# Effects of surface roughness and top layer thickness on the performance of
Fabry-Perot cavities and responsive open resonators based on distributed Bragg
reflectors

K. Papatryfonos  Université Paris-Saclay, CNRS, Centre de Nanosciences et de
Nanotechnologies, 91120 Palaiseau, France  E.R. Cardozo de Oliveira
Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies,
91120 Palaiseau, France  N. D. Lanzillotti-
<EMAIL_ADDRESS>Université Paris-Saclay, CNRS, Centre
de Nanosciences et de Nanotechnologies, 91120 Palaiseau, France

###### Abstract

Optical and acoustic resonators based on distributed Bragg reflectors (DBRs)
hold significant potential across various domains, from lasers to quantum
technologies. In ideal conditions with perfectly smooth interfaces and
surfaces, the DBR resonator quality factor primarily depends on the number of
DBR pairs and can be arbitrarily increased by adding more pairs. Here, we
present a comprehensive analysis of the impact of top layer thickness
variation and surface roughness on the performance of both Fabry-Perot and
open-cavity resonators based on DBRs. Our findings illustrate that even a
small, nanometer-scale surface roughness can appreciably reduce the quality
factor of a given cavity. Moreover, it imposes a limitation on the maximum
achievable quality factor, regardless of the number of DBR pairs. These
effects hold direct relevance for practical applications, which we explore
further through two case studies. In these instances, open nanoacoustic
resonators serve as sensors for changes occurring in dielectric materials
positioned on top of them. Our investigation underscores the importance of
accounting for surface roughness in the design of both acoustic and optical
DBR-based cavities, while also quantifying the critical significance of
minimizing roughness during material growth and device fabrication processes.

##  1 Introduction

Distributed Bragg reflector (DBR) devices based on semiconductor
heterostructures are pivotal components in both fundamental and applied fields
of photonics and nanophononics. A pair of DBRs enclosing an optical spacer
constitutes a Fabry-Perot optical cavity, capable of shaping the optical local
density of states. Over the past three decades, optical cavities have been
used in a plethora of applications encompassing quantum technologies,
optoelectronics, photonics, and spectroscopy [1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12]. Similarly, acoustic cavities utilizing the same DBR configurations
can confine and enhance phononic fields, offering prospects for ultra-high
frequency applications [13, 14, 15]. Furthermore, these cavities were recently
used in optomechanics, and as platforms for simulating solid-state physics
phenomena [16, 17, 18, 19, 20, 21, 22, 23, 24].

The widespread use of DBRs and their resonant structures emphasizes the
importance of developing accurate tools for optimized device design. Current
designs and evaluations of optical and nanophononic devices often focus solely
on the number of DBR periods and materials [25]. However, the reality of
material growth and fabrication always entails some surface roughness, even in
samples produced using cutting-edge techniques. While the role of interface
roughness has been discussed in some cases [26], the impact of surface
thickness variation and surface roughness is commonly overlooked, presumably
due to the assumption that its influence might be negligible. To justify this
conjecture, one would have to assume surface roughness of negligible
magnitude, combined with the fact that the confined mode is mainly localized
within the cavity spacer. However, in reality, some variations of the top
layer thickness often occur during device processing, or due to oxidation of
the top layer of structure. Additionally, some degree of surface roughness is
invariably present even when using state-of-the-art growth techniques [27, 28,
29]. The resulting top layer variations and roughness can significantly affect
the performance of DBR resonators, which rely on precise definition of the
layered structure to define the resonant frequencies.

In this study, we delve into the impact of top layer thickness and surface
roughness on the phonon dynamics and quality factor (Q-factor) in Fabry-Perot
(FP) cavities and nanoacoustic open resonators based on DBRs. Our
investigation centers on GaAs/AlAs DBR superlattices, which are extensively
utilized in both acoustic and optical micro-cavities. We examine both Fabry-
Perot and open-cavity resonators, unraveling the influence that even small
variations in the last layer thickness and roughness can exert on the
Q-factor, which is the metric that quantifies energy dissipation in these
resonators.

Moreover, open resonators are particularly useful for applications that demand
environmental responsivity. What is more, additional materials can be
deposited on them to form an acoustic resonator responsive to external
stimuli. We explore practical scenarios where these cavities function as
sensors with dielectric materials positioned atop them, allowing for a direct
assessment of the impact of roughness on device performance. Specifically, we
incorporate VO2 and mesoporous SiO2 materials, conducting a detailed
exploration of the parameter space to assess the combined influence of
roughness and number of DBR periods on the quality factor. Materials sensitive
to external stimuli, such as mesoporous thin films reacting to humidity
changes [30, 31, 32], and VO2 responding to thermally or optically induced
phase transitions [33, 34], undergo changes in their elastic properties under
such stimuli [35, 36]. For instance, mesoporous thin films are known to adsorb
liquids into their pores, and changes in the relative humidity lead to water
condensation inside the pores, which alters their mechanical properties. On
the other hand, VO2 exhibits two phases -monoclinic and tetragonal-, and phase
transitions can modify their mechanical properties. These changes influence
the speed of sound, affecting the response of such resonators. Such materials
and designs could potentially impact responsive nanoacoustic and nanophotonic
devices.

In Section 2, we analyze the main sources and types of roughness and thickness
variation in the top layer of GaAs/AlAs structures, and present the model that
we employed to analyze their impact. The initial part of the study focuses on
examining the influence of the top layer thickness on the responses of both
Fabry-Perot resonators and open cavities, as elaborated in Section 3. Building
upon this analysis, Section 4 delves into quantifying the impact of layer
thickness and surface roughness on these resonators. In Sections 5 and 6, we
introduce the dielectric materials mesoporous SiO2 and VO2 on top of a DBR and
explore the resonator response as a function of their roughness. Within these
sections, we also quantify the effectiveness of methods used to mitigate the
effects of roughness, such as polishing or planarization.

##  2 Model Implementation

To simulate the structure, we employ a model based on the transfer matrix
method (TMM). We consider free-strain boundary conditions and solve the
standard 1D wave equation. Detailed implementation specifics of the TMM for
studying transmission and reflectivity spectra in multi-layered superlattices
are outlined in [37, 38]. We here extend this method to account for surface
roughness, introducing a simplified model that simulates the effects of
roughness on acoustic FP and open cavity resonators. Surface roughness can
have various sources, resulting in either more (additive roughness) or less
(subtractive roughness) material than originally designed. In GaAs/AlAs
structures additive or subtractive roughness may arise during material growth
or device processing, and as a result, it can vary in terms of its type and
direction.

For example, a relatively small variation on the order of one monolayer (ML)
($\sim$ $0.5\>nm$), is commonly observed in high-quality epitaxially grown
flat layers, such as MBE-grown GaAs/AlAs or InGaAs/InP structures [27, 28,
39]. This roughness tends to be discrete, usually resulting in 1-ML-thick
plateaus in the lateral direction. Non-flat layers, like quantum dot (QD)
nanostructures, frequently integrated within DBRs due to their exceptional
optical and quantum properties, might contribute to nm-scale roughness in the
layers grown on top of them. Larger roughness is expected when an additional
layer is placed atop the MBE-grown cavity, or for samples grown using
different techniques. In such cases, the roughness can vary significantly
based on the specific technique and material, typically ranging from a few
nanometers to a few tens of nanometers. Furthermore, additional roughness or
width variations of the top layer might arise due to oxidation [27], or during
device fabrication processes, such as etching or mask removal steps. This
roughness’ texture is random and continuous in the vertical direction, and it
varies from sample to sample.

Given the diversity of roughness types, and the practical impossibility of
knowing its precise structure in each sample, developing a full atomic-scale
model for roughness becomes highly impractical. For this reason, in this work
we have developed and implemented a model that approximates the roughness of
one sample as a set of samples that exhibit planar surfaces and different
thicknesses, averaging their simulated surface displacements. Specifically, we
model the surface roughness by considering a normal distribution of thickness
variation for the top layer of the structure, with a standard deviation
$\sigma$. For each case, we average the resulting spectra over 2000 iterations
of random thickness distributions with a specific $\sigma$, while keeping the
parameters of the other layers of the DBR constant. Subsequently, we fit the
resonant modes using a Gaussian function and calculate the Q-factor by
computing the ratio between the resonant frequency and the linewidth extracted
from the fitting. Unless explicitly stated differently, we imply this
averaging process when we refer to roughness simulation in this paper.

In the outlined approach, we work under the assumption that the inhomogeneous
broadening approximation remains valid. Specifically, surface roughness
introduces inhomogeneous broadening of the resonant modes, which influences
the resonator’s response and compromises its Q-factor. The studied structures
are based on GaAs/AlAs DBR superlattices with flat interfaces, finalized with
the top layer which contains a rough surface. The periodicity of the
superlattice introduces Brillouin zone folding and miniband openings at
frequencies $\Omega_{m}=m\pi v/d$, where $m\in\mathbb{N}$, and $d$ and $v$
represent the unit cell thickness and phonon group velocity, respectively [40,
41, 15]. Throughout our analysis, we consider appropriate thickness variations
and roughness ranges, taking into account the typical values observed in
practical samples. The material parameters for GaAs, AlAs, mesoporous SiO2,
and VO2, utilized in our analysis, are drawn from relevant literature sources
[27, 42, 43, 35].

##  3 Influence of Top Layer Thickness on Fabry-Perot and Open Resonators

Our study begins with an analysis of how variations in the top layer thickness
impact the response of Fabry-Perot and open resonators. These fluctuations in
top layer thickness are common in practical samples and can arise due to
processes like oxidation of the top layer upon exposure to the atmosphere
[27], or during device fabrication steps such as etching or mask removal [6].
Schematics of the investigated resonators are presented in Figure 1(a,b).

Figure 1: Effects of top layer thickness variation on Fabry-Perot and open-
cavity resonators. (a,b) Schematics of (a) Fabry-Perot resonator with a
16-period top DBR and a 20-period bottom DBR and (b) open cavity with 20 DBR
periods and $\lambda/2$ spacer. (c,d) Colormaps of the optical reflectivity as
a function of the top layer thickness for (c) Fabry-Perot and (d) open cavity.
(e,f) Colormaps of the acoustic displacement as a function of the top layer
thickness for (e) Fabry-Perot and (f) open cavity.

The first structure (Fig. 1(a)), is a conventional FP cavity, incorporating a
$\lambda/2$-cavity between two GaAs/AlAs DBR superlattices, with $\lambda$
representing the acoustic wavelength. This configuration results in a high-
quality cavity for both photons and phonons due to the nearly identical
refractive index contrast and acoustic impedance contrast of GaAs and AlAs
[44]. Moreover, the closely matched lattice constants of these materials
facilitate the growth of thick, high-quality layers using standard epitaxial
techniques, making them ideal for applications involving high Q-factor
microcavities.

The second design under investigation is the open-cavity resonator (Fig.
1(b)), formed by a $3\lambda/2$ low-acoustic-loss spacer cavity on top of a
single DBR. In the absence of surface roughness, in this arrangement, the free
surface acts as a perfect mirror for phonons at the studied frequencies, since
they cannot propagate toward the air. Consequently, when combined with a high-
reflection bottom DBR, this arrangement generates a cavity with a high
Q-factor, rendering it an ideal platform for our investigation.

We performed a comprehensive study of the parameter space to assess the
combined effects of the top layer thickness and DBR periods on the quality
factor. In all configurations, we maintain a constant difference of 4
GaAs/AlAs layer pairs between the top and bottom DBRs. This difference ensures
a symmetric optical cavity due to the higher refractive index contrast at the
DBR/air interface compared to the DBR/substrate interface. Figure 1(c) and (d)
present colormaps of the optical reflectivity as a function of the top layer
thickness, illustrating the dependence on the top layer thickness for both FP
and open cavities, respectively. Figure 1(c) shows a minimal sensitivity of
the FP frequency over the whole thickness range except for $\lambda/2$ which
leads to a small frequency shift. In the open cavity (Figure 1(d)), we observe
a sharp frequency shift for a small thickness change around $\lambda/2$,
corresponding to a slope of 0.34 THz/nm. Additionally, the transmission is
much weaker than the FP case because the optical quality factor is lower in
the open cavity due to the missing upper high-reflectivity DBR.

Figures 1(e) and (f) depict colormaps of the acoustic displacement for a
cavity with the same DBRs as Fig. 1(c) and (d). Notably, as shown in Fig.
1(e), the FP cavity exhibits a range between approximately 30 and 100 nm in
which the mode frequency demonstrates minimal sensitivity to the thickness
change, featuring a slope of $\sim 0.1$ MHz/nm. This linear regime over a
broad thickness range results in a frequency shift of the acoustic resonance
of less than 10 MHz. Conversely, the open cavity exhibits a strong dependence
of the acoustic resonance on the top layer thickness, with the acoustic mode
frequency inversely proportional to the layer thickness. Around the midpoint
of $\lambda/2$ this curve exhibits a slope of $\sim 18$ MHz/nm, significantly
larger than the FP cavity case. In this regime, small fluctuations in the
thickness of a rough open cavity are expected to induce a pronounced
broadening of the acoustic resonance, which we will investigate further in the
subsequent section.

Comparing optical and acoustic resonators, we observe two main differences.
The first one is the anti-crossing observed in the acoustic Fabry-Perot (FP)
resonator at 18.5 GHz (Fig.1(e)), which is not evident in the optical FP
(Fig.1(c)). At a thickness of $\lambda/2$, a second cavity due to reflection
at the surface is formed in both cases. However, the acoustic cavity is much
stronger, thus lifting the degeneracy of the modes. The second difference
concerns the open cavity, where the transmission for the optical mode
(Fig.1(d)) is much weaker than for the acoustic mode (Fig.1(f)). These
differences arise from the fact that the free surface acts as a perfect mirror
for acoustic phonons, whereas this is not the case for photons, as the
electromagnetic field can penetrate into the air, creating a much higher
quality factor cavity for phonons compared to photons.

##  4 Influence of Surface Roughness on the Quality Factor of Fabry-Perot and
Open Resonators

We now delve into the influence of surface roughness on the Q-factor of both
FP and open resonators near their resonant frequencies, considering various
DBR configurations. Figure 2(a) focuses on Fabry-Perot resonators, analyzing
five configurations with varying numbers of periods in the bottom DBR (ranging
from 5 to 25 in 5-bilayer intervals). In all cases, the top DBR comprises 4
periods less than the bottom DBR (1, 6, 11, 15, and 21). Corresponding results
for open cavities are shown in Figure 2(b). The two key parameters —roughness
and the number of DBR layers— exert a pronounced and predictable influence on
the Q-factor.

Figure 2: Effects of roughness on the Q-factor of (a) a $\lambda/2$ cavity
Fabry-Perot resonator, and (b) a 3$\lambda/2$ open-cavity resonator for
various DBR configurations. In both panels, the bottom DBR varies from 5 to 25
bilayers in intervals of 5, while in (a) the top DBR of the Fabry-Perot
comprises 4 bilayers less than the bottom one in all cases.

For structures with flat surfaces ($\sigma=0$ nm), the Q-factor is solely
determined by the number of DBR periods. However, as surface roughness
increases, the effectiveness of stacking additional layers significantly
wanes. This is due to the upper Q-factor limit imposed by increased
fluctuations in resonant frequencies, introduced by surface roughness.
Although negligible for applications where ultrahigh Q-factors are not a
requirement, this effect might have drastic consequences in specific
applications, such as polaritonics and single photon sources [2, 3]. This
effect is considerably more prominent in open-cavity resonators than in Fabry-
Perot ones, as depicted in Figure 2. In essence, higher roughness dictates a
reduced number of DBR periods required to attain the maximum Q-factor in the
structure, consequently leading to diminished maximum Q-factor values.

For instance, an open resonator composed of GaAs-with its characteristic low
roughness (approximately one monolayer)-would benefit from an increased number
of DBRs. Specifically, assuming a 0-nm-roughness GaAs/AlAs acoustic Bragg
mirror consisting of 20 periods achieves a Q-factor of 6459, while 25 periods
a Q-factor of 24711, and continually rising as more layers are added.
Conversely, 1-nm-roughness would result in a Q-factor of 476 for 20 DBR layers
and 505 for 25 DBR layers or more (Q-factor saturates above 25 layers),
demonstrating a significant reduction of the Q-factor even for such minor
roughness. Additionally, when the same resonator possesses a 2 nm roughness on
its top layer, or a material with this roughness is deposited on top of it,
the Q-factor saturates already at 20 DBR layers and adding more layers fails
to increase it, as indicated in Fig. 2(b).

A comparison between the FP and open cavity configurations, reveals that the
open cavity Q-factor is more sensitive to surface roughness than the FP
cavity. Notably, even under relatively high surface roughness conditions (10
nm), the Q-factor of the FP cavity does not saturate; although it decreases to
approximately 50 $\%$ of its maximum value. Overall, our study suggests that
factoring in roughness becomes imperative when engineering high-quality
photonic or acoustic resonators. Doing so, would enable precise Q-factor
evaluations for each design, and prevent superfluous layer stacking. This, in
turn, would improve device design, enhance operational efficiency, and
minimize costs. When working with acoustic resonators at higher frequencies,
and hence thinner layers, the effect of roughness is extremely critical.

Regarding the DBR unit cell layers’ thicknesses, Fig. 2 presents results for
$\lambda/4-3\lambda/4$ (with respect to the acoustic wavelength $\lambda$). It
is worth noting that analogous conclusions hold for $\lambda/4-\lambda/4$
DBRs. Since the overarching findings remain similar, we present here detailed
results solely for the former case. We note two differences when the
$\lambda/4-\lambda/4$ unit cell is used; first, the stop band becomes twice
broader, and second, the mode’s resonant frequency exhibits an increased
dependency on the top layer thickness.

##  5 Influence of surface roughness on resonators responsive to external
stimuli

We now proceed to analyze the effects of roughness on the responsivity of
GaAs/AlAs DBR-based resonators when an additional top layer of a different
material, responsive to external stimuli, is introduced. We focus solely on
open-cavity resonators here, due to their anticipated exceptional responsivity
to external changes. We simulate resonators with two types of spacers;
mesoporous SiO2, or VO2, as their top layer. Our analysis encompasses three
different roughnesses of 0 nm, 2 nm, and 5 nm, and two different phases for
each material: 0 and 100$\%$ relative humidity for the mesoporous cavity, and
monoclinic and tetragonal phase for the VO2 cavity. The selected roughness
values of 2-5 nm were based on minimal (optimal) realistic values achievable
for optimized SiO2 and VO2 materials, while the 0 nm case serves as a
reference point. In all these cases, the open-cavity DBR consists of 15
GaAs/AlAs pairs. This choice strikes a balance between structure complexity
and performance based on the outcomes of Figure 2, for the considered
roughness values. Specifically, for a 5-nm roughness, stacking additional DBR
layers would have a minimal impact on the Q-factor thereby needlessly
increasing the complexity of the structure. Conversely, employing fewer layers
substantially diminishes the Q-factor.

We simulate the humidity in the mesoporous material by considering a weighted
average for the material density and speed of sound, taken from a combination
of dense SiO2, air, and water. As for VO2, the acoustic properties in its two
phases are drawn from Reference [35]. Central findings are illustrated Figure
3, displaying the phonon displacement spectrum of the simulated structures.
Figure 3(a) focuses on the mesoporous material, comparing its response to 0
$\%$ and 100 $\%$ relative humidity, for three different surface roughness
values. As demonstrated, the cavity’s response to humidity change is evident,
with peak amplitude reduction and frequency shift as humidity rises [45, 31].
However, increased roughness leads to broader peaks, potentially making
frequency shifts harder to resolve. For a 5 nm roughness, the largest
considered, the shift remains resolvable. Nonetheless, the trend suggests that
roughness exceeding 5 nm -realistic for non-optimized SiO2 samples- or smaller
humidity changes, could render the system response insufficient.

Figure 3(b) reveals changes in the acoustic displacement spectra of VO2 across
its two phases, given the same three surface roughness values. Results show
that minor roughness —around 2 nm or less— yields well-defined peaks, making
phase changes clearly resolvable. Conversely, roughness around 5 nm leads to
larger broadening, affecting the resolution of frequency shifts. The
broadening here is especially pronounced compared to the mesoporous case. This
is due to the speed of sound being considerably larger in VO2 compared to the
mesoporous material. This means that a similar thickness change causes a
larger frequency shift in VO2, thus enhancing the roughness-induced
inhomogeneous broadening. These outcomes indicate that surface roughness
considerably impacts Q-factors of GaAs/AlAs DBR open cavities utilized for
sensing. Surface roughness reduces the system’s ability to resolve external
changes, even for modest roughness levels, with similar or larger values
expected in realistic samples. The subsequent section will explore methods to
mitigate roughness impact and enhance device sensitivity.

Figure 3: Acoustic displacement spectrum of the structure containing 15
GaAs/AlAs DBR periods and a top layer sensitive to external stimuli. The top
layers considered are (a) mesoporous SiO2 at 0 $\%$ and 100 $\%$ relative
humidity, and (b) VO2 at the monoclinic and tetragonal phases. A roughness of
$\sigma=0$, $2$, and $5$ nm is considered in both cases.

##  6 Strategy to enhance Q-factor

As we have seen in the preceding sections, surface roughness can significantly
reduce the quality factor of the cavity, and consequently its sensitivity to
external changes. Although Section 5 showed that open-cavity resonators with
rough surfaces can still resolve frequency changes induced by external
stimuli, the effective reduction in Q-factor compared to an ideal no-roughness
scenario remains substantial. It would, therefore be highly desirable to
minimize the roughness as much as possible in practical structures, especially
in those designed for sensing applications. [31] The "flattening" of rough
surfaces can be achieved in various ways, most commonly via either polishing
the surface or by depositing additional composite thin films. The latter is
usually a much simpler approach, and it is preferred when adding such a layer
does not have a negative effect on the other properties of the sample, while
the former is preferred otherwise.

The polishing method removes the top rough part of the material by either
rubbing it or by applying a chemical treatment resulting in a smooth surface,
while the planarization method deposits a polymer resist on the top of the
sample. As the polymer composite is spin-coated on the sample in liquid form,
it tends to even out the surface roughness, before it is cured into a solid
polymer with a smooth surface. We have studied the effect of such surface-
reduction methods on our structure, and the results are summarized in Figure
4. To account for both methods, we have simulated two similar structures that
both have an additional layer placed atop. We simulate the planarization
method as follows: first, we consider a rough layer of 243.24 nm of monoclinic
VO2. Then, we add a 20-nm-thick ($d_{2}$) layer of a virtual material. The
roughness of the two top layers is complementary in shape in such a way that
the total thickness $D$ remains constant at 263.24 nm, without any surface
roughness. The added virtual material has the same elastic parameters as the
monoclinic VO2, as shown in the left schematic of Fig. 4. The acoustic
resonance corresponds to a monoclinic-VO2 $\lambda/2$ cavity. In the second
case, we assume the same structure as the previous case, but with the bottom
layer in the tetragonal VO2 phase (different mass density and speed of sound),
as illustrated in the right schematic of Fig. 4).

Figure 4: Acoustic displacement spectrum of an open cavity with 20 GaAs/AlAs
DBR periods, finalized with a rough VO2 and a flat capping layer containing
the elastic properties of VO2 in the monoclinic phase. The black peak
corresponds to the displacement spectrum of monoclinic VO2, whereas the red,
blue, and green peaks, to the tetragonal VO2 with 0-nm, 5-nm, and 20-nm
roughness, respectively.

Figure 4 illustrates the results for both material combinations, for different
values of original roughness (i.e. before adding a planarization layer). Our
original structures consisted of 20 GaAs/AlAs DBR periods with a thin VO2
layer on top. In the case of 0-nm-roughness, both peaks have relatively high
Q-factors. The structure with bottom and top layers with VO${2}$ in the
monoclinic phase is entirely insensitive to roughness as both layers have the
same acoustic properties. For the other structure, as the simulated roughness
progressively increases to 5 nm and 20 nm, the quality factor decreases. Even
though the top surface is flat, there is still an effective roughness
remaining in the second structure, in-between the last two layers, coming from
the fact that the initial VO2 layer was rough, and the second layer had
different properties. However, there is still a clear improvement compared to
the case when the additional layer is not deposited (see Fig 3). The acoustic
mode associated with the monoclinic phase presents an improved quality factor
due to the absence of surface roughness. In the case of the tetragonal phase,
the impedance matching between the air and the two layers also increases the
quality factor of the resonator with respect to the case where the roughness
is directly present at the surface of the device.

##  7 Conclusions

In this study, we conducted an analysis of the impact of top layer thickness
and surface roughness on the quality factor and performance of acoustic and
optical resonators based on GaAs/AlAs DBRs. Our findings reveal that thickness
inaccuracies appreciably influence the mode frequency, while even relatively
small surface roughness, on the order of a few nanometers, significantly
influences the quality factor. Consequently, these effects can have a
substantial impact on the device performance of DBR microcavities. This
important aspect is often overlooked in the design process, making it
essential to consider surface roughness for optimal performance. Notably, our
developed model allows us to determine the maximum achievable quality factor
and the minimum number of DBR layers required to achieve it, for different
surface roughness values. By doing so, we can eliminate the need for growing
unnecessary additional layers that do not contribute effectively, leading to
more efficient and cost-effective designs.

We observed that surface roughness affects acoustic cavities more prominently
than their optical counterparts, although both are significantly influenced.
Additionally, open-cavity designs are much more sensitive to roughness
compared to FP cavities, which is expected due to the closer proximity of
roughness to the cavity region, leading to a more pronounced interaction.
Moreover, we explored practical case studies in which such microcavities were
designed to serve as sensors. Our investigations provided valuable insights as
to how surface roughness impacts these applications, and enabled us to assess
strategies for mitigating its effects. We believe that the insights obtained
in this study have broad implications for the design of efficient
optoelectronic, photonic, and acoustic devices.

##  8 Acknowledgments

The authors acknowledge funding from European Research Council Consolidator
Grant No.101045089 (T-Recs). This work was supported by the European
Commission in the form of the H2020 FET Proactive project No. 824140 (TOCHA).

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