1. Introduction

The localization of electromagnetic waves and energy is very important in science and engineering, which could find novel applications in a variety of optical and microwave devices because of the high quality factor of the localized modes. For example, it can be used to design new narrow-band filters and low-threshold lasers [1]. Conventionally, the localization of electromagnetic and optical waves has been realized using some structural defects introduced to the regular structure of photonic crystals [2], or using a three-dimensional fractal cavity, such as the Menger sponge [3]. Recently, it has been demonstrated that a left-handed medium (LHM) slab could also be used to localize electromagnetic waves and energies [4].

LHM is a new artificial electromagnetic material. As predicted by Veselago in 1968, a LHM slab behaves like a lens which can refocus the electromagnetic waves [5]. If the LHM slab is perfectly matched with the air where the permittivity is -ε ₀ and the permeability is -µ ₀, which is called the anti-vacuum condition, Pendry further indicated that both propagating and evanescent waves produced by a source could be recovered at the image point, and hence such a LHM slab can be made as a perfect lens [6]. However, it has been shown that the lossless perfect lens is not physical since it will lead to infinite energy in the slab and the unavoidable loss in the lens will suppress the evanescent components [7–13]. Actually, there is no amplification at all of evanescent waves inside the perfect lens. What there exists is the excitation of plasmon-polaritons in its back surface, which is physical when there exists either absorption or deviations with respect to that antivacuum condition. If two current sources with the same amplitudes and opposite directions are placed at the image points of the perfect lens, we have shown that all electromagnetic fields are confined in a region between the two sources, and further study has been conducted if there exists a deviation with respect to the anti-vacuum condition [4]. However, the requirement of two opposite-phase sources makes such an energy-localization structure impractical. Also, a small phase shift of the two sources will produce a notable leakage of waves.

In this paper, we propose a new compact structure to localize electromagnetic waves using a thin grounded LHM lens. In the new structure, only single current source is required which is placed in front of the lens at a distance of the slab thickness. It has been shown rigorously that all electromagnetic fields are completely confined between the source and the conducting plane if the anti-vacuum condition is satisfied. We can adjust the thickness of the slab and the electromagnetic energies can be localized in any small regions as required. As mentioned earlier, however, the lossless perfect lens is unphysical and it does not exist in nature. Hence, further study has been conducted for the lossy and retardation effects on the energy localization. Most remarkably, electromagnetic waves remain strongly localized even when small losses are taken into account. The new structure can be directly used as a substrate to develop new high-quality microwave and optical devices.

2. A new compact energy localization structure

Let us consider a linear source I which is located in front of a LHM slab with the relative permittivity ε_r1 and relative permeability µ _r1 at a distance of d ₁. A perfectly electrically conducting (PEC) plane is backed with the LHM slab, as shown in Fig. 1. From the electromagnetic theory, the electric fields in different regions are easily expressed as [14]

where, $k_{\mathit{iz}}=\sqrt{k_i^2-k_y^2}\left(i=0,1\right)$, R is the reflection coefficient of the slab, and $E_1^{+}$ and $E_1^{-}$ are forward and backward transmission coefficients. Such coefficients are given by

in which p_ij =µ_ik_jz /µ_jk_iz , and R_{i j} =(1-p_{i j} )/(1+p_ij ) is the Fresnel reflection coefficient on the slab boundaries. Similarly, the magnetic fields in different regions can also be obtained.

 

Fig. 1. A compact energy-localization system, in which a linear source I is located in front of a grounded LHM slab at a distance of d ₁, the thickness of the slab.

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Now we consider a special case of the perfect lens when ε_r1=-1 and µ_{r 1}=-1. For all propagating waves (|k_y |< k ₀), we have k _1z=-k _0z. Hence, the Fresnel reflection coefficient R ₀₁ is zero, which yields R=-1. For all evanescent waves (|k_y |>k ₀), we have k _1z=k _0z=iα _0z. In such cases, the Fresnel reflection coefficient R ₀₁ approaches to infinity, which still gives R=-1 by simple derivation of Eq. (4). As a consequence, the electric field in the region of z<0 is completely zero based on Eq. (1). Similarly, the magnetic field is also zero in the same region. Hence, we conclude that all electromagnetic waves and energies are confined in a region between the source and the PEC plane (0≤z≤2d ₁), and there is no power radiating outside the region.

Actually, the above conclusion is consistent with that in [4]. Due to the existence of the PEC plane, the image theory gives an equivalent description of the physical problem shown in Fig. 1 in the considered region: the source I is placed in front of a LHM slab with a thickness of 2d ₁, and there is another source -I located at the image point z=4d ₁ of the thick slab. Obviously, the equivalent problem is exactly the same as the problem proposed in [4] for the energy localization. However, the new structure is more compact, and can be directly regarded as a substrate to design new microwave and optical components.

The physical picture of the energy-localization system shown in Fig. 1 could be described as follows. The propagating components of the electromagnetic fields radiated by the source I experience certain phases before entering the LHM slab. Such phases are completely cancelled when they pass through the equal length of the LHM slab. The propagating waves are reflected by the PEC plane to form oppositely-polarizedwaves, which experience negative phases through the LHM slab. Such negative phases are cancelled again at the source location. On the other hand, the evanescent components have a decay before entering the LHM slab. But such a decay will be amplified to the original level when they travel through the LHM slab. Again, the evanescent waves are reflected by the PEC plane to form oppositely-polarized waves, which experience another amplification through the LHM slab. Finally, such an amplification is decayed to the original level at the source position. Therefore, for the left region of the source I, there exists an equivalent source -I at the same position for both propagating and evanescent components. The fields radiated by the two sources cancel each other to yield a zero total field in this region.

The compact structure shown in Fig. 1 with the perfect lens is an ideal energy-localization system. However, the perfect lens is unphysical and it does not exist in nature. All realistic LHM slabs have losses and cannot be as perfect lens. Hence, we have to study the lossy and retardation effects on the energy localization. Consider a general LHM slab where ε_r1=-1+_δ ε+iγ_ε and µ_{r 1}=-1+δ_µ +iγ_µ . In this case, the reflection coefficient R is no longer -1 and there will be a pole along the integration path, which corresponds to the surface modes [12,13]. Our studies have shown that the larger are the loss γ and the retardation δ, the weaker is the ability of energy localization using such a system.

 

Fig. 2. Electric field distributions along the line y=0, where the source is located at z=0, the slab interfaces are d ₁=15 mm and d ₂=2d ₁=30 mm, δ_ε =δ_µ =0, and f=10 GHz. (a) The propagating components (Left). (b) The evanescent components (Right).

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3. Numerical results

Figure 2 demonstrates the electric field distributions along the line y=0 of a thick LHM slab (d ₁=15 mm and d ₂=2d ₁=30 mm) for different losses (γ_ε =γ_µ =10^-4 and 10^-3) when δ_ε =δ_µ =0 and the frequency is 10 GHz. Here, we have separated the propagating and evanescent components of the fields to observe their different features in the localization modes. We remark that the large spike at z=0, the source location, in the evanescent components is due to the singularity of Hankel’s function, where the field values in the tiny vicinity of the the source is not plotted. From this figure, we clearly see that the propagating waves are not sensitive to the losses and nearly all fields are confined in the region between the source and the PEC plane even when the loss is 10^-3, as shown in Fig. 2(a). The evanescent waves behave quite differently. From Fig. 2(b), we can observe the strong surface modes along the air-LHM boundary, which are very sensitive to the losses. Clearly, the evanescent waves are also mainly confined in the same region.

 

Fig. 3. Electric field distributions for a thin LHM slab, where d ₁=1.5 mm, d ₂=2d ₁=3 mm, δ_ε =δ_µ =0, and f=10 GHz. (a) The propagating components along the line y=0 (Left). (b) The total electric field on the yoz plane when γ_ε =γ_µ =10^-3 (Right).

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Actually, the LHM slab can be made as thin as required. Figure 3 illustrates the numerical results for a thin slab when d ₁=1.5 mm and d ₂=3 mm. Here, only the propagating fields are plotted and the evanescent fields have a similar behavior to that in Fig. 2(b). In order to give an overall picture, we have computed the total electric field distribution (including both the propagating and evanescent components) in a yoz plane, as shown in Fig. 3(b). Here, δ_ε =δ_µ =0 and γ_ε =γ_µ =10^-3. From Fig. 3(b), we clearly see that nearly all electromagnetic energies are confined in a very small region 0≤z≤3 mm.

 

Fig. 4. Electric field distributions for a thin LHM slab, where d ₁=1.5 mm, d ₂=2d ₁=3 mm, δ_ε =δ_µ =10^-2, γ_ε =γ_µ =10^-2, and f=10 GHz. (a) The propagating components (Left). (b) The evanescent components (Right).

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Next, we consider a LHM slab with a larger loss and retardation (γ_ε =γ_µ =10^-2 and δ_ε =δ_µ =10^-2). In such a case, the electric field distributions along the line y=0 are illustrated in Fig. 4. Because of the large loss and retardation, some of electromagnetic waves and energies are leaked out of the small region. However, we still observe an obvious energy localization.

Finally, we consider how the degree of localization changes if the distance from the source to the slab is slightly different from the slab thickness. Let the left edge of the LHM slab be located at z=d ₁-δ _d, and the distance from the source to the PEC plate remains unchanged. In another word, the slab becomes thicker when δ _d > 0, and the internal focus is no longer located at the PEC boundary. In such a case, the field distributions of the propagating and evanescent components are plotted in Fig. 5, where the peaks of field values for different δ_d (δ_d =0.1 mm and 0.01 mm) do not exist at the same position due to the change of the slab interface. Note that larger field value is observed in the region of z < 0 when δ _d increases, which indicates more energy flows outside the region between the source and PEC plate. However, the role of energy localization is still obvious. From Fig. 5(b), when δ _d becomes larger, the amplitude of surface polaritons increases. This is consistent with Rao’s analysis in [15].

 

Fig. 5. Electric field distributions along the line y=0, where the source is located at z=0, the slab interfaces are located at z=d ₁-δ_d and z=d ₂=2d ₁, and the PEC plane is placed at z=d ₂. Here, d ₁=1.5 mm, d ₂=3 mm, δ_ε =δ_µ =0, γ_ε =γ_µ =10^-3, f=10 GHz, and δ_d =0.01 mm and 0.1mm, respectively. (a) The propagating components (Left). (b) The evanescent components (Right).

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4. Conclusions

We have proposed a new compact structure in this paper to localize electromagnetic waves and energies using a thin grounded perfect lens. Since only single source is required, the new energy-localization system is more practical and could be directly used as a substrate to develop new high-quality microwave and optical devices. Because the perfect lens does not exist in nature, the lossy and retardation effects are further studied on the energy localization. Numerical results validate the above conclusions