Wolfram Demonstrations Project

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

and a cladding region of thickness

. The length of the optical fiber is

. The thermal conductivity of the inner core is taken to be

and the cladding region is

. The heat density within the inner core is taken to be

, where

is a suitable absorption coefficient. The temperature within the inner core is denoted by

and the temperature in the cladding as

. 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

Newton's law of cooling is also assumed for the ends of the fiber containing cladding. Thus for the cladding region:

and

we specify a heat flux for the inner core:

we assume Newton's law of cooling for the inner core:

This Demonstration finds the temperature distribution both in the cladding and core region for user-set values of the Biot number

, thermal conductivities ratio

, dimensionless heat generation

, and dimensionless absorption coefficient

. In order to obtain this distribution, a finite difference scheme is applied as described in depth in Details.

Contributed by: Brian G. Higgins and Housam Binous

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A schematic of the problem geometry is shown in the first snapshot.

Mathematical formulation

For computational convenience we introduce the following dimensionless coordinates:

, and

Thus the thermal conduction equations for the inner core and cladding are:

for

and

for

and

The quantity

is a dimensionless heat generation term defined as:

while

is the dimensionless absorption coefficient for the light source entering the core region.

Boundary conditions

The boundary conditions for the core region are:

BC1:

BC2:

, 0<ξ<ξ₁, at

BC3:

Here

is a Biot number and

is a ratio of thermal conductivities for the cladding and core regions. Typically the conductivity of the cladding is much smaller than that of the core so that

At the core/cladding interface we must have continuity of heat flux and temperature. These conditions are:

BC4:

BC5:

For the cladding region, we have Newton's law of cooling at all exposed surfaces:

BC6:

, at

BC7:

, ξ₁<ξ<1, at

BC8:

, ξ₁<ξ<1, at

Finite difference formulation

We consider a uniform grid in the

and

directions. Let

be the number of grid points in the

direction,

be the number grid points in the

direction for the core, and

be the number of grid points in the

direction for the cladding region.

We discretize the domain into equal segments and use central differences for the Laplacian operator. The result is:

However at

there is a singularity. Using L'Hopital's rule we have at

Thus the Laplacian at

(i.e.

) is given by

where

is a fictitious node. Then noting that at

(because of symmetry). A central difference approximation of the derivative

gives

, so on eliminating the fictitious node we have:

Boundary conditions

As a rule we use central differences for derivatives at the boundary.

Consider BCs in

direction:

For BC 1 we require that:

BC1:

For BC2 we have:

BC2:

for

For BC3 we have:

BC3:

for

Similar finite difference formulas can be derived for the other BCs. However, for the interface conditions we use first-order differences. This allows us to readily couple the two grid boundaries.

Brian G. Higgins and Housam Binous

"Thermal Distribution in an Optical Fiber with Heat Source"
http://demonstrations.wolfram.com/ThermalDistributionInAnOpticalFiberWithHeatSource/
Wolfram Demonstrations Project
Published: August 2, 2011