Scatterer localization using a left-handed medium

David Karkashadze; Juan Pablo Fernández; Fridon Shubitidze

doi:10.1364/OE.17.009904

1. Introduction

Landmine and improvised-explosive-device (IED) detection [1], breast-cancer research [2], and geophysical prospecting [3] are some of the endeavors in which it is necessary to locate and characterize one or more hidden objects using electromagnetic scattering data. In any of these cases it is difficult to dissociate the pinpointing stage from the characterization, since the targets’ response to an interrogating field depends nonuniquely on both the properties of the objects and—nonlinearly—on their location and orientation relative to the sensor being used. Conventional approaches to scatterer detection rely on the concurrent inversion of collected data for location, orientation, and intrinsic features. The resulting problems are often ill-posed and always computationally expensive.

This paper presents a method that exploits the focusing property of some ideal left-handed (or double-negative) metamaterials [4–6] to locate hidden scatterers. The method is similar in spirit to the backpropagation method used in computer tomography (see e.g. Ref. [7]) but is designed to avoid the iterative optimization procedures that the latter incurs; it also has some similarities to the time-reversal techniques of acoustics (see Ref. [8] for a recent example). As is well known by now [9,10], metamaterials are fabricated by embedding in a dielectric medium a periodic array of conducting inclusions whose response to an incident electromagnetic field influences the bulk behavior of the composite and allows the engineering of its material properties. In particular, left-handed media (LHM) have simultaneously negative electric permittivity and magnetic permeability over a particular range of frequencies, which results in a negative index of refraction [4, 5, 11] and several related properties that have been confirmed experimentally [12, 13]. The name “left-handed” comes from the fact that an electromagnetic wave propagating through an LHM has its electric field E, magnetic field H, and wave vector k forming a left-handed triad, the opposite of what happens in a conventional medium. One consequence is that Snell’s law is reversed: a ray of light incident upon an LHM bends in “U-turn”—or, rather, “V-turn”—fashion at the interface [14]. The case in which the medium has the same wave impedance as—and is thus (anti-)matched to—free space is particularly interesting, since a signal diverging from a radiating or scattering object is refocused within a planar slab of such an LHM, and again, with no losses, upon exit. Furthermore, such a slab behaves like a perfect lens [11]. The same effect occurs for Gaussian beams and other spatially distributed sources [15].

Fig. 1. A visually obscured object responds to the field of a sensor by emitting a measurable scattered field (left). Using a virtual half-space of left-handed material one can effect a numerical refraction of this field to estimate the location of the object (right).

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This refocusing property suggests a procedure, sketched in Fig. 1, for locating hidden objects: Initially, a field of a given frequency (which we label the primary field) is shone over an area in the hope of striking one or more targets. The resulting (secondary) signal due to the currents and magnetic response induced in the scatterers by the primary field is collected at various points on a level surface. We then perform a numerical refraction of the signal during the signal processing. We imagine the measurement surface to lie just below the flat boundary of a fictitious half-space filled with an LHM, upon which the secondary field scattered by the target is incident. We use this information to trace the wave inside the medium in search of the imaging point, whose position is related to the depth of the scatterer. We take the virtual left-handed medium to have relative electric permittivity and magnetic permeability ε/ε ₀=µ/µ ₀=−1, resulting in an index of refraction n=−1. The focusing point is then the mirror image of the target’s location. Moreover, since the medium is matched to free space there is no reflection, and only the incident field and the field within the metamaterial appear in the analysis.

We emphasize that our method is a purely computational maneuver based on the mathematical properties of ideal left-handed media. This allows us to make the foregoing assumptions without regard to the feasibility of making an actual LHM lens [16]. The only hardware the method requires is a sensor that excites the targets and records the secondary field produced by them in response. Neither does the procedure require a characterization of the material in terms of its elementary constituents—host medium and metallic insertions—though we note that these “microscopic” simulations have been widely undertaken and yield results in lockstep with analytic theory and experimental measurement [5, 6, 9].

The results we present in this paper have been obtained using the method of auxiliary sources (MAS) [17–22]. The MAS has proved to be a robust and accurate numerical method of general applicability, and its reduced computational complexity, speed of execution, and ease of implementation make it potentially useful for the simulation of realistically elaborate electromagnetic scenarios involving one or more objects. The MAS represents the electromagnetic fields in each domain of the structure under investigation by a finite linear combination of (analytic) fundamental solutions of the field equations. The “auxiliary sources” giving rise to these fundamental solutions are chosen to be point charges or dipoles—or, as is the case here, current elements—located on fictitious surfaces that usually conform to the domain boundaries and are situated some distance away from them. This last property exempts us from having to consider the difficult “self-energy” terms that appear in the method of moments; moreover, the calculations involve only points on the auxiliary and actual surfaces, so it is not necessary to resort to the detailed mesh structures required by other techniques. To ease the computational burden further we restrict our investigation to two dimensions, with the knowledge that a three-dimensional extension is warranted and not trivial. We use SI units and assume and suppress a time dependence exp(jωt).

Fig. 2. Interface between free space and a left-handed medium showing a negative angle of refraction. The electric field E, the magnetic field H, and the wave vector k form a left-handed triad within the metamaterial. The Poynting vector and the propagation of energy are in the usual direction and satisfy causality.

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This work is organized as follows: After briefly reviewing the relevant theory of metamaterials in Section 2 and the method of auxiliary sources in Section 3, we present in Section 4 some numerical results in the wave regime which show that the method can locate point sources and extended objects. In Section 5 we discuss and conclude.

2. Refocusing of a scattered electromagnetic field using LHM

In this paper we exploit the optical properties of a lossless LH medium matched to free space [4, 23, 24]. In particular, we consider a fictitious metamaterial whose electric permittivity ε ^LHM=−ε ₀ and magnetic permeability µ ^LHM=−µ ₀ are such that the index of refraction n ^LHM=−1. This fact, when combined with the plane-wave relations

K \times E = - ω ∣ μ^{LHM} ∣ H and K \times H = ω ∣ ε^{LHM} ∣ E,

which imply that the wave vector k, the electric field E=E ₀ exp(−j k·r), and the magnetic field H=H ₀ exp(−j k·r) form a left-handed triad, indicates that the wave vector and phase velocity have the opposite sign than in a conventional right-handed medium and that wave fronts travel toward the radiating source. Snell’s law is reversed in cases of oblique incidence [4]. On the other hand, Poynting’s theorem still holds [16], so the electric and magnetic fields and the Poynting vector $S = Re {\frac{1}{2} E \times H^{*}}$ continue to be related through the right-hand rule, as shown in Fig. 2. The ideal metamaterial has a wave impedance identical to that of the vacuum. As a consequence, the Fresnel formulas [15] predict perfect transmission into the medium for a TE-polarized wave (to which we henceforth restrict our attention) whose electric field is perpendicular to the plane of incidence. This combination of negative angle of refraction and perfect transmission into the LHM results in the incident electromagnetic field being refocused back to the radiating source in a sort of mirror image of what occurs in free space [11]. This follows from the boundary conditions for electromagnetic fields at the interface between free space and the LH medium. The tangential components of E, H, and k are always transmitted from one medium to another unaffected: if n̂ is a unit vector normal to the surface, we have

\hat{n} \times (E_{i} + E_{r}) = \hat{n} \times E^{LHM},

where E _i and H i are the incident electric and magnetic fields, E _r and H _r are the reflected electric and magnetic fields, and E ^LHM and H ^LHM are the total electric and magnetic fields inside the LHM. The corresponding conditions for the normal components,

\hat{n} \cdot ε_{0} (E_{i} + E_{r}) = \hat{n} \cdot ε^{LHM} E^{LHM},

show that these undergo a change of sign at the interface. Equations (2) and (3) are of fundamental importance in the method of auxiliary sources, to which we now turn.

Fig. 3. MAS geometry for modelling a uniform half-space.

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3. The method of auxiliary sources for LHM in the EM-wave regime

To detect and locate a permeable conducting object embedded in an opaque medium like soil or breast tissue one can illuminate it with an electromagnetic field established by a sensor. According to Faraday’s Law, this primary field induces electric currents and magnetic response in the target, and these in turn produce a secondary or scattered field in the surrounding space that can be measured and recorded at an array of points. Now assume that the measurement surface is an interface between free space and the fictitious LHM of the preceding section, and that the recorded scattered field is a new primary field incident on the boundary. This field, which we denote by (E ^meas, H ^meas), propagates in the free half-space, as does the reflected field (E _r, H _r). Within the LH medium we have the refracted field (E ^LHM, H ^LHM). The reverse behavior of Snell’s law in a left-handed material leads us to expect that by computing (E ^LHM, H ^LHM) we in effect backpropagate the incident field to its radiating source and thus are able to locate the original scatterer. The MAS lets us find (E ^LHM, H ^LHM) unambiguously starting from (E ^meas, H ^meas).

To apply the MAS to a one-dimensional uniform half-space we require two parallel auxiliary surfaces, one inside and one outside the fictitious metamaterial, as detailed in Fig. 3. The reflected fields outside the half-space (Region 1) are taken to originate from a set of auxiliary y-directed current elements {ξ_1,i} placed inside the half-space on surface S ^aux ₁ and radiating in an unbounded vacuum. The interior fields of Region 2 arise from a set {ξ_2,i} of (also y-directed) filaments placed on surface S ^aux ₂ and radiating in an unbounded LHM.

The electromagnetic field due to each auxiliary source can be found from the vector potential A, which satisfies a vector Helmholtz equation within each medium. At a given angular frequency ω we have

\nabla^{2} A + k^{2} A = μ J,

where µ is the permeability of the medium and k=ωn/c is the wave number. For a y-directed current element

J = I \hat{y} δ^{(2)} (ρ - ρ^{'}),

where ρ′≡x′x̂+z′ẑ is the location of the source and ρ≡x̂x+z′ẑ is any observation point, one finds

A (ρ) = - \frac{jμ}{4} I \hat{y} H_{0}^{(2)} (k ∣ ρ - ρ^{'} ∣) = - \frac{jμ}{4} I \hat{y} H_{0}^{(2)} (k \sqrt{{(x - x^{'})}^{2} + {(z - z^{'})}^{2}})

with H ⁽²⁾ ₀ a Hankel function of the second kind [25, Chap. 12]. (Note the slightly nonstandard convention for the two-dimensional radial vector.) In terms of A the electric and magnetic fields are

E (ρ) = - jω A = - \frac{μω}{4} I \hat{y} H_{0}^{(2)} (k \sqrt{{(x - x^{'})}^{2} + {(z - z^{'})}^{2}})

and

E (ρ) = \frac{1}{μ} \nabla \times A = \frac{jk}{4} I \frac{(x - x^{'}) \hat{Z} - (z - z^{'}) \hat{x}}{\sqrt{{(x - x^{'})}^{2} + {(z - z^{'})}^{2}}} H_{1}^{(2)} (k \sqrt{{(x - x^{'})}^{2} + {(z - z^{'})}^{2}}) .

The reflected field is due to auxiliary sources {ξ1,i} placed on S ^aux ₁ at ρ′=ρ1,_i=x_ix̂+d ₁ ẑ, i=1,2,…,N, and is thus given by the superposition

E_{r} (ρ) = - \frac{μ_{0} ω}{4} \hat{y} \sum_{i} ξ_{1, i} H_{0}^{(2)} (k_{0} \sqrt{{(x - x_{i})}^{2} + {(z - d_{1})}^{2}}),

H_{r} (ρ) = \frac{{jk}_{0}}{4} \sum_{i} \frac{(x - x_{i}) \hat{z} - (z - d_{1}) \hat{x}}{\sqrt{{(x - x_{i})}^{2} + {(z - d_{1})}^{2}}} ξ_{1, i} H_{1}^{(2)} (k_{0} \sqrt{{(x - x_{i})}^{2} + {(z - d_{1})}^{2}}),

with k ₀=ω/c and the current amplitudes ξ1,_i as yet undetermined. The interior field of Region 2 is due to filaments {ξ2,_i} located at ρ′=ρ2,_i=x_ix̂-d ₂ẑ on S ^aux ₂:

E^{LHM} (ρ) = - \frac{μ^{LHM} ω}{4} \hat{y} \sum_{i} ξ_{2, i} H_{0}^{(2)} (k^{LHM} \sqrt{{(x - x_{i})}^{2} + {(z + d_{2})}^{2}}),

H^{LHM} (ρ) = \frac{{jk}^{LHM}}{4} \sum_{i} \frac{(x - x_{i}) \hat{z} - (z + d_{2}) \hat{x}}{\sqrt{{(x - x_{i})}^{2} + {(z + d_{2})}^{2}}} ξ_{2, i} H_{1}^{(2)} (k^{LHM} \sqrt{{(x - x_{i})}^{2} + {(z + d_{2})}^{2}}),

where

k^{LHM} = k_{0} c \sqrt{ε^{LHM}} \sqrt{μ^{LHM}} = - k_{0} .

In the Appendix it is proved that the fields of Eqs. (10) and (11) satisfy causality and correctly incorporate the underlying physics of LHM.

To determine the unknown complex currents ξ_1,i and ξ_2,i we enforce (2) at the collocation points ρ _m=x_mx̂ on S. We obtain the system

μ_{0} \sum_{i} ξ_{1, i} H_{0}^{(2)} (k_{0} \sqrt{{(x_{m} - x_{i})}^{2} + d_{1}^{2}}) - μ^{LHM} \sum_{i} ξ_{2, i} H_{0}^{(2)} (k^{LHM} \sqrt{{(x_{m} - x_{i})}^{2} + d_{2}^{2}}) + E_{y}^{meas} (x_{m})

and

k_{0} \sum_{i} ξ_{1, i} \frac{d_{1}}{\sqrt{{(x_{m} - x_{i})}^{2} + d_{1}^{2}}} H_{1}^{(2)} (k_{0} \sqrt{{(x_{m} - x_{i})}^{2} + d_{1}^{2}})

for m=1,2,3, …,M. In this paper we take d ₁=d ₂≡d, with d=0.9λ for vacuum/LHM interfaces and d=λ/2 around and within the scatterers. The auxiliary sources and the collocation points are always evenly spaced on their respective surfaces; we use 10 sources per wavelength for the LHM boundaries and 5 sources per wavelength for the targets’ surfaces. This is the smallest number of sources that enable us to refocus the targets within the LHM. More sources produce a better field match at the boundaries but result in an essentially identical image of the field distribution. Choosing M=N lets us solve (12)–(13) via standard square-matrix inversion techniques for the 2N unknown current amplitudes ξ_1,i and ξ_2,i (which have absorbed common factors of ω/4 in (12) and j/4 in (13)). We could alternatively select M>N points on the plane, in which case the solution of (2) would be correct in a least-squares sense. We can then readily use expressions (9)–(10) to compute the electromagnetic fields everywhere.

The method does not require that we specify boundary conditions (absorbing or otherwise) at infinity, even though we assume that the surface extends indefinitely in the x- and y-directions. All fields decay very rapidly as we move away from the sources that produce them, and to obtain figures like those in the following section it suffices to use a computational space a few times larger than the region of interest and zoom in at the end.

4. Results

All examples in this section involve targets whose sizes are comparable to the wavelength. We first consider three pointlike sources—unit y-directed current elements of the form (5)—located 5λ away from the vacuum-LHM interface. Figure 4 shows the amplitude of the y-component of the electric field on both sides of the interface (top), as well as its phase distribution (bottom). The sources are separated by about λ. The simulated measurement area is 12λ wide (along x̂), centered on the origin, and straddling the interface, which appears as a white line down the middle. The field in the LHM on the right has smeared peaks at virtually the same distance from the interface as each source on the vacuum at left.

By examining the reconstructed field on the right side of Fig. 4 we can define all the quantities that let us quantify the efficacy of the given approach: image definition, image sharpness, resolving ability (~λ), discrimination (~12λ), focal spot size (<λ) and the fact that the image is free from aberration. The focusing depth and the depth of resolution are particularly important. The field of Fig. 4 is reconstructed to a sufficient degree of approximation only up to the nearest focal spot, as measured from the interface. Beyond that distance the image becomes very blurry, which means that a significant amount of information on the incident field is being lost. It may appear that this is due to the finite size of the simulated measurement area (in this case 12λ along the x-axis), and that is certainly true to some extent. However, the main reason for the information loss is the presence of singularities of the incident field (and, in general, of any scattered field singularities). When the incident and scattered fields have weak singularities, the field inside the LHM can be reconstructed sharply to great depth; on the other hand, strong field singularities preclude good field reconstruction. This is clearly visible in the examples that we present later on (especially in the case of a Gaussian beam illuminating a transparent dielectric scatterer).

Fig. 4. Electric field distribution (top) and corresponding phase (bottom) around three point sources to the left of the interface (left) and MAS-predicted equivalents in the virtual LHM half-space (right).

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Next we simulate an extended source using a Gaussian beam that impinges obliquely on the boundary. The beam source is 4λ wide and is centered at a distance of 10λ from the origin pointing 30° below the horizontal. The wavelength and simulated measurement area are set as in the previous example. Figure 5 depicts the y-component of the electric field distribution and provides confirmation of the lensing properties exhibited by the LHM, which acts to focus the beam. We can see that the diverging beam converges within the left-handed half-space at a point mirroring the location of the source and then diverges again without any losses; this provides an example in which the negative medium performs ideal restoration of the incident field. Therefore, in the following examples we will use as incident field that of a Gaussian source, since it does not distort the field scattered by other objects.

We can now investigate an important question: when the simulated measurement area is finite and the target’s size is comparable to the wavelength, does the reconstructed scattered field contain enough information about the object’s shape, its (in general complex) internal material properties, and its location and orientation?

Fig. 5. Electric field distribution around a Gaussian source in vacuum (left) and MAS-reconstructed distribution in the virtual LHM half-space (right).

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Consider a Gaussian beam shining on two cylinders of differing material properties, as displayed in Fig. 6. In each case, as before, the amplitude of the y-component of the electric field is shown on both sides of the vacuum-LHM interface.

At top the scatterer is a transparent dielectric cylinder of radius R=10λ, with ε_r=2.0 and µ_r=1.0, centered a distance 15λ from the interface and illuminated by a λ -wide Gaussian source located 60λ from the interface. The simulated measurement area is a rectangle of length 140λ along z and width 100λ along x.

In the case shown at bottom the scatterer is again a cylinder, but is now a perfect electric conductor (PEC); its radius is R=5λ and its center is 10λ from the interface. It is illuminated by a 20λ -wide Gaussian source placed 70λ away and making a 20° angle with the interface. The simulated measurement area now has 50λ along z and 50λ along x.

We can see that (a) the image definition is much better and the sharp resolution reaches deeper for the transparent dielectric than for the good conductor. Moreover, (b) the reconstructed field restores nearly the full contour and cross-section of the transparent object, and (c) for the PEC object the image definition and depth of resolution, as measured from the interface, are twice the length of the cylinder’s radius; i.e., the same as the distance from the interface to the mirror image of the cylinder center.

The given result completely agrees with what we found in the previous examples. It is known that every scattered EM field is connected to some unique singularity, a so-called Scattered Field Main Singularity (SFMS). It is in fact the physical essence of the MAS to replace these SFMS by the auxiliary sources, from which the actual scattered field is taken to originate [18]. It is also known that the field scattered by a circular cylinder has a SFMS at the center of the cylinder. The PEC cylinder has a stronger SFMS in comparison to the transparent dielectric one, and that strongly limits the depth of field reconstruction that can be achieved with the MAS.

We next turn our attention to scenarios involving multiple objects. Figure 7 shows a directed source (at a height 20λ) shining electromagnetic radiation into five conducting nonmagnetic cylinders. All targets have radius R=λ and permittivity ε_r=j2.0+j30.0; each cylinder is centered 4λ away from the LHM half-space, and the interobject spacing is 4λ. The cylinders’ locations are distinctly visible in the reconstructed field, despite the fact that their sizes and separations are comparable to the wavelength.

Fig. 6. Reconstruction of the electromagnetic field for the case of a Gaussian beam illuminating two different cylinders, a good dielectric (top) and a perfect conductor (bottom). Electric field distribution around the object (left) and MAS-reconstructed distribution in the virtual LHM half-space (right).

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Fig. 7. Electric-field distribution (left) and corresponding phase (right), in vacuum and in the virtual LHM half-space, when a directed source shines on five nonmagnetic conducting cylinders.

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In Fig. 8 we reconstruct the field due to three objects illuminated by a Gaussian beam. The cylinders have a relative permittivity ε_r=2.0+j100.0 and the elongated body has ε_r=3.0+j100.0. The objects are placed as shown, with the Gaussian source located 14λ away from the interface and shining directly at the objects. The objects and inter-object separations are slightly larger than the wavelength. We see that the LHM reconstruction algorithm is able to infer the locations of the targets closer to the ground and predicts very accurately the field distribution between the free-space/LHM boundary and the objects. Again, the reconstructed fields behind the objects diverge from the actual field; this is because the scattered field at the free-space/LHM interface does not contain much information about the electromagnetic field due to the objects’ invisible parts (see also the explanations to Figs. 4–6).

Figure 9 shows what happens when light from a directed source strikes two objects such that one is completely enclosed by the other. Both objects are nonmagnetic cylinders; the outer one has radius R=λ and permittivity ε_r=2.0 and is centered 3λ away from the interface, while the inner one, of radius λ/4 and ε_r=2.0+j2.0, has its center offset upward and toward the boundary by λ/4 in each direction. The Gaussian source is located 10λ above the center of the outer cylinder and 13λ from the interface. Once again the cross sections of both cylinders, external and internal, are clearly recognizable.

Fig. 8. Electric field distribution for three objects separated by more than one wavelength: actual (in free space, at left) and reconstructed (in the LHM, at right).

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Fig. 9. Electric-field distribution around a compound object in free space (left) and MAS reconstruction within a virtual LHM (right).

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5. Conclusion

We have investigated the possibility of using a virtual double-negative metamaterial to estimate the location and geometric shape of a visually obscured object without resorting to the solution of a traditional ill-posed inverse scattering problem. The analysis was circumscribed to two dimensions and to scatterer sizes comparable to the wavelength. After briefly discussing the theoretical basis of our procedure, and its numerical realization through the Method of Auxiliary Sources, we proceeded to infer the locations of different combinations of targets.

Our results demonstrate that a planar LHM half-space can refocus “back to the origin” a field incident upon it and is thus a potentially useful tool to estimate the location and geometry of a visually obscured target. The results are particularly distinct when the field singularities due to the source and the scatterers are weak, as in the cases of Gaussian light sources and transparent targets.

We are aware that a two-sided LHM slab is needed to achieve perfect focusing, particularly when the electromagnetic parameters of the LHM are not exactly −1. Our main objective here was to determine locations of scatterers, not to image them with perfect resolution. For that reason, and to avoid dealing with an extra variable (the thickness of the slab) that was not essential at this stage, we decided to use a half-space of LHM—i.e., a slab of infinite thickness. Our technique is not limited to half-spaces, though, and could be applied to the case of a slab. We will take up that problem in the future, and expect to see enhanced scatterer images as a result.

The success of the method suggests a number of interesting applications. For example, one could use the fictitious LHM half-space to solve inverse scattering problems in the microwave and optical frequency regimes. These applications could make it worthwhile to extend the analysis to three dimensions and to a wider range of frequencies.

Appendix

The differences between free-space fields and their LHM counterparts are most clearly seen in the asymptotic regime where kρ≫1. The fields established by an auxiliary source become

E = I_{0} H_{0}^{(2)} (kρ) e^{jωt} \hat{y} \sim I_{0} \sqrt{\frac{2}{πkρ}} e^{j (- kρ + ωt + \frac{π}{4})} \hat{y},

H = \frac{jk}{ω μ_{r} μ_{0}} I_{0} H_{1}^{(2)} (kρ) e^{jωt} (\hat{y} \times \hat{ρ}) \sim - \frac{k}{ω μ_{r} μ_{0}} I_{0} \sqrt{\frac{2}{πkρ}} e^{j (- kρ + ωt + \frac{π}{4})} (\hat{y} \times \hat{ρ}),

where $I_{0} = - \frac{1}{4} μ_{r} μ_{0} ω I$ is the amplitude of the auxiliary source in terms of the amplitude I introduced in the main text. For clarity in what follows we have restored the time dependence e^jωt. From Eqs. (14) and (15) we find that the phase velocity v _phase=ω/k. The direction of energy propagation is given by the Poynting vector [16]

S = \frac{1}{2} Re {E \times H^{*}} = \frac{I_{0}^{2} k^{*} e^{2 Im kρ}}{ω μ_{r}^{*} μ_{0} π ∣ k ∣ ρ} \hat{ρ} .

In the case of free space we have k=k₀ and µ_r=1, and thus

v_{phase}^{vac} = \frac{ω}{k_{0}} and S^{vac} = \frac{I_{0}^{2}}{ω μ_{0} πρ} \hat{ρ},

whereas in the case of the LHM, where k=−k₀ and µ_r=−1,

v_{phase}^{LHM} = - \frac{ω}{k_{0}} and S^{LHM} = \frac{I_{0}^{2}}{ω μ_{0} πρ} \hat{ρ} .

By comparing Eq. (17) to Eq. (18) we see that the phase velocity inside the LHM has the same magnitude—but the opposite direction—than in the vacuum, while the energy propagates in the same direction in both media.

Equations (10) and (11) of the main text express the electromagnetic field of an auxiliary source within the LHM in terms of the same Hankel functions of the second kind used for the free-space sources of Eq. (9). This would apparently make the LHM sources behave like sinks rather than radiators, implying that our method violates causality. The discrepancy is resolved if one uses the identity H ⁽²⁾ _ν (ze-^jπ)=−e^jνπH ⁽¹⁾ν(z) from [26, Eq. (9.1.39)] to obtain

E^{LHM} = - I_{0} H_{0}^{(1)} (k_{0} ρ) \hat{y},

H^{LHM} = j \frac{k_{0}}{ω μ_{0}} I_{0} H_{1}^{(1)} (k_{0} ρ) (\hat{y} \times \hat{ρ}),

which show the sources’ “outgoing” behavior more transparently.

Acknowledgments

This work was supported by the Strategic Environmental Research and Development Program under grant # MM-1592.

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Scatterer localization using a left-handed medium

Abstract

1. Introduction

2. Refocusing of a scattered electromagnetic field using LHM

3. The method of auxiliary sources for LHM in the EM-wave regime

4. Results

5. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (9)

Equations (25)

Optics Express