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

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Errata

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

References

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  1. R. K. Luneburg, The Mathematical Theory of Optics (U. California Press, Berkeley, 1964).
  2. S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. 29(9), 1358–1368 (1958).
    [CrossRef]
  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]
  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]
  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 16(15), 11540–11547 (2008).
    [PubMed]
  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]
  7. R. Winston, P. Benítez, and J. C. Miñano, with contributions from N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, Oxford, 2005).
  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]
  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]
  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]
  11. A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, 2nd Ed. (Chapman and Hall/CRC Press, Boca Raton, 2008).
  12. C. T. H. Baker, The Numerical Treatment of Integral Equations (Clarendon Press, Oxford, 1977).
  13. R. Estrada and R. P. Kanwal, Singular Integral Equations (Birkhäuser, Boston, 2000).
  14. L. N. Trefethen, Spectral Methods in Matlab (S.I.A.M., Philadelphia, 2000).
  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]
  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 95(3), 951–956 (2011).
    [CrossRef]

2011

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]

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]

2010

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

2008

2007

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]

2000

1992

1963

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]

1962

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]

1958

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

Babai, D.

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]

Baer, E.

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

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]

Beadie, G.

Feuermann, D.

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]

Fleet, E.

Flores, J. R.

Goldstein, A.

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]

Gómez-Reino, C.

Gordon, J. M.

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]

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]

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

Hiltner, A.

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

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]

Jin, Y.

Kamdar, A. R.

Kazmierczak, T.

Kotsidas, P.

Lane, P. A.

Modi, V.

Morgan, S. P.

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

Phillips, D. L.

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]

Ponting, M.

Rosenberg, A.

Shirk, J. S.

Sochacki, J.

Tai, H.

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]

Twomey, S.

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]

Yang, Y.

Appl. Opt.

J. ACM

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]

J. Appl. Phys.

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

J. Appl. Polym. Sci.

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]

Macromol. Symp.

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

Opt. Express

Sol. Energy Mater. Sol. Cells

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]

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]

Other

R. Winston, P. Benítez, and J. C. Miñano, with contributions from N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, Oxford, 2005).

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

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

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).

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Figures (9)

Fig. 1
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.

Fig. 2
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.

Fig. 3
Fig. 3

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

Fig. 4
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.)

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.)

Fig. 6
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.

Fig. 7
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(θsun))}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 θsun 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.

Fig. 8
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.)

Fig. 9
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.

Equations (20)

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r n ( r ) Sin ( α ) = κ
r * 1 κ d r r ρ 2 κ 2 = f ( κ ) where ρ ( r ) r n ( r ) and f ( κ ) = 1 2 [ Sin 1 κ r o + Sin 1 κ r 1 + 2 Cos 1 κ ] 0 κ 1.
n L u n e b u r g = exp ( ω ( ρ , r o ) + ω ( ρ , r 1 ) ) where ω ( ρ , s ) = 1 π ρ 1 Sin 1 ( κ s ) κ 2 ρ 2 d κ
r * 1 κ d r r ρ 2 κ 2 = f ( κ ) 2 = Sin 1 ( κ r o ) + Sin 1 ( κ r 1 ) + 2 Sin 1 ( κ N ) 2 Sin 1 ( κ ) 0 κ N
κ N g ' ( ρ ) κ d r ρ 2 κ 2 = f ( κ ) 2 which has the solution n ( ρ ) = N exp ( 1 π ρ N f ( κ ) d κ r κ 2 ρ 2 ) .
N κ κ g ' ( ρ ) d ρ ρ 2 κ 2 = { f 1 ( κ ) 2 0 κ A f 1 + ( κ ) 2 A κ N where f 1 ( κ ) = Sin 1 ( κ r o ) + Sin 1 ( κ r 1 ) + 2 Sin 1 ( κ N ) 2 Sin 1 ( κ ) and f 1 + ( κ ) remains to be determined in the analysis that follows .
n ( ρ ) = N exp [ 1 π ρ A f 1 ( κ ) κ 2 ρ 2 d κ + 1 π A N f 1 + ( κ ) κ 2 ρ 2 d κ ] = { N exp [ ω ( ρ , r 1 , A ) + ω ( ρ , r 0 , A ) + 2 ω ( ρ , N , A ) 2 ω ( ρ , 1 , A ) + 1 π A N f 1 + ( κ ) κ 2 ρ 2 d κ ] 0 ρ A N exp [ 1 π ρ N f 1 + ( κ ) κ 2 ρ 2 d κ ] A ρ N .
ln ( n ( ρ ) N ) = 1 π ρ A f 1 ( κ ) κ 2 ρ 2 d κ + 1 π A N f 1 + ( κ ) κ 2 ρ 2 d κ , and, rearranging: 1 π A N f 1 + ( κ ) κ 2 ρ 2 d κ = ln ( n ( ρ ) N ) 1 π ρ A f 1 ( κ ) κ 2 ρ 2 d κ = ln ( 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 ) = 1 π ρ s Sin 1 ( κ A ) κ 2 ρ 2 d κ
f 1 + ( κ ) = i w i φ i ( κ ) .
1 π A N f 1 + ( κ ) κ 2 ρ 2 d κ = 1 π A N i w i φ i ( κ ) κ 2 ρ 2 d κ = 1 π i w i A N φ i ( κ ) κ 2 ρ 2 d κ .
B w = g
w = ( B * B + β H ) 1 B * g
H = [ 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 ] .
f 1 + ( κ ) = κ κ i + 1 κ i κ i + 1 κ κ i + 2 κ i κ i + 2 κ κ i + 3 κ i κ i + 3 w i + κ κ i κ i + 1 κ i κ κ i + 2 κ i + 1 κ i + 2 κ κ i + 3 κ i + 1 κ i + 3 w i + 1 + κ κ i κ i + 2 κ i κ κ i + 1 κ i + 2 κ i + 1 κ κ i + 3 κ i + 2 κ i + 3 w i + 2 + κ κ i κ i + 3 κ i κ κ i + 1 κ i + 3 κ i + 1 κ κ i + 2 κ i + 3 κ i + 2 w i + 3 .
N κ κ g ' ( ρ ) d ρ ρ 2 κ 2 = { f 1 ( κ ) 2 0 κ A 1 f 1 + ( κ ) 2 A 1 κ A 2 f 2 + ( κ ) 2 A 2 κ N
n ( ρ ) = { N exp [ ω ( ρ , r 1 , A 1 ) + ω ( ρ , r 0 , A 1 ) + 2 ω ( ρ , N , A 1 ) 2 ω ( ρ , 1 , A 1 ) + A 1 A 2 f 1 + ( κ ) π κ 2 ρ 2 d κ + A 2 N f 2 + ( κ ) π κ 2 ρ 2 d κ ] 0 ρ A 1 N exp [ ρ A 2 f 1 + ( κ ) π κ 2 ρ 2 d κ + A 2 N f 2 + ( κ ) π κ 2 ρ 2 d κ ] A 1 ρ A 2 N exp [ ρ N f 2 + ( κ ) π κ 2 ρ 2 d κ ] A 2 ρ N
z = ρ 2 ,     t = κ 2 , γ ( t ) = f + ( κ ) κ , F ( z ) = 2 φ ( z ) where φ ( ρ ) = log ( n ( ρ ) N ) ω ( ρ , r 1 , A ) + ω ( ρ , r 0 , A ) + 2 ω ( ρ , N , A ) 2 ω ( ρ , 1 , A ) ,
A 1 N 1 γ ( τ ) τ z d τ = π F ( z ) , where A 1 = A 2 , N 1 = N 2
γ ( t ) = 1 2 π N 1 t t A 1 A 1 N 1 N 1 s s A 1 d d s [ A 1 s F ( u ) s u d u ] d s + 1 2 d d t A 1 t F ( u ) s u d s .
f + ( A ) = Sin 1 ( A r o ) + Sin 1 ( A r 1 ) + 2 Sin 1 ( A N ) 2 Sin 1 ( A ) .

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