Advanced Physics for Brachytherapy 2018

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: ) ,(   m l Y

l

  =

m





m

Y

' ) ,( l Y m   



) ,(' 

dφ dθ sin =

 

 Er

Er

, )ˆ( ) ,(  Y



)ˆ, ,( = 





l

ml

,

l

2

 

l

m

−= l

0





1

m

Er

l Y Er

 d m



) ,( 

  =

)ˆ, ,(

' )ˆ(

Y

' ) ,( l Y m  

) (cos ) ,('   d



=

δ dφ

ml

,

  − 

ll' δ mm'

l

4



2

1

and therefore they can be used to expand any function defined on the surface of a sphere in an infinite series:

Note we have described the energy fluence in spherical coordinates and that we have NOT discretized in direction

l

  =

m

f

C

Y





= ) ,(





, ) ,(



ml

,

l

l

l m

0

=

−

l Y f 

 d m

= mlC

(





) ,

) ,( 

 

,