Abstract

Two important aspects must be considered when optimizing antireflection coatings (ARCs) for multijunction solar cells to be used in concentrators: the angular light distribution over the cell created by the particular concentration system and the wide spectral bandwidth the solar cell is sensitive to. In this article, a numerical optimization procedure and its results are presented. The potential efficiency enhancement by means of ARC optimization is calculated for several concentrating PV systems. In addition, two methods for ARCs direct characterization are presented. The results of these show that real ARCs slightly underperform theoretical predictions.

© 2012 OSA

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References

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  1. A. Luque, Solar cells and optics for photovoltaic concentration (Adam Hilger, 1989).
  2. M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (Version 38),” Prog. Photovolt. Res. Appl. 19(5), 565–572 (2011).
    [CrossRef]
  3. R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).
  4. D. J. Aiken, “Antireflection coating design for series interconnected multi-junction solar cells,” Prog. Photovolt. Res. Appl. 8(6), 563–570 (2000).
    [CrossRef]
  5. C. Algora and V. Díaz, “Modelling of GaAs solar cells under wide angle cones of homogeneus light,” Prog. Photovolt. Res. Appl. 7(5), 379–386 (1999).
    [CrossRef]
  6. V. Dı́az Luque and C. Algora del Valle, “On the effects of tilted light in a global prediction of AlGaAs/GaAs solar cell performance,” Solar Energy & Solar Cells 57(4), 313–322 (1999).
    [CrossRef]
  7. C. E. Valdivia, E. Desfonds, D. Masson, S. Fafard, A. Carlson, J. Cook, T. J. Hall, and K. Hinzer, “Optimization of antireflection coating design for multijunction solar cells and concentrator systems,” Proc. SPIE 7099, 709915, 709915-10 (2008).
    [CrossRef]
  8. S. Wojtczuk, P. Chiu, X. Zhang, D. Derkacs, C. Harris, D. Pulver, and M. Timmons, “InGaP/GaAs/InGaAs 41% concentrator cells using bi-facial epigrowth,” in Proceedings of the 35th IEEE Photovoltaic Specialists Conf. (2010) pp.1259–1264.
  9. A. N. Matveev, Optics (Mir, 1988)
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  13. D. J. Aiken, “High performance anti-reflection coatings for broadband multi-junction solar cells,” Sol. Energy Mater. Sol. Cells 64(4), 393–404 (2000).
    [CrossRef]
  14. M. F. Schubert, F. W. Mont, S. Chhajed, D. J. Poxson, J. K. Kim, and E. F. Schubert, “Design of multilayer antireflection coatings made from co-sputtered and low-refractive-index materials by genetic algorithm,” Opt. Express 16(8), 5290–5298 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-8-5290 .
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  15. D. J. Poxson, M. F. Schubert, F. W. Mont, E. F. Schubert, and J. K. Kim, “Broadband omnidirectional antireflection coatings optimized by genetic algorithm,” Opt. Lett. 34(6), 728–730 (2009).
    [CrossRef] [PubMed]
  16. E. D. Palik, Handbook of Optical Constant of Solids (Academic Press, 1997).
  17. Sopra materials database, www.sopra-sa.com .
  18. D. Redfield, “Method for evaluation of antireflection coatings,” Solar Cells 3(1), 27–33 (1981).
    [CrossRef]
  19. M. Victoria, C. Domínguez, S. Askins, I. Antón, and G. Sala, “Optical characterization of FluidReflex concentrator,” in Proceedings of Int. Conf. on Concentrating Photovoltaic Systems (2010) pp. 118–121.
  20. I. Rey-Stolle and C. Algora, “Optimum antireflection coatings for heteroface AlGaAs/GaAs solar cells-Part II: The influence of uncertainties in the parameters of window and antireflection coatings,” J. Electron. Mater. 29(7), 992–999 (2000).
    [CrossRef]
  21. M. Victoria, C. Domínguez, I. Antón, and G. Sala, “Comparative analysis of different secondary optical elements for aspheric primary lenses,” Opt. Express 17(8), 6487–6492 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-8-6487 .
    [CrossRef] [PubMed]
  22. S. R. Kurtz and M. J. O’Neill, “Estimating and controlling chromatic aberration losses for two-junction, two-terminal devices in refractive concentrator systems,” in Proceedings of the 25th IEEE Photovoltaic Specialists Conf. (1996) pp. 361–364.
  23. L. W. James, “Effects of concentrator chromatic aberration on multi-junction cells,” in Proceedings of the 24th IEEE Photovoltaic Specialists Conf. (1994) pp. 1799–1802.
  24. I. García, C. Algora, I. Rey-Stolle, and B. Galiana, “Study of non-uniform light profiles on high concentration solar cells using quasi-3D distributed models,” in Proceedings of the 33rd IEEE Photovoltaic Specialists Conf. (2008) pp. 1–6.
  25. K. Nishioka, T. Takamoto, and W. Nakajima, “Analysis of triple-junction solar cell under concentration by SPICE,” in Proceedings of the 3rdWorld Conf. on Photovoltaic Energy Conversion (2003) pp. 869–872.
  26. M. Victoria, R. Herrero, C. Domínguez, I. Antón, and S. Askins andG. Sala, “Characterization of the spatial distribution of irradiance and spectrum in concentrating photovoltaic systems and their effect on multi-junction solar cells,” Prog. Photovolt: Res. Appl.(2011) published online DOI: .
    [CrossRef]
  27. J. Jaus, P. Nitz, G. Peharz, G. Siefer, T. Schult, O. Wolf, M. Passig, T. Gandy, and A. W. Bett, “Second stage reflective and refractive optics for concentrator photovoltaics,” in Proceedings of the 33rd IEEE Photovoltaic Specialist Conf. (2008) pp. 1–5.
  28. V. Díaz, J. M. Ruíz, C. Algora, and J. Alonso, “Outdoor characterization of GaAs solar cell under tilted light for its encapsulation inside optic concentrator,” in Proceedings of the 27th Eur.Photovoltaic Sol. Energy Conf., (2001).
  29. C. Domínguez, I. Antón, and G. Sala, “Solar simulator for concentrator photovoltaic systems,” Opt. Express 16(19), 14894–14901 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-19-14894 .
    [CrossRef] [PubMed]
  30. C. Stevenson, P. R. Denton, G. Sadkhin, and V. Fridman, “Stability and Repeatability of 2-Layer Anti-Reflection Coatings,” Denton Vacuum technical paper, http://www.dentonvacuum.com/PDFs/Tech_papers/stab.pdf

2011 (2)

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (Version 38),” Prog. Photovolt. Res. Appl. 19(5), 565–572 (2011).
[CrossRef]

M. Victoria, R. Herrero, C. Domínguez, I. Antón, and S. Askins andG. Sala, “Characterization of the spatial distribution of irradiance and spectrum in concentrating photovoltaic systems and their effect on multi-junction solar cells,” Prog. Photovolt: Res. Appl.(2011) published online DOI: .
[CrossRef]

M. Victoria, R. Herrero, C. Domínguez, I. Antón, and S. Askins andG. Sala, “Characterization of the spatial distribution of irradiance and spectrum in concentrating photovoltaic systems and their effect on multi-junction solar cells,” Prog. Photovolt: Res. Appl.(2011) published online DOI: .
[CrossRef]

2009 (2)

2008 (3)

2004 (1)

2000 (3)

D. J. Aiken, “High performance anti-reflection coatings for broadband multi-junction solar cells,” Sol. Energy Mater. Sol. Cells 64(4), 393–404 (2000).
[CrossRef]

D. J. Aiken, “Antireflection coating design for series interconnected multi-junction solar cells,” Prog. Photovolt. Res. Appl. 8(6), 563–570 (2000).
[CrossRef]

I. Rey-Stolle and C. Algora, “Optimum antireflection coatings for heteroface AlGaAs/GaAs solar cells-Part II: The influence of uncertainties in the parameters of window and antireflection coatings,” J. Electron. Mater. 29(7), 992–999 (2000).
[CrossRef]

1999 (2)

C. Algora and V. Díaz, “Modelling of GaAs solar cells under wide angle cones of homogeneus light,” Prog. Photovolt. Res. Appl. 7(5), 379–386 (1999).
[CrossRef]

V. Dı́az Luque and C. Algora del Valle, “On the effects of tilted light in a global prediction of AlGaAs/GaAs solar cell performance,” Solar Energy & Solar Cells 57(4), 313–322 (1999).
[CrossRef]

1985 (1)

1981 (1)

D. Redfield, “Method for evaluation of antireflection coatings,” Solar Cells 3(1), 27–33 (1981).
[CrossRef]

Aiken, D. J.

D. J. Aiken, “Antireflection coating design for series interconnected multi-junction solar cells,” Prog. Photovolt. Res. Appl. 8(6), 563–570 (2000).
[CrossRef]

D. J. Aiken, “High performance anti-reflection coatings for broadband multi-junction solar cells,” Sol. Energy Mater. Sol. Cells 64(4), 393–404 (2000).
[CrossRef]

Algora, C.

I. Rey-Stolle and C. Algora, “Optimum antireflection coatings for heteroface AlGaAs/GaAs solar cells-Part II: The influence of uncertainties in the parameters of window and antireflection coatings,” J. Electron. Mater. 29(7), 992–999 (2000).
[CrossRef]

C. Algora and V. Díaz, “Modelling of GaAs solar cells under wide angle cones of homogeneus light,” Prog. Photovolt. Res. Appl. 7(5), 379–386 (1999).
[CrossRef]

Algora del Valle, C.

V. Dı́az Luque and C. Algora del Valle, “On the effects of tilted light in a global prediction of AlGaAs/GaAs solar cell performance,” Solar Energy & Solar Cells 57(4), 313–322 (1999).
[CrossRef]

Antón, I.

Askins, S.

M. Victoria, R. Herrero, C. Domínguez, I. Antón, and S. Askins andG. Sala, “Characterization of the spatial distribution of irradiance and spectrum in concentrating photovoltaic systems and their effect on multi-junction solar cells,” Prog. Photovolt: Res. Appl.(2011) published online DOI: .
[CrossRef]

Carlson, A.

C. E. Valdivia, E. Desfonds, D. Masson, S. Fafard, A. Carlson, J. Cook, T. J. Hall, and K. Hinzer, “Optimization of antireflection coating design for multijunction solar cells and concentrator systems,” Proc. SPIE 7099, 709915, 709915-10 (2008).
[CrossRef]

Chhajed, S.

Cook, J.

C. E. Valdivia, E. Desfonds, D. Masson, S. Fafard, A. Carlson, J. Cook, T. J. Hall, and K. Hinzer, “Optimization of antireflection coating design for multijunction solar cells and concentrator systems,” Proc. SPIE 7099, 709915, 709915-10 (2008).
[CrossRef]

Desfonds, E.

C. E. Valdivia, E. Desfonds, D. Masson, S. Fafard, A. Carlson, J. Cook, T. J. Hall, and K. Hinzer, “Optimization of antireflection coating design for multijunction solar cells and concentrator systems,” Proc. SPIE 7099, 709915, 709915-10 (2008).
[CrossRef]

Di´az Luque, V.

V. Dı́az Luque and C. Algora del Valle, “On the effects of tilted light in a global prediction of AlGaAs/GaAs solar cell performance,” Solar Energy & Solar Cells 57(4), 313–322 (1999).
[CrossRef]

Díaz, V.

C. Algora and V. Díaz, “Modelling of GaAs solar cells under wide angle cones of homogeneus light,” Prog. Photovolt. Res. Appl. 7(5), 379–386 (1999).
[CrossRef]

Dobrowolski, J. A.

Domínguez, C.

Dunlop, E. D.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (Version 38),” Prog. Photovolt. Res. Appl. 19(5), 565–572 (2011).
[CrossRef]

Emery, K.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (Version 38),” Prog. Photovolt. Res. Appl. 19(5), 565–572 (2011).
[CrossRef]

Fafard, S.

C. E. Valdivia, E. Desfonds, D. Masson, S. Fafard, A. Carlson, J. Cook, T. J. Hall, and K. Hinzer, “Optimization of antireflection coating design for multijunction solar cells and concentrator systems,” Proc. SPIE 7099, 709915, 709915-10 (2008).
[CrossRef]

Green, M. A.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (Version 38),” Prog. Photovolt. Res. Appl. 19(5), 565–572 (2011).
[CrossRef]

Hall, T. J.

C. E. Valdivia, E. Desfonds, D. Masson, S. Fafard, A. Carlson, J. Cook, T. J. Hall, and K. Hinzer, “Optimization of antireflection coating design for multijunction solar cells and concentrator systems,” Proc. SPIE 7099, 709915, 709915-10 (2008).
[CrossRef]

Herrero, R.

M. Victoria, R. Herrero, C. Domínguez, I. Antón, and S. Askins andG. Sala, “Characterization of the spatial distribution of irradiance and spectrum in concentrating photovoltaic systems and their effect on multi-junction solar cells,” Prog. Photovolt: Res. Appl.(2011) published online DOI: .
[CrossRef]

Hinzer, K.

C. E. Valdivia, E. Desfonds, D. Masson, S. Fafard, A. Carlson, J. Cook, T. J. Hall, and K. Hinzer, “Optimization of antireflection coating design for multijunction solar cells and concentrator systems,” Proc. SPIE 7099, 709915, 709915-10 (2008).
[CrossRef]

Hishikawa, Y.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (Version 38),” Prog. Photovolt. Res. Appl. 19(5), 565–572 (2011).
[CrossRef]

Kim, J. K.

Masson, D.

C. E. Valdivia, E. Desfonds, D. Masson, S. Fafard, A. Carlson, J. Cook, T. J. Hall, and K. Hinzer, “Optimization of antireflection coating design for multijunction solar cells and concentrator systems,” Proc. SPIE 7099, 709915, 709915-10 (2008).
[CrossRef]

Mont, F. W.

Poitras, D.

Poxson, D. J.

Redfield, D.

D. Redfield, “Method for evaluation of antireflection coatings,” Solar Cells 3(1), 27–33 (1981).
[CrossRef]

Rey-Stolle, I.

I. Rey-Stolle and C. Algora, “Optimum antireflection coatings for heteroface AlGaAs/GaAs solar cells-Part II: The influence of uncertainties in the parameters of window and antireflection coatings,” J. Electron. Mater. 29(7), 992–999 (2000).
[CrossRef]

Sala, G.

Schubert, E. F.

Schubert, M. F.

Southwell, W. H.

Valdivia, C. E.

C. E. Valdivia, E. Desfonds, D. Masson, S. Fafard, A. Carlson, J. Cook, T. J. Hall, and K. Hinzer, “Optimization of antireflection coating design for multijunction solar cells and concentrator systems,” Proc. SPIE 7099, 709915, 709915-10 (2008).
[CrossRef]

Victoria, M.

M. Victoria, R. Herrero, C. Domínguez, I. Antón, and S. Askins andG. Sala, “Characterization of the spatial distribution of irradiance and spectrum in concentrating photovoltaic systems and their effect on multi-junction solar cells,” Prog. Photovolt: Res. Appl.(2011) published online DOI: .
[CrossRef]

M. Victoria, C. Domínguez, I. Antón, and G. Sala, “Comparative analysis of different secondary optical elements for aspheric primary lenses,” Opt. Express 17(8), 6487–6492 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-8-6487 .
[CrossRef] [PubMed]

Warta, W.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (Version 38),” Prog. Photovolt. Res. Appl. 19(5), 565–572 (2011).
[CrossRef]

Appl. Opt. (2)

J. Electron. Mater. (1)

I. Rey-Stolle and C. Algora, “Optimum antireflection coatings for heteroface AlGaAs/GaAs solar cells-Part II: The influence of uncertainties in the parameters of window and antireflection coatings,” J. Electron. Mater. 29(7), 992–999 (2000).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Proc. SPIE (1)

C. E. Valdivia, E. Desfonds, D. Masson, S. Fafard, A. Carlson, J. Cook, T. J. Hall, and K. Hinzer, “Optimization of antireflection coating design for multijunction solar cells and concentrator systems,” Proc. SPIE 7099, 709915, 709915-10 (2008).
[CrossRef]

Prog. Photovolt. Res. Appl. (3)

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (Version 38),” Prog. Photovolt. Res. Appl. 19(5), 565–572 (2011).
[CrossRef]

D. J. Aiken, “Antireflection coating design for series interconnected multi-junction solar cells,” Prog. Photovolt. Res. Appl. 8(6), 563–570 (2000).
[CrossRef]

C. Algora and V. Díaz, “Modelling of GaAs solar cells under wide angle cones of homogeneus light,” Prog. Photovolt. Res. Appl. 7(5), 379–386 (1999).
[CrossRef]

Prog. Photovolt: Res. Appl. (1)

M. Victoria, R. Herrero, C. Domínguez, I. Antón, and S. Askins andG. Sala, “Characterization of the spatial distribution of irradiance and spectrum in concentrating photovoltaic systems and their effect on multi-junction solar cells,” Prog. Photovolt: Res. Appl.(2011) published online DOI: .
[CrossRef]

Sol. Energy Mater. Sol. Cells (1)

D. J. Aiken, “High performance anti-reflection coatings for broadband multi-junction solar cells,” Sol. Energy Mater. Sol. Cells 64(4), 393–404 (2000).
[CrossRef]

Solar Cells (1)

D. Redfield, “Method for evaluation of antireflection coatings,” Solar Cells 3(1), 27–33 (1981).
[CrossRef]

Solar Energy & Solar Cells (1)

V. Dı́az Luque and C. Algora del Valle, “On the effects of tilted light in a global prediction of AlGaAs/GaAs solar cell performance,” Solar Energy & Solar Cells 57(4), 313–322 (1999).
[CrossRef]

Other (15)

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

S. Wojtczuk, P. Chiu, X. Zhang, D. Derkacs, C. Harris, D. Pulver, and M. Timmons, “InGaP/GaAs/InGaAs 41% concentrator cells using bi-facial epigrowth,” in Proceedings of the 35th IEEE Photovoltaic Specialists Conf. (2010) pp.1259–1264.

A. N. Matveev, Optics (Mir, 1988)

J. S. Rayleigh, “On the reflection of vibrations at the confines of two media between which the transition is gradual” in Proceedings London Math. Soc.11 (1880) pp. 51–56.

M. Victoria, C. Domínguez, S. Askins, I. Antón, and G. Sala, “Optical characterization of FluidReflex concentrator,” in Proceedings of Int. Conf. on Concentrating Photovoltaic Systems (2010) pp. 118–121.

E. D. Palik, Handbook of Optical Constant of Solids (Academic Press, 1997).

Sopra materials database, www.sopra-sa.com .

J. Jaus, P. Nitz, G. Peharz, G. Siefer, T. Schult, O. Wolf, M. Passig, T. Gandy, and A. W. Bett, “Second stage reflective and refractive optics for concentrator photovoltaics,” in Proceedings of the 33rd IEEE Photovoltaic Specialist Conf. (2008) pp. 1–5.

V. Díaz, J. M. Ruíz, C. Algora, and J. Alonso, “Outdoor characterization of GaAs solar cell under tilted light for its encapsulation inside optic concentrator,” in Proceedings of the 27th Eur.Photovoltaic Sol. Energy Conf., (2001).

C. Stevenson, P. R. Denton, G. Sadkhin, and V. Fridman, “Stability and Repeatability of 2-Layer Anti-Reflection Coatings,” Denton Vacuum technical paper, http://www.dentonvacuum.com/PDFs/Tech_papers/stab.pdf

S. R. Kurtz and M. J. O’Neill, “Estimating and controlling chromatic aberration losses for two-junction, two-terminal devices in refractive concentrator systems,” in Proceedings of the 25th IEEE Photovoltaic Specialists Conf. (1996) pp. 361–364.

L. W. James, “Effects of concentrator chromatic aberration on multi-junction cells,” in Proceedings of the 24th IEEE Photovoltaic Specialists Conf. (1994) pp. 1799–1802.

I. García, C. Algora, I. Rey-Stolle, and B. Galiana, “Study of non-uniform light profiles on high concentration solar cells using quasi-3D distributed models,” in Proceedings of the 33rd IEEE Photovoltaic Specialists Conf. (2008) pp. 1–6.

K. Nishioka, T. Takamoto, and W. Nakajima, “Analysis of triple-junction solar cell under concentration by SPICE,” in Proceedings of the 3rdWorld Conf. on Photovoltaic Energy Conversion (2003) pp. 869–872.

A. Luque, Solar cells and optics for photovoltaic concentration (Adam Hilger, 1989).

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Figures (5)

Fig. 1
Fig. 1

Transmittance for different wavelengths and incidence angles on a solar cell when no ARC is used (up-left), the optimized monolayer is used (up-right), the optimized bilayer is used (down-left) and optimized trilayer is used (down-right). Monolayer shows one maximum in transmittance, bilayer clearly shows two maximum, trilayer improves transmittance mainly at high incidence angles. Only wavelengths useful for top and middle subcells in typical MJ solar cells are shown as bottom subcell generates an excess of current making the ARC transmittance at higher wavelengths not very critical.

Fig. 2
Fig. 2

Average transmission losses when due to manufacturing errors the different ARC layers optical thickness (physical thickness multiplied by refractive index) are not the optimized values. For this configuration, the higher the refractive index of the layer the more sensitive the transmission is to manufacturing errors in the layer.

Fig. 3
Fig. 3

Angular light distribution over the cell created by a 1000X CPV system composed of a Fresnel lens and a SOE, from left to right: bare cell, reflective pyramid, refractive CPC (Compound Parabolic Concentrator), refractive pyramid, and dome. Black columns represent the amount of light impinging at a particular angle and red lines represent accumulated values. A detailed description of the angular transmission curves and irradiance illumination distribution over the cell for the different SOEs can be found in [21]. Angular light distribution created by the refractive pyramid coincides with the previous reported results [7].

Fig. 4
Fig. 4

Total optical efficiency for different wavelengths measured using 80 nm-wide band-pass filters (dots, right vertical axis). Simulated averaged transmission for the same ARC structures (solid lines, left vertical axis).

Fig. 5
Fig. 5

Efficiency for cells with three different ARC structures (bare cell, bilayer and trilayer ARCs) under FluidReflex concentrator system. Masks with several radii are used to reduce the semi-angle of the incidence light cone.

Tables (3)

Tables Icon

Table 1 Ideal Refractive Indexes for 1, 2 and 3 Layers ARC When the Solar Cell Is Surrounded by Air and by a Material with n = 1.5. Dielectric Materials Which Refractive Indexes Are Close To the Ideal Values Are Also Indicated

Tables Icon

Table 2 Thicknesses and Weighted Transmittances When the Optimized Monolayer, Bilayer and Trilayer Are Used in a MJ Solar Cell Under FluidReflex Concentrator

Tables Icon

Table 3 Al2O3/TiO2 Bilayer ARC Optimized Results for a MJ Solar Cell To Be Illuminated By a System Composed of a Fresnel Lens Plus Different SOEs

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

X g e o m sin 2 ( θ entrance ) n 2 2 sin 2 ( θ cell )
n A R C = n m e d i u m n c e l l
d * = λ 4 n A R C cos ( θ r ) d
sin θ r = n m e d i u m n A R C sin θ i
n i = n medium ( N + 1 i N + 1 ) n cell ( i N + 1 )
T w e i g h t e d = J sc J sc ideal = min [ λ 1 λ 2 0 π 2 T ( λ , θ , d i ) T o p t ( λ , θ ) L ( θ ) B ( λ ) S R j ( λ ) d θ d λ ] j = 1 N min [ λ 1 λ 2 0 π 2 T o p t ( λ , θ ) L ( θ ) B ( λ ) S R j ( λ ) d θ d λ ] j = 1 N
η o p t , λ i = X e f f , λ i X g e o = I s c D U T , λ i I s c D U T ( 1 ) I s c R E F ( 1 ) I s c R E F , λ i X g e o
X g e o m = A e n t r a n c e A e x i t

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