A simple method for finding recurrence relations in physical theories: application to electromagnetic                     s...
Electromagnetic scattering - overview      Interest in Raman scattering      Particularly in Surface Enhanced Raman Scatte...
Electromagnetic scattering - overview                                                                                     ...
Electromagnetic scattering - different methods      Discrete Dipole Approximation      Finite Element methods      Mie Theo...
T -matrix - overview  Express fields as a sum of vector spherical harmonics:                                         (1)   ...
T -matrix - history          Introduced by Waterman in 19651          Can be applied to multiple scatterers          Can e...
T-matrix - EBCM  Introduced with T -matrix by Waterman                            T = −RgQ Q−1  Expressions are much simpl...
T -matrix - expressions  We use the expressions2                                      π                        1          ...
Suspect relations  Owing to the relations between Bessel functions, we suspect there  might be some between the integrals ...
Question     Do the integrals have relations, and if so, what are they?                     Walter Somerville   A simple m...
Rank  Rank of a matrix is the number           of linearly independent  rows/columns.                                   ...
Rank – example                                    fn (x)                                          x                       ...
Rank – example                                    fn (x)                                          x                       ...
Examining rank     72 entries of of K1 , K2 , L1 , L2     Rank of 14     Some relations are easy                       Wal...
Easy relations                 L1 − 3L2 = −7.348L1 + 7.071K21                  31    31         11                        ...
Easy relations                              √       √ 2                 L1 − 3L2 = −3 6L1 + 5 2K21                  31    ...
Easy relations                              √       √ 2                 L1 − 3L2 = −3 6L1 + 5 2K21                  31    ...
Dimensionality reduction                  L1 , L2                                            K1 , K2                     ...
Dimensionality reduction                  L1 , L2                                            K1 , K2                     ...
Dimensionality reduction                  L1 , L2                                            K1 , K2                     ...
Example relation  A relation between twelve elements:  α (k + 1) L1         2           1         2             n,k+1 − nL...
Current state/Future work     We have found a relation between four types of integrals     It’s not obvious how to fill the...
Upcoming SlideShare
Loading in …5
×

16.00 o4 w somerville

305 views

Published on

Research 3: W Sommerville

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
305
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
2
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

16.00 o4 w somerville

  1. 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. 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. 3. Electromagnetic scattering - overview λL =633nm Δν =1620cm‐1 log10(F/5) 0.8Extinction [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. 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. 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. 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. 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. 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. 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. 10. Question Do the integrals have relations, and if so, what are they? Walter Somerville A simple method for finding recurrence relations
  11. 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. 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. 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. 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. 15. Easy relations L1 − 3L2 = −7.348L1 + 7.071K21 31 31 11 2 Walter Somerville A simple method for finding recurrence relations
  16. 16. Easy relations √ √ 2 L1 − 3L2 = −3 6L1 + 5 2K21 31 31 11 Walter Somerville A simple method for finding recurrence relations
  17. 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. 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. 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. 20. Dimensionality reduction L1 , L2 K1 , K2    + · + · + · + · · + · + · + · + · + · + · + · + + · + · + · + ·            + · + · + · + ·   · + · + · + · +     · + · + · + · +   + · + · + · + ·     + · + · + · + ·   · + · + · + · +   · + · + · + · + + · + · + · + ·       + · + · + · + · · + · + · + · + Walter Somerville A simple method for finding recurrence relations
  21. 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. 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

×