1. Introduction

The recent development of the fabrication technology has made it possible to realize optical microcavities that exhibit a high Q/V ratio [1]. Whispering gallery mode (WGM) microcavities [2–4] are particularly attractive because they typically have an ultra high Q and a reasonably small V. When an ultra-high Q WGM cavity is employed, the light can be strongly confined inside the cavity and the light-matter interaction is greatly enhanced. Taking advantage of these excellent features, various applications such as nonlinear wavelength conversion [5], lasing [6, 7], all-optical switching [8, 9], cavity quantum electrodynamics [10, 11], opto-mechanics [12, 13] and nano- and micro-particle sensing [14, 15] have already been developed with an ultra-high Q WGM microcavity.

A system composed of coupled microcavities system is currently attracting a lot of attention because it can be employed as a platform for bistable lasing [16], optical isolation [17, 18], all-optical switching [19], and quantum optics [20]. Such systems have been achieved by employing ultra-high Q WGM microcavities [17, 18, 21–23] and other types of cavities such as microrings [24] and photonic crystal nanocavities [16, 19, 25–27]. When the cavities couple strongly, it is well known that the resonance is split in the frequency domain (i.e. Rabi-like splitting). On the other hand, photons are transferred back and forth between two cavities in the time domain, which leads to energy oscillation (i.e. Rabi-like oscillation). Recently, energy oscillation was observed and controlled in coupled photonic crystal nanocavities both in experiments [20,26,27] and calculations [28,29] with the expectation to achieve quantum information processing, on-demand slowing and stopping light and ultra-fast all-optical switching. However, such experiments with ultra-high Q WGM cavities have not yet been reported despite the potential benefit of using an ultra high-Q microcavity.

In this paper, we report the first time-domain observation of energy oscillation between coupled ultra-high Q WGMs. We employed two counter-propagating WGMs with a loaded Q factor of > 10⁷ in a silica toroid microcavity as the platform for our experiments. It is known that the part of the light scattered by the surface couples back into a mode propagating in the opposite direction in an ultra-high Q WGM cavity [30]. This induces coupling between clockwise (CW) and counter-clockwise (CCW) propagating modes. The visibility of the coupling is evaluated by Γ = κ/γ₀, where κ is the coupling rate and γ₀ is the intrinsic cavity decay rate. Γ can be understood as the ratio between the resonance splitting and the cavity linewidth in the frequency domain or the number of times the energy transfers before the light decays in the time domain. The benefits of our system are as follows. (1) A large Γ can be easily achieved. For instance, Γ > 10 was achieved in our experiment thanks to the strong coupling and ultra-high Q factor. This large value is essential if we want to excite and observe the energy oscillation between the two modes. (2) Experiments are performed without using two individual cavities. This greatly simplifies the experimental setup. (3) The resonant frequencies of the two modes are always the same, which enables us to perform the experiment without the need for any of the frequency tuning methods that are usually required in a coupled cavities system. It should be noted that there have already been several studies of coupling between CW and CCW modes in the frequency domain [30–34], but there have been no studies dealing with such coupling in the time domain.

In addition, a drop-port measurement technique [35], where two tapered fibers are employed, is the key to observing the energy oscillation clearly. When a single tapered fiber is employed, the light coupled into the fiber from the cavity is disturbed by interference with the light transmitted through the fiber. This makes it difficult to observe the energy in the cavity mode directly. On the other hand, when the drop-port measurement technique is used, we can observe the light coupled from both the CW and CCW modes because there is no transmitted light in the drop-port tapered fiber. By using this technique, we clearly observed the energy in the CW and CCW modes and the oscillation between them.

This paper is organized as follows. In §2, we develop a numerical model to analyze how to excite time-domain energy oscillation between coupled CW and CCW modes. In §3, we report our experimental results. Finally, we summarize this study and conclude the paper in §4.

2. Numerical analysis

2.1. Developing numerical model

First, we develop a numerical model to analyse the dynamical behavior of coupled WGMs using coupled mode theory (CMT) [17, 36, 37]. A schematic illustration of the developed model is shown in Fig. 1(a) and the corresponding master equations are as follows;

 

Fig. 1 (a) Schematic illustration of the developed numerical model. (b) Calculated transmission (blue) and reflection (red) spectra of the cavity with (κ, γ₀, γ_taper)/2π = (100, 10, 2.5) MHz. (c) and (d) Calculated light energy in the CW (solid blue line) and CCW (solid red line) modes when symmetric (c) and asymmetric (d) excitations are employed. The inset shows the cavity mode (blue) and Fourier transform of a rectangular input pulse with the width of 10 ns (green).

Download Full Size | PDF

As shown in Eqs. (1) and (2), the resonant frequencies of the two modes are the same, which means that the CW and CCW modes are degenerated. In such a situation, it is known that two new supermodes with different resonant frequencies are formed, which are called higher- and lower-frequency modes. When we define the mode amplitudes of higher- and lower-frequency modes as $a_{\text{H}}=\left(a_{\text{CW}}+a_{\text{CCW}}\right)/\sqrt{2}$ and $a_{\text{L}}=\left(a_{\text{CW}}+a_{\text{CCW}}\right)/\sqrt{2}$, Eqs. (1) and (2) can be transformed into the following equations, as,

2.2. Exciting energy oscillation

Next, we show that both higher- and lower-frequency modes must be excited simultaneously to induce energy oscillation between the CW and CCW modes in the time domain. By using the relationships, namely $a_{\text{CW}}=\left(a_{\text{H}}+a_{\text{L}}\right)/\sqrt{2}$ and $a_{\text{CCW}}=\left(a_{\text{H}}-a_{\text{L}}\right)/\sqrt{2}$, we can express the light energy in the CW and CCW modes (U_CW = |a_CW|², U_CCW = |a_CCW|²), as,

Here, we confirm the validity of the above discussion employing time-domain analysis. Figure 1(c) shows the light energy in the CW and CCW modes when we couple a 10-ns rectangular pulse into a cavity with the same parameters as Fig. 1(b). The figure shows clearly that the light energies in both the CW and CCW modes oscillate by turn after the pulse is inputted. In addition, there are complete energy transfer points (i.e. U_CW = 0, U_CCW ≠ 0 or U_CW ≠ 0, U_CCW = 0) in the figure. These are what we want to observe in the experiment. On the other hand, when we input a rectangular pulse whose center frequency is detuned slightly ((ω − ω₀)/2π = 20 MHz), there are no complete-energy-transfer points as shown in Fig. 1(d). This is because the requirement for symmetric excitation is not satisfied in this situation [see the inset of Fig. 1(d)]. Considering this analysis, we employ symmetric excitation with a rectangular pulse to observe energy oscillation between CW and CCW modes.

3. Experimental results

3.1. Fabrication and characterization

Our silica toroid microcavity was fabricated using (1) photolithography, (2) SiO₂ etching, (3) XeF₂ dry etching and (4) laser reflow [3]. According to Gorodetsky et al. [34], higher coupling efficiency between CW and CCW modes can be achieved with a microsphere cavity when the cavity diameter is smaller. So, we fabricated a silica toroid microcavity with a small diameter of around 25 μm. As a result, the microcavity was turned into a spheroid microcavity, namely a spherical microcavity on a Si pillar. [13, 38].

We employed a tapered fiber to couple the light into the microcavity [39]. By heating and stretching a commercial single mode fiber, the diameter of the tapered fiber was reduced to around ∼ 1 μm. The fabricated tapered fiber had a transmittance of over 90%. One of the transmission spectra of the fabricated microcavity is shown in Fig. 2(c). There is clear resonance splitting caused by coupling between the CW and CCW modes. The linewidth of the resonance is 6.3 MHz (Q_load of 3 × 10⁷) and the splitting is 85 MHz. The corresponding visibility Γ is approximately 13.

 

Fig. 2 (a) Experimental setup for the reflection measurement. (b) The reflected signals from the cavity for different modes. Black (solid), gray (solid) and red (dashed) lines are the reflected signals with and without the cavity and the fitting curve, respectively. The input pulse widths were 10, 8 and 5 ns for the top, middle and bottom panels, respectively. It should be noted that the detected signals were offset by the remaining ASE noise and it was removed from the figure. (c) The transmission spectra of the modes that are employed in (b).

Download Full Size | PDF

3.2. Methods to observe energy oscillation

Here, we discuss on a method to observe the energy oscillation between the CW and CCW modes. In general, we observe a transmission (i.e. $s_{\text{out}}^{\text{t}}$) to characterize a microcavity. However, as can be understood from Eq. (3), the transmission is the product of the interference between lights coupled from the CW mode and transmitted through a tapered fiber. In addition, in our experiments, most of the light is transmitted through tapered fiber without coupling with the cavity because only a small part of the input pulse spectrum overlaps the transmission spectrum [see the inset of Fig. 1(c)]. So, the transmission port is not suitable for observing the energy oscillation precisely.

To overcome this problem, we employ two different methods, namely ‘reflection measurement’ and ‘drop-port measurement [35].’ In the former method, the reflection port ( $s_{\text{out}}^{\text{r}}$) is used. From Eq. (4), the reflection is proportional to the light energy in the CCW mode (U_CCW). So it is a replica of the light energy in the cavity. On the other hand, the latter method employs an additional tapered fiber, which is called drop-port tapered fiber. Assuming that the coupling rate between the drop-port fiber and the cavity is ${\gamma }_{\text{taper}}^{\text{drop}}$, the two outputs from the drop-port fiber can be written as follows $s_{\text{out}}^{\text{drop},\text{CW}}={\gamma }_{\text{taper}}^{\text{drop}}{a}_{\text{CW}}$ and $s_{\text{out}}^{\text{drop},\text{CCW}}={\gamma }_{\text{taper}}^{\text{drop}}{a}_{\text{CCW}}$, which are also proportional to the light energy in the cavity. A clear advantage of this method is that the energy in both the CW and CCW modes can be observed simultaneously.

The reflection measurement can be performed with a single tapered fiber, which simplifies the experimental setup. This enables us to obtain data with high stability and accuracy. However, obviously it is difficult to observe the energy in CW modes with a reflection measurement. This means that a drop-port measurement must be used to verify the existence of an energy oscillation between the CW and CCW modes. Considering the characteristics of the two methods, we first employed a reflection measurement to carefully analyze the experimental results and compare them with the simulation results. Then, we confirmed that there is indeed an energy oscillation between the two modes by using a drop-port measurement.

3.3. Reflection measurement

First, we show the results obtained in the reflection measurement. Figure 2(a) is a block diagram of the experimental setup. The continuous light outputted from a tunable laser source (TLS) is turned into rectangular pulses using an electro-optical modulator (EOM). The RF signal driving EOM is generated by a pulse pattern generator (PPG). The trigger signal is inputted into an optical sampling oscilloscope (OSO). A polarization controller (PC) is employed to match the polarization of the input light signal to that of a WGM of interest. An optical circulator is employed to extract reflected light. The reflected light from a microcavity is amplified with an erbium-doped fiber amplifier (EDFA), and then amplified spontaneous emission (ASE) noise is filtered out with a band-pass filter (BPF). The transmitted light is attenuated to avoid reflection at the end of the fiber. Finally, the OSO detects and records the reflected light.

Figure 2(b) shows the reflected signal from the cavity for different modes (different cavities). As can be seen, the reflected signal oscillates periodically when we input the signal having a rectangular shape. This suggests that there is an energy oscillation between the CW and CCW modes although only a CCW mode (a reflected signal) can be observed in this setup. Note that we could reduce the coupling between the microcavity and the tapered fiber (γ_taper) as long as we could detect the reflected signal because the number of energy oscillations depends on the κ/γ = κ/(γ₀ + γ_taper) ratio. The oscillation period in each panel is different due to the difference in the coupling rate κ. Figure 2(c) shows the transmission spectra that correspond to Fig. 2(b). γ and κ that can be estimated from the spectra [shown in Fig. 2(c)] agree well with those calculated from the reflected signals [shown in Fig. 2(b)]. This agreement also suggests that the oscillation in the reflected signal is due to the coupling between the CW and CCW modes.

Next, we discuss the dependence of the reflected signal on the input pulse width. Figure 3(a) shows the reflected signal from the cavity for different input pulse widths. There is a good agreement between the experimental data and the lines calculated by employing the equations in §2. When Δt_pulse ≤ 10 ns, the peak power of the reflected signal (i.e. the peak energy in the CCW mode) increases as the pulse width increases. However, the peak power does not increase when Δt_pulse ≥ 20 ns. Such behavior is explained from the analysis in the frequency domain. Figure 3(b) shows the Fourier transformed spectra of the input pulse and their spectral overlap with the cavity mode. The components of the spectral overlap near the cavity mode (resonant components) increase as the input pulse width becomes longer, until Δt_pulse reaches 10 ns. This is the reason why the peak power of the signal increases as the input pulse width becomes longer. When Δt_pulse ≥ 20 ns, the spectrum of the input pulse overlaps well with the cavity mode. Therefore, the peak power do not vary even when we input pulses with longer width than 20 ns.

 

Fig. 3 (a) The signal reflected from the cavity and the oscillation period for different input pulse widths Δt_pulse. The black solid and red dashed lines are the experimental and numerical results, respectively. The blue dots are the oscillation period. The numerical results were calculated using equations in §2. The fitting parameters are (κ, γ₀, γ_taper)/2π = (75, 6.4, 1.3) MHz. (b) The Fourier transform of the theoretical input pulses (green) and their spectral overlap with the cavity mode (red). All the lines are normalized by the peak power of the Fourier transform of the input signal with Δt_pulse = 100 ns. Note that the gray solid and dashed lines indicate the position of the split cavity mode and the DC component, respectively.

Download Full Size | PDF

As for the oscillation period [shown with blue dots in Fig. 3(a)], it is constant for different input pulse widths when Δt_pulse ≤ 10 ns. It is because the oscillation period is determined only by κ and so should be independent of the input pulse width. However, twice larger oscillation period appears when Δt_pulse ≥ 20 ns. This is also explained by Fig. 3(b). With Δt_pulse ≤ 10 ns there are two peaks in the spectral overlap, which correspond to the split cavity mode. The interference between the two peaks induces a fast oscillation that can be observed in Fig. 3(a) when Δt_pulse ≤ 10 ns. On the other hand, with Δt_pulse ≥ 20 ns, the DC component (around 0 MHz) becomes stronger and interferes with the components in the cavity mode. This interference creates slow (low frequency) oscillation. So, we observe an oscillation with doubled period (i.e. half frequency) when we input longer pulse. Figure 3(a) also indicates that the doubled oscillation period disappears after the input pulse is turned off. This is because the DC component is non-resonant, so the inteference immediately vanishes when the input is turned off.

The above analysis suggests that there is optimal input pulse width both for observing energy oscillation with the correct period and inputting a large amount of energy into the cavity. In the above case, it is around Δt ≈ 10 ns.

3.4. Drop-port measurement

Finally, we describe the experiments based on the drop-port measurement. As mentioned in §3.2, the drop-port measurement is essential if we are to observe the energies in the CW and CCW modes simultaneously and directly. By using this method, we confirmed that there is indeed an energy oscillation between the two modes. The experimental setup is shown in Fig. 4(a). The energies in the CW and CCW modes are extracted directly from the drop-port tapered fiber, and then are detected with the OSO after amplification. Unlike the scheme reported in Monifi et al. [35] where two tapered fibers were fabricated and controlled together, our drop-port tapered fiber is bent [40] and its position is manipulated independently by a nanopositioner. This setup enables us to control the coupling rate between the drop-port tapered fiber and the cavity easily and accurately. Thanks to the high controllability, we can both extract the energy in the cavity mode and minimize the additional coupling loss induced by the drop-port tapered fiber. Note that the tapered fiber has a transmittance of over 80% even after bending.

 

Fig. 4 (a) Experimental setup for the drop-port measurement. (b) Outputted power measured at the drop port. The blue and red solid lines represent the output from the CW and CCW modes, respectively. Note that the timings of the CW and CCW signals were calibrated by measuring the delays between the two signals.

Download Full Size | PDF

Figure 4(b) shows the result of the drop-port measurement. We employed the same modes as in the bottom panel of Figs. 2(b) and 2(c). As seen from the figure, the energy clearly oscillates between the CW and CCW modes. This is direct evidence showing that there is indeed an energy oscillation between the CW and CCW modes. In addition, the oscillation period agrees well with that estimated in Fig. 2(b). Here γ is slightly larger than that in Fig. 2(b). This is believed to be due to the additional coupling loss induced by the drop tapered fiber. From the figure, we can conclude that we observed the energy oscillation between counter propagating WGMs. This is the first observation of coupling between modes with an ultra high Q of over 10⁷ in the time domain.

4. Conclusion

In conclusion, we have reported the first observation of energy oscillation between two coupled ultra-high Q whispering gallery modes in the time domain. We employed CW and CCW modes in a silica toroid microcavity because they provide a good platform for our experiments. Thanks to the large Γ, which is due to the large coupling rate between the two modes and an ultra-high Q factor, clear energy oscillation was excited and observed. By using the reflection measurement, we confirmed that the oscillation periods in the time domain were the same as those inferred from the mode splitting in the frequency domain, and we investigated the dependence of the oscillation on the input pulse width. In addition, we observed the energy oscillation between the CW and CCW modes simultaneously by taking advantage of the drop-port measurement technique. We believe that our results pave the way toward development of a method for controlling the coupling states between ultra high Q microcavities, which is important in terms of achieving quantum information processing. We may be able to tune the coupling strength between the two modes [33] at an arbitrary timing. The relative phase between the two super modes [ϕ_H − ϕ_L in Eps. (7) and (8)] may be controlled by employing an input pulse with a non-flat spectral phase. Such experiments are possible only with the combination of time-domain measurement.