Gradient-index lenses for near-ideal imaging and concentration with realistic materials

Panagiotis Kotsidas; Vijay Modi; Jeffrey M. Gordon

doi:10.1364/OE.19.015584

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 (Δn ≡ 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.

Fig. 1 (a) Sample ray trajectory through a perfect-imaging spherical GRIN lens, from a source point at r _o to a target point at r ₁. r ^* denotes the closest point of approach to the origin. (b) A wavefront from a far-field source (r _o → ∞) traced to a target at focal length F = r ₁.

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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 Δn < 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 Δn 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 10³ 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])

r n (r) Sin (α) = κ

(where 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

\int_{r^{*}}^{1} \frac{κ d r}{r \sqrt{ρ^{2} - κ^{2}}} = f (κ) where ρ (r) \equiv r n (r) and f (κ) = \frac{1}{2} [{Sin}^{- 1} \frac{κ}{r_{o}} + {Sin}^{- 1} \frac{κ}{r_{1}} + 2 {Cos}^{- 1} κ] 0 \leq κ \leq 1.

To solve Eq. (2), one multiplies both sides by dκ/√(κ² - ρ ²), integrates from ρ to 1, and interchanges the order of integration to obtain:

n_{L u n e b u r g} = \exp (ω (ρ, r_{o}) + ω (ρ, r_{1})) where ω (ρ, s) = \frac{1}{π} \int_{ρ}^{1} \frac{{Sin}^{- 1} (\frac{κ}{s})}{\sqrt{κ^{2} - ρ^{2}}} d κ

where it was implicitly assumed that 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

\int_{r^{*}}^{1} \frac{κ d r}{r \sqrt{ρ^{2} - κ^{2}}} = \frac{f (κ)}{2} = {Sin}^{- 1} (\frac{κ}{r_{o}}) + {Sin}^{- 1} (\frac{κ}{r_{1}}) + 2 {Sin}^{- 1} (\frac{κ}{N}) - 2 {Sin}^{- 1} (κ) 0 \leq κ \leq N

(note the revised domain for κ). The last two last terms in Eq. (4) stem from the two extra refractions at the lens surface [3]. Using the substitution d(ln(r)) = -dg(ρ)/dr ≡ -g′(ρ) yields an Abel integral equation:

- \int_{κ}^{N} \frac{g^{'} (ρ) κ d r}{\sqrt{ρ^{2} - κ^{2}}} = \frac{f (κ)}{2} which has the solution n (ρ) = N \exp (\frac{1}{π} \int_{ρ}^{N} \frac{f (κ) d κ}{r \sqrt{κ^{2} - ρ^{2}}}) .

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.

Fig. 2 (a) Luneburg-type lens with the new solution that permits a surface index above unity, for a far-field source, with F = 1.1 and n(1) = N = 1.1. (b) Trace of several paraxial rays.

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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).

Fig. 3 Input parameters for a limited-aperture GRIN lens with a constant-index core.

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The governing equation becomes:

\begin{array}{l} \int_{N}^{κ} \frac{κ g^{'} (ρ) d ρ}{\sqrt{ρ^{2} - κ^{2}}} = {\begin{matrix} \frac{f_{1} (κ)}{2} 0 \leq κ \leq A \\ \frac{f_{1}^{+} (κ)}{2} A \leq κ \leq N \end{matrix} \\ where f_{1} (κ) = {Sin}^{- 1} (\frac{κ}{r_{o}}) + {Sin}^{- 1} (\frac{κ}{r_{1}}) + 2 {Sin}^{- 1} (\frac{κ}{N}) - 2 {Sin}^{- 1} (κ) \\ and f_{1}^{+} (κ) remains to be determined in the analysis that follows . \end{array}

As before, the solution follows from multiplying both sides by dκ/√(κ² - ρ ²), integrating from ρ to N, and interchanging the order of integration:

\begin{array}{l} n (ρ) = N \exp [\frac{1}{π} \int_{ρ}^{A} \frac{f_{1} (κ)}{\sqrt{κ^{2} - ρ^{2}}} d κ + \frac{1}{π} \int_{A}^{N} \frac{f_{1}^{+} (κ)}{\sqrt{κ^{2} - ρ^{2}}} d κ] \\ = {\begin{matrix} N \exp [ω (ρ, r_{1}, A) + ω (ρ, r_{0}, A) + 2 ω (ρ, N, A) - 2 ω (ρ, 1, A) + \frac{1}{π} \int_{A}^{N} \frac{f_{1}^{+} (κ)}{\sqrt{κ^{2} - ρ^{2}}} d κ] 0 \leq ρ \leq A \\ N \exp [\frac{1}{π} \int_{ρ}^{N} \frac{f_{1}^{+} (κ)}{\sqrt{κ^{2} - ρ^{2}}} d κ] A \leq ρ \leq N . \end{matrix} \end{array}

Equation (7) can be recast as

\begin{array}{l} \ln (\frac{n (ρ)}{N}) = \frac{1}{π} \int_{ρ}^{A} \frac{f_{1} (κ)}{\sqrt{κ^{2} - ρ^{2}}} d κ + \frac{1}{π} \int_{A}^{N} \frac{f_{1}^{+} (κ)}{\sqrt{κ^{2} - ρ^{2}}} d κ, and, rearranging: \\ \frac{1}{π} \int_{A}^{N} \frac{f_{1}^{+} (κ)}{\sqrt{κ^{2} - ρ^{2}}} d κ = \ln (\frac{n (ρ)}{N}) - \frac{1}{π} \int_{ρ}^{A} \frac{f_{1} (κ)}{\sqrt{κ^{2} - ρ^{2}}} d κ \\ = \ln (\frac{n (ρ)}{N}) - ω (ρ, r_{1}, A) + ω (ρ, r_{0}, A) + 2 ω (ρ, N, A) - 2 ω (ρ, 1, A) \\ where ω (ρ, A, s) = \frac{1}{π} \int_{ρ}^{s} \frac{{Sin}^{- 1} (\frac{κ}{A})}{\sqrt{κ^{2} - ρ^{2}}} d κ \end{array}

and n(ρ) is constant for the range 0 ≤ ρ ≤ ρ_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

f_{1}^{+} (κ) = \sum_{i} w_{i} φ_{i} (κ) .

(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

\frac{1}{π} \int_{A}^{N} \frac{f_{1}^{+} (κ)}{\sqrt{κ^{2} - ρ^{2}}} d κ = \frac{1}{π} \int_{A}^{N} \frac{\sum_{i} w_{i} φ_{i} (κ)}{\sqrt{κ^{2} - ρ^{2}}} d κ = \frac{1}{π} \sum_{i} w_{i} \int_{A}^{N} \frac{φ_{i} (κ)}{\sqrt{κ^{2} - ρ^{2}}} d κ .

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

B w = g

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

w = {(B^{*} B + β H)}^{- 1} B^{*} g

where * denotes matrix transposition, β is an arbitrary number usually in the range 0 to 1, and the matrix H can have various representations [15]. We present examples with smooth (non-oscillating) solutions by following Phillips’ procedure (and hence H matrix):

H = [\begin{matrix} 1 & - 2 & 1 & 0 & 0 & . & . & . \\ - 2 & 5 & - 4 & 1 & 0 & . & . & . \\ 1 & - 4 & 6 & - 4 & 1 & 0 & . & . \\ 0 & 1 & - 4 & 6 & - 4 & 1 & 0 & . \\ . & . & . & . & . & . & . & . \\ . & . & 0 & 1 & - 4 & 6 & - 4 & 1 \\ . & . & . & 0 & 1 & - 4 & 5 & - 2 \\ . & . & . & 0 & 0 & 1 & - 2 & 1 \end{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]:

\begin{array}{l} f_{1}^{+} (κ) = \frac{κ - κ_{i + 1}}{κ_{i} - κ_{i + 1}} \frac{κ - κ_{i + 2}}{κ_{i} - κ_{i + 2}} \frac{κ - κ_{i + 3}}{κ_{i} - κ_{i + 3}} w_{i} + \frac{κ - κ_{i}}{κ_{i + 1} - κ_{i}} \frac{κ - κ_{i + 2}}{κ_{i + 1} - κ_{i + 2}} \frac{κ - κ_{i + 3}}{κ_{i + 1} - κ_{i + 3}} w_{i + 1} \\ + \frac{κ - κ_{i}}{κ_{i + 2} - κ_{i}} \frac{κ - κ_{i + 1}}{κ_{i + 2} - κ_{i + 1}} \frac{κ - κ_{i + 3}}{κ_{i + 2} - κ_{i + 3}} w_{i + 2} + \frac{κ - κ_{i}}{κ_{i + 3} - κ_{i}} \frac{κ - κ_{i + 1}}{κ_{i + 3} - κ_{i + 1}} \frac{κ - κ_{i + 2}}{κ_{i + 3} - κ_{i + 2}} w_{i + 3} . \end{array}

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 Δn (low enough, in fact, to qualify as a champion design). All three profiles yield the same essentially perfect imaging.

Fig. 4 Three solutions (for the same input parameters) for a lens with a constant-index core and a prescribed surface index. Depending on the calculational method adopted, the GRIN region can exhibit oscillatory or smooth behavior. The initial guess for n(0) markedly influences the solution. (Note the expanded ordinate.)

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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

\int_{N}^{κ} \frac{κ g^{'} (ρ) d ρ}{\sqrt{ρ^{2} - κ^{2}}} = {\begin{matrix} \frac{f_{1} (κ)}{2} 0 \leq κ \leq A_{1} \\ \frac{f_{1}^{+} (κ)}{2} A_{1} \leq κ \leq A_{2} \\ \frac{f_{2}^{+} (κ)}{2} A_{2} \leq κ \leq N \end{matrix}

where the function 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

n (ρ) = {\begin{matrix} N \exp [\begin{array}{l} ω (ρ, r_{1}, A_{1}) + ω (ρ, r_{0}, A_{1}) + 2 ω (ρ, N, A_{1}) - 2 ω (ρ, 1, A_{1}) \\ + \int_{A_{1}}^{A_{2}} \frac{f_{1}^{+} (κ)}{π \sqrt{κ^{2} - ρ^{2}}} d κ + \int_{A_{2}}^{N} \frac{f_{2}^{+} (κ)}{π \sqrt{κ^{2} - ρ^{2}}} d κ \end{array}] 0 \leq ρ \leq A_{1} \\ N \exp [\int_{ρ}^{A_{2}} \frac{f_{1}^{+} (κ)}{π \sqrt{κ^{2} - ρ^{2}}} d κ + \int_{A_{2}}^{N} \frac{f_{2}^{+} (κ)}{π \sqrt{κ^{2} - ρ^{2}}} d κ] A_{1} \leq ρ \leq A_{2} \\ N \exp [\int_{ρ}^{N} \frac{f_{2}^{+} (κ)}{π \sqrt{κ^{2} - ρ^{2}}} d κ] A_{2} \leq ρ \leq N \end{matrix}

and illustrated in Fig. 5 .

Fig. 5 n(r) for a lens that incorporates constant-index regions in both the core and the outer shell. Lens input parameters are F = 1.680, A ₁ = 0.900, A ₂ = 1.423 and N = 1.573. This solution (based on the smoothing calculational method portrayed in Section 4) has n(0) = 1.534 extending over a core radius of 0.33. (Note the expanded ordinate.)

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6. Closed-form solution

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

\begin{array}{l} z = ρ^{2}, t = κ^{2}, γ (t) = \frac{f^{+} (κ)}{\sqrt{κ}}, F (z) = 2 φ (\sqrt{z}) \\ where φ (ρ) = \log (\frac{n (ρ)}{N}) - ω (ρ, r_{1}, A) + ω (ρ, r_{0}, A) + 2 ω (ρ, N, A) - 2 ω (ρ, 1, A), \end{array}

Equation (8) becomes a singular integral equation

\int_{A_{1}}^{N_{1}} \frac{γ (τ)}{\sqrt{τ - z}} d τ = π F (z), where A_{1} = A^{2}, N_{1} = N^{2}

for which the closed-form solution is [13]

γ (t) = - \frac{1}{2 π} \sqrt{\frac{N_{1} - t}{t - A_{1}}} \int_{A_{1}}^{N_{1}} \sqrt{\frac{N_{1} - s}{s - A_{1}}} \frac{d}{d s} [\int_{A_{1}}^{s} \frac{F (u)}{\sqrt{s - u}} d u] d s + \frac{1}{2} \frac{d}{d t} \int_{A_{1}}^{t} \frac{F (u)}{\sqrt{s - u}} d s .

γ(κ) is then found by numerically evaluating the derivatives and integrals in Eq. (19). The singularities that appear in the evaluation of the integral can be treated numerically with Matlab’s quadgk function [17].

An extra condition is needed to produce smooth solutions:

f^{+} (A) = {Sin}^{- 1} (\frac{A}{r_{o}}) + {Sin}^{- 1} (\frac{A}{r_{1}}) + 2 {Sin}^{- 1} (\frac{A}{N}) - 2 {Sin}^{- 1} (A) .

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.

Fig. 6 (a) The new closed-form solution with a constant-index core (here, up to r=0.3), A=0.68 and F=1.5. Note the expanded ordinate. (b) Comparison between our closed-form solution in the full-aperture (A=1) limit and the classic Luneburg solution - both for F=1 (focus on the lens surface) - to highlight the ability of the closed-form solution to lower Δn and raise n _min.

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Although the full-aperture solutions generated with this technique exhibit both a considerably higher n _min and a lower Δn 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.

Fig. 7 (a) GRIN profile for a dual-axis tracking solar concentrator, with F = 1.7 and A = 0.65. (Note the expanded ordinate.) (b) Efficiency-concentration curve characterizing lens performance. The geometric efficiency does not account for material-related Fresnel reflection and absorption, which are case-specific and readily incorporated. The abscissa refers to concentration C relative to the thermodynamic limit [7] C _max = {A/(F Sin(θ_sun))}² which in this case is 5847. A realistic concentrator design that accounts for liberal optical tolerance to off-axis orientation augurs designing for C ≈1500 [9] for which C/C _max ≈0.26 and the geometric collection efficiency is basically 100%. (c) Raytrace simulation (LightTools^®, Synopsys Inc.) with a polychromatic, extended solar source (5 mrad effective solar angular radius θ_sun comprising the intrinsic solar disc convolved with lens inaccuracies), illustrating how a non-full aperture GRIN solution can be “shaved” (dashed lines) at no loss of collectible radiation. 50000 rays uniformly distributed spatially and in projected solid angle were traced for each of 12 wavelengths spanning the solar spectrum.

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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.

Fig. 8 A solution with two GRIN (continuum) regions, for a nominally stationary high-irradiance photovoltaic concentrator [10] with a full acceptance angle of 100°. (Note the expanded ordinate.) F = 1.32 and A = 0.985. Raytracing confirms a geometric collection efficiency of 95% at C = 1500 integrated over the full 100° acceptance angle. (C = 1500 is chosen to provide liberal off-axis tolerance based on C _max = 22,730.)

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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].

Fig. 9 (a) A double-GRIN profile for F = 1.09 and A = 0.99, for which C _max = 33000. (Note the expanded ordinate.) (b) Raytrace for an extended, polychromatic solar source.

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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.

References and links

1. R. K. Luneburg, The Mathematical Theory of Optics (U. California Press, Berkeley, 1964).

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Gradient-index lenses for near-ideal imaging and concentration with realistic materials

Abstract

Corrections

1. Introduction

2. The classic Luneburg solution

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

4. Distributions with a constant-index core

4.1 Derivation

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

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

6. Closed-form solution

7. Sample champion designs for solar concentration

7.1 Truncated lenses of restricted aperture

7.2 Stationary high-irradiance solar concentration

7.3 Ultra-high solar irradiance

8. Conclusions

Acknowledgments

References and links

Cited By

Figures (9)

Equations (20)

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