NLP*2011 1st International Workshop on Nonlinear Photonics, 6-8 September, Kharkov, Ukraine

Fano resonances in two-layered planar nonlinear metamaterial with a fish scale structure

Pavel L. Mladyonov1, Sergey L. Prosvimin1,2, and Vladimir R. Tuz1,2 1 Institute of Radio Astronomy of National Academy of Sciences of Ukraine, 4, Krasnoznamennaya st., Kharkiv 61002, Ukraine

2School of Radio Physics, Karazin Kharkiv National University, 4, Svobody Square, Kharkiv 61077, Ukraine.

Ahstract- We report on a bistable light transmission through two-layered structure which consists of two gratings of planar perfectly conducting infinite strips placed on the both sides of a dielectric slab. It is demonstrated that a switching may be achieved nearly the frequency of the high-quality-factor Fano­shape trapped-mode resonance excitation.

During the past decade there is a growing interest in theo­

retical and experimental studies of different types of resonant

wave phenomena associated with manifestation of the classical

Fano resonance in nanoscale devices [1]. This interest is

because the unique profile of Fano resonance which typically

exhibits a sharp asymmetric line shape. In the optical systems

it appears as the transmission (reflection) spectra varying from

o to lover a very narrow frequency range. Especially much

attention is given to the features of Fano resonances in the

structures which consist of nonlinear components. They can

be use to achieve the bistable transmission at low input powers,

due to a large quality factor of the Fano resonance, which has

been recently demonstrated for all-optical switching operation

in photonic crystal microcavities [2] and plasmonic devices

[3].

The main idea of using Fano resonance for all-optical

switching and other operations based on the bistability lies

in the fact that the introducing an element with nonlinear

characteristic into a system yields the nonlinearity-induced

shift of the resonance. Therefore, due to the nonlinearity, it

is possible to tune the location of the resonance by changing

the intensity of the input waves. The main advantage is that

such form of resonance allows us to obtain great amplitude

of switching since there are gently sloping bands of the

high reflection and transmission before and after the resonant

frequency.

These peculiarities of the nonlinear Fano resonances were

also investigated theoretically in two kinds of planar meta­

material in the forms of both asymmetrically split rings [4]

and two concentric rings [5]. The substrate which carries the

metallic pattern is considered as a Kerr nonlinear dielectric.

In such structures the strong mode of antiphased currents

(trapped-mode), which provide low radiation losses and there­

fore high Q-factor resonance which has a Fano shape can be

excited. The effect of nonlinearity appears as the formation of

a closed loops of bistable transmission within the frequency of

trapped mode resonance. Since the nonlinear response of the

metamaterial operating in the trapped-mode regime extremely

978-1-4577-0479-6/11/$26.00 ©2011 IEEE

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sensitive to the dielectric properties of the substrate it allows

us to control switching operation effectively.

Another type of metamaterials which bears Fano-shape

trapped-mode resonance and is very promising for applications

is a planar metamaterial which consists of equidistant array

of continuous meandering metallic strips on a thin dielectric

substrate (fish scale structure, [6]). In the past this structure

has been investigated both theoretically [7] and experimentally

[6]. It is revealed that the fish scale structure in the case when

the wave is polarized orthogonally to the strips is strongly

transparent across a wide spectral range apart from isolated

wavelength. In the case when the wave is polarized along

the strips and the structure is combined with a homogeneous

metallic mirror it becomes a good broad-band reflector apart

from at an isolated wavelength where reflectivity is small due

to absorption in the substrate. Especially at this wavelength,

there is no phase change of the reflected wave with respect

to the incident wave. The latter phenomenon is known as

a "magnetic mirror". Finally, the structure also acts as a

local field concentrator and a resonant multifold "amplifier"

of losses in the constitutive dielectric.

The Fano-shape trapped-mode resonance appears in such

fish scale structure when the form of strips is slightly different

from straight line in the case when the grating is excited

with the wave polarized orthogonally to the strips. The Q­

factor of this resonance is higher then that ones of convenient

resonances exited by the grating. The less the form of grating

is different form the straight line, the greater the quality

factor of the trapped-mode resonance is. But, unfortunately,

as the quality factor rises, the resonant frequency shifts to

the frequency where the Rayleigh anomaly occurs and the

field localization decreases. This fact reduces advantages of

the structure. Fortunately, the two-layered planar fish scale

metamaterial is deprived of this drawback [8].

Furthermore, in the two layered structure, besides the res­

onances excited by each grating, there are interference reso­

nances which appear in the similar manner as in the structure

of straight line gratings. Therefore such system allows us to

obtain different resonant features. The most important thing is

that in the two-layered structure the trapped-mode resonances

are excited at the frequencies which lie far from the frequency

of the Rayleigh anomaly and these resonances have quality

factor which is sufficiently greater than that ones of the

interference resonances. We should note that the trapped-mode

NLP*2011 1st International Workshop on Nonlinear Photonics, 6-8 September, Kharkoo, Ukraine

Since the substrate permittivity e depends on the averagecurrent value lin, the relation (2) can be rewritten as follows

(3)

(2)lin = 10 , Q(w,c).

intensive light Thus, the intensity-dependent permittivity ofthe substrate is given further as

e = Cl +c211inI2.

If the input intensity 10 of the incident field is high, theappropriate average current magnitude for a given e can befound using (I) as

The expression (3) is a nonlinear equation related to theaverage current value in the metal strips. The input field in­tensity is a parameter of the equation (3). At a fixed frequencyw, the solution of this equation is the average current valuedependent on the intensity of the incident field lin = lin(Io),where the function lin (Io) is presumably multivalued .

On the basis of the current lin(IO) found by a numericalsolution of the equation (3), the new value of permittivity ofthe nonlinear substrate e = Cl + c211in(loW is determinedand the reflection and transmission coefficients are calculated

as the functions of the intensity of the incident field.At the trapped-mode resonance, the flow of electromagnetic

energy is confined to a very small region between the stripsof two closely placed gratings separated with distance h. Thecrucial influence of the permittivity on the system propertiesoccurs in this place . Therefore, the approximation based onthe transmission line theory can be used here to estimatethe field intensity between the strips [5]. From our numericalcalculation it riches about 140 kW/cm2 nearly the resonantfrequency re :::::: 0.58 when the input intensity is about 10 ~

1 kW/cm2•

In the case of the nonlinear permittivity of substrate, de­pendences of the magnitudes of the average current versusthe intensity of the incident field lin = lin(Io) have theform of hysteresis. As a result, at a certain intensity of theincident field, the transmission coefficient stepwise changesits value from small to large level (Fig. 2). The frequencydependences of the transmission coefficient magnitude alsomanifests discontinuous switching from small to large levelwith frequency increasing/decreasing (Fig. 3). This switchingappears closely to the resonant frequencies of the trapped­mode excitation. One of the peculiarities of obtained nonlineartransmission is the presence of a closed loop responsiblefor bistability. This feature is quite unique for this type ofresonances [9].

The distinctive feature of the structure under study is theformation of two closely spaced Pane-shape trapped-moderesonances in the spectra It is due to the fact that nearlythe resonant frequency, the magnitude of currents which flowalong the strips of both gratings are significant, and they are

J = J(w,c), T = T(w,c), R = R(w,c). (I)

To introduce the nonlinearity (the third-order Kerr-effect),let us assume that the permittivity e of the substrate dependson the intensity of the electromagnetic field inside it. In ourapproximate approach to the nonlinear problem solution, firstassume that the inner intensity is directly proportional to thesquare of the current magnitude averaged over a metal patternextent lin ~ J . Secondly, in view of the smallness of thetranslation cell of the array, we suppose that the nonlinearsubstrate remains to be a homogeneous dielectric slab under

Fig. I. Fragment of two-layered planar metamaterial and its unit cell.

resonances exist in a very thin structure which is important forpractical applications.

The goal of the present report is to show promising useof the two-layered planar fish scale metamaterial which bearsthe Fano-shape trapped-mode resonances to obtain all-opticalswitching.

Let us consider a two-layered structure which consists oftwo gratings of planar perfectly conducting infinite stripsplaced on the both sides of a dielectric slab (see Fig. I). Theslab has thickness hand permittivity c. The grating strips havean arbitrary shape within a unit cell and locate at planes z = 0and z = -h. The square unit cell of the structure under studyhas a size d = dx = dy = 800 nm, ~ = 80 nm. The widthof the metal strips is 2w = 40 nm. The array is placed on adielectric substrate with thickness h = 160 nm. Suppose thatthe normally incident field is a plane monochromatic wavepolarized orthogonally to the strips (H-polarization), and theintensity of the primary field is 10.

The algorithm based on the method of moments was pro­posed earlier [7], [8] to study the resonant nature of thestructure response , under the assumption of such a small inputintensity 10 that the dependence of the substrate permittivitye on the field intensity is infinitesimal.

The algorithm requires that, at the first step, the surfacecurrent induced in the metal strips by the field of the incidentwave is to be calculated. The metal pattern is treated as aperfect conductor, while the substrate is assumed to be a lossydielectric . As a result, the magnitude and distribution of thecurrent J along the strips, the reflection R and transmissionT coefficients are determined in the form

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NLP*2011 1st International Workshop on Nonlinear Photonics, 6-8 September, Kharkou, Ukraine

.-~ /.?""-- l, I I

i I / N :I! I 1\\ 'r---- /, = 1\ I

I - 1, = 100 !\ : I ~

(b) \ : I _..... I, = 200 \.- 1, = 300

\ : ' j \,J;: ~ :

~/, = I

- 1, = 100

l r;, ...... I, = 200 ) ~1:' \ - I = 300 \ .

l.~' ~~~ , ,:", \~......' .'.0·' , ....,*,:-::..." \ '. '" ., ' ..... . . . :-- ".

Io- ~ "': :- ' " ".... ..

1\----

. ~ .... ' (\

(a) / I\.. / \V """"'-- ",

6

........- .-.- .- ' ....., ...."'.... ...... .......'. .' ----

/.,.~ ------'. - >~-...,

V .......(a)

". ' ./ i } ),,.

I »>rem 0.585, - ~- re= 0.580.-' •••••- '" = 0.575..' _.- 't '" .. '--' -- ", = 0.572

~ I"- :"""' ·

100I•

10

0.1

0.01

IE·)

IE'"0.50

1.0

ITI0.8

0.6

0.4

0.2

0.00.50

0.55

0.55

0.60

0.60

0.65

'"

0.65re

0.70

0.70

0.75

0.75

0.80

0.80

Fig. 2. Tbe magnitudes of the average current (a) and the transmissioncoefficient (b) versus theintensity of incident field in thecase of the nonlinearpermittivity (£1 = 3.0+0.0li and £2 = 0.005) of the substrate . The value ofdimensionless frequency aJ = d/A are chosen closely to the lowest frequencyof the first trapped-mode resonance.

sufficiently greater then the magnitude of current which flowsin the single grating. Nevertheless, at the resonant frequency,the local minimum of the current magnitude appears due toan interaction between two gratings which are placed closelyone to another. As a result, instead of the typical currentmagnitude resonance which exists in the single grating, thereis two-humped resonance in the two-layered structure [10].These two resonances form the wide band of the reflection,which can find an application in filtering. In our opinion suchclosely spaced resonances are also suitable for realization ofthe multistable response if the parameters of structure willbe appropriately chosen [II]. Also we should note here that,in contrast to the configurations of metamaterials consideredearlier [4], [5], in such structure all the mentioned nonlineareffects can be additionally controlled by the use of gratingsstrips as electrodes to which voltage is applied.

This work was supported by the National Academy ofSciences of Ukraine under the Program "Nanotechnologiesand Nanomaterials", the Project no. 1.1.3.17.

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Fig. 3. Frequency dependences of the average current (a) and the magnitudeof the transmission coefficient (b) in the case of the nonlinear permittivity(£1 = 3.0 + O.Oli and £2 = 0.005) of the substrate.

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