Highly Sensitive Complicated Spectrum Analysis in Micro-Bubble Resonators Using the Orthogonal Demodulation Pound–Drever–Hall Technique

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


Principle and Setup
In classical PDH technique theory, we can obtain the bipolar voltage wavelength discriminating signal (I-term and Q-term signals), and I 0 (ω) and Q 0 (ω) can be expressed as [27]: Q 0 (ω) = kP 0 J 0 (β)J 1 (β)Im[R(ω)R * (ω + ω m ) − R * (ω)R(ω − ω m )] (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; ω m is the angular frequency of the signal input into the phase electro-optical modulator (PM) (ω m = 2πf m , f m = 50 MHz); I 0 (ω) is the in-phase term; Q 0 (ω) is the quadrature term; k represents the scale factor containing the light power-voltage conversion coefficient, the insertion loss in every electronic element, and the gain of the amplifier; P 0 represents the laser power; J 0 and J 1 are the first kind of 0-and 1-order Bessel function; β is the modulation degree (β = πV/V π ); V is the signal of the peak-to-peak value voltage input into the PM; V π is the half-wave voltage of the PM; and R(ω) is the complex spectral response of the resonance. When the bandwidth 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 ω m is negligible, ]ω m ; the I-term and Q-term signals can be expressed as [27]: 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 ± ω m ) ≈ 1 and R*(ω ± ω m ) ≈ 1, and the I-term and Q-term signals can be expressed as [27]: Appl. Sci. 2020, 10, 6256 3 of 11 When the PM modulation frequency is close to the bandwidth of resonance, the I-term and Q-term signals exist simultaneously. The [d|R(ω)| 2 ]/dω]ω m (low Q condition) and 2Im[R(ω)] (high Q condition) are both sharp dispersion curves like the PDH discriminating frequency curve [27]. The PDH system transforms the sharp imaginary information of the resonance and the slope information of the resonance into the I-term and Q-term signals. Any tiny change of resonance will significantly influence the I-term and Q-term signals. Therefore, the I-term and Q-term signals can be used to analyze the complicated resonances that the transmitted spectrum cannot be used to analyze. However, the ϕ is an unknown value 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 and make the OD-PDH spectrum (IQ curve) stable. In sensing applications, this technique can help obtain the highest sensing sensitivity. In this experiment, we use Equation (7): to obtain the IQ curve, which does not relate to ϕ. For a single resonance, R(ω) can be expressed as [33]: where ω is the laser angular frequency, ω 0 is the resonant angular frequency, q 0 is the intracavity Q, and q e is the external cavity Q. When two resonances overlap or weakly couple, R(ω) can be described by [34]: where κ 0n = ω n /(2q 0n ), κ en = ω n /(2q en ), q n = (q 0n −1 + q en −1 ) −1 , X n = −κ 0n − κ en + i(ω − ω n ), ω n = 2πf n (n = 1, 2), and ∆λ d = c/f 1 − c/f 2 ; κ 0n is the intracavity decay rate of resonances; κ en is the external cavity decay rate of the two resonances; f n is the resonant frequency of the two resonances; q 0n is the intracavity Q of the two resonances; q en is the external cavity Q of the two resonances; q n is the resonance total Q of the two resonances; g is the coupling coefficient; ω n is the angular resonant frequency of the two resonances; ω is laser angular frequency; c is the light velocity in a vacuum; and ∆λ d represents the two resonances' distances as wavelengths. The parameters q 0n , q en , f n (n = 1, 2), and g can be obtained by fitting the IQ curve. The other parameters can be derived by the fitting parameters. When two resonances strongly couple, according to the coupled mode theory, the coupled mode equations can be written as [14]: where 4), and ∆λ d2 = c/f 3 − c/f 4 ; E in is the optical field amplitude of the input light; E p (p = 3, 4) is the optical field amplitude of the two resonances; κ 0p is the intracavity decay rate of the two resonances; κ ep is the external cavity decay rate of the two resonances; f p is the resonant frequency of the two resonances; q 0p is the intracavity Q of the two resonances; q ep is the external cavity Q of the two resonances; q p is the resonance total Q of the two resonances; g 1 is the coupling coefficient; and ω p is the angular resonant frequency of the two resonances. The optical field amplitude of the output light (E out ) can be described by: Appl. Sci. 2020, 10, 6256 4 of 11 The R(ω) can be described by: The parameters q 0p , q ep , f p (p = 3, 4), and g 1 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) = (x 1 , y 1 ), (x 2 , y 2 ), . . . , (x m , y m ) were obtained in measurement. The COD is defined as: where y m ,ŷ m , 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] |κ 1 − κ 2 |/2 as the threshold (κ T ) and the regimes where g < κ T and g > κ T as the weak-and strong-driving regimes, respectively; κ q (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.
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.

Experimental Results
Figure 2a plots standard a single optical resonance (blue scattered line), its IQ curve (black scattered 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 tunable laser via a tapered fiber whose diameter was about 2.5 µm. The IQ curve has two peaks that sit symmetrically centered on the optical resonance. The intracavity Q (q 0 = 1.20 × 10 6 ), external cavity Q (q e = 1.14 × 10 6 ), total Q (q a = (q 0 −1 + q e −1 ) −1 = 5.84 × 10 5 ) and resonant frequency (f a = 1.94 × 10 14 Hz) can be obtained by fitting the IQ curve with a single resonance IQ expression. Total Q (q b = 5.84 × 10 5 ) and resonant frequency (f b = 1.94 × 10 14 Hz) can also be obtained by fitting the transmitted spectrum directly. The results agree with each other and prove that the IQ curve can be used to analyze the spectrum. 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 a wall thickness of 4 µm was excited with an 850 nm tunable laser via a tapered fiber whose diameter was about 2.5 µm. However, direct fitting of the spectrum via a two-resonance expression did not derive unique fitting parameters. Figure 2b gives the fitted spectrum with the two group parameters. The two groups' different parameters listed in Table 1, however, provided an excellent fitting. The coefficient of determination parameters (COD, R-Square) were both above 0.980. Thus, we used the OD-PDH technique to analyze the resonance. Figure 2c plots the spectrum (blue scattered line) in Figure 2b and its IQ curve (black scattered line), fitting curve of the two resonances in Figure 2b and its replotted IQ curve (overlapping, red solid line; coupled, yellow dashed line). We used the fitted parameters from Figure 2b to replot the IQ curve. The CODs of the overlapping and coupled replotted IQ curves were 0.981 and 0.937, respectively. Clearly, the overlapping replotted IQ curves agree with the experimental spectrum. This proves that the IQ curve can discriminate asymmetrical resonance. Figure 2d plots the spectrum (blue scattered line) in Figure 2b and  curve can confirm that the overlapping parameters are the correct results. To prove that the spectrum in Figure 2b contains the two resonances, we injected air into the MBR to execute pressure tuning.  Figure 2b,e,f, respectively. The pressure tuning proves that the asymmetrical spectrum contains two resonances. Figure 2f plots 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 × 10 4 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 × 10 4 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 < κ T (κ T = 1.31 × 10 4 ); this reveals that the two resonances are weakly coupled. This process proves the validity of the OD-PDH technique in analyzing weakly coupled resonances.  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 × 10 4 ); this reveals that the two resonances are weakly coupled. This process proves the validity of the OD-PDH technique in analyzing weakly coupled resonances.    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 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, 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 > κ T , 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 > κ T . 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.
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.   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.  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.

Conclusions
In summary, an OD-PDH system was built in this study. Based on this system, a method used to analyze 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 and detection signal, and generate a stable IQ curve. The IQ curve relates to the optical field rather than the light intensity, and contains both the amplitude and the phase information of the complex transmitted spectrum. The OD-PDH system transforms the sharp imaginary information of the resonance and the slope information of the resonance into the IQ curve. Any tiny change of resonance will influence the IQ curves significantly. Thus, the IQ curves can be used to analyze the complicated resonances that the transmitted spectrum cannot be used to analyze. We used the IQ curve to analyze the asymmetrical resonance caused by overlapping and weak-coupling resonances, which cannot be discriminated by the transmitted spectrum. Moreover, we analyzed the strong-coupling resonance, which may be mistakenly treated as an overlapping resonance. The resonance parameters were obtained accurately by fitting the IQ curve and validated by replotting the spectrum. This method can increase the measurement range in differential resonance sensing, thus allowing the differential sensing of overlapped resonances.

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