INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 37 (2004) 362–367

PII: S0022-3727(04)64344-0

Magneto-inductive waves supported by metamaterial elements: components for a one-dimensional waveguide E Shamonina1 and L Solymar2,3 1 2

Department of Physics, University of Osnabr¨uck, D-49069 Osnabr¨uck, Germany Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK

E-mail: [email protected]

Received 4 June 2003 Published 14 January 2004 Online at stacks.iop.org/JPhysD/37/362 (DOI: 10.1088/0022-3727/37/3/008) Abstract The propagation and manipulation of magneto-inductive waves is studied under conditions when there is reflection due to lack of matching, when power is coupled from one waveguide to another, when coupling causes directional properties and when tunnelling occurs between two waveguides. The relationships obtained are compared and contrasted with those occurring in traditional transmission lines.

1. Introduction The properties of guided electromagnetic waves have been investigated in great detail in the last 100 years. There is a wide variety of waveguides in practical use from the coaxial line to optical fibres. Components have been designed for all these waveguides to enable them to fulfill their functions to guide and process the waves. Other kinds of waves have received less attention. There are not many components for plasma waves, or for surface plasmons and for bulk acoustic waves either. On the other hand surface acoustic waves have fared very well. Lots of transducers, resonators, couplers, ordinary filters, matched filters, directional couplers, etc have been designed to exploit their properties. When a new wave is discovered the question about the feasibility of components immediately arises. The new wave in question is the so-called magneto-inductive (MI) wave conceived while investigating the properties of metamaterials, a new discipline initiated mainly by the work of Pendry [1, 2]. Metamaterials have resonant inclusions which may drastically change their effective permeability and permittivity [3–8]. High positive and negative values are possible within narrow frequency bands. A recent experimental study by Wiltshire et al [9] showed that a metamaterial made up of ‘Swiss Rolls’ could successfully carry magnetic information. It was later realized [10–13] that metamaterial elements can affect not only the propagation of transverse electromagnetic waves but may also 3

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be responsible for the emergence of an entirely different kind of wave due to the magnetic coupling between the elements. The first MI waveguide proposed [10] consisted of a one-dimensional array of capacitively loaded metallic loops. A more detailed analysis of the same structure showed that MI waves may also propagate in two and three dimensions [11]. An experimental proof of the existence of these waves was provided by Wiltshire et al [12] who also showed [13] that besides capacitively loaded loops ‘Swiss Rolls’ were also suitable carriers of MI waves. It seems safe to surmise that MI waves will be present whenever there is magnetic coupling between the elements for instance in metamaterials made of split ring resonators [3–8]. Such waves will be considered for practical applications when resonant transmission is a crucial requirement as in magnetic resonant imaging. This paper is concerned with a theoretical study of potential components. The principal aim is to look in more detail at MI waves under conditions when reflections take place and in particular when two different waveguides are coupled in some way to each other. We shall also look at the phenomenon of tunnelling, familiar from other disciplines, and how power can be transferred from one waveguide to another. MI waves and magneto-inductive waveguides (MIWs) will be introduced in section 2, an analysis based on previously published work will be presented in section 3, junctions, couplers and directional couplers will be discussed in sections 4–6 whereas section 7 will be devoted to a comparison between the

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MI waves supported by metamaterial elements

(a)

1

z

3

2

y

formula for self-inductance. Expressions in terms of double integrals are also available for the mutual inductance [14]. By knowing the self-inductance of the loops and the mutual inductances between any two loops it is possible then to calculate the currents in all the elements for a given excitation. In many cases the coupling between elements further away can be neglected. This is what we shall do in the rest of the paper by relying on nearest neighbour interaction only. To keep our analysis simple we shall also neglect losses. We shall therefore assume every element of a radius r0 to have an impedance
 1 Z = j ωL − , (1) ωC

N

N-1

Zt

r0

x d

V

(b) 1

V

3

2

d

N-1

N x

r0 Zt

where a time variation of the form exp( jωt) has been assumed, ω is the frequency, L is the self-inductance and C is the capacitance. The mutual impedance between two elements at a distance d from each other ZM = jωM

(2)

is proportional to the mutual inductance M. The simplified dispersion relationship for the lossless case with only nearest neighbour interactions is then given by Figure 1. A schematic representation (top) and photographs of a section (bottom) of (a) an axial line and (b) a planar line. The photographs are taken from [12].

properties of traditional transmission lines (TTLs) and MIWs. Brief conclusions will be drawn in section 8.

2. The basic set-up Schematic and actual representations of waveguides known to support MI waves are shown in figure 1. As may be seen there are two main configurations: the axial and the planar. In the first one the line connecting the centres of the loops is perpendicular to the plane of the loops (figure 1(a)), in the second one the line is in the same plane as the loops (figure 1(b)). The little blobs shown in the photographs are bulk capacitors which ensure the resonant character of the elements. Wave propagation in both cases is based on resonant magnetic coupling. The main difference between them is the orientation of the magnetic coupling. The magnetic field threading two neighbouring elements is in the same direction for the axial case and in the opposite direction for the planar case. As a consequence the axial configuration carries a forward wave (phase and group velocities in the same direction) whereas for the planar configuration the wave carried is a backward one, phase and group velocities are in opposite directions [11]. In both cases wave propagation is possible only in a narrow passband centred at the resonant frequency of the elements. The width of the passband is proportional to the strength of the coupling.

3. Analysis The analysis is quite straightforward for the case when the dimensions of the waveguide are small relative to the free space wavelength. The magnetic field due to a current carrying loop is available in most undergraduate text books and so is a

ω02 − 1 = κ cosh(γ d), ω2

(3)

√ where ω0 = 1/ LC is the resonant frequency, κ = 2M/L is the coupling constant. γ = α + jβ, α is the attenuation coefficient and β is the propagation wave number. In the passband 1 1 ω