2.1. Summary of Theory
As described in more detail in our theoretical treatment [
15], the fiber that couples light into and out of the microresonator is tapered to a thin waist upon which two modes are excited owing to the nonadiabatic downtaper transition. The incoupling strength and outcoupling loss are assumed to be equal (we showed in [
15] that this assumption is not restrictive) and given by
T1 for the fundamental fiber mode and by
T2 for the higher-order fiber mode. The two fiber modes have the same frequency and couple to a single microresonator WGM that has intrinsic loss
αL, but they have different propagation constants and different amplitudes. The ratio of amplitudes, higher-order to fundamental, is denoted by
m. When the frequency is tuned to WGM resonance, the throughput power depends on the relative phase of the two input modes, which can be selected by moving the microresonator along the fiber waist. The throughput power relative to the input power is given, on resonance, by
where
R+ and
R− denote the relative throughput with the two fiber modes in phase and out of phase, respectively. These throughput powers relative to the input power in the fundamental fiber mode are measured by comparing to the value far off resonance, where no light couples into the WGM; the light in both fiber modes then encounters the adiabatic uptaper in which light cannot couple between modes, so only the light in the fundamental mode gets transmitted into the single-mode untapered fiber, and thus,
R = 1 [
15]. Under the conditions that lead to large sensing enhancement, a shallow dip is observed for in-phase coupling (
R+ slightly less than 1), and a small peak occurs with out-of-phase coupling (
R− slightly greater than 1). This means that
with values of the three parameters increasing from left to right. Sensing is carried out with in-phase coupling, where the relative dip depth is given by
. As a function of the intrinsic loss, the dip depth is
The introduction of an absorbing analyte changes the effective intrinsic loss, resulting in a fractional change in the dip depth. For our case of a hollow bottle resonator [
16,
17] filled with methanol, the effective intrinsic loss is
which, upon introducing the analyte, becomes
where
αi is the intrinsic loss coefficient of the microresonator that includes scattering, absorption, and radiation losses,
αs is the absorption coefficient of the solvent that contains the analyte in a typical sensing experiment, and
αa is the absorption coefficient of the analyte. The microresonator circumference is
L, and
f is the internal evanescent (interacting) fraction [
5] of the WGM that interacts with the solvent and analyte. The change in effective internal loss is due to the addition of the analyte:
dαL =
fαaL.
When the fractional change in the throughput, |Δ
R+/
R+|, is small, the change in
M will be linear in
dαL, and the fractional change in the dip depth is positive (the dip gets deeper with an increasing analyte concentration) and can be written as
This will be the case for two-mode input, where
R+ remains near 1. Note, however, that for one-mode input using an adiabatically tapered fiber with the same waist radius, setting
m = 0 in Equation (1) results in
. For small enough analyte concentrations (as will be seen in the
Section 3), the change in
M will also be linear in the one-mode case, and thus, the sensitivity enhancement will be given by the ratio of Equation (6) with arbitrary
m to Equation (6) with
m = 0:
where
η21 is the sensitivity enhancement factor for two-mode input relative to one-mode input. Given the conditions of Equation (2), this enhancement can be quite large (>1000). The enhancement factor of Equation (7) can be written in more compact form in terms of the values of
R+ and
R−:
If |Δ
R+/
R+| is not small, or if we wish to check the linearity of the dip-depth response, we insert Equation (5) into Equation (3) and replace the
dM/
M found as in Equation (6) for
m ≠ 0 with
Then,
η21 is equal to the ratio of Equation (9) to the
dM/
M of Equation (6) with
m = 0.
To compare the two-mode dip-depth sensing response to the linewidth sensing response, which is the same in the two-mode and one-mode cases, begin with the relationship between the linewidth and the total loss:
where
c is the speed of light,
n is the effective refractive index of the WGM, and
a is the microresonator radius. From this, it is clear that
and so the enhancement of dip-depth sensing relative to linewidth sensing in the two-mode case is given by the ratio of Equation (6) to Equation (11):
which can be on the order of 100.
Finally, we note that neither enhancement factor,
η21 or
ηdl, depends on the quality factor
Q. However, the absolute sensitivity will increase as
Q does, as is evident from the inverse dependence on the total loss in Equations (6) and (11). In [
15], we showed that in the
ideal one-mode case (
m = 0,
T2 = 0), the intrinsic
Q1i would have to be at least ~100 times larger than the loaded two-mode
Q2 to have the same absolute dip-depth sensitivity:
where
x =
T1/
αL. Similarly, for the ideal one-mode case to have a linewidth sensitivity equal to the two-mode dip-depth sensitivity, the criterion is even stronger:
meaning that the loaded one-mode
Q1 needs to exceed the loaded two-mode
Q2 by two orders of magnitude.
2.2. Experimental Setup and Procedure
As noted earlier, we detect the response (change in the dip depth or in the linewidth of a WGM) resulting from absorption by various concentrations of an analyte (SDC 2072 dye from H. W. Sands Corp., Jupiter, FL, USA) dissolved in methanol and interacting with the internal fraction
f of a 1550 nm WGM in a hollow bottle resonator (HBR) [
16,
17]. Before using the microresonator, the absorption coefficients of the solvent and analyte were measured in a 2 mm cuvette. We found that
αs = 8.76 cm
−1, and that
αa = 4.02 cm
−1 at a concentration of 1 micromolar (µM), varying linearly with the concentration. Since this value of
αs would not allow for the observed
Q values on the order of 10
7, the methanol absorption must be saturating in the microresonator. We checked this by focusing the laser beam through the cuvette, producing a peak intensity that we estimated to be less than that of the interacting fraction of the WGM. Under these conditions, we found
αs = 2.2 × 10
–2 cm
−1, confirming the saturation;
αa was unchanged.
An illustration of the experimental setup for dye absorption sensing is shown in
Figure 1. A tunable diode laser spanning a wavelength range from 1508 nm to 1580 nm is used as the light source. A function generator FG is used to scan the laser in frequency. Before the light passes through a set of waveplates (WP), the beam passes through an anamorphic prism (AP) and an optical isolator (OI). The waveplates are used to select WGMs of one polarization. A fiber coupler (FC) is used to couple light into the tapered fiber, and a fiber isolator is used to prevent any back reflections arising from the tapered fiber. The light then travels through the tapered fiber and couples into and out of the microresonator. The signal is extracted at the other end of the tapered fiber and fed into a detector. The power meter receiving the detector signal is coupled to an oscilloscope that is triggered by the synchronization output of FG.
The tapered fiber in
Figure 1 can be asymmetric, with a nonadiabatic downtaper and an adiabatic uptaper, for two-mode input, or symmetric, adiabatically bitapered, for one-mode input. The design, modeling, fabrication, and testing of the asymmetric fiber is described in detail in the next subsection. The symmetric fiber is fabricated to have the same waist radius as the asymmetric fiber.
Initially, an asymmetric tapered fiber was used to couple light into and out of a microresonator filled with methanol, and a WGM that showed a shallow throughput dip with the two input modes in phase and a small peak with the input modes out of phase was selected. Then, with the input modes in phase, an analyte at predetermined concentrations was added to the methanol, and changes in the dip depth and linewidth were recorded. Then, the asymmetric tapered fiber was replaced by a symmetric tapered fiber of the same waist radius, and, for the same WGM, changes in the dip depth and linewidth for different analyte concentrations were recorded for one-mode input. Tightly capped plastic vials were used as reservoirs for the analyte. The two ends of the microresonator were connected to the reservoirs, and the analyte inside one reservoir was pushed through the microresonator using a syringe feed at the top of that reservoir. A photograph of the part of the setup showing the tapered fiber, microresonator (HBR), and reservoirs is shown in
Figure 2.
The ratio of fractional changes in the dip depth, asymmetric (two-mode) to symmetric (one-mode), for the same analyte concentration, gives the value of the enhancement
η21, and the ratio of the fractional change in the dip depth (two-mode) to the fractional change in the linewidth (either case) gives the value of the enhancement
ηdl. The two-mode dip-depth data are
for various values of
αa, which allows the value of the interacting fraction
f to be determined. Note, however, that both enhancement factors are only weakly dependent on
f, unlike the absolute sensitivity, which increases approximately linearly with
f. The internal interacting fraction of a WGM will be, for absorption measurements, somewhat greater than the ratio of the power circulating in the inner region to the total power circulating in the mode [
5]. In addition to having large internal interacting fractions, another advantage of the HBR is that it requires only a very small volume of the solvent and analyte [
17].
2.3. Fabrication, Modeling, and Testing of Asymmetric Tapered Fiber
A tapered fiber is made by stretching a heated optical fiber, and it consists of a thin filament called the taper waist, each end of which is linked to the unstretched fiber by a section known as the taper transition. These taper transitions can be classified as adiabatic or nonadiabatic [
18]. If the light propagating in the tapered fiber remains in the local fundamental mode (HE
11) at all points along the taper transition, the transition is adiabatic; however, in a nonadiabatic taper transition, higher-order fiber modes are also excited, and hence, the light gets distributed among the local fundamental mode and higher-order modes as it travels along the tapered fiber.
In our lab, the well-known “flame brush” technique is used to fabricate both symmetric and asymmetric tapered fibers. An optical fiber (SMF-28) with its jacket removed is attached to two motorized translation stages of a homemade fiber puller. Underneath the stripped area of the fiber, the flame of a hydrogen torch on another translation stage continuously brushes the stripped fiber along its length back and forth over a distance known as the brushing length
L. While fabricating an asymmetric (nonadiabatic) tapered fiber, the two translation stages are pulled at different speeds, whereas a symmetric (adiabatic) tapered fiber is fabricated by pulling the two stages with the same speed. Much previous work has been conducted on the fabrication and characterization of asymmetric tapered fibers [
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29]. A schematic diagram of the asymmetric tapered fiber is shown in
Figure 3, where light propagates from left to right. The downtaper transition (nonadiabatic) is relatively abrupt, whereas the uptaper transition (adiabatic) is more gradual. Thus, both fundamental and higher-order fiber modes will travel down the taper waist. However, the higher-order fiber modes will not survive the adiabatic uptaper and will be lost in the cladding.
To explain the labeling in
Figure 3, first consider a symmetric tapered fiber. The fiber radius in the transition region is given by [
22]
where
r0 is the radius of the untapered fiber,
z is the distance into the transition region, and
is the effective brushing length, i.e., the length of the taper waist. The waist radius is
where
z0 is the transition length; in the symmetric limit of
Figure 3,
z2 =
z1 =
z0. As
Figure 3 indicates, the waist length is not equal to the brushing length (
), and the transition length is not equal to the pulling distance (
z0 ≠
p0). The explanation for these effects is based on the following assumptions [
30]: (i) the hydrogen flame has a finite width and, hence, the heated region
extends 0.26 mm beyond the brushing length
L on each end (
); (ii) the end of the uniform waist is not at the end of the limit of the heated region, but is recessed inside it by a distance estimated to be
s = 0.13 mm for the symmetric taper (so
), also implying that
z0 =
p0 +
s. For a standard symmetric taper,
L = 6.50 mm and
p0 = 26.88 mm, so that
For the asymmetric taper, we further assume [
30] that the recession distance is inversely proportional to the pulling distance, or linearly dependent on the speed at which the other side of the taper is being pulled; this means that
si =
sp0/
pi (
i = 1, 2), and so
and
The two parts of the waist length have the same ratio as the lengths of the corresponding transition regions (
L1/
L2 =
z1/
z2), and so the waist radius is given by
for either transition (
i = 1 or 2). Support for this model is found when it is used in the fabrication of asymmetric and symmetric tapered fibers with the same waist radius, using them to couple light into the HBR. When the asymmetric fiber shows a shallow dip with the incident fiber modes in phase and a small peak with the incident modes out of phase, the symmetric fiber exhibits a resonance dip that goes nearly to zero throughput. Further confirmation through beat length measurements will be given below.
To see how adiabatic and nonadiabatic tapers can be differentiated, consider the local cladding taper angle Ω, defined as the angle between the fiber axis and the tangent to the taper profile at the point of interest:
For a taper transition to be adiabatic, Ω must be less than some maximum value [
20,
31] at all values of the inverse taper ratio
r(
z)/
r0. Strictly speaking, the condition to be satisfied is
for all
z in the taper transition, where
kf and
kh are the propagation constants of the fundamental and the higher-order fiber modes. These propagation constants can be calculated for the fiber (step-index dielectric waveguide) by solving the characteristic equation [
32] at various taper radii in the taper transition.
As mentioned earlier, the relative phase of the fundamental and higher-order fiber modes varies with propagation, making the throughput profile of a nonadiabatic tapered-fiber-coupled microresonator system no longer a symmetric Lorentzian dip [
33,
34,
35,
36,
37,
38]. The distance between successive points where they are in phase is a beat length that can easily be measured by translating the fiber-HBR point of contact and finding the distance from one symmetric throughput dip (or peak) to the next. A practical implementation of Equation (22) is thus found by defining the maximum taper angle for the transition to be adiabatic:
where the beat length
zb is given by:
thus provides a delineation curve with which Ω(
z) can be compared to determine the adiabaticity. Knowing the propagation constants at various taper radii allows us to calculate the beat length
zb.
As light propagates down a taper transition, it makes a transition from core guidance (total internal reflection, TIR, at the core-cladding interface) to cladding guidance (TIR at the cladding-air interface). While it is core-guided, the light remains in the fundamental mode (HE11, also known as LP01); in this single-mode region, the propagation constant of the higher-order mode, kh, is taken to be the vacuum propagation constant times the cladding refractive index. Once the light becomes cladding-guided, we allow three possibilities for the higher-order mode, since the taper waist is multimode with a typical radius of 1.16 µm. These are: first, the LP11 family, consisting of TE01, TM01, and HE21; second, the LP21 and LP02 family, made up of HE31, EH11, and HE12, represented in the calculations by HE12; and third, the LP31 family (EH21 and HE41, represented by HE41).
Since we have three choices for
kh, we will have three delineation curves. To plot each delineation curve, the larger of the Ω
max values for core guidance and cladding guidance at different values of the inverse taper ratio is used. These Ω
max values corresponding to different inverse taper ratios are fitted to polynomials and are shown in
Figure 4, along with the Ω[
r(
z)/
r0] for the adiabatic transition of a typical symmetric tapered fiber with a waist radius of 1.15 μm.
In
Figure 4, the red, green, and yellow lines represent the different delineation curves. The red line is the delineation curve assuming the higher-order mode to be LP
11 under the cladding guidance condition. In contrast, the green and yellow lines are the delineation curves assuming the higher-order mode to be HE
12 and HE
41, respectively. For all values of the inverse taper ratio, the orange line in
Figure 4 representing the adiabatic taper will remain below the delineation curves, whereas for a non-adiabatic taper, the plot of the taper angle will pass above a delineation curve for at least some values of the inverse taper ratio. The correct delineation curve is determined by the experimental measurement of the beat length, as will be described below.
An asymmetric tapered fiber with one transition meant to be nonadiabatic and one meant to be adiabatic was fabricated, and the cladding taper angles are plotted along with delineation curves in
Figure 5. In
Figure 5, the red, green, and yellow lines represent the delineation curves. The blue curve represents the cladding taper angle for the abrupt downtaper transition. Since the blue curve passes above the minima of the delineation curves, the downtaper transition is nonadiabatic. The orange curve representing the taper angle for the uptaper transition passes below the delineation curves, indicating that the uptaper transition is adiabatic.
Figure 5 thus implies that the fabricated asymmetric tapered fiber has a nonadiabatic downtaper and an adiabatic uptaper.
The average beat lengths of several asymmetric-tapered-fiber-coupled microresonator systems were measured using a screw-gauge actuated 3D stage to which the asymmetric fiber is mounted. The distance of translation of the fiber–HBR point of contact from where a symmetric throughput dip is observed to where the next symmetric dip is observed is the beat length. The experimentally measured beat length
zb and the waist radius
rw predicted by the model for three different asymmetric taper profiles are shown in
Table 1. Profile 1 is the fiber whose taper transitions are plotted in
Figure 5.
The experimentally measured beat lengths for the three different asymmetric taper profiles indicate that: (i) the modes responsible for beating are HE
11 and LP
11 and, hence, the delineation curve of interest is the red one in
Figure 4 and
Figure 5; and (ii) the radii predicted by the asymmetric taper fiber model for the three different taper profiles were very close to the radii estimated from the beat length measurements. It is worth noting that in addition to the HE
11 and LP
11 modes, other higher-order modes such as HE
12 and HE
41 may also be excited by light propagating along the nonadiabatic downtaper. Among all the higher-order modes, LP
11 will be the most strongly excited, since the fiber sags during pulling [
39]; this agrees with our beat length measurements shown in
Table 1. Previously, it was shown that [
6] by choosing a particular ratio of the resonator size to the diameter of the tapered fiber, only the HE
11 and LP
11 modes can significantly interact with the WGMs of a microresonator, and therefore, any weak excitation of other higher-order modes can be neglected. Thus, when different asymmetric taper profiles are used to couple light into the microresonator, only two modes, namely, HE
11 and LP
11, will significantly interact with WGMs.