# Design of a passively mode-locking whispering gallery mode microlaser

Tomoki S. L. Prugger Suzuki  1 Ayata Nakashima  1 Keigo Nagashima  1 Rammaru
Ishida  1 Riku Imamura  1 Shun Fujii  1,2 Sze Yun Set  3 Shinji Yamashita  3
and Takasumi Tanabe1  1Department of Electronics and Electrical Engineering,
Faculty of Science and Technology, Keio University, Yokohama, Kanagawa,
223-8522, Japan
2Quantum Optoelectronics Research Team, RIKEN Center for Advanced Photonics,
Saitama 351-0198, Japan
3Research Center for Advanced Science and Technology, The University of Tokyo,
Tokyo 153-8904, Japan<EMAIL_ADDRESS>
###### Abstract

Ultrahigh repetition rate lasers will become vital light sources for many
future technologies; however, their realization is challenging because the
cavity size must be minimized. Whispering-gallery-mode (WGM) microresonators
are attractive for this purpose since they allow the strong light-matter
interaction usually needed to enable mode-locking. However, the optimum
parameter ranges are entirely unknown since no experiments have yet been
conducted. Here, we numerically investigate pulsed operation in a toroidal WGM
microresonator with gain and saturable absorption (SA) to study the
experimental feasibility. We show that dispersion is the key parameter for
achieving passive mode-locking in this system. Moreover, the design guideline
provided in this work can apply to any small resonators with gain and SA and
is not limited to a specific cavity system.

††journal: osajournal††articletype: Research Article

##  1 Introduction

Mode-locked ultrashort pulse lasers are vital light sources for laser
processing[1], optical communication[2, 3], signal processing[4, 5], LiDAR
application, and remote sensing[6, 7]. Glass-based fiber lasers are excellent
platforms for these applications because of their high beam quality,
robustness, and simple configuration[8]. The two key elements needed to build
a mode-locked laser are a gain medium and a mode-locker, and erbium (Er) doped
fiber and carbon nanotubes (CNTs), respectively, are frequently used for these
purposes in a fiber laser.

For the applications mentioned above, the use of a pulsed laser with a high
repetition rate will open a new avenue to faster processing, higher capacity,
and more effective signal acquisition. However, realizing a repetition rate
exceeding 1 GHz remains a challenge[9] since the laser cavity length must be
minimized so that the light pulses are densely packed in time. Multi-gigahertz
operation has been demonstrated by using a 5-mm Fabry-Pérot Er-doped fiber
cavity, where the end surface is coated with CNTs as a saturable absorber
(SA)[10, 11], but further scaling of the repetition rate is not possible
because the high pump power needed to compensate for the low gain damages the
CNTs.

On the other hand, an ultrahigh-$Q$ whispering-gallery-mode (WGM)
microresonator has proven to be an excellent platform that allows light to
interact strongly with materials[12]. A WGM microresonator made of
$\mathrm{SiO_{2}}$ on silicon has exhibited one of the highest $Q$ values of
larger than $\sim 10^{8}$. Since it is made of $\mathrm{SiO_{2}}$, it is also
a good host for Er doping and will eventually provide a good platform on which
to demonstrate a WGM microlaser[13, 14]. Continuous-wave (CW) lasing has been
demonstrated at an ultralow threshold power thanks to the high $Q$.

Moreover, dissipative soliton generation has also been demonstrated with this
type of WGM resonator by applying CW pumping to one of the longitudinal modes.
Four-wave mixing in a high-$Q$ WGM microresonator generates an optical
frequency comb. If a specific condition is met, we can lock all the generated
comb modes and form ultrashort pulses. They are attractive because dissipative
solitons from a WGM microresonator have an extremely high repetition rate
thanks to their small cavity size. However, such solitons sit on top of a CW
background, making it difficult to achieve the pulse amplification that is
often required if we are to use this light source as the seed for high-power
applications. Moreover, sophisticated wavelength sweeping and feedback are
needed to generate and stabilize the soliton pulse[15], and this makes the
system complex and expensive.

If we can directly obtain mode-locked pulses from a small WGM microlaser, we
would achieve background-free operation without using a complex feedback
system. In addition to the CW lasing operation in a silica toroid WGM
microresonator, SA is also confirmed with the same device by using CNTs. So,
the combination of Er-doped $\mathrm{SiO_{2}}$ toroid WGM microresonator with
CNT as the SA is of interest and has great potential in terms of realizing an
ultracompact high-repetition-rate mode-locked pulse laser.

However, it is unclear whether we can demonstrate mode-locked operation in
such a device since no experiments have yet been conducted with
$\mathrm{SiO_{2}}$ WGM microresonators for such a purpose and the optimum
parameter ranges are entirely unknown. Therefore, we aim to reveal the optimum
range of parameters, such as the cavity size, $Q$, Er ion concentration,
dispersion, and nonlinear loss of the CNTs, needed to achieve mode-locking,
and discuss the experimental feasibility.

Similar studies have already been published where researchers have tried to
achieve an integrated mode-locked laser on a silicon substrate, but we would
like to emphasize that we are dealing with different regimes. The
technological advances made with microresonators allow us to use ultrahigh $Q$
and sophisticated Er doping in silica on silicon. We expect the high $Q$ to
compensate for the limited gain of Er hence allow us to make the cavity size
extremely small.

The paper is organized as follows. Section 2 describes the theoretical model
and explains how we determined the parameters that we used in our numerical
analysis. Some of the essential parameters taken from our experiments are
explained in detail. Section 3 is the core part of this paper, where gain,
nonlinear loss, and dispersions are studied as mode-locking parameters. In
particular, we reveal the tradeoff between the gain and larger dispersion when
we increase the cavity size. Additional numerical calculations are performed,
which set design guidelines to overcome the tradeoff. Finally, we propose a
design for passively mode-locking a WGM microlaser.

##  2 Theoretical model and experimental verification of parameters

###  2.1 Model

Conceptually, a WGM microlaser is a miniaturized fiber ring laser. The system
that we studied is shown in Fig. 1. The resonator is made of Er-doped
$\mathrm{SiO_{2}}$ to enable gain and is fabricated by using the sol-gel
method. CNTs are deposited on the surface of the resonator as a saturable
absorber. A continuous-wave pump is injected into the cavity, and the output
light is coupled out by using a tapered fiber. Throughout this paper, we
define the intrinsic and loaded $Q$s as follows. The intrinsic $Q$
($Q_{\mathrm{int}}$) is the $Q$ determined by the losses, such as the
scattering and absorption of the resonator. The excess absorption losses
caused by the CNTs and Er ion doping are normally excluded unless specified
because they are usually considered separately in equations. The loaded $Q$ is
defined as
$Q_{\mathrm{load}}^{-1}=Q_{\mathrm{int}}^{-1}+Q_{\mathrm{ext}}^{-1}$, where
$Q_{\mathrm{ext}}=\omega_{0}\kappa^{-2}$ is the $Q$ value determined by the
amplitude coupling coefficient between the resonator and the waveguide
($\kappa$) and the angular frequency of the cavity resonance ($\omega_{0}$).

Figure 1: Er-doped silica microtoroid resonator coupled with a tapered fiber
that we studied numerically. CNTs are attached to the surface of the resonator
to enable saturable absorption.

We solve the model based on the nonlinear Schrëdinger equation, as given
by[16, 17],

$T_{\mathrm{r}}\frac{\partial}{\partial
T}A\left(t,T\right)=\left(-iL\frac{\beta_{2}}{2}\frac{\partial^{2}}{\partial
t^{2}}+iL\gamma\left|A\right|^{2}\right)A\left(t,T\right)+\left\\{g_{\mathrm{T}_{\mathrm{r}}}\left(T\right)-\left[l_{\mathrm{T}_{\mathrm{r}}}+\alpha_{\mathrm{T}_{\mathrm{r}}}\left(t,T\right)\right]\right\\}A\left(t,T\right)$
(1)

where $A$, $t$, $T$ are the slowly varying field envelope in the microcavity,
short time, and long time, respectively. $L$, $T_{\mathrm{r}}$, $\beta_{2}$,
$\gamma$, $l$, $g_{\mathrm{T}_{\mathrm{r}}}$, and
$\alpha_{\mathrm{T}_{\mathrm{r}}}$ are the cavity length, one roundtrip time,
second-order dispersion, nonlinear coefficient, linear loss per roundtrip
(determined by the loaded cavity $Q$), net-gain of Er per roundtrip, and
nonlinear loss of the CNTs, respectively. Here, the nonlinear coefficient
$\gamma$ was calculated using the nonlinear refractive index of silica
$n_{2}=2.2\times 10^{-20}~{}\mathrm{m^{2}/W}$.

$g_{\mathrm{T}_{\mathrm{r}}}\left(T\right)=g_{0}\left(\frac{1}{1+\frac{\overline{\left|A(t,T)^{2}\right|}}{P_{\mathrm{sat}}^{g}}}\right)\left(1+\frac{1}{{\omega_{\mathrm{g}}}^{2}}\frac{\partial^{2}}{\partial
t^{2}}\right)$ (2)
$\alpha_{\mathrm{T}_{\mathrm{r}}}\left(t,T\right)=\alpha_{\mathrm{ns}}+\alpha_{0}\frac{1}{1+\frac{\overline{\left|A(t,T)^{2}\right|}}{P_{\mathrm{sat}}^{\alpha}}}$
(3)

where $g_{\mathrm{0}}$, $P_{\mathrm{sat}}^{g}$, $\omega_{\mathrm{g}}$,
$\alpha_{\mathrm{ns}}$, $\alpha_{0}$, and $P_{\mathrm{sat}}^{\alpha}$ are the
saturated net gain coefficient per roundtrip (hereafter referred to as gain),
gain saturation power, gain bandwidth, non-saturable loss per roundtrip,
modulation depth per roundtrip, and saturation power, respectively.
($\overline{|A(t,T)^{2}|}$) is the field intensity averaged over a short time
$t$ (i.e., average power in the cavity). We use standard split-step Fourier
method[18] to perform numerical simulations with a step size equal to the
roundtrip time $T_{\mathrm{r}}$, for $1.5\times 10^{6}$ roundtrips.

We must verify our numerical parameters with references and experiments before
performing our numerical simulation.

###  2.2 Parameters

Here we describe the parameters of the system that we are going to use. The
values are summarized in Tab. 1.

Table 1: Parameter values used in simulation Parameter | Variable | Value | Unit | Source
---|---|---|---|---
Nonlinear coefficient | $\gamma$ | $4.2\times 10^{-3}$ | $\mathrm{W^{-1}m^{-1}}$ | $\gamma=n_{2}\omega_{0}/cA_{\mathrm{eff}}$
Second order dispersion | $\beta_{2}$ | var. | $\mathrm{ps^{2}/km}$ | Calculation (Fig. 2)
Quality factor | $Q$ | $10^{6}-10^{8}$ | - | Experiments
Gain | $g_{0}$ | $10^{-4}-10^{-1}$ | /roundtrip | Experiments (Fig. 3)
Gain saturation power | $P_{\mathrm{sat}}^{g}$ | 0.145 | W | Eqs. (7)-(10)
Gain bandwidth | $\omega_{g}/2\pi$ | 2.5 | THz | Ref. [19]
Modulation depth | $\alpha_{\mathrm{T}_{\mathrm{r}}}$ | $10^{-4}-10^{-1}$ | /roundtrip | Experiments (Fig. 4)
Non saturable loss | $\alpha_{\mathrm{ns}}$ | 0 | /roundtrip | N.A.
Saturation power | $P_{\mathrm{sat}}^{\alpha}/A_{\mathrm{eff}}$ | 15 | $\mathrm{MW/cm^{2}}$ | Experiment (Fig. 4)

Among various parameters, dispersion, gain, and modulation depth are critical
and need to be carefully verified since they significantly influence the mode-
locking behavior.

####  2.2.1 Cavity parameters: Dispersion and Q

Dispersion is given when the material and geometry of the cavity are
determined[20]. We use an $\mathrm{SiO_{2}}$ toroid; hence we can calculate
the dispersion by determining the major diameter $D$ and minor diameter $d$
(Fig. 1). The device is usually fabricated by laser reflow, and $D/d$ is
typically kept at about 10. Hence, we assumed $d=D/10$ and obtained the
dispersion, as shown in Fig. 2, and use this value for further calculation.

Figure 2: Second-order dispersion at 1550 nm for an $\mathrm{SiO_{2}}$
microtoroid with a different major diameter $D$. The minor diameter $d=D/10$.
Inset is the dispersion as a function of wavelength.

As we describe in section 2.2.2, the intrinsic $Q$ is usually on the order of
$10^{6}$ to $10^{7}$. Furthermore, a cavity with an intrinsic $Q$ of $10^{8}$
is also possible. Hence, we study in the range from $10^{6}$ to $10^{8}$ for
the intrinsic $Q$ in the $10^{6}$ to $10^{8}$ range.

####  2.2.2 Er ion parameter: Gain

We fabricated Er-doped $\mathrm{SiO_{2}}$ toroids by the sol-gel method[21].
We obtained a measured $Q$ of $\sim 10^{7}$ when no $\mathrm{Er}^{3+}$ was
doped. We made two resonators, both with the same diameter of $D\sim 100$ µm,
but with different $\mathrm{Er}^{3+}$ concentrations of $1.7\times
10^{18}~{}\mathrm{cm}^{-3}$ for device A and $0.85\times
10^{18}~{}\mathrm{cm}^{-3}$ for device B. The measured loaded $Q$s (i.e., in
the presence of the Er absorption effect) of these two cavities were (device
A): $Q_{\mathrm{pump}}=1.4\times 10^{6}$ and $Q_{\mathrm{lase}}=1.1\times
10^{6}$, and (device B): $Q_{\mathrm{pump}}=3.0\times 10^{5}$, and
$Q_{\mathrm{lase}}=1.1\times 10^{7}$, where $Q_{\mathrm{pump}}$ and
$Q_{\mathrm{lase}}$ are the $Q$s of the pump and lasing modes. We pumped the
cavities at $\sim$1485 nm and observed lasing at $\sim$1605 nm, where the
lasing power is shown as a function of the pump power in Fig. 3(a).

When we take account of the coupled-mode theory and upper-state population of
Er ions based on rate equations, the slope $\eta$ of the laser output-input
power curve is theoretically given as[14],

$\begin{split}\eta&=\kappa_{\mathrm{s}}^{2}\left(\frac{\nu_{\mathrm{s}}n_{\mathrm{s}}V_{\mathrm{m}}^{\mathrm{s}}}{\nu_{\mathrm{p}}n_{\mathrm{p}}V_{\mathrm{m}}^{\mathrm{p}}}\right)\left(\frac{\alpha_{\mathrm{p}}\left(\alpha_{\mathrm{s}}+g_{\mathrm{s}}^{\ast}\right)-\left(\alpha_{\mathrm{p}}+g_{\mathrm{p}}^{\ast}\right)\left(\alpha_{\mathrm{s}}+\alpha_{\mathrm{s}}^{\mathrm{passive}}\right)}{\alpha_{\mathrm{s}}^{\mathrm{passive}}\left(\alpha_{\mathrm{s}}+g_{\mathrm{s}}^{\ast}\right)}\right)\\\
&\cdot\left(\frac{4n_{\mathrm{p}}^{2}\kappa_{\mathrm{p}}^{2}\left(\alpha_{\mathrm{s}}+g_{\mathrm{s}}^{\ast}\right)^{2}}{c^{2}\left[\left(\alpha_{\mathrm{p}}+\alpha_{\mathrm{p}}^{\mathrm{passive}}\right)\left(\alpha_{\mathrm{s}}+g_{\mathrm{s}}^{\ast}\right)-\left(\alpha_{\mathrm{p}}+g_{\mathrm{p}}^{\ast}\right)\left(\alpha_{\mathrm{s}}+\alpha_{\mathrm{s}}^{\mathrm{passive}}\right)\right]^{2}}\right)\end{split}$
(4)

where $\nu$, $n$, $V_{\mathrm{m}}$, $\kappa$, and $c$ are the light frequency,
refractive index, mode volume, amplitude coupling coefficient between the
resonator and the waveguide, and vacuum light velocity, respectively. Scripts
p and s denote pump and signal modes, respectively.
$\alpha_{\mathrm{s,p}}^{\mathrm{passive}}$ is the loss (in $\mathrm{m}^{-1}$
units) determined by the loaded $Q$ of a passive cavity without
$\mathrm{Er}^{3+}$ doping for signal and pump modes. The absorption and gain
(both in $\mathrm{m}^{-1}$ units) at a strong pump (i.e.,
$N_{\mathrm{T}}=N_{2}$ and $N_{1}=0$) are theoretically given as,

$\alpha_{\mathrm{s,p}}=\varGamma_{\mathrm{s,p}}\sigma_{\mathrm{s,p}}^{\mathrm{a}}N_{\mathrm{T}}$
(5)
$g_{\mathrm{s,p}}^{\ast}=\varGamma_{\mathrm{s,p}}\sigma_{\mathrm{s,p}}^{\mathrm{e}}N_{\mathrm{T}}$
(6)

where $N_{\mathrm{T}}$, $\sigma^{\mathrm{a}}$, and $\sigma^{\mathrm{e}}$ are
the overlap factor between the optical mode with a normalized
$\mathrm{Er}^{3+}$ distribution, $\mathrm{Er}^{3+}$ concentration, absorption
cross-section, and emission cross-section, respectively. For simplification we
assume $\varGamma_{\mathrm{s,p}}=1$ in our calculations.

By substituting Eqs. (5) and (6) into Eq. (4), we obtain graphs providing the
relationship between $N_{\mathrm{T}}$ and $\eta$ for different $\kappa$
values, as shown in Fig. 3(b). These graphs enable us to estimate the
effective $N_{\mathrm{T}}$ from an experimentally obtained $\eta$.

Once $N_{\mathrm{T}}$ is determined, we can obtain
$g_{\mathrm{T}_{\mathrm{r}}}$ from Fig. 3(c). The gain can be estimated by
solving the rate equation and propagation equation of an erbium-doped
waveguide numerically. Hence, Fig. 3(c) is calculated as[22]

$\frac{\partial P_{\mathrm{s,p}}\left(z,t\right)}{\partial
z}=\varGamma_{\mathrm{s,p}}\left[\sigma_{\mathrm{s,p}}^{\mathrm{e}}N_{2}(t)-\sigma_{\mathrm{s,p}}^{\mathrm{a}}N_{1}(t)\right]P_{\mathrm{s,p}}\left(z,t\right)$
(7) $\frac{\partial N_{2}(t)}{\partial
t}=-\frac{N_{2}(t)}{\tau}-\frac{1}{A_{\mathrm{eff}}}\left(\frac{\partial
P_{\mathrm{s}}\left(z,t\right)}{\partial z}+\frac{\partial
P_{\mathrm{p}}\left(z,t\right)}{\partial z}\right)N_{2}(t)$ (8)
$N_{1}+N_{2}=N_{\mathrm{T}}$ (9)

where $P$, $N_{1}$, $N_{2}$, and $\tau$ are the light energy (in units of
photons), ground-level carrier density, excited-level carrier density, and
carrier lifetime, respectively. The gain saturates as we increase the pump
power. Figure 3(c) is given by plotting the gain $g_{0}$ as a function of
$N_{\mathrm{T}}$, where

$g_{0}=\varGamma_{\mathrm{s}}\left[\sigma_{\mathrm{s}}^{\mathrm{e}}N_{2}(t_{\mathrm{s}})-\sigma_{\mathrm{s}}^{\mathrm{a}}N_{1}(t_{\mathrm{s}})\right]L$
(10)

$N_{1,2}\left(t_{\mathrm{s}}\right)$ are the carrier densities in a steady-
state calculated by using Eqs. (7)-(9), when the pump is sufficiently strong.

Figure 3: (a) Experimentally demonstrated laser output-input power curve for
cavities with two different $\mathrm{Er}^{+3}$ concentrations
$N_{\mathrm{T}}$. Pump power and lasing power are values in tapered fiber. We
estimate $\kappa^{2}$ to be $10^{7}~{}\mathrm{s}^{-1}$ (under-coupling
condition) from the transmittance experiment, with some uncertainness because
we were not able to measure the transmittance simultaneously when performing
the lasing experiment. Inset is a picture of the $\mathrm{Er}^{+3}$ doped
silica toroid microresonator ($D=100$ µm) fabricated by the sol-gel method and
pumped with 1480 nm laser light. (b) $\eta$ versus $N_{\mathrm{T}}$ for
different $\kappa$ values ($\kappa=\kappa_{\mathrm{p}}=\kappa_{\mathrm{s}}$)
at $\alpha_{\mathrm{s,p}}^{\mathrm{passive}}=0.13~{}\mathrm{cm}^{-1}$ cavity
corresponding to $Q=10^{7}$ at 1550 nm. We use previously reported values of
$\sigma_{\mathrm{p}}^{\mathrm{a}}=1.5\times 10^{-21}~{}\mathrm{cm}^{2}$,
$\sigma_{\mathrm{s}}^{\mathrm{a}}=2.8\times 10^{-21}~{}\mathrm{cm}^{2}$,
$\sigma_{\mathrm{p}}^{\mathrm{e}}=0.8\times 10^{-21}~{}\mathrm{cm}^{2}$, and
$\sigma_{\mathrm{s}}^{\mathrm{e}}=4.8\times 10^{-21}~{}\mathrm{cm}^{2}$[22].
(c) Gain $g$ (at saturating pump power) as a function of $N_{\mathrm{T}}$. The
vertical axis on the right is the gain $g_{0}$ when $D=300$ µm. We use
$A_{\mathrm{eff}}=2.1\times 10^{-7}~{}\mathrm{cm}^{2}$ corresponding to a
microresonator with $d=30$ µm.

Figures 3(b) and 3(c) could be used as follows. If $N_{\mathrm{T}}$ is given,
we can immediately obtain a theoretical $g_{0}$ from Fig. 3(c). On the other
hand, Fig. 3(b) allows us to double-check the $N_{\mathrm{T}}$ value if we
perform a lasing experiment, as we demonstrated in Fig. 3(a). Figure 3(a) has
a slope efficiency of $\eta=1\%$ at an experimental $\mathrm{Er}^{3+}$ doping
concentration of $N_{\mathrm{T}}=1.7\times 10^{18}~{}\mathrm{cm}^{-3}$. This
corresponds to an effective $N_{\mathrm{T}}$ of $6.3\times
10^{18}~{}\mathrm{cm}^{-3}$ according to Fig. 3(b) and an experimentally
estimated gain of $g_{0}=2.1\times 10^{-3}~{}\mathrm{cm}^{-1}$ from Fig. 3(c).
On the other hand, the theoretical curve in Fig. 3(b) suggests that we should
obtain a gain of $g_{0}=5.6\times 10^{-4}~{}\mathrm{cm}^{-1}$ at an
$N_{\mathrm{T}}=1.7\times 10^{18}~{}\mathrm{cm}^{-3}$ concentration. Although
we do not know exactly why we obtained a slightly higher effective
$N_{\mathrm{T}}$, we think the values are in reasonably good agreement and
within the experimental error. Taking some experimental uncertainty into
consideration, we decided to investigate the $g_{0}=10^{-4}$ to $10^{-1}$
range.

Equations (7)-(10) also allow us to calculate $g_{0}$ versus $P_{\mathrm{s}}$,
at different $P_{\mathrm{p}}$ values, from which we can obtain
$P_{\mathrm{sat}}\sim 0.184$ W when we assume a 500 mW pump. We use this value
for the calculation.

####  2.2.3 SA parameters: Saturation power and modulation depth

Saturable absorption is achieved by using CNTs. Here we explain how we
verified the SA operation and obtained the SA parameters that we needed for
our calculation. The CNTs are dispersed in polydimethylsiloxane (PDMS) at a
concentration of 0.24 mg/ml. A PDMS/CNT droplet is transferred from a thin
fiber to the $\mathrm{SiO_{2}}$ toroid, as shown in the inset of Fig. 3(b).
Once the PDMS has been cured by applying heat, the absorption coefficient is
measured by a transmittance measurement. The measurement method is described
in further detail elsewhere[23]. The result is shown in Fig. 3(b). The fitting
curve gives us a saturation power of 15 MW/$\mathrm{cm}^{2}$ and a modulation
depth $\alpha_{0}$ of $3.2\times 10^{-3}$. These values are also consistent
with experiments reported by other groups[24]. Therefore, we used
$P_{\mathrm{sat}}^{\alpha}/A_{\mathrm{eff}}=15$ MW/$\mathrm{cm}^{2}$ and
studied a modulation depth range of $10^{-4}$ to $10^{-1}$.

Figure 4: Absorption coefficient measurement. CNTs are dispersed in PDMS, and
a droplet is transferred to the fabricated $\mathrm{SiO_{2}}$ toroid, as shown
in the inset figure.

##  3 Numerical calculation results and discussions

This numerical study aims to reveal the optimum parameters needed to achieve
mode-locking operation in a tiny Er-doped WGM microresonator. When we
monitored the waveform under different conditions, we found four different
states, as shown in Fig. 5. These four states are chaotic pulses (CP),
multiple pulses (MP), a stable mode-locking regime (ML), and a continuous wave
(CW). The regime that we are interested in is the ML regime, where a single
pulse circulates in the WGM resonator.

We investigated the way in which the gain, loss, and dispersion of the cavity
play a role in achieving stable mode-locking.

Figure 5: Operation regimes were identified based on our three criteria:
chaotic pulses (CP), multiples pulses (MP), stable mode-locking (ML), and
continuous wave (CW).

###  3.1 Q factor dependence

In this section, the relationship between gain per roundtrip $g_{0}$ and
nonlinear loss (modulation depth $\alpha_{0}$) is investigated for an
$\mathrm{SiO_{2}}$ toroid microresonator where $D=300$ µm. Figure 5 shows the
simulation results for intrinsic $Q=10^{7}$ and $10^{8}$. The red line
indicates the estimation of $g_{0}$ when $N_{\mathrm{T}}=6.3\times
10^{18}~{}\mathrm{cm}^{-3}$, which we obtained from our lasing experiment
(Fig. 3).

The regimes for the four different states (CP, MP, ML, and CW) are indicated
in the figure. When the gain is low, the loss is larger than the gain, and the
device will not lase. When the gain is greater than the loss, the device
starts to lase. At a higher gain, along with an adequate modulation depth, the
system exhibits ML operation. When we increase the gain even further, the
pulse splits into multiple components and exhibits an MP state. Finally, the
system is in an unstable regime at CP state when the gain is too high.

Figure 6(a) shows that the cavity exhibits CW lasing when $Q$ is $10^{7}$ at
$g_{0}=10^{-3}$, which explains our CW lasing demonstration in Fig. 3(a).
However, it also shows that it is very challenging to achieve the ML state due
to insufficient gain if we use this cavity. This situation will change if we
use a cavity with a higher $Q$ of $10^{8}$, with which we should be able to
reach the ML regime. Although a $Q$ of $10^{8}$ is experimentally feasible, it
is not easy to realize.

Therefore, we will look for another way of achieving ML, such as increasing
the WGM microcavity laser diameter or designing the cavity dispersion. By
increasing the cavity diameter, we expect to obtain increased gain per
roundtrip. On the other hand, we expect that the mode-locking threshold will
decrease if we can make a cavity that has a smaller dispersion. Both
approaches will relax the cavity loss conditions for mode-locking.

Figure 6: Mode-locking investigation of a $D=300$ µm toroidal microresonator
with gain and nonlinear loss as parameters. (a) shows results for $Q=10^{7}$
and (b) for $Q=10^{8}$. The red line shows the position of the theoretical
gain at $N_{\mathrm{T}}=6.3\times 10^{18}~{}\mathrm{cm}^{-3}.$

###  3.2 Size and dispersion dependence

As discussed previously, a sufficiently large $g_{0}$ is crucial for achieving
ML. An alternative way of increasing $g_{0}$ is to increase the size $D$ of
the microresonator. This approach appears to be straightforward, but the
presence of dispersion makes the optimization more complex[20]. Figure 2 shows
that the dispersion is at its minimum at $D\sim 150$ µm, but becomes anomalous
when we increase the cavity size. A smaller dispersion is usually advantageous
for easy mode-locking because of the stronger self-phase modulation that
occurs at a lower intracavity power. The presence of self-phase modulation is
normally needed to achieve mode-locking. On the other hand, we prefer a larger
cavity size because of the larger gain. Therefore, it appears that there is a
tradeoff between gain and dispersion when we change the cavity size, and there
may be an optimum point.

First, we investigated only the dispersion effect. Namely, we assumed that we
can fix the diameter of the toroid microresonator at 300 µm but can change the
dispersion. Figure 7(a) shows the numerical results we obtained when we fixed
the modulation depth $\alpha_{0}$ and $Q$ at $5\times 10^{-4}$ and $10^{7}$,
respectively. The red line again shows an experimental estimation of the
$g_{0}$. As expected, ML is possible even with a small $g_{0}$ when the
dispersion is close to zero. Unfortunately, in this case, however, the
dispersion of a 300 µm diameter toroid is about $-20$ $\mathrm{ps}^{2}$/km,
which is outside the ML regime.

Second, we performed further numerical investigations at different cavity
diameters $D$ for a cavity where $Q$ was fixed at $10^{7}$. The dispersions
for a cavity with different $D$s are taken from Fig. 2. Increasing the
diameter will increase the gain per roundtrip but result in a larger
dispersion. On the other hand, the decreasing diameter will reduce the gain
per roundtrip but achieve a weaker anomalous dispersion. Since these two
approaches appear to have a tradeoff relationship, we confirmed which design
approach was more suitable for passively mode-locking a WGM micro-laser. The
result is shown in Fig. 7(b).

Figure 7: (a) Modelocking investigation of a 300 µm diameter toroidal
microresonator for $Q=10^{7}$, with gain per roundtrip
($g_{\mathrm{T}_{\mathrm{r}}}$) and dispersion ($\beta_{2}$) as parameters.
(b) Modelocking investigation for toroid microcavities ($Q=10^{7}$) with
different diameters $D$ while $D/d=10$ is maintained. Note that the vertical
axis is now gain per second ($g_{\mathrm{t}}$), where
$g_{\mathrm{t}}=g_{0}/T_{\mathrm{r}}$. The value is at
$g_{\mathrm{t}}=2.2\times 10^{8}~{}\mathrm{s}^{-1}$ when $g_{0}=10^{-3}$ and
$D=300$ µm ($T_{\mathrm{r}}=4.5$ ps).

Figure 7(b) shows interesting behavior. As expected, the CW lasing threshold
decreases as we increase the cavity diameter since the gain increases.
However, the ML threshold is not sensitive to the diameter, and it is almost
constant when $D>300$ µm. This is encouraging because it is telling us that ML
is possible even with a small cavity, which is usually more challenging.
Taking into consideration that a larger diameter WGM toroid resonator is often
more difficult to fabricate[25], we concluded that the target diameter of the
cavity is $D=300$ µm, but with a slightly higher intrinsic $Q$ and higher
doping concentration.

Moreover, our results also suggested another approach, namely dispersion
control. If we can shift the null dispersion point at a large $D$, we should
be able to reduce the ML threshold since we can use a larger cavity.
Throughout this study, we kept $D/d=10$, but we might change this ratio (a
larger $D$ with a smaller $d$) by adjusting our laser reflow condition.

Finally, we would like to emphasize that this design guideline is not
especially for this specific system but could be applied to any microcavity
system with gain and saturable absorption.

##  4 Conclusion

In this paper, we numerically investigated the passive mode-locking of a
toroidal WGM microlaser. The small cavity size means that ultrahigh repetition
rates can be achieved; on the other hand, the gain per roundtrip is minimized,
and therefore ultra-high $Q$ microresonators are necessary. Moreover,
dispersion plays an essential role in mode-locking: a weak anomalous
dispersion promotes ML operation with a limited gain.

In addition, the tradeoff relationship between gain and dispersion was
highlighted due to their dependence on microresonator diameter. As a result, a
300 µm diameter WGM microlaser with a $Q$ slightly higher than $10^{7}$ proves
to be a promising platform for a microlaser with an ultra-high repetition rate
exceeding 100 GHz.

Finally, this design guideline can be applied to any microcavity system that
has gain and saturable absorption.

Acknowledgments This work was supported by JSPS KAKENHI (JP18K19036,
JP19H00873), Amada Foundation, and MEXT Q-LEAP.

Disclosures The authors declare no conflicts of interest.

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