07.0310212.m.viitanen

8/3/2019 07.0310212.M.Viitanen

1/11

Progress In Electromagnetics Research, PIER 51, 127137, 2005

SURFACE WAVE CHARACTER ON A SLAB OFMETAMATERIAL WITH NEGATIVE PERMITTIVITYAND PERMEABILITY

S. F. Mahmoud

Electronic Engineering DepartmentKuwait UniversityP.O. Box 5969, Safat, 13060, Kuwait

A. J. Viitanen

Department of Electrical and Communications EngineeringHelsinki University of TechnologyP.O. Box 3000, FIN-02015, HUT, Finland

AbstractCharacteristics of surface wave modes on a grounded slabof negative permittivity and negative permeability parameters areinvestigated. It is shown that, unlike a slab with positive parameters,the dominant mode can have evanescent fields on both sides ofthe interface between the slab and the surrounding air. Detailedcharacteristics of such mode for various combinations of the slabparameters are given.

1 Introduction

2 Surface Wave Modes

3 Modes with Evanescent Fields on both Sides of theInterface

4 Concluding Remarks

References

8/3/2019 07.0310212.M.Viitanen

2/11

128 Mahmoud and Viitanen

1. INTRODUCTION

Recently, there has been a growing interest in metamaterials havingnegative permittivity and negative permeability in some frequencyrange for their potential of novel applications in microwave circuits.Veselago [1] was the first to show that the Poynting vector isantiparallel to the phase progression in such material. Lindell et al.[2, 3] have studied media with negative values of permittivity andpermeability and labeled them as backward wave (BW) media ormedia capable of supporting backward waves. In the presence of aninterface between a BW medium and a regular medium (one withpositive permittivity and permeability), the power flowing laterallyin the BW medium is in opposite direction to that in the regularmedium. Other authors have used other labels to such materials, whichinclude negative refractive index [4], left-handed media [5, 6] or doublenegative parameters media [79]. In this work we shall adopt thelabeling backward wave for materials with negative permittivity andpermeability. Several theoretical and experimental studies have beenconducted on wave reflection and refraction at an interface between aBW medium and a positive parameter medium [7, 10] and on radiationfrom a traveling wave source at such interface [8]. One interestingexample is the concept of a perfect lens proposed in [4]. Characteristicsof surface waves on a slab of BW material have been investigated in[3, 11]. Dispersion relations are determined and mode cutoff effects arestudied. It is found that a band-pass region appears for the first mode.Guided modes in metallic waveguides partially loaded by a slab of BWmaterial have been studied [12].

Although surface wave modes on a BW slab are studied in [3, 11],we reconsider them in this research to study the nature of power carriedby these modes. We also investigate the possibility of supporting amode with evanescent fields on both sides of the interface betweenthe slab and the surrounding (regular) medium. Conditions for theabsence of surface waves in a certain frequency band are sought. Tothis end, we first review the main features of surface waves on agrounded slab of BW material, concentrating on the power carriedby each mode in Section 2. Detailed study of the dominant TE or TMmodes is carried out for various combinations of the permittivity andpermeability parameters of the slab in Section 3. Concluding remarksare given in Section 4.

8/3/2019 07.0310212.M.Viitanen

3/11

Progress In Electromagnetics Research, PIER 51, 2005 129

Slab with negative and

AIR

d

z

x

Figure 1. A grounded BW slab.

2. SURFACE WAVE MODES

Surface wave modes on a BW slab have been studied in [311]. In thissection we review the salient features of surface waves on a groundedslab and then derive the power carried by any surface wave mode.To this end, let us consider a grounded slab of negative relativepermittivity r and negative relative permeability r impeded in airas depicted in Fig. 1. We consider TM to z modes and the TE to zmodes.

TM modes:Considering TM modes with the magnetic field totally along the

y-direction and independent of y, hy takes the form:

hy = cos kxx exp[jz] 0 x d= cos kxxd exp[(x d)] d x < (1)

where: kx = (k20

rr 2)1/2, = (2 k20)1/2, k0 = (00)1/2and a time harmonic variation exp(jt) has been assumed. Using

Maxwell equation to get the electric field component ez as proportionalto 1r hy/x and enforcing the continuity of ez at x = d, we get themodal equation:

[kxd tan(kxd)]/r = d =

V2 k2xd2 (2)where V = k0d

rr 1 is a normalized frequency parameter.

Equation (2) is the same modal equation as that of a regular slab withpositive r and r. However the negative value ofr has a drastic effect

on the solution for the modal phase constant as will be seen. As anexample, the relative phase constant /k0 for the first two TMn modesis plotted versus V in Fig. 2 for a slab having r = 2 and r = 1.5.The first mode TM1 of the BW slab starts at some minimum value

8/3/2019 07.0310212.M.Viitanen

4/11


Modal phase constant;TM modes

1

1,2

1,4

1,6

1,8

0 2 4 6 8 10 12

V number

/k0

Grounded Slab with

r=-2

r=-1.5

TM1 TM2

Figure 2. Relative phase constant of TM modes versus normalizedfrequency V.

Relative SW Power;TM modes

-5

0

5

10

15

0 2 4 6 8 10V number

P

Grounded Slab

r=-2.

r=-1.5

TM1

TM2

Figure 3. Relative Power of TM modes on a grounded backward slabversus V.

of V; say V = Vmin. For Vmin < V < , there are two solutions forthe TM1 mode. One of these has a cutoff at V = and negativegroup velocity. The other solution exists at any value of V > Vmin andrepresents a tightly bounded mode for large V. Similar behavior isdisplayed by the second mode. So, in general the TMn mode has twobranches, the one with the lower has a negative group velocity andsuffers cutoff ( = k0) at V = n. The other branch continues to exist

at higher frequencies and has a positive group velocity.It is interesting at this point to investigate the net modal power ofeach branch of the TMn modes; n > 0. This is shown in Fig. 3 whereit is seen that the modal power is zero at V = Vmin where the mode

8/3/2019 07.0310212.M.Viitanen

5/11


starts to emerge. The power of the mode branch with the lower ispositive (forward) and the power of the mode branch with the higher is negative (backward). It should be noted here that the net power

of a mode is the result of positive power that flows in the air alongthe direction of phase change and negative power, or backward power,flowing in the BW slab in the opposite direction.

TE modes:

Using similar analysis, we get the modal equation of TE surfacewave modes on a BW slab as:

[kxdCot(kxd)]/r = d =

V2 k2xd2 (3)

Again this is the same equation as that for a regular slab, but herethe negative value of r results in a drastic change in the solution forthe phase constant . Considering the lowest order mode; TE0 mode,we notice that when kx = 0 in (3), the LHS tends to (1/|r|) which ispositive and must be equal to V. Actually it can be shown that thereis no solution to (3) with sinusoidal field distribution in the slab forV < (1/|r|). So the TE0 mode starts at a frequency corresponding toV = (1/|r|) whence d = V. As kxd approaches /2, d tends to zeroand the mode cutoffs at V = kxd = /2. Thus the TE0 mode exists

alone in the frequency range given by 1/|r| < V < /2 (providedthat |r| > 2/) and therefore can be considered the dominant modeof the slab. The relative phase constant /k0 of the TE0 mode andthe next mode TE1 mode is plotted against V in Fig. 4 for r = 2and r = 1.5. It is seen that the TE1 mode (and other higher ordermodes) has two branches with opposite group velocities just as the

Modal Phase Constant;TE modes

1

1,2

1,4

1,6

1,8

0 2 4 6 8 10 12

V number

/k0

Grounded Slab

r=-2

r=-1.5

TE0 TE1

Vmin=

1/| |

Figure 4. TE modal phase constant.

8/3/2019 07.0310212.M.Viitanen

6/11


Relative SW Power; TE modes

-5

0

5

10

15

20

0 2 4 6 8 10 12V number

P

Grounded Slab

r=-2

r=-1.5

Vmin

TE0

TE1

Figure 5. Relative powers of TE mode.

TM modes. In general, a higher order TEn mode; n > 0 starts ata Vmin,n such that n < V min,n < (n + 1/2). The mode has twosolutions for in the band Vmin,n < V < (n + 1/2) and one solutionfor V > (n + 1/2). The corresponding relative power carried by eachmode is plotted in Fig. 5. The TE0 mode carries positive or forwardpower which increases indefinitely towards cutoff at V = /2. Eachof the TEn modes with n > 0 has two branches; the one with thelower carries a net positive or forward power and ends in cutoffat V = (n + 1/2). The other branch carries a net negative power,or backward power, and continues at higher frequencies getting morebounded to the slab.

3. MODES WITH EVANESCENT FIELDS ON BOTHSIDES OF THE INTERFACE

So far we have considered modes with oscillating fields inside the slaband decaying fields in the air. The question that rises now is whethermodes with decaying fields on both sides of the interface can exist. It iswell known that a slab with positive parameters will not support modeswith decaying fields on both sides of the interface. For a backward waveslab, however, it turns out that this is not the case. To study thisstatement, let us investigate the modal Equations (2) and (3) wherewe assume that kx is pure imaginary; that is kx = jp, where p is real.

Equations (2) and (3) reduce then to:[pd tanh(pd)]/r = d =

V2 + p2d2 (TM modes) (2)

[pdCoth(pd)]/r = d =

V2 + p2d2 (TE modes) (3)

8/3/2019 07.0310212.M.Viitanen

7/11


We note that d must be positive for a proper surface wave mode.Now if r (or r) is positive, the LHS of (2) (or (3)) is negative andthere is no solution for a valid surface wave. However, when r (or

r) is negative, it is possible to have a surface wave solution for whichthe fields are evanescent away from both sides of the interface. Notealso that for such a mode V2 does not have to be positive, that is,rr can be greater or less than one. It is worth noting here that thesame phenomenon has been reported in metamaterial resonators withcascaded BW slab and a regular slab where a mode with evanescentfields away from the interface is detected [13]. Now let us consider theevanescent TE and TM surface wave mode separately.

TE0 mode; rr > 1 :

For the TE0 mode, Eq. (3) tells us that p = 0 when V = 1/|r|whence /k0 =

rr. At this point, we distinguish between two

cases according to whether |r| is greater or less than one. If |r| > 1,then for V < 1/|r|, p increases and /k0 becomes > rr. Thisgoes on until V tends to zero and both p and tend to infinity.To illustrate this behavior, /k0 of the TE0 mode is plotted for a slabhaving r = 2 and r = 1.5 in Fig. 6 (see solid curve). It is seen thatthe mode exists in the range 0 < V < /2. The mode is evanescent,i.e., /k0 is greater than

rr, in the range 0 < V < 1/

|r|. An

explanation for the field decay on both sides of the interface canbe given based on the boundary condition on hz, which requires the

Relative Phase Constant for the TE0 Mode

1

2

3

4

0 1 2 3 4

Normalized Frequency V

/k0

1

1,2

1,4

1,6

ns

V=1/|

ns

Evanescent Fields

______ =-2, =-1.5

Left scale

-------- =-3, =-0.5

Right scale

r

r

r

r

Figure 6. The phrase constant of the dominant TE0 mode on aBW slab with ns rr > 1 and |r| > 1 (solid curve) andns rr > 1 and |r| < 1 (dotted curve).

8/3/2019 07.0310212.M.Viitanen

8/11

8/3/2019 07.0310212.M.Viitanen

9/11


As an application, a BW slab can thus be used as a substrate inmicrostrip circuits to avoid the excitation of unwanted surface waves.It has been proved that modes with evanescent fields on both sides

of the interface between the slab and air (referred to as evanescentmodes) can exist. Namely, the TE0 mode becomes evanescent in thelow frequency band 0 < V < 1/|r| when ns rr > 1 and |r| > 1.In this case the longitudinal phase constant attains high values thatmakes the structure attractive in building superdirective antennas. Itis interesting to find out that one or more evanescent surface wave modecan exist on the BW slab even when its refractive index is less than thesurrounding medium. Namely when ns is < 1, a TM evanescent modecan propagate at all frequencies if|r| > 1 and in a finite low frequencyif |r| < 1. On the other hand, when ns is < 1, a TE evanescentmode can propagate at all frequencies if |r| > 1 and ceases to exist if|r| < 1.

For future it will be interesting to study the problem of two layerslab, one layer is of positive permittivity and permeability and theother is of negative permittivity and permeability. In such structurebounded by metal plates a memory device behaviour for evanescentfields is observed [13].

REFERENCES

1. Veselago, V. G., The electrodynamics of substances withsimultaneously negative values of permittivity and permeability,Soviet Phys. Uspekhi, Vol. 10, 509514, 1968.

2. Lindell, I. V., S. A. Tretyakov, K. I. Nikoskinen, andS. Ilvonen, BW media-media with negative parameters, capableof supporting backward waves, Micro. Opt. Tech. Lett., Vol. 31,No. 2, 129133, Oct. 2001.

3. Lindell, I. V. and S. Ilvonen, Waves in a slab of uniaxial BWmedium, J. of Electromagn. Waves and Appl. (JEMWA), Vol. 16,No. 3, 303318, 2002.

4. Pendry, J. B., Negative refraction makes a perfect lens, Phys.Rev. Lett., Vol. 85, 39663969, 2000.

5. Smith, D. R. and N. Kroll, Negative refraction index in lefthanded materials, Phys. Rev. Lett., Vol. 85, 29332936, 2000.

6. Shelby, R. A., D. R. Smith, S. C. Nemat-Nasser, and S. Schultz,

Microwave transmission through a two-dimensional, isotropic,left-handed metamaterial, Appl. Phys. Lett., Vol. 78, 489491,2001.

7. Engheta, N., An idea for thin subwavelength cavity resonators

8/3/2019 07.0310212.M.Viitanen

10/11


using metamaterial with negative permittivity and permeability,IEEE Antennas and Wireless Propagation Letters, Vol. 1, 1013,2002.

8. Alu, A. and N. Engheta, Radiation from a traveling wave currentsheet at the interface between a conventional material and ametamaterial with negative permittivity and permeability, Mic.Opt. Tech. Lett., Vol. 35, No. 6, 460463, Dec. 2001.

9. Alu, A. and N. Engheta, Anomalous mode coupling in guidedwave structures containing metamaterials with negative permit-tivity and permeability, Proc. IEEE Nano 2002 Conference,Washington D.C., Aug. 2628, 2002.

10. Lakhtakia, A., M. W. McCall, and W. S. Weiglhofer,

Brief overview of recent developments on negative phase-velocity mediums (alias left-handed materials), Int. J. Electron.Commun. (AEU), Vol. 56, 407410, 2002.

11. Cory, H. and A. Barger, Surface wave propagation along ametamaterial slab, Mic. Opt. Tech. Lett., Vol. 38, No. 5, 392395, Sept. 2003.

12. Alu, A. and N. Engheta, Resonances in sub-wavelengthcylindrical structures made of pairs of double-negative and doublepositive or negative and negative coaxial shells, Proc. of2003 Intl. Conf. On Electromagnetics in Advanced Applications(ICEAA), Sept. 812, 2003.

13. Tretyakov, S. A., S. I. Maslovski, I. S. Nevedov, andM. K. Karkkainen, Evanescent modes stored in cavity resonatorswith backward wave slabs, Microwave and Optical TechnologyLetters, Vol. 38, No. 2, 153157, July 20, 2003.

Samir F. Mahmoud graduated from the Electronic EngineeringDepartment, Cairo university, Egypt in 1964. He received theM.Sc. and Ph.D. degrees in the Electrical Engineering Department,Queens university, Kingston, Ontario, Canada in 1970 and 1973.During the academic year 19731974, he was a visiting researchfellow at the Cooperative Institute for Research in EnvironmentalSciences (CIRES). Boulder, CO, doing research on Communicationin Tunnels. He spent two sabbatical years, 19801982, between QueenMary College, London and the British Aerospace, Stevenage, where

he was involved in design of antennas for satellite communication.Currently Dr. Mahmoud is a full professor at the EE Department,Kuwait University. Recently, he has visited several places includingInteruniversity Micro-Electronics Centre (IMEC), Leuven, Belgium

8/3/2019 07.0310212.M.Viitanen

11/11


and spent a sabbatical leave at Queens University and the royalMilitary College, Kingston, Ontario, Canada in 20012002. Hisresearch activities have been in the areas of antennas, geophysics,

tunnel communication, e.m wave interaction with composite materialsand microwave integrated circuits. Dr. Mahmoud is a Fellow of IEEand one of the recipients of the best IEEE/MTT paper for 2003.

Ari J. Viitanen received the Dipl.Eng., Lic.Tech., and Dr.Tech.degrees in electrical engineering from the Helsinki University ofTechnology, Finland in 1984, 1989, and 1991, respectively. From 1985to 1989 he was a Research Engineer with the Nokia Research Center.Since 1990 he has been with the Electromagnetics Laboratory, HelsinkiUniversity of Technology. His research interests are electromagneticfield theory, complex media and microwave engineering.

07.0310212.m.viitanen

Documents