Waveguides - paginas.fe.up.ptmines/EE/EE_waveguides.pdf · MAPTele – EE Waveguides 2 Guided...

Faculdade de Engenharia

Waveguides

EE – MAPTele 2008/2009

MAPTele – EEWaveguides 2

Faculdade de EngenhariaGuided Propagation

z

y

dielectric 2

dielectric 2

metalic waveguide

y

z

dielectric 1

dielectric waveguide

21 nn >

ci θθ ≥

waveguide study

Maxwell’s equations

boundary conditions 0

0

=⋅∇

=⋅∇

=×∇

−=×∇

H

E

EjH

HjE

r

r

rr

rr

ωε

ωµ

0

022

22

=+∇

=+∇

HH

EErr

rr

µεω

µεω

time-harmonic fields in losslessand source -free simple media

ideal conductor

ideal conductortotal internal reflection


Faculdade de EngenhariaUniform waveguides

x

y

z

can include ideal conductors

propagation along +z

( )µε ,

transverse section doesn’t change with z

waveguides filled with lossless media

infinite length

( ) ( ) zeyxHzyxH γ−= ,,, 0rr

( ) ( ) zeyxEzyxE γ−= ,,, 0rr

nonuniform wave


Faculdade de EngenhariaUniform waveguides – field determination

x

y

z

( ) ( ) zeyxHzyxH γ−= ,,, 0rr

( ) ( ) zeyxEzyxE γ−= ,,, 0rr

0

022

22

=+∇

=+∇

HH

EErr

rr

µεω

µεω

0

00202

0202

=+∇

=+∇

HhH

EhE

xy

xyrr

rr

µεωγ 222 +=h

2

2

2

22

yxxy ∂∂

+∂∂

=∇

2 vector eqs. 6 scalar eqs.

00

00

00

02020202

02020202

02020202

=+∇=+∇

=+∇=+∇

=+∇=+∇

zzxyzzxy

yyxyyyxy

xxxyxxxy

HhHEhE

HhHEhE

HhHEhE

not all independent


Faculdade de EngenhariaUniform waveguides – transverse components

x

y

z HjErr

ωµ−=×∇ EjHrr

ωε=×∇

000

000

000

zxy

yxz

xyz

Hjy

Ex

E

HjEx

E

HjEy

E

ωµ

ωµγ

ωµγ

−=∂

∂−∂

∂

−=−∂

∂−

−=+∂

∂

000

000

000

zxy

yxz

xyz

Ejy

Hx

H

EjHx

H

EjHy

H

ωε

ωεγ

ωεγ

=∂

∂−∂

∂

=−∂

∂−

=+∂

∂

( ) zeyxHH γ−= ,0rr

( ) zeyxEE γ−= ,0rr

∂

∂−∂

∂−=

∂

∂+∂

∂−=

∂

∂+∂

∂−=

∂

∂−∂

∂−=

xH

jy

Eh

E

yH

jx

Eh

E

xE

jy

Hh

H

yE

jx

Hh

H

zzy

zzx

zzy

zzx

00

20

00

20

00

20

00

20

1

1

1

1

ωµγ

ωµγ

ωεγ

ωεγ

transverse components as functionsof the longitudinal components

if 0≠h


Faculdade de EngenhariaDetermination of field inside waveguide

x

y

z

∂

∂−

∂∂

−=

∂

∂+

∂∂

−=

∂

∂+

∂∂

−=

∂

∂−∂

∂−=

xH

jy

Eh

E

yH

jx

Eh

E

xE

jy

Hh

H

yEj

xH

hH

zzy

zzx

zzy

zzx

00

20

00

20

00

20

00

20

1

1

1

1

ωµγ

ωµγ

ωεγ

ωεγ

if 0≠h

2. determine

0

00202

0202

=+∇

=+∇

zzxy

zzxy

HhH

EhEµεωγ 222 +=h

1. solve

3. obtain

( ) ( )( ) ( ) z

z

eyxHzyxH

eyxEzyxEγ

γ

−

−

=

=

,,,

,,,0

0

rr

rr

application of boundary conditions

Note

ß TE waves

ß TEM waves

ß TM waves

0and0 00 ≠= zz HE

0and0 00 ≠= zz EH

0and0 00 == zz EH


Faculdade de EngenhariaCutoff frequency

x

y

z µεωγ 222 +=h

cutoff frequency à

12

2

−=µεω

µεω hµεωγ 22 −= h

µεπ2

hfc =

12

−

=

ffcµεωγ

evanescent mode cff < αγ =( ) ( ) zeyxHzyxH α−= ,,, 0

rr( ) ( ) zeyxEzyxE α−= ,,, 0

rr

propagating mode cff > βγ j=

( ) ( ) zjeyxHzyxH β−= ,,, 0rr

( ) ( ) zjeyxEzyxE β−= ,,, 0rr


Faculdade de EngenhariaPropagating mode characteristics

x

y

z

( )cff >βγ j=

12

−

=

ffcµεωγ

propagating mode:

phase constant à µεωβββ =

−= m

cm f

f,1

2

wavelength àm

m

c

m

ff β

πλ

λλ

2,

12

=

−

=

βπ

λ2

=

if 0≠cf mλλ >

phase constant in an infinitemedium with parameters ( )µε ,


Faculdade de EngenhariaPropagating mode characteristics

x

y

z

( )cff >βγ j=

12

−

=

ffcµεωγ

propagating mode:

phase velocity à

2

1

−=

ffc

mββgroup velocity à

βω

=fv

if 0≠cf mf vv >

phase velocity in an inifinitemedium with parameters ( )µε ,

µε1

,

12

=

−

= m

c

mf v

ff

vv

if guide is filled with air, cvm =

cv f >

2

1

−=

ff

vv cmg

if 0≠cf mg vv <

ωβ ddvg

1=

2mgf vvv =


Faculdade de EngenhariaWave impedance

x

y

z TEM waves propagating along +z in an infinite medium with

( )( )HzE

EzH

rr

rr

×−=

×=

ˆ

ˆ1

η

η

εµη =

waves propagating along +z inside a waveguide

( )( )HzZE

EzZ

Hrr

rr

×−=

×=

ˆ

ˆ1

:wavesTEorTEM

waves TMorTEM :

( )( )yHxHZzEyExE

yExEZ

zHyHxH

xyzyx

xyzyx

ˆˆˆˆˆ

ˆˆ1ˆˆˆ

+−−=++

+−=++ 0=zH

0=zE

TM or TEM waves

TE or TEM waves

wave impedance àx

y

y

x

H

E

HE

Z −==


Faculdade de EngenhariaAverage propagated power

x

y

z

average power à

x

y

y

x

H

E

HE

Z −==

∫ ⋅=A

avav AdSrr

P

zdAAd ˆ=r

{ }*21

HESav

rrr×= Re

{ }∫ −=A

xyyxav dAHEHE **21P Re

∫

+

=

Ayxav dAEE

Z

22121P Re { }∫

+=

Ayx dAHHZ

22

21

Re


Faculdade de EngenhariaAverage stored energy and energy-transport velocity

x

y

z

average stored energy àper unit length

( )∫ +=A

avmaveav dAwwW ,,'

++=⋅= 222

, 4*

4 zyxave EEEEEwεε rr

++=⋅= 222

, 4*

4 zyxavm HHHHHwµµ rr

av

aven W

v'

P=energy-transport velocity à


Faculdade de EngenhariaTEM waves

x

y

z TEM waves à 0== zz HE

∂

∂−∂

∂−=

∂

∂+

∂∂

−=

∂

∂+

∂∂

−=

∂

∂−

∂∂

−=

xHj

yE

hE

yH

jx

Eh

E

xE

jy

Hh

H

yE

jx

Hh

H

zzy

zzx

zzy

zzx

00

20

00

20

00

20

00

20

1

1

1

1

ωµγ

ωµγ

ωεγ

ωεγ

02 =hµεπ2

hfc =

mgf

m

m

vvv

j

==

=

=

=

λλ

ββ

βγ

12

−

=

ffcµεωγ

2

1

−=

ffc

mββ

use Maxwell’s equations

εµωγ j=

µεωβ =m

0=



x

y

z

TEM waves: 0== zz HE

Maxwell’s equations:

000

00

00

=∂

∂−

∂∂

−=−

−=

yE

x

E

HjE

HjE

xy

yx

xy

ωµγ

ωµγ

000

00

00

=∂

∂−

∂∂

=−

=

yH

x

H

EjH

EjH

xy

yx

xy

ωεγ

ωεγ

HjErr

ωµ−=×∇

EjHrr

ωε=×∇

γωµjZTEM =

x

y

y

x

H

E

HE

Z −==

ωεγ

j=

εµ

= η=

εµωγ j=


Faculdade de EngenhariaTM waves

x

y

z TM waves à 0and0 00 ≠= zz EH 00202 =+∇ zzxy EhE

yE

hE

xE

hE

xE

hj

H

yE

hj

H

zy

zx

zy

zx

∂∂

−=

∂∂

−=

∂∂−=

∂∂=

0

20

0

20

0

20

0

20

γ

γ

ωε

ωε

x

y

y

x

H

E

HE

Z −==

ωεγ

jZTM =

ωε

µεω

j

ffc 1

2

−

= 12

−

−=

ff

j cη

12

−

=

ffcµεωγ


Faculdade de EngenhariaTM waves

x

y

z

TM waves à 0and0 00 ≠= zz EH

12

−

−=

ff

jZ cTM η

12

−

=

ffcµεωγ

∫

+

=

A

yxav dAEEZ

22121P Re

evanescent modes à cff <

cff >propagating modes à

TMZ purely imaginary

0P =av

( )21 ffZ cTM −=η

(real and less than )η


Faculdade de EngenhariaTE waves

x

y

z TE waves à 0and0 00 ≠= zz HE

x

y

y

x

H

E

HE

Z −==

12

−

=

ffcµεωγ

00202 =+∇ zzxy HhH

xH

hj

E

yH

hj

E

yH

hH

xH

hH

zy

zx

zy

zx

∂∂=

∂∂

−=

∂∂−=

∂∂−=

0

20

0

20

0

20

0

20

ωµ

ωµ

γ

γ

γωµj

ZTE =

12

−

=

ff

j

c

η


Faculdade de EngenhariaTE waves

x

y

z

TE waves à 0and0 00 ≠= zz HE

12

−

=

ffcµεωγ

∫

+

=

A

yxav dAEEZ

22121P Re

evanescent modes à cff <

cff >propagating modes à

TEZ purely imaginary

0P =av

(real and larger than )η

12

−

=

ff

jZ

c

TEη

( )21 ffZ cTE −=η


Faculdade de EngenhariaWave impedance versus frequency

x

y

z

1

ηZ

evanescent region

cff

2 1

( )21 ffZ cTE −= η

( )21 ffZ cTM −= η

η=TEMZ


Faculdade de EngenhariaParallel-plate waveguide

waveguide filled with lossless dielectric

b

y

z

x

W

( )µε ,

ideal conducting plates ( )∞=σ

infinite length à propagation along +z

bW >> 0=∂∂x

variation of field with x can be neglected


Faculdade de EngenhariaMetallic waveguides – boundary conditions

metallic waveguides à bounded by ideal conductors

0condcond == BE

continuoustanE continuousnormBand

boundary conditions

HB µ=

0normtan == HE in the vicinity of conductors

b

y

z

x

W

0== zx EE

0=yHbyy == and0at


Faculdade de EngenhariaParallel-plate waveguide – field determination

b

y

z

x

W

∂

∂−

∂∂

−=

∂

∂+

∂∂

−=

∂

∂+

∂∂

−=

∂

∂−∂

∂−=

xH

jy

Eh

E

yH

jx

Eh

E

xE

jy

Hh

H

yEj

xH

hH

zzy

zzx

zzy

zzx

00

20

00

20

00

20

00

20

1

1

1

1

ωµγ

ωµγ

ωεγ

ωεγ

(if )0≠h2. determine

0

00202

0202

=+∇

=+∇

zzxy

zzxy

HhH

EhE

µεωγ 222 +=h

1. solve

0=∂∂x

0

0

022

02

022

02

=+

=+

zz

zz

Hhdy

Hd

Ehdy

Ed

yE

hE

yH

hj

E

yH

hH

yE

hj

H

zy

zx

zy

zx

∂∂

−=

∂∂

−=

∂∂

−=

∂∂

=

0

20

0

20

0

20

0

20

γ

ωµ

γ

ωε

ß TE waves

ß TEM waves

ß TM waves

0and0 00 ≠= zz HE

0and0 00 ≠= zz EH

0and0 00 == zz EH

( ) ( )( ) ( ) z

z

eyxHzyxH

eyxEzyxEγ

γ

−

−

=

=

,,,

,,,0

0

rr

rr

0=∂∂x

Note: TEM waves à h=0



b

y

z

x

W

TEM waves à 0and0 00 == zz EH

Maxwell’s equations:

000

00

00

=∂

∂−

∂∂

−=−

−=

yE

x

E

HjE

HjE

xy

yx

xy

ωµγ

ωµγ

000

00

00

=∂

∂−

∂∂

=−

=

yH

x

H

EjH

EjH

xy

yx

xy

ωεγ

ωεγ

HjErr

ωµ−=×∇ EjHrr

ωε=×∇

and 0=hprevious method does not work

0=∂∂ x

000

==dy

dHdy

dE xx 0xE 0

xHand are constants



b

y

z

x

W

0xE 0

xHand are constants0)()0( 00 == bEE xx

00yx HZE =

00 =yH

00xy HZE −=

=0yE

η=TEMZ

constant

00 =xE

yEE ˆ00 =

r

boundary conditions

0== zx EE0=yH

byy == and0at

wave impedance

x

y

y

x

H

E

HE

Z −==

xEH ˆ00

η−=

r

Z

EH y

x

00 −=


Faculdade de EngenhariaTM waves – longitudinal component

b

y

z

x

W

general solution:

0and0 00 ≠= zz EH 0022

02

=+ zz Eh

dyEd

( ) ( ) ( )hyBhyAyEz cossin0 +=

( ) 000 =zE

0=B

L,3,2,1, == nb

nh

π

( ) 0sin =bhA

=

==

byn

AE

nb

nh

nz

π

π

sin

,2,1,

0

L

TM wavesà

boundary conditions

0== zx EE0=yH

byy == and0at

( ) 00 =bEz

( ) ( )hyAyEz sin0 =


Faculdade de EngenhariaTM waves – transverse components

b

y

z

x

W

=

==

byn

AE

nb

nh

nzπ

π

sin

,3,2,1,

0

L

yE

hE

yH

hj

E

yH

hH

yE

hj

H

zy

zx

zy

zx

∂∂

−=

∂∂

−=

∂∂

−=

∂∂

=

0

20

0

20

0

20

0

20

γ

ωµ

γ

ωε

−=

=

bynA

nbE

byn

An

bjH

ny

nx

ππ

γ

ππ

ωε

cos

cos

0

0

0=

0=


Faculdade de EngenhariaTM waves – TMn mode

b

y

z

x

W

TMn mode

−=

=

=

byn

An

bE

byn

An

bjH

byn

AE

ny

nx

nz

ππ

γ

ππ

ωε

π

cos

cos

sin

0

0

0

bn

hπ=

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

y/b

Ez0 /A

n

n=1

n=2

n=3


Faculdade de EngenhariaTE waves – longitudinal component

b

y

z

x

W

TE waves à 0and0 00 ≠= zz HE 0022

02

=+ zz Hh

dyHd

( ) ( ) ( )hyBhyAyH z cossin0 +=

note: it does not exist a boundary condition for 0zH

it is necessary to determine the transverse components in order to apply the boundary conditions

boundary conditions

0== zx EE0=yH

byy == and0at

general solution:


Faculdade de EngenhariaTE waves –Ten mode

b

y

z

x

W

=

=

=

byn

Bn

bjE

bynB

nbH

bynBH

nx

ny

nz

ππ

ωµ

ππ

γ

π

sin

sin

cos

0

0

0

TEn mode

bn

hπ=

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

y/b

Ez0 /A

n

n=1

n=2n=3

n

z

BH 0


Faculdade de EngenhariaParallel-plate waveguide – cutoff frequency

b

y

z

x

W

µεπ2

hf c =

0TEM =h

L,3,2,1,TETM, == nb

nh π

( ) 0TEM =cf

( )µεb

nfc 2TETM, =

dominant mode à mode with lowest cutoff frequency

parallel-plate waveguide à TEM mode is the dominant mode

for given frequency f à only modes with propagate ffc <

à TEM mode is always present( ) 0TEM =cf

increasing f à more modes can propagate


Faculdade de EngenhariaGuided propagation – plane wave interference

metallic waveguide

A

B

C

wave front

C à immediately after reflection

A e C in the same wave front π2ofmultiple=∠−∠ AC

Γ∠+−Γ∠+−∠=∠ BCAB lklkAC

propagation along distance là acquired phase:

in each reflection à acquired phase: Γ∠ π=

1−=Γ

( )BCAB llkA +−+∠= π2

( ) πnllk BCAB 2=+

lk− lεµω−=


Faculdade de EngenhariaGuided propagation – allowed angles

metallic waveguides

A

B

C

( )θcosBClb =

b

( ) πnllk BCAB 2=+

θ

( )θcosb

lBC =

( )θ2cosBCAB ll =( ) ( )θθ

2coscos

b=

( ) ( )( )12coscos

+=+ θθ

bll BCAB ( )θcos2b=

( ) πθ nbk 2cos2 =

( )bk

nπθ =cos

εµωπ

bn

=

not all angles are allowed in order to the wave to be propagated along the guide


Faculdade de EngenhariaGuided propagation – cutoff frequency

A

B

C

bθ

( )θcosεµω

πb

n=

integern( ) 1cos ≤θ

as the frequency increases, more angles are allowed

as the frequency increases, more modes can propagate

cutoff à ( ) 1cos cut =θ 1c

=εµω

πb

nεµ

πω

bn

=c εµππ

bn

f2c =

bn

hπ

=

εµπ2h

=


Faculdade de EngenhariaRectangular waveguides

guide filled with lossless dielectric ( )µε ,

ideal conducting plates ( )∞=σ


b

y z

x a


Faculdade de EngenhariaRectangular waveguides – boundary conditions

0condcond == BE


boundary conditions

HB µ=

0normtan == HE in the vicinity of conductors

b

y z

x a

axxHEE

byyHEE

xzy

yzx

=====

=====

and0at0

and0at0000

000


Faculdade de EngenhariaRectangular waveguides – field determination

∂

∂−

∂∂

−=

∂

∂+

∂∂

−=

∂

∂+

∂∂

−=

∂

∂−∂

∂−=

xH

jy

Eh

E

yH

jx

Eh

E

xE

jy

Hh

H

yEj

xH

hH

zzy

zzx

zzy

zzx

00

20

00

20

00

20

00

20

1

1

1

1

ωµγ

ωµγ

ωεγ

ωεγ


0

00202

0202

=+∇

=+∇

zzxy

zzxy

HhH

EhEµεωγ 222 +=h

1. solve

( ) ( )( ) ( ) z

z

eyxHzyxH

eyxEzyxEγ

γ

−

−

=

=

,,,

,,,0

0

rr

rr

b

y z

x a



TEM waves à

b

y z

x a

000 == zz HE

HErr

and in the xy plane

∫∫ ⋅∂∂

+=⋅S

inP

SdtE

IldHr

rrr

ε

0=⋅∇ Hr

Hr

lines represent closed paths in the transverse section of the guide

0cond =H

lines are closedHr

current inside the guide

surfaced

bounded by P0=inI



TEM waves do not existin rectangular waveguides

b

y z

x a

HErr

and in xy plane

∫∫ ⋅∂∂

+=⋅S

inP

SdtE

IldHr

rrr

ε

surface

bounded by P0=inI

flux of through S is zeroEr

0=⋅∂∂

∫S

SdtE rr

ε 0=⋅∫P

ldHrr

0=Hr

0=Er

TEM waves do not exist in waveguides with single metallic conductor


Faculdade de EngenhariaTM and TE waves – longitudinal components

b

y z

x a

0

00202

0202

=+∇

=+∇

zzxy

zzxy

HhH

EhE

solve

022 =+∇ ψψ hxy

( )yx,ψψ =

2

2

2

22

yxxy ∂∂

+∂∂

=∇

022

2

2

2

=+∂∂

+∂∂

ψψψ

hyx

( ) ( ) ( )yYxXyx =,ψmethod of separation of variables à

022

2

2

2

=++ XYhdy

YdX

dxXd

Y

011 2

2

2

2

2

=++ hdy

YdYdx

XdX


Faculdade de EngenhariaMethod of separation of variables

b

y z

x a

( )( )

( )( )

011 2

2

2

2

2

=++ hdy

yYdyYdx

xXdxX

function of x function of y

previous equation is only satisfied when

( )( )

constant1

2

2

=dx

xXdxX

( )( )

constant1

2

2

=dy

yYdyY

( )( ) 22

21xk

dxxXd

xX−=

( )( ) 22

21yk

dyyYd

yY−=

0222 =+−− hkk yx222yx kkh +=



b

y z

x a

( )( ) 22

21xk

dxxXd

xX−=

( )( ) 22

21yk

dyyYd

yY−=

222yx kkh +=

( )( ) 22

21xk

dxxXd

xX−=

( )( ) 22

21yk

dyyYd

yY−=

( ) ( ) 022

2

=+ xXkdx

xXdx

( ) ( ) 022

2

=+ yYkdy

yYdy

( ) ( ) ( )xkBxkAxX xx cossin +=

( ) ( ) ( )ykDykCyY yy cossin +=

general solution of is

( ) ( ) ( )yYxXyx =,ψ

022

2

2

2

=+∂∂

+∂∂

ψψψ

hyx

( ) ( ) ( )[ ] ( ) ( )[ ]ykDykCxkBxkAyx yyxx cossincossin, ++=ψ


Faculdade de EngenhariaTM waves – longitudinal component

b

y z

x a

TM waves à 00 =zH 00202 =+∇ zzxy EhE

( ) ( )[ ] ( ) ( )[ ]ykDykCxkBxkAE yyxxz cossincossin0 ++=

( )xX ( )yY

boundary conditions

byy

axxEz

=====

and0

and0at00( ) 0,00 =yEz

( ) 0,0 =bxEz( ) 0,0 =yaEz

0=B

( ) ( )xkAxX xsin=

integer, ma

mkx

π= integer, n

bn

kyπ

=

( ) 00,0 =xEz

0=D

( ) ( )ykCyY ysin=

=

byn

axmEE mnz

ππ sinsin,00


Faculdade de EngenhariaTMmn mode

b

y z

x a

=

byn

axmEE mnz

ππ sinsin,00

00,2

00,2

00,2

00,2

sin cos

cos sin

cos sin

sin cos

x mn

y mn

x mn

y mn

j n m x n yH E

h b a bj m m x n y

H Eh a a b

m m x n yE E

h a a b

n m x n yE E

h b a b

ωε π π π =

ωε π π π = −

γ π π π = −

γ π π π = −

222

+

=

bn

amh ππ

notes

2.

0≠h 00 ≠≠ nm or1.

00 == nm or 0== HErr

11 ≥≥ nm and


Faculdade de EngenhariaTEmn mode

b

y z

x a

222

+

=

bn

am

hππ

notes

2.

0≠h 00 ≠≠ nm or1.

00 == nm or

00, cos cosz mn

m x n yH H

a bπ π =

0

0,2

00,2

00,2

00,2

sin cos

cos sin

cos sin

sin cos

x mn

y mn

x mn

y mn

m m x n yH H

h a a b

n m x n yH H

h b a b

j n m x n yE H

h b a b

j m m x n yE H

h a a b

γ π π π =

γ π π π = ωµ π π π =

ωµ π π π = −

it is possible that


Faculdade de EngenhariaRectangular waveguide – cutoff frequency

µεπ2

hf c =

rectangular waveguides à dominant mode is the TE10 mode

22

TETM,

+

=

bn

am

hππ 22

21

+

=

bn

am

f c µε

b

y z

x a

TMmn modes à 11 ≥≥ nm and the TMmn dominant mode is the TM11 mode

TEmn modes à if the TEmn dominant mode is the TE10 mode00 ≠≠ nm or ba >

( ) ( )1110 TMcTEc ff <


Faculdade de EngenhariaCircular waveguides

waveguide filled with lossless dielectric ( )µε ,

ideal conducting surface ( )∞=σ


z

φ

a


Faculdade de EngenhariaCircular waveguides – boundary conditions

0condcond == BE


boundary conditions

HB µ=

0normtan == HE in the vicinity of conductor

cylindrical coordinates à ( ) zzr ezEErEE γ

φ φ −++= ˆˆˆ 000r

( ) zzr ezHHrHH γ

φ φ −++= ˆˆˆ 000r

z

φ

a

arHEE rz ==== at0000φ


Faculdade de EngenhariaCircular waveguides – field determination


µεωγ 222 +=h

1. solve

( ) ( )( ) ( ) z

z

erHzrH

erEzrEγ

γ

φφ

φφ−

−

=

=

,,,

,,,0

0

rr

rr

0

00202

0202

=+∇

=+∇

zzr

zzr

HhH

EhE

φ

φ

2

2

22 11

φφ ∂∂+

∂∂

∂∂=∇

rrr

rrr

z

φ

a

∂

∂−∂

∂−=

∂

∂+∂

∂−=

∂

∂+∂

∂−=

∂∂−

∂∂−=

rHjE

rhE

Hr

jr

Eh

E

rEjH

rhH

Er

jr

Hh

H

zz

zzr

zz

zzr

00

20

00

20

00

20

00

20

1

1

1

1

ωµφ

γ

φωµγ

ωεφ

γ

φωεγ

φ

φ

NoteTEM modes do not propagatein circular waveguides



solve

022 =+∇ ψψφ hr

( )φψψ ,r=

method of separation of variables

0

00202

0202

=+∇

=+∇

zzr

zzr

HhH

EhE

φ

φ

2

2

22 11

φφ ∂∂+

∂∂

∂∂=∇

rrr

rrr

z

φ

a

011 2

2

2

2 =+∂∂

+

∂∂

∂∂

ψφψψ

hrr

rrr

( ) ( ) ( )φφψ Φ= rRr,

( )( )

( )( )

( )( )2

222

2

22 1φ

φφ d

drh

drrdR

rRr

drrRd

rRr Φ

Φ−=++

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 022

2

22

2

=Φ+Φ

+Φ

+Φ φφ

φφφ rRh

dd

rrR

drrdR

rdrrRd



function of r function of φ

previous equation is satisfied when

( )( )

constant1

2

2

=Φ

Φ φφ

φ dd

( )( )

( )( )

( )( )2

222

2

22 1φ

φφ d

drh

drrdR

rRr

drrRd

rRr Φ

Φ−=++

( )( )

( )( )

constant222

22

=++ rhdr

rdRrR

rdr

rRdrR

r

( )( ) 22

2

-1

φφ

φφ

kd

d=

ΦΦ

( )( )

( )( ) 222

2

22

kφ=++ rhdr

rdRrR

rdr

rRdrR

r

z

φ

a



( )( ) 22

2

-1

φφ

φφ

kd

d=

ΦΦ

( ) ( ) 022

2

=Φ+Φ

φφ

φφk

dd

( ) ( ) ( )φφφ φφ kBkA cossin +=Φ

z

φ

a

( ) ( )φπφ Φ=+Φ 2

( ) ( )( ) ( )φπφ

φπφ

φφφ

φφφ

kkk

kkk

cos2cos

sin2sin

=+

=+

integer, nnk =φ

( ) ( ) ( )φφφ nBnA cossin +=Φ ( ) ( )φφ nB cos=Φ


Faculdade de EngenhariaBessel differential equation

z

φ

a

integer, nnk =φ

( )( )

( )( ) 222

2

22

φkrhdr

rdRrR

rdr

rRdrR

r=++

( )( )

( )( ) 222

2

22

nrhdr

rdRrR

rdr

rRdrR

r=++

Bessel differential equation( ) ( ) ( ) ( ) 02222

22 =−++ rRnrh

drrdR

rdr

rRdr

( ) ( ) ( )hrNDhrJCrR nn +=general solution:

Bessel functionsof the 1st kind

Bessel functionsof the 2nd kind


Faculdade de EngenhariaBessel functions of the 1st kind

z

φ

a

for n integer

∑∞

=+

+

+−

=0

2

2

2)!(!)1(

)(m

mn

mnm

n nmmx

xJ

0 2 4 6 8 10 12

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

J0 (x)

J1 (x)J2 (x)

J3 (x) ( )( ) 100

000

=⇒=

=⇒≠

n

n

Jn

Jn

notes

2. oscillating functions,

with decreasing amplitudeand aperiodic zeros

1.


Faculdade de EngenhariaBessel functions of the 1st kind – zeros

z

φ

a

0 2 4 6 8 10 12

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

J0 (x)J1 (x)

J2 (x)J3 (x)

( )xJo ( )xJ1 ( )xJ2( )xJ3

19.409417.959816.470614.93095

16.223514.796013.323711.79154

13.015211.619810.17358.65373

9.76108.41727.01565.52012

6.38025.13363.83172.40481

zero

zeros of Bessel functions of the 1st kind


Faculdade de EngenhariaBessel functions of the 1st kind – first order derivative

z

φ

a

note

oscillating functions,

with decreasing amplitude

and aperiodic zeros

( ) ( ) ( )[ ]xJxJxJ nnn 1121

' +− −=

0 2 4 6 8 10 12-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

)('1 xJ

)('2 xJ

)('3 xJ

)('0 xJ


Faculdade de EngenhariaBessel functions of the 1st kind – zeros of first order derivative

z

φ

a

zeros of derivatives of Bessel functions of the 1st kind

0 2 4 6 8 10 12-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

( )xJ o' ( )xJ 1' ( )xJ 2' ( )xJ 3'

17.788716.347514.863616.47065

14.585813.170411.706013.32374

11.34599.96958.536310.17353

8.01526.70615.33147.01562

4.20123.05421.84123.83171

zero


Faculdade de EngenhariaBessel functions of the 2nd kind

z

φ

a

notes

1. oscillating functions,

with decreasing amplitude

and aperiodic zeros

)sin(

)()cos()(lim)(

ππp

xJpxJxN pp

npn−

→

−=

0 2 4 6 8 10 12-2

-1.5

-1

-0.5

0

0.5

1

)(0 xN

)(1 xN

)(2 xN

)(3 xN 2. become infinite when x=0

( ) ( ) ( )hrNDhrJCrR nn +=

0=D when the region of interest

includes r = 0


Faculdade de EngenhariaCircular waveguides – solution of wave equation

z

φ

a

note

TM waves à

( ) ( )hrJCrR n=

022 =+∇ ψψφ hr

( ) ( ) ( )φφψ Φ= rRr,

( ) ( )φφ nB cos=Φ

( ) ( ) ( ) ( ) ( )φφφψ nhrJCrRr nn cos, =Φ=( )φψ ,0 rE z =

TE waves à ( )φψ ,0 rH z =


Faculdade de EngenhariaTMnp mode – longitudinal component

TM waves à 00 =zH

boundary conditions

z

φ

a


( ) 0,0 == φarEz

and ( ) ( )φnhrJCE nnz cos0 =

( ) 0=haJ n

( )xJ o ( )xJ1 ( )xJ2 ( )xJ3

19.409417.959816.470614.93095

16.223514.796013.323711.79154

13.015211.619810.17358.65373

9.76108.41727.01565.52012

6.38025.13363.83172.40481

zero

L;6537.8

;5201.5

;4048.2

0a

ha

ha

hn ===→=

L;1735.10

;0156.7

;8317.3

1a

ha

ha

hn ===→=

M

aJp

hh nTMnp

ofzeroth==


Faculdade de EngenhariaTMnp modes – transverse components

boundary conditions

z

φ

a


( ) ( )φnhrJCE nnz cos0 =

φγ

γ

ωε

φωε

φ

φ

∂∂

−=

∂∂

−=

∂∂

−=

∂∂

=

0

20

0

20

0

20

0

20

z

zr

z

zr

Erh

E

rE

hE

rE

hj

H

Erh

jH

( ) ( )

( ) ( )

( ) ( )

( ) ( )φγ

φγ

φωε

φωε

φ

φ

nhrJCrhn

E

nhrJCh

E

nhrJCh

jH

nhrJCrhnj

H

nn

nnr

nn

nnr

sin

cos'

cos'

sin

20

0

0

20

=

−=

−=

−=

these components satisfy theboundary conditions

note

aJp

hh nTM np

ofzeroth==


Faculdade de EngenhariaTMnp modes – cutoff frequency

z

φ

a

aJp

h nTM np

ofzeroth=

( )µεπ2np

np

TMTMc

hf =

µεπ a

Jp n

2

ofzeroth=

( )xJ o ( )xJ1 ( )xJ2 ( )xJ3

19.409417.959816.470614.93095

16.223514.796013.323711.79154

13.015211.619810.17358.65373

9.76108.41727.01565.52012

6.38025.13363.83172.40481

zero

lowest zero of à 2.4048 (n=0, p=1)nJ

( )µεπ a

f TMc2

4048.201

=

dominant TM mode à TM01


Faculdade de EngenhariaTEnp modes

TE waves à 00 =zE

boundary conditions

z

φ

a


e ( ) ( )φnhrJCH nnz cos0 =

rH

hj

E

Hrh

jE

Hrh

H

rH

hH

z

zr

z

zr

∂∂

=

∂∂

−=

∂∂

−=

∂∂

−=

0

20

0

20

0

20

0

20

ωµ

φωµ

φγ

γ

φ

φ

( ) ( )

( ) ( )

( ) ( )

( ) ( )φωµ

φωµ

φγ

φγ

φ

φ

nhrJCh

jE

nhrJCrhnj

E

nhrJCrhn

H

nhrJCh

H

nn

nnr

nn

nnr

cos'

sin

sin

cos'

0

20

20

0

=

=

=

−=

( ) 0' =haJ n

aJp

hh nTEnp

'ofzeroth==


Faculdade de EngenhariaTEnp modes – cutoff frequency

z

φ

a

( )µεπ2np

np

TETEc

hf =

µεπ a

Jp n

2

ofzeroth /

=

lowest zero of à 1.8412 (n=1, p=1)/nJ

dominant TE mode à TE11

aJp

hh nTEnp

'ofzeroth==

( )xJ o' ( )xJ 1' ( )xJ 2' ( )xJ 3'

17.788716.347514.863616.47065

14.585813.170411.706013.32374

11.34599.96958.536310.17353

8.01526.70615.33147.01562

4.20123.05421.84123.83171

zero

( )µεπ a

f TEc2

8412.111

=

( ) ( )0111 TMcTEc ff < dominant mode in circular

waveguides is TE11


Faculdade de EngenhariaPlanar dielectric waveguides

lossless materials ( )0=σ


W

z

y

x

b

2n

1n

2n

21 nn >

bW >> 0=∂∂x


Faculdade de EngenhariaPlanar dielectric waveguides – boundary conditions

boundary conditions:

W

z

y

x

b

2n

1n

2n

0=sρ

0=sJr

continuous

continuous

continuous

continuous

tan

tan

H

D

B

E

normal

normal

( )( )( )( ) sn

sn

n

n

JHHa

DDa

BBa

EEa

rrr

rr

rr

rr

=−×

=−⋅

=−⋅

=−×

21

21

21

21

ˆ

ˆ

0ˆ

0ˆ

ρ

dielectric:

continuousandcontinuousand

xz

xz

HHEE

2at

by ±=


Faculdade de EngenhariaPlanar dielectric waveguide – field determination


µεωγ 222 +=h

1. solve

( ) ( )( ) ( ) z

z

eyxHzyxH

eyxEzyxEγ

γ

−

−

=

=

,,,

,,,0

0

rr

rr

0

0

022

02

022

02

=+

=+

zz

zz

Hhdy

Hd

Ehdy

Ed

W

z

y

x

b

2n

1n

2n

0=∂∂x

dydE

hE

dydH

hj

E

dydH

hH

dydE

hj

H

zy

zx

zy

zx

0

20

0

20

0

20

0

20

γ

ωµ

γ

ωε

−=

−=

−=

=

+

+=

2medium,

1medium,

2

22

2

12

2

nc

nch

ωγ

ωγ

21h

22h



0

0

022

02

022

02

=+

=+

zz

zz

Hhdy

Hd

Ehdy

Ed

W

z

y

x

b

2n

1n

2n

=2medium,

1medium,22

212

h

hh

solve

022

2

=+ ψψ hdyd

general solution: 02 >h

νjh =

realh ( ) ( )hyBhyA cossin +=ψ

02 <h yy DeCe ννψ +− +=

medium 1

medium 2

real1h

νjh =222

2 ν−=h

−−=

+=

2

222

2

122

1

nc

nc

h

ωγν

ωγ

( ) 21

22

21

2

hnnc

−−

= ων


Faculdade de EngenhariaLower and upper values of phase constant

W

z

y

x

b

2n

1n

2n

propagating modes à βγ j=

( )( )2

1222

211

221

cn

cnh

ωγν

ωγ

−−=

+=

21

2

1 hnc

−

=

ωβ 2

2

2 νω

+

= n

c

21 nc

nc

ωβ

ω>>1n

cω

β < 2ncω

β >

21 nn >


Faculdade de EngenhariaLongitudinal components determination

W

z

y

x

b

2n

1n

2n

( ) ( )yhByhA 11 cossin +=ψ

yy DeCe ννψ +− +=

medium 1

medium 2

TM waves à ψ=0zE

TE waves à ψ=0zH

boundary conditions

2at continuousand byHE zz ±=

( ) ( ) ( )

−<

≤+

>

=

−

2,

2,cossin

2,

11

byDe

byyhByhA

byCe

y

y

y

ν

ν

ψ

exponential decay in medium 2

2at continuous by ±=ψ

+

−=

+

=

−

−

2cos

2sin

2cos

2sin

112

112

bhB

bhADe

bhB

bhACe

b

b

ν

ν


Faculdade de EngenhariaLongitudinal components determination

W

z

y

x

b

2n

1n

2n

+

−=

+

=

−

−

2cos

2sin

2cos

2sin

112

112

bhB

bhADe

bhB

bhACe

b

b

ν

ν

211

211

2cos

2sin

2cos

2sin

b

b

ebh

Bbh

AD

ebh

Bbh

AC

ν

ν

−

+

−=

+

=

( ) ( ) ( )

−<

+

−

≤+

>

+

=

+

−−

2,

2cos

2sin

2,cossin

2,

2cos

2sin

211

11

211

byebhBbhA

byyhByhA

bye

bhB

bhA

yb

y

by

ν

ν

ψ

( ) ( ) ( )

−<

≤+

>

=

−

2,

2,cossin

2,

11

byDe

byyhByhA

byCe

y

y

y

ν

ν

ψ


Faculdade de EngenhariaEven and odd modes

W

z

y

x

b

2n

1n

2n

( ) ( ) ( )

−<

+

−

≤+

>

+

=

+

−−

2,

2cos

2sin

2,cossin

2,

2cos

2sin

211

11

211

bye

bhB

bhA

byyhByhA

byebhBbhA

yby

by

ν

ν

ψ

TM waves à ψ=0zE

TE waves à ψ=0zH

even modes

( )

−<

≤

>

=

+

−−

2,

2cos

2,cos

2,

2cos

21

1

21

even

bye

bhB

byyhB

bye

bhB

by

by

ν

ν

ψ

odd modes

( )

−<

≤

>

=

+

−−

2,

2sin

2,sin

2,

2sin

21

1

21

odd

bye

bhA

byyhA

bye

bhA

by

by

ν

ν

ψ

0=A 0=B

odd TM modes à odd0 ψ=zE

even TM modes à even0 ψ=zE

odd TE modes à odd0 ψ=zH

even TE modes à even0 ψ=zH


Faculdade de EngenhariaEven TM modes

W

z

y

x

b

2n

1n

2n

( )

−<

≤

>

=

+

−−

2,

2cos

2,cos

2,

2cos

21

1

21

even

byebhB

byyhB

byebhB

by

by

ν

ν

ψ

even TM modes à even0 ψ=zEand00 =zH

dydE

hE

dydH

hj

E

dydH

hH

dydE

hj

H

zy

zx

zy

zx

0

20

0

20

0

20

0

20

γ

ωµ

γ

ωε

−=

−=

−=

=

0=

0=

( )

−<

−

<−

>

=

+

−−

2,

2cos

2,sin

2,

2cos

212

11

1

212

0

bye

bhB

j

byyhB

hj

bye

bhB

j

H

by

by

x

ν

ν

νωε

ωενωε

( )

−<

<

>

−

=

+

−−

2,

2cos

2,sin

2,

2cos

21

11

21

0

bye

bhB

j

byyhB

hj

bye

bhB

j

E

by

by

y

ν

ν

νβ

βνβ

boundary conditions

2at continuous byH x ±=

−=

2sin

2cos 1

1

112 bhB

hjbh

Bj ωενωε


Faculdade de EngenhariaEven TM modes – characteristic relation

W

z

y

x

b

2n

1n

2n

−=

2sin

2cos 1

1

112 bhB

hjbh

Bj ωενωε

−=2

cot 1

1

21

bhh

εε

ν

rrn εµ=

1if =rµ

rε=

−=

2cot 1

2

1

21

bhnn

hν

( ) 21

22

21

2

hnnc

−−

= ων ( )

−=−−

2

cot 11

21

22

21

22

2

1 bhhhnn

cnn ω

characteristic relation

gives the characteristic value h1 as afunction of the waveguide and frequency


Faculdade de EngenhariaEven TM modes – solutions of the characteristic equation

W

z

y

x

b

2n

1n

2n

( )

−=−−

2

cot 11

21

22

21

22

2

1 bhhhnn

cnn ω

characteristic equation

( )BxxxA cot22 −=−

0 5 10 15 20 25 30 35 40 45 50-50

-40

-30

-20

-10

0

10

20

30

40

50

equation of the form:

only 3 solutions!(in this case) notes

2. there is a propagating mode foreach solution

1. finite number of solutions

3. number of propagating modesincreases with the frequency


Faculdade de EngenhariaEven TM modes – example

W

z

y

x

b

2n

1n

2n cmb

GHzf

2

4

25

01

02

====

εεεε

21

1

2

==

nn

3.3051,1 =h

2.8713,1 =h

2.6062,1 =h

( ) ( )112

1

2

01.0cot3

5004 hhh −=−

π



-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Even TM1 mode – example

W

z

y

x

b

2n

1n

2n cmb

GHzf

2

4

25

01

02

====

εεεε

21

1

2

==

nn 3.3051,1 =h 8541 =ν

( ) 21

22

21

2

hnnc

−−

=

ων

BE z0

( )

−<

≤

>

=

+

−−

2,

2cos

2,cos

2,

2cos

21

1

21

0

bye

bhB

byyhB

bye

bhB

Eb

y

by

z

ν

ν

medium 1

exponential decayin medium 2

y


Faculdade de EngenhariaEven modes TM1, TM2 and TM3 – example

W

z

y

x

b

2n

1n

2n

cmb

GHzf

2

4

25

01

02

====

εεεε

2.8713,1 =h

2.6062,1 =h example

8541 =ν

BEz0

BEz0

BEz0

TM1

TM2

TM3

3.3051,1 =h

2523 =ν

5.6742 =ν

decay ratedecreases withincrease of themode order


Faculdade de EngenhariaDependence of the even TM1 mode on the frequency – example

W

z

y

x

b

2n

1n

2n

cmb 24 01

02

===

εεεε

3121,1 =h

3.3051,1 =hexample

5.1201 =ν

BEz0TM1

2641,1 =h

2.36141 =ν

8541 =ν

cutoff frequency

TM1

TM1

GHz8=f

GHz25=f

GHz100=f

BEz0

BEz0

decay ratedecreases with frequency decrease

0=ν wave not confined to the guide

f small


Faculdade de EngenhariaEven TM modes– cutoff frequency

W

z

y

x

b

2n

1n

2n

cutoff condition: 0=ν

2

222

−−= n

cω

γν 0=2

22

−= n

cω

γ

( )22

21

22

1 nnc

h −

=

ω

2

122

1

+= n

ch

ωγ

22

211 nn

ch −=

ω

−=

2cot 1

2

1

21

bhnn

hν 02

cot 1 =

−

bh0= K,2,1,

21

21 =

−= nn

bhπ

( ) K,2,1,21

22

21

=−

−

= nnnb

cnf evenTMc


Faculdade de EngenhariaEven TM modes – dominant mode

W

z

y

x

b

2n

1n

2n

( ) K,2,1,21

22

21

=−

−

= nnnb

cnf evenTMc

cutoff frequency increases as thethickness of the guide decreases

dominant even TM mode à even TM1

( )22

212

1nnb

cf evenTMc

−=

second even TM mode à even TM2

( )22

212

32

nnb

cf evenTMc

−=

propagation of a single even TM mode

( ) ff evenTMc >2

22

212

3

nnf

cb

−<


Faculdade de EngenhariaCutoff frequency of even TM modes – example

W

z

y

x

b

2n

1n

2n

( ) K,2,1,21

22

21

=−

−

= nnnb

cnf evenTMc

cmb

GHzf

2

4

25

01

02

====

εεεε

( ) GHz3.41

=evenTMcf

( ) GHz7.213

=evenTMcf

( ) GHz132

=evenTMcf

M

( ) GHz3.304

=evenTMcf

propagating modes

evanescent modes


Faculdade de EngenhariaOdd TM modes – dominant mode

W

z

y

x

b

2n

1n

2n

dominant odd TM mode à odd TM1 mode

( ) 0oddTM1=cf

next odd TM modeà odd TM2 mode

( )22

21

oddTM2nnb

cfc

−=

only one odd TM mode if ( ) ffc >oddTM2

( ) ( )K,2,1,

122

21

oddTM =−

−= n

nnb

cnfc

always present

note: dominant even TM modeà even TM1 mode

( )22

21

evenTM2

1nnb

cfc

−=

odd TM1 even TM1 odd TM2

1º 2º 3º …

( )

=−−

2

tan 11

21

22

21

22

2

1 bhhhnn

cnn ω characteristic relation

cutoff frequency


Faculdade de EngenhariaPlanar dielectric waveguides – summary

W

z

y

x

b

2n

1n

2n

−=

2cot 1

1

2

1

2 bhh

nn

ν

22

21

21

nnb

cnfc

−

−

=

−=

2cot 1

1

bhhν

=

2tan 1

1

2

1

2 bhh

nn

ν( )

22

21

1

nnb

cnfc

−

−=

=

2tan 1

1

bhhνTE

TM

ODD

TE

TM

EVEN

CUTOFF FREQUENCY

CHARACTERISTIC RELATIONMODES

( ) 21

22

21

2

hnnc

−−

= ων

K,2,1=n


Faculdade de EngenhariaPlanar dielectric waveguides – dominant modes

W

z

y

x

b

2n

1n

2n

odd TM1 even TM1 odd TM2

1º 2º 3º

…

22

21

21

nnb

cnfc

−

−

=

( )22

21

1

nnb

cnfc

−

−=

K,2,1=n

even modesà

odd modesà

odd TE1 even TE1 odd TE2

dominant modes0=cf

always present

22

212 nnb

cfc

−= single-mode regime: 2

2212 nnb

cf

−<

example

cm2

2;1 12

=

==

b

nn GHz17.2<f


Faculdade de EngenhariaNumerical aperture and acceptance angle

W

z

y

x

b

2n

1n

2n

n1

n2

n2

iθ θ φ

z

air

21 nn >

( ) ( )in θφ sincos1 =

≥ −

1

21sinnn

φtotal internal reflection1

2sinnn

≥φ2

1

21cos

−≤

nn

φ

( ) 22

21sin nni −≤θ

notes

1. numerical aperture (NA) à

2. acceptance angle à

22

21 nn −

( )NAA1sin −=θ


Faculdade de EngenhariaTotal internal reflection – field in medium 2

W

z

y

x

b

2n

1n

2n

n1

n2

n2

iθ θ φ

z

air y

( )21 nn >

1

2sinnn

≥φ

tnn φφ sinsin 21 = φφ sinsin2

1

nn

t =

1sin ≥tφ

( )2

2

1 sin1cos

−±= φθ

nn

t 1sin2

2

1 −

±= φ

nn

j

medium 2:

rkjerr

⋅− 2 rakj nterr

⋅−=

ˆ2

zya ttnt ˆsinˆcosˆ φφ +=

( )zykj tteφφ sincos2 +−

=r ( ) ynnkznnkj

ee1sinsin 2

212212 −±−=

φφrr

propagation along +z exponentialdecay

1sincos2

2

1 −

−= φφ

nn

jt


Faculdade de EngenhariaIntermodal distortion

n1

n2

n2

−=

2

11

ff

v cg

εµmodal analysis à different modes propagate with different velocities

propagate in medium 1 withvelocity 1ncv =

geometrical optics à modes are represented by vectors indicatingthe propagation direction of plane waves

propagation alonga shorter distance

propagation alonga larger distance

different propagation timesvlt =

signal components propagated in different modesare received at different time instants

signal distortion

distortion


Faculdade de EngenhariaGraded-index waveguides

n1

n2

n2

y

n n1 n2

“step-index”

n1

n2

n2

y

n n1 n2

“graded-index”

ncv = lower velocity in central region

smaller distance and velocity larger distance and velocity(approximately) equal times

less distortion


Faculdade de EngenhariaTotal internal reflection – allowed modes

W

z

y

x

b

2n

1n

2n

A

B

C

1n

2nwavefronts

C à immediately after reflection

A and C in the same wavefront

π=∠−∠ 2ofmultipleAC

Γ∠+−Γ∠+−∠=∠ BCAB lklkAC 11

propagation along distance là acquired phase:

in each reflectionà acquired phase: Γ∠

( )BCAB llkA +−Γ∠+∠= 12

lk1− lnc 1ω

−=

( ) πω

nllnc BCAB 221 =Γ∠++−


Faculdade de EngenhariaTotal internal reflection – allowed modes

W

z

y

x

b

2n

1n

2n

A

B

C

1n

2n

( )φcosBClb = ( )φcosb

lBC =

( )φ2cosBCAB ll =( ) ( )φφ

2coscos

b=

( ) ( )( )12coscos

+=+ φφ

bll BCAB ( )φcos2b=

bφ

( ) πω

nllnc BCAB 221 =Γ∠++− ( ) πφ

ωnbn

c=Γ∠+− cos1


Faculdade de EngenhariaTotal internal reflection – reflection coefficient phase

A

B

C

1n

2n

b φ

oblique incidence:

ti

ti

nnnn

φφφφ

coscoscoscos

21

21

+−

=Γ⊥

it

it

nnnn

φφφφ

coscoscoscos

21

21| | +

−=Γ

1sincos2

2

1 −

−= φφ

nn

jt

( ) ( )( ) ( ) 2

222

11

22

2211

sincos

sincos

nnjn

nnjn

−−

−+=Γ⊥

φφ

φφ

( ) ( )( ) ( ) 2

222

1122

22

2211

22

| |sincos

sincos

nnjnn

nnjnn

−−

−−−=Γ

φφ

φφ

( )( )

−=Γ∠ −

⊥ φφ

cossin

tan21

22

2211

nnn

( )( )

−+=Γ∠ −

φ

φπ

cos

sintan2 2

2

22

22111

||n

nnn

note:

z

y

⊥ polarization à xEE x ˆ=r

0=zE TE wave

|| polarization à xHH x ˆ=r

0=zH TM wave

TEΓ∠

TMΓ∠


Faculdade de EngenhariaTotal internal reflection – equation for allowed modes

A

B

C

1n

2n

b φ

( )( )

−=Γ∠ −

φφ

cossin

tan21

22

2211

nnn

TE

( )( )

−+=Γ∠ −

φ

φπ

cos

sintan2 2

2

22

22111

n

nnnTM

z

y( ) πφ

ωnbn

c=Γ∠+− cos1

L,2,1=n

( )( ) modes TE,

cossin

tan2cos1

22

2211

1 π=

φ−φ

+φω

− − nn

nnbn

c

( )( )

( ) modes TM,1cos

sintan2cos 2

2

22

22111

1 π−=

φ

−φ+φ

ω− − n

n

nnnbn

c

propagating modes obey these equations


Faculdade de EngenhariaTotal internal reflection – characteristic equations

A

B

C

1n

2n

b φ z

y

( )( ) TE,

cos

sintan2cos

1

22

2211

1 π=

φ

−φ+φ

ω− − n

n

nnbn

c

( )( )

( ) TM,1cos

sintan2cos 2

2

22

22111

1 π−=

φ

−φ+φ

ω− − n

n

nnnbn

c

geometrical optics:

modal analysis:

( )

−=−−

2

cot 11

2

1

221

22

21

2 bhh

nn

hnncω

( )

−=−−

2cot 1

121

22

21

2 bhhhnn

cω

( )

=−−

2

tan 11

2

1

221

22

21

2 bhh

nn

hnncω

( )

=−−

2tan 1

121

22

21

2 bhhhnn

cω

K,2,1=n

even TM

even TE

odd TM

odd TE

it is possible to show that these two sets of equations are equivalent



z

y

Total internal reflection – equivalence of the two approaches

−=

2cot 1

1bhhν

=

2tan 1

1bhhν

even TE

odd TE

( )( ) πφ

φφω n

nnn

bnc

=

−+− −

cossin

tan2cos1

22

2211

1

geometrical optics

modal analysisß coreodd TE modes: ( ) zjx eyhA

hj

E βωµ −−= 11

0 cos

( ) ( )[ ]zyhjzyhjx eeA

hj

E ββωµ +−+−− +−= 11

1

0

2

2cos

θθ

θjj ee −+

=

φ

( )zykzyh φφβ sincos11 +±=+±

φωβ

φω

sin

cos

1

11

nc

nc

h

=

=

( )π

ωβn

h

cnbh =

−+− −

1

22

21

1 tan2

even TE modes:note

( ) zjx eyhB

hj

E βωµ −= 11

0 sin ( ) ( )[ ]zyhjzyhj eeBh

j ββωµ +−+−− −= 11

1

0

2

equation is valid for both even andodd TE modes


Faculdade de EngenhariaTotal internal reflection – equivalence of the two approaches

−=

2cot 1

1bhhν

=

2tan 1

1bhhν

even

odd

( )( ) πφ

φφω n

nnn

bnc

=

−+− −

cossin

tan2cos1

22

2211

1

geometrical optics

modal analysisTE modes

π

ωβ

nh

nc

bh =

−

+− −

1

2

22

11 tan2

πν

nh

bh =

+− −

1

11 tan2

2

222

−= n

cω

βν

+=

22tan 1

1bh

nhπ

ν

+=−=

=

+

12,cot2,tan

2tan

mnmn

nθθ

θπ

−

=νoddn,

2cot

evenn,2

tan

11

11

bhh

bhh

equation from geometrical optics is equivalentto characteristic equations from modal analysis


Faculdade de EngenhariaOptical fibers

z

n1 a

n2

lossless materials


cylindrical geometry à cylindrical coordinates


Faculdade de EngenhariaOptical fibers – boundary conditions

z

n1 a

n2

continuoustanE continuoustanHand

boundary conditions

arHH

arEE

z

z

=

=

φ

φ

atcontinuousnda

tacontinuousand00

00

cylindrical coordinates

( ) zzr ezEErEE γ

φ φ −++= ˆˆˆ 000r

( ) zzr ezHHrHH γ

φ φ −++= ˆˆˆ 000r


Faculdade de EngenhariaOptical fibers – field determination

z

n1 a

n2


µεωγ 222 +=h

1. solve

0

00202

0202

=+∇

=+∇

zzr

zzr

HhH

EhE

φ

φ

2

2

22 11

φφ ∂∂+

∂∂

∂∂=∇

rrr

rrr

∂

∂−∂

∂−=

∂

∂+∂

∂−=

∂

∂+∂

∂−=

∂∂−

∂∂−=

rHjE

rhE

Hr

jr

Eh

E

rEjH

rhH

Er

jr

Hh

H

zz

zzr

zz

zzr

00

20

00

20

00

20

00

20

1

1

1

1

ωµφ

γ

φωµγ

ωεφ

γ

φωεγ

φ

φ

>

+

≤

+

=

arnc

arnch

,

,

2

22

2

12

2

ωγ

ωγ

21h

22h


Faculdade de EngenhariaOptical fibers – wave equation

z

n1 a

n2

solve

0

00202

0202

=+∇

=+∇

zzr

zzr

HhH

EhE

φ

φ

>≤

=arharh

h,,

22

212022 =+∇ ψψφ hr

( ) ( ) ( )φφψ Φ= rRr ,

( ) φφ jnAe=Φ

( ) ( ) ( ) ( ) 02222

22 =−++ rRnrh

drrdR

rdr

rRdr Bessel differential equation

general solution: ( ) ( )hrJArR n= (if region of interest includes r =0) 02 >h realh

02 <h νjh = ( ) ( ) ( )rCKrIBrR nn νν +=

Bessel functionsof the 1st kind

modified Besselfunctions of the1st kind

modified Besselfunctions of the2nd kind


Faculdade de EngenhariaModified Bessel functions of the 1st kind

z

n1 a

n2

n integer

( ) ( ) ( )( )∑

∞

=

+−

+==

0

2

!!2

k

kn

nn

n knkxjxJjxI

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

12

x

0I

1I

2I

3I

( ) ∞=∞→

xInxlim

should not be part of the

solution when the region of

interest includes the infinite

nI



0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

12

x

Modified Bessel functions of the 2nd kind

z

n1 a

n2

n integer

0K

1K

2K

3K( ) ∞=

→xKnx 0

lim

should not be part of the

solution when the region of

interest includes the origin

nK

( ) ( ) ( ) ( )[ ]xIxIp

xK ppnpn −= −→ ππ

sin2lim

( ) 0lim =∞→

xKnx


Faculdade de EngenhariaSolution of the wave equation

z

n1 a

n2

( ) ( )hrJArR n=realh

νjh = ( ) ( ) ( )rCKrIBrR nn νν +=

( ) ( ) ( )φφψ Φ= rRr ,

( ) φφ jnAe=Φ

∞=∞→rlim ∞=→0lim r

guided wave àreal1h

νjh =2

( ) ( )( )

>≤

=arerBK

arerhAJr

jnn

jnn

,

,, 1

φ

φ

νφψ

( ) 21

22

21

2

hnnc

−−

=

ων

22

221

2

1 νωω

β +

=−

= n

chn

c

21 nc

nc

ωβ

ω>> 21 nn >

0lim =∞→r


Faculdade de EngenhariaLongitudinal components

z

n1 a

n2

( ) ( ) ( )φφψ Φ= rRr ,

( ) φφ jnAe=Φ

cladding

( ) ( )( )

>≤

=arerBK

arerhAJr

jnn

jnn

,

,, 1

φ

φ

νφψ

( )( ) φ

φ

jnnz

jnnz

erhBJH

erhAJE

10

10

=

=core

( )( ) φ

φ

ν

νjn

nz

jnnz

erDKH

erCKE

=

=0

0

TM modes à 00 =zH

TE modes à 00 =zE

HE and EH modes à 0and0 00 ≠≠ zz HE

note:

hybrid modes


Faculdade de EngenhariaTransverse components

z

n1 a

n2

( )( ) φ

φ

jnnz

jnnz

erhBJH

erhAJE

10

10

=

=

∂

∂−

∂∂

−=

∂

∂+∂

∂−=

∂

∂+

∂∂

−=

∂

∂−

∂∂

−=

rH

jE

rhE

Hr

jr

Eh

E

rE

jH

rhH

Er

jr

Hh

H

zz

zzr

zz

zzr

00

20

00

20

00

20

00

20

1

1

1

1

ωµφ

γ

φωµγ

ωεφ

γ

φωε

γ

φ

φ

1hh =

φωεβ jn

nnr erhAJr

nrhBJhj

hH

+−= )()('1

11

1121

0

φφ ωε

β jnnn erhAJhjrhBJ

rn

hH

+−−= )(')(1

111121

0

φωµβ jn

nnr erhBJr

nrhAJhj

hE

−−= )()('1

10

1121

0

φφ ωµ

β jnnn erhBJhjrhAJ

rn

hE

−−−= )(')(1

110121

0

νjh =( )( ) φ

φ

ν

νjn

nz

jnnz

erDKH

erCKE

=

=0

0

φνωενβνν

jnnnr erCK

rnrDKjH

+= )()('1 2

20

φφ ννωενβ

νjn

nn erCKjrDKrnH

+−= )(')(1

220

φνωµ

νβνν

jnnnr erDK

rn

rCKjE

−= )()('1 0

20

φφ ννωµνβ

νjn

nn erDKjrCKrnE

−−= )(')(1

020

cladding

core


Faculdade de EngenhariaBoundary conditions

z

n1 a

n2

arHH

arEE

z

z

=

=

φ

φ

at continuousnda

at continuousand00

00

( ) ( ) ( ) ( ) 0'' 021

1

012

1

=+++ aKj

DaKan

CahJh

jBahJ

ahn

A nnnn νν

µων

νβµωβ

( )( )

ν= φ

φ

cladding, core,10

jnn

jnn

z erCKerhAJ

E ( ) ( )aCKahAJ nn ν=1arE z =at continuous0

( ) ( ) 01 =− aCKahAJ nn ν

ß similarly( ) ( ) 01 =− aDKahBJ nn ν

( ) ( ) ( ) ( ) 0'' 221

1

112

1

=−+− aKj

CaKan

DahJh

jAahJ

ahn

B nnnn νν

εων

νβεωβ


Faculdade de EngenhariaBoundary conditions

z

n1 a

n2 ( ) ( ) ( ) ( ) 0'' 0

211

012

1

=+++ aKj

DaKan

CahJh

jBahJ

ahn

A nnnn νν

µων

νβµωβ

( ) ( ) 01 =− aCKahAJ nn ν

in matrix notation

( ) ( ) 01 =− aDKahBJ nn ν

( ) ( ) ( ) ( ) 0'' 221

1

112

1

=−+− aKj

CaKan

DahJh

jAahJ

ahn

B nnnn νν

εων

νβεωβ

( ) ( )( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

0

''

''

0000

021

1

012

1

22

121

11

1

1

1

=

−−

D

CB

A

aKj

aKan

ahJh

jahJ

ahn

aKan

aKj

ahJahn

ahJh

jaKahJ

aKahJ

nnnn

nnnn

nn

nn

νν

µων

νβµωβ

ννβ

νν

εωβεων

ν


Faculdade de EngenhariaCharacteristic relation

z

n1 a

n2 nontrivial solution

( ) ( )( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

0

''

''

0000

021

1

012

1

22

121

11

1

1

1

=

−−

DCB

A

aKj

aKan

ahJh

jahJ

ahn

aKan

aKj

ahJahn

ahJh

jaKahJ

aKahJ

nnnn

nnnn

nn

nn

νν

µων

νβµωβ

ννβ

νν

εωβεων

ν

( ) ( )( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

0

''

''

0000

021

1

012

1

22

121

11

1

1

1

=−−

aKj

aKan

ahJh

jahJ

ahn

aKan

aKj

ahJahn

ahJh

jaKahJ

aKahJ

nnnn

nnnn

nn

nn

νν

µων

νβµωβ

ννβ

νν

εωβεων

ν

2

221

222

11

121

11

12

11)()('

)()('

)()('

)()('

+

=

+

+

νβ

ννν

νννω

han

aKaK

nahJh

ahJn

aKaK

ahJhahJ

c n

n

n

n

n

n

n

n characteristic relation forTM, TE, HE and EH modes


Faculdade de EngenhariaCharacteristic relation – TM and TE modes

z

n1 a

n2

for n=0

( ) ( )( ) ( )

( ) ( )

( ) ( )

0

'0'0

0'0'

0000

00

101

0

02

101

1

010

010

=

−−

DCBA

aKj

ahJh

j

aKj

ahJh

jaKahJ

aKahJ

νν

µωµω

νν

εωεων

ν

( ) ( )( ) ( ) 0'' 0

210

1

1

010

=

−− C

AaKahJ

h

aKahJ

ννεε

ν ( ) ( )( ) ( ) 0'1'1

0101

010

=

D

BaKahJ

h

aKahJ

νν

ν

nontrivial solution B=D=0 allowed

TM modes

nontrivial solution A=C=0 allowed

TE modes

0=zH 0=zE

0)(

)()(

)(

0

122

101

1121 =+

aKaK

nahJh

ahJn

ννν 0

)()(

)()(

0

1

101

11 =+aK

aKahJh

ahJνν

ν


Faculdade de EngenhariaCutoff frequency

z

n1 a

n2

cutoff condition à 0=ν

0)( 10 =ahJ

0)( 11 =ahJ

0)( 1 =ahJn

)(1

)(1 11

1122

21 ahJ

nah

ahJnn

nn −=

+ −HEnp

EHnp

≥ 2

HE1p EH1p1

TE0p TM0p0

cutoffmoden

notes

1. cutoff frequency for HE11 mode =0

2. next modes: TE01 and TM01 (cutoff frequency given by the 1st zero of J0 à 2.4048)


Faculdade de EngenhariaNormalized frequency

z

n1 a

n2

normalized frequency(V parameter)

( ) 2221

2 ahV ν+= ( )22

21

2

nnca

−

=

ω

2

122

1

+= n

ch

ωγ

2

222

−−= n

cω

γν

22

21

0

2 nnaV −=λπ

cutoff à 0=ν ( ) ( )cut1cut ahV =

wavelength in vacuum

011 =→ cfHE

4048.2and 10101 >→ ahTETM

single mode à 4048.2<V

multimode à 4048.2>V

note

propagation of


Faculdade de EngenhariaNormalized constant phase versus normalized frequency

For a multimode fiber, with a large number of propagating modes, the number of propagating modes M is approximately:

note

2

2VM ≅

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