1. Introduction

Strong light-matter interactions in WGM microcavities can be achieved due to inherent ultrahigh Q-factors and small mode volumes of WGMs [1,2]. As a result, the WGM microcavities have great value for fundamental researches and applications, including cavity QED [3–5], optical solitons [6], parity-time symmetry [7,8], WGM microbottle resonators [9,10], and quantum emitters [11,12], ultralow threshold lasers [13,14], Raman lasers [15], Kerr frequency combs [16,17], sensors [18,19], filters [20,21], etc.

Before these studies, coupling light into/out of passive WGM microcavity conveniently and effectively is critical. Heretofore, various coupling schemes have been demonstrated, such as using the fiber taper [22], prism [23], microfiber coupler [24], D-shaped fiber [25], angle-polished fiber [26], etched capillary [27] and so on. The fiber taper coupling is most efficient but fragile, because the taper waist is normally less than 2 μm. For the prism coupling, the coupling distance and the incident angle can be adjusted but it is bulky. The others except etched capillary coupling are the homogeneous coupling schemes of the two coupling schemes. The etched capillary coupling is different from the former because it is reflective type and free of alignment, which is convenient for sensing and promote the practicality of WGM devices. The unique value triggers many homogeneous coupling schemes including etched photonic crystal fiber coupling [28], cone-shaped inwall capillary coupling [29], tapered hollow annular core fiber coupling [30] and dual-core hollow fiber coupling [31]. However, the couplers of etched capillary, etched photonic crystal fiber and cone-shaped inwall capillary need chemical etching, which reduces not only the mechanical robustness but also surface smoothness. The couplers of tapered hollow annular core fiber and dual-core hollow fiber need complex optical fiber structures with special fiber cores, making fabrication process much complex. Besides, these coupling schemes only consider and realize evanescent wave coupling but no free space coupling by deformed microcavities [32] in coupling mechanism.

In this paper, we propose a simple zigzag transmission coupler for WGMs, which is composed of single mode fiber (SMF), microsphere and silica capillary. Firstly, we theoretically study the relation between the position of the second inner reflective region (SIRR) and the device parameters, which will be verified in experiment. The effects of the arc power in fabrication process on the cone-apex angle and length of complete collapse are also investigated in experiment. Besides, the critical cone-apex angles to distinguish between evanescent wave coupling and free space coupling are also obtained by ray optics. Finally, we adopt ray dynamics and tunneling process to investigate different spectral characteristics for evanescent wave coupling and free space coupling. As there are two coupling methods in this simple coupler, it can be adopted as a basic building block for promoting the development of photonic integrated devices.

2. Device structure and theoretical model

The device is composed of a SMF, a barium titanate glass microsphere with a refractive index up to 1.93, which decreases the radiation loss effectively, and a flexible fused silica capillary tube with an inner diameter of 75 μm and an outer diameter of 125 μm (TSP075150, Polymicro Technologies, LLC), as shown in Fig. 1(a). The microsphere is locked firmly nearby the inflection point of the cone (IPC) and deformed slightly due to the stress. The input beam is incident into the SMF and reflected towards the ambient directions beside the cone apex. Due to the central symmetry of the structure, we only draw the beam path diagram in one direction. The beam is reflected again at the outer wall resulting in SIRR, and incident onto the inwall in a form of a zigzag transmission resulting in the first outer reflective region (FORR), as shown in the red light beam whose upper and lower boundary lines are marked with white and yellow, respectively. Then the beam can be coupled into the microsphere by evanescent or free space coupling. Finally, the light returns to the SMF in the opposite way on the other side of the device. An optical sensing analyzer (SM125, Micron Optics, Inc.) is adopted as the light source and optical receiver.

 

Fig. 1. (a) Schematic of the capillary-based microsphere resonator with optical zigzag transmission. (b) Simulated optical zigzag transmission in the capillary-based coupler (θ = 18°, s = 0 μm).

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To effectively couple light into the microsphere, we must select the microsphere with a right size to lock it onto SIRR, which is the first inner reflective region after the one near the cone apex. The reflective region is the overlapping region between the beam and the edge of capillary wall when the beam is reflected. Generally, we do not embed the microsphere at the first inner reflective region nearby the cone apex, because it needs a much smaller microsphere, resulting in a sharp increase in radiation losses of WGMs. Moreover, in fact, the slope of the cone is not constant and its absolute value decreases gradually from the cone apex to the IPC. Hence, the firmness of the microsphere’s lock by the inwall increases gradually from the cone apex to IPC.

The structure and size of the device determines where the SIRR is located relative to the IPC. There can be three basic cases, i.e., before IPC (BIPC), on IPC (OIPC) and after IPC (AIPC). In the BIPC case, the lower boundary line of SIRR is before IPC and then the microsphere usually needs to be located about the middle of the cone for effective excitation of WGMs. However, if the distance between the lower boundary line of SIRR and IPC is less than the length of frontal coupling region h₁, the solution is the same with the OIPC case. In the coupling region, when the microsphere is locked nearby IPC, the distances between the microsphere surface and the inwall of capillary are smaller than one wavelength, resulting into effective coupling by evanescent wave and free space coupling. When this distance is larger than h₁, this special BIPC case is called as before coupling region (BCR). In the OIPC case, the SIRR covers the IPC, which needs the microsphere to be locked nearby the IPC to excite WGMs. Hence, the microsphere should be a little smaller than the inner diameter of the capillary. As previously mentioned, the actual IPC is not an inflection dot but a smooth curve, which is better to lock the microsphere. In the AIPC case, the upper boundary line of SIRR is after the IPC. If the distance between SIRR and IPC is less than the length of posterior coupling region h₂ as shown in Fig. 1(a), the solution is the same with the OIPC case. If not, it is impossible to excite WGMs when the microsphere is locked. This special AIPC is called as after coupling region (ACR). The reflection at the end face of the capillary is not considered, because it leads to Fano resonance [33] and EIT [34], making the structure not pure and basic any more. To sum up, in details, there are 5 cases for the location of the SIRR relative to the IPC and coupling region, i.e., ① BCR, ② BIPC, ③ OIPC, ④ AIPC and ⑤ ACR, which have been numbered.

Figure 1(b) shows simulated optical zigzag transmission in the structure by Rsoft BeamPROP module. The cone-apex angle θ is set as 18 °, while the length of complete collapse s is set 0 μm. The optical zigzag transmission is very clear, while a small amount of light refracts into the air core and the outer air of the capillary due to the diffraction at the cone apex. The diffracted light is ignored in the following discussion as the portion is very small. From the perspective of optical modes, the zigzag transmission is caused by multimode interference. When the light enters from the SMF into the capillary, multiple modes are excited with or without the complete collapse. These modes form the pattern of zigzag transmission along Z axis beside the IPC by interference. To investigate the transition of these five cases, the inequalities to determine SIRR in the OIPC case are obtained conveniently by the ray optics method as follows (Appendix for the derivation process)

The inequalities (1) and (2) versus θ are plotted in Fig. 2. Here, n₀, n₁, n₂ and n₃ are adopted as 1.0, 1.4681, 1.4628 and 1.4441, respectively. Δ is 2.9°. s is 0 μm. r is 37.5 μm. The solid blue and red lines correspond to the inequalities (1) and (2), respectively, where h₁ and h₂ are both 0 μm. The dashed ones correspond to the condition that h₁ and h₂ are both 15 μm. The five cases of SIRR are marked in the corresponding regions with different colors. The AIPC, OIPC and BIPC regions are effective regions where the microsphere is locked nearby the IPC and then WGMs are excited effectively. From inequalities (1) and (2), the solid and dashed red lines have no concern with s. However, the solid and dashed blue lines will move down as s increases, in other words, the whole effective region becomes larger. Meanwhile, the length of SIRR also increases. The length of SIRR in Fig. 1(b) is 22 μm while 32 μm in Fig. 5(a). The dashed purple line indicates the actual value of l as 25 μm. Hence, the boundary values are 14°, 21°, 28.5° and 37°, respectively. Zigzag transmission exists only when the width of the beam is much smaller than that of its transmission medium.

 

Fig. 2. Regions where SIRR is in cases ① ACR, ② AIPC, ③ OIPC, ④ BIPC and ⑤ BCR, respectively, are marked with different colors. And the horizontal dotted line represents the actual value of l as 25 μm.

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3. Fabrication process

The fabrication process is simple and only needs fusion splicing. The capillary with an inner diameter of 75 μm is adopted because it can accommodate the microsphere without inner chemical etching. Moreover, its outer diameter of 125 μm matches the cladding diameter of SMF well. The capillary is spliced with SMF by a commercial fusion splicer (FITEL-S178). We can adjust the splice time and arc power to form a cone between the capillary and SMF, which results from the collapse of the capillary. The cone-apex angle is mainly dependent on the collapse velocity, which is related to the arc power. Figure 3(a) shows the cone-apex angle θ and s versus arc power, where the splice time is 1000 ms. It is observed that as the arc power increases gradually, the cone-apex angle becomes larger and larger in general because the capillary collapes faster and faster. The smallest cone-apex angle formed by only fusion splicing is about 18° because the too small arc power will form a bubble at the splicing interface, and then the cone of 9° needs to be formed via tapering the cone of 20° by 100 μm [30]. Meanwhile, s also becomes larger and larger in general with arc power increasing because the larger arc power leads to the collapse of longer capillary, and its variation trend is consistent with the cone-apex angle.

 

Fig. 3. (a) Cone-apex angle θ and s versus arc power. The cone with the cone-apex angle as (b) 9°, (c) 18°, (d) 25° and (e) 33°. The couplers in (c), (d) and (e) are marked with corresponding yellow rectangle c, d and e in (a). Bit is the unit of ARC power in the welding machine FITEL-S178, and 1 bit represents a certain amount of power corresponding to a fixed melting quantity of the optical fiber end face.

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Figures 3(b)–3(d) show the resulting couplers with θ as 9°, 18° and 25°, which correspond to cases of ACR, AIPC and OIPC, respectively. The couplers with θ of 33°, 37° and 40° are also prepared but they all belong to OIPC, because their s are up to about 270 μm, 340 μm and 390 μm, respectively. Hence, we only consider the coupler of 33°, as shown in Fig. 3(e). Due to tapering, its outer profile has a slight collapse, which makes SIRR move toward the cone apex slightly. Finally, the capillary is cleaved to about 1 mm, and then a barium titanite glass microsphere is inserted into the cone, by the precision fiber alignment stage (M-562, Newport) and fiber taper.

4. Experimental results and analysis

The SMF of the device is connected to an optical sensing analyzer to obtain its reflection spectrum. Figures 4(a)-(d) show the reflection spectra of devices with θ as 9°, 18°, 25° and 33°, respectively. The reflection at the end face of capillary has been eliminated by immersing the end face into the matching liquid. Two sizes of microsphere are inserted into cones. One’s diameter is ∼55 μm, called small microsphere, and another is ∼70 μm, called big microsphere. When they are locked firmly by the cone, the former is located about the middle of the cone and the latter is located nearby IPC. In Fig. 4, the spectra of the former are in blue and those of the latter are in red. The coupler with θ as 9° belongs to ACR so there are no clear resonance peaks in spectra no matter which kind of microsphere is inserted because SIRR is after h₂. Due to slight collapse of the cone, SIRR moves toward the cone apex slightly and then there are some little resonance peaks beside 1585 nm with a big microsphere inserted, as shown in Fig. 4(a). The coupler with θ as 18° belongs to AIPC so there are WGM peaks in spectra with a big microsphere inserted because SIRR overlaps h₂. However, there are no resonance peaks in spectra with a small microsphere inserted because the small microsphere is away from IPC and SIRR, as shown in Fig. 4(b). For 25° and 33°, they both belong to OIPC and their experimental results are similar to 18° because SIRR has overlap with the coupling region of h₁ and h₂, and the small microsphere is also away from IPC and SIRR, as shown in Figs. 4(c)–4(d). Via the fast Fourier transforms of these spectra, corresponding microsphere’s diameter can be obtained by measuring the interval between adjacent peaks. This method has been demonstrated in detailed [29].

 

Fig. 4. Reflectance spectra of the devices with the cone-apex angle of (a) 9°, (b) 18°, (c) 25° and (d) 33°. Insets are corresponding microscopic images of the devices inserted with small and big microspheres, respectively. Each highest Q-factor and FSR are indicated.

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However, the coupler C in Fig. 4(c) has different spectral characteristics from the coupler B in Fig. 4(b). For the coupler B, there is a clear envelope of FSR with dense peaks beside a high-intensity peak forming a group of high-intensity WGM peaks, as indicated in the orange dashed box in Fig. 4(b). Meanwhile, there are only very weak peaks between two groups of high-intensity dense WGM peaks. However, there is no clear envelope of FSR but compact and fairly intensity-uniform resonance peaks for the coupler C. For example, among one FSR between two peaks of the same order, there are some sparse peaks with similar intensities to the former, as indicated in green in Fig. 4(c). The coupler D in Fig. 4(d) has similar spectral characteristics with the coupler C but its peaks have much low intensities. For further exploration on couplers B and C, their light propagations are simulated as shown in Fig. 5(a) and Fig. 5(b). It is observed that there exists total reflection in SIRR of the coupler B while there is refraction in SIRR of the coupler C, because they have different cone-apex angles. In other words, the light in the capillary is coupled into the microsphere by evanescent wave coupling for the coupler B but by free space coupling for the coupler C. α_i is the incident angle of the upper boundary line and β_i is incident angle of the lower boundary line. The subscript i indicates the order of the reflection, as shown in Fig. 1(a). If α_i and β_i are located in the cone, α_i=90-iθ and β_i=90-iθ+Δ. If not, they are equal to the previous angles. In consideration of the critical angle of total reflection, arcsin (n₀/n₃), the critical cone-apex angles θ_c1=23° and θ_c2=24.5° are obtained by solving these two equations with i=2. When θ<23°, the coupling method is evanescent wave coupling. And when θ>24.5°, the coupling method is free space coupling. For θ between 23° and 24.5°, the coupling method may be pure or mixed, which involves many factors and then is not discussed here. This theoretical analysis complies with the simulation.

The coupler C belongs to OIPC but its SIRR doesn’t cover IPC because the light in the front part of its FORR has a smaller angle of reflection and then has a lower reflectivity according to Fresnel formula. This leads to hardly any reflected light in the front part of FORR and then hardly any refracted light in the front part of SIRR. The refracted light in latter part will enter into the microsphere directly by free space coupling. The coupler D has a larger cone-apex angle than the coupler C so its light in FORR has a lower reflectivity regardless of the front part and latter part. Hence, compared with the coupler C, less light enters into the microsphere of the coupler D and then the resonance peaks in its reflectance spectrum have lower intensities. For an ideal circular microcavity, in order to achieve critical coupling by free space coupling, m of the WGM in the cavity must be 30, where m is the number of wavelength in the circumference of the resonator [32]. For example, Vogt et al. have experimentally realized an ideal spherical cavity coupled by free space coupling with a coupling efficiency of 51% in the terahertz (THz) frequency range [35]. There, m is 34, a little bit more than 30, which means that the coupling regime is under coupling regime to obtain a higher Q-factor. In the near-infrared band from 1510 nm to 1590 nm, m of WGMs in our barium titanate glass microspheres with diameters of ∼55 μm and ∼70 μm are ∼215 and ∼273, respectively. They are much larger than 30 so the free space coupling efficiency of the microspheres as ideal spherical microcavities is almost zero and negligible. However, the experiments and simulations show that an effective free space coupling does occur in Fig. 4(c), which means that the microsphere in the device must have deformed due to the stress from the capillary wall, which holds the microsphere firmly in the cone [36].

Ray dynamics and tunneling process [37] are adopted to analyze the deformed microresonator supporting WGMs excited by the chaos-assisted momentum–transformed coupling method [38]. Because WGMs are mainly distributed on the equator plane, the microsphere is simplified as a two-dimension shape limited by its equator. The shape’s boundary is considered as quadrupolar deformation because any oval can be approximated by a quadrupole [39]. The quadrupole is defined in polar coordinates (ρ, φ) by ρ(φ) = R₀(1 + ɛ cos(2φ)) where R₀ is the radius of undeformed microsphere and ɛ is the deformation parameter. The Poincaré surface of section (PSS) can present the features of the inner ray dynamics in the microsphere well. The PSS of the microcavity with a quadrupolar shape with a typical ɛ value of 0.07 is shown in Fig. 5(c). χ is the incident angle of the light reflecting on the inwall of the microcavity. In PSS, there are three kinds of structure including chaotic orbits, islands and Kolmogorov–Arnold–Moser (KAM) curves, as indicated in black, blue and red, respectively. The chaotic orbits are also called chaotic sea. The latter two are both regular orbits. The high-Q WGMs are usually localized in these regular orbits [40]. There exists dynamical tunneling between the regular orbits and neighboring chaotic orbits.

For the evanescent wave coupling of the coupler B, its initial incident rays are located around B in PSS, because the evanescent wave coupling mainly excites WGMs of low orders due to the much stronger evanescent waves of low-order WGMs. The region B is in the upper end of PSS where there are dense regular orbits and almost no chaotic orbits [41], resulting in almost no dynamical tunneling. As a result, there is a clear FSR envelope in the reflectance spectrum of the coupler B, and there are also dense peaks beside a high-intensity peak. From another perspective, the density of WGMs attributes to the breaking of azimuthal degeneracy of WGMs due to deformation [42]. For the coupler C, its initial incident rays are located around C in PSS because the light is refracted into the microcavity and then its initial sin χ with χ of about 30° must lie under the critical line 1/n_s, where n_s is the refractive index of the microsphere equal to 1.93. There are mainly chaotic orbits in the region C and there are sparse regular orbits around the region C. The chaotic sea will couple with the regular orbits by dynamical tunneling, which can be modeled to a two-level system by considering the continuum of chaos to a single state. We obtain the approximate distribution of energy for free space coupling by solving the steady-state solution of the modified equation of motion for the photon counts which is given as follows [38]

5. Conclusion

In summary, we propose a new kind of capillary-based coupler to excite WGMs of the slightly deformed micro-sphere resonator, which is only composed of SMF, silica capillary and barium titanate glass microsphere, and is free of chemical corrosion and complex structures of optical fiber. The beams pass through the cone in a zigzag shape and then the coupling is effective only when SIRR is near IPC, the inequalities for which condition are obtained by ray optics and experimentally verified. When the coupler belongs to effective regions of AIPC and OIPC, the corresponding cone-apex angle and length of complete collapse can be obtained only by adjusting the arc power. By adjusting the cone-apex angle, the device can realize both evanescent wave coupling and free space coupling. The cone-apex angle smaller than 23° corresponds to evanescent wave coupling while that larger than 24.5° corresponds to free space coupling. There is a clear FSR envelope in the reflectance spectrum of the former with dense peaks beside a high-intensity peak. However, there is no clear FSR envelope but compact resonance peaks with fairly uniform intensities in the latter’s, which is interpreted well by ray dynamics and tunneling process. The equation of motion for the photon counts is modified to analyze the energy distribution in modes for the free space coupling. The device is simple and realizes flexible coupling scheme for WGMs, and then can work as a pure and basic building block in photonic integrated devices.

 

Fig. 5. Simulated light propagation of (a) coupler with θ as 18° and s as 98 μm, and (b) coupler with θ as 25° and s as 165 μm. (c) The PSS of the microcavity with a quadrupolar shape with ɛ equal to 0.07. The green dashed line represents the critical line sin χ = 1/n_s. n_s is the refractive index of the microsphere and equal to 1.93. Pink B and C are initial regions of couplers B and C, and their φ are all equal to about 0.55π.

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Appendix

Compared with Fig. 1(a), Fig. 6 only shows the half structure of the device and modifies the values of some quantities to make the structural relationship more distinct. A coordinate system ZOR is set up and its origin is located at the cone apex. Here, the white lower case letters represent the values of line segments or angles while capital letters represent cross points. Meanwhile, the dashed lines represent auxiliary lines and the green line represents the central axis of the device. To understand Fig. 6 more comprehensively, we can refer to Fig. 1(a). For example, MO is s, LU is SIRR, AB is FORR, CE is h₁ and C is IPC. The Z-coordinate values of O, U′, L′, C, U and L are 0, Z_U′, Z_L′, Z_C, Z_U and Z_L, respectively. TW is parallel to QN and CU′. EF is parallel to LA.

 

Fig. 6. Diagram of half structure of the device.

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According to the geometrical relations in Fig. 6, we can obtain

(A1)$$\begin{array}{l} \;\;{Z_C} = r/\tan \theta ,\;{Z_{U^{\prime}}} = r/\tan (2\theta ),\;{\alpha _1} = \pi /2 - \theta ,\;{\beta _1} = \pi /2 - \theta + \Delta ,\;{\alpha _2} = \pi /2 - 2\theta ,\;\\ {\beta _2} = \pi /2 - 2\theta + \Delta ,\;\gamma = 2\theta - \Delta ,\;DM = {r_0}/\tan \Delta ,\;DO = DM + MO = {r_0}/\tan \Delta + s,\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;U^{\prime}U = 2l\tan (\pi /2 - 2\theta ),\;L^{\prime}L = 2l\tan (\pi /2 - 2\theta + \Delta ), \end{array}$$

and then we can obtain the following equation

(A2)$${Z_U} = {Z_{U^{\prime}}} + U^{\prime}U = r/\tan (2\theta ) + 2l\tan (\pi /2 - 2\theta ).$$

We also can obtain the equation

(A3)$$(DO + ON)\tan \Delta = ON\tan \theta .$$

By solving this Eq. (A3), we can obtain

(A4)$$ON = DO\tan \Delta /(\tan \theta - \tan \Delta ),$$

and then

(A5)$$QT = ON\tan \theta = DO\tan \Delta \tan \theta /(\tan \theta - \tan \Delta ).$$

Then, we obtain the following equations

(A6)$$TW = (r - QT)/\tan (\gamma ),\;{Z_{L^{\prime}}} = ON + TW,$$

and then

(A7)$$\begin{array}{l} {Z_L} = {Z_{L^{\prime}}} + L^{\prime}L = ({r_0}/\tan \Delta + s)\tan \Delta /(\tan \theta - \tan \Delta ) + \\ \;\;\;\;\;\;\;(r - ({r_0}/\tan \Delta + s)\tan \Delta \tan \theta /(\tan \theta - \tan \Delta ))/\tan (2\theta - \Delta ) + \\ \;\;\;\;\;\;\;2l\tan (\pi /2 - 2\theta + \Delta ). \end{array}$$

To ensure the SIRR in the OIPC case, it must be satisfied that

(A8)$${Z_L} \ge {Z_C} - CF,$$

(A9)$${Z_U} \le {Z_C} + {h_2}.$$

Because CF is small and slightly larger than CE, i.e., h₁, we use h₁ instead of CF for simplicity without losing effectiveness. Hence, the inequality (A8) can be written as

(A10)$${Z_L} \ge {Z_C} - {h_1}.$$

By solving the inequalities (A10) and (A9), we can obtain the inequalities (1) and (2), respectively.

Funding

National Natural Science Foundation of China (61675126, 61875116); Natural Science Foundation of Shanghai (18ZR1415200); Open Project Program of Wuhan National Laboratory for Optoelectronics (2018WNLOKF014); 111 project (D20031).

Disclosures

The authors declare no conflicts of interest.

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