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The method
Many orthogonal bases exist
In example:
➢
Spherical harmonics,
, are a
series of functions defined on the surface
of a sphere that exhibit orthogonality:
and therefore they can be used to expand
any function defined on the surface of a
sphere in an infinite series:
) ,(
m
l
Y
=
=
−
=
=
d m
l
Y f
mlC
l
m
l
Y
l
l m
ml
C
f
)
,(
) ,
(
,
,
0
) ,(
,
) ,(
'
)ˆ(
4
)ˆ, ,(
) ,(
,
,
0
)ˆ(
) ,(
,
)ˆ, ,(
=
=
−=
=
d m
l
Y Er
Er
ml
l
m
l
Y
l
l
m
Er
ml
Er
ll' δ mm'
δ dφ
2
1
1
)
(cos
) ,('
'
) ,(
dφ dθ sin
2
) ,('
'
) ,(
=
−
=
d
m
l
Y m
l
Y
m
l
Y m
l
Y
Note we have described the energy
fluence in spherical coordinates
and that we have NOT discretized
in direction