Amplified phase shift cavity ring down spectroscopy in linear fiber cavities
for sensing applications at 1550 nm

Ubaid Ullah and M. Imran Cheema ∗

1Department of Electrical Engineering, Syed Babar Ali School of Science and
Engineering, Lahore University of Management Sciences, Lahore 54792 Pakistan

<EMAIL_ADDRESS>

##  1 Abstract

We propose a novel optical solution that exploits the phase shift cavity ring-
down technique to probe samples that induce high background noise at 1550nm.
The proposed setup uses a unique design that is potentially more sensitive
than the traditional fiber cavities. A tapered fiber serves as a sensing head
is inserted inside the all-fiber cavity along with the amplifier. An extensive
theoretical formulation is carried out to explore and optimize every aspect of
the proposed setup and to enhance sensitivity while using off-the-shelf and
easily accessible optical components. A generalized phase delay expression is
derived while assuming tapered fiber is immersed in an analyte of known
refractive index. The theoretical analysis is then followed by careful
experiments. The experimental results are in good agreement with our
theoretical findings. We experimentally measure glucose solution concentration
employing a tapered fiber of length $0.3cm$ and diameter $13\mu m$. The setup
offers the sensitivity of $-0.0695^{\circ}/mM$ and the detection limit $1mM$
for a $0-100mM$ glucose concentration range.

##  2 Introduction

Phase shift-cavity ring down spectroscopy (PS-CRDS) is a multipass sensitive
technique independent of the source intensity fluctuations. In PS-CRDS, a CW
laser source is sinusoidally modulated at an angular frequency, $\omega$, and
is injected into a cavity. The output of the cavity has a phase-shift, $\phi$,
compared to the input. The phase shift is directly related to the cavity ring
down time, $\tau$, as, $tan\phi=-\omega\tau$ [1]. The change in the ring down
time as a function of losses in a sample filled cavity can then be utilized
for various sensing applications. For example, PS-CRDS has been utilized in
numerous sensing applications for detecting gases and particles, including
nitrogen dioxide [2], oxygen [3], ethene [4], acetylene [5], peroxy, organic
nitrates [6], and soot particles ejecting from $V2527$ aircraft engine [7].
All of these applications utilize free space cavities for detection purposes.

Being a highly sensitive technique, PS-CRDS is highly attractive for chemical
identification and biosensing applications. However, free space cavities are
not suitable for liquid phase applications, primarily for a couple of reasons.
One, the cavity becomes highly unstable due to the misalignment of optics,
contamination, and the liquid inside the cavity. Second, the cavity mirrors
degrade over time due to continuous contact with the liquid [8, 9]. To
minimize the impact of these issues, researchers have looked over various
cavity designs in terms of Brewster cavities, thin sample cells, and liquid
jets inside the cavity [10]. Nevertheless, the cavity instability problem is
not entirely solved, and the overall designs are also highly complicated for
developing a viable liquid phase PS-CRDS sensor with free space cavities [9].

The problems mentioned above for liquid phase sensing applications can be
solved by using optical fiber loops. Passive fiber loops in conjunction with
the conventional cavity ring down spectroscopy (CRDS) and PS-CRDS have been
used for various liquid sensing applications [11, 12, 13]. A passive loop
generally contains a tapered fiber [14] or microgap [15] as a sensing head.
The passive loops are typically several tens of meters and require weak
couplers for coupling light into and out of the loop. Liquid phase sensing
applications at 1550nm are highly attractive as various cost-effective, and
off-the-shelf optical components are available to build a viable optical
sensor. One of the major problems with passive loops towards biosensing or
chemical identification applications is that they produce a high background
absorption for aqueous solutions at 1550nm. This problem can be minimized by
using an EDFA amplifier in the fiber loop [16]. CRDS in conjunction with
active fiber loops have been then utilized for various gas and liquid phase
applications [17, 18, 19, 20]. An amplifier in the fiber loop minimizes the
background absorption and increases the overall sensitivity and detection
limit of the sensing system.

Although CRDS in an active loop improves on background liquid absorption at
1550nm, however, gain fluctuations and nonlinearity of the amplifier seriously
impacts the CRDS measurements [21, 22]. It happens because in CRDS, after
every round trip, the pulse strength decreases, and as a result, the amplifier
response varies with varying input powers. This process distorts the
exponential decay curve of the ring down time, and hence noise is added in the
overall measurement.

Compared to fiber loops, a linear cavity is more sensitive for sensing
measurements as the circulating optical field samples an analyte twice in one
round trip, as also clearly shown in Fig. 1. This suggests that if we use PS-
CRDS in linear active cavities, we can not only have a more sensitive approach
at 1550nm but we can avoid the amplifier’s impact on the circulating signal.
This is because in PS-CRDS, not only the signal amplitude remains constant for
all times, but we also primarily measure the phase difference between the
input and output sinusoid and not the signal strength. Therefore, the present
work gives the first and novel scheme of an active linear cavity at 1550nm in
conjunction with PS-CRDS measurements for liquid phase sensing applications.

Figure 1: In one round trip, the propagating optical field in linear cavity
(Fig. (a)) samples the analyte twice but the loop cavity (Fig. (b)) samples it
only once.

This work theoretically determines the optimum tapered fiber diameter in an
active linear cavity to maximize sensitivity for refractometric liquid sensing
applications using PS-CRDS measurements. In literature, various demonstrated
evanescent field sensors use either absorption [23] or refractive index change
[24] of an analyte in response to a sensing event. However, a complete model
will involve both the absorption and refractive index change of the analyte.
Here, we consider the impact of both the absorption and refractive index
changes of the liquid surrounding the tapered fiber on the output PS-CRDS
signal. We experimentally measure glucose solution concentration employing a
tapered fiber of length $0.3cm$ and diameter $13\mu m$. The setup offers the
sensitivity of $-0.0695^{\circ}/mM$ and the detection limit $1mM$ for a
$0-100mM$ glucose concentration range.

##  3 Theoretical Formulation

The schematics of the proposed active linear cavity with PS-CRDS measurements
are shown in Fig. (2). The working principle involves applying a triangular
wave for the current modulating CW tunable laser to linearly sweep the laser
wavelength, ensuring that the cavity resonates periodically. An external Mach-
Zehnder modulator is employed to sinusoidally modulate the laser with a
frequency of $\omega_{0}$ for the PS-CRDS measurements. The modulated light
enters into the optical cavity through FBG 1. The light propagates through the
tapered fiber (sensing head) that induces loss corresponding to the analyte
under test. The forward light encounters the first circulator that guides
light towards a benchtop silicon optical amplifier through port $2\rightarrow
3$. The amplified signal completes half of the round trip after passing
through port $1\rightarrow 2$ of the second circulator and reaches FBG 2. A
small portion of the light is transmitted while a large portion gets
reflected. The reflected light passes through port $2\rightarrow 3$ of
circulator 2, then through port $1\rightarrow 2$ of circulator 1, transverses
tapered fiber, and finally reaches FBG 1, where it completes one round trip.
Upon reflection from FBG 1, the second round trip starts. The transmitted
light enters into the lock-in amplifier that provides the phase shift between
cavity transmission and reference modulation signals. At the resonance, cavity
build-up and relaxation time generate a maximum phase delay, which is measured
as a phase dip and a transmission signal maxima on the lock-in amplifier. The
out-of-resonance signal corresponds to transmission minima and phase maxima.
On increasing the amplifier gain, losses decrease in the cavity, and thus the
phase shift increases. Note that as per the PS-CRDS equation,
$\tan\phi=-\omega t$, the output signal lags the input signal at a resonant
wavelength.

Figure 2: Schematics of the proposed experimental setup.

The signal flow diagram of the field amplitude in the cavity of the proposed
setup is shown in Fig. (3), and the physical parameters of the optical
components used in cavity construction are shown in Tab. (1). The overall
transmission field is the superposition of several transmission terms, which
denote the throughput of the cavity after successive round trips. The
geometric series representing the cavity transmission is given by equations
(1-2).

Figure 3: Signal flow diagram of the field amplified inside the optical cavity. Physical Parameter |  Symbol |  Numerical Value [Unit]
---|---|---
Reflectivity of FBG 1 |  $R_{1}$ |  0.8568
Reflectivity of FBG 2 |  $R_{2}$ |  0.9992
Mode index in tapered fiber |  $n_{tap}$ |  $n_{tap}(n_{sol},r_{t})$
Mode index in optical fiber |  $n_{f}$ |  1.4457
Speed of light in free space |  $c$ |  $2.99792458\times 10^{8}$ [m/sec]
Modulation frequency |  $f_{m}$ |  3 [MHz]
Laser wavelength |  $\lambda_{0}$ |  1550 [nm]
Insertion loss of port 1$\rightarrow$2 (c1) |  $L_{1\rightarrow 2}^{c1}$ |  -0.8778 [dB]
Insertion loss of port 2$\rightarrow$3 (c1) |  $L_{2\rightarrow 3}^{c1}$ |  -0.7857 [dB]
Insertion loss of port 1$\rightarrow$2 (c2) |  $L_{1\rightarrow 2}^{c2}$ |  -0.57 [dB]
Insertion loss of port 2$\rightarrow$3 (c2) |  $L_{2\rightarrow 3}^{c2}$ |  -0.60 [dB]
Fiber connector loss |  $L_{cn}$ |  -0.15 [dB]
Tapered fiber constant loss |  $L_{t}=L_{cons}+L_{ab}$, $L_{cons}=$ |  -2 [dB]
Tapered fiber absorption loss |  $L_{t}=L_{cons}+L_{ab}$, $L_{ab}=$ |  $L_{ab}(\alpha,l_{t})$ [dB]
Length of tapered fiber |  $l_{t}$ |  0.3 [cm]
Length of the fiber in forward direction |  $l_{f}=l_{FBG1}+l_{taper}+l_{2\rightarrow 3}^{c1}+l_{amp}+l_{1\rightarrow 2}^{c2}+l_{FBG2}$ |  883.7 [cm]
Length of the fiber in reverse direction |  $l_{b}=l_{FBG2}+l_{2\rightarrow 3}^{c2}+l_{1\rightarrow 2}^{c1}+l_{taper}+l_{FBG1}$ |  583.7 [cm]
Table 1: Physical properties of the optical cavity involved in the proposed
setup.

$A_{t}=A_{1}+A_{2}+A_{3}+..............$ (1)
$\begin{split}A_{t}=\sqrt{L_{t}L_{cn}^{5}L_{c1}^{2\rightarrow
3}L_{c2}^{1\rightarrow
2}G}t_{1}t_{2}A_{i}e^{-\iota\delta_{f}}\bigg{[}1+\sqrt{L_{t}^{2}L_{cn}^{9}L_{c1}^{1\rightarrow
2}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow 2}L_{c2}^{2\rightarrow 3}G}\\\
r_{1}r_{2}e^{-i(\delta_{f}+\delta_{b})}+\bigg{(}\sqrt{L_{t}^{2}L_{cn}^{9}L_{c1}^{1\rightarrow
2}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow 2}L_{c2}^{2\rightarrow
3}G}r_{1}r_{2}e^{-i(\delta_{f}+\delta_{b})}\bigg{)}^{2}+.....\bigg{]}\end{split}$
(2)

where $\delta_{f}$ denotes phase delay in the forward propagation while
$\delta_{b}$ denotes phase delay in the reverse propagation, which makes a
complete round trip inside an optical cavity. The following expressions define
$\delta_{f}$ and $\delta_{b}$, respectively.

$\begin{split}\delta_{f}=\frac{2\pi n_{f}l_{f}}{\lambda_{0}}+\frac{2\pi
n_{t}l_{t}}{\lambda_{0}}\\\ \delta_{b}=\frac{2\pi
n_{f}l_{b}}{\lambda_{0}}+\frac{2\pi n_{t}l_{t}}{\lambda_{0}}\end{split}$ (3)

We use the geometric series’s following property to simplify and rewrite the
transmission field equation in a closed-form expression.

$1+\chi+\chi^{2}+\chi^{3}+.......=1/\big{(}1-\chi)\hskip 28.45274ptfor\hskip
8.5359pt0<\chi<1$ (4)

The transmission field equation is then simplified to the following form:

$A_{t}=\frac{\sqrt{L_{t}L_{cn}^{5}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow
2}G}t_{1}t_{2}A_{i}e^{-\iota\delta_{f}}}{1-\sqrt{L_{t}^{2}L_{cn}^{9}L_{c1}^{1\rightarrow
2}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow 2}L_{c2}^{2\rightarrow
3}G}r_{1}r_{2}e^{-i(\delta_{f}+\delta_{b})}}$ (5)

where,

$\chi=\sqrt{L_{t}^{2}L_{cn}^{9}L_{c1}^{1\rightarrow 2}L_{c1}^{2\rightarrow
3}L_{c2}^{1\rightarrow 2}L_{c2}^{2\rightarrow 3}G}r_{1}r_{2}$ (6)

Now the transmittance $T$ is defined as the ratio of transmitted intensity
($A_{t}A_{t}^{*}$) to the input intensity ($A_{i}A_{i}^{*}$) and is given by:

$T=\frac{L_{cn}^{5}L_{t}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow
2}G\big{(}t_{1}t_{2}\big{)}^{2}}{\begin{split}\bigg{(}1-\sqrt{L_{t}^{2}L_{cn}^{9}L_{c1}^{1\rightarrow
2}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow 2}L_{c2}^{2\rightarrow
3}G}r_{1}r_{2}\bigg{)}^{2}\\\ +4\sqrt{L_{t}^{2}L_{cn}^{9}L_{c1}^{1\rightarrow
2}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow 2}L_{c2}^{2\rightarrow
3}G}r_{1}r_{2}\sin^{2}((\delta_{f}+\delta_{b})/2)\end{split}}$ (7)

The transmission and reflection coefficients are defined as follows:

$r_{1}=\sqrt{R_{1}},\hskip 8.5359ptr_{2}=\sqrt{R_{2}},\hskip
8.5359pt,t_{1}=\sqrt{1-R_{1}},\hskip 8.5359ptt_{2}=\sqrt{1-R_{2}}$ (8)

We obtain the following final transmission expression by substituting the
above mentioned parameters:

$T=\frac{L_{cn}^{5}L_{t}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow
2}G(1-R_{1})(1-R_{2})}{\bigg{[}\begin{split}\bigg{(}1-\sqrt{L_{t}^{2}L_{cn}^{9}L_{c1}^{1\rightarrow
2}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow 2}L_{c2}^{2\rightarrow
3}G}r_{1}r_{2}\bigg{)}^{2}\\\ +4\sqrt{L_{t}^{2}L_{cn}^{9}L_{c1}^{1\rightarrow
2}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow 2}L_{c2}^{2\rightarrow
3}G}\sqrt{R_{1}R_{2}}\sin^{2}((\delta_{f}+\delta_{b})/2)\bigg{]}\end{split}}$
(9)

We can see that if we assume that gain and losses are equal and
$R_{1}=R_{2}=R$ then Eq. (9) reduces to the following standard Fabry-Perot
equation:

$T=\frac{(1-R)^{2}}{(1-R)^{2}+4R\sin^{2}((\delta_{f}+\delta_{b})/2)}$ (10)

This suggests the correctness of our derived expression. We are also
interested in the full width half maximum (FWHM) of our proposed active cavity
as FWHM will help us determine the phase expression for PS-CRDS calculations.
The maximum possible transmission of the proposed setup is given by the
following equation:

$T_{0}=\frac{L_{cn}^{5}L_{t}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow
2}G(1-R_{1})(1-R_{2})}{\bigg{(}1-\sqrt{L_{t}^{2}L_{cn}^{9}L_{c1}^{1\rightarrow
2}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow 2}L_{c2}^{2\rightarrow
3}GR_{1}R_{2}}\bigg{)}^{2}}$ (11)

From equation (9) and equation (11), the following condition needs to be
satisfied for determining the half width half maximum, HWHM (represented by
subscript 1/4), :

$\frac{L_{cn}^{5}L_{t}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow
2}G(1-R_{1})(1-R_{2})}{2*\bigg{(}1-\sqrt{L_{cn}^{9}L_{t}^{2}L_{c1}^{1\rightarrow
2}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow 2}L_{c2}^{2\rightarrow
3}GR_{1}R_{2}}\bigg{)}^{2}}=\frac{L_{cn}^{5}L_{t}L_{c1}^{2\rightarrow
3}L_{c2}^{1\rightarrow
2}G(1-R_{1})(1-R_{2})}{\begin{split}\bigg{(}1-\sqrt{L_{cn}^{9}L_{t}^{2}L_{c1}^{1\rightarrow
2}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow 2}L_{c2}^{2\rightarrow
3}G}r_{1}r_{2}\bigg{)}^{2}\\\ +4\sqrt{L_{cn}^{9}L_{t}^{2}L_{c1}^{1\rightarrow
2}L_{c1}^{2\rightarrow 3}L_{c2}^{1\rightarrow 2}L_{c2}^{2\rightarrow
3}G}\sqrt{R_{1}R_{2}}\\\
\sin^{2}\bigg{(}\frac{(\delta_{f}+\delta_{b})_{1/4}}{2}\bigg{)}\hskip
85.35826pt\end{split}}$ (12)

It can easily be shown that equation (12) reduces to the following form:

$\sin\bigg{(}\frac{(\delta_{f}+\delta_{b})_{1/4}}{2}\bigg{)}=\frac{1}{2}\bigg{[}\frac{1}{\sqrt{\chi}}-\sqrt{\chi}\bigg{]}$
(13)

where $\chi$ is given by equation (6). In the proposed active Fabry-Perot
cavity, the transmission peaks will be much narrower than the FSR of the
cavity, therefore, assuming
$\dfrac{(\delta_{f}+\delta_{b})_{1/4}}{2}<<\dfrac{\pi}{2}$, then equation (13)
can be written as:

$\bigg{(}\frac{(\delta_{f}+\delta_{b})_{1/4}}{2}\bigg{)}=\dfrac{1}{2}\bigg{[}\frac{1}{\sqrt{\chi}}-\sqrt{\chi}\bigg{]}$
(14)

Compared to HWHM, FWHM (represented by subscript 1/2) is two times in width,
therefore,

$(\delta_{f}+\delta_{b})_{1/2}=\Delta\delta_{1/2}=2\bigg{[}\frac{1}{\sqrt{\chi}}-\sqrt{\chi}\bigg{]}$
(15)

In the wavelength domain, the following expression represents FWHM:

$\Delta\lambda_{1/2}=\frac{\lambda_{0}^{2}\big{(}\frac{1}{\sqrt{\chi}}-\sqrt{\chi}\big{)}}{\pi\big{[}n_{f}(l_{f}+l_{b})+2n_{tap}l_{t}\big{]}}$
(16)

For a standard passive Fabry-Perot cavity (see Fig. 1a) based on two FBGs with
$R_{1}=R_{2}=R$ and without any tapered fiber including $l_{f}=l_{b}=l$, then
equation 16 reduces to the standard FWHM expression:

$\Delta\lambda_{1/2}=\dfrac{\lambda_{0}^{2}(1-R)}{2\pi n_{f}l\sqrt{R}}$ (17)

The loss of the cavity is directly proportional to FWHM. If the gain is equal
to all losses in the cavity and $R_{1}=R_{2}=1$ then it represents a lossless
cavity. FWHM will be zero in this ideal cavity, as can also be seen from
equation (16).

$\Delta\lambda_{1/2}=\frac{\lambda_{0}^{2}(1-1)}{\pi\big{[}n_{f}(l_{f}+l_{b})+2n_{tap}l_{t}\big{]}}=0,\hskip
8.5359pt\textit{As}\hskip 4.26773pt\sqrt{\chi}=1$ (18)

On the other extreme, if we assume that maximum possible loss inside the
cavity, i.e., $R_{1}=R_{2}=0$ , then FWHM will become infinite as can be seen
in the following:

$\Delta\lambda_{1/2}=\infty\hskip 8.5359ptWhen\hskip
8.5359pt\lim_{\sqrt{\chi}\to 0}\frac{1}{\sqrt{\chi}}\to\infty$ (19)

The results in equations (17-19) suggest the correctness of the FWHM
expression in equation (16).

###  3.1 PS-CRDS sensing principle of the proposed cavity

The proposed setup employs an amplitude modulated laser source to excite the
cavity. The light inside the cavity passes through the tapered fiber, optical
circulators, and amplifier multiple times before it decays down to zero. The
decay time is related to the phase delay between the cavity output and the
reference modulating signal according to the following PS-CRDS relationship
[1]:

$\tan\phi=-\omega_{m}\tau$ (20)

where $\omega_{m}$ is the modulation frequency. The amplifier gain is used for
overcoming background absorption that is usually induced by liquids at 1550nm.
The ring down time will increase with an increase in the gain, and hence the
phase, $\phi$, will also increase.

A change in the analyte concentration represents a change in the tapered
fiber’s surrounding refractive index (RI). Therefore, when the concentration
increases, the optical mode is pushed out of the tapered fiber and experiences
a loss. Simultaneously, the analyte concentration will also induce absorption
loss to the evanescent field of the propagating mode. These losses decrease
the cavity’s ring down time, and hence the phase, $\phi$, also reduces. The
ring down time of a cavity is also related to its FWHM, as can be easily shown
from energy conservation principles [25],

$\dfrac{\lambda_{0}}{\Delta\lambda_{1/2}}=\omega_{0}\tau$ (21)

where $\omega_{0}$ is the optical frequency. Substituting $\tau$ from equation
(21) in equation 20, we obtain

$\phi=\tan^{-1}\bigg{(}\frac{f_{m}\lambda_{0}^{2}}{(c/n_{f})\Delta\lambda_{1/2}}\bigg{)}$
(22)

where $\Delta\lambda_{1/2}$ is given by equation (16).

##  4 Simulation results

We assume that a tapered fiber is immersed in aqueous glucose solutions of
various concentrations for generating simulation results. We use the following
expression for determining refractive indices of glucose solutions [26]:

$n_{sol}=x_{0}(\lambda)+y_{0}(\lambda)C_{g}$ (23)

where $x_{0}=1.318$ is the refractive index of DI water at room temperature
and $1550nm$ [27], $y_{0}=6.5998\times 10^{-4}$ denotes a conversion slope
between glucose concentration and solution refractive indices, and $C_{g}$
stands for glucose concentration measured in mM. These values and equation
(23) indicate that the glucose solution’s refractive index increases with an
increase in glucose. Therefore, when we immerse a tapered fiber in the
solution, the tapered fiber’s leaked evanescent field increases as the glucose
concentration increases—consequently, the ring down time and hence phase,
$\phi$ changes.

The tapered fiber diameter also influences the evanescent field strength; the
thinner the tapered fiber, the stronger the evanescent field and vice versa.
This suggests that for achieving maximum sensitivity, there will be an
interplay between the tapered fiber diameter and the analyte concentration as
both impact the evanescent field strength. Hence, an optimum tapered fiber
diameter may be present for achieving high sensitivity for detecting glucose
concentrations. To explore the optimized tapered fiber diameter, let us look
at the fraction of the evanescent field outside the tapered fiber, i.e., the
ratio of evanescent field power to the total optical power in a cross-
sectional area of the tapered fiber, and is given below:

$\Gamma=\frac{\int_{r_{t}}^{\infty}P(r,\phi)rdrd\phi}{\int_{0}^{\infty}P(r,\phi)rdrd\phi}$
(24)

Figure 4: The evanescent field changes for given RI changes as a function of
tapered fibers radii. Each data point is generated for a refractive index
change of ($10^{-4}-10^{-2}$).

We use the finite element method to determine $\Gamma$ of a range of tapered
fiber diameters for various lossless glucose concentration solutions. For each
tapered fiber radius, we obtain a linear curve of $\Gamma$ as a function of
glucose concentrations. We then plot each curve slope against the tapered
fiber’s corresponding radius, as shown in Fig. (4). Each data point is
generated for refractive index changes of $10^{-4}-10^{-2}$. It is clear from
the plot that as we decrease the tapered fiber’s radius from a relatively
larger value, there is a gradual increase in the sensitivity. The sensitivity
increases drastically when the tapered fiber’s diameter is near the source
wavelength. The increase in the sensitivity trend keeps going up to a critical
tapered fiber radius, after which this trend flips, and the sensitivity starts
decreasing with a decrease in the tapered fiber radius. The critical tapered
fiber radius is also specific to the analyte.

Physically, as the tapered fiber’s radius decreases from $10\mu m$ the
evanescent field and the power density inside the tapered fiber increase. The
increase in the power density ensures relatively large changes in $\Gamma$ for
small refractive index changes. Near the critical radius, the power density
increases dramatically as the fiber’s radius becomes comparable to the source
wavelength. At this point, changes in $\Gamma$ due to refractive index changes
are also maximum. The critical radius is $0.55\mu m$ for glucose solution, as
shown in Fig. (4). The propagating mode becomes loosely intact with the
tapered fiber after the critical radius, and hence the change in $\Gamma$ due
to the change in the refractive index of the solution decreases.

After looking at the impact on sensitivity due to analyte concentrations and
tapered fiber diameter, let us look at the influence of glucose absorption
losses on the sensitivity. The tapered fiber loss can be due to static and/or
dynamic losses. Static losses are constant and occur due to structural
deformations and impurities accumulation during the taper drawing process. On
the other hand, the dynamic losses vary with the analyte concentrations. These
include absorption due to the analyte as well as background water. The
following expression gives the total absorption coefficient, $\alpha$, of a
glucose aqueous solution relative to air [28]:

$\alpha=\Gamma\alpha_{w}C_{w}+\Gamma\alpha_{g}C_{g}-\Gamma\alpha_{w}f_{w}^{g}C_{g}$
(25)

where $\alpha_{w}$ is the absorption coefficient of DI water, $\alpha_{g}$
absorption coefficient of glucose, $C_{w}$ is the DI water concentration,
$C_{g}$ is the glucose concentration, and $f_{w}^{g}$ is the water
displacement coefficient of glucose. In equation (25), the first and second
terms represent absorption due to water and glucose. The last term deducts a
portion of water absorption from that solution part, which is replaced by
glucose. It is mandatory to include refractive index changes to understand the
whole picture of a solution’s absorption process. The increase in the
concentration of a solution brings two effects in the picture, i.e., a change
in the refractive index and a change in the solution’s absorption.
Furthermore, the refractive index’s increase enhances the evanescent field and
a corresponding increase in $\Gamma$ of the tapered fiber. The large
evanescent field interacts with a large solution portion, and hence, the
absorption loss increases, as shown in equation (25). In parallel, an increase
in glucose concentration also changes the absorption loss of the solution such
that, if $\alpha_{g}<\alpha_{w}f_{w}^{g}$, there is a decrease in the
absorption loss while for $\alpha_{g}>\alpha_{w}f_{w}^{g}$ the absorption loss
increases. The solution’s net absorption loss is a combined effect of an
increase in the loss due to the evanescent field and a change in the
absorption loss due to increased glucose concentration.

We obtain a linear curve of $\Gamma$ as a function of glucose concentrations.
We then plot each curve slope against the tapered fiber’s corresponding
radius, as shown in Fig. (4). Each data point is generated for refractive
index changes of $10^{-4}-10^{-2}$.

For the current work, we use $\alpha_{w}=1.45\times 10^{-4}(mM)^{-1}cm^{-1}$
[27], $C_{w}=55135mM$, $C_{g}=(0-15.15)mM$, $\alpha_{g}=7.23\times
10^{-4}(mM)^{-1}cm^{-1}$[28], and $f_{w}^{g}=6.245$ [28]. We again employ
finite element method simulations along with equations (24-25) and obtain
linear curves of $\alpha$ as a function of glucose concentrations. We then
plot each curve slope against the tapered fiber’s corresponding radius, as
shown in Fig. (5). The blue plot considers only water absorption loss, i.e.,
an ideal solution that has the same absorption coefficient as water but
induces refractive index changes equivalent to glucose solutions. In contrast,
the orange plot considers both water and glucose absorption losses for glucose
solutions. In the case of glucose solution $\alpha_{g}<\alpha_{w}f_{w}^{g}$,
which indicates that the addition of glucose will reduce the solution’s
absorption loss compared to pure DI water [29]. Numerically, it is also
evident from Fig. (5) inset. In the plot, it is clear that the absorption loss
is always lower when we also include glucose absorption loss as compared to
water with only refractive index changes. We assume a glucose concentration of
$(0-15.15mM)$ in our analysis. As discussed earlier, the absorption loss
coefficient has two sources. In the case of glucose solution, the prominent
contribution is due to the increase in the refractive index with glucose
concentration.

Figure 5: The overall tapered fiber absorption changes (see equation (25)) for
given RI changes as a function of tapered fibers radii. Each data point is
generated for a glucose solution of concentrations $(0-15.15)mM$.

The following expression now represents the overall loss experienced by the
tapered fiber:

$L_{ab}=e^{-\alpha l_{t}}$ (26)

where $l_{t}$ is the tapered fiber length. The overall tapered fiber loss is
represented by $L_{t}=L_{cons}+L_{ab}$. Now using parameters listed in Tab.
(1), previously mentioned finite element method simulations, and equations
(19,22-26), we determine phase, $\phi$, for various glucose concentrations at
different amplifier gains. A representative result of $\phi$ as a function of
glucose concentrations at an amplifier gain is shown in Fig. (6).

Figure 6: Simulated phase shift as a function of glucose concentration for
1.1$\mu m$ diameter tapered fiber at 6dB amplifier gain

We also plot the phase shift for various glucose concentrations against the
amplifier gain, as shown in Fig. (7). We assume the optimum tapered fiber of
radius, $r_{t}=0.5\mu m$ in these simulation results.

Figure 7: Simulated phase shift as a function of the amplifier gain. Each data
point is generated for a glucose solution of concentration $(0-15.15mM)$.

##  5 Experimental Results

For the proposed active cavity setup, this section provides various
experimental results including determining the amount of glucose in aqueous
solutions.

###  5.1 Phase and amplitude peaks as a function of the wavelength scan

A representative amplified PS-CRDS measurement for the active cavity setup
(see Fig. (2)) is shown in Fig. (8).

Figure 8: Representative amplified PS-CRDS experimental data. The black dotted
line is the wavelength scan signal. The blue and red curves are amplitude and
phase data, respectively. The circles on the phase curve are data points for
determining the phase for a glucose concentration.

We notice that resonance peaks are not synchronized with phase dips in
wavelength positions; instead, phase minimum is red and blue-shifted during
the decreasing and increasing wavelength scans, respectively. We will use the
coupled mode (CM) theory to explain this behavior. The fundamental CM equation
for a linear optical cavity is given below [30]:

$\frac{dE_{cav}}{dt}=j\omega_{0m}E_{cav}-\bigg{(}\frac{1}{\tau}\bigg{)}E_{cav}+\kappa
E_{ip}$ (27)

where $E_{cav}$ denotes the cavity field, $\omega_{0}$ represents the cavity
resonance frequency, $\tau$ is the cavity ring down time, $\kappa$ is the
field coupling coefficient at FBG1, and $E_{ip}$ is the cavity input field.
The ring down time, $\tau$, can be represented as:

$\frac{1}{\tau}=\frac{1}{\tau_{0}}+\frac{1}{\tau_{e1}}+\frac{1}{\tau_{e2}}$
(28)

where $\tau_{o}$ represents photon life time due to internal losses of the
cavity, e.g., in our case it will represent losses due to intrinsic SMF-28
fiber attenuation and losses due to the tapered fiber and associated sensing
mechanisms. The coupling life times or photon lifetimes,
$\tau_{e1},\tau_{e2}$, are due to transmission losses at FBG1 and FBG2,
respectively and are given by the following relations:

$\tau_{e1}=\frac{n_{f}l}{c(1-R_{1})}$ (29)

$\tau_{e2}=\frac{n_{f}l}{c(1-R_{2})}$ (30)

where $l$ is the cavity length. The coupling coefficient, $\kappa$, is then
given by the following equation [30]:

$\sqrt{\frac{2}{\tau_{e1}}}$ (31)

In PS-CRDS setup, an optical cavity is excited by a modulated laser light with
modulation frequency $\omega_{m}$ and optical frequency $\omega$. Note that
$\omega$ is the scan frequency of the current tuned laser. The input field,
$E_{ip}$ can then be modeled as the positive optical frequency and positive
modulation frequency component of $E_{in}cos(\omega_{m}t)cos(\omega)$ as given
below:

$E_{ip}=\frac{e^{j(\omega+\omega_{m})t}}{4}E_{in}$ (32)

We can now solve differential equation (27) with the sinusoidal input,
$E_{ip}$, and obtain the following result:

$E_{cav}=\sqrt{\frac{2}{\tau_{e1}}}\frac{e^{j(\omega+\omega_{m})t}-e^{\left(j\omega_{0}-\frac{1}{\tau}t\right)}}{(1+j\tau(\omega+\omega_{m}-\omega_{0}))}E_{in}$
(33)

At the steady state, the cavity field reduces to the following form:

$E_{cav}=\frac{\sqrt{\frac{2}{\tau_{e1}}}\tau
e^{j(\omega+\omega_{m})t}}{4(1+j\tau(\omega+\omega_{m}-\omega_{0}))}E_{in}$
(34)

We can find the cavity output field by multiplying the cavity field, equation
(34) with the FBG2 coupling coefficient and is given below:

$E_{op}=\kappa_{e2}E_{cav}=\sqrt{\frac{2}{\tau_{e2}}}E_{cav}=\frac{\tau
e^{j(\omega+\omega_{m})t}}{2\sqrt{\tau_{e1}\tau_{e2}}(1+j\tau(\omega+\omega_{m}-\omega_{0}))}E_{in}$
(35)

Using equations (29-30), the magnitude of the output field is given by the
following equation.

$\lvert
E_{op}\rvert=\frac{c\tau\sqrt{(1-R_{1})(1-R_{2})}}{2n_{f}l\sqrt{(1+\tau^{2}(\omega+\omega_{m}-\omega_{0})^{2})}}\lvert
E_{in}\rvert$ (36)

We now obtain the PS-CRDS phase delay, $\phi$, by taking the ratio of
imaginary to real parts of equation (35), i.e.,

$tan\phi=-(\omega+\omega_{m}-\omega_{0})\tau$ (37)

At a resonant frequency, $\omega=\omega_{0}$, equations (36)-(36) reduce to
the standard PS-CRDS equations [3]. A physical interpretation of equations
(36-37) explains the red and blue shifts of phase dips relative to resonance
peaks. As shown in equation (36), maximum amplitude $\lvert E_{op}\rvert$
occurs at $\omega+\omega_{m}=\omega_{0}$ for a resonance peak. While it is
evident from equation (37) that a phase dip maxima requires
$\omega+\omega_{m}>\omega_{0}$ condition. In the upward frequency scan, the
condition $\omega+\omega_{m}>\omega_{0}$ is true after the resonance peak
point, i.e., $\omega+\omega_{m}>\omega_{0}$ near the vicinity of the resonance
peak and hence the blue shift in wavelength. Similarly, in the frequency
downward scan, $\omega+\omega_{m}>\omega_{0}$ becomes true before the
resonance peak point, i.e., $\omega+\omega_{0}>\omega_{0}$ and hence the red
shift in wavelength. It should be noted that the trend of wavelength shifts of
phase dips with respect to amplitude peaks is true for both passive and active
cavities in which PS-CRDS measurements are conducted by wavelength tuning the
laser source.

###  5.2 Phase as a function of the amplifier gain

Here, we look at the amplifier gain impact on the PS-CRDS phase measurements.
The increase in the amplifier gain increases the cavity phase delay as the
light will stay longer inside the cavity. Fig. (9) shows the increase in the
phase delay with respect to amplifier gain for a tapered fiber of diameter
$13\mu m$ placed in the open air. The amplifier gain can compensate for losses
inside the cavity and bring the sensing event into the detection limit that
may be masked by the strong background absorption at 1550nm.

Figure 9: Experimentally observed phase shift as a function of the amplifier
gain. We use a tapered fiber of diameter $\approx 13\mu m$ and is placed in
air. All other experimental parameters are tabulated in Tab. (1).

###  5.3 Detection of glucose in aqueous solutions

This section presents our experimental results of sensing glucose in aqueous
solutions using our amplified PS-CRDS setup. First, we prepare various
concentrations of glucose solutions. These solutions are then injected into a
custom-made fluidic cell containing the tapered fiber via a standard syringe.
We wash the tapered fiber and the fluidic cell thoroughly with DI water before
measurements. The laser is current tuned by a triangular signal of amplitude
$100mV_{pp}$ and frequency $10Hz$. The current tuning enables the laser
wavelength to be swept over a range of $45pm$ and centered at $1550nm$ to
obtain the PS-CRDS data, as shown in the representative Fig. (8). We obtain
absolute phase shifts by subtracting the phase minimum and maximum for a
resonance peak as shown with black circles in Fig. (8). The absolute phases
are then averaged to produce a single-phase measurement for an injected
sample. To produce the error bars, we repeat measurements three times for each
glucose concentration. The measured phase delay for various glucose
concentrations is shown in Fig. (10). From the experimental data, we obtain
the sensitivity and detection limit of $-0.0695degree/mM$ and $1mM$,
respectively.

Figure 10: Experimental results for PS-CRDS phase shift as a function of
glucose concentrations.

##  6 Conclusions

We develop novel ideas to enhance the sensitivity of the amplified PS-CRDS
setup employing tapered fiber as a sensing head. The proposed setup allows
light to transverse through the tapered fiber twice in one round trip and
doubles the sensitivity compared to the conventional fiber loops. To further
enhance the sensitivity, we present a theoretical model that accounts for both
absorption and refractive index changes with the analyte concentration
changes. The model can predict an optimum diameter of the tapered fiber,
maximizing the sensitivity for a specific analyte. We anticipate that this
work will find many applications in biosensing and chemical analysis.

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