Page 169 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electromagnetic System
divergence free. Finally, we insert (4.60) into (4.10), and cancel the com-
mon exponential factor to obtain
2 2 2 2
∇ E – [ k – k ε]E = ∇ E – γ E = 0 (4.62)
t t 0 t t t t
⁄
where k = ω c . We have thus obtained an eigenmode equation for the
0
planar transverse electric field vector E yz,( ) . In general, solving (4.62)
t
will yield the eigenpairs γ ( n () , E n () ) that represent the allowable solu-
t
2
tions. From each γ n () = k [ 2 n() + k ε] we obtain the allowable k± n () . We
0
make two observations:
n ()
• Since we obtain both a positive and a negative value for k , this
implies that the solution generates waves that propagate in the posi-
tive and negative directions along the waveguide.
• Equation (4.62) has the exact form of the stationary Schrödinger
equation. The role of the dielectric constant takes on that of the poten-
tial in the Schrödinger equation, and we could use solution methods
already developed for solving potential well problems. We now use
this fact to investigate a classical case.

4.3.1 Example: The Homogeneous Glass Fiber
We consider a circular cross-section glass fiber waveguide with a qua-
2 2
–
dratically varying refractive index n = Abr , with and constant
A
b
2
[4.7]. Since ε = n , this means that we have a spatially linear function
for the dielectric constant . In cylindrical coordinates r θ,( ) , equation
ε
(4.62) now becomes
2 2
∂ E 1 ∂E 1 ∂ E
2 2 2 t t t 2 2
∇ E – [ k – k ε]E = 2 + --- + ---- 2 2 – [ k – k ε]E = 0 (4.63)
0
0
t
∂
t
t
t
r ∂ r r r ∂ θ
The geometry is axially symmetric. We therefore assume that we can de-
couple the radial and angular solution components using
,
(
E r θ) = Rr()Θ θ() . Inserting this into (4.63) we obtain
t
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