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⁄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.
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
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,
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 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
)(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
)()()()( 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
)()()()( 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
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 ,&,,
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
2/12
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
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
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
)()(
)()(
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,
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
2
mVua c
2
2
2
1
2nn
aV
2
2
2
1
2
2nn
am
2
2
2
122
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 λ = 0.90 μm. Calculate the maximum and minimum
values of the propagation constant , β.
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, 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
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
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.
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