## Cooperative electromagnetic interactions between nanoparticles for solar energy harvesting |

Optics Express, Vol. 22, Issue S3, pp. A577-A588 (2014)

http://dx.doi.org/10.1364/OE.22.00A577

Acrobat PDF (1205 KB)

### Abstract

The cooperative electromagnetic interactions between discrete resonators have been widely used to modify the optical properties of metamaterials. Here we propose a general approach for engineering these interactions both in the dipolar approximation and for any higher-order description. Finally we apply this strategy to design broadband absorbers in the visible range from simple n-ary arrays of metallic nanoparticles.

© 2014 Optical Society of America

## 1. Introduction

1. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. **9**, 205–2013 (2010). [CrossRef] [PubMed]

2. 10. P. Bermel, M. Ghebrebrhan, W. Chan, Y. X. Yeng, M. Araghchini, R. Hamam, C. H. Marton, K. F. Jensen, M. Soljacic, J. D. Joannopoulos, S. G. Johnson, and I. Celanovic, “Design and global optimization of high-efficiency thermophotovoltaic systems,” Opt. Express **18**, A314–A334 (2010). [CrossRef] [PubMed]

3. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**, 288–290 (1986). [CrossRef] [PubMed]

4. A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole in hyperbolic media,” Phys. Rev. A **84**, 023807 (2011). [CrossRef]

5. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. **100**, 207402 (2008). [CrossRef] [PubMed]

6. K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. **2**, 517 (2011). [CrossRef] [PubMed]

7. S. D. Jenkins and J. Ruostekoski, “Metamaterial transparency induced by cooperative electromagnetic interactions,” Phys. Rev. Lett. **111**, 147401 (2013). [CrossRef] [PubMed]

8. V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. **104**, 223901 (2010). [CrossRef] [PubMed]

9. P. Ben-Abdallah, R. Messina, S.-A. Biehs, M. Tschikin, K. Joulain, and C. Henkel, “Heat superdiffusion in plasmonic nanostructure networks,” Phys. Rev. Lett. **111**, 174301 (2013). [CrossRef] [PubMed]

10. R. Messina, M. Tschikin, S.-A. Biehs, and P. Ben-Abdallah, “Fluctuation-electrodynamic theory and dynamics of heat transfer in systems of multiple dipoles,” Phys. Rev. B **88**, 104307 (2013). [CrossRef]

## 2. Scattering by nanoparticle networks in the dipolar approximation

**p**

*(*

_{m;A}*A*=

*E*,

*H*) (the higher orders contributions are discussed in the next paragraph). The local electromagnetic field

**A**

*(*

^{ext}**r**

*) at the dipoles location*

_{m}**r**

*results from the superposition of external incident field, the field generated by the others dipoles and the auto-induced field which comes from the interactions with the interfaces. Therefore it takes the form [11*

_{m}11. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. **186**, 705 (1973). [CrossRef]

12. B. T. Draine and P. J. Flateau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A. **25**, 2693 (2008). [CrossRef]

13. A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B **82**, 045404 (2010). [CrossRef]

14. M. S. Tomas, “Green function for multilayers: light scattering in planar cavities,” Phys. Rev. A **51**, 2545 (1995). [CrossRef] [PubMed]

**A**at the position

**r**

*given a*

_{m}**B**-dipole located in

**r**

*. Δ𝔾*

_{n}*defined as*

^{AB}**r̂**

*≡*

_{mn}**r̂**

*/*

_{mn}*r*.

_{mn}**r**

*denotes here the vector linking the center of dipoles m and n, while*

_{mn}*r*= |

_{ij}**r**

*|,*

_{ij}*k*is the wavector, 1 the unit dyadic tensor and

*χ*represents either the vacuum permittivity

_{A}*ε*

_{0}or the vacuum permeability

*μ*

_{0}and

*α⃡*is the free polarizability tensor of

_{i,A}*m*object under the action of field

^{th}**A**. By inserting the external contribution (1) of local field into relation (23) we get the following system which relates all dipole moments Here, where we have introduced the regularized Green tensor

*n*arbitrary dipoles of free electric and magnetic polarizability

*α⃡*

_{m;A}_{=}

*distributed in a unit cell we have, according to the periodicity, the supplementary relations for the incident fields*

_{E,H}**p**

_{j}_{β};

*=*

_{A}**p̃**

*(*

_{β;A}exp*i*

**k**

_{//}.

**r**

*). Here*

_{jβ}**r**

*is the position vector of the*

_{jβ}*β*dipole inside the unit cell

^{th}*j*of lattice and

**k**

_{//}is the parallel component of wavector.

15. P. P. Ewald, “Die berechnung optischer und elektrostatischer gitterpotentiale,” Annalen der Physik , **369**, 253–287 (1921). [CrossRef]

16. F. Capolino, D. Wilton, and W. Johnson, “Efficient computation of the 2-D Green’s function for 1-D periodic structures using the Ewald method,” IEEE Trans. on Antennas and Propaga. **53**, 9 (2005). [CrossRef]

17. E. Castanie, R. Vincent, R. Pierrat, and R. Carminati, “Absorption by an optical dipole antenna in a structured environment,” Int. J. Opt. **2012**, 452047 (2012). [CrossRef]

18. P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett. **107**, 114301 (2011). [CrossRef] [PubMed]

*ω*inside the

*m*resonator is given by the rate of doing work by the electric and mgnetic fields inside the resonator volume

^{th}*V*Here

_{m}**A**denotes either the local electric or magnetic field

**E**and

**H**while

**j**

*and*

_{E}**j**

*are the corresponding local current density. In the dipolar approximation*

_{H}**j**

*= −*

_{m;A}*iω*

**p**

_{m;A}*δ*(

**r**−

**r**

*), expression (12) can be recasted into the discrete form By inverting (1) after having replaced the dipole moments by their expression with respect to*

_{i}**A**

*and explicitely calculate the power dissipated in each object under an external lighting.*

_{inc}*R*the polarizability is straightforwardly derived from the Mie scattering theory [20

20. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (Wiley Science, New York, 1998). [CrossRef]

21. A. B. Evlyukhin, C. Reinhardt, U. Zywietz, and B. N. Chichkov, “Collective resonances in metal nanoparticle arrays with dipole-quadrupole interactions,” Phys. Rev. B **85**, 245411, (2012). [CrossRef]

*n*, are immersed inside a medium of index

_{m}*n*, we have

_{h}*α⃡*=

_{A}*α*1 with Here

_{A}*k*

_{0}is the wavevector inside vacuum and

*ρ*=

_{h}*k*

_{0}

*n*

_{h}*R*and

*ρ*=

_{m}*k*

_{0}

*n*

_{m}*R*,

*n*being the refractive index of resonator. According to Eqs. (13), (16) and (17) it follows that the power dissipated in each particle can be expressed both in term of absorption cross-sections and of incident external field

_{m}## 3. Generalization of scattering problem beyond the dipolar approximation

*n*can be expressed in term of ingoing (−) and outgoing (+) vector spherical wave functions (which form a complete basis) where we have adopted the usual convention for the multipolar index (

_{h}*m*,

*n*) which are replaced by a single index

*p*=

*n*(

*n*+ 1) +

*m*and where

*q*set the polarization state (i.e.

*q*= 1 for

*TE*waves and

*q*= 2 for TM waves).

*e*

^{−iωt}convention) with the source term

*pq*) defined by for the magnetic contributions and by for the electric ones where

*δ*as

*n*and highlighetd by an external electromagnetic field. By definition, this field can be decomposed on the complete basis of

_{h}*p′*= (

*n*, −

*m*)) on the incident field. Then using the fact that

**n**) surrounding the particle. It follows by applying the Lorentz relation with the field

**A**

*that Note that*

_{inc}*I*[

_{ψ}**A**] is the action of the distribution on the test function

*ψ*[22]. Then using the orthogonality relations (26) and according to (24) Then, using the matrix

**T**which relates the vectors

**A**

*of components of incident field to the vector*

_{inc}**A**

*of diffracted field we have Now, interactions between distinct particles dispersed inside a multilayer can be studied using a generalized form of the translation matrix as introduced by Stout et al. [23*

_{diff}23. B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. **49**, 2129–2152 (2002). [CrossRef]

## 4. Broadband absorber design

*d*and compare their absorption spectra with that ones of isolated particles and of homogeneous metallic film. All lattices are immersed in a transparent material of refractive index

*n*= 1.5 and are maintained at a distance

_{h}*h*= 100

*nm*from the surface. The results plotted in Fig. 2 clearly show that the resonance peaks in single particle lattices are essentialy centered at the resonance frequency of free particles. On the other hand the absorption spectrum of nanoparticles lattices is much broder and does not simply consist in a superposition of single particle spectra. Moreover, we see that the cooperative interactions allow increasing the absorption even in diluted lattices where the filling factor

*f*is smaller than 3%. Finally, the comparison of the overall absorption of nanoparticle lattices with that of simple metallic films with a thickness defined, using the effective medium theory, from the nanoparticle filling factors points out the prime importance of cooperative effects to magnify the absorption level. In binary lattices, new configurationnal resonances add up to the resonances of single lattices and naturally enlarge the absorption spectrum.

6. K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. **2**, 517 (2011). [CrossRef] [PubMed]

25. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional ligh absorber,” Appl. Phys. Lett. **95**, 041106 (2009). [CrossRef]

26. A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano. Lett. **10**, 2574–2579 (2010). [CrossRef] [PubMed]

27. N. P. Sergeant, O. Pincon, M. Agrawal, and P. Peumans, “Design of wide-angle solar selective absorbers using aperiodic metal-dielectric stacks,” Opt. Express **17**, 22800–22812 (2009). [CrossRef]

*𝒞*of a two dimensional paving with a certain thickness (see Fig. 3). In the unit cell of a lattice we consider a set of

*n*vectors

**r**

*and*

_{i}*n*positive reals

*R*that represent the location of particles center and the radius of particles, respectively. To avoid the particle interpenetration these vectors must satisfy to the supplemental constraint |

_{i}**r**

*−*

_{i}**r**

*|>*

_{j}*R*+

_{i}*R*.

_{j}29. T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. **109**, 127701 (2012). [CrossRef] [PubMed]

30. J. Drevillon and P. Ben-Abdallah, “Ab initio design of coherent thermal sources,” J. Appl. Phys. **102**, 114305 (2007). [CrossRef]

*λ*;

_{min}*λ*] where we want to increase the absorption. The monochromatic absorption

_{max}*A*(

*λ*) at a given wavelength is simply given by the the sum of power dissipated inside the particles of the unit cell normalized by the incident flux

*ϕ*(

_{inc}*λ*) on its surface

*𝒮*that is The GA consists in maximizing the fitness function of structures (i.e.

*Ā*→

*max*). To do so, we select 90% of the highest fitness as future parents for the next generation of selecting process. Those parents are linear crossed and the new ’child’ generation is completed by new individual structures (randomly generated) to keep the same total number of lattices for any generation. To avoid the convergence toward local extrema, every m (typically 10) generations, we introduce also some mutations that is to say random perturbations with a probability of about 5% on the value of parameters we are optimizing. The results presented in Fig. 4 for superposed binary Au-Ag lattices (with the radius

*r*= 77

_{Au}*nm*and

*r*= 39

_{Ag}*nm*, the separation distances from the surface

*h*= 120

_{Au}*nm*and

*h*= 242

_{Ag}*nm*, the lattice constants

*d*=

_{Au}*d*= 200

_{Ag}*nm*and the off-centring

*e*= 56

_{x}*nm*and

*e*= 10

_{y}*nm*) exhibit a broad absorption band in the visible range. By taking into account the multipolar interactions until the second order (i.e. quadrupolar interactions) we see that the level of aborption becomes close to one. The comparaison of these results with full electromagnetic simulations based on the finite element method shows that the higher order multipole moments do not contribute significantly to the overall absorption. The absorption enhancement can be understood by examinating the electromagnetic cooperative effects inside the structure. These effects are highlighted in Fig. 5 at two different wavelengths by plotting the local losses inside the gold (resp. silver) nanoparticles within the optimized binary lattice in presence or without silver (resp. gold) nanoparticles. At

*λ*= 550

*nm*, that is to say, at the resonance of gold particles (which corresponds to the region where we observe in Fig. 4 an important bump in the absorption spectrum when it is calculated in the dipolar approximation) we see that the presence of silver nanoparticles enhance by 20% the losses inside the gold particles. Similarly, at

*λ*= 650

*nm*, the gold particles enhance by a factor of 60% the dissipation inside the silver particles. However, because of the weakness of intrinsic losses inside isolated Ag particles this cooperative effect is not sufficiently important to increase the overall absorption of the structure. At low wavelength we have checked (not plotted in Fig. 5) that the high absorption levels as shown in Fig. 4 results from cooperative effects between the gold particles themselves. The silver particles do not play any role in the exaltation mechanism.

*nm*, that the discrepancy between the optimal structure and the perturbed ones, given by the mean square error

## 5. Conclusion

## Acknowledgments

## References and links

1. | H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. |

2. | 10. P. Bermel, M. Ghebrebrhan, W. Chan, Y. X. Yeng, M. Araghchini, R. Hamam, C. H. Marton, K. F. Jensen, M. Soljacic, J. D. Joannopoulos, S. G. Johnson, and I. Celanovic, “Design and global optimization of high-efficiency thermophotovoltaic systems,” Opt. Express |

3. | A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

4. | A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole in hyperbolic media,” Phys. Rev. A |

5. | N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. |

6. | K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. |

7. | S. D. Jenkins and J. Ruostekoski, “Metamaterial transparency induced by cooperative electromagnetic interactions,” Phys. Rev. Lett. |

8. | V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. |

9. | P. Ben-Abdallah, R. Messina, S.-A. Biehs, M. Tschikin, K. Joulain, and C. Henkel, “Heat superdiffusion in plasmonic nanostructure networks,” Phys. Rev. Lett. |

10. | R. Messina, M. Tschikin, S.-A. Biehs, and P. Ben-Abdallah, “Fluctuation-electrodynamic theory and dynamics of heat transfer in systems of multiple dipoles,” Phys. Rev. B |

11. | E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. |

12. | B. T. Draine and P. J. Flateau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A. |

13. | A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B |

14. | M. S. Tomas, “Green function for multilayers: light scattering in planar cavities,” Phys. Rev. A |

15. | P. P. Ewald, “Die berechnung optischer und elektrostatischer gitterpotentiale,” Annalen der Physik , |

16. | F. Capolino, D. Wilton, and W. Johnson, “Efficient computation of the 2-D Green’s function for 1-D periodic structures using the Ewald method,” IEEE Trans. on Antennas and Propaga. |

17. | E. Castanie, R. Vincent, R. Pierrat, and R. Carminati, “Absorption by an optical dipole antenna in a structured environment,” Int. J. Opt. |

18. | P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett. |

19. | J. D. Jackson, |

20. | C. F. Bohren and D. R. Huffman, |

21. | A. B. Evlyukhin, C. Reinhardt, U. Zywietz, and B. N. Chichkov, “Collective resonances in metal nanoparticle arrays with dipole-quadrupole interactions,” Phys. Rev. B |

22. | L. Schwartz, Théorie des distributions, Hermann (1951). |

23. | B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. |

24. | E. D. Palik, |

25. | E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional ligh absorber,” Appl. Phys. Lett. |

26. | A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano. Lett. |

27. | N. P. Sergeant, O. Pincon, M. Agrawal, and P. Peumans, “Design of wide-angle solar selective absorbers using aperiodic metal-dielectric stacks,” Opt. Express |

28. | J. H. Holland, |

29. | T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. |

30. | J. Drevillon and P. Ben-Abdallah, “Ab initio design of coherent thermal sources,” J. Appl. Phys. |

31. | L. Landau, E. Lifchitz, and L. Pitaevskii, |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(160.4760) Materials : Optical properties

(220.0220) Optical design and fabrication : Optical design and fabrication

(300.1030) Spectroscopy : Absorption

(300.2140) Spectroscopy : Emission

(350.6050) Other areas of optics : Solar energy

(160.3918) Materials : Metamaterials

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Light Trapping for Photovoltaics

**History**

Original Manuscript: January 14, 2014

Revised Manuscript: February 26, 2014

Manuscript Accepted: February 28, 2014

Published: March 12, 2014

**Citation**

Mathieu Langlais, Jean-Paul Hugonin, Mondher Besbes, and Philippe Ben-Abdallah, "Cooperative electromagnetic interactions between nanoparticles for solar energy harvesting," Opt. Express **22**, A577-A588 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-S3-A577

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### References

- H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9, 205–2013 (2010). [CrossRef] [PubMed]
- 10. P. Bermel, M. Ghebrebrhan, W. Chan, Y. X. Yeng, M. Araghchini, R. Hamam, C. H. Marton, K. F. Jensen, M. Soljacic, J. D. Joannopoulos, S. G. Johnson, and I. Celanovic, “Design and global optimization of high-efficiency thermophotovoltaic systems,” Opt. Express18, A314–A334 (2010). [CrossRef] [PubMed]
- A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett.11, 288–290 (1986). [CrossRef] [PubMed]
- A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole in hyperbolic media,” Phys. Rev. A84, 023807 (2011). [CrossRef]
- N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett.100, 207402 (2008). [CrossRef] [PubMed]
- K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun.2, 517 (2011). [CrossRef] [PubMed]
- S. D. Jenkins and J. Ruostekoski, “Metamaterial transparency induced by cooperative electromagnetic interactions,” Phys. Rev. Lett.111, 147401 (2013). [CrossRef] [PubMed]
- V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett.104, 223901 (2010). [CrossRef] [PubMed]
- P. Ben-Abdallah, R. Messina, S.-A. Biehs, M. Tschikin, K. Joulain, and C. Henkel, “Heat superdiffusion in plasmonic nanostructure networks,” Phys. Rev. Lett.111, 174301 (2013). [CrossRef] [PubMed]
- R. Messina, M. Tschikin, S.-A. Biehs, and P. Ben-Abdallah, “Fluctuation-electrodynamic theory and dynamics of heat transfer in systems of multiple dipoles,” Phys. Rev. B88, 104307 (2013). [CrossRef]
- E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J.186, 705 (1973). [CrossRef]
- B. T. Draine and P. J. Flateau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A.25, 2693 (2008). [CrossRef]
- A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B82, 045404 (2010). [CrossRef]
- M. S. Tomas, “Green function for multilayers: light scattering in planar cavities,” Phys. Rev. A51, 2545 (1995). [CrossRef] [PubMed]
- P. P. Ewald, “Die berechnung optischer und elektrostatischer gitterpotentiale,” Annalen der Physik, 369, 253–287 (1921). [CrossRef]
- F. Capolino, D. Wilton, and W. Johnson, “Efficient computation of the 2-D Green’s function for 1-D periodic structures using the Ewald method,” IEEE Trans. on Antennas and Propaga.53, 9 (2005). [CrossRef]
- E. Castanie, R. Vincent, R. Pierrat, and R. Carminati, “Absorption by an optical dipole antenna in a structured environment,” Int. J. Opt.2012, 452047 (2012). [CrossRef]
- P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett.107, 114301 (2011). [CrossRef] [PubMed]
- J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley, 1999).
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Science, New York, 1998). [CrossRef]
- A. B. Evlyukhin, C. Reinhardt, U. Zywietz, and B. N. Chichkov, “Collective resonances in metal nanoparticle arrays with dipole-quadrupole interactions,” Phys. Rev. B85, 245411, (2012). [CrossRef]
- L. Schwartz, Théorie des distributions, Hermann (1951).
- B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt.49, 2129–2152 (2002). [CrossRef]
- E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, New York, 1998).
- E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional ligh absorber,” Appl. Phys. Lett.95, 041106 (2009). [CrossRef]
- A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano. Lett.10, 2574–2579 (2010). [CrossRef] [PubMed]
- N. P. Sergeant, O. Pincon, M. Agrawal, and P. Peumans, “Design of wide-angle solar selective absorbers using aperiodic metal-dielectric stacks,” Opt. Express17, 22800–22812 (2009). [CrossRef]
- J. H. Holland, Adaptation in Natural and Artificial Systems (MIT Press/Bradford Books Edition, Cambridge, MA, 1992).
- T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett.109, 127701 (2012). [CrossRef] [PubMed]
- J. Drevillon and P. Ben-Abdallah, “Ab initio design of coherent thermal sources,” J. Appl. Phys.102, 114305 (2007). [CrossRef]
- L. Landau, E. Lifchitz, and L. Pitaevskii, Electromagnetics of Continuous Media(Pergamon, Oxford, 1984).

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