Abstract

The linear chromatic aberration (LCA) of several combinations of polycarbonates (PCs) and poly (methyl methacrylates) (PMMAs) as singlet, hybrid (refractive/diffractive) lenses and doublets operating with wavelengths between 380 and 1600 nm – corresponding to a typical zone of interest of concentrated photovoltaics (CPV) – are compared. Those comparisons show that the maximum theoretical concentration factor for singlets is limited to about 1000 × at normal incidence and that hybrid lenses and refractive doublets present a smaller LCA increasing the concentration factor up to 5000 × and 2 × 106 respectively. A new achromatization equation more useful than the Abbé equation is also presented. Finally we determined the ideal position of the focal point as a function of the LCA and the geometric concentration which maximizes the flux on the solar cell.

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References

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  1. C. Algora, “Very high concentration challenges of III–V multijunction solar cells,” in Concentrator Photovoltaics, A. Luque, and V. Andreev, ed. (Springer, 2007), Chap. 5.
  2. Spectrolab datasheet: www.spectrolab.com/DataSheets/PV/CPV/CDO-100-C3MJ.pdf , accessed on 02/06/2011.
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    [CrossRef]
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    [PubMed]
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    [CrossRef]
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    [CrossRef]
  14. V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. 65(6), 556–562 (1997).
    [CrossRef]
  15. D. C. O'Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (Spie Press, 2004), Chap. 4.
  16. B. H. Kleemann, M. Seeßelberg, and J. Ruoff, “Design concepts for broadband high–efficiency DOEs,” J. Eur. Opt. Soc. Rapid Publ. 3, 08015 (2008).
    [CrossRef]
  17. F. Languy, C. Lenaerts, J. Loicq, and S. Habraken, “Achromatization of solar concentrator thanks to diffractive optics,” presented at the 2nd Int’l Workshop on Concentrating Photovoltaic Power Plants, Darmstadt, Germany, 9–10 March 2009, http://www.concentrating-pv.org/darmstadt2009/index.html .
  18. M. Born, and E. Wolf, Principles of Optics, 7th Ed. (Cambridge University Press, 2003), p. 188.
  19. G. K. Skinner, “Design and imaging performance of achromatic diffractive-refractive x-ray and gamma-ray Fresnel lenses,” Appl. Opt. 43(25), 4845–4853 (2004).
    [CrossRef] [PubMed]

2008 (1)

B. H. Kleemann, M. Seeßelberg, and J. Ruoff, “Design concepts for broadband high–efficiency DOEs,” J. Eur. Opt. Soc. Rapid Publ. 3, 08015 (2008).
[CrossRef]

2007 (1)

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007).
[CrossRef]

2004 (1)

2001 (1)

Y. B. Lee and T. H. Kwon, “Modeling and numerical simulation of residual stresses and birefringence in injection molded center-gated disks,” J. Mater. Process. Technol. 111(1-3), 214–218 (2001).
[CrossRef]

2000 (1)

S. Puliaev, J. L. Penna, E. G. Jilinski, and A. H. Andrei, “Solar diameter observations at Observatório Nacional in 1998-1999,” Astron. Astrophys. Suppl. Ser. 143(2), 265–267 (2000).
[CrossRef]

1997 (1)

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. 65(6), 556–562 (1997).
[CrossRef]

1989 (1)

1970 (1)

Andrei, A. H.

S. Puliaev, J. L. Penna, E. G. Jilinski, and A. H. Andrei, “Solar diameter observations at Observatório Nacional in 1998-1999,” Astron. Astrophys. Suppl. Ser. 143(2), 265–267 (2000).
[CrossRef]

Buralli, D. A.

Ivanov, C. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007).
[CrossRef]

Jilinski, E. G.

S. Puliaev, J. L. Penna, E. G. Jilinski, and A. H. Andrei, “Solar diameter observations at Observatório Nacional in 1998-1999,” Astron. Astrophys. Suppl. Ser. 143(2), 265–267 (2000).
[CrossRef]

Kasarova, S. N.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007).
[CrossRef]

Kleemann, B. H.

B. H. Kleemann, M. Seeßelberg, and J. Ruoff, “Design concepts for broadband high–efficiency DOEs,” J. Eur. Opt. Soc. Rapid Publ. 3, 08015 (2008).
[CrossRef]

Kwon, T. H.

Y. B. Lee and T. H. Kwon, “Modeling and numerical simulation of residual stresses and birefringence in injection molded center-gated disks,” J. Mater. Process. Technol. 111(1-3), 214–218 (2001).
[CrossRef]

Lee, Y. B.

Y. B. Lee and T. H. Kwon, “Modeling and numerical simulation of residual stresses and birefringence in injection molded center-gated disks,” J. Mater. Process. Technol. 111(1-3), 214–218 (2001).
[CrossRef]

Moreno, V.

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. 65(6), 556–562 (1997).
[CrossRef]

Morris, G. M.

Nikolov, I. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007).
[CrossRef]

Penna, J. L.

S. Puliaev, J. L. Penna, E. G. Jilinski, and A. H. Andrei, “Solar diameter observations at Observatório Nacional in 1998-1999,” Astron. Astrophys. Suppl. Ser. 143(2), 265–267 (2000).
[CrossRef]

Puliaev, S.

S. Puliaev, J. L. Penna, E. G. Jilinski, and A. H. Andrei, “Solar diameter observations at Observatório Nacional in 1998-1999,” Astron. Astrophys. Suppl. Ser. 143(2), 265–267 (2000).
[CrossRef]

Rogers, J. R.

Román, J. F.

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. 65(6), 556–562 (1997).
[CrossRef]

Ruoff, J.

B. H. Kleemann, M. Seeßelberg, and J. Ruoff, “Design concepts for broadband high–efficiency DOEs,” J. Eur. Opt. Soc. Rapid Publ. 3, 08015 (2008).
[CrossRef]

Salgueiro, J. R.

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. 65(6), 556–562 (1997).
[CrossRef]

Seeßelberg, M.

B. H. Kleemann, M. Seeßelberg, and J. Ruoff, “Design concepts for broadband high–efficiency DOEs,” J. Eur. Opt. Soc. Rapid Publ. 3, 08015 (2008).
[CrossRef]

Skinner, G. K.

Sultanova, N. G.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007).
[CrossRef]

Winston, R.

Am. J. Phys. (1)

V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. 65(6), 556–562 (1997).
[CrossRef]

Appl. Opt. (2)

Astron. Astrophys. Suppl. Ser. (1)

S. Puliaev, J. L. Penna, E. G. Jilinski, and A. H. Andrei, “Solar diameter observations at Observatório Nacional in 1998-1999,” Astron. Astrophys. Suppl. Ser. 143(2), 265–267 (2000).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ. (1)

B. H. Kleemann, M. Seeßelberg, and J. Ruoff, “Design concepts for broadband high–efficiency DOEs,” J. Eur. Opt. Soc. Rapid Publ. 3, 08015 (2008).
[CrossRef]

J. Mater. Process. Technol. (1)

Y. B. Lee and T. H. Kwon, “Modeling and numerical simulation of residual stresses and birefringence in injection molded center-gated disks,” J. Mater. Process. Technol. 111(1-3), 214–218 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Mater. (1)

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007).
[CrossRef]

Other (11)

J. D. Lytle, “Polymeric Optics,” in Handbook of Optics, 3rd Edition, Vol. IV, M. Bass, ed. (McGraw-Hill, 2009), Chap. 3.

ASAPTM optical design software of Breault Research Organization, http://www.breault.com .

C. Algora, “Very high concentration challenges of III–V multijunction solar cells,” in Concentrator Photovoltaics, A. Luque, and V. Andreev, ed. (Springer, 2007), Chap. 5.

Spectrolab datasheet: www.spectrolab.com/DataSheets/PV/CPV/CDO-100-C3MJ.pdf , accessed on 02/06/2011.

Website for NREL’s AM1, 5 Standard Data set: http://rredc.nrel.gov/solar/spectra/am1.5/ASTMG173/ ASTMG173.html , accessed on 02/06/2011.

J. Chaves, Introduction to nonimaging optics (CRC Press, 2008), Chap. 1.

E. Hecht, Optics 4th Ed. (Addison-Wesley, 2002), Chap. 5.
[PubMed]

Fresnel lens brochure of the Fresnel Technologies Inc.: www.fresneltech.com/materials.html , accessed on 02/06/2011.

F. Languy, C. Lenaerts, J. Loicq, and S. Habraken, “Achromatization of solar concentrator thanks to diffractive optics,” presented at the 2nd Int’l Workshop on Concentrating Photovoltaic Power Plants, Darmstadt, Germany, 9–10 March 2009, http://www.concentrating-pv.org/darmstadt2009/index.html .

M. Born, and E. Wolf, Principles of Optics, 7th Ed. (Cambridge University Press, 2003), p. 188.

D. C. O'Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (Spie Press, 2004), Chap. 4.

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Figures (10)

Fig. 1
Fig. 1

Representation of the optical efficiency of two wavelengths having the same LCA in absolute value but having different optical efficiency (yellow part corresponds to losses).

Fig. 2
Fig. 2

Concentration ratio at normal incidence as a function of the LCA* (a) and gain if the detector is moved from the position of the minimum |LCA max| to the ideal position (b).

Fig. 3
Fig. 3

Ideal position of f0) (a) and optical efficiency (b) as functions of the LCA* and the geometrical concentration.

Fig. 4
Fig. 4

Dispersion curves of PMMAs (a) and PCs (b).

Fig. 5
Fig. 5

Relative focal distances for PMMAs (a) and PCs (b).

Fig. 6
Fig. 6

Schematic representation of a kinoform diffractive lens.

Fig. 7
Fig. 7

Schematic representation of a doublet.

Fig. 8
Fig. 8

Doublets with PC #1 (a), PC #2 (b), PC #3 (c) and PC #4 (d).

Fig. 9
Fig. 9

Evolution of the focal distance for hybrid lenses in OP considering the curve minimizing the LCA.

Fig. 10
Fig. 10

Schematic representation of a converging lens with LCA.

Tables (6)

Tables Icon

Table 1 Information about the PMMAs and PCs

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Table 2 Dispersion coefficients some PCs and PMMAs

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Table 3 Data for singlets

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Table 4 Combination of PCs and PMMAs for Achromatic Doublets

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Table 5 Data for achromatized hybrid lenses

Tables Icon

Table 6 Major data about singlets, doublets and hybrid lenses

Equations (49)

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C o p t = η o p t C g e o = Φ ' Φ A ' A .
C o p t max = sin 2 θ .
L C A = f ( λ B ) f ( λ A ) .
1 f r e f ( λ ) ( 1 R 1 1 R 2 ) ( n ( λ ) 1 ) ,
f d i f ( λ ) = λ A λ f ( λ A ) .
L C A r e f * ( λ ) = f ( λ ) f ( λ 0 ) f ( λ 0 ) = n ( λ 0 ) n ( λ ) n ( λ ) 1 ,
L C A d i f * ( λ ) = f ( λ ) f ( λ 0 ) f ( λ 0 ) = ( λ 0 λ 1 ) ,
C g e o max = ( 1 L C A * L C A * ) 2 .
n ( λ ) = 1 + i = 1 m B i λ 2 λ 2 C i ,
n ( λ ) = A 1 + A 2 λ 2 + A 3 λ 2 + A 4 λ 4 + A 5 λ 6 +    .
f ( λ ) R o C e q n ( λ ) 1 ,
R o C e q * = R o C e q f ( λ 0 ) = ( R 1 R 2 R 2 R 1 ) 1 f ( λ 0 ) = ( n ( λ ) 1 ) × 100 % .
v d = ( n d 1 ) / ( n F n C ) ,
v = ( n 990 n m 1 ) / ( n 380 n m n 1600 n m ) .
v d , d i f f = d F C = 3.4518.
η 1 ( λ ) = sinc 2 { 1 λ 0 λ n ( λ ) 1 n ( λ 0 ) 1 } .
f 1 v 1 + f 2 v 2 = 0 ,
f e f f 1 = f 1 1 + f 2 1 .
f 2 ( λ 1 ) = B ± B 2 4 A C 2 A ,
A = ( b f l + d ) ( n 2 ( λ 1 ) 1 n 2 ( λ 2 ) 1 ) ( 1 n 1 ( λ 2 ) 1 n 1 ( λ 1 ) 1 ) , B = b f l { d ( n 2 ( λ 1 ) 1 n 2 ( λ 2 ) 1 n 1 ( λ 2 ) 1 n 1 ( λ 1 ) 1 ) + ( b f l + d ) ( n 1 ( λ 2 ) 1 n 1 ( λ 1 ) 1 n 2 ( λ 1 ) 1 n 2 ( λ 2 ) 1 1 ) } , C = d b f l 2 ( 1 n 1 ( λ 2 ) 1 n 1 ( λ 1 ) 1 ) .
f 1 ( λ 1 ) = d ( b f l ) ( b f l + d ) f 2 ( λ 1 ) ( b f l ) f 2 ( λ 1 ) .
b f l r e f ( λ ) = f 2 ( λ ) ( d f 1 ( λ ) ) d f 1 ( λ ) f 2 ( λ ) .
f r e f ( λ 1 ) = B B 2 4 A C 2 A
A = ( b f l + d ) ( n ( λ 1 ) 1 n ( λ 2 ) 1 ) ( 1 λ 2 λ 1 ) , B = b f l { d ( λ 2 λ 1 n ( λ 1 ) 1 n ( λ 2 ) 1 ) + ( b f l + d ) ( λ 2 λ 1 n ( λ 1 ) 1 n ( λ 2 ) 1 1 ) } , C = d b f l 2 ( 1 λ 2 λ 1 ) .
f d i f ( λ 1 ) = d ( b f l ) ( b f l + d ) f r e f ( λ 1 ) ( b f l ) f r e f ( λ 1 ) .
y 1 ( z ) = r f ( λ 0 ) ( 1 + L C A * ) z + r ,
y 2 ( z ) = r f ( λ 0 ) ( 1 L C A * ) z r .
1 f ( λ 0 ) ( 1 L C A * ) z 1 = 1 f ( λ 0 ) ( 1 + L C A * ) z + 1 ,
z ( 1 + L C A * + 1 L C A * 1 ( L C A * ) 2 ) = 2 f ( λ 0 ) ,
z = f ( λ 0 ) ( 1 L C A * 2 ) .
1 f = 1 s o + 1 s i ,
f ^ = b f l + d .
1 f 1 = 1 s i 1                     w i t h f 1 < 0.
s o 2 = | f 1 | + d .
1 s i 2 = 1 f 2 1 s o 2 = 1 f 2 1 d f 1 = d f 1 f 2 f 2 ( d f 1 ) .
f ^ = s i 2 + d = f 2 ( d f 1 ) d f 1 f 2 + d .
1 f 1 = 1 s i 1                     w i t h f 1 > 0 ,
s o 2 = f 1 d ,
1 s i 2 = 1 f 2 1 s o 2 = 1 f 2 + 1 f 1 d ,
f ^ = s i 2 + d = f 2 ( d f 1 ) d f 1 f 2 + d ,
f ^ ( λ 1 ) = f ^ ( λ 2 ) = f ^ .
f 1 ( λ 1 ) = f 1 ( λ 2 ) ( n 1 ( λ 2 ) 1 n 1 ( λ 1 ) 1 ) = α f 1 ( λ 2 ) ,
f 2 ( λ 2 ) = f 2 ( λ 1 ) ( n 2 ( λ 1 ) 1 n 2 ( λ 2 ) 1 ) = β f 2 ( λ 1 ) .
f i j f i ( λ j )
f 11 = f ^ f 21 b f l d f 21 b f l = α f ^ β f 21 b f l d β f 21 b f l ,
( f ^ f 21 b f l d ) ( β f 21 b f l ) = ( f 21 b f l ) α ( f ^ β f 21 b f l d ) ,
f 21 2 { f ^ β f ^ α β } + f 21 { b f l ( α d + f ^ α β f ^ β d ) } + { d b f l 2 α b f l 2 d } = 0.
f 21 B ± B 2 4 A C 2 A ,
{ A = ( b f l + d ) β ( 1 α ) , B = b f l ( d ( α β ) + ( b f l + d ) ( α β 1 ) ) C = d b f l 2 ( 1 α ) . ,

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