Electro-Optics - Applied Optics Fundamentals and Device Applications - page 91

On the other hand, the frequency is still too fast to interact with the far-
infrared vibrational frequencies. It is further assumed that the frequency is
far from any absorption resonances. This approximation further increases
the symmetry of the problem so as to reduce the number of independent
elements of the susceptibility tensor to three. Generally, this is the situation
when using wavelengths of lower energy than the bandgap in the Double-Y
semiconductor waveguides. It is then found that the nonlinear index may be
expressed [100].

= 12
0

π χ ( )

3

Δ n ( esu ) (2.26)

1122

n

The second case is with the pump frequency close to the semiconductor
band edge frequency of the waveguide. This is the situation when using
GaAs sources with AlGaAs waveguides. In this case, a different derivation
was developed by Jensen and Torabi [101] taking into account photon absorp-
tion and electron-hole recombination processes. It is found that the index
variation goes from the first power dependence on the intensity, described
by Equation 2.22, to a dependence of the form

1 3 /

n n = + n I

0 3

In this case, the nonlinear index is written as

1 3 /

c
z n N

8

α τ

⎛
⎜

⎞
⎟

1 3 / 0 0

n I = Δ n = (2.27)

3

2

3 ω

0 γ

where we use the n 3 to differentiate cases and where c 0 , N γ , α, and τ 0 are
various physical constants of the material. The parameter z is a normalized
frequency.
Finally, as an example, we calculate the optical power necessary for a phase
change of π radians according to Equation 2.24 (assuming a 1 cm interaction
length). A typical value of the nonlinear index for GaAs is n 2 ≈ 2 × 10 −13 cm 2 /W
that indicates a required intensity of approximately 250 MW/cm 2 [102]. For a
typical waveguide cross section of 4 μm 2 , this indicates a required coupled
input power of 10 W. Although this is an extremely high power for use in an
integrated optical circuit, recent work with MQW structures suggests that
the large nonlinearity observed in these materials will enhance the refrac-
tive index nonlinearity. This would conceivably allow the use of a reduced
intensity while still achieving the necessary phase change. Experiment sug-
gests intensities as low as 10 4 W/cm 2 may be effective, which corresponds to
milliwatt power levels in the waveguides.