## Flat Fresnel doublets made of PMMA and PC: combining low cost production and very high concentration ratio for CPV |

Optics Express, Vol. 19, Issue S3, pp. A280-A294 (2011)

http://dx.doi.org/10.1364/OE.19.00A280

Acrobat PDF (1612 KB)

### 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 × 10^{6} 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.

© 2011 OSA

## 1. Introduction

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

3. 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.

## 2. Influence of the LCA on the optical concentration ratio

*A’*and a receiver surface

*A*. The ratio

*A’*/

*A*corresponds to the geometrical concentration factor

*C*. If Φ’ is the flux collected and Φ the flux absorbed, then Φ’/Φ refers to the optical efficiency

_{geo}*η*. Finally, the optical concentration factor

_{opt}*C*is given by Eq. (1).

_{opt}*θ*represents the acceptance half angle of the incoming light [4

4. R. Winston, “Light Collection within the Framework of Geometrical Optics,” J. Opt. Soc. Am. **60**(2), 245–247 (1970). [CrossRef]

5. 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]

*f*=

*f*(

*λ*).

*λ*and

_{A}*λ*, the linear chromatic aberration (LCA) corresponds to the difference of the focal distances:

_{B}*R*and

_{1}*R*changes is given by Eq. (4) [7

_{2}7. E. Hecht, *Optics* 4th Ed. (Addison-Wesley, 2002), Chap. 5. [PubMed]

8. D. A. Buralli, G. M. Morris, and J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt. **28**(5), 976–983 (1989). [CrossRef] [PubMed]

*λ*as a reference:

_{0}*λ*is such that the minimum LCA and the maximum LCA – achieved with

_{0}*λ*and

_{m}*λ*respectively – are equal in absolute value. The system is thus optimised to decrease the maximum longitudinal chromatic aberration in absolute value |LCA

_{M}_{max}| (see Fig. 1a ).

*λ*. The LCA for the refractive and diffractive cases may be thus rewritten respectively as Eq. (6) and Eq. (7). It appears that both depend only on the wavelength. And it can be easily shown, that if the detector is placed in

_{0}*f*(

*λ*), the geometrical concentration allowing for the collection of the whole flux is given by Eq. (8).

_{0}*LCA*

_{max}| but in non-imaging optics, to maximize the amount of collected rays on the collector, minimising the LCA is not sufficient. Figure 1 shows two wavelengths with the same LCA in absolute value but the amount of light collected with λ

_{m}(red part) in Fig. 1a is more important than the amount of light collected with λ

_{M}(green part) in Fig. 1b. It is easy to show that the ideal position of the detector

*z*

_{det}as a function of the

*LCA**corresponds to a parabola of Eq. (30) (see Appendix A). This position allows for higher concentration as represented in Fig. 2a while Fig. 2b shows the gain of geometric concentration factor achieved by moving the detector from

*f*(λ

_{0}) to

*z*

_{det}. In CPV, the maximum concentration is given by the angular size of the sun, but if the

*LCA**is greater than 0.466% then the upper limit is driven by the LCA and becomes lower than 46,000.

*LCA*

_{max}| and geometrical concentration factor, where should

*λ*be focused: before collector position or after? Focusing after the detector increases the LCA but the view angle of fast converging wavelengths decreases, which is favourable for systems with fast converging wavelengths with

_{0}*η*

_{opt}< 1. Focusing before the detector position decreases the LCA but a higher amount of fast converging wavelengths will miss the detector. The ideal position and the optical efficiency as a function of the

*LCA** and the geometrical concentration factor are given in Fig. 3 .

## 3. Dispersion curves

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

*et al.*article [10

10. 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]

^{TM}[12

12. ASAP^{TM} optical design software of Breault Research Organization, http://www.breault.com.

*n*depends on the wavelength

*λ*. The way the refractive index changes with the wavelength might be approximated with several functions. Two popular functions of dispersion are used in this publication. Equation (9) corresponds to Sellmeier's equation and Eq. (10) to Laurent's (also called Schott's) equation. In this article, Sellmeier's equation is limited to

*m*= 3 and Laurent's equation is limited to the term in λ

^{−8}ensuring typically a difference lower than ± 0.001 between interpolated and experimental data [10

10. 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]

13. 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]

*et al.*from measurements going from 435.8 to 1052 nm and the extrapolation outside of this range gives wrong results. We performed another interpolation (PC #1) giving more probable results in the near infrared region. Thus, PC #1 (old) will no longer be considered hereafter.

## 4. Chromatic aberration of single lenses

### 4.1 Refractive lens

*f*may be approximated by Eq. (4) which has been rewritten in Eq. (11):with

*RoC*the equivalent radius of curvature of the lens [7

_{eq}7. E. Hecht, *Optics* 4th Ed. (Addison-Wesley, 2002), Chap. 5. [PubMed]

*LCA*| giving the focal distances of Fig. 5 .

_{max}*λ*) and compares the |

_{0}*LCA**

_{max}| of each material as well as the equivalent radius of curvature. In the case of a singlet, |

*LCA*| is achieved for the two extreme wavelengths:

_{max}*λ*= 380 nm and

_{m}*λ*= 1600 nm. As shown on Fig. 4 and 5, the LCA is more important for PCs than PMMAs. This is due to the high dispersion of the refractive indexes of PCs compared to the refractive indexes of PMMAs. It is common to use the Abbé number

_{M}*v*to measure the dispersion in the visible region.

_{d}*d*,

*F*and

*C*being three Fraunhofer lines in the visible region corresponding respectively to 587.562, 486.134 and 656.281 nm. In order to take into account the wide spectrum band studied, we use the solar Abbé number

### 4.2 Diffractive lens

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]

*h*of few microns. Each zone of the diffractive lens is designed by keeping the optical path length constant all over the zone. Between two adjacent zones, a 2π-phase shift is introduced. In other words, there is no discontinuity in the wavefront and the diffraction efficiency is maximum at the designed wavelength

*λ*

_{0}. This continuity is ensured by a constant thickness of the teeth:

8. D. A. Buralli, G. M. Morris, and J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt. **28**(5), 976–983 (1989). [CrossRef] [PubMed]

*LCA*designs. Moreover, the farther they are used from the design wavelength, the more diffractive lenses suffer from a lack of diffraction efficiency. The diffraction efficiency at the first order

*η*

_{1}is given by Eq. (16). E. g., for PMMA #2, with λ

_{0}= 550nm, the diffraction efficiency at the first order remains above 90% only between 472 and 663 nm.

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.

## 5. Achromatization

*LCA**

_{max}< 1%) since the lens designer may choose two wavelengths (λ

_{1}and λ

_{2}) that will focus at the same point. In general, to create a doublet one uses the well-known Abbé condition given by Eq. (17) [18] in combination with the formula of the effective focal length (Eq. (18)).

*f*

_{1,}

_{d}v_{1,}

*+*

_{d}*f*

_{2,}

_{d}v_{2,}

*= 0 is used,*

_{d}*f(λ*and

_{C})*f(λ*will be the same but only the focal distance of λ

_{F})*is known directly. Moreover nothing proves that having the same focal distance for λ*

_{d}*and λ*

_{F}*gives the smallest*

_{C}*LCA**

*. Therefore, we suggest using a more straightforward formula giving directly the focal distance of two chosen wavelengths λ*

_{max}_{1}and λ

_{2}.

### 5.1 Refractive doublet

*bfl*) – i.e. the distance from the back of the second lens to the focal point (see Fig. 7 ) – the focal distance of the second lens is given by Eq. (19) and (20).with

*d*≪

*bfl*,

*A*and

*C*are always positive while

*B*depends on the order of the materials. The first lens will have another expression of the focal distance

*bfl*, λ

_{1}and λ

_{2}are three parameters. For the other wavelengths in the refractive regime,

*n*(

_{1}*λ*)

*–*has a dispersion higher than

*n*(

_{2}*λ*),

*B*will be negative and thus

*f*

_{2}(λ) will be positive if the plus sign is chosen, which is in accordance with the Abbé condition (Eq. (17)). In Eq. (19), if the minus sign was chosen, then

*f*

_{2}(

*λ*) would have been negative and

*f*

_{1}(

*λ*) positive, which is not in accordance with the Abbé condition. This kind of doublets has a higher LCA than doublets obeying to Abbé conditions and may have LCA more pronounced compared with singlets. Equation (19) to (22) allow for a quicker optimisation of the LCA, when

*bfl*is fixed we have just to find λ

_{1}and λ

_{2}optimizing the LCA. Note that the choice of

*bfl*does not affect the

*LCA** as may be understood from Eq. (6).

### 5.2 Hybrid lens

*d*= 0), simply by summing the refractive and the diffractive profiles [19

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]

*f*given by Eq. (25) corresponds exactly to Eq. (21).

_{dif}*f*and

_{ref}*f*are positive for a converging lens.

_{dif}## 6. Performance

### 6.1 Refractive doublet

_{0}and λ

_{0}’) minimizing the |

*LCA*

_{max}| have been determined with a precision of 1nm for each. For every combination of materials, under the two wavelengths of better achromatization, |

*LCA*

_{max}| is presented followed by radii of curvature of the PC and the PMMA respectively. All those combinations are graphically represented in Fig. 8 : each of the four PCs in combination with all PMMAs is presented. Combining a weakly crown OP (PMMA) with a flint OP (PC) may lead to very different performances. But low LCA may also be achieved with two PMMAs or two PCs. However – as it might be understood from the Abbé condition – this leads to very small radii of curvature. And since |

*LCA**

_{max}| is greater than 1% we no longer have to consider this possibility hereafter.

*LCA**of 0.068 corresponding to a maximal concentration of 2.1 × 10

_{max}^{6}with incoming flux at normal incidence.

### 6.2 Hybrid lens

_{0}and λ'

_{0}) giving the smallest and the equivalent radius of curvature.

## 7. Discussion

*LCA**

_{max}| of the singlet and materials with high

*LCA**

_{max}| of the singlet up to a factor 2 × 10

^{6}while this factor is limited to 5000 × in the case of hybrid lenses. Hybrid lenses have some advantages: they could be manufactured in only one material and have a higher radius of curvature. Moreover studies are still under way to improve hybrid lenses for high concentration systems [17

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.

^{16}.

## 8. Conclusions

*LCA**

_{max}| by about a factor 4.2 compared to the singlets. The maximum geometrical concentration achieved with a hybrid lens corresponds to about 5000 under normal incidence, which lies well under the maximum theoretical concentration of 46,000 under solar angular incidence.

## Appendix A

*2r*and a given

*LCA**around

*f*(λ

_{0}).

## Appendix B

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

*f*is the focal distance,

*s*the lens-object distance and

_{o}*s*the lens image distance. In order to calculate the

_{i}*bfl*, let’s define

### a) Diverging lens first

*f*

_{1}(λ) and a refractive index

*n*

_{1}(λ) – is virtual

*f*

_{2}(λ) and a refractive index

*n*

_{2}(λ):

*s*from the second lens

_{i2}### b) Converging lens first

### c) Expression of the focal distances

*f*

_{1}and

*f*

_{2}depend on the wavelength,

*bfl*depends on the wavelength and thus

*f**also. But we would like to have the same back focal distance for two chosen wavelengths λ

_{1}and λ

_{2}:

*bfl*(λ

_{1}) =

*bfl*(λ

_{2}) =

*bfl*which is equivalent to

*f*

_{1}(λ) and

*f*

_{2}(λ). Considering both lenses as thin lenses the focal distance is given by Eq. (11). Thus

*f*

_{21}

^{2},

*f*

_{21},

*f*

_{21}

^{0}

## Acknowledgements

## References and links

1. | C. Algora, “Very high concentration challenges of III–V multijunction solar cells,” in |

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

3. | 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. |

4. | R. Winston, “Light Collection within the Framework of Geometrical Optics,” J. Opt. Soc. Am. |

5. | 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. |

6. | J. Chaves, |

7. | E. Hecht, |

8. | D. A. Buralli, G. M. Morris, and J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt. |

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

10. | S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. |

11. | J. D. Lytle, “Polymeric Optics,” in |

12. | ASAP |

13. | 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. |

14. | V. Moreno, J. F. Román, and J. R. Salgueiro, “High efficiency diffractive lenses: Deduction of kinoform profile,” Am. J. Phys. |

15. | D. C. O'Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, |

16. | B. H. Kleemann, M. Seeßelberg, and J. Ruoff, “Design concepts for broadband high–efficiency DOEs,” J. Eur. Opt. Soc. Rapid Publ. |

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, |

19. | G. K. Skinner, “Design and imaging performance of achromatic diffractive-refractive x-ray and gamma-ray Fresnel lenses,” Appl. Opt. |

**OCIS Codes**

(160.4760) Materials : Optical properties

(220.1000) Optical design and fabrication : Aberration compensation

(220.1770) Optical design and fabrication : Concentrators

(050.1965) Diffraction and gratings : Diffractive lenses

(080.4298) Geometric optics : Nonimaging optics

**ToC Category:**

Solar Concentrators

**History**

Original Manuscript: March 4, 2011

Revised Manuscript: March 28, 2011

Manuscript Accepted: March 29, 2011

Published: April 8, 2011

**Citation**

Fabian Languy, Karl Fleury, Cédric Lenaerts, Jérôme Loicq, Donat Regaert, Tanguy Thibert, and Serge Habraken, "Flat Fresnel doublets made of PMMA and PC: combining low cost production and very high concentration ratio for CPV," Opt. Express **19**, A280-A294 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-S3-A280

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### References

- 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.
- R. Winston, “Light Collection within the Framework of Geometrical Optics,” J. Opt. Soc. Am. 60(2), 245–247 (1970). [CrossRef]
- 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. Chaves, Introduction to nonimaging optics (CRC Press, 2008), Chap. 1.
- E. Hecht, Optics 4th Ed. (Addison-Wesley, 2002), Chap. 5. [PubMed]
- D. A. Buralli, G. M. Morris, and J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt. 28(5), 976–983 (1989). [CrossRef] [PubMed]
- Fresnel lens brochure of the Fresnel Technologies Inc.: www.fresneltech.com/materials.html , accessed on 02/06/2011.
- 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]
- 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 .
- 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]
- 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]
- 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.
- 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]
- 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.
- 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]

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