Biomedical Engineering Reference
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and
k
is related to the absorption coefficient α by
=
4
vk
c
π
(6.5)
α
where
v
is the light frequency
c
is the speed of light in a vacuum
λ
0
is the vacuum wavelength of the light
The real and imaginary parts of the complex index of refraction are coupled
to each other as described by the Kramer-Kronig relations [4]. This coupling
results in a change in the real part of the index
n
corresponding to any change
in the imaginary part
k
. At wavelengths relatively close to the absorption
edge of the semiconductor (i.e., photon energies close to the bandgap energy
E
g
), the strong dependence of the optical absorption of
E
g
produces a corre-
sponding dependence of
n
on
E
g
. Thus, the dependence of
E
g
on temperature
translates into a dependence of
n
on temperature. It is possible to obtain an
order-of-magnitude estimate of the effect by using some well-known empiri-
cal relations. An empirical relation
4
n E
g
eV
(
)
=
77
(6.6)
which is called Moss's rule has been found to be obeyed by semiconductors
whose value of
n
4
is in the range of 30-440 [5]. Note that for Al
x
Ga
(1−
x
)
As,
with
x
= 0 to
x
= 0.4,
n
4
is in the range of 164-120. An empirical relation for the
bandgap energy of GaAs is given by [6]
1 51 2 67 10
4
−
g
eV
(
)
=
.
−
.
×
(
K
)
(6.7)
E
T
for temperatures above 100 K. Using these relations, one would expect an
increase in index of refraction of Δ
n
~ 0.01 (about 0.3%) for an increase in
temperature from 273 to 350 K. This is a relatively small change in index
compared to the difference in index between the waveguide and confining
layers of an AlGaAs waveguide; hence, little change in the light propaga-
tion characteristics would be expected. Also, the decrease in bandgap energy
from 1.437 to 1.417 eV as the temperature is increased to 350 K should not pro-
duce a significant increase in interband absorption, and free-carrier concen-
tration (and hence absorption) would not change significantly. The change
in index, however, even though relatively small, could be expected to have
a significant effect on interference-based devices such as a Mach-Zehnder
interferometer, and operating temperature would have to be taken into
account in their design. Individual devices should be tested in situ at various
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