# Enhancing the robustness of coupling between a single emitter and a photonic
crystal waveguide

Alexander Shurinov  Quantum Technologies Centre, Lomonosov Moscow State
University, Russia, Moscow, 119991, Leninskie Gory 1 building 35  Ivan
Dyakonov  Quantum Technologies Centre, Lomonosov Moscow State University,
Russia, Moscow, 119991, Leninskie Gory 1 building 35<EMAIL_ADDRESS>Sergei Kulik  Quantum Technologies Centre, Lomonosov Moscow State University,
Russia, Moscow, 119991, Leninskie Gory 1 building 35  Laboratory of quantum
engineering of light, South Ural State University (SUSU), Russia, Chelyabinsk,
454080, Prospekt Lenina 76  Stanislav Straupe  Quantum Technologies Centre,
Lomonosov Moscow State University, Russia, Moscow, 119991, Leninskie Gory 1
building 35  Russian Quantum Center, Russia, Moscow, 121205, Bol’shoy bul’var
30 building 1

(August 27, 2024)

###### Abstract

We present a heuristic mathematical model of the relation between the geometry
of a photonic crystal waveguide and the Purcell enhancement factor at a
particular wavelength of interest. We use this model to propose approaches to
the design of a photonic crystal waveguide maximizing the Purcell enhancement
at a target wavelength. Numerical simulations indicate that the proposed
structures exhibit robustness to fabrication defects introduced into photonic
crystal geometry.

††preprint: APS/123-QED

##  I Introduction

A planar photonic crystal waveguide (PCW) is a rich system finding
applications in diverse areas of optical physics. Among those are slow light
[1], topological photonics [2], chiral photonics [3], cavity quantum
electrodynamics [4] and many others. An attractive feature of the planar
photonic crystal is its flexibility for tailoring dispersion properties of
light. In particular, the dispersion curve of a PCW mode inside the crystal
bandgap can be engineered to reach extremely high values of group velocity at
a target wavelength. This feature allows one to use a PCW as a platform for
travelling-wave cavity quantum electrodynamics. A single emitter generating
photons at wavelength $\lambda$ matched to the large group velocity range of
the PCW mode dispersion curve is strongly coupled to the PCW mode and thus its
emission exhibits a significant Purcell enhancement. This effect enabled the
development of an on-demand single photon source compatible with planar
photonic integrated circuits [5]. Furthermore, the ability to strongly couple
an emitter to a cavity mode while still being able to efficiently excite the
emitter and read-out photons from the cavity boosted the research in nonlinear
light-matter interaction at the single-photon level [6].

Semiconductor quantum dots (QDs) are the most common type of emitters which
can be coupled to a PCW to create a single photon source. Despite being well-
studied, QDs with predefined parameters are still notoriously hard to
fabricate deterministically. The most widespread Stransky-Krastanov growth
process produces QDs with randomly distributed spectral characteristics. The
emission wavelength of QDs typically falls in range of a few nanometers around
the designed center wavelength. The first derivative $d\omega/dk$ of the PCW
dispersion curve gets close to zero in a very narrow wavelength range
$\lambda_{0}\pm\Delta\lambda/2$ and efficient Purcell enhancement is not
guaranteed for most of the fabricated QDs with emission wavelengths missing
the $\Delta\lambda$ region. Furthermore, the fabrication process introduces
defects into the PCW structure which affect the dispersion properties of the
PCW mode. The workaround for this issue is straightforward – an array of
structures is fabricated and only those which meet particular experimental
requirements are selected. Although this method may be satisfactory for
research purposes, the lack of reproducibility in the single-photon source
fabrication is one of the major bottlenecks in contemporary quantum optical
experiments [7]. At the same time current trends in optical quantum computing
demand the development of hybrid integration methods to place single emitters
onto a photonic platform of choice [8].

In this paper we will address the design approaches which mitigate the effect
of fabrication imperfection on the Purcell factor at the source wavelength. We
start with developing a heuristic PCW design approach which significantly
simplifies the selection of a PCW geometric configuration. The theory behind
this approach is based on simple optical phenomena – interference and
diffraction of light scattered inside the PCW membrane and leaking out of the
membrane. The derived equations provide clear guidelines how to choose PCW
geometric parameters in order to set the maximal Purcell enhancement at the
required wavelength and completely eliminate the necessity to evaluate
multiple time consuming 3D FDTD simulations. After the description of the
heuristic PCW theory we address the problem of PCW robustness to fabrication
imperfections. The question of a PCW dispersion curve robustness against
fabrication defects has been previously highlighted in the series of works.
These include studies of fabrication defects’ influence on the quality factors
of photonic crystal microcavities [9, 10] and automated design methods to
optimize the photonic crystal microcavity structure [11, 12]. We focus on the
development of a design approach which increases the robustness of the
coupling between an emitter and a photonic crystal waveguide mode. We propose
two design approaches increasing the robustness of coupling to the fabrication
errors and test them using numerical simulations.

##  II Photonic crystal

Figure 1: a) An overview of a photonic crystal waveguide structure. b) Typical
dispersion curves of PCW eigenmodes inside the photonic bandgap. The example
illustrates the existence of 3 modes (red dashed lines). Blue dashed lines
indicate the PC bandgap prior to hole removal. Green areas indicate PCW bulk
modes and yellow area corresponds to light waves with non-zero wavevector
component orthogonal to the PCW membrane surface. Images c), d) and e)
illustrate the geometric parameters used throughout the paper.

A typical two-dimensional photonic crystal is a periodic arrangement of
circular holes etched in a thin film of a material with high refractive index.
A deleted row of holes forms a photonic crystal waveguide (see illustration in
Fig.1(a)). A characteristic feature of a PCW is the existence of a frequency
range where the group velocity of light decreases significantly. This fact
makes a PCW structure an extremely appealing system for mediating interaction
between light and an isolated dipole. A PCW effectively serves as a
microresonator with small mode volume and high quality factor. These systems
were demonstrated to suite the purpose of integration of $A_{3}B_{5}$ quantum
dot single photon sources in a planar photonic structure [5]. Quantum dot can
be considered as a dipole, which is orientated perpendicular to the waveguide
axis in the PC plane. A PCW microresonator forms an open cavity which can be
smoothly interfaced with other integrated photonic waveguides. In this paper
we study methods to increase robustness of PCW features to fabrication
defects.

We focus our attention on a PCW created by deleting a row of air holes from a
2D triangular array. The host material is chosen to be gallium arsenide (GaAs)
because the target application is a planar semiconductor quantum dot single
photon source. We start with the description of PCW characteristics and
development of its heuristic model. Manga Rao and Hughes [13] derived an
expression for a Purcell factor $F_{p}$ in terms of PCW parameters:

$F_{p}=\frac{3\pi c^{3}a}{V_{eff}\omega_{d}^{2}\epsilon^{3/2}v_{g}},$ (1)

where $a$ is the distance between air holes (if the lattice is triangular
$a=a_{x}$, since we are focused on that type of lattice from now on we will
write $a$ instead $a_{x}$, but for another lattice angle $a_{x}$ is the only
correct option), $V_{eff}$ is the effective mode volume and $v_{g}$ is the
group velocity at the resonant frequency of the dipole $\omega_{d}$. The
formula indicates that the largest $F_{p}$ is achieved when the wavepacket
group velocity reaches zero. Thus the design of a PCW efficiently coupled with
a single emitter resonant at $\omega_{d}$ is equivalent to engineering a PCW
dispersion law to meet the requirement $d\omega/dk(\omega_{d})=0$. Numerical
methods for calculation of the dispersion structure of a PCW are well-known
and straightforward [14] and can be easily applied to a PCW with a defined
geometry. However, there exists no recipe of how to estimate geometrical
parameters of a PCW exhibiting high Purcell factor at a wavelength of
interest. We devise heuristic expressions linking the target wavelength and
the parameters of a hexagonal PCW which stem upon simple optical effects
taking place inside a photonic crystal. Based on these results we introduce
methods to increase the robustness of a PCW structure to fabrication defects.

##  III A PCW Purcell factor heuristics

Figure 1(b) illustrates a typical dispersion structure of a PCW. The
geometrical parameters for this example are as follows: PC hole pattern angle
$\theta_{gr}=60^{\circ}$, period $a=0.238$ $\mu$m, hole radius $r=0.08$ $\mu$m
and membrane thickness $h=0.16$ $\mu$m. These values were chosen to put the
Purcell factor $F_{PCW}$ peak at $925$ nm. A natural question arises whether
this configuration is unique. It turns out that the answer is negative. We
performed an extensive numerical analysis of the Purcell enhancement happening
in different PCW configurations, results are presented in Fig. 2.The 3D FDTD
simulation was carried out in Lumerical FDTD package, the details of the
simulation are specified in Appendix D. We observed a continuous set of
configurations of a triangular PCW with the same lattice angle corresponding
to a peak value of $F_{PCW}$ at a target wavelength. The red line in Fig. 2
illustrates the numerically computed set of $(a,r)$ configurations
corresponding to the most efficient coupling of a PCW mode to dipole radiation
at $925$ nm. The curve in $(a,r)$ space closely follows the function
$a=c_{1}+c_{2}r$, where the coefficients $c_{1}$ and $c_{2}$ are weakly
dependent on $a$ and $r$. The $a(r)$ dependence is finely approximated by a
linear function in the region where the $F_{PCW}$ reaches its highest levels.
The yellow curve represents the values of $a$ and $r$ corresponding to
$F_{PCW}$ peak which are provided by the proposed theoretical description.

In the following subsections we provide a heuristic theoretical description
for the origins of such dependence and the values of $c_{1}$ and $c_{2}$
coefficients.

###  III.1 The slope coefficient $c_{2}$

The Purcell factor $F_{PCW}$ defines the probability

$\beta=F_{PCW}/(1+F_{PCW})$ (2)

of emitting a photon into a PCW mode. The existence of a PCW mode is a purely
interferometric effect hence the probability $p$ should be related to the
geometry of a photonic crystal. We expect to derive the connection between the
geometric parameters which correspond to a configuration of a PCW structure
reaching maximal Purcell factor for the required wavelength. We roughly split
the emitter radiation into three categories: light exiting the PCW plane,
light propagating inside the PCW structure, and light coupled to the PCW mode.
For the light exiting the PCW plane we define the notion of a vertical Fabry-
Perot resonator and the corresponding Purcell factor $F_{FP}$, which is used
to evaluate a portion of light leaking from the PCW membrane. Then the
fraction of light emitted in the PCW itself from the total amount of radiation
which remains inside the crystal can be estimated using the effective angle
$\theta_{wg}$ (see Fig. 1(d)). Under such assumptions we can derive the
following equation:

$\left(1-\frac{F_{FP}}{1+F_{FP}}\right)\frac{\int_{\pi/2-\theta_{wg}}^{\pi/2+\theta_{wg}}\sin^{3}\theta{d}\theta}{\int_{0}^{\pi}\sin^{3}\theta{d}\theta}=\frac{F_{PCW}(a,r)}{1+F_{PCW}(a,r)},$
(3)

where the first term on the left-hand side of the equation denotes the
probability of the photon to stay inside the crystal and the second term
denotes the fraction of the photons emitted into the waveguide mode. Here we
assumed that if the total internal reflection angle is relatively small
(approximately $16^{\circ}$ in case of GaAs to air transition), then the
majority of the photons which leak out of the crystal can be attributed to the
emission into the Fabry-Perot resonator mode. The right-hand side accounts for
the probability of emitting the photon exactly into the waveguide mode using
the $F_{PCW}$ value. Here we have three variables, which we need to calculate:
$F_{FP}$, $F_{PCW}$ and $\theta_{wg}$. We calculate values $F_{FP}$ and
$F_{PCW}$ and use equation 3 to determine the value of $\theta_{wg}$.

The $\theta_{wg}$ is different for each individual crystal configuration. The
expression connecting geometrical parameters of a crystal to the $\theta_{wg}$
value is defined by the crystal configuration. For the $60^{\circ}$ triangular
hole pattern the $\theta_{wg}$ definition is illustrated in Fig. 1(d). For a
triangular lattice, the angle $\theta_{wg}$ is implicitly related to the
period $a$ and the hole radius $r$ (see Appendix A):

$\frac{a}{r}=\frac{\cos(\pi/4-\theta_{wg}/2)-\tan\theta_{wg}\sin(\pi/4-\theta_{wg}/2)}{\tan\theta_{gr}/2-\tan\theta_{wg}/2},$
(4)

where $\theta_{gr}$ is the lattice angle. Once we have estimated
$F_{FP},F_{PCW}$, and $\theta_{wg}$ values we can substitute them into
equation 4 and evaluate the $c_{2}=a/r$ coefficient.

Figure 2: The set of configurations in $(a,r)$ parameter space corresponding
to a peak $F_{PCW}$ at 925 nm. The 2D heatmap represents the $F_{PCW}$ values
experienced by the emitter in a PCW with given $a$ and $r$. The $a$ and $r$
parameter values are unevenly distributed for time-saving purposes. We added
extra points in the maximal $F_{PCW}$ region. The white tiles represent the
points which were not computed. The red curve connects the $F_{PCW}$ maximal
values calculated numerically using 3D FDTD simulation. The yellow curve
contains values estimated by the proposed heuristic theoretical approach.

To calculate $F_{FP}$ we note, that light emitted by a dipole at frequency
falling into the photonic crystal bandgap exhibits propagation through a low-
quality Fabry-Perot resonator between the top and the bottom surfaces of the
PCW membrane. Resonance frequencies $\omega_{m}=(\pi
c/nh)m=\frac{\pi\cdot{c}}{nh}=w_{1}$ and the linewidth
$dw=c(1-R)/(n\sqrt{R}h)$ for m=1 of this Fabry-Perot resonator are expressed
using the membrane thickness $h$ and the refractive index of the material $n$.
Fresnel law yields the reflection coefficient $R=(n-1)^{2}/(n+1)^{2}$ for
light incident normally to the membrane surfaces. A fraction of light emitted
by the dipole into the vertical Fabry-Perot mode can be estimated using a
Purcell factor

$F_{FP}=\frac{3Q(\lambda/n)^{3}}{4\pi^{2}V_{eff}}\cdot\frac{d\omega^{2}}{4(\omega-\omega_{1})^{2}+d\omega^{2}},$
(5)

From this equations we obtain the quality factor $Q=\frac{w_{1}}{dw}$. The
effective mode volume is given by
$V_{eff}=\frac{\int_{V}{\epsilon{E^{2}}}dV}{\underset{V}{\max}(\epsilon{E^{2}})}$
[15], where the integration is carried out over a single unit cell for
periodical structures (for example PCWs) and over the whole possible volume
for non-periodical structures (the period equals to infinity). For such
configuration the effective mode volume can be estimated as (see Appendix B)
$V_{eff}=\frac{2}{3}h^{3}$. The $F_{FP}$ of the vertical mode resonator then
equals to $F_{FP}\approx 0.64$ if we set $\lambda=925$ nm and
$n_{GaAs}(\lambda=925\,\mathrm{nm})=3.46$.

Next we need to express $F_{PCW}(a,r)$. The effective volume of the PCW mode
$V_{eff}$ from Eq. 1 is the last term which is not yet related to $a$ and $r$.
We consider a unit cell as a rectangle with 4 quarter circles (see Fig. 1(d)).
The effective mode volume for the light modes with the free-space dispersion
relation are defined by

$V_{eff}=\frac{\int_{V}IdV}{\underset{V}{\max I}}=\frac{\langle I\rangle
V}{I_{0}},$ (6)

where $I$ denotes the radiation intensity inside the unit cell. We consider a
PCW dispersion curve with $d\omega/dk(\omega_{d})=0$ and focus on the system
behaviour at a frequency $\omega$ slightly smaller than $\omega_{d}$. We
assume that the mode volume of light states at the frequencies
$\omega<\omega_{d}$ is roughly the same as at the resonance frequency
$\omega_{d}$, because the slight change of the light frequency shouldn’t
drastically affect the mode volume. We also assumed that the emitter is
preferentially coupled to the primary mode (mode 1 in Fig. 1b). Light at
frequency $\omega$ can only populate the states satisfying the free space
dispersion relation $\omega=ck/n(\omega)$. The dipole intensity is
proportional to $\frac{1}{r^{2}}$ and the intensity at the ’entrance’ of the
unit cell is $I_{0}$. We can find the function $I(x)=\frac{const}{(x+1)^{2}}$,
where $x$ is the dimensionless coordinate along the axis of the waveguide,
$const$ equals numerically to $I_{0}$, and $x=0$ at the ’entrance’ of the unit
cell. Using $I(x)$ we can easily calculate the average intensity in the unit
cell

$\langle
I(x)\rangle=\frac{1}{a}\int_{0}^{a}\frac{const}{(x+1)^{2}}dx=\frac{const}{(\langle
x\rangle+1)^{2}},$ (7)

where $a$ is the lattice period along the propagation axis. This result
implies, that we need to consider $\langle I\rangle$ as the intensity at the
$\langle x\rangle$ coordinate taking into account diffraction effects at the
unit cell ’entrance’. The estimated number of Fresnel zones (we assume, that
the unit cell is located far from the dipole, so the light propagates almost
parallel to the waveguide axis)

$m=\frac{(a\tan\theta_{gr}-2r)^{2}}{4\lambda<x>}\approx 1.2,$ (8)

which means that the Fresnel approximation is eligible when accounting for the
diffraction effects. Since the unit cell entrance cross-section is
rectangular, we calculate the parameters of the Cornu spiral as

$\begin{array}[]{c}u=\sqrt{2m},\\\
c(u)=\int_{0}^{u}\cos(\frac{\pi}{2}\tau^{2})d\tau,\\\
s(u)=\int_{0}^{u}\sin(\frac{\pi}{2}\tau^{2})d\tau.\end{array}$ (9)

Coefficients $c(u)$ and $s(u)$ allow to evaluate the required ratio

$\frac{\langle
I\rangle}{I_{0}}=\frac{c(w)^{2}+s(w)^{2}}{c(\infty)^{2}+s(\infty)^{2}}.$ (10)

Then the effective mode volume is equal to $V_{eff}=\frac{\langle
I\rangle}{I_{0}}V$, where $V$ is the geometrical volume of the unit cell.

Now we have all the ingredients to specify the explicit relation $a(r)$. The
$F_{PCW}$, and $F_{FP}$ are all expressed as functions of $a$ and $r$ and
after substitution of each one to the equation (3) we get $\theta_{wg}$(a,r)
and equation (4) gives $\frac{a}{r}=c_{2}$. The explicit formulas and a
numerical calculation algorithm are provided in Appendix B.

###  III.2 The constant coefficient $c_{1}$

When the hole radius $r$ is close to zero, the PC becomes analogous to a Bragg
grating (see Fig. 1). When the $F_{PCW}$ reaches its’ peak, the crystal blocks
all the photons, which are emitted outside the waveguide direction. The Bragg
equations define destructive interference criteria for the photons, which are
not emitted to the PCW mode. We will only take into account two types of
reflective surfaces: the parallel and the sloped as shown in Fig. 1(e).These
two sets of surfaces share similar properties: the distance between the
circles along these surfaces is minimal over all other possible surfaces and
equals to the crystal parameter $a$. We do not take into account the
interference effects happening on every other possible set of surfaces due to
the increasing distance between the circles and thus the necessity to take
diffraction effects into consideration. For the sloped surfaces the Bragg
condition is

$\begin{array}[]{c}2dn\cos(\alpha)=\lambda,\\\
\alpha=\theta_{gr}-\alpha_{\mathrm{avg}},\\\
d=a_{slp}\sin(\theta_{gr}),\end{array}$ (11)

where $a_{slp}$ is the lattice period, found using the Bragg condition for
sloped surfaces, and $\alpha_{\mathrm{avg}}$ is the average angle of emission,
which can be determined using the following formula

$\alpha_{\mathrm{avg}}=\frac{2}{\pi}\int_{0}^{\pi/2}\sin^{3}\theta\cos\theta{d}\theta.$
(12)

To obtain this formula we take into account that we need to calculate the
angle corresponding to the average direction of power emission in any quarter
surface. The term $\sin^{2}\theta$ describes the dipole radiation pattern.
Another $\sin\theta$ term arises from the transition to a spherical coordinate
system. Lastly the $\cos\theta$ term implies, that we are interested in a
projection on the $y$-axis, because we look for the destructive interference
condition for light waves propagating perpendicular to the waveguide axis.

The Bragg condition for the parallel surfaces is

$2\frac{a_{par}\cdot\tan{\theta_{gr}}}{2}n\cos\alpha_{\mathrm{avg}}=\lambda,$
(13)

where $a_{par}$ is the lattice period found using the Bragg condition for the
parallel surfaces. Finally we need to estimate the fractions of radiation
which preferentially interfere in the sloped and the parallel Bragg gratings.
These fractions could be obtained using the dipole radiation pattern

$\begin{array}[]{c}\eta_{\mathrm{slp}}=\frac{\int_{\pi/2-\theta_{gr}}^{\pi/2}\sin^{3}\theta{d}\theta}{\int_{0}^{\pi/2}\sin^{3}\theta{d}\theta},\\\
\eta_{\mathrm{par}}=\frac{\int_{0}^{\pi/2-\theta_{gr}}\sin^{3}\theta{d}\theta}{\int_{0}^{\pi/2}\sin^{3}\theta{d}\theta}.\\\
\end{array}$ (14)

Taking these fractions into account, we can now express the total constant
coefficient through $a$ and $r$:

$c_{1}=\eta_{par}a_{par}+\eta_{slp}a_{slp}.$ (15)

##  IV Improvement of the PCW robustness to fabrication imperfections using
composite PCW structures

The first method is based on Eq. 4. We assume, that $\theta_{wg}$ remains the
same for all quarters of the PCW unit cell. However, it turns out that this
angle shouldn’t necessarily be realised by the same lattice period $a$ and
radius $r$ in each quarter. We state that if two PCWs with different lattice
periods $a_{1},a_{2}$ and hole radii $r_{1},r_{2}$, respectively, reveal the
peak of $F_{PCW}$ at the same wavelength, then the compound crystal will also
reveal the peak at the same wavelength (see fig. 3).

Figure 3: Compound PCW composed of two half-PCWs, designed to exhibit peak
value of $F_{PCW}$ at $\lambda_{1}$ and $\lambda_{2}$, which satisfy the
condition $\lambda_{1}+\lambda_{2}=2\lambda_{0}$.

We suggest that the PCW in the center of Figure 3 is more robust to the
fabrication imperfections than the one on the right, and the one on the left.
If we introduce slight random deviations to the values of radius $r$, the
$F_{PCW}$ spectral curve shifts away from the target wavelength and the
coupling of the dipole radiation to the PCW mode decreases substantially. We
do not take into account random deviations of the PCW period $a$ because its
value is several times larger than the deviation due to manufacturing
imprecision introduced in state-of-the-art fabrication lines [16]. It’s also
worth noting that a systematic bias of $a$ and $r$ is easily accounted for by
our theoretical description. If the bias in each of the parameters can be
determined experimentally, the other parameter can be adjusted accordingly
using the formulas for $c_{1}$ and $c_{2}$ coefficients. The idea behind
increased robustness to fabrication errors relies on simple reasoning. The
$\beta$ factor value gets higher than $0.9$ even at moderate $F_{PCW}\approx
10$ or greater, which can already be considered as good coupling of the
emitter radiation to the PCW mode. The width of the $F_{PCW}$ spectral
dependence is extremely narrow due to the mode dispersion curve (see fig 1) of
an ideal PCW. Our goal is to ’spoil’ the PCW structure in order to make the
$F_{PCW}$ spectrally wider at the expense of the $F_{PCW}$ peak value becoming
lower but still sufficient for good coupling.

Figure 4: The graphs shows summarized results of the simulations of the
compound crystals $F_{PCW}$ spectral curves. Each panel depicts the $F_{PCW}$
peak values and FWHMs for each of the 100 curves computed for compound PCWs
with added randomized hole radius error. The left column shows the results for
compound structures composed of two halves of PCW optimized for identical peak
$F_{PCW}$ wavelength $\lambda_{0}=925$ nm. The right column shows the results
for compound structures comprised of two halves of PCWs optimized for
different $\lambda_{1}$ and $\lambda_{2}$ satisfying
$\lambda_{1}+\lambda_{2}=2\lambda_{0}$.The insets in each figure illustrate
the used set of variable parameters.

We prove our point by performing numerical simulations using the Lumerical
software package. The model describes a composite PCW structure comprised of
two half-crystals with different periods and hole radii. Both halves
correspond to the PCWs delivering optimal $\beta$ at the same wavelength
$\lambda=925$ nm. We add random additive $\delta r_{i}$ sampled from the range
of $[-10,10]$ nm to the radius each hole in the numerical PCW model and
perform 100 simulation runs. Figure 4 illustrates the results of the
simulation. The best combination corresponds to the compound crystal with
periods $a_{1}=233$ nm and $a_{2}=238$ nm respectively. The geometry of both
parts of the compound crystal was estimated using our model and each one
reveals the highest value of $F_{PCW}$ at $925$ nm. This compound crystal
configuration demonstrates better performance (Fig. 4, left column, central
panel) with simulated fabrication defects compared to the standard PCW with
identical halves (Fig. 4 top row, left panel). To quantify the performance we
introduce an average Purcell enhancement factor $\tilde{F}_{PCW}$ and a
probability of Purcell enhancement $p$. The target wavelength lies within the
width at half maximum of Purcell enhancement curve in $p\times 100$ instances
of simulations, where $p$ is the probability of Purcell enhancement. The
average $\tilde{F}_{PCW}$ is a mean value of $F_{PCW}$ in these instances. In
the case of the compound crystal with randomized radii the probability of
Purcell enhancement at $925$ nm equals to $p=0.35$ and the average value
$\tilde{F}_{PCW}=8.69$, whereas for the standard crystal the values are 24 out
of 100 and $\tilde{F}_{PCW}=7.62$ respectively, confirming the robustness of
the compound PCW in comparison with the non-compound one.

Another option is to consider a compound PCW made of two parts designed to
exhibit maximal $F_{PCW}$ value at different wavelengths $\lambda_{1}$ and
$\lambda_{2}$. Here the question arises: at which wavelength the maximum
$F_{PCW}$ of a compund crystal will be observed? The answer turns out to be
simple: If the parts of the PCW reveal the peak $F_{PCW}$ value at wavelengths
$\lambda_{1}$ and $\lambda_{2}$ respectively, then the compound PCW reveals
the peak at $\frac{\lambda_{1}+\lambda_{2}}{2}$. One can easily prove this
statement by taking into account that there are two different reflective
surface systems (under an approximation of a small hole radius $r$, see Fig.
1e), and in order to prevent light from propagation in a direction
perpendicular to the waveguide axis, the effective distance between each
surface must be equal to $\lambda/2$, meaning that the total effective
distance between the surfaces on both sides from the emitter should be
$\frac{\lambda_{1}+\lambda_{2}}{2}$.

Configuration |  $\tilde{F}_{PCW}$ @ 925 nm | p | mean FWHM, nm | mean $F_{PCW}$
---|---|---|---|---
$\lambda_{1}=\lambda_{2}=925$ nm and $a_{1}=a_{2}=238$ nm | 7.6 | 0.24 | 3.0 | 24.9
$\lambda_{1}=\lambda_{2}=925$ nm and $a_{1}=233$ nm, $a_{2}=238$ nm | 8.6 | 0.35 | 4.0 | 21.4
$\lambda_{1}=\lambda_{2}=925$ nm and $a_{1}=223$ nm, $a_{2}=238$ nm | 7.0 | 0.57 | 7.4 | 10.2
$\lambda_{1}=920$ nm, $\lambda_{2}=930$ nm and $a_{1}=a_{2}=238$ nm | 6.9 | 0.26 | 3.5 | 23.8
$\lambda_{1}=915$ nm, $\lambda_{2}=935$ nm and $a_{1}=a_{2}=238$ nm | 5.4 | 0.18 | 4.4 | 20.6
$\lambda_{1}=905$ nm, $\lambda_{2}=945$ nm and $a_{1}=a_{2}=238$ nm | 4.6 | 0.44 | 11.1 | 12.5
Table 1: The summary of the combined crystal performance according to
numerical simulations. The mean FWHM and mean $F_{PCW}$ values indicate the
Purcell factor FWHM and maximal value averaged over 100 simulation runs.

The numerical results for this composition method are also obtained using 100
FDTD simulations similar to those used previously. The results are shown in
the right column of Fig. 4. The conclusion for this case is the following: the
greater is the difference between the wavelengths of both parts of a compound
PCW, the less is the maximal value and the greater is the FWHM of the
$F_{PCW}$ spectral curve. The periods for both parts can be the same or
different, but the radius value must be set according to our model, so that
each part of the PCW shows a peak at the required wavelength.

Results of the simulation of the Purcell factor spectral curve in the combined
crystals with randomized radius error $\delta r\in[-10,10]$ nm are summarized
in Table 1. We observe the tendency of the probability $p$ to grow when the
two halves of the PCW are designed with largely different parameters.

##  V Discussion and conclusion

We have established a mathematical connection between geometrical parameters
of a PCW structure and a Purcell enhancement factor at the specific
wavelength. Compound PCW structures which are predicted to provide maximal
enhancement at the target wavelength exhibit stronger robustness to the random
hole radius deviation compared to standard PCW structures. We attribute the
observed effect to the broken symmetry of the crystal. The improvement
manifests itself in higher probability of a dipole emitter to be efficiently
coupled to a PCW mode and the higher average Purcell enhancement factor
$\tilde{F}_{PCW}$. The $\tilde{F}_{PCW}$ values are around $\approx 10$ which
yields the $\beta$ factor to be $\approx 91\%$. Although such coupling
efficiency values cannot be considered as satisfactory for the most demanding
applications like fault-tolerant linear optical quantum computing [17, 18],
they can nevertheless enable near-term experiments with multiple single-photon
sources on an integrated platform.

We would like to draw the readers’ attention to a few interesting features
which were uncovered in course of the simulations with randomized radii
deviations. The configurations designed for identical target wavelengths
$\lambda_{1}=\lambda_{2}=925$ nm with unequal periods $a_{1}\neq a_{2}$ (see
Fig. 4) b and c) both show clustering of points. The configurations in Fig. 4c
and Fig. 4f have the most pronounced clustering along four vertical lines. We
were unable to explain the origin of this behaviour, but we speculate that
this effect might be related to the emergence of a topologically protected
mode inside the PCW bandgap [19]. Another peculiar observation is the tendency
to cluster along the vertical lines separated by an almost equal distance
$\Delta\lambda$. This means that a discrete set of wavelengths exhibits a
highly robust Purcell enhancement in the presence of the randomized hole
radius error.

Our results provide a clear understanding of the PCW parameter interplay and
thus they significantly simplify the initial structure design procedure. They
can also serve for augmenting sophisticated automated optimization design
routines by narrowing down the parameter space or serving as a quick sanity
check avoiding the necessity to run a 3D FDTD simulation task. Our heuristic
model describes a triangular PCW structure only, but we believe that similar
reasoning and mathematical analysis apply to any other photonic crystal
layout.

##  VI Acknowledgements

A. S., I. D. and S. S. acknowledge support by Rosatom in the framework of the
Roadmap for Quantum computing (Contract No. 868-1.3-15/15-2021 dated October
5, 2021 and Contract No.P2154 dated November 24, 2021). S. K. is supported by
the Ministry of Science and Higher Education of the Russian Federation on the
basis of the FSAEIHE SUSU (NRU) (Agreement No. 075-15-2022-1116).

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##  Appendix A The $\frac{a}{r}$ formula

The idea underlying the derivation of the formula for $\theta_{wg}$ is the
following: $\theta_{wg}$ is an angle between the waveguide’s axis and a beam
which is reflected without obtaining a component parallel to the waveguide’s
axis (see Fig. 1d). Then we can use simple geometric relations to get

$\tan\theta_{wg}=\frac{AB}{BC}=\frac{a\tan\theta_{gr}/2-r\cos\alpha}{a/2-r\sin\alpha}$
(16)

and $\alpha=\pi/4-\theta_{wg}/2$. Once we combine these relations we get the
equation

$\tan\theta_{wg}=\frac{a\tan\theta_{gr}/2-r\cos(\pi/4-\theta_{wg}/2)}{a/2-r\sin(\pi/4-\theta_{wg}/2)},$
(17)

which yields the formula in the form of eq. 4:

$\frac{a}{r}=\frac{\tan\theta_{wg}\sin(\pi/4-\theta_{wg}/2)-\cos(\pi/4-\theta_{wg}/2)}{\tan\theta_{wg}/2-\tan\theta_{gr}/2}.$
(18)

##  Appendix B Estimated value of the effective mode volume for a flat surface
Fabry-Perot resonator

We use a simple relation to estimate an effective mode volume
$V_{eff}=h\cdot{S_{eff}}$ of the Fabry-Perot resonator, comprised of the top
and bottom surfaces of the PCW membrane. We take into account the angular
structure of the dipole radiation which has the form of the product of two
functions $f(\theta)f(\phi)$. Based on that we can estimate the effective mode
area $S_{eff}$ as a product of effective mode lengths in both directions
$\theta$ and $\phi$ (see Fig. 1c). The effective mode length for each
direction can be derived as

$l_{eff}=\frac{\int_{L}Idl}{I_{max}}.$ (19)

The intensity value is inversely proportional to the distance between the
source and the observation point and hence we have
$I_{max}\sim\frac{1}{(h/2)^{2}}$ and $I\sim\frac{1}{AC^{2}}$. We use geometric
relations $AC=\frac{h/2}{\sin\theta}$, $DC=h/2\cot\theta$,
$dl=d(DC)=\frac{h/2}{\sin^{2}\theta}d\theta$ and substitute them into equation
(19). We also need to account for the dipole radiation angular dependence
$\sin^{2}\theta$ and the term $\sin\theta$ corresponding to projection on the
$z$ axis, because we are only considering light states propagating
perpendicular to the PCW membrane. The dependence on the azimuthal angle
$\phi$ accumulates only a single $\sin\phi$ term due to the absence of
dependency on $\phi$ in the dipole emission pattern. The final integrals for
both effective lengths are

$\begin{array}[]{c}l_{\theta}\sim\frac{1}{I_{max}}\int_{0}^{\pi}{\frac{\sin^{3}(\theta)}{h/2}d\theta},\\\
l_{\phi}\sim\frac{1}{I_{max}}\int_{0}^{\pi}{\frac{sin\phi
d\phi}{h/2}}.\end{array}$ (20)

The integration yields effective length values $l_{\theta}\sim\frac{2}{3}h$
and $\l_{\phi}=h$. The total effective mode volume then equals to
$V_{eff}=h\cdot\frac{2}{3}h\cdot{h}=\frac{2}{3}h^{3}$.

##  Appendix C The calculation algorithm

One can use the following procedure in order to calculate the value of the
hole radius $r$ given the parameters $h,n,\lambda_{d},a$ :

  1. 1.

Set an initial guess $r_{0}$ for the radius value for the given $a$, it should
be relatively close to the exact value;

  2. 2.

Calculate the $F_{FP}$ using equation (5);

  3. 3.

Calculate the $F_{PCW}$ using equations (6)-(10) and use $r_{0}$ value defined
at step 1. After calculating the PCW mode volume value substitute it into
equation (1);

  4. 4.

Calculate the slope coefficient $c_{2}$ for the given $a$ using equation (4);

  5. 5.

Calculate the $c_{1}$ coefficient using equations (11)-(15);

  6. 6.

The exact radius for the model can now be estimated as
$r=\frac{a-c_{1}}{c_{2}}$;

  7. 7.

Repeat all the steps using $r$ as the new initial value. This should be done
until the initial value of radius set at step 1 equals to the exact value,
which was obtained at step 6.

Such procedure is necessary due to the explicit dependence $r(a)$, and it is
the easiest way to obtain the result.

##  Appendix D The FDTD simulation details

Figure 5: The example of the numerically computed Purcell factor spectral
dependece for a given crystal configuration.

We used a numerical model of a PC waveguide to compare to the predictions of
our heuristic model. We set up the simulation of a PCW structure in Lumerical
FDTD. We set the transverse dimensions of a GaAs membrane to be larger than
the simulation region to avoid the internal reflections inside the crystal. We
use PML boundary conditions along the waveguide axis as well as above and
below the membrane. We do not set any specific Dirichlet boundary conditions
[14], because we are not focused on precise calculation of the field
distribution in the PCW structure. The PML boundary conditions must not
overlay the features of the PCW in order to exclude numerical artifacts.

The PC waveguide dispersion was calculated in the following way. The
simulation uses Bloch boundary conditions in the direction perpendicular to
the waveguide axis. We use a dipole cloud as a source of radiation which
excites multiple possible modes in the system. The randomly positioned
electric field monitors record the local field amplitude versus the simulation
time. The recorded traces are then decomposed as an infinite sum of
exponentially decaying harmonic functions. Frequencies with the least decay
rate are considered to be eigenfrequencies corresponding to a given wavevector
modulus $k$, which is used to specify Bloch boundary conditions.

The Purcell factor $F_{PCW}$ at the wavelength of interest was estimated using
the built-in function of the Lumerical FDTD Software package. The example of
the Purcel factor spectral dependence is illustrated in Fig. 5.