1. Mie-theory for a golden sphere A story of waves PART I

2. Research question? Mie-theory = scattering theory Scattering theory: – How does light react to particles? 2 ?

3. THE MAXWELLS “All of electromagnetism is contained in the Maxwell equations” Richard P. Feynman 3

4. The Maxwells The general (microscopic) Maxwell equations in SI and frequency domain: Gauss’s law Faraday’s law Ampère’s law Gauss’s law of magnetics Continuity equation 4 -

5. In a material Splitting physics: – Bound = ions + electrons in crystal – Cond = electrons in conduction band – Ext = external added charges – Free = charges not bound to the crystal lattice 5

6. In a material Splitting physics (2): – Bound = ions + electrons in crystal – Cond = electrons in conduction band – Ext = external applied currents 6

7. In a material Polarization: Magnetization: Conductivity: Ohm’s law 7

8. The Maxwells in a material Maxwell 1: 8

9. The Maxwells in a material Maxwell 2: no change Maxwell 3: no change 9

10. The Maxwells in a material Maxwell 4: 10

11. The Maxwells in a material In summary we get: These are the macroscopic equations If no external fields and currents => same as vacuum 11

12. The relative permittivity: The relative permeability: Different notations Electric susceptibility Only for metals! Magnetic susceptibility 12 Relative permittivity Relative permeability

13. Homogenity and isotropy Relation in real space: – Isotropy: – Homogenity: 13

14. In a material: nonlinear Actually: and 14

15. In a material: nonlinear Nonlinear terms: – Displacement field – Magnetizing field NONLINEAR OPTICS: Second Harmonics, Third Harmonics, … 15

16. THE VECTOR SPHERICAL HARMONICS “The vectors M and N are obviously appropriate for the representation of the fields E and H, for each is proportional to the curl of the other.” Julius A. Stratton 16

17. Mie-theory: EM field around a sphere Solve the maxwell equations = find E and B Mie did this by expanding in a good basis. – For spherical systems: – Choose boundary conditions – Solve a set of equations Use E and B to calculate the Poynting vector S Use S to calculate the cross section Vector Spherical Harmonics 17

18. Vector spherical harmonics We combine the Maxwell equations: Wave equations 18

19. Vector spherical harmonics The vector wave equation: First we solve the scalar form in spherical coordinates: But we know the solution: 19

20. Vector spherical harmonics Now we make vectors: – LMN method: – Y  method: 20

21. Vector spherical harmonics: LMN Properties: Transverse Longitudinal Never radial This does not mean: 21

22. Vector spherical harmonics: LMN What do they look like? – Combine: – With: LC of spherical Bessels and Neumanns 22

23. Vector spherical harmonics: LMN 23

24. What you should remember In a material: The VSH form a basis for the solution of these equations. 24

25. THAT’S ALL FOR NOW FOLKS “The best is yet to come” Francis A. Sinatra 25

26. Mie-theory for a golden sphere A story of waves PART II

27. What you should remember In a material: The VSH form a basis for the solution of these equations. 27

28. Vector spherical harmonics: LMN 28

29. MIE-THEORY “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen” Gustav Mie 29

30. Ok: we solved the Maxwells Now what? We adapt to a specific system: R Plane wave q 30

31. The three waves We need three waves in order to solve this: – The incoming wave – The field in the particle – The field generated by the particle ? 31

32. Boundary conditions In general we have: In our case we demand at R: 32

33. A plane wave in VSH We consider the following incoming wave: Or expanded in VSH: 33

34. INTERLUDIUM LONGITUDINAL AND TRANSVERSE “We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” James C. Maxwell 34

35. Light VS sound Longitudinal wave: e.g. sound Transverse wave: e.g. light 35

36. The divergence Taking the divergence yields:  Transverse fields have zero divergence 36

37. The curl Taking the curl yields:  Longitudinal fields have zero curl 37

38. Helmholtz decomposition Von Helmholtz said: – “Any well behaving vector field can be decomposed in the gradient of a scalar summed with the curl of a vectorfield.” Thus: – “Any vectorfield can be decomposed in a transverse and longitudinal part.” 38

39. Helmholtz decomposition Von Helmholtz said: – “Any well behaving vector field can be decomposed in the gradient of a scalar summed with the curl of a vectorfield.” Thus: – “Any vectorfield can be decomposed in a transverse and longitudinal part.” 39

40. A plane wave in VSH We consider the following incoming wave: Or expanded in VSH: 40

41. A plane wave in VSH Hence we get: Bohren & Huffman: LC of spherical Bessels and Neumanns Due to orthogonality  polarization “The desired expansion of a plane wave in spherical harmonics was not achieved without difficulty. This is undoubtedly the result of the unwillingness of a plane wave to wear a guise in which it feels uncomfortable; expanding a plane wave in spherical wave functions is somewhat like trying to force a square peg into a round hole. However, the reader who has painstakingly followed the derivation, and thereby acquired virtue through suffering, may derive some comfort from the knowledge that it is relatively clear sailing from here on.” 41

42. The other waves The wave in the sphere: The scattered wave: 42

43. The solutions Using the boundary conditions: 43

44. The electric field 44

45. BEYOND WAVES “What is needed now is some flesh to cover the dry bones of the formal theory.” Bohren & Huffman 45

46. The Poynting vector So we can calculate the E and B field for every frequency But that is not always convenient. The Poynting vector = flow of energy The Poynting vector: 46

47. 47 The Poynting vector

48. The cross section Gives idea of interaction strength: 48

49. The cross section How do we calculate it? – Consider a large sphere around the nanoparticle What energy flows through? A 49

50. The cross section Total = absorption: Scattering: Extinction: The cross section is defined positve 50

51. An example: golden sphere in vacuum The relative cross section: Wavelength (nm) R = 500 nm R = 50 nm 51

52. What did we learn? The connection between D, H, charges and currents. Vector spherical harmonics are solutions to the wave equations and form a basis. Solving a set of equations will solve the scattering problem of a sphere 52

53. THANK YOU FOR YOUR ATTENTION 53