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Energy Express

  • Editor: Bernard Kippelen
  • Vol. 19, Iss. S3 — May. 9, 2011
  • pp: A280–A294
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Flat Fresnel doublets made of PMMA and PC: combining low cost production and very high concentration ratio for CPV

Fabian Languy, Karl Fleury, Cédric Lenaerts, Jérôme Loicq, Donat Regaert, Tanguy Thibert, and Serge Habraken  »View Author Affiliations


Optics Express, Vol. 19, Issue S3, pp. A280-A294 (2011)
http://dx.doi.org/10.1364/OE.19.00A280


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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.

© 2011 OSA

1. Introduction

Nowadays, lots of optical systems like cameras, telescopes, microscopes etc. use an achromatic doublet of glasses, which delivers good image quality but is quite expensive, bulky and heavy. Even for a single glass lens the cost, bulk and weight might be a problem. This is why Fresnel lenses are often used, especially when built in optical plastic, e. g. headlights, solar concentrators, projectors, traffic lights, etc. So, in order to combine small chromatic aberrations with low-cost production, we suggest using achromatic Fresnel doublets for concentrated photovoltaics (CPV). The reduction of the chromatic aberration allows for a higher concentration ratio and a higher efficiency of multijunction photovoltaic cells [1

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.

]. Even if most optical systems work in the visible range, we extended our study to a typical zone of interest of wavelengths for solar concentration: from 380 to 1600 nm. Typically, out of this range the external quantum efficiency of a triple junction cell drops to about 40% [2

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

] and the direct solar flux is also low [3

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

Considering a concentrator with a collector surface A’ and a receiver surface A. The ratio A’/A corresponds to the geometrical concentration factor Cgeo. If Φ’ is the flux collected and Φ the flux absorbed, then Φ’/Φ refers to the optical efficiency ηopt. Finally, the optical concentration factor Copt is given by Eq. (1).

Copt=ηoptCgeo=Φ'ΦA'A.
(1)

For a collector of rotational symmetry, the upper limit of concentration is achieved with an optical efficiency of 100% and for a concentrator where both collector and absorber are immersed in the air is given by Eq. (2) where 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]

].

Coptmax=sin2θ.
(2)

On Earth, for two degrees of concentration, the upper limit of solar concentration is about 46,000 due to a semi-acceptance angle of the sun of ~960” (i.e. 0.267°) [5

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]

]. For a polychromatic source, this concentration may be achieved with reflective surfaces (like the compound parabolic concentrator [6

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

]). But systems suffering from chromatic aberration will not be able to achieve such high concentration ratio. This is the case for lenses since their focal distances depend on the wavelength: f = f(λ).

Considering two wavelengths λA and λB, the linear chromatic aberration (LCA) corresponds to the difference of the focal distances:

LCA=f(λB)f(λA).
(3)

The way the focal distance of an ideal (with only chromatic aberration) thin refractive lens having a front and back radii of curvature R1 and R2 changes is given by Eq. (4) [7

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

] while for a diffractive lens it is given by Eq. (5) [8

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]

].

1fref(λ)(1R11R2)(n(λ)1),
(4)
fdif(λ)=λAλf(λA).
(5)

Hereafter, we arbitrarily chose to take λ0 as a reference: λ0 is such that the minimum LCA and the maximum LCA – achieved with λm and λM respectively – are equal in absolute value. The system is thus optimised to decrease the maximum longitudinal chromatic aberration in absolute value |LCAmax| (see Fig. 1a
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).
).

For easier comparison, the notation with an asterisk as exponent is introduced, corresponding to a normalisation with λ0. 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 f(λ0), the geometrical concentration allowing for the collection of the whole flux is given by Eq. (8).

LCAref*(λ)=f(λ)f(λ0)f(λ0)=n(λ0)n(λ)n(λ)1,
(6)
LCAdif*(λ)=f(λ)f(λ0)f(λ0)=(λ0λ1),
(7)
Cgeomax=(1LCA*LCA*)2.
(8)

Imaging optics tends to decrease the |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 λm (red part) in Fig. 1a is more important than the amount of light collected with λM (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
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).
while Fig. 2b shows the gain of geometric concentration factor achieved by moving the detector from f0) 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.

However lens designers do not always have the choice of detector size and position. For a given |LCA max| and geometrical concentration factor, where should λ0 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 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
Fig. 3 Ideal position of f0) (a) and optical efficiency (b) as functions of the LCA* and the geometrical concentration.
.

A very important point of this section is that the optical concentration factor could be even more limited by the LCA rather than by the acceptance angle.

3. Dispersion curves

PMMA is probably the most common optical plastic (OP) used in solar concentration thanks to its high transmittance and low dispersion curve. Another common OP used for Fresnel lenses is PC, with similar spectral transmission and high impact resistance [9

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

]. Thanks to their high difference of refractive index (about 0.1 at 550 nm), PC and PMMA are good candidates for refractive doublets. Some data about OPs may be found in Kasarova 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]

], Handbook of optical materials [11

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

] and some ray tracing software like ASAPTM [12

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

]. But depending on the supplier and the injection parameters, OPs properties change. This is why we made our own ellipsometric measurements on several samples. Eight OPs were taken, three from the literature and we determined the dispersion curve of five OPs coming from our suppliers. In total five PCs and five PMMAs listed in Table 1

Table 1. Information about the PMMAs and PCs

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were studied.

n(λ)=1+i=1mBiλ2λ2Ci,
(9)
n(λ)=A1+A2λ2+A3λ2+A4λ4+A5λ6+   .
(10)

The dispersion coefficients are presented in Table 2

Table 2. Dispersion coefficients some PCs and PMMAs

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. Note that those coefficients must be used with the wavelength expressed in microns, both Laurent’s and Sellmeier’s equations were used for each sample, but only the one giving the smallest is presented. The curves of the refractive indexes are presented in Fig. 4 (a) and (b)
Fig. 4 Dispersion curves of PMMAs (a) and PCs (b).
for PMMAs and PCs respectively. The variation of refractive index from a sample to another might be explained by the fact that industrial process is not constant and injection parameters might be different for every injection leading to variation in the refractive index [13

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]

]. Nevertheless, the curve “PC #1 (old)” has an abnormal behaviour: above 1200 nm the refractive index drops too rapidly compared to any other PCs. This might be simply explained by the fact that the dispersion coefficients were retrieved by Kasarova 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

For an ideal (with only chromatic aberration) thin refractive lens in paraxial condition, the focal distance f may be approximated by Eq. (4) which has been rewritten in Eq. (11):
f(λ)RoCeqn(λ)1,
(11)
with RoCeq the equivalent radius of curvature of the lens [7

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

]. Following the definition of the asterisk as an exponent

RoCeq*=RoCeqf(λ0)=(R1R2R2R1)1f(λ0)=(n(λ)1)×100%.
(12)

Table 3

Table 3. Data for singlets

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hereunder gives the central wavelength (λ0) and compares the |LCA*max| of each material as well as the equivalent radius of curvature. In the case of a singlet, |LCAmax| is achieved for the two extreme wavelengths: λm = 380 nm and λM = 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 vd to measure the dispersion in the visible region.
vd=(nd1)/(nFnC),
(13)
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 v:

v=(n990nm1)/(n380nmn1600nm).
(14)

4.2 Diffractive lens

Diffractive lenses may be designed in different ways. For a single wavelength, only the kinoform (Fig. 6
Fig. 6 Schematic representation of a kinoform diffractive lens.
) may have one focus and a theoretical diffraction efficiency of 100% [14

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]

]. The ideal kinoform looks like a Fresnel lens, composed of a multitude of zones with a constant thickness 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: h=λ0/(n(λ0)1) [8

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]

].

While illuminating the lens with another wavelength, the focal distance will be modified following Eq. (5), independently of the refractive index . The Abbé number corresponding to this dispersion is thus also independent of the refractive index [15

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.

] and is given by Eq. (15).

vd,diff=dFC=3.4518.
(15)

Similarly, v=0.8115. The Abbé number is negative and has lower absolute value than any refractive material, meaning that for a converging lens, long wavelengths will converge shorter than short wavelengths and that diffractive lenses are more dispersive which is unfavourable for small 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 λ0 = 550nm, the diffraction efficiency at the first order remains above 90% only between 472 and 663 nm.

η1(λ)=sinc2{1λ0λn(λ)1n(λ0)1}.
(16)

Those problems of high dispersion and low diffraction efficiency explain why diffractive lenses are never used alone in systems needing low LCA. Nevertheless the lack of diffraction efficiency may be drastically diminished using multilayers [16

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

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.

] without affecting the focal distance.

5. Achromatization

In order to decrease the LCA, the combination of two lenses might prove to be very powerful (LCA*max < 1%) since the lens designer may choose two wavelengths (λ1 and λ2) 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

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

] in combination with the formula of the effective focal length (Eq. (18)).

f1v1+f2v2=0,
(17)
feff1=f11+f21.
(18)

We have thus two possibilities: combining a diverging and a converging refractive lens or a converging refractive lens with a converging diffractive lens. Which of those combinations gives the best results? Before answering this question, we point out that the Eq. (17) is not fully useful for someone wishing to achromatize its system at a given focal distance. Indeed, if the formula f 1, dv 1, d + f 2, dv 2, d = 0 is used, f(λC) and f(λF) will be the same but only the focal distance of λd is known directly. Moreover nothing proves that having the same focal distance for λF and λC gives the smallest LCA*max. Therefore, we suggest using a more straightforward formula giving directly the focal distance of two chosen wavelengths λ1 and λ2.

5.1 Refractive doublet

It can be shown (Appendix B) that for a given back focal length (bfl) – i.e. the distance from the back of the second lens to the focal point (see Fig. 7
Fig. 7 Schematic representation of a doublet.
) – the focal distance of the second lens is given by Eq. (19) and (20).
f2(λ1)=B±B24AC2A,
(19)
with

A=(bfl+d)(n2(λ1)1n2(λ2)1)(1n1(λ2)1n1(λ1)1),B=bfl{d(n2(λ1)1n2(λ2)1n1(λ2)1n1(λ1)1)+(bfl+d)(n1(λ2)1n1(λ1)1n2(λ1)1n2(λ2)11)},C=dbfl2(1n1(λ2)1n1(λ1)1).
(20)

In this equation, if 0 < dbfl, 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

f1(λ1)=d(bfl)(bfl+d)f2(λ1)(bfl)f2(λ1).
(21)

Those two equations ensure that bfl(λ1)=bfl(λ2)=bfl where bfl, λ1 and λ2 are three parameters. For the other wavelengths in the refractive regime,

bflref(λ)=f2(λ)(df1(λ))df1(λ)f2(λ).
(22)

If the first material – of refractive index n1(λ) has a dispersion higher than n2(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 λ1 and λ2 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

A hybrid lens results in the combination of a refractive and a diffractive lens. In this case, there is no need for a combination of two different optical materials since the focal length of a diffractive lens is independent of the refractive index. A hybrid lens could thus be manufactured in only one piece. Moreover the whole profile could be engraved on one surface (corresponding thus to 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]

]. The focal distance of the refractive lens corresponds to one mathematical solution of a quadratic equation
fref(λ1)=BB24AC2A
(23)
with

A=(bfl+d)(n(λ1)1n(λ2)1)(1λ2λ1),B=bfl{d(λ2λ1n(λ1)1n(λ2)1)+(bfl+d)(λ2λ1n(λ1)1n(λ2)11)},C=dbfl2(1λ2λ1).
(24)

The expression of the focal distance of the diffractive lens fdif given by Eq. (25) corresponds exactly to Eq. (21).

fdif(λ1)=d(bfl)(bfl+d)fref(λ1)(bfl)fref(λ1).
(25)

In the case of the hybrid lens, both fref and fdif are positive for a converging lens.

6. Performance

6.1 Refractive doublet

In Table 4

Table 4. Combination of PCs and PMMAs for Achromatic Doublets

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the two wavelengths (λ0 and λ0’) 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
Fig. 8 Doublets with PC #1 (a), PC #2 (b), PC #3 (c) and PC #4 (d).
: 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.

Best result of achromatization is achieved with a combination of PMMA #3 and PC #4, leading to a LCA*max of 0.068 corresponding to a maximal concentration of 2.1 × 106 with incoming flux at normal incidence.

6.2 Hybrid lens

Since the focal distance of the hybrid lens depends only on the dispersion curve of the refractive part and on the two wavelengths of achromatization, all OPs have been considered in a single plot (Fig. 9
Fig. 9 Evolution of the focal distance for hybrid lenses in OP considering the curve minimizing the LCA.
).

Table 5

Table 5. Data for achromatized hybrid lenses

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presents the two wavelengths of achromatization (λ0 and λ'0) giving the smallest and the equivalent radius of curvature.

7. Discussion

This section shows the performance of achromatization for both hybrid and doublet lenses. Table 6

Table 6. Major data about singlets, doublets and hybrid lenses

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collects most important data to be compared.

Though v is in direct relation with |LCA*max| of the singlet and materials with high v are more suitable to be used in a hybrid lens, there is absolutely no direct relation between Abbé numbers and good achromatization with a refractive doublet. But it clearly appears that even if hybrid lenses have a smaller LCA than singlets, a refractive doublet is even more powerful decreasing the |LCA*max| of the singlet up to a factor 2 × 106 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.

] but at this time, lens designers would probably prefer refractive doublets to avoid the lack of diffraction efficiency and to get a higher achromatization performance.

Note that the maximum concentration is limited by the diffraction limit. For a circular lens with a radius of 10 cm and a focal of 20 cm, the concentration is limited to about 1.5 × 1016.

8. Conclusions

Abbé formula is useful to achromatize at two wavelengths and to see that is preferable to choose two materials with Abbé numbers strongly different from each other in order to get high radii of curvature. But firstly the Abbé does not give any information about the quality of the achromatization. Secondly it does not allow choosing the focal distances of the two achromatized wavelengths. And thirdly this formula does not take into account the distance between the two lenses. Equations (19) to (22) are thus more useful for lens designers.

Not only is the Abbé number not sufficient but also the dispersion curves in the literature are limited to a spectral bandwidth and are dependent on the type of plastic, production conditions etc.

Hybrid lenses allow for fast converging systems since both lenses are converging. Unfortunately they suffer from a lack of diffraction efficiency at the focus due to spurious orders but they reduce the |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.

As for refractive doublets, they allow – at normal incidence – a theoretical concentration ratio up to 50 times higher than 46,000. The linear chromatic aberration of such doublets can thus be regarded as not limiting the concentration ration. Doublets are thus good candidates to achieved very high concentration at low cost.

Appendix A

Considering a lens of diameter 2r and a given LCA* around f0).

Referring to Fig. 10
Fig. 10 Schematic representation of a converging lens with LCA.
, the best position of the detector is given by the intersection of the lines given by Eq. (26) and (27).
y1(z)=rf(λ0)(1+LCA*)z+r,
(26)
y2(z)=rf(λ0)(1LCA*)zr.
(27)
Thus

1f(λ0)(1LCA*)z1=1f(λ0)(1+LCA*)z+1,
(28)
z(1+LCA*+1LCA*1(LCA*)2)=2f(λ0),
(29)
z=f(λ0)(1LCA*2).
(30)

Quod erat demonstrandum.

Appendix B

The notation of this appendix follows Hecht’s books [7

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

] with the thin lens formula given by
1f=1so+1si,
(31)
where f is the focal distance, so the lens-object distance and si the lens image distance. In order to calculate the bfl, let’s definef^.

f^=bfl+d.
(32)

For a doublet, it doesn’t matter if the first lens is the converging one or the diverging one as will be proved in the following subsections.

a) Diverging lens first

If the light coming from infinity strikes the diverging lens first, the image of the first lens – having a focal distance f 1(λ) and a refractive index n 1(λ) – is virtual

1f1=1si1                   withf1<0.
(33)

The image of the first lens becomes the object of the second lens (the converging one) having a focal distance f 2(λ) and a refractive index n 2(λ):

so2=|f1|+d.
(34)

The final image is thus situated at a distance si2 from the second lens

1si2=1f21so2=1f21df1=df1f2f2(df1).
(35)

Thus, from the first lens, the final image is situated at

f^=si2+d=f2(df1)df1f2+d.
(36)

b) Converging lens first

Equation (33) to (36) become now
1f1=1si1                   withf1>0,
(37)
so2=f1d,
(38)
1si2=1f21so2=1f2+1f1d,
(39)
f^=si2+d=f2(df1)df1f2+d,
(40)
which is exactly the same expression as Eq. (36).

c) Expression of the focal distances

Since 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 λ1 and λ2: bfl1) = bfl2) = bfl which is equivalent to

f^(λ1)=f^(λ2)=f^.
(41)

To ensure Eq. (41), let’s determine the focal distances f 1(λ) and f 2(λ). Considering both lenses as thin lenses the focal distance is given by Eq. (11). Thus

f1(λ1)=f1(λ2)(n1(λ2)1n1(λ1)1)=αf1(λ2),
(42)
f2(λ2)=f2(λ1)(n2(λ1)1n2(λ2)1)=βf2(λ1).
(43)

In order to facilitate the reading, we define
fijfi(λj)
(44)
from Eq. (36)
f11=f^f21bfldf21bfl=αf^βf21bfldβf21bfl,
(45)
effectuating the cross product
(f^f21bfld)(βf21bfl)=(f21bfl)α(f^βf21bfld),
(46)
and separating the terms in f 21 2, f 21, f 21 0

f212{f^βf^αβ}+f21{bfl(αd+f^αβf^βd)}+{dbfl2αbfl2d}=0.
(47)

Thus
f21B±B24AC2A,
(48)
with

{A=(bfl+d)β(1α),B=bfl(d(αβ)+(bfl+d)(αβ1))C=dbfl2(1α).,
(49)

Finally, replacing α and β with Eq. (42) and (43), Eq. (49) turns into Eq. (24).

Quod erat demonstrandum.

Acknowledgements

The authors are grateful to the Ministry of the Walloon Region (DGTRE) for the financial support accorded in the framework of the Solwatt program (Convention Solwatt 850552). Thank you too to Amos and Optim Test Centre, societies from Liege for the diamond turning and injection molding of plastic Fresnel doublets. A finally, thank you to all interns of Hololab for their indirect collaboration.

References and links

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.

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. 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]

6.

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

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]

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. 29(11), 1481–1490 (2007). [CrossRef]

11.

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

12.

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

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]

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]

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

  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.
  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. 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]
  6. J. Chaves, Introduction to nonimaging optics (CRC Press, 2008), Chap. 1.
  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]
  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. 29(11), 1481–1490 (2007). [CrossRef]
  11. J. D. Lytle, “Polymeric Optics,” in Handbook of Optics, 3rd Edition, Vol. IV, M. Bass, ed. (McGraw-Hill, 2009), Chap. 3.
  12. ASAPTM optical design software of Breault Research Organization, http://www.breault.com .
  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]
  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]

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