1

Light spins of 2-dimensional

electromagnetic waves

HyoungHyoungHyoungHyoung---- In LeeIn LeeIn LeeIn Lee

A ResearcherA ResearcherA ResearcherA Researcher

Research Institute of MathematicsResearch Institute of MathematicsResearch Institute of MathematicsResearch Institute of Mathematics

Seoul National UniversitySeoul National UniversitySeoul National UniversitySeoul National University

supported by supported by supported by supported by

National Research Foundation of KoreaNational Research Foundation of KoreaNational Research Foundation of KoreaNational Research Foundation of Korea

Grant Numbers: Grant Numbers: Grant Numbers: Grant Numbers:

NRFNRFNRFNRF---- 2011201120112011---- 0023612 & 0023612 & 0023612 & 0023612 &

NRFNRFNRFNRF---- 2015R1D1A1A010566982015R1D1A1A010566982015R1D1A1A010566982015R1D1A1A01056698

1/23

2

2-dimensional cylindrical waves

( ), , , z independt cer enθ −

m = azimuthal mode index (AMI)

ω = frequency

θ

z

r

( )

( ) ( ),

0

,x y

z

r

im i t

k k or k

k

kθ

θ ω

=

−

3

TM (transverse magnetic)

WGM (whispering gallery mode)

Eθ

zH

rE

E || H (E-parallel-H)

all six components exist

Eθ

zE

rE

rH

Hθ

zH

ε µ≠ ε µ=

for any combinations of electric

permittivity and magnetic permeability

for dual field

(electric & magnetic fields on equal footing)

WGM E || H

4

real- vs complex-valued field variables

(real-valued) electric & magnetic fields

0

0

: 0

:

: 0

: 0

Coulomb

Amp ret

Faradayt

Gauss

εε

µµ

∇ ⋅ =

∂ ∇ × = ∂

∂ ∇ × + = ∂

∇ ⋅ =

�

��

��

�

E

EH

HE

H

è

, :

, :E H�

� �

�

E H

(complex-valued) electric & magnetic fields

( )

( )

* *

0

* *

1

2

1Im

2

light RW E E H H

energy c

lightS E E H H

spin

ωε µ

ε µ

≡ ⋅ + ⋅

≡ × + ×

� � � �

� � � � �

( )

( )

0

12

0

1Re exp

1Re exp

E i t

H i i t

ωε

π ωµ

≡ − ≡ −

� �

� �

E

H

WGM

5

specific spin for plasmonics of a metallic cylinder

( )

( )

( )

* *

* *0

22 20

Im

: , ,

21

r z

r

r z

specificE E H HR S

lightc W E E H H

spin

WGM E E H

R S E E

c W E E H

θ

θ

θ

ε µω

ε ε

ω ε

ε µ

× + × ≡ ⋅ + ⋅

⇒ ≡ ≤+ +

� � � ��

� � � �

�

metalmetalmetalmetal

vacuum

Konstantin Y. Bliokh, Franco Nori, “Transverse and longitudinal angular

momenta of light”, Physics Reports 592, 1–38 (2015)

WGM

cylindrical plasmonic wire

( )mJ kr( ) ( )1mH kr

6

distributions of light spins in the radial direction

0

zz

RSs

c W

ω≡

r

Rρ ≡

Hyoung-In Lee and J. Mok, “Cylindrical Electromagnetic Waves with Radiation

and Absorption of Energy”, Pacific J. Mathematics for Industry 8, 1-16 (2016)

WGM

7

(open problem) light spin of combined TM-TE waves

rE

zH

Eθ

TM waveTM waveTM waveTM wave

rH

zE

Hθ

TE waveTE waveTE waveTE wave

yz

xθ r

( )ikzim i tθ ω− −for coherently combined wave

(both TM and TE), it is hard to

prove a generic inequality

( )* *

* *0

Im1

E E H HR S

c W E E H H

ε µω

ε µ

× + ×≡ ≤

⋅ + ⋅

� � � ��

� � � �

Numerically, it is proved for

several conceivable exact

solutions to Maxwell’s

equations.

WGM

8

another problem …

8/23

9

( ) ( )

( ) ( )

( )( ) ( )

( ) ( )

( )

1 2

1 2

*

, exp , e

, exp , e

ˆ ˆsin cos,

ˆcos sin

Re 0

i

i

x x

z

E v r t i r t v

H v r t i r t v

W ky e W kx ev x y

W ky W kx e

E H

Φ

Φ

= Φ =

= Φ =

+=

+ +

⇒ × =

� � � � �

� � � � �

�

� �

Kiyoji Uehara, Toshio Kawai, and

Koichi Shimoda, "Non-Transverse

Electromagnetic Waves with

Parallel Electric and Magnetic

Fields", J. Phys. Soc. Jpn. 58, pp.

3570-3575 (1989)

(free space only)

3-component

2-dimensional

vector field

E || H

10

a relevance

to the

formation of

an optical

vortex

E || H

Hyoung-In Lee and J. Mok, "Two-dimensional models for optical vortices driven

by gain media," J. Opt. Soc. Am. B 31, A24-A30 (2014)

11

an adaptation

to cylindrical

waves

After an angular averaging, we obtain

Bessel beam with a single center.

( ) ( )

( )

( ) ( )

( ) ( )

( )

ˆsin sin cos

ˆ, ; cos sin cos

ˆcos co

, exp ,

s

r

z

r e

A r r

E

e

H A i t

r e

r t r

θ

τ θ τ θ

θ τ τ θ τ θ

τ θ

− −

= − − −

+

= = Φ

−

�

�� � � �

( ) ( ) ( ) ( )2

0 1 0

0

1ˆ ˆ, , ;

2zB r A r d J r e J r e

π

θθ θ τ τπ

= = − +∫��

E || H

12

spin-up &

spin-down

optical

waves

E || H

(+ spin) CCW(+ spin) CCW(+ spin) CCW(+ spin) CCW (counter(counter(counter(counter----clockwise)clockwise)clockwise)clockwise)

zS

zS

((((---- spin) CWspin) CWspin) CWspin) CW (clockwise)(clockwise)(clockwise)(clockwise)

( ) ( ) ( )

( ) ( )( )

( )

2

0

1, exp

0,

1,

, ;2

ˆ ˆ ˆexp

m

mmm m r z

m

m

B r im A r d

dJ rmi i t m i J r e e J r e

r

m od

e n

d

d

m v

r

π

θ

θ τ θπ

χ

χ

τ τ

θ

± =

= − + +

= ==

=

=

∫

∓

��

∓ ∓

azimuthal

averaging

After an angular averaging, we obtained

Bessel beam with a single center.

13

light spin

for this E || B waves

m=2

m=16

m=4

r

m=8

0

zz

RSs

c W

ω≡

( )* *

* *0

Im E E H HR S

c W E E H H

ω × + ×≡

⋅ + ⋅

� � � ��

� � � �E || H

14

E || H

( ) ( ) ( )( )

( )

( ) ( )

( ) ( )

*

*

2

, ,

2 , 0,1,2,3,

ˆ ˆ ˆ

,

,

,

1m mim

m

i t i tm

m m

n zi t

m

m

r

m n n

dJ rme i J r e e J r

e B r e

e

B r

B r B

drB

r

er

rθ

θ

θ

θ

θ

θ

θ

−

±

+

− +

×

× ×

±

≠

= = ⋅ ⋅ ⋅

= − + +

≠

�

�

�

∓

�

�

azimuthal averaging -> CCW or CW waves

( ) ( ) ( ), ,i phasei t

m m

adiabatictye B r e B r t

parameetrθ θ××

± ±

⇒ ⇒ →

� �

15

(interference) sum over all azimuthal modes

( ) ( ) ( ) ( )

( )

( ) ( ) ( )

0

1, 1,

,0 , ,

1, 1,

,

,

cos cos e e

cos cos

: e sin sin cos

n in n inn n n n

n n even n n even

irrotational CCW CWz z m z n

n n even n n even

n inn

n n eve

r n

nn

r J r i J r i J r

r E E E

nradial i i J r r

r

azimu

E

θ θ

θ

θ χ χ

θ

θ θ

∞ ∞−

= = = =

∞ ∞

∞

=

= = = =

∞−

=−∞ =−∞

= + +

= + +

= = −

∑ ∑

∑ ∑

∑∑

( )( ) ( )

( ) ( )

,

,

,

,

: e cos sin cos

: e cos cos

nn in

n n even

n inn

n

z

n

n

n ven

n

n e

dJ rthal i r

dr

naxial iE

E

i J r rr

θ

θ

θ θ θ

θ

∞−

=

∞

=−∞ −∞ =

∞−

=−∞ =

∞

=−∞

= = − = =

∑

∑

∑

∑

E || H

16

A decomposition into

2 rotational & 1 non-rotational waves

x

y

+ 2 x =

x

y

x

y

(a) (c) (b)

( ) ( ){ }0

1cos cos

2r J rθ − ( )cos cosr θ ( )0J r

E || H

17

spectral sum over all azimuthal modes

( ) ( ){ } ( ) ( ){ } ( )

, ,

2 2 2

, , ,

2 2

1 10

sin cos sin sin cos cos cos cos

n r n n

n n n

n n r n n z n

n n n n n

E r E Er r r

W E E E E E

W r r r

θ

θ

θ

θ θ θ θ θ

∞ ∞ ∞

=−∞ =−∞ =−∞

∞ ∞ ∞ ∞ ∞

=−∞ =−∞ =−∞ =−∞ =−∞

∂ ∂∇ ⋅ = + =

∂ ∂

= ⋅ = + +

= + +

∑ ∑ ∑

∑ ∑ ∑ ∑ ∑

�

� �

{ }2

1

=

[1] spectral sum over all rotational modes

satisfies divergence-free condition

[2] spectral sum of 2 field components

satisfies energy conservation( )

2 2 2

, , ,

2 2 2

, , ,

r n n n

r n n n

n

E E E

E E E

θ θ

θ θ

∞

=−∞

+ +

+ +∑

E || H

18

an example of mixing a rotational wave

with a counter-rotational wave E || H

Dmitry A. Kuzmin, I. V.

Bychkov, V. G. Shavrov, V. V.

Temnov, Hyoung-In Lee, and

J. Mok, "Plasmonically

induced magnetic field in

graphene-coated nanowires,"

Opt. Lett. 41, 396-399 (2016)

18/23

19

analogy to

the inter-cell tunneling

on staggered lattice

(rather than to the intra-cell coupling)

( ) ( )

( ) ( ) ( )2

0

, , e :

1 1cos exsin p sin

2 2

i phase i t Hm

m

e B r periodically driven Floquet Hemiltonian

J r m d im i r dr

π π

π

θ

τ τ τ τ τπ π

τ

× − × ×±

−

= − = − × ∫ ∫

�

( ) ( )si inn sm xp t p

momentum

t adiabaticity parameter

m position

momentum

r t

r

ime

τ

τ

θ

τ− → −

→ →

→ →

→

E || H seen as quantum mechanics

( ) ( )1

exp sin2

xJ t ixp i t p dp

π

ππ

−

= − × ∫

20

(analogy) quantum walks on an infinite line

A.M. Childs, R. Cleve, E. Deotto, E.

Farhi, S. Gutmann, and D.A.

Spielman, “Exponential

algorithmic speedup by quantum

walk,”, 2002. lanl-report quant-

ph/0209131.

(column-wise) quantum walks

on an (idealized) infinite line

leads to Bessel function as a

propagator for right- and left-

moving wave packets.

1 1 1 1 1 1 1 1

( )

( ) ( ) ( )

( ) ( ) ( )

( )

2 cos

1 1,

0 1 0 0

1 0 1 0: 4 4

0 1 0 1

0 0 1 0

1, 2cos

2

1, ,

2

,, , 2

2

ipjp

ip k j it piHt

k j

k j

j H j j

example H for

j p e p E p

G j k t k e j e d

G j k t i

k j m

t r

p

J t

p

j

π

π

τ

π

π

θ

ππ

− −−

−

−−

± = − ∞ < < ∞

= ×

= − ≤ ≤ ⇒ =

= =

= −

− ⇒

⇒

⇒

− ∞ < < ∞

∫

E || H

21

( ) ( ) ( ) ( )

( ) ( ) ( )

1† †† † †1 11 1

†

1

si2 n

1

2 cos

m m

m m m m m m j jm mm m m

kkk

H m A c c c c B c c c c C c c

H k A ka iB a c ck

++ ++ +

= + + − + − +

= − + ⋅ ⋅ ⋅

∑ ∑ ∑

∑

SSH (Su-Schrieffer-Heeger) model with sublatticsE || H

inter-cell intra-cell

22

Dirac Hamiltonian

Roger S. K. Mong and

Vasudha Shivamoggi,

“Edge states and the bulk-

boundary correspondence

in Dirac Hamiltonians”,

Phys. Rev. B 83, 125109

(2011

( )

( )( ) ( ) ( )

|| || ||||||

||

† 0 *

1, , 1,,,

†

,

0 *

0 2 cos 2 sin

n k n k n kn kn k

kkk k

ik ik

r i

H b b b

n k

H h k

h k be b b e

h k b b k b k

⊥

⊥ ⊥

− +

⊥

−

⊥ ⊥

= Ψ Γ ⋅ Ψ + Ψ + Ψ

→

= Ψ ⋅ Γ Ψ

= + +

= + +

∑

∑

� � ��

�

��

� �

� �

� � �

�

� �

� � � � �

� � � � � � �

E || H

23

Conclusion:

Light spins illustrated,

and several topological implications deduced

from examining two special configurations of

electromagnetic waves

Thanks for your attention !