We study the oscillatory thermal dynamics of a high-Q PDMS-coated silica microtoroid both experimentally and theoretically. We demonstrate that the competing thermo-optic effects in silica and PDMS lead to thermally-induced self-modulation in the transmission spectra. A dynamical model is built using thermal dynamics and coupled-mode theory to analyze the oscillation behaviors. Effects of input power, taper-cavity air gap and wavelength scanning speed on the oscillation behaviors are investigated with a detailed comparison between theory and experiments.
©2009 Optical Society of America
Recently there has been a growing interest to a type of monolithic resonators, i.e., Whispering Gallery Mode (WGM) resonators, in which light is trapped in circular orbits by continuous total internal reflections from the circular boundary. Silica optical microresonators [1,2], such as microspheres, microrings, and microtoroids, which possess circular boundaries to support WGMs, have been intensively investigated for a variety of applications, including highly sensitive bio/chemical sensors [3–5], low-threshold microlasers , and cavity quantum electrodynamics [7–10]. The combination of ultra-high-quality factor (Q) and small mode volume allows buildup of high light intensity in the resonator, and gives rise to large optical nonlinearity [11–13] and thermal effects [14–18], which subsequently induce optical bistability. Some typical thermal responses, such as distorted asymmetric lineshape and hysteresis induced by thermo-optic and thermal expansion effects have been observed in high-Q silica resonators [15,16]. Oscillations resulting from thermal nonlinearity [14–17] and regenerative pulsation resulting from competition between different nonlinearities  have been observed and studied previously in silica microspheres. Self-modulation due to competing free-carriers and phonon populations has been reported in silicon microresontaors .
Resonators composed of multiple materials can lead to improved performance or novel behaviors not attainable in single-material resonators. It has been demonstrated that silica microcavities with appropriate coating layer can help to increase the sensitivity of resonator-based biosensors [20,21], enlarge the nonlinear coefficient, compensate for the thermal effects [4,22–24], and achieve electromagnetically induced transparency-like effect . Up to date, a systematic investigation of the thermal response of transmission spectra in such a multi-layer-resonator has not been reported. In this paper, we experimentally and theoretically analyze the thermally-induced oscillations in the transmission spectra of a polydimethylsiloxane (PDMS) coated silica toroidal microcavity. We demonstrate that such oscillations are attributed to competition between opposite thermo-optic effects in silica core and PDMS coating layer.
2. Experimental results
The multi-layer-resonator in our experiment is prepared by coating a PDMS layer with thickness t PDMS ~0.2 μm onto the surface of a silica microtoroid with major (minor) diameter of D = 42 μm (d = 6 μm) . PDMS has a negative thermo-optic coefficient, which can compensate for the positive thermal effect of WGMs in silica toroidal resonators and increase the resonance stability of the WGMs. The thickness of the PDMS coating is smaller than the critical value of 0.43 μm required to completely compensate for the positive thermo-optic effect of the fundamental WGMs in silica . Figure 1 shows the experimental setup, in which a tunable external cavity diode laser in 1450 nm band is used to excite WGMs in the PDMS-coated microtoroid. A triangle wave from a function generator is utilized to implement a fine scan of the laser wavelength around the WGMs. A fiber taper is used to efficiently couple the signal light into and out of the microcavity . The transmitted light is detected by a low noise photodetector, which is subsequently connected to an oscilloscope to monitor the transmission spectra of the resonator.
The intrinsic quality factor of the fundamental WGM of the resonator used in this study is 3.8 × 106, measured at a low input power to avoid thermal effects. Figure 2 presents the typical transmission spectra of the PDMS-coated silica microtoroid for two different input powers at 1457 nm. As the laser wavelength approaches the resonance wavelength, light gradually couples into the cavity mode and starts heating the mode volume due to the material absorption. Subsequently, the temperature induced resonance shift leads to distortion of the Lorentzian lineshape of the resonance modes. As shown in the insets of Fig. 2, thermal effects result in a hysteretic response featured by broadening and compression of the resonance line during wavelength up- and down- scans, respectively. During up-scan (Fig. 2(a)), a distorted asymmetric lineshape exhibits a sharp drop followed by a mild recovery in the transmission spectrum due to the fast temperature variation induced refractive index changes of PDMS . Figure 2(b) demonstrates that at the large input power (~3.1mW), a fine structure characterized by oscillations within the linewidth of the resonance shows up. This is attributed to the competition between silica and PDMS due to their opposite thermo-optic effects, different thermal relaxation times and thermal absorption coefficients. Specifically, the transmission spectrum is modulated by the combined effect of the positive thermo-optic effect of silica and negative thermo-optic effect of PDMS within the same cavity. In this work, we consider theoretically and experimentally the behavior of the oscillations in the transmission spectra during wavelength up-scan only, as they exhibit more interesting features.
3. Theoretical model
The dynamic process of the thermal flow and consequently the induced resonance shifts of WGMs in the PDMS-coated silica microtoroid are discussed in this section. The effective refractive index (RI) of the WGM can be expressed as n eff ≅ η 1 n 1 + η 2 n 2, where n 1 and n 2 represent the RI of fused silica and PDMS, respectively, while η 1 and η 2, satisfying η 1 + η 2 = 1, denote the fraction of light energy traveling in silica core and PDMS coating, respectively. In the multi-layer resonator, light absorption induced temperature changes are distinct in silica and PDMS due to their different thermal properties. Assuming that absorption induced temperature changes within the mode volume in silica and PDMS are ΔT 1(t) and ΔT 2(t), respectively, the resonance wavelength λr (t) of the WGM can be written as 11] represents the optical Kerr coefficient of fused silica, A is effective cross sectional area of the WGM, and the power of the circulating field within the cavity is PC(t) = |EC(t)|2 /τr with EC (t) and τr ~n eff πD/c describing the intracavity field and the cavity round-trip time, respectively. The second and third terms in the bracket, respectively, describe the effective changes in the resonance wavelength due to thermo-optic effect of the silica/PDMS resonator and the optical Kerr nonlinearity in silica. Since the thermal expansion coefficient of silica (5.5 × 10−7 K−1) is much smaller than its thermo-optic coefficient and the PDMS coating is not thick enough to induce significant observable changes in the resonance wavelength due to thermal expansion, we have neglected the thermal expansion effects in Eq. (1) and in the following discussions.Equation (2) describes the time evolution of the temperature variations ΔΤι(t) within the mode volume (i = 1: Silica, and i = 2: PDMS) with γ th, i depicting the thermal relaxation rate, and γ abs, i representing the thermal absorption coefficient. The first term on the right side of Eq. (2) describes heat dissipation process, while the second term denotes temperature changes due to absorption of the intracavity field. In Eq. (3), δ 0 = ωr /2Q 0 describes the intrinsic loss rate of the cavity that is inversely proportional to the intrinsic quality factor Q 0 with ωr denoting the resonance frequency. δc = ωr / 2Qc denotes the taper-cavity coupling induced loss rate that is inversely proportional to the coupling quality factor Qc. The total quality factor Q is calculated through 1/Q = 1/Q 0 + 1/Qc. κ = (2δcτr)1/2 denotes the coupling coefficient from the fiber taper to the cavity. Δω(t) ≡ ωs(t) − ωr(t) is the detuning between the excitation scanning frequency ωs and the resonance frequency ωr. Excitation field represented by E in is assumed to be constant during the frequency scan, and it is related to the power launched into the tapered fiber P in through |Ein|2 = P in⋅τr.
In a study of thermal effects in resonators, two important thermal mechanisms should be considered [14,15,17]: A faster one describing the heat dissipation from the mode volume to the rest of the microcavity with a relaxation time of microseconds, and a slower one depicting temperature equalization of the microcavity as a whole with the environment of the cavity with a relaxation time on the order of tens of milliseconds. The WGM we studied in our experiments has a quality factor of Q 0 ~3.8 × 106 at the frequency ωr ~1.3 × 1015 rad/s. For the wavelength scanning speed of 40 nm/s, 80 nm/s, and 120 nm/s used in our experiments, the transition times τ trans for the excitation wavelength scanning through the resonance of the WGM are τ trans ~9.6μs, 4.8μs, and 3.2μs, respectively. This timescale is comparable with the relaxation time of the faster thermal mechanism, but much shorter than the relaxation time of the slower thermal mechanism. Thus, we consider only the thermal dissipation from the mode volume to the rest of the cavity in the theoretical model. Therefore, γ th, i in Eq. (2) is determined by thermal conductivity Ki, heat capacity Ci, density ρi of the cavity materials, geometry of the cavity, and effective mode volumes of the WGM in specified cavity layers V mode ,i. The thermal absorption coefficient can be estimated as γ abs,i ~α abs, icτrηi /(C i ρiV mode ,in eff), where α abs, i is the linear absorption coefficient of cavity material, c is the speed of light in free space. In general, it is not easy to exactly estimate the values of γ th, i and γ abs,i, thus it is a common practice to keep these parameters adjustable to find the best fit to experimental data [16–18].
The resonance building time inside a cavity is related to the homogeneous linewidth of the resonance and it is calculated as τ mode = Q 0/ωr ~2.9 ns for the cavity used in our experiments. Since τtrans >>τ mode is always satisfied in our experiments, the mode has long enough time to fully develop. Therefore, we can analyze Eq. (3) in steady-state by setting the time derivative of the field EC(t) to zero ,Eqs. (1) and (2). Consequently, the normalized transmission is written as T rans = |Eo(t)|2/|Ein|2 to characterize the performance of the resonator.
Parameters used in the simulations are listed in Table 1 . The parameters, A, η 1, η 2, V mode,1 and V mode,2 are obtained by calculating the WGM field and its distribution in the silica core and PDMS coating using Finite Element Method.
4. Comparison of the theoretical model with experimental observations
Here, we give a quantitative description and discussion of the observed phenomena together with the dynamics of the wavelength detuning and the temperature changes within the mode volume in silica core and PDMS coating. Figure 3 compares the experimental data in Fig. 2(b) with the numerical simulation results using the theoretical model. The simulation results are in good agreement with the transmission spectrum obtained in the experiments, as shown in Fig. 3(a). The corresponding temperature variations (in silica and PDMS) and wavelengths changes (resonance λr, scanning λs, and detuning Δλ = λs−λr) in the wavelength up-scan are demonstrated in Figs. 3(b) and 3(c), respectively. Figures 3(d)-3(f) are the enlarged views of the first oscillation cycle marked by the dotted oval in Fig. 3(a), in which four distinct regions (I-IV) in the transmission spectrum reflect four different thermodynamic processes.
To explain the dynamic behaviors of the thermal oscillations in detail, we analyze the four regions (I-IV) marked in Figs. 3(d)-3(f) one by one. It is noted that Kerr effect is two orders of magnitude smaller than the thermo-optic effect in silica, so the resonance wavelength λr is mainly determined by the second term in Eq. (1). If the sum of the thermo-optic effects in silica (η 1⋅dn 1/dT⋅ΔT 1) and PDMS (η 2⋅dn 2/dT⋅ΔT 2) increases with time, the resonance wavelength shows a red shift, otherwise a blue shift.
Region I: When the scanning wavelength λs gets closer to the resonance wavelength λr, light is gradually coupled into the resonator, leading to a gradual decrease in the transmission between points O and A as shown in Fig. 3(d). As a result of the thermal absorption, the WGM experiences an increase in temperature, which subsequently induces refractive index increase within the mode volume in silica and decrease within the mode volume in PDMS. As seen in Fig. 3(e), during this region, temperature increase within the mode volume in PDMS is larger than that in silica, as implied by the larger optical absorption of PDMS. Thus, thermo-optic effect of PDMS layer becomes the dominant factor, inducing a blue shift in λr (green line in Fig. 3(f)). Since λs and λr move in opposite directions, the magnitude of Δλ decreases rapidly (purple line in Fig. 3(f)), resulting in a sharp drop-off in the transmission. At point A, the scanning wavelength is exactly on resonance with the WGM, i.e., Δλ = 0, and the transmission reaches its minimum value determined by the coupling between the fiber taper and the cavity.
Region II: In this region, temperatures of the mode volume in silica and PDMS keep increasing. At the beginning of this region, the influence of the thermo-optic effect of PDMS still dominates, so λr keeps a blue shift, which is opposite to the up-scan direction of λs. As a result of shifts of λr and λs in opposite directions, a quick increase in the magnitude of Δλ accompanied by an increase in transmission is seen in Figs. 3(e) and 3(d). Gradually, the influence of the thermo-optic effect in silica increases, and λr experiences a transition from blue to red shift as the wavelength up-scan continues. Although the red shift allows the resonance wavelength to move in the same direction as the wavelength up-scan, the pace of the red shift in λr cannot catch up with wavelength scan speed, so the transmission increases continuously. Finally at point B, λr exhibits the same red shift speed as the scanning speed of λs. As a result, the magnitude of Δλ reaches a local maximum value, leading to a local maximum in the transmission. The time width of region II is around 2.35 μs.
Region III: After passing point B, the thermo-optic effect of silica prevails since a stable temperature rise has been established in the mode volume in silica. As a result, the red shift of λr becomes faster than that of λs, leading to a gradual decrease in the magnitude of Δλ, which is reflected in the gradual decrease of transmission. At point C, λr finally catches up with λs so that the WGM is exactly on resonance and the transmission reaches its minimum again. In this region, the reduction in the slope of temperature increase within the mode volume in PDMS is attributed to the enhanced heat dissipation from the mode volume in PDMS at high temperature to the rest of PDMS. The time width of region III is around 11.6 μs.
Region IV: After passing point C, the red shift in λr continues and the magnitude of Δλ increases again. This increases the transmitted power and reduces the absorbed power as well as the heat generation in the resonator. When absorption is smaller than dissipation, the temperature of the mode volume begins to decrease. The rate of temperature decrease within the mode volume in PDMS is much faster than that in silica due to the large heat dissipation in PDMS. Therefore, λr feels more influence of the thermal-optic effect of PDMS, which is manifested by the green line in Fig. 3(f) showing the red shift speed of λr much faster than the scanning speed of λs. The fast separation of λr from λs helps to recover the transmission quickly. At point D, λr is already largely detuned from λs, and the transmission returns to its maximum. The time width of region IV is around 7.84 μs. After λs passing through point D, the decrease of the temperature within the mode volume in silica induces a blue shift pushing λr close to λs which leads to light coupling into the microcavity. This starts heating up the PDMS which in turn further enhances the blue shift in λr. Therefore, the blue shift of λr and the up-scan of λr allow another decrease in the magnitude of Δλ, leading to gradual decrease in the transmission again. The cycle repeats until the power absorption cannot balance out the heat dissipation in silica so that finally the red shift of λr lags behind the up-scan of λs.
In Fig. 3, the scanning wavelength increases linearly with time, so the plots in Figs. 3(a)-3(f) exhibit the same patterns when in time domain. It is seen in Fig. 3(a) that as λs increases, the width of AC (Fig. 3(d)) in each cycle increases, while the width of CE decreases. This can be explained as follows: The width of AC (CE) is mainly determined by the temperature increase (decrease) rate of the mode volume in silica, as shown in Fig. 3(b). When the thermally induced oscillation starts, the temperature of the mode volume in silica keeps increasing until the oscillation cycles end. Equation (2) shows that higher temperature causes slower (faster) temperature increase (decrease) rate in AC (CE) region. Moreover, λs and λr are moving in the same direction in AC region with λs > λr, but in opposite directions in CE region with λs < λr. Therefore, as the up-scan goes on, longer time is needed in AC region for λr to catch up with λs, while in CE region, shorter time is expected for blue shift of λr to coincide with λs.
5. Further experiments on oscillation behaviors
In the previous sections, we have shown that the oscillations in the transmission spectra are caused by the opposite thermal drifts of the resonance wavelength in different materials within the same resonator. In the wavelength scanning process, λr exhibits blue and red shifts alternately and gives rise to continuous changes between the positive and negative wavelength detuning, leading to oscillations in the transmission spectra. In this section, we investigate the influence of the input power, taper-cavity air gap, and the wavelength scanning speed on the oscillations in detail. The tested PDMS-coated microtoroid is the same as the one used in Fig. 1, and the measured WGM is around 1448.3 nm.
Figure 4 presents the transmission spectra of the PDMS-coated cavity measured at various input powers in the under coupling region. Numerical simulation of the spectra based on the theoretical model in Section 3 shows good agreement with the experimental results. The corresponding variations of λs and λr are also calculated and shown in the insets to help analyze the oscillation behaviors. It is observed that, for a small input power of 1.11 mW (Fig. 4(a)), no oscillation exists in the transmission spectrum. This is because the temperature induced resonance shift of the WGM cannot catch up with the up-scan of λs at a low power level. In particular, λr of the WGM experiences a blue shift initially, and the following red shift in λr caused by the thermal-optic effect in silica is not fast enough to catch up with λs, leading to the distorted resonance lineshape in Fig. 4(a). However, for larger input powers (Figs. 4(b)-4(d)), once λr is exceeded by λs due to both the up-scan of λs and the blue shift of λr, the succeeding temperature increase within the mode volume in silica allows λr to increase and catch up with λs. The cycle repeats and results in oscillations in the transmission spectra. Equation (2) shows that higher input power could support larger temperature rise in the cavity to sustain more oscillation cycles appearing in the transmission illustrated in Fig. 4. At the same time, the width of region CE (Fig. 3(d)) in the same cycle, but different spectra shown in Fig. 4, increases with increasing power. Since the wavelength detuning in region IV is larger at a higher input power, it takes a longer time, i.e., broader wavelength scanning range, for λr to catch up with λs. In our measurement, a threshold power of 1.33 mW is observed to obtain oscillations in the transmission.
Figure 5 investigates the influence of the air gap between the fiber taper and the PDMS-coated microtoroid on the oscillation behavior. Since the air gap determines the taper-cavity coupling, it has similar effects on the transmission spectra as the input power. When the cavity is in the under-coupling region (Figs. 5(a) and (b)), decreased air gap (closer to the critical resonant condition) will increase the power coupled into the cavity, resulting in more oscillation cycles. When the cavity is in the over-coupling region, taper-cavity coupling induced loss rate δc and coupling coefficient κ become larger compared to those in the under and critical coupling regions. This will degrade the total quality factor Q of the system. Therefore, the accumulation speed of the circulating power PC in the cavity decreases in the beginning of the resonance mode as can be understood from Eq. (3). Consequently, it takes longer time for the thermal effect to manifest itself in silica and PDMS. Since the PDMS layer is not sufficiently thick to compensate for the thermal effect in silica, a red shift of λr is expected in this region. Figure 5(c) demonstrates that in the beginning of the resonance mode, silica induced resonance red shift dominates and no oscillation is observed. As wavelength scan goes on, the circulating power increases and a sufficient power build-up is achieved as a result of the decrease in the magnitude of Δλ. This leads to a fast temperature increase within the mode volume in PDMS, which induces oscillations in the transmission spectrum. When the air gap becomes even smaller (approaching to deep over-coupling region), the thermo-optic effect of PDMS can never compete with that of silica due to the slower accumulation of circulating power in the cavity as shown in Fig. 5(d). As a result, the thermo-optic effect of silica dominates in the whole thermal drift area, causing a continuous red shift in λr. Thus, no oscillation appears in the transmission spectrum.
As we have mentioned in the theoretical model, the transmission spectra are dependent on the wavelength scanning speed. Figure 6 depicts the effect of different scanning speeds (40 nm/s, 80 nm/s, and 120 nm/s) on the transmission spectra obtained in the under coupling region. It is seen that as the scanning speed increases, the number of oscillation cycles decreases. For a larger scanning speed, the scanning wavelength passes through the resonance mode at a higher speed, leading to a shorter material absorption time and lower temperature increase in the mode volume. Therefore, the resonance region is observed to be narrower as the scanning speed increases, which also limits the number of oscillation cycles. In addition, at a higher scanning speed, the red shift of λr in region III (marked in Fig. 3(d)) needs to take longer time, i.e., broader wavelength scanning range, to keep up with the accelerated up-scan of λs. This leads to a broader dip in each cycle and fewer cycles in the whole thermal drift area. Therefore, from Figs. 6(a) to 6(c), the number of oscillations during the resonance transition decreases with the increased scanning speed. At the scanning speed of 120 nm/s (Fig. 6(c)), the WGM experiences only regions I and II. In region III, the red shift of resonance wavelength cannot catch up with the fast wavelength up-scan which results in only one sharp dip.
In summary, we have theoretically and experimentally studied the oscillatory thermal dynamics of a high-Q PDMS-coated silica microtoroid. The input power, taper-cavity gap (coupling efficiency), and wavelength scanning speed are demonstrated to dramatically affect the oscillation behaviors in the transmission spectra. A higher input power, critical taper-cavity gap, and lower scanning speed result in more oscillation cycles during the transition through the resonance. The numerical simulation results agree well with the experimental results. The self-oscillation phenomenon generally exists in resonant systems which hold opposite nonlinear mechanisms, and may be applied in optical modulation, all-optical switching, and optical memory elements
Lina He and Yun-Feng Xiao contributed to this work equally. The authors gratefully acknowledge I-CARES (International Center for Advanced Renewable Energy & Sustainability) and CMI (Center for Materials Innovation) at Washington University for support of this work. Yun-Feng Xiao was also supported by the Research Fund for the Doctoral Program of Higher Education and the Scientific Research Foundation for the Returned Overseas Chinese Scholars.
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