# Integrated table-top facility for the study of Whispering Gallery Modes in
dynamic liquid micro-cavities coupled to sub-micron tapered fibers

Meenakshi Gaira<EMAIL_ADDRESS>C.S. Unnikrishnan  Tata Institute
of Fundamental Research, Mumbai 400005, India

###### Abstract

A complete integrated table-top facility for the study of high-$Q$ Whispering
Gallery Modes in solid and liquid micro-cavities is described, with emphasis
on the in situ fabrication of reliable tapered fibers of sub-micron waist
sizes for coupling light into time-dependent liquid micro-cavities. The
experimental parameters have been chosen to get nearly adiabatic tapers, with
their waist size consistent with the theoretical model. The oscillations in
the transmitted power during fiber tapering are monitored to check the
coupling of higher order modes and identify the point when the fiber is
suitable for pure single mode coupling. The fabricated tapered fibers have
greater than 85% transmission on the average and very good polarization
fidelity. The tapered fibers have been used for efficiently exciting and
detecting WGMs of $Q\sim 10^{7}$ in silica microspheres and of $Q\ \text{up
to}\ 7\times 10^{7}$ in microdrops of silicone oils.

††preprint: AIP/123-QED

## I INTRODUCTION

Whispering Gallery Modes (WGMs) of micro-cavities have applications in non-
linear optics Carmon and Vahala (2007); Beckmann  _et al._ (2011),
optomechanics Kippenberg and Vahala (2007), cavity QED Vahala (2003); Vernooy
_et al._ (1998), biosensing Dantham  _et al._ (2012); Wu  _et al._ (2019),
etc. (see Ref. Ilchenko and Matsko, 2006; Vahala, 2003) because of their high
Quality factor ( $Q$ ) Vahala (2003); Henriet  _et al._ (2014) and confinement
of high optical power in very small mode volume. Tapered optical fibers Cai
(2001); Lin (2010); Rivière  _et al._ (2013) are the most commonly used
devices for coupling light into the microsphere cavities, as compared to other
couplers like prismsBraginsky, Gorodetsky, and Ilchenko (1989); von Klitzing
_et al._ (2001), eroded fibers Dubreuil  _et al._ (1995) and angle polished
fibers Ilchenko, Yao, and Maleki (1999) because they are efficient, compact
and they filter out the higher order modes which otherwise add noise in the
main signal in the fundamental mode. Tapered fibers are also used in fiber
couplers, fiber splitters, sensors Polynkin  _et al._ (2005); Villatoro and
Monzón-Hernández (2005); Zhang  _et al._ (2008); Díaz-Herrera  _et al._
(2003), for coupling light into photonic crystals Sadgrove  _et al._ (2013);
Thompson  _et al._ (2013), in trapping and detecting ultra-cold atomsVetsch
_et al._ (2010); Nayak  _et al._ (2007); Morrissey  _et al._ (2009), etc. (see
Ref. Lou, Wang, and Tong, 2014).

Tapered fibers are widely used for efficiently and selectively exciting WGMs
in solid micro-cavities. However, there are only a few reported worksDahan,
Martin, and Carmon (2016); Jonáš  _et al._ (2012); Hossein-Zadeh and Vahala
(2006); Kaminski, Martin, and Carmon (2015) on using tapered fibers for
coupling light to liquid micro-cavities because of their dynamic geometry that
keeps changing fast with time. Liquid microdrops are time-dependent micro-
cavities with naturally very smooth surface as compared to artificially made
solid micro-cavities. Unlike solid micro-cavities, these cavities are dynamic
as many physical processes, like evaporation and thermal oscillations along
with mechanical deformations and oscillations, continuously take place
depending on the external surrounding conditions. Optical WGMs in liquid
microdrops provide a novel probe for better understanding of dynamics of
liquid drops (like their capillaryMaayani  _et al._ (2016) and acoustic
modesDahan, Martin, and Carmon (2016), fluctuations in evaporation rate,
etc.), properties of liquids (like bulk modulus, surface tension, absorption,
viscosity, etc.), and provide a platform for investigating micro-
optomechanicsDahan, Martin, and Carmon (2016); Maayani  _et al._ (2016); Jonáš
_et al._ (2012). Also, the measured optical $Q$ can be used to estimate a
reliable upper limit of the absorption coefficient of liquids. This could be
advantageous for liquids with very low absorption $\lesssim
10^{-3}~{}\rm{cm}^{-1}$, for which determining absorption becomes difficult
when using standard methods (which use bulk samples) due to systematic errors
of the same order.Gaira and Unnikrishnan (2020)

Here, we report the development of a complete integrated table-top facility
for the study of high-$Q$ WGMs in solid and liquid micro-cavities. Our
motivation for building this set up is to experiment with taper-fiber coupled
liquid microdrops to explore these physics-rich systems for novel studies and
applications. The facility consists of four main units — set up for
fabrication and characterization of tapered fibers, set up for fabrication of
silica microspheres and formation of liquid microdrops, set up for exciting
and detecting WGMs in micro-cavities using tapered fibers, and a vibration-
isolated experimental platform inside a laminar flow enclosure unit for
protecting the tapered fibers and the micro-cavities from dust particles
during experiments. We have compiled in this self-contained paper the
technical details of our experimental facility along with the required
information from the literature, for completeness.

This paper is organized as follows. In section 2, we describe the salient
features of all the units comprising the facility. In section 3, we compare
the tapering process with its theoretical model. In sections 4 and 5, we
explain the propagation of light through a tapered fiber to understand how to
fabricate tapered fibers which have high transmission and which can completely
filter out the unwanted higher order modes. We also discuss the analysis of
transmission data of a tapered fiber, taken while tapering. In section 6, we
briefly describe an experiment to check whether the cylindrical symmetry of
the fibers is maintained while tapering, by comparing their polarization
response before and after tapering. In section 7, we show the observations of
high-$Q$ WGMs in silica microspheres. In section 8, we conclude with the
measurement of the exceptionally high $Q$ values of WGMs in microdrops of
silicone oils, using cavity ring-down interferometryRasoloniaina  _et al._
(2014); Dumeige  _et al._ (2008); Ye, Shen, and Lin (2016).

## II EXPERIMENTAL SETUP AND PROCEDURE

### II.1 Fabrication of solid microspheres and formation of liquid microdrops

Figure 1: Schematic of experimental set up for fabrication of silica
microspheres. L1 and L2: lenses, M: mirror, MS: silica fiber’s tip which has
got re-formed into a microsphere, MIC: microscope with a CCD camera attached
to it, and VLP: visible laser pointer used for optical alignment of
$\rm{CO_{2}}$ laser beam. L1, L2 and MS are on XYZ mounts or stages for
alignments.

We study WGMs in two kinds of micro-cavities — solid microspheres and liquid
microdrops. We fabricate solid microspheres by melting the tips of silica
fibers using two balanced counter-propagating $\rm{CO_{2}}$ laser beams,
generated by retro-reflection of a beam by a Si mirror. Surface tension re-
forms the fiber tips to spheres. A schematic of the set up is shown in Fig. 1.
The balanced beams avoid asymmetric radiation pressure which otherwise
produces bent or asymmetric (about the fiber stem) microspheres. There are two
ZnSe lenses for focusing the beams and the fabrication process is monitored
with a microscope and a CCD camera attached to it. Since $\rm{CO_{2}}$ laser
beam ($10.6$ µm wavelength) is invisible, a visible laser pointer, which
consists of a diode laser ($650$ nm wavelength) and a ZnSe beam
combiner/splitter, has been mounted on the front panel of the $\rm{CO_{2}}$
laser. Its beam is co-aligned with the $\rm{CO_{2}}$ laser beam for optical
alignments. We produce microspheres with diameters in the range 30–300 µm with
this setup (see Fig. 2).

Figure 2: Microscope image of a silica microsphere of diameter $\approx 145$
µm, kept very close (sub-wavelength gap) to a tapered fiber to excite its
WGMs. Figure 3: Microscope image of a silicone oil microdrop of diameter
$\approx 105$ µm around a silica microsphere of diameter $\approx 40$ µm
(bordered by a dashed circle). We form such microdrops by dipping silica
microspheres in silicone oils. Note that the centroid of the liquid microdrop
is below the centroid of the much smaller solid microsphere.

We form a liquid microdrop (see Fig. 3) by dipping a small silica microsphere
inside the liquid. A microdrop gets formed around the solid microsphere due to
cohesive and adhesive forces in and between the liquid and silica. Microdrops
of silicone oils with diameter 50-200 µm are used in our experiments. Silicone
oils have been chosen for the microdrops because it is technically easier to
excite and detect WGMs in these drops. This is due to the following reasons.
a) very low evaporation rate, b) refractive index close to silica which helps
in phase matching,Lin (2010) c) very low optical absorption, and d) good
tendency to form drops around silica microspheres which allows easy
positioning and alignment of the drops. (this tendency depends on the
viscosity, the cohesive forces in the liquid and the adhesive forces between
the liquid and silica).

Figure 4: Schematic of experimental set up for fabrication of tapered fibers.
MTS denotes two identical motorized translation stages. BOF: bare optical
fiber, FT: flame-torch (hydrogen-oxygen torch), FFC: free space fiber coupler,
PD: photo-detector, and MIC: microscope with a CCD camera attached to it.

### II.2 Fabrication of tapered fibers

For efficiently and selectively exciting (and detecting) WGMs, approximately
adiabatic Love  _et al._ (1991); Ravets  _et al._ (2013); Ward  _et al._
(2014) tapers are needed. A tapered fiber is called adiabatic if the taper
angle is small enough everywhere such that negligible power couples from the
fundamental mode to the higher order modes and the radiation modes (which
cause power loss), as it propagates along the taper. For adiabatic tapers of
single mode fibers, the ‘local mode theory’ Love  _et al._ (1991) says that at
every local point of the tapered fiber, the field distribution and propagation
constant can be described by the fundamental mode of a cylindrical guide with
the same geometry as the local point. So, during propagation, the fundamental
mode of one local point transforms to the fundamental mode of successive local
point without any loss of power.

Heat and pull is the most commonly used method for fabricating tapered fibers
of sub-micron waist sizes.Orucevic, Lefèvre-Seguin, and Hare (2007); Grellier
(2000); Hauer  _et al._ (2014); Ward  _et al._ (2006, 2014) A heat source like
$\rm{CO_{2}}$ laser, flame torch or micro-furnace softens a fiber and
simultaneously two motorized translation stages pull the fiber from both the
sides. The advantages and disadvantages of each of these heat sources are
mentioned in Ref. Ward  _et al._ , 2014. We use a hydrogen-oxygen torch as the
heat source. A schematic of the set up is shown in Fig. 4. A hydrogen-oxygen
torch is preferred over other gas torches because it has a clean flame and
leaves no residue other than water vapor, which also disperses in the air
leaving a clean taper. This helps in making tapered fibers with good
performance. Single mode 780 HP fiber from Thorlabs, with $4.4$ µm core
diameter and $125$ µm cladding diameter, is used in our experiments. A long
working distance microscope (from Navitar) with a CCD camera is mounted on one
side for monitoring the whole fabrication process.

The first step in the fabrication of a tapered fiber is splicing a bare
optical fiber with an optical patch cable, which is connected to a free space
fiber coupler that couples the laser light (Toptica DL100 ECDL at 780 nm) into
the fiber. The other end is connected to a photo-detector using Thorlabs BFT
(Bare Fiber Terminator) connector which is like a temporary FC connector for a
bare fiber. Next, the protective coating around the fiber is stripped off over
2 cm near the center of the bare optical fiber. Then the stripped part is
cleaned with lint-free wipes soaked in alcohol. Subsequently, the fiber is
clamped on the two motorized stages using fiber clamps such that the stripped
part comes between the two clamps. After this, the hydrogen-oxygen torch is
moved below the stripped part of the fiber with an air gap of nearly 3-4 mm
between the nozzle and the fiber. In our case, the nozzle diameter is 1mm. The
hydrogen gas flow is set between 25 to 50 ml/min and the flame uses oxygen
from the surrounding atmosphere. All these parameter values are chosen to get
a sufficiently wide and short flame which provides a wide hot zone (part of
the flame which softens the fiber) with low gas pressure and nearly uniform
temperature along its length. Such kind of hot zone is required for getting
adiabatic tapers. This will be discussed in detail in the next sections. With
these settings, 1 to 4 mm hot zone length can be obtained, depending on the
position of the flame with respect to the fiber.

Figure 5: Schematic of experimental set up for exciting and detecting WGMs in
micro-cavities. MC denotes a micro-cavity mounted on a 3-Axis nano-positioning
stage and a goniometer stage. Other elements are TOF: tapered optical fiber,
BS: beam splitter, FPI: Fabry-Pérot interferometer, FFC: free space fiber
coupler, PD: photo-detector, LC: laser controller, and MIC: microscope with a
CCD camera attached to it. Black lines represent co-axial signal cables.

As the torch is moved close to the fiber, the part of the fiber under the
flame starts glowing and then immediately fiber pulling is started using the
two computer-controlled motorized stages. The pulling speed is not very
critical for obtaining good quality tapers. Prior works report pulling speed
values ranging from $20$ µm/s to $200$ µm/s.Orucevic, Lefèvre-Seguin, and Hare
(2007); Ward  _et al._ (2014); Ravets  _et al._ (2013) We chose a slower speed
of $20$ µm/s for each motorized stage to have a longer time span between the
single mode operating point and the breaking point of the fiber (explained in
Sec. 5). This allows us to have better control on stopping the tapering
process. The stages are stopped immediately after getting the desired taper
waist at which almost all higher order modes get filtered out leaving only the
fundamental mode. This stopping point is decided by observing the transmission
through the fiber while tapering. For this, a photo-detector and an
oscilloscope are used. This will be discussed in detail in the following
sections. Immediately after stopping the stages, the torch is moved away from
the fiber. Fig. 6 shows the microscope images of a fiber before and after
tapering.

Figure 6: Microscope images of a fiber (a) before tapering, diameter $\approx$
125 µm, (b) after tapering, diameter $<$ 1 µm.

### II.3 Exciting and detecting WGMs in micro-cavities

After fabrication of a tapered fiber, micro-cavities are brought closer to the
tapered fiber to excite their WGMs. A schematic of the set up for coupling
light into these cavities is shown in Fig. 5. Half of the set up is the same
that is for the fabrication of tapered fibers. Before the fiber coupler, the
laser light is split into two parts using a 70:30 cubic beam splitter. The
major part of the light is coupled to the tapered fiber and the small part
goes to a Fabry-Pérot interferometer (with a detector attached to it) which
provides a relative frequency scale. Two microscopes are used for monitoring
the side view and the top view (not shown in the schematic). Both side view
and top view are important for aligning the position and orientation of the
micro-cavity with respect to the tapered fiber. For alignment, the micro-
cavity is mounted on a 3-Axis piezoelectric nano-positioning stage (PI
P611.3S) and a goniometer (Thorlabs GNL20).

The laser frequency is scanned to excite the WGMs at the resonance frequencies
of the micro-cavities. Two different lasers were used for the observations
reported in this paper. Toptica DL100 (ECDL) was used for experiments with
solid micro-cavities. It has $\sim 1$ MHz linewidth, which is good enough to
resolve resonances with $Q<10^{8}$, when using the method of ‘linewidth
measurement’ for determining the $Q$ values. Its frequency is scanned by
driving the piezo attached to the grating inside it. For liquid microdrops, we
use ‘cavity ring-down’ technique (explained in Sec. 8) to measure the $Q$
values. This technique requires very fast frequency scanning. For this, Sacher
TEC 55 (diode laser with VHG-based external cavity) at 780 nm wavelength is
used . Its frequency is scanned by modulating the injection current using a
ramp signal generator. In our experiments, its frequency has been scanned up
to $10^{6}$ GHz/s speeds.

All the data is acquired and saved using an oscilloscope with up to 4 GSa/s
acquisition rate. For cavity ring-down measurements with liquid microdrops,
which require very fast data detection, the large area photo-detector at the
tapered fiber’s output is replaced by a fast and high gain detector (Thorlabs
APD 430A with 400 MHz bandwidth). High gain is used because experiments for
measuring $Q$ are done with low power $<20$ µW to minimize thermal, non-linear
and radiation pressure effects. Carmon, Yang, and Vahala (2004); Dahan,
Martin, and Carmon (2016); Kippenberg and Vahala (2007); Kippenberg, Spillane,
and Vahala (2004); Agha, Okawachi, and Gaeta (2009)

Figure 7:  Laminar flow assembly (dimensions $2.5~{}\rm{m}\times
1.9~{}\rm{m}\times 2.9~{}\rm{m}$) with the whole set up covered inside it.

### II.4 Enclosure units and vibration isolation

The entire fabrication and experimental facility has been built inside a
laminar flow assembly (shown in Fig. 7) to protect the tapered fibers and the
micro-cavities from dust particles in the laboratory environment. Without this
air filtering assembly, we were not able to use a tapered fiber for more than
an hour or two as many dust particles accumulate on the waist region. Apart
from causing light loss due to evanescent scattering, the dust particles
prevent the clog-free placing of the fiber’s waist near the microsphere for
evanescent coupling of light. Although the optical resolution of the
microscope is $\sim 1$ µm, dust particles of all sizes, from less than 100 nm
to few microns, are visible and identifiable under the microscope as these
scatter the light propagating through the tapered fiber. After installation of
the assembly, we can use a tapered fiber for 2 to 4 days, depending on the
quality of the surrounding air which varies with seasons. This laminar flow
assembly was custom designed with dimensions $2.5~{}\rm{m}\times
1.9~{}\rm{m}\times 2.9~{}\rm{m}$. This assembly provides us a cleanroom
environment of ISO Class 5 standard, which means $\sim 10,000$ particles per
$\rm{m}^{3}$ of size $\geq 0.3$ µm. This kind of assembly is useful in labs
where a fully isolated cleanroom can’t be made due to some technical or
operational reasons. The setups are also surrounded by PVC strip curtains to
prevent them from dust. Additionally, on the optical table also, the setups
have been fully covered by acrylic covers with wooden frames.

While exciting WGMs in micro-cavities, a sub-wavelength gap is to be
maintained between the cavity and the tapered fiber. Therefore, the coupling
of evanescent light from the fiber to the cavity is sensitive to vibration
noise of even tens of nanometers. For this, all the experimental setups have
been built on a single floated optical table. Other than blocking dust
particles, the PVC strip curtains also block external air flows due to ACs and
dehumidifiers which can cause vibrations. All the cables are clamped on the
optical table to damp the vibrations that may get transferred through them.
All the instruments, optics and optomechanics are mounted firmly. Thus, the
possible sources of vibrations, which could be avoided or damped, were taken
care of.

## III SHAPE AND WAIST SIZE OF TAPERED FIBERS

The mathematical profile of a tapered fiber, considering constant and uniform
temperature hot zone length, can be obtained using the simple constraint of
conservation of mass.Birks and Li (1992); Orucevic, Lefèvre-Seguin, and Hare
(2007) According to this model, for a stationary flame, the shape of a tapered
fiber is exponential (schematically shown in Fig. 8) and is given by

$r(z)=\begin{cases}r_{i}\exp\left(\frac{|z|-h/2-L/2}{h}\right)&\text{for \ \ \
}h/2<|z|<(L+h)/2\\\ \\\ r_{i}\exp\left(\frac{-L}{2h}\right)&\text{for \ \ \
}0<|z|<h/2.\end{cases}$ (1)

Here, $r(z)$ is radius of the tapered fiber at axial position $z$, $h$ is the
hot zone length, $L$ is the pulling length, $O$ denotes the center of the hot-
zone (which is origin for the axial axis), $r_{i}$ and $r_{w}$ represent the
radius of the initial fiber and the taper waist respectively.

For pulling time $t$ and constant pulling speed $v$ on both the sides of a
fiber, the pulling/tapering length can be written as

$L=2vt.$ (2)

Using eq. 1 and eq. 2, the waist size as a function of pulling time can be
written as

$r_{w}=r_{0}\exp(-vt/h).$ (3)

Figure 8: Schematic of the exponential-profile of tapered fibers. $r_{i}$ is
the initial radius, $r_{w}$ is the waist radius, $h$ is the hot zone length,
$L$ is the pulling length, z=0 is marked by $O$.  Figure 9: Change in the
waist diameter of a fiber with pulling time while tapering. Red points are the
experimental data points and the blue curve is an exponential fit to eq. 3.
The fitting parameter $h$ for this data was found to be $2.0$ mm. Figure 10:
Theoretically calculated effective index for first few modes in a silica-air
waveguide as a function of the core diameter for wavelength $\lambda=780$ nm.
The waist of a tapered fiber is a silica-air waveguide with core and cladding
refractive index $n_{s}=1.45$ and $n_{a}=1$ respectively. The theoretical cut-
off waist size, below which the waist can support only a single mode, has been
marked by a dashed black line.

A sequence of microscope images of a fiber waist were taken while tapering
(using LabVIEW) and then the size of the fiber waist in all the images was
measured. The data of change in the waist diameter as a function of time was
theoretically fitted with eq. 3 for multiple tapered fibers. The value of
fitting parameter $h$ (which is different for different tapered fibers) was
found to be consistent with the range of hot zone lengths (typically 2 to 3
mm) used in the experiments. The fitted plot for one of the tapered fiber is
shown in Fig. 9. While tapering, the fiber waist moves and oscillates many
times due to the flame pressure. Although small height flame is used to
minimize this pressure effect, we still had to keep moving the microscope
manually while tapering to bring the fiber waist under focus. Many images that
were out of focus were not used for the final data. For this reason, the data
points in Fig. 9 are not at equal time intervals. Also, the images after 5
micron waist size were not used because those images were too fuzzy to measure
the waist size accurately.

## IV PERFORMANCE OF TAPERED FIBERS

As light propagates along a tapered fiber, it passes through different zones
of optical confinement. At the starting part of the tapered section, the core
and cladding are sufficiently thick, therefore, the light is totally confined
in them and this core-cladding region supports only a single mode. A gradual
transition occurs from this as the thickness decreases and the light sees a
core-cladding-air waveguide. This region can also support some higher order
modes. Here, the single mode of light coming from one end of the fiber can
couple to higher order modes, depending on the slope of the taper.
Approximately adiabatic taper Orucevic, Lefèvre-Seguin, and Hare (2007); Love
_et al._ (1991); Ravets  _et al._ (2013) minimizes the coupling from the
fundamental mode to the higher order modes and also dissipative radiation
modes. We can see from eq. 1 that the slope of the exponential profile depends
on the hot zone length. Wider hot zone length gives a more gradual slope.
However, too wide hot zone may affect the uniformity of temperature along its
length, depending on the nozzle design.Ward  _et al._ (2014); Orucevic,
Lefèvre-Seguin, and Hare (2007); Lin (2010) Thus, we choose a nozzle size
(from the available ones), the hydrogen gas flow rate and the position of the
fiber with respect to the flame such that we get a sufficiently wide hot zone
with nearly uniform temperature along its length to get nearly adiabatic
tapers. In our experiments, the hot zone length typically lies between 2 to 3
mm.

Due to small departures from adiabaticity, higher order modes get excited
while the fundamental mode propagates from one end of the taper towards the
taper waist and then also re-couple to the fundamental mode while propagating
from the taper waist to the other end of the taper. These higher order modes
contribute to the noise in the signal. Due to axial symmetry of the tapered
fiber, the fundamental $LP_{01}$($HE_{11}$) mode can only couple to the higher
order modes with the same azimuthal symmetry i.e. $LP_{0m}$($HE_{1m}$).
Therefore, probability of coupling to $LP_{02}$($HE_{12}$) mode is the most.
But, if the tapered fiber is bent or due to some imperfections in the tapering
process, other closer higher order modes with different azimuthal symmetries
can also couple, like $TE_{01}$, $HE_{21}$, $EH_{11}$, etc., depending on
their coupling efficiencies. Orucevic, Lefèvre-Seguin, and Hare (2007); Ravets
_et al._ (2013) During our fabrication process, tapered fibers sometimes
slightly bend upwards due to the flame pressure (which however doesn’t affect
the transmission much). This may also cause coupling to a variety of higher
order modes with different azimuthal symmetries.

When the propagating light reaches near the center/waist of the tapered
section, it sees a cladding-air waveguide because here the core is
insignificant as its size is much smaller than the wavelength of light. The
number of modes that this region can support depends on the size of the
cladding. As the radius of the taper’s waist decreases, the number of modes
supported by it decreases. Higher order modes can be filtered out at the waist
by making it so thin that it can support only a single mode. This cut-off
waist size can be calculated theoretically as briefly explained below,
following the detailed treatment in Ref. Cai, 2001; ofm, .

In a step-index, cylindrical waveguide, the electric or magnetic field
distribution of the guided modes can be found by solving Maxwell’s equations
for the boundary conditions imposed by the cylindrical dielectric core (here
silica) and cladding (here air). The complete solution can be found in many
references (see Ref. Snitzer, 1961; ofm, ; Saleh and Teich, 1991). The
effective propagation constant $\beta_{eff}$ (axial component of the wave
vector) of the guided modes can be found by solving the following
transcendental equation Snitzer (1961); ofm ; Saleh and Teich (1991)

$\begin{gathered}\left(\frac{{J_{l}}^{\prime}(pa)}{paJ_{l}(pa)}+\frac{{K_{l}}^{\prime}(qa)}{qaK_{l}(qa)}\right)\left(\frac{{n_{s}}^{2}{J_{l}}^{\prime}(pa)}{paJ_{l}(pa)}+\frac{{n_{a}}^{2}{K_{l}}^{\prime}(qa)}{qaK_{l}(qa)}\right)\\\
=l^{2}\left[\frac{1}{(qa)^{2}}+\frac{1}{(pa)^{2}}\right]^{2}\left(\frac{\beta_{eff}}{k_{0}}\right)^{2},\end{gathered}$
(4)

with

$q^{2}=\beta_{eff}^{2}-{n_{a}}^{2}{k_{0}}^{2},$ (5)

and

$p^{2}={n_{s}}^{2}{k_{0}}^{2}-\beta_{eff}^{2},$ (6)

where $J_{l}$ and $K_{l}$ are the modified Bessel functions of first and
second kind respectively, $l$ is the mode number which is a non-negative
integer, $a$ is the radius of the core (here silica), $k_{0}$ is the
propagation constant in vacuum, $n_{s}$ and $n_{a}$ are the refractive indices
of silica and air respectively. Here, $n_{s}=1.45$ and $n_{a}=1$ and
wavelength is 780 nm. This equation was solved for first few fiber modes and
their effective index $n_{eff}$ as a function of waist diameter ($2\,a$) is
plotted in Fig. 10. $n_{eff}$ is given by

$n_{eff}=\frac{\beta_{eff}}{k_{0}}.$ (7)

In a thick waist, the light is mostly confined inside the silica, leading to
an effective index close to $n_{s}=1.45$. As the waist size decreases down to
sub-wavelength scale, the evanescent field in the air region increases and
hence the effective index of the taper’s waist decreases from $n_{s}=1.45$
towards $n_{a}=1$. Lin (2010) It is clear from the plot that the last higher
order modes ($TE_{01}$ and $TM_{01}$) cut-off at $\approx 570$ nm waist
diameter. So tapered fibers, with waist diameters less than $570$ nm, can
filter out all the higher order modes at $780$ nm or higher wavelengths. The
optical microscope (with 1 µm resolution) in the set up can’t resolve such
small waist diameters. Therefore, we observe the transmission through the
fiber while tapering to identify the stopping point at which all the coupled
higher order modes cut-off. This is explained in the next section.

## V ANALYSIS OF TRANSMISSION SIGNAL

As mentioned before, the power transmission through the fibers is observed
while heating and pulling the fibers. The changes in output transmission with
pulling time for a nearly adiabatic tapered fiber is shown in Fig. 11. The
changes in the output transmission result from the interference of different
modes supported by the tapered fiber. Lin (2010); Ravets  _et al._ (2013);
Hauer  _et al._ (2014) Different modes propagate with different propagation
constants and hence recombine at the output with different acquired relative
phases (with respect to the phase of fundamental mode), depending on length
and thickness of the tapered fiber. The interference signal oscillates because
the acquired relative phases change with pulling time. Oscillations have
different frequency components because the phases acquired by different modes
change with different rates, causing beating patterns. The amplitude of these
oscillations is a measure of the efficiency of excitation and recombination of
these modes, which change with pulling time. The component frequencies also
change with pulling time because the rates of change of phases are time-
dependent during pulling. A zoomed view of the oscillations and the beating
pattern is shown in the inset plot (on the left) inside Fig. 11.

Figure 11: Transmission through a fiber while tapering as a function of
pulling time. Left inset : A zoomed view of the oscillations and beating
pattern, showing interference between the fiber modes. Right inset : Zoom on
the transition of tapered fiber from multi-mode to single mode (significant
drop in the amplitude of oscillations).

In the plot, the point at which the amplitude of the oscillations dies down is
when the fiber waist becomes single mode. A zoom on the single mode regime is
shown in the inset plot (on the right) inside Fig. 11. The heating and pulling
process is stopped immediately after this point is reached. If the pulling
continues, then the fiber breaks after 10 to 20 seconds of this single mode
operating point (which correspond to just 400 to 800 microns extra pulling
length). As estimated in the previous section, the last higher order modes
($TE_{01}$ and $TM_{01}$) cut-off at $\approx 570$ nm waist diameter.
Therefore, if $TE_{01}$ or $TM_{01}$ modes get coupled along with other higher
modes, while the fundamental mode is propagating down the tapered slope, then
the single mode operating point is reached at nearly $570$ nm waist diameter.
However, if these modes don’t get coupled while other higher modes get
coupled, then the single mode operating point is reached before $570$ nm waist
diameter. The complete analysis has been done in Ref. Ravets  _et al._ , 2013;
Orucevic, Lefèvre-Seguin, and Hare, 2007. The experimental cut-off waist sizes
(at which the last “coupled” higher mode cuts off) of the tapered fibers
fabricated in our set up are consistent with the theoretical estimation. Waist
sizes of most of the tapered fibers were below $570$ nm. An SEM image of a
tapered fiber’s waist is shown in Fig. 12. Its waist size is $460$ nm. The
little extra pulling time, during which the fibers continue to get tapered
even after the single mode operating point is reached, results in reducing the
final waist size below the cut-off waist size.

Figure 12: SEM image of a tapered fiber’s waist.

The tapered fibers fabricated in our set up have greater than $85\%$
transmission on average and the best ones go above $95\%$. We found that
cleaning the fiber using hydrogen-oxygen flame, before starting the pulling
process, significantly reduces the chances of bad tapering (i.e. high power
loss and more coupling to higher order modes).

## VI BIREFRINGENCE AND CROSS-TALK IN A TAPERED FIBER

It is important to characterize a tapered fiber for its polarization response,
to ensure quantitative accuracy in experiments and measurements. In normal
fibers, small amounts of random birefringence, stress and bends cause cross-
talk and phase differences between the two polarization modes (say vertical
and horizontal). Therefore, the polarization state of the output light is
different from the polarization state of the input light. Every fixed
polarization state can be described with its two parameters,
ellipticity/eccentricity and tilt/orientation angle. Here, ellipticity is
defined as

$e=\sqrt{1-\frac{E_{min}^{2}}{E_{max}^{2}}}=\sqrt{1-\frac{P_{min}}{P_{max}}},$
(8)

where $E_{min}$ and $E_{max}$ are the amplitudes of the maximum and minimum
components of the electric field respectively and $P_{min}$ and $P_{max}$ are
the measured power values corresponding to $E_{min}$ and $E_{max}$
respectively. Ellipticity has values from $0$ (for circularly polarized light)
to $1$ (for linearly polarized light).

Fig. 13 shows a plot for ellipticity of the transmitted light vs polarization
angle (with respect to a reference polarization angle) of the input light
before and after tapering of an optical fiber. For this measurement, a
linearly polarized laser beam of 780 nm wavelength was coupled to a single
mode bare fiber clamped on the heat and pull set up (see figure 4). The
polarization angle of the input light was rotated using a half waveplate
before the fiber coupler. The waveplate was rotated manually in steps of 2
degree which rotates the polarization angle by 4 degree. At the output, a
polarizer was kept before the detector and it was rotated by 360 degree to
measure the power components at all the angles. Out of all the power
components, the maximum ($P_{max}$) and minimum ($P_{min}$) components were
selected for calculating ellipticity. The piezo-driven polarizer rotator mount
takes approximately 135 seconds for one complete rotation. Then the same
measurements were taken after tapering the fiber. The plotted data, shown in
Fig. 13, is for a tapered fiber with transmission $>80\%$.

As explained in sections 4 and 5, fibers are tapered down to the waist size at
which all the coupled higher order modes filter out. The only guided mode that
lefts out can be defined as a combination of two orthogonal, quasi-linearly
polarized fundamental modes $HE^{x}_{11}$ and $HE^{y}_{11}$.Lei  _et al._
(2019); Kien  _et al._ (2004) In the plot (Fig. 13), the ellipticity values
are nearly the same before and after tapering. This shows that negligible
birefringence and cross-talks between the two polarization modes were caused
due to tapering, which further implies that the circular symmetry of the fiber
was maintained intact during tapering. While doing this experiment, it was
ensured that the conditions which cause birefringence and cross-talk between
the two polarization modes (like bends and loops in the fiber, pressure due to
the fiber clamps, etc.) remain the same before and after tapering the fiber.
Also, we verified that the ellipticity changes by $\leq 0.05$ due to any
stretching of the fiber from pulling with motorized stages, or due to long
term drifts (1 to 24 hrs observations).

Figure 13: Ellipticity of the transmitted light Vs polarization angle of the
input linearly polarized light for an optical fiber before and after tapering.
The red and the blue lines are guides to the eyes.

The polarization at the output of a single mode optical fiber as a function of
the input polarization has been theoretically modeled in Ref. Chartier  _et
al._ , 2001, along with its experimental verification. In this paper, the
optical fiber is modeled as a series of concatenated linear birefringent
waveplates, with different amounts of phase retardation and different
orientations, neglecting any polarization-dependent losses. They used the
ellipticity angle $\chi$ to represent ellipticity, which is related to the
definition of ellipticity used by us by the relation
$e=\sqrt{1-\tan^{2}\chi}$. Their theoretical model explains the general shape
of the curve in Fig. 13. The dissimilarity between the two minima at
$56^{\circ}$ and $148^{\circ}$ was already present in the fiber used, before
tapering. The asymmetry might be linked to the details of the splicing between
the bare fiber and the patch cable, which is necessary for launching light in
the coupling fiber. In any case, we verified in this experiment that tapering
down the fiber to sub-micron scale did not change its polarization response,
as shown in Fig. 13

We also checked, with a tapered fiber with transmission $>90\%$, if the
dependence of its transmission on the input polarization changes due to
tapering (or if any polarization-dependent losses were introduced due to
tapering). As expected, no change was found due to tapering.

## VII Q-VALUE MEASUREMENTS OF WGMs IN SOLID MICROSPHERES

Figure 14: Transmission spectrum of a microsphere-taper coupled system
(microsphere diameter $\approx 60$ µm). The dips in the signal correspond to
the WGMs of the microsphere. The theoretical Lorentzian fit for one of the
dips has been shown in the inset plot. In the inset plot, the blue marks are
the experiment data points and the red curve is the theoretical fit.

We have used the tapered fibers for efficiently exciting and detecting
high-$Q$ WGMs in silica microspheres of sizes 30 to 300 µm. Fig. 2 shows a
microscope image of a silica microsphere kept very close to a tapered fiber to
excite its WGMs. To couple light in, a microsphere is moved close to a tapered
fiber and a very small gap of sub-wavelength order is maintained between the
two to work in the under-coupled regime. In this regime, the observed/loaded
$Q$ of the cavity tends to the intrinsic/unloaded $Q$ of the cavity because
the external loss due to coupling is minimum and hence the observed $Q$ has
its maximum value in the under-coupled regime (see Ref. Lin, 2010). Fig. 14
shows the transmission spectrum of a microsphere and tapered-fiber coupled
system. The dips in the signal correspond to the WGMs of the microsphere.
Different dips correspond to different WGMs with different field distributions
and angular orientations. Therefore, different modes have different coupling
efficiencies depending on the evanescent field distribution and position and
orientation of the tapered fiber’s waist with respect to the microsphere. The
inset shows a Lorentzian function fitted to one of the dips for measuring the
Quality factor of the mode. We routinely observe $Q$-factors exceeding
$10^{7}$.

## VIII Q-VALUE MEASUREMENTS OF WGMs IN LIQUID MICRODROPS

Figure 15: Cavity ring down signal of a WGM in a silicone oil microdrop.
Fitted parameters, $V$ (frequency scanning speed) = $4.6\times 10^{5}$ GHz/s,
$\tau$ (amplitude lifetime) = 54 ns $\Rightarrow\tau_{photon}$ (photon
lifetime) = $\tau/2$ = 27 ns $\Rightarrow$ $Q=6.5\times 10^{7}$

A microscope image of a silicone oil drop on a silica microsphere is shown in
Fig. 3. We have done experiments with drops of sizes 50 to 200 µm. Unlike the
case of solid cavities, in a liquid drop, thermally induced broad-spectrum
fluctuations of its size and shape affect the resonance signal. The
significantly affecting thermal fluctuations lie in the frequency range from
kHz to 100s of kHz. These thermal fluctuations inhibit the reliable
measurement of the $Q$ values which are greater than or close to the thermal
limits on the measured $Q$ values (see Ref. Lai, Leung, and Young, 1990; Jonáš
_et al._ , 2012; Gaira and Unnikrishnan, 2020; Giorgini  _et al._ , 2017),
when using the standard ‘linewidth measurement’ method. To overcome this, the
laser frequency is scanned fast enough to observe the dynamic interference
between the direct light propagating through the coupling tapered fiber, the
frequency of which is changing very fast, and the out-coupled (stored)
resonant light from the high Q microdrop cavity. For observing dynamic
interference (or cavity ring-down), the laser frequency has to be scanned fast
enough such that the cavity comes and goes out of resonance in duration less
than or close to the decay time of the mode. We use $10^{5}$ to $10^{6}$ GHz/s
scanning speeds for observing the cavity ring-down signals with microdrops of
silicone oils. The ring-down data for a WGM resonance in a silicone oil drop
has been plotted in Fig. 15. The analytical expression for the cavity ring-
down curve of a taper-fiber coupled micro-cavity is given in Ref. Rasoloniaina
_et al._ , 2014; Dumeige  _et al._ , 2008; Ye, Shen, and Lin, 2016. It has
been used for theoretically fitting the experimental data. The fitting gives a
$Q$-value of $6.5\times 10^{7}$.

The observed signal can be explained as follows. As the laser frequency and
the resonance frequency match, the mode builds up in the cavity. Immediately
after that, the laser frequency detunes from the resonance frequency so fast
that we can see the exponential decay of the light stored in the cavity. The
light which is coupling out of the cavity interferes with the laser light
whose frequency is drifting very fast. This results in oscillations in the
transmission signal along with an exponential envelope.

We have made $Q$-measurements for drops of different silicone oils.Gaira and
Unnikrishnan (2020) The plotted data in Fig 15 is with a drop of Sigma-
Aldrich’s product no. 378364 (polydimethylsiloxane with 100 cSt viscosity at
$25^{\circ}C$ ).

## IX SUMMARY

We have described an integrated table-top facility for the study of high-$Q$
WGMs in solid and liquid micro-cavities. The facility is self-contained and
dust-protected with the modular setups for fabrication of micro-cavities and
tapered fibers, and for exciting WGMs in solid and liquid micro-cavities,
which are all built on a single floated optical table and inside a custom-
designed laminar flow assembly. All the required fabrication and experiments
can be done within a small cleanroom-like chamber without the need of
transferring the sensitive and fragile elements from one place to another. We
have explained in detail the fabrication and performance analysis of tapered
fibers of sub-micron waist sizes. The experimental parameters and precautions
that are important for fabricating good quality tapered fibers have been
explained. The adiabaticity of the fibers can be checked while tapering by
observing the transmission signal of the fiber. This also helps in identifying
the stopping point when all the coupled higher order modes cut-off. These cut-
off waist diameters of the tapered fibers are consistent with the theoretical
estimation. The fabricated tapered fibers have greater than 85% transmission
on average and they can completely filter out the unwanted higher order modes.
These tapered fibers have been successfully used for efficiently exciting WGMs
of $Q\sim 10^{7}$ in silica microspheres and of $Q\ \text{up to}\ 7\times
10^{7}$ in microdrops of silicone oils. The $Q$-values of these WGM resonances
were measured using ‘linewidth measurement’ for silica microspheres and using
‘cavity ring-down’ technique for silicone oils’ microdrops. The capability to
track and measure the resonances in microdrops with a wide range of high Q
values promises many interesting studies on these dynamic cavities, using our
integrated experimental facility.

## X ACKNOWLEDGEMENTS

We are thankful to P.G. Rodrigues, P.V. Sudersanan, S.K. Guram and the central
workshop facility in TIFR for the design and fabrication of many components
and units of the setup. M. Gaira thanks S. Selvarajan, N. Jetty and D.B. Dhuri
for useful discussions.

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