SLIDE 1
Multipoles
Electric dipole Magnetic dipole
Experimental demonstration of toroidal dipole
- T. Kaelberer, et al., Science 330, 1510 (2010)
Confusion about the toroidal multipoles
Optical force on toroidal nanostructures: toroidal dipole versus - - PowerPoint PPT Presentation
Optical force on toroidal nanostructures: toroidal dipole versus - - PowerPoint PPT Presentation
Optical force on toroidal nanostructures: toroidal dipole versus renormalized electric dipole Xulin Zhang, HKUST Collaborators: Dr. Shubo Wang, Prof. Zhifang Lin, Prof. C. T. Chan Background Multipoles Electric dipole Magnetic dipole
SLIDE 2
SLIDE 3
Classical textbooks
？ Background
SLIDE 4
Multipole definition
Source-representation
( )
( ) ( )
( ) ( )
( ) 2 ( )
1 , , 2 2 , , 3 1 , , 2 1 , 3 1 ( ) ( ) , 3 1 ( ) 2 10
i i i i V V e m ij i j ij j i V V i i i i V V e ij i j ij V m ij i j j i V i
p r dv m dv q rr dv q r dv p r dv m dv Q rr dv Q r r dv t ρ ρ ρ δ ρ ′ ′ ′ ′ = = × ′ ′ ′ ′ ′ ′ = = × ′ ′ ′ ′ = = × ′ ′ ′ ′ = − ′ ′ ′ ′ ′ = × + × ′ ′ = ⋅ −
∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫
r J r J r J r r J r J r J r
2
,
i V
dv ′ ′
∫
r J
primitive multipoles irreducible multipoles toroidal multipoles
= +
Field-representation
( ) (
)
( ) (
)
( ) (
)
( ) (
)
3 3 , 3 3 ,
, , , , , .
s mn mn mn mn mn n m s mn mn mn mn mn n m
iE a k b k k E b k a k ωµ = + = +
∑ ∑
E N r M r H N r M r
renormalized multipoles irreducible dipoles: p, m toroidal dipole: t renormalized dipoles: p, m
SLIDE 5
Optical force
Two viewpoints of optical forces
Field-representation
2 , , 1 , , 2 1,
2 Re
z mn m n m n mn m n m n n m
F E g a p l q a k πε
∗ ∗ + =
= +
∑
p
( )
2 int , , 2 1,
4 Re
z mn m n m n n m
F E l a b k πε
∗ =
= −
∑
Variation of Maxwell stress tensor Based on vector spherical functions
( ) (
)
( ) (
)
3 3 ,
, ,
s mn mn mn mn mn n m
iE a k b k = +
∑
E N r M r
2 1 2 1
2 Re
z mn m,n m,n mn m,n m,n n ,m
F E g b q l p b k πε
∗ ∗ + =
= +
∑
m
Source-representation
Taylor expansion of vector potential
( ) ( )
/
= / (4 ) /
i R c V
e R dv
ω
µ π ′ ′
∫
A r J r
( ) ( )
1 1 Re Re 2 2
i i ∗ ∗
= ∇ ⋅ = ∇ ⋅
p m
F E p F B m
Lorentz force
4 5 int 2
Re + Im 12 12 k k c c πε πε
∗ ∗
= − × × F p m m t
( )
Im 2
i
k c
∗
= − ∇ ⋅
t
F E t
SLIDE 6
Magnetic dipole
Source-representation Field-representation
Optical force on single helix
The effect from the toroidal dipole is completely masked by the irreducible electric and magnetic dipoles, obscuring any distinction between the irreducible and the renormalized dipoles, as in most conventional structures.
SLIDE 7
Source-representation
The toroidal dipole is necessary for the description of optical forces in the source- representation. How to explain the resonant force in the field- representation？
Optical force on toroidal nanostructures
SLIDE 8
Field-representation
Relationship of the multipoles between source-representation and field-representation
sca sca sca sca sca
, , C C C C C + ≈ ≈ + ≈ ≈
p t p m m p t p m m
F F F F F
, ik c + ≈ ≈ p t p m m
The force contribution of toroidal dipole is replaced by a renormalized electric dipole.
Optical force on toroidal nanostructures
Electric radiation field: (source representation)
SLIDE 9
Detection of the toroidal resonance
Scattering cross section
The total number of photons scattered
Optical force
The number of photons scattered as well as their distributions
( ) ( )
4 2 2 3 2 4 2 3 2 2 2 11 11 01 2 2 2 2 11 11 01 2
3 3 6 + + 6 + +
sca sca sca e
- e
sca e
- e
Z C c E Z ik C c c E C A A A k C B B B k ω πε ω πε π π
+
= = + = =
p p t p m
p p t
Source-representation Field-representation
Detection of toroidal resonance
SLIDE 10
Some other demonstrations
SRR structures
SLIDE 11
Summary
We also show that optical force enables the observation of the toroidal response of a nanostructure even when its effect on scattering power is overwhelmed by the conventional multipoles. Some confusions in the understanding of various multipoles have been clarified by introducing and distinguishing the primitive multipoles, irreducible multipoles, and renormalized multipoles. The toroidal dipole is meaningful and necessary in the source representation while its contribution can be completely incorporated into the renormalized electric dipole in the far field.
- Phys. Rev. A 92, 043804 (2015)
SLIDE 12