Microresonator and Laser Parameter Definition via Self-Injection Locking

Self-injection locking to a high-quality-factor microresonator is a key component of various up-to-date photonic applications, including compact narrow-linewidth lasers and microcomb sources. For optimal construction of such devices, it is necessary to know the parameters of the locking mode, which is a challenging task. We experimentally develop and verify an original technique based on the fundamentals of the self-injection locking effect, which allows determining the key parameters of the mode of the microresonator (quality factor, vertical index) as well as the laser diode parameters. We demonstrate that this method can be used in the spectral ranges where conventional methods are not applicable, for example, in the mid-IR. We stabilize a $2.64\mu \mathrm{m}$ laser diode by a high-$Q$ whispering gallery mode microresonator made of crystalline silicon. Using the elaborated technique, the quality factor of the microresonator is determined to be $5\times {10}^8$.

Figure 1

Experimental setups in the near-IR. First, we measure the dependence of the resonance width on the gap value between the microresonator and the coupler. Second, we measure the $Q$ factor of that mode by the ringdown method. Finally, we verify that the $Q$ factor conserves during measurements by measuring the resonance width another time.

Figure 2

Calculated normalized transmitted power dependence on the detuning of the laser frequency from the WGM frequency for the different locking phases. The sum of the forward and backward locking ranges is measured in the experiments. It is a good approximation of the total locking range $\delta {\omega }_{\mathrm{lock}}$ in the case of a high $Q$ factor, when the inner overlap band $\delta {\omega }_{\mathrm{in}}$ is negligible.

Figure 3

LI curves measured for different gap values. The highlighted regions correspond to the excited modes with different polar indices: $p=0$ (light green), $p=2$ (light blue), and high $p$ (light red). The mode shapes remain constant upon variation of the gap value, which indicates that the locking phase ${\psi }_0$ remains unchanged.

Figure 4

Left panel: $\mathrm{FLR}+\mathrm{BLR}$ dependence for the mode with $p=0$. The transition to the “soft” regime occurs at a gap of around 0.2 $\mu \mathrm{m}$, and the fit with $\delta {\omega }_{\mathrm{FWHM}}$ is valid for gap values less than $0.2\mu \mathrm{m}$. Right panel: mode traces at zero gap (I), the transition point (II; $\mathrm{FLR}+\mathrm{BLR}=4{\kappa }_m$), and the critical coupling point (III; ${\kappa }_{\mathrm{mi}}={\kappa }_{\mathrm{mc}}$). The experimental curves are plotted with dotted lines and the approximation curves generated using the linear SIL model are plotted with solid lines. The high-order mode appeared at overcoupling.

Figure 5

Dependence of $\mathrm{FLR}+\mathrm{BLR}$ on the gap for the ${\mathrm{Mg}\mathrm{F}}_2$ microresonator. The experimental data is denoted using symbols, with the red circles representing an approaching microresonator and the blue squares representing a receding microresonator with respect to the coupler. The yellow solid line is the $\delta {\omega }_{\mathrm{lock}}+\delta {\omega }_{\mathrm{in}}$ curve calculated using Eqs. (1) and (2). The violet solid line is $2\delta {\omega }_{\mathrm{FWHM}}$ calculated using Eq. (3). The extremum takes place at $550\mu \mathrm{m}$, which gives $Q_{\mathrm{int}}=(4.0\pm 0.5)\times {10}^8$ and $p=2$. The ringdown method yields $Q_{\mathrm{int}}=(3.6\pm 0.5)\times {10}^8$; see the insets.

Figure 6

(a) Experimental setup in the mid-IR. (b) Mode scan at critical coupling. (c) Dependence of $\mathrm{FLR}+\mathrm{BLR}$ on the gap for the $\mathrm{Si}$ microresonator at 2639 nm. The blue and red squares represent the experimental data and the yellow solid line is calculated using Eq. (2). The extremum takes place at $270\mu \mathrm{m}$, giving $Q_{\mathrm{int}}=5.0\times {10}^8$ and $p=1$.