1. Introduction

Unusual physical effects in dielectric media with both negative permittivity and negative permeability were first postulated theoretically by Veselago [1] who predicted a number of novel phenomena including, for example, negative refraction of waves. Such media are usually known as 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 demonstrated only recently for a novel class of engineered composite materials [2, 3, 4, 5]. Such LH materials have attracted attention not only due to their recent experimental realization and a number of unusual properties observed in experiment, but also due to the expanding debates on the use of a slab of a LHM as a perfect lens for focusing both propagating and evanescent waves [6].

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

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

The possibility to control the effective parameters of metamaterials using nonlinearity has recently 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 predicted theoretically. The main reason for the expectation of strong nonlinear effects in metamaterials is 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 propagating wave. Moreover, changing the intensity of the electromagnetic wave not only changesthe material parameters, but also allows switching between transparent left-handed states and opaque dielectric states.

 

Fig. 1. (Color online) Geometry of the problem. In our calculations we take the following values: d ₁=1cm, d ₂=1.65cm, ε ₁=4, µ ₁=1, ε ₂=2.25, µ ₂=1, ε ₀=-1, and µ ₀=-1, d_c=0.01cm.

Download Full Size | PDF

Recently, preliminary studies in the calculation of the linear and nonlinear dispersion properties 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 unusual properties of the LHMs, it is important to study the properties of nonlinear type of surface waves that can be excited at the interfaces between a semi-infinite nonlinear uniform LHM and a conventional 1-D PC.

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

2. Nonlinear surface waves

We use the transfer matrix method to describe surface Tamm states that form at the interface between an uniform nonlinearLHM and a conventional semi-infinite one-dimensional photonic crystal. Geometry of our problem is sketched in Fig. 1, where ε ₀ and µ ₀=µ_LH are linear dielectric permittivity and magnetic permeability in nonlinear LHM, respectively. We consider that the LHM medium has a well known intensity dependent Kerr nonlinearity:

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 scalar Helmholtz-type wave equation, which for the case of the TE-polarized wave in nonlinear media can be written as

Here, we supposed that the effective LH material is a homogeneous and isotropic medium, so that ε=ε ₀ and µ=µ ₀. 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 is a self-focusing effect. It will be opposite to those in RH media with positive permeability µ for the same α. It must be noted that in the presented study we suppose that LH material has self-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]

where z ₀ is the center of the sech function and it should be chosen to satisfy the continuity of the tangential components of the electromagnetic fields at the interface. Here c is the speed of light and η=[β ²-ε_LHµ_LH]^1/2, where β is the normalized wavenumber component along the interface.

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

We consider the propagation of TE-polarized waves described by one component of the electric field, E=E_y, and governed by a scalar Helmholtz-type wave equation as Eq. (2). We look 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 interface in both the directions. In the left-side homogeneous LHM medium, the fields are decaying provided $\beta >\sqrt{{\epsilon }_{\mathrm{LH}}{\mu }_{\mathrm{LH}}}$ . In the right-side periodic structure, the waves are the Bloch modes. In the 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 calculation of 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 homogeneous medium and the Bloch modes in the periodic structure and satisfy the conditions of continuity of the tangential components of the electric and magnetic fields at the interface between nonlinear homogeneous medium and periodic structure [24].

3. Results and discussion

We discuss the dispersion properties of the nonlinear Tamm states in the first spectral gap on the plane of the free-space wave number k=ω/c versus the propagation constant β (see Fig. 2). As we know, the Tamm states exist in the gaps of the photonic band gap spectrum which the boundaries of the first band gap are shown by dashed lines in Fig. 2. In Fig. 2 we present dispersion properties of nonlinear surface Tamm states for different values of the dimensionless intensity of light $\tilde{I}=\frac{I}{I_c}$ at the surface of photonic crystal (Ĩ_s), where I ~ |E|², and I_c correspond to the characteristic intensity that α̃=αI_c=-1. As one can see from Fig. 2 there are two branch of dispersion curves for a given intensity of light, which describe two type of nonlinear Tamm states for a thin cap layer, d_c=0.01 cm. The lower (solid) curve corresponds to the dispersion 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-humped structure at the interface. Corresponding values of the intensity Ĩ_s for curves 1–3 are 0.3, 0.6, 0.9, respectively and the curve 0 shows the dispersion of linear regime where Ĩ_s→0. The mode profiles of points (a), (b) of Fig. 2 are shown in Fig. 3, where we ploted the profiles of the two modes having the same longitudinal wave numbers β=1.19, with different frequency k=ω/c. For the mode (a), the energy flow in the metamaterial exceeds that in the periodic structure (slow decay of the field for LHM and fast decay into the periodic structure), thus the total energy flow is backward. For mode (b) we have the opposite case, and the mode is forward.

 

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 of surface modes. The corresponding values of the intensity Ĩ_s for the curves 1–3 are 0.3, 0.6, 0.9, respectively and the curve 0 show the dispersion of linear regime where Ĩ_s→0. Points (a), (b), correspond to the mode profiles shown in Fig.3. The other parameters are the same as the Fig. 1.

Download Full Size | PDF

 

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

Download Full Size | PDF

The energy flow for surface Tamm states with two-humped and one-humped structures have different behavior. To demonstrate this, in Fig. 4 we plot the total energy flow in the modes as a function of the wave number β. We see from Fig. 4 that all surface modes with two-humped structure are backward for lower intensity (Ĩ_s=0.1), whilst there are forward and backward surface modes for one-humped structures, so that in this case backward modes are limited to small β. By increasing intensity one can find backward mode for two-humped types as we can see in Fig. 4. These results confirm our discussion based on the analysis of the dispersion characteristics.

Finally, in Fig. 5 we plot the existence regions of the nonlinear surface modes on the parameter plane (Ĩ_s,β) for one-humped (Fig. 5(a)) and two-humped (Fig. 5(b)) modes. The shaded regions are the areas where the surface modes exist. The region 1 with dark color and the region 2 with light color show the corresponding regions for the backward and forward nonlinear surface Tamm states, respectively.

 

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.

Download Full Size | PDF

 

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

Download Full Size | PDF

4. Conclusion

We have presented a theoretical study of nonlinear electromagnetic surface waves supported by an interface between a nonlinear left-handed metamaterial and a conventional 1-D photonic crystal. We have demonstrated that in the presence of a nonlinear LHM there are two kind of the surface Tamm states with two different structures of one-humped and two-humped types which can be either forward or backward, while for linear regime the Tamm states are always one-humped structure. We have analyzed the existence regions for forward and backward modes with both one and two-humped structures. We believe that our results will be useful for a deeper understanding of the properties of nonlinear surface waves in plasmonic and metamaterial systems.