# Microresonator and Laser Parameters Definition via Self-Injection Locking

Artem E. Shitikov1,2<EMAIL_ADDRESS>Oleg V. Benderov3 Nikita M.
Kondratiev1 Valery E. Lobanov1 Alexander V. Rodin3 Igor A. Bilenko1,2 1Russian
Quantum Center, 143026 Skolkovo, Russia  2Faculty of Physics, Lomonosov Moscow
State University, 119991 Moscow, Russia  3Moscow Institute of Physics and
Technology, 141701, Dolgoprudny, Russia

(August 27, 2024)

###### Abstract

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 developed and verified experimentally 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 laser diode parameters. We
demonstrated that this method can be used in the spectral ranges where
conventional methods are not applicable, for example, in the mid-IR. We
stabilized a 2.64 $\mu$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 was determined to be $5\cdot 10^{8}$.

##  I Introduction

The effect of the self-injection locking (SIL) has been utilized for many
years in radiophysics and microwave electronics to stabilize the devices and
to increase their spectral purity Ohta and Murakami (1968); Ota and Nata
(1970); H.-C. Chang (2003); Choi and Choi (2007); Glyavin  _et al._ (2015);
Zhang  _et al._ (2015a, b). For more than the last three decades, it has also
been studied and actively applied in optics and laser physics Velichanskii
_et al._ (1978); Lang and Kobayashi (1980); Belenov  _et al._ (1983); Patzak
_et al._ (1983); Agrawal (1984); Tkach and Chraplyvy (1986); Dahmani  _et al._
(1987); Hollberg and Ohtsu (1988); Li and Abraham (1989); Hemmerich  _et al._
(1990, 1994); Hjelme  _et al._ (1991). The most interesting results were
demonstrated with whispering gallery mode (WGM) microresonators Braginsky  _et
al._ (1989); Savchenkov  _et al._ (2004); Matsko and Ilchenko (2006);
Savchenkov  _et al._ (2007); Ward and Benson (2011); Lin  _et al._ (2014);
Henriet  _et al._ (2015); Strekalov  _et al._ (2016); Grudinin  _et al._
(2006); Lecaplain  _et al._ (2016); Shitikov  _et al._ (2018), combining a
large quality factor in a wide spectral range with small size and low
environmental sensitivity. Nowadays, this effect is a key ingredient of
various up-to-date photonic applications. First, the Rayleigh backscattering
in optical microresonators Gorodetsky  _et al._ (2000) provides a passive
frequency-selective fast optical feedback to the laser diode resulting in
significant laser phase noise suppression and linewidth reduction. Recent
research has demonstrated passive stabilization of single-frequency Vassiliev
_et al._ (1998, 2003); Liang  _et al._ (2010, 2015); Dale  _et al._ (2016);
Savchenkov  _et al._ (2019a) or even multifrequency Donvalkar  _et al._
(2018); Galiev  _et al._ (2018); Pavlov  _et al._ (2018); Savchenkov  _et al._
(2019b) semiconductor lasers to subkilohertz linewidths with WGM
microresonators in different spectral ranges, from UV to mid-IR. Moreover, it
was shown that such stabilized laser diodes can be used as a pump source for
the generation of the microresonator-based frequency combs Pavlov  _et al._
(2018). High attainable Q-factors and small mode volume of the WGM allows
decreasing the soliton generation power threshold to several $\mu W$, which
opens the way to compact energy-efficient devices. The development of such
compact microcomb sources is one of the hottest topics of modern photonics
which attracts researchers worldwide Pavlov  _et al._ (2018); Raja  _et al._
(2019); Boust  _et al._ (2019); Briles  _et al._ (2019); Shen  _et al._
(2020); Voloshin  _et al._ (2019). Such devices are of paramount importance in
numerous areas of modern science and technology, such as coherent
communication Marin-Palomo  _et al._ (2017); Fülöp  _et al._ (2018), ultrafast
optical ranging and LIDARs Suh and Vahala (2018); Trocha  _et al._ (2018);
Riemensberger  _et al._ (2020), high-precision spectroscopy Suh  _et al._
(2016); Yang  _et al._ (2019), astrophysics Obrzud  _et al._ (2019); Suh  _et
al._ (2019), low-noise microwave synthesis Liang  _et al._ (2015), and optical
clocks Papp  _et al._ (2014); Newman  _et al._ (2019).

Constructing such microresonator-based devices operating in the SIL regime,
one may face a challenging problem of the accurate determination of the
locking microresonator mode parameters. Its quality factor is a crucial
parameter that determines the effectiveness of the SIL and overall device
performance, since it determines the resulting linewidth of the laser source
defining the stabilization coefficient Kondratiev  _et al._ (2017) and the
threshold of the nonlinear effects Kippenberg  _et al._ (2018). The highest
Q-factor can be achieved in WGM crystalline resonators (up to 1011 Savchenkov
_et al._ (2007)) and application of prism couplers made devices with such
resonators perfectly working and robust. There are several well-known ways to
measure the quality factor: the full width at half maximum (FWHM) of the
resonance curve measurement, the ringdown method based on the recording of
free oscillations after pulsed excitation, and evaluation from the linewidth
dependence on the microresonator loading Savchenkov  _et al._ (2018). However,
in the SIL regime, such methods are mostly non-applicable. First, they require
an optical isolator between the laser and the microresonator, which leads to
the need for the disassembly and subsequent assembly of the device or the
setup. Second, a determination of the particular WGM that was used for the SIL
can be a challenging task in the overmoded resonators. Also, in some spectral
ranges, the use of these methods requires the availability of hard-to-find and
expensive instruments.

Figure 1: Experimental setups in the near-IR. First, we measured the
dependence of the resonance width on the gap value between the microresonator
and the coupler. Second, we measured the Q-factor of that mode by the ringdown
method. Finally, we verified that Q-factor conserves during measurements by
measuring the resonance width another time.

In our work, we analyzed in detail the manifestations of the SIL for different
types and parameters of the microresonator modes and showed that SIL can be
used as an effective diagnostic tool. We found out that SIL transmission
curves depend on the microresonator Q-factor in a fashion that allows
extracting it directly from the locking range measurement, which makes it
unnecessary to measure microresonator quality factor from the frequency or
time response of the unlocked resonator. We developed a new, original method
based on the measurement of the SIL bandwidth as a function of the gap between
the coupling prism and the microresonator. The possibility of varying the
coupling rate by changing the size of the gap gives a useful degree of freedom
(inaccessible in the case of Fabry-Perot cavities). The technique also offers
a unique opportunity for determining the intrinsic quality factor of the exact
WGM used in the SIL regime. Besides, the developed method allows determining
the vertical index of the selected mode and provides information about a laser
diode parameter – the output beam coupling rate – which is of particular
importance for accurate SIL modeling and microresonator-based device
construction. This technique is based on the fundamentals of SIL and is valid
for any wavelength and microresonator and was verified at 1550 nm for MgF2 by
the conventional well-adopted ringdown method. We also compared the elaborated
technique with the alternative techniques and showed that it is especially
useful for the WGM Q-factor determination in specific wavelength ranges where
it is difficult to do using conventional methods due to the lack of necessary
devices (high-speed detectors, narrow-linewidth lasers).

Namely, we successfully applied the technique for silicon resonators in the
mid-IR band, avoiding the need to use an expensive narrow-linewidth tunable
laser. We demonstrated the first to our knowledge SIL of a laser diode to a
WGM microresonator made of crystalline silicon at $2.6\,\mu m$ wavelength, and
characterized the effect using the above-mentioned technique. The Q-factor of
the WGM was up to $0.5\cdot 10^{9}$, and the locking width exceeded $0.6$ GHz.
The mid-IR laser diodes’ significant stabilization seems to be very
interesting because there is a shortage of affordable, narrow-linewidth laser
sources in the enormous range from 2$\ \mu$m to 20$\ \mu$m. Commonly used
optical parametric oscillators and quantum cascaded lasers with distributed
feedback are rather expensive and hard-to-obtain products. Sub-kHz lasers have
great potential in the mid-IR range since it covers the molecular fingerprint
region important for high-resolution gas spectroscopy Tokunaga  _et al._
(2013), biosensing Xia  _et al._ (2019), and even fundamental constants
measurements Mejri  _et al._ (2015). It is also promising for free-space
optical communication due to better performance than the near-IR light in fog
conditions because it suffers less from atmospheric turbulence Hao  _et al._
(2017). For optical communication, a narrow-linewidth laser source allows high
order modulation to increase the bit-rate Al-Taiy  _et al._ (2014). The
demonstration of the high-quality factor in silicon microresonator in the mid-
IR is also a new step in silicon photonics. The high nonlinear refractive
index, $n_{2}=4.5\cdot 10^{-18}$ m2/W Bristow  _et al._ (2007), combined with
the absence of the multiphonon absorption up to 8 $\mu$m Hass and Bendow
(1977) and two-photon absorption from 2.3 $\mu$m Bristow  _et al._ (2007)
opens up a new perspective for silicon in the mid-IR since the quality factor
of the commonly used fluoride WGM reference cavities can be limited by the
multiphonon absorption Lecaplain  _et al._ (2016).

##  II SIL-based measurement technique

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 (FLR and BLR) was
measured in experiments. It is a good approximation of the total locking range
$\delta\omega_{\rm lock}$ in the case of high Q-factor, when the inner overlap
band $\delta\omega_{\rm in}$ is negligible.

We consider a typical microresonator experimental setup (see Fig. 1) with
piezo elements to control the microresonator coupling and the laser-
microresonator distance. We studied different types of dependencies of the
transmitted light intensity from the laser diode current (LI-curves), observed
in experiments with a photodiode, which corresponds to a frequency scan of a
laser diode. In case of the SIL the LI-curve shape varies with the phase of
the backscattering wave $\psi_{0}$ gained while travelling from the laser to
the microresonator and back. The locking phase $\psi_{0}$ strongly affects the
laser stabilization and can be controlled by adjustment the distance between
the laser diode and WGM microresonator. In our experiments the locking phase
$\psi_{0}$ was precisely controlled by a piezo element (see Fig. 1).
Calculated dependencies of the transmitted power on the laser detuning from
WGM frequency are shown in Fig. 2. A free resonance is observed in the absence
of backscattering (black dotted line in Fig. 2) and a locked resonance appears
when the backscattering causes SIL (blue dashed line in Fig. 2). In the self-
injection locking regime, one may determine the locking range, that is the
bandwidth on the LI-curve where the frequency change is suppressed by SIL. At
locking phases $\psi_{0}\in[-\pi/2;\pi/2]$, the locking range is bounded by
sharp edges. For non-even locking phases ($\psi_{0}\neq\pi n$), the shape of
the LI-curves during the forward and the backward frequency scans are
different, which can be seen in the experiment. This means that we need both
scans to capture the whole locking band correctly. The theoretical predictions
are shown in Fig. 2 with solid lines with triangle markers. The sum of the
forward locking range (FLR), measured at forward scan, and backward locking
range (BLR), measured at backward scan, is connected with the locking range
$\delta\omega_{\rm lock}$ as ${\rm FLR}+{\rm BLR}=\delta\omega_{\rm
lock}+\delta\omega_{\rm in}$ (see Fig. 2). The FLR-BLR overlap or the inner
band $\delta\omega_{\rm in}$ is significant only for low intrinsic quality
factor $Q_{int}$ or for overcoupled microresonator.

We note that in the locked state the width of the locking range can be
naturally defined using the sharp jumps of the curve. The self-injection
locking theory predicts that this width depends on the loaded quality factor
of the locking WGM $Q_{m}$ ($1/Q_{m}=1/Q_{\rm int}+1/Q_{\rm coupl}$, where
$Q_{\rm coupl}$ represents losses added by the coupler that extracts light
from resonator) Kondratiev  _et al._ (2017). Using this theory, we propose a
technique, based on measuring of ${\rm FLR}+{\rm BLR}$ dependence on a gap
between the cavity and the coupler, which provides information on the locking
mode’s intrinsic quality factor $Q_{\rm int}$ and its vertical index. Knowing
$Q_{\rm int}$ one may estimate $Q_{m}$ for every gap value.

Using the expression for the stationary tuning curve for the zero locking
phase $\psi_{0}=0$ Kondratiev  _et al._ (2017), one may obtain the following
expressions for the total locking range width $\delta\omega_{\rm lock}$ and
for the width of the inner band $\delta\omega_{\rm in}$ in the case of zero
locking phase:

$\displaystyle\delta\omega_{\rm lock}$ $\displaystyle\approx
3\sqrt{3}\gamma\frac{\kappa_{mc}}{\kappa_{m}^{2}}\bar{\kappa}_{do}+\frac{\sqrt{3}}{3}\kappa_{m},$
(1) $\displaystyle\delta\omega_{\rm in}$ $\displaystyle\approx
8\sqrt[4]{\frac{\kappa_{mc}\kappa_{m}\gamma\bar{\kappa}_{do}}{27}}.$ (2)

Here, $\gamma$ is the forward-backward wave coupling rate, $\bar{\kappa}_{do}$
is the laser’s effective (hot) output beam coupling rate and
$\kappa_{m}=\kappa_{mc}+\kappa_{mi}$ is the microresonator’s mode decay rate,
where $\kappa_{mi}$ is determined by the intrinsic losses and $\kappa_{mc}$ is
determined by the coupling losses. The corresponding values of the loaded and
intrinsic quality factor are defined as $Q_{m}=\omega/\kappa_{m}$,
$Q_{int}=\omega/\kappa_{mi}$. One can load a microresonator by approaching it
to the coupler. The microresonator is called overcoupled when $\kappa_{\rm
mc}$ is larger than $\kappa_{\rm mi}$ and undercoupled when $\kappa_{\rm mi}$
is larger than $\kappa_{\rm mc}$. In case of equality of $\kappa_{\rm mi}$ and
$\kappa_{\rm mc}$ the coupling is called critical. It should be noted that the
expressions (1) and (2) provide good estimations for the widths of the
discussed bands for the locking phases $\psi_{0}\in[-\pi/2;\pi/2]$. In an
experiment the locking phase close to zero is the most desirable due to the
best performance Galiev  _et al._ (2020). The second term in (1) becomes a
significant part of the locking range only in close proximity of the coupler,
where $\kappa_{mc}$ is large. However, the locking range (1) is actually
determined at the level higher than 1/2 of maximum intensity and, thus, does
not correspond to the full width at half-maximum (FWHM) which is usually used
for the Lorentzian-like curve characterization and equals to the resonator
loss rate. During the loading, the $Q_{m}$ decreases and at some moment the
sharp edges of the locking band are blurred and after that FWHM should be
measured to characterize the mode. Using the tuning curve expression from
Kondratiev  _et al._ (2017), one can obtain the following expression for the
FWHM in the SIL regime:

$\delta\omega_{\rm
FWHM}=4\gamma\frac{\kappa_{mc}}{\kappa_{m}^{2}}\bar{\kappa}_{do}+\kappa_{m}.$
(3)

It can be seen that $\delta\omega_{\rm FWHM}$ is smaller than the full locking
range $\delta\omega_{\rm lock}$ for high values of the intrinsic quality
factor and can not be observed as it is hidden behind the LI curve jumps, so
(1) should be used to estimate the locking width. If the locking width is
reduced, a transition from the “hard” (with jumps) to “soft” (smooth curve)
regime occurs. The transition point is when all the three widths are equal to
double unlocked microresonator linewidth: $\delta\omega_{\rm
lock}=\delta\omega_{\rm in}=\delta\omega_{\rm FWHM}=2\kappa_{m}$. After this
transition occurs, $\delta\omega_{\rm FWHM}$ becomes bigger than the full
locking width and the curve has no jumps anymore (Eq. (2) becomes
meaningless), making the usage of $\delta\omega_{\rm FWHM}$ reasonable for the
width estimations. When the first term of (3) becomes much smaller than the
second one, the $\delta\omega_{\rm FWHM}$ becomes equal to the microresonator
total loss rate $\kappa_{m}$, the curve shape becomes Lorentzian, and SIL is
totally switched off.

Analyzing the derivative of (1) with respect to the microresonator coupling
rate $\kappa_{mc}$, one may evaluate that its maximum takes place at critical
coupling $\kappa_{mi}=\kappa_{mc}$. Knowing $\kappa_{mc}$ at maximum locking
range one would also know $\kappa_{mi}$. At the same time, $\kappa_{mc}$ can
be evaluated from the value of the gap between the microresonator and the
coupler $d$ Gorodetsky and Ilchenko (1994):

$\displaystyle\kappa_{mc}\approx\frac{\omega}{2}\left(\frac{n^{2}-1}{n}k\right)^{-3/2}\exp{(-2kd\sqrt{n^{2}-1})}\times$
$\displaystyle\times\begin{cases}1/\sqrt{\frac{\pi}{1+\sqrt{n^{2}-1}}},p=0\\\
1/(\sqrt{2}\pi\sqrt{p}),p>0\end{cases}$ (4)

Here both the refractive indices of the coupler and the microresonator are
assumed to be equal to $n$, $\omega$ is the pump frequency, $a$ is the radius
of the microresonator, $k=\omega/c$, $p$ is the vertical index of the mode. It
is assumed that refractive index of the coupler is approximately equal to the
refractive index of the microresonator, which is exactly true in our case of a
Si resonator with a silicon hemisphere coupler and quite acceptable for MgF2
with a glass prism. Thus, the position of the extremum of the curve describing
the dependence of the locking range width (or FLR + BLR in our experiments) on
the gap between microresonator and coupling element provides us the
information about the locking mode’s critical coupling. The accuracy of the
technique depends on the accuracy of the maximum of the locking width position
corresponding to the coupler position measurements. Note, that according to
Galiev  _et al._ (2020) critical coupling provides the optimal laser
stabilization in most cases, and thus the obtained information can be also
used for the fine-tuning of the SIL-based devices using the same piezo
element.

It is worth noting, that according to (II), the coupling-related linewidth of
the microresonator at zero gap in the case of high $Q_{int}$ is mostly
determined by the vertical index $p$ of the WGM (see Table (1) for 2 mm radius
for MgF2 at 1550 nm and 1.25 mm radius for Si at 2639 nm). Thus, knowing
$\kappa_{m}$ at zero gap one can evaluate the vertical index of the mode from
(II). The maximum difference between $\kappa_{mc}(d=0)$ is for $p=0$ and
$p=1$, and becomes lower with increase of $p$.

Table 1: Coupling rate $\kappa_{mc}$ of the WGM at zero gap in MHz | p=0 | p=1 | p=2 | p=3 | p=4 | p=5
---|---|---|---|---|---|---
$MgF_{2}$ 1550 nm | 159 | 52 | 37 | 30 | 26 | 23
$Si$ 2639 nm | 90 | 14 | 10 | 8 | 7 | 6

Summarizing, to determine a WGM’s parameters in the SIL regime, one needs to
measure the dependence of the sum of FLR and BLR on the gap between the
coupler and microresonator. Also, it is necessary to monitor the change of the
LI-curve shape. There can be two cases: the first one is when the "hard" SIL
regime with sharp LI-curve edges is maintained up to the zero gap; the second
case is when the "soft" SIL regime with smooth edges appears in the proximity
of the coupler.

The fit process of the measured dependence was carried out by the brute force
method. In the first case we start with selecting $Q_{\rm int}$ to match the
position of the experimental curve maximum with Eq. (1) and Eq. (2) using Eq.
(II) for several $p$ values. Then we adjust $\gamma\bar{\kappa}_{do}$ to fit
$\rm BLR+FLR$ at critical coupling and choose the best variant at the zero
point over the $p$ indices. In the second case the fit process is almost the
same but the $p$ index can be estimated from the data at zero gap using Table
1. Then we select $Q_{\rm{int}}$ to match the maximum of the experimental
curve with Eq. (1) and Eq. (2) using Eq. (II). Then, we adjust
$\gamma\bar{\kappa}_{do}$ in order to match the locking range value at maximum
point. The part of the measured dependence related to the "soft" regime should
be fitted with $2\delta\omega_{\rm FWHM}$ from Eqs. (3) and (II) using the
parameters determined in previous steps, and good coincidence indicates the
correct choice of the parameters ($p$ in particular). The uncertainty of the
measurements can be added according to the piezo element voltage controller
step.

The only alternative for determining the Q-factor in the SIL regime is a
technique based on the resonance depth analysis Savchenkov  _et al._ (2018).
That technique is based on similar principle of the coupling rate exponential
fitting (II), but with the resonator power transmittance curve. First of all,
the resonance depth measurements are sensitive to the coupling of all light
into the detector (which can be challenging in many cases) and different
detector noises. Secondly, it is necessary to emphasize that it is vital to
determine the vertical index of the mode as described above to calculate
Q-factor correctly. Finally, it is very natural to measure the locking range
in the SIL regime. It can be made very accurately and is not demanding for the
equipment parameters. For a brief estimation of the $Q_{\rm int}$ one doesn’t
need to measure dozens of points: it is necessary to determine the zero-gap
point for the vertical index $p$ determination, and the critical coupling
point to relate the coupling to the intrinsic loss. FLR+BLR at these points
provides enough information to completely fit with (1), (3) and (II).

Once the locking range for the given resonator and coupling rate is obtained,
it can be used to characterize the laser diode. In the case of the high-Q WGM
the second term in (1) is negligible and for the effective (hot) output beam
coupling rate $\bar{\kappa}_{do}$ one may obtain from (1):

$\bar{\kappa}_{d_{o}}=\frac{1}{3\sqrt{3}}\frac{\delta\omega_{\rm
lock}\kappa^{2}_{m}}{\gamma\kappa_{mc}}=\frac{4}{3\sqrt{3}}\frac{\delta\omega_{\rm
lock}}{\Gamma_{m}},$ (5)

where $\Gamma_{m}=4\frac{\gamma\kappa_{\rm mc}}{\kappa_{\rm m}^{2}}$ is the
amplitude reflection coefficient from the WGM cavity at resonance Kondratiev
_et al._ (2017), which can be measured experimentally. On the other hand,
$\bar{\kappa}_{do}=\frac{T_{o}^{2}}{\tau_{d}R_{e}R_{o}^{2}}\sqrt{1+\alpha_{g}^{2}}$,
where $R_{o},R_{e}$ are the amplitude reflection coefficients of the output
and end laser mirrors, $T_{o}=\sqrt{1-R_{o}^{2}}$ is the output amplitude
transmission coefficient, $\tau_{d}$ is the round-trip time of a laser diode,
$\alpha_{g}$ is the Henry factor, which allows taking into account the
refractive index changes due to the injected carriers. So, the determination
of the $\bar{\kappa}_{do}$ provides vital information about the laser diode
itself. That way of $\bar{\kappa}_{do}$ estimation can be used not only for
bulk resonators but also for on-chip ones.

##  III Near-IR experiment

To verify the proposed technique, we performed some measurements of the FLR
and BLR dependencies on the resonator-to-coupler gap (with the control of the
Q-factor at zero-gap to determine the vertical index of the mode) for the MgF2
microresonator at 1550 nm and compared the calculated quality factor with the
result obtained with the well-known ringdown method.

Figure 3: LI-curves measured for different gap values. The highlighted regions
corresponds to the excited modes with different polar indices: light green -
$p=0$, light blue - $p=2$, light red - high $p$. The mode shapes remain
constant upon variation of the gap value, which indicates that the locking
phase $\psi_{0}$ stays unchanged.

The WGMs were excited by a BK7 coupling prism with a DFB laser diode, exact
wavelength was 1549.5 nm and power was below 2 mW. The microresonator was 4 mm
in diameter and with 0.5 mm side curvature radius. The experimental setups and
algorithm are presented in Fig. (1).

Figure 4: Left panel: FLR+BLR dependence for the mode with $p=0$. The
transition to the "soft" regime occurs at a gap of around 0.2 um, and the fit
with $\delta\omega_{\rm FWHM}$ is valid for gap values less than 0.2 $\mu$m.
Right panel: the mode traces at three distinctive points: I - at zero gap, II
- at transition point ($\rm{FLR}+\rm{BLR}=4\kappa_{m}$) and III - at critical
coupling point ($\kappa_{\rm mi}=\kappa_{\rm mc}$). The experimental curves
are presented with dotted lines and the approximation curves made with the
linear SIL model are solid lines. The high-order mode appeared at
overcoupling.  Figure 5: Dependence of FLR+BLR on the gap for the
$\rm{MgF_{2}}$ microresonator. The dots are the experimental data, the red
ones are for approaching, and the blue ones are for receding microresonator
with respect to the coupler. The yellow solid line is the
$\delta\omega_{\rm{lock}}+\delta\omega_{\rm{in}}$ curve calculated from (1)
and (2). The violet solid line is $2\delta\omega_{\rm{FWHM}}$ calculated from
(3). The extremum takes place at 550 $\mu m$, which gives us $Q_{\rm
int}=(4.0\pm 0.5)\cdot 10^{8}$ and $p=2$. The ringdown method provides $Q_{\rm
int}=(3.6\pm 0.5)\cdot 10^{8}$, see the insets.

The translation stage with piezo element (PZT) was previously experimentally
calibrated, and the measured displacement coefficient was 0.27 $\pm$ 0.01
$\mu$m/V. The step size of the PZT was 27 nm with our controller. We measured
the dependence of the locking width on the piezo voltage (which provided us
the gap value) in both cases when frequency increased and decreased. FLR+BLR
was measured using a calibration interferometer with 93 MHz free spectral
range. The calibration interferometer was made with a 90:10 beamsplitter and
mirrors. Observing the interference fringes, one may determine the frequency
scale. The interferometer was continuously available, but during the
measurements it was blocked to make the mode spectra clearer. For the locking
width measurements we used a detector with large sensitive area 19.6 $mm^{2}$
and 460 kHz bandwidth, and for the ringdown measurements the detector with 0.8
$mm^{2}$ area and 700 MHz bandwidth. It is important that the locking phase
doesn’t change during the loading (see Fig. (3), where the transmission
resonance curves are presented). Every line is like an "isogap", whose value
one may find on the y-axis. In the mode traces, the modes with different $p$
and $Q$ are distinguishable. The modes from different WGM families were
excited, so the different locking phases were observed on a trace. Colored
areas in Fig. (3) correspond to different types of the excited modes. The
modes highlighted with red had high $p$ and lower $Q$ so that they appeared
closer to the coupler and remained narrow. The mode highlighted with green had
$p=0$ and $Q_{\rm{int}}=0.8\cdot 10^{8}$, and its FLR+BLR dependence from the
gap is shown in the left panel of Fig. 4. A critical coupling point (maximum)
and transition point ("hard" to "soft" SIL) are highlighted with arrows III
and II correspondingly. There is a pronounced minimum in this case. It appears
between the points where the FLR+BLR is wide due to the self-injection locking
and overcoupled regime of the microresonator. At this point LI curve becomes
narrower transforming to the "soft" regime while $\kappa_{c}$ is not large
yet. We also have to note, that while $\delta\omega_{\rm
lock}>\delta\omega_{\rm FWHM}$ in the "hard" regime, $\delta\omega_{\rm
lock}+\delta\omega_{\rm in}<2\delta\omega_{\rm FWHM}$ and vice versa for the
soft regime. We compare the LI curves at the three mentioned points with a
linear SIL model approximation Kondratiev  _et al._ (2017) in the right panel
of Fig. 4. First, we subtracted the transmission inclination due to the laser
diode power change during the frequency scan from experimental data and
normalized the signal to the off-resonant power. We note, that the
experimental transmission in the critical coupling regime is 74% (see dashed
lines in the right panel of Fig. 4) instead of close to zero value. This is
due to imperfect mode matching in the experiment and the theoretical curves
should be modified accordingly. The residual sinusoidal transmission
oscillations refer to the calibration interferometer. Then we calculated
$Q_{m}$ and $Q_{\rm int}$ for the mentioned points and experimentally measured
$\bar{\kappa}_{do}$. Using this data as input parameters for the modeling we
carried out an approximation (see solid lines in the right panel of Fig. 4)
and get excellent agreement of the resonance curve form. The locking phase was
initially assumed to be zero as forward and backward scans look symmetric
(slight shift to $\pi/10$ makes the coincidence even better). However, after
the alignment of the mode III we had to shift further the positions of the
theoretical curves for the modes I and II to the right by 65 and 50 MHz for
the perfect matching. We attribute this to the thermal effects, not included
into the model, that shift the mode as a whole the more, the deeper is the dip
(corresponding to the higher intracavity power). We also note that the power
in the wide region near the mode drops down slightly. That probably happens
due to the power leakage into some low-Q mode, revealed with loading. To
account for this, we also lower the unity level for the theoretical curves I
and II for 0.015 and 0.02. A mode from another family becomes visible in the
curves I and II to the right from the locked one during the coupling
increasing, while initially in III it was hidden by the locking. It was also
revealed that the mode shape near the coupler depends on the
$\bar{\kappa}_{\rm do}$: the lower it is, the closer the mode shape to
Lorentzian.

In many cases the transmission to "soft" regime (without sharp edges)
appeared, when the resonator was close enough to the coupler. It allows the
direct determination of $p$ by the linewidth at zero gap (see Table (1)). But
in some cases, when the dimensionless coupling rate between counterpropagating
modes and vertical index of the mode are high enough, the SIL regime can be
observed at zero gap. After microresonator reached the surface of the coupler
the resonance stays constant.

The mode highlighted in blue in Fig. 3 had a locking phase near $\pi$. We
changed its the phase to zero by moving the laser and performed the
measurements of its FLR+BLR dependence on the gap value, obtaining Fig. (5).
Here the locking range is much wider and a transition from the "hard" to the
"soft" regime is not easily noticeable. The extremum is at $d=550$ $\mu$m from
the coupler, which corresponds to an internal Q-factor $Q_{\rm{int}}=(4.0\pm
0.5)\cdot 10^{8}$ for $p=2$. One may see that FLR+BLR is a bit wider when the
microresonator recedes from the coupler, which is caused by the hysteresis of
the piezo element. It is worth noting that there is a strong dependence of the
extremum location on $p$, when it changes from $p=0$ to $p>0$. This dependence
is weak for the high $p$ values, and for the $q$ index also Gorodetsky and
Ilchenko (1994). The transition point to "hard" locking regime for the modes
with $p=0$ is reached earlier than for $p>0$, as $\kappa_{c}$ increases
faster, the combined coupling coefficient decreases to values not enough for
the strong SIL.

Figure 6: (a) Experimental setup in the mid-IR. (b) Mode scan at the critical
coupling. (c) Dependence of FLR+BLR on the gap for the Si microresonator at
2639 nm. The blue and red dots are the experimental data and the yellow solid
line is a curve calculated using Eq. (2). The extremum takes place at 270 $\mu
m$ that gives us $Q_{\rm int}=5.0\cdot 10^{8}$ and $p=1$.

Finally, we performed quality-factor measurements of the mode highlighted in
blue in Fig. 3 via the ringdown method. We used an external narrow linewidth
fiber laser with an optical isolator. To excite the same WGM, we determine its
exact frequency by beating transmitted light with laser with isolator at
undercoupled regime. Then, we fix the polarization with a Glan-Thompson prism.
Finally, we adjust the power of the external laser with an isolator to make it
equal to the DFB laser power. Thus, the mode was excited with the external
laser with an isolator at the same frequency, the same polarization and at the
same power. The WGM’s intrinsic quality factor was measured by the ringdown
method in the undercoupled regime as $Q_{\rm{int}}=(3.6\pm 0.5)\cdot 10^{8}$.
To verify that there was no degradation of the $Q_{\rm{int}}$ during the
measurements, we additionally checked that the locking width did not change
after these measurements.

To measure the backscattering we added a 50:50 beamsplitter between the
coupling lens and the external laser with an isolator. Half of the
backscattering was coupled into the photodetector so we measured
$\Gamma_{m}^{2}$. From the previous experiments we may evaluate the
$\gamma\bar{\kappa}_{do}$ as a fit parameter. The backscattering wave
measurements allows us to measure $\gamma$ as a part of $\Gamma_{m}$
separately from $\bar{\kappa}_{do}$. The measured reflected power for the
critical coupling was 1/150 part of the input power ($\Gamma_{m}^{2}=1/150$).
One may calculate $\bar{\kappa}_{d_{o}}/(2\pi)=1.2\cdot 10^{10}$ Hz.

The Q-factor measured with the proposed technique is in good agreement with
the value measured by a conventional ringdown method, which confirms the
applicability of the technique. The proposed technique provides a unique
opportunity to accurately measure the Q-factor via SIL without changing the
setup or introducing additional equipment.

##  IV Mid-IR experiment

The developed technique was applied to determine the Q-factor of a silicon
microresonator at 2639 nm. The microresonator made of crystalline silicon was
polished according to technique described in Shitikov  _et al._ (2018) to
provide low surface losses. First, we determined the internal Q-factor of this
microresonator at 1550 nm as $1-2\cdot 10^{8}$ depending on the mode. The
Q-factor was measured using an external laser with an isolator by measuring
the FWHM.

In the mid-IR experiment light was coupled into microresonator by a hemisphere
made of silicon. The temperature-stabilized 2639 nm DFB diode laser was used
as a pump laser. A temperature stabilized InAs photodiode with the amplifier
with 10 MHz bandwidth and 100 $\mu$W saturation power was used as a detector.
The immersive lens allowed to couple mm-scale beam into detector. Measurements
of the WGM’s Q-factor by the ringdown method are impossible here due to
absence of the detector with high bandwidth, and limitations of the laser
sweep speed. We conducted the experiments with an attenuating plate between
the laser and a coupler to suppress the backscattering (and SIL) to determine
the Q-factor by measuring the FWHM. Accurate measurements of the quality
factor were impossible due to the thermo-optical oscillations and laser noise.
However, we estimated the linewidth of the microresonator at zero gap to be
approximately $15$ MHz, which corresponds to $p>0$. The proposed technique
provides us with a unique opportunity to determine the microresonator’s
parameters via self-injection locking. The thermo-optical oscillations were
drastically suppressed in SIL regime (see Fig. (6) (b)). The internal Q-factor
value of the WGM microresonator was as high as $5.0\pm 0.7\cdot 10^{8}$ and
the vertical index of the mode $p=1$ (see Fig. (6) (c)).

At the zero gap there was a mode with sharp edges having width of about 70
MHz. The increase of the Q-factor by several times compared to the near-IR
indicates that the main mechanism of the losses is the Rayleigh scattering.
This confirms the assumption of attainability of ultra-high Q-factors in
silicon WGM microresonators in the mid-IR. The effect of a wider measured
locking range during taking microresonator from the coupler caused by
hysteresis of the piezo element was also observed. The total locking range for
small $d$ was higher than theory predictions. We suppose that this is due to
the thermal effects in silicon, which are so noticeable because the coupler
and microresonator are both made of silicon, so that the slightest changes of
the refractive index are marked. This effect was reduced, but did not vanish
with the decrease of the incident power and for other modes. The total locking
range exceeded $0.6$ GHz, that for $Q_{\rm{int}}=(5.0\pm 0.7)\cdot 10^{8}$
corresponds to the stabilization coefficient of
$K=\frac{16\delta\omega_{\rm{lock}}}{3\sqrt{3}\kappa_{m}}=4200$.

##  V Conclusion

We developed, verified and implemented an original technique in an experiment
that allowed us to determine the parameters of the reference microcavity in
the SIL regime. The method is applicable for various spectral ranges,
especially where it would be difficult to use established methods. The
determination of the intrinsic Q-factor is achieved by measuring the locking
width evolution upon changing the gap between the coupler and microresonator.
In contrast to the method based on the resonance depth analysis Savchenkov
_et al._ (2018) this technique allows for the Q-factor determination in the
SIL regime, vertical mode index $p$ identification, and is not susceptible to
stray light entering the photo detector. The information on the $p$ index of
the mode increases the accuracy of the measurements. We showed experimentally
that the phase of the SIL stays constant during the variation of the gap
between the microresonator and the coupler. Measuring additionally the
backscattering, we determined the effective output beam coupling rate of the
laser, that is a key parameter for modeling a SIL laser.

The technique was also used to determine the WGMs’ parameters of silicon
microcavity at 2639 nm, that was the first demonstration of the SIL in a
silicon microcavity in the mid-IR. The elaborated approach allowed us to
determine the internal Q-factor of the locking mode that was equal to $(5\pm
0.7)\cdot 10^{8}$. Such a high value of the quality-factor directly observed
in silicon verifies the significant possibilities of silicon microresonators
for mid-IR photonics.

This work was supported by the Russian Science Foundation (project
20-12-00344). AES and VEL acknowledge the Foundation for the Advancement of
Theoretical Physics and Mathematics “BASIS” for personal support.

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