Optical bistability in a nonlinear-shell-coated metallic nanoparticle

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.

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.
Optical bistability has attracted remarkable attention in recent years because of its promising applications in logic functions 1 , metamaterials [2][3][4][5] , all-optical switching 6 and low power lasing 7 . An optical bistable system has two stable transmission states to switch on and off, depending upon the history of input signal [8][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 intensity 11 .
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 scale 12 . 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 ε c and µ c (ε s and µ s ). The surrounding medium has a relative permittivity ε m and a relative permeability µ m . We assume that the incident plane wave propagates in +z direction, with electric field polarized in x direction:  are the expansion coefficients of Debye potentials for TM waves. Detailed expressions can be found in Method. After replacing the linear permittivity ε s in Eq. (6) with the field-dependent nonlinear permittivity ε  s in Eq. (4), we can obtain a bistable relation between electric field intensity of the incident wave E 0 2 and the average electric field intensity in the nonlinear shell 〈| | 〉 E s 2 . The optical bistable response of a coated sphere has been studied previously using quasi-static approximation 12 which shows the relation between the local field average in the shell and the external applied field as where Figure 1. 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 x-direction and propagates along z-direction. The relative permittivity and permeability of the core (the shell) are ε c and µ c (ε s and µ s ). The surrounding medium has a relative permittivity ε m and a relative permeability µ m .
Scientific RepoRts | 6:21741 | DOI: 10.1038/srep21741 Our results of full-wave solutions will be compared with previous results using quasi-static approximation 12 in the next section.
Numerical calculations. In our calculations, we set the linear permittivity of the shell as ε s = 2.2 and the metallic core with a permittivity following Drude model 17 where ε ∞ = 3.7, ω p = 8.9 eV and γ = 0.021 eV. The surrounding medium has a permittivity of ε m = 1. The relative permeability of the metallic core µ c , the shell µ s and the surrounding medium µ m are all set as unit.
In Fig. 2, we show the scattering efficiency of the linear coated nanoparticle, which is defined as 2 (A n TM and A n TE 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 η, 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 η 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.
Next, we introduce weak nonlinearity into the shell. The nonlinear relative permittivity of the shell is set as 18  0 4. Bistable responses can be clearly observed. Take = b 10 nm 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 , 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 ). 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.
Then, we study the influence of the fractional volume of metallic core η on the bistability. We fix the size of the shell = b 10 nm. In Fig. 5(a,d) for λ = 445 nm and ε = 1 m , bistable responses can be clearly seen over the whole range of the fractional volume of metallic core η. However, when the surrounding medium is changed to possess  Fig. 5(b,e), the bistability disappears when the fractional volume of metallic core η goes beyond a critical value η = .
0 53 c . If we keep ε = 1 m and lower the wavelength to λ = 365 nm, as shown in Fig. 5(c,f), we find that the bistability disappears when η decreases below a critical value η = .
0 48 c . 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 λ ε = , = 445 nm 1 m . However, as shown in Fig. 6(b), the maximum critical fractional volume of metallic core η c for λ ε = , = 445 nm 4 m decreases as the shell size increases. As shown in Fig. 6(c), the minimum critical fractional volume of metallic core η c for λ ε = , = 365 nm 1 m increases as the shell size increases. As a consequence, the bistable region becomes broader as the shell size decreases.  In Fig. 5(f), we find that the switching-up threshold field increases and the bistable region widens as the fractional volume η increases when the permittivity of the surrounding medium is ε = 1 m . On the contrary, in Fig. 5(e), the switching-up threshold field decreases and the bistable region becomes narrow as the fractional volume η increases when permittivity of the surrounding medium is ε = 4 m . We thus speculate that there is critical surrounding permittivity ε mc 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 (ε = . , . , . 1 1 2 2 3 3 s ). As can be seen in Fig. 7(a-c), the contour plots of − E up 0 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 ε = .  Fig. 7(g)], the switching-up threshold field is nearly unchanged with increasing the fractional volume η. 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 η. In Fig. 7(j), the switching-up threshold field decreases with increasing the fractional volume η because the surrounding permittivity is more than the critical one. These properties also apply to the cases when ε = . , . , 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 ε = 1 m and 4, 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 = b 5 nm and the switching-up wavelength is about 750 nm for = b 15 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 = b 5 nm and the switching-down wavelength is about 540 nm for = b 15 nm), the electric field amplitude in the shell will discontinuously jump up to the upper stable branch.

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 fields 19 ,    In the core, they should be written as Substituting the boundary conditions into the Debye potentials, we can obtain the coefficients as follows,