## Abstract

A rigorous analysis of the contra-directional coupling between a left-handed waveguide and a right-handed waveguide shows that such a structure supports twin modes with complex (conjugate) propagation constants. Using evanescent coupling or a source inside one of the waveguides, the twin modes can be excited. In this case, light rotates inside the lamellar structure forming an exotic mode : the *light wheel*. The structure can finally be seen as a new type of cavity and can be used for beam reshaping.

©2007 Optical Society of America

Left-Handed Materials (LHM) present simultaneously negative permittivity and permeability. Such materials were a pure theoretical oddity[1] until the recent experimental demonstration of the negative refraction[2]. Left-handed materials can be made using metamaterials, *i.e*. periodical structures with a period smaller than the wavelength, which behave as a homogeneous medium. Following these works, the LHM have aroused huge interest. It clearly appeared that LHM had hardly been studied although they could lead to structures presenting highly nonconventional behaviours : they seemingly allow to overcome the Rayleigh limit[3] and could even lead to invisibility[4].

In a large number of situations, the way LHM behave is exactly the opposite of the way conventional (right-handed) dieletric materials usually behave. This is of course the case for refraction (so that LHM are sometimes said to present a negative index), but for the Goos-Hänchen effect as well[5, 6]. Generally, lamellar structures made with left-handed materials present exotic properties. It has been shown for instance that a LHM slab could support backward or forward guided modes[7] and backward or forward leaky modes [8, 9].

In this paper we present the study of the contra-directional coupling between a guided mode supported by a dielectric slab and a backward mode supported by a left-handed slab[10]. The contra-directional coupling has always been considered in the case of fire-end coupler[11, 12] and has attracted a recent interest[13]. Here we show that the modes of this structure can be excited either using a prism coupler or by a source in one of the waveguides, leading to the formation of a “light wheel” : light rotates inside the lamellar structure, which thus behaves as an edge-less cavity. Such a cavity would be obtained without introducing any type of defect, as in the case of photonic crystals.

The considered structure is presented Fig. 1 : two slabs are surrounded by air and separated by a distance *h*. The upper slab, whose thickness is called *h*
_{1}, presents a relative permittivity of *ε*
_{1} and a relative permeability *μ*
_{1} while the lower slab, whose thickness is *h*
_{2}, is characterized by *ε*
_{2} and *μ*
_{2}. This rather simple structure has already been studied[10], but not with a rigorous complex plane analysis of the contra-directional coupling.

We will search for solutions which present a harmonic time dependence and which do not depend on *y* - just like the geometry of the problem. With these assumptions, the solutions can be either TE polarized (*i.e*. the electric field is polarized along the *y* axis) or TM polarized (the magnetic field being polarized along the *y* axis). In this work, we will only consider the TE case but the same study can be carried out for the TM case, leading to the same conclusions. We will hence seek a solution for which *E _{y}*(

*x*,

*z*,

*t*) =

*E*(

*z*) exp(

*i*(

*αx*-

*ωt*)) with

*ω*=

*k*

_{0}$c=\frac{2\pi}{\lambda}$ where

*λ*is the wavelength in the vacuum. In the following, we will consider that

*λ*is the distance unity (

*i.e*. we take

*λ*= 1).

The function *E*(*z*) in a medium *j* characterized by *ε _{j}* and

*μ*, is in general a linear combination of two exponential terms exp(±

_{j}*iγz*) with

*γ*

_{j}^{2}+

*α*

^{2}=

*ε*

_{j}*μ*

_{j}*k*

^{2}

_{0}. Since we are only interested in guided modes, we will consider that

*α*>

*k*

_{0}so that the electric field is decreasing exponentially in the air (we will note ${\gamma}_{0}=\sqrt{{\alpha}^{2}-{k}_{0}^{2}}$). Above and under the waveguides (in regions I and V) only one decreasing exponential term remains. In this case, the continuity relations at an interface between two different media can be written as a homogeneous system of equations. A non null solution can thus be found only if the determinant of the system is null, which can be written

where ${F}_{j}=\frac{1-{x}_{j}\mathrm{tan}\left({\gamma}_{j}{h}_{j}\right)}{{x}_{j}+\mathrm{tan}\left({\gamma}_{j}{h}_{j}\right)}$ and ${x}_{j}=\frac{{\gamma}_{j}}{{\mu}_{j}{\gamma}_{0}}$ whatever the sign of *ε _{j}* or

*μ*. Relation (1) is the dispersion relation because each zero of the left part is a mode supported by the structure.

_{j}The coupling of the two waveguides appears when the upper and the lower waveguides, taken separately, both support a guided mode for the same constant propagation, called *α*
_{0}. First we have chosen a small thickness for the right-handed slab (*ε*
_{1} = 3, *μ*
_{1} = 1, *h*
_{1} = 0.2*λ*) so that it behaves as a monomode waveguide and we have computed *α*
_{0}. If the two waveguides are identical (both right-handed and with the same thickness) the structure supports two propagating modes with two close and real propagation constants. The excitation of these two modes generates an oscillation of the energy between the two guides as shown Fig.2.

When the lower waveguide is left-handed (with *ε*
_{2} = -3, *μ*
_{2} = -1), its thickness has to be chosen carefully to ensure a perfect coupling with the above slab. Using the dispersion relation of the left-handed waveguide[7] we have chosen the thickness of the LHM slab (*h*
_{2} = .7146*λ*) so that (i) it supports no fundamental mode (which would otherwise propagate forward) and (ii) the only supported mode is more localized in the LHM and hence propagates backward.

*With the above parameters the dispersion relation is verified only for complex values of α*: the two different solutions have then the same real part but opposite imaginary parts (*i.e*. conjugate propagation constants). The solutions are shown in the complex plane Fig. 3 for different values of *h*. Such a behaviour is a characteristic of a perfect contra-directional coupling in the case of transmission lines[11].

Let us now study the characteristics of the solutions, which are shown Fig. 4. Figure 4 shows the modulus, the phase and the time-averaged Poynting vector along the *x* direction for the solution corresponding to the propagation constant with a positive imaginary part with *h* = 0.5. The solution is an hybrid mode, since the field is important in both waveguides (see Fig. 4(a)). The field in the right-handed medium and the field in the left-handed medium are in phase quadrature : there is a phase difference of $\frac{\pi}{2}$ in between as shown Fig. 4(b). Since the propagation constant of the other solution is the conjugate of the one represented Fig. 4, then the solution itself is the conjugate of the first one. Therefore the only difference between the two solutions is the phase, so that in both cases there is a phase quadrature between the two guides. This is very different from what happens with co-directional coupled waveguides in which the two guides are in-phase or out-of-phase, depending on the considered mode (see Fig. 2).

The time-averaged Poynting vector along the *x* direction is shown Fig. 4(c). As expected, it is negative in the LHM and positive elsewhere. It is very interesting to note that *the overall power flux of each mode is found to be null*. This means that all the energy that is sent in one of the waveguides comes back using the other one. This helps to understand the nature of the modes : although they have an important real part of the propagation constant along the *x* axis, they can be considered as evanescent modes because they do not convey any energy. It is then difficult to define a *propagation* direction, but the imaginary part of the propagation constant suggests that the modes actually have a direction in which they *develop*. When the imaginary part is positive, the mode is decreasing when *x* is increasing. It is thus expected to develop towards the right - otherwise it would diverge. This means that the other mode develops in the other direction. This is what happens in transmission lines as well[11].

Let us just underline that these two modes, having the same real part of their propagation constant, can be excited separately in a fire-end coupler only[12]. Otherwise, they may be excited using evanescent coupling, *i.e*. using an incident beam propagating in a high-index medium close to the waveguides. When the incidence angle of the beam is greater than the critical angle, the waveguided modes can be excited. In this case, the two twin modes are excited together. Figure 5 shows such a situation. The field is clearly enhanced in the two waveguides, and the structure of the mode in the left-handed slab is particularly visible. The most striking feature is the fact that there is a dark zone just above the incident beam in the closest slab. It is produced by the superposition of the two modes - and the center of the beam is the only place where it may happen since the modes develop in different directions. Such a dark zone could be expected - after all, this is the case Fig. 2 - but in the lower waveguide. This is actually a characteristic of this particular coupling and it persists even when the two slabs are exchanged.

Figure 6 shows the time-averaged Poynting vector along the x direction for three different transverse cross-sections. If the cross-section is chosen in the center of the incident beam, the dark zone is characterized by a null Poyting vector. Further away from this point, the typical profile of the modes shown Fig. 4 can be recognized, strengthening our interpretation.

Finally, when the two modes are excited they form what we called a *light wheel*: the light is heading to the right in the right-handed waveguide and to the left in the left-handed slab. On the right of the incident beam, the energy flows from the right-handed slab to the left-handed slab. This is the contrary on the left of the beam, so that light globally rotates inside the structure. In the case of an evanescent coupling, the reflected beam is distorted because of the excitation of the light wheel - but it is distorted symetrically if the coupling is good enough. Such a simple structure could then be used to perform beam reshaping[14].

Since non dissipative LHM are unlikely, we show Fig. 7 what happens to the light wheel when the LHM is lossy : the light wheel is still visible, although it is distorted and much narrower, which means that the imaginary part of the modes are higher. This imaginary part is in this case not only due to the contra-directional coupling but to the dissipation as well. Dissipation limits the spatial extension of the light wheel and breaks its symetry but it still can be excited.

The above results suggest that despite its lamellar geometry the structure can be considered *as a cavity*. To ensure that an atom placed inside a waveguide is actually coupled with the light wheel we have computed the Green function of the problem. Figure 8 shows the Green function when a source is place in the middle of the dielectric slab, and the light wheel is clearly excited. More precisely, two contra-rotative light wheels are excited and interfere.

In conclusion, we have shown that in the case of a contra-directional coupling between two optical waveguides, the two conjugate modes of these coupled waveguides could be excited together, leading to a new type of phenomenon : the rotation of light which would then form a light wheel. This phenomenon could be used for beam reshaping, or even as a new type of cavity. It is an original way to suppress any propagative mode in a dielectric slab.

It seems that the contra-directional coupling can be obtained using a simple layer of silver[12]. In this case, however, the structure would still support a guided mode : the odd plasmon-polariton mode cannot be suppressed. With a LHM slab, all the propagating modes of the structure could be suppressed in a way very similar to what happens for transmission lines[11]. This is particularly interesting if the structure is used as a cavity. Let us note that for beam reshaping the left-handed behaviour (with no losses) leading to the contra-directional coupling could be obtained using a photonic crystal as well. We believe anyway that this study brings a new evidence that left-handed materials are renewing the physics of lamellar structures in photonics[15].

The authors would like to thank D. Felbacq and E. Centeno for fruitful discussions. This work is supported by the ANR project POEM.

## References and links

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