OSA's Digital Library

Energy Express

Energy Express

  • Editor: Christian Seassal
  • Vol. 22, Iss. S3 — May. 5, 2014
  • pp: A973–A985
« Show journal navigation

Enhanced light trapping in solar cells with a meta-mirror following generalized Snell’s law

M. Ryyan Khan, Xufeng Wang, Peter Bermel, and Muhammad A. Alam  »View Author Affiliations


Optics Express, Vol. 22, Issue S3, pp. A973-A985 (2014)
http://dx.doi.org/10.1364/OE.22.00A973


View Full Text Article

Acrobat PDF (2788 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

As the performance of photovoltaic cells approaches the Shockley-Queisser limit, appropriate schemes are needed to minimize the losses without compromising the current performance. In this paper we propose a planar absorber-mirror light trapping structure where a conventional mirror is replaced by a meta-mirror with asymmetric light scattering properties. The meta-mirror is tailored to have reflection in asymmetric modes that stay outside the escape cone of the dielectric, hence trapping light with unit probability. Ideally, the meta-mirror can be designed to have such light trapping for any angle of incidence onto the absorber-mirror structure. We illustrate the concept by using a simple gap-plasmon meta-mirror. Even though the response of the mirror is non-ideal with the unwanted scattering modes reducing the light absorption, we observe an order of magnitude enhancement compared to single pass absorption in the absorber. The bandwidth of the enhancement can be matched with the range of wavelengths close to the solar cell absorber band-edge where improved light absorption is required.

© 2014 Optical Society of America

1 Introduction and background

The photovoltaic (PV) devices converts sunlight to electricity in two steps: sunlight is absorbed to create electron-hole pairs within the active layer, and the photo-generated carriers are extracted by the built-in electric field of the device. As solar cell technologies move closer toward the ideal thermodynamic (Shockley-Queisser, S-Q) limit of PV performance [1

1. W. Shockley and H. J. Queisser, “Detailed Balance Limit of Efficiency of p-n Junction Solar Cells,” J. Appl. Phys. 32(3), 510 (1961). [CrossRef]

], the new designs need to be more effective in reducing the remaining loss components without introducing new ones [2

2. X. Wang, M. R. Khan, J. L. Gray, M. A. Alam, and M. S. Lundstrom, “Design of GaAs Solar Cells Operating Close to the Shockley-Queisser Limit,” IEEE J. Photovoltaics 3(2), 737–744 (2013). [CrossRef]

]. Unfortunately, non-ideal material characteristics couple absorption-transport processes in a fundamental way: the finite optical absorption sets the minimum (optical) thickness of the absorber (optical thickness), while finite mobility and radiative/non-radiative recombination set the maximum (electrical) thickness limit of the cell.

The strategies that address the issue of absorption or carrier extraction all rely on decoupling optical and electrical thicknesses of solar cells. For example, solar cell configurations such as radial junction nanowire (NW) PV [3

3. B. M. Kayes, H. A. Atwater, and N. S. Lewis, “Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells,” J. Appl. Phys. 97(11), 114302 (2005). [CrossRef]

,4

4. M. M. Adachi, M. P. Anantram, and K. S. Karim, “Core-shell silicon nanowire solar cells,” Sci. Rep. 3, 1546 (2013).

], and projected electrodes [5

5. M.-H. Hsu, P. Yu, J.-H. Huang, C.-H. Chang, C.-W. Wu, Y.-C. Cheng, and C.-W. Chu, “Balanced carrier transport in organic solar cells employing embedded indium-tin-oxide nanoelectrodes,” Appl. Phys. Lett. 98(7), 073308 (2011). [CrossRef]

,6

6. B. Ray, M. R. Khan, C. Black, and M. A. Alam, “Nanostructured Electrodes for Organic Solar Cells: Analysis and Design Fundamentals,” IEEE. J. Photovoltaics 3, 318–329 (2012).

] focus on decoupling the light absorption from the carrier extraction length by absorbing light vertically, but separating the carrier laterally. These structures provide a shorter and more efficient path for carrier collection. Light trapping (LT) schemes instead focus on collecting the carrier vertically so that electrical path-length is reduced, while reducing reflection and increasing lateral scattering of the light so that optical absorption is enhanced. We can subdivide the LT schemes in two broad categories: (i) nano-structuring the active layer itself, and (ii) redesigning the dielectric environment around it. In the first category, random texturing, NW array, nanocone (NC) structures [7

7. L. Hu and G. Chen, “Analysis of Optical Absorption in Silicon Nanowire Arrays for Photovoltaic Applications,” Nano Lett. 7(11), 3249–3252 (2007). [CrossRef] [PubMed]

9

9. J. Zhu, Z. Yu, G. F. Burkhard, C.-M. Hsu, S. T. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, “Optical Absorption Enhancement in Amorphous Silicon Nanowire and Nanocone Arrays,” Nano Lett. 9(1), 279–282 (2009). [CrossRef] [PubMed]

] reduce reflection of the incoming light and induce scattering for overall improved absorption. Other photonic crystal structures can scatter light inside the dielectric and yield longer light passage in the absorber [10

10. S. Basu Mallick, N. P. Sergeant, M. Agrawal, J.-Y. Lee, and P. Peumans, “Coherent Light Trapping in Thin-Film Photovoltaics,” MRS Bull. 36(06), 453–460 (2011). [CrossRef]

12

12. S. Jeong, M. D. McGehee, and Y. Cui, “All-back-contact ultra-thin silicon nanocone solar cells with 13.7% power conversion efficiency,” Nat Commun 4, 2950 (2013).

]. Also, embedded plasmonic nano-particles [13

13. V. Kochergin, L. Neely, C.-Y. Jao, and H. D. Robinson, “Aluminum plasmonic nanostructures for improved absorption in organic photovoltaic devices,” Appl. Phys. Lett. 98(13), 133305 (2011). [CrossRef]

,14

14. J. Yang, J. You, C.-C. Chen, W.-C. Hsu, H. R. Tan, X. W. Zhang, Z. Hong, and Y. Yang, “Plasmonic Polymer Tandem Solar Cell,” ACS Nano 5(8), 6210–6217 (2011). [CrossRef] [PubMed]

] can create localized high fields which in turn greatly enhance localized absorption. Unfortunately, these strategies also increase the surface area for recombination and the corresponding degradation in charge collection may ultimately negate any gain from light trapping [15

15. X. Wang, M. R. Khan, M. Lundstrom, and P. Bermel, “Performance-limiting factors for GaAs-based single nanowire photovoltaics,” Opt. Express 22(S2), A344–A358 (2014). [CrossRef]

].

In the second category, a separate dielectric or metallic scatterer layer is used to enhance absorption of the incident light inside the un-patterned, planar absorber layer. These approaches include scattering of light from metal particles at the top surface, or generation of surface plasmon polaritons (SPPs) at the back metal grating [16

16. J. N. Munday and H. A. Atwater, “Large Integrated Absorption Enhancement in Plasmonic Solar Cells by Combining Metallic Gratings and Antireflection Coatings,” Nano Lett. 11, 2195–2201 (2010).

,17

17. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]

]. However, at optical frequencies, typical plasmonic materials absorb a fraction of the incident photons, introducing an additional loss mechanism in the device. The patterning could also be done in a separate lossless dielectric to scatter the light going into the absorber. These designs aim at scattering light into higher angles to enhance the light path yielding good absorption [18

18. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107(41), 17491–17496 (2010). [CrossRef] [PubMed]

,19

19. M. A. Green, “Enhanced evanescent mode light trapping in organic solar cells and other low index optoelectronic devices,” Prog. Photovolt. Res. Appl. 19(4), 473–477 (2011). [CrossRef]

]. The key idea here is to couple the normally incident light into guided modes. However, the lossless patterned dielectric scatterers usually have complex designs and could be challenging to implement in practice.

In this paper, we wish to explore the potential of a new approach to dielectric light scattering into lateral guided modes by meta-mirrors. Remarkably, even a nominal design improves light absorption by an order of magnitude. Note that, absorption enhancement limits have been discussed for both geometric [20

20. E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev. 29(2), 300–305 (1982). [CrossRef]

,21

21. E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72(7), 899–907 (1982). [CrossRef]

] and wave optics regime [22

22. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express 18(S3Suppl 3), A366–A380 (2010). [CrossRef] [PubMed]

]. Yablonovitch et al. [20

20. E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev. 29(2), 300–305 (1982). [CrossRef]

,21

21. E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72(7), 899–907 (1982). [CrossRef]

] showed that absorption in a weak absorber can be enhanced up to a factor of 4n2. Our proposed concept suggests that these limits may be exceeded (and the light may be trapped with unit probability) by using a meta-mirror as a back reflector, although the specific mirror design that achieves this perfect absorption is yet to be determined.

Briefly, recent developments in the concepts of meta-surface [23

23. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011). [CrossRef] [PubMed]

29

29. A. Pors, O. Albrektsen, I. P. Radko, and S. I. Bozhevolnyi, “Gap plasmon-based metasurfaces for total control of reflected light,” Sci. Rep. 3, 2155 (2013).

] have shown the possibility of rearranging light propagation in planar structures. In this work, we propose inserting a reflecting meta-mirror at the back of the solar cell to asymmetrically scatter the rays. The asymmetric scattering follows the generalized Snell’s law of reflection [23

23. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011). [CrossRef] [PubMed]

]. In principle, the meta-mirror can be designed to scatter light primarily into a single mode that stays away from the critical angle of the front surface; and thereby ensuring total internal reflection (TIR). As we show in Sec. 2, an idealized meta-mirror that scatters the light into such single mode has the potential to perfectly trap the entire incident light irrespective of the dielectric thickness (i.e., both in geometric or wave optics regime) for any angle of incidence. In practice, exclusive scattering into a single mode may not be possible. Also, a solar cell is a broadband absorber (although absorption close to the band-edge is most critical), and, therefore, wavelength dependence of asymmetric scattering associated the practical design must be explored. Despite these issues, a nominal design discussed in Sec. 3 anticipates an order of magnitude improvement over single pass absorption. We conclude by suggesting that an optimized structure has the potential for considerably higher enhancement of optical light trapping and the bandwidth is always sufficiently large for the scheme to be a practical light trapping scheme for high efficiency solar cells working close to the S-Q limit.

2 Theory and design of meta-mirrors

2.1 Theory of perfect absorption

Fig. 1 (a) The concept of light trapping is shown. A dielectric with refractive index n is placed on top of a meta-mirror which follows the generalized Snell’s law of reflection. The meta-mirror is shown here as a combination of a perfect mirror and a gradient phase adding interface layer. The ideal path for the light ray is shown. (b) The red line shows the law of reflection at the back meta-mirror. The black line represents specular reflection for total internal reflection at the top dielectric-air interface. The shaded region represents the escape cone angles. The blue circles are the incident angles at the meta-mirror after multiple bounces for the example illustrated in (a).
Figure 1(a) shows a light trapping scheme in a dielectric. Assume that the mirror follows generalized Snell’s law of reflection. Light incident from air refracts into the dielectric with some angle θi1 inside the escape cone (see Fig. 1(b)). Thus we have |θi1|<θC (here θC=sin1(1/n) is the critical angle of refraction). Then the plane wave hits the back mirror and reflects back with a larger angle, θr1 as expected from Eq. (3). The mirror is designed under the constraint that, for any |θi1|<θC, the reflected wave assumes an angle θr1>θC. Thus, the light reflected from the mirror will reach the top dielectric-air interface with angle greater than θC—resulting in total internal reflection. The plane wave will travel back to the mirror with new incidence angle of θi2=θr1. Now, the reflected wave will have θr2>θi2, i.e., θr2>θr1>θC. This cycle will continue with the light attaining a higher angle after each bounce off the back mirror. Rays with larger angle of propagation will always stay outside the escape cone and remain trapped inside the dielectric. Note that, after some N bounce cycles, the incident angle on the mirror will be θiNsin1(1δ). In this case, instead of reflecting back into a propagating mode, the light wave will scatter into a surface wave along the dielectric-mirror interface.

The scheme provides means for trapping the incident light indefinitely—a dielectric with the smallest of absorption coefficient will have perfect absorption. Ideally, the structure absorbs 100% of the light, irrespective of how weak the absorption coefficient may be. Thus, the conceptual structure can have an absorption enhancement factor limited only by the inverse of the original optical absorption — for poor absorbers, this breaks the conventional absorption enhancement limit (e.g., the Yablonovitch 4n2 limit [20

20. E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev. 29(2), 300–305 (1982). [CrossRef]

,21

21. E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72(7), 899–907 (1982). [CrossRef]

]). This perfect absorption should not be misunderstood to be unphysical, because detailed balance ensures that at open circuit condition, absorption is counterbalanced by appropriately high radiation [30

30. Y. Luo, R. Zhao, A. I. Fernandez-Dominguez, S. A. Maier, and J. B. Pendry, “Harvesting light with transformation optics,” Sci. China Inf. Sci. 56(12), 1–13 (2013). [CrossRef]

].

Figure 1 shows an example of the path followed by light ray inside the dielectric (n=3.5) of the proposed structure. Here, for the ray tracing calculations, we have assumed δ=sinθC.

2.2 Design rules for meta-mirror

Phases are only defined per modulo 2π, i.e., the added phase of Φ(x) has the same effect as the added phase of Φ(x)+m2π, where m is an integer. Therefore, we can wrap the linear profile of Φ(x) into a periodic triangular form, where the added phase varies linearly from 0 to 2π inside a period. This profile of Φ(x) for the mirror can be created by an extra graded index dielectric on top of a conventional mirror [31

31. S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory,” Opt. Lett. 37(12), 2391–2393 (2012). [CrossRef] [PubMed]

] or by a meta-surface [23

23. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011). [CrossRef] [PubMed]

,27

27. S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D. P. Tsai, “High-Efficiency Broadband Anomalous Reflection by Gradient Meta-Surfaces,” Nano Lett. 12(12), 6223–6229 (2012). [CrossRef] [PubMed]

] [29

29. A. Pors, O. Albrektsen, I. P. Radko, and S. I. Bozhevolnyi, “Gap plasmon-based metasurfaces for total control of reflected light,” Sci. Rep. 3, 2155 (2013).

].

The period Γx of the mirror structure depends on the required slope of Φ(x). Hence Γx will be determined by Δkx=dΦ(x)/dx (or, equivalently by δ=Δkx/k0n). As explained earlier, for the light trapping scheme to work, the value of δ is constrained by θC at a given wavelength λ0=k0/2π. Thus the mirror structure period should be limited by constraints set by θC, or equivalently by the refractive index n of the active layer dielectric and the wavelength λ0. However, as we will show in the following analysis, design of Γx only depends on the wavelength of interest.

While the periodic meta-mirror based on graded index layer explains the basic concepts of asymmetric scattering intuitively, in practice it is easier to design equivalent meta-mirror based on the concept of ‘gap-plasmon’ meta-surface [29

29. A. Pors, O. Albrektsen, I. P. Radko, and S. I. Bozhevolnyi, “Gap plasmon-based metasurfaces for total control of reflected light,” Sci. Rep. 3, 2155 (2013).

], discussed as follows.

3 Gap-plasmon based periodic meta-mirrors

3.1 Properties of designed meta-mirror

Fig. 2 (a) The meta-surface mirror configuration embedded in the dielectric. (b) Scattered Hy field profile for: (top) θi=0,  λ=600nm, (middle) θi=15,λ=600nm, and (bottom) θi=0, λ=700nm. (c) The transfer characteristics (θr vs. θi) are shown as a spectrum of scattered Hy field amplitudes at λ=600nm. (d) The scattered Hy field amplitudes for different θr as a function of λ is shown (at θi=0). Here, the mode marked ( + 1) corresponds to the generalized Snell’s law of reflection.
Figure 2(a) shows a gap-plasmon meta-surface mirror embedded in a dielectric (n=3.5). The various metal strip-sizes in gap-plasmon configuration add gradient phase Φ(x) ranging from 0 to 2π. The details of the meta-mirror design are given in Appendix B. The numerical calculations are done using S4 [32

32. V. Liu and S. Fan, “S4 : A free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183(10), 2233–2244 (2012). [CrossRef]

], which is based on Rigorous Coupled Wave Analysis (RCWA). Metal components have been assumed to be loss-less having refractive index of nmetal=7i. It is also taken as non-dispersive for clear demonstration of the concept. The stability of the results has been checked by testing for higher number of Fourier components. Since RCWA does not automatically yield reliable results for metal inclusions in the structure, we validated the results by using Finite Difference Time Domain (FDTD) simulations in Meep [33

33. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010). [CrossRef]

] as well. We choose period Γx=550nm for the design wavelength λ0=600nm. The structure is uniform along y-direction. And, we only consider Hy-polarization for the incident light. Our design will only focus on one polarization to provide a clear illustration and simple explanation of the design idea. However, it is possible to design a meta-mirror to simultaneously address both polarizations, as explained in Ref [29

29. A. Pors, O. Albrektsen, I. P. Radko, and S. I. Bozhevolnyi, “Gap plasmon-based metasurfaces for total control of reflected light,” Sci. Rep. 3, 2155 (2013).

].

The transfer characteristics (θr vs. θi) of the mirror is presented in Fig. 2(c) as a spectrum of scattered Hy field components. The ( + 1)-diffraction mode (also marked in the figure) corresponds to the generalized Snell’s law of reflection, Eq. (3). The angle of reflection for normal incidence θi=0 can be calculated for the ( + 1)-diffraction mode as follows:
θr(+1)=sin1(δ)=sin1(Δkxnk)=sin1(2π/Γxn2π/λ)=sin1(λ/nΓx).
(5)
For λ=600nm, Γx=550nm, and n=3.5, we find θr(+1)=18.16 which matches with the numerical simulations.

Although the ( + 1)-diffraction mode represents the desired θiθr relationship of the ‘generalized Snell’s law of reflection’, Fig. 2(c) shows that there are other scattered modes as well. This set of parasitic couplings can only contribute to some specific modes as a consequence of the periodicity of the mirror. For example, for θi=0, the scattered angle is expected to assume a value θr=sin1(mΔkx/nk)=sin1(mλ/nΓx) for any structure with period Γx, where the order of the modes are given by m=0,±1,±2,, etc. However, a linear phase profile Φ(x) created by the sub-cells can ensure exclusive coupling into m=+1 mode. Although only m=+1 is desired here, some amplitude contribution in other modes cannot be avoided because of the non-linear phase profile of ‘sub-cells’ in one period (details about sub-cells are in Appendix B), and finite mutual coupling between adjacent sub-cells. This is especially prominent at higher angles of incidence θi on the meta-mirror. For example, as presented in Fig. 2(b), the scattered wave for normal incidence (top profile) at λ=600nm show minimum distortion in the plane wave. However, as the angle of incidence increases (e.g., θi=15), we start to see perturbation in the scattered field profile (middle profile) due to contribution of the unwanted modes. The existence of these parasitic modes degrades light trapping, as they allow the light to exit through the escape cone.

Note that the θiθr relationship shown in Fig. 2(c) predominantly follows the generalized Snell’s law of reflection for low θi (i.e., |θi|<20). However, larger θi reverts to Snell’s Law, i.e., specular reflection (with small contribution from other diffraction modes). For this specific structure, no surface wave is observed for the high incidence angles—this is a deviation from the ideal characteristics. A meta-mirror using a ‘xylophone’-like gap-plasmon configuration, as in [27

27. S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D. P. Tsai, “High-Efficiency Broadband Anomalous Reflection by Gradient Meta-Surfaces,” Nano Lett. 12(12), 6223–6229 (2012). [CrossRef] [PubMed]

], is likely to show much improved characteristics.

Also Eq. (5) predicts that θr(+1) would increase with λ. This is seen in Fig. 2(d). The reflection angle θr is presented as a function of λ (at  θi=0). Note that for λ<Γx, from Eq. (5) we have θr(+1)<θC. As explained earlier in Sec. 2.1, the situation mentioned above breaks the condition for our light trapping scheme. The higher wavelengths gradually allow more contributions from modes other than m=+1 — this is reflected in the perturbed scattered field shown in Fig. 2(b) (bottom profile). Also, note that the larger contribution for the θr=0 mode is observed for λ>800nm. This means that the meta-mirror is deviating from the generalized Snell’s law and showing more specular reflection outside the aforementioned wavelength range. This indicates that light trapping may be degraded for λ>800nm for the structure built using this meta-mirror. We will address the issue further in the following discussions.

3.2 Light-trapping in planar dielectric with back meta-mirror

For the study of absorption, let us choose the absorber thickness L=500nm (which would be typical for thin film solar cell) on top of the mirror. To reduce reflections off the top surface, a quarter wave matched (for λ0=600nm) layer is used as an ARC. We choose refractive index n=3.5 and absorption coefficient α=105/m. Both n and α are assumed to be non-dispersive in the wavelength range of interest (200nm<λ<1000nm) for better comparison of the light trapping properties. Note that the dielectric and metal in the meta-surface structure are also assumed to be non-dispersive.

Fig. 3 (a) The light trapping scheme for thin film solar cell is shown using the meta-surface mirror. (b) The black line represents the absorption in the structure with a conventional mirror at the back. The blue line in (b) shows the absorption in the absorber layer for structures in (a).
The blue line in Fig. 3(b) represents the absorptance in the structure constructed using the meta-mirror configuration at the back (as shown in Fig. 3(a)). The black line shows the absorption in the same cell structure only with a conventional mirror at the back. As expected, the absorption in the planar cell is 2αL=0.1 (black line). For the meta-mirror configuration in Fig. 3(b), there is minimal change in absorptance for λ<Γx=550nm as the design constraint is not satisfied in this wavelength range. Also, note that, the absorption starts to go down for λ>800nm as predicted from the previously studied mirror properties. It is clear that there is an average absorption enhancement of a factor of ~5.75, compared to the conventional back mirror structure, for 550nm<λ<800nm. Obviously, the band of wavelengths where the absorption enhancement occurs can be shifted to arbitrary wavelengths by tailoring the meta-mirrors at appropriate periodicities. The study considered here only focuses on normally incident light. This reflects the operating conditions for solar cells with dual-axis tracking that could experience direct normal incidence throughout the day. Of course, a meta-mirror designed using Γxλ0/2 can ideally trap light for all angles of incidence (see Sec. 2.2 for details).

Although the solar cell is a broadband absorber, the finite bandwidth over which a meta-mirror sustains significant gain is not a concern. For a practical solar cell (e.g., Si), the absorption is strong for high energy photons (lower wavelengths) away from the band-edge. Thus, almost all the high energy photons are absorbed within practical cell thickness—the light trapping schemes are irrelevant. However, the absorption coefficients close to the band-edge are considerably lower and it is within this narrow bandwidth where light-trapping is relevant and important. So long as the meta-surface mirrors are designed to overlap this weak absorption regime, the improvement in light trapping can be significant. Also, the proposed scheme is expected to be fairly insensitive to small changes in the optical real refractive index for practical PV materials.

Finally, we realize that the meta-mirror structure proposed here consists of sub-wavelength components that may be difficult to fabricate. Despite the difficulty, however, recent works presented in Refs [25

25. X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband Light Bending with Plasmonic Nanoantennas,” Science 335(6067), 427 (2012). [CrossRef] [PubMed]

,27

27. S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D. P. Tsai, “High-Efficiency Broadband Anomalous Reflection by Gradient Meta-Surfaces,” Nano Lett. 12(12), 6223–6229 (2012). [CrossRef] [PubMed]

,29

29. A. Pors, O. Albrektsen, I. P. Radko, and S. I. Bozhevolnyi, “Gap plasmon-based metasurfaces for total control of reflected light,” Sci. Rep. 3, 2155 (2013).

]. show that such structures are feasible and can be demonstrated experimentally.

4. Conclusions

In this paper, we have proposed a planar structure for light trapping which utilizes the asymmetric scattering of the meta-mirror at the back. The meta-mirror only reflects light into a single mode following generalized Snell’s law of reflection. The ideal configuration anticipates perfect absorption for all angles of incidence onto the planar absorber. Practical implementation based on gap-plasmon meta-mirrors may inherently allow scattering light into multiple diffraction modes. The undesired modes cause the light to leak out of the structures. Although perfect absorption is no longer expected, an optimized design can lower coupling to unwanted modes and improve overall absorption considerably. Indeed, a nominal design presented here shows an absorption gain of a factor of ~11.5 compared to single pass absorption in the dielectric (i.e., factor of ~5.75 compared to a back mirrored planar structure). Obviously, the gap-plasmon meta-mirror is just one example of a meta-surface being capable of scattering light asymmetrically. It remains an important open question to see if other implementations might provide improved absorption approaching the ideal limit.

Appendices

Appendix A. Graded index mirror

Fig. 4 (a) The graded index mirror configuration embedded in the dielectric. (b) The transfer characteristics (θr vs. θi) are shown as a spectrum of scattered Hy field amplitudes at λ=600nm. (c) The scattered Hy field amplitudes for different θr as a function of λ is shown (at θi=0). Here, the mode marked ( + 1) corresponds to the generalized Snell’s law of reflection.
For a more qualitative understanding, let us consider a hypothetical mirror which can demonstrate the properties discussed in Sec.2.1. The simplest of examples for this purpose would be a graded refractive index layer on top of a conventional mirror (see Fig. 4(a)). Each segment has the same thickness, but different refractive index, and therefore, provides varying optical paths for the incoming wave. The phase added to the light wave by the segments will follow the index gradient; therefore, as the wave reflects from the graded index layer, it will accumulate a spatially gradient phase Φ(x). The graded index layer has periodic and linearly varying indices ranging from nL to nH. For a large number of sub-segments, if the phase added by the dielectric with index nL is ΦL then the phase added by segment nH would be ΦL+2π.

Let us assume that every period consists of NS segments and the refractive index changes by Δn between consecutive segments. The phase added at the end of each period is 2π, or equivalently, the extra optical path length would be λ0 (design wavelength). Thus, for the graded index layer of thickness tGI,
(Δn NS)×tGI×2=λ0.
(6)
Here, Δn NS corresponds to the refractive index variation over one period. The factor of two is due to the fact that the light travels through the layer twice as there is a mirror at the back. Finally, we can write:
Δn=λ02tGI×1NS,
(7)
nGI=nL+βΔn.
(8)
The refractive indices of the graded index layer sub-segments are given by nGI in the above equation; where, β=0,1,2,, (NS1).

Figure 4(a) shows the graded index mirror embedded in a dielectric (n=3.5). We introduce a matching layer on top of the graded index layer to minimize specular reflection. We choose period Γx=550nm for the design wavelength λ0=600nm (as mentioned in Sec. 2.2, the light trapping scheme requires that Γx<λ0). The structure is uniform along y-direction. The design parameters are summarized in Table 1.

Table 1. Design parameters for the graded index mirror structure.

table-icon
View This Table
We only consider Hy-polarization for the incident light.

The transfer characteristics (θr vs. θi) of the mirror is presented in Fig. 4(b). The θiθr relationship predominantly follows the generalized Snell’s law of reflection for low θi (i.e., |θi|<20). However, larger θi reverts to Snell’s Law, i.e., specular reflection (with small contribution from other diffraction modes). No surface wave is observed for the high incidence angles. These characteristics are similar to those of the meta-mirror discussed in Sec.3.1. Note that we have θr(+1)=18.16 for this structure as well.

Figure 4(c) shows the θrλ properties (at θi=0). We see that θr(+1)<θC for λ<Γx which does not support the light trapping constraint for our proposed scheme. Also, observe that a more prominent mode emerges at θr=0 for λ>800nm—similar to the meta-mirror structure.

Fig. 5 (a) The light trapping scheme for thin film solar cell is shown using the graded index mirror. The ARC layer on top of the dielectric is used to minimize reflection of the incident light. (b) The black line shows the absorption in the structure with a conventional mirror at the back. The blue line show the absorption in the absorber layer for structures in (a).
The absorption enhancement using the graded index mirror is presented here for absorber thickness of L=500nm, and absorption coefficient of α=105/m. The study is similar to the one done for the meta-mirror configuration (in Sec.3.1) for 200nm<λ<1000nm. The absorption in the structure shown in Fig. 5(a) is represented by the blue line in Fig. 5(b). The black line is the absorption in a structure having a conventional mirror at the back. As expected, the enhancement of absorptance is minimal for λ<Γx=550nm (also see Sec.2.2). However, absorption peaks are observed in this λ-range. This can be associated with the resonances created by the series of dielectric layers.

Appendix B. Design of gap-plasmon periodic meta-mirror

Fig. 6 (a) Sub-cell of the meta-mirror periodic structure. The various lengths have been defined here. Varying metal strip widths  W are used to form the supercell (single period) of the meta-mirror. (b) The phase added by super-cells as a function of W at normal incidence (blue solid line). 10 components/sub-cells are used in a supercell (red circles). These components have W values as listed in the table (bottom). (c) The 10 sub-cells create a linearly varying (relative) phase profile for normal incidence (θi=0), which is shown for two periods as red circles. However, for θi=40, the linearity of the phase profile is disrupted (see blue crosses).
The meta-mirror has a periodic structure. Each period, i.e., the supercell, is constructed from 10 sub-cells, see Fig. 6(a). Each sub-cell is intended to add increasing phases to get Φ(x) of 0 to 2π within each period. We design the meta-mirror as a ‘gap-plasmon’ meta-surface structure [27

27. S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D. P. Tsai, “High-Efficiency Broadband Anomalous Reflection by Gradient Meta-Surfaces,” Nano Lett. 12(12), 6223–6229 (2012). [CrossRef] [PubMed]

,29

29. A. Pors, O. Albrektsen, I. P. Radko, and S. I. Bozhevolnyi, “Gap plasmon-based metasurfaces for total control of reflected light,” Sci. Rep. 3, 2155 (2013).

]. A dielectric material (refractive index 1.38) is placed between a metal strip of width W and a metal plate. The thicknesses of these layers are marked in Fig. 6(a). Varying W will add different phases to the scattered (reflected) wave. Of-course this response has to be investigated inside the dielectric (active layer of the device) with n=3.5. The metal for the numerical studies is assumed to be loss-less and non-dispersive with refractive index nmetal=7i (i.e., relative permittivity of −49). The phases added to the reflected waves under such constraints are presented in Fig. 6(b) as a function of W.

The design is done under normal incidence condition (blue solid line in Fig. 6(b)). Now, 10 sub-cells that add β×36 phases (here β=0,1,,9) need to be used to construct the supercell. The period of the sub-cell is Γxsub=55nm; thus period of the meta-mirror is Γx=10×Γxsub=550nm. The widths (W) of individual stripes are tabulated in Fig. 6. Note that, as the change in phase added Φ spans from 0 to ~300 for W range of 0 to 55nm, the sub-cells do not contribute to the overall phase equally, see Fig. 6(b). This non-uniform contribution to the overall phase, however, has only a minor effect on the meta-mirror performance for normally incident light. These simulations were done using S4 [32

32. V. Liu and S. Fan, “S4 : A free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183(10), 2233–2244 (2012). [CrossRef]

] as explained in Sec.3.1.

The supercell constructed from the selected sub-cells (as tabulated in bottom of Fig. 6) is expected to show a spatially linear phase profile. It is presented in Fig. 6(c) as a variation in the relative phases. It is clear that, for normal incidence (θi=0) onto the meta-mirror, a linear phase profile is preserved (the red circles). However, the linear phase profile is not observed for oblique incidence (blue crosses, θi=40). The disturbance of the linear phase profile for oblique incidence causes the θiθr characteristics to stray from the generalized Snell’s law of reflection.

Acknowledgments

The authors thank Prof. V. Shalaev and Prof. A. Boltasseva for valuable discussions. This material is based upon work supported as part of the Center for Re-Defining Photovoltaic Efficiency Through Molecule Scale Control, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001085. The computational resources for this work were provided by the Network of Computational Nanotechnology under NSF Award EEC-0228390. This was also supported by the Bay Area PV Consortium, a Department of Energy project with Prime Award number DE-EE0004946.

References and links

1.

W. Shockley and H. J. Queisser, “Detailed Balance Limit of Efficiency of p-n Junction Solar Cells,” J. Appl. Phys. 32(3), 510 (1961). [CrossRef]

2.

X. Wang, M. R. Khan, J. L. Gray, M. A. Alam, and M. S. Lundstrom, “Design of GaAs Solar Cells Operating Close to the Shockley-Queisser Limit,” IEEE J. Photovoltaics 3(2), 737–744 (2013). [CrossRef]

3.

B. M. Kayes, H. A. Atwater, and N. S. Lewis, “Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells,” J. Appl. Phys. 97(11), 114302 (2005). [CrossRef]

4.

M. M. Adachi, M. P. Anantram, and K. S. Karim, “Core-shell silicon nanowire solar cells,” Sci. Rep. 3, 1546 (2013).

5.

M.-H. Hsu, P. Yu, J.-H. Huang, C.-H. Chang, C.-W. Wu, Y.-C. Cheng, and C.-W. Chu, “Balanced carrier transport in organic solar cells employing embedded indium-tin-oxide nanoelectrodes,” Appl. Phys. Lett. 98(7), 073308 (2011). [CrossRef]

6.

B. Ray, M. R. Khan, C. Black, and M. A. Alam, “Nanostructured Electrodes for Organic Solar Cells: Analysis and Design Fundamentals,” IEEE. J. Photovoltaics 3, 318–329 (2012).

7.

L. Hu and G. Chen, “Analysis of Optical Absorption in Silicon Nanowire Arrays for Photovoltaic Applications,” Nano Lett. 7(11), 3249–3252 (2007). [CrossRef] [PubMed]

8.

Q. G. Du, C. H. Kam, H. V. Demir, H. Y. Yu, and X. W. Sun, “Broadband absorption enhancement in randomly positioned silicon nanowire arrays for solar cell applications,” Opt. Lett. 36(10), 1884–1886 (2011). [CrossRef] [PubMed]

9.

J. Zhu, Z. Yu, G. F. Burkhard, C.-M. Hsu, S. T. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, “Optical Absorption Enhancement in Amorphous Silicon Nanowire and Nanocone Arrays,” Nano Lett. 9(1), 279–282 (2009). [CrossRef] [PubMed]

10.

S. Basu Mallick, N. P. Sergeant, M. Agrawal, J.-Y. Lee, and P. Peumans, “Coherent Light Trapping in Thin-Film Photovoltaics,” MRS Bull. 36(06), 453–460 (2011). [CrossRef]

11.

S. B. Mallick, M. Agrawal, and P. Peumans, “Optimal light trapping in ultra-thin photonic crystal crystalline silicon solar cells,” Opt. Express 18(6), 5691–5706 (2010). [CrossRef] [PubMed]

12.

S. Jeong, M. D. McGehee, and Y. Cui, “All-back-contact ultra-thin silicon nanocone solar cells with 13.7% power conversion efficiency,” Nat Commun 4, 2950 (2013).

13.

V. Kochergin, L. Neely, C.-Y. Jao, and H. D. Robinson, “Aluminum plasmonic nanostructures for improved absorption in organic photovoltaic devices,” Appl. Phys. Lett. 98(13), 133305 (2011). [CrossRef]

14.

J. Yang, J. You, C.-C. Chen, W.-C. Hsu, H. R. Tan, X. W. Zhang, Z. Hong, and Y. Yang, “Plasmonic Polymer Tandem Solar Cell,” ACS Nano 5(8), 6210–6217 (2011). [CrossRef] [PubMed]

15.

X. Wang, M. R. Khan, M. Lundstrom, and P. Bermel, “Performance-limiting factors for GaAs-based single nanowire photovoltaics,” Opt. Express 22(S2), A344–A358 (2014). [CrossRef]

16.

J. N. Munday and H. A. Atwater, “Large Integrated Absorption Enhancement in Plasmonic Solar Cells by Combining Metallic Gratings and Antireflection Coatings,” Nano Lett. 11, 2195–2201 (2010).

17.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]

18.

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107(41), 17491–17496 (2010). [CrossRef] [PubMed]

19.

M. A. Green, “Enhanced evanescent mode light trapping in organic solar cells and other low index optoelectronic devices,” Prog. Photovolt. Res. Appl. 19(4), 473–477 (2011). [CrossRef]

20.

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev. 29(2), 300–305 (1982). [CrossRef]

21.

E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72(7), 899–907 (1982). [CrossRef]

22.

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express 18(S3Suppl 3), A366–A380 (2010). [CrossRef] [PubMed]

23.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011). [CrossRef] [PubMed]

24.

N. Yu, P. Genevet, F. Aieta, M. A. Kats, R. Blanchard, G. Aoust, J.-P. Tetienne, Z. Gaburro, and F. Capasso, “Flat Optics: Controlling Wavefronts With Optical Antenna Metasurfaces,” IEEE J. Sel. Top. Quantum Electron. 19(3), 4700423 (2013). [CrossRef]

25.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband Light Bending with Plasmonic Nanoantennas,” Science 335(6067), 427 (2012). [CrossRef] [PubMed]

26.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar Photonics with Metasurfaces,” Science 339(6125), 1232009 (2013). [CrossRef] [PubMed]

27.

S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D. P. Tsai, “High-Efficiency Broadband Anomalous Reflection by Gradient Meta-Surfaces,” Nano Lett. 12(12), 6223–6229 (2012). [CrossRef] [PubMed]

28.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012). [CrossRef] [PubMed]

29.

A. Pors, O. Albrektsen, I. P. Radko, and S. I. Bozhevolnyi, “Gap plasmon-based metasurfaces for total control of reflected light,” Sci. Rep. 3, 2155 (2013).

30.

Y. Luo, R. Zhao, A. I. Fernandez-Dominguez, S. A. Maier, and J. B. Pendry, “Harvesting light with transformation optics,” Sci. China Inf. Sci. 56(12), 1–13 (2013). [CrossRef]

31.

S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory,” Opt. Lett. 37(12), 2391–2393 (2012). [CrossRef] [PubMed]

32.

V. Liu and S. Fan, “S4 : A free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183(10), 2233–2244 (2012). [CrossRef]

33.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010). [CrossRef]

OCIS Codes
(040.5350) Detectors : Photovoltaic
(160.3918) Materials : Metamaterials

ToC Category:
Light Trapping for Photovoltaics

History
Original Manuscript: February 18, 2014
Revised Manuscript: March 28, 2014
Manuscript Accepted: March 31, 2014
Published: April 23, 2014

Citation
M. Ryyan Khan, Xufeng Wang, Peter Bermel, and Muhammad A. Alam, "Enhanced light trapping in solar cells with a meta-mirror following generalized Snell’s law," Opt. Express 22, A973-A985 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-S3-A973


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. W. Shockley and H. J. Queisser, “Detailed Balance Limit of Efficiency of p-n Junction Solar Cells,” J. Appl. Phys.32(3), 510 (1961). [CrossRef]
  2. X. Wang, M. R. Khan, J. L. Gray, M. A. Alam, and M. S. Lundstrom, “Design of GaAs Solar Cells Operating Close to the Shockley-Queisser Limit,” IEEE J. Photovoltaics3(2), 737–744 (2013). [CrossRef]
  3. B. M. Kayes, H. A. Atwater, and N. S. Lewis, “Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells,” J. Appl. Phys.97(11), 114302 (2005). [CrossRef]
  4. M. M. Adachi, M. P. Anantram, and K. S. Karim, “Core-shell silicon nanowire solar cells,” Sci. Rep.3, 1546 (2013).
  5. M.-H. Hsu, P. Yu, J.-H. Huang, C.-H. Chang, C.-W. Wu, Y.-C. Cheng, and C.-W. Chu, “Balanced carrier transport in organic solar cells employing embedded indium-tin-oxide nanoelectrodes,” Appl. Phys. Lett.98(7), 073308 (2011). [CrossRef]
  6. B. Ray, M. R. Khan, C. Black, and M. A. Alam, “Nanostructured Electrodes for Organic Solar Cells: Analysis and Design Fundamentals,” IEEE. J. Photovoltaics3, 318–329 (2012).
  7. L. Hu and G. Chen, “Analysis of Optical Absorption in Silicon Nanowire Arrays for Photovoltaic Applications,” Nano Lett.7(11), 3249–3252 (2007). [CrossRef] [PubMed]
  8. Q. G. Du, C. H. Kam, H. V. Demir, H. Y. Yu, and X. W. Sun, “Broadband absorption enhancement in randomly positioned silicon nanowire arrays for solar cell applications,” Opt. Lett.36(10), 1884–1886 (2011). [CrossRef] [PubMed]
  9. J. Zhu, Z. Yu, G. F. Burkhard, C.-M. Hsu, S. T. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, “Optical Absorption Enhancement in Amorphous Silicon Nanowire and Nanocone Arrays,” Nano Lett.9(1), 279–282 (2009). [CrossRef] [PubMed]
  10. S. Basu Mallick, N. P. Sergeant, M. Agrawal, J.-Y. Lee, and P. Peumans, “Coherent Light Trapping in Thin-Film Photovoltaics,” MRS Bull.36(06), 453–460 (2011). [CrossRef]
  11. S. B. Mallick, M. Agrawal, and P. Peumans, “Optimal light trapping in ultra-thin photonic crystal crystalline silicon solar cells,” Opt. Express18(6), 5691–5706 (2010). [CrossRef] [PubMed]
  12. S. Jeong, M. D. McGehee, and Y. Cui, “All-back-contact ultra-thin silicon nanocone solar cells with 13.7% power conversion efficiency,” Nat Commun4, 2950 (2013).
  13. V. Kochergin, L. Neely, C.-Y. Jao, and H. D. Robinson, “Aluminum plasmonic nanostructures for improved absorption in organic photovoltaic devices,” Appl. Phys. Lett.98(13), 133305 (2011). [CrossRef]
  14. J. Yang, J. You, C.-C. Chen, W.-C. Hsu, H. R. Tan, X. W. Zhang, Z. Hong, and Y. Yang, “Plasmonic Polymer Tandem Solar Cell,” ACS Nano5(8), 6210–6217 (2011). [CrossRef] [PubMed]
  15. X. Wang, M. R. Khan, M. Lundstrom, and P. Bermel, “Performance-limiting factors for GaAs-based single nanowire photovoltaics,” Opt. Express22(S2), A344–A358 (2014). [CrossRef]
  16. J. N. Munday and H. A. Atwater, “Large Integrated Absorption Enhancement in Plasmonic Solar Cells by Combining Metallic Gratings and Antireflection Coatings,” Nano Lett.11, 2195–2201 (2010).
  17. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9(3), 205–213 (2010). [CrossRef] [PubMed]
  18. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A.107(41), 17491–17496 (2010). [CrossRef] [PubMed]
  19. M. A. Green, “Enhanced evanescent mode light trapping in organic solar cells and other low index optoelectronic devices,” Prog. Photovolt. Res. Appl.19(4), 473–477 (2011). [CrossRef]
  20. E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev.29(2), 300–305 (1982). [CrossRef]
  21. E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am.72(7), 899–907 (1982). [CrossRef]
  22. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express18(S3Suppl 3), A366–A380 (2010). [CrossRef] [PubMed]
  23. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science334(6054), 333–337 (2011). [CrossRef] [PubMed]
  24. N. Yu, P. Genevet, F. Aieta, M. A. Kats, R. Blanchard, G. Aoust, J.-P. Tetienne, Z. Gaburro, and F. Capasso, “Flat Optics: Controlling Wavefronts With Optical Antenna Metasurfaces,” IEEE J. Sel. Top. Quantum Electron.19(3), 4700423 (2013). [CrossRef]
  25. X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband Light Bending with Plasmonic Nanoantennas,” Science335(6067), 427 (2012). [CrossRef] [PubMed]
  26. A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar Photonics with Metasurfaces,” Science339(6125), 1232009 (2013). [CrossRef] [PubMed]
  27. S. Sun, K.-Y. Yang, C.-M. Wang, T.-K. Juan, W. T. Chen, C. Y. Liao, Q. He, S. Xiao, W.-T. Kung, G.-Y. Guo, L. Zhou, and D. P. Tsai, “High-Efficiency Broadband Anomalous Reflection by Gradient Meta-Surfaces,” Nano Lett.12(12), 6223–6229 (2012). [CrossRef] [PubMed]
  28. S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater.11(5), 426–431 (2012). [CrossRef] [PubMed]
  29. A. Pors, O. Albrektsen, I. P. Radko, and S. I. Bozhevolnyi, “Gap plasmon-based metasurfaces for total control of reflected light,” Sci. Rep.3, 2155 (2013).
  30. Y. Luo, R. Zhao, A. I. Fernandez-Dominguez, S. A. Maier, and J. B. Pendry, “Harvesting light with transformation optics,” Sci. China Inf. Sci.56(12), 1–13 (2013). [CrossRef]
  31. S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory,” Opt. Lett.37(12), 2391–2393 (2012). [CrossRef] [PubMed]
  32. V. Liu and S. Fan, “S4 : A free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun.183(10), 2233–2244 (2012). [CrossRef]
  33. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun.181(3), 687–702 (2010). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited