Biomedical Engineering Reference
In-Depth Information
(response ~1 s) to fast vibrations (response ~10
−15
s). Drastic differences can
occur as the excitation frequency moves across a resonance. This is clearly
exhibited by excitonic effects in semiconductor quantum well structures.
For third-order effects, we are particularly interested in the term χ
(3)
. The
real and imaginary parts of this term give rise to nonlinear refraction and
absorption, respectively. The third-order contribution to the total polariza-
tion in Equation 2.18 can be written as
=
∑
3
3
P
(
ω
)
χ ω ω ω ω
(
;
,
,
)
E
(
ω
)
E
(
ω
)(
E
)(
ω
)
(2.19)
i
4
ijkl
4
1
2
3
j
1
k
2
l
3
j k l
,
,
In general, the tensor
χ
ijk
3
can have 81 individual elements. Fortunately, many
of the elements are found to be zero or equal to each other due to various
crystal symmetries. In particular, with GaAs being a 4
_
3 m class crystal, the
tensor reduces to 21 nonzero elements of which only 4 are independent. The
frequencies of the four fields in Equation 2.19 may be of any form satisfying
the conservation of energy:
ω
=
ω ω ω
+
+
(2.20)
4
1
2
3
The fields of interest in the Double-Y Mach Zehnder Logic Gate, for exam-
ple, are a strong pump beam of frequency ω
p
, and a weaker probe beam
of frequency ω
pr
. It is found that the problem is simplified as long as the
probe beam is not strong enough to affect the pump beam. Although it
is not necessary for this theoretical development to make a restriction on
the relationship between these frequencies, generally the beams will come
from the same source material so we set ω
p
= ω
pr
. Equation 2.19 can then be
written as
∑
3
3
P
(
ω
)
=
χ
(
−
ω ω ω
;
,
,
−
ω
),
E
(
ω
)
E
(
ω
)
E
(
ω
)
(2.21)
i
pr
ijkl
pr
pr
p
p
j
pr
k
p
i
p
i j k
,
,
,
l
The effect of interest in the Double-Y example is the alteration of the refrac-
tive index Δ
n
by the pump beam intensity in the waveguide that in turn
alters the path length for the weaker probe beam. The subsequent index can
be written in the form
2
n
=
n
+
γ
E
(2.22)
0
or
n n
=
+
n I
0
2
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