Light spins of 2-dimensional electromagnetic waves · 2016. 11. 18. · Amp re t Faraday t Gauss...
Transcript of Light spins of 2-dimensional electromagnetic waves · 2016. 11. 18. · Amp re t Faraday t Gauss...
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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
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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
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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)
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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
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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.
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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
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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××
± ±
⇒ ⇒ →
� �
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(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
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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
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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
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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
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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
−
= − × ∫
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(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 !