[ieee 2013 international conference on advanced optoelectronics and lasers (caol) - sudak, ukraine...

CAOL*2013 International Conference on Advanced Optoelectronics & Lasers, 09-13 September, 2013, Sudak, Ukraine

Electromagnetic wave diffraction by periodicstructures with nonlinear inclusions

(Invited Paper)

Vyacheslav V. Khardikov1,2, Pavel L. Mladyonov1, Sergey L. Prosvirnin1,2, and Vladimir R. Tuz1,2

1Institute of Radio Astronomy of National Academy of Sciences of Ukraine, 4, Krasnoznamennaya st., Kharkiv 61002, Ukraine2School of Radio Physics, Karazin Kharkiv National University, 4, Svobody Square, Kharkiv 61077, Ukraine.

Abstract— The results of the study of the electromagneticwaves resonant reflection from and transmission through planarmetamaterials based on periodic structures, which are composedof some elements consisted of nonlinear materials are presented.

I. INTRODUCTION

Modern nanotechnologies effort an opportunity to structureoptically thin layers of materials with a periodic subwave-length pattern in order to produce planar metamaterials. Theplanar metamaterials are an impressive contemporary object,which is driven by certain fascinating facilities both mostlywell known over the decade and quite novel.

Latest works aim to develop an electromagnetic frameworkfor the design of tiny controlling optical and THz devices byusing novel resonant planar metamaterials with active mediainclusions. These artificial open high-Q periodic structures arepromising to achieve a strong localization and enhancementof internal field which are necessary to produce amplifiers,generators, and tunable metamaterials. Strong intensity ofelectromagnetic field inside these novel planar metamaterialsis achieved by their designing to bear a resonant regime ofexcitation of trapped modes [1], [2].

Recent papers report results of aggregating laser materialswith these planar metamaterials to develop gaining or lasingdevices such as the spaser [3], [4], [5]. Another high promisedphenomenon is an optical bistability or multistability of atransmission response of nonlinear planar metamaterials. Thiseffect is employed to construct optical switches. diodes, tran-sistors, logical elements, and systems of optical storage [6],[7], [8], [9].

In this paper we report our recent results on the theoreticalstudy of trapped mode planar metamaterials with nonlinearinclusions.

II. SATURATION EFFECT IN ACTIVE METAMATERIALS

We have proposed a simple design of all-dielectric silicon-based planar metamaterial [10] manifested an extremely sharpresonant reflection and transmission in the wavelength of about1550 nm due to both low dissipative losses and involving atrapped mode operating method. The quality factor of the res-onance exceeds in tens times the quality factor of resonancesin known plasmonic structures. The designed metamaterialis envisioned for aggregating with a pumped gain medium

to achieve an enhancement of luminescence and to producean all-dielectric analog of a ”lasing spaser”. We report thatan essential enhancement (more than 500 times) of lumines-cence of in a layer contained pumped quantum dots (QD)may be achieved by using the designed metamaterial. Thisvalue exceeds manyfold the known the value of luminescenceenhancement by in known plasmonic planar metamaterials.

900

900

800

260

240

160

210

50

100

YX

Z

Fig. 1. A sketch of the unite cell of the double-periodic planar structure.The all-dielectric array composed of two dielectric bars per a periodic cell isimmersed into the QD layer. All dimension are in nm.

We use the model of gain nonlinear medium by introducingnegative frequency dependent conductivity

σ(ω) =1

1 + I/Is

σ0(1 + iωτ)(1 + ω2

0τ2) + 2iωτ − ω2τ2

where ω0 = 1.26 · 1015 s−1 which corresponds to wave-length λ0 = 1550 nm; τ = 4.85 · 10−15 s; εQD = 2.19which corresponds to refractive index nQD = 1.48 of non-pumped quantum dot laser medium, and σ0 = −500 Sm/mcorresponding to an emission factor tan δe = −0.021 on theanalogy of a lossy factor of media. Small value of τ resultsin a wide-band QD spectral line and it enables to excludefrom consideration the effects caused by displacement of meta-material dissipation peak and maximum of exciton emission

978-1-4799-0018-3/13/$31.00 ©2013 IEEE

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line of QDs. Let us notice that the pump level (parameterσ0) is in one order less than it was needed in the case ofplasmonic metamaterials because of low losses of all-dielectricarray. The factor (1 + I/Is)−1 allows considering the effecton luminescence enhancement of the gain saturation effectinherent in gain media. Here the parameter Is is proportionalto the saturation intensity and it displays effect of inversionpopulation reducing in the gain medium by simulated emissionwhich is proportional to the maximum of the internal field (I).The saturation factor ((1 + I/Is)−1) is calculated separatelyfor each point of the spatial grid that allows considering theinhomogeneous of the QD layer. We should note that the smallvalue of an emission factor results in independence of theQD refractive index from the saturation factor. Thus the effectof saturation on the luminescence enhancement of QD layerhybridized with all-dielectric metamaterial can be consideredunder this model.

The diffraction approach proposed in [11] was used to cal-culate the luminescence enhancement for QD layer hybridizedwith all-dielectric metamaterial. This approach consists ofevaluation of considered structure luminescence through thedifference of energy dissipation in the passive and gain struc-ture. The dissipation energy is calculated from the solutionof corresponding diffraction problem for plane wave. Theluminescence enhancement equals to the ratio of the lumines-cence of the hybrid structure to the luminescence of 210 nmhomogeneous QD layer placed on 50 nm silica substrate. Thewavelength dependencies of the luminescence enhancement ofQD layer hybridized with all-dielectric metametrial for differ-ent value of the saturation intensity are shown in Fig. 2. Thereducing of the luminescence enhancement with decreasing ofthe saturation intensity may be explained by exciting stronglocal field in the hybrid structure which results in decreasingof saturation factor. The distribution of the saturation factorin cross section (z = −155 nm) is shown in Fig. 3. One cansee the burning hole appearance in the distribution (see darkblue areas in Fig. 3). The energy of optical pumping withinthese hole was completely used by simulated emission. Theeffect of gain saturation dose not strongly influence on thephotoluminescence in the system but it needs to be taken intoaccount for modeling of the optical amplifier and lasing spacer.

III. BISTABILITY AND MULTISTABILITY OF RESPONSE OFNONLINEAR PLANAR METAMATERIALS

The phenomenon of optical bistability or multistability is acommon property of nonlinear optical systems with feedback,which means that there are two or more stable states of thesystem corresponding to different amplitudes or polarizationsof the field [12].

A. Magnetically controllable array on nonlinear antiferro-magnetic substrate

The structure under study is shown in Fig. 4. The double-ring (DR) array is placed on a substrate made of antiferro-magnetic film (AF). The metamaterial parameters are d =dx = dy = 0.3 mm a1 = 0.11 mm and a2 = 0.09 mm,

λ, nm

Lu

min

esce

nce

en

ha

nce

me

nt

Fig. 2. The wavelength dependencies of the enhancement of the luminescenceof the QD layer hybridized with all-dielectric metamaterial for different valueof Is. Line 1 - Is = 2.0, line 2 - Is = 0.4, line 3 - Is = 0.04. The saturationintensities present in arbitrary units.

Fig. 3. The distribution of saturation factor in the cross section z = −155nm. λ = 1553 nm, Is = 0.4.

2w = 0.004 mm, and h = 0.05 mm. As a material for thesubstrate, MnF2 antiferromagnetic film is considered [13],[14]. The external static magnetic field (ESMF) is applied tothe system in the Faraday geometry.

For the fixed external magnetic field H0 = 1.0 kG, inthe dispersion dependences of the permeability tensor com-ponents there are two resonant frequencies, æ1 = 0.262(f = 0.2618 THz) and æ2 = 0.268 (f = 0.2678 THz) whereæ = dx/λ.

Under the action of intense light the dynamical magne-tization in AF media is coupled nonlinearly with the wavemagnetic field which leads to the magnetic optical nonlinearity.Remarkably, when the magnetic field strength inside the AFfilm increases both the real and imaginary parts of mag-netic permeability undergo changes. Under a certain thresholdlevel of the input light intensity, this can lead to dispersion-

180


Fig. 4. Fragment of the planar metamaterial and its elementary unit cell.

absorption bistable behavior of the system.Here the way to obtain large magnitude of the magnetic field

strength in the AF film lies in the choice of the parametersof the DR array of the metamaterial to tune the frequency ofthe trapped-mode excitation to be close to the frequency ofantiferromagnetic resonance. Such a situation is illustrated inFig. 5 where the optical response of the metamaterial in thecase when the intensity of input light is small (linear regime)is presented.

Fig. 5. Reflection and transmission spectra of the array placed on anantiferromagnetic substrate, linear regime.

The behavior of the dispersion characteristics of the AFpermeability leads to appearing alternate bands of high trans-mission and absorption in the spectra of metamaterial. In thefrequency band of the trapped-mode excitation there is a peakof current magnitude, but its frequency dependence has a formof alternating maxima and minima due to the strong absorptionin the substrate in the vicinity of the AF resonances nearly æ1

and æ2.Since the magnetic field strength is proportional to the

current and the current magnitude obviously increases whenthe intensity of the incident field rises, under a certain intensityof the incident field the magnetic properties of substrate canchange.

Our calculations [15] show that in case of the nonlinearpermeability of substrate, dependences of the inner fieldintensity versus the incident field intensity have a form ofhysteresis (Fig. 6). Such form of curves is studied quite welland is explained by the nonlinear phase-shift and nonlinearattenuation which appear in the nonlinear system. As theincident field intensity increases, the nonlinear phase shiftrapidly raises and the attenuation decreases that guarantee thepresence of obvious bistable switching.

Fig. 6. Inner field intensity versus incident field intencity.

The frequency dependences of the transmission coefficientmagnitudes also manifest discontinuous switching with fre-quency changing (Fig. 7). Since the dispersion curves ofnonlinear susceptibility have the bands of grow and decay,the real and imaginary parts of the permeability tensor coeffi-cients also increase and decrease with frequency. The bistabletransmission occurs exactly in these frequency bands and ismanifested in the ambiguity of the transmission coefficientmagnitudes at the leading and trailing edges of the resonance.

Thus, the all-optical switching can be realized due to thecapability of a planar DR metamaterial provides the sufficientfield localization inside the thin nonlinear substrate at thefrequency of the trapped-mode excitation.

B. Double array consisted of wavy strips on nonlinear sub-strate

A studied structure consists of two gratings of planarperfectly conducting infinite strips placed on a dielectric slabwith thickness h (see Fig. 8). We assume that this slab isa Kerr-type nonlinear dielectric which permittivity ε linearlydepends on the intensity |E|2 of the electric field. The gratings

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Fig. 7. Transmission spectrum of the array placed on the antiferromagneticsubstrate, nonlinear regime.

with wavy-line strips are located on the both sides of the slabat planes z = 0 and z = −h. The elementary translation cell ofthe structure under study is a square with sides d = dx = dy .The full length of the strip within the elementary translationcell is S. Suppose that the thickness h and size d are muchless then the wavelength λ of the incident electromagneticradiation. The width of the metal strips and their deviationfrom the straight line, respectively, are 2w and ∆. Assumethat the normally incident field is a plane monochromaticwave polarized parallel to the strips (x-polarization), and themagnitude of the primary field is A. We suppose that theintensity of the incident field is enough for the nonlinearityto become apparent, i.e., it is about 1 kW/cm2.

In our publications [16], [17] the detailed description of thenumerical method based on the Method of Moments (MoM) isgiven to study electromagnetic properties of planar fish-scalemetamaterials in both single- and bilayer configurations whenthe intensity of the incident field is small (linear regime). Itinvolves solving the integral equation related to the surfacecurrents which are induced in the metallic pattern by thefield of the incident wave. Remarkably that in the bilayerconfiguration the method of solution rigorously takes intoaccount a coupling between two gratings via evanescent partialspatial waves. Obtained solution allows us calculating themagnitude and distribution of the current J along the strips,the reflection R and transmission T coefficients as functionsof frequency ω, permittivity ε and other parameters of thestructure.

When the studied structure is under an action of the intenselight (in the nonlinear regime), permittivity of the substrateε depends on the intensity of the electromagnetic field insideit, ε = ε(Iin). In [18], [19] an approximate treatment wasproposed to solve the nonlinear problem.

Fig. 8. Fragment of a bilayer planar fish-scale metamaterial and its unit cell.

The origin of trapped-mode resonances is the oppositedirected but being almost equal currents which appear in twoclosely spaced metallic wires. The scattered fields producedin this situation is very weak, and, as a consequence, thecoupling of the metamaterial array to free space is small andtherefore its radiation losses are reduced, which ensures ahigh-Q resonant response.

The first distribution is the antiphased current oscillationsnear point of inflection wavy-line of each grating (see Fig. 9a).The observers structure can be considered as a system oftwo coupled resonators which work on the same frequencybecause the gratings are identical. Obviously that the distanceh between the gratings will strongly effect on the resonantfrequency position since this parameter define the electromag-netic coupling degree. The Q-factor of this resonance is higherin the bilayer structure in comparison with a single-layered onebut their similarity is in the fact that the current magnitude inthe metallic pattern depends relatively weakly on the thicknessand permittivity of the substrate.

Ein

0 SS/2

|J|

0

SS/2

Ein

0

SS/2

0 SS/2

|J|(a)

0 SS/2

|J|

0 SS/2

|J|

(b)æ1

æ2

Fig. 9. The resonant current distribution along the strips in the case ofbilayer structure composed of gratings with wavy-shaped strips. The resonantfrequencies are (a) æ1 = 0.7855, (b)æ2 = 0.82 (æ = d/λ).

The second distribution is the antiphased current oscillationsbetween two adjacent gratings (see Fig. 9b). It is well knownthat the closer are the interacting metallic elements, the higheris the Q-factor of the trapped-mode resonance. Thus varyingthe distance between the gratings or the substrate permittivity

182


changes the trapped-mode resonant conditions and this chang-ing manifests itself in the current magnitude. Remarkablythat in this type of current distribution the field is localizedbetween the gratings, i.e. directly in the substrate, which cansufficiently enhance the nonlinear effects if the substrate ismade of intensity dependent material.

This circumstance is depicted in Fig. 10 where typicalcurves of the inner field intensity and the transmission co-efficient magnitude are shown as functions of the frequencyand incident field intensity in the nonlinear regime.

0.0

0.2

0.4

0.6

0.8

|T |

3

2

0.78 0.79 0.81 0.820.80

1

0.0

0.0050.1

1

10

Iin

1

2

3

0.78 0.79 0.81 0.820.80 æ

æ

æ æ1 2

(a)

(b)

Fig. 10. The frequency dependences of the inner field intensity (on thelogarithmic scale) (a) and the transmission coefficient magnitude (b) versusthe incident field intensity in the case of the nonlinear permittivity (ε =εl + εnlIin, dimension of Iin is in kW/cm2); εl = 3.0, εnl = 5 × 10−3

cm2/kW; curve 1 - A = 1 kW/cm2, curve 2 - A = 200 kW/cm2, curve 3 -A = 300 kW/cm2.

For the nonlinear conditions the curves of the transmissioncoefficient magnitude experience different distortion nearlythe trapped-mode resonance frequencies. At the frequencyæ ≈ 0.78 the antiphased current oscillations localize inarea of each grating and are weakly affected on dielectricsubstrate. In such case the resonances curve transforms into aclosed loop that is typical for the sharp nonlinear Fano-shaperesonance. The second resonance æ ≈ 0.82 is smooth but thecurrent oscillations between two adjacent gratings has lad to

greater concentration of field in dielectric substrate. For thenonlinear conditions the resonance near æ ≈ 0.82 undergoesmore distortion in the wider frequency band, and at a certainincident field intensity this resonance can overlap the firstone (Fig. 10b). Evidently that in this case the transmissioncoefficient has more than two stable states, i.e. the effect ofthe multistability occurs.

C. Polarization bistability in magnetophotonic structures

Magnetophotonic crystals (MPCs) are periodic structuresthat contain magnetic materials and therefore exhibit interest-ing physics arising from the interplay between photonic bandgap (PBG) phenomena and magnetooptical effects [20], [21].Examples include external magnetic tunability of PBGs andstrong enhancement of Faraday polarization rotation. It is evenmore interesting to consider an MPC in presence of opticalnonlinearity. In particular, the interplay between the Faradayeffect (which is associated with optical nonreciprocity) andthe Kerr effect (which is known to result in direction-sensitiveoptical bistability [22], [23], [24]) can give rise to new types ofasymmetric or unidirectional light transmission, and the effectsof this interplay on the polarization of transmitted and reflectedlight still remains to be investigated.

A one-dimensional magnetophotonic crystal with a Kerr-type nonlinear defect placed either symmetrically or asym-metrically inside the structure is considered (see Fig. 11). If such a system is under the longitudinal action of anexternal static magnetic field, the simultaneous effects of time-reversal nonreciprocity and nonlinear spatial asymmetry take aplace. These effects manifest themself in the bistable responseaccompanied by abrupt polarization switching between twocircular or elliptical polarizations for transmitted and reflectedwaves [25].

Fig. 11. A magnetophotonic crystal structure with a nonlinear cavity.

In the underlying structure, as a result of the action of anexternal static magnetic field, the defect resonances split intoZeeman-like doublets (see the black and red lines in Fig. 12and Fig. 13).

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If the incident field is linearly polarized, the reflected

and transmitted fields become elliptically polarized due tothe Faraday rotation and circular dichroism. Directly at theZeeman-like resonances the fields are circularly polarized.Therefore, in the nonlinear regime, it is possible to usemultistability to switch between two distinct polarization statesin the transmitted and/or reflected light (see the dark cyan andpurple lines in Fig. 12 and Fig. 13).

Fig. 12. Frequency dependences (æ = D/λ) of the transmission coefficient(T ) of the LCP (−) and RCP (+) waves for m = n = 5.

Fig. 13. Frequency dependences (æ = D/λ) of (a) the elipticity angle ηand (b) the polarization azimuth θ of the transmitted and reflected fields form = n = 5. The incident light is linearly polarized. The vertical line marksthe bistable polarization switching at æ0.

In conclusion, metamaterials, which bears so-called trapped-mode resonance operation is a promising technique to designall-optical control devices. From our numerical calculations

it seems reasonable to conclude that the bistable responsecan be obtained at the incident power densities of 10–100kW/cm2 with available materials in the considered structureconfiguration.

ACKNOWLEDGMENT

This work was supported by the Ukrainian State Foundationfor Basic Research, the Project no. Φ54.1/004.

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