Abstract— The main goal of this dissertation is to carry out a

modal analysis of waveguides containing DPS and/or DNG media.

The DNG metamaterials are the object of great interest in the

scientific community because of their potential and extraordinary

properties. This kind of materials has several applications, such as

in microwaves, antennas for mobile communications, optical devices

and invisibility cloaking devices.

In the first part of this work propagation in unbounded media is

addressed, with special emphasis on the DNG metamaterials. The

main property of these materials is its negative refraction index.

Another important feature of these materials is the propagation of

backward waves due to the opposite directions of the Poynting and

the wave vectors.

The second part of this work is to develop a modal analysis of

three different structures, including DPS media. All the structures

contain two different dielectrics with distinct refractive indexes. The

structures studied were conductor-backed dielectric slabs, dielectric

shielded slabs, and optical fibers. The modes of propagation in these

structures were studied and their dispersion diagrams were

obtained.

Finally, in the third part, wave propagation in structures

containing DNG media is studied, namely, a DPS-DNG interface

and a DNG slab. For the DPS-DNG interface case, a Lorentz

dispersion model with losses was used, in order to obtain solutions

with physical meaning. In the case of a DNG slab, super slow modes

and slow modes are found to propagate.

Key words— Double positive medium, Double negative

medium, Waveguides, Metamaterials, Modal analysis,

Microwaves and Photonics.

1. INTRODUCTION

n the beginning of the eighteenth century, a phenomenon that

related electricity and magnetism was observed by Oersted.

Nevertheless, he could not find a physical explanation for this

observation. After Oersted’s experiences, André Marie Àmpere

contributed to the progress in the electromagnetics

phenomenology field, presenting several studies at the Academy

of Paris. Àmpere formulated a mathematical description of the

magnetic force between electric currents, known as Àmpere’s

law [1]-[2]. A few years later, Henry and Faraday reported

simultaneously and independently the electromagnetic induction.

Faraday idealized experiences which proved electromagnetic

induction and the lines of force concepts. All of Faraday’s

experiences inspired several researchers, such as James Clark

Maxwell. In 1873, in the publication of A Dynamical Theory of

the Electromagnetic Field, Maxwell unified two major scientific

theories: optics and electromagnetics [3]. In fact, Maxwell

proved that the light is an electromagnetic phenomenon. He

demonstrated that the velocity of the light was defined by electric

permittivity and magnetic permeability. However, in 1905,

Albert Einstein proposed the special relativity theory to

demonstrate that Maxwell’s theory was not enough to describe

all of electromagnetics phenomenona. Einstein assumed that the

velocity of light is constant for every referential. More recently, a

physic and mathematical perspective of the constitutive relations

and their relationship with classical electromagnetic theory was

described in a treaty know as Foundations of Classical

Electrodynamics: Charge, Flux, and Metric [4]. In 1963,

Sheldon Glashow proposed that weak nuclear force, electricity

and magnetism could derive from a partially unified electroweak

theory. Four years later, Abdus Salam and Steven Weinberg

confirmed Glashow’s theory and these three researchers were

awarded the Nobel Prize in 1979 [5].

A dielectric is a nonconductor of electric current. Waveguide is a

structure that guides electromagnetic waves between two points.

There are several kinds of waveguides, such as metallic and

dielectric waveguides and optical fibers. The concept of

waveguides was proposed for the first time in 1893 by J. J.

Thomson. However, it was Oliver Lodge who tested them

experimentally a year later. In 1897, Lord Rayleigh studied the

propagation of electromagnetic waves in a cylindrical metallic

waveguide. Thirteen years later, Hondros and Debye obtained

solutions for the modal wave propagation for a nonmetallic

circular cylindrical dielectric waveguide [6]. In 1936, G. C.

Southworth and W. Baarrow were interested in longer wave

lengths, such as the propagation of microwaves in dielectric

wires and dielectric rod antennas. They were pioneers in getting

exprimental results for waveguides at Bell Labs in Holmdel, the

predecessor of the current Crawford Hill Labs. In the early 40’s,

classical textbooks about electromagnetic waves propagation

were published, such as Electromagnetic Theory and

Electromagnetic Waves by J. A. Stratton and S. A. Schelkunoff,

respectively. These books contained chapters discussing the

detailed solutions for the propagation modes of circular dielectric

guides, as well as metallic waveguides in terms of the Bessel and

Hankel functions. A few years later, hollow metallic waveguides

had a great impact in microwaves applications, such as radars

and microwave radio transmissions [7]. In 1950, dielectric

waveguides started be studied as antennas. Three years later,

Kiely published a theoretical analysis for dielectric antennas. In

this decade, Zucker and Collin also published a book called

Antenna Theory. In the late 50’s, dielectric waveguides and

optical fibers started being considered for optical applications. In

1960, Snitzer and Osterberg observed the mode patterns of fibres

in the visible spectrum. A few years later, Theodore Maiman

demonstrated the operation of a laser. In fact, the laser would

revolutionize telecommunications [8].

In the 60’s, the attenuations of fibre optics were 1000 dB/km

therefore, and therefore these were unavailable for

telecommunications. In 1970, Charles Kao and George Hockman

Modal Analysis of Waveguides Containing DPS

and/or DNG media

João Filipe da Costa Pardal Maurício, Instituto Superior Técnico

I

proposed an optics telecommunication system with losses of less

than 20 dB/km. Since then, low-loss fibres were developed and

optics telecommunications systems are nowadays widely used. In

2010, a 64Tb/s transmission with 320km between optic

amplifiers was achieved [9].

Nowadays, fibre optics telecommunications systems are strongly

consolidated and remain the subject of investigation by the

scientific and technologic communities.

Metamaterial is an artificial material with negative electric

permittivity and magnetic permeability . This class of

materials cannot be found in nature and have extraordinary

electromagnetic proprieties [10]. In 1898, Lagidis Chunder Bose

developed the first microwave experiment on twisted structures.

It was the first time that the “artificial” materials concept was

experimented. Currently, the immersions of these structures in a

host medium are known by artificial chiral medium [11]. In

1914, Karl F. Lindman studied wave interaction with

compilations of randomly oriented small wire helices in order to

create an artificial chiral media [12]. A few years later, Winston

E. Kock made lightweight microwave lens by combination of

conducting spheres, periodical strips and disks. These

metamaterials were planned for lower frequencies and can be

adapted for high frequencies by length scaling. In 1967, Victor

Veselago published theoretical analysis about plane waves

propagation in a material with negative permittivity and

permeability. In this study, Veselago demonstrated that the

direction of the Poynting vector is antiparallel to the direction of

phase velocity for a monochromatic uniform plane wave in this

kind of media [13]. Later in the 90’s, John Pendry and his

colleagues began to produce structures with this kind of

proprieties. Pendry created an array of closely spaced, thin,

conducting elements such as metal hoops, in order to develop

materials with negative permittivity [14]. In 1999, he developed

an array with a periodic array of split-ring resonators (SRRs) that

expressed negative permeability for a certain frequency band

[15]. The combination of the metal hoops array with SRRs

horizontally positioned created the metamaterial DNG. In 2001,

David R. Smith and his colleagues reported the experimental

demonstration of functioning electromagnetic metamaterials

stacking, periodically, split-ring resonators and thin wire

structures [16]. For metamaterials with negative permittivity and

permeability, some designations have been proposed such as

BWM (Backward Wave Media), LFH (Left-Hand media) and

DNG (Double Negative Media). Currently, metamaterials have

aroused much interest for the researchers and scientific

community due to the potentiality of several applications such as

super lens, invisible cloaks, antennas for mobile communications

and some microwaves devices.

2. PROPAGATION IN UNLIMITED MEDIA

The materials’ proprieties can be described by two complex

constitutive parameters, permittivity and permeability . In

general these parameters are function of frequency, i.e.,

( ) and ( ) with , and can be written as

follows:

- ' '', with ', ''i ;

- ' '', with ', ''i .

Figure 1 illustrates the medium classification based on the

relationship between ' and ' [17].

Figure 1. Media classification.

In order to describe the propagation of electromagnetic waves,

Maxwell’s equations are written in the differential form:

t

t

BE

DH

(1)

The two constitutive relations, which describe the response of the

medium to applied fields is expressed by:

0

0

B H

D E (2)

In the time harmonic regime, the following relations between

operators can be derived: t i and i k . In order to

get the orientations of Poynting vector S and wave vector k the

previous relations are used, which set the spatial orientation of E

and H [13]:

0

0

k E H

k H E (3)

Figure 2. Spatial orientation of electric field, magnetic field,

Poynting vector and wave vector in DPS and DNG media.

As illustrated in figure 2, in DNG medium, wave vector k and

Poynting S vector have opposite direction, which creates a

backward wave. It means that electromagnetic waves and

electromagnetic energy have opposite directions. However, in

DPS medium, these vectors have the same direction, providing a

forward wave. On the other hand, in DNG medium, the triplet

vector 0 0[ , , ( )]E H k is left, while, in DPS medium, the triplet

vector is right. However, the triplet vector 0 0[ , , ]E H S is right

in both media. So, DNG media is usually knwon by LHM (Left-

Hand Media) and BWM (Backward Wave Media).

2.1 Dispersion

For a certain isotropic medium characterized by permittivity

and permeability , the mean values of the electric energy and

magnetic energy, respectively, e

W and m

W , which are

given by:

*

0

*

0

1

4

1

4

e

m

W

W

E E

H H

(4)

For a DNG metamaterial with 0 and 0 , the previous

expressions can’t be applied, because they would lead to negative

values for e

W and m

W . However, for a dispersive medium

the expressions defined in (4) should be replaced by the

following:

*0

*0

[ ( )]( ) ( )

4

[ ( )]( ) ( )

4

e

m

W t t

W t t

E E

H H

(5)

When the electric and magnetic fields are almost

monochromatic, such that ( ) exp( )t vt E E and

( ) exp( )t vt H H with slowly variation in the period 2T ,

it means v . Through expressions (5) it is concluded that, a

DNG medium has to be necessarily dispersive, otherwise, the

energy stored in the electromagnetic field would not be positive

[18]. Considering phase velocity p

v and group velocity g

v are

given by:

p

g

vk

vk

(6)

And defining wave number by 0( )k nk n c the following

relation was obtained:

1 1

g f

n

v v c

(7)

The previous expression implies that group velocity and phase

velocity are equal, if the index of refraction n does not vary with

frequency . It means that medium is not dispersive. From the

following equations, obtained for an isotropic media, it is easily

verified, for a DNG medium, that both velocities have opposite

directions, because 0g p

v v . For dispersion study it is important

to introduce a dispersive model, such as the Lorentz model. So:

2

2 2

0

2

2 2

0

( ) 1

( ) 1

pe

e e

pm

m m

i

i

(8)

Where 0 ,e m

represents the resonance frequencies, ,e m

expresses

the collision frequencies and ,pe m

designed the plasma

frequencies. Through this model it is possible to determinate a

certain frequency range 1 , 2 ,

[ , ]e m e m

, where the parameters

( ) and ( ) exhibit a negative real part:

1

2

1

2

e e

e b

m a

m

( ) 0 [ , ]

( ) 0 [ , ]

b

a

(9)

The following figures illustrate the variation of real and

imaginary parts of the index of refraction, ( )n and ( )n , with

the wavelength using Lorentz model. Through figures 3 and 4,

it is verified that the range where index of refraction is negative

does not correspond exactly to the interval where material is

DNG. This means that a NIR (Negative Index of Refraction)

media is not necessarily a DNG media.

Figure 3. Variation of the real part of the index of refraction with

wavelength, using Lorentz model.

Figure 4. Variation of the imaginary part of the index of refraction

with wavelength, using Lorentz model.

2.2 Negative Refraction

The negative refraction phenomenon can be demonstrated by

considering the scattering of a wave that falls obliquely on the

interface DPS-DNG, such as illustrated in figure 5 [10].

Applying Snell law’s on interface at 0z , it is obtained that:

1 1

2sgn( ) sin sin( )

2trans inc

nn

n

(10)

This proves that negative refraction exists in 0x , since

0trans

.

Figure 5. Geometry of a wave that falls obliquely on the interface

DPS-DNG.

The fact that refraction is negative arises from imposition of

electromagnetics boundary conditions on the interface that

separates two medium. The wave vectors and Poynting vectors

associated with this phenomenon are described by:

1

1

2

ˆ ˆ(cos sin )

ˆ ˆ( cos sin )

ˆ ˆ(cos sin )

inc inc inc

refl inc inc

trans trans trans

k

k

k

k z x

k z x

k z x

(11)

2

0

1

2

0

1

2

0

2

1ˆ ˆ(cos sin )

2

1ˆ ˆ( cos sin )

2

1ˆ ˆ(cos sin )

2

inc inc inc

refl inc inc

trans trans trans

E

RE

E

S z x

S z x

S z x

(12)

Where 1

and 2

represents wave impedance in media 1 and

media 2, respectively. If the wave propagates in DPS media,

wave vector and Poynting vector have the same direction.

Nevertheless, if the wave propagates in DNG media, the index of

refraction is negative and from Snell law’s it can be obtained

that:

2ˆ ˆ(cos sin )

trans trans transn

c

k z x (13)

2

0

2

1ˆ ˆ(cos sin )

2trans trans trans

E

S z x (14)

This means that the wave vector and the Poynting vector have

opposite directions. However, these results were obtained for

( ) 0n solution. If ( ) 0n solution was adopted, Snell law’s

for a DNG medium would be the same for a DPS medium. For a

wave vector k , this would mean 0trans

and the power flux in

medium 1 would drive up to the interface.

3. MODAL ANALYSIS OF WAVEGUIDES ON DPS

MEDIA

A DPS (Double Positive) medium corresponds to the situation

that 0 and 0 . In this section three waveguides with

different geometries are analyzed. These structures are: open

dielectric slab, shielded dielectric slab and optical fibers.

Depending on electromagnetic field longitudinal components (

zE and z

H ),which are zero or nonzero, the guide modes can be

classified by:

- TE (Transverse Electric) modes, if 0z

E and 0z

H ;

- TM (Transverse Magnetic) modes, if 0z

E and 0z

H ;

- TEM (Transverse Electromagnetic) modes, if 0z

E and

0z

H ;

- Hybrid modes, if 0z

E and 0z

H .

3.1 Open Dielectric Slab

The open dielectric slab is based on a perfect electric conductor

(PEC) at 0x and considered infinite along positive x-axis as is

shown in figure 6. The medium 1 contain a dielectric with index

of refraction 1

n and the medium 2 contain a dielectric with index

of refraction 2

n , where 1 2

n n .

Figure 6. Geometry of open dielectric slab.

In this type of structure, as well as in shielded dielectric slab

case, the components of electromagnetic field vary:

exp[ ( )]i z t . In the other hand, for an uniform and unlimited

structure according to the y-axis, the components do not vary,

i.e., 0y . Being the longitudinal propagation constant,

the following relationship is obtained:

2 2 2

1 0 2 2 2 2 2

1 2 02 2 2 2

2 0

( )h n k

h n n kn k

(15)

Where h denotes transverse wavenumber, represents

transverse attenuation constant and 0

k the vacuum wavenumber.

2

0

0

( )E

n xt

H

t

H

E

(16)

According to Maxwell equations (16) and considering boundary

conditions imposed at 0x (PEC) and x d (continuity of the

tangential components), obtained for odd modes sin( )y

E hx or

sin( )y

H hx and even modes cos( )y

E hx or cos( )y

H hx ,

in medium 1. Then only TE odd modes and TM even modes

propagate on an open dielectric slab [19]. The following

dimensionless constants are usually introduced:

2 2 2

2 2

0 1 2

u hd

w d u w v

v k d n n

(17)

Where v represents normalized frequency. Then modal equations

can be expressed as:

- For TE modes: cot( )w u u (18)

- For TM modes:

2

2

2

1

tan( )n

w un

(19)

The dispersion diagrams are usually showed by b as function of

v . Where b denotes normalized index of refraction defined by:

2 2 2 2

2

2 2

1 2

1n nu w

bv v n n

(20)

Where n represents modal index of refraction and is expressed

by:

0

nk

(21)

For each superficial mode: 2 0 1 0

n k n k , so, 2 1

n n n . Then

when 0v (cut-off situation) n tends for 2

n . However, when

v (high frequencies limit) n tends for 1

n . The cut-off

occurs for 0w . Intercepting modal equations (18)-(19) with

circumference given by 2 2 2

u w v on ( , )u w plane, cut-off

frequencies for TE and TM modes that propagate in this slab are

obtained. However, only 0w solutions have physical meaning,

because 0 is always a necessary condition. Then, cut-off

frequencies are expressed by:

( ) 2 with 1,2,3,...c m

v TE m m (22)

( ) with 0,2,4,...c m

v TM m m (23)

The dispersion diagram for the first six modes is shown in figure

7. It is possible to note that the curves have a similar behaviour

for all modes, logically, with different cut-offs for each curve. As

shown in figure 7, this waveguide has null frequency cut-off and

operates in monomodal regime until 2v that corresponds of

second mode cut-off, TE1 mode.

Figure 7. Open dielectric slab dispersion diagram obtained with

1 21.6, 1.4 and 1n n d m .

3.2 Shielded Dielectric Slab

This structure is shielded by two PECs at 0x and x D , as is

shown in figure 3. Again, according to Maxwell equations (16)

and seeing boundary conditions at 0x and x D (PECs) and

x d (continuity of the tangential components), it is obtained

that TE and TM modes propagate in this structure. Modal

equations are defined:

- For TE modes:

cot( ) tan( )s u u s (24)

- For TM modes:

2

2

2

1

tan( ) cot( )n

s u u sn

(25)

Where defines dielectric fill coefficient and is expressed by:

a

d (26)

The normalized constants are expressed by the following

relationships:

2 2 2 2 2 2

1 0 2 2 2

2 2 2 2 2 2

2 0

u n k d du hdu s v

s qd s n k d d

(27)

Figure 8. Geometry of shielded dielectric slab.

The cut-off of set mode is obtained for 0 , that corresponds

to 0n . Fundamental mode TM0 has null cut-off frequency.

For both modes that propagate in this slab, the same equation for

cut-off was obtained:

2

2 1

1

tan tan 02 2

c cv vn

n nn

(28)

Where represents dielectric contrast defined by:

2 2

1 2

2

12

n n

n

(29)

All of modes TE and TM except fundamental mode have a fast

area and slow area. This means:

- 2

If quick area 0t

v v n n ;

- 2 1If slow area

tv v n n n .

Where t

v represents normalized transition frequency and is

expressed by:

- For TE modes: tan( )t t

v v ;

- For TM modes: tan( ) 0 , 1,2,3,...t t

v v m m ;

Once 2

w s for slow zone, modal equations set at (24)-(25)

can be rewritten:

- For TE modes: cot( ) tanh( )w u u w ;

- For TM modes:

2

2

2

1

tan( ) coth( )n

w u u wn

.

When a tends to infinity, slow modes of shielded dielectric slab

tend to superficial modes of open dielectric slab. This means:

- For TE modes: cot( ) if w u u a ;

- For TM modes:

2

2

2

1

tan( ) if n

w u u an

.

In the other hand, when a fast modes TE and TM tend to

radiation modes. Fundamental mode is always slow, so, it is

defined by TM slow modes equation. Although, when 0v the

following approximation is valid: tan( )u u . So:

2 2

2 22 2

2 2

21 12 2 2 22

0 2 1 02

1

tan( ) tan( )

( ) ( )

n nw w u u w u

n nn

n n n nn

(30)

Where 0n represents modal index of refraction when normalized

frequency v tends to cut-off and is expressed by:

0 1 2 2 2

2 1

1n n n

n n

(31)

In two limits situations, i.e., when a and

0 0a , for 0

n obtained:

0 1

0

0 2

lim

lim

a

a

n n

n n

(32)

This means that when 0a , fundamental mode TM0 degenerate

into TEM mode of parallel plane lines filled by homogeneous

dielectric with index of refraction 1

n . So for all values, 0v has

been 2

n n . For this structure, the dispersion diagram is

represented by n as a function of frequency v as it is illustrated

in figure 9.

Figure 9. Shielded dielectric slab dispersion diagram obtained with

1 21.5, 1 and 2n n .

From figure 9, it is easi to distinguish between slow area and fast

area. In fast area the modes exhibit a strong growth. In fact this

area is designated by fast area, because the phase velocity of

guided mode verifies the follow condition: 2p

v c n . In

opposition, in the slow area: 1 2p

c n v c n . From the

dispersion diagram it is still possible to observe that for TE2m-1

and TM2m modes with m=1,2,3… the cut-off frequency is the

same. This means that modal bifurcation exists.

3.3 Optical Fibers

The optical fiber is a cylindrical waveguide filled by two

dielectrics with two index of refraction 1n and 2

n , such as

illustrated in figure 10. The inner and outer medium are called

core and cladding, respectively.

Figure 10. Optical fiber with step profile.

TE and TM modes only propagate in optical fiber when they do

not have azimuthal variation, i.e., 0m . When 0m , boundary

conditions only are satisfied if there are linear combinations of

TE and TM modes both for core and for cladding. Accounting

boundary conditions that assured the continuity of the tangential

components at core-cladding interface, the modal equations are

expressed by:

1

2

2 1

2 2 2 2

2

' ( ) ' ( ) ' ( ) ' ( )

( ) ( ) ( ) ( )

1 1 1 1

m m m m

m m m m

J u K w J u K w

uJ u wK w uJ u wK w

mu w u w

(33)

Where m

J and m

K represent Bessel functions and modified

Bessel functions, respectively [20]. The guided modes can be

classified by:

- Index of azimuthal variation m ;

- Index of radial variation n ;

- Null or non-null of components of the electromagnetic

field along longitudinal propagation axis, i.e., designation

mode (TE, TM or hybrid – HE and EH).

For index of azimuthal variation accentuated two cases:

- When 0m : only occurs for guided modes if non nulls

components of electromagnetic field are , z r

E E and H.

It corresponds to TM0n modes, once 0z

H and the

following modal equation is obtained:

0 01

2 0 0

' ( ) ' ( )0

( ) ( )

J u K w

uJ u wK w

(34)

Or if non nulls components of electromagnetic are

, z r

H H and E. It corresponds to TE0n modes, once

0z

E and the following modal equation is obtained:

0 0

0 0

' ( ) ' ( )0

( ) ( )

J u K w

uJ u wK w (35)

- When 0m : only propagates in hybrid modes HE and

EH, once only these satisfied boundary conditions at core-

cladding interface.

The modal analysis of hybrid modes can be simplified for

weakly-guiding fibers, i.e., 1 . So dielectric contrast can

be approximated by:

1 2

1

n n

n

(36)

This approximation corresponds to 1 2 and modal equations

for hybrid modes can be written as follows:

2 2

' ( ) ' ( ) 1 1

( ) ( )

m m

m m

J u K wm

uJ u wK w u w

(37)

The previous equation (37) has two solutions, one positive and

one negative. The first one corresponds to hybrid mode EHmn and

the modal equation is expressed by:

1 1( ) ( )

0( ) ( )

m m

m m

J u K w

uJ u wK w

(38)

The negative corresponds to hybrid mode HEmn and the modal

equation is defined by:

1 1( ) ( )

0( ) ( )

m m

m m

J u K w

uJ u wK w

(39)

Furthermore, modal equations introduced in (38)-(39) can be

reducted to one equation:

1 1( ) ( )

0( ) ( )

p p

p p

uJ u K ww

J u K w

(40)

Where p is defined such that:

- 1p , for TE and TM modes;

- 1p m , for EH modes;

- 1p m , for HE modes.

From equation (40) it is infered that for weakly-guided fibers, all

modes are characterized by a conjunct of common values for p

and n that satisfied the same modal equation. This means that

modes are degenerated. According to Gloge these modes are

almost linearly polarized and designed by LPpn modes [20]. The

relationship between traditional modes (TE, TM, HE and EH)

and LP modes appears on table 1.

The dispersion diagram for the first twelve modes that

propagates in optical fibers is represented in figure 11. For the

hybrid modes, the fundamental mode corresponds to HE11 mode.

For LP modes it corresponds to LP01 mode. The cut-off of LP

modes concur with 0w , which means v u in (40) equation.

So:

1( ) 0

p cJ v

(41)

Observing figure 11 is trivial that LP01 mode does not have cut-

off frequency and number modes increase with v . For

waveguide operating in monomodal regime, it is necessary to

obtain the cut-off frequency for the LP11 mode, that corresponds

to 2.4048v .

Figure 11. Dispersion diagram for the first twelve LP modes.

4. MODAL ANALYSIS OF WAVEGUIDES ON DNG

MEDIA

An isotropic and unlimited DNG medium is characterized by

having different electromagnetics proprieties and, consequently,

producing new electromagnetic effects that are not verified for

DPS media. Then these new effects are analyzed for a DPS-DNG

interface and a DNG slab.

4.1 DPS-DNG interface

A DPS-DNG interface is shown in figure 12. In this structure, the

electric and magnetic fields verified the following condition,

0y and the fields vary as follows:

1

2

exp( ) exp( ) , 0

exp( ) exp( ) , 0

z

y

z

A x ik z xE

B x ik z x

(42)

Where 1

and 2

represent the transversal attenuation constant

in the DPS medium and DNG medium, respectively, which are

given by:

2 2

1 1 1

2 2

2 2 2

eff

eff

n

n

(43)

Figure 12. Geometry of DPS-DNG interface.

Taking into account the boundary conditions at 0x , the modal

equations are expressed by:

2 2

1 1

2 2

1 1

for TE modes

for TM modes

(44)

These modal equations lead to valid propagating solutions, which

does not happen in a DPS-DPS interface. However, if the

medium was ENG or MNG, only one solution was valid for TE

or TM modes, respectively. In order to analyse the solutions of

modal equations, Lorentz’s model was used as a frequency

dispersive model. Figure 13 represents the variation of

permittivity and permeability as function of frequency f

for a lossless dispersive model. On the other hand, the variation

of real part and imaginary part of index of refraction with

frequency is illustrated in figure 14.

Figure 13. Lossless dispersive model for 2 2 and .

Figure 14. Lossless dispersive model for n .

From figures 13 and 14, it is clear that ( )n is only real when

2( ) and

2( ) parameters have simultaneously negative or

positive values. This corresponds to having a DNG medium and

a DPS medium, respectively. In the other hand, when 2( ) is

negative and 2( ) is positive, the index of refraction is purely

imaginary and there is an ENG region. From modal equation (44)

for TE modes, the effective index of refraction can be expressed

by: 2

22 2 1 1

1

2

2

1

( )( ) ( ) ( ) ( )

( )

( )1

( )

effn

(45)

The dispersion diagram for TE modes in the lossless case is

depicted in figure 15. In this case, it can be verified that the

effective index of refraction ( )eff

n only has a real part. The

variation of transverse attenuation constants 1

and 2

, must be

always positive. It is easily understood from equation (42),

because if these constants do not have this variation, electric field

yE does not decrease with distance and tends to infinity.

According to the modal equation expressed in (44) for TE

modes, 1

and 2

change from real to imaginary at the same

point, such as illustrated in figure 16.

Figure 15. Dispersion diagram for TE modes in lossless case.

Figure 16. Variation of

1 2 and with frequency f , for TE modes,

in lossless case.

From modal equations defined in (44), when 2 11 and

2 11 , respectively, for TE and TM modes, the effective

index of refraction eff

n goes asymptotically to infinite at a

frequency which isn’t the resonance frequency. These results do

not have a physical meaning, once null losses were considered.

Considering a lossy dispersive model, with e m

. The

variation of 2( ) and

2( ) for a lossy dispersive model is

represented in figure 17. In the other hand, figure 18 illustrates

the behavior of index of refraction n as function of frequency.

As it is possible to see, the imaginary part of 2 and

2 are not

null: they are positive due to the use of dispersive media.

Figure 17. Lossy dispersive model for

2 2 and .

Figure 18. Lossy dispersive model for n .

From figure 18, it is possible to see a difference with respect to

model losses associated for ENG medium. For this medium, the

real part of index of refraction exhibits negative values. This

means that, although DNG medium has necessarily negative

index of refraction, a medium with negative index of refraction

(NIR) cannot be a DNG medium. The variation of effective

index of refraction, for TE modes, is represented in figure 19. It

is trivial to see that eff

n does not tend asymptotically to infinite.

This means, thus, that these are physical results [21].

Figure 19. Dispersion diagram for TE modes in the lossy case.

The variation of transverse attenuation constants 1 and 2

for

TE modes in loss case are illustrates in figure 20. As was

expected, the real parts of 1 and 2

are both positive, in order

for the electric field y

E to decay with distance. In the other

hand, the imaginary parts of 1

and 2

are both negative. In

fact, this is a necessary condition in order to have propagation

along the z-axis.

Figure 20. Variation of

1 2 and with frequency f , for TE modes,

in loss case.

4.2 DNG DIELECTRIC SLAB

The DNG dielectric slab geometry is represented in figure 21.

For this structure, the solutions of modal equation can be divided

into TE modes and TM modes. Applying boundary conditiona to

the support components y

E and y

H and with the following

relations:

1

2

w d

u h d (46)

The modal equation is expressed by:

1

2

1

2

tan( ) , for TE even modes

cot( ) , for TE odd modes

w u u

w u u

(47)

And,

1

2

1

2

tan( ) , for TM odd modes

cot( ) , for TM even modes

w u u

w u u

(48)

Figure 21. DNG dielectric slab geometry.

4.2.1. Propagation of Surface Modes

The transverse propagation constant 2h exhibits real values and

negative values, if 2 2 and 2 2

, respectively.

Considering U iu , the modal equations previously defined in

(47) can be rewritten such that:

1

2

1

2

tanh( ) , for TE even modes

coth( ) , for TE odd modes

w U U

w U U

(49)

With U iu , the fundamental relationship 2 2 2 w u v

follows:

2 2 2 w U v (50)

In the following figures, figure 22 and figure 23, the curves of

modal equations for DPS slab and DNG slab are depicted. The

modal solutions are given by the interception of modals

equations (49) with the curves expressed by 2 2 2 w u v or

2 2 2 w U v . The positive abscissa semi-axis represents the real

part of u and the negative direction expresses the imaginary part

of u .

Figure 22. Modal solutions for DPS dielectric slab.

Figure 23. Modal solutions for DNG dielectric slab.

According to the u values for each modal solution, the modes

can be classified by slow modes or super slow modes. If the

solution has an imaginary u value, the modes designed by super

slow modes, because phase velocity is below the light velocity in

the medium, i.e., 2 2

pv c . In the other hand, if the solution

has a real u value, the modes called slow modes and the phase

velocity satisfied the follow condition: 2 2 1 1

pc v c .

The dispersion diagram for the TE modes for the DNG slab is

illustrated in figure 24. In this figure, the straight line with a great

slop represents the transition between the super slow modes and

the slow modes. The straight line with a lower slop corresponds

to the cut-off for surface modes, i.e., 0w . The fundamental

mode is the super slow TE odd mode and has null cut-off

frequency.

Figure 24. Dispersion diagram for TE modes for DNG slab with

1 1 2 21 and 2 .

The previously dispersion diagram was obtained to2 2 1 1

with 2 1 . If the condition

2 2 1 1 with 2 1

was to

be considered, the dispersion diagram would be different. It

means that for DNG slab, the dispersion diagram varies

according with the relationship of the media parameters. This

fact does not occur to DPS slab, where dispersion diagram has

only one characteristic behavior. In the case of a less dense inner

medium, the dispersion diagram can be divided in two cases:

- If 2 1 , as was showed in figure 25;

- If 2 1 , as was showed in figure 26.

Figure 25. Dispersion diagram for TE modes for DNG slab with

1 1 2 22, 1, 1 and 1.5 .

Figure 26. Dispersion diagram for TE modes for DNG slab with

1 1 2 22, 2, 1.5 and 1.5 .

REFERENCES

[1] “Hans Christian Ørsted,” IEEE , [Online]. Available:

http://www.ieeeghn.org/wiki/index.php/Hans_Christian_%C3%98rsted.

[Acessed in 2 June 2014].

[2] “Andre-Marie Ampère,” IEEE , [Online]. Available:

http://www.ieeeghn.org/wiki/index.php/Andre-Marie_Amp%C3%A8re.

[Acessed in 2 June 2014

[3] J. C. Maxwell, A Treatise On Electricity And Magnetism. Dover, New

York: Dover Publications, 1954.

[4] R. Lerner and G. Trigg, Encyclopaedia of Physics. Berlin: Wiley-VCH,

2005.

[5] S. Bais, The Equations: Icons of Knowledge. Cambridge: Harvard

University Press, 2005.

[6] K. S. Packard, “The Origin of Waveguides: A Case of Multiple

Rediscovery,” vol. 32, n.º 9, pp. 961-969, September 1984.

[7] C. R. Doerr and H. Kogelnik, “Dielectric Waveguide Theory,” Journal of

Lightwave Technology, vol. 26, n.º 9, pp. 1176-1187, May 2008.

[8] D. F. Welch, “A Brief History of High-Power Semiconductor Lasers,” vol.

6, n.º 8, pp. 1470-1477, November/December 2000.

[9] G. P. Agrawal, Fiber-Optic Communication Systems. Rochester, New York:

A John Wiley & Sons, Inc, 2010.

[10] N. Engheta e R. W. Ziolkowski, Edits., METAMATERIALS Physics and

Engineering Explorations, Piscataway. New Jersey: IEEE Press/Wiley,

2006.

[11] D. T. Emerson, “The work of Jagadis Chandra Bose: 100 years of

millimeter-wave research,” IEEE Trans. Microwave Theory and

Techniques, vol. 45, n.º 12, pp. 2267-2273, 1997.

[12] I. V. Lindell, A. H. Sihvola, J. Kurkijirvi and K. F. Lindman, “The last

Hertzian and a harbinger of electromagnetic chirality,” The Radioscientist,

vol. 3, n.º 2, pp. 38-53, Junho 1992.

[13] V. G. Veselago, “The electrodynamics of substances with simultaneously

negative values of μ and ε,” Soviet Physics Uspekhi, vol. 10, n.º 4, pp. 509-

514, 1968.

[14] J. B. Pendry, A. J. Holden, W. J. Stewart and I. Youngs, “Extremely low

frequency plasmons in metallic mesostructures,” Physical Review Letters,

vol. 76, n.º 25, pp. 4773-4776, 1996.

[15] J. B. Pendry, A. Holden, J. D. Robbins and J. W. Stewart, “Magnetism from

conductors and enhanced nonlinear phenomena,” IEEE Transactions on

Microwave Theory and Techniques, vol. 47, n.º 11, pp. 2075-2084, 1999.

[16] D. R. Smith, W. J. Padilla, D. Vier, S. Nemat-Nasser and S. Schultz,

“Composite medium with simultaneously negative permeability and

permittivity,” Physical Review Letters, vol. 84, n.º 18, pp. 4184-4187, 2000.

[17] C. R. Paiva, Meios Duplamente Negativos (DNG). DEEC IST, 2008.

[18] Jin Au Kong, Electromagnetic Wave Theory. Cambridge, Massachusetts:

EMW publishing, 2008.

[19] S. K. Raghuwanshi, “Comparative study of asymmetric versus symmetric

planar,” Indian J. Phys, vol. 84, n.º 7, pp. 831-846, 2009.

[20] T. Okoshi, Fibers Optics. London: Academic Press, Inc, 1982.

[21] J. R. Canto, C. R. Paiva and A. M. Barbosa, “Dispersion and Losses in

Surface Waveguides,” Progress In Electromagnetics Research, vol. 116,

pp. 409-423, 2009.