1. Introduction

Whispering-gallery mode (WGM) has wide applications in many research areas, including cavity quantum electrodynamics [1], optical frequency comb [2], Raman laser [3] and optical sensing [4]. Droplet resonators are ideal instruments for sensing [4,5] and opto-mechanics [6–8], because of surface tension and the ability to detect in the liquid specimen. The majority of previous works focus on the fundamental WGMs that are lain on the equatorial plane. Other azimuthal modes (high-order azimuthal modes) whose fields are not restricted on the equatorial plane draw less attention. However, the high-order azimuthal modes have quite a few usages. The difference in propagation constant between the azimuthal modes is much larger than the difference between fundamental modes. And the wavelength spacing of azimuthal modes is much smaller than that of fundamental modes. Therefore the high-order azimuthal modes are easier to achieve high-efficient coupling for a probing light with narrow spectral range. Besides the coupling convenience, the high-order azimuthal modes are linked to the shape variation. Since the Q values of azimuthal modes are sensitive to the shape fluctuation, the high-order azimuthal modes have the potential to detect the thermally induced shape fluctuation of droplets [9,10].

Due to a few reasons, the phase matching of high-order azimuthal WGMs has rarely been observed and analyzed. Firstly, the high-order azimuthal modes degenerate in the ideal sphere resonator. The high-order azimuthal modes share the same wavelength with the fundamental modes. For spherical resonators such as solid microspheres and small size droplets whose diameter is less than 1 mm, the wavelength spacing of azimuthal WGMs is less than several pico-meter (pm). It requires high spectral resolution to distinguish specific modes. Even though the phase matching is realized of those WGMs, it is still hard to select specific azimuthal mode and examine the coupling condition of the modes [10]. Some efforts of the azimuthal mode selection are made in solid resonators. A novelty shape WGM resonator called “bottle microresonators” is proposed [11,12]. It utilizes the spheroid shape to separate those azimuthal modes. However, for the liquid resonator, there are few discussions about azimuthal WGMs selection. Secondly, the wavelength of the probing source also affects the wavelength spacing azimuthal modes. The probing source used in previous works of droplet resonators is usually a wavelength-swept laser with the wavelength between 600 nm and 1000 nm [8,10] Because the water absorption is relatively low in that spectral waveband. However, the larger wavelength is beneficial for azimuthal modes separation, although the loss is relatively higher than that of the commonly used wavelength.

In this letter, we observe the optical phase matching of a high-order azimuthal WGM in a water droplet. The droplet is prepared on a hydrophobic surface, which significantly simplifies the platform structure [13]. Due to the surface tension and gravity, the water droplet forms a spheroid with a diameter of ∼1.78 mm. The spheroid is beneficial for separating different azimuthal modes in a droplet. Furthermore, we employ a filtered amplified spontaneous emission (ASE) source, whose center wavelength is 1550 nm and bandwidth is 1 nm, as the probing source. The wavelength at this waveband also helps to enlarge the wavelength spacing of azimuthal modes. In the experiments, we observe the phase matching between the specific high-order azimuthal mode (l-|m|=6), which is not fundamental mode, and the fiber taper mode around 1550 nm for the first time. The phase matching mode has the deepest coupling notch and the maximum coupling coefficient around 1550 nm.

2. Experimental setup

The experimental setup is depicted in Fig. 1(a). The fiber taper serves as a coupler for the WGMs in the droplet and the mode in fiber taper [10,14].

 

Fig. 1. (a) The water droplet coupling experimental setup. Filtered ASE light enters the fiber taper and couples to the droplet. The light passing through the fiber taper is sent to an OSA. (b) Top view of the droplet coupling system. (c) Front view of the droplet coupling system. The equatorial diameter of the droplet in optical micrograph is about 1.8 mm.

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Comparing to other coupling methods, such as prism [15] and angle polished fiber [16], fiber taper coupler has many merits. Firstly the droplet is easy to attach to the coupler surface and deform droplet spheroid shape when coupling. So the prism and angle polished fiber cannot be used in droplet coupling. However the thin waist of the fiber taper prevents the droplet attachment. Another advantage of fiber taper is the adjustability of its propagation constant. The taper helps the fiber mode extend into free space surrounding the taper. By controlling the taper waist in the pulling process, the amount of light extending into free space can be adjusted. Therefore, the effective refractive index and propagation constant of fiber taper mode is tunable. Adjustable propagation constant ensures high-efficient coupling to any mode of interest, in resonators with a broad range of sizes. The fiber taper is made of a single mode fiber (G652D; BOGANG). The fiber is pulled on both sides and heated by oxy-hydrogen flame [17]. If the taper is not tiny enough, the fiber taper will have the interference pattern of multiple transverse modes. In order to avoid the interference pattern, a C band ASE source (LIGHTCOMM) is applied to examine the multiple transverse modes interference. The transmission spectrum is monitored by an optical spectrum analyzer (OSA, AQ6370D, YOKOGAWA) during the pulling process. Due to the interference of transverse modes, some interference ripples can be observed in the transmission spectrum. When a single-mode taper is produced, those interference ripples will gradually disappear. In our experimental setup, we make a fiber taper around 1550 nm, whose waist diameter is 1∼3 µm [Figs. 1(b) and 1(c)]. The accurate measurement of fiber taper could be obtained by SEM. However, we lack a suitable platform to transfer the fragile fiber taper.

The water droplet is dripped onto a super-hydrophobic surface that attached to a computer-controlled three-axis stage. The water droplet on the super-hydrophobic surface forms an oblate spheroid with a 1.8 mm equatorial diameter [Fig. 1(c)] due to the surface tension and gravity. The coupling system is placed in an acrylic chamber to avoid external interference. With the help of a humidifier, the humidity in the chamber is 60%(±5%) to reduce the evaporation. Quick coupling is the other way to reduce evaporation. With the help of the computer-controlled three-axis stage, after the droplet is made and photographed, the droplet could be quickly close to the fiber taper with low movement resolution. Then the droplet gradually approaches with relative high movement resolution (1µm) to coupling. The time for a successful coupling process is within 10 seconds. During this period, water evaporation is small which ensures the effectiveness of diameter measurement and the calculation of the modes. The evaporation problem might be avoided when using a mixture of water and glycerol. However, it is difficult to clean the taper after the mixture attaches to the taper unexpectedly. The fragile taper will break easily. The computer remotely controls the stage to help the water droplet approach and couple to the fiber taper.

In our experiments, the ASE source is applied to probe the coupling transmission spectrum, and the OSA is used to obtain the spectrum. Traditionally, the tunable single-longitudinal-mode laser is used as the probing source. Because of the large coherence length, the single mode laser source helps the optical cavity obtain the high Q value. However the single mode laser source brings unexpected residue fluctuation in the droplet resonator [10]. The possible reason is the thermal capillary oscillations [6,8] which is difficult to suppress. The ASE source, which has small coherence length and large bandwidth, is insensitive to the residue fluctuation, which is beneficial to identify different azimuthal modes. For this reason, we choose the ASE source as the probing source. The spectral resolution of the transmission spectrum is limited by the OSA, which is 20 pm in our experiment. In order to control the optical bandwidth and power coupling to the droplet resonator, an optical band-pass filter (OBPF, XTA-50/W, Yenista) is introduced to limit the bandwidth to 1 nm. Because the OBPF limits the optical power coupling to the droplet resonator, the evaporation of the water droplet is weakened as well.

3. Principle

By analyzing the coupling coefficients that are calculated from the coupling transmission spectrum, the propagation detuning of different azimuthal modes can be evaluated. The coupling coefficient (κ) is related to the coupling gap (D₀) and the propagation detuning (Δβ) between WGMs and the taper (κ²∝κ²(D₀)exp[-(β_f–β_m)²) [18]. Strictly speaking, the gap is not identical for different azimuthal modes. However, according to the theoretical calculation [18], the difference in the gap is small when the l and m is large enough. The field of the azimuthal modes (with a large m and l) is close to (slightly above or below) the equatorial plane. So the gap difference could be ignored and the gaps between the fiber and those azimuthal modes could be regarded as the same. If the gap is basically the same for those azimuthal modes (around the equatorial plane), the difference in coupling coefficients will indicate the propagation detuning of these modes. When the detuning is zero, the coupling coefficients have the maximum value, which is called “phase matching condition”. In order to have an intuitive understanding of WGMs and its propagation constants. We recall a useful visualize interpretation that treats WGM as a “zig-zag” path beneath the sphere surface as Fig. 2 shown [15].

 

Fig. 2. Intensity cross sections for azimuthal mode |m|=l–2 and the schematic of the mode propagation constants along the spherical surface.

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For a sphere with a radius R and an index n_s, the WGM in the sphere is described by four integer eigenvalues p, n, l, m [18–20]. The eigenvalues p represents the polarization (TE mode, TM mode). The eigenvalues n correspond to the radial mode, representing the field extremum in the radial direction. The eigenvalues l counts the half number of extremum in the equatorial plane, corresponding to the angular mode. The eigenvalues m corresponds to the azimuthal mode, which has an angle α between the mode paths with the equatorial plane $\cos (\alpha )= {m \mathord{\left/ {\vphantom {m {\sqrt {l({l + 1} )} }}} \right.} {\sqrt {l({l + 1} )} }}$. The value of l-|m|+1 is equal to the number of the field extremum in the polar direction. If |m|=l, there is only one maximum in the polar direction on the equatorial plane. The mode is called the fundamental mode. When |m|≠l, the modes will have several extremums that are symmetric to the equatorial plane in the polar direction. So the mode is not only restricted on the equatorial plane. These modes are called “high order mode”. For those “high order mode”, if l-|m| is even, there always will be an extremum on the equatorial plane. When the l-|m| is odd, the extremums will appear above and below the equatorial plane. Since those high order modes are not only distributed on the equatorial plane, those modes are more sensitive to the shape fluctuation comparing to the fundamental mode.

The propagation constant (β_l) of WGM is determined by the eigenvalues l. Whereas the projection of propagation constant on the equatorial plane (the equatorial propagation constant, β_m) depends on m as shown in Eq. (1). When the droplet couple to the fiber taper, the difference between the equatorial propagation constant, β_m and the propagation constant of fiber taper mode β_f is an important parameter to evaluate the propagation detuning (Δβ=β_m–β_f) [18] since the coupling occurs in the equatorial plane. The detuning will decrease the coupling coefficient (κ).

In the coupling system, the fiber taper lies on the droplet equatorial plane. Only the even azimuthal modes can be excited [23]. The wavelength spacing between azimuthal modes Δλ=2Δλ_m=eλ²/πn_s is less than the pseudo FSR. But the difference in propagation constant Δβ=2β_m=2/R is larger than that of the fundamental mode Δβ_l=1/R. Therefore, compared to the fundamental mode, the azimuthal mode is easier to achieve phase matching for the probing light with a narrow spectral range.

4. Results and discussion

The coupling results between the fiber taper and water droplet resonator in the under-coupled regime are shown in Fig. 3(a). The pseudo FSR, the wavelength spacing between the deepest notches in the coupling result, of the measured spectrum is 0.33 nm, corresponding to the droplet whose equatorial diameter is ∼1.78 mm. It is close to the result directly measured in the optical micrograph (1.8 mm). Theoretical calculations of the fundamental modes [18] indicate that there are three fundamental modes within the narrow spectral range λ_n=1,l=4724=1550.449nm, λ_n=1,l=4725=1550.122 nm and λ_n=1,l=4726=1549.795 nm.

 

Fig. 3. (a) The experimental coupling spectrum of the water droplet. The FSR of fundamental modes is 0.33 nm. (b) The simulation result of the coupling between the water droplet and fiber taper. (c) The zoom-in result around 1550 nm. The rightmost notch is in agreement with the calculated fundamental mode (l = m = 4725). (d) The zoom-in result around 1550 nm shows the phase matching between the fiber taper and the high-order azimuthal mode (l-|m|=6).

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The pseudo FSR form theoretical calculation is 0.327 nm which is close to the measured result from the spectrum. Other notches show that the azimuthal degeneracy is also lifted and the azimuthal modes are well separated. The wavelength spacing between azimuthal modes is around ∼50 pm, which indicates that the eccentricity e=–0.068. The minus indicates that the spheroid is an oblate spheroid whose polar radius is less than the equatorial radius. To further evaluate the coupling condition of azimuthal modes, the notches around 1550 nm, which share identical angular eigenvalues (l = 4725) is sketched in Fig. 3(c). It is worth noting that the deepest notches in our results are not the fundamental modes. Equation (3) indicates that, for an oblate spheroid whose eccentricity e is negative, the fundamental mode (|m|=l) corresponds to the largest wavelength among the azimuthal modes that share same n, l. The rightmost notch in Fig. 3(b) agrees with the calculated fundamental modes (n = 1, l = m = 4725). In addition, the Q values of the modes in Fig. 3(c) are measured using the Lorentzian fits. The Q value of the azimuthal modes, from left to right, are 2.3×10⁵, 2.3×10⁵, 2.0×10⁵, 1.9×10⁵, 1.9×10⁵, 3.1×10⁵and 3.6×10⁵ respectively. The rightmost modes has the largest Q value among the modes in Fig. 3(c). The Q value also indicates that the fundamental mode is the rightmost one in Fig. 3(c). Therefore, in our result, the mode that has the deepest notch which is high order azimuthal mode, appears in the center between two fundamental modes.

Figures 3(c) and 3(d) are the simulated spectra of a droplet resonator. The simulation is based on the transfer matrix method. Those azimuthal modes are treated as the modes that have different parameters including the perimeter, the propagation constant, and the coupling coefficient, which depends on the azimuthal eigenvalue m. By changing m value, the parameters for different azimuthal modes can be calculated. With the help of the transfer matrix method, the simulated spectrum can be calculated. The modes are painted in different colors to distinguish different azimuthal modes. The simulating results are similar but not identical to the experimental results, partly because of the inaccuracy of parameters (like the coupling gap and inaccurate radius of the taper waist). It is hard to measure the fiber taper accurately in the experiment. The fragile taper cannot be transferred to use SEM. In addition, the evaporation makes the droplet size shrink over time. It is hard measure its size in real-time accurately. However, the simulation is useful to identify the different azimuthal WGMs and evaluates the cross-coupling coefficient |κ| and the propagation detuning. The straight-through intensity transmission (σ²) depends on the inner loss (1-|a|²) and the straight-through coupling coefficient (|t|²=1-|κ|²). When the droplet is under-coupled (t > a), the gradually growing |κ| cause the decrease of t from unity to a. when t approaches to a, the transmission at resonance gradually decreases. Eventually, when t = a, which is called critical coupling, the transmission becomes zero [24]. So, in the undercoupled regime, the lowest the straight-through intensity transmission (σ²) comes from the biggest cross-coupling coefficient |κ|. In our experiment, the fiber taper is located on the equatorial plane. The difference overlap integrals of the fiber mode with different azimuthal modes are (κ²(D₀)) basically identical. Even the fiber taper may be not strictly on the equatorial plane, the difference in the cross-coupling coefficient caused by the integrals is much smaller than the difference caused by the propagation detuning. The gap (overlap integrals) could be considered as the same. The cross-coupling coefficient depends on the propagation detuning (κ²∝exp[-(β_f–β_m)²]). In other words, the phase matching mode has the deepest transmission notch comparing to other mismatch modes. Equation (1) indicates that the propagation constant is linearly changed as the value m. There is only one phase matching mode that has the deepest notch. Therefore the spectrum in our result represents a set of specific high-order azimuthal modes matching with the fiber taper. Those modes have suitable (β_m) that match the propagation constant of the fiber taper (β_f) at the corresponding wavelength.

To further investigate the propagation detuning, the comparison of the cross-coupling coefficients of the results in Figs. 3(b) and 3(d) is sketched in Fig. 4. The solid curve is the simulation result, and the dotted curve with square markers is the experimental result. Equation (1) shows that the equatorial propagation constant (β_m) is linearly changed with m, and there is only one maximum which corresponds to phase matching mode. The maximum coupling coefficient corresponds to the azimuthal mode (l-|m|=6) on both simulation and experiment results. The coupling coefficient of the leftmost point corresponding to the fundamental mode is significantly less than the maximum coefficient. As we mentioned, if the coupling gap is identical for all azimuthal modes, the coupling coefficient difference reveals the propagation detuning between the fiber taper and the high-order azimuthal modes. In our experiment, the high-order azimuthal mode (l-|m|=6) is optical phase-matched to the fiber taper around the wavelength of 1550 nm.

 

Fig. 4. The comparison of measured and simulated coupling coefficient |κ| of different azimuthal modes. The rightmost point corresponds to the fundamental mode. The maximum point of coupling coefficient corresponds to the phase matching azimuthal mode.

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5. Conclusion

We observe the optical phase matching between the fiber taper and a high-order azimuthal WGM, whose l-|m|=6 according to theoretical calculation and the simulation, in a water droplet around 1550 nm for the first time. We utilize a water droplet resonator whose diameter is ∼1.8 mm (Fig. 1(c)). The droplet eccentricity, which calculated from the coupling result e=−0.068. The oblate spheroid resonator separates the high-order azimuthal mode in the spectrum. When the probing light wavelength is 1550 nm, the wavelength spacing is around 50 pm. When the droplet is in the under-coupled regime, the intensity transmission (σ²) depends on the coupling coefficient |κ|, which depends on the propagation detuning and the gap between the fiber taper and the droplet. The intensity transmission (σ²) reveals the propagation detuning of those azimuthal modes (κ²∝exp[-(β_f–β_m)²]). Our coupling results indicate that the azimuthal mode (l-|m|=6) which has the lowest intensity transmission (σ²) and the largest coupling coefficient in the coupling spectrum, is the phase matching mode. The azimuthal modes phase matching ensures efficient coupling for a probing light with a narrow spectral range. It is useful in high sensitivity detection of shape and eccentricity fluctuation and specific modes selection when using the high-order azimuthal modes in the droplet resonator.