## Optimization of broadband optical response of multilayer nanospheres |

Optics Express, Vol. 20, Issue 16, pp. 18494-18504 (2012)

http://dx.doi.org/10.1364/OE.20.018494

Acrobat PDF (1062 KB)

### Abstract

We propose an optimization-based theoretical approach to tailor the optical response of silver/silica multilayer nanospheres over the visible spectrum. We show that the structure that provides the largest cross-section per volume/mass, averaged over a wide frequency range, is the silver coated silica sphere. We also show how properly chosen mixture of several species of different nanospheres can have an even larger minimal cross-section per volume/mass over the entire visible spectrum.

© 2012 OSA

## 1. Introduction

1. Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. **104**, 207402 (2010). [CrossRef] [PubMed]

3. C. Noguez, “Surface plasmons on metal nanoparticles: the influence of shape and physical environment,” J. Phys. Chem. C **111**, 3806–3819 (2007). [CrossRef]

4. S. Alyones, C. Bruce, and A. Buin, “Numerical methods for solving the problem of electromagnetic scattering by a thin finite conducting wire,” IEEE Trans. Antennas Propag. **55**, 1856–1861 (2007). [CrossRef]

5. C. W. Bruce and S. Alyones, “Extinction efficiencies for metallic fibers in the infrared,” Appl. Opt. **48**, 5095–5098 (2009). [CrossRef] [PubMed]

6. P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: Applications in biological imaging and biomedicine,” J. Phys. Chem. B **110**, 7238–7248 (2006). [CrossRef] [PubMed]

8. C. E. Román-Velázquez and C. Noguez, “Designing the plasmonic response of shell nanoparticles: Spectral representation,” J. Chem. Phys. **134**, 044116 (2011). [CrossRef] [PubMed]

9. S. Oldenburg, R. Averitt, S. Westcott, and N. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. **288**, 243 – 247 (1998). [CrossRef]

13. R. Bardhan, S. Mukherjee, N. A. Mirin, S. D. Levit, P. Nordlander, and N. J. Halas, “Nanosphere-in-a-nanoshell: A simple nanomatryushka,” J. Phys. Chem. C **114**, 7378–7383 (2010). [CrossRef]

14. R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A **75**, 053801 (2007). [CrossRef]

17. Z. Ruan and S. Fan, “Design of subwavelength superscattering nanospheres,” Appl. Phys. lett. **98**, 043101 (2011). [CrossRef]

## 2. Calculation of optical response via transfer matrix method

**E**

*= ∇ ×*

_{TE}**r**

*ϕ*. For TM polarization, the magnetic fields can be written as

_{TE}**H**

*= ∇ ×*

_{TM}**r**

*ϕ*. The scalar potential

_{TM}*ϕ*and

_{TE}*ϕ*satisfy the scalar Helmholtz equation ∇

_{TM}^{2}

*ϕ*+

*k*

^{2}

*ϕ*= 0 where

*k*

^{2}=

*ω*

^{2}

*ε*(

**r**). Due to the spherical symmetry,

*ϕ*can be decomposed into a discrete set of spherical modes:

*l*= 0, 1, 2,... and

*m*= −

*l*,...,

*l*. Since

*ε*(

**r**) is a constant

*ε*inside the

_{i}*i*shell,

^{th}*R*(

_{l}*r*) is a linear combination of the first and second kind spherical Bessel functions within the individual shells: The coefficients (

*A*,

_{i}*B*) of adjacent shells are linked by the transfer matrix of the interface: The matrix element is determined by the boundary condition satisfied by

_{i}*R*(

_{l}*r*), which comes from the continuity of the tangent components of

**E**and

**H**across the boundary. For TE polarization,

*rR*(

_{l}*r*) and (

*rR*(

_{l}*r*))′ are continuous across the boundary. By writing the continuity conditions in matrix form, we get:

*rR*(

_{l}*r*) and (

*ε*

^{−1}

*rR*(

_{l}*r*))′ are continuous across the boundary. By writing the continuity conditions in matrix form, we get: The transfer matrix of the whole system can be calculated by cascading the transfer matrices of individual interfaces.

*A*

_{1}= 1 and

*B*

_{1}= 0. So the coefficients of Bessel functions in the surrounding medium are directly given by the transfer matrix element,

*A*

_{n}_{+1}=

*M*

_{11}and

*B*

_{n}_{+1}=

*M*

_{21}. Within the surrounding medium, it is convenient to write the radical function as a linear combination of the spherical Hankel functions: Taking the convention that the fields vary in time as

*e*

^{−iωt},

## 3. Optical response of silver/silica bilayer nanospheres

*ε*= 2.1. There are two configurations of silver/silica bilayer nanoparticle depending on the core material.

*l*= 1 surface plasmon mode at the silver/silica boundary. The peak wavelength only varies slightly when the inner and outer radius change. For instance, consider a silica coated silver sphere suspended in air. Fixing the outer radius at 50nm, the peak wavelength varies from 410nm to 415nm when the inner radius varies from 5nm to 45nm. Fixing the aspect ratio

*R*

_{1}/

*R*

_{2}at 0.8, the peak wavelength varies from 390nm to 480nm when the outer radius varies from 25nm to 75nm.

9. S. Oldenburg, R. Averitt, S. Westcott, and N. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. **288**, 243 – 247 (1998). [CrossRef]

13. R. Bardhan, S. Mukherjee, N. A. Mirin, S. D. Levit, P. Nordlander, and N. J. Halas, “Nanosphere-in-a-nanoshell: A simple nanomatryushka,” J. Phys. Chem. C **114**, 7378–7383 (2010). [CrossRef]

*R*

_{1},

*R*

_{2}] = [5nm, 50nm], the absorption cross-section accounts for 25% of the total cross-section at resonance. For [

*R*

_{1},

*R*

_{2}] = [45nm, 50nm], this percentage rises to 60%. The tunability of the resonance wavelength and the tunability of the total cross-section composition makes silver coated silica sphere a good candidate for achieving broadband optical response.

## 4. Optimization of average cross-sections over wide frequency range

*μ*m). Therefore, this general structure includes structures with fewer layers (one through five layers) as boundary points.

21. S. G. Johnson, “The nlopt nonlinear optimization package, http://ab-initio.mit.edu/nlopt,”.

22. S. Kucherenko and Y. Sytsko, “Application of deterministic low-discrepancy sequences in global optimization,” Comput. Optim. Appl. **30**, 297–318 (2005). [CrossRef]

*σ*(

_{abs}*ω*) =

*ωIm*[

*α*(

*ω*)], where

*α*(

*ω*) is the polarizability [24

24. N. T. Fofang, T.-H. Park, O. Neumann, N. A. Mirin, P. Nordlander, and N. J. Halas, “Plexcitonic nanoparticles: Plasmonexciton coupling in nanoshell j-aggregate complexes,” Nano Lett. **8**, 3481–3487 (2008). [CrossRef] [PubMed]

26. V. S. Lebedev, A. S. Medvedev, D. N. Vasil’ev, D. A. Chubich, and A. G. Vitukhnovsky, “Optical properties of noble-metal nanoparticles coated with a dye j-aggregate monolayer,” Quantum Electron. **40**, 246–248 (2010). [CrossRef]

24. N. T. Fofang, T.-H. Park, O. Neumann, N. A. Mirin, P. Nordlander, and N. J. Halas, “Plexcitonic nanoparticles: Plasmonexciton coupling in nanoshell j-aggregate complexes,” Nano Lett. **8**, 3481–3487 (2008). [CrossRef] [PubMed]

*R*,

*T*] = [60.40nm, 8.68nm] has an average scattering cross-section of 8.65m

^{2}/g over 600–800nm, while its physical cross-section is only 2.07m

^{2}/g. The structure with [

*R*,

*T*] = [18.09nm, 2.00nm] provides an average total cross-section of 17.52m

^{2}/g over 600–800nm, which means that only 1g of such nanoparticles, when fully dispersed, can obscure an area as large as 17.52m

^{2}.

## 5. Optimization of the minimal cross-sections over wide frequency range

*N*species of silver coated silica spheres where

*N*= 1, 2, 3,.... The design parameters are the size parameters of individual species and relative weights (i.e. proportions) of each species in the mixture. The weights represent the relative weights in volume (mass) when the normalization is over volume (mass). When the size parameters of individual species are fixed, the problem of finding the optimal weights turns out to be a Linear Programming (LP) program. Therefore, we employed a two-level optimization structure. In the lower level, we used a standard LP solver to find out the optimal weights given the current size parameters. The resulting FOM as a function of size parameters is further optimized in the upper level with the same nonlinear method we used in the previous section. This separation into linear and nonlinear parts of the original optimization problem reduces the dimension of the parameter space and helps MLSL algorithm to find the global optimal in less iterations.

*l*= 1 resonance can cover the large wavelength region and its

*l*= 2 resonance can cover the small wavelength region. When the number of species increases, the new species try to cover the dips in the original scattering spectra with their resonant peaks. On the other hand, it is relatively difficult to build an absorption plateau because absorption peaks have narrow bandwidth. The FOM of the scattering cross-section is about twice as large as that of the absorption cross-section. The total cross-section can be enhanced significantly (35% for volume normalization and 46% for mass normalization) when

*N*increases from 1 to 2. The enhancement when

*N*increases from 2 to 3 is only moderate. The benefit of adding more species gradually saturates.

## 6. Concluding remarks

## Acknowledgments

## References and links

1. | Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. |

2. | X. Huang, S. Neretina, and M. A. El-Sayed, “Gold nanorods: From synthesis and properties to biological and biomedical applications,” Adv. Mater. |

3. | C. Noguez, “Surface plasmons on metal nanoparticles: the influence of shape and physical environment,” J. Phys. Chem. C |

4. | S. Alyones, C. Bruce, and A. Buin, “Numerical methods for solving the problem of electromagnetic scattering by a thin finite conducting wire,” IEEE Trans. Antennas Propag. |

5. | C. W. Bruce and S. Alyones, “Extinction efficiencies for metallic fibers in the infrared,” Appl. Opt. |

6. | P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: Applications in biological imaging and biomedicine,” J. Phys. Chem. B |

7. | J. Zhu, J. Li, and J. Zhao, “Tuning the wavelength drift between resonance light absorption and scattering of plasmonic nanoparticle,” Appl. Phys. lett. |

8. | C. E. Román-Velázquez and C. Noguez, “Designing the plasmonic response of shell nanoparticles: Spectral representation,” J. Chem. Phys. |

9. | S. Oldenburg, R. Averitt, S. Westcott, and N. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. |

10. | E. Prodan and P. Nordlander, “Structural tunability of the plasmon resonances in metallic nanoshells,” Nano Lett. |

11. | E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science |

12. | R. Bardhan, N. K. Grady, T. Ali, and N. J. Halas, “Metallic nanoshells with semiconductor cores: Optical characteristics modified by core medium properties,” ACS Nano |

13. | R. Bardhan, S. Mukherjee, N. A. Mirin, S. D. Levit, P. Nordlander, and N. J. Halas, “Nanosphere-in-a-nanoshell: A simple nanomatryushka,” J. Phys. Chem. C |

14. | R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A |

15. | Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. |

16. | Z. Ruan and S. Fan, “Temporal coupled-mode theory for fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C |

17. | Z. Ruan and S. Fan, “Design of subwavelength superscattering nanospheres,” Appl. Phys. lett. |

18. | H. C. van de Hulst, |

19. | C. Bohren and D. Huffman, |

20. | E. D. Palik, |

21. | S. G. Johnson, “The nlopt nonlinear optimization package, http://ab-initio.mit.edu/nlopt,”. |

22. | S. Kucherenko and Y. Sytsko, “Application of deterministic low-discrepancy sequences in global optimization,” Comput. Optim. Appl. |

23. | M. J. D. Powell, “The bobyqa algorithm for bound constrained optimization without derivatives,” Tech. rep., Department of Applied Mathematics and Theoretical Physics, Cambridge England (2009). |

24. | N. T. Fofang, T.-H. Park, O. Neumann, N. A. Mirin, P. Nordlander, and N. J. Halas, “Plexcitonic nanoparticles: Plasmonexciton coupling in nanoshell j-aggregate complexes,” Nano Lett. |

25. | A. Yoshida and N. Kometani, “Effect of the interaction between molecular exciton and localized surface plasmon on the spectroscopic properties of silver nanoparticles coated with cyanine dye j-aggregates,” J. Chem. Phys. C |

26. | V. S. Lebedev, A. S. Medvedev, D. N. Vasil’ev, D. A. Chubich, and A. G. Vitukhnovsky, “Optical properties of noble-metal nanoparticles coated with a dye j-aggregate monolayer,” Quantum Electron. |

**OCIS Codes**

(290.2200) Scattering : Extinction

(290.5825) Scattering : Scattering theory

**ToC Category:**

Scattering

**History**

Original Manuscript: June 4, 2012

Revised Manuscript: July 20, 2012

Manuscript Accepted: July 23, 2012

Published: July 27, 2012

**Virtual Issues**

Vol. 7, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Wenjun Qiu, Brendan G. DeLacy, Steven G. Johnson, John D. Joannopoulos, and Marin Soljačić, "Optimization of broadband optical response of multilayer nanospheres," Opt. Express **20**, 18494-18504 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-16-18494

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### References

- Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett.104, 207402 (2010). [CrossRef] [PubMed]
- X. Huang, S. Neretina, and M. A. El-Sayed, “Gold nanorods: From synthesis and properties to biological and biomedical applications,” Adv. Mater.21, 4880–4910 (2009). [CrossRef]
- C. Noguez, “Surface plasmons on metal nanoparticles: the influence of shape and physical environment,” J. Phys. Chem. C111, 3806–3819 (2007). [CrossRef]
- S. Alyones, C. Bruce, and A. Buin, “Numerical methods for solving the problem of electromagnetic scattering by a thin finite conducting wire,” IEEE Trans. Antennas Propag.55, 1856–1861 (2007). [CrossRef]
- C. W. Bruce and S. Alyones, “Extinction efficiencies for metallic fibers in the infrared,” Appl. Opt.48, 5095–5098 (2009). [CrossRef] [PubMed]
- P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: Applications in biological imaging and biomedicine,” J. Phys. Chem. B110, 7238–7248 (2006). [CrossRef] [PubMed]
- J. Zhu, J. Li, and J. Zhao, “Tuning the wavelength drift between resonance light absorption and scattering of plasmonic nanoparticle,” Appl. Phys. lett.99, 101901 (2011). [CrossRef]
- C. E. Román-Velázquez and C. Noguez, “Designing the plasmonic response of shell nanoparticles: Spectral representation,” J. Chem. Phys.134, 044116 (2011). [CrossRef] [PubMed]
- S. Oldenburg, R. Averitt, S. Westcott, and N. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett.288, 243 – 247 (1998). [CrossRef]
- E. Prodan and P. Nordlander, “Structural tunability of the plasmon resonances in metallic nanoshells,” Nano Lett.3, 543–547 (2003). [CrossRef]
- E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science302, 419–422 (2003). [CrossRef] [PubMed]
- R. Bardhan, N. K. Grady, T. Ali, and N. J. Halas, “Metallic nanoshells with semiconductor cores: Optical characteristics modified by core medium properties,” ACS Nano4, 6169–6179 (2010). [CrossRef] [PubMed]
- R. Bardhan, S. Mukherjee, N. A. Mirin, S. D. Levit, P. Nordlander, and N. J. Halas, “Nanosphere-in-a-nanoshell: A simple nanomatryushka,” J. Phys. Chem. C114, 7378–7383 (2010). [CrossRef]
- R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A75, 053801 (2007). [CrossRef]
- Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett.105, 013901 (2010). [CrossRef] [PubMed]
- Z. Ruan and S. Fan, “Temporal coupled-mode theory for fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C114, 7324–7329 (2010). [CrossRef]
- Z. Ruan and S. Fan, “Design of subwavelength superscattering nanospheres,” Appl. Phys. lett.98, 043101 (2011). [CrossRef]
- H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
- C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Songs, 1983).
- E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, New York, 1985).
- S. G. Johnson, “The nlopt nonlinear optimization package, http://ab-initio.mit.edu/nlopt ,”.
- S. Kucherenko and Y. Sytsko, “Application of deterministic low-discrepancy sequences in global optimization,” Comput. Optim. Appl.30, 297–318 (2005). [CrossRef]
- M. J. D. Powell, “The bobyqa algorithm for bound constrained optimization without derivatives,” Tech. rep., Department of Applied Mathematics and Theoretical Physics, Cambridge England (2009).
- N. T. Fofang, T.-H. Park, O. Neumann, N. A. Mirin, P. Nordlander, and N. J. Halas, “Plexcitonic nanoparticles: Plasmonexciton coupling in nanoshell j-aggregate complexes,” Nano Lett.8, 3481–3487 (2008). [CrossRef] [PubMed]
- A. Yoshida and N. Kometani, “Effect of the interaction between molecular exciton and localized surface plasmon on the spectroscopic properties of silver nanoparticles coated with cyanine dye j-aggregates,” J. Chem. Phys. C114, 2867–2872 (2010). [CrossRef]
- V. S. Lebedev, A. S. Medvedev, D. N. Vasil’ev, D. A. Chubich, and A. G. Vitukhnovsky, “Optical properties of noble-metal nanoparticles coated with a dye j-aggregate monolayer,” Quantum Electron.40, 246–248 (2010). [CrossRef]

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