Lecture 18 Chapter XI Propagation and Coupling of Modes in Optical Dielectric Waveguides – Periodic Waveguides Highlights (a) Periodic (corrugated) WG

Lecture 18

Chapter XI Propagation and Coupling of Modes in Optical Dielectric Waveguides –

Periodic WaveguidesHighlights

(a) Periodic (corrugated) WG and filters; (b) Distributed feedback lasers; (c) EO mode coupling; (d) Directional couplers.

Propagation of optical modes in dielectric films with thicknesses comparable to the wavelength

Generate, guide, modulate, and detect light in thin film configuration---- integrated optics

Basic problem of TE and TM mode propagation in slab dielectric waveguides.

A coupled-mode formalism to describe situations involving exchange of power between modes.

§11.1 Waveguide Modes – A General Discussion

0220

2 (r)E(r)E(r) nk 20

220 )/2( k

)(,, ztieyxt )E()E(r

A mode of a dielectric waveguide is a solution of the wave equation

The solutions are subject to the continuity of the tangential components of E and H at the dielectric interfaces.

0,][, 22202

2

2

2

)E((r))E( yxnkyxyx

We will limit our treatment to planar waveguides …

I

II

III

1n

2n

3n

z

tx

y

x0x

0y


0,)(),( 221

202

2

)( yxEnkyxEx

Region I

0,)(),( 222

202

2

)( yxEnkyxEx

Region II

0,)(),( 223

202

2

)( yxEnkyxEx

Region III

132 nnn Assume

Let’s consider the physical nature of solution as a function

of the propagation constant


(a) 20nk

0)/)(/1( 22 xEE

in all 3 regions

)(xE

x0 t

I II III

(b) & (c) 3020 nknk

0)/)(/1( 22 xEEin region II

0)/)(/1( 22 xEEin region I & III

)(xE

x

I II III

x

0 t

(b)

(c)

Exponential behavior

Confined modes is discrete


(d)

(e) 010 nk

0)/)(/1( 22 xEEin all 3 regions

1030 nknk

0)/)(/1( 22 xEEin region II & III

0)/)(/1( 22 xEE

in region I

)(xE

x

I II III

0 t

Substrate radiation modes

)(xE

x

I II III

0 tRadiation modes

is continuous variable in (d) & (e)


Confined Modes in a Symmetric Slab Waveguide

31 nn 12 nn

dxd

From Maxwell’s curl equations

xy Hi

z

E

yzx Hi

x

E

z

E

zy Hi

x

E

t /HE t /EH

xy Ei

z

H

yzx Ei

x

H

z

H

zy Ei

x

H

dx 0xdx

0211 n

0222 n

0211 n

x

zy


Propagate along the z direction iz /

Transverse Electric (TE) modes :

Electric field is restricted to the transverse plane

yE xH zH

Transverse Magnetic (TM) modes :

Magnetic field is restricted to the transverse plane

yH xE zE

xy HE

zy Hi

x

E

xy EH

x

HiE y

z


Let’s consider the TE mode only

Waveguide is symmetric about the plane x=0

Even modes

),,(),,( tzxEtzxE yy Odd modes

),,(),,( tzxEtzxE yy

])|(|exp[ zidxpAE y

)exp()cos( zihxBE y

])|(|exp[ zidxpipA

H z

)exp()sin( zihxihB

H z

dx ||

dx ||

Interface continuous condition at x=d

)cos(hdBA

)sin( hdhBpA )tan( hdhdpd


From wave equations

222

20

2 hnk 22

120

2 pnk 22

021

22

22 )()()( dknnhdpd


Odd modes

])(exp[

])(exp[

zidxpA

zidxpAEy

)exp()sin( zihxBEy

dx

dx ||

dx

Interface continuous condition at x=d

)cot(hdhdpd

§11.3 A perturbation Theory of Coupled Modes in Dielectric Optical Waveguides

Let’s consider the modes coupling, we start with wave equation

)P(r)E(r

E(r) ttt

t,

,2

2

2

2

02

)(rP)(rP)P(r ttt ,,, pert0 )E(rr)(rP tt ,])([, 00

)(rr tPtt

EE y

y ,)( pert2

2

2

22

..)()(2

1),( )()( ccexzAtE ztim

ym

mym r

0)()()( )(2)(22

2

rrrx

my

mym

Assume perturbed field is a superposition of confined modes:

where


Substitute perturbed field into wave equation:

),(

..])2(2

1

))((2

[

pert2

2

)(2

2

)(22

)(2)(2

trPt

ccedz

Ad

dz

dAi

erx

Ae

zimy

mmm

m

zimy

mym

ymmti

m

m

“slow” variation approximationdz

dA

dz

Ad mm

m 2

2

),(.. pert2

2)()( trP

tcce

dz

dAi

m

ztimy

mm

m


dxxtrPt

i

ccedz

dAe

dz

dA

sy

ztisztis mm

)(),(2

..

)(pert2

2

)()(

)()(

Above equation can be used to treat a large variety of mode

interactions. Each physical example involves, in general, a

different perturbation polarization P.

§11.4 Periodic Waveguide

0z Lz 1n

2n

3n

0x

ax

tx

Guiding Layer

Substrate

Applications: Optical filtering, distributed feedback laser, etc.

)()( 20 rnr )()()(' rrr

),)(),)(), 02

pert tntt E(rrE(rr(rP

Corrugation couples only TE to TE and TM to TM modes, but not TE to TM


For TE mode propagation

m

ztimymy ccexA

nt m .].)([

2

)()],[ )()(0

2

pertr

(rP

m

ztisy

mym

ztisztis

ccedxxxzxnAt

i

ccedz

dAe

dz

dA

m

mm

.].)()(),([4

..

)()()(22

20

)()(

)()(

q

zqiqeaxnzxn )/2(22 )(),(

sl /

Assume index changes as:


zlisys

s sedxxxnAi

dz

dA ]2)/2[(2)(2)(0)(

)]()[(4

Both source and wave need to have the same frequency and nearly the same phase dependence.

ss

l

2

)( sm AA lq

zis

s eAdz

dA )(2)()(

zis

s eAdz

dA )(2)()(

*

dxxxn

ai sy

l 2)(20 )]()[(4

0

ss

l

Some General Properties of the Coupled Mode Equations


ziabBe

dz

Ad )(2

ziba Ae

dz

Bd )(2

2||||)(2 lab

*abba 0)|)(||)(|( 22 zAzB

dz

d

If the modes A and B carry power in the same direction

*abba

2||||)(2 lab

0)|)(||)(|( 22 zAzBdz

d

If the modes A and B carry power in opposite direction


Two modes contradirectional coupling


The coupling constant for square-wave corrugation waveguide

l

lzileaxn

zzxnzxn

)(

3sin3

1sin

2

2

1)(),(

2

22

where

elsewhere 0

0 )(

22

212 x-ann

xn

2

So that

0 2

1 0

l

l even

l oddl

i

al


For l odd

dxxxn

ls

y2)(20 )]()[(

4

02

)(sgl

/2)( sg

Guide wavelength of the sth mode

0 2222

21

0 2)(22

21

2)(2

]sin[cos)(

)]([)()]()[(

a ss

sss

a

sy

sy

dxxhh

qxhCnn

dxxnndxxxn

22

23

02

222

21

0 2)(22

21

331

3

4)()]([)(

aqaqs

t

a

knnndxxnn

ssa

sy


21

22

2

221

22

3

2

22

21

22 )/(

4

3/

2

31

)(

3

2

nn

a

nn

a

t

a

n

nn

l

ss

The problem of two-wave coupling by a corrugation has thus

been reduced to a pair of coupled differential equations and

an expression for the coupling constant.

Documents

Lecture 18 Chapter XI Propagation and Coupling of Modes in Optical Dielectric Waveguides – Periodic Waveguides Highlights (a) Periodic (corrugated) WG