# Remote non-invasive Fabry-Pérot cavity spectroscopy for label-free sensing

Abeer Al Ghamdi<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Leeds, Leeds LS2 9JT, United Kingdom  School of Physics and
Astronomy, King Saud University, Riyadh 11362, Saudi Arabia  Benjamin Dawson
School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United
Kingdom  School of Chemical and Process Engineering, University of Leeds,
Leeds LS2 9JT, United Kingdom  Gin Jose  School of Chemical and Process
Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom  Almut Beige
School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United
Kingdom

###### Abstract

One way of optically monitoring molecule concentrations is to utilise the high
sensitivity of the transmission and reflection rates of Fabry-Pérot cavities
to changes of their optical properties. Up to now, intrinsic and extrinsic
Fabry-Pérot cavity sensors have been considered with analytes either being
placed inside the resonator or coupled to evanescent fields on the outside.
Here we show that Fabry-Pérot cavities can also be used to monitor molecule
concentrations non-invasively and remotely, since the reflection of light from
the target molecules back into the Fabry-Pérot cavity adds upwards peaks to
the minima of its overall reflection rate. Detecting the amplitude of these
peaks reveals information about molecule concentrations. By using an array of
optical cavities, a wide range of frequencies can be probed at once and a
unique optical fingerprint can be obtained.

##  I Introduction

Sensors which offer high specificity and reliably monitor substances in
chemical, physical and biological systems play a vital role in many
applications. For example, biosensors (cf. e.g. Refs. [1, 2, 4, 3, 5, 6]) are
used in tissue and cell analysis, microbiological investigations and drug
improvement studies. To answer the ever-growing demand for highly sensitive
and selective measurement devices which do not require large lab-based
equipment, Fabry-Pérot cavity sensors have already attracted considerable
attention [7, 8]. These sensors can be divided into two main categories:
intrinsic and extrinsic Fabry-Pérot cavity sensors. In the case of intrinsic
sensors [9, 10, 11], which are the most common, the sample to be measured is
placed inside the resonator (cf. Fig. 1(a)). In the case of extrinsic sensors
[12, 13, 14, 15, 16, 17, 18, 19], the sample changes the optical properties of
the resonator when in contact with one of its mirrors on the outside (cf. Fig.
1(b)).

Figure 1: [Colour online] Current Fabry-Pérot cavity sensors can be divided
into two main categories. (a) In the case of intrinsic sensors, the target
molecules are placed on the inside, where they alter the effective cavity
length via a change in refractive index. When driven by a laser field, the
resonance frequency of the cavity shifts and the line width of the signal can
broaden. (b) In the case of extrinsic sensors, the target molecules are placed
on the outside of one of the cavity mirrors in order to alter its reflection
rate, thereby also changing the optical properties of the cavity.

Fabry-Pérot cavities are optical resonators with two highly-reflecting mirrors
separated by a gap of length $L_{0}$. When monochromatic light enters the
cavity, it bounces back and forth between the mirrors many times before
eventually leaking out. The amount of light that passes through the resonator
depends on the frequency of the incoming light compared to the distance of the
mirrors. More concretely, the incoming laser light accumulates a phase factor
during each round trip. If the cavity is in resonance, this phase factor is an
integer multiple of $2\pi$. Hence all light travelling in the forward
direction interferes constructively, while light travelling in the backwards
direction interferes destructively, such that all light eventually leaves the
cavity on the opposite side [20]. Moreover, in the case of mirrors with
absorption and with reduced reflection rates, the transmission peak is
broadened, and a wider range of frequencies is transmitted.

The characteristic reflection and transmission spectrum of a Fabry-Pérot
cavity can be characterised for example by the quality factor $Q$ with
$Q=\omega_{\rm cav}/\Delta\omega$, where $\omega_{\rm cav}$ denotes the cavity
resonance frequency and $\Delta\omega$ characterises the line width. In
general, the higher the $Q$ factor of a cavity, the more sensitive are
$\omega_{\rm cav}$ and $\Delta\omega$ to the presence of molecule
concentrations. For example, some Fabry-Pérot cavity sensors take advantage of
the presence of analytes with distinctive optical transitions on mirror
reflection rates or of refractive index changes inside the resonator. In
general, information about concentrations can be deduced by comparing
frequency shifts and the broadening of the line width of the transmitted light
signal to a known baseline [8].

Unfortunately, intrinsic Fabry-Pérot cavity sensors tend to suffer from low
coupling efficiency. In the presence of analytes, the sensor is likely to
require a re-alignment of its mirrors in order to produce reliable results
which is experimentally demanding [9, 10, 11]. Extrinsic Fabry-Pérot cavity
sensors overcome this problem by never changing the inside of the resonator
and have already been used to perform highly sensitive measurements of small
molecules like proteins and DNA [12, 13, 14, 15, 16, 17, 18]. However, like
intrinsic Fabry-Pérot cavity sensors, they are invasive and still need to be
in close contact with the sample to make a measurement which strongly limits
their applications.

Figure 2: [Colour online] In this paper we propose an alternative way of using
Fabry-Pérot cavities in sensing applications. Here the substance which we want
to analyse is some distance away from the resonator. Hence we refer to this
type of sensor in the following as a remote Fabry-Pérot cavity sensor. If the
sample contains atomic particles with optical transitions near the resonance
frequency of the sensor, the coherent back reflection of light affects the
overall transmission rate of the system. The above experimental setup
effectively consists of many mirrors. The measurement signal is an effective
frequency-dependent reflection rate of the molecules and, as we shall see
below, provides information about their concentration.

To overcome this problem, this paper proposes a Fabry-Pérot cavity sensor
which can perform remote measurements of molecule concentrations. As we shall
see below, the sensor can be used to continuously and non-invasively monitor a
wide range of physical, chemical and biological processes. No direct contact
with the target molecules is required as long as they are optically
accessible. Hence remote Fabry-Pérot cavity sensors might even be suitable to
continuously and non-invasively monitor biomolecule concentrations [21, 22,
14, 15]. Alternatively, they could be used to monitor the contents of
transparent containers and bottles without the need to open them, which has
applications, for example, in the food industry.

The basic design of the remote Fabry-Pérot cavity sensor which we propose in
this paper is shown in Fig. 2. In the following, we require that the target
molecules have optical transitions and therefore reflect some of the
transmitted laser light back into the resonator. The environment of the
analytes needs to be transparent—or at least semitransparent—in the relevant
frequency range. In contrast to extrinsic and intrinsic Fabry-Pérot cavity
sensors, remote Fabry-Pérot cavity sensors explore the high sensitivity of
their reflection rate $R(\omega)$ to the presence of stray light to measure
target molecule concentrations. This is possible, since the interference
inside the cavity changes when the analytes reflect some of the already
transmitted light back into the resonator.

Suppose the Fabry-Pérot cavity is ideal and the incoming laser light is on
resonance with the resonator. In this case, all light is eventually
transmitted and the reflection rate $R(\omega)=0$. Hence the only effect that
the stray light coming from the target molecules can have is to increase
$R(\omega)$. In general, if the reflecting molecules are randomly distributed
within the sample and cover an area wider than the wavelength of the reflected
light, the reflected light accumulates equally-distributed random phases and
the resulting $R(\omega)$ depends only on the optical properties of the cavity
mirrors and on the concentration and the optical properties of the reflecting
atomic particles.

As we shall see below, randomly distributed molecules therefore add narrow
upwards peaks to minima of the reflection spectrum of the Fabry-Pérot cavity.
These peaks can be detected and their height reveals information about
molecular concentrations. The higher the concentration of the target
molecules, the larger the amplitude of the upwards peaks that are added to the
minima of the measurement signal. As one might expect, the amplitude of these
peaks is exactly the same as the amplitude of the reflected light in the
absence of the Fabry-Pérot cavity. The main purpose of the resonator is to
filter out one specific frequency component of the reflected light.

Figure 3: [Colour online] Alternative design of a remote sensor with increased
selectivity which allows for different resonant frequencies to be probed at
once. The sensor contains a bundle of optical fibres encased in a protective
sheath and embedded with cavities of different lengths $L_{0}$. By measuring
the resultant change for each resonance frequency from a known baseline, the
type and concentration of the sample to be measured can be determined more
easily.

Notice also that the reflection rate of the target molecules depends strongly
on their atomic level structure. Simultaneously probing the response of a
remote Fabry-Pérot cavity sensor to different frequencies of light can
therefore provide a unique optical fingerprint which increases the selectivity
of the sensor. This is an important feature of remote Fabry-Pérot cavity
sensors, since cavity resonance frequencies can be varied much more easily
than the frequency of particular receptor molecules. For example, this can be
done by varying the length of the cavity or by changing the angle of the
incident light. Moreover, as illustrated in Fig. 3, a single sensor could
contain an array of optical cavities with different resonance frequencies.
Another important advantage of remote Fabry-Pérot cavity sensors is that they
do not need to be adjustable, since they do not need to be stabilised in
direct contact with the target molecules. This means, they can be fabricated
more easily, for example by integrating them into optical fibers [24, 25, 26,
23] while accompanying them by integrated mode-matching optics [27].

This paper contains five sections. Section II reviews the optical properties
of Fabry-Pérot cavities. Afterwards, in Section III, we calculate the overall
reflection rates of mirror arrays which contain at least three mirrors with
the help of a scattering matrix approach [28, 29]. In Section IV, we use the
results obtained in Section III to predict the overall reflection rate of the
proposed remote Fabry-Pérot cavity sensor in Fig. 2 and study the dependence
of this rate on molecule concentrations. Finally, we summarise our findings in
Section V.

##  II The reflection rates of Fabry-Pérot cavities

To be more realistic in our predictions, we consider in the following
asymmetric mirrors with coherent light absorption and allow the media on both
sides of an interface to have a different refractive index. For simplicity, we
only consider light propagating along the $x$-axis. To introduce the notation
for studying light scattering by Fabry-Pérot cavities, we first have a closer
look at the case of a single mirror. As illustrated in Fig. 4, we assume that
the mirror is in contact with air on one side and with a dielectric medium
with a refractive index $n\neq 1$ on the other.

Figure 4: [Colour online] Schematic view of an asymmetric mirror interface
with coherent light absorption. The mirror consists of a reflecting layer
which may be covered on both sides by thin layers of absorbing material. On
the right-hand side, it is attached to a dielectric medium with a refractive
index $n_{1}\neq 1$. On the left, the mirror interface borders on air with a
refractive index $n_{0}=1$. Since light approaching the mirror from different
sides might experience different absorption rates, the overall reflection and
transmission rates of the mirror interface, $r_{a}^{(1)}$, $r_{b}^{(1)}$,
$t_{a}^{(1)}$ and $t_{b}^{(1)}$, are in general not the same, even when
referring to the case with the mirror is placed in air.

###  II.1 The electromagnetic field in air and in a dielectric medium

A change in refractive index alters the electric field amplitudes and the
frequencies of incoming wave packets. However, in this subsection, we show
that it is possible to simply ignore this effect as long as we are only
interested in overall reflection rates. The reason for this is that we can
always replace an experimental setup which contains a medium with one that
only contains air. The predicted interference effects are the same in both
cases as long as the dimensions of the system are changed accordingly. Not
having to pay attention to refractive index changes simplifies the analysis in
the following sections, but we will need to keep in mind that all (complex)
reflection and transmission rates in this paper refer to the case where
mirrors are placed in the air.

As usual, we describe the dynamics of the electromagnetic field in a
dielectric medium with permittivity $\varepsilon$ and permeability $\mu$ in
the absence of any charges and currents by Maxwell’s equations. For light
propagating along the $x$ axis, these predict that

$\displaystyle{\partial^{2}\over\partial x^{2}}E(x,t)$ $\displaystyle=$
$\displaystyle\varepsilon\mu\,{\partial^{2}\over\partial t^{2}}E(x,t)\,,$ (1)

where $E(x,t)$ denotes the electric field amplitude at position $x$ and time
$t$. From classical electrodynamics, we know that the basic solutions of
Maxwell’s equations are plane travelling waves which can be superposed to form
wave packets of any shape that travel at the speed of light $c$ [30],

$\displaystyle c$ $\displaystyle=$ $\displaystyle(\varepsilon\mu)^{-1/2}\,.$
(2)

Suppose $s=\pm 1$ indicates the respective direction of propagation,
$\lambda=1,2$ denotes the polarisation and $k$ is a positive wave number. Then
the basic solutions of Maxwell’s equations for the electric field amplitudes
$E_{sk\lambda}(x,t)$ for given parameters $(s,k,\lambda)$ can be written as

$\displaystyle E_{sk\lambda}(x,t)$ $\displaystyle=$ $\displaystyle E_{0}\,{\rm
e}^{{\rm i}k(x-sct)}+{\rm c.c.}$ (3)

In the following, we refer to $E_{0}$ as the complex electric field amplitude
for left- and for right-moving light. In the case of air, we have
$\varepsilon=\varepsilon_{0}$ and $\mu=\mu_{0}$. However, in a general
dielectric medium, the refractive index $n$,

$\displaystyle n$ $\displaystyle=$
$\displaystyle(\varepsilon\mu/\varepsilon_{0}\mu_{0})^{1/2}\,,$ (4)

is different from $n=1$. In general, the electric field $E(x,t)$ are
superpositions of the above electric field amplitudes $E_{sk\lambda}(x,t)$.

For light propagating along the $x$-axis, we therefore obtain the equivalence
relation [31]

$\displaystyle E_{\rm med}(x,t)$ $\displaystyle=$
$\displaystyle(n^{3}\varepsilon_{0}/\varepsilon)^{1/2}\,E_{\rm air}(nx,t)\,.$
(5)

Using Eq. (1), it is relatively easy to check that if $E_{\rm air}(x,t)$
solves Maxwell’s equations in air, then $E_{\rm med}(x,t)$ in Eq. (5) solves
Maxwell’s equations in a dielectric medium and vice versa. Moreover, suppose
$E_{\rm med}(x,t)$ describes the electric field in a medium of length $L$ and
with an area $A$ around the $x$-axis, while $E_{\rm air}(x,t)$ describes the
electric field in a volume of air of length $nL$ and with an area $n^{2}A$
around the $x$ axis. Then one can show, using Eq. (5), that

$\displaystyle A\int_{0}^{L}{\rm d}x\,\varepsilon\,E_{\rm med}(x,t)^{2}$
$\displaystyle=$ $\displaystyle A\int_{0}^{L}{\rm
d}L\,n^{3}\varepsilon_{0}\,E_{\rm air}(nx,t)^{2}$ (6) $\displaystyle=$
$\displaystyle n^{2}A\int_{0}^{nL}{\rm d}x\,\varepsilon_{0}\,E_{\rm
air}(x,t)^{2}\,.~{}~{}~{}$

This means the factor on the right-hand side of Eq. (5) has been chosen such
that the electric field energy is exactly the same in both cases.

###  II.2 The reflection and transmission rates of a single mirror

The observations in the previous subsection allows us to model the effect of
the two-sided semitransparent mirror in Fig. 4 by simply replacing it with an
analogous mirror placed in air. In the following, we denote the (complex)
transmission and reflection rates of this effective mirror by $t_{a}^{(1)}$,
$t_{b}^{(1)}$, $r_{a}^{(1)}$ and $r_{b}^{(1)}$, respectively. The superscript
(1) helps to distinguish these rates from the rates of other setups with more
than one mirror present. The reflection and transmission rates for light
approaching the mirror from different directions are in general not the same.
As we shall see below, they differ in general by a phase factor. In the
presence of absorption, they differ also in size.

In the following, we describe light scattering by a semi-transparent mirror,
and later also by a Fabry-Pérot cavity and other mirror arrays, by time-
independent operators. Suppose $E_{i}^{\rm in}$ and $E_{i}^{\rm out}$ with
$i=a,b$ are the complex electric field amplitudes of the incoming and of the
outgoing laser light on both sides of an interface which has been placed at
$x=0$. Then, by definition,

$\displaystyle~{}~{}E_{a}^{\rm out}\,{\rm e}^{{\rm i}k(x+ct)}+{\rm c.c.}$ (7)
$\displaystyle=$ $\displaystyle r^{(1)}_{a}E_{a}^{\rm in}\,{\rm e}^{{\rm
i}k(-x-ct)}+t^{(1)}_{b}E_{b}^{\rm in}\,{\rm e}^{{\rm i}k(x+ct)}+{\rm c.c.}\,,$
$\displaystyle~{}~{}E_{b}^{\rm out}\,{\rm e}^{{\rm i}k(x-ct)}+{\rm c.c.}$
$\displaystyle=$ $\displaystyle t^{(1)}_{a}E_{a}^{\rm in}\,{\rm e}^{{\rm
i}k(x-ct)}+r^{(1)}_{b}E_{b}^{\rm in}\,{\rm e}^{{\rm i}k(-x+ct)}+{\rm c.c.}$

This applies for all times $t$ when

$\displaystyle\left(\begin{array}[]{c}E^{\rm out}_{a}\\\ E^{{\rm
out}*}_{b}\end{array}\right)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}r_{a}^{(1)*}&t_{b}^{(1)}\\\
t_{a}^{(1)*}&r_{b}^{(1)}\end{pmatrix}\left(\begin{array}[]{c}E^{{\rm
in}*}_{a}\\\ E^{\rm in}_{b}\end{array}\right)\,.$ (12)

The energy of the incoming and of the outgoing light are only the same when

$\displaystyle|E^{\rm out}_{a}|^{2}+|E^{\rm out}_{b}|^{2}$ $\displaystyle=$
$\displaystyle|E^{\rm in}_{a}|^{2}+|E^{\rm in}_{b}|^{2}\,.$ (13)

This equation only holds for all possible electric field amplitudes when

$\displaystyle\big{|}r_{a}^{(1)}\big{|}^{2}+\big{|}t_{a}^{(1)}\big{|}^{2}=\big{|}r_{b}^{(1)}\big{|}^{2}+\big{|}t_{b}^{(1)}\big{|}^{2}$
$\displaystyle=$ $\displaystyle 1\,,$ $\displaystyle
t_{b}^{(1)}r_{a}^{(1)}+t_{a}^{(1)}r_{b}^{(1)}$ $\displaystyle=$ $\displaystyle
0\,.$ (14)

Hence the phases of the above scattering matrix elements need to be chosen
carefully for energy to be conserved. For example, we could assume that

$\displaystyle r_{b}^{(1)}=-r_{a}^{(1)}\,,~{}~{}t_{b}^{(1)}=t_{a}^{(1)}\,.$
(15)

In the presence of absorption, the energy of the outgoing light must be
smaller than the energy of the incoming light. In this case, the equal sign in
Eq. (13) is replaced by a smaller-equal sign and the reflection and
transmission rates can assume a wider range of values [32, 34, 33]. Mapping
the above outgoing electric field amplitudes onto the corresponding amplitudes
in a medium can be done with the help of Eq. (3).

###  II.3 Fabry-Pérot cavities

Next, we consider two parallel semitransparent mirrors, $M_{1}$ and $M_{2}$,
separated by a distance $L_{0}$. As illustrated in Fig. 5, we denote the
reflection and transmission rates of these mirrors in air by $r_{i}^{(2)}$ and
$t_{i}^{(2)}$ respectively with $i=a,b,c,d$. To determine the effect of both
mirrors, we now need to take into account that the complex electric field
amplitudes $E^{\rm in}_{i}$ with $i=a,d$ of monochromatic light accumulate
phase factors ${\rm e}^{\pm{\rm i}\phi_{0}}$ with

$\displaystyle\phi_{0}=n_{0}L_{0}k=n_{0}L_{0}\omega/c_{0}$ (16)

when travelling the length of the cavity. Here $n_{0}=1$ and $k$ and $\omega$
denote the wave number and the frequency of the incoming light and $c_{0}$ is
the speed of light in air. Which sign applies depends on the respective
direction of travel.

Figure 5: [Colour online] Schematic view of a Fabry-Pérot cavity which
consists of two mirrors, M1 and M2, with a distance $L_{0}$ between them. On
its right-hand side, the cavity borders with a medium with a refractive index
$n_{1}\neq 1$. All other spaces are filled with air. As in Fig. 4, the
$r_{i}^{(2)}$ and $t_{i}^{(2)}$ denote transmission and reflection rates,
while the $E_{i}^{\rm in}$ and $E_{i}^{\rm out}$ with $i=a,b,c,d$ denote
complex electric field amplitudes.

Suppose light approaches the Fabry-Pérot cavity only from the right (i.e.
$E^{\rm in}_{a}=0$) and $E^{\rm out}_{a}$ and $E^{\rm out}_{a}$ are the
complex electric field amplitudes of outgoing light. Then, by definition, we
now have

$\displaystyle E_{d}^{\rm out}\,{\rm e}^{{\rm i}k(x-ct)}+{\rm c.c.}$ (17)
$\displaystyle=$ $\displaystyle r^{(2)}_{d}E_{d}^{\rm in}\,{\rm e}^{{\rm
i}k(2L_{0}-x+ct)}+t^{(2)}_{c}t^{(2)}_{d}r^{(2)}_{b}$
$\displaystyle\times\sum_{N=0}^{\infty}\left(r^{(2)}_{b}r^{(2)}_{c}\right)^{N}E_{d}^{\rm
in}\,{\rm e}^{{\rm i}k(-x-2NL_{0}+ct)}+{\rm c.c.}\,,~{}~{}$

if the two mirrors of the Fabry-Pérot cavity are placed at $x=0$ and at
$x=L_{0}$. This equation holds at all times $t$ when

$\displaystyle E_{d}^{{\rm out}*}$ $\displaystyle=$
$\displaystyle\Bigg{[}r^{(2)}_{d}{\rm e}^{2{\rm
i}\phi_{0}}+t^{(2)}_{c}t^{(2)}_{d}r^{(2)}_{b}$ (18)
$\displaystyle\times\sum\limits_{N=0}^{\infty}\left(r^{(2)}_{b}r^{(2)}_{c}{\rm
e}^{-2{\rm i}\phi_{0}}\right)^{N}\Bigg{]}E_{d}^{\rm in}\,.$

Having a look also at other cases, we therefore find that

$\displaystyle\left(\begin{array}[]{c}E^{\rm out}_{a}\\\ E^{{\rm
out}*}_{d}\end{array}\right)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}S^{(2)}_{11}&S^{(2)}_{12}\\\
S^{(2)}_{21}&S^{(2)}_{22}\end{pmatrix}\left(\begin{array}[]{c}E^{{\rm
in}*}_{a}\\\ E^{\rm in}_{d}\end{array}\right)$ (23)

with the scattering matrix elements

$\displaystyle S^{(2)*}_{11}$ $\displaystyle=$ $\displaystyle
r_{a}^{(2)}+t_{a}^{(2)}t_{b}^{(2)}r_{c}^{(2)}{\rm e}^{2{\rm
i}\phi_{0}}\sum\limits_{N=0}^{\infty}\left(r_{b}^{(2)}r_{c}^{(2)}{\rm
e}^{2{\rm i}\phi_{0}}\right)^{N},~{}~{}~{}$ $\displaystyle S^{(2)}_{12}$
$\displaystyle=$ $\displaystyle
t^{(2)}_{b}t^{(2)}_{d}\sum\limits_{N=0}^{\infty}\left(r^{(2)}_{b}r^{(2)}_{c}{\rm
e}^{-2{\rm i}\phi_{0}}\right)^{N},$ $\displaystyle S^{(2)*}_{21}$
$\displaystyle=$ $\displaystyle
t^{(2)}_{a}t^{(2)}_{c}\sum\limits_{N=0}^{\infty}\left(r^{(2)}_{b}r^{(2)}_{c}{\rm
e}^{2{\rm i}\phi_{0}}\right)^{N},$ $\displaystyle S^{(2)}_{22}$
$\displaystyle=$ $\displaystyle r^{(2)}_{d}{\rm e}^{2{\rm
i}\phi_{0}}+t^{(2)}_{c}t^{(2)}_{d}r^{(2)}_{b}\sum\limits_{N=0}^{\infty}\left(r^{(2)}_{b}r^{(2)}_{c}{\rm
e}^{-2{\rm i}\phi_{0}}\right)^{N}.$

Since the factors $r_{i}^{(2)}r_{j}^{(2)}$ in this equation are in general
smaller than one, the above sums can be simplified using the geometric series
equation. Doing so, we find that

$\displaystyle S^{(2)*}_{11}$ $\displaystyle=$ $\displaystyle
r_{a}^{(2)}+\frac{t_{a}^{(2)}t_{b}^{(2)}r_{c}^{(2)}{\rm e}^{2{\rm
i}\phi_{0}}}{1-r_{b}^{(2)}r_{c}^{(2)}{\rm e}^{2{\rm i}\phi_{0}}}\,,$
$\displaystyle S^{(2)}_{12}$ $\displaystyle=$
$\displaystyle\frac{t^{(2)}_{b}t^{(2)}_{d}}{1-r^{(2)}_{b}r^{(2)}_{c}{\rm
e}^{-2{\rm i}\phi_{0}}}\,,$ $\displaystyle S^{(2)*}_{21}$ $\displaystyle=$
$\displaystyle\frac{t^{(2)}_{a}t^{(2)}_{c}}{1-r^{(2)}_{b}r^{(2)}_{c}{\rm
e}^{2{\rm i}\phi_{0}}}\,,$ $\displaystyle S^{(2)}_{22}$ $\displaystyle=$
$\displaystyle r^{(2)}_{d}{\rm e}^{2{\rm
i}\phi_{0}}+\frac{t^{(2)}_{c}t^{(2)}_{d}r^{(2)}_{b}}{1-r^{(2)}_{b}r^{(2)}_{c}{\rm
e}^{-2{\rm i}\phi_{0}}}\,.$ (25)

In the absence of absorption and gain, the energy of the incoming and of the
outgoing light must be the same. In analogy to Eq. (13), this now applies when

$\displaystyle|E^{\rm out}_{a}|^{2}+|E^{\rm out}_{d}|^{2}$ $\displaystyle=$
$\displaystyle|E^{\rm in}_{a}|^{2}+|E^{\rm in}_{d}|^{2}\,.$ (26)

Substituting Eq. (23) into this equation, we see that the scattering matrix
elements of the Fabry-Pérot cavity must be such that

$\displaystyle\big{|}S^{(2)}_{11}\big{|}^{2}+\big{|}S^{(2)}_{21}\big{|}^{2}=\big{|}S^{(2)}_{12}\big{|}^{2}+\big{|}S^{(2)}_{22}\big{|}^{2}$
$\displaystyle=$ $\displaystyle 1\,,$ $\displaystyle
S^{(2)*}_{11}S^{(2)}_{12}+S^{(2)*}_{21}S^{(2)}_{22}$ $\displaystyle=$
$\displaystyle 0\,,$ (27)

in analogy to Eq. (II.2). Because of Eq. (II.3), these conditions now only
hold when all transmission and reflection rates are real, when $r^{(2)2}_{i}$
and $t^{(2)2}_{i}$ add up to one for all $i$ and when

$\displaystyle
t^{(2)}_{a}=t^{(2)}_{b}\,,~{}~{}t^{(2)}_{c}=t^{(2)}_{d}\,,~{}~{}r^{(2)}_{b}=-r^{(2)}_{a}\,,~{}~{}r^{(2)}_{d}=-r^{(2)}_{c}\,.$

Suppose laser light enters the Fabry-Pérot cavity in Fig. 5 only from the left
and $E^{\rm in}_{d}=0$. Using the above equations, one can then show the
overall reflection rate $R^{(2)}(\omega)$ in the absence of absorption is
simply given by [35]

$\displaystyle R^{(2)}(\omega)$ $\displaystyle=$ $\displaystyle|E^{\rm
out}_{a}|^{2}/|E^{\rm in}_{a}|^{2}$ (29) $\displaystyle=$
$\displaystyle\big{|}S^{(2)}_{11}\big{|}^{2}=\left|{r_{a}^{(2)}+r_{c}^{(2)}{\rm
e}^{2{\rm i}\phi_{0}}\over 1+r_{a}^{(2)}r_{c}^{(2)}{\rm e}^{2{\rm
i}\phi_{0}}}\right|^{2}\,.$

When the distance $L_{0}$ tends to zero, the cavity mirrors turn into a single
mirror interface and $\phi_{0}=0$. For symmetry reasons and since we do not
want $R^{(2)}(\omega)$ to assume a minimum in a case when there is essentially
only a single mirror, the reflection rates $r^{(2)}_{a}$ and $r^{(2)}_{c}$
need to have the same sign. Again, in the presence of absorption, a wider
range of mirror parameters can be taken into account.

Figure 6: [Colour online] The dependence of the overall reflection rate
$R^{(2)}(\omega)$ in Eq. (29) on the on the frequency $\omega$ of the incoming
light. Here, we consider symmetric mirrors without absorption and choose all
reflection and transmission rates as suggested in Eq. (II.3). In addition, we
assume that $|r_{a}^{(2)}|^{2}=|r_{c}^{(2)}|^{2}=0.36$ (blue), $0.49$ (red),
$0.64$ (yellow) and $0.81$ (purple). The figure shows the typical reflection
spectrum of a Fabry-Pérot cavity. At the resonance frequency of the resonator,
$R^{(2)}(\omega)=0$, independent of the reflection rates of the two mirrors.
Increasing the mirror reflection rates, increases the quality factor $Q$ of
the cavity and results in a narrower downwards peak in the cavity resonance
fluorescence spectrum $R(\omega)$.

Fig. 6 shows the overall reflection rate $R^{(2)}(\omega)$ in Eq. (29) as a
function of $\omega$ for different symmetric mirrors without absorption. When
$r_{a}^{(2)}$ and $r_{c}^{(2)}$ are the same, $R^{(2)}(\omega)=0$ when
$\cos(2\phi_{0})=-1$. This applies, for example, when $\phi_{0}$ equals $\pi$
and the length $L_{0}$ of the cavity equals half the wavelength of the
incoming light. In general, these minima occur at angles $\phi_{0}$ which are
exactly $2\pi$ apart. As expected, the spectral response is sharply peaked
about the cavity resonance frequencies when the mirror reflections rates
$|r_{a}^{(2)}|$ and $|r_{c}^{(2)}|$ are close to one. Lower reflections rates
increase the line width of the reflection spectrum. Absorption moreover
decreases the amplitude of the reflection peak but the general shape of the
curves remains the same.

##  III The overall reflection rates of different mirror arrays

In this Section, we study the effect of additional mirrors on the reflection
rate $R^{(2)}(\omega)$ of the Fabry-Pérot cavity in Fig. 5. In the following,
we are especially interested in the case where a relatively large collection
of tiny, randomly-distributed mirrors is placed behind the resonator.

Figure 7: [Colour online] Schematic view of a three-mirror system which
contains two mirrors $M_{1}$ and $M_{2}$ separated by a distance of $L_{0}$
and a third mirror, $M_{3}$, a distance of $L_{1}$ away from the
$M_{1}$-$M_{2}$ cavity. In a realistic scenario, the setup might be attached
to a dielectric medium with a refractive index $n\neq 1$. The figure also
shows the relevant electric field amplitudes $E^{\rm in}_{i}$ and $E^{\rm
out}_{i}$ with $i=a,b,c,d$ near the relevant mirror interfaces. However,
notice that these reflect the case where the medium is replaced by air, as
described in Section II.1.

###  III.1 The overall reflection rates of three-mirror systems

However, first we have a closer look at the three-mirror system with mirrors
$M_{1}$, $M_{2}$ and $M_{3}$ in Fig. 7. To analyse their optical response, we
first replace the mirrors $M_{1}$ and $M_{2}$ by a single effective mirror and
denote the reflection and transmission rates of this effective mirror by
$r^{(3)}_{i}$ and $t^{(3)}_{i}$ with $i=a,b$.111The superscript (3) indicates
that these rates describe an effective mirror in a three-mirror setup. Suppose
the mirrors $M_{1}$ and $M_{2}$ have the same optical properties as the
mirrors $M_{1}$ and $M_{2}$ of the Fabry-Perot cavity in Fig. 5. Then we can
use the results that we obtained in Section II.3 to show that

$\displaystyle
r^{(3)}_{a}=S^{(2)*}_{11}\,,~{}~{}t^{(3)}_{a}=S^{(2)*}_{21}\,,~{}~{}r^{(3)}_{b}=S^{(2)}_{22}\,,~{}~{}t^{(3)}_{b}=S^{(2)}_{12}$

with $S^{(2)}_{11}$, $S^{(2)}_{12}$, $S^{(2)}_{21}$ and $S^{(2)}_{22}$ given
in Eq. (23). Moreover, we denote the reflection rate of $M_{3}$ in the
following by $r_{c}^{(3)}$. As in the previous section, $k$ and $\omega$
denote the wave number and the frequency of the incoming laser light.

Next we notice that the effective mirror and the additional mirror $M_{3}$
form a Fabry-Pérot cavity of length $L_{1}$ which contains a medium with a
refractive index $n_{1}\neq 1$ (cf. Fig. 7). To simplify our discussion, we
further replace this Fabry-Pérot cavity in the following by a Fabry-Pérot
cavity of length $n_{1}L_{1}$ which contains air (cf. discussion in Section
II.1). Now suppose $E^{\rm in}_{i}$ and $E^{\rm out}_{i}$ with $i=a,d$ denote
the incoming and outgoing (complex) electric field amplitudes near the
respective mirror surface in air and the phase $\phi_{1}$ is given by

$\displaystyle\phi_{1}=n_{1}L_{1}k=n_{1}L_{1}\omega/c_{0}\,.$ (31)

Combining Eqs. (II.3) and (III.1), one can now show that

$\displaystyle\left(\begin{array}[]{c}E^{\rm out}_{a}\\\ E^{\rm
out}_{d}\end{array}\right)=\begin{pmatrix}S^{(3)}_{11}&S^{(3)}_{12}\\\
S^{(3)}_{21}&S^{(3)}_{22}\end{pmatrix}\left(\begin{array}[]{c}E^{\rm
in}_{a}\\\ E^{\rm in}_{d}\end{array}\right)$ (36)

with the scattering matrix element $S^{(3)}_{11}$ given by

$\displaystyle S^{(3)*}_{11}$ $\displaystyle=$ $\displaystyle
S^{(2)*}_{11}+\frac{S^{(2)}_{12}S^{(2)*}_{21}r_{c}^{(3)}{\rm e}^{2{\rm
i}\phi_{1}}}{1-S^{(2)}_{22}r_{c}^{(3)}\,{\rm e}^{2{\rm i}\phi_{1}}}\,.$ (37)

When the setup is only driven by monochromatic laser light from the left, the
reflection rate $R^{(3)}(\omega)$ of the three-mirror interferometer therefore
equals

$\displaystyle
R^{(3)}(\omega)=\big{|}S^{(3)}_{11}\big{|}^{2}=\left|S^{(2)*}_{11}+{S^{(2)}_{12}S^{(2)*}_{21}r_{c}^{(3)}{\rm
e}^{2{\rm i}\phi_{1}}\over 1-S^{(2)}_{22}r_{c}^{(3)}\,{\rm e}^{2{\rm
i}\phi_{1}}}\right|^{2}.~{}~{}~{}$ (38)

As long as the reflection rate of the third mirror is relatively small, we
expect that this rate resembles the reflection rate $R^{(2)}(\omega)$ of a
Fabry-Pérot cavity relatively well. How much it changes in the presence of the
third mirror $M_{3}$ depends, for example, on the frequency of the incoming
light and on the distance $L_{1}$ between $M_{2}$ and $M_{3}$.

###  III.2 The effect of a randomly positioned third mirror

In this paper we are especially interested in the case where the distance
$L_{1}$ of the third mirror varies randomly over a range that is much larger
than the wavelength of the incoming laser light. In this case, the
corresponding reflection rate $\overline{R^{(3)}(\omega)}$ of the three-mirror
system in Fig. 7 depends no longer on $L_{1}$. It can be obtained by averaging
over all possible values of $\phi_{1}$,

$\displaystyle\overline{R^{(3)}(\omega)}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}{\rm
d}\phi_{1}\,R^{(3)}(\omega)\,.$ (39)

A closer look at Eq. (38) shows that we do not need to know the phase of the
reflection rates $r_{c}^{(3)}$ of the additional mirror, since the dependence
on this phase disappears when the above average is taken.

Figure 8: [Colour online] To illustrate the effect of randomly-positioned
mirrors on the overall reflection rate of a Fabry-Pérot cavity, the figure
shows the dependence of $\overline{R^{(3)}(\omega)}-R^{(2)}(\omega)$ in Eq.
(42) on the frequency $\omega$ of the incoming light. Here
$|r_{a}^{(2)}|^{2}=|r_{c}^{(2)}|^{2}=0.81$, while $|r_{c}^{(3)}|^{2}=0.01$
(blue), $0.04$ (red), $0.07$ (yellow) and $0.1$ (purple). The difference is a
small and narrow upwards peak of a certain full width at half the maximum
(FWHM) and with an amplitude given by $|r_{c}^{(3)}|^{2}$.

In the absence of absorption, the mirror parameters of the Fabry-Pérot cavity
need to be in agreement with Eqs. (II.3) and (II.3). These imply for example
that $S^{(2)}_{12}=S^{(2)}_{21}$. Eq. (38) can therefore be used to show that

$\displaystyle R^{(3)}(\omega)$ $\displaystyle=$
$\displaystyle\left|S^{(2)*}_{11}+\big{|}S^{(2)}_{12}\big{|}^{2}\,r_{c}^{(3)}{\rm
e}^{2{\rm i}\phi_{1}}\sum_{n=0}^{\infty}\left(S^{(2)}_{22}\,r_{c}^{(3)}{\rm
e}^{2{\rm i}\phi_{1}}\right)^{n}\right|^{2}\,.$

Hence performing the integration in Eq. (39), we obtain the average reflection
rate

$\displaystyle\overline{R^{(3)}(\omega)}$ $\displaystyle=$
$\displaystyle\big{|}S^{(2)}_{11}\big{|}^{2}+\big{|}S^{(2)}_{12}\big{|}^{4}\cdot\big{|}r_{c}^{(3)}\big{|}^{2}\sum_{n=0}^{\infty}\left|S^{(2)}_{22}\,r_{c}^{(3)}\right|^{2n}$
(41) $\displaystyle=$
$\displaystyle\big{|}S^{(2)}_{11}\big{|}^{2}+{\big{|}S^{(2)}_{12}\big{|}^{4}\over
1-\big{|}S^{(2)}_{22}\,r_{c}^{(3)}\big{|}^{2}}\,\big{|}r_{c}^{(3)}\big{|}^{2}\,.$

From Eqs. (II.3) and (II.3), we also see that
$|S^{(2)}_{12}|^{2}=1-|S^{(2)}_{11}|^{2}$ and
$|S^{(2)}_{11}|^{2}=|S^{(2)}_{22}|^{2}$. These relations can be used to
simplify Eq. (41) into

$\displaystyle\overline{R^{(3)}(\omega)}-R^{(2)}(\omega)$ $\displaystyle=$
$\displaystyle{\big{[}1-R^{(2)}(\omega)\big{]}^{2}\over
1-R^{(2)}(\omega)\,\big{|}r_{c}^{(3)}\big{|}^{2}}\,\big{|}r_{c}^{(3)}\big{|}^{2}.~{}~{}$
(42)

For relatively small values of $r^{(3)}_{c}$, the overall reflection rates
$R^{(2)}(\omega)$ and $\overline{R^{(3)}(\omega)}$ given in Eqs. (29) and (42)
are essentially the same. Moreover, it is relatively straightforward to check
that $\overline{R^{(3)}(\omega)}=\big{|}r_{c}^{(3)}\big{|}^{2}$ at the
resonance frequency of the cavity, where all incoming laser light is normally
transmitted and $R^{(2)}(\omega)=0$.

Fig. 8 shows the difference $\overline{R^{(3)}(\omega)}-R^{(2)}(\omega)$ as a
function of the frequency $\omega$ for different values of $|r_{c}^{(3)}|$ and
illustrates clearly that the presence of a third randomly-positioned mirror
adds small upwards peaks to the usual minima of the reflection spectrum of the
Fabry-Pérot cavity. This is not surprising, since, in the absence of
absorption, $R^{(2)}(\omega)$ is zero at the cavity resonance frequencies.
Hence anything that changes the amount of interference between the mirrors can
only have one effect, namely an increase of the overall reflection rate of the
system. As one would expect, these peaks increase in size but also get
slightly broader as $|r_{c}^{(3)}|$ increases.

###  III.3 The effect of a relatively large number of tiny, randomly-
positioned mirrors

In this final subsection, we consider the experimental setup in Fig. 9 which
contains a single Fabry-Pérot cavity as well as a relatively large collection
of $N$ tiny, identical, randomly-positioned, weakly-reflecting mirrors. In
analogy to the previous subsection, we denote the reflection rate of the
additional mirrors by $r_{c}^{(3)}$. Moreover, $\Delta A$ and $\Delta L$
denote the surface area of a single tiny mirror and the optical depth of the
sample. In addition, $\alpha$ denotes the area covered by the incoming laser
field. Hence $V=\alpha\cdot\Delta L$ is the size of the total illuminated and
randomly occupied volume.

Figure 9: [Colour online] Schematic view of an experimental setup which
contains a Fabry-Pérot cavity as well as a group of tiny mirrors which are
randomly positioned in a medium with refractive index $n\neq 1$. These
additional mirrors occupy a volume of length $\Delta L$ and covering an area
$\alpha$ which is placed some distance $L_{1}$ away from the cavity. An
incoming laser with its cross section given by $\alpha$ approaches the cavity
and the additional mirrors from a perpendicular direction. For simplicity, we
assume here that the additional mirrors only cover a relatively small
percentage of the area such that each one of them is likely to be seen by the
incoming laser light. Figure 10: [Colour online] The dependence of the overall
reflection rate $R(\omega)$ in Eq. (47) of the Fabry-Pérot cavity in Fig. 9 on
the frequency $\omega$ of the incoming light. Here the mirror parameters are
$|r_{a}^{(2)}|^{2}=|r_{c}^{(2)}|^{2}=0.81$, while $|r_{c}^{(3)}|^{2}=0.1$. In
addition, $P_{N}(\geq 1)$ equals 0.6 (blue), 0.7 (red), 0.8 (yellow) and 0.9
(purple). The graphs have again been calculated using Eqs. (29) and (42) and
shows that the presence of a relatively large number of tiny, randomly-
positioned mirrors increases the minimum of the reflection rate $R(\omega)$ of
the Fabry-Pérot cavity. Instead of zero, the minimum now equals $R_{\rm min}$
in Eq. (48).

Before calculating the reflection rate $R(\omega)$ of the experimental setup
in Fig. 9, we first consider the case with only one randomly positioned tiny
mirror present in $V$. In this case, the probability $P_{1}(1)$ of finding
this mirror within a small volume $\Delta V=\Delta A\cdot\Delta L$ equals

$\displaystyle P_{1}(1)$ $\displaystyle=$ $\displaystyle\frac{\Delta
A}{\alpha}\,.$ (43)

Hence the probability $P_{1}(0)$ for this $\Delta V$ not to contain the mirror
is

$\displaystyle P_{1}(0)$ $\displaystyle=$ $\displaystyle 1-\frac{\Delta
A}{\alpha}\,.$ (44)

Now suppose there are $N$ identical tiny reflectors in the volume $V$. In this
case, the probability $P_{N}(0)$ for not finding a tiny mirror in a given
volume $\Delta V$ becomes

$\displaystyle P_{N}(0)$ $\displaystyle=$ $\displaystyle\left(1-\frac{\Delta
A}{\alpha}\right)^{N}\,.$ (45)

Hence the overall reflection rate $R(\omega)$ of the experimental setup in
Fig. 9 is given by

$\displaystyle R(\omega)$ $\displaystyle=$ $\displaystyle
P_{N}(0)\,R^{(2)}(\omega)+\left(1-P_{N}(0)\right)\,\overline{R^{(3)}(\omega)}\,.~{}~{}$
(46)

This applies since $P_{N}(0)$ is also the probability for a tiny laser beam
with cross section $\Delta A$ not to encounter a tiny mirror. In this case,
the reflection rate equals $R^{(2)}(\omega)$. Moreover, $1-P_{N}(0)$ is the
probability that the thin laser beam meets a randomly positioned tiny mirror
and that its reflection rate equals $\overline{R^{(3)}(\omega)}$.

Next we notice that $P_{N}(\geq 1)=1-P_{N}(0)$ is the probability for at least
one mirror present in a given volume $\Delta V$. This observation allows us to
write Eq. (46) as

$\displaystyle R(\omega)-R^{(2)}(\omega)$ $\displaystyle=$ $\displaystyle
P_{N}(\geq
1)\left[\overline{R^{(3)}(\omega)}-R^{(2)}(\omega)\right]\,.~{}~{}~{}$ (47)

An analytical expression for $\overline{R^{(3)}(\omega)}-R^{(2)}(\omega)$ can
be found in Eq. (41). For very small reflection rates $r_{c}^{(3)}$, the
reflection rates $R(\omega)$ and $R^{(2)}(\omega)$ are essentially the same.
However, as illustrated in Fig. 10, the reflection rate no longer becomes zero
at the cavity resonance frequencies. Instead, the minimum of the curves is now
given by

$\displaystyle R_{\rm min}$ $\displaystyle=$ $\displaystyle P_{N}(\geq
1)\,\big{|}r_{c}^{(3)}\big{|}^{2}\,.$ (48)

This rate increases, as the reflectivity and the number $N$ of the tiny
mirrors increases until saturation sets in and $P_{N}(\geq 1)$ tends to one.
Notice also that $R_{\rm min}$ is the reflection rate that the tiny, randomly-
positioned mirrors would have in the absence of the Fabry-Pérot cavity. The
main purpose of the resonator is to filter one frequency. As we shall see
below, we can deduce additional information about the optical properties of
the mirrors from the shape of $R(\omega)$ in the presence of the resonator.

##  IV Remote Fabry-Pérot cavity spectroscopy

Figure 11: [Colour online] An alternative view on the remote Fabry-Pérot
cavity sensor in Fig. 2. Here the target molecules are randomly distributed
within a volume $V$ of length $\Delta L$ a distance $L_{1}$ away from the
resonator. Moreover, $\Delta A$ and $\alpha$ denote the cross section of a
single molecule and $\alpha$ is the area that the incoming laser light
excites. Our hypothesis here is that the molecules closely resemble tiny
semitransparent mirrors which suggests the same optical response of the above
experimental setup and the experimental setup shown in Fig. 9.

In this final section, we utilise the above-described interference effects for
label-free sensing. Fig. 11 shows an alternative schematic view of the remote
Fabry-Pérot cavity sensor in Fig. 2. Similar to the experimental setup in Fig.
9, the sensor contains a laser-driven Fabry-Pérot cavity. Its transmitted
light approaches the target molecules which can be located some distance away
from the sensor from the left. Our measurement signal is the overall
reflection rate $R(\omega)$ of the device. A comparison of the experimental
setups in Figs. 9 and 11 shows that both have the same optical response when:

  1. 1.

The target molecules closely resemble tiny, semitransparent mirrors which
reflect at least some of the incoming light back into the Fabry-Pérot cavity
without changing its frequency. This applies to a very good approximation, if
the frequency of the laser falls within their resonance fluorescence spectrum.
As we have seen above, it does not matter, if the reflected light accumulates
a random phase in the reflection process. It anyway accumulates a random phase
due to the randomness of the position of every molecule within the sample.

  2. 2.

Moreover, the environment surrounding the target molecules should be mostly
transparent to the incoming light. If the environment reflects some of the
incoming light even in the absence of the target molecules, the sensor needs
to be more sensitive and needs to be more carefully calibrated before
measurements can be performed.

  3. 3.

The target molecules are randomly distributed within the finite volume $V$, as
it applies for example naturally when they are dissolved in a liquid.

Under these conditions, we can use the reflection rate $R(\omega)$ which we
derived in the previous section to obtain the overall reflection rate
$R(\omega)$ of the remote Fabry-Pérot cavity sensor in Fig. 11 in the presence
of analytes. All we need to do is to replace the variables $N$, $\Delta A$ and
$r_{c}^{(3)}$ by the total number of target molecules in the illuminated
sample, the average scattering cross section of a single target molecule and
its reflection rate, respectively.

###  IV.1 Optical signatures of the presence of target molecules

Section III suggests that the reflection of light from the target molecules
back into the resonator adds small upwards peaks to the minima of the overall
reflection rate $R(\omega)$ of the remote Fabry-Pérot cavity sensor. This can
be detected, since the amplitude of the minima is no longer zero but equals
instead $R_{\rm min}$ in Eq. (48). To produce visible peaks, the frequency of
the driving laser light must lie within the resonance fluorescence spectrum of
the target molecules. If the incoming laser light is not in resonance, the
target molecules become transparent and their reflection rate $r_{c}^{(3)}$
becomes zero. The specificity of remote Fabry-Pérot cavity sensors comes from
the fact that the reflection rate $r_{c}^{(3)}$ depends strongly on the atomic
level structure of the target molecules.

As we can see from Figs. 8 and 10, the amplitude of these additional peaks is
in general relatively small. In addition to measuring $R_{\rm min}$, we
therefore recommend to plot the difference $R(\omega)-R^{(2)}(\omega)$ between
the measured signal $R(\omega)$ and the overall reflection rate
$R^{(2)}(\omega)$ of the Fabry-Pérot cavity in the absence of target molecule
concentrations. This difference has a distinct shape. From Eqs. (42) and (47),
we see that the so-called full width at half the maximum (FWHM) of the upwards
peak near a cavity resonance frequency depends only on the molecule reflection
rate $|r_{c}^{(3)}|^{2}$. It is therefore possible to deduce
$|r_{c}^{(3)}|^{2}$ from the measurement signal and to obtain additional
information about the species of the analytes.

How much light of a given frequency and polarisation is reflected by the
target molecule depends on the strength of their dipole moments and on the
level spacings of their energy eigenstates. Every molecule has its own unique
resonance fluorescence spectrum and therefore also its own frequency-dependent
reflection rate $r_{c}^{(3)}=r_{c}^{(3)}(\omega)$. This observation can be
exploited to further enhance the specificity of remote Fabry-Pérot cavity
sensors. For example, a sensor which simultaneously probes the response of a
sample to several laser frequencies should be enough to distinguish any
species with atomic transitions in the optical regime. One way of implementing
this idea is to incorporate an array of cavities into the sensor design, as
illustrated in Fig. 3.

###  IV.2 The dependence of reflection rates on molecule concentrations

Given the above definitions of the variables $N$, $L$ and $\alpha$, the number
density of the target molecules equals

$C=\frac{N}{V}=\frac{N}{\alpha\cdot\Delta L}$ (49)

when the particles are placed in air. (As discussed in Section II.1, a
correction is needed, if the particles are hosted in a medium with a
refractive index $n$ that is different from one.) Hence the probability
$P_{N}(\geq 1)$ in Eq. (47), which now coincides with the relative amount of
laser light that encounters at least one target molecule within the
illuminated sample, depends in general on $C$. By measuring how the overall
reflection rate $R(\omega)$ of a remote Fabry-Pérot cavity sensor changes at
the resonance frequency, it is therefore also possible to obtain information
about target molecule concentrations. As we can see from Eq. (48), this can be
done by measuring the minimum reflection rate $R_{\rm min}$ of the sensor.

For example, suppose every volume element $\Delta V=\Delta A\cdot\Delta L$
contains in general at most one target molecule. This applies to a very good
approximation if the scattering cross section $\Delta A$ of a single molecule
multiplied by the total number of molecules $N$ within the sample is much
smaller than the laser cross section $\alpha$, i.e.

$\displaystyle N\cdot\Delta A$ $\displaystyle\ll$ $\displaystyle\alpha\,.$
(50)

In this case, the probability $P_{N}(0)$ in Eq. (45) simplifies to
$P_{N}(0)=1-N\Delta A/\alpha$. Hence $P_{N}(\geq 1)=N\Delta A/\alpha$ to a
very good approximation and Eq. (47) becomes

$\displaystyle R(\omega)-R^{(2)}(\omega)$ $\displaystyle=$
$\displaystyle\left[\overline{R^{(3)}(\omega)}-R^{(2)}(\omega)\right]\frac{N\Delta
A}{\alpha}$ $\displaystyle=$
$\displaystyle\left[\overline{R^{(3)}(\omega)}-R^{(2)}(\omega)\right]\,\Delta
A\cdot\Delta L\cdot C$

up to first order in $C$. Since $\overline{R^{(3)}(\omega)}-R^{(2)}(\omega)$
in Eq. (42) does not depend on $C$, this difference depends linearly on the
molecular concentrations.

If we want to measure $C$ with accuracy, the concentration of the target
molecules should therefore not be too high. Ideally, $C$ should be such that a
relatively high percentage of the analytes sees the incoming laser light. If
the concentration $C$ becomes too high, all of the incoming laser field is
reflected by molecules and $P_{N}(\geq 1)=1$. In this case, the sensor
saturates and the overall reflection rate $R(\omega)$ depends no longer on
$C$. In general, remote Fabry-Pérot cavity sensors need to be calibrated
carefully, since $R_{\rm min}$ in Eq. (48) depends also on the molecular
scattering cross section $\Delta A$ and the the optical depth $\Delta L$ of
the sample.

##  V Conclusions

This paper takes advantage of the fact that Fabry-Pérot cavities are very
sensitive to any changes that affect the interference of light inside the
resonator. However, in contrast to intensive and extensive Fabry-Pérot cavity
sensors, we do not rely on refractive index changes or on changes of the
reflection and transmission rates of its mirrors. Instead, our main research
hypothesis is that randomly distributed atomic particles diffract laser light
in a similar fashion as tiny, randomly distributed semitransparent mirrors.
The remote Fabry-Pérot cavity sensor works since its overall reflection rate
$R(\omega)$ changes in a unique way when any of the outgoing light is
reflected back into the cavity. Here the only difference between
semitransparent mirrors and the target molecules is that reflection rates by
the latter is in general weaker and depends more strongly on the frequency and
possibly also on the polarisation of the incoming light.

In the absence of any target molecules, the reflection rate $R(\omega)$ of a
Fabry-Pérot cavity sensor assumes a minimum at the cavity resonance frequency.
For example, in the case of an ideal cavity, all incoming resonant light is
transmitted and $R(\omega)=0$. In the presence of the target molecules, this
minimum is reduced and a small upwards peak is added which can be detected.
The new minimum reflection rate $R_{\rm min}$ depends on the size of the
reflection rate $r_{c}^{(3)}$ and the concentration $C$ of the target
molecules. The strong dependence of $|r_{c}^{(3)}|$ on the resonance
fluorescence spectrum of the target molecules contributes to the selectivity
of the proposed sensing device. Moreover, the size of the upwards peak
provides information about molecular concentrations.

Although this is a theoretical paper, we expect that the proposed remote
Fabry-Pérot cavity sensor can be used for the non-invasive, label-free
detection of molecule concentrations in a wide range of scenarios. For
example, as we have seen above, the sample which contains the target molecules
does not need to be in direct contact with the resonator. Moreover, by probing
a wide range of frequencies with an array of optical cavities, a unique
optical fingerprint of the target molecules can be obtained. However, some
conditions need to hold for remote Fabry-Pérot cavity sensors to work:

  1. 1.

Optical access to the sample which contains the molecules is required.

  2. 2.

The laser frequency and therefore also the resonance frequency of the incoming
light should lie within the resonance frequency spectrum of the molecules such
that they absorb and re-emit light at that frequency with a relatively high
rate.

  3. 3.

The concentrations of the molecules should be neither too low nor too high to
obtain a significant response without saturating the device.

  4. 4.

The positions of the target molecules should be sufficiently random in order
to remove any dependence of the sensor reflection rate $R(\omega)$ on the
exact distances between the Fabry-Pérot cavity and the molecules.

Since these requirements can be met, at least in principle, we are optimistic
that the idea which we present here will find a wide range of applications in
sensing physical, chemical and biological processes in real time, remotely and
without perturbing them. Being limited mainly by their quality factor, remote
Fabry-Pérot cavity sensors might even find applications as difficult as
monitoring molecule concentrations in the human blood [21, 22, 14, 15], if
they can be engineered and calibrated well enough to work in realistic
uncertain environments.
Acknowledgement. A. Al Ghamdi acknowledges financial support from the
Government of the Kingdom of Saudi Arabia. Moreover, this work was supported
in part by the Oxford Quantum Technology Hub NQIT (grant number EP/M013243/1)
and the EPSRC (2115757). Statement of compliance with EPSRC policy framework
on research data: This publication is theoretical work that does not require
supporting research data.

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