## Abstract

We demonstrate the existence of broadband anomalous transmission of electromagnetic (EM) waves through an electrically very narrow aperture in a thin metal plate backed by left-handed media (LHM). It is shown that the incoming energy is simply squeezed through the aperture, without any reflection, regardless of how small the physical aperture is. Analogous to the Venturi effect in fluids, the EM energy behaves as an incompressible fluid, flowing through the constriction (aperture), as if the wave character of EM was lost. This counterintuitive tunnelling effect is not related to resonances and is impossible to achieve with naturally occurring materials. Analysis indicates that a cascaded waveguide implementation, in the form of a slotted metallized LHM wall, retains the exotic broadband transparency character. Applications for a device with these properties abound in the RF/microwave/THz/optical ranges, to which the analysis can be scaled. To complement this analysis, a proposal for achieving broadband LHM is also included.

## 1. Introduction

The early work of Bethe (1944) on a small hole in a thin screen is classical in the field of small apertures, mostly because of applications in the areas of microwave devices, microwave measurements, communications, electromagnetic (EM) interaction and coupling and antennas. Notable also is the work of Bouwkamp (1950), van Bladel (1970) and Harrington (1961, 1968), who found out that a subwavelength aperture can be made to resonate by a nearby conducting body.

More recently, Ebbesen reported enhanced transmission of light through subwavelength holes in an opaque metal film (Ebbesen *et al*. 1998), generating significant activity (López-Rios *et al*. 1998; Porto *et al*. 1999; Popov *et al*. 2000; Martín-Moreno *et al*. 2001) because of applications in microscopy, photolithography and photonic devices. A related idea was adapted to microwaves by Lockyear *et al*. (2005), who experimentally and analytically explored an annular aperture in a thick metal plate, finding transmission at frequencies quantized by the cavity length, and indicated the possibility of transmission by annular apertures of negligible radius.

All schemes for transmission through small apertures seen to date are resonant, i.e. narrowband in nature (even the non-aperture, but related case of a narrow planar waveguide channel of an electric permittivity near zero is a resonant case (Liu *et al*. 2008)). Here we want to exploit metamaterials to induce broadband transmission in remarkably small, subwavelength, apertures. We are referring to the negative-index ‘left-handed’ media (LHM), proposed by Veselago (1968) and characterized by phase velocity and energy flowing in opposite directions. As found by Pendry (2000), an LHM slab can amplify evanescent modes allowing reconstruction of a point source. Successful LHM has been built as a periodic array of elements (Houck *et al.* 2003; Parazzoli *et al*. 2003), typically split ring resonators (SRRs), in combination with wires, of the type proposed by Pendry *et al*. (1999). Given the level of current development of LHM, we can assume it is homogeneous.

Subwavelength focusing has also been observed with photonic bandgap (PBG) materials. The methods of Notomi (2000) and Luo *et al*. (2002) using PBGs are somewhat narrowband in nature. However, Monzon *et al*. (2006) analysed a particular conformation of the equifrequency surface contours leading to negative refraction (NR) insensitive to frequency, and providing broadband focusing. This was tested experimentally (Monzon *et al*. 2006), leading to three-dimensional focusing over a 1.5:1 bandwidth (analytically capable of 2 : 1). More recently (Yao *et al*. 2008), silver nanowire metamaterials have been proposed to produce NR over a broad spectral range in the optical range. Similarly, a fishnet structure (Valentine *et al*. 2008) (akin to the three-dimensional structure in Monzon *et al*. 2006) was tested in the optical range and was found to possess NR. At the end of this paper, we include a basic analytical demonstration of the feasibility of broadband LHM of index −1. Hence, it is reasonable for the analysis to assume the existence of such materials.

## 2. Analysis

Consider an electrically small slot aperture of width *s*, located in the *z*=0 plane, with axes aligned with the *x*-direction (figure 1*a*), and let the incoming magnetic field be *x*-directed (transverse magnetic, TM). The screen is perfectly conducting, and separates free space from LHM. We know from refraction characteristics of an LHM-vacuum interface that in the very neighbourhood of the aperture, the axially directed *k*_{z} will be asymmetric about the interface plane *z*=0, whereas the transversally directed *k*_{y} will be asymmetric (index −1) (Veselago 1968). That is an experimental fact (Houck *et al.* 2003; Parazzoli *et al*. 2003). By analytical continuation, this will hold irrespective of whether *k*_{z} and *k*_{y} are real or complex (evanescent field content, as here we will take the limit as losses are vanishingly small). This is based on the principle of plane wave expansion decomposition and will hold for any aperture size. As in LHM power flows in a direction opposite to that of phase propagation, the end result of this elementary picture is a Poynting vector field that flows past the screen fluid-like, as it is squeezed by the small aperture (figure 1*b*). This is nothing like what we see with ordinary materials, where the impinging bundle of ray representation results in invalid ray crossing at the aperture. A detailed analysis follows.

In the usual treatment of an aperture, we replace the aperture fields by equivalent magnetic currents , and electric currents. The use of the equivalence theorem (Harrington 1961) allows us to place the equivalent currents on top of the short-circuited aperture, where in view of the electric nature of our screen, only the magnetic currents radiate. This way the problem is split in two. The screen is further eliminated by using images, which results in multiplying by a factor of 2, and using the short-circuited fields , which are due to external sources, if any (in our case we only have it on the vacuum side). Matching of the tangential magnetic field at the aperture, as produced by and (or on the LHM side because of the sign reversal of , as it must point into the pertinent space), results in an integral equation for . As the field is TM, is a collection of *x*-directed magnetic line sources, each of which produces an aperture magnetic field of the form , where *k* is the wavenumber and η the corresponding wave impedance (a time dependence is assumed and suppressed from now on), we obtain upon matching of the aperture tangential magnetic fields
2.1
Here the subscript LHM is used to denote the limiting case as the losses are vanishingly small
2.2
From the circuit relations about the branch cut of the Hankel functions, we find (Magnus & Oberhettinger 1949)
2.3
As *ν*→0, and *p*=+1 (as *δ* must be positive), we find . Hence, as *δ*→0, equation (2.1) reduces to the non-singular integral equation
2.4
It should be noted that a non-singular kernel in diffraction theory is very unusual.

We next find the field behaviour in the neighbourhood of the aperture edges. Upon extending Meixner’s analysis (Meixner 1972) (corresponding to the field singularities for the case of two dielectric wedges with a common face in contact with a metal wedge), to the case of LHM. This results in a dominant edge electric field in the transversal to *x*=0 plane of the form
2.5
where *ρ* is the radial distance to the common apex, and *ν* is given by the solutions of
2.6
For *β* and *β*_{LHM} the angles of the dielectric wedges of permittivity are *ϵ* and *ϵ*_{LHM}, respectively. In the present case, *β*→*β*_{LHM}→*π*, as *ϵ*_{LHM}→−*ϵ*→−*ϵ*_{0}. In the limit, we find that *ν*→1/2 results in the ratio of two small numbers of the right-hand side of equation (2.6), which can accommodate any value on the left-hand side, including LHM. Hence, just in the symmetric case *β*→*β*_{LHM}→*π*, the exponent *ν*=1/2 becomes independent of the electrical characteristics of the materials. And just like in the regular dielectric case, the *E* field singularity is of the form . Hence, for very small apertures, we can approximate *m* by
2.7
Using equation (2.7) in equation (2.4), and keeping only leading terms in *k*_{0}*s*, results in . The axial magnetic field at the aperture *H*_{a}(*y*) is then given by either side of equation (2.1), namely
2.8
The Poynting vector on the aperture is *z*-directed and is given by , which simplifies to *S*=Re{*m*(*y*)*H*_{a}(*y*)*}. Using equation (2.7) and upon retaining only the leading term in *k*_{0}*s* from equation (2.8), and using the fact that the short-circuited magnetic fields are twice those of the incident field, we find the Poynting vector on the electrically very narrow aperture
2.9
for *P*_{inc}, the external power density incident on the screen. Upon integration over the aperture, we find the total power crossing the aperture,
2.10

Thus, we find that the effective width of an electrically small aperture backed by LHM is *λ*_{0}/*π*. Hence, within a cylindrical radius *R*=*λ*_{0}/2*π*, all incident power will go through, i.e. transmission both ways (because of reciprocity), and irrespective of how small the aperture is. This is a counterintuitive result, in complete discord with the behaviour of apertures backed by ordinary materials, where a strong resonance is required to get significant transmission. The energy is squeezed through the aperture akin to the Venturi effect in fluids, the constriction being the aperture. Note however that this surprising result is consistent with the sketch based on refraction of wave vectors shown in figure 1*b*.

The effect is counterintuitive because the present transmission problem is non-resonant, yet it is related to the problem of small radiators, the fundamental limitations of which were theoretically established by Chu (1948) and Wheeler (1959), according to which a small radiator must have a high-quality factor *Q* (because of the large reactive energy stored in the antenna vicinity compared with the radiated power), which results in a very narrow bandwidth. Note that although our problem is not resonant, there is a large reactive energy stored in the vicinity of the aperture (evidence of this are the Hankel functions singular terms). The radius *R*=*λ*_{0}/2*π* is not surprising, as it is known as the radiansphere (Wheeler 1959), defined as the distance at which a transition occurs between a radiator’s energy storing near field and its radiating far field. The validity of the radiansphere concept is an experimental fact and forms the basis of a well-established measurement of small antenna radiation efficiency (Wheeler 1959; Newman *et al*. 1975).

The above result (equation (2.10)) is quite interesting, as it implies that the effective aperture width (a fixed fraction of a wavelength) does not depend on how small the physical aperture is. The end result is evocative of the case of a loss-free small dipole, loaded with an inductor to make it resonate, offering a backscattering cross section , regardless of dipole shape or size (Harrington 1982). It is also reminiscent of the case of a small slot aperture on a thin screen (under plane wave normal incidence), which is made to resonate by connecting a capacitor, yielding a transmission cross section , independent of aperture size and shape (Harrington 1962). This is another experimentally verified concept, apparently little known in some circles, as it was recently rediscovered and presented with plenty of experimental support (Suckling *et al*. 2005).

For validation, we resorted to numerical simulations. A waveguide with a slotted conducting screen and backed by LHM appears suitable. However, for practical reasons, the LHM layer must be finite, but that deviates from the above geometry because of the free LHM-vacuum boundary (plasmon polariton excitation at that boundary is a strong possibility, which will tend to mask the tunnelling effect). For that reason, a slightly more demanding geometry was selected: that of a metal-coated LHM layer with electrically narrow slot apertures on both sides.

The selected geometry is presented in figure 2*a*, where the waveguide height is ‘*a*’, and the slot openings are 0.5 mm, which is *λ*_{0}/200 at the lowest frequency of the band of interest (which spans the 5 : 1 range 3–15 GHz). For reference, we present in figure 2*b* the transmission characteristics of the structure when vacuum is used instead of LHM. The figure shows a Fabry–Perot resonance at 6 GHz, and a monotonic decay in transmission thereafter, reaching −25 dB at 15 GHz. According to the above analysis (which is approximate for the numerical test geometry), the transmission through the LHM-filled wall should be almost an ideal 100 per cent for *a*<2*R*.

The simulations have been performed using the high-frequency structure simulator (HFSS) (with *ϵ*_{LHM}=−1.01(1−i10^{−3}) and *μ*_{LHM}=−0.99(1−i10^{−3})), and the representative results at 8 GHz are presented in figure 3, which include (figure 3*a*) electric field amplitude distribution snapshots at equal time intervals over half a period, as measured by the phase of the incident field, (figure 3*b*) and the corresponding Poynting vector field. Both graphs indicate no reflections, and that the incident energy is simply squeezed as it flows past the apertures, as if we were dealing with a fluid, just as sketched in figure 1*b*.

Broadband simulation data over 3–5 GHz (5 : 1 band) are shown in figure 4, which include transmission decibel (figure 4*a*) and reflection decibel (figure 4*b*). Surprisingly, the worst insertion loss is less than one-tenth of a decibel, i.e. we lose less than 2 per cent of the power in the worst case, while the corresponding free space fixture exhibits 25 dB insertion loss (figure 2*b*). The broadband impedance match to extremely narrow apertures, as afforded by the LHM layer, is impressive, and there is nothing of its kind possible with ordinary materials. Note that up to 10 GHz the waveguide height is less than twice the effective width of the aperture *a*<2*R*, for which equation (2.10) indicates excellent transmission, agreeing with the simulation. At the high end of the band, equation (2.10) is not representative of the simulated geometry. The analysis is next adapted to the waveguide, and, as will be seen shortly, the analysis further validates the almost ideal numerical transmission.

Using all aperture images, and following the analysis that led to equation (2.4), we find
2.11
Using the same singular behaviour in equation (2.7), leads to a new factor *A*_{WG}=*A*/*G*, for . As the field in the aperture is now given by
2.12
repeating the power calculation that led to equation (2.10) results in the power transmission coefficient
2.13
Calculation and analysis (using closed-form finite sums (Magnus & Oberhettinger 1949) for *G*) show that *T*_{WG}=1 up to frequencies for which *k*_{0}*a*=2*π*, in perfect agreement with the above simulations.

Note that we are talking about an electrically very small aperture, for which *k**s*→0, and for which aside from *s* there is no other geometrical parameter that can be used to define a frequency characteristic for the Venturi flow-like behaviour of the system (which is the single-perforated screen problem). Hence, just based on this elementary concept, we do expect broadband performance using LHM. That very broadband LHM may not be experimentally available at the present time (assuming that is the case, as nowadays metamaterials breakthrough occurs daily) is another issue, and we will get back to this in the next section, where we expand on the feasibility of broadband LHM. Note also that because there is no other parameter, we only have the wavelength to specify the performance of the system. Hence, it is not surprising that the radiansphere appears naturally as the effective width of a non-resonant electrically small aperture backed by LHM, just as it does in the case of a resonant aperture in ordinary media.

To assess the effect of loss, as it could be found on a perhaps more realistic LHM practical implementation, we have to run the same geometry but with loss tangents 100 and 500 times that employed in the previous simulation. The results are presented in figure 5, which show the Poynting vector field snapshots for the corresponding amounts of loss, from which it is clear that even in the presence of significant losses, the incident field flux is simply squeezed as it flows past the apertures.

To assess how forgiving the effect is with respect to parameter deviations from ideal LHM, as it could be found on a perhaps more realistic LHM practical implementation, we have run the same geometry but with *ϵ*=−1.5(1−i10^{−3}) and *μ*=−2(1−i10^{−3})/3. The idea is that if for some design constrain we cannot match the permeability and permittivity to that of ideal LHM, it is reasonable to attempt to match the index. The results are presented in figure 6, which show the Poynting vector field snapshots, from which it is clear that even in the presence of significant material deviations but provided the negative unit index is maintained, the incident field flux is simply squeezed as it flows past the apertures, just as it happens in ideal LHM.

We have also investigated the effect of losses on transmission. Using the geometry and parameters of figure 2, with and , FEM simulations were performed at 10 GHz assuming equal amounts of electric and magnetic loss, namely , for in the range 0.0–0.5. As losses affect all devices without exception, it is fair to compare the loss through the system with the loss that would be observed if we just had the slab of lossy LHM (i.e. no waveguide discontinuities). This way, we will be able to quantify the net effect of the Venturi energy squeezing on loss. The data are presented in figure 7, which also includes the analytical LHM slab. Although the net insertion loss could be as large as 6 dB for the large values of chosen, the net effect could be explained as a small fractional increase in the effective thickness of the slab. This is due to the fact that for a well-matched slab, the insertion loss in decibels is a linear function of , whereas for the Venturi effect, the insertion loss in decibel is almost linear with . This is not surprising in view of the Poynting vector field snapshots that show curved trajectories in the slab, each of which has a path length longer than *d*, hence the extra path length means extra loss.

Regarding the practical aspects nature of the experimental characterization of the effect, it is expected that an EM near-field scanner setup will allow complete near-field characterization of the transmitted electric field profile at all points in the principal plane, just as was done in (Monzon *et al*. 2006), which included the measurement of the NR and focusing of broadband microwave beams by a photonic structure. Aside from a synthesized generator, and a spectrum analyser, miniature dipole probes can be used as both sources and sensors, the latter being scannable in a rectangular grid in the near field of the device.

### (a) On the feasibility of broadband left-handed media

Successful gain-assisted dispersion management in infrared-active negative-index metamaterials was shown in Govyadinov *et al*. (2007), where their approach was illustrated with a nanowire-based optical NIM system, resulting in an explicit demonstration of broadband dispersionless index and impedance matching. We feel the method can be used to make the effective LHM permittivity and permeability broadband by assisting them with gain as follows. Considering Pendry wires, characterized by a loss factor *γ* resulting in , and incorporating gain in the form of an alternating fractional section *β* of the filaments described by a negative loss factor −*ξ*, resulting in , subject to *α*+*β*=1. By selecting *ξ*≈*γ*, and constraining *γ* and *ω*_{p}, so that over the band of interest , then we find *ϵ*≈−1+i 2*γ*(*α*−*β*)/*ω*. It is clear that by suitably choosing the filaments to be 50 per cent active (*α*=*β*), we have good control over the dispersion, rendering the description of the composite array of wires essentially flat *ϵ*≈−1+i*O*(*δ*), over the band of interest.

It should be noted that the introduction of negative loss is akin to the insertion of negative resistance and that the same method can be used with other types of metamaterial periodic inclusions, such as, for instance, Pendry’s SRR. Thus, it is in principle possible to achieve broadband characteristics; it is just a technological challenge.

## 3. Summary

We demonstrated both theoretically and through numerical simulations an anomalous transmission of microwaves through a very narrow aperture in a thin metal plate backed by LHM. This new effect is not resonant, and analogous to the Venturi effect in fluids. It is shown that the incoming energy is simply squeezed through the aperture, without any reflection, as if we were not dealing with a wave, and in principle, regardless of how small the aperture is. The analysis and simulations dealing with a cascaded waveguide implementation indicate that the principle presented here can be used to produce more complex broadband transparent structures. A proposal for achieving broadband LHM is also included. The results can be scaled to RF/microwave/THz/IR, and potentially the optical range.

## Footnotes

- Received February 19, 2009.
- Accepted June 9, 2009.

- © 2009 The Royal Society