16.00 o4 w somerville
DESCRIPTION
Research 3: W SommervilleTRANSCRIPT
A simple method for finding recurrence relations inphysical theories: application to electromagnetic
scattering
Walter Somerville Eric Le Ru
The MacDiarmid Institute for Advanced Materials and NanotechnologySchool of Chemical and Physical Sciences
Victoria University of Wellington
Oct 17, 2011
Walter Somerville A simple method for finding recurrence relations
Electromagnetic scattering - overview
Interest in Raman scattering
Particularly in Surface Enhanced Raman Scattering (SERS)
Requires knowledge of electric field near to the surface ofmetallic nanoparticles
Walter Somerville A simple method for finding recurrence relations
Electromagnetic scattering - overview
500 600 700 8000.0
0.2
0.4
0.6
0.8
Ext
inct
ion
[cm
-1]
Wavelength [nm]
log10(F/5)
λL=633nmΔν=1620cm‐1
θ
Walter Somerville A simple method for finding recurrence relations
Electromagnetic scattering - different methods
Discrete Dipole Approximation
Finite Element methods
Mie Theory
T -matrix
Walter Somerville A simple method for finding recurrence relations
T -matrix - overview
Express fields as a sum of vector spherical harmonics:
EInc(r) = E0
∑n,m
anmM(1)nm(kM , r) + bnmN
(1)nm(kM , r).
Relate incident and scattered field with the T -matrix,(pq
)= T
(ab
)
Walter Somerville A simple method for finding recurrence relations
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
1Waterman, P. C. (1965) Proc. IEEE 53, 805–812Walter Somerville A simple method for finding recurrence relations
T-matrix - EBCM
Introduced with T -matrix by Waterman
T = −RgQQ−1
Expressions are much simpler when particle has a symmetry ofrevolution
Walter Somerville A simple method for finding recurrence relations
T -matrix - expressions
We use the expressions2
K 1nk =
∫ π
0dθ xθmdndkξnψ
′k
K 2nk =
∫ π
0dθ xθmdndkξ
′nψk
L1nk =
∫ π
0dθ sin θxθτndkξnψk
L2nk =
∫ π
0dθ sin θxθdnτkξnψk
ξ, ψ ∼ spherical-Bessel functions, dn, dk spherical harmonics.
2Somerville, W. R. C., Auguie, B., and Le Ru, E. C. Sep 2011 Opt. Lett.36(17), 3482–3484
Walter Somerville A simple method for finding recurrence relations
Suspect relations
Owing to the relations between Bessel functions, we suspect theremight be some between the integrals
ψn−1(z) + ψn+1(z) =2n + 1
zψn(z)
There are also relations between the angular functions
n cos θ dn(θ)− sin θ τn(θ) =√
n2 −m2dn−1(θ)
Walter Somerville A simple method for finding recurrence relations
Question
Do the integrals have relations, and if so, what are they?
Walter Somerville A simple method for finding recurrence relations
Rank
Rank of a matrix is the number of linearly independentrows/columns.
rank
1 2 34 5 65 7 9
= 2
A non-maximum rank indicates that there are some linear relations.
Walter Somerville A simple method for finding recurrence relations
Rank – example
fn(x)
x1 −→ 5
1 1 1 1 1 1
1 2 3 4 5n ↓ 2 3 4 5 6
3 5 7 9 115 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
Rank – example
fn(x)
x1 −→ 5
1 1 1 1 1 1
1 2 3 4 5n ↓ 2 3 4 5 6
3 5 7 9 115 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
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
Easy relations
L131 − 3L2
31 = −7.348L111 + 7.071K 2
21
L213 − 3L1
13 = −3√
6L111 + 5
√2K 1
12
Walter Somerville A simple method for finding recurrence relations
Easy relations
L131 − 3L2
31 = −3√
6L111 + 5
√2K 2
21
L213 − 3L1
13 = −3√
6L111 + 5
√2K 1
12
Walter Somerville A simple method for finding recurrence relations
Easy relations
L131 − 3L2
31 = −3√
6L111 + 5
√2K 2
21
L213 − 3L1
13 = −3√
6L111 + 5
√2K 1
12
Walter Somerville A simple method for finding recurrence relations
Dimensionality reduction
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·
· + · + · + · +
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·· + · + · + · +
L1,L2 K1,K2
L151 −
85
29L2
51 = −s 45
29√
2
(L1
42 − L242
)+
7× 23
29√
10
(L1
31 +17
23L2
31
)For spheroid only
Walter Somerville A simple method for finding recurrence relations
Dimensionality reduction
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·
· + · + · + · +
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·· + · + · + · +
L1,L2 K1,K2
(L1
42 − 2L242
)=
1
s√
5
[L1
31 − (4− 30s2)L231
]+√
15(2sK 1
32 + K 232
)−√
3
s
(2K 1
41 + K 241)
Walter Somerville A simple method for finding recurrence relations
Dimensionality reduction
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·
· + · + · + · +
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·· + · + · + · +
+ · + · + · + ·· + · + · + · +
L1,L2 K1,K2
Walter Somerville A simple method for finding recurrence relations
Example relation
A relation between twelve elements:
α((k + 1) L1
n,k+1 − nL2n,k+1
)− β
(kL1
n,k−1 + nL2n,k−1
)=[
−n (1 + 2k){k4 + 2k3 +
((1− n2
)s2 − 1
)k2 +
((1− n2
)s2 − 2
)k+(
n2 − 1)s2}]
K 1n,k
+ [(1 + 2k) (n − 1) ks (n + 1) (k + 1)]K 2n,k
+ [ns (n + 1)α]K 1n−1,k+1 + [(n + 1) (k + 1)α]K 2
n−1,k+1
+ [ns (n + 1)β]K 1n−1,k−1 + [−k (n + 1)β]K 2
n−1,k−1
+ [−s (n + 1) (1 + 2k) k (k + 1)] L1n−1,k
+[−s (n + 1) (1 + 2k)
(k2 + k − n2s2 + s2
)n]L2n−1,k
where
α = k2(k2 + s2 − n2s2 − 1), β = (k + 1)2(k2 + s2 − n2s2 + 2k).
Walter Somerville A simple method for finding recurrence relations
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 fastercalculations of the T -matrix
Walter Somerville A simple method for finding recurrence relations