## Abstract

Within an effective medium theory, we numerically study by means of a finite element method the transmission properties of prisms and slabs of media with negative refractive index. The constitutive parameters employed are similar to those of recent experiments that confirmed the existence of negative refraction as well as the focusing property of flat slabs. In this way, we further analyze in detail the influence of diffraction and scattering due to the large wavelength of the radiation in use, and its suppression by employing waveguide configurations with absorbing walls. Also, we address the effects of different amounts of absorption on both the angle of refraction, (for which we derive a new refraction law in prisms), and on the position, resolution and isoplantism of the focus produced by flat slabs.

©2004 Optical Society of America

## 1. Introduction

Recently, there has been a great deal of effort in studying both experimentally [1],[2] and theoretically [3]–[9] the possibility of obtaining negative refraction at microwave frequencies in so-called negative index media or left-handed materials (LHM), characterized by a negative permittivity *ε*, permeability *μ* and refractive index *n*[10]. Since these media do not occur in nature, composite structures of metallic wires and rings of millimeter size were constructed [1],[2] to demonstrate their effect in the microwave region. To this end, a sample with a prism shape was employed. Initial questions [7],[8] on the possible role of absorption of these structures and measurements in the near field region, both of which could affect or even obliterate the observation of a neat refraction, have subsequently been addressed by detection farther from samples and by either attempting to reduce absorption effects [11],[12], or by making separate measurements at two different angled prisms [13].

In addition to absorption, the experiments have been carried out with LHM samples whose sizes (some centimeters) are not much larger than the wavelength (about 3 cm). Therefore, to a certain extent they behave in free space like particles for the microwaves and, hence, scattering and diffraction effects, should be important. Thus, in principle, one should expect that the transmitted wavefront will have a complicated structure in which not a single beam direction of propagation can be assessed. However, introducing these samples in a waveguide configuration with absorbing walls, like in the experiments of Refs. [1] and [13] should significantly reduce these effects, thus conveying a cleaner main propagation direction to the transmitted beam. However, in previous works these questions, as well as the role of different amounts of absorption, have not been fully addressed.

Another characteristic that has been put forward of LHMs is the property of flat slabs of index inverse to that of the surrounding medium, to produce focusing [10]. This has also been the subject of discussion after it was proposed [14] that such a LHM slab could focus with resolution beyond the half wavelength limit through amplification of evanescent waves. Limitations to this effect were subsequently argued [8],[15]–[21]. However, the observation of focusing in a LHM slab has so far appeared swamped by high absorption [13] and thus the obtained focus had not a sharp pattern. In addition, the position of the focus did not correspond to what theory would predict for a real refractive index and thus it was questioned the role that either absorption or measurements conditions might have in this respect [22].

In this paper we shall address the diffraction and scattering effects versus those of refraction in prisms like those used in Ref. [13]. In this way, we analyze further the consequences of that experiment of Ref. [13]. We shall show the influence of scattering on the difficulty to observe a refraction direction when those prisms are in a free space environment, and then, by contrast, we shall show how these effects are minimized when the prisms are placed in a waveguide enviroment with absorbing walls like in configuration used in [13]. Also, the presence of losses in the LHM prisms are accounted for in establishing the refraction law. We shall see that, in general, these losses introduce directions of phase propagation and directions of constant amplitude which are perpendicular to each other in the air. However, we shall show that for the refractive index of the samples used in experiments such as those of Ref. [13], the deviation introduced by the imaginary parts of the refractive index upon the lossless Snell’s law are not appreciable.

Further, we shall analyze the effect on focusing of the refractive index of the slabs used in the experiments of Ref. [13]. Thus, we shall study the deviations obtained from those samples of the ideal surface impedance matching with vacuum (i.e., of refractive index -1). In this respect, we shall also address the effect of absorption upon the position, size and intensity of the focus, and hence the aberrations and focus blurring due to this mismatch of the slab surface impedance matching conditions.

## 2. Numerical method

We solve Maxwell’s equations by means of a finite element method, using FEMLAB, a commercial software. The solution domain is divided into triangles forming an initial mesh. Upon analyzing several configurations, we have chosen a grid with a mesh growth rate of 1.55 and a mesh curvature factor of 0.65, where the geometrical resolution parameters consist of 10 vertices per edge by default. This configuration is adapted to the geometry and optimizes the convergence of the solution. The final mesh approximately has 50,000 elements and 20,000 nodes. To solve the elliptic Helmholtz equation, the Good Broyden iterative solver is employed. The convergence of this method is ensured on comparison of the nth-step solution in a new refined mesh with that of the previous iteration. We also verify with the tangential and normal boundary conditions of the magnetic field **H** [23] and the conservation of energy. Furthermore, we have checked calculations on comparing with the distribution of transmitted and reflected waves in cases already known of both left and right-handed materials. This determination of convergence is global, however the local errors are distinguishable by changing the number of elements in the mesh like for instance as reported in Ref. [19].

We have considered a two-dimensional geometry for waves in TE polarization (wavelength in vacuum *λ*= 3*cm*). This 2-D configuration carries most of the information on the transmission characteristics of the system dealt with in [13]. Within an effective medium scheme (valid when the lattice constant of the metallic elements, 6 mm, is much smaller than the wavelength, 3 cm, as is the case in the experiments of Ref. [13]), we have used a fixed value, for a single frequency, of both the negative permittivity *ε* and the magnetic permeability *μ*.

## 3. Refraction from an absorbing LHM prism

We first perform numerical experiments in which a plane wave at a frequency *ν*= 10.5*GHz* incides from the bottom onto a prism, either lossless or absorbing, of LHM in free space [see Figs. 1(a) and (c)]. We consider the LHM effective complex refractive index *n̂*_{2} = *n*
_{2}(1 +*i*
*κ*
_{2}), *κ*
_{2} being the attenuation index [24], and we assume *n*
_{2} = -0.35, a value similar to that of the samples used in [13].

Since the dimensions of the prism and those of the wavelength are comparable, diffraction and scattering effects are large, so that it is impossible to distinguish a neat refracted beam propagation direction. This situation is improved by placing the prism between absorbing walls in a waveguide configuration (Figs. 1(b) and 1(d) illustrate the difference with the previous free-space configuration). On the other hand, Figs. 1(c) and 1(d) show the large losses of the transmitted wave occurring in the absorbing samples, even within distances that are not much longer than the wavelength.

Neglecting a non-instantaneous response of the medium to excitations, a movie of the evolution with time of the time-harmonic electric field *E* = ∣*E*∣ cos (*ϕ* + *ωt*) (*ω* = 2*πν*) is shown in Fig. 2. *ϕ* represents the phase of *E*.

Next, we address the effect of prism losses on the refraction law at the tilted exit face. In the following analysis we aim to understand the purely refractive effect, aside from diffraction and scattering phenomena associated to the finite size of the prism faces. This refraction is better observed with the prism in the absorbing wall waveguide configuration. For a normally incident plane wave on the entrance interface of the prism, it results a transmitted homogeneous wave propagating in the same direction. Then arriving at the tilted face at an angle of incidence *θ*_{i}
, which is equal to the real angle of the prism *φ*. Therefore, if *k*(**r** ∙ **s**^{t}
) is the phase of the plane wave transmitted into the vacuum, (*k*
**s**^{t}
denotes the wavevector), we have at the exit interface:

Equation (1) is the refraction law. It is convenient to write[24]:

Then:

So that:

From Eq. (6) we see that the planes of constant phase are:

while the planes of constant amplitude are given by:

*c*
_{1} and *c*
_{2} being real constants. Due to Eq. (5), the planes of Eqs. (7) and (8) are orthogonal to each other, and the normal to the planes of constant phase makes an angle *θ*′_{t} with the interface
normal:

Therefore, we observe that this angle of refraction *θ*′_{t} for the phase of the transmitted wave implies a refractive index *n*′ = *n*
_{2}/(${n}_{2}^{2}$ sin^{2}
*θ*
^{2} + *q*
^{2} cos^{2}
*γ*)^{1/2} which depends on the incidence angle *θ*_{i}
or, consistently, on the angle of the prism *φ*. An increase of the losses involves a decrease in the absolute value of the refraction angle *θ*′_{t}/(Fig. 3). This figure shows that, within the range of constitutive parameters used here, this variaton of *θ*′_{t} is low and its change due to *n*
_{2}
*κ*
_{2} is within the systematic error of the refraction direction estimation.

Figures 4(a) and 4(b) show the lines of constant amplitude ∣*E*∣ and those of constant real part of electric field *E* = ∣*E*∣cos (*ϕ* + *ωt*) at a time *t* = *t*
_{0} for the absorbing prism within the waveguide configuration. Figure 4(a) shows the difficulties in estimating the refraction angle from the lines of constant amplitude, due to the cross section of the wave in the guide configuration and to edge effects. This direction is better estimated from those planes of constant phase *ϕ*, (Fig.4.(b)).

## 4. Imaging with a rectangular LHM slab

We next consider a rectangular slab of lossy LHM , (*n*
_{2}
*κ*
_{2} = -0.35) of thickness 6 cm with a point source at 2 cm from its first interface (cf. Fig. 5), like in the experiment of Ref.[13]. In that reference, the role of the losses in the focusing process was not discussed we shall do it next. From Fig. 1 of Ref. [13] we estimate those losses approximately: *n*
_{2}
*κ*
_{2} = 0.25.

The configuration employed is with matched boundary conditions at the vertical boundaries of the slab, which is equivalent to sandwiching the slab across a waveguide of absorbing walls.

We next calculate how the imaginary part of the slab refractive index *n*
_{2}
*κ*
_{2} affects the direction of propagation of the refracted wave, and in turn to the location of the focus. Let a plane wave be incident on the slab:

in our case *n*
_{1} = 1, and *θ*_{i}
is the angle of incidence. The transmitted wave into the LHM slab is:

where **a**_{t}
is the attenuation vector and **k**_{t}
is the phase vector in the slab, whose components derived from the boundary conditions are:

*n̂*_{2} = *n*
_{2}(1+*iκ*
_{2}) is the complex refractive index of the LHM slab. Then *k*_{t}
becomes:

Equation (12) constitutes the law of refraction for the phase vector **k**_{t}
. From this law, we infer that increasing the losses involves a decrease in the refraction angle into the slab *θ*′_{t} in absolute value, (Fig. 6), and thus the focus approaches the slab as *n*
_{2}
*κ*
_{2} increases. This is shown by numerical simulations in Figs. 7 and 8. In figure 7 we also show the ray tracing diagram on passing through the LHM slab, taking into account the refraction law given by Eq. (12). Thus, we show how the caustic of the focus approaches the slab as absorption increases, in accordance with the numerical simulations of Fig. 8. The position of the focus in Fig. 7(b) is very similar to that obtained in Fig. 3(b) of Ref. [13].

Figures 7(a),(b), also show the appearance of a depth of focus. The full width at half maximum (FWHM) of the image does not vary with losses. This FWHM is 1.71*λ*, thus showing no superresolution, as expected.

The decrease of the peak intensity of the focus normalized to the intensity of the source as the losses increase is shown in Fig. 9.

We have made numerical simulations by fixing the real component of the refractive index to *n*
_{2} = -0.35 and taking *n*
_{2}
*κ*_{2} either 0.05, 0.10, 0.15. Furthermore, we consider the permeability real: *μ̂* = *μ* and we vary it. For each *n̂*_{2} and *μ* we have a value of permittivity *ε̂*. So that we get to vary the normalized surface impedance *ẑ* = (*μ̂*/*ε̂*)^{1/2} = (*μ*/*n̂*_{2}). We analyze the resulting image (Fig.10).

We observe that as the permeability *μ* increases, the distance of the focus to the slab decreases. If we consider the transmission factor of the slab at TE polarization *t*_{TE}
for the second interface, which is given by:

where the angles are given by Fig. 5, it is clear that as *μ* increases, the modulus of transmission factor ∣*t*_{TE}
∣ decreases. This effect is more critical at high angles and, therefore, the focus is made up mainly of paraxial rays and, consistently, decreasing the distance of the focus to the slab.

## 5. Analysis of the slab isoplanatism

Spatial invariance, or isoplanatism, is a central property of lenses and optical systems. We next investigate the degree of isoplanatism of a LHM slab with different refractive indices.

We simulate a configuration for a slab with the properties of the experiment of Ref. [13] in front of an object constituted by two slits of width *λ*/20 (*λ* = 3*cm*)(Fig. 11). In a first simulation, the slit centers are separated 10cm, we assume that from both slits emerge the same intensity (Fig. 12(c)) and compare a lossless LHM slab (*n̂* = -0.35) [Figs. 12(a) and 12(d)] with that of *n̂* = -0.35 + 0.25*i* [Figs. 12(b) and 12(e)]. Then, in a second more stringent test, we consider in one slit an absorber so that different intensities emerge from these slits (Fig. 13). Accordingly, in Figs. 12 and 13 we observe the formation of an image of the slits, which as explained above, approaches the slab as the losses increase. There is a considerable limitation of resolution, and also a lot of intensity is lost in other directions. In Fig. 12, we also observe additional structures in other regions above the slab.

Simulation with slits separated 5 cm shows the difficulty to resolve the two slits(Figs. 14 and 15), even though in the case in which they emit with different intensity, there is a recovery of their image. In both cases of objects, isoplanatism is slightly lost as higher resolution of details is sought. Namely, Fourier optics loses validity as the dimensions of the slab and those of the wavelength become comparable, as expected.

Finally, we show by contrast with the former cases a more favorable situation of a slab with a refractive index *n̂*_{2} = -1.0454 + 0.0041*i* such as for the material used in the experiment of Ref. [11]. This case would be very similar to the one originally considered by Veselago [10] (see also Ref. [14]). The small imaginary part of *n̂*_{2} regularizes divergencies at the exit interface of the slab that otherwise would exist for the refractive index strictly being *n̂*_{2} = -1. Now, we consider that the slits are separated either 10cm (3.33*λ*) or 3cm (*λ*). In this case, (see Figs. 16–19), the two slits are sharply resolved, even achieving some degree of superresolution, although the peak positions in the image appear slightly altered with respect to those of the object slits, this is again indicative of some loss of isoplanatism due to diffraction.

Additionally, Fig. 19(b) shows the image from finite width slabs of 40, 25 and 15cm. Thus, we see that there is not an appreciable resolution loss by reducing the width of the slab within these values, even though in the cases of 25 and 15cm the edges of the slab are not in contact with the waveguide absorbing walls.

## 6. Conclusion

Since due to the large wavelength in the microwave region, prisms samples of LHMs used in experiments are unavoidably of particle size for the incident wave, scattering and diffraction phenomena are of importance. We have studied these effects and shown how sandwiching the sample within a waveguide configuration of absorbing walls, as done in the experiments of Ref. [13], dramatically suppresses these effects, thus allowing to distinguishing a neat refraction direction within some estimation error.

We have established a refraction law for these prisms at the exit interface, which shows that, due to absorption, the refracted wave into the air is in general inhomogeneous with the phase and amplitude planes normal to each other. However, for the constitutive parameters used in the experiments so far, the alteration of the usual Snell’s law caused by absorption is almost negligible. Nonetheless, we believe that the special characteristics of our new refraction law due to absorption are of interest for future experiments with different constitutive parameters. Absorption, on the other hand, is shown to be quite large in these prisms, within ranges comparable to the wavelength. It is desirable that these losses can be overcome in future designs of LHMs.

This absorption has important effects on both the position and focus size produced by flat slabs. We have clarified why the focus appears closer to the slab than what one could expect from the value of the real part of the refractive index. Finally, we have analyzed the isoplanatism characteristics of these slabs.

## Acknowledgments

Work supported by the Spanish MCyT and by the European Union.

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