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Nonlocality-Broaden Optical Bistability in a Nonlinear Plasmonic Core-Shell Cylinder Yang Huang, Ya Min Wu, and Lei Gao J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b12546 • Publication Date (Web): 06 Apr 2017 Downloaded from http://pubs.acs.org on April 11, 2017

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The Journal of Physical Chemistry

Nonlocality-Broaden Optical Bistability in a Nonlinear Plasmonic Core-Shell Cylinder Yang Huang†, Ya Min Wu†, and Lei Gao‡ †

School of Science, Jiangnan University, Wuxi 214122, China College of Physics, Optoelectronics and Energy of Soochow University, & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China

‡

ABSTRACT With the self-consistent mean-field method in the framework of full-wave nonlocal scattering theory, we theoretically investigate the optical bistability in the nonlocal metallic nano-cylinder coated with Kerr-type nonlinear shell. Nonlocality enhanced Fano profile is found for this coated cylinder in the linear limit. We illustrate the relation between the linear plasmonic resonant wavelength and the viable parameters for optical bistability in parameter space. It is found that nonlocality will lead to impressive blue shift of the resonant wavelength, hence dramatically increase the bistable region in the parameter space of incident wavelength and

geometrical

factor.

We

demonstrate

the

input-field-controllable

and

input-wavelength-controllable scatterings in the nonlinear case, respectively. It indicates that nonlocal effects show opposite influences on these two nonlinear scattering processes, and the bistability in the scattering spectrum is weaken by nonlocality. Our study reveals that this self-tunable optical resonant scatters can be used as all-optical switches and might provide flexible possibility in the design of optical bitable device. INTRODUCTION Optical bistability (OB) is a typical feature of nonlinear effects1-2, which can provide the optical structures the function to control two distinguishing stale states with the history of the input light. Due to its ability of controlling light with light, OB, as a remarkable all-optical signal processing, has a lot of potential applications in optical communications and computing 3-5

. A typical OB configuration require a light intensity dependent refractive index so that the

optical signal show nonlinear response 3. In this connection, introducing the Kerr nonlinear material

6

is one way to realizing OB. To achieve significant strong OB at small excitation

power, high Kerr nonlinearity is required. Note that conventional Kerr-type materials generally have very weak nonlinear response. Therefore, the certain mechanism of near field enhanced process is adopted to boost the nonlinear response inside the Kerr medium. For this purpose, surface plasmonic resonance7 is usually employed to provide an ultrahigh feedback of the local field intensity, hence enhance the Kerr nonlinearity8. Along this line, with developed self-consistent mean-field approximation and spectral representation method we investigated the OB in nonlinear plasmonic composites9. Most recently, the graphene wrapped nonlinear composite was proposed and we achieved the OB in its near field as well as in the transmission spectra10. Despite the spherical graphene coat, the nanostructured graphene nanoribbons exhibit giant Kerr nonlinearity and plasmonic bistability11.

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Enhanced field intensity by plasmonic resonances however has its limit when the electron-electron interaction in the plasmonic material cannot be neglected12. This happens once the dimension of a single plasmonic structure13-14 or the gap between plasmonic elements15-16 is small. Nonlocality arises from the electron-electron interactions and results in the spatial dispersion of the dielectric response in the material, consequently it will play a further role in the surface plasmons. One simple approach to describe the nonlocal nature of the conduction electrons in plasmonic medium is the hydrodynamic theory (HT)15, 17-19 where the conduction electrons are regarded as constrained gas at the boundary of the plasmonic structure. In fact, more modern approaches to assessing the electron response in plasmonic structure would be the density functional theory (DFT). Recently, a time-dependent density functional theory (TD-DFT) approach has been applied to provide exact calculation of plasmon by employing the jellium model20-23. However, TD-DFT has less numerical efficiency than HF and becomes computationally impossibility for larger systems. Although HT gains its advantage in qualitatively agreement with experimental results15 and could give analytical results16, 24-25, it admits the disadvantage in the absence of spill-out effects electrons and the size-dependent plasmon damping resulting from the surface electron diffusion. To overcome, some extended versions of hydrodynamic model which includes the surface spill-out conduction electrons26-29 and charge diffusion kinetics30-32 have been proposed. On the other hand, nonlocality have showed the reduced field enhancements20, 24, 30, 33-35, so that the nonlinear processes in such plasmon system would also be affected. Actually, influence of surface electrons interaction on the nonlinear current in the gap of the two dimers was reported36. Recently, studies of the nonlocal effects on third harmonic generation in other gap nanostructure are carried out with either TD-DFT37 or hydrodynamic model involving nonlinear polarizability terms38-39 . In addition, experimental evidence has demonstrated the giant Kerr nonlinearity in a nanometer-scale gold quantum well40. We also have performed the theoretical investigation of the nonlocal effects on the OB in a two-dimensional dielectric/metal

composite

within

quasistatic

approximation41,

and

found

the

nonlocality-enhanced OB. In this paper, beyond quasistatic limit, we introduce the self-consistent mean-field approximation in the framework of extended nonlocal electromagnetic theory42 to study the OB in a coated sphere consisting of nonlocal core and Kerr nonlinear shell. We demonstrate a broader parameter space for the OB in this structure due to the nonlocality and study the OB in both near field and far field. THEORETICAL FORMULATION First of all, let us consider the infinite core-shell cylinder consisting of nonlocal metallic core and dielectric shell with inner radius a and outer radius b in vacuum, as shown in Fig.1. The geometric aspect ratio is defined as   a / b , and the shell has a permittivity  s . To precisely model the nonlocal metallic medium, some researchers27-28,

30

have proposed

modified expressions of dielectric response based on hydrodynamic model with semi-classical method. Here, for the sake of simplicity, we adopt conventional linear


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hydrodynamic model as a way to take into account the nonlocal nature of the conductive electrons in the metal, due to the fact that model we present here is out of range of electron spill-out effect and results are qualitatively same. Hence the metallic core possess a Drude permittivity  T for the transverse electric fields and a spatially dispersive permittivity  L for the longitudinal electric fields inside the medium, which are written as 14,

 T ( )= g  p2 / [ (  i )],  L ( , k )= g  p2 / [ (  i )- 2 k 2 ],

(1)

where  g is the background permittivity of the metal corresponding to the interband transition, which consists of the dielectric function of nonlocal metal, p and  denote the plasma frequency and the damping constant of the metal respectively.  denotes the pressure term of the electron gas which is proportional to the Fermi velocity. In addition, the transverse field satisfies the conventional dispersion law kT2  ( / c) 2  T ( ) , while the wave vector of longitudinal electric wave will be determined by the equation  L ( , kL )  0 . Note that, unlike the spherical case43, the excitation of longitude electric mode inside the infinite nonlocal metallic cylinder particularly depends on the polarization of the incident field. Only the transverse magnetic (TM) wave (electric polarization direction is perpendicular to the axis of the cylinder) can excite the longitudinal mode44. The incident electric field can be expanded as

EI  E0



 [i

n2

n

n 

  J n (k0 r ) exp(in )e r  i n 1 J n' (k0 r ) exp(in )e ] , (2) k0 r

and the scattering field is given by

ER  E0



 [i

n 

n2

an n

  H n ( k0 r ) exp(in )e r  i n 1an H n' (k0 r ) exp(in )e ] . (2) k0 r

where J n ( x)  or hn ( x)  is the Bessel (or Hankel) function of the first kind, and

k0   / c indicate the wave vector in the vacuum. Similarly, the field inside the dielectric region is written as,



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ES  E0



 J n ( ks r ) N (k r )  d n n s ]exp(in )er ks r ks r . (3)  ' ' n 1  i [cn J n (ks r )  d n N n (ks r )]exp(in ) e

 i

n 

n2

n[cn



N n ( x) denotes the Bessel function of the second kind, and ks  ( / c)  s is the wave vector in the dielectric shell. The electric transverse and longitudinal waves excited in the nonlocal core are yielded respectively as,

ET  E0

EL  E0



 [i

n2

n 

gn n

  J n ( kT r ) exp(in )e r  i n 1 g n J n' (kT r ) exp(in )e ] , (4) kT r

  J n (kL r ) n 1 ' n2 [ i h J ( k r ) exp( in ) e i h n exp( in ) e ] , (5)      r  n n n L kL r n  

Here, kT (or kL ) is the wave vector of transverse (or longitudinal) wave in the nonlocal core. The corresponding magnetic fields can be directly derived by Maxwell equations and are not presented here. Note that, for the specialty of longitudinal mode, there is no magnetic field related to it.

an , cn , d n , g n , and hn are five unknown coefficients to be determined by imposing the conventional boundary conditions and an additional one, i.e., the vanishing normal component of the exciton polarization vector at the interfaces14. Therefore, the electric fields throughout the whole core-shell structure under TM wave lighting can be achieved. The scattering cross section efficiency of the core-shell cylinder is defined as,

Qsca 

2  2 an . (6)  k0b n 

Next, we would like to focus on the nonlinear field inside the shell region. Eq. (3) gives the general expression of the electric field inside shell region in the linear limit. And the related coefficients we derived is dependent on the linear permittivity  s of the dielectric shell. If the shell is considered as a Kerr-type medium whose permittivity has a field-intensity dependent dielectric function as 2

s   s   s Es , (7)


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then the electric field intensity in the nonlinear shell is complicated and not easy to achieve. To solve this problem, we adopt the self-consistent mean-field approximation9 and present an alternative expression for this dielectric constant function of the Kerr-medium as,

s   s  s E where

E

2 s

2 s

, (8)

is the average of the field intensity in the linear shell and would be

determined by the following 2D volume integral

E

2 s



1 Ss



Ss

2

( Es )dS . (9)

Es in Eq. (3) shows

E

2 s

dependent since s is the function of

E

2 s

. Besides that,

Es has E0 factor in its field expression as well. So that Eq. (9) indicates the self-consistent nonlinear equation with respect to

E

2 s

and E0 in such coated cylinder, which has the

following relation,

E

2 s

 E0

2

f( E

2 s

) . (10)

Eq. (10) reveals the nonlinear dependence of the mean-field

E

2 s

on the incident field

intensity beyond quasistatic approximation. On the other hand, within the quasistatic approximation, Eq. (10) will reduce to

E

2

E0 

2

2 s 2

( A  B ) 2

, (11)

where A and B correspond to the coefficients associated with the general field expressions of the core-shell cylinder in the quasistatic approximation, whose electric potentials has the following expressions,

Vc (r )   E0Cr cos    Vs (r )   E0 ( Ar  B / r ) cos  . (12) V (r )   E  r  D / r  cos  0  h Besides the near field, scattering efficiency in Eq. (6) will be independent on the incident field intensity if the Kerr medium is introduced. In the following part, we will give some calculations based on above derivations to illustrate the optical bistability in this core-shell structure.



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Fig. 1. Scattering efficiency spectra with   0.1 under nonlocal (red) and local (black) descriptions, respectively. The outer radius of the core-shell cylinder is b  10 nm . The insert diagram shows the schematic figure of the present model. RESULTS AND DISCUSSIONS we are now in a position to present the calculation results. Some parameters are employed as follows, the background permittivity  g =3.7 , the plasma frequency p  1.348  1016 s 1 . And the decay length of longitudinal plasmons into metal   (  / c)p with p  c / p is introduced to measure the nonlocal degree24. For the sake of simplicity, we fix the outer radius of the core-shell cylinder as b  10 nm . The linear permittivity of the nonlinear shell is  s =2.2 and the third-order nonlinear coefficient is  s  4.4  1020 m 2 / V 2 . To begin, linear scattering efficiency spectrum of the core-shell cylinder is investigated as shown in Fig. 1 with the aspect ratio  =0.1 . Viewing Fig. 1, sharp Fano profile is found whose curve goes from the dip to the peak within a very narrow band. Similar Fano curves could been seen in metallic nano shells with small inner radius45, which raised from the coupling between the narrow-band dipole dark mode and broad-band off-resonant dipole bright mode. And a smaller inner radius would lead to a sharper Fano curve. For the present core-shell cylinder with metallic core and dielectric shell, Fano curve exists as well but with different coupling mechanism. We use quasistatic approximation as examples to illustrate. Coefficient D in Eq. (12) is proportional to the polarization  of the core-shell cylinder. The surface plasmon resonance is mathematically determined by the denominator of  equaling to zero. On the contrary, numerator of  tending to zero indicates non-polarization and shows cloaking state. When the aspect ratio  0 , the resonant condition and the non-resonant (cloaking) condition will be simultaneously satisfied by


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 s   c  0 . (13) In this small aspect ratio limit, the clocking and resonant states will degenerate, hence forming a sharp Fano profile in its scattering efficiency spectrum as shown in Fig. 1. And this Fano resonance arises from two interference states, one is the dipole mode associated with the surface plasm resonance at the inner interface, the other is the bulk resonance of the dielectric shell46. Recently, another kind of Fano profile which arise from the coupling of quadrupolar mode and the gap mode was found in nonlocal dimer structure with sub-nanometer gap47. As a conventional nonlocal effect, nonlocality of the metallic core would result in the dramatic blue shift of the Fano peak in its spectrum. And this blue-shift will become more obvious when the nonlocal core is small since the nonlocal nature of the conductive electron inside the metallic core begin to dominate. Hence, unlike the local case, the resonant frequency shows non-monotonous dependence on the aspect ratio as shown in Fig. 2. Nonlocality in plasmonic structure generally come along with the diminished near-field enhancement and weaken far-field scattering. However, Fig. 1 reveals an enhanced scattering efficiency spectrum in the nonlocal description which can be used to boost the nonlinear effects in the shell.

Fig. 2. log10 (Qsca ) as functions of incident wavelength and aspect ratio for core-shell cylinder in (a) nonlocal and (b) local case. White and dark curves indicate the maximal and minimal values respectively, denoting the cloaking and resonant states.



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Fig. 3. Dependence of the electric field intensity Es inside the shell on the external field intensity E0 with (a)   0.1 and

(b)  =0.6 . The incident wavelength   345 nm ,

and degree of nonlocality is   3.5 103 p a value appropriate to silver. Next, we investigate the OB in near field, i.e., the nonlinear dependence of the field intensity in shell region on the incident field based on Eq. (10). Fig. 3 illustrate the bistable curves in nonlocal and local cases with different aspect ratios. The bistability in quasistatic limit is plotted as well to make a comparison. In the case of low aspect ratio (  =0.1 ), the difference between nonlocal and local bistable curves are dramatic when they share the same incident wavelength. And the quasistatic approximation fits well with the local case since the size of the nonlocal core is in deep subwavelength regime. If the aspect ratio increases, quasistatic approximation shows obvious deviation from the full wave description as shown in Fig. 3(b). The quasistatic approximation fails to give accurate predictions even when the total size of the core-shell cylinder is still in deep subwavelength. It reveals the importance of adopting full wave theory beyond the quasistatic approximation to study the OB in such compact core-shell structure. As to the influence of nonlocality, high aspect ratio seems to reduce the deviation between nonlocal and local results, however it is not a linear dependence which we shall demonstrate in the following apart. Generally, nonlocality will give rise to higher switching-up and switching-down threshold fields, and provide broader bistable region in E0 . Besides that, it results in higher upper branch Es in which we might achieve a very high field inside the shell region with a low external field. In Fig. 4, bistable curves with different degrees of nonlocality are plotted. We found heavy degree of nonlocality exhibits high threshold bistability.


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Fig. 4. Dependence of the electric field intensity Es inside the shell on the external field intensity E0 with various nonlocal degree  .

  0.1 and   345 nm .

In above discussions, we choose the some specific physical and geometrical parameters as examples for nonlocal and local cases to show the general nonlocal effects on OB in near field. It should be emphasized that OB request the parameters satisfying the specific conditions48. Otherwise, it wouldn’t occur. In our model, nonlinear equation in Eq. (10) contains a 2D integral, and it is difficult to present its fully analytical expression. To give a qualitative illustration, we will adopt the quasistatic version of the nonlinear equation in Eq. (11). By Expanding Eq. (11), it shows a seven-order equation with respect to Es , which is different from the

cubic one in the previous work48. The valid parameter region for

bistability can be achieved in Fig. 5 with numerical methods. Besides that, Fig. 5 also shows the results of the nonlocal quasistatic case which we report it elsewhere49. It demonstrates that the valid parameters space for OB in nonlocal case is much broader than the local one, especially in the low  region. Although derived by the quasistatic approximation, its derivation from full wave calculation is not very large at low  , thus Fig. 5 provides a clear diagram indicating the bistable region in the parameter space for both nonlocal and local cases. If we compare Fig. 5 and Fig. 2, one might find the valid parameters in Fig. 5 lie on the right side of the bright line in Fig. 2. And the critical boundary for bistability have almost same position as the surface plasma resonant lines. That means only the incident field with lower photonic energy than these would excite the SPRs could have the opportunity to show OB. Hence it provides us a way to search the potential OB in this kind of core-shell cylinder by checking its resonant wavelength. Similar conclusion was made in the graphene-coated dielectric particle10 with a different configuration.



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Fig. 5. Valid Regions for OB in parameter panels in (a) nonlocal and (b) local cases, filled with red and black colors respectively. To further demonstrate, we plot the switching threshold fields as the function of aspect ratio at two different incident wavelength in Fig. 6. The upper dotted linear of each part denotes the switching-up threshold field and the lower one is the switching-down threshold field. The gap between these two lines indicates the bistable region. We find two branches for bistable region in the nonlocal case when  =348nm , however the local case only possess one. When the incident wavelength is increased, two branches touch each other and forms a continuous band [see Fig. 6(b)]. It confirms the conclusion we made in the previous part that nonlocality give rise to a broader bistable region in the geometrical parameter space. The main advantage in the nonlocal case is the appearance of the additional bistable region in low aspect ratio which could not be found in the local case. On the other hand, however, the switching threshold fields under nonlocal description are very high, especially in low aspect ratio, which requires higher input field intensity to exhibit OB.

Fig. 6. Bistable region in the parameter panel of aspect ratio with incident wavelength (a)

  328nm and (b)   335nm respectively. Upper boundary of each bistable region


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denotes the switching up threshold field and the lower one indicates the switching down threshold field. We introduce Kerr-type medium as the core material in the nonlocal core-shell cylinder to achieve tunable near field intensity inside the shell with bistability. Note that once Kerr medium is adopted, the scattering coefficient should be influenced, and the scattering efficiency in Eq. (6) shows field intensity dependence as well. These two processes happen simultaneously and the nonlinear relation between the near fields in Eq. (11) is the key to derive the tunable scattering controlled by incident field intensity. In Fig. 7, we plot the scattering efficiency as a function of the incident field intensity. The nonlinear process in scattering signal shows a more complicated variation on E0 which indicates very different physical properties in the core-shell cylinder. With the present incident wavelength, these nonlinear curves only exist in the nonlocal cases. It can be demonstrated in Fig. 6 that with

 =0.1 and   328nm (or   335nm ), no bistability occurs in near field for local case, hence no multi-solutions correspond to one input intensity. Consequently, local curve in scattering spectrum merely exhibit linear dependence on E0 . However, if we increase the aspect ratio, nonlinear dependence of scattering on incident field would occur even in local case, although not shown here. In view of the technological application, this founds might be useful in the optical tunable sensors.

Fig. 7. Scattering efficiency versus incident field intensity with (a)   328nm and (b)

  335nm . The aspect ratio here is chosen as  =0.1 . In the end, dependence of the far-field scattering property on the incident wavelength in nonlinear case is investigated as shown in Fig. 8. We choose the Fano profiles in Fig. 1 to illustrate the influence of the Kerr nonlinearity on the scattering singles. Like the nonlinear scattering efficiency various with the incident field intensity at a fixed wavelength, scattering



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spectrum also show complicated nonlinear loops in both nonlocal and local cases when E0 is high. Although nonlocality could offer more parameter space for bistability as we demonstrated in previous part, here, nonlocal scattering curves in spectrum seems to need higher input field intensity to exhibit nonlinear dependence on the incident wavelength. For instance, it starts to show nonlinear loops in the local case when E0  2 108 V / m , however, in nonlocal case it doesn’t. Moreover, for larger intensity, the range of nonlocal scattering loops are smaller than the local ones as expected [see the blue and red curves in Fig. 8]. Besides that, we found that increasing E0 would merely make the nonlinear loops broaden but the maximal scattering signal is not affected. This phenomenon is quite different from the case of Kerr core-nonlocal shell cylinder41, whose maximal scattering value in nonlinear scattering spectrum is decreased with an increased E0 . Unlike the broaden Fano peak, the Fano dip shows slight variation with E0 . Fig. 8 indicate that, besides the input intensity, the operation of the wavelength could as well provide nonlinear switches of scattering signals from two different states in such core-shell cylinder.

Fig. 8. Scattering efficiency versus incident wavelength at different incident field intensities in (a) nonlocal and (b) local cases. the aspect ratio is   0.1 . It should be remarked that in the present work, we only consider the core-shell cylinder with nonlocal core and Kerr dielectric shell. The nonlocal core is described with a spatial dispersive dielectric function in the linear regime, and the dielectric shell has nonlinear response but without any nonlocality. We neglect the nonlinear hydrodynamics of the conduct electrons inside nonlocal metal which could result in the second harmonic generation and wave-mixing effects of the electromagnetic responses50-51. If the nonlinear polarizability terms


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of the electronic hydrodynamic function is taken into account, the optical bistability might exist in both fundamental and second-harmonic wavelengths as expected. But it needs further demonstrate which is beyond what we talk about in this article. As to its inverse geometry where the Kerr material is contained inside a thin metallic tube, whose linear model is similar to the structure in Ref.52, its optical bistability in near and far fields was studied partially in our previous work41 but within the quasistatic approximation. Due to the geometric difference, Fano profiles in these two cases have different origins. However the nonlocality of the metal in two geometries leads to the same blue-shift resonant wavelength, hence dramatically increase the bistable region in the parameter space of incident wavelength and geometrical factor. Moreover, the bistability in the scattering spectrum of both cases is weaken by nonlocality.

Fig. 9. log10 (Qsca ) as functions of incident wavelength and aspect ratio for core-shell cylinder in (a) nonlocal and (b) local cases with size-dependent damping. (c) and (d) show the valid regions for OB in parameter panels in nonlocal and local cases with size-dependent damping, filled with red and black colors respectively. In the above calculations, we neglect the size-dependent damping of the metal. To illustrate the size-dependent damping effects on the OB of core-shell cylinder, we replace the damping constant  of the metal with the size-dependent one, i.e.,  +A F / a where A is geometric constant30. It is shown that the size-dependent damping will result in the broadening of the resonant peaks in the scattering efficiency as shown in Fig. 9 (a) and (b). Because the size-dependent damping is proportional to 1/ a , the broadening effect is dramatic especially in the regime with small aspect ratio. It will lead to the weak resonant scattering efficiency and the reduced parameter space for OB which is illustrated in Fig. 9 (c) and (d). In this case, the additional branch of bistable region shown in Fig. 6 with small aspect ratio will disappear. Nevertheless, the bistable curves in scattering efficiency versus incident field intensity and in scattering efficiency versus incident wavelength still exist in the valid regions for OB in parameter panels. The conclusions of nonlocal effects on the optical bistability we make in the previous parts remain the same. In general, the broadening effects



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of the size-dependent damping will result in larger switching thresholds and smaller bistable region the due to the large damping in small scale as shown in Fig. 10.

Fig. 10. Dependence of the electric field intensity Es inside the shell on the external field intensity E0 with (red solid) and without (black solid size-dependent damping. The incident wavelength   345 nm , aspect ratio  =0.6 and degree of nonlocality is   3.5 103 p . CONCLUSIONS To conclude, we incorporate the full-wave extended nonlocal scattering theory with the self-consistent mean-field approximation to study the optical bistability of a nonlocal core-Kerr shell nano-cylinder in both near-field and far-field. The nonlinear equation which describe the dependence of the average field in the nonlinear shell on the external field is derived, with which we establish the nonlinear relation between the scattering efficiency and the incident field. In the linear limit, we find the Fano profile in the scattering spectrum of this coated cylinder and the nonlocality enhanced surface plasmon resonance which can boost the Kerr nonlinearity in the shell region. By comparing results of the quasistatic approximation and the present method, we found deviation still exist in the bistable near field even the core-shell cylinder is in deep subwavelength range. And this deviation become more obvious when size of nonlocal core is increased. Under the same parameter, nonlocality will generally give rise to high switching threshold fields and wide hysteresis loop in bistable near field. And this tendency will become dramatic with higher degree of nonlocality. For the sake of simplicity, we numerically demonstrate the valid parameters region for bistablity in the parameter space of incident wavelength and aspect ratio by quasistatic model. It is found that these bistability could only exist with a longer wavelength than its linear resonant wavelength. Therefore, the blue shift of the resonant wavelength caused by nonlocality will dramatically increase the bistable region in the geometrical parameter space, providing more flexible possibility in the design of optical bistable device. Besides the bistable near field, scattering efficiency shows the nonlinear dependence on input field


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intensity as well. Due to the adoption of Kerr medium, scattering efficiency is determined by the mean-field in the shell, hence the criterion of exhibiting bistability is valid for bistable far field. Next, We investigate the nonlinear loop of scattering efficiency in the function of incident wavelength. Unlike the variation in the input field, Fano profile in wavelength spectrum is easy to show nonlinear bistable loop in local case than that in nonlocal one, indicating the nonlocality weaken bistability. In the end, we find the size-dependent damping will result in larger switching thresholds and smaller bistable region. The additional branch of bistable region in small aspect ratio regime will disappear due to the large damping in small scale. As to a practical relevance, namely, damage thresholds, it is not easy to achieve although evidence shows that technologically the threshold can be up to ~1010 V/m 11. Nevertheless, the method we discussed and the structure we investigated in this work still provides a new way to achieve nonlinear switches of scattering signals. AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] *E-mail: [email protected] Notes The authors declare no competing financial interest. ACKNOWLEDGMENTS This paper was supported by the National Natural Science Foundation of China (No. 11374223, No. 61378037,), National Science of Jiangsu Province (Grant No. BK20161210), Qiang Lan project, “333” project (Grant No. BRA2015353), and the Fundamental Research Funds for the Central Universities (Grant No. JUSRP11724). REFERENCES (1) Shen, Y. R. Recent Advances in Optical Bistability. Nature 1982, 299, 779-780. (2) Wurtz, G. A.; Pollard, R.; Zayats, A. V. Optical bistability in nonlinear surface-plasmon polaritonic crystals. Phys. Rev. Lett. 2006, 97, 057402. (3) Hu, X.; Jiang, P.; Ding, C.; Yang, H.; Gong, Q. Picosecond and low-power all-optical switching based on an organic photonic-bandgap microcavity. Nat. Photonics 2008, 2, 185-189. (4) Sheng, J.; Khadka, U.; Xiao, M. Realization of all-optical multistate switching in an atomic coherent medium. Phys. Rev. Lett. 2012, 109, 223906. (5) Yuan, H.; Jiang, X.; Huang, F.; Sun, X. Ultralow threshold optical bistability in metal/randomly layered media structure. Opt. Lett. 2016, 41, 661.



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