WOLFRAM|DEMONSTRATIONS PROJECT

Thermal Distribution in an Optical Fiber with Heat Source

​
temperature distribution
problem geometry
Biot number Bi
3
thermal conductivities ratio β
0.1
dimensionless absorption coefficient γ
0.3
dimensionless heat generation Ω
20
The power efficiency of fiber lasers is often determined by the core temperature distribution in the fiber. The optical fiber is modeled as a doped inner core of radius
R
0
and a cladding region of thickness
h=
R
1
-
R
0
. The length of the optical fiber is
L
. The thermal conductivity of the inner core is taken to be
k
0
and the cladding region is
k
1
. The heat density within the inner core is taken to be
q(z)=Ω
-αz

, where
α
is a suitable absorption coefficient. The temperature within the inner core is denoted by
T
1
and the temperature in the cladding as
T
2
. The thermal boundary conditions on the optical fiber are as follows. The outer surface of the cladding is subject to Newton's law of cooling with a specified heat transfer coefficient
h
:
∂
T
2
∂r
+
h
k
1
(
T
2
-
T
0
)=0
.
Newton's law of cooling is also assumed for the ends of the fiber containing cladding. Thus for the cladding region:
∂
T
2
∂r
±
h
k
1
(
T
2
-
T
0
)=0
at
z=0
and
z=L
.
At
z=0
we specify a heat flux for the inner core:
∂
T
1
∂r
=-
q
0
R
0
k
0
at
z=0
.
At
z=L
we assume Newton's law of cooling for the inner core:
∂
T
1
∂r
+
h
k
0
(
T
1
-
T
0
)=0
at
z=L
.
This Demonstration finds the temperature distribution both in the cladding and core region for user-set values of the Biot number
Bi
, thermal conductivities ratio
β
, dimensionless heat generation
γ
, and dimensionless absorption coefficient
Ω
. In order to obtain this distribution, a finite difference scheme is applied as described in depth in Details.