett, J. A.

J. A. Bennett, “Peter Dolland answers Jesse Ramsden,” Sphaera 8, 5 (1998).

Brister, A.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

Brocker, D. E.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
[Crossref]

Buff, A.

Buralli, D. A.

Campbell, S. D.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
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Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design, Part One (Dover Publications, Inc., 1985).

Corsetti, J. A.

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

Dupuy, C.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
[Crossref]

Fang, K.

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

Fleet, E. F.

Flynn, R. A.

George, N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd Ed. (Roberts & Company Publishers, 1988).

Harmon, P.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
[Crossref]

Houde-Walter, S. N.

Ji, S.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

Kalinowski, M.

Kang, M.

Kirk, A.

Mackey, M.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

Mait, J.

G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Ternary versus binary material systems for gradient index optics,” Proc. SPIE 10181, 1018108 (2017).
[Crossref]

G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” Proc. SPIE 9822, 98220Q (2016).
[Crossref]

Mait, J. N.

Markel, V. A.

Mayer, T. S.

McCarthy, P.

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

Melvin, M.

Milojkovic, P.

G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Ternary versus binary material systems for gradient index optics,” Proc. SPIE 10181, 1018108 (2017).
[Crossref]

G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” Proc. SPIE 9822, 98220Q (2016).
[Crossref]

J. N. Mait, G. Beadie, R. A. Flynn, and P. Milojkovic, “Dispersion design in gradient index elements using ternary blends,” Opt. Express 24, 29295–29301 (2016).
[Crossref] [PubMed]

J. N. Mait, G. Beadie, P. Milojkovic, and R. A. Flynn, “Chromatic analysis and design of a first-order radial GRIN lens,” Opt. Express 23, 22069–22086 (2015).
[Crossref] [PubMed]

Moore, D. T.

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

S. N. Houde-Walter and D. T. Moore, “Gradient-index profile control by field-assisted ion exchange in glass,” Appl. Opt. 24, 4326–4333 (1985).
[Crossref] [PubMed]

Morris, G. M.

Musgraves, J. D.

Park, S.-K.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
[Crossref]

Ponting, M.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

Richardson, K.

Rivero-Baleine, C.

Sands, P. J.

Schmidt, G. R.

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

Shirk, J. S.

Sisken, L.

Smith, C.

Stone, T.

Visconti, A. J.

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

Wachtel, P.

Werner, D. H.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
[Crossref]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, Inc., 1988).

Yin, K.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (2)

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52, 112105 (2013).
[Crossref]

A. J. Visconti, K. Fang, J. A. Corsetti, P. McCarthy, G. R. Schmidt, and D. T. Moore, “Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece,” Opt. Eng. 52, 112107 (2013).
[Crossref]

Opt. Express (3)

Opt. Mater. Express (1)

Proc. SPIE (2)

G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Materials figure of merit for achromatic gradient index (GRIN) optics,” Proc. SPIE 9822, 98220Q (2016).
[Crossref]

G. Beadie, J. Mait, R. A. Flynn, and P. Milojkovic, “Ternary versus binary material systems for gradient index optics,” Proc. SPIE 10181, 1018108 (2017).
[Crossref]

Sphaera (1)

J. A. Bennett, “Peter Dolland answers Jesse Ramsden,” Sphaera 8, 5 (1998).

Other (6)

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, Inc., 1988).

“Optical glass - overview,” https://www.us.schott.com/advanced_optics/english/download/index.html .

“Mgrinx64_radial_v102.dll,” https://github.com/GrinLens/Zemax-Radial-and-Ball-GRIN/releases .

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injket-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.
[Crossref]

J. W. Goodman, Introduction to Fourier Optics, 3rd Ed. (Roberts & Company Publishers, 1988).

A. E. Conrady, Applied Optics and Optical Design, Part One (Dover Publications, Inc., 1985).

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Figures (10)

Fig. 1
Fig. 1 Achromatic lenses. (a) Wood. (b) Hybrid diffractive-Wood. (c) GRIN (refractive-Wood).
Fig. 2
Fig. 2 Dispersion map representation of binary and ternary material blends. Binary blends exist on the periphery of the region indicated. Ternary blends exist within its interior. The red vertical line indicates a graded ternary blend whose index slope is constant at the design wavelength.
Fig. 3
Fig. 3 Dispersion map representation of ternary material blends for quantized volume ratios.
Fig. 4
Fig. 4 Histogram of triplet combinations for achromatic Wood lenses from the coarse search, plotted on a semi-log10 scale. Vertical dashed lines indicate boundaries for the top 10 and top 100 FOM values, out of the 357,760 glass combinations.
Fig. 5
Fig. 5 Index versus index slope representation of selected Wood lenses. The base glasses form vertices of triangles, with the lens solution forming a line that transects the triangle. The best-performing material triplet is indicated by its solid black perimeter. Labels indicate the triplet’s rank in Table 1.
Fig. 6
Fig. 6 Wood lens designs based on two different material systems: (top) Material system from simple triangle search, (bottom) Top-ranked solution from Table 1. Lens design details discussed in the text.
Fig. 7
Fig. 7 Histogram of triplet combinations for achromatic diffractive Wood lenses from the coarse search, plotted on a semi-log10 scale. Vertical dashed lines indicate boundaries for the top 10 and top 100 FOM values, out of the 357,760 glass combinations.
Fig. 8
Fig. 8 Index versus index slope representation of selected diffractive Wood lens FOM solutions. The base glasses form vertices of triangles, with the lens solution forming a line which transects the triangle. The best-performing solution is outlined by the triangle with solid lines. Each triangle is labeled with its solution’s corresponding rank in Table 3.
Fig. 9
Fig. 9 Histogram of triplet combinations for achromatic GRIN singlet lenses from the coarse search, plotted on a semi-log10 scale. Vertical dashed lines indicate boundaries for the top 10 and top 100 FOM values, out of the 357,760 glass combinations.
Fig. 10
Fig. 10 Index versus index slope representation of the best-performing blend for an achromatic GRIN singlet. The vertices of the triangle are defined by the base glasses. The optimal blend for the lens is defined by a line that transects the triangle. The blend is labeled with the triplet’s rank in Table 5.

Tables (6)

Tables Icon

Table 1 Material Triplet Rankings for an Achromatic Wood Lens.

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Table 2 Parameters for Achromatic Wood Lenses.

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Table 3 Material Triplet Rankings for an Achromatic Diffractive Wood Lens.

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Table 4 Parameters for Achromatic Diffractive Wood Lenses.

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Table 5 Material Triplet Rankings for an Achromatic GRIN Singlet Lens.

Tables Icon

Table 6 Parameters for Achromatic GRIN Singlet Lenses.

Equations (79)

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ϕ R ( λ ) = ( 1 R f 1 R b ) [ n ( λ ) 1 ] ,
d ϕ R ( λ ) d λ = ϕ ˙ R ( λ ) , = ϕ R ( λ ) V R ( λ ) ,
V ( λ ) = ϕ ( λ ) ϕ ˙ ( λ ) .
V R ( λ ) = n ( λ ) 1 n ˙ ( λ ) .
ϕ D ( λ ) = 8 M λ D 2 .
V D ( λ ) = λ .
ϕ G ( λ ) = ϕ R ( λ ) C [ Φ ( λ ) ] + [ ϕ W ( λ ) + ϕ R 2 ( λ ) ] Snc [ Φ ( λ ) ] .
Φ ( λ ) = ( 8 t 2 D 2 ) [ | n edg ( λ ) n ctr ( λ ) | n ctr ( λ ) ] .
n ( r , λ ) = n ctr ( λ ) + [ n edg ( λ ) n ctr ( λ ) ] ( r D ) 2 .
ϕ G ( λ ) = ϕ R ( λ ) + ϕ W ( λ ) ,
ϕ ˙ G ( λ ) = ϕ R ( λ ) V R ( λ ) + ϕ W ( λ ) V W ( λ ) ,
ϕ R ( λ ) = ( 1 R f 1 R b ) [ n ctr ( λ ) 1 ] ,
ϕ W ( λ ) = ( 8 t D 2 ) [ n ctr ( λ ) n edg ( λ ) ] .
V W ( λ ) = n ctr ( λ ) n edg ( λ ) n ˙ ctr ( λ ) n ˙ edg ( λ ) .
ϕ G ( λ ) = ϕ ctr ( λ ) + ϕ edg ( λ ) ,
ϕ ctr ( λ ) = [ 1 R f ( 1 R b 8 t D 2 ) ] [ n ctr ( λ ) 1 ] ,
ϕ edg ( λ ) = [ ( 1 R b 8 t D 2 ) 1 R b ] [ n edg ( λ ) 1 ] .
n 2 ( r , λ ) = n a 2 ( λ ) + γ b ( r ) [ n b 2 ( λ ) n a 2 ( λ ) ] + γ c ( r ) [ n c 2 ( λ ) n a 2 ( λ ) ] ,
γ a ( r ) + γ b ( r ) + γ c ( r ) = 1 .
n ˙ ( r , λ ) = [ 1 n ( r , λ ) ] { n a ( λ ) n ˙ a ( λ ) + γ b ( r ) [ n b ( λ ) n ˙ b ( λ ) n a ( λ ) n ˙ a ( λ ) ] + γ c ( r ) [ n c ( λ ) n ˙ c ( λ ) n a ( λ ) n ˙ a ( λ ) ] } .
n ctr ( λ ) = { n a 2 ( λ ) + γ b ( 0 ) [ n b 2 ( λ ) n a 2 ( λ ) ] + γ c ( 0 ) [ n c 2 ( λ ) n a 2 ( λ ) ] } 1 / 2 ,
n edg ( λ ) = { n a 2 ( λ ) + γ b ( D 2 ) [ n b 2 ( λ ) n a 2 ( λ ) ] + γ c ( D 2 ) [ n c 2 ( λ ) n a 2 ( λ ) ] } 1 / 2 ,
n ˙ ctr ( λ ) = [ 1 n ctr ( λ ) ] { n a ( λ ) n ˙ a ( λ ) + γ b ( 0 ) [ n b ( λ ) n ˙ b ( λ ) n a ( λ ) n ˙ a ( λ ) ] + γ c ( 0 ) [ n c ( λ ) n ˙ c ( λ ) n a ( λ ) n ˙ a ( λ ) ] } .
n ˙ edg ( λ ) = [ 1 n edg ( λ ) ] { n a ( λ ) n ˙ a ( λ ) + γ b ( D 2 ) [ n b ( λ ) n ˙ b ( λ ) n a ( λ ) n ˙ a ( λ ) ] + γ c ( D 2 ) [ n c ( λ ) n ˙ c ( λ ) n a ( λ ) n ˙ a ( λ ) ] } .
ϕ ( λ ) = ϕ 1 ( λ ) + ϕ 2 ( λ ) ,
ϕ ˙ ( λ ) = ϕ ˙ 1 ( λ ) + ϕ ˙ 2 ( λ ) , = ϕ 1 ( λ ) V 1 ( λ ) + ϕ 2 ( λ ) V 2 ( λ ) .
ϕ ˙ 1 ( λ ) = ϕ ˙ 2 ( λ ) ,
ϕ ( λ ) = C 1 h 1 ( λ ) + C 2 h 2 ( λ ) .
C 2 h ˙ 2 ( λ 0 ) = C 1 h ˙ 1 ( λ 0 ) .
C 1 = h ˙ 2 ( λ 0 ) ϕ ( λ 0 ) h 1 ( λ 0 ) h ˙ 2 ( λ 0 ) h 2 ( λ 0 ) h ˙ 1 ( λ 0 ) ,
C 2 = h ˙ 1 ( λ 0 ) ϕ ( λ 0 ) h 1 ( λ 0 ) h ˙ 2 ( λ 0 ) h 2 ( λ 0 ) h ˙ 1 ( λ 0 ) .
ϕ ( λ ) ϕ ( λ 0 ) = h 1 ( λ ) h ˙ 2 ( λ 0 ) h 2 ( λ ) h ˙ 1 ( λ 0 ) h 1 ( λ 0 ) h ˙ 2 ( λ 0 ) h 2 ( λ 0 ) h ˙ 1 ( λ 0 ) .
h 1 ( λ ) = h 2 ( λ ) + Δ n .
ϕ ( λ ) = C 1 Δ n .
ϕ ( λ ) = 8 t D 2 n ctr ( λ ) 8 t D 2 n edg ( λ ) .
C 1 = 8 t D 2 ,
C 2 = 8 t D 2 ,
h 1 ( λ ) = n ctr ( λ ) ,
h 2 ( λ ) = n edg ( λ ) .
n ctr ( λ ) = n edg ( λ ) + Δ n .
ϕ ( λ ) = 8 t Δ n / D 2 .
ϕ ( λ ) = 8 M λ D 2 + 8 t D 2 [ n ctr ( λ ) n edg ( λ ) ] .
C 1 = 8 M λ 0 D 2 ,
C 2 = 8 t D 2 ,
h 1 ( λ ) = λ λ 0 ,
h 2 ( λ ) = n ctr ( λ ) n edg ( λ ) .
ϕ ( λ ) = ( 1 R f 1 R b ) [ n ctr ( λ ) 1 ] + 8 t D 2 [ n ctr ( λ ) n edg ( λ ) ] .
C 1 = 1 R f ( 1 R b 8 t D 2 ) ,
C 2 = ( 1 R b 8 t D 2 ) 1 R b ,
h 1 ( λ ) = n ctr ( λ ) 1 ,
h 2 ( λ ) = n edg ( λ ) 1 .
m 12 = [ h 1 ( λ 0 ) h 2 ( λ 0 ) h ˙ 1 ( λ 0 ) h ˙ 2 ( λ 0 ) ] 2 σ [ h 1 ( λ ) h 2 ( λ ) h ˙ 1 ( λ 0 ) h ˙ 2 ( λ 0 ) ] + δ σ , ϕ ( λ 0 ) σ [ ϕ ( λ ) ϕ ( λ 0 ) ] ,
σ [ g ( λ ) ] = { ( 1 N λ ) i = 1 N λ [ g ( λ i ) g ¯ ] 2 } 1 / 2 ,
g ¯ = ( 1 N λ ) i = 1 N λ g ( λ i ) ,
n ^ = ( 1 N ) i = 1 N [ | n a ( λ i ) n b ( λ i ) | + | n ˙ a ( λ i ) n ˙ b ( λ i ) | + | n ¨ a ( λ i ) n ¨ b ( λ i ) | ] .
m continuum = m 12 ( γ 1 + γ ) ,
γ = { [ γ b ( 0 ) γ b ( D / 2 ) ] 2 + [ γ c ( 0 ) γ c ( D / 2 ) ] 2 0.05 } 5 .
m WL = [ n ctr ( λ 0 ) n edg ( λ 0 ) ] 2 σ [ n ctr ( λ ) n edg ( λ ) ] .
m DWL = [ 1 n ctr ( λ 0 ) n edg ( λ 0 ) λ 0 [ n ˙ ctr ( λ 0 ) n ˙ edg ( λ 0 ) ] ] 2 σ [ λ λ 0 n ctr ( λ ) n edg ( λ ) λ 0 [ n ˙ ctr ( λ 0 ) n ˙ edg ( λ 0 ) ] ] .
m GSL = [ ( n ctr ( λ 0 ) 1 ) n ˙ ctr ( λ 0 ) n ˙ edg ( λ 0 ) ( n edg ( λ 0 ) 1 ) ] 2 σ [ ( n ctr ( λ ) 1 ) n ˙ ctr ( λ 0 ) n ˙ edg ( λ 0 ) ( n edg ( λ ) 1 ) ] .
ϕ G ( λ ) = ϕ R ( λ ) C [ Φ ( λ ) ] + [ ϕ W ( λ ) + ϕ R 2 ( λ ) ] Snc [ Φ ( λ ) ] ,
Snc [ Φ ( λ ) ] = S [ Φ ( λ ) ] Φ ( λ ) .
s G = sgn [ n ctr ( λ ) n edg ( λ ) ] ,
C [ Φ ( λ ) ] = cos Φ ( λ ) ,
S [ Φ ( λ ) ] = sin Φ ( λ ) ,
C [ Φ ( λ ) ] = cosh Φ ( λ ) ,
S [ Φ ( λ ) ] = sinh Φ ( λ ) .
ϕ R ( λ ) = ( 1 R f 1 R b ) [ n ctr ( λ ) 1 ] .
ϕ R 2 ( λ ) = [ n ctr ( λ ) 1 ] 2 t R f R b n ctr ( λ ) ,
ϕ W ( λ ) = ( 8 t D 2 ) [ n ctr ( λ ) n edg ( λ ) ] .
Φ ( λ ) = ( 8 t 2 D 2 ) [ | n edg ( λ ) n ctr ( λ ) | n ctr ( λ ) ] ,
ϕ ˙ G ( λ ) = { C [ Φ ( λ ) ] ϕ ˙ R ( λ ) + ϕ R ( λ ) C ˙ [ Φ ( λ ) ] } + { Snc [ Φ ( λ ) ] [ ϕ ˙ W ( λ ) + ϕ ˙ R 2 ( λ ) ] + [ ϕ W ( λ ) + ϕ R 2 ( λ ) ] S ˙ nc [ Φ ( λ ) ] } = ϕ R ( λ ) [ C [ Φ ( λ ) ] V R ( λ ) + C ˙ [ Φ ( λ ) ] ] + ϕ R 2 ( λ ) [ Snc [ Φ ( λ ) ] V R 2 ( λ ) + S ˙ nc [ Φ ( λ ) ] ] + ϕ W ( λ ) [ Snc [ Φ ( λ ) ] V W ( λ ) + S ˙ nc [ Φ ( λ ) ] ] , = ϕ R ( λ ) V Rth ( λ ) + ϕ R 2 ( λ ) V R 2 th ( λ ) + ϕ W ( λ ) V Wth ( λ ) ,
V Rth ( λ ) = V R ( λ ) C [ Φ ( λ ) ] s G V R ( λ ) Φ 2 ( λ ) Snc [ Φ ( λ ) ] [ Φ ˙ ( λ ) / Φ ( λ ) ] ,
V R 2 th ( λ ) = V R 2 ( λ ) Snc [ Φ ( λ ) ] + V R 2 ( λ ) { C [ Φ ( λ ) Snc [ Φ ( λ ) ] } [ Φ ˙ ( λ ) / Φ ( λ ) ] ,
V Wth ( λ ) = V W ( λ ) Snc [ Φ ( λ ) ] + V W ( λ ) { C [ Φ ( λ ) ] Snc [ Φ ( λ ) ] } [ Φ ˙ ( λ ) / Φ ( λ ) ] ,
V R ( λ ) = n ctr ( λ ) 1 n ˙ ctr ( λ ) ,
V R 2 ( λ ) = { n ctr ( λ ) [ n ctr ( λ ) 1 ] n ctr ( λ ) + 1 } [ 1 n ˙ ctr ( λ ) ] ,
V W ( λ ) = n ctr ( λ ) n edg ( λ ) n ˙ ctr ( λ ) n ˙ edg ( λ ) .
Φ ˙ ( λ ) Φ ( λ ) = ( 1 2 ) [ n edg ( λ ) n ctr ( λ ) n edg ( λ ) ] [ n ˙ ctr ( λ ) n ctr ( λ ) n ˙ edg ( λ ) n edg ( λ ) ] .

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