ABSTRACT
In order to understand and design optical circuits, light beam
propagation in optical waveguides that are the basic element of
the optical circuits should be analyzed. The waveguide operation
can easily be understood by considering the electron confinement
in quantum wells. Figures 3.1 (a) and (b), respectively, show
the analogy between the electron confinement and the photon
confinement, and a schematic illustration of light beam confinement
into an optical waveguide. It is known that the wavefunction of
electron ψ and the electric field of light waves E are determined by the following wave equations:(
− 2
2m ∇2 + V (x)
) ψ(x) = Eψ(x) (3.1)
(∇2 + (n (x) k0)2)E(x) = 0. (3.2) Here, m, V and E are mass, potential energy, and total energy of an electron, respectively; is Plank constant divided by 2π ; n is refractive index and k0 is wavenumber in vacuum. Equations (3.1) and (3.2) can be written as follows:(
− 2 2m∇2 − (E − V (x))
) ψ(x) = 0(
− 2 2m∇2 −
2 (x) ) E(x) = 0
⎫⎬ ⎭ (3.3)
Comparing the two equations in (3.3), it is found that the following
relationships exist:
ψ(x) ↔ E (x) (3.4)
E − V (x) ↔ 2k20 2m
n2 (x) (3.5)
It is known that an electron tends to be confined in a region with
small potential energy V (x) as shown in Fig. 3.1(a). The relationship of Eq. (3.5) indicates that “–V (x)” corresponds to “n2(x).” This suggests that a photon tends to be confined in a region with high
refractive index. Therefore, by constructing line-shaped regions
with higher refractive index in a planar substrate, light beams
are confined in the line as shown in Fig. 3.1(b). The line is the
core of the optical waveguide, and the surrounding area is clad.
By constructing higher-refractive-index regions in a medium with
designated patterns, light beams can be guided in the designated
routing to make optical circuits.
