Abstract

Luminescent solar concentrators (LSCs) are devices theoretically able to condense both direct and diffuse solar radiation into thin dielectric layers with extremely high efficiencies. A theory based on thermodynamic principles was developed in the past to estimate the concentration limits that can be achieved with an LSC and facilitate researchers’ efforts to predict the potential of their designs to convert optical to electrical power. However, while concentration efficiencies of thousands or even tens of thousands of suns are supported by this model, values of only a fraction of those have ever been recorded experimentally. This is because in the calculation of the thermodynamic limits the quantum yield of the luminophores is assumed to be equal to unity and any processes that quench the intensity of the trapped field are completely ignored. In an attempt to better match theory with reality and provide more accurate performance estimates, we have revised the limits of concentration based on a statistical optics framework. The new model gives insight into the main mechanisms inhibiting the concentration of LSCs and can be used to extract design rules for efficient LSCs. Comparisons between the method presented in this paper and results obtained with Monte Carlo ray-tracing simulations demonstrate excellent agreement between the two. Finally, we discuss the conditions for validity of the thermodynamic limits, and we show that in some circumstances these can actually be surpassed.

© 2015 Optical Society of America

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References

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2016 (1)

C. Tummeltshammer, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Losses in luminescent solar concentrators unveiled,” Sol. Energy Mater. Sol. Cells 144, 40–47 (2016).

2014 (4)

C. Tummeltshammer, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Homeotropic alignment and Forster resonance energy transfer: the way to a brighter luminescent solar concentrator,” J. Appl. Phys. 116, 173103 (2014).
[Crossref]

S. R. Wilton, M. R. Fetterman, J. J. Low, G. You, Z. Jiang, and J. Xu, “Monte Carlo study of PbSe quantum dots as the fluorescent material in luminescent solar concentrators,” Opt. Express 22, A35–A43 (2014).
[Crossref]

F. Meinardi, A. Colombo, K. A. Velizhanin, R. Simonutti, M. Lorenzon, L. Beverina, R. Viswanatha, V. I. Klimov, and S. Brovelli, “Large-area luminescent solar concentrators based on ‘Stokes-shift-engineered’ nanocrystals in a mass-polymerized PMMA matrix,” Nat. Photonics 8, 392–399 (2014).

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, “Zero-reabsorption doped-nanocrystal luminescent solar concentrators,” ACS Nano 8, 3461–3467 (2014).
[Crossref]

2013 (2)

S. Woei Leow, C. Corrado, M. Osborn, M. Isaacson, G. Alers, and S. A. Carter, “Analyzing luminescent solar concentrators with front-facing photovoltaic cells using weighted Monte Carlo ray tracing,” J. Appl. Phys. 113, 214510 (2013).
[Crossref]

C. Tummeltshammer, M. S. Brown, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Efficiency and loss mechanisms of plasmonic luminescent solar concentrators,” Opt. Express 21, A735–A749 (2013).
[Crossref]

2012 (1)

M. G. Debije and P. P. C. Verbunt, “Thirty years of luminescent solar concentrator research: solar energy for the built environment,” Adv. Energy Mater. 2, 12–35 (2012).
[Crossref]

2011 (1)

D. Sahin, B. Ilan, and D. F. Kelley, “Monte-Carlo simulations of light propagation in luminescent solar concentrators based on semiconductor nanoparticles,” J. Appl. Phys. 110, 033108 (2011).
[Crossref]

2010 (4)

G. V. Shcherbatyuk, R. H. Inman, C. Wang, R. Winston, and S. Ghosh, “Viability of using near infrared PbS quantum dots as active materials in luminescent solar concentrators,” Appl. Phys. Lett. 96, 191901 (2010).
[Crossref]

C. L. Mulder, P. D. Reusswig, A. M. Velázquez, H. Kim, C. Rotschild, and M. A. Baldo, “Dye alignment in luminescent solar concentrators: I. Vertical alignment for improved waveguide coupling,” Opt. Express 18, A79–A90 (2010).
[Crossref]

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nat. Mater. 9, 239–244 (2010).
[Crossref]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref]

2009 (3)

P. P. C. Verbunt, A. Kaiser, K. Hermans, C. W. M. Bastiaansen, D. J. Broer, and M. G. Debije, “Controlling light emission in luminescent solar concentrators through use of dye molecules aligned in a planar manner by liquid crystals,” Adv. Funct. Mater. 19, 2714–2719 (2009).
[Crossref]

M. G. Hyldahl, S. T. Bailey, and B. P. Wittmershaus, “Photo-stability and performance of CdSe/ZnS quantum dots in luminescent solar concentrators,” Solar Energy 83, 566–573 (2009).
[Crossref]

L. R. Wilson and B. S. Richards, “Measurement method for photoluminescent quantum yields of fluorescent organic dyes in polymethyl methacrylate for luminescent solar concentrators,” Appl. Opt. 48, 212–220 (2009).
[Crossref]

2008 (2)

B. Rowan, L. Wilson, and B. Richards, “Advanced material concepts for luminescent solar concentrators,” IEEE J. Sel. Top. Quantum Electron. 14, 1312–1322 (2008).
[Crossref]

K. Catchpole and A. Polman, “Design principles for particle plasmon enhanced solar cells,” Appl. Phys. Lett. 93, 191113 (2008).
[Crossref]

2007 (2)

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

V. Sholin, J. D. Olson, and S. A. Carter, “Semiconducting polymers and quantum dots in luminescent solar concentrators for solar energy harvesting,” J. Appl. Phys. 101, 123114 (2007).
[Crossref]

2006 (1)

T. Markvart, “Detailed balance method for ideal single-stage fluorescent collectors,” J. Appl. Phys. 99, 026101 (2006).
[Crossref]

2005 (1)

A. Mansour, H. Killa, S. Abd El-Wanees, and M. El-Sayed, “Laser dyes doped with poly(ST-Co-MMA) as fluorescent solar collectors and their field performance,” Polym. Test. 24, 519–525 (2005).
[Crossref]

1997 (1)

1990 (1)

G. Smestad, H. Ries, R. Winston, and E. Yablonovitch, “The thermodynamic limits of light concentrators,” Sol. Energy Mater. 21, 99–111 (1990).
[Crossref]

1983 (2)

1982 (3)

1981 (1)

1980 (1)

1977 (1)

1975 (1)

G. Reynolds and K. Drexhage, “New coumarin dyes with rigidized structure for flashlamp-pumped dye lasers,” Opt. Commun. 13, 222–225 (1975).
[Crossref]

1969 (1)

R. S. Knox, “Thermodynamics and the primary processes of photosynthesis,” Biophys. J. 9, 1351–1362 (1969).
[Crossref]

1967 (1)

R. T. Ross, “Some thermodynamics of photochemical systems,” J. Chem. Phys. 46, 4590–4593 (1967).
[Crossref]

Abd El-Wanees, S.

A. Mansour, H. Killa, S. Abd El-Wanees, and M. El-Sayed, “Laser dyes doped with poly(ST-Co-MMA) as fluorescent solar collectors and their field performance,” Polym. Test. 24, 519–525 (2005).
[Crossref]

Alers, G.

S. Woei Leow, C. Corrado, M. Osborn, M. Isaacson, G. Alers, and S. A. Carter, “Analyzing luminescent solar concentrators with front-facing photovoltaic cells using weighted Monte Carlo ray tracing,” J. Appl. Phys. 113, 214510 (2013).
[Crossref]

Atwater, H. A.

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nat. Mater. 9, 239–244 (2010).
[Crossref]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref]

Bailey, S. T.

M. G. Hyldahl, S. T. Bailey, and B. P. Wittmershaus, “Photo-stability and performance of CdSe/ZnS quantum dots in luminescent solar concentrators,” Solar Energy 83, 566–573 (2009).
[Crossref]

Baldo, M. A.

Barnham, K. W. J.

A. Chatten, D. Farrel, C. Jermyn, P. Thomas, B. Buxton, A. Buchtemann, R. Danz, and K. W. J. Barnham, “Thermodynamic modelling of luminescent solar concentrators,” in Conference Record of the Thirty-First IEEE Photovoltaic Specialists Conference, Lake Buena Vista, Florida (2005), pp. 82–85.

Bastiaansen, C. W. M.

P. P. C. Verbunt, A. Kaiser, K. Hermans, C. W. M. Bastiaansen, D. J. Broer, and M. G. Debije, “Controlling light emission in luminescent solar concentrators through use of dye molecules aligned in a planar manner by liquid crystals,” Adv. Funct. Mater. 19, 2714–2719 (2009).
[Crossref]

Batchelder, J. S.

Beverina, L.

F. Meinardi, A. Colombo, K. A. Velizhanin, R. Simonutti, M. Lorenzon, L. Beverina, R. Viswanatha, V. I. Klimov, and S. Brovelli, “Large-area luminescent solar concentrators based on ‘Stokes-shift-engineered’ nanocrystals in a mass-polymerized PMMA matrix,” Nat. Photonics 8, 392–399 (2014).

Boettcher, S. W.

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nat. Mater. 9, 239–244 (2010).
[Crossref]

Boyd, W. R.

W. R. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).

Bradshaw, L. R.

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, “Zero-reabsorption doped-nanocrystal luminescent solar concentrators,” ACS Nano 8, 3461–3467 (2014).
[Crossref]

Briggs, R. M.

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nat. Mater. 9, 239–244 (2010).
[Crossref]

Broer, D. J.

P. P. C. Verbunt, A. Kaiser, K. Hermans, C. W. M. Bastiaansen, D. J. Broer, and M. G. Debije, “Controlling light emission in luminescent solar concentrators through use of dye molecules aligned in a planar manner by liquid crystals,” Adv. Funct. Mater. 19, 2714–2719 (2009).
[Crossref]

Brovelli, S.

F. Meinardi, A. Colombo, K. A. Velizhanin, R. Simonutti, M. Lorenzon, L. Beverina, R. Viswanatha, V. I. Klimov, and S. Brovelli, “Large-area luminescent solar concentrators based on ‘Stokes-shift-engineered’ nanocrystals in a mass-polymerized PMMA matrix,” Nat. Photonics 8, 392–399 (2014).

Brown, M. S.

Buchtemann, A.

A. Chatten, D. Farrel, C. Jermyn, P. Thomas, B. Buxton, A. Buchtemann, R. Danz, and K. W. J. Barnham, “Thermodynamic modelling of luminescent solar concentrators,” in Conference Record of the Thirty-First IEEE Photovoltaic Specialists Conference, Lake Buena Vista, Florida (2005), pp. 82–85.

Buxton, B.

A. Chatten, D. Farrel, C. Jermyn, P. Thomas, B. Buxton, A. Buchtemann, R. Danz, and K. W. J. Barnham, “Thermodynamic modelling of luminescent solar concentrators,” in Conference Record of the Thirty-First IEEE Photovoltaic Specialists Conference, Lake Buena Vista, Florida (2005), pp. 82–85.

Carter, S. A.

S. Woei Leow, C. Corrado, M. Osborn, M. Isaacson, G. Alers, and S. A. Carter, “Analyzing luminescent solar concentrators with front-facing photovoltaic cells using weighted Monte Carlo ray tracing,” J. Appl. Phys. 113, 214510 (2013).
[Crossref]

V. Sholin, J. D. Olson, and S. A. Carter, “Semiconducting polymers and quantum dots in luminescent solar concentrators for solar energy harvesting,” J. Appl. Phys. 101, 123114 (2007).
[Crossref]

Catchpole, K.

K. Catchpole and A. Polman, “Design principles for particle plasmon enhanced solar cells,” Appl. Phys. Lett. 93, 191113 (2008).
[Crossref]

Chang, Y.-H.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Chatten, A.

A. Chatten, D. Farrel, C. Jermyn, P. Thomas, B. Buxton, A. Buchtemann, R. Danz, and K. W. J. Barnham, “Thermodynamic modelling of luminescent solar concentrators,” in Conference Record of the Thirty-First IEEE Photovoltaic Specialists Conference, Lake Buena Vista, Florida (2005), pp. 82–85.

Chattopadhyay, S.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Chen, K.-H.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Chen, L.-C.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Cole, T.

Colombo, A.

F. Meinardi, A. Colombo, K. A. Velizhanin, R. Simonutti, M. Lorenzon, L. Beverina, R. Viswanatha, V. I. Klimov, and S. Brovelli, “Large-area luminescent solar concentrators based on ‘Stokes-shift-engineered’ nanocrystals in a mass-polymerized PMMA matrix,” Nat. Photonics 8, 392–399 (2014).

Corrado, C.

S. Woei Leow, C. Corrado, M. Osborn, M. Isaacson, G. Alers, and S. A. Carter, “Analyzing luminescent solar concentrators with front-facing photovoltaic cells using weighted Monte Carlo ray tracing,” J. Appl. Phys. 113, 214510 (2013).
[Crossref]

Cusso, F.

Danz, R.

A. Chatten, D. Farrel, C. Jermyn, P. Thomas, B. Buxton, A. Buchtemann, R. Danz, and K. W. J. Barnham, “Thermodynamic modelling of luminescent solar concentrators,” in Conference Record of the Thirty-First IEEE Photovoltaic Specialists Conference, Lake Buena Vista, Florida (2005), pp. 82–85.

Debije, M. G.

M. G. Debije and P. P. C. Verbunt, “Thirty years of luminescent solar concentrator research: solar energy for the built environment,” Adv. Energy Mater. 2, 12–35 (2012).
[Crossref]

P. P. C. Verbunt, A. Kaiser, K. Hermans, C. W. M. Bastiaansen, D. J. Broer, and M. G. Debije, “Controlling light emission in luminescent solar concentrators through use of dye molecules aligned in a planar manner by liquid crystals,” Adv. Funct. Mater. 19, 2714–2719 (2009).
[Crossref]

Drake, J. M.

Drexhage, K.

G. Reynolds and K. Drexhage, “New coumarin dyes with rigidized structure for flashlamp-pumped dye lasers,” Opt. Commun. 13, 222–225 (1975).
[Crossref]

El-Sayed, M.

A. Mansour, H. Killa, S. Abd El-Wanees, and M. El-Sayed, “Laser dyes doped with poly(ST-Co-MMA) as fluorescent solar collectors and their field performance,” Polym. Test. 24, 519–525 (2005).
[Crossref]

Erickson, C. S.

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, “Zero-reabsorption doped-nanocrystal luminescent solar concentrators,” ACS Nano 8, 3461–3467 (2014).
[Crossref]

Farrel, D.

A. Chatten, D. Farrel, C. Jermyn, P. Thomas, B. Buxton, A. Buchtemann, R. Danz, and K. W. J. Barnham, “Thermodynamic modelling of luminescent solar concentrators,” in Conference Record of the Thirty-First IEEE Photovoltaic Specialists Conference, Lake Buena Vista, Florida (2005), pp. 82–85.

Fetterman, M. R.

Gamelin, D. R.

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, “Zero-reabsorption doped-nanocrystal luminescent solar concentrators,” ACS Nano 8, 3461–3467 (2014).
[Crossref]

Ghosh, S.

G. V. Shcherbatyuk, R. H. Inman, C. Wang, R. Winston, and S. Ghosh, “Viability of using near infrared PbS quantum dots as active materials in luminescent solar concentrators,” Appl. Phys. Lett. 96, 191901 (2010).
[Crossref]

Gilbertson, J. D.

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, “Zero-reabsorption doped-nanocrystal luminescent solar concentrators,” ACS Nano 8, 3461–3467 (2014).
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L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2008).

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P. P. C. Verbunt, A. Kaiser, K. Hermans, C. W. M. Bastiaansen, D. J. Broer, and M. G. Debije, “Controlling light emission in luminescent solar concentrators through use of dye molecules aligned in a planar manner by liquid crystals,” Adv. Funct. Mater. 19, 2714–2719 (2009).
[Crossref]

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Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
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Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
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Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[Crossref]

Hyldahl, M. G.

M. G. Hyldahl, S. T. Bailey, and B. P. Wittmershaus, “Photo-stability and performance of CdSe/ZnS quantum dots in luminescent solar concentrators,” Solar Energy 83, 566–573 (2009).
[Crossref]

Ilan, B.

D. Sahin, B. Ilan, and D. F. Kelley, “Monte-Carlo simulations of light propagation in luminescent solar concentrators based on semiconductor nanoparticles,” J. Appl. Phys. 110, 033108 (2011).
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Inman, R. H.

G. V. Shcherbatyuk, R. H. Inman, C. Wang, R. Winston, and S. Ghosh, “Viability of using near infrared PbS quantum dots as active materials in luminescent solar concentrators,” Appl. Phys. Lett. 96, 191901 (2010).
[Crossref]

Isaacson, M.

S. Woei Leow, C. Corrado, M. Osborn, M. Isaacson, G. Alers, and S. A. Carter, “Analyzing luminescent solar concentrators with front-facing photovoltaic cells using weighted Monte Carlo ray tracing,” J. Appl. Phys. 113, 214510 (2013).
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Jen, Y.-J.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
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A. Chatten, D. Farrel, C. Jermyn, P. Thomas, B. Buxton, A. Buchtemann, R. Danz, and K. W. J. Barnham, “Thermodynamic modelling of luminescent solar concentrators,” in Conference Record of the Thirty-First IEEE Photovoltaic Specialists Conference, Lake Buena Vista, Florida (2005), pp. 82–85.

Jiang, Z.

Kaiser, A.

P. P. C. Verbunt, A. Kaiser, K. Hermans, C. W. M. Bastiaansen, D. J. Broer, and M. G. Debije, “Controlling light emission in luminescent solar concentrators through use of dye molecules aligned in a planar manner by liquid crystals,” Adv. Funct. Mater. 19, 2714–2719 (2009).
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D. Sahin, B. Ilan, and D. F. Kelley, “Monte-Carlo simulations of light propagation in luminescent solar concentrators based on semiconductor nanoparticles,” J. Appl. Phys. 110, 033108 (2011).
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M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nat. Mater. 9, 239–244 (2010).
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C. Tummeltshammer, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Losses in luminescent solar concentrators unveiled,” Sol. Energy Mater. Sol. Cells 144, 40–47 (2016).

C. Tummeltshammer, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Homeotropic alignment and Forster resonance energy transfer: the way to a brighter luminescent solar concentrator,” J. Appl. Phys. 116, 173103 (2014).
[Crossref]

C. Tummeltshammer, M. S. Brown, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Efficiency and loss mechanisms of plasmonic luminescent solar concentrators,” Opt. Express 21, A735–A749 (2013).
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A. Mansour, H. Killa, S. Abd El-Wanees, and M. El-Sayed, “Laser dyes doped with poly(ST-Co-MMA) as fluorescent solar collectors and their field performance,” Polym. Test. 24, 519–525 (2005).
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Klimov, V. I.

F. Meinardi, A. Colombo, K. A. Velizhanin, R. Simonutti, M. Lorenzon, L. Beverina, R. Viswanatha, V. I. Klimov, and S. Brovelli, “Large-area luminescent solar concentrators based on ‘Stokes-shift-engineered’ nanocrystals in a mass-polymerized PMMA matrix,” Nat. Photonics 8, 392–399 (2014).

Knox, R. S.

R. S. Knox, “Thermodynamics and the primary processes of photosynthesis,” Biophys. J. 9, 1351–1362 (1969).
[Crossref]

Lee, C.-S.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
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Lesiecki, M. L.

Lewis, N. S.

M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nat. Mater. 9, 239–244 (2010).
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Lifante, G.

Liu, T.-A.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
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Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
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F. Meinardi, A. Colombo, K. A. Velizhanin, R. Simonutti, M. Lorenzon, L. Beverina, R. Viswanatha, V. I. Klimov, and S. Brovelli, “Large-area luminescent solar concentrators based on ‘Stokes-shift-engineered’ nanocrystals in a mass-polymerized PMMA matrix,” Nat. Photonics 8, 392–399 (2014).

Low, J. J.

Mansour, A.

A. Mansour, H. Killa, S. Abd El-Wanees, and M. El-Sayed, “Laser dyes doped with poly(ST-Co-MMA) as fluorescent solar collectors and their field performance,” Polym. Test. 24, 519–525 (2005).
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T. Markvart, “Detailed balance method for ideal single-stage fluorescent collectors,” J. Appl. Phys. 99, 026101 (2006).
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McDowall, S.

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, “Zero-reabsorption doped-nanocrystal luminescent solar concentrators,” ACS Nano 8, 3461–3467 (2014).
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F. Meinardi, A. Colombo, K. A. Velizhanin, R. Simonutti, M. Lorenzon, L. Beverina, R. Viswanatha, V. I. Klimov, and S. Brovelli, “Large-area luminescent solar concentrators based on ‘Stokes-shift-engineered’ nanocrystals in a mass-polymerized PMMA matrix,” Nat. Photonics 8, 392–399 (2014).

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Mulder, C. L.

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2008).

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V. Sholin, J. D. Olson, and S. A. Carter, “Semiconducting polymers and quantum dots in luminescent solar concentrators for solar energy harvesting,” J. Appl. Phys. 101, 123114 (2007).
[Crossref]

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S. Woei Leow, C. Corrado, M. Osborn, M. Isaacson, G. Alers, and S. A. Carter, “Analyzing luminescent solar concentrators with front-facing photovoltaic cells using weighted Monte Carlo ray tracing,” J. Appl. Phys. 113, 214510 (2013).
[Crossref]

Pan, C.-L.

Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
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C. Tummeltshammer, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Losses in luminescent solar concentrators unveiled,” Sol. Energy Mater. Sol. Cells 144, 40–47 (2016).

C. Tummeltshammer, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Homeotropic alignment and Forster resonance energy transfer: the way to a brighter luminescent solar concentrator,” J. Appl. Phys. 116, 173103 (2014).
[Crossref]

C. Tummeltshammer, M. S. Brown, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Efficiency and loss mechanisms of plasmonic luminescent solar concentrators,” Opt. Express 21, A735–A749 (2013).
[Crossref]

Patrick, D. L.

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, “Zero-reabsorption doped-nanocrystal luminescent solar concentrators,” ACS Nano 8, 3461–3467 (2014).
[Crossref]

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Y.-F. Huang, S. Chattopadhyay, Y.-J. Jen, C.-Y. Peng, T.-A. Liu, Y.-K. Hsu, C.-L. Pan, H.-C. Lo, C.-H. Hsu, Y.-H. Chang, C.-S. Lee, K.-H. Chen, and L.-C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
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M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nat. Mater. 9, 239–244 (2010).
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H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
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K. Catchpole and A. Polman, “Design principles for particle plasmon enhanced solar cells,” Appl. Phys. Lett. 93, 191113 (2008).
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M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nat. Mater. 9, 239–244 (2010).
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B. Rowan, L. Wilson, and B. Richards, “Advanced material concepts for luminescent solar concentrators,” IEEE J. Sel. Top. Quantum Electron. 14, 1312–1322 (2008).
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Richards, B. S.

Ries, H.

G. Smestad, H. Ries, R. Winston, and E. Yablonovitch, “The thermodynamic limits of light concentrators,” Sol. Energy Mater. 21, 99–111 (1990).
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H. Ries, “Thermodynamic limitations of the concentration of electromagnetic radiation,” J. Opt. Soc. Am. A 72, 380–385 (1982).
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Ross, R. T.

R. T. Ross, “Some thermodynamics of photochemical systems,” J. Chem. Phys. 46, 4590–4593 (1967).
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Rowan, B.

B. Rowan, L. Wilson, and B. Richards, “Advanced material concepts for luminescent solar concentrators,” IEEE J. Sel. Top. Quantum Electron. 14, 1312–1322 (2008).
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D. Sahin, B. Ilan, and D. F. Kelley, “Monte-Carlo simulations of light propagation in luminescent solar concentrators based on semiconductor nanoparticles,” J. Appl. Phys. 110, 033108 (2011).
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Shcherbatyuk, G. V.

G. V. Shcherbatyuk, R. H. Inman, C. Wang, R. Winston, and S. Ghosh, “Viability of using near infrared PbS quantum dots as active materials in luminescent solar concentrators,” Appl. Phys. Lett. 96, 191901 (2010).
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V. Sholin, J. D. Olson, and S. A. Carter, “Semiconducting polymers and quantum dots in luminescent solar concentrators for solar energy harvesting,” J. Appl. Phys. 101, 123114 (2007).
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F. Meinardi, A. Colombo, K. A. Velizhanin, R. Simonutti, M. Lorenzon, L. Beverina, R. Viswanatha, V. I. Klimov, and S. Brovelli, “Large-area luminescent solar concentrators based on ‘Stokes-shift-engineered’ nanocrystals in a mass-polymerized PMMA matrix,” Nat. Photonics 8, 392–399 (2014).

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G. Smestad, H. Ries, R. Winston, and E. Yablonovitch, “The thermodynamic limits of light concentrators,” Sol. Energy Mater. 21, 99–111 (1990).
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M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nat. Mater. 9, 239–244 (2010).
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Swartz, B. A.

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C. Tummeltshammer, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Losses in luminescent solar concentrators unveiled,” Sol. Energy Mater. Sol. Cells 144, 40–47 (2016).

C. Tummeltshammer, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Homeotropic alignment and Forster resonance energy transfer: the way to a brighter luminescent solar concentrator,” J. Appl. Phys. 116, 173103 (2014).
[Crossref]

C. Tummeltshammer, M. S. Brown, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Efficiency and loss mechanisms of plasmonic luminescent solar concentrators,” Opt. Express 21, A735–A749 (2013).
[Crossref]

Thomas, P.

A. Chatten, D. Farrel, C. Jermyn, P. Thomas, B. Buxton, A. Buchtemann, R. Danz, and K. W. J. Barnham, “Thermodynamic modelling of luminescent solar concentrators,” in Conference Record of the Thirty-First IEEE Photovoltaic Specialists Conference, Lake Buena Vista, Florida (2005), pp. 82–85.

Thomas, W. R.

Tummeltshammer, C.

C. Tummeltshammer, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Losses in luminescent solar concentrators unveiled,” Sol. Energy Mater. Sol. Cells 144, 40–47 (2016).

C. Tummeltshammer, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Homeotropic alignment and Forster resonance energy transfer: the way to a brighter luminescent solar concentrator,” J. Appl. Phys. 116, 173103 (2014).
[Crossref]

C. Tummeltshammer, M. S. Brown, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Efficiency and loss mechanisms of plasmonic luminescent solar concentrators,” Opt. Express 21, A735–A749 (2013).
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M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nat. Mater. 9, 239–244 (2010).
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Velizhanin, K. A.

F. Meinardi, A. Colombo, K. A. Velizhanin, R. Simonutti, M. Lorenzon, L. Beverina, R. Viswanatha, V. I. Klimov, and S. Brovelli, “Large-area luminescent solar concentrators based on ‘Stokes-shift-engineered’ nanocrystals in a mass-polymerized PMMA matrix,” Nat. Photonics 8, 392–399 (2014).

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M. G. Debije and P. P. C. Verbunt, “Thirty years of luminescent solar concentrator research: solar energy for the built environment,” Adv. Energy Mater. 2, 12–35 (2012).
[Crossref]

P. P. C. Verbunt, A. Kaiser, K. Hermans, C. W. M. Bastiaansen, D. J. Broer, and M. G. Debije, “Controlling light emission in luminescent solar concentrators through use of dye molecules aligned in a planar manner by liquid crystals,” Adv. Funct. Mater. 19, 2714–2719 (2009).
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F. Meinardi, A. Colombo, K. A. Velizhanin, R. Simonutti, M. Lorenzon, L. Beverina, R. Viswanatha, V. I. Klimov, and S. Brovelli, “Large-area luminescent solar concentrators based on ‘Stokes-shift-engineered’ nanocrystals in a mass-polymerized PMMA matrix,” Nat. Photonics 8, 392–399 (2014).

Wang, C.

G. V. Shcherbatyuk, R. H. Inman, C. Wang, R. Winston, and S. Ghosh, “Viability of using near infrared PbS quantum dots as active materials in luminescent solar concentrators,” Appl. Phys. Lett. 96, 191901 (2010).
[Crossref]

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M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nat. Mater. 9, 239–244 (2010).
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B. Rowan, L. Wilson, and B. Richards, “Advanced material concepts for luminescent solar concentrators,” IEEE J. Sel. Top. Quantum Electron. 14, 1312–1322 (2008).
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Wilson, L. R.

Wilton, S. R.

Winston, R.

G. V. Shcherbatyuk, R. H. Inman, C. Wang, R. Winston, and S. Ghosh, “Viability of using near infrared PbS quantum dots as active materials in luminescent solar concentrators,” Appl. Phys. Lett. 96, 191901 (2010).
[Crossref]

G. Smestad, H. Ries, R. Winston, and E. Yablonovitch, “The thermodynamic limits of light concentrators,” Sol. Energy Mater. 21, 99–111 (1990).
[Crossref]

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M. G. Hyldahl, S. T. Bailey, and B. P. Wittmershaus, “Photo-stability and performance of CdSe/ZnS quantum dots in luminescent solar concentrators,” Solar Energy 83, 566–573 (2009).
[Crossref]

Woei Leow, S.

S. Woei Leow, C. Corrado, M. Osborn, M. Isaacson, G. Alers, and S. A. Carter, “Analyzing luminescent solar concentrators with front-facing photovoltaic cells using weighted Monte Carlo ray tracing,” J. Appl. Phys. 113, 214510 (2013).
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Yablonovitch, E.

You, G.

Zewail, A. H.

ACS Nano (1)

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, “Zero-reabsorption doped-nanocrystal luminescent solar concentrators,” ACS Nano 8, 3461–3467 (2014).
[Crossref]

Adv. Energy Mater. (1)

M. G. Debije and P. P. C. Verbunt, “Thirty years of luminescent solar concentrator research: solar energy for the built environment,” Adv. Energy Mater. 2, 12–35 (2012).
[Crossref]

Adv. Funct. Mater. (1)

P. P. C. Verbunt, A. Kaiser, K. Hermans, C. W. M. Bastiaansen, D. J. Broer, and M. G. Debije, “Controlling light emission in luminescent solar concentrators through use of dye molecules aligned in a planar manner by liquid crystals,” Adv. Funct. Mater. 19, 2714–2719 (2009).
[Crossref]

Appl. Opt. (5)

Appl. Phys. Lett. (2)

K. Catchpole and A. Polman, “Design principles for particle plasmon enhanced solar cells,” Appl. Phys. Lett. 93, 191113 (2008).
[Crossref]

G. V. Shcherbatyuk, R. H. Inman, C. Wang, R. Winston, and S. Ghosh, “Viability of using near infrared PbS quantum dots as active materials in luminescent solar concentrators,” Appl. Phys. Lett. 96, 191901 (2010).
[Crossref]

Biophys. J. (1)

R. S. Knox, “Thermodynamics and the primary processes of photosynthesis,” Biophys. J. 9, 1351–1362 (1969).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

B. Rowan, L. Wilson, and B. Richards, “Advanced material concepts for luminescent solar concentrators,” IEEE J. Sel. Top. Quantum Electron. 14, 1312–1322 (2008).
[Crossref]

J. Appl. Phys. (5)

V. Sholin, J. D. Olson, and S. A. Carter, “Semiconducting polymers and quantum dots in luminescent solar concentrators for solar energy harvesting,” J. Appl. Phys. 101, 123114 (2007).
[Crossref]

D. Sahin, B. Ilan, and D. F. Kelley, “Monte-Carlo simulations of light propagation in luminescent solar concentrators based on semiconductor nanoparticles,” J. Appl. Phys. 110, 033108 (2011).
[Crossref]

S. Woei Leow, C. Corrado, M. Osborn, M. Isaacson, G. Alers, and S. A. Carter, “Analyzing luminescent solar concentrators with front-facing photovoltaic cells using weighted Monte Carlo ray tracing,” J. Appl. Phys. 113, 214510 (2013).
[Crossref]

T. Markvart, “Detailed balance method for ideal single-stage fluorescent collectors,” J. Appl. Phys. 99, 026101 (2006).
[Crossref]

C. Tummeltshammer, A. Taylor, A. J. Kenyon, and I. Papakonstantinou, “Homeotropic alignment and Forster resonance energy transfer: the way to a brighter luminescent solar concentrator,” J. Appl. Phys. 116, 173103 (2014).
[Crossref]

J. Chem. Phys. (1)

R. T. Ross, “Some thermodynamics of photochemical systems,” J. Chem. Phys. 46, 4590–4593 (1967).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

H. R. Stuart and D. G. Hall, “Thermodynamic limit to light trapping in thin planar structures,” J. Opt. Soc. Am. A 14, 3001–3008 (1997).
[Crossref]

H. Ries, “Thermodynamic limitations of the concentration of electromagnetic radiation,” J. Opt. Soc. Am. A 72, 380–385 (1982).
[Crossref]

Nat. Mater. (2)

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Supplementary Material (1)

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Figures (7)

Fig. 1.
Fig. 1. Geometry of the investigated LSC: an LSC slab of thickness d and refractive index n is immersed into diffuse electromagnetic radiation characterized by isotropic radiance S(ω1;Ts). Also shown: energy levels of the fluorescent system under consideration indicating the possible electronic transitions.
Fig. 2.
Fig. 2. Fields created inside the LSC due to incident radiation: the fields S1±(θ;z) generated inside the LSC by the external radiation So can be deduced by application of flux conservation (incoming flux=outgoing flux) at the air–dielectric interface after considering the reflections at the boundaries and the attenuation imposed onto the traveling beams by the absorptivity of the fluorophores. θo is the polar angle measured from the LSC normal toward the air, while θ is the polar angle measured toward the dielectric. No azimuthal dependence is exhibited due to the symmetry of the problem.
Fig. 3.
Fig. 3. Incident radiation generates three types of sources that traverse each infinitesimal slice extending between [z,z+Δz]: A, radiation transmitted directly into the LSC with a photon flux per unit incident solid angle S1±(θ;z); B, collective radiation contributed by all the LSC regions below z and above z+Δz; C, feedback source S2±(z) due to reflections on LSC–air interfaces. The three sources (A–C) are responsible for generating a Lambertian emission profile emerging from each elementary slice with an angularly independent radiance L±(z).
Fig. 4.
Fig. 4. Normalized concentration ratio as a function of luminophore concentration × LSC thickness (cm×d) for various quantum yield values. The thick black line defines the values for which thermodynamic and geometrical optic limits coincide. For optically thick LSCs where the thermodynamic limits are valid, it is always CRq<CLT.
Fig. 5.
Fig. 5. Absorption and emission spectra for Coumarin 6 dye (spectra obtained from [31]). Also shown, typical histogram of photons received at the edges of the LSC as calculated by Monte Carlo, ray-tracing method. Inset: absorption and emission spectra for PbS quantum dots (spectra obtained from [32]).
Fig. 6.
Fig. 6. Comparison between Monte Carlo, weighted average, and single frequency models. Average values for ϵ(ω2)¯ are calculated over the entire region where the absorption and emission spectra of Coumarin 6 overlap in the full-spectrum weighted average model. In contrast, only the spectra of those photons that are delivered to the sides of the LSC are used in the restricted spectrum model. Values used to obtain the results: full-spectrum weighted average frequency model, ϵ(ω1)¯=52731(M·cm)1, ϵ(ω2)¯=5420(M·cm)1. Restricted spectrum weighted average frequency model, ϵ(ω1)¯=52731(M·cm)1, ϵ(ω2)¯=271(M·cm)1.
Fig. 7.
Fig. 7. Heatmap of maximum concentration ratio as a function of absorption and quantum yield calculated by the weighted average model. Black dotted line corresponds to the CRq=CLT boundary. White bar shows the range of concentrations that could be achieved by five experimental systems selected from the literature. Star sign signifies the position of optimum concentration for q=0.78.

Equations (18)

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CLT=(nno)2(ω2ω1)2eh(ω1ω2)/KTo,
S(ω;Ts)=ω22π2c21ehω/KTs1.
S1(θ;z)=S1+(θ;z)+S1(θ;z)=S1(θ)(eα1z+eα1(dz)).
(1Ro(θo))SodΩocosθo=[S1+(θ;z=0+)R(θ)S1(θ;z=0+)]dΩcosθ.
S1(θ)=So(nno)21R(θ)1eα1dR(θ).
S2(z)=Loζ(eα2z+eα2(dz))
L+(z+Δz)dΩcosθ=α2q2Δz[L+(z)+L(z+Δz)]dΩcosθ+(1α2Δz)L+(z)dΩcosθ+q2Δz[S2(z)α2+(ω2ω1)2eh(ω2ω1)/KToS1(θ;z)α1].
dL+(z)dz=α2[q2L(z)(1q2)L+(z)]+q2π[S2(z)α2+(ω2ω1)2eh(ω2ω1)/KToS1(θ;z)α1]dL(z)dz=α2[q2L+(z)(1q2)L(z)]+q2π[S2(z)α2+(ω2ω1)2eh(ω2ω1)/KToS1(θ;z)α1],
u(z)=(L(z)L+(z))andv(z)=(L(z)+L+(z)),
du(z)dz=α2(1q)v(z)qπ[S2(z)α2+(ω2ω1)2eh(ω2ω1)/KToS1(z)α1],
dv(z)dz=α2u(z).
L±(z)=qπ(H±(z)H+(0)S2+(0)S2±(z))+q2πα12α12(1q)α22·(ω2ω1)2eh(ω1ω2)/KTo·[(1+α2α1)(H±(z)H+(0)S1+(0)S1±(z))+(α2α11)(H±(z)H+(0)S1(0)S1(z))]
Φtot±(z)=[πL±(z)+S2±(z)]dΩcosθ.
Φtot(z)=Φtot+(z)+Φtot(z).
CRqΦtot(z)2πSo.
Lo=S1(θ)(1eα1d)πζ(1eα2d)(ω2ω1)2eh(ω1ω2)/KTo.
CR(q=1)Φtot(z)2πSo=(nno)2(ω2ω1)2eh(ω1ω2)/KTo·12π{(π+ζ)+α2α1(πζ)(1eα1zeα1(dz))}.
Φtot2πSo=n2no2,

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