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Micropillar single-photon source design for simultaneous
near-unity efficiency andindistinguishability
Wang, Bi-Ying; Denning, Emil Vosmar; Gür, Uur Meriç; Lu,
Chao-Yang; Gregersen, Niels
Published in:Physical Review B
Link to article, DOI:10.1103/PhysRevB.102.125301
Publication date:2020
Document VersionPeer reviewed version
Link back to DTU Orbit
Citation (APA):Wang, B-Y., Denning, E. V., Gür, U. M., Lu, C-Y.,
& Gregersen, N. (2020). Micropillar single-photon sourcedesign
for simultaneous near-unity efficiency and indistinguishability.
Physical Review B, 102(12), [125301
].https://doi.org/10.1103/PhysRevB.102.125301
https://doi.org/10.1103/PhysRevB.102.125301https://orbit.dtu.dk/en/publications/41bcdad6-1531-4e99-9309-494847455feahttps://doi.org/10.1103/PhysRevB.102.125301
-
Micropillar single-photon source design for simultaneous
near-unity efficiency andindistinguishability
Bi-Ying Wang,1, 2 Emil V. Denning,2 Uğur Meriç Gür,3
Chao-Yang Lu,1 and Niels Gregersen2, ∗
1Hefei National Laboratory for Physical Sciences at
Microscale,University of Science and Technology of China, Hefei,
Anhui, 230026, China
2DTU Fotonik, Department of Photonics Engineering,Technical
University of Denmark, Ørsteds Plads,
Building 343, DK-2800 Kongens Lyngby, Denmark3DTU Elektro,
Department of Electrical Engineering,
Technical University of Denmark, Ørsteds Plads,Building 348,
DK-2800 Kongens Lyngby, Denmark
(Dated: September 16, 2020)
We present a numerical investigation of the performance of the
micropillar cavity single-photonsource in terms of collection
efficiency and indistinguishability of the emitted photons in the
presenceof non-Markovian phonon-induced decoherence. We analyze the
physics governing the efficiency us-ing a single-mode model, and we
optimize efficiency ε and the indistinguishability η on an
equalfooting by computing εη as function of the micropillar design
parameters. We show that εη islimited to ∼ 0.96 for the ideal
geometry due to an inherent trade-off between efficiency and
indis-tinguishability. Finally, we subsequently consider the
influence of realistic fabrication imperfectionsand Markovian pure
dephasing noise on the performance.
I. INTRODUCTION
Within the fields of optical quantum computation1
andcommunication2, a key component is the single-photonsource3,4
(SPS) capable of emitting single indistinguish-able photons on
demand. The figures of merit include theefficiency ε defined as the
number of detected photons inthe collection optics per trigger as
well as the indistin-guishability η of consecutively emitted
photons, as mea-sured in a Hong-Ou-Mandel experiment, and their
prod-uct εη governs the success probability of the
multi-photoninterference experiment5,6. The spontaneous
parametricdown-conversion process7 has for a long time been
theworkhorse for single-photon generation within the quan-tum
optics community. While simple to implement ex-perimentally, its
main drawback is its probabilistic naturepreventing multi-photon
interference experiments withmore than a handful of photons.
As alternative, the semiconductor quantum dot (QD)embedded in a
host material8 has emerged as a promis-ing platform for scalable
optical quantum informationprocessing. As a two-level system, the
QD allows fordeterministic emission of single photons and also
entan-gled photon pairs9 through the spontaneous emission pro-cess.
However, for QDs in a bulk material the high in-dex contrast at the
semiconductor-air interface reducesthe collection efficiency to a
few percent. The collec-tion can be improved by placing the QD in a
structuredenvironment3,4 directing the light towards the
collectionoptics. Design schemes allowing for vertical emissionwith
high efficiency include the photonic nanowire10–13
design with ε of ∼ 0.7 and the ”bullseye”9,14–17 planarcircular
Bragg grating design with ε of ∼ 0.8516. Thesedesigns feature
broadband (> 50 nm) spectral operationwhich allows for easy
spectral alignment of the QD andthe cavity lines. However, a
drawback of the broadband
approach is the presence of the phonon sideband in thephoton
emission spectrum arising due to interaction withthe solid-state
environment even at zero temperature.These photons are
distinguishable and constitute around10 % of the total emission,
and while they can be filteredout, this occurs at the price of
reduced efficiency. An-other drawback is the proximity of the QD to
the fluc-tuating charge environment18 of semiconductor-air
sur-faces, which again compromises the indistinguishability.
The most successful SPS design so far is the micro-cavity
pillar17,19–21 featuring a QD in a vertical λ cav-ity sandwiched
between two distributed Bragg reflec-tors (DBRs). Almost 20 years
ago, Barnes et al. pro-posed a model22 to quantify the efficiency
in terms ofthe Purcell factor, from which one finds that the
effi-ciency can be improved either by enhancing the cavityQ factor
or by reducing the mode volume. A practicalchallenge for the
QD-based SPS is the random spatial andspectral distribution of
typical Stranski-Krastanov-grownQDs, and important technological
developments were thedemonstrations of in-situ lithography23 and QD
imag-ing techniques15 allowing for deterministic QD positioncontrol
as well as Stark tuning21 to ensure the spectralalignment. The
micropillar SPS then relies on narrow-band Purcell enhancement to
ensure efficient emissioninto the cavity mode, where the narrow
bandwidth (∼ 0.2nm) enables a funneling of photons into the
zero-phononline (ZPL) and suppression of the phonon sideband
al-lowing for high indistinguishability without any filter-ing.
Additionally, the QD is less subject to chargenoise due to a larger
separation between the QD andthe surfaces. These assets have led to
the simultaneousdemonstrations20,21 of pure (g2(0) < 0.01)
single photonemission from micropillars combining near-unity (>
0.98)indistinguishability with extraction efficiency of ∼
0.65.While, for circular micropillars half of the emitted light
-
2
is lost in standard resonant excitation
cross-polarizationsetups20,21 where polarization filtering is used
to sup-press the laser background, this loss can be avoided us-ing
more advanced techniques like spatially orthogonalexcitation and
collection24 or two-colour resonant25 orphonon-assisted26
excitation.
The question now arises of how the micropillar SPSextraction
efficiency can be improved while maintain-ing high
indistinguishability. Whereas charge noisecan be effectively
suppressed using metal contacts21,27,phonon-induced decoherence is
not avoided even at zerotemperature28 and must be taken into
account in theoptimization. The efficiency can be improved by
increas-ing the mirror reflectivities leading to a larger Q
factorand in turn an increased Purcell factor. However, evenif one
disregards unavoidable fabrication imperfectionslimiting the
maximum achievable Q, a fundamental ob-stacle is encountered
arising from the interaction withthe phonon bath: While the cavity
improves the indis-tinguishability in the weak coupling regime,
increasingthe Purcell factor to improve the efficiency will
even-tually bring the QD-cavity system into the strong cou-pling
regime, which is detrimental to the indistinguisha-bility.
Specifically, in the strong coupling system, the sys-tem undergoes
vacuum Rabi oscillations29, which causesthe emission spectrum to
split into two hybrid polari-ton states. However, in the presence
of phonon coupling,the phonon-induced transitions between the upper
po-lariton state and the lower polariton state strengthenthe
decoherence of the system, and hence decrease
theindistinguishability30. There is thus an inherent trade-off31,32
between the achievable efficiency and indistin-guishability in the
presence of phonon-induced decoher-ence. Consequently, an
optimization of the micropillarSPS design requires accurate
modeling of the efficiencyas well as the indistinguishability
subject to phonon-induced decoherence on an equal footing, and such
astudy, to our knowledge, is still absent in the literature.
In this work, we perform an investigation of the achiev-able
performance of the micropillar geometry in termsof efficiency and
indistinguishability. Our theoreticalmethod is based on two steps:
First, we use a Fouriermodal method33,34 to rigorously calculate
the efficiencyallowing for direct insight into governing physics
de-scribed here using a single-mode model. Subsequently,we use the
calculated optical properties of the cav-ity as parameter inputs to
a non-Markovian masterequation, which is used for evaluation of the
photonindistinguishability31,32. This master equation accountsfor
non-Markovian phonon scattering as well as dephas-ing from charge
noise. The influence of the sidewallroughness is also included as
an additional phenomeno-logical scattering rate. For realistic
parameters of de-phasing and sidewall roughness, we provide design
spec-ifications for an optimized micropillar SPS with perfor-mance
significantly improved compared to state-of-the-art.
This article is organized as follows: We introduce the
FIG. 1. Illustration of the micropillar SPS design consistingof
a λ cavity sandwiched between two DBRs. The QD, rep-resented by the
red dot, is placed at the center of the cavity.The coupling
parameters β and γ of the single-mode model,cf. Section III A, are
schematically shown.
micropillar geometry and the design recipe in Section II.The
details of the optical and microscopic modeling areprovided in
Section III. We present our analysis of themicropillar performance
and our results in Section IV,which we discuss further in Section V
followed by a con-clusion. The Appendix contains additional details
of themodeling.
II. MICROPILLAR GEOMETRY
In this work, we investigate the micropillar cavity il-lustrated
in Fig. 1 consisting of a QD placed in the centerr0 of a λ cavity.
Light is confined vertically by the DBRsand laterally by total
internal reflection. We consider anasymmetric structure with larger
number nbot of layerpairs in the bottom DBR than the number ntop in
thetop DBR to suppress leakage of light into the substrate.In the
following, we fix the number of layer pairs nbot =40 and study the
SPS performance as function of pillardiameter d and of ntop. We
choose GaAs for the cavitymaterial and GaAs/Al0.85Ga0.15As for the
DBR pairs asthis choice avoids the oxidation of pure AlAs. For
ourdesign wavelength of λ0 = 895 nm, we use a refractiveindex for
GaAs of 3.5015 (T = 4 K) and an index forAl0.85Ga0.15As of
2.9982
35. For each pillar diameter con-sidered, the thickness of the
layers is chosen36 such thatthe resonance matches the design
wavelength λ0. Thusthe cavity thickness is given by λ0/neff ,
whereas the DBRlayer thicknesses are chosen as λ0/(4neff), where
neff isthe diameter-dependent effective index of the fundamen-tal
mode of the section. For pillar diameters above ∼ 2.5µm, the layer
thicknesses are very close to those of the1D geometry and are given
by 64 nm, 75 nm and 256 nm,respectively, for the GaAs and
Al0.85Ga0.15As layers inthe DBR and the GaAs cavity.
-
3
III. NUMERICAL MODEL
Our numerical model consists of an optical simulationperformed
using a Fourier modal method to determinethe efficiency combined
with an open quantum systemsmaster equation approach to evaluate
the indistinguisha-bility of the emitted photons in the presence of
realisticdecoherence effects.
A. Optical modeling
We model the QD as a classical point dipole pδ(r−r0)with
in-plane orientation and harmonic time dependencewith frequency ω
in resonance with the cavity mode. Todetermine the efficiency, we
compute the total emittedpower P = ω2 Im [p
∗ ·E(r0)] as well as the power coupledto the lens PLens, where
we take into account an overlapwith a Gaussian mode to model the
coupling to a single-mode fiber, as discussed in Appendix B. The
collectionefficiency ε is then determined as
ε = ICPLensP
, (1)
where IC is a correction factor (cf. Section III C) takinginto
account the presence of fabrication imperfections.
While the collection efficiency is rigorously describedby Eq.
(1), this expression does not provide detailedunderstanding of the
governing physics. For this reason,we also introduce a single-mode
model4,22 to determinethe single-mode efficiency εs. The model
describes theefficiency in terms of the spontaneous emission β
factorgiven by the power PC coupled to the cavity mode overthe
total emitted power P = PC + PR, where PR is thepower coupled to
all other modes. In the weak couplingregime, the β factor can be
written in terms of the Purcellfactor Fp as
β =PC
PC + PR=
FpFp + PR/P0
, (2)
where P0 is the power emission for a bulk medium. FromEq. (2) it
is clear that β can be improved by increas-ing the Purcell factor,
which can be achieved by enhanc-ing the cavity quality factor Q or
by reducing the cavitymode volume. The second parameter of the
model is thetransmission γ = PLens,C/PC, where PLens,C is the
powercoupled to the lens from the cavity mode alone again tak-ing
into account an overlap with a Gaussian profile. Oursingle mode
model now describes the efficiency simply asthe product εs = βγ as
illustrated in Fig. 1.
The calculations of β, γ and ε are performed usingan
open-geometry Fourier modal method33,34. Based onan eigenmode
expansion of the optical field, the methodprovides direct access to
the cavity mode contribution tothe electric field strength EC(r)
allowing for immediateevaluation of PC and PLens,C, see Appendix A
for furtherdetails. Finally, the spontaneous emission rate into
the
cavity ΓC is computed from the classical power PC us-ing the
relationship37 ΓC/Γ0 = PC/P0, where Γ0 is thespontaneous emission
rate in a bulk medium.
B. Microscopic modeling
We describe the QD in a solid-state environment as atwo-level
system, with a ground state, |0〉, and an excitedstate, |e〉, coupled
to both photon and phonon environ-ments. The cavity mode is
described by a bosonic anni-hilation operator, a, coupled to the QD
with a rate, g.The Hamiltonian describing the dynamics of the
emitterand cavity is thus given by
H0 = ~ωe |e〉〈e|+ ~ωca†a+ ~g(a†σ + aσ†), (3)
where ωe and ωc are the QD transition frequency and thecavity
resonance, respectively, and σ = |0〉〈e| is the dipoleoperator of
the QD. The cavity loss due to coupling withelectromagnetic modes
outside of the cavity is describedby the Hamiltonian
Hc,EM = ~∑l
(ωA,lA
†lAl + hA,la
†Al + H.c.), (4)
where Al is the bosonic annihilation operator of the
lthenvironmental mode in the cavity loss channel, with fre-quency
ωA,l and coupling strength hA,l to the cavitymode. Using the cavity
escape rate κc = ωc/Q describingthe Markovian cavity decay into all
Al modes, we com-pute the QD-cavity coupling strength as g = 12
√ΓCκc.
Similarly, the coupling of the QD to optical radiationmodes is
described by
He,EM = ~∑l
(ωB,lB
†lBl + hB,lσ
†Bl + H.c.), (5)
where Bl is the bosonic annihilation operator of the
lthenvironmental mode in the radiation reservoir of the QD,with
frequency ωB,l and coupling strength to the emitterhB,l. The
emitter–phonon coupling is described by
HP = ~∑k
[νkb†kbk + gk |e〉〈e| (bk + b
†k)], (6)
where bk is the annihilation operator for the phononmode with
momentum k, frequency νk and couplingstrength to the emitter
gk.
The total Hamiltonian is thus given by
H = H0 +Hc,EM +He,EM +HP. (7)
1. Indistinguishability
The indistinguishability of the photon is calculatedthrough the
two-colour spectrum as31
I = P−2S
∫ ∞−∞
dω
∫ ∞−∞
dν|S(ω, ν)|2. (8)
-
4
This two-colour spectrum is defined as
S(ω, ν) =
∫ ∞−∞
dt
∫ ∞−∞
dt′〈E†(t′)E(t))
〉ei(ωt−νt
′), (9)
where E is the operator corresponding to the detectedelectric
field. The frequency-integrated spectral power,PS, is defined
as
PS =
∫ ∞−∞
dωS(ω, ω). (10)
To calculate the two-colour spectrum, we use a polaronmaster
equation, which is described in the following.
2. Master equation
Following previous work31,32,38,39, we make use of
theBorn-Markov master equation in the polaron frame. As-suming that
the emitter and cavity are resonant (ωe =ωc), the polaron master
equation for the reduced densityoperator of the cavity–emitter
system, ρ, is (in a refer-ence frame rotating with the resonance
frequency, ωe)
∂tρ = −i[grX̂, ρ] + κcD(a) + ΓD(σ)+ 2γdD(σ
†σ) +Kph(ρ).(11)
Here, gr = gB is a renormalised emitter–cavity cou-pling rate.
The renormalisation factor is given by
B = 〈B±〉0 and B± = exp[±∑k g
2k/νk(b
†k + bk)
], where
〈x〉0 = Z−1 Tr[xe−
∑k ~νk/(kBT )b
†kbk]
denotes the expec-
tation value with respect to the thermal phonon state,where T is
the temperature, kB is the Boltzmann con-
stant and Z = Tr[e−
∑k ~νk/(kBT )b
†kbk]
is the partition
function. The symbol D[x] = xρx†−{x†x, ρ}/2 denotes theLindblad
dissipator; Γ represents the Markovian emissionrate into the
radiation modes, Bl, and γd is the pure de-phasing rate. The phonon
dissipator is written as31
Kph =−g2(
[X̂, X̂ρ]χX+[Ŷ , Ŷ ρ]χY+[Ŷ , Ẑρ]χZ+H.c.), (12)
with X̂ = σ†a+ σa†, Ŷ = i(σ†a− σa†) and Ẑ = σ†σ − a†a.The
coefficients are
χX =
∫ ∞0
dτΛXX(τ),
χY =
∫ ∞0
dτcos(2grτ)ΛY Y (τ),
χZ = −∫ ∞
0
dτsin(2grτ)ΛY Y (τ),
(13)
with the phonon correlation functions ΛXX(τ) =〈BX(τ)BX〉 =
B2[sinhϕ(τ)−1] and ΛY Y (τ) = 〈BY (τ)BY 〉 =B2 coshϕ(τ) with BX =
(B+ + B− − 2B)/2 and BY =i(B+ −B−). Here, ϕ(τ) is given by38
ϕ(τ) =∫∞
0dω J(ω)ω2
[coth
(~ω
2kBT
)cos(ωτ)− i sin(ωτ)
], (14)
where the phonon spectral density is J(ν) =∑k |gk|2δ(ν − νk) =
αν3 exp
(−ν2/ν2c
)with α being
exciton-phonon coupling strength and νc =√
2vc/L de-noting the cutoff frequency40, where vc and L are
thespeed of sound and the size of the QD, respectively. Here,the
exciton-phonon coupling strength α depends on theQD material and on
the cutoff frequency νc, which itselfis related to the size of the
QD41. The Franck-Condonfactor, B2, can be written compactly in
terms of thisfunction as B2 = e−ϕ(0).
The master equation, Eq. (11), allows us to calculatethe
two-colour spectrum, S(ω, ν). This is done by firstcalculating the
dipole spectrum,
S0(ω, ν) =
∫ ∞0
dt
∫ ∞0
dτei(ω−ν)te−iωτ
×GP(τ)〈σ†(t+ τ)σ(t)
〉,
(15)
where GP(τ) = B2eϕ(τ) is the free phonon Green’s
function and the two-time dipole correlation function,〈σ†(t+
τ)σ(t)
〉, is evaluated using the quantum regres-
sion theorem with Eq. (11). From this dipole spectrum,the
detected field spectrum is calculated as31
S(ω, ν) = GF(ω, ν)S0(ω, ν) +G∗F(ν, ω)S
∗0 (ν, ω), (16)
where
GF(ω, ν) =(κc/2)
2
(iω − κc/2)(−iν − κc/2)(17)
is an optical two-colour Green’s function connecting thedipole
with the field emitted from the cavity. In thePurcell-regime, where
g � κc, the indistingusihabilitycan be approximated by the
expression
I ' ΓtΓt + 2γt
[B2
B2 + F (1−B2)
]2, (18)
where Γt = 4g2/κc + Γ is the total Purcell-enhanced
spontaneous emission rate of the QD, γt = γd +2π(gr/κc)
2J(2gr) coth(~gr/kBT ) is a phonon-enhancedpure dephasing rate
and F is the fraction of photons inthe phonon sideband not
eliminated by the cavity31.
Through narrowband Purcell enhancement, the mi-cropillar SPS
ensures an efficient emission into the cavitymode, where the narrow
bandwidth (∼ 0.2 nm) enablesa funneling of photons into the ZPL and
a suppressionof the phonon sideband allowing for high
indistinguisha-bility without any filtering. To illustrate this
influence,we show the spectra S(ω, ω) for a low Q and a highQ
micropillars in Fig. 2 both exhibiting a narrow peakand a broadband
base corresponding to the ZPL and thephonon sideband, respectively.
We observe that the highQ micropillar leads to a broadened ZPL as a
result ofthe large Purcell factor and a suppression of the
phononsideband.
-
5
10-7
10-4
10-1
102
105
-5 -2.5 0 2.5 5
S(ω
,ω)
ω-ωe (meV)
Q=16000
Q=400
FIG. 2. Emission spectrum showing the zero-phonon line andthe
phonon sidebands at T = 4 K. Dashed curve: Q = 400and Fp = 2; solid
curve: Fp = 20 and Q = 16000. Parameters:α = 0.03 ps−2 and ~νc =
1.4 meV following Ref. 31.
C. Modeling of fabrication imperfections
While the figures of merit of the micropillar SPS canbe computed
accurately using the formalisms presentedabove, the influence of
unavoidable fabrication imperfec-tions leading to discrepancy42,43
between the predictionof the modeling and the experimentally
measured perfor-mance should be taken into account. The
imperfectionscan be described phenomenologically by introducing
ad-ditional loss channels due to sidewall roughness and ma-terial
absorption44. The quality Q factor is then writtenas19,45
1
Q=
1
Qint+
1
Qedge+
1
Qabs, (19)
where the intrinsic Qint is computed for the perfectstructure43,
and Q−1edge and Q
−1abs represent the contribu-
tions to loss from sidewall scattering and material ab-sorption,
respectively. Since Fp is linearly proportionalto Q, the
introduction of imperfections leads to a cor-rected Purcell factor
given by F cp = Fp(Q/Qint) and inturn a corrected β factor βc = F
cp/(F
cp+PR/P0). We then
model the correction IC to the efficiency in the presenceof
imperfections as IC = β
c/β. Moreover, the correctedindistinguishability is obtained by
evaluating the decayrate κc = ω/Q in Eq. (11) using the Q given by
Eq. (19).
We model the sidewall scattering using a phenomeno-logical
model44 discussed in Appendix D, where κ is aphenomenological
constant proportional to the sidewallscattering rate Q−1edge. The
effect of material absorption
loss can be modeled either using Q−1abs in Eq. (19) orby
introducing an imaginary part43,46 to the refractiveindex. However,
the effect of material loss is typicallynegligible43,46 in
micropillars with Q factors below ∼ 105and is thus ignored in this
work.
IV. RESULTS
In this section, we first present calculations of thesingle-mode
model parameters to elucidate the physicsgoverning the collection
efficiency, after which we presentresults from full simulations of
the efficiency and indis-tinguishability taking into account
fabrication imperfec-tions. All calculations are performed at a
numerical aper-ture (NA) of 0.82 for the first lens. Following Ref.
31,we use the parameters ~νc = 1.4 meV and α = 0.03
ps2corresponding to a typical QD of size 25 nm41.
A. Analysis using the single-mode model
We first present the cavity Q factor as function of thepillar
diameter d and of the number of layer pairs ntopin the top DBR in
Fig. 3(a) for the ideal structure. Anincrease in ntop leads to a
higher reflectivity resultingin an increased cavity Q factor. For
large diameters,the Q factor approaches the value of a planar
cavitywhereas for short diameters it drops due to poor modaloverlap
of the HE11 mode in the cavity section and thecorresponding Bloch
mode in the DBR sections. Oscil-lations are observed for the
largest values of ntop arisingfrom coupling36,43,47 with
higher-order Bloch modes ofthe DBRs. This behavior is also observed
for the Purcellfactor Fp illustrated in Fig. 3(b), where the
reductionof the pillar diameter d and in turn the mode
volumeadditionally leads to an overall increase in Fp until thedrop
of the Q at around d ∼ 1.2 µm sets in. As ex-pected, the increase
in the Purcell factor Fp is directlyreflected in the calculated
spontaneous emission β factorpresented in Fig. 3(c). However, the
observed oscilla-
103
104
105
1 2 3 4
(a)
Q
d (µm)
ntop=172125
0
180
360
540
1 2 3 4
(b)
Fp
d (µm)
ntop=172125
0.76
0.84
0.92
1
1 2 3 4
(c)
β
d (µm)
ntop=172125
0
0.5
1
1 2 3 4
(d)
γ
d (µm)
ntop=172125
FIG. 3. Computed cavity Q factor (a), Purcell factor Fp (b)and
spontaneous emission β factor (c) as well as the trans-mission γ
(d) to the lens as a function of pillar diameter dfor different
numbers of layer pairs ntop in the top DBR. Theideal structure with
γd = κ = 0 is considered.
-
6
tions in β are not originating from variations in Q butrather
from diameter-dependent variations in the emis-sion PR to
background modes, and they are reduced inmagnitude with increasing
Fp in agreement with Eq. (2).While the best coupling β to the
cavity is obtained forthe diameter of ∼ 1.1 µm corresponding to the
maximumPurcell factor in Fig. 3(b), the transmission γ presentedin
Fig. 3(d) for this d is modest due to its highly diver-gent output
beam profile48. As the diameter increases,the divergence is reduced
and γ is improved. Also, whereβ is maximum for ntop = 25, an
increasing top mirrorreflectivity leads to an increasing leakage of
light intothe substrate and thus a reduction of γ for ntop = 25
asshown in Appendix E.
We thus observe that there exists a trade-off betweenthe
coupling β to the cavity and the transmission γ to thelens. Since
the efficiency is the product of the two, onemay ask if the overall
performance can be optimized bychoosing a large diameter and by
increasing the reflectiv-ity of both DBRs while keeping nbot
comfortably largerthan ntop to avoid leakage of light into the
substrate. Inthe following, we will see that this is not the
case.
B. Analysis using the full model
Having established our understanding of the variationsof β and
γ, we now present the collection efficiency com-puted using the
full model Eq. (1). As shown in Ap-pendix C, excellent agreement
between the single-modemodel and the full model is observed, and
the variationsin the efficiency can thus be fully understood from
thebehavior of β and γ. The collection efficiency ε = βγis
presented in Fig. 4(a) for the ideal structure withoutfabrication
imperfections. The reduction in ε for low di-ameters due to the
drop in γ is clearly observed, and weobserve that an efficiency of
∼ 0.95 is obtained in a broadregime for diameters above ∼ 2 µm and
ntop ∼ 21.
The computed indistinguishability η in the presenceof
phonon-induced decoherence for the otherwise idealstructure is
shown in Fig. 4(b). The computed indistin-guishability initially
increases from 17 to 21 layer pairsas the influence of dephasing is
overcome by the in-crease in Fp in agreement with Eq. (18).
However, forntop ≥ 21 the high Q factor results in significant
dropsin η for decreasing pillar diameter d due to the onsetof the
strong coupling regime (4g & κc) detrimental tothe
indistinguishability31. This drop becomes more pro-nounced as ntop
increases and the decay rate κc decreases.
To answer our previous question, we realize that theincrease of
bottom mirror nbot layer pairs to increase thetransmission γ for
ntop ≥ 25 may improve the efficiencybut will simultaneously bring
us even further into theregime of strong coupling detrimental to η.
On the otherhand, a good performance with η > 0.995 is obtained
ina regime at the boundary of strong coupling, in the caseof ntop =
21 for d larger than ∼ 2 µm. Overall, η is in-creased towards unity
by decreasing κc and by increasing
0.1
0.4
0.7
1(a)
ε ntop=17
21
25
0.93
0.96
0.99
1 2 3 4
(b)
ηd (µm)
ntop=17
21
23
25
FIG. 4. Computed efficiency ε (a) and indistinguishability η(b)
versus pillar diameter d for different layer pairs ntop inthe top
DBR for the ideal geometry with γd = κ = 0.
the diameter, which should be chosen sufficiently large toensure
weak coupling. However, as shown in Figs. 3(b,c)the increase in d
ultimately leads to a reduction in effi-ciency. An optimization of
the micropillar performancein terms of ε and η is thus necessary to
identify the op-timal performance.
C. Optimization
To optimize the efficiency and indistinguishability onan equal
footing, we consider as figure of merit the prod-uct εη presented
for the ideal case without charge noiseand sidewall roughness in
Fig. 5(a). Here, by directlyreading the maximum of εη in the
numerical simula-tion, the best performance of εη ∼ 0.95 is
obtained forntop = 21 at various peak d positions in the interval
[2,4]µm, where the observed variations are direct reflectionsof
those for the β factor in 3(b). In this regime, we haveε ∼ 0.95 and
η ∼ 0.997, in good agreement with theperformance optimum identified
in Ref. 31.
We now introduce imperfections initially in the formof a finite
dephasing rate ~γd = 0.086 µeV49 due tonon-phononic Markovian
decoherence mechanisms, e.g.charge noise due to an unstable charge
environment18.Compared to the ideal case, the indistinguishability
isreduced by the additional charge noise decoherence mech-
-
7
0.6
0.8
1(a)
ideal
εη
ntop=17
21
25
0.6
0.8
1(b)
CN
εη
0.4
0.6
0.8
1
1.6 2.4 3.2 4
(c)
SR+CN
εη
d (µm)
0.9
0.95
1
1 2 3 4
ntop=21
η
d (µm)
CNSR+CN
0.96
0.98
1
1 2 3 4
ntop=21
η
d (µm)
idealCN
FIG. 5. The computed figure of merit εη as function of
pillardiameter d for different layer pairs ntop in the top DBR
for(a) the ideal case, (b) in the presence of charge noise (CN)only
and (c) with both charge noise and sidewall roughness(SR) present.
The insets compare the indistinguishability inthe three cases for
ntop = 21. Parameters: ~γd = 0.086 µeV;κ = 0.2 µm.
anism. As shown in the inset of Fig. 5(b), the reductionin η
increases with d due to the corresponding reductionin the Purcell
factor in agreement with Eq. (18). Theresulting figure of merit εη
is shown in Fig. 5(b), whereεη is reduced to ∼ 0.94 for ntop = 21
and d ∈ [2,3.25]µm, where ε ∼ 0.95 and η ∼ 0.99.
As a second element of imperfection, we now also takesidewall
roughness into account by introducing a rela-tively large
phenomenological constant κ = 0.2 µm19,and the results are plotted
in Fig. 5(c). We observe thata maximum εη of∼ 0.93 is obtained for
ntop = 21 and d ∈
[3.05,3.9] µm, where ε ∼ 0.945 and η ∼ 0.98. Whereasthe charge
noise only influenced the indistinguishability,the sidewall
imperfections are detrimental to both ε andη through the reductions
in IC and Fp due to the de-crease in Q. The reduction of η is shown
in the insetof Fig. 5(c), where the impact of the increase in
sidewallscattering Q−1edge with decreasing diameter is directly
ob-served.
While charge noise can be overcome by increasing Fp,this
ultimately will lower η due to the onset of strongcoupling.
Similarly, while the effect of sidewall imperfec-tions is reduced
for increasing pillar diameter, this alsoleads to a decrease in Fp
and in turn the efficiency. Thedetrimental effects of charge noise
and sidewall imperfec-tions thus cannot be completely overcome in
the designprocess alone, and they should be addressed in the
fab-rication, e.g. by introducing metal contacts21 for
chargestabilization and using a reduced Al content in the DBRsto
avoid oxidation.
V. DISCUSSION
While our simulations identify geometrical parametersleading to
predicted performance significantly beyondstate-of-the-art, the
exact performance obtained exper-imentally will depend on unknown
experimental param-eters. Even though the exciton-phonon coupling
strengthα and the cutoff frequency νc are QD-dependent, previ-ous
investigations have revealed that the optimum cavitylinewidth for
maximum indistinguishability is indepen-dent of the QD size in the
10-40 nm range41. Futhermore,the magnitude of the Markovian noise
term as well asthe unavoidable fabrication imperfections influence
theperformance. Nevertheless, our study reveals that thediameter
range from 2.5 to 4 µm for the ntop = 21 con-figuration leads to
the optimal micropillar SPS perfor-mance, and in our study, this
conclusion is again notaltered by the presence of realistic pure
dephasing orfabrication imperfections, which instead simply lead
toan overall degradation of the performance by a few per-cent. In
the ideal performance regime, εη in Fig. 5 dis-play variations of ∼
5 % as function of d. Interestingly,these variations are not caused
by the well-establisheddiameter-dependent variations43,47 in Q due
to couplingwith higher-order Bloch modes, and an adiabatic
cavitydesign50 will thus not eliminate the variations. Rather,Figs.
3(c,d) show that the εη variations appear due tostrong variations
with d of the background emission ratePR/P0, which is often simply
assumed equal to unity. Abetter physical understanding of the
background emis-sion rate in micropillars is thus desirable but
beyond thiswork.
However, even with such understanding, the micropil-lar geometry
does not provide a mechanism to completelysuppress the background
emission. The consequence isthat the trade-off between efficiency
and indistinguisha-bility is unavoidable, and a figure of merit εη
for the mi-
-
8
cropillar above ∼ 0.96 is out of reach. While the micropil-lar
geometry presently remains the champion device forefficient
generation of single indistinguishable photons,this limitation
motivates further investigation of alter-native design strategies
such as the photonic nanowireSPS51,52, where the background
emission can be con-trolled using dielectric screening
effects53.
Whereas we predict an εη ∼ 0.96, experimentallydemonstrated
figures of merit20,21 are significantly be-low this value. This
discrepancy is partly due to thecross-polarization setup in
resonant excitation used inthese works, reducing the efficiency by
a factor of 2.This reduction is not a fundamental limitation and
canbe avoided using e.g. phonon-assisted excitation26 forthe
rotationally symmetric micropillars considered inthis work or by
the introduction of an elliptical cross-section17. Even so, these
methods will not overcome theinherent trade-off between efficiency
and indistinguisha-bility of the micropillar SPS. Nevertheless, an
optimiza-tion of also the elliptical micropillar design in terms
ofefficiency and indistinguishable polarized photon emis-sion
remains of interest but is beyond this work.
VI. CONCLUSION
In summary, by applying a Fourier modal method anda polaron
master equation approach, we have numericallycalculated the
efficiency and indistinguishability of themicrocavity pillar
single-photon source. To elucidate themechanism governing the
efficiency, we have presented asingle-mode model, which fully
captures the physics interms of the initial spontaneous emission to
the cavitymode and the subsequent transmission to the first lens.By
varying the number of DBR layer pairs and the diam-eter, we have
observed the onset of the trade-off betweenefficiency and
indistinguishability predicted previously,and we have identified
design parameters leading to amaximum figure of merit εη of the
product of the effi-ciency ε and the indistinguishability η of ∼
0.96 for theideal geometry. In addition, we have investigated the
in-fluence of pure dephasing as well as sidewall scattering onthe
performance. Using realistic values for these mech-anisms, we have
shown that, while the performance isdegraded by imperfections, the
optimum design parame-ters are unchanged.
ACKNOWLEDGEMENT
The authors thank Jean-Michel Gérard for fruitful dis-cussions.
EVD and NG were supported by the Inde-pendent Research Fund
Denmark, grant No. DFF-4181-00416 and DFF-9041-00046B. BW
acknowledges supportfrom the China Scholarship Council. This work
wassupported by the Danish National Research Foundationthrough
NanoPhoton - Center for Nanophotonics, grantno. DNRF147.
APPENDIX
In the Fourier modal method, the geometry is dividedinto layers
uniform along a propagation z axis, and theelectric field in each
layer is expanded upon the forwardand backwards propagating
eigenmodes e±j (r⊥)e
±iβjz
that are solutions to the wave equation for the layer as-suming
uniformity along z. The fields on each side of theinterface between
layers q and q+1 are related by inter-
face reflection ¯̄Rq,q+1 and transmission¯̄Tq,q+1 matrices
determined from the boundary conditions at the inter-face, and
the propagation through a layer of length L isdescribed by a
propagation PL matrix with elements e
iβjL
in its diagonal. The total reflection matrix ¯̄Rt (¯̄Rb) for
the top (bottom) DBR mirror is obtained from interfaceand
propagation matrices in an iterative manner usingthe S matrix
formalism. The total field at the positionz0 + δz just above the QD
(δz → 0) is given by
E(r⊥, z0 + δz) =∑
j
[aje
+j (r⊥) + bje
−j (r⊥)
], (20)
where the field amplitude vectors ā and b̄ represent thefield
expansion just above the QD. The forward propa-gating field vector
ā is computed from the point dipolepδ(r − r0) using the
reciprocity theorem and takinginto account multiple round trips
inside the cavity34.The backwards propagating field is obtained as
b̄ =¯̄PL/2
¯̄Rt¯̄PL/2ā.
A. Cavity contribution to the field
To determine the contribution EC from the cavitymode to the
total field given in Eq. (20), we define
the eigenvectors c̄+k of the roundtrip matrix¯̄R+ =
¯̄PL/2¯̄Rb
¯̄PL¯̄Rt
¯̄PL/2 as solutions to the eigenproblem
¯̄R+c̄+k = λ+k c̄
+k , (21)
where λ+k are eigenvalues with |λ+k | < 1. For normalized
eigenvectors, the cavity roundtrip eigenvector c̄+M
corre-sponding to the fundamental cavity mode is easily identi-fied
as the one with the first element having a maximumvalue such that
|c+1M| ≥ |c
+1k|. For the forward propagat-
ing field in (20), we have∑j
aje+j (r⊥) =
∑j,k
c+jkαke+j (r⊥), (22)
where ᾱ = (¯̄c+)−1ā, whereas for the backward propagat-ing
field, we have∑
j
bje−j (r⊥) =
∑j,k
γ−jkαke−j (r⊥), (23)
where ¯̄γ− = ¯̄PL/2¯̄Rt
¯̄PL/2¯̄c+. The total field can then be
written as
E(r⊥, z0) =∑j,k
c+jkαke+j (r⊥) +
∑j,k
γ−jkαke−j (r⊥), (24)
-
9
and the cavity mode contribution is given by
EC(r⊥, z0) =∑
j
c+jMαMe+j (r⊥) +
∑j
γ−jMαMe−j (r⊥).
(25)
B. Far field calculation
The electric field at the surface z = zS of the micropil-lar is
computed from the electric field expansion Eq. (20)using the total
transmission matrix of the top DBR. Wethen determine the far field
EF(θ, φ) on the surface ofa sphere S of radius R using a standard
near field tofar field transformation54. To take into account the
lim-ited NA of the lens, we set EF(θ > θMax, φ) = 0 withθMax =
sin
−1(NA). Following the procedure in Refs. 12and 13, we then
consider a Gaussian beam in the paraxiallimit in a focal plane z =
zF given by
Eg(x, y, zF) =
√2cµ0πw20
exp
(−x
2 + y2
w20
)x̂, (26)
Hg(x, y, zF) =1√cµ0
ẑ×Eg, (27)
where w0 is the beam waist and the integrated power inthe plane
z = zF is normalized to 1. We then computethe associated far field
EgF and H
gF on the surface S, and
the overlap with the Gaussian beam is then given as
O(w0, zF) = R2∫ (
E∗F,θHgF,φ − E∗F,φH
gF,θ
)sin θdθdφ, (28)
where the subindices θ and φ indicate the field compo-nents in
spherical coordinates. Finally, the transmissionto the lens taking
into account the coupling to the Gaus-sian profile is given by
PLens =1
2Max|O(w0, zF)|2. (29)
The power PLens,C coupled to the lens from the cavity isobtained
by calculating EF(θ, φ) using the cavity contri-bution to the
electric field given by Eq. (25) instead.
C. Comparison of efficiency and the one fromsimplified model
We compare the calculated efficiency ε obtained fromthe full
model Eq. (1) and the efficiency εs from thesingle-mode model in
Fig. 6. Excellent agreement be-tween the two models is observed,
indicating that thecontribution from radiation modes to the
collection oflight in a Gaussian mode at the first lens is
negligible.
D. Modeling of sidewall roughness
The sidewall scattering Q−1edge can be computed using
a model44 taking into account the ratio of the mode in-
0.1
0.4
0.7
1
1 2 3 4
d (µm)
ntop=17 ε
εs
ntop=25 ε
εs
FIG. 6. The full efficiency ε and the single-mode efficiency
εscomputed using ntop = 17 and 25 top DBR layer pairs. Theideal
structure with κ = 0 is considered.
102
103
104
105
1 2 3 4
d (µm)
ntop=17 QintQ
ntop=21 QintQ
ntop=25 QintQ
FIG. 7. The intrinsic cavity quality factor Qint and the Qfactor
taking into account sidewall scattering as a function ofpillar
diameter d for κ = 0.2 µm.
tensity at the micropillar edge and its diameter d as
Q−1edge = κJ20 (ktd/2)
d/2. (30)
Here, κ is a phenomenological constant, J0 is theBessel function
of the first kind, kt is defined askt =
√n2k20 − β21 with n being the refractive index
of GaAs, k0 = ω/c and β1 denoting the propagatingconstant of the
fundamental HE11 mode in the cavitysection. The Q factor in the
presence of sidewall rough-ness is then determined from Eq. (30)
and the intrinsicquality factor Qint using Eq. (19) and is
presented inFig. 7. The sidewall roughness represents an
additionaldecay channel of the light in the cavity leading to
areduction of Q, which becomes more significant for highintrinsic
quality factors. As the pillar diameter increases,this influence
decreases as the relative overlap of theHE11 mode with the boundary
is reduced.
-
10
0.82
0.88
0.94
1
15 17 19 21 23 25 27 29
ntop
ε
β
γ
FIG. 8. The efficiency ε, the spontaneous emission β factorand
the transmission γ as a function of top DBR layer pairsntop for a
pillar diameter d = 3.2 µm with κ = 0.
E. Dependence on the top DBR thickness
The efficiency, the spontaneous emission β factor andthe
transmission γ are presented in Fig. 8 as function ofthe number of
top DBR layer pairs. As ntop increases,the cavity Q factor and in
turn the Purcell factor areimproved leading to an increasing β.
However, as thetop mirror reflectivity becomes comparable to that
of thebottom DBR, an increasing fraction of light penetratesinto
the substrate reducing the transmission γ. We thushave a trade-off
between β and γ, where an optimumefficiency of ε ∼ 0.95 is obtained
for ntop ∼ 21.
∗ [email protected] J. L. O’Brien, A. Furusawa, and J.
Vučković, Nat. Pho-
tonics 3, 687 (2009).2 J. W. Pan, Z. B. Chen, C. Y. Lu, H.
Weinfurter,
A. Zeilinger, and M. Zukowski, Rev. Mod. Phys. 84,
777(2012).
3 N. Gregersen, D. P. S. McCutcheon, and J. Mørk, inHandbook of
Optoelectronic Device Modeling and Simula-tion Vol. 2, edited by J.
Piprek (CRC Press, Boca Raton,2017) Chap. 46, pp. 585–607.
4 N. Gregersen, P. Kaer, and J. Mørk, IEEE J. Sel. Top.Quantum
Electron. 19, 9000516 (2013).
5 H. Wang, J. Qin, X. Ding, M.-C. Chen, S. Chen, X. You,Y.-M.
He, X. Jiang, L. You, Z. Wang, C. Schneider, J. J.Renema, S.
Höfling, C.-Y. Lu, and J.-W. Pan, Phys. Rev.Lett. 123, 250503
(2019).
6 J. Renema, V. Shchesnovich, and R.
Garcia-Patron,arXiv:1809.01953.
7 P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A.
V.Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337 (1995).
8 A. J. Shields, Nat. Photonics 1, 215 (2007).9 H. Wang, H. Hu,
T. H. Chung, J. Qin, X. Yang, J. P. Li,
R. Z. Liu, H. S. Zhong, Y. M. He, X. Ding, Y. H. Deng,Q. Dai, Y.
H. Huo, S. Höfling, C. Y. Lu, and J. W. Pan,Phys. Rev. Lett. 122,
113602 (2019).
10 J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P.
Jaffrennou,N. Gregersen, C. Sauvan, P. Lalanne, and J. M.
Gérard,Nat. Photonics 4, 174 (2010).
11 M. E. Reimer, G. Bulgarini, N. Akopian, M. Hocevar,M. B.
Bavinck, M. A. Verheijen, E. P. Bakkers, L. P.Kouwenhoven, and V.
Zwiller, Nat. Commun. 3, 737(2012).
12 M. Munsch, N. S. Malik, E. Dupuy, A. Delga, J. Bleuse,J. M.
Gérard, J. Claudon, N. Gregersen, and J. Mørk,Phys. Rev. Lett.
110, 177402 (2013).
13 M. Munsch, N. S. Malik, E. Dupuy, A. Delga, J. Bleuse,J. M.
Gérard, J. Claudon, N. Gregersen, and J. Mørk,Phys. Rev. Lett.
111, 239902 (2013).
14 S. Ates, L. Sapienza, M. Davanco, A. Badolato, andK.
Srinivasan, IEEE J. Sel. Top. Quantum Electron. 18,1711 (2012).
15 L. Sapienza, M. Davanço, A. Badolato, and K. Srinivasan,Nat.
Commun. 6, 7833 (2015).
16 J. Liu, R. Su, Y. Wei, B. Yao, S. F. C. da Silva, Y. Yu,J.
Iles-Smith, K. Srinivasan, A. Rastelli, J. Li, andX. Wang, Nat.
Nanotechnol. 14, 586 (2019).
17 H. Wang, Y. M. He, T. H. Chung, H. Hu, Y. Yu, S. Chen,X.
Ding, M. C. Chen, J. Qin, X. Yang, R. Z. Liu, Z. C.Duan, J. P. Li,
S. Gerhardt, K. Winkler, J. Jurkat, L. J.Wang, N. Gregersen, Y. H.
Huo, Q. Dai, S. Yu, S. Höfling,C. Y. Lu, and J. W. Pan, Nat.
Photonics 13, 770 (2019).
18 A. Berthelot, I. Favero, G. Cassabois, C. Voisin, C.
Dela-lande, P. Roussignol, R. Ferreira, and J.-M. Gérard,
Nat.Phys. 2, 759 (2006).
19 S. Reitzenstein and A. Forchel, J. Phys. D: Appl. Phys.43,
033001 (2010).
20 X. Ding, Y. He, Z. C. Duan, N. Gregersen, M. C. Chen,S.
Unsleber, S. Maier, C. Schneider, M. Kamp, S. Höfling,C. Y. Lu,
and J. W. Pan, Phys. Rev. Lett. 116, 020401(2016).
21 N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo,M. P.
Almeida, G. Hornecker, S. L. Portalupi, T. Grange,C. Antón, J.
Demory, C. Gómez, I. Sagnes, N. D.Lanzillotti-Kimura, A.
Lemáıtre, A. Auffeves, A. G. White,L. Lanco, and P. Senellart,
Nat. Photonics 10, 340 (2016).
22 W. L. Barnes, G. Björk, J. M. Gérard, P. Jonsson, J.
A.Wasey, P. T. Worthing, and V. Zwiller, Eur. Phys. J. D18, 197
(2002).
23 A. Dousse, L. Lanco, J. Suffczyński, E. Semenova, A. Mi-ard,
A. Lemâıtre, I. Sagnes, C. Roblin, J. Bloch, andP. Senellart,
Phys. Rev. Lett. 101, 267404 (2008).
24 S. Ates, S. M. Ulrich, S. Reitzenstein, A. Löffler,A.
Forchel, and P. Michler, Phys. Rev. Lett. 103, 167402(2009).
25 Y. M. He, H. Wang, C. Wang, M. C. Chen, X. Ding, J. Qin,Z. C.
Duan, S. Chen, J. P. Li, R. Z. Liu, C. Schneider,M. Atatüre, S.
Höfling, C. Y. Lu, and J. W. Pan, Nat.
-
11
Phys. 15, 941 (2019).26 S. E. Thomas, M. Billard, N. Coste, S.
C. Wein, Priya,
H. Ollivier, O. Krebs, L. Tazäırt, A. Harouri, A. Lemaitre,I.
Sagnes, C. Anton, L. Lanco, N. Somaschi, J. C. Loredo,and P.
Senellart, arXiv:2007.04330.
27 J. Houel, A. V. Kuhlmann, L. Greuter, F. Xue, M. Poggio,B. D.
Gerardot, P. A. Dalgarno, A. Badolato, P. M. Petroff,A. Ludwig, D.
Reuter, A. D. Wieck, and R. J. Warburton,Phys. Rev. Lett. 108,
107401 (2012).
28 P. Kaer, N. Gregersen, and J. Mørk, New J. Phys. 15,035027
(2013).
29 K. Roy-Choudhury and S. Hughes, Phys. Rev. B 92,205406
(2015).
30 E. V. Denning, M. Bundgaard-Nielsen, and J.
Mørk,arXiv:2007.14719.
31 J. Iles-Smith, D. P. McCutcheon, A. Nazir, and J. Mørk,Nat.
Photonics 11, 521 (2017).
32 E. V. Denning, J. Iles-Smith, A. D. Osterkryger,N. Gregersen,
and J. Mork, Phys. Rev. B 98, 121306(2018).
33 T. Häyrynen, J. R. de Lasson, and N. Gregersen, J. Opt.Soc.
Am. A 33, 1298 (2016).
34 A. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt,and T.
Søndergaard, Numerical Methods in Photonics(CRC Press, 2014).
35 S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres,A.
Vonlanthen, and H. Sigg, J. Appl. Phys. 87, 7825(2000).
36 P. Lalanne, J. P. Hugonin, and J. M. Gérard, Appl.
Phys.Lett. 84, 4726 (2004).
37 L. Novotny and B. Hecht, Principles of nano-optics
(Cam-bridge university press, 2012).
38 I. Wilson-Rae and A. Imamoğlu, Phys. Rev. B 65,
235311(2002).
39 K. Roy-Choudhury and S. Hughes, Phys. Rev. B 92,205406
(2015).
40 A. Nazir and D. P. McCutcheon, Journal of Physics: Con-densed
Matter 28, 103002 (2016).
41 E. V. Denning, J. Iles-Smith, N. Gregersen, and J. Mørk,Opt.
Mater. Express 10, 222 (2020).
42 G. Lecamp, J. P. Hugonin, P. Lalanne, R. Braive,S. Varoutsis,
S. Laurent, A. Lemâıtre, I. Sagnes, G. Patri-arche, I.
Robert-Philip, and I. Abram, Appl. Phys. Lett.90, 091120
(2007).
43 N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss,C.
Schneider, S. Höfling, L. Worschech, A. Forchel, T. R.Nielsen, J.
Mørk, and J. M. Gérard, IEEE J. QuantumElectron. 46, 1470
(2010).
44 T. Rivera, J. P. Debray, J. M. Gérard, B. Legrand,L.
Manin-Ferlazzo, and J. L. Oudar, Appl. Phys. Lett.74, 911
(1999).
45 C. Schneider, P. Gold, S. Reitzenstein, S. Höfling, andM.
Kamp, Appl. Phys. B 122, 19 (2016).
46 M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, andM.
Hetterich, Opt. Express 17, 1144 (2009).
47 S. Reitzenstein, N. Gregersen, C. Kistner, M. Strauss,C.
Schneider, L. Pan, T. R. Nielsen, S. Höfling, J. Mørk,and A.
Forchel, Appl. Phys. Lett. 94, 061108 (2009).
48 N. Gregersen, T. R. Nielsen, J. Claudon, J.-M. Gérard,and J.
Mørk, Opt. Lett. 33, 1693 (2008).
49 A. Greilich, R. Oulton, E. A. Zhukov, I. A. Yugova,D. R.
Yakovlev, M. Bayer, A. Shabaev, A. L. Efros, I. A.Merkulov, V.
Stavarache, D. Reuter, and A. Wieck, Phys.Rev. Lett. 96, 227401
(2006).
50 M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein,S.
Höfling, J. Mørk, L. Worschech, M. Kamp, andA. Forchel, Phys. Rev.
Lett. 108, 057402 (2012).
51 N. Gregersen, D. P. S. McCutcheon, J. Mørk, J.-M.Gérard, and
J. Claudon, Opt. Express 24, 20904 (2016).
52 A. D. Osterkryger, J.-M. Gérard, J. Claudon, andN.
Gregersen, Opt. Lett. 44, 2617 (2019).
53 J. Bleuse, J. Claudon, M. Creasey, N. S. Malik, J. M.Gérard,
I. Maksymov, J. P. Hugonin, and P. Lalanne,Phys. Rev. Lett. 106,
103601 (2011).
54 C. A. Balanis, Advanced Engineering Electromagnetics, 1sted.
(Wiley, 1989).