metamaterial waveguides

© 2005, it - instituto de telecomunicações. Todos os direitos reservados.

António L. Topa

Instituto de Telecomunicações

Instituto Superior Técnico

Portugal

Metamaterial Waveguides

2

6º Congresso do Comité Português da URSI

| Lisboa, 16 de Novembro 2012

Outline

• Introduction to metamaterials and metamaterial waveguides

• New features of closed metamaterial waveguides

• New features of open metamaterial waveguides

• Influence of metamaterial loss and dispersion

• Conclusions

3

I. Introduction



Negative refraction

4

I. Introduction



Viktor G. Veselago

5

Negative index of refraction



2n

n

6

What are Metamaterials?



Metamaterials are artificial engineered composite structures that can be

designed to exhibit specific electromagnetic properties not observed in the

constituent materials and not commonly found in nature.

7

Types of Metamaterials



• Chiral media

• Omega media

• Wire media

• Single negative media

• Double negative media

• Indefinite media

8

I. Introduction

Three-Dimensional Waveguides

Closed Waveguides Open Waveguides

Fully analytical methods Semi-analytical methods

Metamaterials

Non-dispersive

Lossless

Dispersive

lossless

Dispersive

Lossy 3D Metamaterial

Waveguides



9

w

t1

x

t2 y , < 0

DNG Ridge Waveguide

Example of a metamaterial topology

I. Introduction (DNG Metamaterial Waveguides)

DNG H-Guide

y

2l

b

x

< 0

< 0



10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5

b /l

b

3.0 cmb

II. Closed Metamaterial Waveguides

y

2l

b

x

0 <

< 0

Dispersion diagram

DPS

DNG

The lossless DNG H-guide



11

Other H-guide Geometries

Double-slab H-guide

H-guide directional coupler

l1 l2

b

l l s

b



12

l1 l2

z = b/l

b

x = l2/l1 l = l1 + l2

<

< <

< <

1

1 1

2 2

2

1

0( )

0

1

r

x l

l xx

x l

x l

<

< <

< <

1

1 1

2 2

2

1

0( )

0

1

r

x l

l xx

x l

x l

Geometric parameters:

Constitutive parameters:

y

x

Double Slab H-guide

DPS DNG



13

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2

l /l

b

b/l = 0.5

0

0.2

0.4

0.6

0.8

1

0.25 0.35 0.45 0.55 0.65 0.75

b /l

l /l

x 0

x 0

Operational and Dispersion Diagrams



14

l1 2s l2

z = b/l b

x = l2/l1

l = l1 + l2

h = s/l

<

< <

< < < <

1

1 1

2 2

2

1

( ) 1

1

r

x l s

l s x s

x s x s

s x s l

x s l

<

< <

< < < <

1

1 1

2 2

2

1

( ) 1

1

r

x l s

l s x s

x s x s

s x s l

x s l

Constitutive parameters:

The Contra-directional Coupler

DPS DNG



15

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2

l /l

b

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2

b /l

b

b/l = 1.0

b/l = 0.5

Dispersion Diagrams



16

• The transverse resonance method is used in the characterization of the

guided-wave propagation of open DNG waveguides.

III. Open Metamaterial Waveguides

• The assessment of the method is done by comparison with the effective

index method.

• Proper leaky modes propagate in open three-dimensional double-

negative (DNG) metamaterial waveguides.



17

w

t1

x

t2 y

d

PEC plane

, < 0

Discretization of the Continuous Spectrum:

Discrete Spectrum

Continuous Spectrum

Surface Modes

Leaky Modes

Radiation Modes

Evanescent Modes



III. Open Metamaterial Waveguides

18

z

y

x

y t1/l

t2/l

I

1b

II

2b

I

1bI

1b

I

2b

I

2b

I

1

I

1

II

1

II

2

II

2

II

2

I Isini ib b

I I Icosi i iq b

II II IIcosi i iq b

II IIsini ib b

Oblique Incidence



19

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

i r tx x x

i r tx x x

i r tz z z

i r tz z z

E d E d E d

H d H d H d

E d E d E d

H d H d H d

I II

1 1

I II

1 1

I I I I II II II II

1 1 1

I I I II

1 1

( )

( )

( )cos ( )sin cos sin

( )cos ( )sin cos

n p

n p

n n p p

n n

n n x p x

n p

n n x p x

n p

n n n u n n n v p p u p v

n n p

I

n n n u n n n v p p

n n

a b E c E

a b H c H

a b E a b E c E E

a b H a b H c H

II II II

1

sinp pu p v

p

H

b a

LSE and LSM Modes

Boundary Conditions

Scattering Matrix

Mode Matching Technique



20

w

w

2wx 2

wx

a1

b1

(1)

(N)

Γ Γ

Transverse Resonance Method



21

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.3 2.4 2.5 2.6 2.7

b

w /d

Even

Odd

DNG Ridge Waveguide: Numerical Results



22

1effn2effn

w

t1

x

t2 y

t1

t2

w

2effn

The Effective Index Method

+



23

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.3 2.4 2.5 2.6 2.7

b

w /d

• There are some limitations to

neglect the TE–TM coupling at

the sides of the DNG

waveguide.

• The EIM is also applicable to

DNG waveguides (excluding

the super-slow modes and far

from the cutoff region), but

only when approximate values

are desired.

Numerical Results and Assessment



24

S.-T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of open

dielectric waveguides: Part I – Mathematical formulations,” IEEE Trans. Microwave

Theory Tech., vol. MTT-29, pp. 843-855, Sep. 1981.

Improper Leaky Modes (DPS waveguide)



25

At a certain point of operation, the modes propagating in the inner region of the

waveguide become backward modes, exhibiting a power flux which is opposite to

that of the modes propagating in the outer region, therefore generating proper

leaky modes.

S1

S2 S2

Proper Leaky Modes (DNG waveguide)



26

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

2.22 2.25 2.28 2.31 2.34 2.37 2.4

b

a

0.35

0.45

0.55

0.65

0.75

0.85

2.2 2.3 2.4 2.5 2.6 2.7

b

w /d

Even

Odd

Series8Series9Series10

H2

H3

H4

Leaky Modes

H2

H3

H4



27

0

4

8

12

16

20

0.35 0.45 0.55 0.65 0.75 0.85

w /d

a H4

H3

H2

Leakage Constant



28

IV. Metamaterial Loss and Dispersion

• Most work on waveguiding structures ignores metamaterial loss

and dispersion.

• Can losses be seen as just a small perturbation of the lossless

case?

• How does the presence of dispersion and losses affect the

performance of the waveguides involving metamaterials?

• Are there any physical effects arising from metamaterial loss and

dispersion that can be used in the design of new devices?



29

The Lorentz model (I)

0

8 1

3.8 GHz

3.0 GHz

4.0 10 rad s

e

ep

e

f

f

0 :

:p

2

2 2

0

( ) 1 e

e

p

ej

2

2 2

0

( ) 1 m

m

p

mj

0

8 1

4.0 GHz

4.0 GHz

3.0 10 rad s

e

ep

e

f

fLoss coefficients

Resonance frequencies

Plasma frequencies

:

Numerical data:

2

200

lim ( ) 1 m

m

p

lim ( ) lim ( ) 1

2

200

lim ( ) 1 e

e

p



30

-12

-10

-8

-6

-4

-2

0

2

2 3 4 5 6

f [GHz]

Im{...}

e

m

n

-4

-3

-2

-1

0

1

2

3

4

5

6

7

2 3 4 5 6

f [Ghz]

Re{...}

n

The Lorentz model (II)

NRI

DNG

The DNG range is not identically to the NRI range.



31

The lossy DNG H-guide

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

3 3.5 4 4.5 5 5.5 6 6.5 7

f [GHz]

-5

-4

-3

-2

-1

0

1

2

3

4

5

3 3.5 4 4.5 5 5.5 6 6.5 7

f [GHz]

All the modal solutions in the DNG H-guide become complex.

y

2l

b

x

< 0

< 0



32

0

50

100

150

200

250

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5

f [GHz]

a [dB/l]

LSM01

LSM11

LSM21

Attenuation constant

0

50

100

150

200

250

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5

f [GHz]

a [dB/l]

LSM01

LSM11

LSM21

LSM31

Lossless case Lossy case



33

The DNG H-guide filter (I)

0

1

2

3

4

5

6

7

8

9

10

4.5 5 5.5 6 6.5 7 7.5

f [GHz]

a [dB/l]

LSM01

0

5

10

15

20

25

30

4.4 4.5 4.6

f [GHz]

a [dB/l]

LSM11



34

The DNG H-guide filter (II)

0

5

10

15

20

25

4.4 4.45 4.5 4.55 4.6

f [GHz]

a [dB/l]

LSM11

8 10.2 10 rad s

8 10.5 10 rad s

8 11.0 10 rad s

8 12.0 10 rad s



35

V. Conclusions

• In addition to strongly dispersive media, DNG metamaterials must be also

considered as lossy, since both dispersion and losses avoid unphysical

meaningless solutions.

• Full-wave analyses (analytical or semi-analytical) provide a physical interpretation

for all the modal solutions of DNG three-dimensional waveguides.

• A comparative study between the lossy and lossless cases proves that losses are

more than a perturbation strongly affecting the performance of the waveguide.

• Methods commonly used for conventional three-dimensional waveguides, can be

easily generalized to DNG metamaterial waveguides.



metamaterial waveguides

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