Graphics Reference
In-Depth Information
Thin slab approximation.
Consider first the generalization of the dipole model
to a thin strip of width
d
. This introduces a second interface at the plane
z
=
d
.
(The coordinate system used for the multipole model has the plane
z
0atthe
surface of the object, with the positive
z
-axis pointing into the object.) As in
the dipole model, the boundary condition at the top surface is that the net inward
flux is zero:
=
F
dr
L
d
(
r
,
ω
)(
−
n
·
ω
)
d
ω
=
L
d
(
r
,
ω
)(
n
·
ω
)
d
ω
at
z
=
0
,
Ω
+
Ω
−
(this is essentially the same as Equation 4.8 above). The condition at the lower
interface is similar,
F
dr
,
ω
)(
·
ω
)
d
ω
=
,
ω
)(
−
·
ω
)
d
ω
L
d
(
r
n
L
d
(
r
n
at
z
=
d
.
Ω
−
Ω
+
These reduce to the corresponding partial differential equations in the fluence
φ
(
r
)
:
2
A
3
σ
t
·
∂φ
(
r
)
φ
(
r
)
−
=
0at
z
=
0
,
(4.15)
∂
z
2
A
3
σ
t
·
∂φ
(
r
)
φ
(
)+
=
=
,
r
0at
z
d
(4.16)
∂
z
where
A
as in Equation (4.10).
The condition at the upper boundary can be satisfied, approximately, as in the
dipole model by placing virtual positive and negative point light sources so that
the energy balances at the plane
z
=(
1
+
F
dr
)
/
(
1
−
F
dr
)
=
−
z
b
. This is illustrated in Figure 4.17; the
dashed line marked
A
is at
z
b
, and the two sources are positioned at
z
v
,
0
and
z
r
,
0
.
The positive and negative sources are thus placed on opposite sides of the plane
at a distance 1
−
/
σ
tr
, the length of one mean free path. Note that this only works if
the thickness
d
of the slab is larger than the mean free path.
Satisfying the boundary condition at the lower interface (
z
d
) requires a
third virtual point source, a negative source placed so that the energy balances
at
z
=
d
. That is, the third source is placed at
z
v
,
1
,whichis
z
r
,
0
mirrored
about the dashed line
B
in Figure 4.17. Unfortunately, the presence of this new
source breaks the boundary condition at the top surface. This is repaired by
adding another positive source at
z
r
,−
1
, which is the mirror image of the bottom
source in the plane
z
=
z
b
+
z
b
(the dashed line
A
in the figure). Yet another positive
source mirrored in the plane
z
=
−
d
has to be added to counteract this new pos-
itive source (at
z
r
,−
1
in the figure). Then a new negative source has to be added
(at
z
v
,−
1
in the figure) and so the process goes.
An infinite number of sources is theoretically necessary to satisfy the bound-
ary conditions. However, the contribution of each source decreases exponentially
=
z
b
+
