Quasi-normal mode approach to the local-field problem in quantum optics

0.22 MB·English

by  Rong-Chun Ge

Quasi-normal mode approach to the local-field problem in quantum
optics
Rong-Chun Ge1,∗, Jeff F. Young2, and S. Hughes1
1Department of Physics, Engineering Physics and Astronomy, Queen’s University, Kingston, ON, Canada K7L 3N6
2 Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC, Canada
5 V6T 1Z1
1 ∗Corresponding author: [email protected]
0
2
CompiledApril14,2015
r
p Thelocal-field(LF) problem of afinite-sizedipole emit-ter radiating insidea lossyinhomogeneous structure
A is a long-standing and challenging quantum optical problem, and it now is becoming more important due to
rapid advances in solid-state fabrication technologies. Here we introduce a simple and accurate quasi-normal

3 mode(QNM)techniquetosolvethisproblemanalyti-callybyseparatingthescatteringproblemintocontribu-
1 tions from the QNM and an image dipole. Using a real- cavity model to describe an artificial atom inside a
lossyanddispersivegoldnanorod,weshowwhenthecontri-butionoftheQNMtoLFswilldominateoverthe
homo- geneous contribution. We also show how to accurately describe surface scattering for real cavities that
]
l are close to the metal interface and explore regimes when the surface scattering dominates. Our results offer
l
a an intui- tive picture of the underlying physics for the LF problem and facilitate the understanding of novel
h photonsourceswithinlossystructures. (cid:13)c 2015 OpticalSocietyofAmerica
- OCIS codes: 240.6680, 160.4236,270.5580.
s
e
m
The local-field (LF) problem in quantum optics has might be interstitially or substitutionally located. Two
.
t a long, rich history and is now becoming more impor- distinct models were developed, both of which isolated
a
m tantduetothedevelopmentofadvancednanofabrication the dipole “defect” within a cavity that occupied some
technologiesfor opticalstructures andnanophotonicde- volume in the vacuum between host atoms. The virtual
-
d vices.Asiswellknown,thelocalphotondensityofstates cavity(VC)andrealcavity(RC)models differedinhow
n (LDOS) largely controls light-matter interactions, in- the LF is treated [7]: in the VC model, the LF is ob-
o
cluding the spontaneous emission (SE) of a dipole emit- tainedbyaveragingthemacroscopicfieldinsidethecav-
c
[ ter [1] located in an arbitrary scattering environment. ity, whereas in the RC model, the LF is obtained by
TheLDOSisrelatedtothephotonGreenfunction(GF) self-consistentlyincluding the scatteringfromthe cavity
2
of the medium, throughLDOS∝Im[G(r,r′ =r;ω)] [2], boundary.Relevantexperiments,thoughlimited,suggest
v
whereG(r,r′;ω)istheelectric-fieldresponseatrfroma that for ion-doped lossy materials, the RC model works
9
5 dipoleemitteratr′.Forexample,theIm(G)insideaho- well [6], however comparison between experiment and
8 mogeneouslosslessdielectricissimplythevacuumLDOS modelreliesonfittingtheeffective”cavityradius”ofthe
5 scaledbytherefractiveindexofthedielectric.Aplethora ion, a rather ill-defined physical quantity. A significant
0
of novel linear and nonlinear optical effects have been practical and conceptual advantage of nanofabricated
.
1 predicted and demonstrated in nanophotonic structures structures that surround finite-size nano-dipoles [4,5,8]
0
engineered to have an LDOS that differs substantially isthatthereistypicallynoambiguityassociatedwiththe
5
from that in uniform lossless media [3]. The vast ma- size and shape of the RC in which the dipole is located,
1
: jority of the experimentally verified LDOS effects have and it is manifestly clear that the RC, versus the VC
v
been obtained when dipole emitters are located in or in modelshouldapplywhendealingwiththeall-important
i
X proximity to lossless dielectric structures, or in proxim- LF problem.
r ity to lossy (metallic) struc- tures. In all of these cases, MostworksontheRCmodeltodatedealwiththeLF
a the Im(G) is well-behaved and there is typically good inside a lossy homogeneous material. It has been shown
agreement between experimental results of the modified theBornapproximationcouldbeusedtoaddressthear-
LDOS and model calculations using Im(G). Rapid ad- guablymoreimportantLFprobleminaninhomogeneous
vances in experimental nanoplasmonics mean that it is lossystructure[9],thoughithasbeenconfinedtospheri-
nowpossible to embeddipole emitters within inhomoge- calcavitystructures.Recently,theLFeffectinthestrong
neous metallic microcavities [4,5]. This raises the chal- coupling regime for anatomin a sphericalstructure has
lenging theoretical problem of how to deal with the fact alsobeen studied [10]. For non-sphericalgeometries,the
that the LDOS is infinity inside lossy, dispersive struc- finite-difference time-domain (FDTD) [11] algorithm is
tures for whichthe imaginarypartofthe permittivity is effectiveforcomputingtheLF;however,thesimulations
non-zero,i.e., Im[ε(ω)]6=0. are verycomputational-time expensive andit is difficult
Related theoretical work was motivated by dispersive to understand the underlying physics of the scattering
lossy materials in which dipole-emitting atoms or ions problem. Thus one would like a more efficient theoreti-
1
cal approachto describing the SE rate from emitters lo-
catedinsidearbitrarilyshaped,lossymaterialstructures.
It would be especially useful to have a formalism that
facilitates quick exploration of parameter space, e.g., to
explore the role of RC size and position, on the overall
deviceperformance.Suchmodelsarealsorelevanttothe
study of spasers [12,13] and quantum plasmonics [14].
In this Letter, we introduce a QNM technique devel-
oped for metallic nanoresonator (MNRs) [15–18] to ad-
dress the LF problem with a RC model wherein the ra-
(a) (b) (c)
dius of the RC is directly associated with the volume of
Fig. 1. (a) Schematic of a 3D gold nanorod in a homogeneous
theemitter,e,g.,anartificialatomorquantumdot(QD).
background (nb =1.5). (b-c) Spatial profile of the QNM,˜fc with
TheMNRproblemisparticularlychallengingfromanu-
ωc/2π = 324.981 THz (1.344 eV); (b) indicates the LF problem
merical perspective because of the large complex per- withaRCnearthecenterofthenanorodalongthey-axis,and(c)
mittivity, requiring very small (sub-nm) computational shows the same RC but 1.5 nm away from the top surface. The
gridding and tediously long simulation times. However, radius and permittivity of the cavity are rc and εc, respectively
(seetext).
we show that the LF can be solvedsemi-analytically for
lossy cavity structures, evenfor complex-shapedMNRs,
parameters:ω =1.26×1016rad/s,γ =1.41×1014rad/s.
if one adopts a QNM approach. The QNMs are the dis- p
In Ref. [22], Tomaˇs showed that the LF could be de-
crete modes of an open system, and localized surface
scribed by a combination of both the volume averaged
plasmons (LSPs) may be directly understood as QNMs
homogeneous contribution (using the homogeneous GF
of the MNRs [15,18], defined as the frequency domain
for gold) Gh (r ,r ;ω) (r is the position of an electric
solutionstothe waveequationwith openboundarycon- loc d d d
dipoleatthecenteroftheRC)andthescatteringcontri-
ditions [19,20]. The QNM has been widely exploited in
bution, Gsc (r ,r ;ω), due to the inhomogeneity of the
various field of physics [19,20], and a QNM approach loc d d
lossy structure; so the LF GF is given by [22]
is particularly useful when the response of the system
is dominated by one or several discrete resonances. In
G (r ,r ;ω)=Gh (r ,r ;ω)+Gsc (r ,r ;ω), (1)
this work, we study the example of a RC inside a gold loc d d loc d d loc d d
nanorod. To check the accuracy of our theory, we first 2
where Gsc (r ,r ;ω) = 3ε Gsc(r ,r ;ω) and
specify a RC embedded at the center of the nanorod, loc d d (cid:16)2ε+εc(cid:17) d d
and compute the SE enhancement factor using an es- Gsc(r ,r ;ω) is the scattered GF without the RC. Us-
d d
tablished FDTD technique [11] and our semi-analytical ing a Born-expansion method, Dung et al., [9] showed
method. We find an excellent agreement between both direct agreement with the Tomaˇs LF formula. The ho-
methodsfortheLFcomputation.Ourapproachnotonly mogeneouscontributionisobtainedanalyticallywiththe
offers orders-of-magnitude improvement in the required two-layersphericalmodel,yieldinga1/r3 dependenceat
c
computationaltime,buthelpstoidentifytheunderlying asmallcavityradius:Im[nˆ·Gh ·nˆ]≈ 9Im[ε] 1 .Thus
physics in an intuitive and easy-to-understand way. Us- one recognizes that the LF prloocblem is|2eεs+seεcn|t2ia6πllryc3solved
ing this semi-analytictechnique, wethen investigatethe if one can obtain Gsc , though this is far from trivial,
loc
role of emitter size, and find that, for large RC emitters except for very simple shapes.
such as a QDs, the contribution of the QNM can domi- With regards to MNRs, it has been shown the scat-
nate over the homogeneous contribution of the gold. In tering behaviour can be accurately described in terms
addition,wealsoexploretheSErateforRCsplacednear of the QNMs [16–18], and a mode expansion technique
the MNR surface (inside the lossy structure), and show forthe GFhasbeendevelopedandconfirmedtobevery
that, ignoring quantum tunnelling effects, the physics accurate[16,17].TheQNMsofthesystem,˜f ,havecom-
µ
can be accurately captured using an image dipole tech- plex eigenfrequenciesω˜ =ω −iγ ,andarenormalized
µ µ µ
nique. We again explore regimes when the surface scat- as follows: [19,23]
tering can be the dominant mechanism for SE enhance-
ment. 1 ∂(ε(r,ω)ω2)
hh˜f |˜f ii= lim ˜f (r)·˜f (r)dr
ouIrnsttehreeoftoylploicwailnagrbwiteracroilnys-isdhearpaed3lDossgyomldantearniaolr.oGdoalds µ ν V→∞ZV (cid:18)2ω ∂ω (cid:19)ω=ω˜µµ ν
ic
MNRs can achieve LSP responses in the visible. Other + ε(r)˜f (r)·˜f (r)dr=δ . (2)
µ ν µν
2ω˜ Z
geometries including patch antennas have experimen- µ ∂V p
tally been probed to achieve enhanced SE of QDs [21]. In practice this normalization is calculated within a
Figure 1(a) shows a schematic of the gold rod with a finite-size computational domain, large enough to cap-
length 100 nm, radius ra = 15 nm, lying along y-axis ture the evanescent field contribution of the QNM [17].
withabackgroundindexnb =1.5(εb =n2b).Forthegold Analternativederivationofthe normalizationschemeis
MNR, we use the Drude model, ε(ω)=1− ωp2 with presentedinRef.[18].ThecorrespondingGFwithequal
ω2+iωγ
space arguments within the MNR can then be written
2
as an expansion of the QNMs [16,17], with the image dipole GF G (r,−r ;ω) =
d d
GT(r,r;ω)= ω2 ˜fµ(r)˜fµ(r). (3) ±tio4nπεr(εbiεn+bsε−i)d|εred+thr|e3(l3o(srs+|yrr+dla)r(dyr|e+2rrd()ε),−wh1e)r,e ±forcorarensyponpdostio-
2ω˜ (ω˜ −ω)
Xµ µ µ thedipolepolarizednormalandparalleltotheinterface,
Forthegoldnanorodofinterest,weactuallyonlyrequire respectively; and 1 is the unit dyadic. Below, we will
the dominant LSP mode, andthus obtain a single mode concentrateon a y-polarizedelectric dipole near the top
version of Eq. (3) as GT(r,r;ω)= ω2 ˜f (r)˜f (r). surface, so the + sign will be used in Eq. (6), and the
c 2ω˜c(ω˜c−ω) c c enhanced SE rate from the image dipole is given by
One problem with the single mode expansion is that
multi-modal contributions near the surface are not cap- c 3 (ε −ε)
tured.Sucheffectsbecomesimportantwhenoneisafew Fd =9ε Im b , (7)
y B(cid:18)2y ω(cid:19) (cid:20)(ε +ε)(ε +2ε)(cid:21)
nmfromalossystructureinterface,causingadivergence d b c
in the LDOS due to quasi-static coupling (Ohmic heat- where y is the distance from the center of the cavity to
d
ing). However, boundary effects can be well described theflatsurface.Asimilarimagedipolemethodhasbeen
by an image dipole [2,24]. Thus, we separate the to- usedrecently[24]toaccountfortheboundaryscattering
tal scattered contribution into the contributions of the effect of the two layer dielectric structure, though we
QNM and image dipole: Gslocc =Gqloscc+Gdloscc. For cavity find that our method is more accurate as is shown in
positions sufficiently far from an interface, e.g. 10 nm, Fig. 3(a).
thenonecansafelyuseGslocc =Gqloscc fortheLFproblem. We now have, in analytic form, the total LF GF for
For convenience, we define the total enhancement of SE the RC inside the lossy MNR,
(LDOS)rateasF =FS+Fd+F0,whereFS/d/0arethe
QNM, image dipodle andd homdogendeouscontrdibutions,re- Galoncal =Ghloc+Gqloscc+Gdloscc, (8)
spectively, and are given by Fα = Im{nd·Gα(rd,rd;ω)·nd},
d Im{nd·G0(rd,rd;ω)·nd} whichseparatesthecontributionsinto(i)ahomogeneous
with GS/d = Glqoscc/dsc, and G0 = Ghloc, respectively (nd contribution, (ii) a QNM contribution, and (iii) a sur-
is the unit vector of polarization). Here G0 is the free face contribution. Subsequently, we have an analytical
space GF, and Im[nd·G0(rd,rd;ω)·nd]=ω3/(6πc3). prescription for obtaining the total local-field Fy.
Given sufficient computational resources, it has been
shown that FDTD works very well when dealing with x 104
1000
the LFP using a grid size of (4πrc3)1/3 [11]; the SE rate 8.5 (a) (b)
3 8
enhancement can be obtained by injecting an electric
dipole at the center of the RC, and computing the to- Fy7.5 FSy 500
7
tal response of the dipole. This yields the total LF SE
6.5 0
emission rate from FDTD, 1.2 1.3 1.4 1.5 1.2 1.3 1.4 1.5
(eV) (eV)
Im n ·GFDTD(r ,r ;ω)·n 3 1500
FdFDTD(ra,ω)= I(cid:8)m{dnd·G0(rd,rdd;ωd)·nd}d(cid:9). (4) 0y 2 (c) (d)
F
y1000
Our goal is to derive Fd analytically, and to first con- Sy 1 F
firm its accuracy against a brute-force numerical simu- F
lation of FFDTD. Then we use our analytical approach 0 500
d 1 2 3 4 1.2 1.3 1.4 1.5
to explore some new physics related to LF effects with r (nm) (eV)
c
metallic nanorods.
With a RC, whose radius is much less than the scale
Fig. 2. LF effect: a y-polarized dipole at 0.5 nm away from the
of the nanorod (for which its effect on the QNM is neg- center of the nanorod as is shown in Fig. 1(b); rc = 0.62 nm for
ligible), then the QNM contribution is given by [22] (a-b),andrc=3nmfor(d).(a)FullFDTDdipolecalculationsof
FFDTD(redsolid),andtheanalyticalresultofF0(greendashed);
y y
3ε 2 (b) Difference between FFDTD and F0 (orange solid), and the
Gqsc(r ,r ;ω)= GT(r ,r ;ω), (5) y y
loc d d (cid:18)2ε+εc(cid:19) c d d QNM contribution. (c) Ratio of the FyS to the Fy0 versus cavity
radius: on resonance (ω =ωc, red solid), redshifted (ω =1.3 eV,
whichisobtainedanalyticallyfromEq.(3)withthenor- orange dot-dashed) , and blueshifted (ω = 1.4 eV, blue dashed).
malizedQNM˜f .Sonumerically,onemustfirstcompute (d)Fy (redsolid)andtheFy0 (bluedashed).
c
the QNM,butthen,unlikeFDTD, Gqsc is knownforall
loc InFig.1(b/c)weshowthespatialprofileofthedipole
spatial points within the scattering geometry (MNR).
mode of the LSP for the MNR with resonance around
Furthermore,usingaquasistaticapproximation-thevari-
1.34 eV [17], and we are interested in the frequency
ation of electric field produced by the image dipole is
regime around the LSP. The FDTD [26] calculation of
negligible over the region of the RC, the effect of image
the homogeneous contribution is described in Ref. [11],
dipole is given by [2,25]
and it can also be calculated analytically for a spherical
3ε
Gdsc(r ,r ;ω)= G (r ,−r ;ω), (6) RC [27,28]. In Fig. 2(a), the homogeneous contribution
loc d d 2ε+εc d d d to the enhancedSE rate (i.e., Fy0) is shownby the green
3
dashed line for a cavity with radius r = 0.62 nm and x 104 x 104
c 10
refractive index nc =3 (we use this value for the rest of 8.5 (a) (b)
ourpaper,butitcaneasilybechanged).TheFDTDcal- 8 9
cthuelastaiomneocfatvhietytolotaclateendha0n.5cenmmenatwFayyFDfrToDm(rtehdescoelnidte)rfoorf Fy7.5 Fy 8 rc
7
the nanorod along the y-axis (Fig. 1(b)) shows a bump
6.5 7
aroundtheLSP;itisfoundthatthedepolarizationeffect 1.2 1.3 1.4 1.5 0 1 2 3
oftheimagedipole,Fd,isnegligibleatthisposition.Due ω (eV) hy (nm)
y
to the position of the resonance bump in F , we expect
y
it should be an indication of the LSP. An independent Fig. 3. LF SE effect for rc = 0.62 nm. (a) Enhancement at
1.5 nm away from the top surface as is shown in Fig. 1(c):
calculation of the scattering contribution via the QNM
magenta dashed/green solid/orange dashed/cyan circles/ are
usingEq.(5)(bluedashedlineinFig.2(b))indeedshows Fy0/FyFDTD/Fy iva Eq. (7)/Fy but using Fyd given in [24], re-
excellent agreement with the difference between the to- spectively; (b) enhancement at ωc versus the distance hy (from
tal and homogeneous contribution of the enhancement, thetopsurfacetothecenteroftheRC):theredsolid/bluedashed
areF0andthetotalcalculatedviaEq.(8)respectively;thevertical
FFDTD−F0. y
y y dashedlineistheregimeof(a).
InFig.2(c)weshowtheratioofFS toF0asafunction
y y
of the radius of the RC. It can be seen as r increases,
c
the QNM contribution of the LSP begins to dominate accuracy of this approach is first confirmed by compar-
over the homogeneous contribution. Figure 2(d) shows ing with an establishedFDTD technique. The QNM LF
both F and F0 for a RC with radius r = 3 nm; the approach enabled us to identify when one gets a signif-
y y c
contribution of the LSP is now larger than the homoge- icant contribution to the LF from the LSP resonance.
neousone,anditalsoshowsastrongasymmetryaround We have also introduced an image dipole technique to
the LSP resonance. obtain surface contributions for RCs that are near lossy
The above observation shows that our formulation, surface.TheseLFmodelsareoffundamentalimportance
Eq. (8), works well for computing the LF effect, when in quantum optics and could have applications for LSP
the RC is at the center of the rod. But for a truly ro- lasing and spasing.
bust technique, it should also work near a surface. Thus
in Fig. 3(a), we show the enhanced SE rate for a RC Funding Information
inside the nanorod but only 1.5 nm away from the top
surface along the y-axis (Fig. 1(c)): the green solid line Natural Sciences and Engineering Research Council of
is the FDTD result FFDTD, and the magenta dashed Canada.
y
is F0, which is seen to underestimate the total SE en-
y
hancement. Detailed calculation shows that the QNM References
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0
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