Professor Notes:
Optoelectronics Waveguide Elements
Vers 1.1
Waveguides
obey Maxwell equations, which for
simple,
isotropic dielectri
c material with no free
charges are
:
t
B
E
(2.1a)
0
.
D
(2.1b)
(
Faraday
law, Gauss law)
t
D
H
(2.1c)
0
.
B
(2.1d)
(
Ampere law, Gauss law
)
And the
relationships between field
types
(for simple, isotropic dielectric material with no free charges)
are
:
E
D
(2.2a)
H
B
(2.2b)
And if we put all of these equations together
(vector analysis)
we end up with the wave equation
:
2
2
2
t
E
E
(2.3a)
which is the same
for the magnetic field
:
2
2
2
t
H
H
(2.3b)
Electromagnetic fields that originate from tuned circuits or from molecular vibrations will be of the
wave form
)
(
0
)
,
(
z
t
j
e
y
x
E
E
(2.4a)
)
(
0
)
,
(
z
t
j
e
y
x
H
H
(2.4b)
and are consistent with
equations (2.3a) and (2.3b). These equations are harmonic in time and space
.
The subscr
ipt indicates amplitude
of the field
defined at a reference
t = 0
and sometimes is omitted.
The use of
rectangular coordinates
(x,y,z)
as our basis
lets us choose
the z

direction as the direction of
propagation
. T
he result (2.4a) and (2.4b)
then
shows that
for unconfined electromagnetic waves the
oscillating field
E
=
E
(x,y)
and
H
=
H
(x,y)
lie in
direction
s that are
transverse
to the direction of
propagation.
I
f
the waves are confined and
defined by a conduit then the internal reflections will generate
components
in the direction of propagation
.
There are two types of conduits that must be considered with optoelectronic systems (1) rectangular and
(2) cylindrical
. The rectangular geometries are associated with sources, for which a layer is confined in
a later
al direction and
results i
n a rectangular cross

section.
And then there is the optical fiber, which is almost always cylindrical.
The rectangular
cross

sec
tion has
the simplest mathematics. The wave equation in rectangular
coordinates is
E
2
2
2
2
2
2
2
2
2
t
E
z
E
y
E
x
E
(2

5
)
which,
using
)
(
0
)
,
(
z
t
j
e
y
x
E
E
,
becomes
0
2
0
2
2
0
2
2
0
2
E
E
y
E
x
E
(2

6
a)
or
0
0
2
2
2
0
2
2
0
2
E
y
E
x
E
(2

6
b)
or
(simpler)
0
0
2
2
0
2
2
0
2
E
q
y
E
x
E
(2

6
c)
And similarly for
H
0
.
There is still plenty of mathematical overhead since the vector quantity
0
E
will have components
E
x
, E
y
and
E
z
all
of which have dependence on (
x,y
). When
these are subject to the Maxwell equations (2

1a) and (2

1c)
(Faraday
law and Ampere law)
we end up with six equations
.
T
hey
will
resolve into recognizable relationships between
E
x
,
E
y
,
E
z
and
H
x
, H
y
,
H
z
as follows
y
H
j
x
E
j
q
E
z
z
x
2
1
(2

7
a)
x
H
j
y
E
j
q
E
z
z
y
2
1
(2

7
b)
x
H
j
y
E
j
q
H
z
z
x
2
1
(2

7
c)
y
H
j
x
E
j
q
H
z
z
x
2
1
(2

7
d
)
These equations still make no claims to a conduit unless we impose one, for which the solutions would
then have boundary conditions. If there are no boundary
conditions one of the options would be a free

space wave
. And if that is the case,
E
z
and
H
z
= 0 and
therefore it is necessary that
q
2
= 0
. A
nd then
0
2
2
(2

8a
)
And thus for free space
, the propagation coefficient
2
2
1
f
c
(2

8b)
w
hich is usually given the nomenclature
k =
(free space)
wave
number
, i.e.
2
k
(2

8c)
If a (rectangular) geometry is imposed as a conduit for the waves, as represented by figure 2.2, then the
reflections from the boundaries will insist that
E
z
and
H
z
, one or both, are non

zero, and
solutions must
be of harmonic form (sin
and cos
),
and
for which
0
2
q
Figure 2

1
. Rectangular cross

section of a lightwave
conduit (waveguide), dimensions
a
x
b
(from
http://www.scribd.com/doc/30920866/Wave

Guides
)
If
0
2
q
then e
quation
s
(2

7
a
)
thru(2

7d) identifies
transverse
(x,y)
components of
E
and
H
in terms of
the components in the longitudinal (z) direction.
As
guided waves within a conduit (waveguide) of
rectangular cross

section
these equations
identify relationships
between
the
transverse fields
(
E
x
, E
y
,
H
x
,
H
y
)
and the longitudinal fields
(
E
z
,
H
z
)
of the following types:
Transverse Electric (TE) modes:
0
z
E
,
0
z
H
Transverse magnetic (TM) modes:
0
z
E
,
0
z
H
Hybrid modes (EH, HE):
0
z
E
,
0
z
H
As an example
,
for
E
z
= 0 (TE modes),
and harmonic
(sin
and cos
)
solutions of the
wave equation (2

6c),
then
E
and
H
field components that satisfy the
conduit
boundary conditions
of
figure 2

1 are:
E
z
= 0
(2

8a)
H
z
= H
0
cos
(
m
x/a
) cos(
n
y/b
)
= H
0
cos
(
k
x
x
)
cos
(
k
y
y
)
(2

8b)
With transverse components
H
x
= H
0
sin
(
k
x
x
)
cos
(
k
y
y
)
(2

9a)
H
y
= H
0
cos
(
k
x
x
)
sin
(
k
y
y
)
(2

9b)
E
x
=

E
0
cos
(
k
x
x
)
sin
(
k
y
y
)
(2

9c)
E
y
=
E
0
sin
(
k
x
x
)
cos
(
k
y
y
)
(2

9d)
Obviously we have skipped some fun
mathematical overhead
and accepted that of others who like such
things. But the key context of the results is that there are two wave numbers, one for each geometrical
dimension:
m
= mode number for x

direction
= number of ½
within boundaries
x
= [0,
a
]
n
= mode number for y

direction = number of ½
within boundaries
y
= [0,
b
]
and the specification for TE modes will be of the form TE
mn
.
Typical end

view representations of some of these modes is shown by
figure 2.2
Figure 2

2:
TE modes, rectangular waveguide (
copy from the site
http://www.doe.carleton.ca/Courses/ELEC4503/Lectures_2010/Lectures_part3.pdf
)
These mode numbers also define wave numbers
k
x
and
k
y
for which
2
2
2
y
x
k
k
q
(2

10a)
But
p
also relates to propagation of waves according to
2
2
2
2
2
k
q
(2

10b)
(
where
k
= wavenumber for free

space propagation.
)
So that the propagation coefficient
in the waveguide
is
2
2
2
y
x
k
k
k
2
2
2
1
1
C
c
k
k
k
k
(2

10c)
If the imposed wavelength
is equal to the
c
specified by (2.10c) then propagation goes to zero (and
propagation is cut
–
off). Propagation
therefore requires hat
<
c
, an
d we can specify this as a cutoff
condi
tion, with
c
being specified by the conduit geometry according to
2
2
2
b
n
a
m
C
(2

11)
If this condition is extended to three dimensions (i.e. a boundary condition
imposed on the z

direction)
then equation (2

11)
becomes
2
2
2
2
c
p
b
n
a
m
C
(2

12)
This is the type of condition
associated with
semiconductor
sources
which
(as we will see later)
have a
rectilinear geometry
.
Equation (2

12) is also
the
resonance
condition for a cavity resona
tor. S
ources
should
therefore
be exp
ected have
resonant conditions and TE, TH and hybrid (EH and HE) modes
On the other hand t
he geometry
of
the
optical
fiber will be cylindrical
and use
coordinates (
r
,
z
)
So for
propagation in the z

direction the transverse fields will be of the form
)
(
0
)
,
(
z
t
j
e
r
E
E
(2.
1
4a)
)
(
0
)
,
(
z
t
j
e
r
H
H
(2.
1
4b)
If we identify E in terms of its components
E
r
, E
,
and
E
z
and H in terms of its components
H
r
,
H
,
and
H
z
,
Maxwell
equations (2.1a) and (2.1c) in
cylindrical coordinates
give
r
z
H
j
E
jr
E
r
1
(2

15a)
H
j
r
E
E
j
z
r
(2

15b
)
z
r
H
j
E
rE
r
r
1
(2

15
c
)
a
nd
r
z
E
j
H
jr
H
r
1
(2

1
5d
)
E
j
r
H
H
j
z
r
(2

1
5e
)
z
r
E
j
H
rH
r
r
1
(2

1
5f
)
More fun with math
will
resolve equations (2.15a thru 2.15f) in terms of
E
z
and
H
z
as follows:
z
z
r
H
r
r
E
q
j
E
2
(2

1
6a
)
r
H
E
r
q
j
E
z
z
2
(2

16b)
z
z
r
E
r
r
H
q
j
H
2
(2

16c)
r
E
H
r
q
j
H
z
z
2
(2

16
d
)
w
here
2
2
2
q
2
2
k
(2

17)
since
2
2
k
(free space).
The
wave equation parameter
q
2
is the same as for rectilinear coordinates
. T
herefore
much of the same
interpretation
can be made, except
the
wave numbers
will be
associated with
(r,
)
instead of
(x,y).
If equations (2

16c) and (2

16d) are substituted into equation (2

15f) then
0
1
1
2
2
2
2
2
2
z
z
z
z
E
q
E
r
r
E
r
r
E
(2

18)
The procedure for solving an equation of this form is to use the separation of variables method for which
)
(
)
(
2
1
F
r
AF
E
z
.
The symmetry of the cylindrical geometry suggests that
F
2
(
)
should be of the (sin,
cos) form
F
2
(
)
=e
j
The
cylindrical wave equation then gives a differential equation in
F
1
of the form
0
1
1
2
2
2
1
2
1
2
F
r
q
r
F
r
r
F
(2

19)
Equation (2

19) is
of the form of
Bessel’s equation
which is
0
1
1
2
2
2
2
y
x
x
y
x
x
y
(2

20
)
Equation (2

20) is the same as equation (2

19) with
the substitution
x = qr
.
S
olutions
are
either
called
Bessel functions
or
cylindrical functions
and
are of the designation
of either
J
(x)
or K
(x)
,
where
is an index number
associated with the
order
of the Bessel function. It is
not
unlike the
index number
m
of
the
sin(
m
x) cos
(
m
x) functions associated with the rectilinear geometries
.
J
(x)
not a closed function
but one
generated by an infinit
e series.
Both kinds of Bessel functions are
shown by Figure 2

5, plots taken from
http://en.wikipedia.org/wiki/Bessel_function
(
Figure 2

4.
Extract from the textbook)
Both kinds of Bessel functions are shown by figure 2

5
, plots taken from
http://en.wikipedia.org/wiki/Bessel_function
Inside the core the wave solutions must be finite for
r
0
and so the solutions must be of the form of
Bessel functions of the first kind
J
(x)
.
For these functions
2
2
2
k
q
must be
> 0, and therefore
<
(
k
= k
1
) is required, where
1
1
1
2
2
n
k
For
wave equations outside the core the wave solutions must be finite for
r
infinity and the solutions
must be of the form of Bessel functions of the second kind
K
(x)
.
This type of Bessel function re
quires
that
q
2
< 0
and therefore
> (
k = k
2
) is required, where
2
2
2
2
2
n
k
So wave propagation is bounded by the core
and the cladding as identified by the propagation
condition
k
2
<
< k
1
(2

21)
This condition and a lot of fun mathematics leads to cutoff conditions that relate to both kinds of Bessel
functions and their derivatives
and are defined in terms of the roots (= zeros) of this condition, for which
there will be a finite number =
m
, according to the bounds of the geometry
. Otherwise we find that there
will be cylindrical modes of the type TE
m
, TM
m
, EH
m
and HE
m
. Emphasis should be made on the
indices, where the first index corresponds to the azmuthal coordinate as well as the Bessel order, and the
second condition corresponds to the root of the boundary condition. The four lowest order modes are
shown by the
textbook figure 2.22.
(
Figure 2

6
.
Extract from the textbook)
For the dielectric fiber, all modes
are hybrid modes
except those
for which
= 0
. So there will not be
any modes of the form TE
11
, etc as occur for the rectangular geometries.
A summary of
these conditions is represented by the (textbook) Table 2

3
Inside the core
2
2
2
1
2
u
k
q
Inside the
cladding
2
2
2
2
2
w
k
q
. At the boundary, for
which
r = a
the cylindrical functions
J
(ua)
and
(wa)
must match each other.
Each of
these functions
will realize a set of modes
associated with
x = ua
and
x = wa
, respectively
The
number of modes
associated with the
boundary
condition is unified by
a single parameter =
V
called the ‘
normalized
frequency
’ and is the sum
2
2
2
2
a
w
u
V
2
2
2
2
2
1
2
2
2
NA
a
n
n
a
so that
NA
a
V
2
(2

22)
Parameter
V
is a dimensionless number that determines how many modes a fiber can support.
The relationship between propagation condition 2

21 and the
n
can be represented graphically by taking
the mathematical boundary conditions for the functions and iterating
them, for which the result is
shown
by (extracted) book figure 2.23.
The number of modes that can exist in a cylindrical waveguide is also identified by means of a quantity
called the
normalized propagation constant
= b
, defined as
2
2
1
V
au
b
2
2
2
1
2
2
n
n
k
w
2
2
2
1
2
2
2
2
2
2
1
2
2
2
2
/
n
n
n
k
n
n
k
k
so that
2
2
2
1
2
2
2
/
n
n
n
k
b
(2

23)
Figure 2.23.
of textbook, using the Bessel function boundary conditions to define
/k
vs
V
.
The figure shows that the HE11 mode has no cutoff
because
its
cutoff condition is
J
1
(x) = 0
which is a
condition met at
r = 0
. The figure also shows that for
NA
a
V
2
< 2.405
(2

2
4
)
corresponding to
J
0
(x) = 0
,
TE
0m
and TM
0m
and other HE
modes
are cut off. So equation 2

23 is the
condition for a
single

mode
fiber.
The number of modes per solid angle for electromagnetic radiation of wavelength
is given by
Modes
/Solid angle
M/
=
2
x
(
Area/
2
)
(2

2
5
)
w
here
the area is the cross

section of the waveguide
.
Since the solid angle associated with a nume
rical aperture is approximately
2
2
sin
NA
Then
2
2
2
2
2
2
2
2
V
NA
a
Area
M
(2

2
6
)
Equation (2

26) identifies how many modes the fiber will accommodate.
From figure 2

23
it is evident that some waveguide modes are alike in their
behavior with respect to the
normalized frequency V. Since the refractive index of the cladding is very close to that of the core the
distinction between these modes becomes blurred and so they are nearly degenerate (realize more than
one solution to the
underlying mathematics). A synopsis of the groupings (and degeneracies) are given
by (textbook) table 2.4
These groupings suggest that a
different
set of modes need to be defined
that are superpositions of these
mode groupings. They are given the ident
ification as
linearly

polarized
modes
because the
superposed
E

fiel
ds are linear and exist in modes orthogonal to one another. This characteristic is consistent with
the definition of linear polarization.
The azmuthal index
is then replaced by an index
j
that follows the rule noted by the grouping, as
follows
j = 1 for TE and TM modes
j =
+1 for EH modes
(2

27)
j =

1 for HE modes
The factor 2 that is evident for each of the
LP
jm
modes in table 2

4 is a consequence of the fact that each
and
every one of these modes have an orthogonal form of the same field cross

section, except
orthogonal. This is represented by figures 2

20 and 2

21 for the LP11 mode, which has a 4

fold
degeneracy.
Consistent with this definition is the behavior of the LP modes with respect to the normalized
propagation factor
as given by (textbook) figure 2

24
This figure exhibits much the same behavior as figure
2

23. Figure 2

23
displays the result
s
for
stiff
geometric
boundary conditions as specified by the cylindrical differential equations.
Figure 2

24
displays the results for
soft
geometric boundary conditions associated with two refractive indices that are
close in value.
The single

mode fiber is of
spec
ial
interest
. I
ts two independent degenerate modes are of the
orthogonal
form
s
shown
by (textbook) figure 2

29
Both polarizations states can be transmitted without interfering with each ot
her.
If there is any cross

section
flattening, whether intention
al or induced,
then the degeneracy is lifted and
the modes will
propagate with different phase velocitie
s.
And they will interfere with one another.
The difference
between the
effective refractive indices,
n
x
, and
n
y
for these (orthogonal) modes
is called the
fiber
birefringence
, and is given by
y
x
f
n
n
B
(2

28)
This causes a delay in phase between one mode and the other
as they propagate down the fiber and will
therefore manifest its phase difference
in terms of a
n
interference
fiber beat length
2
P
L
(2

29)
where
f
B
2
Typical
B
f
are 10

3
for a high

birefringence fiber to 10

8
for a low

birefringence
fiber.
http://www.optics.arizona.edu/kost/OPTI%20515%20Web%20Site/Lecture%20Notes/Lecture%2016%2
0Fiber%20Mode%20Pr
ofiles%20&%20LP%20Modes.pdf
http://cc.ee.ntu.edu.tw/~jfkiang/electromagnetic%20wave/demonstrations/demo_35/im2005_demo_35.h
tm
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