TE Radiation Mode Derivation
Figure 1: Schematic of a dielectric slab waveguide
Propagation of electromagnetic waves in dielectric media is governed by Maxwell equations
E =
H =+
----------- (1)
--------- (2)
. B = 0 ------------------- (3)
. D = ------------------- (4)
E Electric Field Strength
H Magnetic Field Strength
B Magnetic Flux Density
D Electric Displacement
J Electric Current Density
Electric charge Density
B, D, E and H are vectors in 3D space and are generally functions of both time and space.
Let us consider,
D = E = n2E --------- (5)
B = H ----------------- (6)
J = 0 -------------------- (7)
= 0 -------------------- (8)
1
Wave propagation in slab waveguide
In this section we will find general solutions to Maxwells equations for wave propagation in
slab waveguide. The geometry of an arbitrary slab waveguide is shown in the Figure and is
characterized by conductor boundaries that are parallel to the z-axis. These structures are
assumed to be uniform in shape and dimension in the z direction and infinitely long. The
conductors will initially be assumed to be perfectly conducting.
The wave propagation can be written in the following format for ease.
= + +
-------- (9)
=
+
+
--------- (10)
From Maxwell equation,
E =
--------- (11)
) --------- (12)
| | = ( + +
] [
[
]+
] = [
] + [
[ ]------]+
(13)
For slab waveguide condition of,
= 0, which mean there is no variation in y-direction
------- (14)
------- (15)
-------- (16)
From above equations, 2, 5 and 7 we will get
H = 2 ------- (17)
Also, this equation (17) can be expanded into previous form of E as equation (12)
) -------- (18)
|
| = 2 ( + +
] [
[
]+
] = [2
] + [2
[2 ] -------]+
(19)
Then applying the condition of slab waveguide, we get
= 2
= 2
= 2
------- (20)
-------- (21)
-------- (22)
When we consider strictly time harmonic fields and we are supposed to obtain normal modes in
the slab waveguide, we assume the z dependence of the mode fields which are given by and
respectively. So, if we combine them we get,
() --------- (23)
TE modes have only three field components: Ey, Hx and Hz. The slab waveguide for our case is
shown as in fig.1.
Assumption relating to physical condition,
= 0 () ------- (24)
= 0 () ------- (25)
= 0 () ------- (26)
When we perform partial differentiation of equations 24, 25 and 26
= &
= &
= &
= ------- (27)
= ------- (28)
= -------- (29)
The main fields in TE mode are Ey, Hx and Hz so, using equations 14, 16 and 21 we will get,
= ------- (30)
= --------- (31)
= 2 --- (32)
Substituting 30 and 31 into 32, we get,
2
2
+ ( 2 2 2 ) = 0 ----- (33)
Where, =
= ---- (34)
k : Free Space Wave Number
: Propagation Constant
n : Material Refractive Index
Now, we need to find the solutions of the differential Equation of order 2 eq. 33 and obtain
magnetic and electric field components from eq. 30 and eq. 31. We at this point should
remember that asymmetric waveguide have two types of radiation modes.
We, at this point should remember that asymmetric waveguide have two types of radiation
modes. A wave impinging from region n2 to the core can be in total internal reflection at the
interface of regions 1 and 3. This condition will rise to an evanescent field in region 3 and a
standing wave in the core and in region 2.
The range of values that belongs to this type of radiation modes follows Snells law. The
smallest = n2k and the largest = n3k. So, the range of for the radiation mode is n3k
n2k. This type of radiation mode with exponentially decaying fields on one side of the core is
peculiar to asymmetric slab waveguide. If, n2 = n3 then the range will be zero, i.e. for symmetric
slab waveguide.
Now, the solution of differential equation of order 2 eq. 33 for electrical component in ydirection is as follows.
= { +
( + ) + ( + )
>0
< < 0 ---------------- (35)
<
The magnetic component can be found using eq. 35 and eq. 31
( )
= (
) [ + ]
(
) [( + ) + ( + )]
{
>0
< < 0 (36)
<
The refractive index n in eq. 33 assumes the value n3 in region 3, for x > 0; n1 in region 1, for d
< x < 0; and n2 in region 2, for x < -d. The parameters, and are defined by the equations
= (32 2 2 )1/2
= (12 2 2 )1/2
= (22 2 2 )1/2
Now, applying the boundary conditions at x = 0 and x = -d for each electric and magnetic
component along y and z respectively.
For Ey, at x = 0
0 = 0+ 0
-----------=
(37)
For Ey, at x = -d
+ = ( + ) + ( + )
(38)
=
For Hz, at x = 0
( ) 0 = ( ) [ 0 + 0]
(39)
For Hz, at x = -d
( ) [ + ] = ( ) [ ( + ) + ( + )]
(40)
= ( + )
Thus, we are able to write all the constants that exist in electrical and magnetic components in
terms of C as follows.
=
=
= + (
=
( (
) )
