BITS Pilani Pilani Campus

Wave Propagation In Planar Waveguides

RAHUL SINGHAL

Plane TEM Wave

polarized is xE

polarized is yH

Transverse means E & H are perpendicular

to direction of propagation

The net electric flux through any closed

surface is equal to 1 times the net electric charge enclosed within that closed surface.

It states that the magnetic field B has

divergence equal to zero, in other

words, it is equivalent to the

statement that magnetic monopoles

do not exist.

Ampre's law with Maxwell's addition states that

magnetic fields can be generated in two ways:

by electrical current (this was the original

"Ampre's law") and by changing electric fields

(this was "Maxwell's addition").

The induced electromotive force in

any closed circuit is equal to the

negative of the time rate of change of

the magnetic flux through the circuit.

22

2

A/mdensity current Electric;J

A/mIntensity Field Magnetic

)Sec./m-Vor (Telsa TDensity Flux Magnetic ;

C/mDensity Flux Electric;

V/mIntensity Field Electric

E

H

HB

ED

E

0.

.

B

D

t

DJH

t

BE

MAXWELL EQUATIONS

zk

yj

xi

Del or Nabla Operator

0.

0.

B

D

t

DH

t

BE

t

B

t

BE

)(

2

2)(

t

D

t

D

tt

H

27

00 /104; ANr

2212

00 /10854.8; NmCr

Inside an ideal dielectric, = 0; = 0.

From Eq based on Gauss law,

EEE 2).(

But,

2

2

2

2

2

22

zyx

where,

0)/.(. DE&,

2

2

2

2

2

22

t

E

t

E

t

DE

02

22

t

EE

02

22

t

HH

Similarly,

02

22

t

1pv

01

2

2

2

2

tvp

02

2

2

22

tc

n

)(exp0 zti

pv

m/s 1031

that so

,1 space, freeor In vacuum

8

cvp

rr

ncv

n

p

rr

/Therefore,

1 with /

medium, isotropican For

0

Solution

0..).( 0 EED r

0.).(0 EE rr EE rr

).(1

.

2

2

002

2

t

E

t

DE r

2

2

00

2).(t

EEE r

0).(2

2

00

2

t

EEE r

0)).(1

(2

2

00

2

t

EEE rr

r

0)()(1

(2

2

00

2

t

HHH rr

r

Similarly,

SOLUTION IN INHOMOGENEOUS MEDIUM

For an isotropic, linear, non-conducting, non-magnetic, but

inhomogenoeous (or heterogeneous) medium,

From curl of Gauss Law,

Rearranging,

Substituting,

0. as medium shomogeneoufor ,0 E

EED

r

rr

0

00 1

)(exp)(E ztixE jj

)(exp)(H ztixH jj

t

H

t

BE

0

y

E

x

Ek

x

E

z

Ej

z

E

y

Ei x

yzxyz

zyx HkHjHi

t

0

t

En

t

E

t

DH r

200

y

H

x

Hk

x

H

z

Hj

z

H

y

Hi x

yzxyz

zyx EkEjEi

tn 20

)(22 xnnr Assume n do not vary

in y and z directions

y- & z- dependence of fields,

will be, in general, of form, )(exp zyi

zoryxj ,,,

=0 without any loss of generality

)()()()( ztiyztizztixztiy eEx

keEx

jeEz

jeEz

i

zero are terms all ,ith not vary w do & As,y

yHE

)()()(

0 zti

z

zti

y

zti

x eHt

keHt

jeHt

i

x

Ek

x

EEijEii

yzxy

zyx HikHijHii )(

)()( 000

Comparing on both sides,

xy

xy

HE

HiEi

0

0

or,

)(

yz

x Hix

EEi )(0

z

yHi

x

E)(0

)()()()( ztiyztizztixztiy eHx

keHx

jeHz

jeHz

i

zero are terms all Again,y

)(2

0

)(2

0

)(2

0 zti

z

zti

y

zti

x eEt

kneEt

jneEt

in

x

Hk

x

HHijHii

yzxy

zyx EnikEnijEnii )()()( 20

2

0

2

0

Comparing on both sides,

xy

xy

EnH

EniHi

2

0

2

0

or,

)(

yz

x Enix

HHi )( 20

z

yEni

x

H 20

xy HE 0

yz

x Hix

EEi 0

z

yHi

x

E0

xy ExnH )( 2

0

yz

x Exnix

HHi )(20

z

yExni

x

H )( 20

TE Modes, Ey component

zxyyzx HHEHEE ,&,,only involves and zero are ,&,,

TM Modes, Hy component

zxyyzx EEHEHH ,&,,only involves and zero are ,&,,

PLANAR OPTICAL WAVEGUIDE

Planar Optical Waveguide

z

y

x

x

z

sin1x

cos10 z-a

+a

x = 0

kn11

n1

n2

knn

m

11

1

22

1nm

2k

sinsin 11 knx

cos1 z

knn

m

11

1

22

1

2cossinn

nmc

22

1

211min cos kn

n

nm

k

2

sin

2

m

x

1nm

mi ia sin4

2/122211max4sin4sin4

nnaana

iM m

m

m

2/122214

nna

i

ia x 24

z

sin1x

cos10 z-a

+a

x = 0

kn11

n1

n2 a + a 2a +

sin

2

m

x

... ,3 ,2 ,1 ,0iEach value of i, corresponds to a

particular mode

with its own i,

in z-direction.

Mode corresponding imax is refracted at interface,

& may propagate in cladding is called a radiation mode

TE Modes of a Symmetric

Step-Index Planar Waveguide

)()( 22 xnxn

yyy ExniExix

Ei )(1 2

0

00

)(xEE yy

0)(22002

2

2

yyy

ExnEdx

Ed

kcc

Exnkdx

Edy

y

;&1

where,

0)(

200

222

2

2

0221222

yy

Enkdx

Ed

2

1)(

n

nxn

ax for

ax for

ax for

0222222

yy

Enkdx

Ed ax for

Let, 221

22

1

22 nku2

2

22

2

222 nkw02

2

2

yy

Eudx

Edax for

022

2

yy

Ewdx

Edax for

&,

22222221221 nknk

)()(

)()(

xExE

xExE

yy

yy

Anti-symmetric Modes

Symmetric Modes

xwyCe

uxAxE

cos)(

ax for

ax for

waCeuaA )cos(

wawCeuauA )sin(

wuau )tan(

wauaua )tan(

222122222221222 nnknknkwu

222122

2

2

2

1

2222 2 nnannakwaua

222122 2

nna

wauaV

2/122)tan( uaVuaua

Symmetric Modes

The continuity of Ey(x) and dEy/dx at x=a gives,

Anti-symmetric Modes

xwy De

x

xuxB

xE

sin

)(

ax for

ax for

wauaua )cot(

2/122)cot( uaVuaua

From Fig. in next slide,

2

122

2

mVmm Anti-symmetric Modes

(m+1)Symmetric Modes

2

222

12

mVm(m+1) Anti-symmetric

Modes

(m+1) Symmetric Modes

... ,2 ,1 ,0m

VM

2

wauahere ; ,

22

2

2

2

1

2

2

2

1

V

wa

V

uab

m = 0

m = 1 m = 2

2mVua c

22212

nna

V

22212

2nn

am

222122

nn

ma

Problem 3.9 What should be the maximum thickness of the

guide slab of a symmetrical SI planar waveguide so that it

supports only the first 10 modes? Take n1 = 3.6, n2 = 3.58,

and = 0.90 m. Calculate the maximum and minimum values of the propagation constant , .

Power Distribution And Confinement Factor

The power flow is given by Poynting vector defined by

S = E H where E and H are expressed in complex form but the actual fields are the real part of the complex form. Thus taking Time Average of the

Poynting vector,

*Re2

1ReRe HEHES

where H* is complex conjugate of H. Thus, time average of S along z-

direction will be given by

)(2

1yxyxz EHHES

For TE modes

yx EH

0

therefore, 2

02

1yz ES

For a particular mode, the power associated per unit area per unit

length in the y-direction will thus be given by

dxEPx

y

2

02

1

The power inside the guide layer (core)

and the power inside the cladding (or outside the guide layer)

dxEP

a

ax

yin

2

02

1

dxEdxEPax

y

a

x

yout

22

02

1

For symmetric TE modes,

,

,cos)(

axCe

axuxAxE xwy

Thus