Optical bistability involving planar metamaterial with a broken structural symmetry

Vladimir Tuz1,2, Sergey Prosvirnin1,2, Lyudmila Kochetova1

1Institute of Radio Astronomy of NASU, Kharkov, Ukraine2V.N. Karazin Kharkov National University, Kharkov, Ukraine

Abstract: We report on a bistable light transmission through a planar metamaterial consisted of a metal pattern of weakly asymmetric elements placed on a nonlinear substrate which enables sharp resonance response by excitation of a trapped mode. A feedback required for bistability is provided by the coupling between strong trapped mode resonance antiphased currents exited on the metal elements and an intensity of inner field in the nonlinear substrate.

Optical transmission bistability is a phenomenon whereby a system changes its transmission from one value to another in response to properties of a light passing through [1]. The switch between the two stable states of transmission is usually induced by an intensity of the incident light, and luckily the switch may exhibit hysteresis. The light intensity driven systems must have an intrinsic nonlinear susceptibility and feedback to provide an optical bistability. Issue of the day is reducing the size of such all-optical devices, decreasing switching times and the required power of light.

Typically all-optical devices include an electromagnetic cavity (e.g. a Fabry-Perot resonator) to provide a feedback, although systems without a cavity can operate successfully too. An example of such structures is the devices based on surface plasmon-polaritons in metal nanostructures [2]. Because the surface plasmons can cause significant electric-field confinement and enhancement in subwavelength structures, they are helpful to stronger nonlinear effects and reduce the size of all-optical devices.

There is another promising way to develop the small-size all-optical switching devices without using of electromagnetic cavities and surface plasmons. Here the case in point is the planar metamaterials (also known as metafilms) with active constituents. Typically these systems are surfaces which consist of some metal or dielectric resonance elements arranged in periodic array and placed on a layer with thickness small in comparison to the wavelength. One approach to obtain the nonlinear response of the planar metamaterial consists in introducing nonlinearity in the properties of individual resonance elements of the structure. Thus in [3] the elements are made tunable and nonlinear by the insertion of diodes with voltage-controlled capacitance. However, in the optical range the manufacturing of such structures is associated with considerable technological difficulties. Other simpler way is to arrange convenient resonance elements on a nonlinear substrate.

The main feature of the planar metamaterials essential for optical switching applications is a resonance character of their

transmission and reflection spectra. The excitation of high-quality factor trapped mode resonances in planar double-periodic structures with a broken symmetry was shown both theoretically [4] and experimentally [5] in microwaves. In particular, these typical peak-and-trough Fano spectral profile resonances are excited in the periodic structure consisted of asymmetrically split metal rings. A small asymmetry of the metal elements of such structure results in excitation of the strong mode of anti-phased currents, which provides low radiation losses and therefore high Q-factor resonances. Recently, the trapped mode resonances were investigated in similar planar structures in the near-IR range [6]. It was shown that the special choice of geometry parameters of the structure enables to increase the Q-factor of the trapped mode resonance by several times in comparison with the ordinary plasmon-polariton resonance. Such high-Q resonance regime is promised to observe bistability in the near-IR range if the structure will include nonlinear material.

In this report, we propose and study an all-optical switching device based on the bistability in a planar metamaterial made of complex shaped resonance particles placed on a substrate of nonlinear material.

Fig. 1. Fragment of planar metamaterial and its elementary cell. The size of the square translation cell is x yd d d

600 nm . The asymmetrically split metal rings are placed on the top face of 0.1h d thick nonlinear dielectric layer. The radius and the width of metal rings are 0.4a d and 2 0.06w d respectively. The angle sizes of ring splits are

01 15 and 0

2 25 .

The studied structure consists of identical subwavelength metal inclusions structured in the form of asymmetrically split

978-1-4244-6997-0/10/$26.00 ©2010 IEEE

LFNM*2010 International Conference on Laser & Fiber-Optical Networks Modeling, 12-14 September, 2010, Sevastopol, Ukraine

107

rings (ASRs), which are arranged in a periodic array and placed on a thin nonlinear dielectric substrate (see Fig. 1). Each ASR contains two identical strip elements opposite one to another. The right split between the strips 1 is a little different from the left split 2 , so that the unit cell is asymmetric with regards to the y-axis. Suppose y-polarized plane wave normally impinges on the structure. The incident field has a real amplitude A and a frequency .

The algorithm based on the method of moments was proposed earlier [4] to study the resonance nature of the structure response under the assumption so small A that a dependence of the substrate permittivity on the field intensity is vanishingly small.

The algorithm is constructed in such a way that, at the first step, the surface current induced in the metal pattern by the field of the incident wave is calculated. The metal pattern is treated as a perfect conductor, while the substrate is assumed to be a lossy dielectric. In particular, assuming 1A the magnitude of the average current I along the single element can be determined as some function

( , )I Q . (1) At the second step, the found surface current distribution is used to calculate transmission and reflection coefficients as

( , )t t , ( , )r r . (2) To introduce the nonlinearity (the third-order Kerr-effect),

let us assume that permittivity of the substrate depends on the intensity of the electromagnetic field inside it. The inner intensity, in its turn, is directly proportional to the square of the average current magnitude in the metal pattern. Thus the substrate permittivity is further given as

21 2 | |I . (3)

Note, since the substrate permittivity is proportional to the current value in the metal pattern, the nonlinearity effect reaches the maximum under the ASR resonance condition.

If the amplitude A of the incident field differs from the unity, the corresponding average current magnitude for a given

can be found using the expression (2) as ( , )I A Q . (4)

Since the substrate permittivity is dependent on the average current value I , the relation (4) can be rewritten as follows

21 2( , | | )I A Q I . (5)

The expression (5) is a nonlinear equation related to the average current value in the metal pattern. The incident field magnitude is a parameter of the equation (5). At the fixed frequency , the solution of this equation is the average current value in the dependence on the magnitude of the incident field

( )I I A , (6) where the function ( )I A presumably is multivalued.

On the basis of the found current ( )I A , it is possible now to determine the permittivity of the nonlinear substrate

21 2 | ( ) |I A , (7)

and to calculate the reflection and transmission coefficients

21 2

21 2

( , | ( ) | ),

( , | ( ) | ),

t t I Ar r I A

(8)

as the functions of the magnitude of the incident field.

Fig. 2. The frequency dependence of the magnitude of the transmission coefficient (a) and the square of the average current magnitude (a.u.) in the metal pattern (b) in the case of the linear permittivity ( 2 0 ) of the substrate.

At first we consider a transmission through an array of ASRs placed on a linear substrate (Fig. 2) [4], [5]. If a normal incident wave is polarized in y-direction, at the dimensionless frequency nearly æ 0.3xd a sharp reflection resonance occurs. This resonance corresponds to a trapped mode because equal and opposite directed currents in the two arcs of each complex particle of array radiate a little in free space. The resonance has high quality factor, and the current magnitude reaches the maximum at this frequency. With increasing the permittivity of the substrate, the resonance frequency is shifted to low values. Note if the incident field is x-polarized or splits between the strips are the same, a symmetric current mode is excited only. The corresponding resonance has low quality factor, and this configuration is not suitable to observe a bistability because the current magnitude is small at this frequency.

LFNM*2010 International Conference on Laser & Fiber-Optical Networks Modeling, 12-14 September, 2010, Sevastopol, Ukraine

108

Fig. 3. The along split ring average of the square current magnitude in a.u. (a) and the magnitude of the transmission coefficient (b) versus the incident field magnitude in the case of the nonlinear permittivity ( 1 4.0 0.02i , 3

2 5 10 ) of the substrate.

Fig. 4. The frequency dependence of the transmission coefficient magnitude in the case of the nonlinear permittivity ( 1 4.0 0.02i , 3

2 5 10 ) of the substrate.

Suppose that the trapped mode resonance frequency is slightly higher than the incident field frequency. As the intensity of the incident field rises, the magnitude of currents on the metal elements increases. This leads to the increasing the field strength inside the substrate and increasing its permittivity. As a result the frequency of the resonant mode decreases and shifts toward the

frequency of incident wave which, in its turn, further enhance the coupling between current modes and the inner field intensity in the nonlinear substrate. This positive feedback increases the slope of the rising edge of the transmission spectrum as compared to the linear result. As the frequency extends beyond the resonant mode frequency the inner field magnitude in the substrate decreases and the permittivity back towards its linear level, and this negative feedback keeps the resonant frequency close to the incident field frequency.

The curves in Fig. 3a are a plot of the along ASR element average of the square current magnitude versus the incident field magnitude at a fixed frequencies. As an example we consider the incident field frequency æ 0.301 (the solid line). As the incident field amplitude increases, the current magnitude (and proportionally inner field intensity) gradually increases along the bottom branch of the curve until it reaches about 2| | 2I . At this point, the current magnitude jumps to around 2| | 5.5I due to the instability of the interior branch of the curve. Decreasing of the incident field magnitude results the square current magnitude follows the upper branch of the curve down to

2| | 4.5I , and drops to value at about 2| | 0.5I . This dramatic current variation from high to low level produces switching from high to low transmission (see Fig. 3b)

The frequency dependence of transmission coefficient magnitude also manifests impressive discontinuous switches to different values of transmission with increasing and decreasing frequency in the resonance range at the sufficiently large intensity of the incident wave (see Fig. 4).

In conclusion, a planar nonlinear metamaterial consisted of a metal array placed on a nonlinear dielectric substrate which enables sharp resonance response by excitation of trapped mode due to broken symmetry of the pattern is challenging object for all-optical switching applications.

REFERENCES

[1] H. M. Gibbs, Optical bistability: controlling light with light, Academic Press, Orlando, Fla., 1985.

[2] C. Min, et al., “All-optical switching in subwavelength metallic grating structure containing nonlinear optical materials”, Opt. Lett., vol. 33, no. 8, pp. 869-871, 2008.

[3] D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear electric metamaterials”, Appl. Phys. Lett.,vol. 95, 084102, 2009.

[4] S. Prosvirnin, S. Zouhdi, “Resonances of closed modes in thin arrays of complex particles”, in Advances in Electromagnetics of Complex Media and Metamaterials,edited by S. Zouhdi et al., Kluwer Academic Publishers, the Netherlands, pp. 281-290, 2003.

[5] V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry”, Phys. Rev. Lett., vol. 99, 147401, 2007.

[6] V. V. Khardikov, E. O. Iarko, and S. L. Prosvirnin, “Trapping of light by metal arrays”, J. Opt., vol. 12, 045102, 2010.

LFNM*2010 International Conference on Laser & Fiber-Optical Networks Modeling, 12-14 September, 2010, Sevastopol, Ukraine

109