# Hybrid photonic-plasmonic cavity design for very large Purcell factors at
telecom wavelengths

Angela Barreda  Friedrich Schiller University Jena, Institute of Solid State
Physics, Helmholtzweg 3, 07743 Jena, Germany  Friedrich Schiller University
Jena, Institute of Applied Physics, Abbe Center of Photonics, Albert-Einstein-
Str. 15, 07745 Jena, Germany  Laura Mercadé  Nanophotonics Technology Center,
Universitat Politècnica de València, Camino de Vera s/n, 46022, Valencia,
Spain  Departament d’Enginyeria Electrónica i Biomédica, Facultat de Física,
Universitat de Barcelona.  Mario Zapata-Herrera  Materials Physics Center
CSIC-UPV/EHU, 20018 Donostia-San Sebastian, Spain  Javier Aizpurua  Materials
Physics Center CSIC-UPV/EHU, 20018 Donostia-San Sebastian, Spain  Donostia
International Physics Center DIPC, 20018 Donostia-San Sebastian, Spain
Alejandro Martínez  Nanophotonics Technology Center, Universitat Politècnica
de València, Camino de Vera s/n, 46022, Valencia, Spain

###### Abstract

Hybrid photonic-plasmonic cavities can be tailored to display high $Q$-factors
and extremely small mode volumes simultaneously, which results in large values
of the Purcell factor, $F_{\mathrm{P}}$. Amongst the different hybrid
configurations, those based on a nanoparticle-on-a-mirror (NPoM) plasmonic
cavity provide one of the lowest mode volumes, though so far their operation
has been constrained to wavelengths below 1 $\upmu$m. Here, we propose a
hybrid configuration consisting of a silicon photonic crystal cavity with a
slot at its center in which a gold nanoparticle is introduced. This hybrid
system operates at telecom wavelengths and provides high $Q$-factor values
($Q\approx 10^{5}$) and small normalized mode volumes ($V_{\mathrm{m}}\approx
10^{-4}$), leading to extremely large Purcell factor values,
$F_{\mathrm{P}}\approx 10^{7}-10^{8}$. The proposed cavity could be used in
different applications such as molecular optomechanics, bio- and chemo-
sensing, all-optical signal processing or enhanced Raman spectroscopy in the
relevant telecom wavelength regime.

††preprint: APS/123-QED

##  I Introduction \\\

The local density of optical states (LDOS) quantifies the available amount of
electromagnetic states that can be occupied by a photon in a certain position
of a system. A high local value of the LDOS will result in enhanced light-
matter interaction there. This way, the environment of an emitter can be
tailored to maximize the LDOS, for instance, using optical cavities. At
resonance, the LDOS gives us the Purcell factor, $F_{\mathrm{P}}$, which
originally accounted for the enhancement of the spontaneous emission of an
atom in a cavity [1, 2, 3]. By definition, the Purcell factor is proportional
to the ratio between the $Q$-factor and the mode volume $V_{\mathrm{m}}$ as
[4, 5]:

$F_{\mathrm{P}}=\frac{3}{4\pi^{2}}\frac{Q}{V_{\mathrm{m}}}$ (1)

where $V_{\mathrm{m}}$ is the mode volume normalized by the wavelength
($\lambda$) over the local refractive index ($n$) cubed $(\lambda/n)^{3}$, and
$Q$ is the quality factor. This suggests a route to design photonic systems
that exhibit simultaneously high $Q$-factor and ultra small $V_{\mathrm{m}}$
to maximize $F_{\mathrm{P}}$ and enhance light-matter interaction.

Dielectric cavities can greatly enhance the Purcell factor since they can have
high $Q$-factor values as a result of the negligible absorption loss [6].
However, the diffraction limit usually prevents them to achieve subwavelength-
scale mode volumes, so $V_{\mathrm{m}}\approx 1$ unless special “slotted”
configurations are used [7, 8, 9]. On the contrary, plasmonic cavities formed
by metallic nanoparticles (NPs) can overcome the diffraction limit leading to
ultrasmall $V_{\mathrm{m}}$ values[10]. However, Joule’s losses in the metal
reduces the reachable $Q$-factor to values $Q\approx 10$ [11, 12].

In the last few years, hybrid photonic-plasmonic cavities have been introduced
as novel systems providing large $F_{\mathrm{p}}$ as they can potentially
combine the best of both worlds: high $Q$-factor values due to the dielectric
photonic cavity and small mode volumes thanks to the metallic NP [11, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22]. Indeed, by modifying the coupling between
photonic and plasmonic modes, the $Q$-factor and $V_{\mathrm{m}}$ can be tuned
between the values of the bare NP and the cavity [17]. Different
configurations of hybrid photonic-plasmonic cavities have been suggested so
far. Some of the most attractive ones consist of a bow-tie nanoantenna placed
on a dielectric photonic crystal nanobeam cavity operating at transverse-
electric polarization [17] ($F_{\mathrm{p}}\approx 1\cdot 10^{4}$). The main
limitation of this configuration is that the plasmonic gap has to be defined
lithographically reaching minimum values of several nm, which prevents
reaching the nm-scale (and below) gaps that leads to extreme plasmonic
localization [23, 24, 25].

Nanoparticle-on-a-mirror (NPoM) cavities are far superior to achieve the
smallest possible mode volumes, as plasmonic gaps around 1 nm and below can be
created. The reason is that NPs can be deposited on top of self-assembled
monolayers (SAMs) formed on metallic surfaces so the plasmonic gap widths
equal the SAM thickness [26, 27, 28, 29, 30]. Recently, a hybrid system
combining a NPoM configuration with GaP photonic crystal cavities - operating
under transverse-magnetic polarization at visible wavelengths - has been
introduced [31]. Thanks to the NPoM-based design, nm- and sub-nm-scale
plasmonic gaps potentially can be achieved and combined with relatively large
$Q$-factors. By means of this novel hybrid system, the Purcell factor values
are enhanced one order of magnitude with respect to the hybrid configurations
based on bow-tie nanoantennas ($F_{\mathrm{p}}\approx 1\cdot 10^{5}$) [31].

Most of the previous works on hybrid photonic cavities have been focused on
the visible wavelength range due to the huge interest in enhancing light-
matter interaction in that band [17]. However, at the telecom wavelengths
($\lambda\approx 1550$ nm) is highly interesting. First, because of the
multiple available applications (mostly related to high-speed optical
communications and data processing) as well as advanced available equipment
(laser, amplifiers and high-speed photodetectors); and, second, because of the
emergence of the silicon photonics area, which has become the mainstream
technology for photonic integrated circuits. Indeed, there have been many
demonstrations of silicon cavities operating at telecom wavelengths with $Q$
factors over $10^{4}$ using lithographically-defined 1D
($F_{\mathrm{p}}\approx 2.7\cdot 10^{6}$) [32] or 2D ($F_{\mathrm{p}}\approx
1.7\cdot 10^{5}$) [33] photonic crystal cavities whilst showing
$V_{\mathrm{m}}\approx 1$. Reaching $V_{\mathrm{m}}<1$ becomes feasible by
defining thin slots in silicon and taking advantage of the discontinuities of
the normal electric field [34, 7], but with the minimum slot size imited by
lithography. In previous work [8], complex subwavelength structuring of
silicon (down to $\approx 10$ nm) enabled to get also $V_{\mathrm{m}}<2\cdot
10^{-4}$.

Owing to this technological interest, hybrid systems have also been recently
proposed in silicon technology for telecom wavelengths. In [19], a hybrid
photonic-plasmonic nano-cavity, constituted by an L3 photonic crystal nano-
cavity and a plasmonic bow-tie nanoantenna was shown to have an ultrahigh
figure of merit $Q/V$ of $8.4\times 10^{6}$ $(\lambda/n)^{-3}$
($F_{\mathrm{p}}\approx 6.3\cdot 10^{5}$). However, either in the only-silicon
cavity [8], or in the hybrid approach [19] the extreme localization enabled by
the NPoM is still missing.

In this work, we propose a hybrid photonic-plasmonic cavity that combines
silicon and the NPoM configuration to get extreme field localization at
resonance. This hybrid consists of a silicon photonic cavity with a slot at
its center [34, 9], operating at transverse-electric polarization at telecom
wavelengths, and a gold NP, which is located at the slot’s cavity. The NP can
be potentially separated from the silicon walls of the slot by a SAM, thus
reaching the nm- and even sub-nm-scale localized fields at resonance. Our
design achieves $Q\approx 1\cdot 10^{5}$ and $V_{\mathrm{m}}\approx 1\cdot
10^{-4}$, giving rise to Purcell factor values of
$F_{\mathrm{P}}\approx{1\cdot 10^{7}}-1\cdot 10^{8}$, which are two-three
orders of magnitude larger than those reported previously based on NPoM
cavities combined with photonic crystals and two orders of magnitude larger
than those corresponding to hybrid photonic-plasmonic cavities operating at
telecom wavelengths. This is mainly due to the nanoscale confinement provided
by the slot-nanoparticle combination and the radiation suppression resulting
from the photonic crystal structure.

We also suggest a route to fabricate - combining top-down and bottom-up
approaches - the hybrid cavity, which may be used to improve the performance
of current silicon photonic devices as well as to envisage new applications in
the relevant telecom-wavelength domain.

##  II Hybrid system description

Figure 1 depicts the proposed hybrid photonic-plasmonic configuration. A gold
NP is located at the center of the slot of a 1D silicon photonic crystal
cavity operating under transverse-electric polarization at telecom
wavelengths. The distance between the NP and the photonic crystal wall (gap,
$d$) is $d=1$ nm, which approximately models the thickness of a SAM. The
silicon wall behaves here as the mirror, though with lower reflectance than
for a metallic one. Such a small gap, which is not defined lithographically
but after depositing the NP after the silicon cavity is fabricated, should
allow for extreme light confinement, thus giving rise to small
$V_{\mathrm{m}}$. In addition, high $Q$-factors can be achieved due to the
photonic bandgap of the dielectric cavity.

Figure 1: Scheme of the proposed hybrid system. (a) Geometry of the 1D slotted
photonic crystal cavity. (b) Top view of the cavity’s center, where the slot
of width $W$ can be distinguished. A gold spherical NP is located inside the
slot. The distance between the NP and the wall of the cavity in the
$y$-direction is $d=1$ nm as shown in the close view marked with dashed-dotted
line.

##  III Methods

The eigenfrequencies and LDOS calculations shown in this work were obtained by
means of the Finite Element Method (FEM), implemented in the commercial
software COMSOL Multiphysics [35]. In particular, the Radio Frequency (RF)
Module was used. The optical constants for the gold NP were taken from Ref.
[36]. The refractive index for silicon was considered as $n=3.48$. To attain
the LDOS, the structure was illuminated by an electric point-dipole source,
whose dipole moment was considered along the $y$-axis (i.e. along the axis
connecting the sphere and the cavity). The normalized LDOS is defined as

$LDOS=\frac{P_{\mathrm{rad}}+P_{\mathrm{no-rad}}}{P_{\mathrm{rad\ vacuum}}}$
(2)

where $P_{\mathrm{rad}}$ is the radiative power density and $P_{\mathrm{no-
rad}}$ is the total power loss density. The numerator corresponds to the
radiative and nonradiative power emitted by the dipole coupled to the
structure. However, the denominator contains the radiative power emitted by
the dipole in a vacuum. The dipole was located at the center of the structure
and in the middle of the gap between the NP and the cavity. Experimentally,
this spacer would be filled by a SAM. The hybrid geometry was surrounded by a
cylindrical air region of radius 9 $\mu$m. An additional smaller cylinder with
a radius of 5 $\mu$m and made of air was placed in the center of the larger
cylinder. The scattered power ($P_{\mathrm{rad}}$) was calculated at the
boundaries of the smaller cylinder, whereas the total power loss density was
obtained by means of the volume integration of the losses in the metallic NP.
The dipole source was surrounded by a sphere with a diameter that is equal to
the gap size ($d=1$ nm) to ensure a sufficient fine grid in close proximity to
the dipole source. The mesh of the surrounding air medium was chosen to be
smaller than 600 nm. The mesh of the gold sphere was smaller than 6 nm. The
chosen sizes for the mesh of the slot, cavity surrounding the slot, holes and
cavity surrounding the holes were smaller than 15 nm, 22.5 nm, 45 nm and 52.5
nm, respectively. The mesh of the sphere surrounding the dipole source was
smaller than 0.2 nm. The cylindrical region of air was surrounded by a
perfectly matched layer (PML) with a thickness of 3 $\mu$m.

The near-field plots at the eigenfrequencies of interest were attained with
the eigenmode solver of COMSOL Multiphysics. The same mesh was used for the
LDOS and eigenmode calculations and for the different analyzed structures:
dielectric (no NP) and hybrid cavity. In all the cases, the same geometry was
considered, and only optical constants of the materials (if present or not)
were changed appropriately.

##  IV Results

###  IV.1 Photonic band diagram

The 1D silicon photonic crystal cavity consists of a silicon waveguide of
width $h=550$ nm and thickness $t=220$ nm patterned with holes and with a slot
in the defect section, as depicted in Figure 1(a). The beam is drilled by a
set of circular holes to produce a transverse-electric (TE) band gap around
$\lambda=1550$ nm. Figure 2(a) shows the TE-like band diagram when the holes
are spaced by a distance $a=580$ nm and have a radius $r=0.365a$, showing a
wide bandgap (shadowed region) as expected. To build the cavity, we include
two photonic crystal mirrors with the previous dimensions at each side of the
cavity whilst the period is adiabatically reduced from $a=580$ nm down to
$a=331.5$ nm (keeping constant the $r/a$ ratio) when moving to the cavity
center. This permits to blue-shift the dielectric band and get a confined mode
in the band gap, as depicted by the red dashed line in Fig. 2(a). Finally, a
slot section of width $w=40$ nm and length $l=547$ nm is introduced between
the central holes of the cavity to enhanced the transverse electric field.
Notice that this slot width is achievable using standard silicon
nanofabrication [9].

Figure 2(b) represents the near-field map at the eigenfrequency of the
confined mode ($\lambda=1613$ nm) in the 1D photonic crystal cavity following
the previous design. It can be clearly seen that the electric field is well
confined in the slot placed at the center of the cavity. It is worth
mentioning that the cavity resonance is red-shifted with respect to the gold
NP resonance. In previous works, it was evidenced that this condition is
necessary to improve the performance of the hybrid cavity with respect to the
bare components [17, 11, 18].

Figure 2: a) Photonic band diagram of the mirror unit cell. In blue (green)
the TE-like (TM-like) bands are depicted. The shadowed region corresponds to
the TE-like quasi bandgap where the photonic crystal cavity mode confined in
the middle region is located (depicted with a red dashed line). b) Mode
profile of $|\mathrm{E}|$ (X-Y crosscut) normalized to its maximum value for
the slotted photonic crystal without the spherical gold nanoparticle.

###  IV.2 Comparison $Q$-$V_{\mathrm{m}}$ values for the bare cavity and
hybrid system

In Figs. 3(a) and (b), we represent the calculated LDOS for the bare cavity
(slot width $w=40$ nm) and the hybrid (Hyb) system composed of the silicon
cavity plus a gold NP of radius $R=19$ nm, respectively. For the bare cavity,
the $Q$-factor is retrieved from the fitting of LDOS to a Lorentzian function.
Once the $Q$-factor is known, the mode volume is retrieved by the inversion of
Eq. 1. The obtained $Q$-factor of the bare cavity is $Q_{\mathrm{c}}=1.6\cdot
10^{5}$, whilst the mode volume value is $V_{\mathrm{c}}=4\cdot 10^{-2}$. As
expected, high $Q$ values are achieved, and could even be higher by a more
meticulous design of the bare cavity. However, the mode volume is still large
in comparison to the values that can be obtained by considering metallic
nanostructures ($V_{\mathrm{NP}}\approx 1\cdot 10^{-6}$). When the NP is
introduced in the gap, the $Q$-factor of the hybrid system is only slightly
diminished ($Q_{\mathrm{Hyb}}=8.3\cdot 10^{4}$) but there is a two orders of
magnitude reduction of $V_{\mathrm{m}}$ down to $V_{\mathrm{Hyb}}=3.2\cdot
10^{-4}$ as a result of the hybridization of the metallic NP with the photonic
cavity. For the hybrid cavity a Fano fitting is necessary due to the
hybridization of two different modes: the corresponding to the cavity and that
of the NP.

If we observe the near-field maps of the electric field in the slot for the
bare cavity and the hybrid system (see Fig. 3(c) and (d), respectively), we
see that the gold NP squeezes the slot field into a nm-scale region, which was
the purpose of introducing the NPoM approach. Therefore, the hybrid system
provides a way for extreme localization of telecom-wavelength fields whilst
keep large $Q$ values.

Figure 3: Normalized LDOS, $Q$-factor and $V_{\mathrm{m}}$ for a) the bare
silicon photonic crystal ($Q_{\mathrm{c}}$, $V_{\mathrm{c}}$) and b) the
hybrid plasmonic-photonic cavity ($Q_{\mathrm{Hyb}}$, $V_{\mathrm{Hyb}}$). The
slot size is $w=40$ nm and the radius of the gold NP is $R=19$ nm. Mode
profile of $|\mathrm{E}|$ (X-Y crosscut) normalized to its maximum value in
the slot for c) the bare silicon photonic crystal and d) the hybrid photonic-
plasmonic cavity. The inset figure in panel b) shows a close view of the
confined mode profile $|\mathrm{E}|$ of the hybrid photonic-plasmonic cavity
in the cavity’s center.

###  IV.3 Evolution of $Q$-$V_{\mathrm{m}}$ values with the slot size

From an experimental point of view, some differences in the size of the slot
can be expected with respect to the nominal parameters due to imperfections
during the fabrication process. In this section, we analyze the influence of
the slot width in the $Q$-factor and $V_{\mathrm{m}}$ values. Figure 4(a)
shows the $Q$-factor and $V_{\mathrm{m}}$ values as a function of the slot
size for the bare cavity and the hybrid system. We observe that as the size of
the slot is modified (both increased or decreased) with respect to the optimum
value $w=40$ nm, the $Q$-factor decreases almost one order of magnitude.
However, $V_{\mathrm{m}}$ values are not so affected due to changes in $W$,
especially for the bare cavity. In all the cases, the gap distance between the
gold NP and the wall of the cavity is $d=1$ nm, which means that the NP radius
is modified accordingly for each $w$ value.

Figure 4: (a) $Q$-factor (blue markers) and $V_{\mathrm{m}}$ (red markers)
values for the bare cavity (points) and hybrid system (squares) as a function
of the slot size $W$. The radius of the gold NP has been varied accordingly to
keep the gap distance between the NP and the wall of the photonic crystal to
$d=1$ nm. (b) $Q$-factor (blue markers) and $V_{\mathrm{m}}$ (red markers)
values for the hybrid system as a function of the gap size. The size of the
slot was kept constant to $W=40$ nm. The $y$-axis is represented in
logarithmic scale.

###  IV.4 Evolution of $Q$-$V_{\mathrm{m}}$ values with the NP-cavity gap

Small imperfections in the chemical production of the NP can also generate
imprecision in its size, provoking uncertainty in the gap distance between the
NP and the wall of the photonic cavity. To analyze this effect, we have
represented in Fig. 4(b) the $Q$-factor and $V_{\mathrm{m}}$ values for
different gap sizes for a slot cavity of $W=40$ nm. In particular, the size of
the NP was changed between $R=19$ nm and $R=10$ nm, leading to variations in
the gap from $d=1$ nm to $d=10$ nm. As the gap increases, both
$Q_{\mathrm{Hyb}}$ and $V_{\mathrm{Hyb}}$ increase. This effect is due to the
lower confinement of the electromagnetic radiation in the gap.

##  V Conclusions

In this work we have proposed a novel design of a hybrid photonic-plasmonic
cavity, which combines a 1D silicon photonic cavity with a slot in its center,
operating under transverse-electric polarization at telecom wavelengths, with
a gold nanoparticle, located at the slot’s center. This configuration shows
how to merge the NPoM approach with silicon photonic crystal cavities in order
to achieve extreme light confinement in nm-scale gaps. Through this
nanostructure we have obtained $Q$-factor values for the hybrid system larger
than $Q_{\mathrm{Hyb}}=1\cdot 10^{5}$ and mode volumes smaller than
$V_{\mathrm{Hyb}}=1\cdot 10^{-4}$. This gives rise to Purcell factor values of
$F_{\mathrm{P}}\approx 1\cdot 10^{7}-1\cdot 10^{8}$, which are two or three
orders of magnitude higher than those obtained in previous NPoM hybrid
photonic-plasmonic cavities [31] or metallic bow-ties antennas coupled
photonic crystals operating under transverse-electric polarization [17]. If we
extend the comparison to hybrid cavities operating at telecom wavelengths, we
can conclude that our hybrid cavity provides two orders of magnitude larger
than hybrid photonic-plasmonic nano-cavity reported in [19]. This is mainly
due to the nanoscale confinement provided by the slot-nanoparticle combination
and the radiation suppression resulting from the photonic crystal structure.

From a practical perspective, the 1D silicon photonic crystal cavity can be
fabricated using standard silicon technology [9, 8]. It is also possible to
release the silicon beam from the substrate [37] as in the configuration in
this work. NPs can be later deposited by drop casting, potentially falling
into the slot, as recently demonstrated for plasmonic nanoslits [38]. The use
of optical forces could even be used to trap the NPs inside the slot, as
demonstrated in slot waveguides [39]. Moreover, some techniques would enable
the transfer of isolated gold NPs to the slot. If the beam is not released
from the substrate, the photonic bandgap would be somewhat reduced but
high-$Q$ modes are still possible and the NPs could be deposited on the silica
substrate at the slot bottom. Thus, we foresee that a combination of top-down
(for the photonic crystal cavity) and bottom-up (for the gold NPs deposition)
techniques would make possible the experimental realization of this cavity,
enabling molecular-scale light confinement together with large $Q$-factors.

Building hybrid cavities that operate at telecom wavelengths does not only
offers new perspectives in applications already existing in that spectral
regime (optical communications, photonic biosensing, nonlinear signal
processing) but could also enable the transfer of (near-)visible-regime
applications (such as surface-enhanced Raman spectroscopy [40] or
optomechanically-driven frequency conversion [38, 41]) to a domain where a
huge amount of high-quality instrumentation (tunable lasers, high speed photo-
detectors, amplifiers) is available, whilst being implementable in silicon
compatible chips.

###### Acknowledgements.

A. B. thanks the financial support by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) through the International Research Training
Group (IRTG) 2675 “Meta-ACTIVE”, project number 437527638. L.M. thanks
financial support from the Next generation EU program, Ministerio de
Universidades (Gobierno de España). A. M. acknowledges funding from H2020
European Commission (THOR-H2020-EU-829067); and Generalitat Valenciana
(PROMETEO/2019/123). M. Z. H. and J. A. acknowledges Spanish Ministry of
Science (Project No. PID2019-107432GB-I00) for financial support,

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