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In this paper, we demonstrate a cone-shaped inwall coupler for excitation of the whispering-gallery modes (WGMs) of a microsphere resonator. The coupler is composed of a single mode fiber (SMF) and a capillary with an inner diameter of 5 μm. After immersing the capillary front end vertically into Hydrofluoric acid to obtain a cone inside the capillary, light in the SMF couples into the capillary efficiently while the hollow core is wide enough for a microsphere to be inserted. Because the front end face of the capillary acts as a reflector, a Fano resonance with an asymmetric line shape and a Q-factor of 2.57 × 104 is observed in the reflection spectrum using a barium titanite glass microsphere with a diameter of 45 μm. The integrated resonator structure has the advantages such as the reflective type, alignment-free and mechanically robust, making it have great potential in sensing applications and optical switching.
© 2017 Optical Society of America
1. Introduction
The whispering-gallery modes (WGMs) describe the electromagnetic waves trapped in microcavities by continuous inner total reflection and circling along the inwall of the microcavity. Optical WGMs can have extremely high Q-factors and small modal volume, resulting in high sensitivity to environmental change and low threshold for lasing [1,2]. These advantages have enabled WGM resonators for broad applications including sensors [3], lasers [4,5], filters [6], tunable delay lines [7], frequency combs [8] and beam splitters [9].
Easily and efficiently coupling light into and out of the microsphere is the key step in the practical application of WGMs. Many coupling methods based on phase-matching the evanescent fields and the WGMs have been investigated. The two most typical methods adopt prisms [10] and fiber tapers [11]. The former is to place the microsphere near a prism surface where the WGM is excited by the evanescent field. The distance between the prism and microsphere can be adjusted to optimize the coupling efficiency. However, it is bulky and not easy to be configured for sensing applications. Similar to prisms in principle, side-polished optical fibers [12] and angle-polished fiber tips [13] have also been utilized to excite WGMs, while they have low coupling efficiency. The fiber taper is the most commonly used WGM coupler because of its highest coupling efficiency [14]. Unfortunately, the fiber taper coupler is efficient only when its waist diameter is less than 2μm [11], which makes the fiber taper fragile. The methods mentioned above share a common disadvantage that they need align the microsphere with the relevant component precisely, which is difficult to fulfill in many practical applications. Recently, two new alignment-free methods have been demonstrated, i.e., using the etched photonic crystal fiber (PCF) [15] and fiber pigtailed thin wall capillary [16]. However, the Etched PCF is very fragile because the freestanding solid core is easy to break at the bottom. Moreover, there are two reflectors in the scheme of Ref [15], which means that there is an envelope of F-P cavity in the reflective spectrum. Ref [16]. adopts a multimode optical fiber whose core diameter is 62.5 μm, to obtain a sufficiently light coupling between the fiber and the capillary. However, the multimode optical fiber is sensitive to environmental perturbations, resulting in the instability of the coupling between the optical fiber and the capillary and then leads to the instability of the reflective spectrum. In addition, there might be interferences among the multiple modes that are simultaneously coupled from the multimode fiber to the capillary, resulting in further instability of the resonance.
In the paper, we propose a cone-shaped inwall coupler for the excitation of the WGMs of a microsphere resonator. The coupler is composed of an optical fiber and a capillary with a small inner diameter. Note that the optical fiber adopted in our scheme is the single mode fiber to avoid multimodal interference and improve the stability. We obtain a long and sharp cone-shaped inwall by chemical etching, which has a higher coupling efficiency between the optical fiber and capillary than Ref [16], with the end face of the capillary as a reflector. As a result, the traditional Lorentz-shape resonances change to the Fano resonances [17–19]. Theoretical simulations and experimental results confirm the Fano resonances. The proposed structure is alignment-free, mechanically robust and operating in a reflection mode, making it very attractive in many applications.
2. Cone-shaped inwall capillary based microsphere resonator
Figure 1(a) shows the schematic structure of the cone-shaped inwall capillary based microsphere resonator. The device is composed of a single mode fiber (SMF), a capillary and a microsphere. The diameter of the hollow core of the capillary is 5 μm before etching. By etching the capillary, the hollow core becomes a long cone of which the front end has enough space to accommodate the microsphere with a diameter of 45 μm. The input beam enters the etched capillary wall from the SMF as shown in Fig. 1(a). One beam couples into and out of the microsphere exciting WGMs in the clockwise direction. Finally, it comes back as is represented by E6. Here, the model uses two coupling points to represent the 3D situation in 360 degrees. There exists another beam that transmits across the microsphere, reflects at the end face of the capillary, then transmits across the microsphere again in the counterclockwise direction, as is represented by E12 in Fig. 1(b), interfering with the former beam E6. Hence, traditional Lorentz-shape resonances of the WGMs are transformed into Fano resonances [20].
Fig. 1 (a) Schematic of the cone-shaped inwall capillary coupled microsphere resonator. (b) Schematic of the propagation of light at the front end of the capillary with Em (m = 1-13) representing the optical fields.
3. Fabrication process of the cone-shaped inwall capillary coupler
A flexible fused silica capillary tube (TSP005150, Polymicro Technologies, LLC) is taken to fabricate the coupler. After stripping off the coating layer of the capillary, it has an inner diameter of 5 μm and an outer diameter that matches the cladding diameter of SMF very well. The capillary is fusion spliced with a standard SMF and then cleaved to a desirable length, i.e., 2000 μm. The splice time and power are adjusted to form a cone shape at the connection between the SMF and the capillary, resulting from the incomplete collapse of the capillary, as shown in Fig. 2(a). In the etching process, the resulting capillary as shown in Fig. 2(b) is vertically immersed in 40% hydrofluoric (HF) solution. The HF solution enters into the hole of the capillary gradually, etching the capillary from outside and inside simultaneously, resulting into a long hollow cone in the capillary with a big opening at the front end, as shown in Fig. 2(c). We establish the coordinate system of which the origin is located at the center of the front end of the capillary and the horizontal axis z is along the central axis of the hollow and points to the SMF, as shown in Fig. 2(c). r(z) represents the radius of the hollow core along its axis. As the HF solution also etches the inside surface of the capillary, the diameter of the hollow part at the front end of the capillary enables the microsphere with smaller diameter to be inserted into the capillary.
Fig. 2 Microscopic images of the coupler at various fabrication steps. (a) The splicing joint between the SMF and capillary. (b) The front end of the capillary. (c) The front end of the capillary after etching. (d) Thickness of the capillary front end and entering depth as a function of the etching time. (e) Radius of etched hollow core versus z.
From the Fig. 2(d), we find the thickness of the capillary front end decreased linearly with the increase of the etching time. We use a to express the etching rate along the r direction, which is half of the slope of the thickness of the capillary front end versus the etching time. a is calculated to be 0.86 μm/min from Fig. 2(d). We also introduce the concept of the depth to indicate the length of the HF solution entering into the hollow core, with measured results also shown in Fig. 2(d). The shape of the etched hollow core can be described by the equation dr(z)/dz = -a/v(z), where v(z) is the velocity along the z direction that the front of the HF solution goes across the location z. The hollow core is formed by one-time erosion of 34 min with a final entering depth of about 571 μm, as shown in Fig. 2(c). The average v(z) is calculated to be 16.79 μm/min. The calculated result agrees with the experimentally measured shape well, as shown in Fig. 2(e). Experimentally, the thickness of the wall of the front end can be eroded to be about 5 μm to obtain an approximately optimal coupling efficiency between the capillary and microsphere, corresponding etching time set to be about 32 min. This fabrication process can be further optimized to investigate the improvement of the loss and the coupling efficiency of the coupler. Finally, we insert a barium titanite glass microsphere with a diameter of 45 μm and the refractive index of 1.93 into the etched hollow core, by using the precision fiber alignment stage (M-562, Newport) and fiber taper. The microspheres need to be small so that they can be inserted into the structure. Microspheres of a higher refractive index would have a higher Q-factor because of the smaller radiation loss. We tested two types of small microspheres with the indices of 1.48 and 1.93, respectively. The results indicated that the high index microsphere indeed had a higher Q-factor. Therefore, we used the microspheres with the index of 1.93.
Using the commercial software (Rsoft Beamprop), a simulation model with parameters measured from Fig. 2(c) is established. The working wavelength is chosen as 1550 nm, with the optical field propagation distribution as shown in Fig. 3(a). The cone formed by etching, which corresponds to the part on the left of the dashed white line, is sufficiently long so that the light can propagate along the capillary wall without obvious loss. The coupling between capillary and SMF is affected by the diameter of the hollow core, namely, inner diameter. Figure 3(b) shows the relation between the power contained in the capillary at the white dashed line and the inner diameter of the capillary, which represent the coupling efficiency between SMF and capillary. As the inner diameter increases from 5 μm to 45 μm, the coupling efficiency decreases slightly, because of the SMF-capillary coupling part having a tapered transmission structure. When the inner diameter increases further, the coupling efficiency decreases sharply. Note that the inner diameter of the capillary used in Ref [16]. is 75 μm. Hence, to obtain a sufficiently high coupling efficiency, Ref [16]. has to adopt the multimode optical fiber with a core diameter up to 62.5 μm, while the coupling efficiency is still not ideal. Moreover, the multimode optical fiber is sensitive to the environmental perturbations such as bending. This ultimately results in the instability of the reflective spectrum, which limit its practical application. This disadvantage is overcome in our scheme because we adopt a standard single mode fiber.
Fig. 3 (a) Rsoft Beamprop simulation of the cone-shaped inwall capillary coupler. (b) The power contained in the capillary wall at the white dashed time versus the inner diameter of the capillary.
4. Experiment
The input light emitted from an optical sensing analyzer (si725, Micron Optics, Inc.) is incident into the SMF and then enters into the etched capillary wall. Figure 4 shows the reflective spectra when the front end is located in air and immersed into the matching liquid whose refractive index is matched with the capillary’s, respectively, with the device shown in Fig. 4(c). In the reflective spectrum of the former situation, there are Fano resonances of two orders, as shown in Fig. 4(a) with the orange and green dashed frames, where i is the order number and l is the mode number [21]. However, when we immersed the front end of the capillary into the matching liquid, it means that the front end face of the capillary doesn’t work as a reflector any more. And the Fano resonances transform into traditional Lorentz-shape resonances which are grouped into the same orders. The resonances in the Fig. 4(b) have a red shift comparing with the resonances in Fig. 4(a), because the microsphere is so close to the front end face that the matching liquid touches the microsphere partly. Moreover, this touch also reduces the contrast of the resonances due to the induced loss.
Fig. 4 Reflective resonance spectra (a) in air and (b) in the matching liquid. (c) Microscopic images of the front end after a microsphere is inserted into it. (d) and (e) are the FFTs of spectra (a) and (b), respectively.
By examining with the equation ng = λ1λ2/(2πR(λ1-λ2)) [16], Fano resonances in the orange frames are of one order and the Fano resonances in the green frames are of another order as shown in Fig. 4(a). λ1 and λ2 are wavelengths of the adjacent resonance peaks, respectively. R is the radius of the microsphere. ng is the group index of the WGM that is nearly equal to the refractive index of the microsphere material. The free spectral range of WGMs is almost independent of the orders [22]. The shapes of the Fano resonances are various, which are caused by phase differences and will be explained in the next section. Figures 4(d) and 4(e) are the fast Fourier transforms (FFTs) of the spectra of Figs. 4(a) and 4(b), respectively. Each peak in Figs. 4(d) and 4(e) corresponds to a certain interfering between two beams from the microsphere, with the optical path difference (OPD) replaced by OPD/πng. Generally, the original horizontal axis of one peak is the OPD of corresponding two beams. Because the WGMs of different orders have nearly equal optical path that light propagates around the microsphere one time [22], the difference of OPDs of adjacent peaks, namely, the interval of the peaks, is equal to the optical path that one beam propagates around the microsphere one time. Hence, when the horizontal axis OPD is replaced with OPD/πng, the interval of adjacent peaks is the diameter of the microsphere directly as shown in Figs. 4(d) and 4(e), which are close to the practical microsphere diameter 45 μm.
5. Theoretical analysis
Figure 1(b) shows the schematic diagram of the front end of the capillary that illustrates the propagation of light. When the input beam E1 coupling from the SMF passes the microsphere, one part of E1 couples into the microsphere and then couples out into the other side of the capillary (E6). Another part of E1 transmits across the microsphere, reflects at the end face of the capillary and transmits across the microsphere again (E12). The interference of E6 and E12 forms asymmetric Fano resonances in the reflective spectrum [23]. In order to understand the formation of Fano resonances in the coupling system, the transfer matrix method (TMM) [24] is applied in the cone-shaped inwall capillary coupled WGM resonator. Note that our theoretical mode considers factors which form ideal Fano resonances, i.e., the interference of E6 and E12, while in the case of a real microsphere, the overlapping of a set of resonance peaks may result in distorted lineshapes. The normalized reflection can be calculated as
where t and k are the transmission coefficient and coupling coefficient satisfying the equation k2 + t2 = 1. τ is the round-trip transmission coefficient. p is halfway phase factor satisfying p = exp(jθ/2). θ is the normalized frequency satisfying θ = 4π2neffR/λ. neff is the effective index. R is the radius of the microsphere. λ is the wavenumber in vacuum. δ and L are the phase difference and distance from the coupling region between the microsphere and capillary to the front end face of the capillary, with δ = βL. β is the propagation constant. r is the amplitude reflectivity at the capillary-air interface satisfying r = (ncapillary-nair)/(ncapillary + nair). ncapillary and nair are the refractive indices of capillary and air, respectively. Simulation parameters are chosen as k = 0.14 and τ = 1. Using Eq. (1), we obtain the reflective spectrum of the cone-shaped inwall capillary coupled WGM resonator. From Eq. (1), the period of the Fano resonance is π. Moreover, the waveforms in one period are anti-symmetric, as showed in Fig. 5(a). Simulation shows that as δ increasing from 0 to 0.5π, the depth of the dips varies continuously from the minimum at δ = 0, to the maximum at δ≈0.28π, and to the minimum at δ = 0.5π again. The situation is on the opposite when δ increases from 0.5π to π. In Figs. 5(b)-5(d), we plot three typical spectra calculated with different phase differences δ, which is corresponding to the three white dashed lines in the Fig. 5(a). Meanwhile, we pick out corresponding Fano resonances from Fig. 4(a) and fit them with Eq. (1), which means the simulation agrees well with the experiment, as shown in Figs. 5(e)-5(g). An asymmetric Fano resonance with a Q-factor of 2.57 × 104 and a slope of 67.8dB/nm is obtained, as shown in Fig. 5(g).Fig. 5 (a) Simulation of Fano resonances versus wavelength detuning and δ (The reflection is normalized). (b), (c) and (d) are three typical simulation Fano resonances corresponding to the three values of δ, 0.72π,0.5π and 0.28π as marked with white dashed lines in (a). (e), (f) and (g) are the corresponding experimentally measured Fano resonances in red and the fitting curves in blue.
The microsphere is in direct contact with the inwall of the capillary circularly. The contact loss contributes to the low Q-factor. Besides, there exist many possible orbital coupling planes (or paths) where light can be coupled into the microsphere, and the microsphere usually has an imperfect sphericity. As a result, there could be multiple sets of resonance peaks as observed by other researchers [16,25]. These multiple resonance peaks could overlap and broaden and thus lower the observed Q-factor. Furthermore, the reflective spectrum is the combination of the ideal lineshapes of Fano resonances obtained by Eq. (1) and overlapping of adjacent resonance peaks, which leads to the transformation of the ideal lineshapes of Fano resonances, especially in fringes. Moreover, we expect that the laser induced self-heating of the resonator can be observed when the Q-factor is improved. As a fact, the research on the thermal responses to self-heating have proven feasible and meaningful for whispering gallery mode microresonators [26,27].
6. Conclusion
In summary, we demonstrated a new coupler to excite the WGMs in microsphere resonators, which is composed of standard SMF and capillary with an inner diameter of 5μm. A long and sharp cone in the capillary is formed via one-time erosion. A microsphere is inserted into the cone and stops at the front end meaning the half of the microsphere is exposed to the outer space, which is convenient for sensing applications. The light reflected at the end face transforms traditional Lorentz-shape resonances into Fano resonances. An asymmetric Fano resonance with a Q-factor of 2.57 × 104 and a slope of 67.8 dB/nm is observed. The device is compact, robust and alignment-free, which makes it have great potential in sensing applications and optical switching.
Funding
National Natural Science Foundation of China (NSFC) (61675126, 61377081, 61007035 and 61635006); Chen Guang project by Shanghai Municipal Education Commission and Shanghai Education Development Foundation (12CG48).
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