This document presents a summary of research on finding recurrence relations in electromagnetic scattering calculations. Recurrence relations can reduce the dimensionality and complexity of calculations using the T-matrix method. The researchers examined integrals of vector spherical harmonics (K1, K2, L1, L2) involved in T-matrix calculations and found relationships between them. These relationships were expressed as equations relating a combination of the integrals. Future work aims to leverage these relations to more efficiently compute T-matrices by reducing the number of integrals that must be directly evaluated.

1. A simple method for finding recurrence relations in
physical theories: application to electromagnetic
scattering
Walter Somerville Eric Le Ru
The MacDiarmid Institute for Advanced Materials and Nanotechnology
School of Chemical and Physical Sciences
Victoria University of Wellington
Oct 17, 2011
Walter Somerville A simple method for finding recurrence relations

2. Electromagnetic scattering - overview
Interest in Raman scattering
Particularly in Surface Enhanced Raman Scattering (SERS)
Requires knowledge of electric field near to the surface of
metallic nanoparticles
Walter Somerville A simple method for finding recurrence relations

3. Electromagnetic scattering - overview
λL =633nm
Δν =1620cm‐1
log10(F/5)
0.8
Extinction [cm ]
-1
0.6
0.4
θ
0.2
0.0
500 600 700 800
Wavelength [nm]
Walter Somerville A simple method for finding recurrence relations

4. Electromagnetic scattering - different methods
Discrete Dipole Approximation
Finite Element methods
Mie Theory
T -matrix
Walter Somerville A simple method for finding recurrence relations

5. T -matrix - overview
Express fields as a sum of vector spherical harmonics:
(1) (1)
EInc (r) = E0 anm Mnm (kM , r) + bnm Nnm (kM , r).
n,m
Relate incident and scattered field with the T -matrix,
p a
=T
q b
Walter Somerville A simple method for finding recurrence relations

6. T -matrix - history
Introduced by Waterman in 19651
Can be applied to multiple scatterers
Can easily handle orientation averaging
Used in
Astrophysics
Aerosols
Acoustic scattering
Plasmonics
1
Waterman, P. C. (1965) Proc. IEEE 53, 805–812
Walter Somerville A simple method for finding recurrence relations

7. T-matrix - EBCM
Introduced with T -matrix by Waterman
T = −RgQ Q−1
Expressions are much simpler when particle has a symmetry of
revolution
Walter Somerville A simple method for finding recurrence relations

8. T -matrix - expressions
We use the expressions2
π
1
Knk = dθ xθ mdn dk ξn ψk
0
π
2
Knk = dθ xθ mdn dk ξn ψk
0
π
L1 =
nk dθ sin θxθ τn dk ξn ψk
0
π
L2 =
nk dθ sin θxθ dn τk ξn ψk
0
ξ, ψ ∼ spherical-Bessel functions, dn , dk spherical harmonics.
2
Somerville, W. R. C., Augui´, B., and Le Ru, E. C. Sep 2011 Opt. Lett.
e
36(17), 3482–3484
Walter Somerville A simple method for finding recurrence relations

9. Suspect relations
Owing to the relations between Bessel functions, we suspect there
might be some between the integrals
2n + 1
ψn−1 (z) + ψn+1 (z) = ψn (z)
z
There are also relations between the angular functions
n cos θ dn (θ) − sin θ τn (θ) = n2 − m2 dn−1 (θ)
Walter Somerville A simple method for finding recurrence relations

10. Question
Do the integrals have relations, and if so, what are they?
Walter Somerville A simple method for finding recurrence relations

11. Rank
Rank of a matrix is the number of linearly independent
rows/columns.
 
1 2 3
rank  4 5 6  = 2
5 7 9
A non-maximum rank indicates that there are some linear relations.
Walter Somerville A simple method for finding recurrence relations

12. Rank – example
fn (x)
x
1 −→ 5
1 1 1 1 1 1
1 2 3 4 5
n ↓2 3 4 5 6
 
3 5 7 9 11
 
5 5 8 11 14 17
Walter Somerville A simple method for finding recurrence relations

13. Rank – example
fn (x)
x
1 −→ 5
1 1 1 1 1 1
1 2 3 4 5
n ↓2 3 4 5 6
 
3 5 7 9 11
 
5 5 8 11 14 17
f0 (x) = 1
f1 (x) = x
fn+2 (x) = fn+1 (x) + fn (x)
Walter Somerville A simple method for finding recurrence relations

14. Examining rank
72 entries of of K1 , K2 , L1 , L2
Rank of 14
Some relations are easy
Walter Somerville A simple method for finding recurrence relations

15. Easy relations
L1 − 3L2 = −7.348L1 + 7.071K21
31 31 11
2
Walter Somerville A simple method for finding recurrence relations

16. Easy relations
√ √ 2
L1 − 3L2 = −3 6L1 + 5 2K21
31 31 11
Walter Somerville A simple method for finding recurrence relations

17. Easy relations
√ √ 2
L1 − 3L2 = −3 6L1 + 5 2K21
31 31 11
√ √ 1
L2 − 3L1 = −3 6L1 + 5 2K12
13 13 11
Walter Somerville A simple method for finding recurrence relations

18. Dimensionality reduction
L1 , L2 K1 , K2
  
+ · + · + · + · · + · + · + · +
· + · + · + · + + · + · + · + ·
  
  
  

 + · + · + · + · 
 · + · + · + · + 


 · + · + · + · + 
 + · + · + · + · 


 + · + · + · + · 
 · + · + · + · + 

· + · + · + · + + · + · + · + ·
  
  
+ · + · + · + · · + · + · + · +
85 2 45 7 × 23 17 2
L1 −
51 L = −s √ L1 − L2 + √ L1 + L
29 51 29 2
42 42
29 10
31
23 31
For spheroid only
Walter Somerville A simple method for finding recurrence relations

19. Dimensionality reduction
L1 , L2 K1 , K2
  
+ · + · + · + · · + · + · + · +
· + · + · + · + + · + · + · + ·
  
  
  

 + · + · + · + · 
 · + · + · + · + 


 · + · + · + · + 
 + · + · + · + · 


 + · + · + · + · 
 · + · + · + · + 

· + · + · + · + + · + · + · + ·
  
  
+ · + · + · + · · + · + · + · +
1
L1 − 2L2 = √ L1 − (4 − 30s 2 )L2
42 42 31 31
s 5
√
√ 1 2 3
+ 15 2sK32 + K32 − 2K41 + K 2 41
1
s
Walter Somerville A simple method for finding recurrence relations

20. Dimensionality reduction
L1 , L2 K1 , K2
  
+ · + · + · + · · + · + · + · +
· + · + · + · + + · + · + · + ·
  
  
  

 + · + · + · + · 
 · + · + · + · + 


 · + · + · + · + 
 + · + · + · + · 


 + · + · + · + · 
 · + · + · + · + 

· + · + · + · + + · + · + · + ·
  
  
+ · + · + · + · · + · + · + · +
Walter Somerville A simple method for finding recurrence relations

21. Example relation
A relation between twelve elements:
α (k + 1) L1 2 1 2
n,k+1 − nLn,k+1 − β kLn,k−1 + nLn,k−1 =
−n (1 + 2k) k 4 + 2k 3 + 1 − n2 s 2 − 1 k 2 + 1 − n2 s 2 − 2 k+
n2 − 1 s 2 1
Kn,k
2
+ [(1 + 2k) (n − 1) ks (n + 1) (k + 1)] Kn,k
1 2
+ [ns (n + 1) α] Kn−1,k+1 + [(n + 1) (k + 1) α] Kn−1,k+1
1 2
+ [ns (n + 1) β] Kn−1,k−1 + [−k (n + 1) β] Kn−1,k−1
+ [−s (n + 1) (1 + 2k) k (k + 1)] L1
n−1,k
+ −s (n + 1) (1 + 2k) k 2 + k − n2 s 2 + s 2 n L2
n−1,k
where
α = k 2 (k 2 + s 2 − n2 s 2 − 1), β = (k + 1)2 (k 2 + s 2 − n2 s 2 + 2k).
Walter Somerville A simple method for finding recurrence relations

22. Current state/Future work
We have found a relation between four types of integrals
It’s not obvious how to fill the matrices using this information
We aim to solve these problems, allowing much faster
calculations of the T -matrix
Walter Somerville A simple method for finding recurrence relations