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

Optics Express, Vol. 19, Issue 16, pp. 15584-15595 (2011)

http://dx.doi.org/10.1364/OE.19.015584

Acrobat PDF (1184 KB)

### 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

## 1. Introduction

*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.

*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

2. S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. **29**(9), 1358–1368 (1958). [CrossRef]

*n*(

*r*) can relax the former constraint – also achieved differently by Sochacki [3

3. J. Sochacki, J. R. Flores, and C. Gómez-Reino, “New method for designing the stigmatically imaging gradient-index lenses of spherical symmetry,” Appl. Opt. **31**(25), 5178–5183 (1992). [CrossRef] [PubMed]

*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

4. Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bioinspired lenses with a gradient refractive index,” J. Appl. Polym. Sci. **103**(3), 1834–1841 (2007). [CrossRef]

6. M. Ponting, A. Hiltner, and E. Baer, “Polymer nanostructures by forced assembly: process, structure and properties,” Macromol. Symp. **294**(1), 19–32 (2010). [CrossRef]

4. Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bioinspired lenses with a gradient refractive index,” J. Appl. Polym. Sci. **103**(3), 1834–1841 (2007). [CrossRef]

6. M. Ponting, A. Hiltner, and E. Baer, “Polymer nanostructures by forced assembly: process, structure and properties,” Macromol. Symp. **294**(1), 19–32 (2010). [CrossRef]

*n*< 0.13 and

*n*

_{min}values that must exceed 1.4, as well as necessitating a constant-index spherical core.

- (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.

*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

4. Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bioinspired lenses with a gradient refractive index,” J. Appl. Polym. Sci. **103**(3), 1834–1841 (2007). [CrossRef]

6. M. Ponting, A. Hiltner, and E. Baer, “Polymer nanostructures by forced assembly: process, structure and properties,” Macromol. Symp. **294**(1), 19–32 (2010). [CrossRef]

*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.

8. J. M. Gordon, “Spherical gradient-index lenses as perfect imaging and maximum power transfer devices,” Appl. Opt. **39**(22), 3825–3832 (2000). [CrossRef] [PubMed]

*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

9. A. Goldstein and J. M. Gordon, “Tailored solar optics for maximal optical tolerance and concentration,” Sol. Energy Mater. Sol. Cells **95**(2), 624–629 (2011). [CrossRef]

^{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

10. P. Kotsidas, V. Modi, and J. M. Gordon, “Nominally stationary high-concentration solar optics by gradient-index lenses,” Opt. Express **19**(3), 2325–2334 (2011). [CrossRef] [PubMed]

**103**(3), 1834–1841 (2007). [CrossRef]

**294**(1), 19–32 (2010). [CrossRef]

## 2. The classic Luneburg solution

*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

8. J. M. Gordon, “Spherical gradient-index lenses as perfect imaging and maximum power transfer devices,” Appl. Opt. **39**(22), 3825–3832 (2000). [CrossRef] [PubMed]

*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

^{2}-

*ρ*

^{2}), integrates from ρ to 1, and interchanges the order of integration to obtain: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)

*N*≡

*n*(1), one rewrites Eq. (2) as(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

3. J. Sochacki, J. R. Flores, and C. Gómez-Reino, “New method for designing the stigmatically imaging gradient-index lenses of spherical symmetry,” Appl. Opt. **31**(25), 5178–5183 (1992). [CrossRef] [PubMed]

*r*)) = -dg(ρ)/d

*r*≡ -g

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

*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

3. J. Sochacki, J. R. Flores, and C. Gómez-Reino, “New method for designing the stigmatically imaging gradient-index lenses of spherical symmetry,” Appl. Opt. **31**(25), 5178–5183 (1992). [CrossRef] [PubMed]

*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

**103**(3), 1834–1841 (2007). [CrossRef]

**294**(1), 19–32 (2010). [CrossRef]

*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).

^{2}-

*ρ*

^{2}), integrating from ρ to

*N*, and interchanging the order of integration:

_{o}(the constant-index core) where ρ

_{o}≤

*A*.

*w*

_{i}in Eq. (9).

15. S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. ACM **10**(1), 97–101 (1963). [CrossRef]

16. D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. ACM **9**(1), 84–97 (1962). [CrossRef]

*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

15. S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. ACM **10**(1), 97–101 (1963). [CrossRef]

16. D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. ACM **9**(1), 84–97 (1962). [CrossRef]

*H*can have various representations [15

15. S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. ACM **10**(1), 97–101 (1963). [CrossRef]

*H*matrix):

*n*(

*r*) for a given value of

*r*.

*w*, as well as

_{i}*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*).

*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

*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

*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.

*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

*quadgk*function [17

17. Matlab v. 7.9 and online documentation: http://www.mathworks.com/help/techdoc/ref/quadgk.html (MathWorks Inc., Natick, MA, 2003).

*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.

*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.

*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

10. P. Kotsidas, V. Modi, and J. M. Gordon, “Nominally stationary high-concentration solar optics by gradient-index lenses,” Opt. Express **19**(3), 2325–2334 (2011). [CrossRef] [PubMed]

- 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
**19**(3), 2325–2334 (2011). [CrossRef] [PubMed] - c) achieve a flux concentration ≈30000 (previously deemed unachievable with a single lens).

**103**(3), 1834–1841 (2007). [CrossRef]

**294**(1), 19–32 (2010). [CrossRef]

### 7.1 Truncated lenses of restricted aperture

### 7.2 Stationary high-irradiance solar concentration

**19**(3), 2325–2334 (2011). [CrossRef] [PubMed]

*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

*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

18. J. M. Gordon, D. Babai, and D. Feuermann, “A high-irradiance solar furnace for photovoltaic characterization and nanomaterial synthesis,” Sol. Energy Mater. Sol. Cells **95**(3), 951–956 (2011). [CrossRef]

## 8. Conclusions

*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.

*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

## References and links

1. | R. K. Luneburg, |

2. | S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. |

3. | J. Sochacki, J. R. Flores, and C. Gómez-Reino, “New method for designing the stigmatically imaging gradient-index lenses of spherical symmetry,” Appl. Opt. |

4. | Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bioinspired lenses with a gradient refractive index,” J. Appl. Polym. Sci. |

5. | G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express |

6. | M. Ponting, A. Hiltner, and E. Baer, “Polymer nanostructures by forced assembly: process, structure and properties,” Macromol. Symp. |

7. | R. Winston, P. Benítez, and J. C. Miñano, with contributions from N. Shatz and J. Bortz, |

8. | J. M. Gordon, “Spherical gradient-index lenses as perfect imaging and maximum power transfer devices,” Appl. Opt. |

9. | A. Goldstein and J. M. Gordon, “Tailored solar optics for maximal optical tolerance and concentration,” Sol. Energy Mater. Sol. Cells |

10. | P. Kotsidas, V. Modi, and J. M. Gordon, “Nominally stationary high-concentration solar optics by gradient-index lenses,” Opt. Express |

11. | A. D. Polyanin and A. V. Manzhirov, |

12. | C. T. H. Baker, |

13. | R. Estrada and R. P. Kanwal, |

14. | L. N. Trefethen, |

15. | S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. ACM |

16. | D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. ACM |

17. | Matlab v. 7.9 and online documentation: http://www.mathworks.com/help/techdoc/ref/quadgk.html (MathWorks Inc., Natick, MA, 2003). |

18. | J. M. Gordon, D. Babai, and D. Feuermann, “A high-irradiance solar furnace for photovoltaic characterization and nanomaterial synthesis,” Sol. Energy Mater. Sol. Cells |

**OCIS Codes**

(110.2760) Imaging systems : Gradient-index lenses

(350.6050) Other areas of optics : Solar energy

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: June 16, 2011

Manuscript Accepted: July 20, 2011

Published: July 28, 2011

**Citation**

Panagiotis Kotsidas, Vijay Modi, and Jeffrey M. Gordon, "Gradient-index lenses for near-ideal imaging and concentration with realistic materials," Opt. Express **19**, 15584-15595 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-16-15584

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### References

- R. K. Luneburg, The Mathematical Theory of Optics (U. California Press, Berkeley, 1964).
- S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. 29(9), 1358–1368 (1958). [CrossRef]
- J. Sochacki, J. R. Flores, and C. Gómez-Reino, “New method for designing the stigmatically imaging gradient-index lenses of spherical symmetry,” Appl. Opt. 31(25), 5178–5183 (1992). [CrossRef] [PubMed]
- Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bioinspired lenses with a gradient refractive index,” J. Appl. Polym. Sci. 103(3), 1834–1841 (2007). [CrossRef]
- G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express 16(15), 11540–11547 (2008). [PubMed]
- M. Ponting, A. Hiltner, and E. Baer, “Polymer nanostructures by forced assembly: process, structure and properties,” Macromol. Symp. 294(1), 19–32 (2010). [CrossRef]
- R. Winston, P. Benítez, and J. C. Miñano, with contributions from N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, Oxford, 2005).
- J. M. Gordon, “Spherical gradient-index lenses as perfect imaging and maximum power transfer devices,” Appl. Opt. 39(22), 3825–3832 (2000). [CrossRef] [PubMed]
- A. Goldstein and J. M. Gordon, “Tailored solar optics for maximal optical tolerance and concentration,” Sol. Energy Mater. Sol. Cells 95(2), 624–629 (2011). [CrossRef]
- P. Kotsidas, V. Modi, and J. M. Gordon, “Nominally stationary high-concentration solar optics by gradient-index lenses,” Opt. Express 19(3), 2325–2334 (2011). [CrossRef] [PubMed]
- A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, 2nd Ed. (Chapman and Hall/CRC Press, Boca Raton, 2008).
- C. T. H. Baker, The Numerical Treatment of Integral Equations (Clarendon Press, Oxford, 1977).
- R. Estrada and R. P. Kanwal, Singular Integral Equations (Birkhäuser, Boston, 2000).
- L. N. Trefethen, Spectral Methods in Matlab (S.I.A.M., Philadelphia, 2000).
- S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. ACM 10(1), 97–101 (1963). [CrossRef]
- D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. ACM 9(1), 84–97 (1962). [CrossRef]
- Matlab v. 7.9 and online documentation: http://www.mathworks.com/help/techdoc/ref/quadgk.html (MathWorks Inc., Natick, MA, 2003).
- J. M. Gordon, D. Babai, and D. Feuermann, “A high-irradiance solar furnace for photovoltaic characterization and nanomaterial synthesis,” Sol. Energy Mater. Sol. Cells 95(3), 951–956 (2011). [CrossRef]

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