Abstract
We provide a self-consistent mean field approximation in the framework of Mie scattering theory to study the optical bistability of a metallic nanoparticle coated with a nonlinear shell. We demonstrate that the nanoparticle coated with a weakly nonlinear shell exhibits optical bistability in a broad range of incident optical intensity. This optical bistability critically relies on the geometry of the shell-coated nanoparticle, especially the fractional volume of the metallic core. The incident wavelength can also affect the optical bistability. Through an optimization-like process, we find a design with broader bistable region and lower threshold field by adjusting the size of the nonlinear shell, the fractional volume of the metallic core and the incident wavelength. These results may find potential applications in optical bistable devices such as all-optical switches, optical transistors and optical memories.
Similar content being viewed by others
Introduction
Optical bistability has attracted remarkable attention in recent years because of its promising applications in logic functions1, metamaterials2,3,4,5, all-optical switching6 and low power lasing7. An optical bistable system has two stable transmission states to switch on and off, depending upon the history of input signal8,9,10. As a conventional type of nonlinear materials, Kerr materials are widely studied in optical bistability. In view of its weak nonlinearity, sophisticated structural design is needed in Kerr nonlinear devices, aiming for faster switching speed and broader range of operation for incident intensity11.
In this reports, we study the optical bistability of a metallic nanoparticle coated with a nonlinear shell based on a self-consistent mean field approximation in the framework of Mie scattering theory. We decompose the scattered fields of the coated nanosphere into spherical waves with Debye potentials in order to establish the relationship between the local field of the shell and the incident field. The self-consistent mean field approximation is then adopted to study the optical bistability. Our results match well with the previous quasi-static solution of the Laplace equations at a deep-subwavelength scale12. On the other hand, our full-wave solutions are valid in a more strict sense and thus are more suitable for structural design of optical bistable devices, in which various parameters need to be varied in a wide parameters space.
Results
Theoretical development
We first consider electromagnetic wave scattering from the coated metallic nanoparticle. The coated particle, as shown in Fig. 1, has a metallic core with radius a and a shell with outer radius b. The relative permittivity and permeability of the core (the shell) are 
and 
(
and 
). The surrounding medium has a relative permittivity 
and a relative permeability 
. We assume that the incident plane wave propagates in 
direction, with electric field polarized in x direction:
Geometry of scattering of a plane wave by a coated sphere.
The radius of the core is a and the outer radius is b. The incident plane wave is polarization along the 
-direction and propagates along 
-direction. The relative permittivity and permeability of the core (the shell) are 
and 
(
and 
). The surrounding medium has a relative permittivity 
and a relative permeability 
.
where
In time-harmonic cases, Maxwell’s equations in the core, shell and surrounding media can be written as 
where 
, denote ‘core’, ‘shell’ and ‘surrounding medium’, respectively. When all media are linear, fields can be expressed with Debye potentials. By matching the boundary conditions at r = a and r = b, the scattered field in the surrounding medium and the fields in the core and shell can all be solved for both transverse-electric (TE) and transverse-magnetic (TM) waves (see details in Methods). Since the nanoparticle under consideration is much smaller than the wavelength, the 1st order TM wave is generally sufficient for numerical calculation. More orders can be included if more accurate results are required.
Now, we consider the case in which the coated shell is a Kerr material with weak nonlinearity. The electric displacement 
and the electric field 
in the shell can be written as:
Here 
is the nonlinear permittivity of the shell, which is related to the linear permittivity 
, the nonlinear susceptibility 
and the local electric field intensity 
of the coated shell. To solve the field in the nonlinear shell, we adopt self-consistent mean field approximation12,13,14. The nonlinearity of the shell is very weak, meaning that the linear part 
is much larger than the nonlinear part χs 
. Thus, the nonlinear permittivity of the shell can be expressed as:
where 
corresponds to the average of the field intensity in a linear shell. It can be calculated as:
Following Refs 15 and 16, we obtain,
where 
is the fractional volume of the metallic core, 
. 
and 
are the expansion coefficients of Debye potentials for TM waves. Detailed expressions can be found in Method. After replacing the linear permittivity 
in Eq. (6) with the field-dependent nonlinear permittivity 
in Eq. (4), we can obtain a bistable relation between electric field intensity of the incident wave 
and the average electric field intensity in the nonlinear shell 
.
The optical bistable response of a coated sphere has been studied previously using quasi-static approximation12 which shows the relation between the local field average in the shell and the external applied field as
where
Our results of full-wave solutions will be compared with previous results using quasi-static approximation12 in the next section.
Numerical calculations
In our calculations, we set the linear permittivity of the shell as 
= 2.2 and the metallic core with a permittivity following Drude model17
where 
= 3.7, 
= 8.9 eV and 
= 0.021 eV. The surrounding medium has a permittivity of 
= 1. The relative permeability of the metallic core 
, the shell 
and the surrounding medium 
are all set as unit.
In Fig. 2, we show the scattering efficiency of the linear coated nanoparticle, which is defined as 
(
and 
are scattering coefficients in Method). It can be seen that resonance enhancement can be found when the size of the coated nanoparticle b, or the fractional volume of the core 
, changes. As shown in Fig. 2(a), the enhanced resonance is red-shifted when the size of the coated nanoparticle increases (with a fixed η). In Fig. 2(b), the enhanced resonance is blue-shifted when 
increases (with a fixed size b). The near field properties of the linear coated nanoparticle at different wavelengths are shown in Fig. 3. The excited surface plasmons bring out enhanced field at the resonant wavelength in the shell, justifying the consideration of nonlinearity in the shell.
The scattering efficiency vs wavelength diagrams when (a) 
is fixed at 0.5 and (b) 
is fixed at 10 nm. The permittivity of the media 
.
Distributions of the electric field for (a) 
, (b) 
and (c) 
. The other parameters are
and 
.
Next, we introduce weak nonlinearity into the shell. The nonlinear relative permittivity of the shell is set as18 
, where 
= 2.2, 
= 
. The relations between the electric field amplitude of the incident wave 
and the average electric field amplitude in the nonlinear shell 
for different shell sizes (
) and the relation calculated with quasi-static approximation in different surrounding media (
= 1 and 4) are shown in Fig. 4. The fractional volume of the core is fixed as 
. Bistable responses can be clearly observed. Take 
as an example. The electric field in the shell first increases as the incident field increases from zero. When the incident field amplitude reaches the switching-up threshold field 
(
) for 
(
), the electric field amplitude in the shell will discontinuously jump to the upper stable branch. If the incident field is decreased back from a large value to zero, the electric field in the shell will first decrease continuously and then jump down to the lower stable branch when the incident field amplitude reaches the switching-down threshold field 
(
) for 
(
). Comparing the relation curves for different shell sizes and the curve calculated with quasi-static approximation, we find that the bistable region (the difference value between the switching-up threshold and the switching-down threshold) in terms of the range of input power decreases and the switching-up threshold become lower as the size of the sphere increases. The results for very small nanoparticles match well with those of quasi-static approximation, but some deviation starts to occur when the size of the nanoparticle increases.
The average local field Es as a function of the incident field E0 for various sizes of the sphere.
The relevant parameters are (a) 
(b) 
. And the core-shell-volumes ratio 
and the incident wavelength 
(the resonance wavelength for 
, 
and 
).
Then, we study the influence of the fractional volume of metallic core 
on the bistability. We fix the size of the shell 
. In Fig. 5(a,d) for 
and 
, bistable responses can be clearly seen over the whole range of the fractional volume of metallic core 
. However, when the surrounding medium is changed to possess 
, as shown in Fig. 5(b,e), the bistability disappears when the fractional volume of metallic core 
goes beyond a critical value 
. If we keep 
and lower the wavelength to 
, as shown in Fig. 5(c,f), we find that the bistability disappears when 
decreases below a critical value 
.
The average local field Es as a function of the incident field E0 for various sizes of the sphere.
The size of the coated sphere
, the incident wavelength 
and the permittivity for the media (a) 
(b) 
. In (c), 
and 
. The switching-up and switching-down threshold fields as a function of 
for (d) 
(e) 
and (f) 
.
The shell size can also affect the bistable behavior. As shown in Fig. 6(a), the switching-up and switching-down threshold fields are almost unchanged as the shell size increases for 
. However, as shown in Fig. 6(b), the maximum critical fractional volume of metallic core 
for 
decreases as the shell size increases. As shown in Fig. 6(c), the minimum critical fractional volume of metallic core 
for 
increases as the shell size increases. As a consequence, the bistable region becomes broader as the shell size decreases.
The switching-up and switching-down threshold fields as a function of 
for (a) 
(b) 
and (c) 
.
In Fig. 5(f), we find that the switching-up threshold field increases and the bistable region widens as the fractional volume 
increases when the permittivity of the surrounding medium is 
. On the contrary, in Fig. 5(e), the switching-up threshold field decreases and the bistable region becomes narrow as the fractional volume 
increases when permittivity of the surrounding medium is 
. We thus speculate that there is critical surrounding permittivity 
for such a transition. We plot the electric field amplitude in the shell versus the incident field amplitude for different linear part of the shell permittivity (
). As can be seen in Fig. 7(a–c), the contour plots of 
are nearly trapezoids. Thus, at the critical surrounding permittivity, the contour of the switching-up threshold fields should be vertical. In Fig. 7(a), the critical surrounding permittivity nearly equals 2.4 when 
. When the surrounding permittivity 
[Fig. 7(g)], the switching-up threshold field is nearly unchanged with increasing the fractional volume 
. When the surrounding permittivity is less than the critical one, as shown in Fig. 7 (d), the switching-up threshold field increases with increasing the fractional volume 
. In Fig. 7(j), the switching-up threshold field decreases with increasing the fractional volume 
because the surrounding permittivity is more than the critical one. These properties also apply to the cases when 
, as shown in the lower two rows of panels in Fig. 7.
(a–c) Contour plots of switching-up threshold 
as function of 
and 
. The dark gray region represents that the values are less than the color scale, while those in the light gray region surpass the color scale. (d–l)The average local field 
as a function of the incident field 
. The size of the sphere
and the incident wavelength
. The linear relative permittivity 
for the first row of panels, 
for the second row of panels and 
for the third row of panels.
At last, we fix the incident field amplitude 
and study how the shell size affects the relationship between the average local field in the shell and the incident wavelength. It can be seen from Fig. 8 that, for 
and 
, the switching-up wavelength at the fixed input power blue-shifts when the shell sizes increase, while the switching-down wavelength at the fixed input power red-shifts when the shell sizes increase. In Fig. 8(b), when the wavelength reaches about 775 nm (the switching-up wavelength for 
nm and the switching-up wavelength is about 750 nm for 
nm), the electric field amplitude in the shell will discontinuously jump to the lower stable branch. However, if one decreases the wavelength to about 520 nm (the switching-down wavelength for 
nm and the switching-down wavelength is about 540 nm for 
nm), the electric field amplitude in the shell will discontinuously jump up to the upper stable branch.
The average local field 
versus wavelength for different sizes and the permittivity of the surrounding medium (a) 
(b) 
.
Conclusions
In this reports, we adopt self-consistent mean field approximation within the framework of Mie scattering theory to study the optical bistability of a nonlinear coated metallic nanoparticle. Introducing weak nonlinearity to the shell, we demonstrate numerically that the metallic nanoparticle coated with a nonlinear shell has broad bistable region. We study the effect of the size of the coated spheres, the fractional volume of the metallic core, the permittivity of the surrounding medium, as well as the incident wavelength on the hysteresis loops and the switching-up and switching-down threshold fields. While our results match well with the previous quasi-static results, our full-wave solutions based on Mie scattering are valid in a much wide range of parameters and thus are more suitable for design of optical bistable devices in optimization.
Methods
Debye potentials
We express the Debye potentials of the incident fields19,
The Debye potentials of the scattering wave are
Then, in the shell, the Debye potentials are
In the core, they should be written as
where 
, 
and 
are the Ricatti-Bessel functions and they can be defined by 
, 
and 
. Here, 
, 
and 
are the Bessel functions , Neumann functions and the first-kind Hankel functions. 
are the associated Legendre polynomials. In addition, we denote 
, 
and 
.
Boundary conditions and expansion coefficients of Debye potentials
To solve the coefficients, we apply the boundary conditions on 
and 
. They are
and
Substituting the boundary conditions into the Debye potentials, we can obtain the coefficients as follows,
and
Additional Information
How to cite this article: Chen, H. et al. Optical bistability in a nonlinear-shell-coated metallic nanoparticle. Sci. Rep. 6, 21741; doi: 10.1038/srep21741 (2016).
References
Hurtado, A., Nami, M., Henning, I. D., Adams, M. J. & Lester, L. F. Bistability patterns and nonlinear switching with very high contrast ratio in a 1550 nm quantum dash semiconductor laser. Appl. Phys. Lett. 101, 161117 (2012).
Litchinitser, N. M., Gabitov, I. R., Maimistov, A. I. & Shalaev, V. M. Effect of an optical negative index thin film on optical bistability. Opt. Lett. 32, 151–153 (2007).
Litchinitser, N. M., Gabitov, I. R. & Maimistov, A. I. Optical bistability in a nonlinear optical coupler with a negative index channel. Phys. Rev. Lett. 99, 113902 (2007).
Chen, P. Y., Farhat, M. & Alù, A. Bistable and self-tunable negative-index metamaterial at optical frequencies. Phys. Rev. Lett. 106, 105503 (2011).
Tuz, V. R., Prosvirnin, S. L. & Kochetova, L. A. Optical bistability involving planar metamaterials with broken structural symmetry. Phys. Rev. B 82, 233402 (2010).
Mazurenko, D. A., Kerst, R., Dijkhuis, J. I., Akimov, A. V., Golubev, V. G., Kurdyukov, D. A., Pevtsov, A. B. & Sel’kin, A. V. Ultrafast Optical Switching in Three-Dimensional Photonic Crystals. Phys. Rev. Lett. 91, 213903 (2003).
Almeida, V. R. & Lipson, M. Optical bistability on a silicon chip. Opt. Lett. 29, 2387–2389 (2004).
Dai, X. Y., Jiang, L. Y. & Xiang, Y. J. Low threshold optical bistability at terahertz frequencies with graphene surface plasmons. Scientific Reports 10, 1038 (2015).
Amin, M., Farhat, M. & Bağcı, H. A nonlinear plasmonic resonator for three-state all-optical switching. Opt. Express 22, 6966–6975 (2014).
Ibrahim, A. B. M. A. On the intrinsic optical bistability of a single Kerr nonlinear dielectric layer. J. Appl. Phys. 117, 184101 (2015).
Dai, X. Y., Jiang, L. Y. & Xiang, Y. J. Tunable optical bistability of dielectric/nonlinear graphene/dielectric heterostructures. Opt. Express 23, 6497–6508 (2015).
Gao, L. Optical bistability in composite media with nonlinear coated inclusions. Phys. Lett. A 318, 119–125 (2003).
Yu, K. W., Hui, P. M. & Lee, H. C. Decoupling approximation for strongly nonlinear composites. Phys. Lett. A 210, 115–120 (1996).
Gao, L. & Li, Z. Y. Self-consistent formalism for a strongly nonlinear composite: comparison with variational approach. Phys. Lett. A 219, 324–328 (1996).
Chern, R. L. & Chen, Y. T. Effective parameters for photonic crystals with large dielectric contrast. Phys. Rev. B 80, 075118 (2009).
Chern, R. L. & Liu, X. X. Effective parameters and quasi-static resonances for periodic arrays of dielectric spheres. J. Opt. Soc. Am. B 27, 488–497 (2010).
Kolwas, K., Derkachova, A. & Shopa, M. Size characteristics of surface plasmons and their manifestation in scattering properties of metal particles. J Quant Spectrosc Ra 110, 1490–1501 (2008).
Argyropoulos, C., Chen, P. Y., Monticone, F., D’Aguanno, G. & Alù, A. Nonlinear Plasmonic Cloaks to Realize Giant All-Optical Scattering Switching. Phys. Rev. Lett. 108, 263905 (2012).
Gao, L., Fung, T. H., Yu, K. W. & Qiu, C. W. Electromagnetic transparency by coated spheres with radial anisotropy. Phys. Rev. E 78, 046609 (2008).
Acknowledgements
This work was supported by the NNSF of China No. 11374223, the National Basic Research Program (No. 2012CB921501), the Ph.D. Program Foundation of the Ministry of Education of China (Grant No. 20123201110010) and PAPD of Jiangsu Higher Education Institutions. Y.Z. and B.Z. acknowledge the support from NTU NAP Start-Up Grant, Singapore Ministry of Education under Grant No. MOE2015-T2-1-070 and MOE2011-T3-1-005.
Author information
Authors and Affiliations
Contributions
L.G. conceived the idea. H.C. performed most theoretical and numerical calculations. H.C., Y.Z. and B.Z. analyzed the data. All authors joined discussion extensively and revised the manuscript before the submission.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Chen, H., Zhang, Y., Zhang, B. et al. Optical bistability in a nonlinear-shell-coated metallic nanoparticle. Sci Rep 6, 21741 (2016). https://doi.org/10.1038/srep21741
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/srep21741
This article is cited by
-
Impact of geometry and magneto-optical properties on field enhancement and optical bistability in core–shell nanoparticles
Journal of Theoretical and Applied Physics (2019)
-
Influence of Anisotropy on Optical Bistability in Plasmonic Nanoparticles with Cylindrical Symmetry
Plasmonics (2018)
-
Terahertz optical bistability of graphene-coated cylindrical core–shell nanoparticles
Journal of Theoretical and Applied Physics (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.
