Performance of the nanopost single-photon source: beyond the single-mode model
---
Martin Arentoft Jacobsen,∗a Yujing Wang,a Luca Vannucci,a Julien Claudonb,
Jean-Michel Gérard,b and Niels Gregersena
We present a detailed analysis of the physics governing the collection
efficiency and the Purcell enhancement of the nanopost single-photon source.
We show that a standard single-mode Fabry-Pérot model is insufficient to
describe the device performance, which benefits significantly from scattering
from the fundamental mode to radiation modes. We show how the scattering
mechanism decouples the collection efficiency from the Purcell enhancement,
such that maximum collection efficiency is obtained off-resonance. Finally, we
discuss how this scattering mechanism can be beneficial for future single-
photon source designs.

##

††footnotetext: a DTU Electro, Department of Electrical and Photonics
Engineering, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark.
E-mail<EMAIL_ADDRESS>b Univ. Grenoble Alpes, CEA, Grenoble INP,
IRIG, PHELIQS, “Nanophysique et Semiconducteurs” Group, F-38000 Grenoble,
France.

##  1 Introduction

The construction of scalable optical quantum technologies 1, 2 relies on the
development of sources of single indistinguishable photons 3, 4, 5, 6 and of
entangled photon pairs 7. The ideal single-photon source (SPS) should be
deterministic and feature pure emission of single photons. The main figure of
merit 6 is the collection efficiency $\varepsilon$ defined as the number of
photons detected in the out-coupling channel per trigger. In a multi-photon
interference experiment 8 with $N$ photons, the success probability $P$ scales
as $P=\varepsilon^{N}$, and increasing $\varepsilon$ towards 1 is thus
critical to achieve scalable optical quantum information processing. The
spontaneous parametric down conversion process 9 is a straight-forward
technique widely used within the quantum optics community for production of
highly indistinguishable photons, however its probabilistic nature limits the
efficiency of pure photon emission to a few percent.

For this reason, the community has turned its attention towards two level
systems, in particular the semiconductor quantum dot 3, 4, 10 (QD), capable of
deterministic emission of single photons. For a QD in a bulk material,
$\varepsilon$ is limited to a few percent this time due to the large index
contrast at the semiconductor-air interface. It is thus necessary to place the
QD inside a photonic nanostructure 5, 6 directing the light towards the
collection optics. A main strategy for controlling the light emission is to
place the QD inside a micro cavity and exploit cavity quantum electrodynamics
(CQED) in the weak coupling regime to selectively enhance the light emission
into the optical mode of the microcavity using the Purcell effect 11. Detailed
understanding of the CQED physics governing the collection efficiency can be
obtained using a single-mode Fabry-Pérot description 12, 13, 14 of the light
emission. Here, the spontaneous emission $\beta$ factor describes the emission
rate $\Gamma_{\rm C}$ of the QD into a fundamental HE11 cavity mode divided by
the total emission rate $\Gamma_{\rm T}=\Gamma_{\rm C}+\Gamma_{\rm B}$
including a contribution $\Gamma_{\rm B}$ to background radiation modes. The
rate $\Gamma_{\rm C}$ into the cavity mode normalized to the rate $\Gamma_{\rm
Bulk}$ in a bulk medium is quantified by the Purcell 15 factor $F_{\rm
p}=\Gamma_{\rm C}/\Gamma_{\rm Bulk}=\frac{3}{4\pi^{2}}\frac{Q}{V_{\rm n}}$ at
resonance, where $Q$ is the cavity quality factor and $V_{\rm n}$ is the mode
volume in units of material cubic wavelengths $(\lambda/n)^{3}$. The
spontaneous emission $\beta$ factor can then be written in terms of the
Purcell factor as

$\displaystyle\beta=\frac{\Gamma_{\rm C}}{\Gamma_{\rm C}+\Gamma_{\rm
B}}=\frac{F_{\rm p}}{F_{\rm p}+\Gamma_{\rm B}/\Gamma_{\rm Bulk}}.$ (1)

Furthermore, we define the transmission $\gamma$ as the fraction of power in
the cavity mode detected by the collection optics. Finally, we can then define
a single-mode Fabry-Pérot model (SMM) $\varepsilon_{\rm s}$ for the efficiency
as $\varepsilon_{\rm s}=\beta\gamma$. From Eq. (1), we observe that increasing
the Purcell factor $F_{\rm p}$ will improve the collection efficiency, and
maximum efficiency is thus expected for a QD on resonance with the cavity.

Indeed, this design paradigm that Purcell enhancement is beneficial for
achieving high collection efficiency is well-established within SPS
engineering: The most succesful SPS design strategies today include the
microcavity pillar 16, 14, 17, 18 and the open cavity approach 19
demonstrating up to $\varepsilon\sim$ 0.6 into a first lens 16 and into a
fiber 19, respectively, combined with highly indistinguishable photon
emission. These narrowband approaches, for which the single mode model
$\varepsilon_{\rm s}=\beta\gamma$ is an excellent approximation 14, rely
critically on resonant Purcell enhancement and thus on control of the spectral
alignment 17 to achieve high efficiency. On the other hand, broadband
approaches including the photonic nanowire 20, 13, 21, 22, 23 and the photonic
crystal waveguide 24, 25, 26 designs exploit suppression of the background
emission rate using the dielectric screening effect 20, 24, 25 to non-
resonantly maximize the $\beta$ factor. Even so, these broadband approaches
also benefit from resonant cavity 13, 21 and slow-light 24, 25 effects to
further improve the efficiency, confirming again that Purcell enhancement is
beneficial in the SPS engineering.

Fig. 1: Sketch of the "nanopost" nanowire optical nanocavity and the
geometrical parameters. The resulting in-plane electrical field profile of a
QD placed inside the nanopost is shown as an inset. The length of the white
scale bar in the inset is $100\text{\,}\mathrm{n}\mathrm{m}$.

However, this paradigm has been challenged by new broadband SPS geometries,
such as the circular Bragg grating or "bullseye" design 27, 28, 16, for which
high collection efficiency is obtained in a wavelength range significantly
broader 27 ($\sim$ 100 nm) than the typical resonance linewidth ($\sim$ 10
nm). Similar characteristics were observed very recently for the nanowire
optical nanocavity or "nanopost" design 29 shown in Fig. (1), for which a
significant Purcell factor $F_{\rm p}$ of 5.6 enabled by the ultrasmall mode
volume of the nanocavity was experimentally demonstrated. Additionally, a
surprisingly high collection efficiency of 0.35 was measured 29, which was
attributed to a breakdown 30 of the single mode model $\varepsilon_{\rm
s}=\beta\gamma$ for the efficiency.

In this work, we investigate this surprising breakdown by performing a
detailed quantitative analysis of the physics governing the nanopost geometry.
We show that the single mode model fails to describe the physics of both the
Purcell enhancement and the collection efficiency due to a decoupling between
the two: The computed efficiency is significantly higher than the prediction
of the single-mode model thanks to additional transmission channels to the
far-field, whose beneficial contributions are dominating over the resonant
cavity effect. We show not only that maximum Purcell enhancement and maximum
collection efficiency are obtained for entirely different design parameters,
but also that maximum efficiency is obtained off-resonance. The analysis is
performed using a Fourier Modal Method 31, allowing for direct insight into
the beneficial interplay beyond the single-mode model with the continuum of
radiation modes.

This article is organized as follows: In Section 2, we present the nanopost
and its performance in terms of $F_{\rm p}$ and $\varepsilon$, and we
demonstrate the breakdown of the single-mode model. In Section 3, we present
our theoretical framework based on the Fourier Modal Method, which we
subsequently use to analyze the complex interplay with radiation mode channels
in Section 4 and its influence on the collection efficiency and the Purcell
factor. In Section 5 we put the nanopost physics into perspective and discuss
its impact on SPS engineering, followed by our conclusion. Additional
simulation results are presented in the Supplementary Information.

##  2 The nanopost geometry and the breakdown of the single-mode Fabry-Pérot
model

The nanopost shown in Fig. (1) consists of a truncated GaAs nanowire with
diameter $D$ on top of a SiO2-Au mirror. The top of the nanowire is flat, and
the surrounding medium is air. The SiO2 layer, located between the nanowire
and the gold, has a thickness indicated by $t_{\rm SiO_{2}}$. The QD, modelled
as a dipole, is placed on-axis inside the nanowire at a position $h_{\rm b}$
from the bottom interface and $h_{\rm t}$ from the top interface. The
refractive indices of the materials are chosen as $n_{\rm GaAs}=3.46$, $n_{\rm
SiO_{2}}=1.5$ and $n_{\rm Au}=0.201+5.85i$ at
$\lambda=$930\text{\,}\mathrm{n}\mathrm{m}$$ and assumed to be constant as a
function of wavelength. In the inset of Fig. (1), a dipole with an emission
wavelength of $930\text{\,}\mathrm{n}\mathrm{m}$ is placed inside the
nanopost, and the resulting in-plane electrical field, simulated using the
FMM, is shown. Three antinodes can be seen in the field profile, corresponding
to the order 3 cavity mode, and they are enumerated from the bottom mirror as
the 1st, 2nd and 3rd antinode. The field profile generated by the dipole is
independent of the vertical position of the dipole, only the intensity
changes. The intensity is not the same at the three antinodes due to the
breakdown of the SMM.

We now present the performance of the nanopost as a function of the diameter,
$D$, and the silica layer thickness, $t_{\rm SiO_{2}}$. We have scanned the
parameter ranges $D=$196\text{\,}\mathrm{n}\mathrm{m}$$ to
$D=$300\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$0\text{\,}\mathrm{n}\mathrm{m}$$ to $t_{\rm
SiO_{2}}=$25\text{\,}\mathrm{n}\mathrm{m}$$ and chosen a design wavelength of
$\lambda_{\rm d}=$930\text{\,}\mathrm{n}\mathrm{m}$$. The height of the
structure and the position of the QD are dynamically changed to keep the order
3 cavity mode resonance at $\lambda_{\rm
r}=$930\text{\,}\mathrm{n}\mathrm{m}$$ for the QD at the 2nd antinode. The
required procedure is presented in Supplementary 7.1.

Fig. 2: Purcell factor, $F_{\rm p}$, for a QD at the 2nd antinode (a), 1st
antinode (b) and any antinode using the SMM (c) as a function of the diameter,
$D$, and the silica layer thickness, $t_{\rm SiO_{2}}$.

The first quantity of interest is the spontaneous emission rate $\Gamma_{\rm
T}$. For the high-$\beta$ structures investigated here, the total normalized
rate $\Gamma_{\rm T}/\Gamma_{\rm Bulk}$ and the Purcell factor $F_{\rm p}$ are
similar, and we will in the following refer to the normalized total rate as
the "Purcell factor $F_{\rm p}$". In Fig. (2a,2b) the Purcell factor is shown
as a function of the diameter and the silica layer thickness for a QD placed
in the 2nd and 1st antinode. In the entire parameter space, the Purcell factor
is larger for the 2nd antinode and a maximum value of $F_{\rm p}=7.9$ is
reached at $D=$250\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$13\text{\,}\mathrm{n}\mathrm{m}$$. This discrepancy between the two
antinodes also demonstrates the deviations of the SMM. Overall, the tendency
of the Purcell factor is similar at the 2 antinodes with one peak value. The
minimum is located in the corner of no silica and the smallest diameter. In
Fig. (2c) the Purcell factor is now shown using the SMM. Comparing the SMM to
the full model for the two antinodes, there are both positive and negative
deviations across most of the parameter space. Compared to the 2nd antinode,
the SMM also predicts a slightly lower value for the maximum Purcell factor of
$F_{\rm p}=7.5$, but at a very different position of
$D=$242\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$8\text{\,}\mathrm{n}\mathrm{m}$$. However, compared to the 1st
antinode the SMM predicts a larger value for the maxmimum Purcell factor. This
is an invitation to obtain a better description and understanding of the
physics responsible for the Purcell factor, which we will provide in this
paper.

Fig. 3: Collection efficiency, $\varepsilon$ ($NA=0.75$), for a QD at the 2nd
antinode (a), 1st antinode (b) and any antinode using the SMM (c) as a
function of the diameter, $D$, and the silica layer thickness, $t_{\rm
SiO_{2}}$.

The source collection efficiency, $\varepsilon$, for the 2nd antinode, the 1st
antinode and the SMM are shown in Fig. (3) for a numerical aperture of
$\mathrm{NA}=0.75$. Here the overall performance for the 2nd antinode is
slightly better than for the 1st antinode, and a maximum value of
$\varepsilon=0.69$ is reached at $D=$238\text{\,}\mathrm{n}\mathrm{m}$$ and
$t_{\rm SiO_{2}}=$0\text{\,}\mathrm{n}\mathrm{m}$$. Despite the very simple
geometry of the nanopost, a surprisingly high collection efficiency of
$\varepsilon=0.69$ is achievable. In general, it can be ascertained that no
silica layer is more beneficial for the efficiency, which is surprising
compared to the traditional photonic nanowire, where the silica layer enhances
the modal reflectivity 22 and thus the collection efficiency. Now comparing
the efficiency of the SMM to the full model, there are substantial
differences. This is unlike the micropillar and the photonic nanowire
geometries for which the single-mode model $\varepsilon_{\rm s}=\beta\gamma$
is an excellent 12, 13, 14 approximation. For the nanopost, the SMM efficiency
is much smaller in the entire parameter space except for the smallest
diameters. This clearly shows that there are important physical mechanisms
which are not accounted for in the SMM. Furthermore, comparing the figures for
the Purcell factors and the efficiencies, there seems to be no apparent
correlation between the two. This is also a surprising result compared to
traditional Fabry-Pérot cavities and indicates that there are different
physical mechanisms at play which govern the Purcell factor and the
efficiency.

In Supplementary 7.2, we vary the numerical aperture and present its influence
on the collection efficiency. In Supplementary 7.3, we present the collection
efficiency taking into consideration the overlap with a Gaussian profile.

##  3 Theory

###  3.1 Method

We use an eigenmode method combined with a standard scattering matrix
formalism 32 in the frequency domain. In this method, the structure is divided
into layers of uniform permittivity along the propagation direction. In each
layer, the electrical field is expanded on the eigenmodes, and the scattering
matrices are used to connect the eigenmodes at the interfaces between the
layers. The eigenmodes are obtained using the Fourier modal method (FMM) with
open boundary conditions 31, which provides direct access to the modes needed
to understand the physics. The nanopost structure is split into four layers:
the bottom gold substrate, the silica layer, the nanowire and the top air
region. Due to the cylindrical symmetry of the nanowire, a cylindrical
coordinate system is used. The QD is modelled as a classical point dipole, and
we make use of the relationship $\Gamma/\Gamma_{\rm Bulk}=P/P_{\rm Bulk}$ to
calculate the Purcell factor 33. $P$ is the emitted power of the dipole, and
$P_{\rm Bulk}$ is the power emitted in a bulk medium, and thus the Purcell
factor is defined as $F_{\rm p}=P_{\rm T}/P_{\rm Bulk}$. The second quantity
of interest is the collection efficiency defined as $\varepsilon=P_{\rm
collected}/P_{\rm T}$, where $P_{\rm collected}$ is the power collected in the
far-field of a lens with numerical aperture $\rm NA$.

####  3.1.1 Eigenmodes

The electrical field for an eigenmode has the following expression:

$\textbf{E}_{j}(r,\phi,z)=\textbf{e}_{j}(r,\phi)\exp{(i\beta_{j}z)},$ (2)

where $j$ refers to eigenmode index, $\textbf{e}_{j}(r,\phi)$ is the mode
profile for the given eigenmode and $\beta_{j}$ is the propagation constant.
For three out of the four layers, the permittivity profile is constant over
the entire plane. In this case, the eigenmodes are simply cylindrical plane
waves. These eigenmodes exist as a continuum where the propagation constant
takes the value $\beta=\sqrt{(n_{\rm layer}k_{0})^{2}-k_{\perp}^{2}}$, where
$n_{\rm layer}$ is the refractive index, $k_{0}$ is the free-space wavenumber
and $k_{\perp}$ is the in-plane $k$-value. $k_{\perp}$ can take any value
$k_{\perp}\in[0,\infty]$ and eigenmodes exist for all the values. For each
$\beta$, two orthogonal solutions exist, which can be separated into TE modes
($e_{z}=0$) and TM modes ($h_{z}=0$). For layers with real-valued refractive
index, this continuum can be separated into radiation modes satisfying
$0<(\beta)^{2}\leq(n_{\rm layer}k_{0})^{2}$ and evanescent modes
$(\beta)^{2}<0$ which decays exponentially and carry no power according to the
Poynting vector. Specifically for the air layer, the propagation constant of
the radiation modes can directly be interpreted as the propagation angle with
respect to the z-axis using the expression $\theta=\arccos{(\beta/k_{0})}$.
Expanding a forward propagating electrical field on the eigenmodes will have
the following expression:

$\textbf{E}(r,\phi,z)=\sum_{s=1}^{2}\int_{0}^{\infty}a_{s}(k_{\perp})\textbf{e}_{s}(r,\phi,k_{\perp})\exp{(i\beta(k_{\perp})z)}dk_{\perp},$
(3)

where $s$ refers to the two solutions and $a_{s}(k_{\perp})$ is the expansion
coefficient. Numerically, the continuum is discretized into $N$ modes and
truncated with a cut-off value for $k_{\perp}$ 34, 31, which will lead to the
following expression:

$\textbf{E}(r,\phi,z)=\sum_{j=1}^{N}a_{j}\textbf{e}_{j}(r,\phi)\exp{(i\beta_{j}z)},$
(4)

where $s$ is absorbed into $j$, and the $\Delta k_{\perp}$ that would appear
due to the discretization is absorbed into definition of the eigenmode
profile.

The first class of eigenmodes for the nanowire are the guided modes for which
the propagation constants satisfy the condition $(n_{\rm
air}k_{0})^{2}<(\beta_{j})^{2}<(n_{\rm GaAs}k_{0})^{2}$. The guided mode is
confined to the core of the nanowire, and outside the nanowire, the field
decays. There is a finite number of guided modes, and there will always be at
least one guided mode, the fundamental $\mathrm{HE}_{11}$ mode. The
description of guided modes can be found in various textbooks 35. The second
class of eigenmodes is the background continuum, very similar to the continuum
of the air layer. These eigenmodes can be viewed as perturbed versions of the
cylindrical plane waves of the air layer and be separated into radiation and
evanescent modes the exact same way. There are also two orthogonal solutions,
but these can no longer be separated into pure TE and TM modes. Studies of
this class of eigenmodes are plentiful in the literature 36, 37, 38, 39, 40,
41, 42, 43, 44, 45.

####  3.1.2 Dipole emission in an infinite structure

The QD is modelled as a classical point dipole with in-plane orientation and
harmonic time dependence at the frequency $\omega$. The corresponding current
density is
$\mathbf{J}(\mathbf{r})=-i\omega\mathbf{p}\delta(\mathbf{r}-\mathbf{r}_{\rm
d})$, where $\mathbf{r}_{\rm d}$ is the position of the QD and $\mathbf{p}$ is
the dipole moment. The emitted power of the dipole can be calculated as 33:

$\begin{split}P=-\frac{1}{2}\int_{V}{\rm Re}[{{\bf J}^{*}({\bf r})\cdot{\bf
E}(\bf r)}]dV=\frac{\omega}{2}{\rm Im}[{{\bf p}\cdot{\bf E}(\bf r_{\rm
d})}].\end{split}$ (5)

The total emitted power, $P_{\rm T}$, can be calculated by evaluating the
total field, $\mathbf{E}_{\rm T}(r_{\rm d})$, but also the power into
individual modes, $P_{j}$, by evaluating $\mathbf{E}_{j}(r_{\rm d})$. Placing
the dipole on-axis inside an infinitely long nanowire with position $z_{\rm
J}$ will result in the following electrical field:

$\begin{split}\mathbf{E}_{\rm T}(\mathbf{r})=\sum_{j}a_{j}^{\rm
J}\mathbf{e}_{j}^{+}(\mathbf{r}_{\perp})\exp(i\beta_{j}(z-z_{\rm J}))\
(z>z_{\rm J})\end{split}$ (6) $\begin{split}\mathbf{E}_{\rm
T}(\mathbf{r})=\sum_{j}b_{j}^{\rm
J}\mathbf{e}_{j}^{-}(\mathbf{r}_{\perp})\exp(-i\beta_{j}(z-z_{\rm J}))\
(z<z_{\rm J}),\end{split}$ (7)

where the superscript $+$ refers to forward propagating and $-$ refers to
backward propagating. For the electrical fields, the forward and backwards
propagating fields are identical:
$\mathbf{e}_{j}=\mathbf{e}_{j}^{+}=\mathbf{e}_{j}^{-}$. $a_{j}^{\rm J}$ and
$b_{j}^{\rm J}$ are the field expansion coefficients and in an infinite
structure, such as the nanowire, have the following simple expressions 32:

$\begin{split}a_{j}^{\rm
J}=-\frac{-i\omega\mathbf{p}\cdot\mathbf{e}_{j}^{+}(\bf r_{\rm
d})}{2}\end{split}$ (8) $\begin{split}b_{j}^{\rm
J}=-\frac{-i\omega\mathbf{p}\cdot\mathbf{e}_{j}^{-}(\bf r_{\rm
d})}{2}.\end{split}$ (9)

The expansion coefficients can then be represented by vectors:

$\begin{split}\mathbf{a}_{\infty\rm NW}=\begin{bmatrix}a_{1}^{\rm
J}&a_{2}^{\rm J}&\cdots&a_{N}^{\rm J}\end{bmatrix}\end{split}$ (10)
$\begin{split}\mathbf{b}_{\infty\rm NW}=\begin{bmatrix}b_{1}^{\rm
J}&b_{2}^{\rm J}&\cdots&b_{N}^{\rm J}\end{bmatrix},\end{split}$ (11)

where subscript $\infty\rm NW$ refers to the infinite nanowire.

####  3.1.3 Multilayered structures and scattering matrices

The reflection and transmission matrices are used to connect the field at the
interfaces between the layers. These matrices are derived from the boundary
condition that the tangential components of the electric and magnetic field
are continuous across an interface 32. The coefficients in the matrices
describe how a given mode is transmitted or reflected into another mode. A
reflection matrix is shown in Eq. (LABEL:refmatrix).

$\displaystyle\begin{split}\mathbf{R}&=\begin{bmatrix}r_{11}&r_{12}&r_{13}&\dots&r_{1N}\\\
r_{21}&r_{22}&r_{23}&\dots&r_{2N}\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
r_{N1}&r_{N2}&r_{N3}&\dots&r_{NN}\end{bmatrix}.\\\ \end{split}$ (12)

The $r_{11}$ coefficient represents the reflection of the fundamental mode
back into itself, while the remaining part of the first column represents the
reflection of the fundamental mode into all other modes and so forth.
Propagation matrices are used to propagate the field inside a layer and are
defined in the following way:

$\displaystyle\begin{split}\mathbf{P}(z)&=\begin{bmatrix}e^{i\beta_{1}z}&0&0&\dots&0\\\
0&e^{i\beta_{2}z}&0&\dots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\dots&e^{i\beta_{N}z}\end{bmatrix}.\\\ \end{split}$ (13)

The total field inside the structure can then be calculated by taking into
account the round-trips which the initially emitted light takes inside the
cavity. Above the emitter, the field takes the following expression 32:

$\begin{split}\mathbf{E}_{\rm T}(\mathbf{r})&=\sum_{j}a_{\mathrm{tot},j}^{\rm
J}\mathbf{e}_{j}^{+}(\mathbf{r}_{\perp})\exp(i\beta_{j}(z-z_{\rm J}))\\\
&+b_{\mathrm{tot},j}\mathbf{e}_{j}^{-}(\mathbf{r}_{\perp})\exp(-i\beta_{j}(z-z_{\rm
J}))\quad(z>z_{\rm J}),\end{split}$ (14)

where the new expansion coefficients are calculated using the following
equation 32:

$\begin{split}\mathbf{a}_{\rm tot}^{\rm J}&=(\mathbf{I}-\mathbf{P}(h_{\rm
b})\mathbf{R}_{\rm bot}\mathbf{P}(h_{\rm b})\mathbf{P}(h_{\rm
t})\mathbf{R}_{\rm top}\mathbf{P}(h_{\rm t}))^{-1}\\\ &(\mathbf{a}_{\infty\rm
NW}+\mathbf{P}(h_{\rm b})\mathbf{R}_{\rm bot}\mathbf{P}(h_{\rm
b})\mathbf{b}_{\infty\rm NW})\end{split}$ (15)

and

$\begin{split}\mathbf{b}_{\rm tot}=\mathbf{P}(h_{\rm t})\mathbf{R}_{\rm
top}\mathbf{P}(h_{\rm t})\mathbf{a}_{\rm tot}^{\rm J}.\end{split}$ (16)

Now the Purcell factor can be calculated by evaluating the total field at the
dipole position using Eq. (5).

####  3.1.4 Far-field and efficiency

To obtain the field in the air above the structure, we apply the propagation
and transmission matrix on the forward propagating light in the cavity and
thus obtain the expansion coefficients.

$\begin{split}\mathbf{a}_{\rm air}=\mathbf{T}_{\rm top}\mathbf{P}(h_{\rm
t})\mathbf{a}_{\rm tot}^{\rm J}.\end{split}$ (17)

To calculate the collected power in a lens with some numerical aperture, a
near- to far-field transformation is used 46. The far-fields
$\mathbf{E}_{\textrm{FF}}(R,\theta,\phi)$ and
$\mathbf{H}_{\textrm{FF}}(R,\theta,\phi)$ are calculated on the surface of a
sphere with radius R, and the radial component of the resulting Poynting
vector is:

$\begin{split}S_{\textrm{FF}}(R,\theta,\phi)=(E^{*}_{\textrm{FF},\theta}H_{\textrm{FF},\phi}-E^{*}_{\textrm{FF},\phi}H_{\textrm{FF},\theta}).\end{split}$
(18)

The collected power in the far-field is then:

$\begin{split}P_{\textrm{FF}}(\textrm{NA})&=\frac{1}{2}R^{2}\int_{0}^{2\pi}\int_{0}^{\theta_{\textrm{NA}}}S_{\textrm{FF}}(R,\theta,\phi)\sin(\theta)d\theta
d\phi\\\
&=\int_{0}^{2\pi}\int_{0}^{\theta_{\textrm{NA}}}p_{\textrm{FF}}(\theta,\phi)\sin(\theta)d\theta
d\phi,\\\ \end{split}$ (19)

where $p_{\textrm{FF}}(\theta,\phi)$ is the power per unit solid angle in the
far-field and $\theta_{\textrm{NA}}$ is determined by the $\mathrm{NA}$
($\textrm{NA}=\sin(\theta_{\textrm{NA}})$). The $R$ dependence cancels out as
the Poynting vector scales as $1/R^{2}$.

####  3.1.5 Single-mode Fabry-Pérot model

When calculating the Purcell factor using the SMM, we only consider the
fundamental mode. The SMM equations equivalent to Eq. (15-16) are

$\begin{split}\mathrm{a}_{\rm tot,SMM}^{\rm J}&=a_{1}^{\rm J}\frac{1+r_{11,\rm
bot}\mathrm{e}^{i2h_{\rm b}\beta_{1}}}{1-r_{11,\rm bot}r_{11,\rm
top}\mathrm{e}^{i2h_{\rm total}\beta_{1}}}\end{split},$ (20)

and

$\begin{split}\mathrm{b}_{\rm tot,SMM}&=\mathrm{a}_{\rm tot,SMM}^{\rm
J}r_{11,\rm top}\mathrm{e}^{i2h_{\rm t}\beta_{1}}\end{split}.$ (21)

The collected power in the far-field is then calculated by inserting Eq. (20)
into Eq. (17), and the SMM efficiency is then defined as
$\varepsilon_{s}=P_{\rm collected,SMM}/P_{\rm T}$, equivalent to the
definition in the introduction.

###  3.2 Mode-coupling and emission channels

Important coupling effects take place at the top and bottom interfaces of the
nanowire. At both interfaces, all the modes couple to each other, i.e. all the
elements in the reflection matrices are non-zero; however, some modes and
elements are more important than others. In Fig. (4), different examples of
mode coupling are shown along with the emission channels that will contribute
to the far-field. The sketch is divided into two parts: the main channels and
the background channels. The main channels consist of all the light that
originated as the fundamental mode, $\alpha_{\rm t}$ and $\alpha_{\rm b}$,
which is indicated by the red arrowheads. This is the propagating mode that
experiences sufficiently large reflections at both interfaces such that it is
Purcell enhanced.

Fig. 4: Sketch of the nanopost and the different emission channels and
examples of mode-coupling. The emission channels are separated into the main
channels and the background channels. The red color corresponds to the
fundamental mode, the blue color corresponds to radiation modes and the green
color corresponds to evanescent modes. Some arrows have two colors, where the
color of the arrowhead corresponds to the original channel, but the 2nd color
on the shaft signifies the current mode classification. As an example consider
$c_{1}$ which is the transmission of the fundamental mode into the air. It has
a partly blue shaft as it is now classified as radiation, but it originated as
the fundamental mode (red arrowhead). See main text for more information.

####  3.2.1 Path of the fundamental mode

Let us now follow the path of the fundamental mode. The light emitted into the
fundamental mode will propagate upwards and downwards indicated by
$\alpha_{\rm t}$ and $\alpha_{\rm b}$. At the top interface the fundamental
mode is:

  * •

Transmitted into the air indicated by $c_{1}$.

  * •

Reflected, indicated by $\alpha_{\rm t,r}$ (the channel responsible for
Purcell enhancement).

  * •

Scattered into radiation that propagates downwards indicated by the red/blue
arrow pointing towards the bottom mirror. This radiation will then be
reflected by the bottom mirror and then be transmitted into the air indicated
by $c_{2}$.

  * •

Coupled to evanescent modes indicated by $\alpha_{\rm t,\gamma}$.

At the bottom interface the fundamental mode is:

  * •

Transmitted into the mirror and lost indicated by the green/red arrow pointing
downwards at the very bottom.

  * •

Reflected, indicated by $\alpha_{\rm b,r}$ (the channel responsible for
Purcell enhancement).

  * •

Scattered into radiation that propagates towards the air indicated by $c_{3}$.

  * •

Coupled to evanescent modes indicated by $\alpha_{\rm b,\gamma}$.

As we will demonstrate in the following, the three radiation channels $c_{1}$,
$c_{2}$ and $c_{3}$ are the main channels that will contribute to the far-
field.

####  3.2.2 Path of the background emission

Let us now consider the background emission channels. First, we have the light
directly emitted into radiation, indicated by the blue arrows of $\beta_{\rm
t}$ and $\beta_{\rm b}$. The radiation can both be emitted upwards or
downwards and then reflected by the mirror. At both interfaces, a small part
of the radiation modes can also couple to the fundamental mode indicated by
the blue/red arrows of $\beta_{\rm t,\alpha}$ and $\beta_{\rm b,\alpha}$. We
also have light coupled to the evanescent modes, which is indicated by the
long green arrows pointing upwards and downwards of $\gamma_{\rm t}$ and
$\gamma_{\rm b}$. These modes do not propagate in a traditional sense, but at
the interfaces, they can scatter into the fundamental mode indicated by the
green/red arrows of $\gamma_{\rm t,\alpha}$ and $\gamma_{\rm b,\alpha}$ at the
top and bottom interfaces. At the top interface, the evanescent modes can also
couple to radiation and be transmitted indicated by the green/blue arrow
pointing upwards, $\gamma_{\rm t,\beta}$.

##  4 Analysis of the Purcell factor and efficiency

Fig. 5: Power emission in the infinite nanowire as a function of the diameter,
$D$. A sketch of the emission in an infinite nanowire with an embedded QD is
shown in the right part of the figure.

The starting point of the analysis is the emission rates in the infinitely
long GaAs nanowire, which directly represent the initial coefficients of Eq.
(10-11) through Eq. (5-7). In Fig. (5), the emission rates for the present
guided modes, the radiation modes and the total emission are shown as a
function of the nanowire diameter. The black dotted vertical lines represent
the interval which is used in the full simulations of the nanopost. In this
interval, the infinite nanowire only contains one guided mode, the fundamental
$\mathrm{HE}_{11}$ mode, and most of the power is emitted into this mode. The
emission into radiation is suppressed in most of the interval and only begins
to increase when the diameter reaches $300\text{\,}\mathrm{n}\mathrm{m}$
before the $\mathrm{EH}_{11}$ mode appears. The emission rates thus show that
the radiation background channel (the initial coefficients for the radiation)
presumably only has a minor influence on the Purcell factor and efficiency as
long as the cavity does not suppress the fundamental mode. Fig. (5) cannot be
used to quantify the importance of the evanescent background channels, as the
field components of the evanescent eigenmodes only have a real part. Later on,
in the third subsection, models will be used to quantify the effect of the
evanescent background channels the finite-length nanopost structure.

###  4.1 Scattering of the fundamental mode at the interfaces

Here, we will study reflection and transmission at the top interface between
the GaAs nanowire and the air above. The fundamental mode is launched towards
the interface, and then the reflection and transmission coefficients are
calculated. The power reflection coefficient for reflection of mode $n$ into
mode $m$ is $R_{m,n}=|r_{m,n}|^{2}$, and the power transmission coefficients
are defined similarly. The total power reflection of the fundamental into
radiation is then defined as
$R_{\mathrm{rad},1}=\sum_{s=1}^{2}\int_{0}^{k_{0}}|r_{s,1}(k_{\perp})|^{2}dk_{\perp}$,
and in the discretized regime
$R_{\mathrm{rad},1}=\sum_{n=2}^{N_{k_{0},\mathrm{wire}}+1}|r_{n,1}|^{2}$,
where the index $N_{k_{0},\mathrm{wire}}$ corresponds to the total number of
radiation modes in the nanowire. Thus the total power reflection of the
fundamental mode is $R_{\mathrm{total},1}=R_{1,1}+R_{\mathrm{rad},1}$. The
total power transmission of the fundamental mode is then defined as
$T_{\mathrm{total},1}=\sum_{n=1}^{N_{k_{0},\mathrm{air}}}|t_{n,1}|^{2}$ and
due to power conservation we have
$R_{\mathrm{total},1}+T_{\mathrm{total},1}=1$.

Fig. 6: Reflection and transmission of the fundamental mode as a function of
the diameter, $D$, at the top interface. A sketch of top interface and the
reflection and transmission of the fundamental mode is seen to the right.

In Fig.(6), the power reflections, along with the power transmission of the
fundamental mode, are shown as a function of the nanowire diameter $D$. By
comparing the magnitudes of the modal reflection ($R_{1,1}$) and the
reflection into radiation ($R_{\mathrm{rad},1}$), it is clear that the
reflection into radiation ($c_{2}$ in Fig.(4)) is essential for the far-field.
This indicates why the SMM fails to describe the efficiency. This mechanism
will have a much smaller influence on the Purcell factor as the radiation
modes only has a small field amplitude at the center of the nanowire shown in
the previous section (Fig. (5)). However, the reflection matrix is
approximately symmetric, such that the first column and the first row is
identical, $r_{1,n}\approx r_{n,1}$. This means that a small part of the
radiation will actually scatter back into the fundamental mode at the top
interface, and this will have an influence on the Purcell factor, which we
will show later. In general for Fig. (6), we observe small modal reflections
for small diameters and this will lead to a limited cavity effect and thus a
lower Purcell factor.

We now consider the reflections at the bottom interface between the GaAs
nanowire and the silica-gold mirror. Compared to the top interface, there is
now an additional parameter, namely the thickness of the silica layer, $t_{\rm
SiO_{2}}$. The purpose of the silica layer is to increase the reflection of
the fundamental mode and avoid coupling to surface plasmons which would
decrease the reflection 47. In Fig. (7), the bottom reflection coefficients
are shown as a function of the diameter and the silica layer thickness. In the
parameter ranges where the modal reflection is large $\sim 0.9$ (strong cavity
effect), the reflection into radiation is small $\sim 0.01$. Here we do not
expect a significant contribution of the scattering into radiation, $c_{3}$.
However, as the modal reflection decreases, the reflection into radiation
increases to larger values $\sim 0.1$, and here $c_{3}$ will contribute to the
far-field. For small diameters and low values of the silica layer thickness,
the modal reflection is small (weak cavity effect) and the scattering into
radiation is very large $\sim 0.2$. This behavior has been described in the
literature47.

Fig. 7: (a) Modal reflection, $R_{1,1}$ and (b) reflection into radiation of
the fundamental mode, $R_{rad,1}$, as a function of the diameter, $D$, and
silica layer thickness, $t_{\mathrm{SiO_{2}}}$, at the bottom interface. A
sketch of the bottom interface is shown in the top part of the figure.

Due to the scale invariance of Maxwell’s equations, both the top and bottom
reflections are broadband, which gives the potential for broadband Purcell
enhancement.

###  4.2 Enhanced efficiency

In the first part of this subsection, we present different methods to model
the efficiency to show how important the different emission channels are. Then
we will apply the modelling methods on the structure with the largest
efficiency, namely $D=$238\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$0\text{\,}\mathrm{n}\mathrm{m}$$ with a QD placed at the 2nd
antinode.

####  4.2.1 Efficiency contributions of the emission channels

We wish to separate and quantify the efficiency contributions of the main
channels and the background channels shown in Fig. (4). We also wish to
separate and quantify the direct emission of $c_{1}$ and the scattered
channels of $c_{2}$ and $c_{3}$ shown in Fig. (4), and therefore we need two
different methods. Recall that the efficiency is calculated as
$\varepsilon=P_{\rm collected}/P_{\rm T}$. For both methods, $P_{\rm T}$ is
calculated using the full model, but $P_{\rm collected}$ is calculated such
that we can either 1. separate the main channels and the background channels
or 2. separate the direct emission of $c_{1}$ and the scattered channels of
$c_{2}$ and $c_{3}$.

In the first method, the reflection and transmission matrices are unchanged;
however, a varying number of the initial coefficients, $\mathbf{a}_{\infty\rm
NW}$ and $\mathbf{b}_{\infty\rm NW}$ of Eq. (10-11), are included when
$\mathbf{a}_{\rm tot}^{\rm J}$ (Eq. (15)) and thus $P_{\rm collected}$ is
calculated. For instance, if we only include the first element of the initial
coefficients, $a_{1}^{\rm J}$ and $b_{1}^{\rm J}$, and put the remaining
elements to zero, then we only include the main channels in $P_{\rm
collected}$, which originated as the fundamental mode. As we increase the
number of initial coefficients included, the background channels are added
starting from the first radiation mode until the last evanescent mode.
$\varepsilon$ can then be plotted as a function of the included initial
coefficients and if this curve is flat, then the main channels dominate the
efficiency.

In the second method, the reflection and transmission matrices are also
unchanged, all initial coefficients are included, but instead, a varying
number of the final coefficients, $\mathbf{a}_{\rm tot}^{\rm J}$ of Eq. (15),
are included when $P_{\rm collected}$ is calculated. The first element of
$\mathbf{a}_{\rm tot}^{\rm J}$ represents all the light that ended up in the
fundamental mode, where the main contribution is from the fundamental mode
itself, but also includes contributions of the background channels which have
scattered into the fundamental mode such as $\beta_{\rm t,\alpha}$,
$\beta_{\rm b,\alpha}$, $\gamma_{\rm t,\alpha}$ and $\gamma_{\rm b,\alpha}$
seen in Fig. (4). However, by using the first method, we can quantify how
strong these contributions are. If these contributions are weak, then the main
channels are dominating. Thus if only the first element of the final
coefficients is included in $P_{\rm collected}$, then only the direct
transmission of the fundamental mode is included in the far-field, namely
$c_{1}$ in Fig.(4). Then as we increase the number of elements of the final
coefficients, the contributions of $c_{2}$ and $c_{3}$ are included. In
principle, channels such as $\beta_{\rm t}$ and $\beta_{\rm b}$ are also
included, but again we use the previous method to quantify these.
$\varepsilon$ can then be plotted as a function of the included final
coefficients, and if this curve is increasing, then the channels $c_{2}$ and
$c_{3}$ are contributing to the efficiency.

Finally, we wish to visualize the interference between the direct transmission
of the fundamental mode and the scattered channels by calculating the
transmission of the fundamental mode and the entire background continuum
separately:

$\begin{split}\mathbf{a}_{\mathrm{air},\mathrm{HE}_{11}}=\mathbf{T}_{\rm
top}\mathbf{P}(h_{\rm t})\mathbf{a}_{\mathrm{tot},\mathrm{HE}_{11}}^{\rm
J}\end{split}$ (22)

and

$\begin{split}\mathbf{a}_{\rm air,BG}=\mathbf{T}_{\rm top}\mathbf{P}(h_{\rm
t})\mathbf{a}_{\rm tot,BG}^{\rm J},\end{split}$ (23)

where

$\begin{split}\mathbf{a}_{\mathrm{tot},\mathrm{HE}_{11}}^{\rm
J}=\begin{bmatrix}a_{\rm tot,1}&0&\cdots&0\end{bmatrix}\end{split}$ (24)

and

$\begin{split}\mathbf{a}_{\rm tot,BG}^{\rm J}=\begin{bmatrix}0&a_{\rm
tot,2}&\cdots&a_{\mathrm{tot},N}\end{bmatrix}.\end{split}$ (25)

Then the phase difference between the two contributions can be calculated:

$\begin{split}\Delta\phi=\arg(\mathbf{a}_{\mathrm{air},\mathrm{HE}_{11}})-\arg(\mathbf{a}_{\rm
air,BG}).\end{split}$ (26)

The phase difference will be separated into TE and TM modes. Along with the
phase difference, the far-field plots of the direct transmission of the
fundamental mode ($\mathbf{a}_{\mathrm{air},\mathrm{HE}_{11}}$), the entire
background continuum ($\mathbf{a}_{\mathrm{air},\mathrm{BG}}$) and the total
field ($\mathbf{a}_{\rm air}$) will be shown.

####  4.2.2 Influence of scattered radiation on collection efficiency

Fig. 8: (a) Efficiency, $\varepsilon$ ($\mathrm{NA}=0.75$), as a function of
the initial coefficients expressed with the propagation constant
$(\beta/k_{0})^{2}$. (b) Efficiency, $\varepsilon$ ($NA=75$), as a function of
the final coefficients expressed with the propagation constant
$(\beta/k_{0})^{2}$.

We will now apply the modelling methods for the structure with the largest
efficiency, $D=$238\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$0\text{\,}\mathrm{n}\mathrm{m}$$. In Fig. (8) the efficiency is
shown as a function of the initial coefficients (Fig. (8a)) and the final
coefficients (Fig. (8b)), expressed with the propagation constant
$(\beta/k_{0})^{2}$. This corresponds to using the two different methods for
calculating the efficiency presented in the previous subsection. The very
first red point in Fig. (8a) corresponds to the fundamental mode (main
channels) and includes all scattering channels of the fundamental mode
($c_{1}$, $c_{2}$ and $c_{3}$). This is sufficient to describe most of the
efficiency. Then the initial background radiation modes (blue) are added one
by one, starting from larger values of $(\beta/k_{0})^{2}$, i.e. from
predominantly vertical emission. This part of the curve is very flat, which
means that the initial background radiation modes (channels originating from
$\beta_{t}$ and $\beta_{b}$) are not crucial for the efficiency, which was
also indicated by the low emission rates of the radiation modes in the
infinite nanowire. Finally, the evanescent background modes (green) are
included from small negative values of $(\beta/k_{0})^{2}$, i.e. slowly
decaying evanescent modes, to large negative values of $(\beta/k_{0})^{2}$,
i.e. fast decaying evanescent modes. Here there is a small increase due to the
slowly decaying evanescent modes. Now in Fig. (8b), the very first red point
also corresponds to the fundamental mode but only includes the direct
transmission, $c_{1}$. Here the efficiency is only $\sim 0.2$, much smaller
than the total efficiency. Then the scattered radiation modes (blue)
corresponding to $c_{2}$ and $c_{3}$ are added one by one, also starting from
larger values of $(\beta/k_{0})^{2}$, i.e. from predominantly vertical
emission. Here there is a massive increase in the efficiency, which proves the
importance of $c_{2}$ and $c_{3}$ to the efficiency. At some point, there is a
kink in the blue part of the curve, which is due to the limited numerical
aperture, as an $\rm NA$ of 0.75 corresponds to $(\beta/k_{0})^{2}=0.36$.
However, the remaining part of the curve is not completely flat, and this is
due to the non-perfect transmission of the radiation modes of the nanowire to
the radiation modes in the air. The radiation modes of the nanowire mainly
transmit into radiation modes with the same value of $\beta$, but there is
some scattering into the other radiation modes. Finally, there is a tiny
decrease due to the evanescent modes, as very few of the evanescent modes
transmit into radiation at the top interface. By comparing the evanescent
parts of Fig. (8a) and Fig. (8b), we see that the effect of the evanescent
modes is mainly back scattering into other modes at the interfaces rather than
direct transmission scattering. To summarize, the key point in the comparison
between Fig. (8a) and Fig. (8b) is that the scattering into radiation of the
fundamental mode is crucial for the efficiency.

Fig. 9: (a) The phase difference between the direct transmission of the
fundamental mode and the background continuum for TE and TM modes as a
function of the propagation constant, $\beta/k_{0}$. (b) The far-field of the
fundamental mode. (c) The far-field of the background continuum. (d) The total
far-field. The white dotted line indicates $\mathrm{NA}=0.75$. Be aware of the
different color scales that have been used for the far-fields.

Now we will visualize the interference between the direct transmission of the
fundamental mode and the scattered channels by inspecting the phase changes
and the far-fields. In Fig. (9a), the phase difference in the air layer
between the direct transmission of the fundamental mode and the entire
background is shown as a function of the propagation constant for TE and TM
modes. For the light that propagates vertically, the phase difference is close
to zero, such that there is constructive interference between the direct
transmission and the background radiation. For the light that propagates
horizontally, the phase difference is closer to $\pi$, and thus there is
destructive interference. In Fig. (9b), Fig. (9c) and Fig. (9d) the far-fields
of the direct transmission of the fundamental mode, the background radiation
and the total field are shown. Here we can directly observe the effect caused
by the phase difference. In the center of the total far-field, the field is
enhanced due to the constructive interference, but for the light that
propagates horizontally there is destructive interference. As such, the
interference between the direct emission and the radiation focuses the far-
field.

A similar analysis of the efficiency for the structure with the largest
Purcell factor is included in Supplementary 7.4.

###  4.3 Enhanced Purcell factor

As shown in Fig. (2), there are deviations between the full model and the SMM
for the Purcell factor, and we wish to understand where these deviations
appear. Therefore we will introduce a model which can identify where these
deviations appear and apply the model on the structure with the largest
Purcell factor, namely $D=$250\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$13\text{\,}\mathrm{n}\mathrm{m}$$.

####  4.3.1 Purcell factor contributions from the emission channels

To gain physical insight into the physics of the Purcell factor, we will use a
model which stepwise increases the complexity. At each step the Purcell factor
is calculated $F_{\rm p}=P_{\rm T}/P_{\rm Bulk}$ along with the power into the
fundamental mode, $P_{\mathrm{HE}_{11}}/P_{\rm Bulk}$. The starting point is
the SMM, where only the fundamental mode is included; then, in seven steps,
the complexity increases until the full model is reached. Specifically, the
initial inputs ($\mathbf{a}_{\infty\rm NW}$ and $\mathbf{b}_{\infty\rm NW}$)
and the top and bottom reflections ($\mathbf{R}_{\rm top}$ and
$\mathbf{R}_{\rm bot}$) will be manipulated at each step. Each step has a
direct physical interpretation. We will now list the 7 steps in the model and
for each step, write up the physical effect that is now included. This means
that for each step in the model, all previous effects are also included.

  1. 1.

SMM

  2. 2.

Scattering of the fundamental mode at the top interface

  3. 3.

Scattering of the fundamental mode at the bottom interface

  4. 4.

Back-scattering of the background continuum to the fundamental mode at the top
interface

  5. 5.

Back-scattering of the background continuum to the fundamental mode at the
bottom interface

  6. 6.

Scattering of the background continuum to itself at both interfaces

  7. 7.

Including initial background continuum

Steps number 4 and 5 correspond to the process
$\textrm{HE}_{11}\rightarrow\textrm{radiation/evanescent}\rightarrow\textrm{HE}_{11}$
which is a recycling effect, and we will show the importance of this process
for the Purcell factor. Step number 6 corresponds to the process
$\textrm{radiation/evanescent}\rightarrow\textrm{radiation/evanescent}$, which
also opens up for further scattering channels such as
$\textrm{radiation/evanescent}\rightarrow\textrm{radiation/evanescent}\rightarrow\textrm{HE}_{11}$.

The 7 steps can then be translated to the initial inputs and the top and
bottom reflections in the following schematic way:

$\begin{split}\mathbf{a}_{\infty\rm
NW}^{(\alpha)}&=\begin{bmatrix}1&7&\cdots&7\end{bmatrix}\\\ \end{split}$ (27)
$\begin{split}\mathbf{b}_{\infty\rm
NW}^{(\alpha)}&=\begin{bmatrix}1&7&\cdots&7\end{bmatrix}\\\ \end{split}$ (28)
$\begin{split}\mathbf{R}_{\rm top}^{(\alpha)}&=\begin{bmatrix}1&4&4&\dots&4\\\
2&2&6&\dots&6\\\ 2&6&2&\dots&6\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
2&6&6&\dots&2\end{bmatrix}_{\rm top}\\\ \end{split}$ (29)
$\begin{split}\mathbf{R}_{\rm bot}^{(\alpha)}&=\begin{bmatrix}1&5&5&\dots&5\\\
3&2&6&\dots&6\\\ 3&6&2&\dots&6\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
3&6&6&\dots&2\end{bmatrix}_{\rm bot}\end{split}$ (30)

Here, the superscript $\alpha$ represents the seven complexity steps. For a
given step $\alpha$, each matrix entry $>\alpha$ is set to zero, while entries
$\leq\alpha$ keep their original value. We will use the step $\alpha=4$ as an
example:

$\begin{split}\mathbf{a}_{\infty\rm
NW}^{(4)}&=\begin{bmatrix}a_{1}&0&\cdots&0\end{bmatrix}\end{split}$ (31)
$\begin{split}\mathbf{b}_{\infty\rm
NW}^{(4)}&=\begin{bmatrix}b_{1}&0&\cdots&0\end{bmatrix}\end{split}$ (32)
$\begin{split}\mathbf{R}_{\rm
top}^{(4)}&=\begin{bmatrix}r_{11}&r_{12}&r_{13}&\dots&r_{1N}\\\
r_{21}&r_{22}&0&\dots&0\\\ r_{31}&0&r_{33}&\dots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
r_{N1}&0&0&\dots&r_{NN}\end{bmatrix}_{\rm top}\\\ \end{split}$ (33)
$\begin{split}\mathbf{R}_{\rm bot}^{(4)}&=\begin{bmatrix}r_{11}&0&0&\dots&0\\\
r_{21}&r_{22}&0&\dots&0\\\ r_{31}&0&r_{33}&\dots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
r_{N1}&0&0&\dots&r_{NN}\end{bmatrix}_{\rm bot}\end{split}$ (34)

At each step in the previous model building, the entire background continuum
was added, i.e. the entire column or row was added at once. As such, it is
difficult to quantify which part of the background continuum that is
important. Therefore we will also present the models where the elements for
the background continuum are increased one at a time. In this way it can be
quantified how the different parts of the background continuum contribute to
the Purcell factor.

####  4.3.2 Influence of radiation modes on the Purcell factor

Fig. 10: Purcell factor, $F_{\rm p}$, computed using the full model and SMM as
a function of the dipole position from the bottom interface, $h_{\rm b}$.
$D=$250\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$13\text{\,}\mathrm{n}\mathrm{m}$$.

We will now apply the model for the Purcell factor for the structure with the
largest Purcell factor, $D=$250\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$13\text{\,}\mathrm{n}\mathrm{m}$$. In Fig. (10), the Purcell factor
is shown as a function of the dipole position from the bottom, $h_{\rm b}$.
The agreement between the SMM and the full model for the 1st antinode is good,
but there are deviations for the 2nd and 3rd antinodes. Nevertheless, we will
focus on the analysis of the 1st and the 2nd antinode.

Fig. 11: Purcell factor ($P_{\rm T}/P_{\rm Bulk}$) and fundamental mode
enhancement ($P_{\mathrm{HE}_{11}}/P_{\rm Bulk}$) for the 1st (a) and 2nd (b)
antinode as a function of the model complexity progression. In (c) and (d) the
background continuum is continuously included between each model complexity.

In Fig. (11a) and Fig. (11b), the Purcell factor and the power enhancement of
the fundamental mode are shown as a function of the model complexity
progression for the 1st and 2nd antinode (model complexity number $\alpha$
will be shortened n. $\alpha$). Evidently, the analysis of the Purcell factor
is complicated due to contributions of the entire background continuum,
multiple scattering channels and feedback mechanism. Therefore there are
changes in the Purcell factor for all steps in the model complexity, which
makes it challenging to model the Purcell factor using only a few modes. The
continuum of radiation modes can be modelled using leaky modes, which can
enable the modelling using only a few modes. This has been demonstrated in
photonic crystal microcavities where strong feedback mechanisms also were
present 48. However, by using the presented model, we will obtain an in-depth
physical insight.

The Purcell factor starts at the same value with the SMM for both antinodes.
By including scattering of the fundamental mode at the top interface (n. 2)
there is a significant increase for the 2nd antinode but a very small decrease
for the 1st antinode. Already now, the deviations compared to the SMM have
started to appear. By including the scattering at the bottom interface (n. 3)
the 2nd antinode is almost unaffected, but a small decrease appears for the
1st antinode. Now including the back-scattering at the top interface (n. 4),
there is a large decrease in the Purcell factor at both antinodes. This
decrease is directly represented in $P_{\mathrm{HE}_{11}}/P_{\rm Bulk}$.
Interestingly, when including the back-scattering at the bottom interface (n.
5), there is now a large increase in the Purcell factor at both antinodes,
which is also represented in $P_{\mathrm{HE}_{11}}/P_{\rm Bulk}$. This also
shows that the recycling effect
$\textrm{HE}_{11}\rightarrow\textrm{radiation/evanescent}\rightarrow\textrm{HE}_{11}$
can provide both negative and positive contributions. When the background is
allowed to scatter to itself, i.e.
$\textrm{radiation/evanescent}\rightarrow\textrm{radiation/evanescent}$ (n.
6), there is an increase for both antinodes. This increase is directly
represented in $P_{\mathrm{HE}_{11}}/P_{\rm Bulk}$, which in fact means that
the process
$\textrm{radiation/evanescent}\rightarrow\textrm{radiation/evanescent}\rightarrow\textrm{HE}_{11}$
is dominating compared to
$\textrm{radiation/evanescent}\rightarrow\textrm{radiation/evanescent}$
itself. Finally, by including the initial background (n. 7), there is an
increase for the 2nd antinode but a decrease for the 1st antinode. These
changes also correspond to the change in $P_{\mathrm{HE}_{11}}/P_{\rm Bulk}$,
which means it is the process of
$\textrm{radiation/evanescent}\rightarrow\textrm{HE}_{11}$ that is important
for the initial background.

The main differences between the two antinodes appear, when the fundamental
mode scatters at the top interface (n. 2) and when the initial background (n.
7) is included. To better understand which part of the background continuum is
important, we will also consider the continuous steps of the model complexity.
This is now shown in Fig. (11c) and Fig. (11d). Here we observe that the main
positive contributions at the 2nd antinode are due to the slowly decaying
evanescent modes. This is the process of
$\textrm{HE}_{11}\rightarrow\textrm{evanescent}$ at the top interface and the
process of the initial background
$\textrm{evanescent}\rightarrow\textrm{HE}_{11}$. For the back-scattering at
the interfaces (n. 4 and n. 5), we observe that the propagating radiation
modes can also significantly affect the Purcell factor.

An additional important observation is that $P_{\mathrm{HE}_{11}}/P_{\rm
Bulk}$ exceeds $P_{\rm T}/P_{\rm Bulk}$ for the first antinode due to the
negative contributions. This would in fact result in $\beta$ factors above 1
using the definition $\beta_{\mathrm{HE}_{11}}=P_{\mathrm{HE}_{11}}/P_{\rm
T}$. This indicates that the $\beta$ factor might not be a suitable figure of
merit for structures where the SMM breaks down, or at least one should be very
careful in the definition of the $\beta$ factor. Alternatively, the typical
interpretation of the $\beta$ factor as a standard power fraction should be
reconsidered in the regime of the SMM breakdown.

The analysis of the Purcell factor for the structure with the largest
efficiency, $D=$238\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$0\text{\,}\mathrm{n}\mathrm{m}$$, is included in Supplementary 7.5.

###  4.4 Wavelength dependence

In this section, we will present the broadband performance of the nanopost.
The focus will be on the designs with the largest Purcell factor and
efficiency, respectively.

Fig. 12: (a) and (b) Purcell factor, $F_{\rm p}$, of the two antinodes for the
two structures as a function of wavelength, $\lambda$. (c) and (d) efficiency,
$\varepsilon$ ($\mathrm{NA}=0.75$), of the two antinodes for the two
structures as a function of wavelength, $\lambda$. The numerical uncertainty
is represented by the thickness of the curves.

In Fig. (12a), the Purcell factor is shown as a function of the wavelength for
the 1st and 2nd antinode of the structure with the largest Purcell factor. The
Purcell factor of the 2nd antinode performs better than the 1st antinode close
to the resonance wavelength, and the spectral width at FWHM (full width half
maximum) of the antinodes are approximately
$\Delta\lambda_{\mathrm{2nd}}=$27\text{\,}\mathrm{n}\mathrm{m}$$ and
$\Delta\lambda_{\mathrm{1st}}=$28\text{\,}\mathrm{n}\mathrm{m}$$ showcasing
the broadband performance. The spectrum for the two antinodes is not
completely symmetric, and the two curves for the antinodes also cross further
away from the resonance. In Fig. (12b), the Purcell factor is shown as a
function of the wavelength for the 1st and 2nd antinode of the structure with
the largest efficiency (at resonance). Here the Purcell factor is much lower
and the spectral width much broader at
$\Delta\lambda_{\mathrm{2nd}}=$52\text{\,}\mathrm{n}\mathrm{m}$$ and
$\Delta\lambda_{\mathrm{1st}}=$56\text{\,}\mathrm{n}\mathrm{m}$$. Furthermore,
the resonance wavelength for the 1st antinode is slightly shifted to
$\lambda_{\rm r}=$931.5\text{\,}\mathrm{n}\mathrm{m}$$. This can be explained
by the low Q factor and the neighbouring low Q factor cavity modes, i.e. the
cavity modes with 2 and 4 antinodes. Due to the low Q factor, there is a small
spectral overlap causing the slight shift. In the Supplementary 7.6, we
present a nanopost design where the resonance wavelength shift is more
pronounced between the two antinodes.

In Fig. (12c) and Fig. (12d) the efficiencies ($\mathrm{NA}=0.75$) are shown
as a function of the wavelength for the 1st and 2nd antinode for the two
structures. The main characteristic of the efficiency is that it does not
follow the Purcell factor, which is surprising compared to traditional Fabry-
Pérot cavities. Instead, the efficiency changes roughly linearly across the
resonance, and in general, the slope for the 2nd antinode is negative and
positive for the 1st antinode. This means that the maximum efficiency is in
fact achieved off-resonance, $\varepsilon_{\rm
2nd,NA=0.75}(\lambda=$880\text{\,}\mathrm{n}\mathrm{m}$)=0.71$, but at a much
smaller Purcell factor.

####  4.4.1 Analysis of the broadband collection efficiency

In the previous work on the nanopost29, the broadband efficiency was
attributed to the broadband $\beta$ factor, which again was attributed to the
dielectric screening effect. This can also be seen in Fig. (5), where the
emission into radiation modes is suppressed in most of the interval. Here we
will define the $\beta$ factor for the fundamental mode as
$\beta_{\mathrm{HE}_{11}}=P_{\mathrm{HE}_{11}}/P_{\rm T}$, even though it
exceeds 1 as we have already shown. The focus is on the structure with the
largest efficiency at resonance, $D=$238\text{\,}\mathrm{n}\mathrm{m}$$ and
$t_{\rm SiO_{2}}=$0\text{\,}\mathrm{n}\mathrm{m}$$.

Fig. 13: $\beta_{\mathrm{HE}_{11}}=P_{\mathrm{HE}_{11}}/P_{\rm T}$ as a
function of the wavelength, $\lambda$, for the 2nd and 1st antinode and the
infinite nanowire. The numerical uncertainty is represented by the thickness
of the curves.

In Fig. (13), $\beta_{\mathrm{HE}_{11}}$ is shown as a function of the
wavelength for the 2nd and 1st antinode and the infinite nanowire.
$\beta_{\mathrm{HE}_{11}}$ is close to 1 (both above and below) in the entire
interval, which indicates that the fundamental mode is still the dominating
contribution. Due to the broadband $\beta_{\mathrm{HE}_{11}}$ of the infinite
nanowire, it is not required to be on resonance to obtain large values of
$\beta_{\mathrm{HE}_{11}}$, and the QD would need to be very close to a node
before $\beta_{\mathrm{HE}_{11}}$ would decrease. Regardless, the fundamental
mode scatters into radiation which affects $P_{\rm T}$ such that
$\beta_{\mathrm{HE}_{11}}$ does not follow a Lorentzian curve like the Purcell
factor.

Fig. 14: Efficiency, $\varepsilon$ ($\mathrm{NA}=0.75$), as a function of the
wavelength, $\lambda$, for the 2nd antinode (a) and 1st antinode (b). The full
model is compared to only using $c_{1}$, $c_{2}$ and $c_{3}$.
$D=$238\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$0\text{\,}\mathrm{n}\mathrm{m}$$. The width of the curves represents
the numerical uncertainty.

To quantify how dominating the fundamental mode is for the efficiency, we will
use the first method presented in Section 4.2.1 to separate the main channels
($c_{1}$, $c_{2}$ and $c_{3}$) from the background channels. In Fig. (14), we
compare the efficiency of the full model to only including $c_{1}$, $c_{2}$
and $c_{3}$ for both antinodes. By comparing to Fig. (13) we observe that the
curves for the main channels directly follow the trend of
$\beta_{\mathrm{HE}_{11}}$. However, there is a discrepancy between the
efficiency of the full model and only using the main channels of $c_{1}$,
$c_{2}$ and $c_{3}$. This discrepancy is not at a minimum on resonance
($\lambda=$930\text{\,}\mathrm{n}\mathrm{m}$$), but almost at the minimum when
$\beta_{\mathrm{HE}_{11}}=1$ for both antinodes. We observe that for
$\beta_{\mathrm{HE}_{11}}<1$, the background channels provide a positive
contribution to the efficiency, while for $\beta_{\mathrm{HE}_{11}}>1$ the
background channels provide a negative contribution. This shows that the
$\beta$ factor is still useful in the analysis but not necessarily a figure of
merit for SPSs in the breakdown of the SMM. Furthermore, Fig. (14) shows that
the background channels still interfere with the main channels causing this
discrepancy and even changing the slope of the curves for the efficiency. As
an example we will consider two wavelengths for the 2nd antinode and use the
first method presented in Section 4.2.1:

In Fig. (15) the efficiency is shown as a function of the initial coefficients
expressed with the propagation constant $(\beta/k_{0})^{2}$ for the two
wavelengths $\lambda=$890\text{\,}\mathrm{n}\mathrm{m}$$ and
$\lambda=$970\text{\,}\mathrm{n}\mathrm{m}$$ at the 2nd antinode. Again the
first red point corresponds to including all the main channels, and then
background channels are added. For
$\lambda=$890\text{\,}\mathrm{n}\mathrm{m}$$, the initial propagating
background radiation provides a significant increase to the efficiency, while
the evanescent modes provide no change. On the other hand, for
$\lambda=$970\text{\,}\mathrm{n}\mathrm{m}$$, the initial propagating
background radiation provides a significant decrease to the efficiency, while
the evanescent modes provide a positive increase to the efficiency. This also
showcases the complex interplay between the main channels and the background
channels.

Fig. 15: Efficiency, $\varepsilon$ ($\mathrm{NA}=0.75$), as a function of the
initial coefficients expressed with the propagation constant
$(\beta/k_{0})^{2}$ for $\lambda=$890\text{\,}\mathrm{n}\mathrm{m}$$ (a) and
$\lambda=$970\text{\,}\mathrm{n}\mathrm{m}$$ (b) at the 2nd antinode.

##  5 Perspective

We have shown that contributions from multiple scattering channels influence
both the Purcell factor and especially the collection efficiency. Unlike
traditional Fabry-Pérot cavities, where scattering of the light is viewed
simply as a loss mechanism, this scattering is in fact beneficial for the
performance of the nanopost SPS. Importantly, this scattering mechanism
decouples the efficiency from the Purcell factor directly challenging a well-
known design paradigm that maximum collection efficiency is obtained on
resonance. The identification of this mechanism opens a door to unconventional
SPS design approaches, especially in the non-resonant regime where the
scattering coefficients are no longer analysed and optimized with respect to
the fundamental HE11 mode alone, and where definitions of fundamental
performance parameters such as the spontaneous emission $\beta$ factor need to
be revisited.

For the nanopost itself, this invites to a new optimization of the collection
efficiency with respect to all geometrical parameters. The maximum collection
efficiency will be obtained for a reduced Purcell factor due to the new trade-
off between efficiency and Purcell enhancement. Potential future work on the
nanopost design could also be to explore the properties of cavity modes with
different orders than 3. Additionally, structuring the bottom mirror could
also lead to increased performance. Adding rings around the nanopost could
positively alter the scattering mechanism while also bridging the gap to the
closely related bullseye design 27, 28, 16, which also features broadband
collection efficiency independently of the Purcell factor. Despite flourishing
literature on the bullseye design, the physical mechanisms underlying the
performance is still unclear. The analysis of the bullseye will be more
challenging as the inner mesa/nanowire features larger diameters resulting in
additional guided modes. The rings around the inner mesa will also heavily
influence the radiation modes and their mode profiles. This will, in turn,
lead to changes in the emission rates and the reflection matrices and, thus,
the scattering channels. In typical bullseye structures27, 28, 16, which are
numerically optimized, the silica layer is hundred of nanometers thick. Here
we anticipate that such large thicknesses will also result in increased mode
coupling as light diverges when propagating in the silica.

##  6 Conclusions

We have shown that the traditional Fabry-Pérot single-mode model, which
typically provides an excellent description 12, 13, 14 of the physics for
cavity-based single-photon sources, significantly underestimates the
achievable performance of the nanopost structure. Using a modal expansion
method, we have performed a detailed analysis of the emission channels. We
have shown that in particular the collection efficiency benefits significantly
from a contribution from light scattered to radiation modes, which often is
simply considered a loss mechanism. This scattering into radiation modes not
only allows for improved collection efficiency but also decouples the
collection efficiency from the Purcell factor, such that optimum performance
is obtained off-resonance. Our parameter scan of the nanopost structure
reveals an achievable Purcell factor $F_{\rm p}$ of 7.9 or a collection
efficiency $\varepsilon$ of $0.69$ obtained for two very different parameter
sets. Our work invites further exploration of unconventional SPS design
mechanisms, especially in the non-resonant regime.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

We thank Battulga Munkhbat for his assistance in creating the sketch of the
nanopost. This work is funded by the European Research Council (ERC-CoG
“UNITY,” Grant No. 865230), the French National Research Agency (Grant No.
ANR-19-CE47-0009-02), the European Union’s Horizon 2020 Research and
Innovation Programme under the Marie Skłodowska-Curie Grant (Agreement No.
861097), and by the Independent Research Fund Denmark (Grant No.
DFF-9041-00046B).

## References

  * Pan _et al._ 2012 J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger and M. Zukowski, _Rev. Mod. Phys._ , 2012, 84, 777–838.
  * O’Brien _et al._ 2009 J. L. O’Brien, A. Furusawa and J. Vučković, _Nat. Photonics_ , 2009, 3, 687–695.
  * Arakawa and Holmes 2020 Y. Arakawa and M. J. Holmes, _Appl. Phys. Rev._ , 2020, 7, 021309\.
  * Aharonovich _et al._ 2016 I. Aharonovich, D. Englund and M. Toth, _Nat. Photonics_ , 2016, 10, 631–641.
  * Gregersen _et al._ 2013 N. Gregersen, P. Kaer and J. Mørk, _IEEE J. Sel. Top. Quantum Electron._ , 2013, 19, 9000516.
  * Gregersen _et al._ 2017 N. Gregersen, D. P. S. McCutcheon and J. Mørk, _Handbook of Optoelectronic Device Modeling and Simulation Vol. 2_ , CRC Press, Boca Raton, 2017, ch. 46, pp. 585–607.
  * Huber _et al._ 2018 D. Huber, M. Reindl, J. Aberl, A. Rastelli and R. Trotta, _J. Opt._ , 2018, 20, 073002.
  * Wang _et al._ 2019 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._ , 2019, 123, 250503.
  * Kwiat _et al._ 1995 P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko and Y. Shih, _Phys. Rev. Lett._ , 1995, 75, 4337–4341.
  * Shields 2007 A. J. Shields, _Nat. Photonics_ , 2007, 1, 215–223.
  * Gérard _et al._ 1998 J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard and V. Thierry-Mieg, _Phys. Rev. Lett._ , 1998, 81, 1110–1113.
  * Friedler _et al._ 2009 I. Friedler, C. Sauvan, J. P. Hugonin, P. Lalanne, J. Claudon and J. M. Gérard, _Opt. Express_ , 2009, 17, 2095–2110.
  * Gregersen _et al._ 2016 N. Gregersen, D. P. S. McCutcheon, J. Mørk, J.-M. Gérard and J. Claudon, _Opt. Express_ , 2016, 24, 20904–20924.
  * Wang _et al._ 2020 B.-Y. Wang, E. V. Denning, U. M. Gür, C.-Y. Lu and N. Gregersen, _Phys. Rev. B_ , 2020, 102, 125301.
  * Purcell 1946 E. M. Purcell, _Phys. Rev._ , 1946, 69, 681.
  * Wang _et al._ 2019 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_ , 2019, 13, 770–775.
  * Somaschi _et al._ 2016 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. Lemaítre, A. Auffeves, A. G. White, L. Lanco and P. Senellart, _Nat. Photonics_ , 2016, 10, 340–345.
  * Ding _et al._ 2016 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._ , 2016, 116, 020401.
  * Tomm _et al._ 2021 N. Tomm, A. Javadi, N. O. Antoniadis, D. Najer, M. C. Löbl, A. R. Korsch, R. Schott, S. R. Valentin, A. D. Wieck, A. Ludwig and R. J. Warburton, _Nat. Nanotechnol._ , 2021, 16, 399–403.
  * Bleuse _et al._ 2011 J. Bleuse, J. Claudon, M. Creasey, N. S. Malik, J.-M. Gérard, I. Maksymov, J.-P. Hugonin and P. Lalanne, _Phys. Rev. Lett._ , 2011, 106, 103601.
  * Gaál _et al._ 2022 B. Gaál, L. Vannucci, M. A. Jacobsen, J. Claudon, J.-M. Gérard and N. Gregersen, _Applied Physics Letters_ , 2022, 121, 170501\.
  * Claudon _et al._ 2010 J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne and J.-M. Gérard, _Nat. Photonics_ , 2010, 4, 174–177.
  * Claudon _et al._ 2013 J. Claudon, N. Gregersen, P. Lalanne and J.-M. Gérard, _ChemPhysChem_ , 2013, 14, 2393–2402.
  * Lecamp _et al._ 2007 G. Lecamp, P. Lalanne and J. P. Hugonin, _Phys. Rev. Lett._ , 2007, 99, 023902.
  * Manga Rao and Hughes 2007 V. S. C. Manga Rao and S. Hughes, _Phys. Rev. B_ , 2007, 75, 205437\.
  * Arcari _et al._ 2014 M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe and P. Lodahl, _Phys. Rev. Lett._ , 2014, 113, 093603.
  * Yao _et al._ 2018 B. Yao, R. Su, Y. Wei, Z. Liu, T. Zhao and J. Liu, _J. Korean Phys. Soc._ , 2018, 73, 1502–1505.
  * Liu _et al._ 2019 J. Liu, R. Su, Y. Wei, B. Yao, S. F. C. da Silva, Y. Yu, J. Iles-Smith, K. Srinivasan, A. Rastelli, J. Li and X. Wang, _Nat. Nanotechnol._ , 2019, 14, 586–593.
  * Kotal _et al._ 2021 S. Kotal, A. Artioli, Y. Wang, A. D. Osterkryger, M. Finazzer, R. Fons, Y. Genuist, J. Bleuse, J.-M. Gérard, N. Gregersen and J. Claudon, _Appl. Phys. Lett._ , 2021, 118, 194002.
  * Østerkryger 2018 A. D. Østerkryger, _PhD thesis_ , Technical University of Denmark, 2018\.
  * Gür _et al._ 2021 U. M. Gür, S. Arslanagić, M. Mattes and N. Gregersen, _Phys. Rev. E_ , 2021, 103, 033301.
  * Lavrinenko _et al._ 2014 A. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt and T. Søndergaard, _Numerical Methods in Photonics_ , CRC Press, 2014\.
  * Novotny and Hecht 2012 L. Novotny and B. Hecht, _Principles of Nano-Optics_ , Cambridge University Press, 2012.
  * Häyrynen _et al._ 2016 T. Häyrynen, J. R. de Lasson and N. Gregersen, _Journal of the Optical Society of America A_ , 2016, 33, 1298.
  * Yariv and Yeh 2007 A. Yariv and P. Yeh, _Photonics: Optical Electronics in Modern Communications_ , Oxford University Press, 6th edn, 2007.
  * Snyder 1971 A. W. Snyder, _IEEE Transactions on Microwave Theory and Techniques_ , 1971, 19, 720–727.
  * Sammut 1975 R. A. Sammut, _PhD thesis_ , The Australian National University, 1975.
  * Nyquist _et al._ 1981 D. P. Nyquist, D. R. Johnson and S. V. Hsu, _J. Opt. Soc. Am._ , 1981, 71, 49–54.
  * Vassallo 1981 C. Vassallo, _J. Opt. Soc. Am._ , 1981, 71, 1282–1282.
  * Sammut 1982 R. A. Sammut, _J. Opt. Soc. Am._ , 1982, 72, 1335–1337.
  * Vassallo 1983 C. Vassallo, _J. Opt. Soc. Am._ , 1983, 73, 680–683.
  * Snyder and Love 1983 A. W. Snyder and J. D. Love, _Optical waveguide theory_ , Springer New York, NY, 1983.
  * Tigelis _et al._ 1987 I. Tigelis, C. N. Capsalis and N. K. Uzunoglu, _International Journal of Infrared and Millimeter Waves_ , 1987, 8, 1053–1068.
  * Morita 1988 N. Morita, _Journal of Electromagnetic Waves and Applications_ , 1988, 2, 445–457.
  * Alvarez and Xiao 2005 L. Alvarez and M. Xiao, _Journal of Electromagnetic Waves and Applications_ , 2005, 19, 933–951.
  * Balanis 1989 C. A. Balanis, _Advanced Engineering Electromagnetics_ , Wiley, New York, 1st edn, 1989.
  * Friedler _et al._ 2008 I. Friedler, P. Lalanne, J. P. Hugonin, J. Claudon, J. M. Gérard, A. Beveratos and I. Robert-Philip, _Opt. Lett._ , 2008, 33, 2635–2637.
  * Lalanne _et al._ 2004 P. Lalanne, S. Mias and J. P. Hugonin, _Optics Express_ , 2004, 12, 458.
  * Munsch _et al._ 2013 M. Munsch, N. S. Malik, E. Dupuy, A. Delga, J. Bleuse, J.-M. Gérard, J. Claudon, N. Gregersen and J. Mørk, _Phys. Rev. Lett._ , 2013, 110, 177402.
  * de Lasson _et al._ 2015 J. R. de Lasson, P. T. Kristensen, J. Mørk and N. Gregersen, _Opt. Lett._ , 2015, 40, 5790–5793.

##  7 Supplementary

###  7.1 Height adjustment procedure

In principle, the dipole position $h_{\rm t}$ and $h_{\rm b}$ would also be
free parameters if we wanted to perform a complete optimization of the
nanocavity. However, this would be a very demanding task. Instead, we have
chosen to study cavities with 3 antinodes as our test simulations have shown
that the 2nd antinode in a 3 antinode cavity provides a good performance. Thus
$h_{\rm t}$ and $h_{\rm b}$ will be fixed to ensure a cavity with 3 antinodes
and a resonance wavelength of $\lambda_{\mathrm{r}}=930\mathrm{nm}$ at the 2nd
antinode. A procedure is required such that $h_{\rm t}$ and $h_{\rm b}$
satisfies these conditions. The first step is to use the SMM (phase conditions
of the fundamental mode) to determine the initial total height $h$:

$h=(2\pi-\arg(r_{\rm bot,11}))/(2\beta_{1})+(2\pi-\arg(r_{\rm
top,11}))/(2\beta_{1}).$ (35)

This is under the conditions $\arg(r_{\rm bot,11})<0$ and $\arg(r_{\rm
top,11})>0$. Then we place the dipole in the second antinode from the bottom:

$h_{\rm b}=(2\pi-\arg(r_{\rm bot,11}))/(2\beta_{1})$ (36)

and

$h_{\rm t}=(2\pi-\arg(r_{\rm top,11}))/(2\beta_{1}).$ (37)

However, due to the background continuum and mode-coupling, this method does
not ensure that the dipole is placed exactly at an antinode nor that the
resonant wavelength of the cavity corresponds exactly to the design
wavelength, $(\frac{\mathrm{d}F_{\rm
p}}{\mathrm{d}\lambda})_{\lambda=\lambda_{d}}=0$. To solve this problem, the
dipole is first adjusted to the exact position of the antinode by plotting
$|E_{r}(z)|^{2}$ and locating the peak. Then the height is slightly adjusted
$h=h\pm\delta h$, while the dipole is moved to the exact position of the
antinode for each adjustment until $(\frac{\mathrm{d}F_{\rm
p}}{\mathrm{d}\lambda})_{\lambda=\lambda_{d}}=0$ is satisfied. Finally, the
position of the 1st antinode (from the bottom) can also be identified by
plotting $|E_{r}(z)|^{2}$.

Finally, we will provide the total height, $h_{\rm total}$ of the structure
along with the height deviation between the initial height obtained from the
SMM and the final height, $h_{\rm diff}=h_{\rm total}-h_{\rm initial}$.

Fig. 16: (a) Total height of the structure, $h_{\rm total}$, as a function of
the diameter, $D$, and the silica layer thickness, $t_{\rm SiO_{2}}$. (b) The
height difference between the initial height and the final height, $h_{\rm
diff}$, as a function of the diameter, $D$, and the silica layer thickness,
$t_{\rm SiO_{2}}$.

In Fig. (16a), the total height of the structure is shown as a function of the
diameter and the silica layer thickness. The total height mostly depends on
the diameter as the diameter determines the propagation constant $\beta_{1}$
and thus the primary influence of the phase. In Fig. (16b), the height
difference of the final total height compared to the SMM is shown. For most of
the parameters, the difference is small in the range
$-5\text{\,}\mathrm{n}\mathrm{m}$ to $5\text{\,}\mathrm{n}\mathrm{m}$.
However, for the small diameters and thin silica layer thicknesses, there is a
very large difference up to $50\text{\,}\mathrm{n}\mathrm{m}$. This is the
same parameter region where the modal reflection at the bottom interface is
small. Thus the phase of the fundamental mode is less dominating compared to
the contributions of the radiation and evanescent modes. Oscillations can also
be observed in the height difference due to numerical noise. The exact
resonance wavelength is sensitive to the height of the structure, but these
oscillations are on the scale of less than $1\text{\,}\mathrm{n}\mathrm{m}$,
and the uncertainty in the resonance wavelength will be on a similar scale.

###  7.2 Influence of the numerical aperture

In all the simulations of the efficiency a numerical aperture of
$\mathrm{NA}=0.75$ has been used. Here we will investigate the influence of
varying the numerical aperture.

Fig. 17: Efficiency, $\varepsilon$, at the 2nd antinode for a numerical
aperture of $\mathrm{NA}=0.4$ (a) and $\mathrm{NA}=1.00$ (b). (c) Efficiency,
$\varepsilon$, as a function of the numerical aperture, $\mathrm{NA}$, for
three different parameters at the 2nd antinode. (d) Efficiency, $\varepsilon$,
as a function of wavelength, $\lambda$, for four different values of the
$\mathrm{NA}$.

In Fig.(17a) and Fig.(17b) the efficiency is shown as a function of the
diameter and the silica layer thickness similar to Fig. (3a), but for a
numerical aperture $\mathrm{NA}=0.4$ and $\mathrm{NA}=1$. Lowering the
numerical aperture to $\mathrm{NA}=0.4$ drastically reduces the efficiency and
the maximum is barely above $\varepsilon=0.2$. This shows that a large
numerical aperture is crucial for the good performance of the nanopost. By
increasing the numerical aperture from $\mathrm{NA}=0.75$ ($\theta\approx
49^{\circ}$) to $\mathrm{NA}=1.00$ ($\theta=90^{\circ}$) there is roughly a
$20\%$ increase in the efficiency, so there is still some light lost at angles
above $\theta\approx 49^{\circ}$. Furthermore, for a numerical aperture of
$\mathrm{NA}=1$, the efficiency directly represents the losses to the bottom
mirror. For diameters above $D=$210\text{\,}\mathrm{n}\mathrm{m}$$, an
increased silica layer thickness increases the losses to the bottom mirror
even though the Purcell factor increases. In Fig. (17c) the efficiency for the
structures with the largest efficiency and Purcell factor, at the 2nd
antinode, are shown as a function of the numerical aperture. The steepest part
of the curves is roughly in the interval $\mathrm{NA}=0.4$ to
$\mathrm{NA}=0.75$, which is the reason for the huge difference in efficiency
between $\mathrm{NA}=0.4$ and $\mathrm{NA}=0.75$. The curves also start to
flatten out as the $\mathrm{NA}$ reaches $1$. In Fig. (17d) the efficiency is
plotted for four different values of the numerical aperture as a function of
the wavelength. The numerical aperture does not influence the curvature of the
efficiency as a function of the wavelength. This means that being on resonance
does not focus the far-field compared to being off resonance.

###  7.3 Gaussian collection efficiency

Fig. 18: Gaussian collection efficiency, $\varepsilon_{g}$, as a function of
the diameter, $D$, and the silica layer thickness, $t_{\rm SiO_{2}}$, for the
2nd antinode.

So far the efficiency has been evaluated by calculating the total power
collected in the lens with some numerical aperture. However, in many
applications the light will couple to a fiber afterwards. Therefore we have
also calculated the power overlap between the emitted far-field and the far-
field of a Gaussian representative for the fundamental mode in many single-
mode fibers 49. The applied method is identical to the one presented in the
appendix of 14, and the Gaussian collection efficiency is defined as
$\varepsilon_{g}=P_{\rm collected,Gaussian}/P_{\rm T}$, where $P_{\rm
collected,Gaussian}$ is defined as the overlap with a Gaussian profile. In
Fig. (18) the Gaussian efficiency is shown for the 2nd antinode. Compared to
the standard efficiency in Fig. (3a), the difference is approximately $0.1$
over the entire parameter space, showcasing the Gaussian shaped profile of the
far-field.

###  7.4 Efficiency analysis for the structure with maximum Purcell factor

We will now apply the efficiency analysis for the structure with the largest
Purcell factor with the parameters $D=$250\text{\,}\mathrm{n}\mathrm{m}$$ and
$t_{\rm SiO_{2}}=$13\text{\,}\mathrm{n}\mathrm{m}$$ and an efficiency of
$\varepsilon=0.41$.

In Fig. (19) the efficiency is shown as a function of the initial coefficients
(Fig. (19a)) and the final coefficients (Fig. (19b)), expressed with the
propagation constant $(\beta/k_{0})^{2}$, just as for the structure with
maximum efficiency. Again the curve in Fig. (19a) is flat and the channels of
the fundamental mode dominates the efficiency. The efficiency increase by
adding the final coefficients, i.e. $c_{2}$ and $c_{3}$, seen in Fig. (19b),
is still significant, but much smaller compared to the structure with maximum
efficiency. Here the increase is from approximately $\varepsilon=0.3$ to
$\varepsilon=0.41$. The curve in Fig. (19b) also flattens out completely due
to the numerical aperture.

Fig. 19: (a) Efficiency, $\varepsilon$ ($\mathrm{NA}=0.75$), as a function of
the initial coefficients expressed with the propagation constant
$(\beta/k_{0})^{2}$. (b) Efficiency, $\varepsilon$ ($NA=75$), as a function of
the final coefficients expressed with the propagation constant
$(\beta/k_{0})^{2}$. Fig. 20: (a) The phase difference between the direct
transmission of the fundamental mode and the background continuum for TE and
TM modes as a function of the propagation constant. (b) The far-field of the
fundamental mode. (c) The far-field of the background continuum. (d) The total
far-field. The white dotted line indicates $\mathrm{NA}=0.75$. Be aware of the
different color scales that have been used for the far-fields.

In Fig. (20a) the phase difference in the air layer between the direct
transmission of the fundamental mode and the entire background is shown as a
function of the propagation constant for TE and TM modes. We observe similar
features as before, i.e. constructive (destructive) interference for light
propagating vertically (horizontally). Though at $\beta/k_{0}=1$, the phase
difference is larger compared to the previous structure. In Fig. (20b), Fig.
(20c) and Fig. (20d) the far-fields of the direction transmission of the
fundamental mode, the background radiation and the total field is shown.
Compared to the structure with maximum efficiency, the far-field of the
background radiation is significantly different. Here, the far-field is mainly
focused towards horizontal angles and the intensity is much smaller compared
to the far-field of $\mathrm{HE}_{11}$. As such the constructive contribution
at smaller angles is not as significant and less of the radiation will be
captured by the lens, due to the numerical aperture. This explains why the
efficiency increase in Fig. (19b) is much smaller compared the structure with
the maximum efficiency. However, there is still destructive interference for
the light that propagates horizontally. As such the interference between the
direction emission and the radiation focuses the far-field, but not to the
same degree as for the structure with the maximum efficiency.

###  7.5 Purcell factor analysis for the structure with maximum collection
efficiency

We will now apply the model for the Purcell factor for the structure with the
largest efficiency, $D=$238\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$0\text{\,}\mathrm{n}\mathrm{m}$$ and $F_{\mathrm{p}}=4.8$.

Fig. 21: Purcell factor, $F_{\rm p}$, computed using the full model and the
SMM as a function of the dipole position from the bottom interface, $h_{\rm
b}$. $D=$238\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$0\text{\,}\mathrm{n}\mathrm{m}$$.

In Fig. (21) the Purcell factor is shown as a function of the dipole position
throughout the cavity, both the full model (n. 7) and the SMM (n. 1) are used.
Here the 3 antinodes can be observed and the SMM predicts a larger Purcell
factor for all 3 antinodes compared to the full model. The positions of the
antinodes are almost identical between the full model and the SMM. The Purcell
factor increases drastically when the dipole is placed close to the metal
mirror due to non-radiative decay processes33.

In Fig. (22a) and Fig. (22b) the Purcell factor and the power enhancement of
the fundamental mode is shown as a function of the model complexityfor the 1st
and 2nd antinode. Compared to the structure with
$D=$250\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$13\text{\,}\mathrm{n}\mathrm{m}$$, there are a few differences. The
SMM predicts a smaller Purcell factor, which is simply caused by the lower
modal reflection at the bottom. There is a large negative contribution when
including the back-scattering at the bottom interface (n 5.) at both
antinodes. This is caused by the change of the silica layer thickness and as
seen in Fig. (22c) and Fig. (22d). The propagating radiation modes are
responsible for this decrease. Furthermore, by including the scattering of the
background to itself (n. 6), there is now a small decrease for both antinodes.
These are the differences between the two structures. The differences between
the 1st and 2nd antinode are exactly the same for the two structures, where
the scattering into evanescent modes at the top interface (n. 2) and the
initial evanescent modes provide a positive contribution at the 2nd antinode.

Fig. 22: Purcell factor ($P_{\rm T}/P_{\rm Bulk}$) and fundamental mode
enhancement ($P_{\mathrm{HE}_{11}}/P_{\rm Bulk}$) for the 1st (a) and 2nd (b)
antinode as a function of the model number. In (c) and (d) the background
continuum is continuously included between each model number.

###  7.6 Asymmetric wavelength dependence for the two antinodes

To further study the resonance shift between the 1st and 2nd antinodes, we
choose a nanopost design of $D=$202\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$5\text{\,}\mathrm{n}\mathrm{m}$$, where the shift is more
pronounced.

Fig. 23: Purcell factor, $F_{\rm p}$, as a function of wavelength, $\lambda$,
for the two antinodes. The parameters are
$D=$202\text{\,}\mathrm{n}\mathrm{m}$$ and $t_{\rm
SiO_{2}}=$5\text{\,}\mathrm{n}\mathrm{m}$$.

In Fig. (23) the Purcell factor is shown for the two antinodes as a function
of the wavelength. The peak positions of the Purcell factors (resonance
wavelength) are $\lambda_{2nd,r}=$930\text{\,}\mathrm{n}\mathrm{m}$$ and
$\lambda_{1st,r}=$933\text{\,}\mathrm{n}\mathrm{m}$$. By observing the curve
for the 1st antinode, this shift is caused by another broad resonance at
approximately $\lambda=$1050\text{\,}\mathrm{n}\mathrm{m}$$. To gain further
insight into the resonances of the structure and verify our results, we have
performed a quasi-normal mode (QNM) simulation50 of the nanopost. In this
simulation 15 QNMs are found and the complex eigenfrequencies,
$\tilde{\omega}_{\mu}=\omega_{\mu}-\mathrm{i}\gamma_{\mu}$, of the 3 important
QNMs are $\tilde{\omega}_{\rm
QNM_{1}}=$2.0237\text{\times}{10}^{15}$-\mathrm{i}$4.4904\text{\times}{10}^{13}$\mathrm{Hz}$,
$\tilde{\omega}_{\rm
QNM_{2}}=$1.7595\text{\times}{10}^{15}$-\mathrm{i}$1.4654\text{\times}{10}^{14}$\mathrm{Hz}$
and $\tilde{\omega}_{\rm
QNM_{3}}=$2.2809\text{\times}{10}^{15}$-\mathrm{i}$2.0703\text{\times}{10}^{13}$\mathrm{Hz}$.
The corresponding real parts of the complex wavelength are $\lambda_{\rm
QNM_{1}}=$930.3\text{\,}\mathrm{n}\mathrm{m}$$, $\lambda_{\rm
QNM_{2}}=$1063.2\text{\,}\mathrm{n}\mathrm{m}$$ and $\lambda_{\rm
QNM_{3}}=$825.8\text{\,}\mathrm{n}\mathrm{m}$$. The Q factors of the QNMs can
also be calculated using $Q_{\mu}=\omega_{\mu}/(2\gamma_{\mu})$ 50, and we
obtain $Q_{\rm QNM_{1}}=22.5$, $Q_{\rm QNM_{2}}=6.0$ and $Q_{\rm
QNM_{3}}=55.1$.

Fig. 24: Comparison of the Purcell factor between the FMM and the QNM
simulation for the 1st antinode (a) and the 2nd antinode (b). In-plane
electrical field profiles of the 3 QNMs at their resonance wavelengths are
shown in (c), (d) and (e). The green star corresponds to the position of the
1st antinode, and the red star corresponds to the position of the 2nd
antinode. The white scale bar in (a) corresponds to
$100\text{\,}\mathrm{n}\mathrm{m}$. The intensity is scaled in each field plot
and should not be used for comparison.

In Fig. (24a,24b), the comparison of the Purcell factor between the FMM and
the QNM simulation is shown for the two antinodes. Overall, the quantitative
agreement between the FMM and QNM simulations is good with some small
deviations. In Fig. (24a), the individual contributions of 3 QNMs are plotted
along with their sum and the result of the FMM for the 1st antinode. These 3
QNMs provide a good description of the overall Purcell factor and they
directly correspond to the peaks in the spectrum. QNM1 and QNM2 also overlap
in the spectrum due to the low Q factor of QNM2, which slightly shifts the
peak position of the total Purcell factor. In Fig. (24b), the individual
contributions of 2 QNMs are plotted along with their sum and the result of the
FMM for the 2nd antinode. Here $\rm QNM_{1}$ is almost sufficient to describe
the entire spectrum, and we do not observe any other peaks than the one at
$\lambda=$930\text{\,}\mathrm{n}\mathrm{m}$$, besides a small bump at longer
wavelengths. Now, consider the in-plane electrical field profiles of the 3
QNMs shown in Fig. (24c,24d,24e). $\rm QNM_{1}$ has 3 antinodes, $\rm QNM_{2}$
has 2 antinodes and $\rm QNM_{3}$ has 4 antinodes. The green star corresponds
to the position of the 1st antinode, where the QD is placed, and this position
is very close to an antinode for $\rm QNM_{2}$ and $\rm QNM_{3}$. Therefore
the contributions of these QNMs appear in the spectrum. However, the position
of the 2nd antinode (red star) is much closer to a node for $\rm QNM_{2}$ and
$\rm QNM_{3}$, and therefore they do not influence the spectrum.