Planar refraction and lensing of highly confined polaritons in anisotropic media

ARTICLE
Plana e ac ion and lensing o highly confined
pola i ons in aniso opic media
J. Duan1,2,12, G. Ál a ez-Pé ez 1,2,12, A. I. F. T esgue es-Ma a 1, J. Taboada-Gu ié ez 1,2, K. V. Vo onin 3,
A. Bylinkin 4,5, B. Chang6, S. Xiao 7, S. Liu8, J. H. Edga 8, J. I. Ma ín1,2, V. S. Volko 3,9,
R. Hillenb and 10,11, J. Ma ín-Sánchez 1,2, A. Y. Niki in 5,10 & P. Alonso-González 1,2✉
Re ac ion be ween iso opic media is cha ac e ized by ligh bending owa ds he no mal o
he bounda y when passing om a low- o a high- e ac i e-index medium. Howe e ,
e ac ion be ween aniso opic media is a mo e exo ic phenomenon which emains ba ely
in es iga ed, pa icula ly a he nanoscale. He e, we isualize and comp ehensi ely s udy he
gene al case o e ac ion o elec omagne ic wa es be ween wo s ongly aniso opic
(hype bolic) media, and we do i wi h he use o nanoscale-confined pola i ons in a na u al
medium: α-MoO
3
. The e ac ed pola i ons exhibi non-in ui i e di ec ions o p opaga ion as
hey a e se plana nanop isms, enabling o un eil an exo ic op ical e ec : bending- ee
e ac ion. Fu he mo e, we de elop an in-plane e ac i e hype lens, yielding oci as small as
λ
p
/6, being λ
p
he pola i on wa eleng h (λ
0
/50 compa ed o he wa eleng h o ee-space
ligh ). Ou esul s se he g ounds o plana nano-op ics in s ongly aniso opic media, wi h
po en ial o e ec i e con ol o he flow o ene gy a he nanoscale.
h ps://doi.o g/10.1038/s41467-021-24599-3 OPEN
1Depa men o Physics, Uni e si y o O iedo, O iedo, Spain. 2Cen e o Resea ch on Nanoma e ials and Nano echnology, CINN (CSIC-Uni e sidad de
O iedo), El En ego, Spain. 3Cen e o Pho onics and 2D Ma e ials, Moscow Ins i u e o Physics and Technology, Dolgop udny, Russia. 4CIC nanoGUNE
BRTA, Donos ia–San Sebas ian, Spain. 5Donos ia In e na ional Physics Cen e (DIPC), Donos ia/San Sebas ián, Spain. 6Na ional Cen e o Nano Fab ica ion
and Cha ac e iza ion, Technical Uni e si y o Denma k, Lyngby, Denma k. 7DTU, Fo onik, Depa men o Pho onics Enginee ing and Cen e o
Nanos uc u ed G aphene, Technical Uni e si y o Denma k, Lyngby, Denma k. 8Tim Taylo Depa men o Chemical Enginee ing, Kansas S a e Uni e si y,
Manha an, KS, USA. 9G apheneTek, Skolko o Inno a ion Cen e , Moscow, Russia. 10 IKERBASQUE, Basque Founda ion o Science, Bilbao, Spain. 11 CIC
nanoGUNE BRTA and Depa men o Elec ici y and Elec onics, UPV/EHU, Donos ia - San Sebas ian, Spain.
12
These au ho s con ibu ed equally: J. Duan,
G. Ál a ez-Pé ez. ✉email: [email p o ec ed]
NATURE COMMUNICATIONS | (2021) 12:4325 | h ps://doi.o g/10.1038/s41467-021-24599-3 | www.na u e.com/na u ecommunica ions 1
1234567890():,;
Hype bolic elec omagne ic wa es a ise as a consequence o
he in insic aniso opy o he c ys al la ice in na u al
media1–3and o he a ificially enginee ed aniso opic
dielec ic en i onmen in me ama e ials4–12, which leads o a
me allic-like esponse (nega i e pe mi i i y) along one ( wo) o
he p incipal axes in such media and a dielec ic-like esponse
(posi i e pe mi i i y) along he o he wo (one). Despi e hei
undamen al in e es and hei po en ial o he de elopmen o
new op ical applica ions, hese exo ic wa es a e s ill sca cely
explo ed. In pa icula , e ac ion o hype bolic wa es has only
been s udied o he case in which he inciden beam comes om
an iso opic medium, ypically ee space5–7. As such, he gene al
case o e ac ion, in ol ing hype bolic wa es in which bo h he
inciden and he e ac ed wa es exhibi non-collinea wa e ec o
kand ene gy flux S, emains expe imen ally unexplo ed, pa i-
cula ly a he nanoscale, whe e he specific case o nega i e
e ac ion o highly confined pola i ons has only ecen ly been
heo e ically p oposed13,14. The s udy o he gene al case o
e ac ion could ex end ou capabili ies o con ol he flow o
ligh .
Impo an ly, he ecen disco e ies o phonon pola i ons
(PhPs) in an de Waals c ys als15,16 wi h hype bolic dispe sion,
such as h-BN17–19,α-MoO
3
20–25, and α-V
2
O
5
26, ha e p o ided
unique ma e ial pla o ms o s udy op ical phenomena27,28 wi hin
s ongly aniso opic na u al media. In pa icula , PhPs in α-
MoO
3
ea u e in-plane hype bolic p opaga ion, ul a-low losses,
and s ong confinemen , o e ing he possibili y o isualize
e ac ion di ec ly on he c ys al su ace and a he nanoscale,
which can open new ou es in plana nano-op ics29.
He e, we heo e ically and expe imen ally demons a e he
gene al case o e ac ion a he in e ace be ween wo s ongly
aniso opic (hype bolic) media. Impo an ly, we do i a he
nanoscale and in a low-loss na u al medium by isualizing he
p opaga ion o pola i ons as hey a e se plana nanop isms
ailo ed on he su ace o α-MoO
3
. Ou images show non-
in ui i e di ec ions o p opaga ion and s ong confinemen o he
e ac ed wa es, enabling o un eil an exo ic op ical e ec :
bending- ee e ac ion, which ex ends he cu en capabili ies o
con ol he p opaga ion o ligh a he nanoscale. Fu he mo e, we
de elop an in-plane e ac i e hype lens, yielding oci as small as
λ
p
/6, being λ
p
he pola i on wa eleng h (λ
0
/50 wi h espec o he
wa eleng h o ligh in ee space). Ou findings p o ide unda-
men al knowledge and an e ec i e s a egy o he manipula ion
o pola i ons in aniso opic media, pa ing he way o in eg a ed
fla subwa eleng h op ics.
Resul s and Discussion
Theo y o e ac ion be ween hype bolic media. The unique
p ope ies o pola i ons in hype bolic media can be be e
unde s ood by analyzing hei iso equency cu e (IFC), a slice o
he pola i on dispe sion su ace in momen um- equency space
(kx,ky;ω) by a plane o cons an equency ω0. The IFCs o
pola i ons in wo di e en hype bolic media a e illus a ed in
Fig. 1a, b. Fo con enience, we conside hese wo media o be
defined by he same hype bolic slab (wi h ep esen a i e pe -
mi i i y εx=−5, εy=1, εz=5, see ‘Me hods’) placed on wo
di e en dielec ic subs a es (wi h pe mi i i ies εsub =1 and
εsub =5). In bo h cases, he IFCs desc ibe open hype bolas (black
and g ay cu es, espec i ely). As a esul , no all wa e ec o s k
a e allowed in hese media, which implies ha pola i ons
canno p opaga e along all in-plane di ec ions in eal space.
In ac , p opaga ion is only allowed wi hin he sec o s
j anðkx=kyÞj<ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
εy=εx
qlimi ed by he asymp o es o he hype -
bola in he ðkx;kyÞplane (see Fig. 1a, b). Addi ionally, he
Poyn ing ec o S—which de e mines he p opaga ion di ec ion
o he pola i on and is no mal o he IFC30,31—is no in gene al
collinea wi h k, as indica ed in Fig. 1a ( hey a e collinea only
along he x-axis in Fig. 1a). As such, he p ope ies o p opaga ing
pola i ons in hype bolic media a e di e en o hose in iso opic
media, whe e he IFCs a e ci cula (see dashed cyan cu e in
Fig. 1a) and pola i ons, as is well-known, a e allowed o p opaga e
along all in-plane di ec ions in eal space wi h he same absolu e
alue o he wa e ec o , k, which is always collinea o S.
Impo an ly, such p ope ies o pola i ons p opaga ing in
hype bolic media ha e a d ama ic e ec when hey e ac a a
bounda y be ween wo di e en hype bolic media. Pa icula ly,
momen um conse a ion a he bounda y implies ha he p o-
jec ion kko he inciden and e ac ed wa e ec o s ( kin and kou ,
espec i ely) mus be conse ed (wi h kk¼kinsinφ, whe e φis
he angle o he bounda y as shown in Fig. 1a), gi ing ise o he
Fig. 1 Schema ics o e ac ion o pola i ons be ween wo hype bolic
media. a Iso equency cu es o pola i ons p opaga ing in a hype bolic slab
(wi h ε
x
=−5; ε
y
=1; ε
z
=5) placed on wo di e en semi-infini e
subs a es wi h εsub =1 (black cu e) and εsub =5 (g ay cu e) ha define
wo di e en hype bolic media (medium 1 and 2, espec i ely). The inciden
wa e in medium 1 is cha ac e ized by collinea kin and Sin (as in an iso opic
medium, indica ed by a dashed cyan ci cle). Upon e ac ion in o medium 2,
momen um conse a ion a he bounda y (o ange line), kjj (kk¼kin sinφ,
whe e φis he angle o he bounda y), is ulfilled by non-collinea kou and
Sou . The dashed o ange lines ep esen he no mal o he bounda y. bThe
gene al case o e ac ion be ween wo hype bolic media is ep esen ed by
an inciden wa e om medium 1 wi h non-collinea kin and Sin (no mal o
he iso equency cu e). When he wa e e ac s in o medium 2,
momen um conse a ion a he bounda y (o ange line) is ulfilled by non-
collinea kou and Sou . The dashed o ange lines ep esen he no mal o he
bounda y. cReal-space illus a ion o e ac ion be ween wo hype bolic
media shown in awhe e he inciden wa e exhibi s collinea kin and Sin, i.e.
θink¼θinS, gi ing ise o non-collinea kou and Sou , i.e. θou S≠θou k.d
Real-space illus a ion o he gene al case o e ac ion be ween wo
hype bolic media shown in bwhe e bo h he inciden and he ou going
wa e exhibi s non-collinea kand S, i.e. θink≠θinSand θou k≠θou S. The
angen s pa allel o bo h hype bolas gi e ise o bending- ee e ac ion, i.e.
θinSθou S. The o ange dashed lines in c,d ep esen he no mal o he
bounda y. The whi e and g ay egions in c,dco espond o α-MoO
3
/ai and
α-MoO
3
/SiO
2
, espec i ely.
ARTICLE NATURE COMMUNICATIONS | h ps://doi.o g/10.1038/s41467-021-24599-3
2NATURE COMMUNICATIONS | (2021) 12:4325 | h ps://doi.o g/10.1038/s41467-021-24599-3 | www.na u e.com/na u ecommunica ions
gene alized Snell’s law32:
kin sin θink

¼kou sin θou k

;ð1Þ
whe e θinkand θou ka e he angles ha kin and kou o m wi h
he no mal o he bounda y, espec i ely (see Fig. 1c, d). The
p opaga ion di ec ions o he inciden and e ac ed wa es a e
hen gi en by Sin and Sou , espec i ely, i.e. he di ec ions no mal
o he hype bolic IFCs o each case, which a e in gene al non-
collinea wi h kin and kou (Fig. 1b–d), and hus e ac ion can
occu a angles θou S ha can be di e en om θou k. This
beha io is in s a k con as o ha in iso opic media, in which
momen um conse a ion a he bounda y implies ha e ac ion
occu s always a θou k=θou S.
A pa icula case o e ac ion be ween hype bolic media o an
inciden wa e wi h collinea kin and Sin is shown in Fig. 1a. We
obse e ha he e ac ed wa e (Sou ) bends away om he
di ec ion o Sin owa ds he bounda y (Fig. 1c), in con as o
wha is expec ed in iso opic media o a wa e passing om a low
e ac i e index o a high e ac i e index, whe e Sou bends
owa ds he no mal o he bounda y (see Supplemen a y Fig. 8).
Also, he modulus o kou is much la ge han ha o kin, showing
a s ong wa eleng h educ ion.
The gene al case o e ac ion occu s when kin and Sin o he
inciden wa e a e no collinea . This case, which has no been
ackled expe imen ally o da e, is ske ched in Fig. 1d, whe e a
wa e impinges a a il ing angle θinSon he bounda y be ween
wo hype bolic media. The bounda y is also il ed a gi en angle
wi h espec o he c ys al axes. In e es ingly, due o he simila
shapes o he IFCs in he conside ed hype bolic media (α-MoO
3
/
ai and α-MoO
3
/SiO
2
, black and g ay cu es in Fig. 1b,
espec i ely), he Poyn ing ec o s o he inciden and e ac ed
wa es a e pa allel o almos any kin and bounda y angle φ,
especially in he egion whe e he a ms o bo h hype bolic IFCs
a e s aigh . Hence, he e ac ed wa e p opaga es almos pa allel
o he inciden wa e (i.e. θinSθou S), as i he inciden wa e
had been ansmi ed di ec ly wi hou any change in i s
p opaga ion di ec ion (black and blue a ows in Fig. 1d). This
ea u e opens he doo o he ealiza ion o bending- ee
e ac ion a a bi a y inciden angles in aniso opic media,
which is no possible in iso opic media. No e, howe e , ha he
modulus and di ec ion o kou is e y di e en om he modulus
and di ec ion o kin, which opens he doo o di ec enginee ing
o he wa eleng h and wa e on wi hou modi ying he di ec ion
o p opaga ion o he wa e.
Nanoimaging o e ac ion o in-plane hype bolic pola i ons.
We expe imen ally demons a e and comp ehensi ely s udy he cha -
ac e is ics o e ac ion o pola i ons p opaga ing in hype bolic media.
This p o ides he demons a ion o hese e ec s a he nanoscale and/o
in a na u al medium. To do so, we design and ab ica e plana p isms
(“Me hods”) in a slab o he na u ally hype bolic an de Waals c ys al α-
MoO
3
. We hen isualize he p opaga ion o hype bolic phonon pola -
i ons (HPhPs) passing h ough hem by pola i on wa e on mapping
using a sca e ing- ype scanning nea -field op ical mic oscope (s-SNOM,
see “Me hods”). To define he p isms, we e ch away iangula egions in
a silica (SiO
2
) subs a e on op o which we place a 160-nm- hick α-
MoO
3
slab, hus o ming a egion α-MoO
3
/ai wi h a di e en e ac i e
index, and hus di e en pola i onic dispe sion, wi h espec o he egion
α-MoO
3
/SiO
2
. The di e en pola i onic dispe sions in hese wo egions
a e clea ly co obo a ed by he nea -field image o Fig. 2a, aken a an
inciden wa eleng h λ0=11.3 μm. Specifically, we obse e HPhPs
launched inside he p ism (highligh ed by whi e dashed lines) by he flake
edge (see “Me hods”and Supplemen a y Fig. 4), which p opaga e wi h
collinea kin and Sin and longe wa eleng h λin (whi ea ow),i.e.smalle
wa e ec o kin
=2π/λin (black a ow), han ou side he p ism (λp, ed
a ow), hus e ealing he lowe e ac i e index o he p isms. We also
obse e in he same nea -field image ha when he HPhPs each a
bounda y o he p ism il ed a an angle θin~ 55°, hey e ac in o he α-
MoO
3
/SiO
2
egion (no e ha eflec ion is expec ed o be e y weak, see
Supplemen a y No e 8) along a di e en di ec ion (Sou exp , blue a ows)
wi h espec o which hei wa e on s a e il ed (kou exp, g een a ows).
Thus, he e ac ed ene gy flux and wa e ec o a e no collinea , in
ag eemen wi h ou p edic ions o hype bolic e ac ion in Fig.1a.
Impo an ly, while he wa e ec o , kou exp, e ac s owa ds he no mal
(no e ha pola i on launching by he p ism bounda y can be uled ou ,
see Supplemen a y No e 3), he ene gy flux, Sou exp,bendsaway omi .
This is in s a k con as o wha is expec ed o a wa e p opaga ing om
a lowe e ac i e index egion o a highe e ac i e index egion in
iso opic media. In addi ion, he modulus o he e ac ed wa e ec o
kou exp (~6.48 μm−1) is conside ably la ge han ha o he inciden
wa e ec o kin (~2.09 μm−1) and ha o he pola i on wa e ec o along
he x-di ec ion ou side he p ism kp(~4.08 μm−1), e ealing a s ong
wa eleng h educ ion esul ing om e ac ion in hype bolic media.
These findings a e in pe ec ag eemen wi h he case p edic ed in
Figs. 1a–c o hype bolic e ac ion conside ing collinea kin and Sin .
To unambiguously e i y he ea u es o e ac ion be ween
hype bolic media, we ca y ou ull-wa e nume ical simula ions
which mimic ou expe imen s33 (see “Me hods”). The esul ing
spa ial dis ibu ion o he ou -o -plane componen o he elec ic
field, Re Ezx;y

;is plo ed in Fig. 2b, clea ly showing e ac ion
o bo h he ene gy flux (Sou ) and he wa e ec o (kou ), in excellen
quali a i e and quan i a i e ag eemen ( kou
~6.35μm−1)wi h he
expe imen al image in Fig. 2a. In addi ion o nume ical simula ions,
we also alida e ou expe imen al esul s by pe o ming analy ical
calcula ions34 (Fig. 2c) analogous o hose shown in Fig. 1.Namely,
we calcula e he IFCs o HPhPs in α-MoO
3
/ai (g ay cu e), and α-
MoO
3
/SiO
2
(black cu e) egions a λ0=11.3 μm, and, applying he
condi ion o momen um (wa e ec o ) conse a ion a he bounda y
(o ange line), we ex ac he Poyn ing ec o and wa e ec o o he
e ac ed pola i ons. Again, we obse e e ac ion o he ene gy flux
(Sou , blue a ow) wi h a il ed wa e ec o (kou ,g eena ow)in
excellen ag eemen wi h he expe imen , as well as wi h he ull-
wa e nume ical simula ions.
To u he analyze e ac ion in hype bolic media, we also
pe o m expe imen s a a di e en illumina ing wa eleng h λ0
(di e en il ing angles o he p ism bounda y θa e also shown in
Supplemen a y Fig. 2), as shown in Fig. 2d– o λ0=11.1 μm.
In e es ingly, we obse e ha , in his case, bo h he angula
sepa a ion be ween kou exp and Sou exp and he confinemen
e ec a e la ge , being he e ac ed wa e (Sou exp) almos pa allel
o he p ism bounda y and he modulus o kou exp (12.47 μm−1)
abou ou imes la ge han ha o kin (~3.14 μm−1) and
2.1 imes la ge han ha o kp(~5.92 μm−1).
Al oge he , hese esul s demons a e he e ficien e ac i e
na u e o ou plana p isms, enabling us o isualize in eal-space
h ee impo an ea u es o highly confined pola i ons e ac ed a
he bounda y be ween wo hype bolic media: (i) la ge il ing o
hei wa e on s (gi en by kou exp) wi h espec o hei
p opaga ion di ec ion (gi en by Sou exp), (ii) coun e -in ui i e
di ec ions o p opaga ion, and (iii) subwa eleng h confinemen
(wi h espec o pola i ons along he same c ys al axis in he same
medium).
Sub-di ac ional plana lensing o hype bolic pola i ons. Such
unique ea u es o e ac ed pola i ons in na u ally in-plane
hype bolic media open he doo o ocus ul a-confined
NATURE COMMUNICATIONS | h ps://doi.o g/10.1038/s41467-021-24599-3 ARTICLE
NATURE COMMUNICATIONS | (2021) 12:4325 | h ps://doi.o g/10.1038/s41467-021-24599-3 | www.na u e.com/na u ecommunica ions 3
pola i ons in a plana geome y. To demons a e his possibili y,
we design and ab ica e a plana lens in α-MoO
3
(Fig. 3a). As
no ed abo e (Fig. 2), when HPhPs wi h collinea kin and Sin ( ed
a ow in Fig. 3b) e ac a a bounda y wi h ano he hype bolic
medium wi h highe e ac i e index (such as when passing om
α-MoO
3
/ai o α-MoO
3
/SiO
2
), hey bend away om he no mal
o he bounda y (blue a ow in Fig. 3b). This means ha , in he
case o conside ing a p ism o iangula shape, as ha shown in
Fig. 3a, all he e ac ed pola i ons (blue a ows) can con e ge
in o a single spo , and hus he p ism ac s as a ocusing lens o
highly confined pola i ons (see Supplemen a y No e 9 o he
analy ical desc ip ion o gene al shapes o lenses ocusing
hype bolic wa es). Mo e impo an ly, as such a lens is based on
e ac ion o HPhPs, he e ac ed wa es can po en ially ea u e
infini ely la ge wa e ec o s when he bounda y is pe pendicula
o he asymp o e o he hype bolic IFC, which would yield deeply
sub-di ac ional oci sizes. Howe e , as HPhPs decay exponen-
ially, he in ensi y a a dis an ocus can be weak due o p opa-
ga ion losses, which would be mo e no able o la ge wa e ec o s
app oaching he asymp o e o he hype bolic IFC. Consequen ly,
we designed ou iangula lens looking o a comp omise
be ween a la ge e ac ed wa e ec o and a long p opaga ion
leng h o he e ac ed pola i ons. Acco ding o ou heo e ical
calcula ions (Supplemen a y Fig. 7), his comp omise is ob ained
o an angle o he in-plane wa e ec o o abou 62°. The e o e,
we ab ica e a iangula p ism wi h bounda ies pe pendicula o
his angle (Fig. 3a). The expe imen al and simula ed nea -field
images o his lens design upon illumina ion a 11.16 μm a e
shown in Figs. 3c, d, espec i ely. In bo h images, we obse e
e ac ion o inciden HPhPs wi h collinea kin and Sin ( ed
a ows) a he lens bounda ies (black dashed con ou ) esul ing in
HPhPs wi h non-collinea kou and Sou (g een and blue a ows,
espec i ely) p opaga ing along di ec ions almos pa allel o he
bounda ies (blue a ows), which e en ually con e ge, esul ing in
a ocus. This esul is in s a k con as o ha obse ed in a
simila lens based on e ac ion o highly confined pola i ons in
an iso opic ma e ial, such as h-BN, in which e ac ed pola i ons
bend owa ds he no mal (Supplemen a y Fig. 8), making all
hem o di e ge (Fig. 3e). The nea -field images o ou hype bolic
lens e eal ha he wa e ec o kou o he e ac ed HPhPs is
much la ge han he wa e ec o kin o he inciden HPhPs (no e
ha in he expe imen al image he e is a con ibu ion o ip-
launched HPhPs wi h wo di e en kin wa e ec o s). Mo e
impo an ly, he wa e ec o kou is also much la ge han he
wa e ec o kpo HPhPs p opaga ing along he x-di ec ion in he
flake on op o SiO
2
(black a ows). In o de o a oid he influ-
ence om he pe mi i i y o subs a e, we e alua e he ocusing
esolu ion by compa ing he ull-wid h a hal -maximum
(FWHM) wi h he wa eleng h o pola i ons p opaga ing in
MoO
3
/SiO
2
(λp¼2π=jkpj). Since he ocus shows a FWHM o
~240 nm ( ed do s and g ay cu e in Fig. 3 , o expe imen al and
simula ed line p ofiles, espec i ely), we ob ain a ocus ha is
much smalle han he pola i on wa eleng h (λp) along he x-
di ec ion, o he ee-space illumina ion (λ0), namely o ~λp=6, o
λ0=50. This esolu ion e eals ha a di ac ion-limi ed op ical
sys em in hype bolic media can show a ocus ha is much smalle
han he inciden pola i onic wa eleng h, which again eflec s he
unique beha io o elec omagne ic wa es in hype bolic media.
Mo eo e , his esul confi ms ou plana lens based on e ac ion
o HPhPs as a nano-op ical elemen ha g ea ly exceeds he
ocusing esolu ion o any lens based on e ac ion o highly
confined pola i ons in iso opic media35–38.
Visualiza ion o he gene al case o e ac ion. So a , we ha e
isualized e ac ion in hype bolic media o he case in which
14
7
0
-7
-14
k
y
(μm
-1
)
-14 -7 0 7 14
k
x
(μm
-1
)
a
λ0~11.1μm
λ0~11.3μm
2μm
θin
-30 -15 0 15 30
20
10
0
-10
-20
k
y
(μm-1)
2μm
kinǁSin
kinǁSin
kou
kin
kou
kin
Expe imen Simula ion Theo y
+
−
Exp. Re(σ3)
+
−
Sim. Re(Ez)
2μm
kinǁSin
Sp
Sou
kou
2μm
Sou
kou
Sp
kou -exp
kou -exp
Sou -exp
Sou -exp
Sou
Sou
λp
λp
λp
λp
λin
λin
Bounda y
Bounda y
x
y
kp
θin
kinǁSin
kp
bc
de
Fig. 2 Real-space isualiza ion o e ac ion be ween wo aniso opic media using highly confined pola i ons wi h collinea inciden k and S. a,b
Expe imen al Reðσ3ðx;yÞÞ (a) and simula ed ReðEzðx;yÞÞ (b) nea -field images o HPhPs p opaga ing in a 160-nm- hick α-MoO
3
flake a λ0=11.3 μm. The
whi e and black dashed lines ma k iangula p isms ab ica ed by e ching an ai ca i y on he SiO
2
subs a e below he α-MoO
3
flake. Sin and kin display
he di ec ion o p opaga ion and he wa e ec o o inciden pola i ons in α-MoO
3
/ai , espec i ely. Ho izon al p opaga ion o non- e ac ed pola i ons is
ma ked as kpand Sp. Upon e ac ion a a bounda y o he p ism wi h an angle θin ~ 55°, HPhPs bend away om he no mal, Sou exp (blue a ow), wi h a
il ed wa e ec o kou exp (g een a ow). Compa ed o non- e ac ed HPhPs, indica ed by λp, he e ac ed HPhPs a e s onge confined (wi h a wa eleng h
abou 1.6 imes sho e . cAnaly ic IFCs o α-MoO
3
/SiO
2
(black hype bolas) and α-MoO
3
/ai (g ay hype bolas) e ec i e media in a, b, and conside ing
momen um conse a ion a he bounda y (o ange line), he ex ac ed wa e ec o and di ec ion o he e ac ed pola i ons, kou and Sou , espec i ely, a e in
good ag eemen wi h bo h expe imen and simula ion. d,eExpe imen al Reðσ3ðx;yÞÞ (d) and simula ed ReðEzðx;yÞÞ (e) nea -field images o HPhPs
p opaga ing in a 160-nm- hick α-MoO
3
flake a λ0=11.1 μm. The e ac ed HPhPs p opaga e almos pa allel o he bounda y wi h a wa eleng h 2.1 imes
smalle han λp. Analy ic IFCs o α-MoO
3
/SiO
2
(black hype bolas) and α-MoO
3
/ai (g ay hype bolas) e ec i e media in d,e. The o ange dashed lines in
a– ep esen he no mal o he bounda y.
ARTICLE NATURE COMMUNICATIONS | h ps://doi.o g/10.1038/s41467-021-24599-3
4NATURE COMMUNICATIONS | (2021) 12:4325 | h ps://doi.o g/10.1038/s41467-021-24599-3 | www.na u e.com/na u ecommunica ions
0.01.02.03.0
Dis ance (
μm
)
)m on( y
isne nI
FWHM~240 nm
~λp/6
~λ0/50
Exp
Sim
Bounda y
bSou
kou
02040-20-40
0
15
30
-15
-30
k
x
(μm
-1
)
k
y
(μm
-1
)
kinǁSin
Sou
MoO3/Ai
Focus
-MoO
3
SiO
2
α
akou
kinǁSin
x
y
+
-
Exp. Re(σ3)
Focus
kpSp
c
kou -exp
Sou -exp
1μm
MoO3/Ai
kinǁSin
x
y
Focus
kpSp
d
Sim. Re(Ez)
kou
Sou
1μm
MoO3/Ai
-MoO
3
SiO
2
α
kinǁSin
x
y
+
-
kpSp
e
h-BN/Ai
1μm
h-BN
SiO
2
kou Sou
Di e ging
kinǁSin
Sim. Re(Ez)
+
-
Fig. 3 Sub-di ac ional plana lens based on e ac ion o HPhPs. a Schema ics o a e ac i e hype bolic lens ab ica ed by e ching a iangula ai ca i y
on he SiO
2
subs a e below he α-MoO
3
flake. The op and bo om bounda ies o he lens ha e he same slope as he bounda y in b. Upon e ac ion a he
bounda ies (o ange con ou line), pola i ons bend a away om he no mal, Sou (blue a ows), wi h a il ed wa e ec o kou (g een a ows), con e ging a
a ocal spo ( ed do ). Sin and kin display he di ec ion o p opaga ion and he wa e ec o o inciden pola i ons in α-MoO
3
/ai , espec i ely. bAnaly ic
iso equency cu es (IFCs) o pola i ons p opaga ing in α-MoO
3
/SiO
2
(black hype bola) and α-MoO
3
/ai (g ay hype bola). When he bounda y (o ange
line) is nea ly pe pendicula o he asymp o e o he open hype bolic IFC, he e ac ed pola i ons p opaga e (Sou , blue a ows) almos pa allel o he
bounda y wi h la ge non-collinea wa e ec o (kou , g een a ows). cExpe imen al nea -field image o he e ac i e plana hype lens (black dashed line)
o pola i ons in a 170-nm- hick α-MoO
3
slab, a λ0=11.16 μm. The pola i ons con e ge upon e ac ion a he iangula bounda y. Compa ed o non-
e ac ed pola i ons, indica ed by kpand Sp(black a ows) he e ac ed pola i ons, Sou exp (blue a ows), p opaga e nea ly pa allel o he bounda y. d
Simula ed nea -field image o he e ac i e plana hype lens (black dashed line) conside ed in band isualized in c.e, Simula ed nea -field image o a
e ac i e lens o in-plane iso opic pola i ons in a 170-nm- hick h-BN slab, a λ0=6.5 μm. Upon e ac ion a he iangula bounda y (black dashed line),
pola i ons bend owa ds he no mal, Sou (blue a ow), wi h collinea wa e ec o kou (g een a ow), yielding a di e ging e ec . Ho izon al p opaga ion o
non- e ac ed pola i ons is ma ked as kpand Sp. Nea -field in ensi y p ofiles (g ay line and ed do s) ex ac ed h ough he ocus spo along he e ical
di ec ion in dand c, espec i ely. Bo h cu es a e no malized o he nea -field in ensi y a away om he lens and flake edges. Confinemen ac o s as la ge
as ~λp/6 and ~λ0/50 a e ob ained wi h espec o he pola i on and ee-space ligh wa eleng hs.
NATURE COMMUNICATIONS | h ps://doi.o g/10.1038/s41467-021-24599-3 ARTICLE
NATURE COMMUNICATIONS | (2021) 12:4325 | h ps://doi.o g/10.1038/s41467-021-24599-3 | www.na u e.com/na u ecommunica ions 5

collinea kin and Sin e ac in o pola i ons wi h non-collinea kou
and Sou . Howe e , he gene al phenomenon o e ac ion
in ol es he inciden pola i ons exhibi ing non-collinea kin and
Sin (as ske ched in Fig. 1b, d). In he ollowing, we s udy his
undamen al phenomenon in hype bolic media (Fig. 4). To do
his, we again ab ica e p isms in α-MoO
3
( ollowing he same
s uc u e design as in Fig. 2) and isualize (by s-SNOM) he
p opaga ion o HPhPs e ac ing upon hem. As shown in he
nea -field image o Fig. 4a, we obse e ha HPhPs launched by
he edge o he flake (black a ows) e ac a bounda y-1 (Fig.
4c), and he ou coming HPhPs p opaga e wi h non-collinea kR
and SRinside he p ism (o ange a ows). As such, hese pola -
i ons can now be used o isualize he gene al case o e ac ion a
ano he bounda y o he p ism. Howe e , o ca y ou his
expe imen success ully, we need o ensu e ha : (i) he angle o
he bounda y allows e ac ion o pola i ons acco ding o
momen um conse a ion, and (ii) he HPhPs can each his
bounda y wi hin a easonable p opaga ion dis ance. We ulfill
hese condi ions by conside ing a ho izon al bounda y (bound-
a y-2) ollowed by a e ical bounda y (bounda y-3) in he i-
angula p ism, as shown in he nea -field image o Fig. 4a. In
pa icula , we obse e ha HPhPs wi h non-collinea kRand SR
a e eflec ed a bounda y-2 (Fig. 4d), yielding pola i ons wi h
non-collinea kin and Sin (g een a ows) p opaga ing di ec ly
owa ds bounda y-3, and eaching i wi hin a easonably sho
dis ance. Consequen ly, hese pola i ons e ac a bounda y-3,
which esul s in pola i ons wi h non-collinea kou and Sou ( iole
a ows), as p edic ed by momen um conse a ion (Fig. 4e), and
a
-10 0 10
-10
0
10
+
-
Exp. Re(σ3)
+
-
Sim. Re(Ez)
k
ou
S
ou
k
R
SRkin
Sin
kou Sou
kR
1μm
k
in
S
ou
k
ou
1μm
Bounda y-2
Bounda y-1
Bounda y-3
Bounda y-2
Bounda y-1 Bounda y-3
Non-collinea inciden and e ac ed k and S (a bounda y-3)
S
R
S
in
kS
kS
x
y
b
S
k
k
y
(μm-1)
Sin
SR
cde
-10 0 10
k
x
(μm
-1
)
-10 0 10
k
x
(μm
-1
)k
x
(μm
-1
)
Re ac ion a Bounda y-1 Re lec ion a Bounda y-2 Re ac ion a Bounda y-3
k
R
S
R
k
in
S
in
kin
kR
Fig. 4 Real-space isualiza ion o he gene al case o e ac ion be ween wo aniso opic media using nanoscale-confined HPhPs passing h ough a
bending- ee plana p ism. a Expe imen al Re σ3x;yðÞ

nea -field images o pola i ons p opaga ing in a 231-nm- hick α-MoO
3
flake a λ0=11.0 μm. The
black con ou line ma ks a iangula p ism ab ica ed by e ching an ai ca i y on he SiO
2
subs a e below he α-MoO
3
flake. A fi s e ac ion akes place
a bounda y-1 (o ange solid line) o inciden pola i ons wi h collinea kand S(black a ows), yielding e ac ed pola i ons wi h non-collinea kRand SR
(o ange a ows). These pola i ons hen eflec a bounda y-2 (g een solid line), yielding pola i ons wi h non-collinea kin and Sin (g een a ows). A second
e ac ion a bounda y-3 ( iole solid line) yields pola i ons wi h non-collinea kou and Sou ( iole a ows). bSimula ed ReðEzðx;yÞÞ nea -field images o
HPhPs o he case shown in a. Dashed lines in expe imen al and simula ed nea -field images indica e he wa e on s o pola i ons as hey pass h ough he
p isms. c–eAnaly ic IFCs o HPhPs in MoO
3
/ai (g ay cu e) and MoO
3
/SiO
2
(black cu e), p edic ing he di ec ions o e ac ion o eflec ion o HPhPs a
bounda y-1 (c), bounda y-2 (d), and bounda y-3 (e) based on momen um conse a ion. The o ange (c), g een (d), and pu ple (e) solid lines ep esen he
bounda y-1, bounda y-2, and bounda y-3, while he co esponding dashed lines ep esen he no mal o he bounda y.
ARTICLE NATURE COMMUNICATIONS | h ps://doi.o g/10.1038/s41467-021-24599-3
6NATURE COMMUNICATIONS | (2021) 12:4325 | h ps://doi.o g/10.1038/s41467-021-24599-3 | www.na u e.com/na u ecommunica ions
in good ag eemen wi h nume ical simula ions mimicking he
expe imen (Fig. 4b). This esul unambiguously demons a es
e ac ion o wa es whose ene gy flux and wa e ec o di ec ions
a e non-collinea , and hus cons i u es he eal-space isualiza ion
o he gene al case o e ac ion be ween wo hype bolic media,
which, in addi ion, we demons a e a he nanoscale and in a
na u al medium. Fu he mo e, we highligh ha , since kin and
kou a e close o he asymp o es o he IFCs, whe e he angen s
o bo h hype bolas a e pa allel, Sin and Sou a e almos pa allel
a e e ac ion a bounda y-3. As a esul , pola i ons e ac
upon bounda y-3 beha e as i hey had been di ec ly ansmi ed
wi hou any change in hei di ec ion o p opaga ion, despi e he
wa e ec o does change upon his e ac ion phenomenon.
The e o e, e ac ion upon his p ism is bending- ee, in
excellen ag eemen wi h ou heo e ical p edic ion shown in
Fig. 1d. Such bending- ee e ac i e p ism showed in ou wo k
will open possibili ies o enginee pola i onic wa e on s a he
nanoscale wi hou he need o changing hei di ec ion o
p opaga ion.
Conclusions
In summa y, ou wo k explo es he cha ac e o e ac ion
be ween wo s ongly aniso opic media, which, despi e i s un-
damen al impo ance, has emained elusi e up o now. Ou
obse a ions o e ac ion o s ongly confined pola i ons in a
hype bolic biaxial an de Waals c ys al e eal an exo ic op ical
e ec : bending- ee e ac ion, which opens he doo o on-
demand s ee ing o ligh a he nanoscale and in na u al media.
Fu he mo e, ou demons a ion o a subwa eleng h plana lens
a he nanoscale based on hype bolic e ac ion pa es he way o
he de elopmen o plana nano-op ical elemen s in aniso opic
media. Al oge he , ou esul s open new a enues o in eg a ed
fla op ics, di ec ional ene gy ans e , and hea managemen
applica ions, as well as o mid-in a ed (bio) sensing.
Me hods
Sample ab ica ion and cha ac e iza ion. The α-MoO
3
and h-BN nanome e -
hick flakes we e ob ained by mechanical ex olia ion using a Ni o ape (Ni o
Denko Co., SPV 224P) om comme cial α-MoO
3
bulk c ys als and iso opically
en iched h-BN (10B) c ys als g own ia he a mosphe ic p essu e flux g ow h
me hod19. Fi s , he bulk c ys als we e hinned down employing he Ni o ape and
ans e ed on an op ically anspa en polydime hylsiloxane (PDMS) s amp.
Second, selec ed flakes wi h sha p edges we e iden ified by an op ical mic oscope
and ans e ed on he a ge subs a e by he d y- ans e echnique, which allows
a p ecise posi ioning and alignmen o he flakes on op o he iangula ai egions
(2D p isms) ab ica ed on a 500-nm- hick SiO
2
laye g own by we oxida ion on a
Si subs a e. Fo an e ficien ans e , he subs a e was hea ed up o 200 °C39.
The iangula ai egions we e ab ica ed in a wo-s eps p ocess: (i) he ai
s uc u ed pa e ns we e defined using a di ec w i ing sys em (MLA 100,
Heidelbe g Ins umen ) equipped wi h a 365 nm LED ligh sou ce on a 1.5-µm-
hick posi i e one pho o esis (AZ MiR 701, Mic oChemicals) p e iously
deposi ed on he SiO
2
/Si subs a e; (ii) pa o he SiO
2
laye was e ched away by a
fluo ine-based plasma using an induc i ely coupled plasma sys em (Ad anced
Oxide E che , SPTS), gene a ing gaps wi h a dep h o a ound 350 nm in o he SiO
2
laye . To ge id o pho o esis esidues and con aminan s, he e ched subs a es
we e cleaned wi h a plasma ashing sys em (PVA TePla 300, PVA TePla AG) be o e
ans e ing α-MoO
3
and h-BN flakes.
In a ed nea -field nanoimaging. In a ed nanoimaging was pe o med wi h a
comme cial sca e ing- ype scanning nea -field op ical mic oscope (s-SNOM)40,41
om Neaspec GmbH. A unable DRS Dayligh Solu ions quan um cascade lase
( om 880 o 1100 cm−1) was used as exci a ion sou ce by ocusing he ligh on o a
me al-coa ed (P /I ) AFM (a omic o ce mic oscope) ip oscilla ing a a apping
equency o ∼280 kHz wi h a apping ampli ude ∼100 nm. We illumina e he α-
MoO
3
flakes wi h s-pola ized in a ed ligh and keep he pola iza ion di ec ion o
he inciden elec ic field (E
inc
) pe pendicula o he flake edges o launch HPhPs.
The flakes we e as e scanned and he ip-sca e ed field E
sca
was eco ded wi h a
pseudo-he e odyne Michelson in e e ome e 35 and de ec ed using a liquid ni o-
gen cooled HgCdTe (MCT) de ec o . To supp ess a -field backg ound signals, he
de ec ed signal was demodula ed a he n h ha monics o he ip oscilla ing e-
quency (n¼3 in ou wo k), yielding he complex- alued nea -field signals
σn¼sneiφn, wi h snbeing he nea -field ampli ude and φnbeing he nea -field
phase. Th oughou he manusc ip , we show he eal pa o he nea -field signal,
Re(σ3ðx;yÞ), as a unc ion o he ip posi ion ðx;yÞ. Edge-launched pola i ons yield
inges wi h spacing λp, whe e λpis he pola i onic wa eleng h, while ip-launched
pola i ons p oduce inges wi h pe iod o λp=2, exis ing close o he flake edges (see
Figs. 2and 3in he main ex ).
Full-wa e nume ical simula ions using fini e-elemen me hods. We simula ed
s-SNOM nea -field images using he fini e-elemen -me hod nume ical so -
wa e COMSOL MULTIPHYSICS.α-MoO
3
slabs we e placed on op o SiO
2
subs a es
whe e we defined he same geome y o he ai s uc u es as in he expe imen s.
The hickness o he slabs was se o he alue ex ac ed om AFM measu emen s
o he α-MoO
3
flakes in he co esponding expe imen al s-SNOM images. The
whole sys em was illumina ed wi h plane wa es wi h s-pola iza ion pe pendicula
o he flake edges. Gi en ha he s-SNOM signal can be app oxima ed by he
e ical componen o he elec ic field42, we calcula ed he eal pa o zcompo-
nen he o he nea -field signal a 50 nm abo e he slab su ace, Re Ezx;y

.
Meshing ypes and sizes we e op imized o ensu e a good con e gence o simula ed
esul s. The expe imen al s-SNOM images we e well ep oduced by ou simula ed
images (see Figs. 2–4in he main ex ). Ye , since ou nume ical simula ions do no
conside he influence o he ip, only he wa e on o edge-launched pola i ons is
seen (no e he di e ence wi h he measu emen s close o he flake edges, whe e
dense inges appea due o ip-launched pola i ons). The dielec ic pe mi i i y o
α-MoO
3
and iso opically en iched h-BN we e aken om e e ences19,33. The eal
pa o he dielec ic unc ion o SiO
2
used h oughou his wo k was ex ac ed by
fi ing he alues epo ed in e e ences43–45 o expe imen al HPhPs dispe sions
wi h a Lo en zian model.
HPhPs dispe sion and IFC om analy ical calcula ions. The analy ically calcu-
la ed dispe sion o pola i ons in a biaxial slab embedded be ween wo semi-infini e
media34 is gi en by
kðωÞ¼ρ
da c an ε1ρ
εz

þa c an ε3ρ
εz

þπl

;l¼0;1;2¼;
whe e kis he in-plane momen um, i.e. k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2
xþk2
y
q,ε1and ε3a e he pe mi -
i i y o he supe s a e and subs a e, espec i ely, dis he hickness o biaxial slab,
ρ¼iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
εz=ðεxcos2αþεysin2αÞ
q, and whe e αis he angle be ween he x-axis and k.
Based on he equa ion abo e, we calcula e he IFC o HPhPs by a ying α om 0°
o 360° o a fixed inciden equency ω. In all cases we show p opaga ing modes,
i.e. hose o which jRe kðÞj>jIm kðÞj. In addi ion, since he pola i onic wa e ec o
depends on ε3(ε1=1 all h oughou his wo k as he supe s a e is ai ), di e en
subs a e pe mi i i ies can gi e ise o di e en e ec i e hype bolic media.
Fo he gene al case discussed in Fig. 1in he main ex , we selec ep esen a i e
pe mi i i y alues: εx¼εy¼εz¼−3 o he iso opic slab and εx¼5;εy¼
1;εz¼5 o he hype bolic slab, espec i ely, since a leas one nega i e componen is
necessa y o he exis ence o pola i ons. The slab hickness is se o 100 nm in bo h
cases. The pe mi i i y o he subs a es ε3a e se o 1 and 5, o he medium in which
pola i ons a e inciden and e ac ed, espec i ely (whi e and g ay egions in
Fig. 1c, d).
Fo he specific case discussed in Figs. 2–4we calcula e he ICFs in α-MoO
3
slabs on op o ai and SiO
2
using he pe mi i i y alues desc ibed in he sec ion
“Full-wa e nume ical simula ions using fini e-elemen me hods”. The hickness o
α-MoO
3
flakes was se o he alue ex ac ed om opog aphy measu emen s.
Da a a ailabili y
All da a ha suppo he findings o his s udy a e a ailable om he co esponding
au ho upon easonable eques
Recei ed: 29 Ma ch 2021; Accep ed: 15 June 2021;
Re e ences
1. Landau, L. D. e al. Elec odynamics o con inuous media. Vol. 8 (else ie ,
2013).
2. Sun, J., Li chini se , N. M. & Zhou, J. Indefini e by na u e: om ul a iole o
e ahe z. ACS Pho on. 1, 293–303 (2014).
3. Ko zeb, K., Gajc, M. & Pawlak, D. A. Compendium o na u al hype bolic
ma e ials. Op . Exp. 23, 25406–25424 (2015).
4. Poddubny, A., Io sh, I., Belo , P. & Ki sha , Y. Hype bolic me ama e ials. Na .
Pho on. 7, 948–957 (2013).
5. Xu, T., Ag awal, A., Abashin, M., Chau, K. J. & Lezec, H. J. All-angle nega i e
e ac ion and ac i e fla lensing o ul a iole ligh . Na u e 497,470–474 (2013).
NATURE COMMUNICATIONS | h ps://doi.o g/10.1038/s41467-021-24599-3 ARTICLE
NATURE COMMUNICATIONS | (2021) 12:4325 | h ps://doi.o g/10.1038/s41467-021-24599-3 | www.na u e.com/na u ecommunica ions 7
6. Ho man, A. J. e al. Nega i e e ac ion in semiconduc o me ama e ials. Na .
Ma e . 6, 946–950 (2007).
7. Yao, J. e al. Op ical nega i e e ac ion in bulk me ama e ials o nanowi es.
Science 321, 930–930 (2008).
8. Gomez-Diaz, J. & Alu, A. Fla land op ics wi h hype bolic me asu aces. ACS
Pho on. 3, 2211–2224 (2016).
9. Shalae , V. M. Op ical nega i e-index me ama e ials. Na . Pho on. 1,41–48
(2007).
10. Lu, D. & Liu, Z. Hype lenses and me alenses o a -field supe - esolu ion
imaging. Na . Commun. 3, 1205 (2012).
11. K ishnamoo hy, H. N., Jacob, Z., Na imano , E., K e zschma , I. & Menon,
V. M. Topological ansi ions in me ama e ials. Science 336, 205–209 (2012).
12. Co es, C., Newman, W., Molesky, S. & Jacob, Z. Quan um nanopho onics
using hype bolic me ama e ials. J. Op . 14, 063001 (2012).
13. Lin, X. e al. All-angle nega i e e ac ion o highly squeezed plasmon and
phonon pola i ons in g aphene–bo on ni ide he e os uc u es. P oc. Na .
Acad. Sci. 26, 6717–6721 (2017).
14. Jiang, J. e al. B oadband nega i e e ac ion o highly squeezed hype bolic
pola i ons in 2D ma e ials. Resea ch 2532819 (2018).
15. Low, T. e al. Pola i ons in laye ed wo-dimensional ma e ials. Na . Ma e . 16,
182–194 (2017).
16. Baso , D., Fogle , M. & De Abajo, F.G. Pola i ons in an de Waals ma e ials.
Science 354 (2016).
17. Dai, S. e al. Tunable phonon pola i ons in a omically hin an de Waals
c ys als o bo on ni ide. Science 343, 1125–1129 (2014).
18. Caldwell, J. D. e al. Pho onics wi h hexagonal bo on ni ide. Na . Re . Ma e .
4, 552–567 (2019).
19. Giles, A. J. e al. Ul alow-loss pola i ons in iso opically pu e bo on ni ide.
Na . Ma e . 17, 134–139 (2018).
20. Ma, W. e al. In-plane aniso opic and ul a-low-loss pola i ons in a na u al
an de Waals c ys al. Na u e 562, 557–562 (2018).
21. Zheng, Z. e al. A mid-in a ed biaxial hype bolic an de Waals c ys al. Sci.
Ad . 5, eaa 8690 (2019).
22. Duan, J. e al. Twis ed Nano-op ics: Manipula ing Ligh a he Nanoscale wi h
Twis ed Phonon Pola i onic Slabs. Nano. Le . 20, 5323–5329 (2020).
23. Zheng, Z. e al. Phonon pola i ons in wis ed double-laye s o hype bolic an
de Waals c ys als. Nano. Le . 20, 5301–5308 (2020).
24. Hu, G. e al. Topological pola i ons and pho onic magic angles in wis ed α-
MoO3 bilaye s. Na u e 582, 209–213 (2020).
25. Chen, M. e al. Configu able phonon pola i ons in wis ed α-MoO3. Na .
Ma e . 19, 1307–1311 (2020).
26. Taboada-Gu ié ez, J. e al. B oad spec al uning o ul a-low-loss pola i ons
in a an de Waals c ys al by in e cala ion. Na . Ma e . 19, 964–968 (2020).
27. Peining, L. e al. Hype bolic phonon-pola i ons in bo on ni ide o nea -field
op ical imaging and ocusing. Na . Commun. 6, 7507 (2015).
28. Dai, S. e al. Subdi ac ional ocusing and guiding o pola i onic ays in a
na u al hype bolic ma e ial. Na . Commun. 6, 6963 (2015).
29. Kildishe , A. V., Bol asse a, A. & Shalae , V. M. Plana pho onics wi h
me asu aces. Science 339, 6125 (2013). 1232009.
30. Macedo, R., Dumelow, T. & S amps, R. L. Tunable ocusing in na u al
hype bolic magne ic media. ACS Pho on. 3, 1670–1677 (2016).
31. Guo, Z., Jiang, H. & Chen, H. Hype bolic me ama e ials: F om dispe sion
manipula ion o applica ions. J. Appl. Phys. 127, 071101 (2020).
32. Yu, N. e al. Ligh p opaga ion wi h phase discon inui ies: gene alized laws o
eflec ion and e ac ion. Science 334, 333–337 (2011).
33. Ál a ez‐Pé ez, G. e al. In a ed Pe mi i i y o he Biaxial an de Waals
Semiconduc o α-MoO3 om Nea - and Fa -Field Co ela i e S udies. Ad .
Ma e . 32, 1908176 (2020).
34. Ál a ez-Pé ez, G., Vo onin, K. V., Volko , V. S., Alonso-González, P. &
Niki in, A. Y. Analy ical app oxima ions o he dispe sion o elec omagne ic
modes in slabs o biaxial c ys als. Phys. Re . B 100, 235408 (2019).
35. Alonso-González, P. e al. Con olling g aphene plasmons wi h esonan me al
an ennas and spa ial conduc i i y pa e ns. Science 344, 1369–1373 (2014).
36. Chaudha y, K. e al. Pola i on nanopho onics using phase-change ma e ials.
Na . Commun. 10, 4487 (2019).
37. Hube , A., Deu sch, B., No o ny, L. & Hillenb and, R. Focusing o su ace
phonon pola i ons. Appl. Phys. Le . 92, 203104 (2008).
38. Zhao, C., Liu, Y., Zhao, Y., Fang, N. & Huang, T. J. A econfigu able
plasmofluidic lens. Na . Commun. 4, 2305 (2013).
39. Cas ellanos-Gomez, A. e al. De e minis ic ans e o wo-dimensional
ma e ials by all-d y iscoelas ic s amping. 2D Ma e . 1, 011002 (2014).
40. Fei, Z. e al. Ga e- uning o g aphene plasmons e ealed by in a ed nano-
imaging. Na u e 487,82–85 (2012).
41. Chen, J. e al. Op ical nano-imaging o ga e- unable g aphene plasmons.
Na u e 487,77–81 (2012).
42. Niki in, A. e al. Real-space mapping o ailo ed shee and edge plasmons in
g aphene nano esona o s. Na . Pho on. 10, 239–243 (2016).
43. Palik, Edwa d D., ed. Handbook o op ical cons an s o solids. Vol. 3.
Academic p ess, 1998.
44. Ki amu a, R., Lau en , P. & Jonasz, M. Op ical cons an s o silica glass om
ex eme ul a iole o a in a ed a nea oom empe a u e. Appl. Op . 46,33
(2007).
45. Kischka , J. e al. Mid-in a ed op ical p ope ies o hin films o aluminum
oxide, i anium dioxide, silicon dioxide, aluminum ni ide, and silicon ni ide.
Appl. Op . 51, 28 (2012).
Acknowledgemen s
G.Á.-P. and J.T.-G. acknowledge suppo h ough he Se e o Ochoa P og am om he
Go e nmen o he P incipali y o As u ias (nos. PA-20-PF-BP19-053 and PA-18-PF-
BP17-126, espec i ely). S.X. acknowledges he suppo om Independen Resea ch Fund
Denma k (P ojec No. 9041-00333B). B.C. acknowledges he suppo om VILLUM
FONDEN (No. 00027987). The Cen e o Nanos uc u ed G aphene is sponso ed by he
Danish Na ional Resea ch Founda ion (P ojec No. DNRF103.) K.V.V. and V.S.V.
g a e ully acknowledge he financial suppo om he Minis y o Science and Highe
Educa ion o he Russian Fede a ion (Ag eemen No. 075-15-2021-606). J.M.-S.
acknowledges financial suppo h ough he Ramón y Cajal P og am om he Go -
e nmen o Spain (RYC2018-026196-I). A.Y.N. and J.I.M. acknowledge he Spanish
Minis y o Science, Inno a ion and Uni e si ies (na ional p ojec s MAT201788358-C3-
3-R and PID2019-104604RB/AEI/10.13039/501100011033). R.H. acknowledges financial
suppo om he Spanish Minis y o Science, Inno a ion and Uni e si ies (na ional
p ojec RTI2018-094830-B-100 and he p ojec MDM-2016-0618 o he Ma ie de Maez u
Uni s o Excellence P og am) and he Basque Go e nmen (g an No. IT1164-19). A.Y.
N. also acknowledges he Basque Depa men o Educa ion (g an no. PIBA-2020-1-
0014). P.A.-G. acknowledges suppo om he Eu opean Resea ch Council unde
s a ing g an no. 715496, 2DNANOPTICA and he Spanish Minis y o Science and
Inno a ion (S a e Plan o Scien ific and Technical Resea ch and Inno a ion g an
numbe PID2019-111156GB-I00).
Au ho con ibu ions
P.A.-G. and J.D. concei ed he s udy. P.A.-G. supe ised he p ojec . J.D. ca ied ou he
nea -field imaging measu emen s wi h he help o A.B. J.D., A.I.F.T.-M., S.X., B.C., and
J.I.M. con ibu ed o sample ab ica ion. S.L. and J.H.E. p o ided he iso opically en i-
ched bo on ni ide. G.Á.-P. and K.V.V. ca ied ou he analy ical calcula ions and A.I.F.
T.-M. pe o med he nume ical simula ions wi h he help o G.Á-P. and supe ised by
J.M.-S. and A.Y.N. P.A.-G., J.D., G.Á.-P., J.T.-G., V.S.V., A.Y.N., R.H., and J.M.-S. pa -
icipa ed in da a analysis. P.A.-G., J.D., and G.Á.-P. w o e he manusc ip wi h inpu
om he es o au ho s.
Compe ing in e es s
R.H. is co ounde o Neaspec GmbH, a company p oducing sca e ing- ype nea -field
scanning op ical mic oscope sys ems, such as he one used in his s udy. The emaining
au ho s decla e no compe ing in e es s.
Addi ional in o ma ion
Supplemen a y in o ma ion The online e sion con ains supplemen a y ma e ial
a ailable a h ps://doi.o g/10.1038/s41467-021-24599-3.
Co espondence and eques s o ma e ials should be add essed o P.A.-G.
Pee e iew in o ma ion Na u e Communica ions hanks he anonymous e iewe s o
hei con ibu ion o he pee e iew o his wo k. Pee e iewe epo s a e a ailable.
Rep in s and pe mission in o ma ion is a ailable a h p://www.na u e.com/ ep in s
Publishe ’s no e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in
published maps and ins i u ional a filia ions.
Open Access This a icle is licensed unde a C ea i e Commons
A ibu ion 4.0 In e na ional License, which pe mi s use, sha ing,
adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e
app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e
Commons license, and indica e i changes we e made. The images o o he hi d pa y
ma e ial in his a icle a e included in he a icle’s C ea i e Commons license, unless
indica ed o he wise in a c edi line o he ma e ial. I ma e ial is no included in he
a icle’s C ea i e Commons license and you in ended use is no pe mi ed by s a u o y
egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission di ec ly om
he copy igh holde . To iew a copy o his license, isi h p://c ea i ecommons.o g/
licenses/by/4.0/.
© The Au ho (s) 2021
ARTICLE NATURE COMMUNICATIONS | h ps://doi.o g/10.1038/s41467-021-24599-3
8NATURE COMMUNICATIONS | (2021) 12:4325 | h ps://doi.o g/10.1038/s41467-021-24599-3 | www.na u e.com/na u ecommunica ions