Abstract

The design of single element planar hemispherical gradient-index solar lenses that can accommodate the constraints of realistic materials and fabrication techniques are presented, and simulated with an extended and polychromatic solar source for concentrator photovoltaics at flux concentration values exceeding 1000 suns. The planar hemispherical far-field lens is created from a near-field unit magnification spherical gradient-index design, and illustrated with an f/1.40 square solar lens that allows lossless packing within a concentrator module.

© 2012 Optical Society of America

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References

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  1. R. Winston, P. Benítez, and J. C. Miñano, with contributions from N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005).
  2. J. M. Gordon, Opt. Express 18, A41 (2010).
    [CrossRef]
  3. 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, Opt. Express 16, 11540 (2008).
  4. M. Ponting, A. Hiltner, and E. Baer, Macromol. Symp. 294, 19 (2010).
    [CrossRef]
  5. P. Kotsidas, V. Modi, and J. M. Gordon, Opt. Express 19, 2325 (2011).
    [CrossRef]
  6. P. Kotsidas, V. Modi, and J. M. Gordon, Opt. Express 19, 15584 (2011).
    [CrossRef]
  7. J. C. Maxwell, Q. J. Pure Appl. Math. 2, 233 (1854).
  8. R. K. Luneburg, The Mathematical Theory of Optics(University California Berkeley, 1964).
  9. M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge University, 1999).
  10. J. M. Gordon, D. Feuermann, and P. Young, Opt. Lett. 33, 1114 (2008).

Baer, E.

Beadie, G.

Benítez, P.

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge University, 1999).

Bortz, J.

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

Feuermann, D.

Fleet, E.

Gordon, J. M.

Hiltner, A.

Jin, Y.

Kamdar, A. R.

Kazmierczak, T.

Kotsidas, P.

Lane, P. A.

Luneburg, R. K.

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

Maxwell, J. C.

J. C. Maxwell, Q. J. Pure Appl. Math. 2, 233 (1854).

Miñano, J. C.

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

Modi, V.

Ponting, M.

Rosenberg, A.

Shatz, N.

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

Shirk, J. S.

Winston, R.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge University, 1999).

Yang, Y.

Young, P.

Macromol. Symp.

M. Ponting, A. Hiltner, and E. Baer, Macromol. Symp. 294, 19 (2010).
[CrossRef]

Opt. Express

Opt. Lett.

Q. J. Pure Appl. Math.

J. C. Maxwell, Q. J. Pure Appl. Math. 2, 233 (1854).

Other

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

M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge University, 1999).

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

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Figures (4)

Fig. 1.
Fig. 1.

Creating a planar hemispherical GRIN lens for a far-field source from a near-field unit magnification spherical GRIN lens. (a) The spherical lens (of unit radius) has equal object (source) and image (absorber) focal length F (measured from the sphere’s center), a core of constant index no, a homogeneous outer shell of index N=n(1), and a GRIN continuum n(r) between them (r denoting radial position), including a trace of several rays. A nonfull aperture is allowed such that rmax<1 (vide infra). (b) When the lens is split and only the hemisphere retained, it becomes a far-field planar aplanatic concentrator. This illustration was purposely chosen to highlight the potential for short focal lengths, but does not fulfill the severe material and manufacturing constraints noted above. Because the illustrative example below does impose these limitations, it possesses physically admissible solutions only for noticeably larger F.

Fig. 2.
Fig. 2.

Refractive index distribution for a planar hemispherical GRIN lens with F=1.94 and rmax=0.693 (f/1.40, sin(θout)=0.5). The constant-index core, with no=1.573, extends up to r=0.1. A single GRIN continuum connects the core and outer shell. Although the GRIN region exhibits extrema for n(r), the natural variable in the governing equation, ρ=rn(r), is actually a monotonically increasing function of r, as required by the formalism [6]. The corresponding Cmax is 10,000 [Eq. (2)], and the lens is intended for CPV at C=1100.

Fig. 3.
Fig. 3.

Geometric collection efficiency as a function of flux concentration C [relative to the thermodynamic limit Cmax, Eq. (2)] evaluated by ray tracing with an extended source of 5 mrad angular radius, for both monochromatic (red) and polychromatic (AM1.5D spectrum) light. C=1100, common to current CPV, is indicated by the vertical dashed line.

Fig. 4.
Fig. 4.

Ray trace (with a polychromatic and extended sun) illustrating the performance of a set of planar lenses that comprise part of the module’s protective glazing, with lossless packing due to a square entry allowed by the truncated design.

Equations (3)

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n(ρ)={Nexp[2ω(ρ,F,A)+2ω(ρ,N,A)2ω(ρ,1,A)+AAsf1+(κ)πκ2ρ2dκ+AsNf2+(κ)πκ2ρ2dκ],0ρANexp[ρAsf1+(κ)πκ2ρ2dκ+AsNf2+(κ)πκ2ρ2dκ],AρAsNexp[1πρNf2+(κ)κ2ρ2dκ],AsρN,
r(ρ)=ρ/n(ρ),ω(ρ,r,F)=ρFsin1(κ/r)πκ2ρ2dκ
Cmax={sin(θout)/sin(θsun)}2=A2/{Fsin(θsun)}2

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