Backward nonlinear surface Tamm states in left-handed metamaterials

Backward nonlinear surface Tammstates in left-handed metamaterials

Abdolrahman Namdar1, Samad Roshan Entezar2,3, Habib Tajalli2,3,and Zahra Eyni2,3

1Physics Department, Azarbaijan University of Tarbiat Moallem, Tabriz, Iran

2Physics Faculty, University of Tabriz, Tabriz, Iran

3Research Institute for Applied Physics and Astronomy, Tabriz, Iran

[email protected]

Abstract: We addressed the existence of the nonlinear electromagneticsurface waves, the so-called Tamm states, that form at an interface sepa-rating a nonlinear uniform left-handed metamaterial and a conventionalone-dimensional photonic crystal. We found two types of nonlinear surfacewaves, one with a hump at the interface and the other one with two humps.It was demonstrated that the nonlinear metamaterial can support the bothtype Tamm states with a backward energy flow and allows for a flexiblecontrol of the dispersion properties of surface states. We also, describedthe intensity-dependent properties of surface Tamm states for a nonlinearself-focusing medium.

© 2008 Optical Society of America

OCIS codes: (240.6690) Surface waves;(160.5298) Photonic crystals; (160.3918) Metamateri-als; (190.0190) Nonlinear optics.

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#96064 - $15.00 USD Received 12 May 2008; revised 18 Jun 2008; accepted 18 Jun 2008; published 30 Jun 2008

(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10543

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Opt. 8, 630-638 (2006).

1. Introduction

Unusual physical effects in dielectric media with both negative permittivity and negative per-meability were first postulated theoretically by Veselago [1] who predicted a number of novelphenomena including, for example, negative refraction of waves. Such media are usually knownas left-handed metamaterials (LHMs) since the electric and magnetic fields form a left-handed(LH) set of vectors with the wave vector. The physical realization of such LH media was demon-strated only recently for a novel class of engineered composite materials [2, 3, 4, 5]. Such LHmaterials have attracted attention not only due to their recent experimental realization and anumber of unusual properties observed in experiment, but also due to the expanding debateson the use of a slab of a LHM as a perfect lens for focusing both propagating and evanescentwaves [6].

Optical surface waves, i.e., light wave packets localized at the interface between two mediawith different properties, are a topic of continually increasing interest because of their funda-mental properties as well as potential applications, e.g., in sensing, trapping, and imaging, basedon near-field techniques. Research has been focused mostly on surface waves and resonancesthat form at metalodielectric interfaces, so-called surface plasmons, [7] and more recently onphotonic crystals (PCs) [8].

Photonic crystals are artificial materials with a periodic modulation in the dielectric constantwhich can create a range of forbidden frequencies called a photonic band gap [9]. Photons withfrequencies within the band gap cannot propagate through the medium. This unique feature canalter dramatically the properties of light, enabling control of spontaneous emission in quantumdevices and light manipulation for photonic information technology [10].

The possibility to control the effective parameters of metamaterials using nonlinearity hasrecently been suggested in Refs. [11, 12] and developed extensively in Refs. [13, 14, 15, 16,17, 18, 19, 20, 21] where many interesting nonlinear metamaterials effects have been predictedtheoretically. The main reason for the expectation of strong nonlinear effects in metamaterialsis that the microscopic electric field in the vicinity of the metallic particles forming the left-handed structure can be much higher than the macroscopic electric field carried by the propa-gating wave. Moreover, changing the intensity of the electromagnetic wave not only changesthe material parameters, but also allows switching between transparent left-handed states andopaque dielectric states.



• • •

Nonlinear

LHM RH

1

RH2

d1 d

2 d

c

dt

μ0

μ1

μ2

ε0

ε1

ε2

z

yx

Fig. 1. (Color online) Geometry of the problem. In our calculations we take the followingvalues: d1 = 1cm, d2 = 1.65cm, ε1 = 4, μ1 = 1, ε2 = 2.25, μ2 = 1, ε0 = −1, and μ0 =−1,dc = 0.01cm.

Recently, preliminary studies in the calculation of the linear and nonlinear dispersion prop-erties of surface modes localized at the interface separating a LHM and a right-handed (RH)medium [13], and the electromagnetic surface waves localized at an interface separating a one-dimensional(1-D) PC and LHM are reported [22]. To more understanding of other unusualproperties of the LHMs, it is important to study the properties of nonlinear type of surfacewaves that can be excited at the interfaces between a semi-infinite nonlinear uniform LHM anda conventional 1-D PC.

The paper is organized as follows. In Sec. II we discuss how to calculate the nonlinear surfacemodes localized at the interface between semi-infinite nonlinear LHM and a conventional 1-Dmedia. The discussion of the dispersion properties of the nonlinear Tamm states in the firstspectral gap on the plane of the free-space wave number versus the propagation constant andexistence regions for the nonlinear surface modes are presented in Sec. III. Finally, Sec. IVconcludes the paper.

2. Nonlinear surface waves

We use the transfer matrix method to describe surface Tamm states that form at the interfacebetween an uniform nonlinear LHM and a conventional semi-infinite one-dimensional photoniccrystal. Geometry of our problem is sketched in Fig. 1, where ε 0 and μ0 = μLH are lineardielectric permittivity and magnetic permeability in nonlinear LHM, respectively. We considerthat the LHM medium has a well known intensity dependent Kerr nonlinearity:

εLH = ε0 + α|E|2, (1)

where parameter α describes Kerr type nonlinearity where can be positive or negative.The propagation of monochromatic waves with the frequency ω is governed by the scalarHelmholtz-type wave equation, which for the case of the TE-polarized wave in nonlinear mediacan be written as

∂ 2E∂ z2 +

∂ 2E∂x2 +

ω2

c2 (εμ + μα|E|2)E = 0. (2)

Here, we supposed that the effective LH material is a homogeneous and isotropic medium, sothat ε = ε0 and μ = μ0. In Eq. (2) the sign of the product μα characterizes the type of self-



action nonlinearity. For example in a LH medium with negative α the nonlinear property isa self-focusing effect. It will be opposite to those in RH media with positive permeability μfor the same α . It must be noted that in the presented study we suppose that LH material hasself-focusing properties, i.e., α < 0.

By considering the stationary solutions of Eq. (2) in the form E(x,z) = Ψ(z)exp(i(ω/c)βx)one can find the profiles of the spatially localized wave envelopes Ψ(z) as [13]

Ψ(z) = (η√

2/αμ)sech[(ω/c)η(z− z0)], (3)

where z0 is the center of the sech function and it should be chosen to satisfy the continuity ofthe tangential components of the electromagnetic fields at the interface. Here c is the speed oflight and η = [β 2 − εLH μLH ]1/2, where β is the normalized wavenumber component along theinterface.

We assume that the terminating layer (or a cap layer) of the periodic structure has the widthdifferent from the width of other layers of the structure. First author has shown the effect of thewidth of this termination layer on the surface states in Ref. [22]. Here, we study the effect ofthe nonlinearity of homogenous LHM on the surface Tamm states and a possibility to controlthe dispersion properties of surface waves by adjusting the nonlinearity parameter α .

We consider the propagation of TE-polarized waves described by one component of theelectric field, E = Ey, and governed by a scalar Helmholtz-type wave equation as Eq. (2). Welook for stationary solutions propagating along the interface with the characteristic dependence∼ exp [−iω(t −βx/c)].

Surface modes correspond to localized solutions with the field E decaying from the interfacein both the directions. In the left-side homogeneous LHM medium, the fields are decayingprovided β >

√εLH μLH . In the right-side periodic structure, the waves are the Bloch modes. Inthe periodic structure the waves will be decaying provided that Bloch wave number is complex;and this condition defines the spectral gaps of an infinite photonic crystal. For the calculationof the Bloch modes, we use the well-known transfer matrix method [23].

To find the nonlinear Tamm states, we take solutions of Eq. (2) in a nonlinear homogeneousmedium and the Bloch modes in the periodic structure and satisfy the conditions of continuity ofthe tangential components of the electric and magnetic fields at the interface between nonlinearhomogeneous medium and periodic structure [24].

3. Results and discussion

We discuss the dispersion properties of the nonlinear Tamm states in the first spectral gapon the plane of the free-space wave number k = ω/c versus the propagation constant β (seeFig. 2). As we know, the Tamm states exist in the gaps of the photonic band gap spectrum whichthe boundaries of the first band gap are shown by dashed lines in Fig. 2. In Fig. 2 we presentdispersion properties of nonlinear surface Tamm states for different values of the dimensionlessintensity of light I = I

Icat the surface of photonic crystal ( Is), where I ∼ |E|2, and Ic correspond

to the characteristic intensity that α = αIc = −1. As one can see from Fig. 2 there are twobranch of dispersion curves for a given intensity of light, which describe two type of nonlinearTamm states for a thin cap layer, dc = 0.01 cm. The lower (solid) curve corresponds to thedispersion of the Tamm states with the maximum amplitude at the interface (one-humped type)and the upper (dashed) curve corresponds to dispersion of the Tamm states with two-humpedstructure at the interface. Corresponding values of the intensity Is for curves 1-3 are 0.3, 0.6,0.9, respectively and the curve 0 shows the dispersion of linear regime where Is → 0. The modeprofiles of points (a), (b) of Fig. 2 are shown in Fig. 3, where we ploted the profiles of the twomodes having the same longitudinal wave numbers β = 1.19, with different frequency k = ω/c.



1.1 1.3 1.50.6

1

1.4

1.8

β

k[1/

cm]

102 3

•

•

(b)

(a)

Fig. 2. (Color online) Dispersion properties of the Tamm states in the first spectral gap.The solid and dashed curves correspond to the one-humped and two-humped structures ofsurface modes. The corresponding values of the intensity Is for the curves 1-3 are 0.3, 0.6,0.9, respectively and the curve 0 show the dispersion of linear regime where Is → 0 . Points(a), (b), correspond to the mode profiles shown in Fig.3. The other parameters are the sameas the Fig. 1.

−10 0 10 20 −1

0

1

2

z [cm]

Re

[E(z

)]

(a)

−10 0 10 20

−1

0

1

z [cm]

Re

[E(z

)]

(b)

Fig. 3. (Color online) Examples of the nonlinear surface Tamm states; Is = 0.3, β = 1.19;(a) Backward nonlinear two hump LHM mode, k = 1.141 cm−1; (b) Forward nonlinearone hump LHM mode, k = 0.879 cm−1 ; Modes (a), (b) correspond to the points (a), (b) inFIG. 2. The other parameters are the same as the Fig. 1.

For the mode (a), the energy flow in the metamaterial exceeds that in the periodic structure (slowdecay of the field for LHM and fast decay into the periodic structure), thus the total energy flowis backward. For mode (b) we have the opposite case, and the mode is forward.

The energy flow for surface Tamm states with two-humped and one-humped structures havedifferent behavior. To demonstrate this, in Fig. 4 we plot the total energy flow in the modes asa function of the wave number β . We see from Fig. 4 that all surface modes with two-humpedstructure are backward for lower intensity ( Is = 0.1), whilst there are forward and backwardsurface modes for one-humped structures, so that in this case backward modes are limited tosmall β . By increasing intensity one can find backward mode for two-humped types as wecan see in Fig. 4. These results confirm our discussion based on the analysis of the dispersioncharacteristics.

Finally, in Fig. 5 we plot the existence regions of the nonlinear surface modes on the para-meter plane (Is,β ) for one-humped (Fig. 5(a)) and two-humped (Fig. 5(b)) modes. The shaded



1.1 1.3 1.5−2

−1

0

1.0

βE

nerg

y flo

w (

arb.

units

)

Is = 0.5

Is = 0.1

Fig. 4. (Color online) Total energy flow in nonlinear one-humped (dashed) and two-humped(solid) structures of surface Tamm modes vs β for different intensity.

1.1 1.40

1

2

β

Is

2

1

(a)

1.1 1.40

1

2

β

Is

1

2

(b)

Fig. 5. (Color online) Existence regions for the nonlinear surface Tamm modes ; the modesexist in the shaded regions. the region 1 with dark color show the corresponding regions forthe backward nonlinear surface Tamm states whilst the region 2 with light color show thecorresponding regions for the forward nonlinear surface Tamm states. (a) and (b) show themodes with one-humped and two-humped structures, respectively. The other parametersare the same as the Fig. 1.

regions are the areas where the surface modes exist. The region 1 with dark color and the re-gion 2 with light color show the corresponding regions for the backward and forward nonlinearsurface Tamm states, respectively.

4. Conclusion

We have presented a theoretical study of nonlinear electromagnetic surface waves supportedby an interface between a nonlinear left-handed metamaterial and a conventional 1-D photoniccrystal. We have demonstrated that in the presence of a nonlinear LHM there are two kind ofthe surface Tamm states with two different structures of one-humped and two-humped typeswhich can be either forward or backward, while for linear regime the Tamm states are al-ways one-humped structure. We have analyzed the existence regions for forward and backwardmodes with both one and two-humped structures. We believe that our results will be useful fora deeper understanding of the properties of nonlinear surface waves in plasmonic and metama-terial systems.



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Backward nonlinear surface Tamm states in left-handed metamaterials