## Abstract

Fundamentally new classes of spherical gradient-index lenses with imaging and concentration properties that approach the fundamental limits are derived. These analytic solutions admit severely constrained maximum and minimum refractive indices commensurate with existing manufacturable materials, for realistic optical and solar lenses.

© 2011 OSA

## Corrections

Panagiotis Kotsidas, Vijay Modi, and Jeffrey M. Gordon, "Gradient-index lenses for near-ideal imaging and concentration with realistic materials: errata," Opt. Express**20**, 338-338 (2012)

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-20-1-338

## 1. Introduction

Can gradient-index (GRIN) lenses capable of perfect imaging and maximum flux concentration be realized for optical and solar frequencies with *real* materials and fabrication techniques? The appropriate GRIN profiles derived to date - *n*(*r*), spherically symmetric in lens radial coordinate *r* - require refractive index values for which transparent, manufacturable materials do not exist. The purpose of this paper is to derive basically new classes of GRIN lens solutions that surmount previous limitations and identify viable devices for optical and solar lenses with imaging and concentration near the fundamental limits.

The first derivation of refractive index profiles *n*(*r*) that produce perfect imaging for a general near-field source and target (Fig. 1
) was published by Luneburg (although a specific solution was provided only for a far-field source and the focus on the lens surface) [1]. Luneburg’s derivation assumed that *n*(*r*) is an invertible monotonic function, devoid of discontinuities. His solution was viewed as unrealizable for optical frequencies because it required (a) a minimum index *n*
_{min} of unity at the lens surface, and (b) a large index gradient (Δ*n* ≡ *n*
_{max}-*n*
_{min} > 0.4). Morgan [2] demonstrated how introducing a discontinuity in *n*(*r*) can relax the former constraint – also achieved differently by Sochacki [3] by limiting (stopping down) the lens effective aperture.

Indeed, perfect imaging does *not* limit solutions to a single continuum GRIN distribution; rather, it only requires that some finite region of the sphere must comprise a continuous gradient index. Other regions of the lens can be arbitrarily chosen, e.g., a core or shells of constant index, or with the index being a specified function of *r* (linear, parabolic, etc.). Since the lens is actually fabricated from discrete shells, the paucity of continuity poses no problem in lens manufacture [4–6].

A principal aim here is to expand the realm of physical solutions to those that can be produced for visible and solar radiation, given the severe restrictions on realistic candidate materials and manufacturing methods, by analyzing GRIN lenses that incorporate these extra degrees of freedom. Recent research advances in transparent polymers [4–6] have spawned materials and production techniques for the requisite ultra-thin spherical lens layers, but impose Δ*n* < 0.13 and *n*
_{min} values that must exceed 1.4, as well as necessitating a constant-index spherical core.

Here, we present several new classes of solutions:

- (a) an extension of Luneburg’s derivations that can accommodate arbitrary refractive index at the sphere’s surface,
- (b) GRIN profiles that allow a constant-index spherical core,
- (c) a technique expanding the realm of solutions for limited (non-full) apertures, and
- (d) solutions that allow a combination of regions of constant or prescribed index, plus one or more continuum (GRIN) regions.

Each additional degree of freedom creates more flexibility in accommodating arbitrary *n*
_{min} and *n*
_{max}. We define champion designs as those that can satisfy the limitations of off-the-shelf polymer technology with feasible scale-up [4–6]: *n*
_{min} = 1.44 and *n*
_{max} = 1.57. The challenge is heightened by the observation that reducing Δ*n* by as little as 0.01 can make the difference between viable vs. unphysical solutions.

All solutions are derived with full mathematical rigor. However the illustrations are restricted to cases of practical interest for sunlight – previously deemed unattainable with existing, readily manufacturable, transparent materials. Examples are presented for spherical GRIN lenses that nominally attain perfect imaging (for monochromatic radiation, in the geometrical optics limit) – relevant here because perfect imaging also implies attaining the thermodynamic limit to flux concentration [7,8]. The latter means that such Luneburg-type solar lenses would constitute *single-element* concentrators that approach the fundamental maximum for acceptance angle - and for optical tolerance to off-axis orientation - at a prescribed concentration (or *vice versa*) [7–9]. This also relates to averaged irradiance levels of the order of 10^{3} now common in concentrator photovoltaics. Moreover, such GRIN lenses offer a unique solution for achieving nominally stationary high-irradiance solar concentration, as recently demonstrated in [10]. These points are elaborated and illustrated in Section 7.

In some instances (Sections 3 and 4), a single extra degree of freedom proves inadequate to yield champion designs, although the associated solutions expand the possibilities well beyond prior findings. When multiple degrees of freedom can engender champion solutions (Sections 5-7), raytrace simulation results are provided that confirm lens performance near the thermodynamic limit of solar concentration even accounting for the extended and polychromatic character of the solar source (based on the dispersion properties of polymers from which the lenses can currently be fabricated [4–6]).

## 2. The classic Luneburg solution

The goal is to derive *n*(*r*) for a spherical lens of unit radius (0 ≤ *r* ≤ 1), in air (*n*(*r*>1) = 1), that perfectly images an object comprising part of a spherical contour with radius *r*
_{o} to a spherical contour image of radius *r*
_{1} (Fig. 1). Snell’s law (equivalent here to the conservation of skewness κ for a given ray along its entire trajectory [7,8])

*α*is the polar angle along the ray), is combined with Fermat’s principle of constant optical path length to obtain the governing integral equation

To solve Eq. (2), one multiplies both sides by dκ/√(κ^{2} - *ρ*
^{2}), integrates from ρ to 1, and interchanges the order of integration to obtain:

*n*(

*r*) is continuous and invertible, with

*n*(1) = 1. The explicit solution cited by Luneburg was for

*r*

_{o}→ ∞ and

*F*= 1:

*n*(

*r*) = √(2 –

*r*

^{2}).

## 3. Extension of Luneburg’s solution to an arbitrary surface index *n*(1)

To accommodate an arbitrary lens surface index *N* ≡ *n*(1), one rewrites Eq. (2) as

*r*)) = -dg(ρ)/d

*r*≡ -g

**′**(ρ) yields an Abel integral equation:

An example of this solution for a lens with *N* = 1.4, *F* = 1.1 and a far-field source is shown in Fig. 2
. Although the derivations presented here relate to the general near-field problem (arbitrary *r*
_{o} and *r*
_{1}), all the illustrative examples pertain to the far-field problem, prompted by solar concentrator applications.

## 4. Distributions with a constant-index core

#### 4.1 Derivation

Sochacki [3] considered Luneburg-type lenses of limited (non-full or stopped-down) aperture, but his requiring *n*(*r*) to be a smooth function, and his exploring only a narrow parameter space that excluded full-aperture lenses, severely restricted the available solutions. Relaxing these constraints permits solutions that smoothly transition from non-full to full aperture as well as a constant-index core that extends over a non-negligible radius. The latter point is particularly germane for current GRIN manufacturing techniques where precise, robust profiles require fabrication around a sizable homogeneous core [4–6].

With the boundary condition *n*(1), a value for the effective aperture *A* is selected (*A* ≤ 1 denoting the irradiated fraction of the sphere’s radius that produces perfect focusing – see Fig. 3
), along with the desired values of *F* and *n*(0).

The governing equation becomes:

As before, the solution follows from multiplying both sides by dκ/√(κ^{2} - *ρ*
^{2}), integrating from ρ to *N*, and interchanging the order of integration:

Equation (7) can be recast as

_{o}(the constant-index core) where ρ

_{o}≤

*A*.

Hence, one needs to solve an integral equation with constant limits of integration, known as a Fredholm integral equation of the first kind. Solutions are difficult to find because such integral equations are commonly ill-posed and singular [11,12]. Before invoking methods that might yield a closed-form solution [13], we first attempt to solve Eq. (8) numerically. One starts by assuming the solution can be represented as

(In the examples here, Lagrange polynomials are employed, but the selection can be expanded to other representations [14].) Substitution of Eq. (9) into Eq. (8) yields

Upon proper discretization of the domain of κ, Eq. (10) becomes a system of linear equations:

where the unknowns are the weights*w*

_{i}in Eq. (9).

As noted by Twomey [15], Phillips [16] demonstrated that “the ‘exact’ solution obtained when *Bw = g* is solved is almost always poor and often disastrously so - in the sense that the solution oscillates or displays some other feature which conflicts with *a priori* knowledge.” Accordingly, we adopt the numerical techniques suggested by Twomey and Phillips [15,16], with the solution given by

*H*can have various representations [15]. We present examples with smooth (non-oscillating) solutions by following Phillips’ procedure (and hence

*H*matrix):

Only physically inadmissible solutions are rejected, e.g., multi-valued functions with more than one value of *n*(*r*) for a given value of *r*.

We present the derivation required for a third-order Lagrange polynomial, but the technique can be expanded to any order polynomial approximation or alternative interpolation technique (splines, Hermite polynomials, etc.) [14]:

Equation (14) is inserted into Eq. (10) and integrated over κ. A proper discretization of the free variable ρ and the dummy variable κ results in an algebraic system of equations in the form of Eq. (11) from which one then retrieves the factors *w _{i}*, as well as

*f*

_{1}

^{+}(κ) through Eq. (9). Finally, inserting

*f*

_{1}

^{+}(κ) into Eq. (7), a smooth

*n*(

*r*) is obtained. Alternatively, the matrix

*B*can be directly inverted (actually, pseudo-inverted due to its poor rank) to obtain oscillatory solutions. Then, with Luneburg’s basic integral equation transformation [1], one finally emerges with the corresponding

*n*(

*r*).

The solutions are not *exactly* constant, but rather oscillate with a magnitude of order 10^{−5} to 10^{−3} around the nominally constant *n*(0). Raytracing verifies that the solutions for the core can basically be treated as constant values. Finally, observing that the solution in Eq. (7) is everywhere continuous implies *f*
_{1}
^{+}(*B*) = *f*
_{1}(*B*) - a condition that needs to be implemented in the solution of Eqs. (11)-(12). Note that the actual *n*(0) and the core’s radial extent emerge as part of the solution. Namely, an initial guess of *n*(0) serves as an input parameter, but the solution iterates to a different final value.

#### 4.2 Example for an extensive constant-index core and a prescribed surface index

The aim is to achieve a constant-index core that comprises a substantial fraction of the lens radius, with a given surface index *N* = 1.555, *A* = 0.97, *F* = 1.71 and ρ_{o} = 0.12 (with grid linear partitions of 18 nodes for κ, 15 nodes for ρ, and β = 1). Three distinct solutions for the *same* input parameters are shown in Fig. 4
, and underscore the influence of (a) the initial guess for *n*(0), and (b) the smoothed vs. oscillatory calculational procedure. The solution based on the pseudo-inverse of the matrix *B* in Eq. (11) exhibits oscillatory behavior that would render lens fabrication problematic (the other two solutions were generated with the smoothing technique depicted above), but has the advantage of admitting a lower Δ*n* (low enough, in fact, to qualify as a champion design). All three profiles yield the same essentially perfect imaging.

## 5. Constant index in both the core and the outer layer

Inspection of Eq. (7) reveals that if *f*
_{1}
^{+}(κ) = *const*, then *n*(ρ) = *const* for *A*
_{2} ≤ ρ ≤ *N*. Hence, imposing this condition in the numerical solution presented in the previous section can yield solutions with *both* a constant-index core *and* a constant-index outer layer.

The governing equation is rewritten as

*f*

_{1}

^{+}(κ) is determined as part of the solution, and the function

*f*

_{2}

^{+}(κ) follows from the prescribed outer shell (e.g., for a constant-index shell,

*f*

_{2}

^{+}(κ)=0, which yields

*n*(ρ)=

*N*for

*A*

_{2}≤ ρ ≤

*N*). Equation (15) is solved by

## 6. Closed-form solution

The Fredholm Eq. (8) possesses a closed-form solution. Applying the transformation

Equation (8) becomes a singular integral equation

*quadgk*function [17].

An extra condition is needed to produce smooth solutions:

In the discretized calculational grid, the first two values of *f*
^{+} might need to be equated to *f*
^{+}(*A*), or a similar heuristic scheme can be found to produce a solution that is smooth and physically admissible.

A sample solution for a far-field source, *F* = 1.5 and *A* = 0.75 with a constant-index core up to *r* = 0.3 is graphed in Fig. 6a
. If one requires a full effective aperture *A*→1, then *N* needs to be raised substantially (*N* ≥ 2) in order to maintain a constant-index core.

Although the full-aperture solutions generated with this technique exhibit *both* a considerably higher *n*
_{min}
*and* a lower Δ*n* than the original Luneburg method (Fig. 6b), they need a coarse discretization in evaluating the integrals in Eq. (19). For this specific example, a 3-point equal-spacing discretization was used in the numerical integrations.

## 7. Sample champion designs for solar concentration

GRIN lenses are promising candidates for high-irradiance photovoltaic concentrators with liberal optical tolerance (with a single optical element) [10]. The examples that follow subsume champion designs (with realistic materials and fabrication techniques even when the polychromatic and extended solar source is accounted for) that can:

- a) offer square truncated lenses that obviate packing losses in solar modules, without introducing incremental losses in collection efficiency;
- b) be used in nominally stationary high-irradiance solar concentrators [10]; and
- c) achieve a flux concentration ≈30000 (previously deemed unachievable with a single lens).

Dispersion losses (due to a wavelength-dependent refractive index) were evaluated based on an AM1.5D solar spectrum and a Cauchy-type dispersion relation for the measured properties of representative polymer materials [4–6], for lenses designed based on the refractive indices at a wavelength of 633 nm.

#### 7.1 Truncated lenses of restricted aperture

Designing for *A* < 1 allows truncation of the lens in a form that basically eliminates packing losses in a typical rectangular module, at no incremental loss in collection efficiency - suitable for dual-axis tracking concentrator photovoltaics. Figure 7
presents one champion design (from among many) that also includes a constant-index core.

#### 7.2 Stationary high-irradiance solar concentration

Spherical GRIN lenses are especially well suited to nominally stationary high-irradiance photovoltaic concentrators [10]. Figure 8
shows a champion design based on a double-GRIN profile (*F* = 1.32 and *A* = 0.985, the latter incurring a 3% loss of collectible radiation because of collector stationarity). Geometric collection efficiency = 95% at *C* = 1500 (suited to current concentrator photovoltaics) integrated over the full 100° acceptance angle, *including* losses due to the polychromatic and extended solar source. If for convenience one approximates *n*(*r*) as constant over 0 ≤ *r* ≤ 0.15, then lens performance is essentially unaffected.

#### 7.3 Ultra-high solar irradiance

Figure 9
presents *n*(*r*) for a double-GRIN lens that generates a solar flux concentration exceeding 30000 at the center of its focal spot – an irradiance level heretofore deemed unattainable with a single lens for broadband radiation. Although dispersion losses result in some of the radiation falling outside the ultra-high irradiance region, the point here is to demonstrate that such enormous flux densities can be produced at all - of value in nanomaterial synthesis and concentrator solar cell characterization [18].

## 8. Conclusions

A variety of previously unrecognized GRIN spherical lens solutions that achieve perfect imaging for arbitrary focal length, effective aperture, severely restricted maximum and minimum refractive indices, and can admit uniform spherical cores have been derived and illustrated. The champion designs can be manufactured with existing techniques, and their performance as high-irradiance solar concentrators has been evaluated via raytracing, including accounting for the polychromatic and extended character of the solar source. Designing for a non-full aperture further allows lenses of arbitrary aperture shape (which can obviate packing losses) without incurring an incremental efficiency loss.

All the solutions derived here (also rigorous for the corresponding 2D cylindrically symmetric GRIN lenses) provide essentially perfect imaging and approach the thermodynamic limit to concentration for monochromatic light (in the geometric optics limit). In this sense, raytrace substantiation (for monochromatic radiation) is redundant, but was performed for categorical verification. Raytrace results for *solar* lenses additionally confirm that dispersion incurs only modest losses for current high-irradiance solar designs. The issues of GRIN lens design and performance in the physical (diffraction-limited) optics limit, as well as an analytic method for mitigating chromatic aberration (in analogy to the design of achromats for conventional lenses), remain topics for future investigations.

The new classes of GRIN solutions permit realizable optical and solar lenses with existing materials and fabrication procedures. Perhaps no less significantly, however, they point to an infinite number of previously unidentified solutions that can now realistically be implemented for optical frequencies. Namely, each additional degree of freedom allows one to more easily satisfy the demands of limited *n*
_{min} and *n*
_{max}, with the extra degrees of freedom comprising more shells of constant or prescribed-function refractive index, as well as interspersing more GRIN (continuum) layers. These options require a straightforward but tedious application of the formalism depicted herein. Basically, only the associated challenges of lens fabrication and calculation time limit their realization. The flexibility of accommodating ranges of refractive index previously viewed as intractable based on existing GRIN optical analyses could also open new vistas in infrared imaging and concentration at such time as manufacturable materials that also allow continuum GRIN profiles become available.

## Acknowledgments

This research was funded by the Defense Advanced Research Programs Agency, under the M-GRIN program, Contract No. HR0011-10-C-0110. We thank Prof. Eric Baer and his group at Case Western Reserve University for enlightening discussions regarding the state-of-the-art in manufacturable gradient-index polymers and fabrication methods for visible light applications. PK thanks Prof. Marc Spiegelman of Columbia University for fruitful discussions regarding numerical solutions of Fredholm integral equations. JMG expresses his gratitude to Columbia University’s Mechanical Engineering Department for its generous hospitality during part of this research program.

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