# Achievements and perspectives of optical Fiber Fabry-Perot Cavities

H. Pfeifer<EMAIL_ADDRESS>Institute of Applied Physics,
University of Bonn, Wegelerstr. 8, 53115 Bonn, Germany  L. Ratschbacher
Institute of Applied Physics, University of Bonn, Wegelerstr. 8, 53115 Bonn,
Germany  Intelligent Wireless Systems, Silicon Austria Labs, Altenberger
Strasse 66c, Linz, Austria  J. Gallego  Institute of Applied Physics,
University of Bonn, Wegelerstr. 8, 53115 Bonn, Germany  current address:
Fremont, CA, USA  C. Saavedra  Institute of Applied Physics, University of
Bonn, Wegelerstr. 8, 53115 Bonn, Germany  División de Ciencias e Ingenierias
Universidad de Guanajuato, Mexico  A. Faßbender  Institute of Physics,
University of Bonn, Nussallee 12, 53115 Bonn, Germany  A. von Haaren
Institute of Physics, University of Bonn, Nussallee 12, 53115 Bonn, Germany
W. Alt  Institute of Applied Physics, University of Bonn, Wegelerstr. 8, 53115
Bonn, Germany  S. Hofferberth  Institute of Applied Physics, University of
Bonn, Wegelerstr. 8, 53115 Bonn, Germany  M. Köhl  Institute of Physics,
University of Bonn, Nussallee 12, 53115 Bonn, Germany  S. Linden  Institute of
Physics, University of Bonn, Nussallee 12, 53115 Bonn, Germany  D. Meschede
Institute of Applied Physics, University of Bonn, Wegelerstr. 8, 53115 Bonn,
Germany

###### Abstract

Fabry-Perot interferometers have stimulated numerous scientific and technical
applications ranging from high resolution spectroscopy over metrology, optical
filters, to interfaces of light and matter at the quantum limit and more. End
facet machining of optical fibers has enabled the miniaturization of optical
Fabry-Perot cavities. Integration with fiber wave guide technology allows for
small yet open devices with favorable scaling properties including mechanical
stability and compact mode geometry. These Fiber Fabry-Perot Cavities (FFPCs)
are stimulating extended applications in many fields including cavity quantum
electrodynamics, optomechanics, sensing, nonlinear optics and more.

Here we summarize the state of the art of devices based on Fiber Fabry-Perot
Cavities, provide an overview of applications and conclude with expected
further research activities.

Fiber-end cavities and Mechanical Stability and Photothermal Effects and
Quantum Interfaces

##  I Introduction

For this interim review we summarize the state of the art of Fiber Fabry-Perot
Cavities (FFPCs), i.e. Fabry-Perot resonators integrated on optical fibers. We
discuss their basic properties and applications (with focus on gaseous media)
and we conclude with an outlook into future perspectives. The article may also
serve as a resource letter through its collection of references with relevance
for FFPCs.

###  I.1 Fabry-Perot interferometers

Ever since their invention in the late 19${}^{\text{th}}$ century Fabry-Perot
interferometers (FPIs) [1] have played an important role in advancing the
resolution as well as the applications of optical spectrometers. In the
simplest case they consist of two parallel plane and partially transmitting
mirrors spaced by a length $\ell$. Multiple reflections of a light beam back
and forth between its mirrors cause interference, where for normal incidence
sequentially reflected beams have a path difference $2\ell$. Hence,
constructive interference occurs when an integer number of wavelengths
$\lambda$ covers the round trip path, $N\cdot\lambda=2\ell$ [2, 3]. The
integer number $N$ is called the order of the interferometer and is typically
large, of order $N\approx 10^{3}-10^{6}$. The path difference is cast into a
resonance condition for the frequency of a light field ($\nu=c/\lambda$) by

$\nu_{N}=N\cdot\Delta\nu_{\text{FSR}}\ ,$ (1)

where $\Delta\nu_{\text{FSR}}=c/2\ell$ is the free spectral range of the FPI
with $\ell$ the cavity length. The width of an isolated resonance is
determined by the reflectivities $R$ of the mirrors. A conveniently accessible
measure for the spectral quality of the FPI is given by the ratio of the free
spectral range and the Lorentzian line width $\Delta_{\text{1/2}}$ (FWHM) of a
single spectral line, which is called finesse ${\cal F}$. In the simplest case
of a symmetric cavity with two mirrors of identical reflectivity and
negligible losses $A\ll 1{-}R$, it reads

${\cal{F}}=\frac{\Delta_{\text{FSR}}}{\Delta_{\text{1/2}}}=\frac{\pi\sqrt{R}}{1-R}\
,$ (2)

where for the frequent situation $R\to 1$ we have ${\cal{F}}\simeq\pi/(1-R)\gg
1$.

With this, FPIs have offered the highest spectroscopic resolution in classic
optical instrumentation before the advent of laser spectroscopy, acting
essentially as narrow band optical filters with resolution
$\nu_{N}/\Delta_{\text{1/2}}=N\cdot{\cal{F}}$.

###  I.2 Fabry-Perot cavities

The light field circulates back and forth between the mirrors with a round
trip time $\tau_{\text{circ}}=1/\Delta_{\text{FSR}}$. FPIs have therefore
given birth to the notion of Fabry-Perot resonators or Fabry-Perot cavities
(FPCs), where the focus is on the properties of the internally stored light
field rather than the transmitted or reflected intensity. An initially stored
light field is attenuated as a consequence of the finite mirror transmissivity
$(1{-}R)$ and other losses $A$. For small losses and transmission, the
relaxation rate $1/\tau_{\text{cav}}$ per round trip is approximately given by
$1/\tau_{\text{cav}}=(1{-}R)/\tau_{\text{circ}}$ which translates into the
mean number of round trips

$n_{\text{round}}=\frac{\tau_{\text{cav}}}{\tau_{\text{circ}}}\simeq\frac{{\cal{F}}}{\pi}\
.$ (3)

The decay constant $\kappa$ of the amplitude of of the stored light field
(energy relaxation rate $2\kappa$) is the relevant rate describing dynamic
properties of any FPC. A spectrum of an FPC shows a Lorentzian line width
(FWHM = $\Delta_{\text{1/2}}$), which is related to the relaxation rate by
$\Delta_{\text{1/2}}=\kappa/\pi$. FPCs have strongly inspired the advent of
the laser – an FPI/FPC containing an amplifier for optical waves – rendering
one of the most important tools in contemporary science and technology.

The geometry of the coherent light fields provided by lasers are usually
described in terms of Gaussian field modes, which are extensively covered in
textbooks such as [3].

Figure 1: Basic geometric parameters of a TEM${}_{\text{00}}$ mode supported
by an optical fiber Fabry-Perot cavity (FFPC). Wavefronts of the cavity mode
field match the radii of curvature of the integrated mirrors. The coupling
strength with incoming light fields is determined by the overlap of the
travelling guided and the cavity modes.

The most widely used Gaussian mode is the so-called TEM${}_{\text{00}}$, which
is schematically shown in Fig. 1. The properties of the fundamental cavity
modes can be expressed in terms of the dimensionless cavity parameters $G_{i}$
(mirrors $i=1,2$), which are related to the mirror geometry and the cavity
length $\ell$,

$G_{i}=1-\ell/r_{i}.$ (4)

The stability condition binds the individual $G_{i}$ values according to
$0\leq G_{1}G_{2}\leq 1$ (i.e. eigensolutions of the paraxial Helmholtz
equation, see [3]), with special cases $G_{1}=G_{2}=1$ (plane-plane cavity), 0
(confocal) and -1 (concentric). A stable cavity mode shows wave fronts
matching the curvature of the mirrors with radii $r_{i}$. Most FPCs are
arranged to have the focus, the position of highest field strength, within its
volume. The waist is conveniently calculated from

$w_{0}^{2}=\frac{\ell\lambda}{\pi}\sqrt{\frac{G_{1}G_{2}(1-G_{1}G_{2})}{(G_{1}+G_{2}-2G_{1}G_{2})^{2}}}\,,$
(5)

and the resonator mode size at the position of mirror M1 is given by

$w^{2}_{M_{1}}=\frac{\ell\lambda}{\pi}\sqrt{\frac{G_{2}}{(G_{1}(1-G_{1}G_{2})}}\,,$
(6)

where $\lambda=c/\nu_{\text{cav}}$ is the resonant wavelength of the cavity.
To illustrate physical properties, the symmetric case is sufficient in most
cases. With $G=G_{1}=G_{2}$ Eq. 5 reduces to

$w_{0}^{2}=\frac{\ell\lambda}{2\pi}\sqrt{\frac{1+G}{1-G}}=\frac{\ell\lambda}{2\pi}\sqrt{\frac{r}{\ell/2}-1}\,,$
(7)

where it is easy to see that the mirrors of the FPC have to fulfill the
condition $r>\ell/2$ for their radii of curvature. The smallest waist radii
$w_{0}$ are obtained in the limit $r\to\ell/2$. In this case of the concentric
cavity, however, the paraxial approximation of Gaussian beams is no longer
valid, and diffraction limits the minimal waist to a radius of order
$w_{0}\sim\lambda/2$. For the most relevant Gaussian TEM${}_{\text{00}}$ mode
of the FPC, the resonance condition Eq.1 is slightly modified by the so-called
Guoy shift [3],

$\nu_{N}=\frac{c}{2\ell}\left(N+\frac{\arccos{(G)}}{\pi}\right).$ (8)

For all practical purposes discussed here, this small correction
($\arccos{G}/\pi\ll N$) is included in an effective cavity length
$\ell=\ell_{\text{eff}}$, which also accounts for the penetration of the
cavity field into the stack of dielectric layers of the mirror coating.

FPCs are frequently used to maximize light matter coupling, and hence atoms,
membranes, or spectroscopic samples are positioned at the waist of the cavity
mode with diameter $2w_{0}$. The field strength of the cavity field can be
considered a function of the total energy $U$ stored in the FPC. An important
measure for the ratio of the stored electromagnetic energy to the maximum
field strength is given by the mode volume $V_{\text{mode}}$. It is calculated
by spatially integrating the field distribution inside the cavity normalized
to the maximum field strength $\vec{{\cal{E}}}(w_{0})$, resulting in

$V_{\text{mode}}=\frac{\pi}{4}w_{0}^{2}\ell=V_{\lambda}\cdot\left(\frac{\ell}{\lambda}\right)^{2}\left(\frac{r}{\ell/2}-1\right)^{1/2},\
$ (9)

with minimal mode volume $V_{\lambda}=(\lambda/2)^{3}$ for a cubic cavity
resonant at wavelength $\lambda$. The total electromagnetic energy $U$ stored
in the FPC relates to the maximum field amplitude
$|\vec{{\cal{E}}}_{\text{max}}|$ usually at the waist through

$U=\frac{1}{2}\epsilon_{0}c^{2}\int_{V_{\text{cav}}}\text{d}V\,|\vec{{\cal{E}}}(\vec{r})|^{2}=\frac{1}{2}\epsilon_{0}c^{2}|\vec{{\cal{E}}}_{\text{max}}|^{2}\cdot
V_{\text{mode}}.$ (10)

##  II Fiber Fabry-Perot cavities

One route of the present evolution of photonics is the miniaturization of
optical devices. In the past FPCs have successfully been integrated into
optical ﬁbers through the inscription of Bragg mirrors directly into the glass
fiber material, serving e g. as optical ﬁlters. The Fiber Fabry-Perot Cavities
(FFPCs) introduced by [4] and discussed here are intrinsically fiber
connected, too. However, as scaled-down versions of classic FPCs with mirrors
spaced by an empty volume, they enable versatile access to the ﬁeld mode
stored in the cavity, which allows applications as outlined in Sec. III.

While the formal treatment of FFPCs is not different from the conventional
description of macroscopic FPCs outlined above in terms of Gaussian beams and
field modes, it is the limit of small typical length scales $\ell\sim
10-$1000\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, which makes FFPCs a special
limiting case.

A breakthrough of FFPCs was achieved when D. Hunger, J. Reichel and colleagues
[4, 5] developed the integration of mirrors with high optical quality on the
end facets of optical fibers. With their continued work on FFPCs, it became
clear that FFPCs offer several technical advantages over macroscopic devices
due to their compactness and robustness (Fig. 2), including:

  * •

High field concentration

  * •

Integration with optical fibers

  * •

High optical quality

  * •

Small foot print

  * •

Open geometry

  * •

Integration with other functional components

The following sections will briefly summarize the physical quantities
associated with these aspects.

Figure 2: Concept of a Fiber Fabry-Perot Resonator (FFPC) integrated with
optical fiber links and functional components.

###  II.1 FFPCs for high field concentration

The realization of high local field strengths is at the heart of many FFPC
applications. For all Gaussian modes sustained by FFPCs the
TEM${}_{\text{00}}$-mode offers the highest fields at its waist with diameter
$2w_{0}$. According to Eqs. 9 and 10 short cavity lengths $\ell$ and
$\ell/r\to 2$ (cf. Eq. 7) approaching the concentric cavity case let the mode
volume decrease and the local field strength ${\cal{E}}_{\text{cav}}$
increase.

The global geometric scale of FFPCs for the relevant parameters like the
cavity length $\ell$ and the radii of mirror curvatures $r_{i}$ is set by the
standard diameter of optical fibers of $125\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ ($D$ in Fig. 1). Choosing a small waist $w_{0}$ for high field
concentration also means high divergence of the field mode. The divergence
angle of a Gaussian mode is $\Theta_{\text{div}}=\lambda/(\pi w_{0})$, and
hence in order to avoid clipping losses by the fiber end facets we need to
make sure that $D/2>\Theta_{\text{div}}\cdot(\ell/2)$ yielding

$\ell<\pi w_{0}D/\lambda\ ,$ (11)

in agreement with the Rayleigh focusing limit for an aperture of diameter $D$.
We use Eq. 9 to estimate the range of mode volumes available for FFPCs with

$V_{\text{mode}}/V_{\lambda}=\frac{\pi}{4}\left(\frac{w_{0}}{\lambda/2}\right)^{3}\left(\frac{\ell/2}{w_{0}}\right)$
(12)

For realistic focusing conditions $5\leq 2w_{0}/\lambda\leq 50$ we therefore
obtain a wide range of possible mode volumes ranging from
$V_{\text{mode}}/V_{\lambda}\sim 10^{2}\cdot(2\ell/w_{0})$ to $\sim
10^{5}\cdot(2\ell/w_{0})$, where Eq. 11 sets an upper limit. All experimental
realizations of FFPCs indeed fall into this regime, where the factor
$2\ell/w_{0}$ may approach values below 10 when favorable small mode volumina
are of interest.

As the mode volume reduces with the radii of curvature of the mirrors, high
field concentrations are reached for small $r$ that, however, still have to
fulfill the condition $r>\ell/2$. For strongly focusing mirrors the radii $r$
will, therefore, typically be on the order of $\ell$, i.e. in the range of
some 10 to $1000\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

###  II.2 Mode matching for FFPCs

Successful use of FFPCs implies efficient coupling of the cavity mode to the
waveguide modes propagating within the fibers. Assuming perfect collinear
alignment of the cavity axis and fiber axis the coupling efficiency into the
cavity can be written as [4]

$\epsilon_{\text{TEM00}}=\frac{4}{(\frac{w_{\text{f}}}{w_{\text{m}}}+\frac{w_{\text{m}}}{w_{\text{f}}})^{2}+(\frac{\pi
n_{\text{f}}w_{\text{f}}w_{\text{m}}}{\lambda r})^{2}}$ (13)

with $w_{\text{m}}$, $w_{\text{f}}$, $n_{\text{f}}$ the waist of the cavity
mode and of the fiber mode at the mirror, respectively, and the index of
refraction of the fiber [4]. The mode field waist of a standard single mode
fiber, for instance, allows for a maximum coupling value of $\epsilon\approx
0.78$, considering a cavity for $\lambda=$780\text{\,}\mathrm{nm}$$ with
$\ell=$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
$r=r_{1}=r_{2}=$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and thus
$w_{0}\approx$3.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
$w_{m}=$4.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, as well as
$w_{f}=$2.53\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and $n_{f}=1.46$. The
mismatch in mode geometry of the incoming guided mode and the mode sustained
by the FFPC can be mitigated by stacking fibers of different types to achieve
mode shaping inside the fiber (cf. Sec. II.6 on GRIN fibers).

In practice, imperfect centering of the mirror curvature on the fiber center
during fabrication and/or cavity misalignment lead to degraded coupling
efficiencies. Detailed mode overlap calculations for misaligned cavity and
fiber modes have been performed in Refs. [6, 7, 8] providing an explanation of
asymmetric cavity line shapes as pure interference effects. The interplay of
direct single mode fiber and cavity mode coupling results in alignment-
dependent cavity line shapes in the reflected side at the incoupling fiber
that are well described by the superposition of a Lorentzian and a dispersive
component [6].

###  II.3 Finesse of FFPCs

In addition to determining the fiber to cavity coupling, the alignment-
dependent cavity mode properties also influence the FFPC finesse due to mirror
clipping losses, cf. Eq. 11. The reduction in cavity finesse for longer
cavities prior to the stability limit has been well explained by considering
the cavity mode footprint at the position of the fiber mirrors to be clipped
by mirrors with effective finite diameters [4].

More recently, sharp variations in finesse at specific cavity lengths have
been well modelled in position and magnitude as mirror-profile-dependent
crossings of the fundamental cavity mode with higher order transversal modes.
Deviations of the cavity mirrors from the idealized spherical mirror surface
have also been identified as the origin of the birefringence, which are
observed in many fiber cavity systems [9, 10, 11].

###  II.4 Polarization properties and birefringence

The dominant contribution for frequency splitting of polarization eigenmodes
in fiber Fabry–Perot cavities has been shown to result from the elliptic
shapes of cavity mirrors forming the FPC. The lifting of the degeneracy of
polarization modes expected from scalar field theory (cf. Eq. 8) can be
explained by extensions to vector theory that allows for field components
along the cavity axis [12, 13]. For many applications cavities with low
birefringence are desirable and mirror manufacturing techniques (cf. Sec.
II.5), including e.g. the rotation of the fiber during multi-shot sequences
[14], have been introduced for this purpose.

Recently, however, it was also shown that the birefringence of optical
cavities can be turned into an advantage by making more field modes available
for controlled light matter interaction [15, 16], e.g for further enhancing
the Purcell effect, see Sec. III.1.

###  II.5 Fabrication of high quality mirrors for FFPCs

The small-radii of curvature required for cavities with small mode volumes are
not within the reach of traditional polishing techniques. Successful
techniques to fabricate the required small, smooth depressions (depth of order
a few µm) on glass surfaces are surface etching methods and laser ablation.
Wet or ion-based etching has proven to be a useful tool for providing radii of
curvature below $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$, but with surface
roughness at the nanometer scale [17, 18, 19], which cause limits in the
achievable mirror reflectivities.

The most successful technique to date for producing curved mirror shapes with
ultra-low surface roughness is $\mathrm{CO_{2}}$-Laser ablation (Fig. 3 (a))
[4], [20], [21]. The strong absorption of fused silica ($\mathrm{SiO_{2}}$)
for light between $9-$11\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ wavelength is
used to strongly heat the top layer of the fiber end surface by illumination
with a focused $\mathrm{CO_{2}}$ laser beam. For suitable intensity, focus and
pulse duration the Gaussian laser intensity profile melts and evaporates part
of the glass surface and produces a depression with a similar profile [4, 22,
14]. The spherical part at the bottom of the resulting structure exhibits a
low roughness comparable to those of superpolished mirrors.

Figure 3: Treatment of fiber end facets for mirror fabrication. (a) Thermal
ablation by pulsed CO2 laser illumination. (b) Phase shift interferometry of
the fiber end facet for quality control. (c) Numerical reconstruction of the
fiber end facet surface through phase unwrapping.

In order to characterize the ablation process and to ensure the desired shape
and alignment of manufactured depressions, the fiber mirrors are optically
inspected. A typical setup uses a high numerical aperture microscope to image
an LED illuminated fiber end in a Michelson Interferometry configuration to
obtain interferogram images of the fiber surfaces with high spatial resolution
(see Fig. 3 (b) and (c)).

The ablation technologies have proven to be suitable for producing a wide
range of fiber end microstructures, from micrometer-long cavities [23] to
elaborate geometric shooting patterns that allow the creation of fiber-
resonators in the millimeter range [24] and customized mirror ellipticities
for tailored cavity birefringence [13].

Subsequently to the mirror surface molding, multilayer dielectric coatings are
applied to realize low-loss, high-reflectivity mirrors in the visible,
infrared or ultraviolet domain. For this purpose up to tens of alternate
layers of materials with high ($\mathrm{Ta_{2}O_{5}}$ (n=2.10) and low
($\mathrm{SiO_{2}}$ n=1.45) index of refraction and individual thicknesses of
$\lambda$/4 are deposited on the fiber surface using ion-beam sputtering
techniques [20]. The resulting fiber-based mirrors with the shape of the
original fiber surface depression can feature transmission, scattering and
absorption losses (after annealing [25]) down into the few parts per million
range, respectively and are ready to be aligned to build high-finesse, low-
mode-volume resonators.

###  II.6 FFPC devices

Depending on the specific application, the geometries of FFPC implementations
can strongly vary resulting in a large range of reachable specifications with
regard to cavity and mode sizes, mode properties, degree of accessibility and
stability of the devices.

FFPC devices can consist of either two opposing fiber mirrors or a single
fiber mirror together with a usually flat, macroscopic mirror substrate [5].
As shown in Fig. 4 (a) and (b), the implementations differ with respect to the
position of the smallest cavity waist that is either located between the fiber
mirrors or on the macroscopic mirror. Solid state quantum emitters on
substrates are interfaced using the latter cavity geometry whereas experiments
with trapped atoms or ions usually employ a symmetric cavity geometry in order
to maximize the coupling to the emitter.

Figure 4: (a) shows a hemi-cavity, where the smallest mode waist is located on
a flat mirror substrate. The amber colored fiber in the image is the
reflection of the fiber mirror approached from the left. (b) in contrast is a
symmetric cavity with the smallest mode waist located in the center. The first
is used to interface structures on the surface of the flat mirror, while the
latter is used to interact with gases, atoms or ions that can be trapped in
the center of the cavity mode volume. (c) shows a schematic of mode-matching
optics using a graded index fiber (GRIN) piece. The stack of different fiber
pieces allows to shape the mode incident on the mirror from the fiber side and
match it to the mode of the cavity [8].

As many applications benefit from high $Q$ optical resonators, the finesse
$\mathcal{F}=Q/N$ is one of the key specifications for FFPCs. In accordance
with Eq. 2, it is determined by the losses $\mathcal{A}$ during a cavity round
trip, with $\mathcal{F}=2\pi/\sum_{i}\mathcal{A}_{i}$. In FFPCs, the reachable
finesse is usually limited by the quality of the reflective coating, the
surface quality and the size of the fiber mirrors. Absorption in ion beam
sputtered layer systems with high reflectivity usually occurs on the order of
some parts per million. Annealing of these mirrors at moderate temperatures
($\leq$350\text{\,}\mathrm{\SIUnitSymbolDegree}$$) can reduce the amount of
absorption even further. Mirror carving using laser ablation creates smooth
surfaces, but is hard to control close to the rim of the facet reducing the
usable mirror size. Small effective mirror diameters however lead to clipping
losses that increase with the separation of the fiber mirrors and reduce the
finesse. Record finesses for FFPCs on the order of some 100000 have been
reported [4, 26].

The cavity length $\ell$ of FFPC devices can be customized depending on the
target specification for mode volume and linewidth (cf. Sec. II.1).
Furthermore, it is a key parameter to set the resonator’s free spectral range
$\Delta\nu_{\text{FSR}}$ that requires consideration, if multiple cavity
resonances are employed. The realization of specific cavity lengths requires
the fabrication of tailored fiber mirrors with respect to the usable mirror
size and the radius of curvature of the employed fiber mirrors. Record long
FFPCs reach lengths on the order of $1\text{\,}\mathrm{mm}$ [24]. The length
of the cavity is thereby limited by the size of the mode on the mirror with
respect to the usable area, the ultimate limit being the size of the optical
fiber (cf. Eq. 11). For mirrors fabricated using laser ablation the usable
mirror size is usually much smaller than the full fiber facet. Mode waists
larger than the spherical part of the mirror lead to clipping losses and hence
strongly reduced finesse values.

Another challenge especially for long FFPCs is mode matching of the fiber-
guided and the cavity mode at the in-coupling mirror as the cavity mode size
is increasingly large compared to usual fiber guided mode field diameters. For
extremely short FFPCs with small mirror radii of curvature the mismatch of the
wavefront curvature leads to a similar effect. To compensate for this effect
in large cavities either fibers with a large mode-field diameter [24] or
fiber-integrated mode matching optics [8] are required. The latter uses graded
index (GRIN) fiber pieces of well defined length that are spliced to the in-
coupling fiber. The mirror fabrication is then performed on the last surface
of the fiber stack. This approach uses the re-focusing properties of GRIN
fiber to shape the fiber-sided mode. A schematic depiction is shown in Fig. 4
(c).

###  II.7 Tunable and stable FFPCs

A key parameter in any application of FFPCs is their frequency stability and
their ability to be tuned and locked to a certain frequency reference.
Frequency noise in high-finesse FFPCs arises from thermal drifts, picked-up
acoustic noise from the environment, electric noise in the cavity resonance
tuning and from the intrinsic mechanical noise at non-zero temperature caused
by vibration modes [27, 28, 29]. The first three can be reduced by appropriate
thermal and acoustic isolation and by using low-noise electrical components in
the experiment. How strongly ambient acoustic noise is picked-up by the device
and the influence of the vibration modes onto the device stability is largely
determined by the design geometry [27].

As FFPCs are miniaturized devices, their intrinsic vibration modes can occur
at comparably high acoustic frequencies ($>$1\text{\,}\mathrm{kHz}$$). Whilst
in macroscopic cavities low frequency vibration modes can be actively damped
[30], passive stability in FFPC devices is reached by pushing the lowest order
vibration modes to higher frequency reducing the amount of thermal vibration
noise in the system. It also allows for high locking bandwidths to actively
stabilize against environmental disturbances through an electric actuation of
the cavity length e.g. via piezo-electric elements [6, 28, 27]. This design
paradigm is achieved through rigid device geometries, where the opposing
mirrors of the FFPC are supported in close vicinity to a common base [27, 28,
31].

Aside from electronic locking also other mechanisms for example through the
optomechanical spring effect of an integrated membrane [32] or via thermal
locking [33, 6] can be realized. The latter is based on a self locking
mechanism, where thermal drifts caused by the heat induced through the intra-
cavity field lead to a stabilization of the cavity resonance. Similarly, these
thermal nonlinearities can even lead to deformation of the FFPC mirrors or
drive the FFPC resonance into photothermal self-oscillations [34, 35].

###  II.8 FFPC integration and microstructured mirrors

To increase the functionality of FFPCs other elements such as mechanical or
electrical components can be integrated into the FFPC geometry or the FFPC
itself is embedded into miniaturized experiment designs e.g. chip-based ion
trap experiments [31]. In the particular example of ion traps the advantage of
small mode-volumes of FFPCs is usually not fully exploited because of
uncontrolled charging of the dielectric fiber mirrors. Usually, this is
mitigated by employing long cavities, however an alternative approach to
reduce the effect of charging is to integrate a microstructure such as a
metallic shield directly on the fiber mirror. This metallic mask is deposited
on the end facet of the fiber leaving out the central part around the core.

As an example for direct microstructuring of FFPC mirrors, a possible
procedure to fabricate a metallic mask is detailed in the following. In the
first step, a polymer cover (see Fig. 5 (a)) is printed on top of the fiber-
end facet by three-dimensional laser lithography [36]. Subsequently, the fiber
as well as the polymer cover are metallized by thermal evaporation in a
deposition chamber. In order to achieve a uniform coating, the fiber is
rotated during the deposition. In the last fabrication step, the polymer cover
is removed with a micromanipulator. Fig. 5 (b) exemplifies a fiber-end facet
with a copper mask that we fabricated using this procedure. In order to show
that the fabrication does not deteriorate the optical properties of the fiber
mirrors, we characterized the FFPC before and after the application of the
metallic masks. The data shown in Fig. 5 (c) demonstrates that the metallic
mask does not have a negative effect on the cavity finesse.

Figure 5: (a) CAD drawing of the polymer cover printed on top of the fiber
end-facet. (b) Optical micrograph of the end facet of a metallized fiber. (c)
Finesse of a FFPC before and after application of a metallic mask on the end-
facet and the cladding of the fibers.

As shown here, the integration of microstructures into FFPCs can expand their
capabilities, add new functions and at the same time maintain a miniaturized
geometry. Future FFPC device developments are expected to more heavily make
use of such additional integrated elements and embed FFPCs into other
miniaturized experimental platforms.

##  III Applications

All applications of FFPCs make use of the miniaturized geometry leading to a
small mode volume and a small footprint, see Fig. 2. Here we concentrate on
three aspects: Small mode volumes allow to enhance the electric field
strengths of a single photon to the level, where strong coupling with single
atoms realizes the quantum regime (Sec. III.1); short cavities increase the
interaction with miniaturized mechanical resonators that are easily introduced
due to the open geometry (Sec. III.2); and FFPCs also allow to construct very
compact spectrometers for applications in sensing (Sec. III.3).

###  III.1 FFPCs for Cavity QED: The Quantum Limit of Light Matter Interaction

While the radiative decay of excited atoms appears to be an unchangeable
property of nature, it is not. It was realized early on by E. Purcell [37] in
the context of the invention of NMR that electromagnetic cavities would allow
to manipulate the coupling strength of light matter interaction by controlling
the density of electromagnetic states coupling to a quantum emitter. Later on
D. Kleppner [38] coined the phrase turning off the vacuum, which put forward
the idea of cavity QED or cQED to investigate the quantum limit of light
matter interaction, i.e. quantum emitters such as single atoms coupled to the
electromagnetic vacuum or few photons. An extensive amount of information
about the general status of cavity supported interaction of atoms and fields
at the quantum level after more than four decades of experimenting is found in
[39], and focusing on more recent applications as light matter interfaces in
future quantum networks in [40].

Today, miniaturized optical cavities integrated with optical fibers, the
subject of this article, have opened yet another experimental avenue to
control light matter interaction at the full quantum level.

####  III.1.1 The role of the FFPC in atom-field coupling

FFPCs have typically small waist diameters $2w_{0}$ and accordingly small mode
volumes Eq.9. In a simplified rate model, we may characterize the interaction
of light and matter by considering the absorption cross section
$\sigma=3\lambda^{2}/2\pi$ for atomic quantum emitters at resonance
wavelengths $\lambda$ and the waist area $A=\pi w_{0}^{2}$ of a photon
propagating with a Gaussian beam shape (Fig. 6).

Figure 6: Simplified parameters for light matter interaction at the atom-
photon level inside a cavity. The most interesting strong coupling limit
occurs when the cross sections for absorption and mode at the interaction
point are matched.

The cavity recycles the travelling photon $n_{\text{round}}=\cal{F}/\pi$ times
with $\cal{F}$ the finesse from Eq. 2. The number of chances that the photon
experiences to be absorbed by the atom is then estimated by the ratio of
$\sigma$ and $A$ times the number of round trips, i.e.

$C=\frac{\sigma}{\pi w_{0}^{2}}\cal{F}\ .$ (14)

One can show that this definition of the cooperativity $C$ is consistent with
the more fundamental definition

$C=\frac{g^{2}}{2\kappa\gamma}=\frac{1}{2}\cdot\left(\frac{g}{\gamma}\right)^{2}\left(\frac{\kappa}{\gamma}\right)^{-1}\
,$ (15)

where the cooperativity $C$ compares the rate of evolution $g$ of the combined
atom-field system coupling with the dynamic parameters governing its
constituents: the free space decay rate $\gamma$ of the atomic dipole of the
relevant transition ($2\gamma$ for the excitation energy), and the relaxation
rate $\kappa$ of the cavity field amplitude ($2\kappa$ for the cavity field
energy). Parameters in publications are typically given in terms of sets for
$\\{g,\kappa,\gamma\\}$-values.

For an initially excited atom, the cooperativity $C$ in Eq. (15) determines
the ratio of the emission rate into the cavity field mode with respect to the
global free space emission rate by the enhancement factor

$\eta=\frac{2C}{1+2C}.$ (16)

For large $\cal{F}$ though small $2C=(2\sigma/(\pi
w_{0}^{2}))\cdot{\cal{F}}\ll 1$ the probability of photon emission into the
cavity field is already strongly enhanced by the finesse factor $\cal{F}$ in
comparison to free space. However, with $\eta\ll 1$ (Eq. 16) the total decay
rate of the atom experiences little modification, giving rise to the so-called
bad cavity regime,

$\gamma^{\prime}=\gamma(1+2C).$ (17)

The realm of cavity QED has also been extended to so-called artificial atoms,
a term, which describes all quantum emitters with two and three-level quantum
systems as well as good coupling to photon fields and thus resembling simple
atoms. Ranging from molecules to quantum dots and more, artificial atoms offer
technical advantages such as solid state samples, see Sec. III.1.3.

####  III.1.2 FFPCs for strong atom-field coupling

The most interesting regime (the so-called strong coupling regime) occurs for
$C>1$ when atom-field coupling becomes so strong that the coupled systems
undergo joint evolution. Coherent coupling of two quantum oscillators, atoms
and fields, indeed leads to a spectrum showing a splitting as a signature of
the strongly coupled system, see Fig. 7.

Figure 7: (a) The spectrum of the empty cavity is taken in reflection with a
strong dip at resonance. (b,c) Reflection spectra for the cavity frequency set
at different detunings from the atom transition frequency (dotted lines
corresponding to the spectrum at resonance in (a), too). (d-f) Reflection
spectra with parameters as in (a-c) but with a single atom resonantly coupled
to the cavity. The spectrum of the atom-cavity system shows a splitting which
indicates the strong coupling regime. The gap width at resonance is given by
$2g$, the atom-photon coupling rate, also called vacuum Rabi splitting. [41]

To achieve this regime, the small mode volumes of FFPCs offer excellent
conditions: The coupling strength $g$ of atoms, or artificial atoms, with the
photon field is given by the product of the atomic transition dipole moment
$d$ and the local cavity field strength (Eq. 10). It is calculated for the
energy of a single cavity photon $\hbar\omega_{\text{cav}}$ from
${\cal{E}}_{\text{cav}}=\sqrt{2\hbar\omega_{\text{cav}}/\epsilon_{0}V_{\text{mode}}}$
per photon resulting in

$g=\frac{1}{\hbar}d\cdot{\cal{E}}_{\text{cav}}=d\sqrt{\frac{2\omega_{\text{cav}}}{\hbar\epsilon_{0}V_{\text{mode}}}}\
.$ (18)

For $2C\gg 1$, Eq. 16 now reads $\eta\to 1$, that is, emission into free space
is almost switched off and the field energy is mostly deposited into the
cavity field. Since we cannot change the free space decay rate $\gamma$ of
atoms, we can experimentally only control the ratio $g/\gamma$ via the field
concentration caused by a small mode volume. The field relaxation rate
$2\kappa$ caused by the outcoupling rate from the resonator, however, is a
function of mirror transmissivity, which is determined by the mirror coatings.
Hence we can technically also choose the ratio $\kappa/\gamma$ in order to
control the dynamic properties of the coupled system of atoms and fields.

Figure 8: A non-exhaustive survey of normalized parameter sets
{$g,\kappa,\gamma$} and $C=g^{2}/2\kappa\gamma$ used in cavity QED experiments
with fiber based Fabry-Perot resonators (FFPCs sorted from left to right: [42,
43, 44, 45, 46, 47, 48, 49, 50]) with respect to performance in atom field
coupling strength $g/\gamma$ vs. resonator field fiber coupling rate
$\kappa/\gamma$. For comparison a small set of bulk cavity system parameters
is shown, too, from [51, 52, 53] sorted from top to bottom.

In Fig. 8 we compare atom-cavity systems prepared by experimenters for the
strong coupling regime over the past decades with respect to their $g/\gamma$
and $\kappa/\gamma$-values for both bulk cavity experiments [51, 52, 53] and
FFPCs [42, 43, 44, 45, 46, 47, 48, 49, 50]. Selected sets of parameters for
the ($g,\kappa,\gamma)$-values used in different laboratories show a clear
tendency to take advantage of FFPC properties especially in the fast cavity
regime, i.e. with $\kappa/\gamma\gg 1$.

In the fast cavity domain, the energy of an initially excited atom is radiated
into the cavity field at the strongly enhanced rate according to Eq. 17. The
cavity field in turn (i.e. the stored photons) is then rapidly coupled to the
fiber port, too. This is experimentally advantageous since the photons leaving
the cavity carry information about the quantum state of the atom-cavity
system. Calling the emission of a photon from the FFPC a ”read” process (since
the information on the photon needed to be present in the cavity before its
release), it is clear, that also the inverse ”write” processes are possible
and hence FFPCs coupled to quantum emitters are prime candidates for quantum
memories or also certain quantum gates in future quantum networks [40]. While
it is impossible to give an exhaustive survey of the applications of the
different platforms of quantum emitters now coupled to FFPCs (atoms, ions,
color centers, quantum dots, $\dots$), we indicate the origin of these
research lines and selected current results.

####  III.1.3 Quantum emitters coupling to FFPC fields

As outlined above, FFPCs are favorable components to enhance the interaction
of light fields with quantum emitters. Conceptually neutral atoms (and ions)
are the simplest and best understood quantum light sources. With neutral atoms
– both transiting a cavity or trapped inside – experimental progress in cavity
QED was pioneered ranging from the microwave to the optical domain.

##### FFPCs for analyzing and preparing cold atom ensembles.

The rise of FFPCs for cavity QED was indeed triggered by experiments with a
degenerate (ultra cold) system of atoms (a Bose Einstein condensate) [5, 47]:
Fiber optical channels were integrated with a so-called atom chip for
analyzing and controlling properties of the atomic ensemble. The experiment
showed that a system of $N$ identical atoms further increases the coupling
strength (Eq. 18) to $g_{\text{N}}\propto\sqrt{N}\cdot g$. Also, quantum
projection measurements (see below) allowed to create entanglement within such
ultra cold atom ensembles [54].

##### FFPCs for controlling single and few atom dynamics.

With many atoms both the atom ensemble and the cavity field resemble classical
harmonic oscillators since both of them exhibit the well known linear energy
spectrum. The spectrum of the combined system shows of course a splitting into
two modes, but the global dynamic behaviour is dominated by what is well known
from two coupled classical pendulums.

This situation is very different with a single or few atoms: In the strong
coupling case the atom becomes easily saturated by the field corresponding to
a single photon or more. Atom-field coupling in this regime is highly non-
linear with respect to the field strength that is proportional to
$g_{n_{\text{ph}}}=\sqrt{n_{\text{ph}}}g$, the number of photons in the
cavity. The quantum nature of this situation may be illustrated with the
detection scheme for the quantum state of an atom coupled to the FFPC: from
Fig. 7 it is clear that an incoming light signal resonant with the empty
cavity is reflected depending on the state of the cavity-atom system: For an
atom in an uncoupled state, the cavity looks empty and the light field is
transmitted; for a coupled state, the signal is reflected. The reflection
signal thus discriminates with high efficiency an uncoupled from the coupled
atomic state [55] which amounts to a so-called quantum non-demolition
measurement.

The majority of cQED experiments with FFPCs has focused on coupling a single
or few atoms (or artificial atoms, see below) to the cavity field [56, 46, 43,
57]. For applications, e.g. the creation of single photons on demand, the
dynamics of the atom-cavity must be externally controlled. This can be
accomplished by using an (artifical) atom component offering a third level
which couples to the quantum states of the strongly interacting atom-cavity
system. Driving the transitions involving this atomic third quantum state, the
dynamics of the strongly coupled atom-field evolution can be manipulated and
functional components for quantum technology such as single photon sources
[46] and quantum memories [43] can be constructed.

We finally note that trapped neutral atoms require laser cooling processes,
where the fast cavity domain renders the method of cavity cooling [58]
inoperative and hence alternative schemes must be used [59]. With respect to
atom trapping the cavity may also be used to provide a dipole trap field by
engineering a resonance with a suitable second wavelength [60].

##### Matching trapped ions with FFPCs.

It is natural to consider trapped ions for cQED experiments with FFPCs.
However, the small scale of the FFPCs and their dielectric nature are subject
to uncontrolled charging under vacuum conditions. Therefore, ion trap
experiments with FFPCs are using relatively long cavities (up to a millimeter)
which increases the mode volume. Successful experiments are now carried out
based on Yb+ [44] and Ca+-ions [42] with wavelengths 370 and
$854\text{\,}\mathrm{nm}$, respectively. Especially the short UV wavelength of
Yb+ adds to the technical challenges in maintaining high performance mirror
coatings [61]. At the present time investigations are underway with respect to
numerous technical improvements especially with regard to ion coupled FFPCs:
Unwanted stray fields are to be controlled by means of integrated electrodes
(see Sec. II.8); a natural extension of FFPCs is integration with miniaturized
ion traps [44, 48, 42]; improved mirror fabrication for long cavities can
mitigate problems with charging of dielectric surfaces [31].

##### FFPCs with color centers.

While atoms and ions are efficient and simple quantum emitters the need for
trapping devices based on laser radiation or radiofrequency fields also causes
much technical overhead. Thus there are strong efforts to replace atoms by
artificial atoms, which are realized with solid state host materials and
cannot run away in order to simplify the systems towards applications.
Prominent examples of artificial atoms – quantum objects exhibiting atom like
energy structures with transitions in the optical domain – include color
centers defects or rare earth ions in transparent materials. With nitrogen
vacancies in diamond (NV centers), one of the most widespread artificial
atoms, FFPCs have been shown to make NV centers at the quantum level available
by e.g. demonstrating single photon light sources [62]. A review focusing on
the operation of color centers with FFPCs is found in [63]. Rare-earth ion
doped crystals offer another type of artificial atoms which are well known
from rare earth ion doped laser crystals. They offer alternative wavelengths
and coupling with FFPCs has been demonstrated to modify their radiative
properties, another important step towards making photon light sources for
future quantum networks at the telecom wave length around
$1550\text{\,}\mathrm{nm}$ available [64].

##### FFPCs with molecules.

The high spectral selectivity of FFPCs allows to also control the emission
properties of molecules for selected transitions out of complex structure and
at the single emitter level [65]. The enhancement of molecule-light field
interaction in a FFPC based microcavity has been shown to realize nonlinear
wave mixing processes at the single photon level [66, 45].

##### FFPCs for semiconductor quantum emitters

Yet another natural choice for artificial atoms is given by semiconductor
nanostructures, where quantum wells and quantum dots exhibit atom like energy
structures which can also be engineered over a large range of wavelengths. In
contrast to color centers, quantum dots hold the promise to be pumped by
simple electric currents. Within the semiconductor world the integration of
all basic opto-electronic components in terms of systems is routine. However,
the realization of a resonant and strongly coupled quantum dot cavity system
based on quantum emitters, cavities, and waveguides remains challenging [67].
Hence the application of FFPCs offers attractive alternatives for e.g. single
photon generation [68] with respect to simplicity of fabrication or
tunability. The feasibility of this approach has been shown in [69, 70], where
the substrate holding the quantum dots was directly integrated with DBR
reflectors (DBR = Distributed Bragg Reflector). Finally, we mention that FFPCs
may help to further enlarge the toolbox for processing photons with more
exotic objects such as carbon nano-tubes [71].

##### FFPCs for hybrid systems

Another interesting application of FFPCs with cQED concepts is concerned with
the distribution of photonic quantum information by means of optical fiber
links. Future quantum networks could profit from components (single-photon
sources, quantum memories, $\dots$) built from different physical platforms
(”hybrid systems”) depending on their functional advantages. Then it is of
interest to e.g. match different wavelengths as well as to control the shape
of photons to allow efficient exchange between e.g. quantum dot single-photon
sources and ion or atom-based quantum memories or other systems[72, 57].

###  III.2 Cavity optomechanics in FFPCs

Cavity optomechanical experiments investigate the interaction of optical
cavity modes with the motion of mechanical elements [73]. These can be
mechanical resonators inside the optical cavity or, in the simplest case, one
of the optical mirrors that is suspended and able to vibrate. The displacement
of the mechanical resonator is thereby coupled to the number of photons in the
optical mode. On one side, this leads to a push of the mechanical resonator
through the radiation pressure and on the other to a detuning of the optical
mode frequency $\nu$ by the mechanical resonator displacement $x$. In the
simplest case the detuning $\Delta\nu$ is inversely proportional to the
unperturbed length of the cavity $\ell$ as $\Delta\nu\approx\nu_{0}\cdot
x/\ell$ [73]. This inverse scaling with $\ell$ holds true for all
optomechanical experiments, where a mechanical mode locally interacts with a
distributed optical cavity mode. The most common examples for this type are
suspended mirrors [74] or membranes in cavities [75]. In other optomechanical
setups, where the optical and mechanical modes are of comparable size and
interacting within their full mode volume, this is typically replaced by the
overlap of the two modes, as for example in the case of photoelastic coupling
in optomechanical crystals [76].

Figure 9: Cavity optomechanical experiments can be realized with a broad
variety of mechanical resonators inside FFPCs. The displacement of the
mechanical resonator thereby modulates the optical cavity resonance frequency
e.g. by a position-dependent modulation of the effective refractive index of
the optical cavity mode. (a) shows a schematic of a membrane-in-the-middle
setup [75]. The optical field extends on both sides of the mechanical
oscillator, which is clamped outside the mode volume of the optical mode.
Examples for this geometry are [77, 26, 32, 78]. (b) depicts a setup using
density waves in liquid helium as the mechanical resonance as demonstrated in
[79, 80]. The standing wave inside the liquid helium is similarly as the
optical field confined between the two mirror surfaces. (c) depicts an
experiment using a nanowire as the mechanical resonator as used in [81].

FFPCs have been used in optomechanical experiments because of their ability to
realize miniaturized cavities corresponding to small $\ell$ that at the same
time still allow for a simple integration of mechanical elements, because of
the accessible cavity volume. Most of the realizations using FFPCs can be
attributed to the so-called membrane-in-the-middle (MIM) type experiments [77,
26, 32, 78] (see Fig. 9 (a)), where a partially transmitting element divides
the cavity into two sub-cavities. The small mode waist that can be realized in
FFPCs also allows to reduce the size of the mechanical resonator in these
experiments. This can be used to increase the mechanical resonator frequency
$\Omega$ and decrease its effective mass $m_{\text{eff}}$. Pushing towards
higher mechanical frequencies allows to enter the sideband resolved regime
($\Omega>\kappa$), where the cavity can be used to e.g. selectively enhance
amplification or dampening of the mechanical resonator. Smaller effective
masses enable larger zero point motions
$x_{\text{ZPM}}=\sqrt{\hbar/2m_{\text{eff}}\Omega}$ and correspondingly a
larger vacuum optomechanical coupling strength
$g_{0}/2\pi=x_{\text{ZPM}}\cdot\partial\nu/\partial x$ that characterizes the
interaction strength on a single-photon level. The reduction of the mechanical
oscillator size has led to realizations using on-chip $\mathrm{SiN}$\- [26]
and nanowire oscillators [81] (see Fig. 9 (c)), with the latter being used to
map out the optical mode, position-dependent coupling and scattering. Apart
from classical MIM-type experiments other realizations in FFPCs have used
sound-waves of liquid helium filling the cavity volume as mechanical modes
[79, 82, 80] (see Fig. 9 (b)). The distributed mechanical mode interacts with
the optical mode through the modulation of the refractive index by the density
variations in the acoustic standing wave pattern.

An overview of the reached parameters in FFPC-based optomechanical experiments
is shown in Fig. 10. As for cavity optomechanical experiments in general,
except those using cold atoms as mechanical oscillators, only moderate values
of $g_{0}/\kappa$ are reached. This keeps nonlinear quantum effects using
single photons caused by the optomechanical interaction yet not realized.
However, FFPC-based systems still reach decent parameters due to their small
cavity lengths and corresponding large vacuum coupling strength and the
possible high finesse. The single-photon cooperativity
$C_{0}=4g_{0}^{2}/\kappa\Gamma$ can still reach favorable values as mechanical
resonators with high quality factor can be realized.

Figure 10: Comparison of optomechanical experiments using FFPCs. (a), (b) show
the reached values of vacuum coupling strength $g_{0}$ and mechanical
frequency $\Omega$ vs the cavity linewidth $\kappa$. The two sideband resolved
realizations [vi, vii] use density waves in liquid helium as the mechanical
resonance [79, 80]. The conventional membrane realizations [i-iv] are [77, 26,
32, 78] and the experiment using nanowire as the mechanical resonator [81].
(c) shows the performance of the different experiments in terms of the
mechanical and optical loss rates $\Gamma$ and $\kappa$ in relation to
$g_{0}$.

A benefit of FFPCs in these experiments compared to even more miniaturized
geometries like whispering gallery mode resonators [83] or optomechanical
crystals [84] lies in the tolerance to large optical drive powers. Whilst
FFPCs like macroscopic optical cavities can make use of large intracavity
photon numbers $n_{\text{c}}$ (up to Watts of optical power in high finesse
cavities) to boost the linearized optomechanical coupling strength
$g=\sqrt{n_{\text{c}}}\cdot g_{0}$, other platforms experience limitations
caused by heating and thermal instabilities. As FFPCs can be integrated to on-
chip platforms [31], be used to interface very miniaturized mechanical
elements, and as they are able to reach large finesse together with strong
optomechanical coupling strengths [26], they continue to be a favorable
platform for cavity optomechanical experiments, especially superior where
large intracavity optical powers are required.

###  III.3 Cavity-enhanced sensing and other applications

Aside their utilization as an interface for quantum emitters or mechanical
oscillators, FFPCs have been used for a large variety of different sensing
tasks. Scanning cavity microscopes [85] (see Fig. 11 (a)), use a fiber mirror
as a scanning probe above a reflective substrate. They can be facilitated to
map the surface [9, 11] or particles located on it [85]. Measurements
including multiple higher-order cavity modes are thereby used to increase the
spatial resolution below the mode waist size of the fundamental mode.

Another application of FFPCs as a cavity-enhanced sensor is for trace gas
sensing [86] (see Fig. 11 (d)). There, the high intracavity intensity along
with the preferred emission into cavity modes is used to enhance the emission
of Raman scattering from atmospheric gases. Fiber-based sensing of substances
is however not limited to gaseous media, but can also be used with liquid
solutions in absorption-measurement-based [87] or refractometer-based schemes
[88].

As the resonance frequency of FFPCs is sensitive to external mechanical
perturbation, it can also be used for its detection. Fiber cavity based force
sensors [89] (see Fig. 11 (e)), and fiber based sensors for strain [90] and
vibrations [13] have been demonstrated.

Strong fields are a prerequisite for enhancing nonlinear optical effects. As
they can be provided by optical cavities, also FFPCs have been facilitated to
demonstrate and make use of optical nonlinearities. Examples include the
generation of photon pairs [91, 92] (see Fig. 11 e)) or in a little different
cavity geometry even the realization of a photonic flywheel, which using
dissipative Kerr solitons may lead to the FFPC-based frequency comb devices
[93] (see Fig. 11 (b)).

At last, also traditional interferometer applications like optical filters can
be realized in fiber-based platforms. Early developments in this field even
date back far before the first realization of high finesse FFPCs [94, 95].

Figure 11: Compilation of various applications and experiments using FFPCs for
sensing or nonlinear devices. (a) Scanning cavity microscope, adapted from
Ref. [85] (CC BY 4.0). (b) Dissipative Kerr solitons and frequency comb,
adapted with permission from Ref. [93]. Copyrighted by the American Physical
Society. (c) Photon pair generation in FFPCs using the optical nonlinearity of
the mirror coating, adapted with permission from Ref. [91]. Copyrighted by the
American Physical Society. (d) Trace gas sensing via enhanced Raman
scattering, adapted with permission from Ref. [86]. Copyrighted by the
American Physical Society. (e) Force sensing using FFPCs, adapted with
permission from Ref. [89].

##  IV Conclusion and prospects

Our summary shows that following pioneering experiments [47, 5] the improved
understanding, characterization and manufacturing techniques developed for
manufacturing FFPCs with excellent and robust optical and mechanical
properties have opened the route for manifold applications in quantum
technology, spectroscopy, and beyond. The already wide breadth of current
applications also points to a potentially even larger success in the future.
FFPCs may continue to offer advantages over alternative schemes: FFPCs can
overcome the low tunabilities characteristic for rigid high finesse cavities
such as micro toroids etc. while the accessibility of the cavity field is
unrivaled; also with respect to fully integrated photonic crystal cavities the
tunability and versatility of FFPCs offer advantages.

We expect experiments and applications using FFPCs to further diversify and
demonstrate their flexibility for tackling challenges in different fields like
platforms for controlled light-matter interaction, microscopy, sensing or
nonlinear devices for applications at extremely low light levels. This
flexibility together with their integration into fiber optics is expected to
also create economic impact beside its already tremendous scientific value.

###### Acknowledgements.

The authors acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation, SFB/TR185 OSCAR) under Germany’s Excellence
Strategy – Cluster of Excellence Matter and Light for Quantum Computing (ML4Q)
EXC 2004/1 – 390534769, as well as by the Bundesministerium für Bildung und
Forschung (BMBF), projects Q.Link.X and FaResQ.

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