C3 Planar Optical Waveguide

30
BITS Pilani Pilani Campus Wave Propagation In Planar Waveguides RAHUL SINGHAL

description

Slides on Optical Waveguide from BITS Pilani. Course - Fiber optics and Optoelectronics

Transcript of C3 Planar Optical Waveguide

Page 1: C3 Planar Optical Waveguide

BITS Pilani Pilani Campus

Wave Propagation In Planar Waveguides

RAHUL SINGHAL

Page 2: C3 Planar Optical Waveguide

Plane TEM Wave

polarized is xE

polarized is yH

Transverse means E & H are perpendicular

to direction of propagation

Page 3: C3 Planar Optical Waveguide

The net electric flux through any closed

surface is equal to 1⁄ε 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.

Ampère's law with Maxwell's addition states that

magnetic fields can be generated in two ways:

by electrical current (this was the original

"Ampère'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.

Page 4: C3 Planar Optical Waveguide

2

2

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

ˆˆˆ

‘Del’ or Nabla Operator

Page 5: C3 Planar Optical Waveguide

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, ρ = 0;σ = 0.

From Eq based on Gauss law,

Page 6: C3 Planar Optical Waveguide

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,

Page 7: C3 Planar Optical Waveguide

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

Page 8: C3 Planar Optical Waveguide

0..).( 0 EED r

0.).(0 EE rr EE r

r

).(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

μ

r

rr

0

00 1

Page 9: C3 Planar Optical Waveguide

)(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 xyzxyz ˆˆˆ

zyx HkHjHi

tˆˆˆ

0

t

En

t

E

t

DH r

2

00

y

H

x

Hk

x

H

z

Hj

z

H

y

Hi xyzxyz ˆˆˆ

zyx EkEjEi

tn ˆˆˆ2

0

)(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 ,,,

γ =0 without any loss of generality

Page 10: C3 Planar Optical Waveguide

)()()()( ˆˆˆˆ zti

y

zti

z

zti

x

zti

y eEx

keEx

jeEz

jeEz

i

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

yHE

)()()(

0ˆˆˆ zti

z

zti

y

zti

x eHt

keHt

jeHt

i

x

Ek

x

EEijEii

yzxy

ˆˆˆ zyx HikHijHii )(ˆ)(ˆ)(ˆ

000

Comparing on both sides,

xy

xy

HE

HiEi

0

0

or,

)(

yz

x Hix

EEi )(0

z

yHi

x

E)(0

Page 11: C3 Planar Optical Waveguide

)()()()( ˆˆˆˆ zti

y

zti

z

zti

x

zti

y eHx

keHx

jeHz

jeHz

i

zero are terms all Again,y

)(2

0

)(2

0

)(2

0ˆˆˆ zti

z

zti

y

zti

x eEt

kneEt

jneEt

in

x

Hk

x

HHijHii

yzxy

ˆˆˆ

zyx EnikEnijEnii )(ˆ)(ˆ)(ˆ 2

0

2

0

2

0

Comparing on both sides,

xy

xy

EnH

EniHi

2

0

2

0

or,

)(

yz

x Enix

HHi )( 2

0

z

yEni

x

H 2

0

Page 12: C3 Planar Optical Waveguide

xy HE 0

yz

x Hix

EEi 0

z

yHi

x

E0

xy ExnH )( 2

0

yz

x Exnix

HHi )(2

0

z

yExni

x

H )( 2

0

TE Modes, Ey component

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

TM Modes, Hy component

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

Page 13: C3 Planar Optical Waveguide

PLANAR OPTICAL WAVEGUIDE

Page 14: C3 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

Page 15: C3 Planar Optical Waveguide

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

Page 16: C3 Planar Optical Waveguide

mi ia sin4

2/12

2

2

11

max

4sin4sin4nn

aanaiM m

m

m

2/12

2

2

1

4nn

ai

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 θi,

corresponds to a

particular mode

with its own βi,

in z-direction.

Mode corresponding imax is refracted at interface,

& may propagate in cladding – is called a radiation mode

Page 17: C3 Planar Optical Waveguide

TE Modes of a Symmetric

Step-Index Planar Waveguide

)()( 22 xnxn

yyy ExniExix

Ei )(1 2

0

00

)(xEE yy

0)(22

00

2

2

2

yy

yExnE

dx

Ed

kcc

Exnkdx

Edy

y

;&1

where,

0)(

200

222

2

2

Page 18: C3 Planar Optical Waveguide

022

1

2

2

2

y

yEnk

dx

Ed

2

1)(

n

nxn

ax for

ax for

ax for

022

2

2

2

2

y

yEnk

dx

Ed ax for

Let, 22

1

22

1

22 nku

2

2

22

2

222 nkw02

2

2

y

yEu

dx

Edax for

02

2

2

y

yEw

dx

Edax for

&,

2

2

22

2

22

1

22

1 nknk

Page 19: C3 Planar Optical Waveguide

)()(

)()(

xExE

xExE

yy

yy

Anti-symmetric Modes

Symmetric Modes

xwyCe

uxAxE

cos)(

ax for

ax for

waCeuaA )cos(

wawCeuauA )sin(

wuau )tan(

wauaua )tan(

2

2

2

1

22

2

2222

1

222 nnknknkwu

2

2

2

1

2

2

2

2

2

1

2222 2nnannakwaua

2

2

2

1

22 2nn

awauaV

2/122)tan( uaVuaua

Symmetric Modes

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

Page 20: C3 Planar Optical Waveguide

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

Page 21: C3 Planar Optical Waveguide

wauahere ; ,

Page 22: C3 Planar Optical Waveguide

22

2

2

2

1

2

2

2

1

V

wa

V

uab

m = 0

m = 1 m = 2

Page 23: C3 Planar Optical Waveguide

2

mVua c

2

2

2

1

2nn

aV

2

2

2

1

2

2nn

am

2

2

2

122

nn

ma

Page 24: C3 Planar Optical Waveguide

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 λ = 0.90 μm. Calculate the maximum and minimum

values of the propagation constant , β.

Page 25: C3 Planar Optical Waveguide

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

Page 26: C3 Planar Optical Waveguide

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

Page 27: C3 Planar Optical Waveguide

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

Page 28: C3 Planar Optical Waveguide

For symmetric TE modes,

,

,cos)(

axCe

axuxAxE xwy

Thus, dxuxAdxuxAP

aa

in 0

2

00

2

0

)2cos1(2

1cos2

2

1

or,

)2sin

2

1

22sin

2

1 2

00

2

0

uau

aAuxu

xAP

a

in

Page 29: C3 Planar Optical Waveguide

Similarly,

a

xw

a

xw

out ew

CdxCeP2

12

2

1 2

0

2

0

or,

wa

out ew

CP 22

0 2

1

Total Power, P

wa

outin ew

Cuau

aAPPP 22

0

2

0 2

12sin

2

1

2

wae

wA

Cua

uaA 2

2

2

0

12sin

2

1

2

Page 30: C3 Planar Optical Waveguide

Substitute uaeAC wa cos/

ua

wua

uaAP 22

0

cos1

2sin2

1

2

w

ua

u

uaua

waA

22

0

sin2cossin222

4

waAuauawa

uwa

uaua

waA

1

2tan

cossin222

4

2

0

2

0

wauaua tan Since,

A similar expression can be calculated for asymmetric modes.