Ring cavities are ubiquitous in integrated photonic systems. Optical resonances of ring cavities are fully modeled by three parameters: (i) the resonance frequency $\omega$; (ii) the total linewidth $({\gamma _i} + {\gamma _c})$; and (iii) the coupling condition (${\gamma _c}/{\gamma _i} \gt 1$ or ${\lt}1$), where ${\gamma _i}$ and ${\gamma _c}$ are the intrinsic loss rate and cavity–waveguide coupling rate, respectively. Power transmission measurement using an evanescently coupled waveguide has been the standard method to characterize integrated ring cavities. With this method, the resonance frequency and total linewidth can be accurately determined. However, there is still no reliable method to determine the coupling condition, as the power transmission spectrum remains unchanged by switching the intrinsic loss rate ${\gamma _i}$ and the cavity–waveguide coupling rate ${\gamma _c}$.

In previous works, the extinction ratio change with wavelengths has been widely used to infer the coupling condition [1,2]. This approach relies heavily on prior simulation results and the assumption that all resonances have the same quality. Here, we show unambiguously that this approach cannot determine the coupling condition reliably. We further propose and demonstrate a robust method to determine the coupling condition of individual optical resonances without any prior knowledge using binary phase modulation in power transmission measurement. Moreover, no modification of photonic integrated circuits is required.

Our new approach is to introduce a fast $\pi$-phase shift (much faster than $1/({\gamma _i} + {\gamma _c})$) to the input light and observe the transient response of the output light [Figs. 1(a) and 1(b)]. Before the $\pi$-phase shift ($t \lt 0$), the cavity is at the on-resonance steady state with transmission ${P_{{\rm out}}}(t{\lt}0)/{P_{{\rm in}}}= |({\gamma _i} - {\gamma _c})/({\gamma _c} + {\gamma _i}{)|^2}$. After the $\pi$-phase shift ($t \gt 0$), the transmitted light is determined by the interference between the intra-cavity field accumulated before the $\pi$-phase shift and the input field after the $\pi$-phase shift:

 

Fig. 1. (a) Schematic to determine optical resonance coupling condition. (b) Binary phase modulation of the input light (right: measured electrical signal). (c) Simulated transient responses of over-coupled (purple) and under-coupled (green) resonances using Eq. (1) with 1 GHz photodetector bandwidth (right: zoom-in with log-scale).

Download Full Size | PDF

While the transmission always decays to the steady-state value, the transient behavior depends on the coupling condition. For under-coupled resonances (${\gamma _c} \lt {\gamma _i}$), the field accumulated before the $\pi$-phase shift always has a smaller amplitude than the input field after the $\pi$-phase shift. Therefore, the output power decays monotonically to the steady-state value [green in Fig. 1(c)]. For over-coupled resonances (${\gamma _c} \gt {\gamma _i}$), the field accumulated before the $\pi$-phase shift is initially larger and becomes smaller than the input field after the $\pi$-phase shift. Therefore, the output power first drops to zero and then increases to the steady-state value [purple in Fig. 1(c)].

We test our approach on a SiN-integrated photonic ring cavity (thickness 750 nm, width 2.0 µm, radius 80 µm), 20 µm long pulley-coupled to the bus waveguide (thickness 750 nm, width 1.9 µm) with a 500 nm gap. In the power transmission spectrum, the extinction ratio first increases and then decreases with respect to the wavelength [dashed black arrows in Fig. 2(b)]. Such extinction ratio change is used to be interpreted as the transition from the under-coupled regime (below 1520 nm) to the over-coupled regime (above 1520 nm) [3,4], by assuming that ${\gamma _c}$ increases with the wavelength while ${\gamma _i}$ is a constant.

 

Fig. 2. (a) Measured transient responses (gray) of optical resonances. Purple and green lines are the simulated curves for over- and under-coupled conditions, using Eq. (1) with 1 GHz photodetector bandwidth. (b) Slow-scanned power transmission spectrum (background subtracted). Black arrows: extinction ratio trend. Green and purple: under- and over-coupled resonances determined by our method, respectively. (c) Coupling rate (gray) and intrinsic loss (red). Error bars represent fitting errors between the slow-scanned transmission and the model.

Download Full Size | PDF

The setup of our approach follows the schematics in Fig. 1(a). The CW laser wavelength is tuned on-resonance with the cavity. Then, light goes through a phase modulator driven by an electrical square wave. The square wave has a period of 200 ns and a rising edge of around 50 ps. The voltage is adjusted to launch a $\pi$-phase shift to the light. At the output, transmitted light is detected by a high-speed photodetector with 1 GHz bandwidth (Newport 1611FC) and recorded by an oscilloscope with 4 GHz bandwidth at 20 GS/s.

We apply our method to all resonances individually. As exhibited in Fig. 2(a), we can clearly distinguish the different transient responses of under-coupled and over-coupled resonances. Unlike the result inferred from the extinction ratio change, we observe that most resonances are over-coupled across the whole spectrum [purple in Fig. 2(b)]. There are also a few under-coupled resonances [green in Fig. 2(b)], whose coupling conditions are difficult to determine from the extinction ratio change. The values of ${\gamma _c}$ and ${\gamma _i}$ are summarized in Fig. 2(c). We find that the coupling rate ${\gamma _c}$ increases with the wavelength. However, the intrinsic loss rates ${\gamma _i}$ also show strong wavelength dependence. The photonic cavity has significantly higher loss near 1505 and 1553 nm wavelengths. The fact that excessive losses only happen within a small wavelength range indicates that they are caused by surface roughness or scattering particles [5,6].

We then demonstrate that this method is robust against experimental imperfections of non-zero laser frequency detunes. The frequency detune introduces an extra optical phase and decreases the amplitude of the intra-cavity field accumulated before the $\pi$-phase shift. Although the output power of over-coupled resonances can no longer reach zero, it still drops far below the steady-state value in a large frequency detune range [Fig. 3(a)]. For under-coupled resonances, the frequency detune has minimal influence [Fig. 3(b)].

 

Fig. 3. Simulated and measured transient responses with respect to detune of (a) over-coupled resonance at 1551.7 nm and (b) under-coupled resonance at 1504.7 nm.

Download Full Size | PDF

In summary, we have proposed and demonstrated a new method to unambiguously distinguish under- and over-coupled conditions of individual optical resonances by introducing binary phase modulation in power transmission measurement. This method is precise and robust, as it does not rely on any assumptions of the cavity or post-processing of the results [7]. Moreover, it offers the key advantage that no additional interference circuit [8] is needed, which is critical for testing individual photonic cavity components in large-scale integrated systems. The capability to accurately determine coupling conditions of individual resonances at different wavelengths can benefit the design of cavity–waveguide coupling, as well as the optimization of device fabrication to minimize optical losses introduced by surface roughness and particles. This will be critical for emerging applications such as quantum state generation [9–11], quantum transduction [12], and photonic sensing [13,14].