Enhancing the robustness of coupling between a single emitter and a photonic crystal waveguide Alexander Shurinov and Ivan Dyakonov

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Enhancing the robustness of coupling between a single emitter and a photonic

crystal waveguide

Alexander Shurinov and Ivan Dyakonov

Quantum Technologies Centre, Lomonosov Moscow State University,

Russia, Moscow, 119991, Leninskie Gory 1 building 35

Sergei Kulik

Quantum Technologies Centre, Lomonosov Moscow State University,

Russia, Moscow, 119991, Leninskie Gory 1 building 35 and

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 and

Russian Quantum Center, Russia, Moscow, 121205, Bol’shoy bul’var 30 building 1

(Dated: October 17, 2022)

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.

I. INTRODUCTION

A planar photonic crystal waveguide (PCW) is a

rich system ﬁnding applications in diverse areas of op-

tical physics. Among those are slow light [1], topolog-

ical 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 partic-

ular, 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 λ

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 Pur-

cell 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 pro-

cess produces QDs with randomly distributed spec-

tral characteristics. The emission wavelength of QDs

typically falls in range of a few nanometers around

the designed center wavelength. The ﬁrst derivative

dω/dk of the PCW dispersion curve gets close to zero

in a very narrow wavelength range λ0±∆λ/2 and eﬃ-

cient Purcell enhancement is not guaranteed for most

of the fabricated QDs with emission wavelengths miss-

ing the ∆λregion. Furthermore, the fabrication pro-

cess 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 experi-

ments [7]. At the same time current trends in optical

quantum computing demand the development of hy-

brid 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 ap-

proach 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 wave-

length and completely eliminate the necessity to eval-

uate multiple time consuming 3D FDTD simulations.

After the description of the heuristic PCW theory we

address the problem of PCW robustness to fabrica-

tion 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

arXiv:2210.07355v1 [quant-ph] 13 Oct 2022

quality factors of photonic crystal microcavities [9, 10]

and automated design methods to optimize the pho-

tonic crystal microcavity structure [11, 12]. We fo-

cus 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 robust-

ness of coupling to the fabrication errors and test them

using numerical simulations.

II. PHOTONIC CRYSTAL

A typical two-dimensional photonic crystal is a pe-

riodic 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 mi-

croresonator with small mode volume and high qual-

ity factor. These systems were demonstrated to suite

the purpose of integration of A3B5quantum dot sin-

gle 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 in-

tegrated 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 delet-

ing 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 semiconduc-

tor quantum dot single photon source. We start with

the description of PCW characteristics and develop-

ment of its heuristic model. Manga Rao and Hughes

[13] derived an expression for a Purcell factor Fpin

terms of PCW parameters:

Fp=3πc3a

Veff ω2

d3/2vg

,(1)

where ais the distance between air holes (if the lat-

tice is triangular a=ax, since we are focused on that

type of lattice from now on we will write ainstead

ax, but for another lattice angle axis the only cor-

rect option), Vef f is the eﬀective mode volume and

vgis the group velocity at the resonant frequency of

the dipole ωd. The formula indicates that the largest

Fpis achieved when the wavepacket group velocity

reaches zero. Thus the design of a PCW eﬃciently

coupled with a single emitter resonant at ωdis equiva-

lent to engineering a PCW dispersion law to meet the

requirement dω/dk(ωd) = 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 expres-

sions 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 exam-

ple are as follows: PC hole pattern angle θgr = 60◦,

period a= 0.238 µm, hole radius r= 0.08 µm and

membrane thickness h= 0.16 µm. These values were

chosen to put the Purcell factor FP CW 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 Pur-

cell enhancement happening in diﬀerent PCW conﬁg-

urations, results are presented in Fig. 2.The 3D FDTD

simulation was carried out in Lumerical FDTD pack-

age, the details of the simulation are speciﬁed in Ap-

pendix D. We observed a continuous set of conﬁgura-

tions of a triangular PCW with the same lattice an-

gle corresponding to a peak value of FP CW at a tar-

get wavelength. The red line in Fig. 2 illustrates the

numerically computed set of (a, r) conﬁgurations cor-

responding 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=c1+c2r,

where the coeﬃcients c1and c2are weakly dependent

on aand r. The a(r) dependence is ﬁnely approx-

imated by a linear function in the region where the

FP CW reaches its highest levels. The yellow curve rep-

resents the values of aand rcorresponding to FP CW

peak which are provided by the proposed theoretical

description.

In the following subsections we provide a heuristic

theoretical description for the origins of such depen-

dence and the values of c1and c2coeﬃcients.

A. The slope coeﬃcient c2

The Purcell factor FP CW deﬁnes the probability

β=FP CW /(1 + FP CW ) (2)

of emitting a photon into a PCW mode. The existence

of a PCW mode is a purely interferometric eﬀect hence

the probability pshould be related to the geometry of

a photonic crystal. We expect to derive the connec-

tion between the geometric parameters which corre-

spond to a conﬁguration of a PCW structure reaching

maximal Purcell factor for the required wavelength.

We roughly split the emitter radiation into three cate-

gories: light exiting the PCW plane, light propagating

inside the PCW structure, and light coupled to the

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EnhancingtherobustnessofcouplingbetweenasingleemitterandaphotoniccrystalwaveguideAlexanderShurinovandIvanDyakonovQuantumTechnologiesCentre,LomonosovMoscowStateUniversity,Russia,Moscow,119991,LeninskieGory1building35SergeiKulikQuantumTechnologiesCentre,LomonosovMoscowStateUniversity,Russia,Moscow,119...

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Enhancing the robustness of coupling between a single emitter and a photonic crystal waveguide Alexander Shurinov and Ivan Dyakonov

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