Design of an efficient single photon source from a metallic nanorod dimer: a quasinormal mode finite-difference time-domain approach

0.33 MB·English

by  Rong-Chun Ge

Design of an efficient single photon source from a metallic nanorod
dimer: a quasinormal mode finite-difference time-domain approach
Rong-Chun Ge∗ and S. Hughes
Department of Physics, Engineering Physics and Astronomy, Queen’s University, Kingston, Ontario, Canada K7L 3N6
∗Corresponding author: [email protected]
5
1 CompiledJanuary26,2015
0
Wedescribehowthefinite-differencetime-domain(FDTD)techniquecanbeusedtocomputethequasinormal
2
mode (QNM) for metallic nano-resonators, which is important for describing and understanding light-matter

n interactions innanoplasmonics. WeusetheQNMtomodel theenhanced spontaneous emissionratefordipole
a emitters near a gold nanorod dimer structure using a newly developed QNM expansion technique. Significant
J enhanced photon emissionfactors of around 1500 are obtained withlarge output β-factors of about 60%. (cid:13)c
2015 OpticalSocietyofAmerica
3
OCIS codes: 240.6680, 160.4236,270.5580.
2

]
l
l Resonantcavity structures have the ability to trap light time-domainapproachcanalsouse PMLs [16].For met-
a
in very small spatial volumes which has a wide range als, a major problem occurs when using the usual mode
h
- of applications in nanophotonics [1]. Various miniatur- calculation approach with a dipole excitation source,
s izedcavitystructureshavebeendevelopedovertheyears since the extracted mode depends sensitively on the
e
m to manipulate light at the subwavelength scale, and ex- dipole position [17], and is therefore incorrect. Thus it
tremenanoscaleconfinementisnowpossiblewithmetal- has been common practice to excite the MNR with a
.
t lic nano-resonators(MNRs). In a frequency regime near plane wave source and obtain the scattered field. How-
a
m a localized surface plasmon (LSP) [2], the local density ever,thisscatteredfieldisnotthesamefieldastheQNM
of optical states (LDOS) can be increased dramatically. anditcannotbeproperlynormalizedforuseinquantum
-
d Consequently, the spontaneous emission (SE) rate of a optics,e.g.,forobtainingthePurcellfactorandeffective
n dipole emitter can be enhanced via the Purcell effect. mode volume [12]—two well known quantities that help
o
TheresonantenhancementfromMNRshasapplications describe the underlying physics of cavity light-matter
c
[ in chemical sensing [3], high resolution imaging [4,5], interactions. While some frequency-domain techniques
optical antennas [6] and single photon emission [7]. exist for computing the QNMs of MNRs [18,19], it is
1
While the optical properties of MNRs are being ac- highly desirable to be able to compute the QNMs using
v
tively pursued, numerically modelling of the basic cav- the commonly employed and general FDTD technique.
8
6 ity physics is extremely demanding, and analyticalsolu- TheFDTDmethodisalreadywidelyusedbytheplas-
8 tions of the modes only exists for very simple structures monics community, and its accuracy for obtaining the
5 such as spheres.For resonantcavity structures, the nat- enhanced field has been verified against other numeri-
0
ural modes of the system are called quasinormal modes caltechniquessuchasthemultipoleexpansiontechnique
.
1 (QNMs)[8,9],definedasthefrequencydomainsolutions [20].Inaddition,theLDOScanbecalculatedbyemploy-
0
tothewaveequationwithopenboundaryconditions(the ing a dipole excitation source [21–23], which can also
5
Silver-Mu¨ller radiation condition). Kristensen et al. [10] model local field effects, e.g., associated with finite-size
1
: first used the QNMs to introduce a rigorous definition photon emitters inside a MNR [21]. While direct dipole
v
of the “generalized effective mode volume” and Purcell calculations are feasible, they are very time consuming
i
X factor [11], and applied these results to photonic cavity and require a new simulation (which may require many
r structures. For MNRs, the QNMs also form the natural hoursofcomputationaltime)foreachspatialpositionof
a starting pointfor developing analyticaltheoriesof light- the dipole emitter; thus a QNM picture would be much
matter interactions in nanoplasmonics [12–14]. more efficient, since it allows one to simulate dipole re-
One of the most common numerical techniques for sponses both as a function of space and frequency as
obtaining the cavity mode for dielectric cavities is the soon as the QNM is obtained and properly normalized.
finite-difference time-domain (FDTD) technique [15]. In this Letter we first describe how the FDTD tech-
The FDTD technique allows one to simulate open nique[24]canbeefficientlyemployedtoobtainQNMsof
boundary conditions with “perfectly matched layers” a MNR by filtering the scattered field with a temporal
(PMLs) located at the leaky mode region outside the window function. We compute the spatial dependence
cavity.Fordielectriccavities,thisopen-boundaryFDTD of the QNM and the effective mode volume, and show
approach has been shown to yield excellent agreement excellent agreement with full-dipole calculations for the
with direct integral equation methods [10]. Other time- enhanced SE factor of a dipole emitter. We then show
domain techniques such as the discontinuous Galerkin how a gold nanorod dimer can act as an efficient single
1
photon source with large Purcell factors (1500) and im- band frequencies and various spatial positions).
pressive output β-factors (around 60%). In contrast to Next we discuss how FDTD is usually used to obtain
sphericaldimerstructures[25],wefindthatthenanorod resonator cavity modes and highlight the main problem
dimeractstoincreasetheβ-factorforgoodphotonemit- formetallicstructures.Asadirectspace-timesimulation
ters (in comparison to a single resonator), and has the method, FDTD can obtain the spectral response of sys-
additional advantage of yielding resonant frequencies in tem froma Fourier transformof its time response.Thus
the visible spectrum. for any incident field with a finite bandwidth, the com-
TheQNM˜f hasacomplexeigenfrequency,ω˜ =ω putedscatteredfield,ES(ω),willcontaintheresonantre-
µ µ µ
−
iγ , with a quality factor Q = ω /2γ . The QNM is sponsesofthesystem,e.g.,spectralpeakscorresponding
µ µ µ
normalized through [8,9] to the QMNs. Defining the total electric field as Etot(ω)
(including the incident source), then the scattered field
1 ∂(ǫ(r,ω)ω2)
˜f ˜f = lim ˜f (r) ˜f (r)dr is simply ES(ω)=Etot(ω) Eh(ω), where Eh(ω) is the
µ ν µ ν
hh | ii V→∞ZV (cid:18)2ω ∂ω (cid:19)ω=ω˜µ · homogeneoussolutionwith−outthe MNR(i.e.,no scatte-
ic ring). For a typical dielectric cavity with a high Q res-
+ ǫ(r)˜f (r) ˜f (r)dr=δ . (1)
2ω˜µ Z∂V p µ · ν µν onance, almost any incident field or dipole source field
can be used to obtain the mode of interested if one ap-
Since the eigenfrequencies are complex, the QNMs di-
plies a temporal window to subtract off the excitation
verge in space and each part of Eq. (1) diverges, but
source. A common method of choice is to use a dipole
the total sum converges quickly in space [12,14]. This
excitationsourceandbeginthe Fouriertransformofthe
convergence occurs approximately when the outgoing
scatteredfieldafterthe excitationpulse hasgone,which
field becomes purely oscillatory,rather than evanescent.
hasbeenshowntoyieldgoodagreementwithothermode
For spatial points near the resonator, the transverse
solving techniques [10]. However,for a metal, the dipole
part of the photon Green function can be expanded
source does not efficiently excite the QNM because of
as[9,14]GT(r ,r ;ω)= ω2 ˜f (r )˜f (r ).One
1 2 µ 2ω˜µ(ω˜µ−ω) µ 1 µ 2 large losses,and extra care is needed in applying a tem-
can then derive the effecPtive mode volume V and en-
eff poralwindow function to compute the QNM ofinterest.
hanced SE factor F (r ,ω), where r is the spatial po-
a a a Below we describe how the FDTD technique can obtain
sition of a dipole emitter. For dipole positions near the
the true QNM, if the QNM is spectrally well isolated.
resonator,these quantities are defined through [10,14]
To cover a broad spectral range, we use a plane-wave
1 1 ˜f ˜f excitationsourcewitha6fstimeduration.Toefficiently
c c
Veff =Re(cid:26)vQ(cid:27), vQ = εhBh˜fc2|(ri0i), (2) efixeclditetothbeeQpNaMra,llweletcohotohseeethxepepcotleadrizpaotliaornizoaftitohnesoofutrhcee
and QNM (e.g., along the rod axis for a nanorod). After ob-
taining the scattered field, we compute the QNM from
F (r ,ω)=F η(r ,n ;ω)+1, (3)
a a P a a
where FPη(ra)= 43πQ2nλ3B3c ωc2ωγcImhna·εBω˜˜fcc((rω˜ac)−˜fcω(r)a)·nai is the ˜fc(r;ωµ)=Z0tendES(r,t)eiωcte−(2t(−τwtoifnf))22dt, (5)
Purcell factor, F , multiplied by a factor η to account
P where ω is the real part of the eigenfrequency
foranydeviationsatr fromthefieldmaximumr ,cav- c
a 0 ω˜ , 2√ln2τ gives the FWHM (full-width at half-
itypolarization,andcavityresonantfrequency[14];here c win
maximum)ofthetimewindow,andt isthetimeoffset
n is the unit vector in the direction of the dipole emit- off
a
from the center of the source pulse to the center of the
ter, n is the background-medium refractive index in
B time windowing function. One criterionfor the selection
whichtheMNRisembeddedandλ isthecorresponding
c of time window parameters is to set ω τ2 /2t Q,
wavelength.Equation(3) assumes a single QNM expan- c win off →
while keeping τ as large as possible. To understand
sion,whichisvalidfortheMNRlight-matterinteraction win
whythisworks,considertheidealcaseforwhichtheinte-
regimes that we study below.
gralinEq.(5)iscarriedoutfrom to ,andassume
SETfhaectoFrDdTirDecttleyc,hbnuiqtuoenlcyaantathlseodoipbotaleinpothsietioenn,hwanhciechd the scattered field is given by ES−(t)∞= ∞iEie(−iωi−γi)t
canbeusedtocheckthenumericalaccuracyoftheQNM ((ωi,γi)6=(ωj,γj)ifi6=j);afterperformPingtheintegra-
tion,theamplitudeoftheithcomponentisproportional
equations. Specifically, one can define [26]
Im n GFDTD(r ,r ;ω) n to |Eie(i∆ωi−∆2γi)2τw2in|e−(τtwofifn)2 with ∆ωi = ωi−ωc and
FaFDTD(ra,ω)= Im(cid:8) {an·a·GB(ra,ara;aω)·n·a}a(cid:9), (4) w∆eγlils=epγair−atteodff,/sτow2tihn.atIf|∆thωei|rτewsionn≫anc1e,saonfdt|h∆eωsiy|s≫tem|∆aγrie|
where GB is the known homogeneous medium Green for i 6= c, then the QNM resonance, ω˜c = ωc−iγc, will
function [26], and GFDTD(r ,r ;ω) is obtained from betheonlysurvivingtermafterapplyingthewindowing
a a
FDTD at the dipole source position [22]. Note that our function. This is achieved by having toff/τw2in = γµ for
full-dipole FDTD calculations have been shown else- the QNM of interest; however,due to the factor e−τtwofifn,
where to yield good agreement with analytical results t should not be too large,otherwise the spectrum will
off
formetalspheres[27](within 5%agreementoverbroad- betooweakandinfluencedbynumericalerrors.Wehave
2
4
2
)
m
0
µ
(
−2
y
−4
−6 −3 0 3 6 −6 −3 0 3 6
x (µm) x (µm)
Fig. 1. Spatial profile of the dipole mode of a gold
nanorod dimer, showing both the scattered field and
QNM. Left: ES(x,y,0;ω ), and right: ˜f(x,y,0;ω ) at
c c
| | | |
ω =291.06 THz. The excitation source is a y-polarized
c
plane wave incident in the x direction.
foundthatagoodchoiceistosimplychooset =4π/γ , Fig. 2. Close up view of the QNM profile of the gold
off c
i.e., two times the lifespan of the QNM. dimer. Left: ˜f(x,y,0;ωc), middle: ˜fy(x,y,0;ωc), and
| | | |
Next we apply the above technique to investigate a right: ˜f (x,y,0;ω ) at ω =291.06 THz. A y-polarized
x c c
| |
golddimerstructuremadeupoftwoidenticalnanorods. dipoleatthecenterofthegapisshownbythewhitedot
We choose a rod radius ra = 15 nm with an axis and double arrow in the left panel.
length l = 100 nm (along y). We use the Drude model,
ω2
ε(ω) = 1 p , with parameters similar to gold,
− ω(ω+iγ)
with ω = 1.26 1016 rad/s (plasmon frequency), and hasadifferentsignfromthechargeatthetopofthelower
p
×
γ = 1.41 1014 rad/s (collision rate). In order to get a nanorod, which makes the localized surface plasmons of
×
larger enhancement of the SE for a quantum dipole, we theindividualnanorodscoupleeffectively.Consequently,
consider two nanorodsthat areparallelwith eachother. the fieldinsidethe cavity(gap)is significantlyenhanced
The separation gap is set to 20 nm, which helps mini- and y-polarized. The middle and right panels in Fig. 2
mizesnonradiativedecayandalsoallowssufficientspace showthey andxcomponentsoftheQNMatz =0.The
toembedaquantumemittersuchasaquantumdot(see node of the y-component sits around both ends of the
dipole arrow in Fig. 2). From FDTD analysis, we find nanorods, slightly inside this metal. Similar nodal lines
the dipole mode is around ω˜ /2π=291.06 i20.28THz appear for an electric dipole composed of charge q.
c
− ∓
(ω 1.2 eV), which is redshifted (by about 34 THz) We remark that for these calculations it is important
c
≈
withrespectto a singlenanorod[14]due to the bonding to choose a y-polarizedplane wave, which efficiently ex-
effect;thecorrespondingqualityfactorisQ 7.2,which citestheQNMofinterest.Ifweuseanx-polarizedplane
≈
is smaller than that for a single nanorod (Q 10). wave,with the parameters same as above,then a rather
single
≈
To obtain the scattered field and the QNM, a y- strange pseudo-mode is obtained. The failure of using
polarizedplanewavewithfrequencyω/2π=291.06THz x-polarized plane wave to obtain the QNM can be ex-
is employed, incident in the x direction. The simu- plained by the fact that the QNM can not be efficiently
lation domain size is 12 8 6 µm3 (Fig. 1) and excited since the dipole moment lies along the y direc-
× ×
2.4 2.4 2.4 µm3 (Fig. 2), and we use a conformal tion.Sosomecareisneededinchoosingthecorrectexci-
× ×
meshing scheme with a maximum step size 40 nm in all tationsource,thoughin practicethis is easy to do when
directions;asmallerrefinedmeshof1nmisusedaround excitingthe QNMofinterest.Wehavealsocheckedthat
themetaldimer;100layersofPMLhavebeenusedwith our mode calculation approach successfully reproduces
symmetric (antisymmetric) boundary condition in z (y) the mode profile and QNMs for dielectric cavity struc-
directionforbothsimulationandthe timestepis1.8875 tures, e.g., in Ref. 10 and also for other MNRs.
as. We use a time window with parameters t =100 fs Single photon enhancement factors. As a possible ap-
off
and τ = 23.3/√ln2 fs. The left and right panels of plication of using MNRs for single photon emitters, a
win
Fig. 1 show the scattered field and QNM, respectively. large enhancement of the SE factor is desired. This can
As anticipated, the QNM is seen to have an increasing beachievedbyhavingaquantumdipoleatthegapcenter
fieldvalueforpositionsfurtherawayfromtheresonator, ofthedimer.InFig.3,theenhancedSEfactor,F ,ofay-
y
whichiscausedbytheoutgoingboundaryconditionsand polarizeddipoleemitteratthecenterofthegapisshown
thecomplexeigenfrequency.Aclose-upviewoftheQNM by the blue (dashed) curve using a full-dipole numerical
near the resonator is shown in the left panel of Fig. 2, calculationwithno approximations,i.e., FFDTD [22,23].
y
and the correspondent mode volume is calculated to be Next we use the QNM to obtain the enhanced SE fac-
V 1.9 10−4(λ /n )3 (whichisaboutdoublethatof tor [14]. The result is shown by the orange (solid) curve
eff c B
≈ ×
the single nanorod). The oscillating charge distribution in Fig. 3; it agrees extremely well with the full-dipole
is of opposite sign at both ends of each nanorod; also, FDTDcalculations.Indeed,theagreementalsoconfirms
theoscillatingchargeatthebottomoftheuppernanorod that the entire response is dominated by a single QNM
3
1600 References
1. R. K. Chang and A. J. Campillo, Optical Proesses in
Microavities, (World Scientific,1996).
2. S.A.Maier,Plasmonics:FundamentalsandAppliations
y 800
F (Springer, 2007).
3. K. A. Willets and R. P. Van Duyne, Ann. Rev. Phys.
Chem. 58, 267 (2007).
4. K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I.
0
0.8 1.2 1.6 Itzkan, R. R. Dasari, and M. S. Feld, Phys. Rev. Lett.
ω 78, 1667 (1997).
(eV)
5. R.Zhang,Y.Zhang,Z.-C.Dong,S.Jiang,C.Zhang,L.
G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Luo, J.-L.
Fig. 3. Enhanced SE factor for the gold nanorod dimer Yang, and J.-G. Hou,Nature 498, 82 (2013).
structure, with the blue (dashed) curve given by direct 6. P. Bharadwaj, B. Deutsch, and L. Novotny, Adv. Opt.
FDTDcalculation(Eq.(4))andtheorange(solid)curve Photon. 1, 438 (2009).
given by the mode expansion technique (Eq. (3)). The 7. C. Belacel, B. Habert,F. Bigourdan, F. Marquier,J.-P.
emissiondipole is y-polarizedat the center of the gapof Hugonin, S. M. de Vasconcellos, X. Lafosse, L. Coolen,
the dimer (see left panel of Fig. 2.) C. Schwob, C. Javaux, B. Dubertret, J.-J. Greffet, P.
Senellart, and A. Maitre, NanoLett. 13, 1516 (2013).
8. P.T.Leung,S.Y.Liu,andK.Young,Phys.Rev.A49,
3982 (1994).
9. K. M. Lee, P. T. Leung, and K. M. Pang, J. Opt. Soc.
which is highly desired for single photon source appli- Am. B 16, 1409 (1999).
cations. From our results, we obtain impressive peak 10. P. T. Kristensen, C. Van Vlack, and S. Hughes, Opt.
Lett. 37, 1649 (2012).
emissionfactors of around1500.Note that althoughthe
11. E. M. Purcell, Phys.Rev.69, 674 (1946).
Q/V factor of the dimer structure is about three times
12. P.T. Kristensen andS. Hughes,ACSPhotonics 1, 2-10
smaller than the single nanorod, the emission factor of
(2014).
the dimer is around three times larger than for a single
13. C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P.
nanorodatanequivalentdipole position.Thusthe Q/V
Lalanne, Phys.Rev.Lett. 110, 237401 (2013).
is clearly not the main metric to describe the enhanced
14. R.-C. Ge, P. T. Kristensen, Jeff. F. Young, S. Hughes,
emission factors for dipole away from the field antin-
e-print:arXiv:1312.2939 [quant-ph]
ode; note the reasonfor not choosing the field antinode,
15. A. Taflove and S. C. Hagness, Computational Electro-
is that the emission would be severely quenched from dynamics, Artech House, 2005.
Ohmic heating and a single mode Purcell factor ceases 16. e.g., see K. Stannigel, M. K¨onig, J Niegemann, and K.
to have any meaning [14]. In this regard, the other im- Busch, Optics Express 17, 14934 (2009).
portantfigure-of-meritforemitting singlephotonsisthe 17. KevinC.Y.Huang,YoungChulJun,Min-KyoSeo,and
output-couplingβ factor(probabilityofphotonemission Mark L. Brongersma, Opt.Express 19, 19084 (2011).
via radiative decay in the far field) of the dipole emit- 18. Q. Bai, M. Perrin, C. Sauvan, J-P Hugonin, and P.
ter, which is computed here to be β 58%; this again Lalanne, Opt.Express 22 , 27371 (2013).
≈
is better than the single gold nanorod, by about 10%; 19. J. R. deLasson, J. Mørk, and P.T. Kristensen, J. Opt.
with respect to single nanorod, the electric field of the Soc. Am. B 30, 19967 (2013).
20. A. Dhawan, S. J. Norton, M. D. Gerhold, and T. Vo-
dimer is repelled outside the lossy nanorods (especially
Dinh,Opt. Express17, 9688 (2009).
aroundthehotspots)duetooppositechargedistribution
21. C.VanVlackandS.Hughes,Opt.Lett.37,2880(2012).
between them. This leads an overall nonradiative decay
22. P. Yao, V.S.C. Manga Rao, and S. Hughes, Laser and
reduction ( Im[ǫ(r,ω)]Etot(r,ω)2dV), even though
∝ | | Photonics Reviews 4, 499 (2010).
the volume ofRthe integral is doubled. 23. V.S.C.MangaRaoandS.Hughes,Phys.Rev.Lett.99,
In summary, we have described how the commonly
193901 (2007).
employedFDTDmethodcanbeusedtoefficientlyobtain
24. We useFDTD Solutions: www.lumerical.com.
the QNMsformetallic resonatorsbyapplyingafiltering 25. G.SunandJ.B.Khurgin,Appl.Phys.Lett.98,113116
function to the scattered field. These calculations are (2011).
verified by comparing the resulting SE emission factor 26. L. Novotny and B. Hecht, Principles of Nano Optics,
with full-dipole FDTD numericalcalculations,where we Cambridge UniversityPress, 2006.
find excellent agreement. Using this technique, we have 27. C. Van Vlack, P. T. Kristensen, and S. Hughes, Phys.
proposed a metal nanorod dimer structure that acts an Rev.B 85, 075303 (2012).
efficient single photon source, yielding a large Purcell
factor of 1500 and an output β factor of around 60%.
This workwassupportedby the NaturalSciences and
EngineeringResearchCouncilofCanada.We thank Jeff
Young and Philip Kristensen for useful discussions.
4