optical force on toroidal nanostructures: toroidal dipole...
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Xulin Zhang, HKUST Collaborators: Dr. Shubo Wang, Prof. Zhifang Lin, Prof. C. T. Chan
Optical force on toroidal nanostructures: toroidal dipole versus renormalized electric dipole
Multipoles
Electric dipole Magnetic dipole
Experimental demonstration of toroidal dipole
T. Kaelberer, et al., Science 330, 1510 (2010)
Confusion about the toroidal multipoles
Toroidal dipole
Background
Classical textbooks ?
Background
Multipole definition
Source-representation
( )
( ) ( ) ( )
( )
( ) 2
( )
1, ,2
2, ,3
1, ,2
1 ,3
1 ( ) ( ) ,3
1 ( ) 210
i i i iV V
e mij i j ij jiV V
i i i iV V
eij i j ijV
mij i j j iV
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 ,iVdv′ ′ ∫ r J
primitive multipoles
irreducible multipoles
toroidal multipoles
= +
Field-representation
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3 3
,
3 3
,0
, , ,
, , .
s mn mn mn mn mnn m
s mn mn mn mn mnn m
iE a k b k
k E b k a kωµ
= +
= +
∑
∑
E N r M r
H N r M r
renormalized multipoles
irreducible dipoles: p, m toroidal dipole: t renormalized dipoles: p, m
Optical force
Two viewpoints of optical forces
Field-representation
200 , , 1 , ,2
1,
2 Rez mn m n m n mn m n m nn m
F E g a p l q akπε ∗ ∗
+=
= + ∑p
( )2int 00 , ,2
1,
4 Rez mn m n m nn m
F E l a bkπε ∗
=
= − ∑
Variation of Maxwell stress tensor
Based on vector spherical functions
( ) ( ) ( ) ( )3 3
,, ,s mn mn mn mn mn
n miE a k b k = + ∑E N r M r
200 12
1
2Rez mn m,n m,n mn m,n m,n
n ,mF E g b q l p b
kπε ∗ ∗
+=
= + ∑m
Source-representation
Taylor expansion of vector potential
( ) ( ) /0= / (4 ) /i R c
Ve R dvωµ π ′ ′ ∫A r J r
( ) ( )1 1Re Re2 2i i
∗ ∗ = ∇ ⋅ = ∇ ⋅ p mF E p F B m
Lorentz force
4 5int
20 0
Re + Im12 12
k kc cπε πε
∗ ∗ = − × × F p m m t
( )Im2 ikc
∗ = − ∇ ⋅ tF E t
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.
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
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)
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
( )( )
420
230 0
240
230 0
2 2 211 11 012
2 2 211 11 012
3
3
6 + +
6 + +
sca
sca
sca e o e
sca e o e
ZC
c E
Z ikCcc E
C A A Ak
C B B Bk
ωπε
ωπε
π
π
+
=
= +
= =
p
p t
p
m
p
p t
Source-representation
Field-representation
Detection of toroidal resonance
Some other demonstrations
SRR structures
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)
Thanks for your attention!
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