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Parametric amplification of magnetoinductive waves

supported by metamaterial arrays

Oleksiy Sydoruk1

, Victor Kalinin2

, and Ekaterina Shamonina1

1

Department of Physics, University of Osnabrück, 49069 Osnabrück, Germany

2

Transense Technologies Ltd., Upper Heyford, Bicester, Oxon OX25 5HD, United Kingdom

Received 18 September 2006, revised 16 November 2006, accepted 17 November 2006

Published online 22 March 2007

PACS 41.20.–q, 81.05.Zx, 84.30.Bv, 84.30.Le

A theoretical study of parametric amplification of magnetoinductive waves propagating on metamaterial

arrays is presented. It is shown that phase matching conditions in an array can be fulfilled provided

that the near-field magnetic coupling between the neighbouring elements is sufficiently high. Equations

governing wave propagation are derived and the effect of losses on parametric amplification is analyzed.

phys. stat. sol. (b) 244, No. 4, 1176–1180 (2007) / DOI 10.1002/pssb.200674502

phys. stat. sol. (b) 244, No. 4, 1176–1180 (2007) / DOI 10.1002/pssb.200674502

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Parametric amplification of magnetoinductive waves

supported by metamaterial arrays

Oleksiy Sydoruk1, Victor Kalinin2, and Ekaterina Shamonina*, 1

1 Department of Physics, University of Osnabrück, 49069 Osnabrück, Germany 2 Transense Technologies Ltd., Upper Heyford, Bicester, Oxon OX25 5HD, United Kingdom

Received 18 September 2006, revised 16 November 2006, accepted 17 November 2006

Published online 22 March 2007

PACS 41.20.–q, 81.05.Zx, 84.30.Bv, 84.30.Le

A theoretical study of parametric amplification of magnetoinductive waves propagating on metamaterial

arrays is presented. It is shown that phase matching conditions in an array can be fulfilled provided

that the near-field magnetic coupling between the neighbouring elements is sufficiently high. Equations

governing wave propagation are derived and the effect of losses on parametric amplification is analyzed.

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The idea that nonlinear insertions into elements constituting an effective medium at microwaves can

allow realization of a number of nonlinear effects known in optics was first suggested in a paper pub-

lished in the early ’90s [1]. Recently this idea has been revised [2–4] with regard to the rapidly develop-

ing field of metamaterials [5]. In particular, second harmonic generation [6, 7] and parametric amplifica-

tion in metamaterials [7, 8] were studied theoretically at the microwave and optical frequency regions.

This paper presents a study of parametric amplification of magnetoinductive (MI) waves [9] propagating

on arrays of metamaterial elements. Applications of MI waves include signal guiding [10] and process-

ing [11, 12], where parametric amplification can be used for compensation of ohmic losses. In an MRI

detection system [13] parametric amplification of a weak signal can lead to an improvement of its signal-

to-noise ratio. We shall first discuss the problem of fulfilling phase matching conditions for MI waves

and then develop a theory of parametric amplification for arrays of metamaterial elements and analyse

the effect of losses.

2 Phase matching conditions for metamaterial arrays

In the simplest case parametric amplification involves two waves, the signal and the pump, that are phase

matched [14]. Perfect phase matching assumes that both the frequency, p

ω , and the wavenumber, p

β , of

the pump wave are twice as large as those of the signal one, s

ω and s

β ,

p s p s

2 2ω ω β β= , = . (1)

In the case of MI waves, which propagate around the resonant frequency of elements, this implies that

both the signal and the pump waves are allowed by the dispersion of a metamaterial array. In the ap-

proximation of the nearest neighbour interactions the width of the passband of MI waves is determined

by the coupling strength between the elements. Therefore to achieve wide passbands the coupling coeffi-

* Corresponding author: e-mail: ekaterina.shamonina@uos.de

phys. stat. sol. (b) 244, No. 4 (2007) 1177

www.pss-b.com © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

a)

C=C0+γ U C=C0+γ U C=C0+γU

M M

R

L L L

R R

b)

Fig. 1 a) Schematic presentation and b) equivalent circuit of a metamaterial array supporting magneto-

inductive waves. The capacitance in b) is voltage-dependent.

cient, 2M Lκ = / (L is the selfinductance of the elements and M is the mutual inductance between them),

should be as large as possible. A practical realization of elements with sufficiently strong coupling has

been recently demonstrated in Ref. [15]. For a given frequency of the signal, s

ω , it is then possible to

design a metamaterial array that would allow propagation of both the signal and the pump MI waves as

well as phase matching between them. We demonstrate this by the following example. We take the fre-

quency of the signal to be s(2π) 63 87ω / = . MHz, corresponding to the frequency of precession of a pro-

ton in a magnetic field of 1.5 T. Disregarding losses we can then write a system of equations for the

signal and the pump assuming that they both satisfy the dispersion relation [9]

2 2

0 s s

2 2

0 s s

1 cos 0

1 (4 ) cos 2 0

a

a

ω ω κ β

ω ω κ β

- / + =ÏÌ

- / + = ,Ó (2)

where 0

ω is the resonant frequency of the elements and a is the period of the array. Choosing the cou-

pling between the elements to be 0 9κ = . we can solve Eq. (2) for the resonant frequency and propaga-

tion constant, giving respectively 0(2π) 73 44ω / = . MHz and

s0 38πaβ = . . For elements with inductance

50L = nH this value of the resonant frequency can be achieved by loading the elements by capacitances

of 0

94C = pF. A schematic presentation of the array is shown in Fig. 1a and the corresponding disper-

sion of MI waves in Fig. 2. It can be seen that both the signal and the pump waves can propagate and are

phase-matched, and therefore parametric amplification of the signal is possible, provided that the meta-

material elements have nonlinear character.

3 Theory of parametric amplification of MI waves

The easiest way to realize the desired nonlinearity would be to load the elements with varactor diodes

whose capacitance is voltage-dependent. For small amplitudes it can be written as 0

C C Uγ= + , where γ

characterizes nonlinearity and U is the voltage across the nonlinear capacitance. An equivalent circuit of

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 2 (online colour at: www.pss-b.com) Dispersion curve

of magnetoinductive waves. Both the signal s s

( )aβ ω, and the

pump s s

(2 2 )aβ ω, waves can propagate on the metamaterial

array.

1178 O. Sydoruk et al.: Parametric amplification of magnetoinductive waves

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

a metamaterial array with nonlinear capacitance is shown in Fig. 1b. Assuming for the voltage across the

capacitance two propagating waves, the signal and the pump, of the form

{ }

{ }

1s s s s2

1p p s s2

( ) ( ) exp [ ( )] c c

( ) ( ) exp [ (2 2 )] c c

U n u n j t n a

U n u n j t n a

ω β

ω β

= - + . .

= - + . .

(3)

and neglecting higher harmonics we can write the current, nI , flowing in the n-th element,

d( )

dn

CUI

t= ,

separately at the signal and pump frequencies. The current at the signal frequency is

s s p s( ) ( ) 2 ( ) ( ) c cI n u n u n u nσ ρ *= + + . . , (4)

and that at the pump frequency is

2

p p s( ) 2 ( ) 2 ( ) c cI n u n u nσ ρ= + + . . , (5)

where * denotes complex conjugate, s 0

2j Cσ ω= / and s

4jρ ω γ= / .

Kirchhoff’s law for the n-th element of the array has the form

[ ]d d ( )

( 1) ( 1) ( ) ( ) 0d d

I nM I n I n L RI n U n

t t+ + - + + + = . (6)

3.1 Discrete model

Substituting expressions for the voltages, Eq. (3), and currents, Eqs. (4) and (5), into Eq. (6) we obtain

two equations governing the voltages across the nonlinear capacitance on the signal frequency, s

ω , and

pump frequency, s

2ω . Kirchhoff’s equations written separately for the signal, s

ω , and pump, s

2ω , fre-

quencies have then the form

s s p s s

s p s s

s s p s s

{[ ( 1) 2 ( 1) ( 1)] exp ( )

[ ( 1) 2 ( 1) ( 1)] exp ( )}

( ) [ ( ) 2 ( ) ( )] 1 2 ( ) 0

j M u n u n u n j a

u n u n u n j a

j L R u n u n u n u n

ω σ ρ β

σ ρ β

ω σ ρ

*+ + + + -

*+ - + - -

*+ + + + / =

and

2 2

s p s s p s s

2

s p s p

2 {[ ( 1) ( 1)] exp ( 2 ) [ ( 1) ( 1)] exp (2 )}

(2 ) [ ( ) ( )] 1 4 ( ) 0

j M u n u n j a u n u n j a

j L R u n u n u n

ω σ ρ β σ ρ β

ω σ ρ

+ + + - + - + -

+ + + + / = ,

respectively.

3.2 Continuous model

Equations (7) and (8) constitute a system of difference equations that can be solved numerically for the

case of a finite array of N elements for the signal, s( )u n , and pump,

p( )u n , voltages ( 1n N= ... ). It is,

however, always desirable to have an analytical tool for studying the process in contrast to a purely nu-

merical solution. For this purpose, assuming that the signal and the pump wave amplitudes vary slowly

(7)

(8)

phys. stat. sol. (b) 244, No. 4 (2007) 1179

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Original

Paper

along the array, we formally expand the terms s,p( 1)u n + and

s,p( 1)u n - into the Taylor series restricting

ourselves only to the first two terms of the expansion

s,p

s,p s,p

d( 1) ( )

d

u

u n u n

n

± = ± .

Substituting this into Eqs. (7) and (8) and neglecting small terms we finally obtain for the amplitudes of

the signal and pump waves

ss p s s s

p 2

p s p p

d ( )( ) ( ) ( )

d

d ( )( ) ( ) ,

d

u zjg u z u z u z

z

u zjg u z u z

z

α

α

*= - -

= - -

(9)

where

s p2 2 2 2

0 s 0 s

and2 sin 16 sin 2

g ga C M a a C M a

γ γ

ω β ω β= = (10)

characterize nonlinearity and

s p

s s s s

and2 sin 4 sin 2

R R

a M a a M aα α

ω β ω β= = (11)

are the absorption coefficients of MI waves for the case of low losses [13]. Finally, z na= is a continu-

ous coordinate. In the lossless case when s,p

0α = , Eq. (9) are analogous to the corresponding equations

for parametric amplification and second harmonic generation derived in optics [14].

0 50 100 150 200 250 3000

0.5

1

1.5

element number

volta

ge [V

]

signal wave, continuous modelpump wave, continuous modelsignal wave, discrete modelpump wave, discrete model

a) 0 50 100 150 200 250 300

0

0.5

1

1.5

element number

volta

ge [V

]

signal wave, continuous modelpump wave, continuous modelsignal wave, discrete modelpump wave, discrete model

b)

0 50 100 150 200 250 3000

0.5

1

1.5

element number

volta

ge [V

]

signal wave, continuous modelpump wave, continuous modelsignal wave, discrete modelpump wave, discrete model

c)

Fig. 3 (online colour at: www.pss-b.com) Ampli-

tudes of the signal (red) and pump (blue) waves a) in

the absence of losses, b) for low and c) for high losses.

The quality factor of the elements is Q = 2000 in the

case b), and Q = 200 in the case c). Periodic exchange

of power in the cases a) and b) disappears for the case

of high losses c).

1180 O. Sydoruk et al.: Parametric amplification of magnetoinductive waves

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

In Fig. 3a the amplitudes of the signal and the pump are shown for a lossless array of 300 elements

with the parameters taken from the previous example and the nonlinearity 0

0 05Cγ / = . V–1. It is assumed

that the first element of the array is excited with two voltage sources, one at the signal frequency with

amplitude 0.05 V and the other one at the pump frequency with amplitude 0.25 V. It can be seen that the

signal and pump waves periodically exchange power, the first maximum of the signal amplitude is

achieved at a length of 75 elements.

Now we shall look at the effect of losses. In Fig. 3b the amplitudes of the signal and the pump waves

are shown for the same array with elements having small losses characterized by a quality factor,

0Q L Rω= / , of 2000. It can be seen that both amplitudes retain their periodic character slowly declining

along the array. If the elements are highly lossy then the amplitude decline is so fast that the pump can-

not effectively transfer its energy to the signal. As can be seen in Fig. 3c for the elements with 200Q =

there is a single pronounced maximum of the signal wave amplitude with no periodic power exchange. It

is gratifying to note that the agreement between solutions of the discrete (denoted by crosses), Eqs. (7)

and (8), and of the continuous (denoted by dashed lines), Eq. (9), models is very good for each case of

Fig. 3.

4 Conclusions

Phase matching of MI waves has been theoretically demonstrated for metamaterial arrays whose ele-

ments are strongly coupled to each other. This can be practically achieved for axially configured arrays

[15]. For planar arrays, where the coupling is essentially weaker, “biatomic” configurations with ele-

ments of two alternating resonant frequencies can be used [16].

We have developed a theoretical model of parametric amplification, both in discrete and continuous

form. The effect of losses on parametric amplification has been studied and the magnitude and position

of the first maximum have been determined.

Acknowledgements We thank one of the reviewers for pointing out an error in the calculations. The authors wish

to thank L. Solymar, M. Shamonin, R. R. A. Syms and C. J. Stevens for a number of stimulating discussions. Sup-

port of the DFG is gratefully acknowledged.

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