Mirror-Assisted Self-Injection Locking of a Laser to a Whispering-Gallery-Mode Microresonator

Self-injection locking is a dynamic phenomenon, which provides passive stabilization of a laser emission frequency via resonant optical feedback. The stabilization coefficient depends on the level of the feedback and quality factor of the external resonant structure creating the feedback. A conventional self-injection locked laser (SIL) based on a dielectric cavity involves barely tunable resonant Rayleigh scattering (RRS). In this work we study theoretically a scheme of a SIL via a high-$Q$ microresonator with a drop-port coupled mirror, in which the optical feedback level is optimally adjusted by tuning the drop-port mirror coupling. We show that the additional reflector can improve the laser stabilization and power handling efficiency if compared with the classic RRS-based scheme.

Figure 1

Scheme of a SIL laser based on a microresonator with a mirror placed in the drop port. The amplitudes at the input-port-coupling point: $B_{\mathrm{in}}$ is the amplitude of the pump; $B_t$ and $B_r$ are the amplitudes transmitted and reflected at the coupler; $A_{+}$ and $A_{-}$ are the amplitudes of the counter-rotating modes in the microresonator. The amplitudes at the drop-port-coupling point: $C_m$ is amplitude reflected from the mirror back to the drop-port coupler; $C_{\mathrm{out}}$ is the output amplitude from the drop port. $T$ and $T^{{}^{\prime }}$ are the amplitude transmittance coefficient of the in-port coupler and drop-port coupler, respectively.

Figure 2

Illustration of the spectral characteristics of the ring microcavities used in a SIL system. The resonance splitting of reflectance $\mathrm{\Gamma }$, transmittance $T_{\mathrm{out}}$, and drop-port output amplitude $C_{\mathrm{out}}$ are presented in (a), (b), and (c), respectively. The blue lines correspond to the classic SIL scheme based on RRS: ${\delta }_c/{\delta }_0+{\delta }_c=0.1$; ${\delta }_m=0$ and $\gamma /{\delta }_0+{\delta }_c=2$. The orange lines correspond to the case of no backscattering: ${\delta }_c/{\delta }_0+{\delta }_c=0.1$; ${\delta }_m/{\delta }_0+{\delta }_c=2$ and $\gamma /{\delta }_0+{\delta }_c=0$. The green lines correspond to the general case when both the RRS and the mirror are presented: ${\delta }_m/{\delta }_0+{\delta }_c=2$; $\gamma /{\delta }_0+{\delta }_c=2$ and ${\delta }_c/{\delta }_0+{\delta }_c=0.1$.

Figure 3

Optimal values of ${\delta }_c$ for the low-Rayleigh-scattering case $\gamma =0$ (blue line) and for the classical SIL case ${\delta }_m=0$ (red line). The other optimal parameters for the low-scattering case and for the classical SIL case are presented in Eqs. (22) and (23), respectively.

Figure 4

Comparison of the optimal stabilization coefficients (solid lines—left $y$ axis) and the reflectance $|\mathrm{\Gamma }|$ (dashed lines—right $y$ axis) for the case of negligible Rayleigh scattering (red line) and for the classical SIL case—no drop-port mirror coupling (blue line). The optimal parameters for the low-scattering case and for the classical SIL case are presented in Eqs. (22) and (23), respectively.

Figure 5

Comparison of the transmission coefficient $T_{\mathrm{out}}$ (solid lines—left $y$ axis) and losses (dashed lines—right $y$ axis) for the case of negligible Rayleigh scattering (red line) and for the classical SIL case—no drop-port mirror coupling (blue line). The parameters are optimized [see Eqs. (22) and (23)].

Figure 6

Comparison of the normalized forward wave $A_{+}/A_{0+}$ (solid lines—left $y$ axis) and backward wave $A_{-}/A_{0-}$ (dashed lines—right $y$ axis) amplitudes for the case of negligible Rayleigh scattering (red line) and for the classical SIL case—no drop-port mirror coupling (blue line). The parameters are chosen to be optimal [see Eqs. (22) and (23)], $A_{0\pm }$ are the amplitudes $A_{\pm }$ at $\gamma /{\delta }_0=1.16$ and ${\delta }_m=0$, which corresponds to the classical SIL optimum.

Figure 7

The map of the backward reflection $|\mathrm{\Gamma }|{|}_{\gamma =0}^{|\tilde{R}|=1}$ [see Eq. (A3)] under the condition of the optimum $\underset{\tilde{\psi },\mathrm{\Delta }\omega ,{\tau }_s}{max}K{|}_{\gamma =0}^{|\tilde{R}|=1}$ [see Eq. (17)]. The dark gray line corresponds to $|\mathrm{\Gamma }|=0.1$.

Figure 8

Map of the optimal stabilization coefficient $\underset{\tilde{\psi },\mathrm{\Delta }\omega ,{\tau }_s}{max}K{|}_{\gamma =0}^{|\tilde{R}|=1}$ [see Eq. (17)]. The dark gray line corresponds to $|\mathrm{\Gamma }|=0.1$. The purple and green lines corresponds to the $\arg max(K){|}_{|\mathrm{\Gamma }|\ll 1}$ [see Eq. (B6)]. The yellow line corresponds to the $\arg max|C_{\mathrm{out}}|$ [see Eq. (B7)].