††thanks: Contributed Equally to the Work

# Environmental Sensitivity of Fabry-Perot Microcavities Induced By Layered
Graphene-Dielectric Hybrid Coatings

Rui Peixoto  Centro de Física das Universidades do Minho e Porto and Faculty
of Sciences, University of Porto, 4169-007 Porto, Portugal  J. P. Santos Pires
<EMAIL_ADDRESS>Centro de Física das Universidades do Minho e Porto and
Faculty of Sciences, University of Porto, 4169-007 Porto, Portugal  Catarina
S. Monteiro<EMAIL_ADDRESS>Institute for Systems and
Computer Engineering, Technology and Science (INESC TEC) and Department of
Physics and Astronomy, Faculty of Sciences, University of Porto, Rua do Campo
Alegre 687, 4169-007 Porto, Portugal  Faculty of Engineering, University of
Porto, R. Dr. Roberto Frias, 4200-465 Porto, Portugal  Maria Raposo
Laboratory of Instrumentation, Biomedical Engineering and Radiation Physics
(LIBPhys-UNL), Department of Physics, NOVA School of Science and Technology,
NOVA University Lisbon, 2829-516 Caparica, Portugal  Paulo A. Ribeiro
Laboratory of Instrumentation, Biomedical Engineering and Radiation Physics
(LIBPhys-UNL), Department of Physics, NOVA School of Science and Technology,
NOVA University Lisbon, 2829-516 Caparica, Portugal  Susana O. Silva
Institute for Systems and Computer Engineering, Technology and Science (INESC
TEC) and Department of Physics and Astronomy, Faculty of Sciences, University
of Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal  Orlando Frazão
Institute for Systems and Computer Engineering, Technology and Science (INESC
TEC) and Department of Physics and Astronomy, Faculty of Sciences, University
of Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal  J. M. Viana
Parente Lopes<EMAIL_ADDRESS>Centro de Física das Universidades do Minho e
Porto and Faculty of Sciences, University of Porto, 4169-007 Porto, Portugal

(August 28, 2024)

###### Abstract

We propose a fiber-based environmental sensor that exploits the reflection
phase shift tunability provided by the use of layered coatings composed of
dielectric slabs spaced by conducting membranes. A transfer matrix study is
done in a simplified theoretical model, for which an enhanced sensitivity of
the reflection interference pattern to the output medium is demonstrated, in
the typical refractive index range of liquid media. An experimental
configuration using a cascaded Fabry-Perot microcavity coated by a graphene
oxide/polyethylenimine (GO/PEI) multilayered structure is demonstrated. Its
cost effective chemical production method makes graphene oxide-based hybrid
coatings excellent candidates for future real-life sensing devices.

##  I Introduction

One of the main technological application of optical fibers is to fabricate
sensing devices that can detect the tiniest changes in physical or chemical
parameters. Such measurements can be very precise but are usually indirect and
exploit the effect of small parametric changes in the propagation of guided
lightwaves. The accuracy of these sensors relies on interferometry that is
made possible, for instance, by building hollow core microcavities (as small
as $30\text{\textmu m}$ [1, 2]) using capillary tubes or photonic crystal
fibers (PCFs) at end of a cleaved optical fiber [3, 4, 5, 6, 7, 8, 9, 10, 1,
11, 12, 13, 14, 15, 16, 17], that act as miniature Fabry-Perot
interferometers. Fabry-Perot cavities in optical fibers were firstly
fabricated by Sirkis et al. for strain measurement [3] and, since then,
similar interferometric sensors have been proposed for curvature [12],
temperature [11], pressure [5], humidity [15], acoustic sensing [17], among
others. Besides, fiber optic sensors based on Fabry-Perot interferometers have
also attracted much attention in other fields of expertise, ranging from
medicine [10] to engineering [4, 6].

The periodicity of the interference fringes can be made dependent on the
refractive index of the environment ($n_{\text{out}}$). This can be achieved
by endowing the output interface with a reflection phase shift sensitive to
the wavelength ($\lambda$), an effect that can be produced by a thin coating
which may feature an internal layered structure built at the nanoscale [18].
Different kinds of coatings have been used over the years, including
dielectric multilayered structures [19, 20, 21], porous metal-oxide films [22]
and metallic coatings based on thin metal films [23, 24], plasmonic
nanostructures [25] or embedded nanoparticles [26, 27]. More recently, it has
been reported [28] that hybrid coatings made of lossy dielectrics (transparent
narrow-gap semiconductors) deposited on very thin highly conducting films can
show nontrivial internal interference effects that allow for an effective
control over the surface reflection coefficient of light at sub-wavelength
scales. With the advent of graphene [29, 30] and other atomically thin planar
conductors (ATPC), it became possible to realize similar ideas in hybrid
optical coatings comprised of dielectric-ATPC-dielectric multilayered
structures. In effect, if two-dimensional graphene precursors are employed,
such hybrid coatings can be produced by using layer-by-layer chemical methods
[31, 32] that take advantage of intermolecular interactions between oppositely
charged electrolytes to self-assemble a multilayered structure [33]. This
inexpensive thin-film assembly already proved to be a viable technique for
real-life optical sensors of pH [34], oxygen detection [35] and fluorescence
[36].

In this paper, we propose a fiber-based environmental sensor that consists of
a Fabry-Perot microcavity coated by a hybrid structure made of dielectric
polymer slabs separated by conducting two-dimensional membranes. This sensor
makes use of the aforementioned nontrivial interference effects inside hybrid
coatings, significantly enhancing the sensitivity of the reflection
interference fringes to the output refractive index, in a range typical of
liquid media. For concreteness, our theoretical analysis considers the
conducting membranes to be pristine graphene sheets, whose optical
conductivity is real and roughly $\lambda$-independent in the mid- to near-
infrared range [37, 38, 39] (in the sensor’s operation regime). However, the
precise nature of these membranes is insubstantial and the conclusions
presented here only require them to be two-dimensional materials with
sufficiently large real optical conductivities. Hence, more practical
realizations of these devices can use hybrid optical coatings based on
graphene-oxide (GO) sheets [40, 41, 42], instead of actual graphene
monolayers. As referred, layered structures based on these graphene precursors
can be grown by inexpensive chemical processes in controlled layer-by-layer
[33, 43, 32] procedures. Here, we employ this technique to experimentally
realize the proposed sensor in a cascaded Fabry-Perot interferometer at the
tip of a single-mode fiber that was coated by stacked
polyethyleneimine/graphene-oxide (PEI/GO) layers. Such devices have been
recently proposed by some of us [44] although analyzed from a different
viewpoint.

The paper is organized as follows: In Sec. II, we review the optical
properties of pristine graphene and highlight the similarities with existing
studies on GO lattices. In Sec. III, we devise our theoretical model for the
sensor, as well as the transfer matrix formalism used to investigate the
reflection properties of the hybrid dielectric-ATPC-dielectric coatings, as a
function of the output refractive index and the system’s parameters. We also
present the theory of interference in a low-finesse Fabry-Perot cavity, which
is a key ingredient to understand the physics behind the sensor’s operation.
In Sec. IV, we present our main theoretical predictions regarding the
operation and performance of our idealized environmental sensor. We also point
out the chief advantages of using graphene-based hybrid coatings over
alternative setups. An experimental realization of this sensing device is
obtained in Sec. V, including measurements that demonstrate the operation
predicted by our theoretical analysis. Finally, in Sec. VI, we sum up our
conclusions and speculate on future applications of this technology.

##  II Optical Response of Graphene and Graphene-Oxide

The crucial ingredient of this sensor is the possibility to coat the outward
surface of the Fabry-Perot microcavity by a layered structure of dielectric
slabs interfaced by conducting two-dimensional membranes. As explained in the
introduction, the detailed nature of these membranes is not important for our
conclusions, which only depend on a dissipative optical response, that is
sufficiently strong but weakly dependent on the wavelength. In other words,
the membrane must behave as a two-dimensional metal in the sensor’s
operational range (i.e., the mid- to near-infrared spectral range).
Nevertheless, the simplest albeit paradigmatic example of such a material is
monolayer graphene, whose optical response will be concisely reviewed here. In
addition, recent studies on the optical response of (more complicated) GO will
also be referred, highlighting the similarities with graphene that justify a
direct comparison between theory and experiments is this paper.

Figure 1: Scheme of our fibre-based sensor using a three-wave micro-metric
Fabry-Perot cavity with an outside multilayered hybrid surface coating (shown
in green) made of conducting membranes spaced by ideal dielectrics. We also
define the dimensions of the Fabry-Perot microcavity ($D$ and $d$) and depict
the three primary waves reflected off the interfaces of the cavity (of
intensity $\mathcal{I}_{1}$, $\mathcal{I}_{2}$ and $\mathcal{I}_{3}$). (color
online)

When working around the near- to mid-infrared regime
($\lambda\\!\gtrsim\\!800\text{nm}$), it is known that the linear optical
response of graphene is isotropic
($\sigma_{\lambda}^{xx}\\!=\\!\sigma_{\lambda}^{yy}\\!=\\!\sigma_{\lambda}$
are the only non-zero terms of the conductivity tensor) and well described by
continuum limit calculations done within a low-energy model having two
uncoupled Dirac-cones [37, 45, 46]. More realistic modeling was also done [47,
48, 49] and confirm this approximation as appropriate. Under this
approximation, the low temperature optical conductivity of graphene can be
shown to have two contributions,

$\sigma_{\lambda}\\!=\\!\sigma_{\lambda}^{\text{intra}}\\!+\\!\sigma_{\lambda}^{\text{inter}},$
(1)

which stem from electronic intra- and inter-band transitions, respectively.
Both contributions can be analytically evaluated and read [45],

$\sigma_{\lambda}^{\text{intra}}\\!=\\!\frac{e^{2}}{h}\left(\frac{2\gamma\lambda^{2}\left|E_{\text{F}}\right|}{h^{2}c^{2}\\!+\\!\gamma^{2}\\!\lambda^{2}}\\!+\\!i\frac{2hc\lambda\left|E_{\text{F}}\right|}{h^{2}c^{2}\\!+\\!\gamma^{2}\\!\lambda^{2}}\right)$
(2a)

$\displaystyle\sigma_{\lambda}^{\text{inter}}\\!$
$\displaystyle=\\!\frac{e^{2}}{2h}\arctan^{2}\left[2\left|E_{\text{F}}\right|-\frac{h}{\lambda}\gamma\right]$
(2b)
$\displaystyle\qquad+i\frac{e^{2}}{4h}\ln\left[\frac{\gamma^{2}\lambda^{2}+\left(ch-2\left|E_{\text{F}}\right|\lambda\right)^{2}}{\gamma^{2}\lambda^{2}+\left(ch+2\left|E_{\text{F}}\right|\lambda\right)^{2}}\right],$

where $\gamma$ is a phenomenological scattering parameter, $c$ is the speed of
light in the vaccum, $h$ is Planck’s constant and $E_{\text{F}}$ is the Fermi
energy of graphene. For this approximate calculations to hold, one must always
consider that $\left|E_{\text{F}}\right|\lesssim 0.3\text{eV}$, corresponding
to a slightly doped graphene monolayer. In Fig. 2, we show some representative
plots of these conductivities where it is evident that
$2\\!\left|E_{\text{F}}\right|$ acts as an effective “optical gap” that marks
a boundary between two response regimes. For $\lambda\\!>\\!\pi\hbar
c/\\!\left|E_{\text{F}}\right|$, the dissipative inter-band response is
suppressed by Pauli-blocking and a mostly reactive current appears on the
system. In contrast, for $\lambda\\!<\\!\pi\hbar
c/\\!\left|E_{\text{F}}\right|$, the imaginary part of the conductivity is
almost zero and the response is mostly due to inter-band transitions. Then,
the optical response of graphene becomes purely dissipative with a
$\lambda$-independent universal conductivity [38]. The latter is the regime
where our proposed sensor will operate, such that we may take the optical
conductivity to be $\sigma_{\lambda}\\!\approx\\!e^{2}/4\hbar$.

As noted in the introduction, although pristine graphene could be used to
realize our proposal, cheaper and more practical alternatives are provided by
other two-dimensional materials, such as graphene-oxide monolayers. These
functionalized versions of graphene have considerably more complex structures
[41] and simple expressions cannot be readily obtained for their electronic or
optical response properties. However, due to its interest for optoelectronic
applications [50], many Density Functional Theory (DFT) studies have been
published on the linear optical properties of oxidized graphene sheets [51,
52, 53, 42]. Most of the results demonstrate that, at least for moderately
oxidized graphene, the optical conductivity retains a purely dissipative
plateau in the mid-infrared [53] with a value approximating the universal
conductivity found in pristine graphene. Based on this, we argue that the
theory we will develop for coatings based off pristine graphene membranes,
also holds for GO, provided the value of the universal optical conductivity is
adjusted as a phenomenological parameter.

##  III Theoretical model for the proposed sensor

We propose a fiber-based interferometric sensor (see Fig. 1a) that operates on
the basis of a reflection interference pattern that is sensitive to its
environment. Any changes in physical or chemical properties of this
surrounding medium typically translate into slight alterations in its
refractive index. For a bare output interface, the sensor would only be
affected by variations in the absolute reflectivity of the last interface, as
the phase introduced upon reflection is independent 111In these conditions, it
can only change from $0$ to $\pi$ if the output refractive index
($n_{\text{out}}$) exceeds that of silica. of the incident wavelength. This
implies that the interference pattern in the reflected spectrum will be
affected solely in the relative strength of its Fourier components, but not in
their periods. To allow a shift in the fringe periods by environmental
changes, one must engineer the output interface so as to give it a
$\lambda$-dependent complex reflection coefficient that also depends
parametrically on the refractive index of the output medium. Such properties
can be embedded into the optical interface by deposing a nano-structured
coating onto the cavity’s outwards surface, as described in the introduction.

Figure 2:  Plots of the complex optical conductivity of a doped pristine
graphene sheet in the infrared (see Eqs. (2a)-(2b)). The plots consider the
Fermi energy of $E_{\text{F}}\\!=\\!0.25eV$ and a relaxation scale of
$\gamma\\!=\\!0.001eV$. The optical gap and the operational regime for our
sensor are indicated. (color online)

Here, we present the basic theoretical model employed to simulate the
operation of our device proposal. Before proceeding with details, it is
important to highlight that the sensor is naturally split into two main parts:
i) the hybrid coating on the output interface and ii) the Fabry-Perot
microcavity at fiber’s termination. Both components work together to produce
the interference effects on which the sensor is based, however they act at
different length scales. The hybrid coating is structured at the nanoscale,
with typical distances of $\approx\\!10\text{nm}$ between successive
conducting membranes. These lengths are shorter than optical wavelengths and
thus responsible for the internal interference that causes nontrivial phase
shifts upon reflection. In addition, one also has the optical interfaces
forming the Fabry-Perot microcavity. These surfaces are distanced by
tens/hundreds of µm, scales that far exceed the ones present in the internals
of the hybrid coating. The purpose of this cavity is simply to generate a
basic three-wave interference pattern in the reflected spectrum which will
then be shaped by variations in the output refractive index. In the following,
we shall consider these two main parts separately.

###  III.1 Optical Modeling of a Hybrid Nano-structured Coating

The layer-by-layer assembly method creates optical coatings where several
(roughly parallel) conducting membranes are separated by nanoscopic slabs of
identical dielectric material. In practice, the control over orientation,
flatness and distances between sheets is not perfect but, for a modeling
purpose, we shall consider the conducting membranes as mutually parallel
planes as shown in Fig. 3a. Moreover, since our proposal is based on a single-
mode optical fiber, it is appropriate to consider that the guided light-wave
has normal incidence on the coating, which simplifies the following
calculations. In short, our aim will be to evaluate the complex
reflection/transmission coefficients of a hybrid coating featuring a given
number ($N$) of conducting membranes spaced by a set of dielectric slabs
characterized by the widths, $\\{d_{1},\\!\cdots\\!,d_{N\\!-\\!1}\\!\\}$. This
problem can be neatly solved by a Transfer Matrix Method (TMM). With no loss
of generality, we consider the membranes to be pristine graphene monolayers
working at the universal optical conductivity regime, i.e.
$\lambda\\!\approx\\!1500\text{nm}$, while the dielectric slabs will be taken
as made up of the same material (of refractive index $n_{\text{pol}}$).

To devise a TMM that evaluates the complex transmission
($t_{\\!\text{$\lambda$}}$) and reflection coefficients
($r_{\\!\text{$\lambda$}}$) of the entire coating, we begin by considering a
monochromatic wave at normal incidence onto a single conducting plane
(depicted in Fig. 3b). In this case, both the electric and magnetic fields are
parallel to the graphene sheet and the solution of the four-wave scattering
problem boils down to imposing appropriate boundary conditions at the
interface. Since the interface is conducting, it supports surface currents
driven by the electric field of the crossing wave which, in turn, alter the
usual boundary conditions associated to a purely dielectric interface. The
general electromagnetic boundary conditions in the presence of a surface
conductivity are derived in the Appendix A. Meanwhile, the electric field of
the partial waves in Fig. 3b can be written as

$\\!\\!\mathbf{E}^{1}\\!\left(\mathbf{r},t\right)\\!=\\!\\!\left(\\!E_{\lambda}^{{\scriptscriptstyle(\\!1,\\!+\\!)}}\\!e^{i\frac{2\pi
n_{1}z}{\lambda}}\\!+\\!E_{\lambda}^{{\scriptscriptstyle(\\!1,\\!-\\!)}}\\!e^{-i\frac{2\pi
n_{1}z}{\lambda}}\\!\right)\\!e^{i\frac{2\pi c}{\lambda}t}\,\hat{\mathbf{x}}$
(3)

in the input medium (refractive index $n_{1}$) and

$\\!\\!\mathbf{E}^{2}\\!\left(\mathbf{r},t\right)\\!=\\!\\!\left(\\!E_{\lambda}^{{\scriptscriptstyle(2,\\!+\\!)}}\\!e^{i\frac{2\pi
n_{2}z}{\lambda}}\\!\\!+\\!E_{\lambda}^{{\scriptscriptstyle(2,\\!-\\!)}}\\!e^{-i\frac{2\pi
n_{2}z}{\lambda}}\\!\right)\\!e^{i\frac{2\pi c}{\lambda}t}\,\hat{\mathbf{x}}$
(4)

in the output one (refractive index $n_{2}$). The magnetic components of the
wave are also harmonic and can be readily obtained from Faraday’s law —
$\mathbf{B}^{{\scriptscriptstyle(\pm)}}\\!\\!=\\!\\!\pm\hat{\mathbf{z}}\\!\times\\!\mathbf{E}^{{\scriptscriptstyle(\pm)}}\\!/v_{m}$—
where $v_{m}\\!=\\!c/n_{m}$ is the speed of light in the corresponding medium.
These magnetic fields read

$\mathbf{B}^{1}\\!\\!\left(\mathbf{r},\\!t\right)\\!=\\!\\!\left(\\!E_{\lambda}^{{\scriptscriptstyle(\\!1,\\!+\\!)}}\\!e^{i\frac{2\pi
n_{1}z}{\lambda}}\\!\\!\\!\\!-\\!\\!E_{\lambda}^{{\scriptscriptstyle(\\!1,\\!-\\!)}}\\!e^{-i\frac{2\pi
n_{1}z}{\lambda}}\\!\right)\\!\\!e^{i\frac{2\pi
ct}{\lambda}}\\!\frac{c\,\hat{\mathbf{y}}}{n_{1}}$ (5)

$\mathbf{B}^{2}\\!\\!\left(\mathbf{r},\\!t\right)\\!=\\!\\!\left(\\!E_{\lambda}^{{\scriptscriptstyle(\\!2,\\!+\\!)}}\\!e^{i\frac{2\pi
n_{2}z}{\lambda}}\\!\\!\\!\\!-\\!\\!E_{\lambda}^{{\scriptscriptstyle(\\!2,\\!-\\!)}}\\!e^{-i\frac{2\pi
n_{2}z}{\lambda}}\\!\right)\\!\\!e^{i\frac{2\pi
ct}{\lambda}}\\!\frac{c\,\hat{\mathbf{y}}}{n_{2}},$ (6)

with their orientation depicted also in Fig. (3)b.

Figure 3: (a) Toy-model used to investigate the optical coefficients of
layered hybrid coating. For concreteness, we have considered the conducting
membranes as doped pristine graphene monolayers operating way above the
optical gap. (b) Scheme of a perpendicular scattering setup for a single
graphene interface between two bulk dielectrics. (color online)

Since the wave is at normal incidence, all the fields involved are parallel to
the conducting interface. This leads to two important simplifications in the
boundary conditions at $z\\!=\\!0$: i) Only the conditions on field components
parallel to the surface are of importance; and ii) Despite generating surface
currents, the time-varying electric fields do not induce any superficial
charge density waves. In this case, the boundary conditions of the problem
reduce to the following simpler equations:

$\mathbf{E}_{\lambda}^{1}=\mathbf{E}_{\lambda}^{2}$ (7a)

$\mathbf{B}_{\lambda}^{1}\\!=\\!\mathbf{B}_{\lambda}^{2}\\!+\\!\mu_{0}\sigma_{\lambda}\mathbf{\hat{z}}\\!\times\\!\mathbf{E}_{\lambda}^{2},$
(7b)

where $\mathbf{E}_{{\scriptscriptstyle 1/2}}$ and
$\mathbf{B}_{{\scriptscriptstyle 1/2}}$ are the monochromatic vector
amplitudes defined in Eqs. (3)-(6), $\sigma_{\lambda}$ is the optical surface
conductivity of the planar membrane and $\mu_{0}$ is the vaccum magnetic
permeability 222We will always assume the magnetic effects to be irrelevant
for the optical properties of our system.. By expressing Eqs. (7a) and (7b) in
terms of only incoming/outgoing electric field amplitudes,
$E_{\lambda}^{{\scriptscriptstyle(\\!1/2,\\!\pm\\!)}}$, we cast the scattering
problem in the matrix form,

$\left(\\!\\!\begin{array}[]{c}E_{\lambda}^{{\scriptscriptstyle(\\!2,\\!+\\!)}}\\\
E_{\lambda}^{{\scriptscriptstyle(\\!2,\\!-\\!)}}\end{array}\\!\\!\right)\\!=\\!\mathbb{T}_{1,2}^{\text{G}}\\!\left(\lambda\right)\cdot\left(\\!\\!\begin{array}[]{c}E_{\lambda}^{{\scriptscriptstyle(\\!1,\\!+\\!)}}\\\
E_{\lambda}^{{\scriptscriptstyle(\\!1,\\!-\\!)}}\end{array}\\!\\!\right),$ (8)

where the transfer matrix of a single conducting interface between dielectrics
of refractive indices $n_{1}$ and $n_{2}$ is given by

$\mathbb{T}_{{\scriptscriptstyle
1,2}}^{\text{G}}\\!(\lambda)\\!=\\!\\!\left(\\!\begin{array}[]{cc}\\!\frac{n_{2}+n_{1}}{2n_{2}}\\!-\\!\frac{\mu_{0}c\sigma_{\lambda}}{2n_{2}}&\\!\frac{n_{2}-n_{1}}{2n_{2}}\\!-\\!\frac{\mu_{0}c\sigma_{\lambda}}{2n_{2}}\\\
\\!\frac{n_{2}-n_{1}}{2n_{2}}\\!+\\!\frac{\mu_{0}c\sigma_{\lambda}}{2n_{2}}&\\!\frac{n_{2}+n_{1}}{2n_{2}}\\!+\\!\frac{\mu_{0}c\sigma_{\lambda}}{2n_{2}}\end{array}\\!\\!\right).$
(9)

Having the linear relation of Eq. (8), it is trivial to generalize the result
and relate the input to the output field components in the whole structure of
Fig. 3b. For that, we recognize that a complex phase factor — $\exp\left(\pm
i2\pi n_{i}d_{i}/\lambda\right)$ — is accumulated by right- and left-traveling
components, respectively, when traversing the dielectric slab between layers
$i$ and $i\\!+\\!1$. Hence, the electric field amplitudes at the input and
output media can be related as

$\left(\\!\\!\begin{array}[]{c}E_{\lambda}^{{\scriptscriptstyle(\\!in,\\!+\\!)}}\\\
E_{\lambda}^{{\scriptscriptstyle(\\!in,\\!-\\!)}}\end{array}\\!\\!\right)\\!=\\!\mathcal{T}_{\lambda}\\!\cdot\\!\left(\\!\\!\begin{array}[]{c}E_{\lambda}^{{\scriptscriptstyle(\\!out,\\!+\\!)}}\\\
E_{\lambda}^{{\scriptscriptstyle(\\!out,\\!-\\!)}}\end{array}\\!\\!\right),$
(10)

where $\mathcal{T}_{\lambda}$ is a $2\\!\times\\!2$ transfer matrix
characterizing the transmission of monochromatic electromagnetic waves across
the entire coating. This matrix is defined in terms of the interface matrices
— $\mathbb{T}_{{\scriptscriptstyle i,i+1}}^{\text{G}}\\!(\lambda)$ — defined
in Eq. (9), and the matrices —
$\mathbb{P}_{i}\\!=\\!\text{diag}\left(\exp\left(i2\pi
n_{i}d_{i}/\lambda\right),\exp\left(-i2\pi n_{i}d_{i}/\lambda\right)\right)$ —
describing the phase-accumulation in each dielectric slab. The general
expression then reads,

$\mathcal{T}_{\lambda}\\!\\!=\\!\mathbb{T}_{{\scriptscriptstyle
0,1}}^{\text{G}}\\!(\lambda)\\!\cdot\\!\mathbb{P}_{1}\\!\cdot\\!\mathbb{T}_{{\scriptscriptstyle
1,2}}^{\text{G}}\\!(\lambda)\\!\cdot\,\cdots\,\cdot\\!\mathbb{P}_{N-1}\\!\cdot\\!\mathbb{T}_{{\scriptscriptstyle
N-1,N}}^{\text{G}}\\!(\lambda),$ (11)

where $n_{0}\\!=\\!n_{\text{in}}$ and $n_{N}\\!=\\!n_{\text{out}}$.

Before moving on, it is worthy to remark that the TMM is suitable for
extremely efficient numerical evaluation, involving only successive product of
very small ($2\\!\times\\!2$) matrices. In our case, by specifying
$n_{\text{in}}$, $n_{\text{out}}$, the list of dielectric widths
$\left\\{d_{1},\cdots,d_{N-1}\right\\}$ and the corresponding refractive
indices $\left\\{n_{1},\cdots,n_{N-1}\right\\}$, the transfer matrix of the
whole coating can be computed in a matter of seconds in the average laptop.
The complex reflection and transmission coefficients can then be extracted
from its matrix elements:

$r_{\lambda}\\!=\\!\frac{E_{\lambda}^{{\scriptscriptstyle(\\!in,\\!-\\!)}}}{E_{\lambda}^{{\scriptscriptstyle(\\!in,\\!+\\!)}}}\\!=\\!-\frac{\left[\mathcal{T}_{\lambda}\right]_{12}}{\left[\mathcal{T}_{\lambda}\right]_{22}}$
(12a)

$t_{\lambda}\\!=\\!\frac{E_{\lambda}^{{\scriptscriptstyle(\\!out,\\!+\\!)}}}{E_{\lambda}^{{\scriptscriptstyle(\\!in,\\!+\\!)}}}\\!=\\!\left[\mathcal{T}_{\lambda}\right]_{11}\\!\\!-\\!\frac{\left[\mathcal{T}_{\lambda}\right]_{12}\left[\mathcal{T}_{\lambda}\right]_{21}}{\left[\mathcal{T}_{\lambda}\right]_{22}}.$
(12b)

Recapping, the TMM provides a convenient and fast method to evaluate the
reflection coefficient of an hybrid coating, having any number of conducting
membranes with given interlayer separations, and functioning at a given
wavelength. Hence, in what follows we may take $r_{\lambda}$ as a known
quantity, which one can readily evaluate for a specific case, and that fully
characterizes the reflection properties of the output coated surface.

###  III.2 Reflection Interference Pattern in a Three-Wave Fabry-Perot Cavity

Figure 4: Intensity reflected by the sensor as a function of the incident
wavelength. The spectrum was obtained for a simulated microcavity
($D\\!=\\!100\mu\text{m}$ and $d\\!=\\!40\mu\text{m}$) coated by $N\\!=\\!20$
graphene monolayers spaced by $20\text{nm}$ dielectric slabs
($n_{\text{pol}}\\!=\\!1.56$). In blue, we have the exact interference
pattern, obtained from a transfer-matrix calculation on to the whole system
(cavity + coating), while in red and black dashed we show the three-wave
approximation (see Eq. (13)) and its linearized version (see Eq. (15)) around
$\lambda_{c}\\!=\\!1550\text{nm}$, respectively. (color online)

Besides the sub-wavelength processes within the hybrid coating, there is also
interference happening at larger scales, between the waves reflected off the
three interfaces that make up the low-finesse Fabry-Perot microcavity at the
fiber’s termination (see cartoon in Fig. 1). Dealing with this µm-scale
interference is not fundamentally different from our previous analysis.
However, in the interest of earning a greater insight on the sensor’s
operation, we approach the problem from a different angle and approximate the
reflected signal by the sum of the three primary reflections (the ones shown
in Fig. 1). This approximation is justified by the low reflectivity of all the
surfaces in the cavity and further confirmed by a direct comparison with the
reflected spectra obtained by the TMM applied to the whole system (see Fig.
4). Considering a three-wave approximation, the intensity reflected by the
microcavity is then given as

$\displaystyle\mathcal{I}_{r}^{\lambda}\\!$
$\displaystyle=\\!\mathcal{I}_{1}\\!+\\!\mathcal{I}_{2}\\!+\\!\mathcal{I}_{3}^{\lambda}\\!-2\sqrt{\mathcal{I}_{1}\mathcal{I}_{2}}\cos\\!\left(\\!\frac{4\pi
D}{\lambda}\\!\right)$ (13)
$\displaystyle\qquad\qquad-2\sqrt{\mathcal{I}_{2}\mathcal{I}_{3}^{\lambda}}\cos\\!\left(\\!\frac{4\pi
n_{s}d}{\lambda}\\!+\\!\theta_{c}^{\lambda}\\!\right)$
$\displaystyle\qquad\qquad+2\sqrt{\mathcal{I}_{1}\mathcal{I}_{3}^{\lambda}}\cos\\!\left(\\!\frac{4\pi(D\\!+\\!n_{s}d)}{\lambda}\\!+\\!\theta_{c}^{\lambda}\\!\\!\right),$

where $n_{s}$ is the refractive index of the silica, $d$ and $D$ are the
cavity widths (as defined in Fig. 1), and $\mathcal{I}_{1,2,3}$ are the
intensities reflected by each interface. These intensities can be obtained
from Fresnel laws at normal incidence, reading

$\displaystyle\sqrt{\mathcal{I}_{1}}$
$\displaystyle\\!=\\!\left|\frac{n_{s}\\!-\\!1}{n_{s}\\!+\\!1}\right|\\!,\
\sqrt{\mathcal{I}_{2}}\\!=\\!\frac{4n_{s}\left|n_{s}\\!-\\!1\right|}{\left(n_{s}\\!+\\!1\right)^{3}}$
(14) $\displaystyle\qquad\qquad\qquad\qquad\,\text{and
}\sqrt{\mathcal{I}_{3}}\\!=\\!\frac{16n_{s}^{2}R_{c}^{\lambda}}{\left(n_{s}\\!+\\!1\right)^{4}}.$

As anticipated, effects from the output coating in the interference pattern
are encapsulated in its complex reflection coefficient, specified by the
corresponding reflectivity ($R_{\\!c}^{\lambda}$) and reflection phase shift
($\theta_{c}^{\lambda}$) in all previous expressions. These parameters are
crucially dependent on the incident wavelength ($\lambda$) and can be
calculated by the TMM described in Subsect. III. As we shall see, the
$\lambda$-dependence of the reflection phase shift plays a pivotal role in the
operation of our sensor and is, in effect, the central concept of this paper.

Regardless of coating details, one can point out some generic features of the
interference fringes. As evident from Eq. (13), even within a three-wave
approximation, a Fabry-Perot interferometer does not yield a periodic
interference pattern upon reflection. Nevertheless, if only a small
neighborhood of a central wavelength ($\lambda_{\text{c}}$) is observed, one
can consider a lowest-order expansion around $\lambda_{c}$ and a three-
component periodic pattern emerges,

$\displaystyle\mathcal{I}_{r}^{{\scriptscriptstyle\lambda_{\text{c}}\\!+\\!\Delta\lambda}}$
$\displaystyle\approx\mathcal{I}_{1}+\mathcal{I}_{2}+\\!\mathcal{I}_{3}^{\lambda_{\text{c}}}\\!$
(15) $\displaystyle-$ $\displaystyle
2\sqrt{\mathcal{I}_{1}\\!\mathcal{I}_{2}}\cos\\!\left(\\!\frac{2\pi\Delta\lambda}{\Lambda_{1}}\\!-\\!\frac{4\pi
D}{\lambda_{\text{c}}^{2}}\\!\right)\\!$ $\displaystyle-$ $\displaystyle
2\sqrt{\mathcal{I}_{2}\mathcal{I}_{3}^{\lambda_{\text{c}}}}\cos\\!\left(\\!\frac{2\pi\Delta\lambda}{\Lambda_{2}}\\!-\\!\frac{4\pi
n_{\text{s}}d}{\lambda_{\text{c}}^{2}}\\!-\\!\theta_{c}^{\lambda_{\text{c}}}\\!\right)$
$\displaystyle+$ $\displaystyle
2\sqrt{\mathcal{I}_{1}\mathcal{I}_{3}^{\lambda_{\text{c}}}}\cos\\!\left(\\!\frac{2\pi\Delta\lambda}{\Lambda_{3}}\\!-\\!\frac{4\pi\left(D\\!+\\!n_{\text{s}}d\right)}{\lambda_{\text{c}}^{2}}\\!-\\!\theta_{c}^{\lambda_{\text{c}}}\\!\right).$

This pattern features three effective periods ($\Lambda_{1,2,3}$) that arise
from the interference of any combination of the three partial waves of Fig. 1.
The three periods are then given as:

$\displaystyle\Lambda_{1}\\!=$
$\displaystyle\frac{\lambda_{\text{c}}^{2}}{2D}\,,\,\Lambda_{2}\\!=\frac{2\pi\lambda_{\text{c}}^{2}}{4\pi
n_{\text{s}}d\\!+\\!\lambda_{\text{c}}^{2}\left.\nicefrac{{d\theta_{c}^{\lambda}}}{{d\lambda}}\right|_{\lambda_{\text{c}}}}\,$
(16)
$\displaystyle\qquad\text{and}\,\Lambda_{3}\\!=\frac{2\pi\lambda_{\text{c}}^{2}}{4\pi\left(D\\!+\\!n_{\text{s}}d\right)\\!+\\!\lambda_{\text{c}}^{2}\,\left.\nicefrac{{d\theta_{c}^{\lambda}}}{{d\lambda}}\right|_{\lambda_{\text{c}}}}.$

With no surprise, Eqs. (16) demonstrate that the effect of coating the output
interface is a slight change in $\Lambda_{2}$ and $\Lambda_{3}$, relative to
their bare values. These are the components of the pattern which arise from
interference with the partial wave reflected off the output interface.
Interestingly, they also show that the shift in the periods is due to a
$\lambda$-dependent reflection phase shift, something which would be absent in
a bare dielectric-to-dielectric interface. These shifts may increase (positive
response) or a decrease (negative response) the periods relative to their bare
values, depending on the sign of
$\left.\nicefrac{{d\theta_{c}^{\lambda}}}{{d\lambda}}\right|_{\lambda_{\text{c}}}$
. As we shall see, both the sign and magnitude of this derivative can be tuned
by changes in the refractive index of the output medium, thus providing the
operational principle behind the cavity’s sensitivity to the environment. Note
that, in practice, the individual periods of the interference pattern can be
studied by band-filtering the Fourier transform of the reflected signal within
a narrow enough spectral window [44].

##  IV Nontrivial Reflection Phase Shifts and Enhanced Response to Liquid
Environments

The dependence of the reflected interference fringes on the output medium is
determined by the phase shifts introduced when the wave is reflected off the
last interface of the microcavity. More precisely, Eqs. (16) show that a
$\lambda$-dependent phase shift changes the periods in the interference
pattern, with respect to their bare values (determined by the microcavity’s
geometry). If the tip of the sensor was a bare dielectric-to-dielectric
interface, then the reflection dephasing would be independent of both
$\lambda$ and $n_{\text{out}}$. In this case, any changes to the periodicity
of the fringes could only arise by geometrical alterations of the microcavity
itself. However, such situation does not hold if a thin dielectric coating (of
index $n_{\text{pol}}$ and width $\Delta$) is placed on top of the output
surface. Then, interference within the coating can bring about nontrivial
phase shifts on the last partial wave, that usually depend on the wavelength.
A straightforward transfer matrix analysis of this simple setup reveals the
following complex reflection coefficient:

$r_{\\!\text{c}}^{{\scriptscriptstyle\lambda}}\\!\\!=\\!\frac{(\frac{n_{\text{s}}}{n_{\\!\text{pol}}}\\!\\!-\\!\\!1)(1\\!\\!+\\!\frac{n_{\\!\text{out}}}{n_{\\!{\scriptscriptstyle\text{pol}}}})\\!+\\!(1\\!\\!-\\!\frac{n_{\\!\text{out}}}{n_{\\!\text{pol}}})(1\\!\\!+\\!\\!\frac{n_{\text{s}}}{n_{\\!\text{pol}}})e^{\frac{4i\pi
n_{\\!\text{pol}}\\!\Delta}{\lambda}}}{(\frac{n_{\text{s}}}{n_{\\!\text{pol}}}\\!\\!+\\!\\!1)(1\\!\\!+\\!\frac{n_{\\!\text{out}}}{n_{\\!\text{pol}}})\\!-\\!(1\\!\\!-\\!\frac{n_{\\!\text{out}}}{n_{\\!\text{pol}}})(1\\!\\!-\\!\\!\frac{n_{\text{s}}}{n_{\\!\text{pol}}})e^{\frac{4i\pi
n_{\\!\text{pol}}\\!\Delta}{\lambda}}}.$ (17)

From Eq. (17), one extracts the reflectivity and reflection phase shift as
functions of the incident wavelength. For our purposes, what really matters is
to analyze the variation of these quantities with $n_{\text{out}}$, provided
the coating is nearly anti-reflecting (i.e.
$\Delta\\!\approx\\!n_{\text{pol}}\lambda/2$). In Fig. 5a, we present such an
analysis for a dielectric coating ($n_{\text{pol}}\\!=\\!1.6$), near anti-
reflecting conditions, i.e. with the parameter $x\\!=\\!2\Delta
n_{\text{pol}}/\lambda$ valued close to $1$. The main feature to be
highlighted happens for an index $n_{\text{out}}^{*}$, for which the absolute
reflectivity of the output surface vanishes and is accompanied by an abrupt
(“resonant-like”) behavior of
$\left.\nicefrac{{d\theta_{c}^{\lambda}}}{{d\lambda}}\right|_{\lambda_{\text{c}}}\\!\\!$.
In the absence of conducting membranes, this behavior appears around
$n_{\text{out}}^{*}\\!\\!=\\!n_{\text{s}}$ which signals that, even for a
coating with precise dimensions for destructive interference of the
backscattered waves (i.e. $x\\!=\\!1$), the interference is only partial
unless the reflectivity is similar on both ends of the coating. This implies
opposite discontinuities of the refractive index in the two interfaces. If the
dimensions of the coating slightly depart from this $\lambda/2$-condition (or
$x\\!\approx\\!1$) then a nonzero reflectivity is obtained, accompanied by a
broadening of the “resonant-like” response of
$\left.\nicefrac{{d\theta_{c}^{\lambda}}}{{d\lambda}}\right|_{\lambda_{\text{c}}}\\!\\!$
to $n_{\text{out}}$ (as depicted in Fig. 5a).

Figure 5:  (a) Simulated plots of $\lambda d\theta_{\\!\text{c}}\\!/d\lambda$
(controlling the shift of the reflected fringes) and reflectivity (in the
inset), as a function of the output refractive index
($n_{{\scriptscriptstyle\\!\text{out}}}$) for a purely dielectric and nearly
anti-reflecting coating of index
$n_{{\scriptscriptstyle\\!\text{pol}}}\\!=\\!1.615$ and total width
$\Delta\\!=\\!x\lambda/2n_{{\scriptscriptstyle\\!\text{pol}}}$. (b) Scheme of
the deformed intensity profile within the coating induced by the conducting
membranes. Simulated data is shown for a plain dielectric coating (bottom) and
the same coating intersected by $15$ graphene sheets (top). To aid
visualization, the vertical axis is to be taken as proportional to the actual
reflected intensity. (c) Simulated environmental dependence of $\lambda
d\theta_{\\!\text{c}}\\!/d\lambda$ for hybrid coatings with $N\\!\\!=\\!0$
-$20$ graphene sheets, operating at a central wavelength of
$\lambda_{\text{c}}\\!=\\!1550\text{nm}$. The resonant response is shifted to
lower $n_{{\scriptscriptstyle\\!\text{out}}}$, as $N$ is increased. (color
online)

The “resonant-like” behavior described above is of the utmost importance for
this work. In fact, Eqs. (16) directly imply that such a strong dependence of
$\left.\nicefrac{{d\theta_{c}^{\lambda}}}{{d\lambda}}\right|_{\lambda_{\text{c}}}\\!\\!$
on $n_{\text{out}}$ leads to an enhanced sensitivity of the periods in the
interference fringes, to the environment in which the Fabry-Perot microcavity
is placed. In passing, it is also relevant to note that
$\left.\nicefrac{{d\theta_{c}^{\lambda}}}{{d\lambda}}\right|_{\lambda_{\text{c}}}\\!\\!$
increases with $n_{\text{out}}$ in this regime, meaning that the periods of
the interference fringes would have a negative environmental response.
However, as the closing of the microcavity is always silica
($n_{\text{s}}\\!\approx\\!1.45$), any purely dielectric coating would only
enhance its environmental sensitivity for refractive indices
$n_{\text{out}}^{*}\\!\approx\\!1.45$, which are typical of transparent solid
media. In order to operate this sensor on fluid environments, one must be able
to shift this “resonant-like” behavior towards lower values of
$n_{\text{out}}^{*}$. As will be demonstrated, this is the advantage of using
a hybrid coating with embedded conducting membranes, instead of a dielectric
one.

Figure 6: (a) Plots of the reflectivity minima for simulated coatings having
an increasing number of conducting membranes. The position of the minimum
coincides with the index $n_{{\scriptscriptstyle\text{\\!out}}}$ around which
the sensor’s response to environmental changes is enhanced. (b) Examples of
calibration curves that show the number of graphene-dielectric bilayers needed
to obtained an enhanced environmental response around a particular refractive
index. The different curves refer to using different dielectric matrices in
the coating and relevant range to study typical liquid media is shaded in
blue. (color online)

Inside a hybrid coating there are conducting membranes that act as
segmentations of the bulk dielectric matrix in which they lie embedded.
Effectively, these membranes split the dielectric coating into chunks of
average width $\delta$, performing as conducting interfaces. The presence of
these interfaces greatly affect the profile of a wave traveling inside, as
they introduce additional internal reflections and also localized dissipation
of field energy. The TMM devised in Sec. III takes all these effects into
account and, as shown in Fig. 5b, they lead to a non-homogeneous field
intensity that decreases along the width of the coating. This deformation of
the field profile leads to a change in the condition for the absence of a
reflected wave, thus requiring a larger discontinuity in the refractive index
at the output boundary. As shown in Fig. 5c, the main consequence is a
downshift of $n_{\text{out}}^{*}$ which now enhances the environmental
sensitivity at refractive indices within the range of $1.3\\!-\\!1.4$.

Before moving on to the experimental proof-of-concept, it is important to
recognize that the shift in $n_{\text{out}}^{*}$ depends on the parameters of
the coating. Namely, we show in Fig. 5c that $n_{\text{out}}^{*}$ decreases
monotonically with the number of conducting membranes, provided the full width
of the coating is kept roughly the same. Instead of looking at the resonance,
the changes in $n_{\text{out}}^{*}$ can be conveniently and fully
characterized by tracking the reflectivity minimum. According to our previous
interpretation of this “resonance-like” behavior, a minimum in the
reflectivity must always accompany the resonance in
$\left.\nicefrac{{d\theta_{c}^{\lambda}}}{{d\lambda}}\right|_{\lambda_{\text{c}}}\\!\\!$.
This analysis is done in Fig. 6a and, for low enough conductivity of the
membranes, the minima depend on the coating parameters as,

$n_{\text{out}}^{*}\\!\\!=\\!\sqrt{\frac{n_{\text{s}}^{2}-\mu_{0}c\sigma_{\lambda}n_{\text{s}}(N\\!+\\!1)}{1-\mu_{0}c\sigma_{\lambda}n_{\text{s}}(N\\!-\\!1)/n_{\text{pol}}^{2}}},$
(18)

where $n_{\text{s}}$ is the refractive index of silica, $\mu_{0}$ the vaccum
magnetic permeability, $c$ the speed of light and $N$ the total number of
conducting membranes. Finally, this expression can be used to create
approximate calibration curves (see Fig. 6b) that serve to guide the
engineering of a coating configuration that is appropriate to enhance the
environmental sensitivity around a pre-established value of the output
refractive index.

##  V Experimental Demonstration

The theoretical proposal made in previous sections is now put to the test on
experimental grounds. Our prototypical device consists of a single-mode
optical fiber (SMF) terminated by a cascaded Fabry-Perot microcavity
interferometer (FPMI). A photograph obtained from an optical microscope is
shown in Fig. 7. In order to assess the influence of an hybrid coating on the
environmental sensitivity of its interference pattern, we perform a
comparative study using two sensing devices: i) an FPMI sensor coated by a
single layer of bulk dielectric polymer and ii) a similar sensor having a
multilayered PEI/GO hybrid coating.

In the following, we begin by detailing technical aspects of the device’s
production method and measurement techniques. Afterwards, we present
experimental measurements and compare them to the predictions of our
theoretical framework.

####  V.0.1 Fabrication of the Fabry-Perot Microcavity Interferometer

To build the FPMI, a silica capillary tube with an internal (external)
diameter of 75µm (125µm) was fusion spliced (Sumitomo Type-72C) to the SMF’s
termination and then cleaved to the desired length ($D$), using a fiber
cleaver (Sumitomo FC-6RS). To close the FPMI, a second SMF was fusion spliced
to the free end face of the capillary and then cleaved to a length $d$. The
splicing processes were all made by using electric arcs centered far away from
the capillary region. This way, we could avoid its collapse and thus keep
straight enough sides in order to minimize transverse optical losses.

####  V.0.2 Production of the PEI and PEI/GO coatings

For producing the optical coatings, we employed polyethylenimine (PEI) as our
dielectric medium ($n_{\\!\text{out}}\\!=\\!1.615$). This polymer was acquired
from Sigma-Aldrich (catalog number P3143) as a water-based solution with a
$50\%$ concentration ($w/v$). The PEI solution was further diluted using
ultra-pure water (Milli-Q water), achieving a lower concentration of
$2\\!\times\\!10^{-2}\text{\\!}M$. For producing the multilayered hybrid
coatings, we used graphene-oxide obtained from a commercial GO solution
(Sigma-Aldrich, catalog number 777676) with a concentration of 4 mg/mL. The
latter was then diluted in ultra-pure water to match the same concentration of
the diluted PEI solution.

All FPMIs used in this study were coated using a layer-by-layer dip coating
method. For purely dielectric (PEI) coatings, the fiber was dipped into the
diluted PEI solution for 1 min, followed by a brief rinse in ultra-pure water
to remove non-adsorbed residues. This process was repeated 10 times, in order
to reach a significant film thickness. In the case of the PEI/GO coatings, a
similar procedure was followed, with the FPMIs being dipped alternately into
the PEI and the GO solution, with a rinse and drying step in between. Here the
repetition went until the desired number of bilayers ($N\\!=\\!10$) was
attained.

Figure 7: Experimental setup composed by an amplified spontaneous emission
(ASE) broadband source, an optical circulator, and an optical spectral
analyzer (OSA). Microscope photograph of the fabricated Fabry-Perot
interferometer, with an air cavity length of 258 µm and a silica slab length
of 64 µm. (color online)

####  V.0.3 Optical Measurement Setup

The experimental setup used for the characterization of the FPMIs is
schematically depicted in Fig. 7. It consisted of an amplified spontaneous
emission (ASE) broadband source centered at
$\lambda_{\text{c}}\\!=\\!1550\text{nm}$ with a bandwidth of (approximately)
100 nm, an optical circulator, and an optical spectrum analyzer (OSA -
Yokogawa AQ6370C).

###  V.1 Experimental Results

Figure 8: (a) Measured reflected spectrum before and after the application of
a coating with 10 PEI/GO bilayers. The arrow makes evident the overall shift
of the interference pattern (in wavelength) caused by introducing the output
coating. (b) Reflected spectrum from the FPMI coated by 10 PEI/GO, measured
with the cavity immersed by liquid environments having different refractive
indices ($n_{{\scriptscriptstyle\\!\text{out}}}\\!\\!=1.3329\\!-\\!1.3603$).
The sensitivity of the pattern is to be contrasted with (c), where an
identical analysis was done using a cavity coated by PEI with roughly the same
width. All the curves appear superposed. (color online)

The influence of coating the output surface of an FPMI was studied by
comparing the reflected spectrum before and after the dip coating process. In
the presence of a PEI coating, no significant shifts or intensity changes were
observed in the reflected spectrum, as obtained within an air environment. In
contrast, the PEI/GO FPMI showed visible changes in the interference fringes,
which appeared shifted in wavelength, with decreased intensity but a larger
visibility. In Fig. 8a, we plot the reflected signal measured before and after
a sensor was coated with 10 PEI/GO bilayers. The shift in the fringes is
clearly marked.

Besides comparing the operation of coated and uncoated FPMIs, we focus our
analysis on the influence of such coatings in their sensitivity to
environmental changes. Namely, we characterize the dependence of the reflected
pattern on the refractive index of the output medium by immersing the sensors
in calibrated liquids, with indices ranging from $n_{\text{out}}\\!=\\!1.3327$
to $n_{\text{out}}\\!=\\!1.3830$ 333These indices were determined using an
Abbe refractometer (Atago DR-A1), with a light source with a wavelength
approximately equal to sodium D-line (589.3 nm).. The spectra of reflected
signals obtained with a PEI/GO-coated sensor are shown in Fig. 8b, where it is
evident that changing the output medium induces measurable variations in both
phase and intensity of the interferences fringes. No such response was
obtained using a PEI-coated FPMI, as shown in Fig. 8c, which indicates an
increased environmental sensitivity of the system in the presence of a hybrid
coating.

The variations in the reflected signal of the PEI/GO sensor can be more finely
evaluated by performing a spectral decomposition 444The FFT was calculated
after converting the dB-reflected signal into linear optical power (W).
through a fast Fourier transform (FFT). The result is shown in Fig. 9a, where
the FFT amplitudes appear plotted as functions of vaccum optical cavity
lengths. From our previous theoretical analysis, it is not surprising that the
FFT features exactly three peaks, associated to the three Fourier components
of Eq. (15). The first peak ($\Lambda_{1}$), at a cavity length of $91.2$ µm
(corresponding to a $d\\!=$63 µm length for
$n_{\text{s}}\\!\\!=\\!\\!1.4444$), is related to the optical cavity composed
by fused silica and PEI/GO coating. The second peak ($\Lambda_{2}$), related
to the air cavity, is located at a cavity length of $D\\!=\\!258.3$ µm, while
the third peak ($\Lambda_{3}$) is related to the optical cavity composed by
air, silica slab, and the PEI/GO coating. Note that the estimated cavity
length values are in full accordance with the ones determined by inspection of
the microscope photograph in Fig. 7. Furthermore, small intensity variations
are observed in Fig. 9a for the first and third peaks, while the air cavity
peak remains unaffected by changes in the output refractive index. These
results are consistent with the understanding that only $\Lambda_{1}$ and
$\Lambda_{3}$ (as defined in Eq. (15)) can be affected by the reflection
coefficient of the output coating and, thus being sensitive to the output
medium. As an example, we also plot the FFT peak intensity of $\Lambda_{3}$ as
a function of $n_{\text{out}}$ in Fig. 9b.

The changes in the shape of the interference pattern depicted in Fig. 8b,
point towards an environmentally-induced variation not only of the FFT peak
intensities, but also of their locations. Such changes in the periodicity of
the pattern are consistent with our earlier theoretical predictions and
therefore provide the sought after connection between the experiments and
theory presented here. In order to analyze these shifts in the Fourier
components, we begin by isolating one of the sensitive peaks using a band-pass
filter in the FFT. For concreteness, we isolate the first peak ($\Lambda_{1}$)
and obtain the filtered spectrum presented in the inset of Fig. 10. The
filtered signal now exhibits an evident phase variation which actually amounts
to a shift of the period $\Lambda_{1}$ induced by an increase of the output
refractive index. In Fig. 10, the shifts in $\Lambda_{1}$ are represented
against our theoretical predictions for two FPMI sensors coated with
$N\\!=\\!10$ and $N\\!=\\!40$ conducting membranes (GO) embedded into a
dielectric (PEI) matrix ($n_{\text{pol}}\\!=\\!1.617$ PEI [58]). Surprisingly,
by using the simplified model of Sec. III and a real optical conductivity
slightly smaller than pristine graphene ($\xi\\!=\\!0.0188$ 555As opposed to
$\xi\\!\approx\\!0.0229$ for pristine graphene sheets working in the mid- to
near-infrared regime.), the agreement between experimental data and the
theoretical curves is reasonably good, for both cases. The average distance
between consecutive GO sheets was slightly adjusted for the fit
($\delta\\!=\\!95.8$ nm and $\delta\\!=\\!95.6$ nm, respectively) but the
values are close enough to indicate a good consistence in the production of
the multilayered coating. Given the imperfections expected to be present in a
GO/PEI coating produced using a chemical layer-by-layer technique, we can
argue our experimental results to be consistent with the downshift of a
“resonant-like” behavior in the reflection phase shift at the output surface
of the FPMI, as theoretically predicted for hybrid coatings in Sec. IV.

Figure 9: (a) Fourier transform (FFT) of the reflected spectra measured with
the sensor inside outer liquid media with varying refractive indices, coated
by 10 PEI/GO bilayers. (b) Study of the dependence of the intensity in third
peak on the output refractive index. The error bars are due to intrinsic
spectral resolution of the OSA which appear as random fluctuations in
measurements made over time, using the same device. (color online)

##  VI Conclusions and Outlook

Figure 10: Fit of the spectral shifts in the shortest period ($\Lambda_{1}$)
measured from the filtered FFT of the reflected fringes to the theoretical
model. Two data sets are shown for identical sensors coated by $10$ (blue
points) and $40$ GO/PEI bilayers (orange points). The solid lines represent
the corresponding theoretical curves, obtained for reasonable values of the
free parameters in the model — $\xi\\!=\\!0.0188$,
$n_{\text{pol}}\\!=\\!1.617$ — and a distance between consecutive GO
monolayers of $\delta\\!=\\!95.8$ nm and $\delta\\!=\\!95.6$ nm, respectively.
For a comparison, the corresponding curves in the absence of GO membranes
(i.e. $\xi\\!=\\!0$) are shown as dashed lines. The inset represents the band-
filtered signal reconstructed by an inverse FFT for the coating of $10$
bilayers. The error bars are due to intrinsic spectral resolution of the OSA
which appear as fluctuations in measurements made over time. (color online)

We have proposed and experimentally demonstrated an interferometric sensor,
based on a Fabry-Perot microcavity built at the end of a single-mode optical
fiber and covered by a thin multilayered hybrid coating made of two-
dimensional (graphene-like) conducting membranes embedded within a dielectric
matrix. Unlike more conventional Fabry-Perot sensors, the proposed device is
able to sense changes in the refractive index of the output medium by
translating them to measurable shifts in the Fourier periods that compose the
reflected spectrum. Our theoretical study has shown that the aforementioned
changes in the interference pattern can be attributed to an enhanced
environmental sensitivity of the reflection phase shift introduced by the
output coating. The latter is caused by complex internal reflection,
interference and energy-loss processes among the different conducting surfaces
contained within the coating. Moreover, as the number of layers in the coating
increases, the sensitivity is enhanced for lower values of the output
refractive index, which allows the engineering of a sensor that works within
range of indices typically found on liquid media ($n_{\text{out}}\\!\approx
1.3\\!-\\!1.4\\!$). The theoretical predictions were further tested against
reflected spectral measurements done on an experimental realization of this
device that employed GO/PEI multilayered structures produced using a cost
effective layer-by-layer chemical dip coating process. Despite inherent
imperfections on the coating’s structure and the possible influence of other
external physical parameters (e.g. thermal sensitivity), a sensible agreement
was obtained between measurements and theoretical predictions. Besides
demonstrating the concept, this agreement further encourages investment on
these types of devices for future real-life sensing applications.

As a closing remark, we comment on the technological applicability of the
sensing devices described in this work. In reality, for these hybrid coatings
to be of practical use, a more consistent production method that creates more
resilient/durable hybrid coatings is imperative. In particular, the
interference phenomenon described here is very sensitive to imperfections in
the layered structure, namely the film roughness shown in Fig. 11. This
roughness is a typical feature known to occur in polymeric films produced
using layer-by-layer techniques [60, 61, 62] and can eventually be controlled.
Surely, the use of vapor phase deposition techniques could potentially allow
more perfect structured coatings to be made. However, such methods would be
highly inefficient, and likely more expensive, ways to produce these
multilayered coatings. Nevertheless, the basic concepts underlying the
operation of these hybrid coatings are arguably more general than our current
experimental realization and this work is to serve as a spark for future
technological developments.

Figure 11: Optical microscopy images of a film of 10 PEI/GO bilayers deposited
on glass. The roughness of the polymeric film is clear from the picture.

###### Acknowledgements.

The authors acknowledge financial support by the Portuguese Foundation for
Science and Technology (FCT) within the Strategic Funding UIDB/04650/2020 and
COMPETE 2020 program in FEDER component (European Union), through Projects No.
POCI-01-0145-FEDER-028887 and UltraGraf (M-ERA-NET2/0002/2016). J.P.S.P. and
C.S.M were further supported by FCT Ph.D. grants PD/BD/142774/2018 and
SFRH/BD/135820 /2018, respectively. J.P.S.P. thanks S. M. João, C. D.
Fernandes, Prof. J. M. B. Lopes dos Santos and Prof. N. M. R. Peres for
profitable discussions and useful comments on this work.

##  Appendix A Electrodynamic boundary conditions in conducting interfaces

The existence of sharp interfaces separating different dielectric media are
known to introduce discontinuities on components of electric and/or magnetic
fields. In the simplest case, these interfaces are electrically neutral and
cannot support charge currents. For this work, however, we must consider cases
in which this last condition does not hold, because the interfaces are two-
dimensional conducting membranes (e.g. graphene monolayers). In this Appendix,
we will derive the general boundary conditions for such a planar interface,
assuming that the two dielectrics have refractive indices $n_{1}$ and $n_{2}$,
while the interface is a linear conducting membrane described by a complex
conductivity
$\sigma_{\lambda}\\!=\\!\sigma^{\prime}_{\\!\lambda}\\!+\\!i\sigma^{\prime\prime}_{\\!\lambda}$.

To obtain these boundary conditions, we begin by considering the interface to
be planar. Despite not involving loss of generality, this geometrical
arrangement allows the construction of the usual pillbox gauss surface (Fig.
12a) and Ampère square circuit (Fig. 12b) that allows us to apply Maxwell’s
equations in integral form:

Figure 12: (a) Pillbox Surface traversing the interface between the two
dielectric media. (b) Square circuit used to derive the boundary conditions
for the parallel components. (color online)

$\varoiint_{S_{\mathcal{V}}}\\!\\!\\!\\!\mathbf{D}\\!\left(\mathbf{r},t\right)\\!\cdot\\!\mathbf{dS}\\!=\\!\\!\iiint_{\mathcal{V}}\\!\\!\\!\rho\\!\left(\mathbf{r},t\right)d\mathbf{r}$
(19a)

$\varoiint_{S_{\mathcal{V}}}\\!\\!\\!\\!\mathbf{B}\\!\left(\mathbf{r},t\right)\\!\cdot\\!\mathbf{dS}\\!=\\!0$
(19b)

$\ointclockwise_{\mathcal{C}}\\!\\!\mathbf{E}\\!\left(\mathbf{r},t\right)\\!\cdot\\!\mathbf{dS}\\!=\\!-\varoiint_{S_{\mathcal{C}}}\\!\\!\left[\frac{\partial}{\partial
t}\mathbf{B}\\!\left(\mathbf{r},t\right)\right]\\!\cdot\\!\mathbf{dS}$ (19c)

$\displaystyle\ointclockwise_{\mathcal{C}}\\!\\!\mathbf{B}\\!\left(\mathbf{r},t\right)\\!\cdot\\!\mathbf{dS}\\!$
$\displaystyle=\\!\mu_{0}\varoiint_{S_{\mathcal{C}}}\\!\\!\\!\mathbf{J}\\!\left(\mathbf{r},t\right)\\!\cdot\\!\mathbf{dS}$
(19d)
$\displaystyle\quad-\mu_{0}\varoiint_{S_{\mathcal{C}}}\\!\\!\left[\frac{\partial}{\partial
t}\mathbf{D}\\!\left(\mathbf{r},t\right)\right]\\!\cdot\\!\mathbf{dS},$

where $\mathbf{E}$, $\mathbf{B}$ and $\mathbf{D}$ are the electric magnetic
and electric displacement fields, respectively. In order to keep the
discussion as focused as possible, we will assume that all these fields are
harmonic in space-time, e.g.
$\mathbf{E}(\mathbf{r},t)\\!=\\!\mathbf{E}_{0}\exp\left(i\mathbf{k}\\!\cdot\\!\mathbf{r}\\!-\\!i\omega
t\right)$, and that the interface is aligned with the plane $z\\!=\\!0$.
Assuming an wavelength $\lambda\\!=\\!2\pi/\left|\mathbf{k}\right|$, the ohmic
surface currents at the interfaces are simply

$\mathbf{K}\\!(\mathbf{r},t)\\!=\\!\mathbf{K}_{0}\exp\left(i\mathbf{k}^{\parallel}\\!\cdot\\!\mathbf{r}\\!-\\!i\omega
t\right),$ (20)

where $\mathbf{K}_{0}\\!=\\!\sigma_{\lambda}\\!\mathbf{E}_{0}^{\parallel}$.
Hereafter, we will use $\parallel$ ($\perp$) to represent vectors which are
parallel (perpendicular) to the interface. Since the current of Eq. (20) is
time-dependent, the condition of local current conservation implies that a
time-dependent change density wave is also usually generated. The latter can
be obtained from the two-dimensional continuity equation,

$\frac{\partial\rho_{\text{s}}(\mathbf{r},t)}{\partial
t}=\nabla_{xy}\mathbf{K}\left(\mathbf{r},t\right),$ (21)

which yields
$\rho_{\text{s}}\\!(\mathbf{r},t)\\!=\\!\rho_{\text{s}}^{0}\exp\left(i\mathbf{k}^{\perp}\\!\cdot\\!\mathbf{r}\\!-\\!i\omega
t\right)$, where

$\rho_{\text{s}}^{0}\\!=\\!\frac{\sigma_{\lambda}}{\omega}\mathbf{E}_{0}^{\parallel}\\!\cdot\\!\mathbf{k}.$
(22)

If we now use the surface current and charge densities of Eqs. (20) and (22)
and apply the integral Maxwell’s equations to the constructions of Fig. 12, we
obtain the following general conditions for the fields in the two sides of the
planar interface:

$\displaystyle\begin{array}[]{cc}\mathbf{E}_{0,1}^{\parallel}&\\!=\\!\mathbf{E}_{0,2}^{\parallel}\end{array}$
(23b)

$\varepsilon_{1}\mathbf{E}_{0,1}^{\perp}\\!=\\!\varepsilon_{2}\mathbf{E}_{2}^{\perp}\\!+\\!\frac{\sigma_{\lambda}}{\omega}\mathbf{E}_{0,1}^{\parallel}\\!\cdot\\!\mathbf{k}$
(23c)

$\displaystyle\begin{array}[]{cc}\mathbf{B}_{0,1}^{\perp}&\\!=\\!\mathbf{B}_{0,2}^{\perp}\end{array}$
(23e)

$\mathbf{B}_{1}^{\parallel}\\!=\\!\mathbf{B}_{2}^{\parallel}+\mu_{0}\sigma_{\lambda}\\!\mathbf{\hat{n}}\\!\times\\!\mathbf{E}_{0,1}^{\parallel},$
(23f)

where $\varepsilon_{1}$ are the dielectric constants of the two media.
Finally, if we consider the case of perpendicular incidence, then the
perpendicular components are zero and
$\mathbf{\hat{n}}\\!\times\\!\mathbf{E}_{0,1}^{\parallel}\\!=\\!\mathbf{E}_{0,1}$.
This way, we recover the special case quoted in Eqs. (7a)-(7b).

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