Biomedical Engineering Reference
In-Depth Information
For Ti diffusion into LiNbO
3
the temperature dependent
D
has been found
exponentially to be [59]:
[
−
T T
/ ]
D D e
=
(5.29)
0
y0
where
T
is the absolute temperature
T
0
= 2.9 × 10
4
D
y0
= 7 × 10
−3
cm
2
/s
The form appears logical since
E
can be expected to be quite low at the sur-
face, yet increasing with diffusion depth. For this type of diffusion, depth
dependent diffusion coefficients are being pursued via mathematical mod-
eling [60]. These are particularly valuable in explaining the lateral surface
diffusion which is often observed in the fabrication of these devices [61].
One particular phenomena which greatly benefits these devices results from
the lower observed lateral diffusion rates. It has been noted that the diffu-
sion of Ti down (vertical) into the LiNbO
3
crystal is about 1.5 times the hor-
izontal diffusion rate (dependent upon the orientation of the crystal) [62].
This enhances the quality of the channels being formed and ultimately the
devices being fabricated.
As seen from the equation the Ti diffused into the crystals is a linear
function of Ti thickness; thus the Ti thickness can be used to determine the
refractive index change. This assumes that all of the Ti is indiffused during
the diffusion time (
t
). A proposed equation for this relation is [63]
⎛
⎜
⎞
⎟
2
d
d
n
c D
τ
Δ
n
=
α
(5.30)
p
where
d
n
/d
c
is the change in refractive index with concentration
α is a proportionality constant
This change has been found to be d
n
e
/d
c
= 0.47 and d
n
0
/d
c
= 0.625, where
n
e
and
n
0
are extraordinary and ordinary refractive indices. For a specific wave-
length the diffusion depth (
D
*) is of the form
⎡
⎣
)
.
0 5
⎤
⎦
(
−
T T
D
*
=
2
D te
0
y
0
)
.
0 5
D
*
=
2
(
Dt
(5.31)
This shows that all the important parameters for defining the waveguide
may be controlled within fabrication. Waveguide width is defined via
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