FI 3221 ELECTROMAGNETIC INTERACTIONS IN MATTER

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1 6/0/06 FI 3 ELECTROMAGNETIC INTERACTION IN MATTER Alexander A. Ikandar Phyic of Magnetim and Photonic CATTERING OF LIGHT Rayleigh cattering cattering quantitie Mie cattering Alexander A. Ikandar Electromagnetic Interaction in Matter

2 6/0/06 REFERENCE Main C. Bohren and D. Huffman, Aborption and cattering of Light by mall Particle, iley-vch Alexander A. Ikandar Electromagnetic Interaction in Matter 3 HAT I CATTERING? Imagine a particle under illumination of electromagnetic wave Alexander A. Ikandar Electromagnetic Interaction in Matter 4

3 6/0/06 HAT I CATTERING? Electric charge inide the particle will experience ocillatory motion, and hence generate econdary EM wave Alexander A. Ikandar Electromagnetic Interaction in Matter 5 HAT I CATTERING? Thi econdary wave i commonly known a cattered wave Alexander A. Ikandar Electromagnetic Interaction in Matter 6 3

4 6/0/06 ABORPTION ome part of the electromagnetic energy can be tranformed into other form (ex : thermal energy) Thi proce i called aborption Alexander A. Ikandar Electromagnetic Interaction in Matter 7 RAYLEIGH CATTERING Rayleigh cattering applie when light i cattered by mall particle By aying mall we mean particle with ize much maller than wavelength of the ident light d < 0 λ Alexander A. Ikandar Electromagnetic Interaction in Matter 8 4

5 6/0/06 RAYLEIGH THEORY OF OPTICAL CATTERING mall particle (a << l): quai-tatic approximation E 0 (contant) e e a r q z The initially uniform field (potential co i Er 0 q) will be ditorted by the introduction of the phere The field inide (E ) and outide (E ) the phere may be found from the calar potential r and r q E E q,, Alexander A. Ikandar Electromagnetic Interaction in Matter 9 A PHERE IN A UNIFORM TATIC FIELD: THE OLUTION Laplace equation in ource-free domain: 0 0 r a r a Boundary condition: r a r r r a e e r a q q lim Ercoq r 0 olution: 3e 0 e e Ercoq 0 ae Ercoq e e coq 3 0 e e r Alexander A. Ikandar Electromagnetic Interaction in Matter 0 5

6 6/0/06 A PHERE IN A UNIFORM TATIC FIELD: THE OLUTION Electric potential inide and outide the phere: 3e 0 e e Ercoq e e coq E r coq a E e e r A phere in an tatic field i equivalent to an ideal dipole Dipole moment: p ee 0 3 e e e e Dipole polarizability: 4a 3V e e e e Alexander A. Ikandar Electromagnetic Interaction in Matter RAYLEIGH CATTERING (a << l) Total cattered power (linearly polarized light): 4 P r in E H * d e e e q q k a E e e cattering cro-ection and cattering efficiency Q : 8 ee l 0 e P E k a 3 e e 8 ee 4 4 Q a k a (dimenionle number) 3 e e hort wavelength light i preferentially cattered Alexander A. Ikandar Electromagnetic Interaction in Matter 6

7 6/0/06 RAYLEIGH CATTERING INTENITY cattering intenity of mall particle i inverely proportional with the fourth power of wavelength of the ident light I ~ λ 4 Q ca 8 ( ka) 3 Thi implie that for hort wavelength the cattering will be trong 4 n n A imilar derivation for a mall cylinder yield I ~ λ 3 Q ca 3 n ( ka) 4 n Alexander A. Ikandar Electromagnetic Interaction in Matter 3 BLUE KY : RAYLEIGH CATTERING It explain why the ky i blue Alexander A. Ikandar Electromagnetic Interaction in Matter 4 7

8 6/0/06 MIE CATTERING How to analye cattering by larger object? Ue integrated Poynting vector. Alexander A. Ikandar Electromagnetic Interaction in Matter 5 CATTERING PARAMETER The field in the region outide of the catterer can be written a E E i E Alexander A. Ikandar Electromagnetic Interaction in Matter 6 8

9 6/0/06 9 The Poynting vector in the region outide of the catterer can be written a Alexander A. Ikandar Electromagnetic Interaction in Matter 7 CATTERING PARAMETER ca i i i i i i H E H E H E H E H H E E H E * * * * * * Re Re Re Re Re The power aborb by the catterer can be evaluated a Alexander A. Ikandar Electromagnetic Interaction in Matter 8 CATTERING PARAMETER ca A ab d a

10 6/0/06 CATTERING PARAMETER The power aborb by the catterer can be evaluated a da where with A ab da, ca A Re Re ca Re A Alexander A. Ikandar Electromagnetic Interaction in Matter 9 ca da, ca * E i Hi * E H * * E H E H i i A da CATTERING PARAMETER ab da Obviouly, for a cloed urface are, A ca A da 0 Alexander A. Ikandar Electromagnetic Interaction in Matter 0 0

11 6/0/06 CATTERING PARAMETER ab da Obviouly, for a cloed urface are, Hence, A ca A da 0 ab ca ca ab Alexander A. Ikandar Electromagnetic Interaction in Matter CATTERING PARAMETER A cro ection i defined a the normalized power with repect to oming wave, hence ca ab ca, ab, Ii Ii Ii From thi cro ection, we can define efficiencie a, A Q, A Q, cat eff cat ab eff ab A eff Q Alexander A. Ikandar Electromagnetic Interaction in Matter

12 6/0/06 MULTIPOLE EXPANION METHOD To find the exact wave olution, we write that all field a a linear combination of ome orthogonal bae function ( ) () (3) E( r) f ( r)ˆ e b f ( r)ˆ e c f ( r) eˆ n an n n n n n 3 ( ) The orthogonal bae function, ( r i f n ), i choen according to the (ymmetry) of the problem. It could be a combination of pherical harmonic and pherical Beel function for pherical ymmetric catterer, or it could be Beel function for cylindrical ymmetric catterer. Alexander A. Ikandar Electromagnetic Interaction in Matter 3 CATTERING BY INFINITE CYLINDER e we are working with a cylindrical haped object, it would be more convenient to ue the curl operator in the form of cylindrical coordinate E = ρ ρ ρφ z = iωμ( ρh ρ + φh φ + zh z ) ρ φ z E ρ ρe φ E z H = ρ ρ ρφ z = iωε( ρe ρ + φe φ + ze z ) ρ φ z H ρ ρh φ H z Alexander A. Ikandar Electromagnetic Interaction in Matter 4

13 6/0/06 CATTERING BY INFINITE CYLINDER Conider a problem a depicted in the figure A cylinder i illuminated by an electromagnetic plane wave The plane wave i propagating on x-y plane toward negative y-axi (no z-dependence) The magnetic field i polarized parallel with the z- axi Alexander A. Ikandar Electromagnetic Interaction in Matter 5 CATTERING BY INFINITE CYLINDER From the Maxwell equation, we obtain H z = (ρe φ ) E ρ iρωμ ρ φ E ρ = E φ = iωε H z ρ olving for the magnetic field, ρ H z ρ with + ρ H z ρ + k ρ H z = H z φ k = μεω Alexander A. Ikandar Electromagnetic Interaction in Matter 6 3

14 6/0/06 CATTERING BY INFINITE CYLINDER olving the previou partial differential equation by eparation of variable method, the radial part yield ρ R z (kρ) ρ + ρ R z (kρ) ρ + k ρ ν R z kρ = 0 hich i the Beel equation, with olution can be a combination of :. Beel function J ν (kρ). Neumann function Y ν (kρ) 3. Hankel function of the firt kind H ν (kρ) 4. Hankel function of the econd kind H ν (kρ) Alexander A. Ikandar Electromagnetic Interaction in Matter 7 CATTERING BY INFINITE CYLINDER Later, it will be hown that we need only Beel function J ν kρ and Hankel function of the firt kind H ν kρ The olution of the field will take the following form χ ρ, φ = M ν J ν kρ + N ν H ν (kρ) e iνφ ν= χ ρ, φ can be the electric field or the magnetic field M ν and N ν are the unknown coefficient yet to be determined Alexander A. Ikandar Electromagnetic Interaction in Matter 8 4

15 6/0/06 CATTERING BY INFINITE CYLINDER e our problem i invariant in z direction, we may conider only the cro-ectional ituation of the problem a depicted below In region, there i an oming plane wave approaching a cylinder Region Region ν J ν k ρ e iνφ Alexander A. Ikandar Electromagnetic Interaction in Matter 9 CATTERING BY INFINITE CYLINDER The cylinder will generate econdary electromagnetic field N ν H ν k ρ e iνφ Region Region ν J ν k ρ e iνφ Alexander A. Ikandar Electromagnetic Interaction in Matter 30 5

16 6/0/06 CATTERING BY INFINITE CYLINDER In region, a tanding wave enue N ν H ν k ρ e iνφ Region M ν J ν k ρ e iνφ Region ν J ν k ρ e iνφ Alexander A. Ikandar Electromagnetic Interaction in Matter 3 CATTERING BY INFINITE CYLINDER Applying the boundary condition, we obtain the following value for the coefficient M ν and N ν N ν = ν nj ν k a J ν k a J ν k a J ν k a nh ν k a J ν k a H ν k a J ν k a M ν = ν nj ν k a H ν k a nj ν k a H ν k a nj ν k a H ν k a J ν k a H ν k a where n = ε ε Alexander A. Ikandar Electromagnetic Interaction in Matter 3 6

17 6/0/06 INTENITY AND POER OF THE CATTERED AVE Intenity of the cattered wave can be derived from the definition of Poynting vector, more pecifically, the time-averaged Poynting vector ca = Re E ca ca ( ) H thu, the radial component of the Poynting vector i ca ρ = Re E φ ca ca ( ) H z Inerting the field component yield ρ ca = ωε πa N ν ρ Alexander A. Ikandar Electromagnetic Interaction in Matter 33 INTENITY AND POER OF THE CATTERED AVE To determine the power of the cattered wave, we apply an imaginary urface around the cylinder and calculate the urface integral below P ca = ρ ca. da a or, P ca = ωε πa N ν ρ. RL dφρ or, P ca = L N ωε ν Alexander A. Ikandar Electromagnetic Interaction in Matter 34 7

18 6/0/06 CATTERING CRO ECTION cattering Cro ection C ca i the ratio of P ca with repect to the intenity of the oming plane wave C ca = P ca I Intenity of an oming plane wave with the amplitude of it magnetic field i unity can be written a I = μ 0c Thu C ca = 4L N Alexander A. Ikandar Electromagnetic k ν Interaction in Matter 35 CATTERING EFFICIENCIE cattering Efficiency Q ca i defined a Q ca = C ca G G i the particle cro-ectional area a viewed by the oming wave, thu for our cae G = al Thi yield Q ca = k a N ν Alexander A. Ikandar Electromagnetic Interaction in Matter 36 8

19 6/0/06 EXAMPLE ilver nanocylinder i illuminated by EM plane wave Alexander A. Ikandar Electromagnetic Interaction in Matter 37 EXAMPLE ilver nanophere i illuminated by EM plane wave Alexander A. Ikandar Electromagnetic Interaction in Matter 38 9

20 6/0/06 APPLICATION ε = ilver 75 nm ε = 4 00 nm Alexander A. Ikandar Electromagnetic Interaction in Matter 39 APPLICATION Alexander A. Ikandar Electromagnetic Interaction in Matter 40 0

21 6/0/06 APPLICATION Alexander A. Ikandar Electromagnetic Interaction in Matter 4 APPLICATION Alexander A. Ikandar Electromagnetic Interaction in Matter 4