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Abstract
We demonstrate an approach for the realization of coupled-mode induced transparency (CMIT) in a hybrid polydimethylsiloxane (PDMS)-coated silica microbubble resonator, with an Au microwire inserted in the hollow channel. Owing to the large negative thermo-optics coefficient of PDMS, different radial order modes with opposite thermal sensitivities can coexist in this hybrid microcavity. By applying a current through the Au microwire, which acts as a microheater, the generated Ohmic heating could thermally tune the resonance frequencies and the frequency detuning of the coupled mode to achieve controllable CMIT. This platform offers an efficient and convenient way to obtain controllable CMIT for applications, such as label-free biosensing and quantum information processing.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the last thirty years, research on the whispering gallery mode (WGM) microcavity has become more relevant. Because the WGM circulates in the microcavity for a long period, the interactions between light and matter are greatly enhanced [1]. This process can be extensively applied to sensing [2,3], nonlinear optics [4,5], tunable optofluidic microlaser [6], and cavity quantum electrodynamics [7,8]. Recently, an all-optical analogy to electromagnetically induced transparency (EIT), caused by destructive interference between different optical paths in a microcavity, has drawn considerable attention. Many schemes were proposed to realize EIT, such as coupled-mode induced transparency (CMIT) [9–15], optomechanically or Brillouin scattering induced transparency [16–18], and nonlinear optically induced transparency [19].
To realize CMIT, two optical modes, either from directly and indirectly coupled microcavities [12,20] or from single microcavity [14], must be involved. Further, their resonant wavelengths must be matched well by various tuning methods, such as thermal tuning [11,12], strain mechanical tuning [13,14] or internal aerostatic pressure [13,21], electro-thermal tuning [22–25], and all-optical-thermal tuning [7]. The resonant wavelength is directly proportional to the refractive index and radius of the microcavity. For mechanical (strain or internal aerostatic pressure) tuning, geometric effect is dominated as compared with photo-elastic effect and it changes the radius of the microcavity [13,14]. All the modes shift in the same direction (longer wavelength or shorter wavelength). For thermal tuning, thermo-optic effect is dominated as compared with thermal-expansion effect and it changes the refractive index of the microcavity [12,24]. All the modes shift in the same direction for the pure microcavity with only one material. Different order modes exhibit different response rates to the mechanical or thermal tuning, based on which one mode can catch up with another mode and EIT lineshape is formed. However, these methods are less efficient to achieve EIT and also uncontrollable. Constructing hybrid microcavity with distinct thermo-optics coefficient (TOC) is an efficient way to enhance the wavelength tuning and obtain EIT easily. For example, coating a thin polymer (negative TOC) layer on the outer surface of the silica (positive TOC) microcavity lead to different order modes with distinct thermal response [11]. Under thermal tuning, different order modes encounter and EIT lineshape is formed efficiently.
In this paper, we propose and demonstrate a novel scheme using electrical tuning to realize CMIT in a hybrid polydimethylsiloxane (PDMS)-coated silica microbubble resonator (MBR), with an Au microwire inserted in the hollow channel. Due to the large negative TOC of PDMS, different radial order modes, with opposite thermal sensitivities, can coexist in this hybrid microcavity. These modes have distinct thermal responses, thus, it is easier to tune these modes on-resonance. A novel thermal tuning method is adopted — applying a current on the Au microwire to obtain controllable CMIT. The generated Ohmic heating changes the temperature of the microcavity, and hence, the resonance wavelength.
2. Theoretical model and sensitivity analysis
The proposed hybrid PDMS-coated MBR structure is displayed in Fig. 1(a). The hybrid microcavity consisted of an outer silica shell, an inner coated PDMS layer, and an Au microwire. The outer radius and thickness of the silica wall are denoted as r and t, respectively. The refractive indices of PDMS, silica, and air are denoted as n1, n2, and n3, respectively. The Au microwire acted as a heat source. By applying current on the Au microwire, Ohmic heating can tune the temperature of the microcavity and the modal resonant wavelength. A photograph of a hybrid microcavity is shown in Fig. 1(b). Here, the PDMS is selected for its negative TOC, opposite to that of silica. Thus, different radial order modes (denoted as p) exhibit different, and even opposite, thermal sensitivities ST, depending on the field distributions in the silica and PDMS layer [26–28].
Fig. 1. (a) Schematic of the hybrid microcavity: PDMS-coated MBR with an Au microwire. The outer radius and silica wall thickness are denoted as r and t, respectively. (b) Optical image of the fabricated hybrid microcavity taken by a camera.
The ST can be written as [29]:
Fig. 2. (a) The thermal sensitivities for the first seven radial order modes (p = 1–7) as a function of the silica wall thickness. Horizontal and vertical lines represent ST = 0 and t = 6 µm, respectively. The electromagnetic field distributions for (b) p = 2 mode and (c) p = 4 mode; here, r = 215 µm and t = 6 µm.
3. MBR fabrication and experimental setup
The silica MBR was fabricated by the fuse-and-blow method, and the details for fabrication can be found in previous studies [31,32]. An Au microwire (diameter of 20 µm) was inserted in the hollow channel of the MBR with a tweezer. The PDMS solution was pumped into the hollow channel by a syringe, for approximately 10 minutes; when it filled the entire MBR, air was subsequently pushed out for approximately 30 minutes. By this method, a uniform, soft PDMS layer, with a thickness of 25 µm, coated the inner surface of the MBR [33]. Lastly the hybrid PDMS-coated MBR was placed in a drying oven, to cure the soft PDMS layer for 2 hours at 120 °C. Thus, a solid, hybrid PDMS-coated MBR, with stable mechanical properties, was formed.
The schematic of the experimental setup is illustrated in Fig. 3. An optical tapered fiber, with diameter of 2–3 µm, drawn by a flaming machine, was used to couple a tunable laser (Toptica CTL 1550) in and out of the hybrid microcavity. The tunable laser passed through the polarizers, which were used to set the polarization of the light for maximal coupling with the WGMs. An attenuator was used to control the power of the probe laser. The tunable laser was then transmitted through the tapered fiber, and detected by a photodetector (PD), which was subsequently connected to an oscilloscope, to monitor the transmission spectra of the MBR. The transmitted light was also recorded by a data acquisition (DAQ) card, controlled by LabVIEW code, to monitor the transmission spectra of the MBR in real-time. The current that flowed through the Au microwire was provided by a current source controller, with resolution of 1 mA.
Fig. 3. Experimental setup diagram with a hybrid PDMS-coated MBR system. The system consisted of a hybrid PDMS-coated MBR and a tapered fiber, which were placed inside a large enclosure. The red + denotes the positive pole and the black – denotes the negative pole. PD: photodetector; DAQ: data acquisition. A resistor with R0 = 6.7 Ω was inserted in the circuit.
4. Experimental results and theoretical analysis
A typical transmission spectrum, shown in Fig. 4(a), that consisted of a low Q mode in the vicinity of high Q modes, was measured in a hybrid microcavity, with r = 215 µm and t = 6 µm. This silica wall thickness can guarantee that different radial order modes, with positive and negative thermal sensitivities, can coexist in this hybrid microcavity. The thickness of the PDMS layer in this sample was approximately 25 µm, which was measured by an optical microscope. The loaded Q factors of the resonant modes, exhibited in Fig. 4(a), were QA ∼ 4.1×105, QB ∼ 4.4×106, and QC ∼8.0×106, respectively. We first demonstrated the wavelength tuning ability via Ohmic heating, by transmitting a current through the Au microwire. The ends of the Au microwire were connected to a power controller under current mode, with resolution of 1 mA. When the current was transmitted through the Au microwire, for heating time t, the generated Ohmic heating (or called as Joule heating) W can be expressed as W = I2Rt, where R is the resistance of the Au microwire. The heat generated by the Ohmic heating was partially absorbed by the resonator, and the other part was dissipated through air convection from the two ports of the resonator, and the temperature change of the microcavity δT is written as δT = W/K = I2Rt/K, where K is the coefficient of thermal conductivity. Finally, the wavelength shift δλ can be calculated by δλ = ST × δT = STI2Rt/K. Thus, δλ was deduced to be proportional to I2.
Fig. 4. (a) The normalized transmission of modes A, B, and C. These three modes were fitted with a Lorentz function. Redshift and blueshift are distinguished by red and blue arrows; red and blue Lorentz fitting lines. (b) - (d) The resonant wavelength shift of modes A, B, and C. All the data points fit well with parabolic equations, as the panel shows.
Experimentally, the resonant wavelengths of mode A, B, and C were measured while the current applied to the Au microwire was varied and the relative wavelength shifts of these modes are plotted in Figs. 4(b)–4(d). The measured data (dots) can be well fitted with parabolic equations, indicating that δλ was proportional to I2, as predicted. In addition, we found that mode A exhibited blueshift and modes B and C exhibited redshift, which indicated that mode A possessed negative sensitivity (ST < 0) and modes B and C possessed positive thermal sensitivities (ST > 0). The sensitivity ratio of modes A and B was equal to the ratio of the coefficient of the quadratic term, fitted in Figs. 4(b) and 4(c). Thus, the sensitivity ratio of these two modes was -0.0036/0.0021 ∼-1.7. This ratio was in good agreement with the calculated sensitivity ratio of the p = 4 and p = 2 modes, of -15.16 pm°C-1/11.16 pm°C-1 ∼ -1.4 (see Fig. 2(a)). It was confirmed that the radial orders of modes A and B were 4 and 2, respectively. Mode C had approximately the same thermal response rate as B (Figs. 4(c) and 4(d)), thus, mode C had the same radial order as B, but they had different axial orders [31]. It was therefore demonstrated that the hybrid microcavity possessed a different radial order mode, with opposite thermal sensitivities, and it would be easier to tune them coupling indirectly with each other through the tapered fiber. Namely, with the increase of current, a high Q mode in silica wall exhibited a redshift while a low Q mode in PDMS layer exhibited a blueshift. When the two modes are tuned to the same wavelength, a transparent window may appear.
By tuning the current applied to the Au microwire to thermally change the transmission spectra among these three modes, two EIT windows were successfully realized at zero-detuning in resonance frequencies. The corresponding transmission spectra for a current range of 13–37 mA (from top panel to bottom panel), at intervals of 3 mA, are plotted in Fig. 5. Initially, the resonant frequencies between the high Q modes and a low Q mode were far from each other; there were no coupling effects and their resonant dips were Lorentz-like (Fig. 5(a)). When the current increased to thermally tuning resonant frequencies, the frequency detuning between high Q modes and a low Q mode decreased, due to their opposite thermal sensitivities, and a Fano-like line shape appeared (Fig. 5(c)). When modes A and B were tuned into resonance (i.e., their frequency detuning was zero), an EIT-like window (Fig. 5(d)) was realized. When the current was further increased, two Fano-like line shapes simultaneously emerged (Fig. 5(f)), which were induced by the coupling of a low Q mode with two high Q modes, respectively. Another EIT-like window (Fig. 5(g)) was realized when modes A and C were tuned into resonance.
Fig. 5. Normalized experimental transmission spectra (black lines) and theoretical fittings (red lines) showing the progress of forming EIT-like windows. Blue arrows: blueshift of mode A. Red arrows: redshift of mode B. Magenta arrows: redshift of mode C. Black arrows: two modes resonate in the same frequency. From the top panel to the bottom panel, the current ranged from 13 mA to 37 mA, at intervals of 3 mA. The fitting parameters used in the fitting were (a) [κA (γA), κB (γB), κC (γC)] = [85 (330), 0.8 (22), 0.2 (20)] MHz; (b) [κA (γA), κB (γB), κC (γC)] = [80 (320), 0.9 (18), 0.2 (20)] MHz; (c) [κA (γA), κB (γB), κC (γC)] = [75 (305), 2 (30), 0.2 (19)] MHz; (d) [κA (γA), κB (γB), κC (γC)] = [30 (32.5), 130(220), 0.3 (20)] MHz; (e) [κA (γA), κB (γB), κC (γC)] = [60 (240), 4 (50), 0.5 (20)] MHz; (f) [κA (γA), κB (γB), κC (γC)] = [70 (268), 2 (35), 1 (20)] MHz; (g) [κA (γA), κB (γB), κC (γC)] = [30 (32), 2 (60), 130 (260)] MHz; (h) [κA (γA), κB (γB), κC (γC)] = [75 (277), 1 (45), 2 (35)] MHz; (i) [κA (γA), κB (γB), κC (γC)] = [75 (277), 0.5 (35), 2 (60)] MHz, θ =π/3, and φi = 0.
To fit the experimental results, the coupled-mode equations were used to explain the mode transmissions of the optical microcavities, written as [14]
The EIT line shapes can be manipulated by changing the value of κA (γA) and κi (γi). As shown in the Fig. 5, the experimental data can be fitted with the theoretical line shape (red lines). From the theoretical fitting parameters, we also found that all three modes are undercoupled (κ<γ). This was a result of the tapered fiber, with a larger diameter waist, selected to attach to the hybrid MBR, for a stable coupling condition and cleaner transmission spectrum (fewer modes were excited). The mismatch between the theoretical lines and experimental data may be caused by other nearby modes, therefore, the fitting parameters of (κ and γ) were adjusted to fit the experimental data.
Additionally, controllable electromagnetically induced absorption (EIA), which is a result of constructive interference between different optical paths, was also obtained (Fig. 6). The loaded Q factors of the two involved resonant modes were QA ∼ 7.8×106 and QB ∼ 7.9×107. However, the coupling depth of the high Q mode B was larger than that of the low Q mode A (Fig. 6(a)). Ohmic heating was also done to tune the resonant frequencies. When the current was increased, modes A and B exhibited blueshift and redshift, respectively, indicating that these two modes possessed opposite thermal sensitivities. When these two modes were tuned into resonance, the EIA window was formed and verified by the theoretical fittings (red line in Fig. 6(e)). Modes A and B were also undercoupled because κ<γ.
Fig. 6. Normalized experimental transmission spectra (black lines) and theoretical fittings (red lines) showing the progress of forming an EIA-like window. Blue arrows: blueshift of mode A. Red arrows: redshift of mode B. Black arrow: two modes resonate in the same frequency. From the top panel to the bottom panel, the current ranged from 10 mA to 26 mA, at intervals of 2 mA. The fitting parameters used in the fitting were (a) [κA (γA), κB (γB)] = [15 (300), 3.5 (29)] MHz; (b) [κA (γA), κB (γB)] = [15 (300), 3.5 (29)] MHz; (c) [κA (γA), κB (γB)] = [12 (230), 3.5 (30)] MHz; (d) [κA (γA), κB (γB)] = [15 (280), 2.5 (27)] MHz; (e) [κA (γA), κB (γB)] = [12 (270), 2.5 (27)] MHz; (f) [κA (γA), κB (γB)] = [10 (290), 3.5 (27)] MHz; (g) [κA (γA), κB (γB)] = [12 (300), 3.5 (25)] MHz; (h) [κA (γA), κB (γB)] = [12 (320), 3 (22)] MHz; (i) [κA (γA), κB (γB)] = [12 (320), 3.5 (24)] MHz, and θ =π/3, and φ = 0.
In general, the difference between EIT and EIA phenomena is observed as transparency peaks or absorption dips [34] as shown in Fig. 5(d) and Fig. 6(e). The two effects depend on the different coupling strengths of incident light [35,36] from the tapered fiber into silica wall and into PDMS layer, respectively. By adjusting excitation wavelength of tunable laser or positons between the MBR and the tapered fiber, it can be seen that a low Q mode of deep coupling (shallow coupling) and a high Q mode of shallow coupling (deep coupling) are always come into being an EIT (EIA) window in Fig. 5 (Fig. 6). Theoretically, we can convert the behaviors between EIT and EIA by adjusting the coupling parameters (κ and γ) artificially, for example, by using a nanotips.
5. Conclusion
In brief, an optical analogy of EIT/EIA windows were demonstrated in a single PDMS-coated silica MBR with an Au microwire. Because of the large negative TOC of PDMS, different radial order modes with opposite thermal sensitivities can coexist in this hybrid microcavity. By applying current to Au microwire, the generated Ohmic heating changes the temperature of the hybrid microcavity, and hence, thermally tunes the resonant frequency. Different radial modes, with opposite sensitivities, can easily encounter and couple with each other to form controllable EIT/EIA or Fano line shapes. The EIT/EIA or Fano line shapes transmission spectra fit well with the theoretical analysis, based on couple mode equations. Finally, this Ohmic heating tuning mechanism provided a very efficient and convenient way to control the mode coupling and transmission spectra, in a single microcavity with an optofluidic channel, which allows this coupling scheme to be further used in label-free biosensing and quantum information processing.
Funding
National Natural Science Foundation of China (61705039); National Key Research and Development Program of China (2015CB352006); China Postdoctoral Science Foundation (2017M610389, 2019T120553); Fujian Provincial Program for Distinguished Young Scientists in University; Changjiang Scholar Program of Chinese Ministry of Education (IRT_15R10); Special Funds of the Central Government Guiding Local Science and Technology Development (2017L3009).
Disclosures
The authors declare no conflicts of interest.
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