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# Analysis Tools for Optical Circuits

DOI link for Analysis Tools for Optical Circuits

Analysis Tools for Optical Circuits book

# Analysis Tools for Optical Circuits

DOI link for Analysis Tools for Optical Circuits

Analysis Tools for Optical Circuits book

## 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.