applied sciences

Article

Highly Sensitive Complicated Spectrum Analysis inMicro-Bubble Resonators Using the OrthogonalDemodulation Pound–Drever–Hall Technique

Jinliang Hu 1, Sheng Liu 1, Biao Li 1, Zhihe Guo 1 , Xiang Wu 1, Liying Liu 1 and Lei Xu 1,2,*1 Key Lab for Micro and Nanophotonic Structures (Ministry of Education), Department of Optical Science and

Engineering, School of Information Science and Engineering, Fudan University, Shanghai 200433, China;jlhu14@fudan.edu.cn (J.H.); 14110720003@fudan.edu.cn (S.L.); 17110720018@fudan.edu.cn (B.L.);17110720004@fudan.edu.cn (Z.G.); wuxiang@fudan.edu.cn (X.W.); lyliu@fudan.edu.cn (L.L.)

2 Department of Physics, Fudan University, Shanghai 200433, China* Correspondence: lei_xu@fudan.edu.cn

Received: 13 August 2020; Accepted: 6 September 2020; Published: 9 September 2020�����������������

Featured Application: The presented method can increase the measurement range in differentialresonance sensing, allowing differential sensing of overlapped resonances.

Abstract: Whispering gallery mode micro-bubble optical cavities are asymmetrical ellipsoids inexperimental settings, which makes their modes nondegenerate. A complicated dense spectrumis thus generated. Overlapping and coupled resonances exist in this dense spectrum. In thisstudy, we determined that the orthogonal demodulation Pound-Drever-Hall technique can be usedto analyze complicated resonances. Using this method, overlapping weak and strong couplingresonances can be analyzed. Compared to spectrum simplification and the ab initio theory of Fanoresonances, this method is repeatable, simple, sensitive, and accurate. The method can increasethe measurement range of differential resonance sensing, thus allowing the differential sensing ofoverlapped resonances.

Keywords: whispering-gallery mode (WGM) microcavity; micro-bubble resonator; Pound-Hall-Drever technique; complicated spectrum analysis

1. Introduction

During recent decades, whispering gallery mode optical cavities [1] have attracted substantialattention because of their benefits of high-quality factors, small mode volumes, and high energy density.They have been widely utilized in many fields, such as cavity quantum electrodynamics (CQED) [1],nonlinear optics [2], microcavity sensors [3–7], optomechanics [8], microcavity lasers [9], optical signalprocessing [10], and single molecule detection [3–5,11]. Micro-bubble resonators (MBRs) [12] havebecome an especially important type of optical microcavities. These sensors facilitate optofluidic [6]and pressure sensing [13] due to their hollow structure. However, MBRs are asymmetrical ellipsoidsin experiments, which causes their resonances to nondegenerate. As a consequence, a complicateddense spectrum is generated, and the resonances may overlap or couple with each other [14]. The tworesonances have three relations in the dense spectrum according to the relationship between thecoupling coefficient (g) and the critical coupling strength (κT): overlap (g = 0), weak coupling (g < κT,Fano resonance [15,16] and electromagnetically induced transparency resonance [17,18]), and strongcoupling (g > κT, mode splitting [19]). A dense spectrum benefits cavity quantum electrodynamics [1]and nonlinear optics [2]. However, tracking and identifying the resonances in sensing applications arean essential task. Many efforts have been made, such as mode simplification [20–24] and the ab initio

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theory of Fano resonances [25], to simplify and analyze the spectrum. However, mode simplification isunrepeatable, costly, and complex. The ab initio theory of Fano resonances only utilizes the amplitudeof resonance, and can only discriminate simple resonances. Thus, analysis of the dense and complicatedspectrum remains a challenge.

In this work, we propose a new method based on the orthogonal demodulatedPound-Drever-Hall [26,27] (OD-PDH) technique to analyze the dense and complicated spectrumdirectly. The Pound-Drever-Hall (PDH) technique can be used in laser frequency stabilization [26,27]and highly sensitive sensing applications, e.g., nanoparticles [11], pressure [13], angular velocity [28],acceleration [29], and relativity applications [30]. The discriminating signal [27,28] from the PDHsystem depends on an optical field rather than light intensity, so contains both amplitude and phaseinformation of the resonance. The PDH system transforms the sharp imaginary information of theresonance and the slope information of the resonance into discriminating signals. Any tiny change ofresonance will significantly influence the discriminating signals. Therefore, it can be used in highlysensitive resonance analysis. The OD-PDH technique [31,32] is an improved PDH technique thatcan eliminate the phase difference between local oscillation and detection signal, and stabilize thediscriminating signal. Using this method, the overlapping weak and strong coupling resonanceparameters can be obtained accurately. Compared to other methods, this method is repeatable, simple,sensitive, and accurate. It can also increase the measurement range in differential resonance sensing,thus allowing differential sensing of overlapped resonances.

2. Principle and Setup

In classical PDH technique theory, we can obtain the bipolar voltage wavelength discriminatingsignal (I-term and Q-term signals), and I0(ω) and Q0(ω) can be expressed as [27]:

I0(ω) = kP0 J0(β)J1(β)Re[R(ω)R∗(ω+ωm) −R∗(ω)R(ω−ωm)] (1)

Q0(ω) = kP0 J0(β)J1(β)Im[R(ω)R∗(ω+ωm) −R∗(ω)R(ω−ωm)] (2)

where the phase difference (ϕ) between the detection signal and the local oscillation signal is 0 and π/2,respectively. ω is the laser angular frequency; ωm is the angular frequency of the signal input into thephase electro-optical modulator (PM) (ωm = 2πf m, f m = 50 MHz); I0(ω) is the in-phase term; Q0(ω)is the quadrature term; k represents the scale factor containing the light power-voltage conversioncoefficient, the insertion loss in every electronic element, and the gain of the amplifier; P0 representsthe laser power; J0 and J1 are the first kind of 0- and 1-order Bessel function; β is the modulationdegree (β = πV/Vπ); V is the signal of the peak-to-peak value voltage input into the PM; Vπ is thehalf-wave voltage of the PM; and R(ω) is the complex spectral response of the resonance. When thebandwidth of the resonance is much larger than the PM modulation frequency (i.e., low quality factor,Q), the sidebands can be modulated by the resonance, and ωm is negligible, thus R*(ω + ωm) − R*(ω) ≈[dR*(ω)/dω]ωm, R(ω) − R(ω − ωm) = [dR(ω)/dω]ωm; the I-term and Q-term signals can be expressedas [27]:

I0(ω) ≈ kP0 J0(β)J1(β)[d∣∣∣R(ω)∣∣∣2/dω ]ωm, (3)

Q0(ω) ≈ 0. (4)

By comparison, when the bandwidth of resonance is much less than the PM modulation frequency(i.e., high Q), the sideband light is totally transmitted, thus R(ω ± ωm) ≈ 1 and R*(ω ± ωm) ≈ 1, and theI-term and Q-term signals can be expressed as [27]:

I0(ω) ≈ 0, (5)

Q0(ω) ≈ 2kP0 J0(β)J1(β)Im[R(ω)]. (6)

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When the PM modulation frequency is close to the bandwidth of resonance, the I-term and Q-termsignals exist simultaneously. The [d|R(ω)|2]/dω]ωm (low Q condition) and 2Im[R(ω)] (high Q condition)are both sharp dispersion curves like the PDH discriminating frequency curve [27]. The PDH systemtransforms the sharp imaginary information of the resonance and the slope information of the resonanceinto the I-term and Q-term signals. Any tiny change of resonance will significantly influence the I-termand Q-term signals. Therefore, the I-term and Q-term signals can be used to analyze the complicatedresonances that the transmitted spectrum cannot be used to analyze. However, the ϕ is an unknownvalue in the experiment, and the phase difference will influence the I-term and Q-term signals [31].Here, we use the orthogonal demodulation technique [31,32] to eliminate the phase difference andmake the OD-PDH spectrum (IQ curve) stable. In sensing applications, this technique can help obtainthe highest sensing sensitivity. In this experiment, we use Equation (7):

IQ(ω) = I20(ω) + Q2

0(ω) (7)

to obtain the IQ curve, which does not relate to ϕ. For a single resonance, R(ω) can be expressed as [33]:

R(ω) = 1−2ω0/qe

ω0/q0 +ω0/qe + 2i(ω−ω0), (8)

where ω is the laser angular frequency, ω0 is the resonant angular frequency, q0 is the intracavity Q,and qe is the external cavity Q. When two resonances overlap or weakly couple, R(ω) can be describedby [34]:

R(ω) = 1 + 2κ01X2 + κ02X1 + 2Re(g)

√κe1κe2

X1X2 − g2 , (9)

where κ0n = ωn/(2q0n), κen = ωn/(2qen), qn = (q0n−1 + qen

−1)−1, Xn = −κ0n − κen + i(ω − ωn), ωn = 2πfn(n = 1, 2), and ∆λd = c/f 1 − c/f 2; κ0n is the intracavity decay rate of resonances; κen is the externalcavity decay rate of the two resonances; fn is the resonant frequency of the two resonances; q0n isthe intracavity Q of the two resonances; qen is the external cavity Q of the two resonances; qn is theresonance total Q of the two resonances; g is the coupling coefficient; ωn is the angular resonantfrequency of the two resonances; ω is laser angular frequency; c is the light velocity in a vacuum;and ∆λd represents the two resonances’ distances as wavelengths. The parameters q0n, qen, fn (n = 1,2), and g can be obtained by fitting the IQ curve. The other parameters can be derived by the fittingparameters. When two resonances strongly couple, according to the coupled mode theory, the coupledmode equations can be written as [14]:

X3E3 − g1E4 + 2i√κe3Ein = 0, (10)

X4E4 − g1E3 = 0, (11)

where κ0p = ωp/(2q0p), κep = ωp/(2qep), qp = (q0p−1 + qep

−1)−1, Xp = i(κ0p + κep) + (ω − ωp), ωp = 2πf p

(p = 3, 4), and ∆λd2 = c/f 3 − c/f 4; Ein is the optical field amplitude of the input light; Ep (p = 3, 4) is theoptical field amplitude of the two resonances; κ0p is the intracavity decay rate of the two resonances;κep is the external cavity decay rate of the two resonances; fp is the resonant frequency of the tworesonances; q0p is the intracavity Q of the two resonances; qep is the external cavity Q of the tworesonances; qp is the resonance total Q of the two resonances; g1 is the coupling coefficient; and ωp isthe angular resonant frequency of the two resonances. The optical field amplitude of the output light(Eout) can be described by:

Eout = Ein +√κe3E3. (12)

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The R(ω) can be described by:

R(ω) =Eout

Ein= 1 +

2iκe3X4

g21 −X3X4

. (13)

The parameters q0p, qep, fp (p = 3, 4), and g1 can be obtained by fitting the IQ curve. The otherparameters can be derived by the fitting parameters. The final IQ curve expression (Equation (7))can be obtained by inserting Equation (8) (single resonance), Equation (9) (overlap and weak couple),or Equation (13) (strongly couple) into Equations (1) and (2), and then inserting Equations (1) and (2)into Equation (7). The resonance expression can be obtained by calculating the square modulus ofEquations (8) (single resonance), (9) (overlap and weak couple), or (13) (strong couple). The coefficientof determination (COD, R-square) was used to evaluate the fitting results. The data (x, y) = (x1, y1), (x2,y2), . . . , (xm, ym) were obtained in measurement. The COD is defined as:

COD =∑

m(ym − y)2/

∑m

(ym − y)2 , (14)

where ym, ym, and y represent the experimental data, fitting data, and average of the experimental data,respectively. The COD ranges from 0 to 1. A larger COD indicates better fitting results. We define [14]|κ1 − κ2|/2 as the threshold (κT) and the regimes where g < κT and g > κT as the weak- and strong-drivingregimes, respectively; κq (q = 1, 2) is the total decay rate of the two resonances.

In the experiment, the MBR was fabricated by heating pressurized silica capillaries [7]. Figure 1provides a schematic of the orthogonal demodulation PDH technique. A tunable laser (1550 nm,Anritsu Tunics Plus CL; 850 nm, New Focus TLB 6716) was used to excite the MBR resonances. Afterpassing through a polarization controller and a PM (850 nm, iXBlue NIR-MPX800-LN-P-P-FA-FA;1550 nm, Thorlabs LN53-10-S-F-F-BNL), the laser beam was coupled into and out of the MBR via ataper fiber. A photoelectric detector (PD; 850 nm, Thorlabs PDA10A2; 1550 nm, Thorlabs PDA10CF-CE)connected to a data acquisition (DAQ, NI PCIe6351) card and IQ demodulation circuits (ADI LT5584)was used to detect the transmission light. A syringe pump was used to inject air into the MBR, and apressure sensor (0.25%; full scale, 2 bar) monitored the MBR’s inner pressure. A high frequencyfunction signal generator (Keysight 33622A) generated the sine wave signal to drive the PM and thesignal was also used as the local oscillation signal to demodulate the detected signal from the PD.The DAQ card recorded the signal from the PD (transmitted spectrum), IQ demodulation circuits (theI-term and Q-term signal), and pressure sensor (pressure), and generated a triangular wave signal tosweep the laser frequency simultaneously. In the experiment, the laser was used to sweep the laserfrequency linearly as a triangular wave signal. The PD transformed the light signal to a voltage signal.The DAQ card recorded the signal as the laser frequency changed. Then, we obtained the transmittedspectrum. The IQ signal was obtained after the signal was processed by the IQ-demodulation circuits.

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out e3 42

in 1 3 4

21( ) E i XE g X X

Rκω =−

= + . (13)

The parameters q0p, qep, fp (p = 3, 4), and g1 can be obtained by fitting the IQ curve. The other parameters can be derived by the fitting parameters. The final IQ curve expression (Equation (7)) can be obtained by inserting Equation (8) (single resonance), Equation (9) (overlap and weak couple), or Equation (13) (strongly couple) into Equations (1) and (2), and then inserting Equations (1) and (2) into Equation (7). The resonance expression can be obtained by calculating the square modulus of Equations (8) (single resonance), (9) (overlap and weak couple), or (13) (strong couple). The coefficient of determination (COD, R-square) was used to evaluate the fitting results. The data (x, y) = (x1, y1), (x2, y2), …, (xm, ym) were obtained in measurement. The COD is defined as:

2 2ˆ= ( ) ( )m mm m

COD y y y y− − , (14)

where my , my , and y represent the experimental data, fitting data, and average of the experimental data, respectively. The COD ranges from 0 to 1. A larger COD indicates better fitting results. We define [14] |κ1 − κ2|/2 as the threshold (κT) and the regimes where g < κT and g > κT as the weak- and strong-driving regimes, respectively; κq (q = 1, 2) is the total decay rate of the two resonances.

In the experiment, the MBR was fabricated by heating pressurized silica capillaries [7]. Figure 1 provides a schematic of the orthogonal demodulation PDH technique. A tunable laser (1550 nm, Anritsu Tunics Plus CL; 850 nm, New Focus TLB 6716) was used to excite the MBR resonances. After passing through a polarization controller and a PM (850 nm, iXBlue NIR-MPX800-LN-P-P-FA-FA; 1550 nm, Thorlabs LN53-10-S-F-F-BNL), the laser beam was coupled into and out of the MBR via a taper fiber. A photoelectric detector (PD; 850 nm, Thorlabs PDA10A2; 1550 nm, Thorlabs PDA10CF-CE) connected to a data acquisition (DAQ, NI PCIe6351) card and IQ demodulation circuits (ADI LT5584) was used to detect the transmission light. A syringe pump was used to inject air into the MBR, and a pressure sensor (0.25%; full scale, 2 bar) monitored the MBR’s inner pressure. A high frequency function signal generator (Keysight 33622A) generated the sine wave signal to drive the PM and the signal was also used as the local oscillation signal to demodulate the detected signal from the PD. The DAQ card recorded the signal from the PD (transmitted spectrum), IQ demodulation circuits (the I-term and Q-term signal), and pressure sensor (pressure), and generated a triangular wave signal to sweep the laser frequency simultaneously. In the experiment, the laser was used to sweep the laser frequency linearly as a triangular wave signal. The PD transformed the light signal to a voltage signal. The DAQ card recorded the signal as the laser frequency changed. Then, we obtained the transmitted spectrum. The IQ signal was obtained after the signal was processed by the IQ-demodulation circuits.

Figure 1. Setup schematic of the orthogonal demodulated Pound–Drever–Hall technique system. TL, tunable laser; PC, polarization controller; PM, phase electro-optical modulator; MBR, micro-bubble resonator; PD, photoelectric detector; DAQ, data acquisition card; IQ, I-term and Q-term signal;

Figure 1. Setup schematic of the orthogonal demodulated Pound-Drever-Hall technique system. TL,tunable laser; PC, polarization controller; PM, phase electro-optical modulator; MBR, micro-bubbleresonator; PD, photoelectric detector; DAQ, data acquisition card; IQ, I-term and Q-term signal; HFSG,high frequency function signal generator; PS, pressure sensor; SP, syringe pump; DS, detection signal;LS, local oscillation signal. IQ DC, IQ demodulation circuit. The inset (a) illustrates the IQ demodulationcircuit detail. The inset (b) provides an image of the MBR.

3. Experimental Results

Figure 2a plots standard a single optical resonance (blue scattered line), its IQ curve (blackscattered line), its fitting IQ curve (red solid line), and its replotted resonance (magenta solid line).An MBR with a diameter of 220 µm and a wall thickness of 17 µm was excited with 1550 nm tunablelaser via a tapered fiber whose diameter was about 2.5 µm. The IQ curve has two peaks that sitsymmetrically centered on the optical resonance. The intracavity Q (q0 = 1.20 × 106), external cavity Q(qe = 1.14 × 106), total Q (qa = (q0

−1 + qe−1)−1 = 5.84 × 105) and resonant frequency (f a = 1.94 × 1014 Hz)

can be obtained by fitting the IQ curve with a single resonance IQ expression. Total Q (qb = 5.84 × 105)and resonant frequency (f b = 1.94 × 1014 Hz) can also be obtained by fitting the transmitted spectrumdirectly. The results agree with each other and prove that the IQ curve can be used to analyze thespectrum. Figure 2b plots an asymmetrical spectrum (black scattered line) and its fitting resonances(overlapping, red solid line; coupled, yellow dashed line). An MBR with a diameter of 276 µm and awall thickness of 4 µm was excited with an 850 nm tunable laser via a tapered fiber whose diameter wasabout 2.5 µm. However, direct fitting of the spectrum via a two-resonance expression did not deriveunique fitting parameters. Figure 2b gives the fitted spectrum with the two group parameters. The twogroups’ different parameters listed in Table 1, however, provided an excellent fitting. The coefficientof determination parameters (COD, R-Square) were both above 0.980. Thus, we used the OD-PDHtechnique to analyze the resonance. Figure 2c plots the spectrum (blue scattered line) in Figure 2b andits IQ curve (black scattered line), fitting curve of the two resonances in Figure 2b and its replotted IQcurve (overlapping, red solid line; coupled, yellow dashed line). We used the fitted parameters fromFigure 2b to replot the IQ curve. The CODs of the overlapping and coupled replotted IQ curves were0.981 and 0.937, respectively. Clearly, the overlapping replotted IQ curves agree with the experimentalspectrum. This proves that the IQ curve can discriminate asymmetrical resonance. Figure 2d plots thespectrum (blue scattered line) in Figure 2b and its IQ curve (black scattered line), fitted IQ curve andits replotted two resonances (overlapping, red solid line; couple, yellow dashed line). To verify that theIQ fitting is valid, we used the two-resonant IQ expression to fit the IQ curve. The IQ curve could bewell fitted by the two groups’ different parameters listed in Table 1 (CODs > 0.996). Then, we usedthe fitting parameters to replot the resonances. The COD of the overlapping replotted resonance was0.968, while the COD of the coupled parameters was only 0.790. Similarly, the overlapped replottedcurve better agrees with the experimental spectrum. These fitting and replotting curves prove the IQ

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curve can confirm that the overlapping parameters are the correct results. To prove that the spectrumin Figure 2b contains the two resonances, we injected air into the MBR to execute pressure tuning.Figure 2e,f shows the resonance (blue scattered line) gradually separating and eventually becomingtwo resonances, and its IQ curves (black scattered line). Different orders of resonances have differentpressure sensitivity levels; the resonances shift through different wavelengths and separate gradually asthe MBR’s internal pressure changes. The pressure values are 0.387, 0.411, and 0.579 bar in Figure 2b,e,f,respectively. The pressure tuning proves that the asymmetrical spectrum contains two resonances.Figure 2f plots the two separate resonances (blue scattered line), the IQ curves (black scattered line),fitted IQ curves, replotted resonances (red solid line), fitted resonances, and replotted IQ curves (yellowdashed line). To verify the fitting results above, we fitted the IQ curve, replotted the resonance, fittedthe resonance, and replotted the IQ curve. The fitting and replotting results (g = 0 MHz) listed inTable 1 agree with the previous results. The CODs of the IQ and the transmitted replotted curve are0.862 and 0.863, respectively. The IQ curve of the low-quality resonance features noise, so its CODvalue is not significantly better. The process proves the validity of the OD-PDH technique in analyzingoverlapping asymmetrical spectra. Note that we obtained g = 0 MHz. This naturally reveals that thetwo resonances simply overlap.Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 10

(a) (b) (c)

(d) (e) (f)

Figure 2. (a) Standard single resonance transmitted spectrum (blue scattered line), its IQ curve (black scattered line), fitted IQ curve (red solid line) and replotted resonance (magenta solid line). (b) Experimental (black scattered line) and fitted (red solid line (g = 0), yellow dashed line (g = 1.19 × 104 MHz)) spectrum at inner pressure of 0.387 bar. (c) Experimental spectrum (blue scattered line) and IQ curve (black scattered line), fitted resonances, and replotted IQ curves (red solid line (g = 0 MHz) and yellow dashed line (g = 1.19 × 104 MHz)). (d) Experimental spectrum (blue scattered line) and IQ curve (black scattered line), fitted IQ curves and replotted resonances (red solid line (g = 0 MHz), and yellow dashed line (g = 3.76 × 103 MHz)). (e) and (f) Experimental resonances (blue scattered line) and IQ curve (black scattered line) at inner pressure of 0.411 bar and 0.579 bar, respectively, fitted and replotted curves (red solid line and yellow dashed line).

Table 1. Overlapping resonances fitting results.

Fitting Object q1 q2 ∆λd/pm g/MHz FCOD 3 RCOD 4 Figure 2b Res. 1 (g = 0) 1.45 × 106 4.58 × 105 0.346 0 0.980 0.981 Figure 2b Res. (g ≠ 0) 3.50 × 105 1.24 × 104 9.39 1.19 × 104 0.998 0.937 Figure 2d IQ. 2 (g = 0) 1.46 × 106 4.80 × 105 0.340 0 0.998 0.968 Figure 2d IQ. (g ≠ 0) 5.10 × 105 7.54 × 104 3.00 3.76 × 103 0.996 0.790

Figure 2f IQ. 1.19 × 106 5.72 × 105 2.44 0 0.956 0.862 Figure 2f Res. 1.48 × 106 4.62 × 105 2.42 0 1.00 0.863

1 Res., Resonance. 2 IQ., IQ curve. 3 FCOD, Fitting coefficient of determination (Fitting R-Square). 4 RCOD, Replotted coefficient of determination (Replotted R-Square).

Figure 3 gives another example of the asymmetrical mode (black scattered line). Again, we can fit the asymmetrical resonance (overlapping, red solid line; coupled, yellow dashed line) in Figure 3a with the two-resonance expression and derive two group parameters as listed in Table 2 (CODs > 0.991). Figure 3b plots the spectrum in Figure 3a (blue scattered line) and its IQ curve (black scattered line), fitted resonances and replotted IQ curves (overlapping, red solid line; coupled, yellow dashed line). Using the two group parameters to reproduce the IQ curve, the group of parameters with g = 1.20 × 104 MHz in Figure 3b fit the spectrum well (replotted coupled curve COD, 0.997; replotted overlapping curve COD, 0.645), which means that the two modes couple. Figure 3c plots the spectrum in Figure 3a (blue scattered line) and its IQ curve (black scattered line), fitted IQ curves and replotted resonances (overlapping, red solid line; coupled, yellow dashed line). We can also fit the IQ curves and generate two groups of parameters (COD > 0.998), and then use those parameters to reproduce the spectrum in Figure 3c. This result also demonstrates that the group of parameters with g = 1.20 ×

Figure 2. (a) Standard single resonance transmitted spectrum (blue scattered line), its IQ curve(black scattered line), fitted IQ curve (red solid line) and replotted resonance (magenta solidline). (b) Experimental (black scattered line) and fitted (red solid line (g = 0), yellow dashed line(g = 1.19 × 104 MHz) spectrum at inner pressure of 0.387 bar. (c) Experimental spectrum (blue scatteredline) and IQ curve (black scattered line), fitted resonances, and replotted IQ curves (red solid line(g = 0 MHz) and yellow dashed line (g = 1.19 × 104 MHz). (d) Experimental spectrum (blue scatteredline) and IQ curve (black scattered line), fitted IQ curves and replotted resonances (red solid line(g = 0 MHz), and yellow dashed line (g = 3.76 × 103 MHz). (e) and (f) Experimental resonances(blue scattered line) and IQ curve (black scattered line) at inner pressure of 0.411 bar and 0.579 bar,respectively, fitted and replotted curves (red solid line and yellow dashed line).

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Table 1. Overlapping resonances fitting results.

Fitting Object q1 q2 ∆λd/pm g/MHz FCOD 3 RCOD 4

Figure 2b Res. 1 (g = 0) 1.45 × 106 4.58 × 105 0.346 0 0.980 0.981Figure 2b Res. (g , 0) 3.50 × 105 1.24 × 104 9.39 1.19 × 104 0.998 0.937Figure 2d IQ. 2 (g = 0) 1.46 × 106 4.80 × 105 0.340 0 0.998 0.968Figure 2d IQ. (g , 0) 5.10 × 105 7.54 × 104 3.00 3.76 × 103 0.996 0.790

Figure 2f IQ. 1.19 × 106 5.72 × 105 2.44 0 0.956 0.862Figure 2f Res. 1.48 × 106 4.62 × 105 2.42 0 1.00 0.863

1 Res., Resonance. 2 IQ., IQ curve. 3 FCOD, Fitting coefficient of determination (Fitting R-Square). 4 RCOD,Replotted coefficient of determination (Replotted R-Square).

Figure 3 gives another example of the asymmetrical mode (black scattered line). Again, we can fitthe asymmetrical resonance (overlapping, red solid line; coupled, yellow dashed line) in Figure 3a withthe two-resonance expression and derive two group parameters as listed in Table 2 (CODs > 0.991).Figure 3b plots the spectrum in Figure 3a (blue scattered line) and its IQ curve (black scattered line), fittedresonances and replotted IQ curves (overlapping, red solid line; coupled, yellow dashed line). Usingthe two group parameters to reproduce the IQ curve, the group of parameters with g = 1.20 × 104 MHzin Figure 3b fit the spectrum well (replotted coupled curve COD, 0.997; replotted overlapping curveCOD, 0.645), which means that the two modes couple. Figure 3c plots the spectrum in Figure 3a(blue scattered line) and its IQ curve (black scattered line), fitted IQ curves and replotted resonances(overlapping, red solid line; coupled, yellow dashed line). We can also fit the IQ curves and generatetwo groups of parameters (COD > 0.998), and then use those parameters to reproduce the spectrum inFigure 3c. This result also demonstrates that the group of parameters with g = 1.20 × 104 MHz fits thespectrum. (replotted coupled transmitted spectrum COD, 0.994; replotted overlapping transmittedspectrum COD, 0.893). To prove that the asymmetrical resonance is a coupled resonance, as we didabove, we injected air into the MBR to tune the resonance; Figure 3d plots the resonance when thepressure ranged from 0.215 to 0.519 bar. A single example of asymmetrical resonance may be coupledand overlapping resonance. After pressure tuning, the resonance did not separate and the resonanceshifted and maintained its asymmetry. These phenomena prove that the resonance is coupled witha low-Q resonance. Note that g > 0, which satisfies g < κT (κT = 1.31 × 104); this reveals that thetwo resonances are weakly coupled. This process proves the validity of the OD-PDH technique inanalyzing weakly coupled resonances.

Table 2. Weakly coupled resonance fitting results.

Fitting Object q1 q2 ∆λd/pm g/MHz κT/MHz FCOD 3 RCOD 4

Figure 3a Res. 1 (g , 0) 1.08 × 106 4.09 × 104 173 1.20 × 104 1.31 × 104 0.997 0.997Figure 3a Res. (g = 0) 9.22 × 105 6.10 × 105 0.390 0 309 0.991 0.645Figure 3b IQ. 2 (g , 0) 1.06 × 106 4.08 × 104 173 1.20 × 104 1.31 × 104 0.998 0.994Figure 3b IQ. (g = 0) 1.80 × 106 9.78 × 105 0.259 0 260 0.999 0.8931 Res., Resonance. 2 IQ., IQ curve. 3 FCOD, Fitting coefficient of determination (Fitting R-Square). 4 RCOD,Replotted coefficient of determination (Replotted R-Square).

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104 MHz fits the spectrum. (replotted coupled transmitted spectrum COD, 0.994; replotted overlapping transmitted spectrum COD, 0.893). To prove that the asymmetrical resonance is a coupled resonance, as we did above, we injected air into the MBR to tune the resonance; Figure 3d plots the resonance when the pressure ranged from 0.215 to 0.519 bar. A single example of asymmetrical resonance may be coupled and overlapping resonance. After pressure tuning, the resonance did not separate and the resonance shifted and maintained its asymmetry. These phenomena prove that the resonance is coupled with a low-Q resonance. Note that g > 0, which satisfies g < κT (κT = 1.31 × 104); this reveals that the two resonances are weakly coupled. This process proves the validity of the OD–PDH technique in analyzing weakly coupled resonances.

(a) (b)

(c) (d)

Figure 3. (a) Experimental (black scattered line) and fitted spectrum (red solid line, g = 0; yellow dashed line, g = 1.20 × 104 MHz). (b) Experimental spectrum (blue scattered line) and IQ curve (black scattered line), fitted spectrum and replotted IQ curves (red solid line, g = 0 MHz; yellow dashed line g = 1.20 × 104 MHz). (c) Experimental spectrum (blue scattered line) and IQ curve (black scattered line), fitted IQ curve and replotted resonances (red solid line g = 0 MHz; yellow dashed line g = 1.20 × 104 MHz). (d) Asymmetrical resonance shifts as the MBR internal pressure (0.215, 0.320, 0.416, and 0.519 bar) increases.

Table 2. Weakly coupled resonance fitting results.

Fitting Object q1 q2 ∆λd/pm g/MHz κT/MHz FCOD 3 RCOD 4 Figure 3a Res. 1 (g ≠ 0) 1.08 × 106 4.09 × 104 173 1.20 × 104 1.31 × 104 0.997 0.997 Figure 3a Res. (g = 0) 9.22 × 105 6.10 × 105 0.390 0 309 0.991 0.645 Figure 3b IQ. 2 (g ≠ 0) 1.06 × 106 4.08 × 104 173 1.20 × 104 1.31 × 104 0.998 0.994 Figure 3b IQ. (g = 0) 1.80 × 106 9.78 × 105 0.259 0 260 0.999 0.893

1 Res., Resonance. 2 IQ., IQ curve. 3 FCOD, Fitting coefficient of determination (Fitting R-Square). 4 RCOD, Replotted coefficient of determination (Replotted R-Square).

Figure 4a shows two resonances (blue scattered line) that are symmetrical and appear to be two independent resonances, and their IQ curves (black scattered line), when the pressure is 0.229 bar. We used a strongly coupled expression to fit both the IQ curve (red solid line) and the spectrum (yellow dashed line) (IQ COD, 0.985; spectrum COD, 0.996), and used the fitted parameters to reproduce the spectrum (red solid line) and IQ curve (yellow dashed line) (replotted IQ curve COD,

Figure 3. (a) Experimental (black scattered line) and fitted spectrum (red solid line, g = 0; yellowdashed line, g = 1.20 × 104 MHz). (b) Experimental spectrum (blue scattered line) and IQ curve(black scattered line), fitted spectrum and replotted IQ curves (red solid line, g = 0 MHz; yellowdashed line g = 1.20 × 104 MHz). (c) Experimental spectrum (blue scattered line) and IQ curve (blackscattered line), fitted IQ curve and replotted resonances (red solid line g = 0 MHz; yellow dashed lineg = 1.20 × 104 MHz). (d) Asymmetrical resonance shifts as the MBR internal pressure (0.215, 0.320,0.416, and 0.519 bar) increases.

Figure 4a shows two resonances (blue scattered line) that are symmetrical and appear to be twoindependent resonances, and their IQ curves (black scattered line), when the pressure is 0.229 bar.We used a strongly coupled expression to fit both the IQ curve (red solid line) and the spectrum (yellowdashed line) (IQ COD, 0.985; spectrum COD, 0.996), and used the fitted parameters to reproduce thespectrum (red solid line) and IQ curve (yellow dashed line) (replotted IQ curve COD, 0.983; replottedtransmitted spectrum COD, 0.933). The results are summarized in Table 3. The spectrum and IQ curvewere accurately reproduced. Note that only one group of fitting parameters can be derived, with g> κT, revealing that the two resonances are strongly coupled. To check that the two resonances arecoupled, we injected air into the MBR to tune the resonance. Figure 4b plots the successive changeof resonances as the pressure increases from 0.229 to 0.522 bar. Repeating the above fitting process,we found that the two resonances become closer to each other when g decreases and g > κT. Thisimplies that the two resonances are most likely strongly coupled. This process proves that the OD-PDHtechnique is valid for strongly coupled resonances.

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0.983; replotted transmitted spectrum COD, 0.933). The results are summarized in Table 3. The spectrum and IQ curve were accurately reproduced. Note that only one group of fitting parameters can be derived, with g > κT, revealing that the two resonances are strongly coupled. To check that the two resonances are coupled, we injected air into the MBR to tune the resonance. Figure 4b plots the successive change of resonances as the pressure increases from 0.229 to 0.522 bar. Repeating the above fitting process, we found that the two resonances become closer to each other when g decreases and g > κT. This implies that the two resonances are most likely strongly coupled. This process proves that the OD–PDH technique is valid for strongly coupled resonances.

(a) (b)

Figure 4. (a) Experimental spectrum (blue scattered line), its IQ curve (black scattered line), IQ fitting, replotted spectrum (red solid line), resonance fitting, and replotted IQ curve (yellow dashed line). The pressure is 0.229 bar. (b) The experimental spectrum shift as the MBR internal pressure (0.229, 0.327, 0.424, and 0.522 bar) increases.

Table 3. Strongly coupled resonance fitting results.

Fitting Object q1 q2 ∆λd/pm g/GHz κT/MHz FCOD 3 RCOD 4 Figure 4a Res. 1 (0.229 bar) 1.03 × 106 1.67 × 106 1.41 1.43 208 0.996 0.933

Figure 4a IQ. 2 9.00 × 105 1.64 × 106 1.41 1.34 279 0.985 0.983 Figure 4b Res. (0.327 bar) 9.69 × 105 1.71 × 106 1.24 1.37 249 0.997 - Figure 4b Res. (0.424 bar) 9.71 × 106 1.79 × 106 1.23 1.31 263 0.998 - Figure 4b Res. (0.522 bar) 9.71 × 106 1.87 × 106 1.15 1.25 276 0.998 -

1 Res., Resonance. 2 IQ., IQ curve. 3 FCOD, Fitting Coefficient of determination (Fitting R-Square). 4 RCOD, Replotted Coefficient of determination (Replotted R-Square).

Finally, we provide a discussion to explain which model should be used. Strong coupling will lead to mode splitting and generate two separate resonances. Thus, the asymmetrical single resonance can only be an overlapping or weak-coupling resonance. The weak-coupling model should be used. Two separate resonances may originate from two strong-coupling and overlapping resonances and will generate four peaks in the IQ curve. We can roughly determine the coupling status by the IQ curve. If the two center peaks are lower than the two side peaks in the IQ curve, the two resonances are either simply overlapping or weakly coupled. The weak-coupling model should be used. If the two center peaks are higher than the two side peaks in the IQ curve, the two resonances are strongly coupled. Therefore, the strong-coupling model should be used. Most importantly, if the weak (strong) coupling model is used for strongly (weakly) coupled resonance, the experimental curves cannot be well fitted.

Figure 4. (a) Experimental spectrum (blue scattered line), its IQ curve (black scattered line), IQ fitting,replotted spectrum (red solid line), resonance fitting, and replotted IQ curve (yellow dashed line).The pressure is 0.229 bar. (b) The experimental spectrum shift as the MBR internal pressure (0.229,0.327, 0.424, and 0.522 bar) increases.

Table 3. Strongly coupled resonance fitting results.

Fitting Object q1 q2 ∆λd/pm g/GHz κT/MHz FCOD 3 RCOD 4

Figure 4a Res. 1 (0.229 bar) 1.03 × 106 1.67 × 106 1.41 1.43 208 0.996 0.933Figure 4a IQ. 2 9.00 × 105 1.64 × 106 1.41 1.34 279 0.985 0.983

Figure 4b Res. (0.327 bar) 9.69 × 105 1.71 × 106 1.24 1.37 249 0.997 -Figure 4b Res. (0.424 bar) 9.71 × 106 1.79 × 106 1.23 1.31 263 0.998 -Figure 4b Res. (0.522 bar) 9.71 × 106 1.87 × 106 1.15 1.25 276 0.998 -

1 Res., Resonance. 2 IQ., IQ curve. 3 FCOD, Fitting Coefficient of determination (Fitting R-Square). 4 RCOD,Replotted Coefficient of determination (Replotted R-Square).

Finally, we provide a discussion to explain which model should be used. Strong coupling willlead to mode splitting and generate two separate resonances. Thus, the asymmetrical single resonancecan only be an overlapping or weak-coupling resonance. The weak-coupling model should be used.Two separate resonances may originate from two strong-coupling and overlapping resonances and willgenerate four peaks in the IQ curve. We can roughly determine the coupling status by the IQ curve.If the two center peaks are lower than the two side peaks in the IQ curve, the two resonances are eithersimply overlapping or weakly coupled. The weak-coupling model should be used. If the two centerpeaks are higher than the two side peaks in the IQ curve, the two resonances are strongly coupled.Therefore, the strong-coupling model should be used. Most importantly, if the weak (strong) couplingmodel is used for strongly (weakly) coupled resonance, the experimental curves cannot be well fitted.

4. Conclusions

In summary, an OD-PDH system was built in this study. Based on this system, a method used toanalyze the complicated resonances in MBR was presented. This method is repeatable, simple, sensitive,and accurate. The OD-PDH system can eliminate the phase difference between the local oscillation anddetection signal, and generate a stable IQ curve. The IQ curve relates to the optical field rather than thelight intensity, and contains both the amplitude and the phase information of the complex transmittedspectrum. The OD-PDH system transforms the sharp imaginary information of the resonance and theslope information of the resonance into the IQ curve. Any tiny change of resonance will influence theIQ curves significantly. Thus, the IQ curves can be used to analyze the complicated resonances that thetransmitted spectrum cannot be used to analyze. We used the IQ curve to analyze the asymmetricalresonance caused by overlapping and weak-coupling resonances, which cannot be discriminated by thetransmitted spectrum. Moreover, we analyzed the strong-coupling resonance, which may be mistakenlytreated as an overlapping resonance. The resonance parameters were obtained accurately by fitting the

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IQ curve and validated by replotting the spectrum. This method can increase the measurement rangein differential resonance sensing, thus allowing the differential sensing of overlapped resonances.

Author Contributions: Conceptualization, J.H. and L.X.; data curation, S.L. and B.L.; formal analysis, J.H., B.L.and L.X.; funding acquisition, L.L. and L.X.; investigation, J.H., B.L. and L.X.; methodology, J.H., L.L. and L.X.;project administration, L.L. and L.X.; resources, L.L. and X.W.; software, J.H.; supervision, L.L.; validation, S.L.and Z.G.; visualization, J.H. and Z.G.; writing—original draft, J.H.; writing—review and editing, L.X. All authorshave read and agreed to the published version of the manuscript.

Funding: This research was funded by National Natural Science Foundation of China (NSFC), grantnumber 11874122, 11474070, 61327008 and 11074051 and the Doctoral Program of Higher Education, grantnumber 20130071130004.

Conflicts of Interest: The authors declare no conflict of interest.

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