1

ELECTROMAGNETIC EFFECTS OF METAMATERIALS WITH NEGATIVE PARAMETERS

Filipa Isabel Rodrigues Prudêncio

Instituto Superior Técnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal

E-mail: pipapru@hotmail.com

ABSTRACT

Unusual propagation characteristics of metamaterials are studied, namely the negative refraction, the antiparallel poynting vector and wave vector, creating backward waves and the anomalous dispersion in DNG metamaterial interfaces.

The guided electromagnetic propagation in planar structures containing both DPS and DNG materials is approached. The DNG interface will logically require a dispersive model, such as the Lorentz model, which is also discussed. Neglecting losses in the dispersive model may lead to unphysical solutions. The numerical results for negative refractive index media show that the dispersive model for the permeability and the permittivity must include losses. The DNG dielectric slab shows the existence super slow modes and mode bifurcation.

Equally interesting, the electromagnetic wave propagation in DNG waveguiding structures based on the H-guide are also introduced in the present dissertation. Finally, the double-slab DNG/DPS H-guide and the H-guide directional coupler are addressed, exhibiting both co-directional and contra-directional coupling effects. Keywords: Double negative media, Complex media, Metamaterials, Negative refraction, Planar waveguides, H-guides

1. INTRODUCTION

Nowadays, artificial electromagnetic materials with effective negative permeability and permittivity have attracted the attention of the electromagnetic community. This new class of composite materials with extraordinary electromagnetic properties cannot be found in nature but can be artificially achieved.

In 1968, the concept of a bianisotropic medium [1] was coined by Cheng and Kong [2]-[3] defining a medium with the most general linear constitutive relations. In microwaves, bianisotropic media [4] are conceived as artificial structures.

The history of complex media, with negative permittivity and permeability, starts with the concept of “artificial” materials in 1898, when Lagidis Chunder Bose developed the first microwave experiment on twisted structures. Currently, these elements immersed in a host medium are denominated by artificial chiral medium. Karl Ferdinand Lindman, in 1914, had studied the wave interaction with collections of randomly - oriented small wire helices, in order to create an artificial chiral media.

Later in 1948, Kock, made lightweight microwave lenses by combination of conducting spheres, strips periodically and disks. These metamaterials, built for lower frequencies, can be designed for higher frequencies by length scaling. During the 1960s, the idea of negative refraction first arose when a physicist, Veselago, considered the optical properties of an imaginary material. In 1967, he investigated the plane wave propagation in a material which permittivity and permeability were simultaneously negative [5].

In 1999, Pendry described how he adjusted the array’s properties and he developed an array with negative permeability. This structure consisted of periodic array of split-ring resonators (SRRs) [6] that expressed negative effective permeability over a narrow frequency band. For metamaterials with negative permittivity and permeability, some terminologies have been proposed, such as “left-handed” media, “back-ward wave media”(BW media) and “double negative (DNG)”, just to name a few.

In April 2001, Smith and his colleagues constructed a composite medium for the microwave regime, and announced the experimental evidence of an unusual form of refraction.

Many research groups all over the world are now studying various aspects of this class of metamaterials and suggesting future applications. Perfect lenses, or the creation of acoustic metamaterials are just some examples. Recently, there has been a growing interest in the theoretical and experimental study of metamaterials.

A recent interest in the design of metamaterials is the creation of an object invisible to radar to be experimentally demonstrated in the near-term. Physicists and engineers know that the ability to control the properties of metamaterials can be exploited to develop the refractive index profile needed to make an object invisible by bending the electromagnetic radiation.

2

Pendry thought to make things invisible [7]-[8] as a rubber sheet around the object to be concealed. The permittivity and permeability, in tangential directions, along the surface of the cloak, remain finite, then, the electromagnetic waves have no problem passing around the object [9]-[10]. Several aspects of metamaterials have been published [11]-[16] in the literature and new suggestions are being studied. This means that the march of scientific progress will lead to further advances.

2.CHARACTERISTICS OF METAMATERIALS

Considering a DNG metamaterial characterized by two complex constitutive parameters, permittivity ε, and permeability µ, which are described by the following relations

€

ε = ε'+iε′′ (1)

€

µ = µ'+iµ′′ (2)

with

€

ε′,ε′′,µ′,µ′′ ∈ ℜ . The left-hand rule can be easily understood from the Maxwell’s equations in the differential form,

€

∇×E =∂B∂t

∇×H = −∂D∂t

, (3)

and the two constitutive relations, which describe the response of the medium to the applied fields,

€

B = µ0µ H

D = ε0ε E, (4)

In the time-harmonic regime, the following relations

between operators can be derived:

€

∂

∂t→−iω and

€

∇→ ik .

The signs of the Poynting vector,

€

S, and of the wave vector,

€

k , are computed using the following relations which govern the spatial orientation of the electric and magnetic vectors, E and H, as it is shown in the Figure 1,

€

k ×E =ωµ0µ H

k ×H = −ωε0ε E. (5)

Figure 1 Triplet vectors

€

E0 ,H0 ,ℜ(k)[ ] and

€

E0 ,H0 ,Sω[ ] to the

DPS and DNG media.

According to the triplet vectors in a DNG medium, vectors

€

k and

€

S have opposite orientations, which creates a backward wave. It means that the electromagnetic wave and the electromagnetic energy have opposite directions. In the same way, it easily to see that, in a DPS medium, the vectors

€

k and

€

S have the same orientation, providing a forward wave. On the DNG medium the triplet vector

€

E0 ,H0 ,ℜ(k)[ ] is left, while, in the DPS medium, the same triplet vector is right. On the other hand, the triplet vector

€

E0 ,H0 ,Sω[ ] is right in both media.

2.1. Dispersion

Electromagnetic theory shows that, for a certain isotropic medium, characterized by permittivity ε and permeability µ, the mean values of the electric and magnetic energy densities,

€

We and

€

Wm are, respectively, given by

€

We =14εε0E ⋅E * , (6)

and,

€

Wm =14

µµ0H ⋅H * . (7)

Since ε and µ are both negative in the case of DNG metamaterials, it is easily proven that the previous expressions should not be used, because the DNG medium is necessarily dispersive. Therefore, the previous time-domain electric and magnetic energy densities, (6) and (7), become invalid and new expressions have been proposed for lossless dispersive media,

€

We =∂D∂t′−∞

t∫ Edt′ , (8)

and,

€

Wm =∂B∂t′−∞

t∫ Hdt′ , (9)

where the electric and magnetic fields are almost monochromatic, such that,

€

E(t) = E exp(−vt) and

€

H(t) = H exp(−vt) with a slowly variation in the period

€

T =2πω

, it means

€

v <<ω . The previous expressions, (8)

and (9), show that

€

We and

€

Wm are positive.

€

E0

€

E0

€

Sω

€

Sω

€

H0 €

ℜ k}{

€

H0

€

E0

€

ℜ k}{

€

E0

3

2.1. Negative refraction

As already seen, metamaterials must exhibit complex permittivity and permeability, and these contitutive parameters may be written as

€

ε = ε eiθε , (10)

and

€

µ = µ eiθµ , (11)

where the permittivity and the permeability phase are confined to

€

0 <θε < π and

€

0 <θµ < π .

The refractive index,

€

n = εµ has also a complex form

€

n = n eiθn (12)

with

€

0 <θn < π . As one well knows, in DNG metamaterials the real parts of

ε and µ are both negative. In the passive materials the

imaginary part of the refractive index

€

n′′ , permittivity

€

ε′′and permeability

€

µ′′, are positive, as depicted in the

Figure 2.

Figure 2 Graphical interpretation of the negative refraction. Combining the last three equations, the refractive index phase assumes the following expression

€

θn =12θε +θµ + 2πm( ) , (13)

where

€

m can be zero (solid line), or one (dashed line) – see Figure 2. Only negative

€

n′ satisfies the positive

€

n′′ , in order to obtain a physical solution. The negative real part of refractive index is one of the most unusual properties of the DNG metamaterials. Also, the real part of wave impedance,

€

z′, must be positive to lead a physical condition. From another point of view, and using the

equations, (1), (2) and

€

n = εµ , the negative sign of the real part of the refractive index can be proved as follows.

€

n2 = ε µ( )2 = εµ = (ε′+ iε′′)(µ′+ iµ′′) =

= ε′µ′+ iε′µ′′+ iε′′µ′ − ε′′µ′′. (14)

On the other hand,

€

n2 = n′+ in′′( )2 = (n′)2 + (in′′)2 + 2(in′n′′) . (15)

Comparing the last two equations, one can obtain

€

i(ε′µ′′+ ε′′µ′) = 2i(n′n′′) . (16)

Now assuming negative real parts

€

ε′ and

€

µ′ , and the positive imaginary parts

€

ε′′ ,

€

µ′′ and

€

n′′ , then the sign of the real part of refractive index,

€

n′, must be negative, as shown before. 3. PROPAGATION OF ELECTROMAGNETIC

WAVES IN DNG GUIDES

The modal characterization of some planar structures involving metamaterials DNG is explained, such as DPS-DNG interface and DNG dielectric slab. Using the propagation of complex structures that containing DNG and DPS media, new physical effects are obtained, which develop new solutions.

3.1. DPS-DNG interface

An interface between the DPS medium and the DNG medium is analyzed, where the

€

z -axis represents the direction of the propagation,

€

x -axis the transverse direction and

€

y -axis the transverse infinite direction where the electric

and the magnetic fields verify the condition,

€

∂

∂y= 0 .

From unlimited media, the transverse electric field

€

Ey , for the TE modes, can be written as

€

Ey =Aexp(−α1x)exp(ikzz), x > 0 Bexp(α2x)exp(ikzz), x < 0

(17)

where

€

α1 and

€

α2 represent, respectively, the transversal attenuation constant in the DPS medium and in the DNG medium, which are given by the following relations

€

α12 = neff

2 − ε1µ1, (18)

and,

€

α22 = neff

2 − ε2µ2 . (19)

Using now, the boundary conditions, the modal equation for the TE modes is expressed as

€

α2α1

= −µ2µ1 .

(20)

Using a similar procedure for the TM modes, it results

ε

µ €

ℑ

€

ℜ

z

n

4

€

α2α1

= −ε2ε1 .

(21)

Therefore, valid propagating solutions are obtained from both modal equations, which does not happen in a DPS/DPS interface. Moreover, when using the Lorentz model as a frequency dispersive model, the modal equation exhibits more than one single solution. In this model, the variation of the permittivity and permeability as a function of frequency, assume the following relations

€

ε(ω) = 1+ω pe2

ω0e2 − iΓeω −ω

2 (22)

and,

€

µ(ω) = 1+ω pm2

ω0m2 − iΓmω −ω

2

(23)

Figure 3 shows the lossless dispersive model. It expresses the magnetic permeability variation

€

µ2 (ω) and the electric permittivity variation

€

ε2 (ω) . Also, the refractive index variation

€

n(ω) is depicted in Figure 4. It is clear that

€

n(ω) is only real when the magnetic permeability

€

µ2 (ω) and the electric permittivity

€

ε2 (ω) have negative or positive values, simultaneously. Note that, there is a negative refraction in the same frequency range, where the parameters

€

ε2 (ω) and

€

µ2 (ω) are both negative.

There is a ENG region in the lossless dispersive model, where the permeability is positive and the permittivity is negative. The refractive index becomes purely imaginary.

Figure 3 Lossless dispersive model for

€

ε and

€

µ .

Figure 4 Lossless dispersive model for

€

n .

In Figure 5 the dispersion model in the lossless case is depicted. The real part and the imaginary part of effective refractive index,

€

neff (ω) , for TE mode is addressed.

€

neff ω( ) =

ε2 ω( )µ2 ω( )−µ2 ω( )µ1 ω( )

2

ε1 ω( )µ1 ω( )

1−µ2 ω( )µ1 ω( )

2 (24)

In this case, one easily verify that the effective index of refraction,

€

neff (ω), only has real part.

From equation (17), the transverse attenuation constants

€

α1 and

€

α2 , must be always positive, otherwise, the electric field

€

Ey , tend to infinite with an increasing distance to the interface, which is unphysical. Using the modal equation (20), the variables

€

α1and

€

α2 change from real to imaginary exactly at the same point, as shown in Figure 6. From equation (18), the constant

€

α1 is real when the effective index of refraction,

€

neff (ω) , is greater than one. In the same way,

€

α1 is purely imaginary when the effective index of refraction,

€

neff (ω) , is lower then one.

DNG ENG DPS

DNG ENG DPS

5

Figure 5 Dispersion diagram for TE mode in the lossless case.

Figure 6 Variation of

€

α1 and

€

α2 as a function of frequency, for TE mode. Then, from modal equations (20) and (21), when

€

ε2 ω( )ε1 ω( )

= −1 and

€

µ2 ω( )µ1 ω( )

= −1 , for TE mode and TM mode,

respectively, the effective index of refraction,

€

neff (ω), goes to infinite at a frequency which is not the resonance frequency. This is an unphysical behavior.

Considering now, a lossy dispersive model, with

€

Γe = Γm . The Figure 7 shows the magnetic permeability

variation

€

µ2 (ω), the electric permittivity variation,

€

ε2 (ω) for a lossy dispersive model, while the refractive index variation

€

n(ω) for the same model is depicted in Figure 8. The imaginary part of

€

µ2 and

€

ε2 are positive, due to using dispersive media.

Figure 7 Lossy dispersive model for

€

ε and

€

µ .

Figure 8 Lossy dispersive model for

€

n . For TE mode, the index of refraction

€

n(ω) does not tend to infinite, and so, there are physical results, as shown in the Figure 9.It is important to note that the imaginary part of

€

neff exhibits a considerable increase in the frequency range where, before, there was an unphysical solution. In the same way,

€

leff has a negligible values in the frequency range where, previously, there was a physical solution.

Figure 9 Dispersion diagram for TE mode in the lossy case.

Results for the attenuation constants

€

α1 and

€

α2 , for both TE modes, are depicted in Figure 10.

DNG ENG DPS

DNG ENG DPS

6

Figure 10 Variation of

€

α1 and

€

α2 as a function of frequency, for TE mode.

For TM modes, one uses the same procedure to construct the dispersion diagram and the variation of the attenuations constants as a function of frequency.

3.2. DNG dielectric slab

The modal solution of this structure can be divided into TE and TM modes. Applying the boundary conditions to the tangential field components, the following relation can be derived for the even TE mode

€

w =µ1µ2

u tan(u) , (25)

and also for TE odd mode,

€

w = −µ1µ2

ucot(u) , (26)

where

€

w = α1d and

€

u = k2d . On the other and, the relation between the normalized wavenumbers is given by

€

u2 + w2 = v2 , (27)

where

€

v = k0d ε2µ2 − ε1µ1 (28)

The TM modes can be defined in the same way. Then, the corresponding expressions for TM odd mode

€

w =ε1ε2u tan(u) , (29)

and for TM even mode

€

w = −ε1ε2ucot(u). (30)

3.2.1 Surface modes

Considering in this subsection,

€

u = −iU , with

€

U ∈ ℜ , where the phase velocity is defined by

€

vp =ω

β (31)

is smaller than the speed of light in the outer unlimited medium. In this case, the modes are termed as super slow modes. Replacing the variable,

€

u = −iU , in the equation (27), yields to

€

w2 =U2 + v2 . (32)

Also, rewriting the modal equations (25) and (26), one gets

€

w = −µ1µ2

U tanh(U) , (34)

and,

€

w = −µ1µ2

U coth(U) . (35)

In the Figure 11, the curves for the modal equations of the TE modes for a DNG dielectric slab in and the curve of equation (27) and (32) are depicted. The modal solutions are given by the interception of these two graphics. The positive abscissa semi-axis expresses the transversal wavenumber,

€

u , as real, while in the negative direction the imaginary part of,

€

u , is represented.

Figure 11 The modal solutions, where

€

ε2 = −2,µ2 = −2 .

One should note that, due to the DNG medium, the sign of the right hand side of the TE modal equations changes due to

€

µ2 < 0 . This causes a change in the slope of the branches of the tangent and cotangent function. Also, this inversion inserts two solutions for a slow mode. The respective dispersion diagram for the TE modes for the DNG dielectric slab is shown in Figure 12. The electric fields

€

Ey of the even slow modes vary according to cosine, while those of the odd slow modes vary according to a sine function. However, if the transverse wavenumber is imaginary, these trigonometric functions become hyperbolic functions, respectively. Note that, when if the outer medium is less dense that inner medium there are slow modes and, given a DNG metamaterial inner medium also arise super slow modes

€

w

€

u

€

U

TE even mode TE odd mode TE odd mode

€

k0d = 2.7k0d = 0.5

7

Figure 12 Dispersion diagram for TE modes of a DNG dielectric slab characterized by

€

ε2 = −2,µ2 = −2 . The straight line, with a greater slop, corresponding to

€

k0d =βd

ε2µ2 expresses the transition between the super

slow modes and the slow modes. On the other hand,

€

w = 0 ,

it corresponds to the cutoff, or

€

k0d =βd

ε1µ1.

In the case of a less dense inner medium, the dispersion diagrams can be divided in two different cases:

€

µ2 > µ1, in Figure 13 and

€

µ2 < µ1 , in Figure 14.

Figure 13 Dispersion diagram for TE mode characterized by

€

ε1 = 2,µ1 = 1,ε2 = −0.8,µ2 = −1.3 .

Figure 14 Dispersion diagram for TE mode characterized by

€

ε1 = 2,µ1 = 2,ε2 = −1.5,µ2 = −1.5 .

The first figure shows an even super slow mode and an even slow mode, with a null cutoff frequency. In second case, only the even super slow mode propagates, which is limited by frequency. In this frequency range, there are always two solutions that tend to same point.

4. PROPAGATION OF ELECTROMAGNETIC WAVES IN H-GUIDES AND H-GUIDE COUPLERS

The electromagnetic wave propagating in waveguiding structures H-guide is addressed, containing both DPS and DNG materials. Assuming that these materials are homogeneous, isotropic and exhibit losses.

4.1. The DNG H-guide

The propagation characteristics of an H-guide are analyzed. Considering the constant b as the parallel plat separation

while 2l is the slab thickness and

€

ζ =b

l is the aspect ratio.

The operation of the H-guide is divided in two

regimes: the closed-waveguide regime, when

€

b

λ< 0.5 and

the open-waveguide regime, when

€

b

λ> 0.5. The modes

propagating are hybrid and have five field components to the DNG or DPS media. Classifying these modes in two families: longitudinal-section electric (LSE) modes and longitudinal-section magnetic (LSM) modes. The modal equations for the even and odd

€

LSMmn modes, respectively, can be written as

€

εα + h tan(hl) = 0 , (36)

and

€

k0d

€

βd

€

k0d

€

βd

€

βd

€

k0d

8

€

εα − h cot(hl) = 0 , (37)

where

€

α2 = k2 + ky2 − k0

2 ,

€

h2 = εµk02 − k2 − ky

2 and also

€

k0 =ω ε0µ0 ,

€

ky = nπ

b, with

€

n being an integer. The

mode cutoff in the open-waveguide regime is obtained by

€

α = 0 , while in the closed-waveguide, it is obtained by

€

β = 0. The operational diagram can be shown in Figure 15.

Figure 15 Operational diagram for a H-guide, containing both DNG and DPS materials. The curves of the DNG H-guide and DPS H-guide do not

have significant differences, when

€

b

λ> 0.5, that is, in the

open-waveguide regime. However, in the closed-waveguide regime, one finds multiple points of operation for the same mode, for the DNG H-guide. The corresponding dispersion

diagram is shown in Figure 16. The case

€

b

λ= 0.4 , includes

the closed-waveguide regime.

Figure 16 Dispersion diagram for

€

b

λ= 0.4 .

There is a dashed line for

€

h = 0 (upper line). It is easy to verify, in the dispersion diagrams, how many modes are

propagating above cutoff for certain value of

€

l

λ.

Also, when the value of

€

l

λ is increased above

some critical point, where the derivate of

€

β is infinity, two solutions arise. Also, the mode bifurcation is verified. The electromagnetic field inside the DNG slab and the field outside are such that the total power flowing along the waveguide is null.

4.2. Double-slab DNG/DPS H-guide

Two juxtaposed dielectric slab, a conventional DPS slab and

a DNG slab form the double-slab DNG/DPS H-guide. The

total thickness of this structure,

€

l = l1 + l2 , is formed by the

DPS slab thickness

€

l1, and the DNG slab thickness

€

l2 ,

while

€

ξ =l2l1

. Having

€

h1 = h2 = h , the modal equation can

be factorized as

€

h + εα tan hl2

εα − h tan

hl2

= 0 . (38)

Note that, there are no even or odd modes propagating.

In Figure 17, the operational diagram is shown for two

different values of

€

ξ .

Figure 17 Operational diagram for

€

ξ = 0.25 and

€

ξ = 4 . The double-slab DNG/DPS H-guide has

€

ε1 = µ1 = 2 to the DPS slab and

€

ε1 = µ1 = 2 to the DNG slab. The last one, in the closed-waveguide regime, is the main influence in the

9

structure if the value of

€

ξ is grater than one. Otherwise, is

€

ξ is lower than one, the main influence is the DPS slab.

4.3. DNG/DPS H-guide directional coupler

The DNG/DPS H-guide directional coupler is addressed. The DPS slab and the DNG slab separated by a distance

€

2s form that structure. In Figure 18, the operational diagram is depicted for a DNG/DPS H-guide directional coupler, having

€

ε1 = µ1 = 2

and

€

ε2 = µ2 = −2 , for

€

ξ = 2 and

€

sl

= 0.4 .

Verifying the coupling regions in the crossing points between the dispersion curves of the isolated DPS and DNG H-guides.

Figure 18 Operational diagram for the DNG/DPS H-guide directional coupler.

5. CONCLUSIONS

The possibility of using complex media, such as DNG metamaterials, in these structures may generate new electromagnetic properties that can be applied in the design of new microwave and millimeter-wave devices.

In fact, the consequences of having negative values for the real part of the permittivity ε, and the permeability µ in DNG metamaterials, provides new properties such as the negative real part of refractive index and the anomalous dispersion. In a DNG medium, the electromagnetic wave phase and energy have opposite direction, which creates a backward wave. This concept cannot be verified in any conventional DPS media. In this case, the electromagnetic wave phase and energy have the same orientations and forward waves are presented.

The anomalous dispersion in DNG medium is also studied. Using the general electromagnetic theory, the expressions for the mean values of the electric and magnetic energy densities must be corrected for a DNG medium. The

conventional energy relations must be replaced because the DNG medium is necessarily dispersive, and the electromagnetic energy densities would be negative. The study of the DPS-DNG interface reveals a new form of propagation: an interface mode. This structure provides the existence of the super slow modes, where can exhibits a double solution in certain limited frequency ranges. When analysing the DNG dielectric slab dispersion diagrams, several important effects can be reported. In the case where the slab is less dense then the outer medium,

€

µ2 < µ1, there are only super slow modes propagating in the structure, odd and even, in an unlimited frequency band. In the reverse case,

€

µ2 > µ1, when the slab is more dense then the outer medium, common slow mode solutions start to exist. Moreover, there may be single super slow mode solutions and double solutions. In the final section, the structures H-guide are studied, using both DNG and DPS materials. Several effects, where the common DPS slab is replaced by a DNG slab, were put in evidence, as well, the mode bifurcation, anomalous dispersion and the existence of super-slow modes.

In the double-slab DNG/DPS H-guide, one concludes that, in the closed waveguide regime, the thicker slab dictates the dispersion characteristic.

Also, the co-directional and contra-directional mode coupling in DNG/DPS H-guide directional couplers is verified. It can be applied in the design of new devices.

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