Page 374 - Electrical Properties of Materials

356 Optoelectronics

A 10 is the amplitude of the wave at A 1 = A 10 exp ik 1 z. (13.13)
z =0.
This may be described by the differential equation,

dA 1
=ik 1 A 1 . (13.14)
dz
Another wave propagating in the same direction with a wavenumber, k 2 , would
then analogously be described by the differential equation,

dA 2
=ik 2 A 2 . (13.15)
dz
Let us identify now waves 1 and 2 with those propagating in waveguides 1
and 2. Next, we shall have to take into account coupling between the wave-
guides. In order to do so, we may advance the following argument. If there is
coupling between the two waveguides, then the rate of change of the amplitude
of the wave in waveguide 1 will also depend on the amplitude of the wave in
waveguide 2, and the higher the coupling, the larger is the effect of wave 2. In
mathematical form,

dA 1
=ik 1 A 1 +iκA 2 . (13.16)
dz
And, similarly, the rate of change of the amplitude of wave 2 is

The coupling coefficient has been dA 2
=iκA 1 +ik 2 A 2 . (13.17)
taken as iκ. dz
Note that we have met this type of coupled differential equation before when
discussing quantum-mechanical problems in Chapters 5 and 7. The solution
is not particularly difficult; I shall leave it as an exercise (13.7) for the reader.
I shall give here only the solution for the case when k 1 = k 2 = k, and when all
the input power appears at port 1 with an amplitude A. We obtain then for the
two outputs,
A 1 = A 10 exp(ikz) cos κz (13.18)

and

A 2 =iA 10 exp(ikz)sin κz. (13.19)
It may be clearly seen that the amount of power transfer depends on the
length of the coupler section. When z = L = π/2κ, all the power from wave-
guide 1 can be transferred to waveguide 2. With L = π/κ, the power launched
in waveguide 1 will first cross over into waveguide 2, but it will then duly
return. At the output, all the power is in waveguide 1.
This exchange of power may take place when k 1 = k 2 , that is, when the velo-
cities are identical. If we apply a voltage, the velocity of propagation increases
in one waveguide and decreases in the other one. In the absence of synchron-
ism the amount of power transferable may be shown to decrease. When the
velocities in the two waveguides are radically different, then they simply ig-
nore each other; there is no power transfer from one to the other irrespective of
the amount of coupling.