## Abstract

This paper proposes a super resolution near-field **radio frequency** focusing device consisting of a thin planar layer of a particular ferrite characterized by negative permeability. Radiation focusing is investigated and it is established that the resulting non-structured lens is characterized by a resolving power 2–3 times the lens thickness, regardless of the wavelength. The resulting near field lens can be used as a magnetic field device for imaging inside non-magnetic objects.

Forty years ago Veselago (1968) established that a flat slab of left-handed media (LHM), characterized by index *n*=−1, could focus rays from a nearby source. Veselago’s concept was dormant for about 30 years, until Pendry (2000) demonstrated analytically that not only the propagating components were recovered in the image of the left-handed slab, but also the evanescent components, i.e. unlike diffraction-limited optics, the image contained all details of the source. The LHM slab then became a perfect lens, giving rise to a flurry of research activity in this area, which gained significant momentum because of Pendry’s *et al.* (1999) prior introduction of an array of split-ring resonators (SRR) exhibiting *μ*<0 at microwave frequencies. Since an array of wires was known to possess *ε*<0, combination with the SRR leads to experimental demonstration of LHM at microwave frequencies (Smith & Kroll 2000; Houck *et al*. 2003; Parazzoli *et al*. 2003).

Pendry’s (2000) seminal paper also included a calculation of focusing in the optical regime, using a silver slab. The analysis was in the quasistatic limit, with all dimensions much smaller than the wavelength, and demonstrated capability to discriminate in the object plane, details roughly a quarter of a wavelength apart.

To date, a lot has been said about negative refraction in the microwave range using structured metamaterials (Smith & Kroll 2000). These result in element size comparable to the wavelength, as is the case of phonic band gap inclusions (Notomi 2000; Luo *et al*. 2002*a*,*b*), or resonant elements (Smith & Kroll 2000; Markos & Soukoulis 2001; Houck *et al*. 2003; Parazzoli *et al*. 2003; Ziolkowski 2003; Lagarkov & Kissel 2004) (*a la* SRR). Since the precise condition *n*=−1 is sought for unique operation (Loschialpo *et al*. 2004; Monzon *et al*. 2004*a*,*b*; Schelleng *et al*. 2004), the schemes are narrow band, and inclusions of size comparable to a wavelength impose significant limitations on resolution, as clearly pointed out by Smith *et al*. (2003). Losses, as well as finite cell size (e.g. structure, such as encountered when employing the RSS and wire combination proposed by Pendry to make electrically small cells), and any other deviation from the exact condition *n*=−1 also pose resolution limitations Smith *et al*. (2003). It is the intention of this paper to fill in a significant gap; we present a non-structured lens capable of producing near-field subwavelength focusing in the GHz range.

The permeability of ferrite granular composite materials has been studied at microwave frequencies for a number of years; however, the interest was mainly in obtaining high permeability. LHM has changed the range of what are acceptable electric and magnetic material properties. In particular, a negative permeability in Permalloy has been recently reported (Kasagi *et al*. 2006) above 5 GHz. This is however not surprising, as the low values of permeability (zero or negative) in the higher portions of the spectrum were just not of any practical interest a decade or two ago. See for instance Tsutaoka *et al*. (1997), which deals with the high-frequency permeability of Mn–Zn Ferrite, where although no reference is made in the text, it is observed in Tsutaoka *et al*. (1997; fig. 3*a*) that practically all the MnZn ferrite–PPS (polyphenylene sulphide resin) composite materials reported have a negative permeability above 1 GHz at room temperature.

We found that certain ferrites can function in a manner analogous to silver in the optical. In fact, they can be dual of each other (while silver is purely electrical, a ferrite is eminently magnetic, with the proviso that in the microwave regime we have a dielectric constant to contend with). The relative permeability of some unbiased ferrites can be represented by a Lorentzian shape (the time dependence is assumed and suppressed).
1
Here is the asymptotic high-frequency value of the permeability, *μ*_{DC} is the low-frequency value, *f*_{0} is the resonant frequency, and *G* is a measure of the losses. According to equation (1), for some particular set of parameters, the permeability can assume negative values over some frequency range. In particular, we will consider a ferrite with the following parameters *μ*_{DC}=39, , *f*_{0}=1.82 GHz, *G*=3.2 GHz, conductivity *σ*=0.006, and *ε*≈11 over the 2–10 GHz range (these parameters for an unbiased ferrite are representative in view of the data reported in Kasagi *et al*. 2006). The magnetic properties of this lossy ferrite are shown in figure 1, where at 7.5 GHz we observe, *μ*≈−1+0.9i, which is to be compared with Ag in the optical where *ε*≈−1+0.4i.

Pendry’s silver foil focusing device works in the optical range only. If we could produce a device analogous to Pendry’s but working in the microwave region, it would be very desirable, and unique. We next demonstrate that the above special ferrite fulfils all the requirements. The mathematical details of evanescent fields are subtle (Pendry 2000), and some divergent ideas (Garcia & Nieto-Vesperinas 2002) may arise if not rigorously treated. The dual of this mechanism seems to be the physical basis for Pendry’s silver lens. The effect of the losses, as a diffusion process, is just to smear the focal point. Just as in the case of structured LHM, losses appear as a limiting factor in order to achieve ideal resolution, except that the structured LHM are further plagued by the limiting effect of the periodicity (Smith *et al*. 2003).

A mathematical analysis analogous to Pendry (2000) follows. For S polarized fields, the transmission through a material (*μ*,*ε*) slab of thickness *D* is given by
2
where the normal to the interfaces is *z* directed, and the interfaces coincide with the *x*–*y* plane. The primes denote wavenumber components in the material, and phase continuity across the interfaces indicate that *k*_{x}=*k*_{x}′ and *k*_{y}=*k*_{y}′. In addition, for *k*_{0} the free space wavenumber,
3
In the quasistatic regime, the transversal wavenumber relate to the small features of the geometry, which are exceedingly smaller than a wavelength (in the material or in free space). According to equation (3) this results in *k*_{z}′≈*k*_{z}, which implies that the permittivity *ε* plays no role in the transmission, which reduces to
4
Hence, regardless of the value of *ε*, for *μ*=−1, the transmission becomes . Since the incident field is of the form , and (purely imaginary for the evanescent waves), it follows that any evanescent decay introduced in reaching the interface is followed by a commensurate amplification thereby leading to perfect focusing.

For P polarization on the other hand, *μ* will play no significant role in the quasistatic approximation, and since *ε* is not −1, we will have no focusing. This is in analogy with the case of silver, as we have verified numerically. It is important to mention that the condition for surface plasmons for S polarized fields is *μ*=−1, which is also our condition for focusing, hence, we should expect the focusing behaviour of our ferrite to be accompanied by surface plasmons. P polarized excitation of plasmons on the other hand occurs when *ε*=−1 (Pendry 2000), which is not considered here. Hence for P polarization we will have no focusing effect and no surface plasmons. The coexistence of focusing and surface plasmons indicates once again the crucial role played by surface plasmons in the generation of focused fields.

Note that the dielectric constant plays no role in this description, which is remarkable as the dimensions with significant field deposition are much smaller than a wavelength. The material can even be dielectrically anisotropic provided it is electrically thin. Compatible with a biased ferrite, anisotropic materials can be employed, provided the normal component of *μ*=−1.

We are ultimately interested in the manner in which waves emanate from a current source and then propagate in a finite ferrite lens, and have used finite difference time domain (FDTD) to analyse thin ferrites at 7.5 GHz. A Snell’s law construct is employed in figure 2 to present the geometries employed in the FDTD simulations (the ray picture is used as a convenience to portray focusing).

First we consider a 4 mm thick, 4 cm long ferrite slab sample excited by a 7.5 GHz electric line source placed 2 mm away from the slab (see figure 2*a*). An FDTD snapshot of the electric field distribution is presented in figure 3, which through examination of the transmitted minimum amplitude contours clearly establishes the existence of a transmitted wave of similar characteristics of the source. In fact, a wave appears to emanate from the ideal image plane (i.e. parallel to the interface, and here, at a distance *d* from the slab). Surface plasmon excitation is evident from the figure. The snapshot was taken after sufficient time was allowed for the time–harmonic regime to be established.

The next example deals with a 4 mm thick, 8 cm long ferrite slab illuminated by a 7.5 GHz electric line source placed 1 mm away from the slab (see figure 2*b*). A set of four FDTD electric field snapshots separated by roughly a quarter period is presented in figure 4, which shows a transmitted wave of similar characteristics of the source, but appearing to emanate at a distance of 3 mm from the slab, i.e. at the ideal image plane, as sketched in figure 2*b*. Surface plasmon excitation is also evident from the figure.

We have analysed lower loss ferrites in order to elucidate the performance achievable with YIG ferrites, which although very thin, could be used in a multiple layer configuration. The lower loss parameters are *ε*≈11+0.01i and *μ*≈−1+0.09i. The geometry follows figure 2*a* with a 2 mm thick slab of length 4 cm. Figure 5 shows an electric field snapshot, where the lower losses translate into a cleaner definition of the transmitted equiamplitude contours which appear as a semi-circle centred 2 mm behind the slab. Although not shown in the figure because of the small length of the slab, surface plasmon excitation is stronger as the losses go down, just as in the case of LHM.

We have also investigated a 4 mm thick ferrite lens with two line sources 12 mm apart and 2 mm away from the slab. The slab was however infinite in extent, and the calculations were made using high frequency structure simulator (HFSS) at 7.5 GHz. Figure 6 consists of magnetic field (root mean square (r.m.s.) values) data involving the lower loss ferrite with one-tenth the losses of the original ferrite. A snapshot of r.m.s. *H* is included, together with the observation at an image plane (which is transversal to the direction of the thickness) of r.m.s. |*H*|^{2}, and tangential |*H*_{x}|^{2} to facilitate the observation of the resolving power of the lens. The total length of the image plane window is 7 cm (normalized). The image plane is located 4 mm away from the interface. The figure offers a better look at the reconstructive effect of the lens (the transmission losses of the original ferrite lens are a reasonable 10 dB). From the figure we can conclude that the resolving power of the low loss ferrite lens is at most three times the lens thickness. The peak separation for the lower loss ferrite is significantly accurate, with an almost exact 12 mm peak separation in both |*H*|^{2} and |*H*_{x}|^{2}.

The HFSS calculation has been repeated with the same configuration, but employing the original lossy ferrite with *ε*≈11+0.1i, *μ*≈−1+0.9i. Snapshots of r.m.s. magnetic field |*H*|^{2}, and tangential |*H*_{x}|^{2} are presented in figure 7, and reconstruction in the image plane is presented in figure 8, which includes three image planes, located 2, 3 and 4 mm away from the interface. With an almost exact 12 mm peak separation, it appears that for the lossy ferrite case the resolving power for |*H*|^{2} is slightly better than that of |*H*_{x}|^{2} which has a 14 mm peak separation. It should be mentioned that as losses increase we depart from the conditions of the analysis and the definition of an image is not so clear, and possibly influenced by the losses themselves to smear the focusing effect and endow it with imaging features, in consonance with a recently observed imaging effect on thin lossy films (Monzon 2009).

We have done other simulations for smaller thicknesses (not included), and in every case found a resolution power of the order of two to three times the lens thickness, which indicates super resolving power in the near field. We have also done FDTD simulations of the Pendry silver slab (not shown) with a finite length, and found a resolution of about three times the slab thickness. Pendry (2000) found the resolving power for a silver slab being about twice the slab thickness. Although Pendry used an idealized electrostatic model, whereas ours is finite and numerically exact, it is possible that the variance in resolving power we observe is attributable to the finiteness of the slab in our model (which results in standing wave surface plasmons), or perhaps the need for fine-tuning of the parameters.

To summarize, we have demonstrated through numerical simulations and first principles, that a thin ferrite slab of negative permeability offers the possibility of RF near-field focusing with super resolution. The resulting near-field lens can be used as a magnetic field device for imaging inside non-magnetic objects. It should be mentioned that a highly absorbent ferrite lens has a resolution commensurate with two to three times the lens thickness, and is not the only material that makes this workable, as lower loss YIG ferrites, although thin and anisotropic, could be layered to produce the focusing effect we are interested in.

## Footnotes

- Received July 21, 2009.
- Accepted September 25, 2009.

- © 2009 The Royal Society