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!