Integrated Optics - Applied Optics Fundamentals and Device Applications - page 211

5

Integrated O ptics

5.1 PlanarOpticalWaveguideTheory
We first discuss the case of the planar waveguide in terms of the carrier com-
pensation process. If the waveguide core layer is assumed homogeneous,
then the amount of free-carrier compensation necessary for waveguiding,
that is, at least for the fundamental transverse electric mode (TE 0 ) to propa-
gate, can be calculated. This cutoff condition for TE 0 to propagate is [1]

2 2

2 π

λ

d n −

−

n

2 2 2 3

n − n ≥ arctan (5.1)

1 2

n

2

n

2

0 1 3

Since n 2 ≫ n 3 and n 1 ≥ n 2 , the argument of the inverse tangent is much greater
than unity but is not always large enough in the cases considered for arctan( x )
to be approximated by π/2. Therefore, the simple mathematical identify
arctan( x ) = π/2—arctan(1/ x ) allows an accurate Taylor series approximation
to be made. Note that

3 5

m m

− arctan( m = m − + ≅ m (5.2)

3 5

and that

n

2

− n

2

≅ n Δ n

2 (5.3) 1 2 2

results in the following condition, which is the minimum index change nec-
essary for waveguiding:

2

2

⎛
⎜

⎞
⎟

π 1

n − n = Δ n ≥ (5.4)

1 2

8 n 2 d / 1 /( n

2

n

2

)

π λ + −

2 0 2 3

Expressing Δ n in terms of the free-carrier concentrations in both the core and
substrate regions gives the carrier compensation necessary for waveguiding
to occur: