Influence of spatial dispersion in metals on the optical response of deeply subwavelength slit arrays

1.1 MB·English

by  Mathieu Dechaux

Influence of spatial dispersion in metals on the optical response of deeply
subwavelength slit arrays
Mathieu Dechaux,1,2 Paul-Henri Tichit,1,2 Cristian Cirac`ı,3 Jessica Benedicto,1,2
R´emi Poll`es,1,2 Emmanuel Centeno,1,2 David R. Smith,4 and Antoine Moreau1,2,4
1Clermont Universit´e, Universit´e Blaise Pascal, Institut Pascal, BP 10448, F-63000 Clermont-Ferrand, France
2CNRS, UMR 6602, IP, F-63171 Aubi`ere, France
3Instituto Italiano di Tecnologia (IIT), Center for Biomolecular Nanotechnologies, Via Barsanti, I-73010 Arnesano, Italy
4Center for Metamaterials and Integrated Plasmonics,
Duke University, Durham, North Carolina 27708, USA
In the framework of the hydrodynamic model describing the response of electrons in a metal, we
showthatarraysofverynarrowandshallowmetallicslitshaveanopticalresponsethatisinfluenced
6 by the spatial dispersion in metals arising from the repulsive interaction between electrons. As a
1
simple Fabry-Perot model is not accurate enough to describe the structure’s behavior, we propose
0
to consider the slits as generalized cavities with two modes, one being propagative and the other
2
evanescent. This very general model allows to conclude that the impact of spatial dispersion on

n the propagative mode is the key factor explaining why the whole structure is sensitive to spatial
a dispersion. As the fabrication of such structures with relatively large gaps compared to previous
J experiments is within our reach, this work paves the way for future much needed experiments on

2 nonlocality.
2

] I. INTRODUCTION tor in vacuum in most cases. This is why, as is usual for
s
cavities, the thickness of the grating is roughly half the
c
i wavelength in vacuum for the fundamental resonance20,
t Drude’s model1 has proven unbelievably accurate
p except for exotic cases21. The same mechanism explains
throughout the twentieth century to describe the opti-
o the absorption by subwavelength grooves, the difference
cal response of metals, despite extensive studies in the
s. seventies and eighties to find its limits2,3. Recent ex- beingthat,sincethecavityisnowclosedononeend(see
c Fig. 1), the depth of the grooves is only a quarter of the
si perimentshavehowevershownthatthebehaviorofreso- wavelength in vacuum14. In all cases, an extraordinar-
nancesinsub-nanometer-sizedgapscannotbeexplained
y ily strong funneling effect explains the way the energy is
h withDrude’smodelalone4,5,makingitnecessarytotake literally sucked up into the slits14,19.
p into account the repulsion between electrons inside the
As long as the metal can be considered as almost per-
[ metal in the framework of a hydrodynamic model6–8. In
fect (in the IR12,14 or THz22 or even microwave23 range)
that case, the metallic response is spatially dispersive
2 orwhentheslitislargerthan50nmintheopticalrange,
and since it cannot be reduced to a single permittivity
v the above physical picture is fully accurate. However,
5 depending on the frequency alone, it is often said to be
below that 50-nm threshold for the slit width in the op-
6 non-local. Further experiments would however be wel-
tical range, the guided mode is more localized in the
7 come because of the sub-nanometer dimensions of the
metal than in the slits because the skin depth δ is typ-
5 considered gaps9,10.
0 ically around 25 nm for noble metals. The mode then
. In the present work, we explain why the non-local re- experiences what can be called a plasmonic drag: as
1
sponse of metals can be expected to have an impact on the width of the gap decreases, it becomes slower and
0
5 deeply subwavelength metallic gratings, a structure that slower, its group velocity goes to zero and its wavevec-
1 has been extensively studied in the past decade for its tor k diverges24. Its effective wavelength, defined as
z
: extraordinary transmission11,12 and absorption13–16. We λ = 2π/k thus becomes very small. The depth
v eff z
underline that these effect will be clear for grooves that of the cavity being actually proportional to this effec-
i
X areaslargeasafewnanometersandthatthefabrication tive wavelength25, the thickness of the grating can be
r of such still very narrow slits seems totally within our made extremely small and the slits will still constitute
a reach15–17. a resonator13. This is the regime we are interested in,
In the first part of this article, we will focus on the because it is one of the rare structures in which a plas-
physicalanalysisoftheabsorptionbythemetallicslitsar- monic guided mode with a very high wavevector can be
raywhenspatialdispersionisneglected. Itisnowwellac- excited7. We show however that the physics of these
ceptedthattheextraordinaryopticaltransmissionofslit resonators is slightly more complicated that previously
arraysisduetotheexcitationofcavityresonancesinside thought13: the slits are so shallow that the guided mode
the slits11,18, even if non-resonant mechanism allow for is not the only channel for light to reach the bottom of
a high transmission for very thin structures19. The only the structure. A one-mode model11,18 is thus not suf-
guided mode propagating in the slits in p-polarization ficient to describe the cavity accurately. We give here
has actually no cut-off and can propagate whatever the a generalized Fabry-Perot formula to better predict the
slitwidth,withawavevectork closetok ,thewavevec- behavior of the resonances.
z 0
2
where κ2 = k2 − (cid:15)k2 and κ2 = k2 − k2, (cid:15) being the
t n 0 n 0
permittivity of the metal and k =2π/λ.
0
Thedifferentmodesareindexedbynandcharacterized
by a propagation constant k along the z axis, and a
z
magnetic field profile Hn(x). The magnetic field in layer
y
II can thus be written as a sum of modes propagating
upward or downward
H (x,z)=(cid:88)Hn(x)(cid:2)A e−iknz+B e+iknz(cid:3)e−iωt.
y y n n
n
(2)
Asexpected,thereisonlyonepropagatingBlochmode
(presenting a propagation constant k with a dominant
1
real part). All the other modes are evanescent and thus
attenuated in the z direction, with essentially imaginary
propagation constants k . The propagative mode is re-
n
flected back and forth in the slits, thus producing the
resonance. The actual reflection coefficients (r for the
1
interface between the grating and air, rb for the bottom
1
of the grooves) can be computed using RCWA27,28, as
is quite common for metallic gratings18,21. We under-
linethatthecomputationofthereflectioncoefficientsre-
quires the computation of many evanescent modes. One
couldexpect,fromthevastliteratureonthesubject,that
thereflectioncoefficientofthewholestructurecansimply
FIG. 1. (Color online) Representation of the grooves carved
be written using a Fabry-Perot formula:
in metal, illuminated from above by a TM-polarized plane
wlaayveer.cWonetadinisitninggtuhieshsltihtsr.eedifferentlayers,layerIIbeingthe r =r0+ 1r1−t0r11tr101bee22iikk11hh, (3)
where t is the transmission coefficient from the incom-
01
In a second part, we take into account spatial disper- ing plane wave to the propagating mode in the slits and
sion in the framework of the hydrodynamic model with t10 thetransmissioncoefficientfromthemodeintheslits
realistic parameters7,26, and show that the response of to the outgoing plane wave, that are computed using
the structure is influenced by nonlocality. Moreover, us- RCWA. While such an approach has proved extremely
ing the generalized Fabry-Perot formula, we show that accurateinthepastfortheExtraordinaryOpticalTrans-
the influence of nonlocality on the wavevector of the mission(EOT)11,18,21,29,here,quiteunexpectedly,itfails
guided mode explains almost totally why the structure to predict the position of the resonance given by local
is so sensitive to spatial dispersion in metals, completing RCWA full simulations (see Fig. 2).
the physical picture. The assumption underlying (3) is that only the prop-
agative mode is able to reach the bottom of the grooves.
However,becausethedepthofthegratingissmallerthan
the skin depth, not all the evanescent modes are atten-
II. GENERALIZED CAVITY MODEL
uated enough to be neglected. To be more precise, one
mode in particular, although it is not propagative in the
The structure considered here13,15,16 is presented in z direction, presents an attenuation constant that is so
Figure 1. It is an infinite array of deeply subwavelength low that it is still significantly strong at the bottom of
grooves of width a ranging 2 to 5 nm and depth h, from the slits.
15 to 30 nm typically. The period d is ranging from The profile of this mode is shown Fig. 2 and it is rela-
30 to 50 nm in the present study. Since the structure tivelyflat. Thismodeisthusveryclosetotheattenuated
is periodic with a period of the order of twice the skin wavethatisexcitedinthemetalwhenaplanewaveisre-
depth, the slits must be considered as coupled, and the flected by a metallic screen. Moreover, each reflection of
modes propagating in the slits can be found for normal the propagating mode at the top or at the bottom of the
incidence by solving the classical dispersion relation for grooves generates this mode too, so that it fully partici-
metallo-dielectric structures13 pates to the resonance. We propose here a very general
two-modesapproachtomodelthereflectioncoefficientof
the structure.
cosh(κa)cosh(κ (d−a))
t
In this framework, we call A and B (resp. A and
1(cid:20)κ (cid:15)κ(cid:21) 1 1 2
+ 2 (cid:15)κt − κt sinh(κa)sinh(κt(d−a))−1=0, (1) Bce2n)t)armeotdhee atrmavpeliltinugde(roefspth.eaptrtoenpuagaatitningg) d(roewspn.weavrdanseosr-
3
The downward amplitudes can be written as a result of
the direct excitation by the incoming plane wave, in ad-
dition to the reflection of the modes inside the slits
(cid:26)A =t +r B +r B
1 01 1 1 21 2
A =t +r B +r B
2 02 2 2 12 2
and the equivalent is obtained at the bottom of the
grooves for the upward amplitudes
(cid:40)
B e−ik1h =rbA eik1h+rb B eik2h
1 1 1 21 2
B e−ik2h =rbA eik1h+rb A eik2h.
2 2 1 12 2
This forces us to introduce all the above new reflection
and coupling coefficients, that can be computed using
RCWA. This approach is inspired by mode recycling in
photonic crystal cavities30, generalized here to (i) a non
symmetricalplasmoniccavityand(ii)thecasewherethe
evanescent mode can be directly excited by the incident
wave.
We first use the relation above to eliminate B from
2
the whole system, yielding
 r =r +t B +t (rbA e2ik2h+rb A ei(k1+k2)h)
A =t0 +r10B1 +r20 (r2bA2 e2ik2h+r12b A1 ei(k1+k2)h)
1 01 1 1 21 2 2 12 1
BA2 ==tr0b2A+er21ik(1rh2bA+2reb2ikA2he+i(kr11b+2kA21)hei(k1+k2)h)+r12B1
1 1 1 21 2
We finally get from these equations an expression for
A
2
A =t(cid:48) +cA +r(cid:48) B
2 02 1 12 1
where
t
t(cid:48) = 02
FIG. 2. (Color online) (a) Representation of the general- 02 1−r2r2be2ik2h
izedcavitymodelwiththedifferentreflectioncoefficients. (b) c= r2r1b2ei(k2+γ1)h
Tfirospt:ePvarnofiesleceonfttmheodper(orpeadgactuirnvge)m. oBdoet(tbomlac:kDciuvrevreg)inagndefftehce- 1−r2r2be2ik2h
r
tive index of the propagating mode as a function of the gap r(cid:48) = 12 .
width(solidline)andthemodeguidedinasingleisolatedslit, 12 1−r2r2be2ik2h
thegap-plasmonpolariton(dottedline). (c)Reflectioncoeffi-
cientofthewholestructure,consideringonlythefundamental WecanthusreplaceA2intheaboveequations,toyield
Blochmode(blackcurve),a2modesmodel(redcurve)anda
complete simulation (green curve) for different groove depth  r =r(cid:48) +t(cid:48) B +αA
(Cleofmt:so7lnmm;uclteinptheyrs:ic3snomf;thrieghmt:ag2nnemtic).fiIenlsdeta:nsdimthuelatPiooynnwtiinthg A1 =t001+r101B11+r21(1r2b(t(cid:48)02+cA1
vector at the entrance of the slits. B +=rr1(cid:48)b2AB1e)2eik2i1kh2h++rbr1b(2tA(cid:48) 1+ei(ckA1+k+2)hr)(cid:48) B )ei(k1+k2)h
1 1 1 21 02 1 12 1
upwards (see Fig. 2). The reflection coefficient of the
where
whole structure can then be written as the result of the
reflection on the metallic plane plus the light that comes r(cid:48) =r +t rbt(cid:48) e2ik2h
from the two modes inside the grating layer 0 0 20 2 02
t(cid:48) =t +t rbr(cid:48) e2ik2h
10 10 20 2 12
r =r0+t10B1+t20B2. (4) α=t20r1b2ei(k1+k2)h+t20r2bce2ik2h
4
And finally, the system reduces to Thenthereflectioncoefficientcanbeputunderaform
similar to the Fabry-Perot formula
 r =r(cid:48) +t(cid:48) B +αA
BA1 ==rr10b(cid:48)(cid:48)BA1+e120itk(cid:48)0111h+t(cid:48)(cid:48)1 r =reff + 1t−effr1r(cid:48)r1b(cid:48)1be(cid:48)e2i2ki1kh1h. (12)
1 1 1 02
ThisformulacanbeconsideredasageneralizedFabry-
where Perot formula. The agreement between this formula and
a full RCWA simulation (see Fig. 2 (c)), that can be
r(cid:48) = r2br21r1(cid:48)2e2ik2h considered as a multi-mode model, is excellent. The two
1 1−r21r1b2ei(k1+k2)h modes are thus the only ones that are responsible for
t(cid:48) =t +t(cid:48) r rbe2ik2h the resonance. More precisely, the propagating mode is
01 01 02 21 2
responsiblefortheresonanceandtheevanescentmodeis
rb(cid:48) = r1br2b1cei(k2−γ1)h responsible for a shift of this resonance compared to the
1 1−r2b1r1(cid:48)−2ei(k2−k1)h one-mode model.
t(cid:48)(cid:48) = t(cid:48)02r2b1ei(k2+γ1)h The effective reflection coefficients are easy to com-
02 1−r2b1r1(cid:48)−2ei(k2+k1)h pscuattet,eroinngcemtahterirceeaslocfotehffieciinentetrsfaacreesebxettrwaecetendthfreomdifftehre-
It is not obvious yet that these equations can be used ent space regions21. The resonance condition now reads
to yield a generalized Fabry-Perot formula, because two
terms have appeared that do not exist in the classical arg(r1(cid:48))+arg(r1b(cid:48))+2(cid:60)(k1)h=2mπ (13)
one-mode model. More precisely, to retrieve the exact
where m is a relative integer. Fig. 3 shows the phase of
same equations, t(cid:48)(cid:48) and α would have to vanish. This is
02 theeffectivereflectioncoefficients,comparedtothephase
usuallythecaseinphotoniccrystals-butphysicallyhere,
of the real reflection coefficients. There is essentially a
theevanescentmodeisdirectlyandefficientlyexcitedby
shift of the phase of r(cid:48) compared to the one of r . This
the incoming wave so that t(cid:48)(cid:48) is not negligible. 1 1
02 totally explains why, compared to the one-mode model
We now replace A by its expression in the reflexion
1 prediction, the resonance is shifted since the shift in the
coefficient’s formula, to yield
phase has a direct impact on the resonance condition.
 (cid:48)(cid:48) (cid:48)(cid:48) Thefactthattheslopeofthephaseisnotchangedmeans
r =r +t B
 0 10 1 thatthereisalmostnoimpactonthequalityfactorofthe
B =rb(cid:48)e2ik1hA e2ik1h+t(cid:48)(cid:48) resonance, so that the only impact of the second mode
1 1 1 02
A =r(cid:48)B +t(cid:48) ontheresonanceistoshiftitwithoutchangingitswidth.
1 1 1 01
where new effective coefficients are introduced: -1.4
-1.6
r(cid:48)(cid:48) =r(cid:48) +αt(cid:48) (5) -1.8
0 0 01 -2
-2.2
-2.4
-2.6
t(cid:48)(cid:48) =t(cid:48) +αr(cid:48) (6) -2.8
10 10 1
-3
400 450 500 550 600 650 700 750 800
Finally, A can be written
1
1.6
1.4
t(cid:48) +t(cid:48)(cid:48) r(cid:48) 1.2
A = 01 02 1 (7) 1
1 1−r1(cid:48)r1b(cid:48)e2ik1h 00..68
0.4
and the reflection coefficient, by eliminating B gives 0.2
1 0
400 450 500 550 600 650 700 750 800
r =r(cid:48)(cid:48) +t(cid:48)(cid:48) rb(cid:48)A e2ik1h+t(cid:48)(cid:48) t(cid:48)(cid:48) (8) Wavelength (nm)
0 10 1 1 10 02
leading to the desired result FIG. 3. (Color online) Phase of the reflection coefficients.
Top: Phaseofr andr(cid:48) (blackandgreencurvesrespectively).
1 1
r =r(cid:48)(cid:48) +t(cid:48)(cid:48) t(cid:48)(cid:48) + r1b(cid:48)e2ik1ht(cid:48)1(cid:48)0(t(cid:48)10+t(cid:48)0(cid:48)2r1(cid:48)). (9) Bottom: Phase of r1b and r1b(cid:48) (black and green lines resp.).
0 10 02 1−r1(cid:48)r1b(cid:48)e2ik1h
If we call
(cid:48)(cid:48) (cid:48)(cid:48) (cid:48)(cid:48) III. IMPACT OF SPATIAL DISPERSION
r =r +t t (10)
eff 0 10 02
Deeplysubwavelengthgratingsareveryinterestingbe-
t =t(cid:48)(cid:48) (t(cid:48) +t(cid:48)(cid:48) r(cid:48)) (11) cause the absorbing resonance is the sign that a guided
eff 10 10 02 1
5
mode with a very high wavevector has been efficiently to consider the impact of nonlocality on the propagation
excited in the structure. Such modes are likely to be constants of the two modes that are involved in the res-
influenced by the repulsion between electrons7 because onance of the structure - both the propagating and the
their effective wavelength is so small that is approaches leastevanescentoftheremainingmodes,asshownFigure
the mean free path of electrons in the metal31, which is 4. The figure actually shows that the propagating mode
therelevantscalefornonlocality. Thatisthereasonwhy significantly more sensitive to nonlocality because of its
the spatial dispersion is expected to have an impact on large wavevector k . A high wavevector actually means
1
the slit array’s response. a large value for Ω and thus a noticeable effect on the
We use here the hydrodynamic model7,8 to take the dispersion relation.
intrinsicnonlocalityofthemetallicresponseintoaccount.
ThecurrentsJcorrespondingtothemovementofthefree 14
electronstrappedinthemetalcanbetakenintoaccount 12
as an effective polarization Pf defined by P˙f = J. The part 1 80
electron gas can be considered as a fluid6, leading to the al 6
e
following linearized equation R 4
2
−β2∇(∇.P )+P¨ +γP˙ =(cid:15) ω2E (14) 0
f f f 0 p 400 450 500 550 600 650 700 750 800 850
where γ is the damping factor, ω the plasma frequency
p 6
andβ (cid:39)1.35.106m/s. Alltheseparameters(exceptβ),as art 5
p
well as the dispersive susceptibility χb due to interband nary 34
transitions are taken from careful fits of the available agi 2
experimental data using a Drude and Brendel-Bormann Im 1
0
model32. This allows us to clearly distinguish between 400 450 500 550 600 650 700 750 800 850
the response of the jellium and the response of the back- Wavelength (nm)
ground, that we assume is local. Maxwell’s equations
then reduce to7 FIG. 4. (Color Online) Impact of nonlocality on the effec-
tiveindex(phaseindex)ofthepropagatingmode(solidline)
(cid:26) ∇×E=iωµ0H and on the evanescent mode (dashed line). The real (top)
∇×H=−iω(cid:15) (1+χ )E+P and imaginary (bottom) parts of the propagation constant
0 b f
are shown for a purely local approach (in black) and in the
where, thanks to (14) the polarization can be written as framework of the hydrodynamic model (red).
P = (cid:15)0.ωp2 (cid:18)E−(1+χ )β2∇(∇.E)(cid:19). (15) In order to assess the impact of nonlocality on the
f ω2+iγω b ω2
p whole structure, we have used COMSOL simulations, as
The resolution of such equations in a multilayered a full nonlocal modal method is still beyond our reach.
structure requires the introduction of an additional Asexpected,theresonanceisblue-shiftedcomparedtoa
boundary condition (ABC). The most obvious ABC is local calculation (from 15 nm for 3 nm wide slits, see 5
in that case to consider that no electrons are allowed to (b) to 24 nm for 2 nm wide slits, see Fig. 5 (a)). Inter-
get out of the metal25, which means J.n = P .n = 0, estingly, using a two-mode model but changing only the
f
where n is the normal to the interface - and it turns out propagationconstantk1 ofthefundamentalmodeascom-
tobeoneofthemostconservativeABC,sothatnonlocal puted using the nonlocal dispersion relation (16) allows
effects are not likely to be overestimated7. toaccountformostoftheshift(seeFig. 5). Thisdemon-
In this framework the dispersion relation giving the strates that the major reason why the whole structure is
propagation constant of the modes is modified26 and be- sensitivetothespatiallydispersiveresponseofthemetal
comes is that nonlocality has an impact on the wavevector of
the single mode propagating in the slits. Its wavevector
(cid:20) (cid:21)
1− (cid:15)Ωsinh(κte) =cosh(κa)cosh(κ e) is actually not as high as predicted by the local theory,
k sinh(κ e) t thus leading to a blue shift of the resonance.
z l
(cid:20) (cid:18) (cid:19)(cid:21)
1 κ (cid:15)κa (cid:15)
+ t − +Ω2 sinh(κa)sinh(κ e)
2 (cid:15)κa 1−κ βκ e t
t t IV. CONCLUSION
(cid:20)
Ω sinh(κa)
+ (1−cosh(κ e)cosh(κ e)−
sinh(κle) β t l The mode propagating in a single extremely thin slit
(cid:19)(cid:21)
(cid:15)sinh(κ e) is often called a gap-plasmon polariton, to distinguish
t cosh(κa)cosh(κ e) (16)
κ t it from the surface plasmon polariton. The propagating
t
mode considered here is of course closely related and is
wheree=d−a,Ω= kz2(1− 1 ),κ2 =k2+(cid:16)ωp2(cid:17)(cid:16) 1 + submittedtothesamephysicaleffect: thepresenceofthe
(cid:17) κl (cid:15) 1+χb l z β2 χf metalmakesthemodeslow,withaverylargewavevector.
1+1χb , κ2t = kz2 −(cid:15)k02 and κ2 = kz2 −k0. This allows This allows to understand (i) that deeply subwavelength
6
1 structures, sometimes smaller than the skin depth, can
0.9
e 0.8 still be considered as cavities and (ii) why, as we have
c 0.7
an 0.6 shown here, the smallest resonators are likely to be in-
ct 0.5
efle 00..34 fluenced by spatial dispersion in metals. This class of
R 0.2 resonators are called gap-plasmon resonators and it has
0.1
0 recently been demonstrated experimentally that these
400 450 500 550 600 650 700 750 800
Wavelength (nm) resonators could constitute extraordinarily efficient con-
1 centrators and absorbers15,16,33–35 and produce totally
00..89 unprecedented Purcell enhancements36,37. Theoretical
ance 00..67 studies show that they have a lot of potential as scat-
Reflect 0000....2345 ttehraetrsthfeorvelirgyhgteenxetrraalctthioenorteotoic3a8,l3t9o.oWlsewaerheatvheuisnctorondfiudceendt
0.1 here can be useful to study these structures and help de-
0
400 450 500 550 600 650 700 750 800 sign them in the future, as they reach sizes that are well
below the size of conventional cavity resonators.
FIG. 5. (Color Online) Reflection coefficient computed using
COMSOL (fully nonlocal calculation, blue curves), using a
local2-modesmodel(redcurves)andusinganonlocalversion
ACKNOWLEDGMENTS
of the 2-modes model (green curves) for 3 nm wide grooves
(top) and 2 nm wide grooves (bottom).
ThisworkhasbeensupportedbytheAgenceNationale
de la Recherche (ANR), project ANR-13-JS10-0003.
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