1. Introduction

The derivation of the well-known fact that electromagnetic waves propagating in ordinary isotropic media are transverse requires that the dielectric permittivity ε and the magnetic permeability µ never vanish anywhere in space. In inhomogeneous media where ε or µ vanishes in some regions of space, therefore, longitudinal electromagnetic waves can be excited locally. A well-known example is the mode conversion of electromagnetic waves into longitudinal plasma waves at resonance points where ε vanishes in inhomogeneous unmagnetized plasmas [1–6]. It has been believed that the irreversible transfer of wave energy to the mode conversion region associated with more general kinds of mode conversion in magnetized plasmas plays a crucial role in a wide range of phenomena in plasma physics, including the heating of tokamak plasma, ionosphere and solar corona [7–11].

In the present paper, we are interested in a new kind of inhomogeneous media inside which both ε and µ vanish. Except for some magnetic materials, µ is usually equal to the vacuum value and does not play any significant role in the optical properties. In recent years, however, artificially fabricated structures termed magnetic metamaterials have been shown to possess both nontrivial dielectric and magnetic properties [12, 13]. In so-called negative-index media where both ε and µ take negative values, it has been demonstrated that the refractive index should also take the negative value given by $n=-\sqrt{\epsilon \mu }$ [14, 15]. A large number of studies have been devoted to exploring the physical properties of these media and their applications to useful nanophotonic devices [16, 17].

Let us consider the situation where a slab made of a negative-index medium is in contact with a slab made of a positive-index medium. We assume that a transition layer, inside which both ε and µ change continuously from positive to negative values, is formed at the boundary between these slabs. Then, it is obvious that there should be planes where ε and µ vanish. We believe it is possible to fabricate a system with transition layers experimentally by a careful design of metamaterials. Due to severe discreteness of metamaterials available in current experiments, our system needs to be large and its spatial variations have to be much smaller than the wavelength. Electromagnetic waves incident on this system are expected to show mode conversion into longitudinal modes, which will be manifested as resonant absorption of wave energy and heating in a narrow region close to resonant points. This is a new kind of mode conversion which has never been studied before. The aim of the present paper is to explore this phenomenon by calculating the mode conversion coefficient, which measures the wave absorption, and the electromagnetic field profile in a numerically exact manner. We note that a similar problem was considered in [18], but no calculation of the mode conversion coefficient was performed there.

We will find that unlike in the case of inhomogeneous plasma, the mode conversion in the transition layer we are considering occurs for both s and p waves. When ε and µ change linearly near the resonant point where both quantities vanish, we will find a universal curve for the mode conversion coefficient. In more general cases, the strength of mode conversion depends on the spatial profiles of ε and µ very sensitively. We will also find that in the case where there are two transition layers at nearby positions, the mode conversion can be strongly enhanced or suppressed depending on the parameters due to the interference between the mode conversion phenomena occurring at the two points. The mode conversion coefficient is shown to be as high as 0.8.

Since mode conversion is associated with the singularity of wave functions at resonance points, theoretical studies meet great difficulties and often adopt approximate methods such as the WKB method [1]. In complicated cases where several different wave modes are strongly coupled, these methods are often unable to produce sufficiently accurate results. Recently, two of us have developed a new theoretical approach to mode conversion based on the invariant imbedding method [19–23], using which we have succeeded in obtaining numerically exact results for various wave propagation characteristics and the electromagnetic field distribution, for a variety of mode conversion problems [6, 9]. This method will be used in the present paper.

2. Model

We consider the propagation of a plane electromagnetic wave of vacuum wave number k ₀ in inhomogeneous metamaterial media. The wave is incident from a uniform region with ε=ε ₀ and µ=µ ₀ onto a stratified medium, where both ε and µ vary only along the z axis, and transmitted to another uniform region with ε=ε ₀ and µ=µ ₀. The inhomogeneous medium lies in 0≤z≤L and the wave is incident from z>L and propagates in the xz plane. When θ is the angle of incidence, the x component of the wave vector, q, is equal to $\sqrt{{\epsilon }_0{\mu }_0{k}_0}\sin \theta .$.

For obliquely incident waves, we distinguish two types of linear polarization. In the s wave case, the electric field vector is perpendicular to the xz plane and the complex amplitude of the electric field satisfies

Once we obtain E(z) by solving this equation, we can easily calculate the magnetic field components using

In the p wave case, the magnetic field vector is perpendicular to the xz plane. There exists a simple symmetry between the s and p waves. All results for the p wave case can be obtained by replacing ε, µ, E and H with µ, ε, -H and E in the equations for the s wave case.

Mode conversion of transverse electromagnetic waves into longitudinal plasma oscillations may occur where either ε or µ is zero. In order to study this phenomenon occurring in a transition layer between positive-index and negative-index media, we consider three simple models, which we call models A, B and C. In model A, we assume that both ε and µ depend linearly on z and vanish at the same value of z inside the transition layer, as shown in Fig. 1(a):

where ε̃=ε/ε ₀, µ̃=µ/µ ₀, L=4h and z _R (=3h) is the resonance point where both ε and µ vanish and mode conversion for both s and p waves occurs. η is a small positive constant that describes damping.

 

Fig. 1. Profiles of the dielectric permittivity and the magnetic permeability inside metamaterial media in (a) model A, (b) model B, and (c) model C. The squares designate the positions where both ε and µ vanish. At the position marked by a filled (empty) circle, only ε (µ) vanishes. The vertical dashed lines indicate the region where 0≤z≤L. The wave is assumed to be incident from the right side where z>L and transmitted to the left side where z<0.

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In model B, ε and µ depend linearly on z, but vanish at different values of z, as in Fig. 1(b):

where L=4h and z _{R 1} (=8h/3) and z _{R 2} (=10h/3) are the resonance points where ε and µ vanish respectively. At z=z _{R 1}, incident p waves will be mode-converted, whereas at z=z _{R 2}, s waves will be mode-converted.

In model C, there are two transition layers as shown in Fig. 1(c):

where L=6h and z _{r 1} (=h) and z _{r 2} (=5h) are the resonance points where both ε and µ vanish.

3. Invariant imbedding equations

We consider an s wave of unit magnitude Ẽ(x,z)=E(z)exp(iqx)=exp[ip(L-z)+iqx], where $p=\sqrt{{\epsilon }_0{\mu }_0{k}_0}\cos \theta$, incident on the medium. The quantities of main interest are the reflection and transmission coefficients, r=r(L) and t=t(L), defined by

Using the invariant imbedding method, we have derived exact differential equations satisfied by r and t [22]:

These are supplemented with the initial conditions r(0)=0 and t(0)=1. For given values of k ₀ and θ and for arbitrary functions ε̃(l) and µ̃(l), we integrate Eq. (7) from l=0 to l=L and obtain r(L) and t(L).

We have also derived the equation for the electric field amplitude inside the inhomogeneous medium:

where E is considered to be a function of z and l. For a given z, E(z,L) is obtained by integrating this equation, together with the equation for r(l), from l=z to l=L using the initial condition E(z,z)=1+r(z).

If mode conversion occurs, the energy of the incident wave is absorbed into the medium, even when the damping parameter η is vanishingly small [1, 5, 6, 9]. In our models, the mode conversion coefficient A is obtained by A=1-R-T, where R=|r|² and T=|t|².

4. Results

We first consider the results for model A. We have calculated the mode conversion coefficient A as a function of the incident angle θ for several values of the parameter $\zeta \left(\equiv \sqrt{{\epsilon }_0{\mu }_0}{k}_{0}h\right).$. When ζ≫1 and η→0, we find numerically that A is a universal function of the parameter Q≡ζsin² θ. That this has to be the case can be seen by rewriting our wave equation in the transition region using the variables ẑ≡ζ^1/2(z-z _R)/h and η̃≡ζ^1/2 η:

In the inset of Fig. 2, we plot A versus θ for ζ=10π, 25π, and 50π and for η=10^-8. If we plot A versus Q, we find that these curves fall into the same universal curve as shown in Fig. 2. Due to the symmetry between ε and µ in model A, this result applies to both s and p waves. We find that as Q increases, the mode conversion coefficient initially increases until it reaches the maximum value of approximately 0.5 at Q=0.44. After that, A decreases rapidly and becomes almost zero when Q>4.

 

Fig. 2. Universal curve for the mode conversion coefficient A for model A plotted versus Q=ζsin²θ. The points B and C marked by square dots correspond to Q=0.44 and π respectively. Inset: Mode conversion coefficient versus incident angle for ζ=10π, 25π and 50π and for η=10^-8.

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Fig. 3. Electric field intensity for model A plotted versus ζ^1/2(z-z _R)/h, when (a) Q=0.44 and (b) Q=π. An s wave is assumed to be incident from the right side and η=10^-5.

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In Fig. 3, we show the universal spatial distributions of the electric field intensity |E|² corresponding to the two values of Q marked as B and C in Fig. 2, as a function of ẑ. We assume that an s wave is incident from the right side. When Q is 0.44 corresponding to the maximum mode conversion, the field intensity remains finite at and beyond the resonance point ẑ=0. On the contrary, when Q is equal to π corresponding to a very small value of A, the field intensity is almost zero for ẑ≤0. In the latter case, we notice that the wave is almost completely reflected before it reaches the resonance point, whereas in the former case, the wave tunnels through the resonance point and propagates in the negative index region. We also point out that the magnetic field profile will look identical to Fig. 3 when p waves are incident.

 

Fig. 4. Mode conversion coefficient for model B (solid line) versus incident angle, when ζ=25π and η=10^-8 and when an s wave is incident. A is almost zero for all θ when a p wave is incident. The result for model A (dashed line), when ζ=25π and η=10^-8, is shown for comparison.

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Next we consider model B. In Fig. 4, we plot A versus θ, when ζ=25π and η=10^-8 and when an s wave is incident. The dependence of A on polarization is extreme in this case. When the incident wave is p-polarized, it is totally reflected and A is almost zero for all θ. This is because in model B, the wave becomes evanescent in the wide region z _{R 1}<z<z _{R 2} for all θ. The probability for a p wave to tunnel through this region and be mode-converted is vanishingly small. The result for model A when the incident wave is either s- or p-polarized is shown for comparison.

Finally, we consider model C. In Fig. 5, we plot R and A versus θ, when ζ=10π and η=10^-8 and when an s or p wave is incident. The result for model A is also shown for comparison. We find that for most incident angles, the mode conversion coefficient for model C is larger than that for model A. In particular, the maximum value of A is greatly enhanced. This enhancement is due to the interference between the mode conversion phenomena occurring at z=z _{r 1} and z=z _{r 2}. The same interference can cause an enhancement of the reflectance and a suppression of A at several discrete incident angles. In the present case, this intriguing phenomenon occurs at θ=10.53° and 15.04°.

In Fig. 6, we plot the spatial distributions of the electric field intensity for model C corresponding to the points P and Q in Fig. 5(b). An s wave is assumed to be incident from the right. The magnetic field profiles will look identical to Fig. 6 when a p wave is incident. In the θ=6.53° case corresponding to the maximum mode conversion, the electric field remains to be finite at the resonance points z=h and 5h and also in the region where z<h. On the other hand, in the θ=10.53° case corresponding to the suppression of mode conversion, the electric field vanishes at the two resonance points and a standing wave is formed between the two points.

 

Fig. 5. (a) Reflectance and (b) mode conversion coefficient for model C (solid lines) versus incident angle, when ζ=10π and η=10^-8. The points P and Q correspond to θ=6.53° and 10.53° respectively. The results for model A (dashed lines), when ζ=10π and η=10^-8, are shown for comparison.

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Fig. 6. Spatial distributions of the electric field intensity for model C, when (a) θ=6.53° and (b) θ=10.53°. An s wave is assumed to be incident from the right side and η=10^-5.

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5. Discussion

So far we have considered only the field components in the direction perpendicular to the incident plane. All other field components can be obtained easily using Eq. (2) in the s wave case and

in the p wave case. We note that these equations have either µ or ε in their denominators. This implies that the electric field intensity diverges at the resonance point where µ=0 in the s wave case and the magnetic field intensity diverges at the resonance point where ε=0 in the p wave case. In real materials, there always exists nonzero damping, which makes the field components finite. Nevertheless, if η is sufficiently small, we expect the electric or magnetic field intensity depending on wave polarization to be very strong at the resonance points. The enhanced field intensity is likely to cause various nonlinear optical effects which are ordinarily very small to be strongly amplified inside the transition layers.

In models A and B, there are discontinuous jumps of ε and µ at z=0. The discontinuity in model A does not play any role because the wave is not reflected at that boundary. The discontinuities in model B are also unimportant since the waves are almost completely attenuated before reaching z=0. The jump in the imaginary part of ε and µ, which we call η, at z=0 and z=L can cause an additional reflection. But for very small values of η used in our calculations, its effect is too small to be noticed.

Finally, we want to point out an intriguing numerical agreement between the maximum absorptance by thin metallic layers obtained in [24] and [25] and the maximum value of the mode conversion coefficient for our model A.