waveguides planarslabs
TRANSCRIPT
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 28
Waveguide basicsImportant normalized quantities
n=1
ncl
ncocrit
crit
z
x
a a kz
a kx
clna
0
2
cona
0
2
clcoclcocritcocrit nnnNA === 22sinsin
Numerical aperture of guide measured in air
Phase shift of critical wave across guide face
22
000 sin clcocritcritx nnakaNAkakakV ===
Ray view of guiding in a slab waveguide. The most
extreme ray is trapped via total internal reflection at
the core/cladding boundary.
Modal analysis of planar slab waveguidesNormalized quantities
Phase shift of particular mode across guide face22
0
22
0 sin NnakkkaakakU cozcocox === Exponential loss of particular mode in cladding across guide face
22
0
22
clclz nNakkkaaW ==
Snyder and Love
ncl
222 WUV +=Note that:Cutoff: U=V, W=0
Light line: W=V, U=0
V
UW
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TE modal analysis of
symmetric slab waveguides
P. Yeh, Optical Waves in Layered Media, Chapter 11
( )( ) 0,
2
2
2
22 =
zxE
tc
rn rr
Substitute a TEx-dependent mode into 3D wave equation:
( ) ( ) zjeyxEzxE = ,r
( ) ( ) 0222022
=
+ xExnkdx
d
yieldingncl
ncoz
ncl
x
Ifn is constant in each layer l
( ) xnkjxnkjlll eCeCxE
2220
2220
21
++=
A propagating mode must have
coclcoclnNnnknk
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 30
Apply boundary conditions
TE
( )
( )
( )( )WWDUUBUUAWDUBUA
WWCUUBUUA
WCUBUA
=+=+
=
=+
expsincosexpcossin
expsincos
expcossin
Rearranging slightly by adding and subtracting equations:
( ) ( )
( ) ( )
( ) ( )( ) ( )WDCWUUB
WDCUB
WDCWUUA
WDCUA
+=+=
=
=
expsin2expcos2
expcos2
expsin2
Modal analysis of planar slab waveguidesTE modes
x
EjH
y
z
=
1
Tangential components of fields (Ey andHz) must be
continuous. Since
This is equivalent to field and its slope are continuous. Yieldsfour conditions for five unknowns (A,B,C,D,), leaving peakamplitude as a free parameter. The conditions are:
E|| continuous atx = +a
dE||/dx continuous atx = +a
E||
continuous atx = -a
dE||/dx continuous atx = -a
Then divide equation 2 by 1 and 4 by 3.
DCBWUU
DCAWUU
=
=
,0tan
,0cot
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 31
1 2 3 4 5 6
u
1
2
3
4
5
6
v
TE SolutionsTwo classes (symmetric and anti-symmetric)
WUUDCA === tan,,0
WUUDCB === cot,,0
Can rewrite the anti-sym equation to make it look similar to sym:
UUW tan=
( )2
tancot +== UUUUW
( )22222
0
22
VnnakWU clco ==+
No cutoff for lowest mode
New mode at V>m /2, m=0,1..
Sym/antisym modes alternate
Observations
U
W
# TE modes = Int(V//2) + 1
Transcendental eigenvalue equations
Modal analysis of planar slab waveguidesTE modes
( ) WUUUU =+= 2tancot
Blue lines are TE solutions
Green lines are TM solutions
Always TM mode with TE
V
cutoff, W=0
lightline,U=0
N=ncl
N=nco
Graphical solution of slab transcendental eigenvalue equations:
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 32
TM modes
We follow the same approach but now for Hy
( )
>
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 33
Power carried by modeTE
Modal analysis of planar slab waveguidesPower
From Faradays law we can findHx fromEy :
[W/m]*21
= dxHEPxyz
y
yyc
yyx
xy
EN
ENEN
ENk
EH
HjEj
t
0
0
0
00
0
0
0
=
====
=
=
BE
r
r
From the Poynting vector, we can
calculate the power carried by this mode.
Since the guide is infinite iny, we will
calculate the power per unit length in they
direction:
Average power per unity inz
Faradays Law
From known tandz dependence
Impedance of mode
Giving power in the mode as
o Impedance of free space = sqrt(0/0)
= [W/m]22 0
dxEP yN
z TE
z
x
Er
Hr
TE
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Robert R. McLeod, University of Colorado 34
Power carried by modeTM
Modal analysis of planar slab waveguidesPower
From Amperes law we can findEx fromHy :
[W/m]*
21
+= dxHEP yxz
( )
( ) ( ) ( ) ( )
( ) y
yyc
yyx
xy
Hx
N
Hx
NH
x
NH
x
NkH
xE
ExjHj
t
0
0
0
00
0
0
0
=
====
=
=
DH
r
r
Amperes Law
From known tandz dependence
Impedance of mode
Giving power in the mode as
o Impedance of free space = sqrt(0/0)
( ) ( )
== [W/m]22
1212 0
0 dxExdxHP xNyxN
z
TM
From the Poynting vector, we can
calculate the power carried by this mode.
Since the guide is infinite iny, we will
calculate the power per unit length in they
direction:
Average power per unity inz
z
xEr
Hr
TM
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ECE 4006/5166 Guided Wave Optics
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Power confinement
Modal analysis of planar slab waveguidesPower
What fraction of the power is in the core?
( )
( )
=
==
dxxH
dxxH
P
P
dxE
dxE
dxE
dxE
P
P
y
a
a
y
Tot
co
y
a
a
y
yN
a
a
yN
Tot
co
2
2
2
2
2
2
2
2
0
0
TE
TM
a/0=1/1.5, nco=3, ncl=1.5
The trend is that the
fractional power in the
core always decreases
for increasing mode
number. It approaches
1 for tightly bound
modes with high indexcontrast.
a/0=1/1.5, nco=3, ncl=1.5
TETM
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado
Approximate expression forN
36
( )Tot
coclcocl
Tot
clcl
Tot
coco
P
Pnnn
P
Pn
P
PnN 222222 +=+
It can be shown from variational techniques that the effective index can
be approximately calculated as (first line TE, second TM):
The derivative term is small for the fundamental modes of weakly-
confined modes, leading to a useful conceptual formula:
Modal analysis of planar slab waveguidesPower
( ) ( )
( )
( ) ( )( ) ( )
( )Tot
yclclcoco
y
yy
Tot
yclclcoco
y
yy
PdxxndxdHkPnPn
dxxnxH
dxxndxdHkxH
P
dxdxdEkPnPn
dxxE
dxdxdEkxExnN
+==
+=
22
2
0
22
22
22
2
0
2
22
0
22
2
22
0
22
2
In general Step index slab
Nthus approaches ncofor the fundamental
mode and decreases
towards ncl with
increasing mode
number. Nis less than
ncl for radiation
modes,N= ncl cos()
a/0=1/1.5, nco=3, ncl=1.5
Snyder and Love section 15-1
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 37
Ray analysis of step-
index guides
Ray analysis of waveguidesForm of waves in step index
z
2
clcl n=
2
coco n=
2
clcl n=
k
r kz
k
cln0
2
con
0
2
crit
0kNA
Range of allowed kin core
( )
=
+
+
claddingin the
corein the,
220
220
zNxNnjk
zNxNnjk
cl
co
e
ezxE
( )
[ ]zNxNnkzkxknk
zkxkk
zkxkk
zz
zz
zx
22
0
22
0
22
+=
+=
+=
+=r
N Effective index = kz / k0
Notes:
1) Only one free parameter,N
2) ncl < N < nco are guided rays
2)N < ncl are not guided
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 38
Discrete bound modesSymmetric guide (1/2)
Ray analysis of waveguidesDiscrete bound modes
Hunsperger, Section 2.2.2
zx
2a
2
clcl n=
2
cocon=
2
clcln=
makak pcoxpcox 22222 =+++
0 1 2 3 4 5
z@mmD
-15
-10
-5
0
5
10
15
r
mm
( ) ( ) ( )zxjkzkxkj eezxE zx cossin, ++ == Form of field in core
NOTE MEASURED FROM || TO BOUNDARY!
cox
clx
cl
co
cox
clz
cl
co
co
clco
cl
coTM
cox
clx
cox
clz
co
clco
TE
kn
n
k
kk
n
n
n
nn
n
n
kk
kk
n
nn
=
=
=
=
=
=
2
222
2
2222
2
2
22222
sin
costan
sin
costan
First lets translate the Goos-Hanchen shifts into this coordinatesystem and write them in terms of convenient quantities:
Then find phase accumulated in round trip of ray from lower to
upper and back to lower boundary at single plane z:
p = TE or TM
Decay in x in cladding
Wave # in x in core= -
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 39
Discrete bound modesSymmetric guide (2/2)
( )clxpcoxcox
cox
clxpcox
pcox
makk
mk
ak
mak
=
=
=+
2
1
tan
tan22
22
Ray analysis of waveguidesDiscrete bound modes
Simplify slightly
Substitute GH phase shift
and rewrite.
=TM
TE12
cl
cop
n
nwhere handles both polarizations.
m Mode number. Quantizes bound spectrum.
0.2 0.4 0.6 0.8 1
H2 p a cL w = HaL k0
1
2
3
4
5
cokr
clkr
xk
zk
cokr
xk
zk
6
1
=
=
co
cl
n
n
Not allowed
Radiation
Bound
a / c = a k0
Same transcendental equation as modal derivation (good!).
Plot of solutions found numerically
akz=ak0N
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 40
Discrete bound modesAsymmetric guide (1/2)
Ray analysis of waveguidesDiscrete bound modes
makak pscoxpclcox 22222 ,, =+++ p = TE or TM
zx
2a
2
clcln=
2
cocon=
2
ssn=
mNnaNn
nN
Nn
nN
makkk
mkk
ak
mak
co
co
s
p
co
cl
p
cox
cox
sxp
cox
clxp
cox
sxp
cox
clxpcox
pspclcox
=
+
=
+
=
=++
22
22
22
1
22
22
1
11
11
,,
2tantan
2tantan
tantan2
2
Common due to fabrication technology. Typical example is air cladding.
Following previous, track phase from edge to edge of core.
substrate
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 41
Discrete bound modesAsymmetric guide (2/2)
Ray analysis of waveguidesDiscrete bound modes
cokr
xk
zk n(x)
xN
ns
ncl
nco
6
2
1
=
=
=
co
s
cl
n
n
nNot allowed
Comparison of symmetric (ns=ncl) and asymmetric fundamental modes.
Substrate & cladding radiation
kz= kcl orN = ncl
kz = ks orN = ns
m=0 asymmetric
m=0 symmetric
a / c = a k0
akz=ak0N
Note analogy to
particle in potential
well
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ECE 4006/5166 Guided Wave Optics
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0 1 2 3 4 5
z@mmD
-15
-10
-5
0
5
10
15
0 1 2 3 4 5
z@mmD
-15
-10
-5
0
5
10
15
0 1 2 3 4 5
z@mmD
-15
-10
-5
0
5
10
15
0 1 2 3 4 5
z@mmD
-15
-10
-5
0
5
10
15
Ray analysis of waveguidesForm of waves in step index
z
kz
k
z
kz
k
z
kz
k
z
kz
k
cok
r
cok
r
cok
r
co
kr
cl
cl
cl
cl
Guided modes
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 43
Radiation modesSamples of mode continuum
z
kz
k
Ray analysis of waveguidesForm of waves in step index
cokr
clkr
z
kz
kco
kr
clkr
z
kz
kco
kr
clk
r
0 1 2 3 4 5
z@mmD
-15
-10
-5
0
5
10
15
r
mm
0 1 2 3 4 5
z@mmD
-15
-10
-5
0
5
10
15
r
mm
0 1 2 3 4 5
z@mmD
-15
-10
-5
0
5
10
15
mm
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 44
BackgroundRay and eikonal equations
( ) ( )rnds
rdrn
ds
d rr
r
=
Eikonal
( ) ( )
zs
nrn
zzr
=+=
rr
rr
Cylindrical coordinates
Waveguide that is invariant alongz
Paraxial (small angle) approximation
( ) ( )
r
r
r
ndz
dn =
2
2
Paraxial eikonal for waveguide inz
UFdt
dm ==
r
r
2
2 Newtons law for particle inpotential well U
Particle in potential wellAnalogy with ray optics
Approximations
( )
( ) ( )
r
r
r
=
==
2
2
2
2
2
dz
d
nnn Express in terms of, not n
Paraxial eikonal for waveguide
inz, in terms of, not n
Note similarity with F = m A
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 45
Particle in potential wellApplication to waveguides
crit
critz
kz
k
cln
0
2
con
0
2
crit
clcoclconn
=
0
22
0
22
max
2222
sinsin =====
clcoclcon
nn
cocritcocrit nnnnNAco
clco
Numerical aperture of guide measured in air
Ray view of guiding in a slab waveguide. The most
extreme ray is trapped via total internal reflection at
the core/cladding boundary.
( ) ( ) ( )[ ] ( )[ ]
( ) ( )
( )
22
22
22
2
2
2
2
sin
sin
sin
2
1
2
clco
clco
cl
nn
nnn
n
dz
d
UmV
dz
d
=+==
r
r
r
r
rrr
r
r
Kinetic energy < potential to remain bound
2
clcln=
2coco n=
12 == airair n
Using analogy
Express d/dz as sin
By Snells Law at exit of guide
Ray analysis of waveguidesNumerical aperture
T.I.R.
Can drop uniform
2
clcln=
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 46
Parabolic gradient-index guide
Solution via eikonal (1/2)
( )
=
m
co ann
1
( ) ( )
( ) ( )
+=
=
zdz
dn
dz
d
dz
rdn
dz
d
ds
rdn
ds
dn
r
r
( ) ( )
dz
dn
dz
dn
d
d
Power-law radial index distribution
< a
Plug into eikonal
Paraxial approx.
Simplify with known dependencies
Ray analysis of waveguidesGrin guides
-2 -1 0 1 2
ra
-1
-0.8
-0.6
-0.4
-0.2
0
d
nD
Index distribution vs radius
( )( )
( )
=
m
an
d
d
nn
d
d
ndz
d
1
1102
2
Subst. n()
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 47
Ray analysis of waveguidesGrin guides
Parabolic gradient-index guide
Solution via eikonal (2/2)
( )( )
( )
( )
11
2
2
1
111
=
=
mm
co
m
co
aa
m
aa
mn
n
an
d
d
nn
d
d
ndz
d
22
2 2=
aadz
d
Take deriv.,
assume n in
denom ~ nco
Special case m=2
( ) ( ) ( )zzz sincos 00
+= Solution for ray trajectory
from prev.
z=0 z=/
z
-a
a
0
Rays escape when radial coordinate goes beyond =a, which gives NA:
( )
( ) +==
=====
====
221
2sinsin
22
sinmax
22222
0
0000
cocococlco
cocococritcocrit
nnnnnNA
nndz
dnnNA
aa
z
Ray slope that hits
edge
Defines NA
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado
Dispersion
Material
Modal
Waveguide
Polarization
48
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 49
Origin of waveguide group delay
from eigenvalue equation
Modal analysis of planar slab waveguidesGroup velocity
( ) ( )[ ] ( ){ }zjktt zetEztE= 0,, 1 FF
( ) ( ) ( ) K+
+
+=
00
02
22
002
1
zzzzkk
kk
( )zkj
z zezk
tEztE 0
0
0,,
=
Consider how a temporal pulse that excites a single mode of the guide will
propagate. We found the solution to the scalar wave equation for a
sinusoidal temporal excitation at frequency :
( ) ( ) ( )zktj zeyxEzxtE = ,,r
whereE(x) satisfied the DE and kz = N k0 =simultaneously satisfied thetranscendental characteristic equation. For an aribitrary temporal signal,
we must decompose it into its frequency components and apply this
transfer function to every frequency independently:
Let us now expand the propagation constant kz as a Taylor series aroundsome central carrier frequency 0:
The first term causes a frequency-independent phase shift. The second
term is a linear phase shift which, by the Fourier shift theorem, is
equivalent to a shift in the time domain:
Group delay [ ]mz
g
ks
0
Pollock and Lipson Ch 6
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 50
Group index& velocity vs. delay
Modal analysis of planar slab waveguidesGroup velocity
( )c
N
d
dNN
ccN
d
d
d
dk gzg
+=
=
1
0
Write propagation constant in terms of effective index:
It is sometimes convenient to have the derivative vs. wavelength:
( )
d
df
d
dfc
d
df
d
d
d
df
d
df=== 2 2
0dk
dk
d
dNN
d
dNNN zg ==+=
thus the effective group index is
I find group velocity to be a) confusing and b) generally useless, so we
will attempt to avoid this concept.
Pollock and Lipson, Section 6.3.2 and 6.3.3
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ECE 4006/5166 Guided Wave Optics
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Group velocity dispersion
second and higher terms
Modal analysis of planar slab waveguidesGroup velocity dispersion
Consider a pulse with a finite bandwidth / = /. The latency due topropagation delay of the pulse will be
( )[ ]s0 L
c
Ng =
The pulse will spread due to the variation of latency across the spectrum:
( ) ( )[ ]
=
=
=
=
2
2
2
2
21
d
Nd
c
L
dNd
ddN
ddN
cL
d
dNN
d
d
c
L
d
dN
c
L
NNc
L
g
gg
It is useful to define the GVD parameterD as
=
kmnmpsor
ms
22
2
dNd
cD
LD
Note that exactly the same derivation would hold for propagation in a bulk
material withNreplaced by the normal refractive index, n.
Typical value:D = 17 ps/(nm km)
for SMF-28 at = 1.5 m
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ECE 4006/5166 Guided Wave Optics
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ExampleSlab waveguide
Modal analysis of planar slab waveguidesGroup velocity dispersion
Anomalous dispersion
Radiation
Not allowed6
1
=
=
co
cl
n
n
Bound
m=0m=1 m=2 m=3
1=cln
6=con
Modes emerge
from radiation
cut-off,
approach light
line.
Zero dispersion
band near
cutoff, then
again when
tightly bound.
Ng = ncl at cutoff
and approaches
nco from above.
Normal dispersion
D
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 53
Material dispersionHow to include
2
0
22
0 NakNnakU coco +=
( ) += clcl NaknNakW2
0
22
0
Modal analysis of planar slab waveguidesGroup velocity dispersion
In many cases, the core and the cladding are made from similar
materials, for example, the core is a doped version of the cladding or a
denser version of the cladding. In this case the dispersion of the two
materials is very nearly the same. Consider how this impacts the
effective index:
( ) WUUUU =+=2
tancot
Thus if the effective index is changed like
+22
NNThen U and W are unchanged, the characteristic equation is still
satisfied:
thus the mode shape does not change.
Thus to a good approximation in most cases, material and waveguide
dispersion can be treated as independent, additive quantities.
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 54
Orthogonality of modes
Modal analysis of planar slab waveguidesOrothogonality
It is possible to show directly from the wave equation that, if the
materials are lossless (no free current and thus no Joule losses),
Where Pm is the power carried by the mode (this depends on the
particular amplitude). The delta functions must be carefully defined:
( ) ( ) ( )
==
=
nm
nm
afdxaxxf
nm0
1,
-
Dirac delta function.
Note (x) has units of 1/x.
Discrete delta function.Note m is unitless.
The modes are not only orthogonal, they are complete, allowing us to write
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
+=
+=
ml
zmljzj
mlml
zmlj
ml
zj
mlml
dmdlemlyxHmlaeyxHazyxH
dmdlemlyxEmlaeyxEazyxE
ml
ml
,
,
,,
,
,
,,
,;,,,,,
,;,,,,,
,
,
rrr
rrr
If we could find the coefficients a, we would know how arbitrary fields
propagate. Note al,m and a(l,m)dldm are unitless ifEl,m are not normalized.
Bound modes,
l,m discrete
Radiation modes,
l,m continuous, e.g. kx, ky
( )
( )
( ) ( )modesBound
modesRadiation
2
2
2
1
,
*
,,,10
*
,,
0
*
=
=
=
nmm
nymyyx
nymy
nm
P
nmmP
dydxHHN
dydxEEN
dydxzHE
rr
General:
TE:
TM:
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 55
Expanding an arbitrary
field as a sum of modes
Modal analysis of planar slab waveguidesOrothogonality
( ) ( ){ }
( ) ( ) ( ) ( )
+
=
dydxzyxHdmdlmlyxEmlayxEa
dydxzyxHyxE
pn
ml
mlml
pn
,,;,,,
2
1
,0,,2
1
*
,
,
,,
*
,
rrr
rr
Start with the assumed expansion
Since we only need to perform the expansion at onez plane which will
then determineEin all ofz, choosez = 0.
Now cross with theH* field from a bound or radiation mode n,p, take the
dot product inz and integrate overx andy:
We can swap the order of thexy integral and the sum/integral over l,m:
( ) ( ) ( ) ( )( ) ( )
+= mlamlPaPdmdlpmnlmlPmlaPa
mlml
ml
pmnlmlml,,or,,
,,
,
,,,,
Thexy integrals collapse via the orthogonality relations:
( ) ( )[ ]
( )( )
( ) ( )[ ] dydxzmlyxHyxEmlP
mla
dydxzyxHyxEP
a mlml
ml
,;,0,,,2
1,
,0,,2
1
*
*
,
,
,
=
=
rr
rr
( ) ( )[ ]
( ) ( ) ( )[ ]
+
=
dmdldydxzyxHmlyxEmla
dydxzyxHyxEa
pn
ml
pnmlml
,,;,,2
1
,,2
1
*
,
,
*
,,,
rr
rr
( ) ( ) ( ) ( ) ( ) += dmdlemlyxEmlaeyxEazyxE zmlj
ml
zj
mlmlml ,
,
,, ,;,,,,,,
rrr
( ) ( ) ( ) ( ) += dmdlmlyxEmlayxEazyxE mlmlml ,;,,,,,
,
,,
rrr
Substitute assumed expansion
al,m unitless
a(l,m) dldm unitless
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 56
Application to TE modes
of a slab waveguide
Modal analysis of planar slab waveguidesOrothogonality
z
x
Er
Hr
TE( ) ( )= dxxHxEP
a xnyn
n
*
,0,2
1
Start with the mode expansion equation
and plug in TE mode. Collapse to 1D.
( ) ( )= dxxExEPN
a ynyn
n
*
,
0
0,2
1
use the relationship betweenHandEfor a TE mode to get
First lets assume that the mode n is bound.
( ) ( )
( )
( ) ( )
( )
==
dxxE
dxxExE
dxxEN
dxxExEN
a
yn
yny
yn
yny
n 2
,
,
2
,
0
,
00,
2
0,2
Now assume n is a radiation mode. Lets write out the orthogonality
condition. Assume that a mode m in the cladding has amplitude Cm
( ) ( ) ( )nmCNdxeCNeCdydxzHE mxjk
n
xjk
mnmxnxm =
=
+
2
0
*
0
*
22
1
2
1,,
rr
( )( ) ( )
( )2
,
,
cladding
0,
yn
yny
E
dxxExEna
=
P(n)
Therefore
( ) ( ){ } = dydxzyxHyxEPa n
n
n,0,,
2
1 *rr
Can drop cc on flat mode,
but I dont recommend it.
an is still
without
units
a(n) dn unitless =
a(n) has units ofx
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 57
Application to TM modes
of a slab waveguide
Modal analysis of planar slab waveguidesOrothogonality
( ) ( )= dxxExHPb xny
n
n ,
* 0,2
1
Start with the mode expansion equation
and plug in TM mode. Collapse to 1D.
( )( ) ( )= dxxHxHxP
Nb yny
n
n ,
*0 0,1
2
use the relationship between H and E for a TM mode to get
First lets assume that the mode n is bound.
( )( ) ( )
( )( )
( )( ) ( )
( )( )
==
dxxHx
dxxHxHx
dxxHx
N
dxxHxHx
N
b
yn
yny
yn
yny
n2
,
,
2
,0
,0
*
1
0,1
1
2
0,1
2
Now assume n is a radiation mode. Lets write out the orthogonality
condition. Assume that a mode m in the cladding has amplitude Cm
( ) ( ) ( )nmCNdxeCeCNdydxzHE mcl
xjk
m
xjk
n
cl
nmxnxm =
=
20*0*
22
1
2
1,,
rr
( )( ) ( ) ( )
( )2
,1
,1
*
cladding
0,
yn
ynyx
H
dxxHxHnb
cl
=Therefore
z
Er
Hr
TM
x
( ) ( ){ } = dydxzyxHyxEPb n
n
n0,,,
2
1 *rr
( )( ) ( )= dxxHxHxP
Nb yny
n
n
*
,0* 0,
1
2
or
bn is still
without
units
P(n)
b(n) dn unitless =
b(n) has units ofx
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 58
Mode normalization
Modal analysis of planar slab waveguidesOrothogonality
It is convenient to scale the mode such that we can drop all of the
constants in the previous expressions. This is our last free parameter of
the mode shapes as derived from the transcendental equation.
( ) ( )
( ) ( )
( )( )
( )
( )
=
=
radiationcladding
bound
2
1
0,2
1
,
,
2
,
,
,
0
,
,
0
yn
yn
yn
yn
yn
n
yn
yny
n
n
E
xE
dxxE
xE
xEP
NxE
dxxExEP
Na
( )( ) ( )
( ) ( )
( )
( )( )
( )
( )
=
=
radiationcladding
bound1
2
0,1
2
,1
,
2
,
,
,0
,
,0*
ynn
yn
yn
yn
yn
n
yn
yny
n
n
H
xH
dxxHx
xH
xHP
NxH
dxxHxHxP
Nb
cl
The hat symbol will be used for normalized modes and modal amp. This normalizes the modes so that the transverse integral of the |field|2 = is
1 or (n=0) (radiation). In 2D, mode amplitudes al.m and a(l,m)dldm now carry units ofE= V/m In 2D, mode amplitudes bl,m and b(l,m) dldm now carry units ofH= A/m
TE:
TM:
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Robert R. McLeod, University of Colorado 59
SummarizingMode orthogonality for the slab waveguide
Modal analysis of planar slab waveguidesOrothogonality
( ) ( ) ( ) ( ) ( )
( ) ( )[ ] ( ) ( ) ( )[ ] ( )
++++
+=
m
zmj
zx
m
zj
zmxmm
m
zmj
y
m
zj
ymm
dmezmxExmxEmbezxExxEb
ydmemxEmaexEazxE
TMTMm
TETEm
;;
;,
,
,
,,
,
r
We may calculate the field at any point down the
guide. Using modes with arbitrary amplitude:
TE
TM
z
xAssume we have a field incident on our slabwaveguide, given by its tangentialEatz = 0. We
may calculateH fromE.
Using normalized modes:
This is the complete solution to the waveguide boundary value
problem, containing all the physics with no approximations!
( ) ( ) ( )= dxxEdxxExEa ymymym2
,,0,And amplitudes (e.g. TE bound):
( ) ( ) ( ) ( ) ( )
( ) ( )[ ] ( ) ( ) ( )[ ] ( )
++++
+=
m
zmj
zx
m
zj
zmxmm
m
zmj
y
m
zj
ymm
dmezmxExmxEmbezxExxEb
ydmemxEmaexEazxE
TMTMm
TETEm
;;
;,
,
,
,,
,
r
TE
TM
( ) ( )=dxxExEa
ynyn ,
0,And amplitudes (e.g. TE bound):
( ) ( ) ( ) dxxExExE ymymym2
,,,with normalized modes:
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 60
Rewrite TM expansionin terms of E, not H
( ) ( ){ }
( ) ( )
( ) ( ) ( )
=
=
=
dxxExExPN
dxxHxE
P
P
dydxzyxHyxEb
xnx
n
ynx
n
n
n
n
*
,
0
*
,
*
21
0,2
1
0,
2
1
,0,,
rr
Modal analysis of planar slab waveguidesOrothogonality
( ) yxH
x
NE 0
=
TM mode
Modal expansion,
modes not normalized
( ) ( )
( )
( ) ( )
( )
( )
== radiation
cladding
bound
2
1normalized
,
,
2
,
,
,
0
,
xncl
xn
xn
xn
xn
n
xn
En
xE
dxxEx
xE
xEPN
xE
( )
= [W/m]22
1
0dxExP xNz From previous derivation
( ) ( ) ( )= dxxExExb xnxn*
,0, ( ) ( )= dxxExEa ynyn
*
,0,
It is useful and instructive to recast the TM expansion in terms of E fields.
We can normalize as usual
Now the two coefficients a and b clearly refer to they andx E-fieldpolarizations (which are the TE and TM modes):
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado
Units and values
61
1D (slab)
2D (everything else)
Not normalized Normalized
Bound
Radiatio
n
W/m
W/m2
V/m, A/m
V/m, A/m
V/m1/2 or A/m1/2
1/m1/2
Units are for quantities in bold face (i.e. P but not delta function).
Blank indicates no units. Radiation mode labels (l,m) have units of 1/meters.
Normalized Poynting vector listed is for TE modes. See pg 34 for TM.
llb,a
ll HE,
= dxHE ll 21lP [ ]12 0N dxEl
2 1
llb,a
llHE ,
= dxHE ll21lP dxEl
2V2/m
Modal analysis of planar slab waveguidesOrothogonality
( ) ( ) dllbdl,la ( ) ( )lHlE ,
( ) ( ) == dxHEl210lP
( ) dxlE2
V2/m
V/m or A/m( ) ( )
dllbdl,la( ) ( )lHlE ,
( ) ( ) == dxHEl 0 21lP( ) dxlE
2 (l=0) [m]
[ ]12 0N
Not normalized Normalized
Bound
Radiation
W
W/m2
V/m, A/m
V/m, A/m
V/m or A/m
1/m
l,ml,m b,a
l,ml,m HE,
dydxHEml,ml, = 21l,mP [ ]12 0N
dxdyEml,2
1
l,ml,m b,a
l,ml,m HE ,
dydxHEml,ml, = 21l,mP
dydxEml,2
V2
( ) ( ) dldmml,bdldm,ml,a
( ) ( )ml,Hml,E ,
( ) ( )0, =mlml,P
( ) dydxml,E2
V2/m
V/m or A/m( ) ( ) dldmml,bdldm,ml,a
( ) ( )ml,Hml,E ,
( ) ( )0, =mlml,P
( ) dydxml,E2
(l,m=0) [m2]
[ ]12 0N
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 62
Power carried by a modeNot normalized
Modal analysis of planar slab waveguidesOrothogonality
( ) ( ) ( ) ( ) ( )
( ) ( )[ ] ( ) ( ) ( )[ ] ( )
++++
+=
m
zmj
zx
m
zj
zmxmm
m
zmj
y
m
zj
ymm
dmezmxExmxEmbezxExxEb
ydmemxEmaexEazxE
TMTMm
TETEm
;;
;,
,
,
,,
,
r
TE
TM
( )
( ) ( )[ ] ( ) ( ) ( )[ ] ( )
( ) ( ) ( ) ( )
++
+++
=
m
zmj
y
m
zj
ymm
m
zmj
zx
m
zj
zmxmm
dmeymxHmbeyxHb
dmezmxHxmxHmaezxHxxHa
zxH
TMTMm
TETEm
;
;;
,
,
,
,
,,
r
TE
TM
Lets calculate the Poynting vector using the modal field expansion. First we need to writedown bothEandH. TheHfields of the TE modes and theEfields of the TM modes must
be calculated through the curl equations with proper units so that their Poynting vector is
correct. The b coefficients can be calculated either fromEorHfields .
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
++
+
+
=
=
dxdmemxHmaexHa
dmemxEmaexEa
dxHEP
m
zmj
x
m
zj
xmm
m
zmj
y
m
zj
ymm
xyz
TETEm
TETEm
;
;
[W/m]
***
,
*
,
21
*
21
,
,
Note first the the cross product in Poyntings theorem will generate zero
forETE HTM andETM HTE so we can treat the two polarizations
independently. Then:
TE (TM very similar)
This looks terrible, but note that we can reverse the order of integrations,
giving us our orthogonality relations, so the entire thing becomes
[ ] ( ) ( ) ( ) ( )[ ]dmmPmbmPmaPbPaPm
TETE
m
TMmmTEmmz +++=22
,
2
,
2
Bound and rad
modes orthogonal,
so no cross terms.
So |a|2 represents the power in the mode, scaled by the power the mode
caries via the arbitrary amplitudes of the modes.
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ECE 4006/5166 Guided Wave Optics
Robert R. McLeod, University of Colorado 63
Modal analysis of planar slab waveguidesOrothogonality
( ) ( ) ( ) ( ) ( )
( ) ( )[ ] ( ) ( ) ( )[ ] ( )
++++
+=
m
zmj
zx
m
zj
zmxmm
m
zmj
y
m
zj
ymm
dmezmxExmxEmbezxExxEb
ydmemxEmaexEazxE
TMTMm
TETEm
;;
;,
,
,
,,
,
r
TE
TM
( )
( ) ( )[ ] ( ) ( ) ( )[ ] ( )
( ) ( ) ( ) ( )
++
+++
=
m
zmj
y
m
zj
ymm
m
zmj
zx
m
zj
zmxmm
dmeymxHmbeyxHb
dmezmxHxmxHmaezxHxxHa
zxH
TMTMm
TETEm
;
;;
,
,
,
,
,,
r
TE
TM
In the case of normalized modes, the Poynting vector will equal N/(2 0) times a deltafunction (radiation modes).
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
++
+
+
=
=
dxdmemxHmaexHa
dmemxEmaexEa
dxHEP
m
zmj
x
m
zj
xm
m
zmj
y
m
zj
ymm
xyz
TETEm
TETEm
;
;
[W/m]
****
,
21
*
21
,
,
Note first the the cross product in Poyntings theorem will generate zero
forETE HTM andETM HTE so we can treat the two polarizations
independently. Then:
TE (TM very similar)
Again, this generates the orthogonality relationship, simplifying to:
( ) ( ) dmmbmabaN
Pmm
mmz
++
+= 0
2
0
2
0
2
0
2 11
2
Bound and rad
modes orthogonal,
so no cross terms.
Power carried by a modeNormalized
The TM terms assumes b has been calculated from the normalized H as per
page 57 and H fields normalized as per page 58.
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ECE 4006/5166 Guided Wave Optics
Coupling efficiency
Modal analysis of planar slab waveguidesOrothogonality
One of the most common questions asked about a waveguide is the
efficiency of coupling from an external source such as a focused beam or
a previous waveguide. We now have the tools to answer this question.
( ) ( )
[ ] ( ) ( ) ( ) ( )[ ]
+++
++
+
=
dmmPmbmPmaPbPa
dmmbmabaN
P
m
TETE
m
TMmmTEmm
mm
mm
z22
,
2
,
2
0
2
0
2
0
2
0
2 11
2
Normalized
Not
External fields can be decomposed intoEy, Hx which couple into the TE
modes andEx, Hy which couple into the TM and thus the two can be
considered separately. Assuming the incident field is in a uniform
material of index ninc, an external TE field will have z-directed power:
( )
= [W/m]0,2
2, 0dxxEP y
n
inczinc
The expressions after the brace are the normalized mode amplitudes2 soin the normalized case:
Assuming appropriate index matching or anti-reflection coating is
employed such that the difference between ninc andNcan be ignored,
then (for TE), in the non-normalized case, the coupling efficiency is
( ) ( ) ( )
=
dmEmadxEa
dxE
m
ym
ymm
y
m 2
,
2
2
,
2
2
cladding0
1
incidentPower
modeinPower
Bound
Radiation( ) ( ) ( )
=
= dmmaa
dmma
a
dxE
mm
y
m 2
2
2
2
2
0
1