## Tailoring optical properties of surface charged dielectric nanoparticles based on an effective medium theory |

Optics Express, Vol. 21, Issue 17, pp. 20387-20393 (2013)

http://dx.doi.org/10.1364/OE.21.020387

Acrobat PDF (774 KB)

### Abstract

We propose an effective-medium theory (EMT) for the surface charged dielectric nanoparticles (CDNPs) in the long wavelength limit, in which a homogenous CDNP is demonstrated to be equivalent to a conventional absorbing neutral particle of the same size but with different constitutive parameters (effective permittivity *ε _{e}* and effective permeability

*μ*). It is found that while the surface charge induces negligible change of magnetic permeability in particle, it gives rise to a significant change of electric permittivity. The change in permittivity depends on the charge, the particle size, and the working frequency but is independent of the constituent material. In infrared frequencies, both the real and imaginary parts of the particle permittivity may be changed considerably by surface charging. At higher frequency, the surface charge can lead to a remarkable decrease in the real part of the permittivity while keeping its imaginary part nearly unchanged. Therefore, based on EMT we can tailor the optical properties of CDNPs by optimizing their parameters, allowing for many exotic phenomena, such as vanishing scattering efficiency, great enhancement of light absorption efficiency, and surface charge induced surface plasmon resonance.

_{e}© 2013 OSA

## 1. Introduction

*i. e.*, the dusty plasmas in the solar system [1

1. D. A. Mendis, “Progress in the study of dusty plasmas,” Plasma Sources Sci. Technol. **11**, A219–A228 (2002). [CrossRef]

3. I. Mann, “Interplanetary medium - A dusty plasma,” Adv. Space Res. **41**160–167 (2008). [CrossRef]

4. R. Aveyard, B. P. Binks, J. H. Clint, P. D. I. Fletcher, T. S. Horozov, B. Neumann, V. N. Paunov, J. Annesley, S. W. Botchway, D. Nees, A. W. Parker, A. D. Ward, and A. Burgess, “Measurement of long-range repulsive forces between charged particles at an oil-water interface,” Phys. Rev. Lett. **88**, 246102 (2002). [CrossRef] [PubMed]

6. M. Megens and J. Aizenberg, “Like-charged particles at liquid interfaces,” Nature **424**, 1014 (2003). [CrossRef]

8. A. Heifetz, H. T. Chien, S. Liao, N. Gopalsami, and A. C. Raptis, “Millimeter-wave scattering from neutral and charged water droplets,” J. Quant. Spectrosc. Radiat. Transf. **111**, 2550–2557 (2010). [CrossRef]

9. E. Rosenkrantz and S. Arnon, “Enhanced absorption of light by charged nanoparticles,” Opt. Lett. **35**, 1178–1180 (2010). [CrossRef] [PubMed]

10. E. Rosenkrantz and S. Arnon, “Resonance frequencies of electrically charged nanoparticles,” IEEE Photonics Journal **3**, 82–88 (2011). [CrossRef]

11. X. C. Li, L. Xie, and X. J. Zheng, “The comparison between the Mie theory and the Rayleigh approximation to calculate the EM scattering by partially charged sand,” J. Quant. Spectrosc. Radiat. Transf. **113**, 251–258 (2011). [CrossRef]

12. G. Mie, “Beiträge zur Optik träber Medien speziell kolloidaler Metaläsungen,” Ann. Phys. **25**, 377–445 (1908). [CrossRef]

14. C. F. Bohren and A. J. Hunt, “Scattering of electromagnetic waves by a charged sphere,” Can. J. Phys. **55**, 1930–1935 (1977). [CrossRef]

17. R. L. Heinisch, F. X. Bronold, and H. Fehske, “Mie scattering by a charged dielectric particle,” Phys. Rev. Lett. **109**, 243903 (2012). [CrossRef]

18. Y. Wu, J. S. Li, Z. Q. Zhang, and C. T Chan, “Effective medium theory for magnetodielectric composites: Beeyond the long-wavelength limit,” Phys. Rev. B **74**, 085111 (2006). [CrossRef]

21. Y. X. Ni, L. Gao, and C. W. Qiu, “Achieving invisibility of homegeneous cylindrically anisotropic cylinders,” Plasmonics **5**, 251–258 (2010). [CrossRef]

9. E. Rosenkrantz and S. Arnon, “Enhanced absorption of light by charged nanoparticles,” Opt. Lett. **35**, 1178–1180 (2010). [CrossRef] [PubMed]

21. Y. X. Ni, L. Gao, and C. W. Qiu, “Achieving invisibility of homegeneous cylindrically anisotropic cylinders,” Plasmonics **5**, 251–258 (2010). [CrossRef]

25. A. Alù and N. Engheta, “Multifrequency optical cloaking with layered plasmonic shells,” Phys. Rev. Lett. **100**, 113901 (2008). [CrossRef]

## 2. Theoretical analysis

*R*, composed of a homogeneous material with permittivity

*ε*and permeability

_{s}*μ*, surrounded by free space with constitutive parameters

_{s}*ε*

_{0}and

*μ*

_{0}. The circular frequency of the incident wave is

*ω*, and the wavenumbers outside and inside the scatter are

14. C. F. Bohren and A. J. Hunt, “Scattering of electromagnetic waves by a charged sphere,” Can. J. Phys. **55**, 1930–1935 (1977). [CrossRef]

16. J. Klačka and M. Kocifaj, “On the scattering of electromagnetic waves by a charged sphere,” Prog. Electromagn. Res. **109**, 17–35 (2010). [CrossRef]

*x*=

*kR*,

*y*=

*k*are the size parameters,

_{s}R*ψ*(

_{n}*x*) =

*xj*(

_{n}*x*),

*σ*is the effective surface conductivity [14

_{s}14. C. F. Bohren and A. J. Hunt, “Scattering of electromagnetic waves by a charged sphere,” Can. J. Phys. **55**, 1930–1935 (1977). [CrossRef]

15. J. Klačka and M. Kocifaj, “Scattering of electromagnetic waves by charged spheres and some physical consequences,” J. Quant. Spectrosc. Radiat. Transf. **106**, 170–183 (2007). [CrossRef]

*e/m*is the charge-to-mass ratio of electron, Φ = |

_{e}*η*|

*R/ε*

_{0}is the electrostatic potential at the surface with

*η*the static surface charge density,

*γ*≈

_{s}*k*[16

_{B}T/h̄16. J. Klačka and M. Kocifaj, “On the scattering of electromagnetic waves by a charged sphere,” Prog. Electromagn. Res. **109**, 17–35 (2010). [CrossRef]

*k*the Boltzmann constant,

_{B}*T*the thermodynamic temperature, and

*h̄*the Plank constant. In all our calculations,

*T*= 300 K. It should be noted that

*η*and elementary charge

*e*always have the same sign. Therefore, only the absolute value of the surface charge takes effect, independent of its sign. Equation (1) reduces to the Mie coefficients of a neutral spherical particle [13] when

*g*= 0.

*i. e.*,

*x*≪ 1 and

*y*≪ 1, the high order Mie coefficients are expected to be negligible and the scattering effect of the charged sphere is dominated by

*n*= 1 terms. Within the coherent-potential approximation [18

18. Y. Wu, J. S. Li, Z. Q. Zhang, and C. T Chan, “Effective medium theory for magnetodielectric composites: Beeyond the long-wavelength limit,” Phys. Rev. B **74**, 085111 (2006). [CrossRef]

21. Y. X. Ni, L. Gao, and C. W. Qiu, “Achieving invisibility of homegeneous cylindrically anisotropic cylinders,” Plasmonics **5**, 251–258 (2010). [CrossRef]

*ε*and permeability

_{e}*μ*of the charged sphere when treated it as a neutral one. The idea of the EMT is as follows. Suppose we have a homogenous background medium with constitutive parameters

_{e}*ε*and

_{e}*μ*, then we can calculate the scattering efficiency

_{e}*C*when putting the considered charged particle inside it. If

_{sca}*C*tends to be zero, we can judge that

_{sca}*ε*and

_{e}*μ*are the effective constitutive parameters for the equivalent neutral particle. Mathematically, we can just replace

_{e}*k*,

*ε*

_{0},

*μ*

_{0}, and

*x*in Eq. (1) and

*g*with

*ε*,

_{e}*μ*, and

_{e}*k*respectively, the vanishing of the Mie coefficient indicates

_{e}x/k*C*= 0. In the long wavelength limit, only

_{sca}*a*

_{1}= 0 and

*b*

_{1}= 0 should be considered. Moreover, in the limit of

*x*→ 0, we can approximate the Ricatti-Bessel functions and their derivatives by

*ψ*

_{1}(

*x*) ≅

*x*

^{2}/3,

*ψ′*

_{1}(

*x*) ≅ 2

*x*/3,

*ξ*

_{1}(

*x*) ≅

*x*

^{2}/3 −

*i/x*, and

*ξ′*

_{1}(

*x*) ≅ 2

*x*/3 +

*i/x*

^{2}. With these approximations, one arrives at two equations for effective constitute parameters where

*g*is given by Eq. (2). If

*g*= 0, the equations reduce to

*ε*=

_{e}*ε*,

_{s}*μ*=

_{e}*μ*, corresponding to the neutral particle case as expected. The differences between the effective and the original constitutive parameters are denoted as Δ

_{s}*ε*=

*ε*−

_{e}*ε*and Δ

_{s}*μ*=

*μ*−

_{e}*μ*, respectively. From Eq. (2) and Eq. (3) and considering

_{s}*ε*for a CDNP of the size parameter

*x*= 0.05 as function of circular frequency. The results are given in Fig. 1, where the negative real part and the positive imaginary part of Δ

*ε*are indicated by red solid line and blue dashed lines, respectively. It can be observed that with the increase of the circular frequency, the decrement of the real part increases fast in the first stage and then comes to a saturation value 2

*f*Φ

*ε*

_{0}/

*x*

^{2}quickly, while the increment of the imaginary part exhibits an obvious increase at a lower frequency and then vanishes at higher frequency. With these intriguing properties, one can significantly decrease the effective permittivity of a CDNP by adjusting its surface potential, especially at high frequency, meanwhile keep the imaginary part of the effective permittivity nearly unchanged. In particular, we can tailor the optical properties of a CDNP by designing its effective relative permittivity to be 1, zero, or even negative through controlling its surface potential. With this special handle, many exotic scattering effects can be realized. In the following, we will present some typical examples.

## 3. Numerical results and discussion

9. E. Rosenkrantz and S. Arnon, “Enhanced absorption of light by charged nanoparticles,” Opt. Lett. **35**, 1178–1180 (2010). [CrossRef] [PubMed]

15. J. Klačka and M. Kocifaj, “Scattering of electromagnetic waves by charged spheres and some physical consequences,” J. Quant. Spectrosc. Radiat. Transf. **106**, 170–183 (2007). [CrossRef]

*Q*=

_{abs}*Q*−

_{ext}*Q*. For a neutral scatter with the permittivity and permeability the same as those of the background medium, the scattering efficiency vanishes. While for a CDNP, even if its permittivity is different from that of the background medium, the scattering efficiency can still approach to zero when its effective permittivity is equal to the background permittivity. The reason lies in that the effective permittivity of a CDNP can be decreased by electrically charging, resulting in the same permittivity for the CDNP and the background medium. According to Eq. (4), the effective permittivity of a CDNP equals to

_{sca}*ε*

_{0}at room temperature and around the optical frequency, when the surface potential is located at Φ ≈ (

*ε*−

_{s}*ε*

_{0})

*x*

^{2}/2

*fε*

_{0}. This corresponds to the vanishing of the scattering efficiency of the CDNP. In Fig. 2, we present the scattering efficiency

*Q*and the real part of effective permittivity Re{

_{sca}*ε/ε*

_{0}} as the functions of the surface potential Φ. While the imaginary part of the effective permittivity is nearly zero as can be expected from the results shown in Fig. 1 at high frequency. It is evident that the scattering efficiency vanishes as the effective permittivity

*ε*approaches to

_{e}*ε*

_{0}as indicated by the blue dashed line. The surface potential of the CDNP for the vanishing scattering efficiency is at Φ ≈ 637 V, in accord with Φ ≈ (

*ε*−

_{s}*ε*

_{0})

*x*

^{2}/2

*fε*

_{0}.

*ε*= −2

_{s}*ε*

_{0}. Rosenkrantz and Arnon [9

**35**, 1178–1180 (2010). [CrossRef] [PubMed]

*ε*

_{0}, resulting in the appearance of the SPR. According to Eq. (3), the condition for the occurrence of SPR is

*x*for a CDNP with the radius 5 nm. The peak of the absorption efficiency indicates the occurrence of the SPR as can be observed from the red solid line, which appears exactly at

*ε*= −2

_{e}*ε*

_{0}, as marked by the black dash-dot line. In addition, under different surface potentials the SPR can still be observed but at different circular frequencies, which is clearly demonstrated in Fig. 3(b). As a result, charging a neutral particle can provide an alternative way to tune the SPR frequency of NPs. Besides measuring the frequency of the SPR, our theory may in turn serve to determine the surface potential or surface charge of the CDNP.

## 4. Conclusion

## Acknowledgments

## References and links

1. | D. A. Mendis, “Progress in the study of dusty plasmas,” Plasma Sources Sci. Technol. |

2. | O. Ishihara, “Complex plasma: dusts in plasma,” J. Phys. D |

3. | I. Mann, “Interplanetary medium - A dusty plasma,” Adv. Space Res. |

4. | R. Aveyard, B. P. Binks, J. H. Clint, P. D. I. Fletcher, T. S. Horozov, B. Neumann, V. N. Paunov, J. Annesley, S. W. Botchway, D. Nees, A. W. Parker, A. D. Ward, and A. Burgess, “Measurement of long-range repulsive forces between charged particles at an oil-water interface,” Phys. Rev. Lett. |

5. | M. G. Nikolaides, A. R. Bausch, M. F. Hsu, A. D. Dinsmore, M. P. Brenner, C. Gay, and D. A. Weitz, “Electric-field-induced capillary attraction between like-charged particles at liquid interfaces,” Nature |

6. | M. Megens and J. Aizenberg, “Like-charged particles at liquid interfaces,” Nature |

7. | H. Y. Li, Z. S. Wu, and L. Bai, “Scattering for charged multisphere structure located in plane wave/Gaussian beam,” J. of Electromagn. Waves and Appl. |

8. | A. Heifetz, H. T. Chien, S. Liao, N. Gopalsami, and A. C. Raptis, “Millimeter-wave scattering from neutral and charged water droplets,” J. Quant. Spectrosc. Radiat. Transf. |

9. | E. Rosenkrantz and S. Arnon, “Enhanced absorption of light by charged nanoparticles,” Opt. Lett. |

10. | E. Rosenkrantz and S. Arnon, “Resonance frequencies of electrically charged nanoparticles,” IEEE Photonics Journal |

11. | X. C. Li, L. Xie, and X. J. Zheng, “The comparison between the Mie theory and the Rayleigh approximation to calculate the EM scattering by partially charged sand,” J. Quant. Spectrosc. Radiat. Transf. |

12. | G. Mie, “Beiträge zur Optik träber Medien speziell kolloidaler Metaläsungen,” Ann. Phys. |

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

14. | C. F. Bohren and A. J. Hunt, “Scattering of electromagnetic waves by a charged sphere,” Can. J. Phys. |

15. | J. Klačka and M. Kocifaj, “Scattering of electromagnetic waves by charged spheres and some physical consequences,” J. Quant. Spectrosc. Radiat. Transf. |

16. | J. Klačka and M. Kocifaj, “On the scattering of electromagnetic waves by a charged sphere,” Prog. Electromagn. Res. |

17. | R. L. Heinisch, F. X. Bronold, and H. Fehske, “Mie scattering by a charged dielectric particle,” Phys. Rev. Lett. |

18. | Y. Wu, J. S. Li, Z. Q. Zhang, and C. T Chan, “Effective medium theory for magnetodielectric composites: Beeyond the long-wavelength limit,” Phys. Rev. B |

19. | S. Y. Liu, W. K. Chen, J. J. Du, Z. F. Lin, S. T. Chui, and C. T. Chan, “Manipulating negative-refractive behavior with a magnetic field,” Phys. Rev. Lett. |

20. | J. F. Jin, S.Y. Liu, Z. F. Lin, and S. T. Chui, “Effective-medium theory for anistropic magnetic metamaterials,” Phys. Rev. B |

21. | Y. X. Ni, L. Gao, and C. W. Qiu, “Achieving invisibility of homegeneous cylindrically anisotropic cylinders,” Plasmonics |

22. | R. A. Shelby, D. R. Smith, S. Shultz, and S. C. Nemat-Nasser, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. |

23. | A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express |

24. | A. Alù and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express |

25. | A. Alù and N. Engheta, “Multifrequency optical cloaking with layered plasmonic shells,” Phys. Rev. Lett. |

**OCIS Codes**

(290.4020) Scattering : Mie theory

(290.5850) Scattering : Scattering, particles

(260.2065) Physical optics : Effective medium theory

**ToC Category:**

Scattering

**History**

Original Manuscript: June 21, 2013

Revised Manuscript: August 2, 2013

Manuscript Accepted: August 12, 2013

Published: August 22, 2013

**Virtual Issues**

Vol. 8, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Neng Wang, Shiyang Liu, and Zhifang Lin, "Tailoring optical properties of surface charged dielectric nanoparticles based on an effective medium theory," Opt. Express **21**, 20387-20393 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-17-20387

Sort: Year | Journal | Reset

### References

- D. A. Mendis, “Progress in the study of dusty plasmas,” Plasma Sources Sci. Technol.11, A219–A228 (2002). [CrossRef]
- O. Ishihara, “Complex plasma: dusts in plasma,” J. Phys. D40, R121–R147 (2007). [CrossRef]
- I. Mann, “Interplanetary medium - A dusty plasma,” Adv. Space Res.41160–167 (2008). [CrossRef]
- R. Aveyard, B. P. Binks, J. H. Clint, P. D. I. Fletcher, T. S. Horozov, B. Neumann, V. N. Paunov, J. Annesley, S. W. Botchway, D. Nees, A. W. Parker, A. D. Ward, and A. Burgess, “Measurement of long-range repulsive forces between charged particles at an oil-water interface,” Phys. Rev. Lett.88, 246102 (2002). [CrossRef] [PubMed]
- M. G. Nikolaides, A. R. Bausch, M. F. Hsu, A. D. Dinsmore, M. P. Brenner, C. Gay, and D. A. Weitz, “Electric-field-induced capillary attraction between like-charged particles at liquid interfaces,” Nature420, 299–301 (2002). [CrossRef] [PubMed]
- M. Megens and J. Aizenberg, “Like-charged particles at liquid interfaces,” Nature424, 1014 (2003). [CrossRef]
- H. Y. Li, Z. S. Wu, and L. Bai, “Scattering for charged multisphere structure located in plane wave/Gaussian beam,” J. of Electromagn. Waves and Appl.24, 2037–2047 (2010).
- A. Heifetz, H. T. Chien, S. Liao, N. Gopalsami, and A. C. Raptis, “Millimeter-wave scattering from neutral and charged water droplets,” J. Quant. Spectrosc. Radiat. Transf.111, 2550–2557 (2010). [CrossRef]
- E. Rosenkrantz and S. Arnon, “Enhanced absorption of light by charged nanoparticles,” Opt. Lett.35, 1178–1180 (2010). [CrossRef] [PubMed]
- E. Rosenkrantz and S. Arnon, “Resonance frequencies of electrically charged nanoparticles,” IEEE Photonics Journal3, 82–88 (2011). [CrossRef]
- X. C. Li, L. Xie, and X. J. Zheng, “The comparison between the Mie theory and the Rayleigh approximation to calculate the EM scattering by partially charged sand,” J. Quant. Spectrosc. Radiat. Transf.113, 251–258 (2011). [CrossRef]
- G. Mie, “Beiträge zur Optik träber Medien speziell kolloidaler Metaläsungen,” Ann. Phys.25, 377–445 (1908). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1983).
- C. F. Bohren and A. J. Hunt, “Scattering of electromagnetic waves by a charged sphere,” Can. J. Phys.55, 1930–1935 (1977). [CrossRef]
- J. Klačka and M. Kocifaj, “Scattering of electromagnetic waves by charged spheres and some physical consequences,” J. Quant. Spectrosc. Radiat. Transf.106, 170–183 (2007). [CrossRef]
- J. Klačka and M. Kocifaj, “On the scattering of electromagnetic waves by a charged sphere,” Prog. Electromagn. Res.109, 17–35 (2010). [CrossRef]
- R. L. Heinisch, F. X. Bronold, and H. Fehske, “Mie scattering by a charged dielectric particle,” Phys. Rev. Lett.109, 243903 (2012). [CrossRef]
- Y. Wu, J. S. Li, Z. Q. Zhang, and C. T Chan, “Effective medium theory for magnetodielectric composites: Beeyond the long-wavelength limit,” Phys. Rev. B74, 085111 (2006). [CrossRef]
- S. Y. Liu, W. K. Chen, J. J. Du, Z. F. Lin, S. T. Chui, and C. T. Chan, “Manipulating negative-refractive behavior with a magnetic field,” Phys. Rev. Lett.101, 157407 (2008). [CrossRef] [PubMed]
- J. F. Jin, S.Y. Liu, Z. F. Lin, and S. T. Chui, “Effective-medium theory for anistropic magnetic metamaterials,” Phys. Rev. B80, 115101 (2009). [CrossRef]
- Y. X. Ni, L. Gao, and C. W. Qiu, “Achieving invisibility of homegeneous cylindrically anisotropic cylinders,” Plasmonics5, 251–258 (2010). [CrossRef]
- R. A. Shelby, D. R. Smith, S. Shultz, and S. C. Nemat-Nasser, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett.78, 489–491 (2001). [CrossRef]
- A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express14, 1557–1567 (2006). [CrossRef] [PubMed]
- A. Alù and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express15, 3318–3332 (2007). [CrossRef] [PubMed]
- A. Alù and N. Engheta, “Multifrequency optical cloaking with layered plasmonic shells,” Phys. Rev. Lett.100, 113901 (2008). [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.