Nonlinea Op ical Response o a Plasmonic Nanoan enna o
Ci cula ly Pola ized Ligh : Ro a ion o Mul ipola Cha ge Densi y
and Nea -Field Spin Angula Momen um In e sion
Published as pa o he ACS Pho onics i ual special issue “F on ie s and Applica ions o Plasmonics and
Nanopho onics”.
Ma ina Quijada, An on Babaze, Ja ie Aizpu ua,*and And ei G. Bo iso *
Ci e This: ACS Pho onics 2023, 10, 3963−3975
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ABSTRACT: The spin and o bi al angula momen um ca ied by
elec omagne ic pulses open new pe spec i es o con ol nonlinea
p ocesses in ligh −ma e in e ac ions, wi h a weal h o po en ial
applica ions. In his wo k, we use ime-dependen densi y
unc ional heo y (TDDFT) o s udy he nonlinea op ical
esponse o a ee-elec on plasmonic nanowi e o an in ense,
ci cula ly pola ized elec omagne ic pulse. In con as o he well-
s udied case o he linea pola iza ion, we ind ha he n h
ha monic op ical esponse o ci cula ly pola ized ligh is
de e mined by he mul ipole momen o o de no he induced
nonlinea cha ge densi y ha o a es a ound he nanowi e axis a
he undamen al equency. As a consequence, he equency
con e sion in he a ield is supp essed, whe eas elec ic nea ields a all ha monic equencies a e induced in he p oximi y o he
nanowi e su ace. These nea ields a e ci cula ly pola ized wi h handedness opposi e o ha o he inciden pulse, hus p oducing an
in e sion o he spin angula momen um. An analy ical app oach based on gene al symme y cons ain s nicely explains ou
nume ical indings and allows o gene aliza ion o he TDDFT esul s. This wo k hus o e s new insigh s in o nonlinea op ical
p ocesses in nanoscale plasmonic nanos uc u es ha allow o he manipula ion o he angula momen um o ligh a ha monic
equencies.
KEYWORDS: nonlinea op ics, ligh pola iza ion, ci cula ly pola ized ligh , high-ha monic gene a ion, plasmonic nanos uc u e,
ime-dependen densi y unc ional heo y
■INTRODUCTION
Mode n echnologies enable he design and nano ab ica ion o
pho onic de ices o manipula ion o op ical ields on spa ial
scales much smalle han he wa eleng h o ligh .
1
In pa icula ,
he esonan coupling o pho ons wi h collec i e elec onic
exci a ions in me als and wo-dimensional (2D) ma e ials, i.e.,
plasmons, can be used o enginee s ongly enhanced nea
ields con ined o he a omic scale.
2−5
Nea - ield enhancemen
boos s he nonlinea op ical esponse so ha plasmonic
sys ems ind p ac ical applica ions no only in he linea
5−7
bu
also in he nonlinea
8−10
egime. Nonlinea me ology,
11
nonlinea sensing,
12,13
ul a as spec oscopy,
14−16
and non-
linea in eg a ed pho onic ci cui s ope a ion
17−19
exempli y
a ious ields ha ake ad an age o he nonlinea op ical
esponse o plasmonic sys ems.
The ecen in e es in he use o s uc u ed ligh
20
has
dynamized esea ch on plasmonic me asu aces exploi ing spin-
con olled nonlinea op ical p ocesses o ob ain beam shaping
h ough manipula ion o o bi al angula momen um (OAM)
and spin angula momen um (SAM) o ligh .
21−30
Along wi h
gas-phase echniques,
31−34
he use o nonlinea me asu aces
35
o he gene a ion o acuum ul a iole (VUV) and ex eme
ul a iole (XUV) cohe en ligh ha ca ies angula
momen um opens exci ing pe spec i es in ul a as spec os-
copies and ime- esol ed expe imen s o p obe chi al sys ems.
The de elopmen o de ices o on-chip con ol o nonlinea
ields equi es knowledge o he nonlinea op ical esponse o
indi idual plasmonic nanopa icles and plasmonic molecules,
which a e he building blocks o such nonlinea de ices. To
his end, he hyd odynamic desc ip ion adop ed o add ess he
Recei ed: June 9, 2023
Published: Oc obe 24, 2023
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nonlinea esponse o conduc ion elec ons, and i s applied o
cha ac e ize he second-ha monic gene a ion om me als and
me al su aces,
36−41
has been u he de eloped ecen ly.
E icien nume ical app oaches o add ess he nonlinea i y o
plasmonic nanopa icles ha e been hus p oposed.
9,42−48
In
his con ex , he si ua ion whe e he undamen al wa e is
linea ly pola ized has been s udied bo h heo e ically and
expe imen ally, p o iding a deep unde s anding abou he main
p ocesses ha con ol he second-o de
45−47,49−57
and he
hi d-o de
42,58−63
esponse o plasmonic nanoan ennas and
subnanome ic plasmonic gaps p one o sus ain op ically
assis ed unneling.
64−66
Howe e , wi h he excep ion o chi al
sys ems
9,67−70
(whe e one is na u ally in e es ed in he
nonlinea ac i i y igge ed by SAM-ca ying inciden ields),
he case o a ci cula ly pola ized undamen al wa e in e ac ing
wi h ypical plasmonic nanoan ennas
71
has ecei ed less
a en ion o nonlinea plasmonic applica ions.
In his wo k, we add ess he nonlinea op ical esponse o a
plasmonic nanos uc u e o a SAM-ca ying inciden ield. We
use ime-dependen densi y unc ional heo y (TDDFT) o
s udy he dynamics o conduc ion elec ons igge ed by an
in ense elec omagne ic pulse in a ee-elec on cylind ical
nanowi e. The elec ic ield o he pulse is ci cula ly pola ized
in he ans e sal plane o he nanowi e (see Figu e 1). As a
e e ence, we also pe o m calcula ions o linea ly pola ized
undamen al ield as s udied in de ail in p e ious
wo ks.
43,47,72−75
Wi hou any a p io i assump ions, ou
TDDFT esul s e eal ha he op ical esponse a he n h
ha monic o he ci cula ly pola ized undamen al wa e is
de e mined by he mul ipole momen o o de no he induced
nonlinea cha ge densi y ha o a es a ound he nanowi e axis
a he undamen al equency. In pa icula , he induced nea
ield is ci cula ly pola ized a all ha monics o he undamen al
equency and e eals an SAM in e sion. Mo eo e , he
equency con e sion in he a ield is supp essed o ci cula ly
pola ized inciden pulses. We u he demons a e ha hese
esul s a e a di ec consequence o he symme y o he sys em
as can be ully desc ibed and unde s ood wi hin an analy ical
app oach. The e o e, ou indings a e quali a i ely obus and
p o ide a new pa adigm o he design o nonlinea nanoscale
op ical de ices.
Unless o he wise s a ed, a omic uni s (a.u.) a e used
h oughou he pape .
■METHODS
The de ails on he me hod including he modeling o
plasmonic nanopa icles and he eal- ime TDDFT calcula ions
o he elec on dynamics can be ound in p io wo ks.
76,77
Thus, only he aspec s speci ic o his s udy will be desc ibed
he e. We conside a plasmonic nanowi e ep esen ed as a ee-
elec on me al cylinde o adius Rc, in ini e along he z-axis
(see Figu e 1). The nanowi e is desc ibed using he s abilized
jellium model
78
cha ac e ized by he Wigne −Sei z adius o
gold (and sil e ), s= 3.02 a0(a0= 0.0529 nm is he Boh
adius), and a wo k unc ion o 5.49 eV. The adius o he
nanowi e is se o Rc= 66.4 a0(≈3.5 nm), which is a good
comp omise be ween he easibili y o he TDDFT calcula ions
and a su icien ly small alue o he su ace Landau
damping
79,80
so ha well- esol ed localized plasmon eso-
nances can be obse ed in he linea op ical esponse.
The ee-elec on model is well sui ed o quan i a i ely
add ess he linea and nonlinea op ical esponse o nano-
pa icles o med by p o o ype me als such as alkali me als and
aluminum. Fo noble me als, as a as he undamen al
equency is below he onse o in e band ansi ions in ol ing
localized d-elec ons, he symme y-p o ec ed aspec s o he
nonlinea op ical esponse can be nicely unde s ood wi hin he
amewo k o he ee-elec on model, conside ing ha he
nonlinea cu en s a e c ea ed by he quasi- ee conduc ion-
band elec ons.
9,37,38,44,47,51
The con ibu ion o d-elec ons o
he dynamical sc eening o he undamen al and ha monic
ields can be ea ed in a model way;
37,38,81−84
howe e , a ully
quan i a i e assessmen o linea and nonlinea p ope ies o
noble me al nanopa icles would equi e u he s udies.
The nonlinea op ical esponse o he nanowi e is igge ed
by a Gaussian pulse o elec ic ield E( ) le -handed ci cula ly
pola ized (SAM o 1) in he (x,y)-plane
E E e e( ) cos( ( )) sin( ( )) e
x y
0 0 0
0
p
2
= [ + ]
i
k
j
j
j
j
y
{
z
z
z
z
(1)
whe e e
x(e
y) s ands o he uni leng h ec o along he x- (y-)
axis, E0is he ield ampli ude, Ωis he undamen al equency,
pis he du a ion o he pulse, and 0is he delay ime. We also
pe o m e e ence calcula ions o a linea ly pola ized
undamen al ield gi en by
E E e( ) cos( ( )) e
x
0 0
0
p
2
=
i
k
j
j
j
j
y
{
z
z
z
z
(2)
In his wo k, we use Ω= 1.5 eV,
5 570 a.u.
p
2
= =
(≈
13.7 s), and E0= 0.064 ×10−2−1.07 ×10−2a.u.
co esponding o an a e age powe o he ci cula ly pola ized
pulse o 3.6 ×1010 −1×1013 W/cm2, and wice smalle
a e age powe o he linea ly pola ized pulse. To analyze he
nonlinea op ical esponse o he nanowi e o le -handed
ci cula ly pola ized (SAM = 1) illumina ion, i is con enien o
ew i e eq 1 in he o m
Figu e 1. Ske ch o he s udied sys em. (a) Plasmonic nanowi e o
adius Rc= 66.4 a0(≈3.5 nm) in ini e along he z-axis. The ex e nal
ield is indica ed wi h a blue a ow. I is ci cula ly pola ized in he
(x,y)‑plane and o a es an iclockwise (le -handed) wi h a
undamen al equency Ω. The ed do loca ed a he x-axis a a
dis ance d om he su ace o he nanowi e indica es he posi ion a
which we calcula e he induced nea ield. (b) C oss-sec ion o he
nanowi e in he (x,y)-plane, de ini ion o he cylind ical coo dina es
used in he pape .
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Ei i
Ei
i e
E e e e e
e e
e e
( ) 2( )e ( )e
e
2( )e e
( )e e
x y
i
x y
i
i i
i i
0 ( ) ( )
0 ( )
( )
0 0
0
p
2
0
0
0
p
2
= { + + }
·
= { +
+ }
i
k
j
j
j
j
y
{
z
z
z
z
i
k
j
j
j
j
y
{
z
z
z
z
(3)
whe e he second exp ession is gi en in cylind ical ( ,φ)
coo dina es, and e
and e
φa e cylind ical uni ec o s (see
Figu e 1b).
The undamen al ield gi en by eq 1 o eq 3 can be ob ained,
o ins ance, using p-pola ized op ical pulses whe e one pulse
(wi h ca ie -en elope phase equals o ze o) is p opaga ing
along he y-axis, and ano he pulse (wi h ca ie -en elope
phase equals o
2
) is p opaga ing along he x-axis.
Fu he mo e, he p esen model applies o a si ua ion in
which a ci cula ly pola ized lase pulse impinges on a
plasmonic nanowi e along i s symme y axis. In his case, he
heigh ho he nanowi e has o be small compa ed o he
wa eleng h so ha he e is no e ec o plasmon p opaga ion
along he z-axis. Mo eo e , hhas o be signi ican ly la ge han
he Fe mi wa eleng h o elec ons and la ge enough o he op
and bo om su ace e ec s o be neglec ed.
76
The dynamics o he elec on densi y wi hin he nanowi e in
esponse o he op ical exci a ion is ob ained om eal-space,
eal- ime TDDFT calcula ions wi hin he Kohn−Sham (KS)
scheme.
85,86
Since he sys em is in a ian wi h espec o a
ansla ion along he z-coo dina e, he ime-dependen
elec on densi y is sough in he o m
n( , ) = ∑jχj|ψj( , )|2, whe e ψj( , ) a e he KS o bi als
o he nonin e ac ing elec on sys em, and = (x,y) is he 2D
posi ion ec o . The sum uns o e he occupied KS o bi als
wi h g ound-s a e ene gies
jF
(
F
is he Fe mi ene gy),
and he s a is ical ac o s
jj
2 2
F
=
accoun o spin
degene acy as well as o he elec on mo ion along he z-axis.
The o bi als ψj( , ) e ol e in ime acco ding o he 2D ime-
dependen KS equa ions, whe e he non e a ded app oxima-
ion is used consis en wi h he small ele an dimensions o
he sys em
i T V n V n V
V
( , ) ( , ) ( , ) ( )
( , ) ( , )
j
j
H xc s
= [ + [ ] + [ ] +
+ ]
(4)
The ini ial condi ions ψj( , = 0) co espond o he KS o bi als
o he g ound-s a e sys em. In eq 4,Tis he kine ic-ene gy
ope a o , VH[n( , )] is he Ha ee po en ial, Vxc[n( , )] is he
exchange−co ela ion po en ial, and Vs ( ) is he s abiliza ion
po en ial. The ime e olu ion o he elec on densi y, n( , ),
in oduces a ime dependence o he Ha ee and exchange−
co ela ion po en ials. The ke nel o mula ed by Gunna sson
and Lundq is
87
wi hin he adiaba ic local-densi y app ox-
ima ion (ALDA)
86
is used in his wo k o compu e Vxc[n( , )].
Finally, V( , ) = ·E( ) is he po en ial o he op ical ield.
F om he ime e olu ion o he elec on densi y, we ob ain
all he ime-dependen quan i ies o in e es such as he
induced cha ge densi y
n n ( , ) ( , ) ( )
0
= [ ]
(5)
and he induced elec ic nea ield
V E ( , ) ( , )
ind ind
=
(6)
whe e n0( ) is he elec on densi y o he g ound s a e o he
sys em, and Vind( , ) = −{VH[n( , )]−VH[n0( )]} is he
induced po en ial. We also calcula e he mul ipole momen s
Qm( ) o he cha ge densi y induced in he nanowi e pe uni
leng h in z
Q m
R m ( ) 1d e ( , ), 0
m
m
im2
c
=
| |
| |
i
k
j
j
j
j
j
y
{
z
z
z
z
z
(7)
This exp ession is gi en in cylind ical ( ,φ) coo dina es. F om
he cha ge neu ali y o he sys em, he monopole momen is
ze o, Q0( ) = 0. The induced dipole momen pe uni leng h
can be ound om
Q Q i Q Q p e e( ) ( ( ) ( ) ( ) ( ) )
R
x y
21 1 1 1
c
= [ + ] + [ ]
. Fo
he sake o compac ness, we use he e m “mul ipole momen s”
below and unde s and ha he co esponding quan i ies a e
calcula ed pe uni leng h along he z-coo dina e.
Any equency- esol ed magni ude
( )
epo ed in his
wo k is ob ained om he ime-dependen esul
( )
using
he ime- o- equency Fou ie ans o m. In wha ollows,
wi hou loss o gene ali y, we conside posi i e equencies,
ω> 0, and
( )e i
as he ime dependence o he spec al
componen s. Along wi h he ime- o- equency Fou ie ans-
o m o Qm( ) gi en by eq 7, he mul ipole momen s Qm(ω) a
ha monics ω=nΩo he undamen al equency can also be
ob ained om he co esponding spec al componen s o he
induced cha ge densi y as
Qm
R
m
R
( ) 1d e ( , )
1d ( , )
m
m
im
m
m
2
c
c
=
| |
=
| |
| |
| |
i
k
j
j
j
j
j
y
{
z
z
z
z
z
i
k
j
j
j
j
j
y
{
z
z
z
z
z
(8)
The coe icien s δϱm( ,ω) a e de ined in he angula eimφbasis
using a ep esen a ion o he induced cha ge densi y
( , ) 1
2( , )e
m
m
im
=
(9)
The mul ipole momen s Qm(ω) gi en by eq 8 can be used o
exp ess he induced po en ial
V Q R
( , , ) ( ) e
m
m
m
imind c
=
| |
i
k
j
j
jy
{
z
z
z
(10)
and he nea ield
Q m R
i
Qm R
i
E e e
e e
( , , ) ( ) e
( ) e
mm
m
m
im
mm
m
mx y
i m
ind c
1
c
1
( 1)
=| | [ ]
=| | [ ]
| |
| |+
| |
| |+
+
(11)
induced a ha monic equencies ω=nΩo he undamen al
ield.
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■RESULTS AND DISCUSSION
Linea Response. P io o he discussion o he nonlinea
op ical esponse, we analyze he linea op ical esponse and
plasmon esonances o he nanowi e using TDDFT.
Consis en wi h he symme y o he sys em, he plasmon
modes s udied he e co espond o he cylinde plasmons
88−90
in he limi o ze o wa e ec o along he z-axis. These plasmon
modes a e localized mul ipola plasmons and can be iden i ied
wi h he mul ipole o de massocia ed wi h he eimφ
dependence o he plasmon-induced elec on densi y,
po en ial, and nea ield. The ±mplasmon modes a e
degene a e in equency. The e o e, below we conside only
posi i e alues o m. The localized mul ipola plasmons e ol e
in he (x,y)-plane and can be unde s ood as su ace plasmons
sus ained along he ci cum e ence o he nanowi e, which leads
o a quan ized wa enumbe
qm
m
Rc
=
.
77,91
The exci a ion o localized mul ipola plasmons in he
nanowi e is e ealed by a esonan p o ile in he equency
dependence o he mul ipola pola izabili ies αm(ω) calcula ed
wi h TDDFT as shown in Figu e 2a (see SI o he de ini ion
o αm(ω)). The analysis o he esonances yields he
equencies ωmand li e imes τm= 1/Γmo he unde lying
plasmon modes. He e, Γmis he ull wid h a hal maximum o
he esonance. The esonan p o ile o Im{αm(ω)} is o en
pe u bed by addi ional ea u es ela ed o he decay o
plasmons in o elec on−hole pai exci a ions.
79,92
We he e o e
de ine ωmno as he equency a which Im{αm(ω)} is
maximum bu as i s mean equency.
The dipola plasmon mode (m= 1) a ω1= 6.30 eV is
sligh ly edshi ed om he classical non e a ded su ace
plasmon equency ωs=ωp/
2
= 6.35 eV (ωp= 8.98 eV is
he bulk plasma equency) because o he spill-ou o he
induced elec on densi y.
81,93,94
No e ha he plasmon
equencies o he nanowi e conside ed he e a e signi ican ly
highe han hose o ac ual noble me al nanos uc u es, since
he dynamical sc eening associa ed wi h d-band elec ons is
no conside ed in ou jellium model.
81,93
Howe e , he
di e ences in he nonlinea esponse o a plasmonic sys em
induced by a SAM-ca ying undamen al ield o by a linea ly
pola ized ield s em om obus symme y p ope ies. Thus,
he in luence o bound elec ons on he ield sc eening and on
he plasmon esonances does no al e he quali a i e indings
epo ed he e.
In e es ingly, while he classical non e a ded heo y p edic s
ωm=ωsi espec i e o m, he equencies o he plasmon
modes calcula ed wi h TDDFT depend on m. As shown in
Figu e 2b, ωm i s edshi s wi h inc easing mup o m= 3 and
hen mono onically blueshi s o la ge m⩾4. This inding
can be unde s ood conside ing he pic u e o localized
mul ipola plasmons as su ace plasmons wi h quan ized
wa enumbe
qm
m
Rc
=
p opaga ing along he ci cum e ence o
he c oss-sec ion o he nanowi e in he (x,y)-plane.
77,91
The
dispe sion o ωmwi h mcan be hen associa ed wi h he well-
known dispe sion ela ionship o he su ace plasmon
equency on a plana ee-elec on me al su ace as a unc ion
o he wa enumbe qpa allel o he su ace.
82
Finally, Figu e
2c nicely illus a es he mul ipola su ace cha ac e o he m=
1−5 plasmon modes. The induced cha ge densi y is mainly
loca ed a he me al− acuum in e ace and ea u es he
cha ac e is ic eimφangula dependence.
Nonlinea Response. We s a he discussion o he
nonlinea op ical esponse o he nanowi e wi h he analysis o
he induced dipole momen p, which de e mines he e iciency
o he equency con e sion in he a ield. Figu e 3
Figu e 2. Linea op ical esponse o a me allic nanowi e calcula ed wi h TDDFT. (a) Imagina y pa o he mul ipola pola izabili ies o o de mpe
uni leng h, Im{αm(ω)}. Resul s a e shown as a unc ion o equency. Each spec um is e ically o se o cla i y. (b) F equency ωmo he
localized mul ipola plasmon esonance sus ained by he me allic nanowi e as a unc ion o he mul ipole o de m. (c) The induced cha ge densi y
δϱ gi en by he cohe en supe posi ion o he ±mmul ipola plasmon modes, o di e en mas labeled abo e each in e pola ed image. Resul s a e
shown in he (x,y)-plane, and he sys em is ansla ionally in a ian wi h espec o he z-axis. The ed (blue) colo co esponds o posi i e
(nega i e) alues o δϱ as indica ed wi h a colo ba .
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summa izes ou esul s ob ained o linea ly (panel a) and
ci cula ly (panel b) pola ized undamen al ield. The
undamen al equency Ω= 1.5 eV used he e is a om he
plasmon esonances o he sys em so ha no plasmon inging
is p oduced in he esponse. The ime dependence o he
induced dipole momen , p( ), shown in he inse o Figu e 3a,b
ollows he ime dependence o he undamen al elec ic ield
E( ). Fo ci cula ly pola ized illumina ion, his esul s in a π/2
phase shi be ween he px( )- and py( )- componen s o he
induced dipole, as shown in he inse o panel b. We use o -
esonance inciden ield on pu pose o a oid s ong ene gy
deposi ion in o he nanowi e ha would lead o an e icien
elec on exci a ion and e en ually o an elec on emission wi h
cha ging o he nanowi e. We ha e explici ly checked ha wi h
he p esen choice o he pump pulse he calcula ed quan i ies
show he expec ed scaling wi h he nonlinea i y o de nand
wi h he ampli ude o he undamen al elec ic ield. Wi h his
o - esonance condi ion, howe e , we can only obse e he
esonan enhancemen o he nonlinea op ical e-
sponse
46,53,54,60,74,75,95−99
due o he esonance be ween he
ha monic equency nΩand he mul ipola plasmon equency
ωmand no due o he esonance be ween he undamen al
equency Ωand, e.g., he dipola plasmon equency ω1o he
nanowi e.
The equency spec a o he induced dipole shown in
Figu e 3 e eal ha he a - ield emission a ha monic
equencies ω=nΩ(n> 1) is only e icien o a linea ly
pola ized undamen al ield. Consis en ly wi h he symme y o
he nanowi e, he TDDFT esul s in Figu e 3a show ha a
nonlinea dipole is induced a odd ha monics, allowing he
emission in o he a ield o linea ly pola ized illumina-
ion.
100,101
In con as , o ci cula ly pola ized illumina ion
(Figu e 3b), he induced dipole is supp essed a all ha monics
o he undamen al equency so ha he equency con e sion
in o he a ield is quenched.
The nonlinea elec ic nea ield calcula ed wi h TDDFT a a
dis ance d= 1.3 nm om he su ace o he nanowi e is
analyzed in Figu e 4a,b. In con as o he a - ield emission
analyzed in Figu e 3, we ind ha all ha monics (odd and
e en) a e p esen in he nea ield bo h o linea (panel a) and
ci cula (panel b) pola iza ions o he undamen al ield. Thus,
he ha monic decomposi ion o he induced nea ield poin s
o he p esence o highe -o de mul ipole momen s (m> 1) o
he induced nonlinea cha ge densi y. Fu he mo e, o le -
handed ci cula ly pola ized inciden ield (SAM = 1) he
p ojec ions o he induced nonlinea nea ield on x- and y-
axes sa is y he ela ion Ey
ind(ω) = −iEx
ind(ω) (see he −π/2
phase be ween Ey
ind(ω) and Ex
ind(ω) o each ha monic
equency in he inse o Figu e 4b, ep esen ed wi h he
blue do s). This ela ion be ween x- and y- ield componen s
co esponds o igh -handed ci cula pola iza ion (SAM = −1)
o he induced nonlinea nea ield, which is opposi e o ha o
he undamen al ield. The in e sion o he SAM be ween he
undamen al ield and he nonlinea nea ield holds o all
ha monic equencies encompassed he e. We demons a e
below wi h an analy ical app oach ha his SAM in e sion
s ems om he symme y o he sys em and ha i is no
speci ic o a gi en obse a ion poin bu a he a gene al
p ope y o he nea ield.
The exci a ion o nonlinea mul ipole momen s a ha monic
equencies nΩis e idenced wi h he esul s shown in Figu e
4c,d. In his igu e, we show he spec al analysis o he ime-
dependen mul ipole momen s Qm( ) o he induced cha ge
densi y δϱ( , ) calcula ed wi h TDDFT using eq 7. The
mul ipole momen s Qm(ω), ob ained om he ime- o-
equency Fou ie ans o m o Qm( ), ea u e well- esol ed
ha monic con ibu ions a ω=nΩ. A a ixed ha monic
equency nΩ, o a linea ly pola ized inciden ield (Figu e 4c)
one o se e al mul ipole momen s Qm(nΩ) o he nonlinea
induced cha ge densi y a e exci ed wi h ±mdegene acy, and
|m|=n−2j(j= 0, 1..., and 2j<n). In pa icula , mul ipole
momen s Q±1(ω) o o de m=±1 (and hus a dipole momen
p(ω) gi en by hei linea combina ion) a e p esen a odd
ha monics, and absen a e en ha monics, consis en wi h he
esul s epo ed in Figu e 3a. Fo a ci cula ly pola ized inciden
ield (Figu e 4d), a quali a i ely di e en nonlinea esponse is
ob ained. Namely, only mul ipole momen s Qn(nΩ) wi h
posi i e m=na e exci ed in his si ua ion, i.e., he nonlinea
induced cha ge densi y is cha ac e ized by he mul ipole
momen o he same o de mas he ha monic o de n.
I is also wo h no ing ha he ou h ha monic o he
undamen al equency Ω= 1.5 eV o e laps he localized
mul ipola plasmon m= 4 o he nanowi e. This occu s
because o he ini e wid h, Γm, o he la e (see Figu e 2a,b)
Figu e 3. Dipola esponse o he nanowi e o a linea ly (a) and ci cula ly (b) pola ized undamen al ield wi h equency Ω= 1.5 eV. The TDDFT
calcula ions a e pe o med o an ampli ude o he undamen al ield E0= 5.2 ×10−3a.u. co esponding o an a e age powe o he ci cula ly
pola ized op ical pulse o 2.4 ×1012 W/cm2. The spec a |px(ω)|2and |py(ω)|2o he scala p ojec ions o he induced dipole momen , p( ), on he
x- (px) and y- (py) di ec ions a e shown as a unc ion o equency measu ed in uni s o he undamen al equency. The blue ( ed do ed) line is
used o px(py). Fo he case o linea pola iza ion, he elec ic ield is x-pola ized so ha py= 0. All he quan i ies a e calcula ed pe uni leng h o
he nanowi e.
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allowing o he condi ion
4
m m
2 2
m m
+
o be
ul illed. The m-o de mul ipola plasmon is cha ac e ized by
he eimφangula dependence o he su ace cha ges (see Figu e
2c). The componen o he induced nonlinea cha ge densi y
δϱm( ,ω) eimφ(see eq 9) which con ibu es o he mul ipole
momen Qm(see eq 8) has he same angula dependence. This
leads o a esonan enhancemen o he mul ipole momen
Q4(4Ω) o he ci cula ly pola ized undamen al ield. Indeed,
he ampli ude o Q4(4Ω) s ands o he gene al end in Figu e
4d ha shows a dec easing sequence o |Qn(nΩ)|wi h
inc easing n. The esonance enhancemen is also obse ed in
Figu e 4c o he mul ipole momen s Q±2(4Ω) and Q±4(4Ω)
exci ed by linea ly pola ized undamen al ield (see also e 101
o he discussion o he esonan enhancemen o he ou h-
ha monic gene a ion in a pola ized sphe ical Al nanopa icle).
Analy ical In e p e a ion o he TDDFT Resul s. The
main physics behind he TDDFT esul s can be unde s ood
using an app oach
102,103
o en e oked in he con ex o
nonlinea me ama e ials,
22−24,35
which is based on he
symme y o he sys em and Neumann’s p inciple o
enso s.
104,105
To his end, i is con enien o use cylind ical
coo dina es and o in oduce he basis o he an iclockwise
(SAM = +1) and clockwise (SAM = −1) o a ing wa es
de ined as
i ie e e e e
1
2
1
2
e
x y
i
1= [ ± ] = [ ± ]
±
±
(12)
The undamen al ield E(Ω) can be exp essed in he e
±1basis
as ollows:
EE e( ) ( )
1
=
=±
(13)
which is posi ion-independen in he absence o e a da ion
e ec s. Fo ci cula ly pola ized illumina ion, eq 13 esul s in
E(Ω) = E±1(Ω)e
±1, whe e ±1 s ands o he SAM o he
undamen al ield. Fo linea ly pola ized illumina ion E(Ω) =
E+1(Ω)e
+1 +E−1(Ω)e
−1, whe e E−1(Ω) = E+1(Ω) o an x-
pola ized inciden ield and E−1(Ω) = −E+1(Ω) o a y-
pola ized inciden ield. Fo ci cula and linea pola iza ions,
E(−Ω) = [E(Ω)]*, whe e [Z]*s ands o he complex
conjuga e o a complex numbe Z.
Using he basis o o a ing wa es, we in oduce he nonlinea
mul ipola hype pola izabili ies αm;μd
1...μd
n
(n)as
Q n E E( ) ( )... ( )
mm
n
0
,...,
; ...
( )
n
n n
1
1 1
=
(14)
whe e Qm(ω=nΩ) is he m h o de mul ipola momen o he
cha ge densi y induced in he nanowi e pe uni leng h (eq 8),
and indexes μ1,...,μncan ake alues o ±1 independen ly. He e,
we conside he smalles possible nonlinea o de in he ield
and neglec he con ibu ion o he highe -o de e ms such as
∝Eμd
1(Ω)...Eμd
n(Ω)Eμd
n+1(Ω)Eμd
n+2(−Ω). Le us pe o m an an i-
clockwise o a ion o he x- and y- axes by an angle o βa ound
he nanowi e axis. In cylind ical coo dina es, his o a ion is
Figu e 4. Spec al analysis o nonlinea nea ields (a, b) and mul ipole momen s (c, d) induced by he linea ly (a, c) and ci cula ly (b, d) pola ized
ield wi h a undamen al equency Ω= 1.5 eV. Resul s a e shown as a unc ion o he equency measu ed in uni s o he undamen al equency Ω.
The TDDFT calcula ions a e pe o med o an ampli ude o he undamen al ield E0= 5.2 ×10−3a.u. co esponding o an a e age powe o he
ci cula ly pola ized op ical pulse o 2.4 ×1012 W/cm2. (a, b) F equency- esol ed x- and y- componen s o he induced nea ield calcula ed a he x-
axis a a dis ance d= 1.3 nm om he su ace o he nanowi e (see geome y in Figu e 1). Fo he colo code, see he inse o panel (a). The inse o
panel b shows he phase be ween Exand Eycomponen s o he nea ield calcula ed a ha monic equencies ω=nΩ(blue do s). The ed do
indica es he co esponding phase o he undamen al ield. (c, d) F equency- esol ed mul ipole momen s |Qm(ω)|o he induced cha ge densi y.
The colo code co esponds o he mul ipole o de mas explained in he inse s. Fo he x-pola ized undamen al ield he esul s ob ained o ±m
a e degene a e. Fo a undamen al ield wi h SAM = 1, he Qm(ω) a e ze o o nonposi i e m≤0.
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exp essed as φ=φ′+β, whe e he a iable wi h an apos ophe
e e s o he o a ed coo dina e sys em. Consequen ly, he
ec o componen s a e ans o med as Eμ
′(ω) = e+iμβEμ(ω), he
mul ipola momen s a e ans o med as Qm
′(ω) = e+imβQm(ω),
and he mul ipola hype pola izabili ies a e ans o med as
e
m
n i m
m
n
; ...
( ) ( ... )
; ...
( )
n
n
n1
1
1
{ } =
(15)
Fo a nanowi e wi h axial symme y, he Neumann’s
p inciple
104,105
implies ha a o a ion by any angle mus
p ese e he o m o he mul ipola hype pola izabili y enso .
The ollowing selec ion ule is hen ob ained
m...
n1
= + +
(16)
The hype pola izabili ies αm;μd
1...μd
n
(n)a e nonze o only i eq 16 is
ul illed.
Conside i s a linea ly pola ized undamen al ield, which
con ains bo h componen s E±1(Ω) equal in absolu e alue in
he basis o e
±1. The condi ion gi en by eq 16 can be ul illed
by se e al combina ions o (m;μ1, ..., μn) wi h m=± |n−2j|
(j= 0, 1, ..., and 2j<n). Consequen ly, only he co esponding
Qm(nΩ) mul ipole momen s a e exci ed. Fo example, o he
second ha monic n= 2, we ha e (±2; ±1, ±1) leading o he
o ma ion o a quad upole momen . As ano he example, o
he hi d ha monic n= 3, we ha e (±3; ±1, ±1, ±1), leading
o an oc upole momen , and (±1; ±1, ±1, ∓1), leading o a
dipole momen . No ice ha he nonlinea dipole momen
(m=±1), and hus he equency con e sion in o he a ield,
is possible only o odd ha monics consis en wi h he
symme y o he sys em, as shown in he TDDFT esul s in
Figu e 3a.
Conside now a le -handed ci cula ly pola ized undamen al
ield (SAM = 1). The only non anishing componen in he e
±1
basis is E+1(ω). The e o e, μ1=μ2= ... = μn= 1, and
μ1+ ... + μn=n. I hus ollows om eq 16 ha , in his
si ua ion, he nonlinea esponse a he n h ha monic o he
undamen al equency exclusi ely allows o he o ma ion o
an n-o de mul ipole momen Qn(nΩ) o he induced cha ge
densi y δϱ( ,nΩ). Fo n≥2, he nonlinea dipole momen is
ze o so ha he equency con e sion in o he a ield is
supp essed, as shown in he TDDFT esul s in Figu e 3b. The
induced nonlinea nea ield can be ob ained om eq 11, and i
is gi en by
( )
n n Q n iE e e( , ) ( )e
R
n
i n
x y
ind ( 1)
n
n
c
1
= [ ]
+
+
.
Tha is, o he ci cula ly pola ized undamen al ield wi h SAM
= +1, he induced ield is ci cula ly pola ized wi h SAM = −1.
The SAM in e sion is ob ained in he nea ield i espec i e o
he posi ion and o all ha monics.
The gene al consequences o he symme y o he sys em
deduced abo e explain he TDDFT esul s epo ed in Figu e
3and in Figu e 4 when changing om linea pola iza ion o he
undamen al ield o ci cula pola iza ion, namely: (i)
supp ession o he equency con e sion in o he a ield,
(ii) SAM in e sion in he nonlinea nea ield o all ha monics
o he undamen al equency, and (iii) exclusi e o ma ion o
Qn(nΩ) mul ipole momen o he nonlinea cha ge densi y a
nΩha monic equency o ci cula ly pola ized illumina ion.
Ro a ing Nonlinea Cha ge Densi y. To gain deepe
insigh in o he nonlinea esponse o he sys em and o
alida e he heo e ical analysis p esen ed abo e, we show in
Figu e 5 he maps o he cha ge densi y Re{δϱ( ,nΩ)}
induced a he undamen al equency Ω(n= 1; linea
esponse), and a highe ha monic equencies nΩ(n= 2, 3, 4;
nonlinea esponse). Re{Z} s ands o he eal pa o he
complex numbe Z. The TDDFT esul s a e shown as a
unc ion o x- and y-coo dina es in he ans e sal plane o he
nanowi e o linea (uppe ow o panels) and ci cula (lowe
ow o panels) pola iza ions o he undamen al ield. The
dipola cha ac e o he induced densi y cha ac e izing he
linea esponse (n= 1) can be clea ly seen o bo h
Figu e 5. Maps o he nonlinea cha ge densi y Re{δϱ( ,nΩ)}, n= 1, 2, 3, 4, induced in a cylind ical nanowi e a a undamen al equency Ω= 1.5
eV, and a highe ha monics. Re{Z} s ands o he eal pa o he complex numbe Z. The nonlinea cha ge densi y is shown as a unc ion o x- and
y-coo dina es in he ans e se plane o he nanowi e. Resul s a e no malized independen ly o each panel such ha he a ia ion is con ained
wi hin he [−1, + 1] in e al. The colo scale is de ined wi h colo ba s a he igh o he igu e. The uppe (lowe ) ow o panels show esul s
ob ained wi h linea (ci cula ) pola iza ion o he undamen al ield. The TDDFT calcula ions we e pe o med o an ampli ude o he undamen al
ield E0= 1.1 ×10−2a.u. co esponding o an a e age powe o he ci cula ly pola ized pulse 1013 W/cm2.
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pola iza ions. Ob iously, o ci cula ly pola ized undamen al
ield, he induced dipole mus o a e ollowing he di ec ion
gi en by he elec ic ield ec o .
In he absence o e a da ion e ec s, o linea ly pola ized
illumina ion, he nonlinea esponse o he ee-elec on
homogeneous nanowi e should be d i en by he su ace
pola iza ion a e en ha monics, and p edominan ly by he bulk
pola iza ion a odd ha monics
9,38,39,44,46,51
(see discussion in
he SI). We indeed obse e his end o he second (n= 2)
and hi d (n= 3) ha monics. Howe e , o he ou h ha monic
(n= 4), he bulk con ibu ion o δϱ( , 4Ω) is also clea ly
isible in addi ion o he nonlinea cha ges gene a ed a he
su ace. Simila ly, o ci cula ly pola ized illumina ion, he
nonlinea pola iza ion o a homogeneous nanowi e is possible
only a he su ace owing o he symme y b eak (see he
discussion in he SI). This is consis en wi h he esul s
ob ained o n= 2. Howe e , o highe ha monics, δϱ( ,nΩ)
has a s ong bulk componen . We a ibu e his e ec o he
b eak o he homogeneous app oxima ion because o he
F iedel oscilla ions o he g ound-s a e cha ge densi y. As we
show in he SI, F iedel oscilla ions pe sis in he bulk o he
nanowi e because o i s ela i ely small adius (see also
discussion o he cu en s induced inside a plasmonic
nanopa icle in e 71).
The mos impo an in o ma ion s ems, howe e , om he
symme y o he cha ge densi y maps. Fo linea ly pola ized
undamen al ield (uppe panels in Figu e 5), Re{δϱ( ,nΩ)} is
symme ic wi h espec o he x-axis, and om he symme y
wi h espec o he y-axis i ollows ha he dipole momen is
o med only a odd ha monics. In shee con as , o ci cula ly
pola ized undamen al ield (lowe panels in Figu e 5), he
induced cha ge densi y Re{δϱ( ,nΩ)} a ha monic equency
ω=nΩpossesses a well-de ined axial symme y o o de nwi h
espec o he nanowi e axis z. Using he many-body esponse
heo y, we show in he SI ha because o he axial symme y o
he nanowi e, only he e m wi h m=nis nonze o in eq 9. The
nonlinea cha ge densi y induced a ha monic equency can
be hus exp essed as
n n ( , ) ( , )e
n
in
1
2
=
so ha
only Qn(nΩ) mul ipola momen can be exci ed in ull
ag eemen wi h he conclusions de i ed om Neumann’s
p inciple. I is howe e impo an o ealize ha he in e se is
no ue, i.e., he selec ion ule Qm≠n(nΩ) = 0 can also be
sa is ied i ∫ |m|+1d δϱm( ,nΩ) = 0 (see eq 8) and does no
necessa ily equi e ha δϱm≠n( ,nΩ) = 0.
The einφangula dependence o he nonlinea cha ge densi y
a ha monic no he undamen al equency ob ained o
ci cula ly pola ized illumina ion has appealing consequences
o he dynamics o he sys em. Indeed, in his si ua ion, he
ime e olu ion o he nonlinea cha ge densi y o ha monic nis
gi en by
Figu e 6. Time e olu ion o he mul ipola momen s Qm
(n)( ) o linea ly and ci cula ly pola ized inciden ields (le and igh column o panels,
espec i ely). He e, ms ands o he o de o he mul ipole, and ns ands o he ha monic o de . The Qm
(n)( ) a e ob ained using he equency-
esol ed maps o he induced cha ge densi y δϱ( ,nΩ) (see Figu e 5). Resul s a e shown as a unc ion o ime o he linea esponse (n= 1) as
well as o n= 2, 3, 4 ha monics o he undamen al equency Ω= 1.5 eV. In he case o ci cula pola iza ion, we indica e wi h dashed lines he ime
dependence o he co esponding mul ipola momen s calcula ed in a ame o a ing a ound he z-axis an iclockwise wi h angula equency Ω. Fo
u he de ails, see he ex o he pape .
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n
n
( , ) Re ( , )e
Re 1
2( , )e
n in
n
in
( )
( )
{ }
= { }
=
(17)
Rega dless o he ha monic o de n, he ime e olu ion o he
nonlinea induced cha ge densi y δϱ(n)( , ) can be seen as a
o a ion a ound he z-axis o he igid mul ipola cha ge
dis ibu ion calcula ed o = 0,
nRe ( , )e
n
1
2
in
{ }
. The
angula equency o his o a ion equals he undamen al
equency Ω, and he di ec ion o he o a ion is he same as
ha o he ci cula ly pola ized undamen al ield. Conse-
quen ly, he oscilla ion equency o he nonlinea mul ipole
momen s, nea ields, and o he physical quan i ies a he n h
ha monic equency s em om he symme y o his o a ing
cha ge dis ibu ion, whe e he same spa ial p o ile is e ie ed
in a ixed e e ence ame n imes pe undamen al pe iod.
Fo linea ly x-pola ized undamen al ield, he induced
nonlinea cha ge densi y can be exp essed as
n m ( , ) Re 1
2e ( , ) cos( )
n
m
in
m
( )
{ }
=
(18)
whe e m=n−2j(j= 0, 1, ..., and 2j<n). Equa ion 18 hus
e lec s he supe posi ion o se e al cha ge dis ibu ions wi h
angula dependencies cos(mφ), each o hem oscilla ing a
ha monic equency nΩ. The mo ies p esen ed in he SI nicely
demons a e he s iking di e ence in he dynamics o he
nonlinea cha ges induced a ha monic equencies by
ci cula ly (SAM = 1) and linea ly pola ized undamen al ield.
To u he es he alidi y and consis ency o ou analysis,
we use δϱ( ,nΩ) ob ained om TDDFT o calcula e he
mul ipola momen s Qm(nΩ) using eq 8. The ime e olu ion o
he mul ipole momen s a ha monic equencies nΩo a
monoch oma ic undamen al ield is gi en by Qm
(n)( )≡
Re{Qm(nΩ)e−inΩ }, and i is shown in Figu e 6. These esul s
a e in ull ag eemen wi h he esul s o he Fou ie analysis o
he ime-dependen quan i ies de ined by eq 7 and calcula ed
using a Gaussian en elope o he undamen al ield (Figu e
4c,d). Thus, while se e al mul ipole momen s a e p esen a
ha monic equency nΩ o linea ly pola ized illumina ion (le
panels), o ci cula ly pola ized illumina ion ( igh panels), he
only nonze o mul ipole momen o he nonlinea cha ge
densi y a n h ha monic is Qn
(n)( ). When he mul ipola
momen s a e calcula ed no in a ixed e e ence ame bu in a
o a ing ame designed o accompany he nonlinea induced
cha ges, hei ime dependence is en i ely emo ed o he
ci cula ly pola ized undamen al ield (dashed lines in he
lowe ow o panels in Figu e 6). This inding con i ms ou
conclusion on he dynamics o he induced nonlinea cha ge
densi y gi en by he an iclockwise o a ion o he igid
mul ipola densi y dis ibu ion a ound he symme y z-axis
wi h he angula equency Ω.
■SUMMARY AND CONCLUSIONS
In summa y, we used TDDFT calcula ions o s udy he
nonlinea op ical esponse o a ee-elec on plasmonic
nanowi e a ew nanome e s in diame e o a s ong op ical
ield ci cula ly and linea ly pola ized wi hin he plane
pe pendicula o he nanowi e axis. We add essed he
dependence o he nonlinea esponse on he pola iza ion o
he undamen al ield and he possibili y o gene a ing SAM-
ca ying ha monics o he undamen al equency. An analy ical
app oach based on he symme y o his homogeneous sys em
and Neumann’s p inciple o he enso s
103−105
was used o
elucida e he main physics behind he TDDFT esul s ob ained
wi hou any ap io is ic assump ions.
In ull ag eemen , he analy ical app oach and he TDDFT
simula ions e eal ha , in his sys em, he nonlinea op ical
esponse o a ci cula ly pola ized undamen al ield a he n h
ha monic o he undamen al equency Ωis d i en by a
nonlinea induced cha ge densi y wi h disc e e n- old
symme y wi h espec o he nanowi e axis. This cha ge
densi y o a es a he undamen al equency Ωa ound he
nanowi e axis, and i s only nonze o mul ipole momen is he n-
o de mul ipole Qn(nΩ). As a consequence, o a ci cula ly
pola ized undamen al ield:
•The induced densi y, induced ield, and induced
po en ial a n h ha monic equency display a
cos[n(φ− Ω ) + ζ] dependence on ime and azimu hal
angle φin cylind ical coo dina es, wi h z-axis along he
nanowi e axis (ζis some cons an ).
•The equency con e sion in o he a ield is o bidden
o all ha monics, in con as o he case o linea ly
pola ized illumina ion whe e he symme y cons ain s
pe mi he equency con e sion in o he a ield a odd
ha monics.
•All ha monic equencies a e p esen in he nea ield.
•Rega dless o he posi ion, he nonlinea nea ield a
ha monic equencies is ci cula ly pola ized in he plane
ans e sal o he nanowi e axis, wi h SAM opposi e o
ha o he undamen al ield. In o he wo ds, we ob ain
a SAM in e sion a all ha monic equencies.
One o he main ake-home messages o he p esen wo k is
he s iking di e ence in he o igin o he ime dependence o
he physical quan i ies a he ha monics o he undamen al
equency, depending on he pola iza ion o he undamen al
ield. We demons a ed ha , in he case o ci cula pola iza ion,
he ime dependence a he n h ha monic equency s ems
om he o de nmul ipola symme y o he nonlinea cha ge
densi y, which o a es wi h angula equency Ω( undamen al
equency) a ound he nanowi e axis. The same spa ial
dis ibu ion o hese cha ges is hen e ie ed in a ixed
e e ence ame, n- imes pe undamen al pe iod. In he case o
linea pola iza ion, he ime dependence comes om se e al
in e e ing cha ge dis ibu ions oscilla ing a he ha monic
equency nΩ.
Ob iously, he quan i a i e esul s epo ed he e depend on
he speci ic cha ac e is ics o he ee-elec on plasmonic
nanowi e used in ou wo k. Ne e heless, i is impo an o
no e ha he quali a i e conclusions ob ained a e de i ed om
he symme y o he sys em, and hus, hey can be applied o a
a ie y o canonical plasmonic nanoan ennas. Fo a ci cula ly
pola ized undamen al ield, he SAM in e sion in he nea
ield should be obse ed o sys ems wi h cylind ical geome y,
while he axial symme y o he plasmonic nanoobjec is he
only equi emen o he o ma ion o a o a ing nonlinea
cha ge dis ibu ion wi h a mul ipola symme y o de equal o
he o de o he equency ha monic.
This wo k ge s along he lines o ac i e esea ch de o ed o
he manipula ion and con ol o ligh pulses ca ying spin and
angula momen um. In pa icula , ou esul s pa e he way
owa d he use o nanosou ces o ci cula ly pola ized high-
ha monic nea ields o on-chip nonlinea applica ions.
ACS Pho onics pubs.acs.o g/jou nal/apchd5 A icle
h ps://doi.o g/10.1021/acspho onics.3c00783
ACS Pho onics 2023, 10, 3963−3975
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