## Scattering and absorption from strongly anisotropic nanoparticles |

Optics Express, Vol. 21, Issue 20, pp. 23181-23187 (2013)

http://dx.doi.org/10.1364/OE.21.023181

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### Abstract

Strongly anisotropic particles with hyperbolic dispersion that are small compared with the wavelength show strong resonance in the near infrared. The unique resonance modes are insensitive to the host refractive index and independent of particle size. In addition, the far-field direction of scattering does not depend on incident angle. Because the strength of resonance is comparable to a plasmonic nanoparticle in the visible region, a hyperbolic-dispersed particle is a promising scatterer as well as local heater in the near infrared.

© 2013 OSA

## 1. Introduction

3. J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A. **108**(28), 11327–11331 (2011). [CrossRef] [PubMed]

6. W. T. Chen, M. L. Tseng, C. Y. Liao, P. C. Wu, S. Sun, Y.-W. Huang, C. M. Chang, C. H. Lu, L. Zhou, D.-W. Huang, A. Q. Liu, and D. P. Tsai, “Fabrication of three-dimensional plasmonic cavity by femtosecond laser-induced forward transfer,” Opt. Express **21**(1), 618–625 (2013). [CrossRef] [PubMed]

4. X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics **6**(7), 450–454 (2012). [CrossRef]

## 2. Model and methodology

7. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express **14**(18), 8247–8256 (2006). [CrossRef] [PubMed]

10. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science **315**(5819), 1699–1701 (2007). [CrossRef] [PubMed]

11. S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. **34**(7), 890–892 (2009). [CrossRef] [PubMed]

13. S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photonics Rev. **7**(2), 265–271 (2013). [CrossRef]

14. Z. Jacob, J. Y. Kim, G. Naik, A. Boltasseva, E. Narimanov, and V. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B **100**(1), 215–218 (2010). [CrossRef]

16. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science **336**(6078), 205–209 (2012). [CrossRef] [PubMed]

*ε*= diag(

*ε*

_{x},

*ε*

_{x},

*ε*

_{z})) is calculated from the effectively medium theory (EMT) [17],

*r*is the volume fraction of the metal and

*ε*

_{m}and

*ε*

_{d}are the permittivties of metal (silver) and dielectric (silicon), respectively. Here, the optical axis as well as the z-axis is perpendicular to the multilayers. Note that although non-locality is not included in this second-order EMT, it serves as a reasonable estimate under the condition, |

*ε*

_{m}(

*ω*)| >>

*ε*

_{d}. To simplify the situation, the form of the permittivity of silver used in our work derives from the Drude-Lorentz model [18] whose parameters were obtained by a fit to the experimental data [19

19. P. B. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

20. M. W. Knight and N. J. Halas, “Nanoshells to nanoeggs to nanocups: optical properties of reduced symmetry core–shell nanoparticles beyond the quasistatic limit,” New J. Phys. **10**(10), 105006 (2008). [CrossRef]

21. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. **330**(3), 377–445 (1908). [CrossRef]

*Q*

_{scat}) and absorption efficiency (

*Q*

_{abs}) are calculated by dividing the scattering and absorption cross-sections by the cross-section of the particle, respectively.

## 3. Results and discussion

*r*) is two third. Within the wavelength range of the plot,

*ε*

_{x}< 0 and

*ε*

_{z}>0, hence dispersion is hyperbolic. Next, an HMM nanosphere of 60 nm radius is placed in air where the optical axis is aligned with the z-axis. The incident electromagnetic wave is linearly polarized along the x-axis and propagates in the z-axis.

*r*) is sufficiently small with respect to the resonance wavelength in free space (

*λ*

_{0}), which gives

*r/ λ*

_{0}= 0.038. To see the modes inside the HMM sphere, the magnetic as well as electric field amplitudes at 1564 nm have been plotted in Fig. 1(c). The magnetic field clearly shows the lowest resonant mode and is similar to the magnetic resonance of a high-refractive index nanoparticle [22]. Similar resonance modes are observed in coupled metallic nanostrips [23]. The detail analysis in terms of magnetic resonances will be discussed elsewhere.

*r/ λ*

_{0}= 0.038. The magnetic field amplitudes as well as electric field amplitudes of the silver sphere at 360 nm are plotted in Fig. 1(e) to visualize the difference between the HMM sphere. By comparing Fig. 1(b) and (d), we see that the scattering and absorption efficiencies for both the HMM sphere and the silver sphere having the identical radius-to-wavelength ratio are of the same order. These results show that the HMM spheres can perform as good scatterers as well as local heaters [24

24. J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Nanoscale radiative heat transfer between a small particle and a plane surface,” Appl. Phys. Lett. **78**(19), 2931–2933 (2001). [CrossRef]

26. O. Neumann, A. S. Urban, J. Day, S. Lal, P. Nordlander, and N. J. Halas, “Solar vapor generation enabled by nanoparticles,” ACS Nano **7**(1), 42–49 (2013). [CrossRef] [PubMed]

*r*= 48, 60, and 72 nm) are plotted in Fig. 2(b). Except for the radii, all conditions are the same as for Fig. 1(b). From Fig. 2(b), the resonance wavelength shifts by more than 100 nm when the radius is increased or decreased by 20%. The resonance mode which is similar to a dielectric cavity mode makes the HMM spheres sensitive to the radii. As a comparison, the radius dependence is also calculated for the silver sphere. Even though the radius is increased or decreased by 20%, the shift of the resonance wavelength is around 1 nm (figure not shown). It is well known that the plasmonic resonance becomes independent of the sphere radius in the limit of sufficiently small radius; this is known as the Rayleigh scattering condition [1].

*θ*is defined in the x-z plane and the magnetic field is always parallel to the y-axis. The other angle

*ϕ*is defined in the y-z plane and the electric field is always parallel to the y-axis (see Fig. 3(a)). The

*θ*- and

*ϕ*-dependencies of the absorption efficiencies are plotted in Fig. 3(b). The absorption efficiency, while hardly depending on

*θ*, largely depends on

*ϕ*. The far-field amplitudes of the electric fields are plotted with respect to

*θ*and φ, in Fig. 3(c)–(f). Note that there are far-fields patterns parallel to the incident plane (|E

_{Π}|) and perpendicular to the incident plane (|E

_{⊥}|) for each

*θ*and

*ϕ*plot. It is important to mention that the maximum angle of the far-field patterns is always 180 degrees (forward direction) and does not depend on

*θ*or

*ϕ*. Such angular-independent scattering properties could be beneficial for unidirectional light extraction and sensing. Obviously, the parallel components of the scattered far-field patterns of the silver sphere rotate as the incident angle becomes larger (see Fig. 3(g) and (j)). Because the radius of the silver sphere is sufficiently small compared with the wavelength, the perpendicular components of the scattered far-field patterns are nearly circular and show little angular dependence (see Fig. 3(h) and (i)).

*θ*= 0 for the HMM sphere (electric field parallel to the x-axis), the shapes of the field patterns in the x–z plane (parallel plane) and in the y–z plane (perpendicular plane) are circular and figure-of-eight (8) in form, respectively (see Fig. 3(c) and (d)). It is well known that when the size of the isotropic sphere is sufficiently small, its parallel and perpendicular components exhibit patterns similar to a figure-of-eight and a circle, respectively (see Fig. 3(g) and (h)) [1]. Thus, the far-field patterns of an HMM sphere and an isotropic sphere are 90 degrees rotated with respect to the incident direction (z-axis in Fig. 3).

## 4. Summary

## Acknowledgments

## References and links

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

2. | S. A. Maier, |

3. | J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A. |

4. | X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics |

5. | D. Li, L. Qin, X. Xiong, R.-W. Peng, Q. Hu, G.-B. Ma, H.-S. Zhou, and M. Wang, “Exchange of electric and magnetic resonances in multilayered metal/dielectric nanoplates,” Opt. Express |

6. | W. T. Chen, M. L. Tseng, C. Y. Liao, P. C. Wu, S. Sun, Y.-W. Huang, C. M. Chang, C. H. Lu, L. Zhou, D.-W. Huang, A. Q. Liu, and D. P. Tsai, “Fabrication of three-dimensional plasmonic cavity by femtosecond laser-induced forward transfer,” Opt. Express |

7. | Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express |

8. | A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B |

9. | Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science |

10. | I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science |

11. | S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. |

12. | S. Ishii, V. P. Drachev, and A. V. Kildishev, “Diffractive nanoslit lenses for subwavelength focusing,” Opt. Commun. |

13. | S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photonics Rev. |

14. | Z. Jacob, J. Y. Kim, G. Naik, A. Boltasseva, E. Narimanov, and V. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B |

15. | T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. |

16. | H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science |

17. | S. M. Rytov, “Electromagnetic Propeties of a Finely Stratified Medium,” Sov. Phys. JETP |

18. | X. Ni, Z. Liu, and A. V. Kildishev, Photonics DB: Optical Constants (DOI: 10.4231/D3FT8DJ4J) (2008). |

19. | P. B. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

20. | M. W. Knight and N. J. Halas, “Nanoshells to nanoeggs to nanocups: optical properties of reduced symmetry core–shell nanoparticles beyond the quasistatic limit,” New J. Phys. |

21. | G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. |

22. | A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep. |

23. | A. Yariv and P. Yeh, |

24. | J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Nanoscale radiative heat transfer between a small particle and a plane surface,” Appl. Phys. Lett. |

25. | P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-Body Radiative Heat Transfer Theory,” Phys. Rev. Lett. |

26. | O. Neumann, A. S. Urban, J. Day, S. Lal, P. Nordlander, and N. J. Halas, “Solar vapor generation enabled by nanoparticles,” ACS Nano |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(290.5850) Scattering : Scattering, particles

(160.3918) Materials : Metamaterials

**ToC Category:**

Scattering

**History**

Original Manuscript: July 15, 2013

Revised Manuscript: September 7, 2013

Manuscript Accepted: September 9, 2013

Published: September 24, 2013

**Citation**

Satoshi Ishii, Shin-ichiro Inoue, and Akira Otomo, "Scattering and absorption from strongly anisotropic nanoparticles," Opt. Express **21**, 23181-23187 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23181

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### References

- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983).
- S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Science + Business Media, 2007).
- J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A.108(28), 11327–11331 (2011). [CrossRef] [PubMed]
- X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics6(7), 450–454 (2012). [CrossRef]
- D. Li, L. Qin, X. Xiong, R.-W. Peng, Q. Hu, G.-B. Ma, H.-S. Zhou, and M. Wang, “Exchange of electric and magnetic resonances in multilayered metal/dielectric nanoplates,” Opt. Express19(23), 22942–22949 (2011). [CrossRef] [PubMed]
- W. T. Chen, M. L. Tseng, C. Y. Liao, P. C. Wu, S. Sun, Y.-W. Huang, C. M. Chang, C. H. Lu, L. Zhou, D.-W. Huang, A. Q. Liu, and D. P. Tsai, “Fabrication of three-dimensional plasmonic cavity by femtosecond laser-induced forward transfer,” Opt. Express21(1), 618–625 (2013). [CrossRef] [PubMed]
- Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14(18), 8247–8256 (2006). [CrossRef] [PubMed]
- A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74(7), 075103 (2006). [CrossRef]
- Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007). [CrossRef] [PubMed]
- I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science315(5819), 1699–1701 (2007). [CrossRef] [PubMed]
- S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett.34(7), 890–892 (2009). [CrossRef] [PubMed]
- S. Ishii, V. P. Drachev, and A. V. Kildishev, “Diffractive nanoslit lenses for subwavelength focusing,” Opt. Commun.285(16), 3368–3372 (2012). [CrossRef]
- S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photonics Rev.7(2), 265–271 (2013). [CrossRef]
- Z. Jacob, J. Y. Kim, G. Naik, A. Boltasseva, E. Narimanov, and V. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B100(1), 215–218 (2010). [CrossRef]
- T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett.99(15), 151115 (2011). [CrossRef]
- H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science336(6078), 205–209 (2012). [CrossRef] [PubMed]
- S. M. Rytov, “Electromagnetic Propeties of a Finely Stratified Medium,” Sov. Phys. JETP2, 466–475 (1956).
- X. Ni, Z. Liu, and A. V. Kildishev, Photonics DB: Optical Constants (DOI: 10.4231/D3FT8DJ4J) (2008).
- P. B. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972). [CrossRef]
- M. W. Knight and N. J. Halas, “Nanoshells to nanoeggs to nanocups: optical properties of reduced symmetry core–shell nanoparticles beyond the quasistatic limit,” New J. Phys.10(10), 105006 (2008). [CrossRef]
- G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys.330(3), 377–445 (1908). [CrossRef]
- A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep. 2: 492(2012).
- A. Yariv and P. Yeh, Optical waves in crystals (Wiley New York, 1984), Vol. 5.
- J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Nanoscale radiative heat transfer between a small particle and a plane surface,” Appl. Phys. Lett.78(19), 2931–2933 (2001). [CrossRef]
- P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-Body Radiative Heat Transfer Theory,” Phys. Rev. Lett.107(11), 114301 (2011). [CrossRef] [PubMed]
- O. Neumann, A. S. Urban, J. Day, S. Lal, P. Nordlander, and N. J. Halas, “Solar vapor generation enabled by nanoparticles,” ACS Nano7(1), 42–49 (2013). [CrossRef] [PubMed]

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