# Theory of Linewidth-Narrowing in Fano Lasers

Yi Yu<EMAIL_ADDRESS>Aref Rasoulzadeh Zali  Jesper Mørk<EMAIL_ADDRESS>DTU
Electro, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
NanoPhoton - Center for Nanophotonics, Technical University of Denmark,
Ørsteds Plads 345A, DK-2800 Kgs. Lyngby, Denmark

###### Abstract

We present a general theory for the coherence of Fano lasers based on a bound
state in the continuum. We find that such lasers enable orders of magnitude
reduction of the quantum-limited linewidth, and by introducing mirror symmetry
breaking, the linewidth can be further reduced, In contrast to ordinary
macroscopic lasers, though, the linewidth may re-broaden due to optical
nonlinearities enhanced by the strong light localization. This leads to the
identification of optimal material systems. We also show that the coherence of
this new type of microscopic laser can be understood intuitively using a
simple, effective potential model. Based on this model, we examine the laser
stability and deduce the dependence of the laser linewidth on the general Fano
lineshape. Our model facilitates the incorporation of other degrees of design
freedom and can be applied to a general class of lasers with strongly
dispersive mirrors.

##  I Introduction

Realizing ultra-coherent nanoscale lasers is important for numerous
applications, including on-chip communications [1], programmable photonic
integrated circuits [2], bio/chemical sensing [3], and quantum and
neuromorphic computing [4, 5]. However, since the quantum-limited laser
linewidth scales inversely with the number of photons in the cavity, the small
mode volume of nanolasers poses a fundamental challenge in realizing highly
coherent microscopic lasers [6, 7, 8, 9, 10]. Indeed, ultra-small metallic
lasers show linewidths of tens of GHz [9], while state-of-the-art
semiconductor lasers [11], albeit with millimeter lengths, can achieve
linewidths less than 1 kHz [12]. Nevertheless, a linewidth of a few MHz was
recently demonstrated in a microlaser by taking advantage of the unusual
properties of a bound state in the continuum (BIC) [13]. One of the laser
mirrors thus uses a Fano resonance to implement the highly dispersive
characteristic of a BIC. This dispersion reduces the laser linewidth by
several orders of magnitude without introducing a large cavity. The concept of
such a Fano laser is generic [14], and in this paper, we present a general
theory for the quantum-limited linewidth of this new type of laser. Compared
to Ref. [13], in which the response function of the Fano mirror is a symmetric
Lorentzian corresponding to the specific case of a Fano-shape parameter $q$=0
[15], we here extend the theory to account for general Fano lineshapes [16,
17, 18]. The new theory can be cast in a form, where the instantaneous laser
frequency behaves analogously to a particle moving in a potential and exposed
to a randomly fluctuating force, representing quantum noise. Such analogy was
initially developed for external cavity lasers [19, 20], which today represent
the most important type of narrow-linewidth laser, but is here extended to the
generic case of a Fano laser. We predict that by introducing mirror symmetry
breaking, further linewidth reduction by a factor of four is possible.
Surprisingly, we find that optical nonlinearities may give rise to a
fundamental limitation to the linewidth.

Our theory not only offers a simple and intuitive explanation of the physics
of linewidth reduction in Fano lasers, but also highlights the qualitative
differences between Fano lasers and external cavity lasers and may inspire
future innovations.

##  II Structure and theoretical model

The Fano laser (Figs. 1(a, b)) is constructed by coupling a discrete mode with
a (quasi-) continuum of modes, which can be implemented in various
configurations. For example, it can be realized in an in-plane design (see
Fig. 1(a)) by a photonic crystal membrane with a line-defect semi-open
waveguide (WG) and a right mirror realized by a nanocavity adjacent to the WG
[21, 22, 23, 24, 25]. A field propagating to the right in the WG can take two
paths. One path follows the WG, while the other comprises tunneling through
the nanocavity. These two paths interfere destructively around the nanocavity
resonance wavelength, leading to a high reflectivity within a narrow
bandwidth, which we refer to as a Fano mirror. A virtual cavity (Fano cavity)
can now be formed between the left mirror and the Fano mirror and has the
characteristics of a BIC. Thus, the mode only forms if the phase condition is
fulfilled (the roundtrip phase change must be an integer multiple of 2$\pi$)
at the resonance of the nanocavity, where the Fano mirror reflectivity is
high. This leads to a sensitive geometry-dependency of the quality factor
($Q$-factor) of the cavity mode, which is a hallmark of a BIC [26]. Such a
Fano laser has been realized using buried-heterostructure technology [13],
where the active material is embedded only in the WG, however, the laser is of
a general nature and can also be realized in other platforms using various
technologies, such as monolithic integration [27, 28], heterogeneous
integration [29], or hybrid integration [30, 31]. In addition to the in-plane
configuration, the Fano laser can also be realized in a vertical configuration
(see Fig. 1(b)), e.g., using a dielectric slab as a broadband mirror and a
photonic crystal slab as the narrowband Fano mirror [32]. As our interest is
on-chip applications, we here focus on the in-plane configuration.

Figure 1: (a) Fano laser realized in an in-plane geometry. A partially
transmitting element (PTE), with a field reflectivity of $r_{B}$, is placed in
the region hosting a continuum of modes. The gain material of the laser is
placed in the continuum region, while the region hosting the discrete mode is
passive. (b) Fano laser realized in a vertical configuration, where a photonic
crystal slab is used as the narrowband Fano mirror, and serves as the
combination of the PTE and discrete state in the photonic crystal band
diagram. The orange shadings illustrate the lasing field, with the darker area
corresponding to stronger field intensity. (c, d) Phase potential $V$ as a
function of instantaneous frequency $\Delta$ for (c) a conventional external
cavity laser and (d) the Fano laser. The laser experiences phase noise
represented by a Langevin force ${{F}_{\Delta}}$. The black curves represent
(c) the external cavity laser and (d) the Fano laser, and the orange curves
represent the Fabry-Perot laser counterpart. For the Fano laser, the solid
(dashed) black curve corresponds to the case without (with) a PTE. Here,
${{\gamma}_{in}}/{{\gamma}_{D}}$ takes the values of 5000 and 78 for (c) and
(d), respectively.

The dynamics of the Fano laser can be described by coupled-mode equations [33]
combined with conventional rate equations [21, 13]. By assuming that the left
mirror has unity reflectivity, we have

$\displaystyle\frac{d}{dt}{{A}^{+}}\left(t\right)$
$\displaystyle=\left(\left(1-j\alpha\right){{G}_{N}}\Delta
N\left(t\right)-{{\gamma}_{in}}\right){{A}^{+}}\left(t\right)$
$\displaystyle+{{\gamma}_{in}}{{A}^{-}}\left(t\right)/{{r}_{R}}\left({{\omega}_{r}}\right)+{{F}_{{{A}^{+}}}}\left(t\right),$
(1) $\displaystyle\frac{d}{dt}N\left(t\right)$
$\displaystyle=R-{{\gamma}_{N}}N\left(t\right)$
$\displaystyle-{{G}_{N}}\left(N\left(t\right)-{{N}_{0}}\right)I\left(t\right)/{{V}_{a}}+{{F}_{N}}\left(t\right),$
(2)

where ${{A}^{+}}\left(t\right)$ (${{A}^{-}}\left(t\right)$) is the slowly
varying complex amplitude of the forward (backward) propagating field,
$\alpha$ is Henry’s factor [34], ${{G}_{N}}$ is the modal gain factor,
$\omega$ (${{\omega}_{r}}$) is the lasing frequency (reference frequency,
e.g., the steady-state lasing frequency excluding quantum noise),
$N\left(t\right)$ is the carrier density in the Fano cavity, ${{N}_{0}}$ is
the carrier density at transparency, $\Delta
N\left(t\right)=N\left(t\right)-{{N}_{s}}$ is the carrier density deviation
from its steady-state value ${{N}_{s}}$, and ${{\gamma}_{N}}$ is the carrier
decay rate. Furthermore, $R$, ${{V}_{a}}$, and ${{\gamma}_{in}}$=1/$\tau_{in}$
denote the pumping rate, the active volume, and the round-trip rate of the
Fano cavity, respectively. The number of photons stored within the Fano cavity
is related to the complex field,
$I(t)={{\varsigma}_{s}}\left({{\omega}_{r}}\right){{\left|{{A}^{+}}\left(t\right)\right|}^{2}}$,
with the conversion factor obtained from the steady-state solution [35],
${{\varsigma}_{s}}\left({{\omega}_{r}}\right)=\left(1-{{\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|}^{2}}\right)/\left(2\hbar{{\omega}_{r}}{{\gamma}_{in}}\ln\left\\{1/\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|\right\\}\right)$.
The backward field is obtained from the nanocavity field,
${{A}_{c}}\left(t\right)$, which in turn is driven by the forward-propagating
field

$\displaystyle{{A}^{-}}\left(t\right)={{r}_{B}}{{A}^{+}}\left(t\right)+\sqrt{2{{\gamma}_{1}}}{{A}_{c}}\left(t\right),$
(3) $\displaystyle\frac{d}{dt}{{A}_{c}}\left(t\right)$
$\displaystyle=\left(-j\left(\delta_{0}+{{\delta}_{NL}}\left(t\right)\right)-{{\gamma}_{t}}\right){{A}_{c}}\left(t\right)$
$\displaystyle+\sqrt{2{{\gamma}_{1}}}{{e}^{2j{{\theta}_{1}}}}{{A}^{+}}\left(t\right)+{{F}_{{{A}_{c}}}}\left(t\right).$
(4)

The nanocavity field is normalized such that
${{\left|{{A}_{c}}\left(t\right)\right|}^{2}}$ is the energy stored in the
nanocavity. The parameter ${{r}_{B}}$ (${{t}_{B}}=\sqrt{1-r_{B}^{2}}$) is the
reflectivity (transmissivity) of the partially transmitting element (PTE in
Fig. 1(a)), which will be discussed later. In addition,
$\delta_{0}={{\omega}_{0}}-{{\omega}_{r}}$ is the detuning between the
nanocavity resonance ${{\omega}_{0}}$ and ${{\omega}_{r}}$,
${{\delta}_{NL}}\left(t\right)$ is the complex change of the nanocavity
resonance due to optical nonlinearities leading to intensity-dependent
detuning and loss [36]. Furthermore, ${{\gamma}_{v}}$ is the nanocavity
intrinsic decay rate, and ${{\gamma}_{1}}$ (${{\gamma}_{2}}$) is the
nanocavity coupling rate to the left (right) side of the WG (Fig. 1(a)), with
${{\gamma}_{c}}={{\gamma}_{1}}+{{\gamma}_{2}}$ and
${{\gamma}_{t}}={{\gamma}_{v}}+{{\gamma}_{c}}$. These (field amplitude) decay
rates are related to the nanocavity intrinsic, coupling, and total $Q$-factors
as $Q_{v}=\omega_{0}/(2\gamma_{v})$, $Q_{c}=\omega_{0}/(2\gamma_{c})$, and
$Q_{t}=\omega_{0}/(2\gamma_{t})$. In Eqs. (1)-(4), $F_{A^{+}}\left(t\right)$,
$F_{A_{c}}\left(t\right)$, and $F_{N}\left(t\right)$ are the Langevin noise
terms of the fields and carrier density. The coefficient
${{e}^{2j{{\theta}_{1}}}}$, depending on the coupling phase, can be derived by
exploiting energy conservation and time-reversal symmetry [17, 37], leading to
$\cos\left(2{{\theta}_{1}}\right)={{\gamma}_{2}}t_{B}^{2}/\left(2{{\gamma}_{1}}{{r}_{B}}\right)-t_{B}^{2}/\left(2{{r}_{B}}\right)-{{r}_{B}}$
and
$\sin\left(2{{\theta}_{1}}\right)=-P{{t}_{B}}\sqrt{4{{\gamma}_{1}}{{\gamma}_{2}}-t_{B}^{2}\gamma_{c}^{2}}/\left(2{{\gamma}_{1}}{{r}_{B}}\right)$.
It should be noted that the transmission through a linear Fano mirror follows
the general expression for a Fano lineshape characterized by the shape
parameter $q$ [38]:
$T\left(\delta_{0}\right)=t_{B}^{2}{{\left(q+\delta_{0}/{{\gamma}_{t}}\right)}^{2}}/\left(1+{{\left(\delta_{0}/{{\gamma}_{t}}\right)}^{2}}\right)$.
Here, $q=-\tan\left(\vartheta\right)$ with
$\vartheta=-P{{\cos}^{-1}}\left(-{{t}_{B}}{{\gamma}_{c}}/\left(2\sqrt{{{\gamma}_{1}}{{\gamma}_{2}}}\right)\right)$.

##  III Potential picture of the Fano laser frequency

For linewidth narrowing, we exploit that the BIC has a large fraction of its
electromagnetic energy stored in the nanocavity. Now, if the gain material of
the laser is excluded from the nanocavity, the field stored in the nanocavity
is not exposed to spontaneous emission and acts as a restoring force that
counteracts the random phase changes induced by spontaneous emission in the
spatial regions containing gain. In this case, by defining the instantaneous
laser frequency $\Delta=\left({{\omega}_{r}}-\omega\right){{\tau}_{D}}$, where
${{\tau}_{D}}$ is the time delay caused by the right mirror, and separating
the amplitude and phase of the laser fields, one can derive from Eqs. (1)-(4)
(see Appendix A) that the laser frequency behaves analogously to a particle
moving with strong friction in a potential $V$ and exposed to random kicks
representing spontaneous emission, cf. Figs. 1(c, d). For a Fano laser, we
have

${{V}}={{V}_{FP}}+{{V}_{FM}},$ (5)

where ${{V}_{FP}}={{\Delta}^{2}}/\left(2{{\tau}_{D}}\right)$ is the potential
of an equivalent Fabry-Perot (FP) laser with cavity length equal to the Fano
cavity, and frequency-independent mirror reflectivities. The other term in Eq.
(5) is ${{V}_{FM}}={{\gamma}_{in}}\ln\left(1+{{\Delta}^{2}}\right)/2$ and
originates from the frequency-dependency of the Fano mirror. The laser
linewidth ($\Delta v$) is represented by the frequency spread resulting from
the random excursions around the potential minimum and is inversely
proportional to the square of the potential curvature, i.e., $1/\Delta
v\propto{{\left({{d}^{2}}V/d{{\Delta}^{2}}\right)}^{2}}$.

This is similar to the case where an FP laser is coupled to a (large) passive
external cavity (Fig. 1(c)), where $V={{V}_{FP}}+{{V}_{EX}}$ and
${{V}_{EX}}=-\kappa{{\gamma}_{in}}\cos\left({{\theta}_{0}}+\Delta\right)$ [20]
with $\kappa$ being the feedback fraction. As seen, an external cavity
introduces a cosine function adding to the original parabolic function of the
FP laser [20]. As a consequence, several potential valleys appear,
corresponding to different external cavity modes that the laser jumps between
on time scales determined by the height of the potential barriers [19]. In
contrast, the Fano laser has a single minimum corresponding to the single BIC.
Furthermore, compared to the cosine function of ${{V}_{EX}}$, ${{V}_{FM}}$ is
a logarithmic function and does not depend on $\kappa$, which is usually much
smaller than unity [20, 39]. Therefore, the Fano laser configuration is much
more effective in narrowing the laser linewidth, and furthermore accomplishes
it without introducing a large external cavity or a large secondary resonator
[29], which would severely increase the footprint of the laser.

If the symmetry of the Fano resonance is changed, which can be achieved by
adding a PTE to change the amplitude and phase of the continuum path [40],
additional linewidth narrowing can be achieved but at the prize of sacrificing
the monostability of the laser (asymmetric potential in Fig. 1(d)). The
detailed picture is that, compared to the case without a PTE, the presence of
the PTE lowers $V$ on one side. This means that the laser tends to have
multiple solutions, rather than being monostable, e.g., the laser can jump to
another solution as the right potential barrier is passed. This new solution
corresponds to $2\omega
nL/c+\arg\left\\{{{r}_{R}}\left(\omega\right)\right\\}=2m\pi$ where $n$ and
$L$ are the refractive index and length of the Fano cavity, respectively, and
$m$ is an integer different from the value corresponding to the original Fano
mode. When the laser oscillation frequency moves away from the nanocavity
resonance, $\left|{{r}_{R}}\left(\omega\right)\right|$ decreases toward 0
($r_{B}$) when the PTE is absent (present), corresponding to a diverging
(finite) laser threshold. Therefore, the PTE increases the chance of mode
hopping into another longitudinal mode of the Fano cavity [24]. However, it
should be noted that the multiple longitudinal modes appearing in a Fano laser
with a PTE cannot be captured here, since our model is derived based on the
expansion around a single longitudinal mode [22].

The field reflectivity of the Fano mirror ${{r}_{R}}\left(\omega\right)$, with
the presence of a PTE, has the general form [37]

$\displaystyle{{r}_{R}}\left(\omega\right)=\left|{{r}_{R}}\left(\omega\right)\right|{{e}^{j{{\phi}_{R}}\left(\omega\right)}}$
$\displaystyle=\frac{jr_{B}^{2}\delta+\left({{\gamma}_{2}}-{{\gamma}_{1}}\right)+r_{B}^{2}{{\gamma}_{v}}-jP{{t}_{B}}\sqrt{4{{\gamma}_{1}}{{\gamma}_{2}}-t_{B}^{2}\gamma_{c}^{2}}}{{{r}_{B}}\left(j\delta+{{\gamma}_{t}}\right)}.$
(6)

Here, $\delta={{\delta}_{0}}+{{\delta}_{NL}}$, with
${{\delta}_{0}}={{\omega}_{0}}-\omega$. The parity of the nanocavity mode with
respect to the mirror plane, cf. Fig. 1(a), is accounted for by the
coefficient, $P$, with $P$=1 (-1) corresponding to an even (odd) mode of the
nanocavity and leading to a red (blue) parity for an asymmetric Fano resonance
with ${{r}_{B}}\neq 0$ [41, 37]. Compared to the case without the PTE, where
the decay ratio ${{R}_{12}}={{\gamma}_{1}}/{{\gamma}_{2}}=1$, the PTE can
break the nanocavity mirror symmetry (${{\gamma}_{1}}/{{\gamma}_{2}}\neq 1$),
e.g., by incorporating a hole in the WG away from the mirror plane [42]. This
can enlarge the phase slope of ${{r}_{R}}\left(\omega\right)$ without
sacrificing the mirror reflectivity
$\left|{{r}_{R}}\left(\omega\right)\right|$ at the nanocavity resonance (see
Fig. 4 in Appendix A), indicating that the laser linewidth can be further
reduced by the inclusion of a PTE.

From the laser model, the lasing frequency $\omega$ fulfills the relation
${{\omega}_{r}}=\omega-\alpha{{\gamma}_{in}}+\alpha\operatorname{Re}\left\\{K{{r}_{R}}\left(\omega\right)\right\\}+\operatorname{Im}\left\\{K{{r}_{R}}\left(\omega\right)\right\\}$
with $K={{\gamma}_{in}}/{{r}_{R}}\left({{\omega}_{r}}\right)$. Based on the
perturbation approach [39], one can derive the Fano laser linewidth above the
threshold

$\displaystyle\Delta{{v}_{FL}}\left({{\omega}_{r}}\right)=\Delta{{v}_{FP}}\left({{\omega}_{r}}\right)/{{\eta}^{2}},$
$\displaystyle\eta=1+{{\gamma}_{in}}{{\left.\left(\alpha\frac{1}{\left|{{r}_{R}}\left(\omega\right)\right|}\frac{\partial}{\partial\omega}\left|{{r}_{R}}\left(\omega\right)\right|+\frac{\partial}{\partial\omega}{{\phi}_{R}}\left(\omega\right)\right)\right|}_{\omega={{\omega}_{r}}}},$
(7)

where
$\Delta{{v}_{FP}}\left({{\omega}_{r}}\right)=\frac{\left(1+{{\alpha}^{2}}\right){{\left({{\gamma}_{i}}+{{\gamma}_{in}}\ln\left\\{1/\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|\right\\}\right)}^{2}}{{C}_{sp}}}{R-{{\gamma}_{N}}{{N}_{th}}\left({{\omega}_{r}}\right)}$
is the linewidth of the equivalent FP laser and $\eta$ is the linewidth
reduction factor. Here, ${{\gamma}_{i}}$ is the loss rate of the active WG,
and ${{C}_{sp}}={{n}_{sp}}/\left(\pi{{V}_{a}}\right)$ is a coefficient
depending on the population inversion factor, ${{n}_{sp}}$. The threshold
carrier density is given by
${{N}_{th}}\left({{\omega}_{r}}\right)={{N}_{0}}+\left({{\gamma}_{i}}+{{\gamma}_{in}}\ln\left\\{1/\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|\right\\}\right)/{{G}_{N}}$.
As seen from Eq. (7), the $\alpha$-parameter can also contribute to linewidth
narrowing, provided it has the same sign as
$\partial\left|{{r}_{R}}\left(\omega\right)\right|/\partial\omega$, which
leads to negative feedback between the laser frequency change and the
refractive index change induced by the change of free carrier density [43].
Eq. (7) is in accordance with the previous result [44] and can be related to
the effective $Q$-factor of the Fano laser.

To evaluate the effective $Q$-factor (${Q}_{FL}$) of the Fano laser, we
consider at the electromagnetic energy stored in the entire Fano laser,
$E_{FL}$, which consists of the energy stored in the Fano cavity, ${{E}_{F}}$,
and the energy stored in the nanocavity, ${{E}_{nc}}$. By neglecting the
waveguide dispersion, the stored energies are proportional to the group delay
of each part [45], ${{\tau}_{in}}$ and ${{\tau}_{D}}$, as

$\displaystyle{{E}_{FL}}={{E}_{F}}+{{E}_{nc}}={{E}_{FP}}\frac{{{\tau}_{in}}+{{\tau}_{nc}}}{{{\tau}_{in}}}$
$\displaystyle={{E}_{FP}}{{\left.\left(1+\frac{\partial}{\partial\omega}\arg\left({{r}_{R}}\left(\omega\right)\right)/\frac{\partial}{\partial\omega}\left(\frac{2L\omega
n}{c}\right)\right)\right|}_{\omega={{\omega}_{r}}}}.$

Here, ${{E}_{FP}}$ is the energy stored in a FP cavity with a dimension
equivalent to the Fano cavity (FP counterpart).
${{\left.\frac{\partial}{\partial\omega}\left(\frac{2L\omega
n}{c}\right)\right|}_{\omega={{\omega}_{r}}}}={{\tau}_{in}}$. The time
averaged power dissipation $P_{s}$ is caused by the propagation loss in the
Fano cavity and the power leakage from the right mirror. The value of this
quantity is the same for both the Fano laser and the FP counterpart, leading
to

$\displaystyle{{Q}_{FL}}={{\omega}_{r}}\frac{{{E}_{FL}}}{P_{s}}={{Q}_{FP}}\left({{\left.1+{{\gamma}_{in}}\frac{\partial}{\partial\omega}\arg\left({{r}_{R}}\left(\omega\right)\right)\right|}_{\omega={{\omega}_{r}}}}\right).$

Here, ${Q}_{FP}$ is the $Q$-factor of the FP cavity laser. Eq. (LABEL:eq:s117)
slightly deviates from the one derived in Ref. [13]. We think this is because
we here take into account the energy (usually negligible) stored in the WG on
the right side of the left reference plane of the Fano mirror, which is also
part of the lasing mode.

In the following, we investigate how the response function of the Fano
resonance affects the laser linewidth. The simulation parameters, unless
specified, are kept fixed: ${{Q}_{c}}$=1000, ${{\omega}_{0}}$=$2\pi
c/{{\lambda}_{0}}$ with ${{\lambda}_{0}}$=1550 nm, $\alpha$=3,
${{\gamma}_{i}}$=4.7$\times{{10}^{10}}\ $s-1, ${{\gamma}_{in}}$=9.5×1012 s-1,
${{\gamma}_{N}}$=4.2$\times$108 s-1, ${{G}_{N}}$=5.2$\times$10-13 m3/s,
$N_{0}$=0.4$\times$1024 m-3, $C_{sp}$=1.1$\times$1020 m-3. These values are in
accordance with the results in Ref. [22, 13]. For simplicity, we assume
negligible intrinsic losses of the isolated nanocavity, corresponding to
${{Q}_{v}}$→$\infty$ (a finite value, $Q_{v}\geq$105, does not change the
picture). In addition, we chose $P$=1, i.e., the Fano resonance has blue
parity, which leads to narrower laser linewidth for the common case in
semiconductors, where $\alpha$ is positive [29, 46].

##  IV Fano laser linewidth in the linear case

We first focus on the linear case where ${{\delta}_{NL}}=0$. As seen from Eq.
(7) and the expression for ${{N}_{th}}\left({{\omega}_{r}}\right)$, the Fano
laser linewidth can, for fixed pumping, be reduced by either reducing the
linewidth of the solitary FP laser by increasing the Fano mirror reflectivity
${{\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|}^{2}}$ (to lower the
laser threshold), or by increasing the linewidth reduction factor by
increasing the (absolute) value of the (normalized) amplitude differential
${{L}_{1}}\left(\delta\right)=2\pi\alpha{{\left|{{r}_{R}}\left(\omega\right)\right|}^{-1}}{{\left.\left(\frac{\partial}{\partial\omega}\left|{{r}_{R}}\left(\omega\right)\right|\right)\right|}_{\omega={{\omega}_{r}}}}$,
or the phase differential
${{L}_{2}}\left(\delta\right)=2\pi{{\left.\left(\frac{\partial}{\partial\omega}{{\phi}_{R}}\left(\omega\right)\right)\right|}_{\omega={{\omega}_{r}}}}$.
All three quantities can be controlled by $\delta$ and ${{R}_{12}}$. In
general, for a given ${{R}_{12}}$, large values for
$\left|{{L}_{1}}\left(\delta\right)\right|$
($\left|{{L}_{2}}\left(\delta\right)\right|$) are found on the low (high)
reflectivity side of the Fano mirror (see Fig. 4 in Appendix A). Examples of
the Fano laser linewidth variation with laser operation frequency are shown in
Fig. 2(a). A pump power of $R={{10}^{35}}$m-3s-1 is applied, which is $\sim$37
dB above the lowest laser threshold obtained for ${{R}_{12}}=1$. Such a high
pump power is beyond (about 10 dB higher than) our current experimental
possibilities, and is not a requirement for achieving narrow linewidth but is
chosen to ensure that lasing can occur even around the minimum reflectivity
coefficient of the Fano mirror. Sub-MHz linewidths can already be achieved at
$R<2\times 10^{33}$m-3s-1, even with a relatively low nanocavity $Q$-factor of
1000.

For $\alpha\neq 0$, the laser linewidth exhibits two local minima. The right
minimum, with a sharp spectral feature, corresponds to the reflectivity
minimum, and the linewidth narrowing here is due mainly to the amplitude-phase
coupling, where a large ${{L}_{1}}\left(\delta\right)$ (because of a large
value of $\partial\left|{{r}_{R}}\left(\omega\right)\right|/\partial\omega$
accompanied by a small value of $\left|{{r}_{R}}\left(\delta\right)\right|$)
provides negative feedback, suppressing laser frequency fluctuations. The left
local linewidth minimum, with a broader spectral feature, occurs close to the
reflectivity maximum of the Fano mirror and is due to a combination of low
threshold (high $\left|{{r}_{R}}\left(\omega\right)\right|$) and prolonged
photon storage in the passive nanocavity (large
${{L}_{2}}\left(\delta\right)$). For this case, the influence of
${{L}_{1}}\left(\delta\right)$ is negligible.

The laser linewidth also assumes a local maximum, which is blue or red detuned
with respect to the linewidth minimum, depending on the sign of $\alpha$. Such
a maximum corresponds to $\eta=0$, where
$L_{1}\left(\delta\right)=-\left(2\pi/{{\gamma}_{in}}+{{L}_{2}}\left(\delta\right)\right)$.

Figure 2: (a) Calculated Fano laser linewidths $\Delta{{v}_{FL}}$ (black, red,
and blue curves) versus (normalized) lasing frequency for different values of
the $\alpha$-parameter. The frequency dependence of the Fano mirror
reflectivity and phase are also shown (upper gray and green curves). The solid
(dashed) parts of the curves represent stable (unstable) solutions. Here,
$r_{B}^{2}$=0.8 and $R_{12}$=1. (b) Phase potential $V$ as a function of
instantaneous frequency $\Delta$ (left) and the resulting temporal trajectory
of the complex field $A^{+}$ of the Fano laser (right), corresponding to the
spectral positions I, II, and III for $\alpha$=3 in (a). The trajectory of
$A^{+}$ in case II will evolve into a ”fuzzy” circle as the simulation time is
prolonged. The trajectory of $A^{+}$ in case III corresponds to self-
pulsations. (c) The minimum Fano laser linewidth and (d) the corresponding
lasing frequency $\omega_{m}$ as a function of $r_{B}$ and $R_{12}$ for
$\alpha$=3. The grey parameter regions are not accessible. In all the cases, a
high pump power of $R={{10}^{35}}$m-3s-1 is assumed.

Here, we will focus on the good-cavity case, where the $Q$-factor is high, so
that the Peterman-factor [47] is close to unity and the dispersion of the gain
material can be neglected; these effects will become important in the bad-
cavity case and lead to correction factors of Eq. (7) [48]. It should also be
pointed out that Eq. (7) only is valid for stable solutions. The laser
stability can be checked by a linear stability analysis or direct numerical
simulations including Langevin terms (see Appendix C), and can for certain
kinds of instability (saddle-point instabilities) also be inferred from the
laser potential [20]. Large fluctuations of the Langevin force may cause the
laser to switch from one local potential minimum to another minimum or cause
switching to another solution, which is not described by the same effective
potential, similar to the case of external cavity lasers [49]. Considering
that the right local minimum corresponds to a very large laser threshold and
can lead to laser instabilities, we in the following focus only on the left
minimum.

Figs. 2(c, d) show, in color, the minimum linewidth and the corresponding
lasing frequency ${{\omega}_{m}}$ versus $r_{B}$ and $R_{12}$. The grey
regions cannot be accessed since $R_{12}$ is bounded by
${{R}_{\min}}\leq{{R}_{12}}\leq{{R}_{\max}}$, where
${{R}_{\max}}=\left(1+{{r}_{B}}\right)/\left(1-{{r}_{B}}\right)$ and
${{R}_{\min}}=1/{{R}_{\max}}$ [50]. As seen, the smallest linewidth always
occurs at ${{\lambda}_{m}}\approx{{\lambda}_{0}}$ with
${{R}_{12}}={{R}_{\max}}$, corresponding to the upper edge of the contour of
Fig. 2(c) (and $q$ = 0 for the general Fano formula). This reflects that the
main factors contributing to the laser linewidth are
$\left|{{r}_{R}}\left(\delta\right)\right|$ and
${{L}_{2}}\left(\delta\right)$, which are large at $\delta=0$ in the case of
${{R}_{12}}={{R}_{\max}}$ (see Fig. 4 in Appendix A).

Next, we focus on highly asymmetric structures, i.e., the upper edge of the
contour of Fig. 2(c). Based on Eq. (7), utilizing $\delta\ll{{\gamma}_{c}}$
and ${{R}_{12}}={{R}_{\max}}$, a simple expression for the Fano laser
linewidth can be derived

$\Delta{{v}_{FL}}\left({{\omega}_{m}}\right)={{\left(\frac{1}{1+\left(1+{{r}_{B}}\right)\left({{\gamma}_{in}}/{{\gamma}_{t}}\right)}\right)}^{2}}\Delta{{v}_{FP}},$
(9)

where

${{\omega}_{m}}={{\omega}_{0}}\left(1-\frac{\alpha{{\gamma}_{i}}}{t_{B}^{2}{{\omega}_{0}}\left(1+\left(1+{{r}_{B}}\right)\left({{\gamma}_{in}}/{{\gamma}_{t}}\right)\right)}\right)\approx{{\omega}_{0}}.$
(10)

Eq. (9) shows that compared to the corresponding FP laser, with a linewidth of
$\Delta{{v}_{FP}}\approx\left(1+{{\alpha}^{2}}\right){{C}_{sp}}\gamma_{i}^{2}/R$
under high-pumping assumption, where
$R-{{\gamma}_{N}}{{N}_{th}}\left({{\omega}_{r}}\right)\approx R$, the Fano
mirror can significantly improve the laser coherence. Such an improvement
becomes more pronounced as ${{\gamma}_{in}}$ (${{\gamma}_{t}}$) increases
(decreases), e.g., by reducing the Fano cavity size or improving the
nanocavity $Q$-factor. Compared to conventional external cavity lasers [39,
51], which are based on weak feedback with a linewidth of
$\Delta{{v}_{ext}}={{\left(1/\left(1+\kappa\left({{\gamma}_{in}}/{{\gamma}_{t}}\right)\right)\right)}^{2}}\Delta{{v}_{FP}}$,
where $\kappa\ll 1$, the Fano laser intrinsically operates in the regime of
strong feedback and thereby enables a much larger linewidth reduction without
significantly increasing the size of the laser. At the same time, the Fano
laser does not experience any modal doublets, in contrast to ordinary strong
injection-locking or feedback systems [46].

For ultrasmall lasers, where ${{\gamma}_{in}}/{{\gamma}_{T}}\gg 1$, Eq. (9)
shows that by breaking the mirror symmetry,
$\Delta{{v}_{FL}}\left({{\omega}_{m}}\right)$ can be further reduced by a
(maximum) factor of four, compared to the ordinary symmetric case
(${{R}_{12}}=1$). This is because a higher $r_{B}$ enables a larger
${{R}_{\max}}$ and thus a larger ${{\gamma}_{1}}$ for a fixed $Q_{c}$, leading
to a larger $\left|{{A}_{c}}\right|$ [52] (compared to the case of $r_{B}$=0,
the nanocavity field gets doubled under the condition of $r_{B}$=1 and
${{R}_{12}}={{R}_{\max}}$). This can also be understood in another way: to
achieve an effective Fano destructive interference at the output of the WG,
the decay of the nanocavity field to the right side,
$\left|\sqrt{2{{\gamma}_{2}}}{{A}_{c}}\right|$, should be balanced by the
field $\left|j{{t}_{B}}{{A}^{+}}\right|$ transmitted directly through the WG.
Therefore, a smaller ${{\gamma}_{2}}$, a larger $\left|{{A}_{c}}\right|$, or a
stronger field localization in the passive nanocavity region. This enhances
the laser’s composite $Q$-factor. This is consistent with Figs. 4(a, b), in
which ${{\left|{{r}_{R}}\left({{\omega}_{0}}\right)\right|}^{2}}$ approaches
unity with the frequency slope doubled as ${{R}_{12}}\to{{R}_{\max}}$.

In the case of a high PTE reflectivity, the Fano laser may appear to be
equivalent to a system of two coupled cavities. However, in contrast to the
case of two coupled cavities, the Fano laser mode still bears the
characteristics of a BIC, even for $r_{B}$=1, where the PTE completely blocks
the right end of the WG. This can be concluded by analyzing the Fano mode as a
superposition of two coupled modes (see Appendix B).

The linewidth expressed by Eq. (9) is identical to the result derived using
the Langevin approach (see Appendix C), where stochastic Langevin noises are
introduced for the Fano cavity field ${{A}^{+}}\left(t\right)$. The absolute
output power is, of course, also important. It can be shown that the external
quantum efficiency is much higher for the cross-port [24] and the left mirror
rather than the port involving transmission through the Fano mirror (see
Appendix D).

##  V Fano laser linewidth in the nonlinear case

The theory and results presented so far assumed the nanocavity to have a
linear response, i.e., ${{\delta}_{NL}}\left(t\right)$=0. However, the
spatially localized field in the nanocavity of the Fano laser can induce a
large power-dependent change in the laser output by spectrally shifting or
changing the amplitude of ${{r}_{R}}\left({{\omega}_{r}}\right)$ through
optical nonlinearities [36]. We incorporate nonlinear absorption and index
changes by taking

$\displaystyle{{\delta}_{NL}}\left(t\right)=\left({{K}_{K}}-j{{K}_{T}}\right){{\left|{{A}_{c}}\left(t\right)\right|}^{2}}+\left({{K}_{D}}-j{{K}_{A}}\right){{N}_{c}}\left(t\right).$
(11)

Here, ${{K}_{K}}{{\left|{{A}_{c}}\left(t\right)\right|}^{2}}$ and
${{K}_{D}}{{N}_{c}}\left(t\right)$ account for the nanocavity resonance shifts
due to Kerr and free-carrier effects, with ${{K}_{K}}$ being the Kerr
coefficient, and ${{K}_{D}}$ accounting for free-carrier dispersion and
bandfilling [36]. The terms
${{K}_{T}}{{\left|{{A}_{c}}\left(t\right)\right|}^{2}}$ and
${{K}_{A}}{{N}_{c}}\left(t\right)$ account for absorption, with ${{K}_{T}}$
being the two-photon absorption (TPA) coefficient and ${{K}_{A}}$ the free
carrier absorption coefficient. Furthermore, ${{N}_{c}}\left(t\right)$ is the
mode-averaged free carrier density generated by TPA in the nanocavity,
governed by

$\displaystyle\partial{{N}_{c}}\left(t\right)/\partial
t=-{{\gamma}_{nc}}{{N}_{c}}\left(t\right)+{{G}_{T}}{{\left|{{A}_{c}}\left(t\right)\right|}^{4}},$
(12)

where ${{\gamma}_{nc}}$ is the effective carrier decay rate in the nanocavity
and ${{G}_{T}}$ is the free carrier generation coefficient due to TPA. The
coefficients ${{K}_{K}}$, ${{K}_{T}}$, ${{G}_{T}}$ and ${{\gamma}_{nc}}$
depend on the nonlinear optical mode volumes of the nanocavity [36]. We
compare photonic crystal L7 nanocavities made of InP, Si, and SiN working at
$\sim$1.55 µm, and choose ${{\gamma}_{in}}$=1.9$\times$1012 s-1 and
${{\gamma}_{nc}}=2\times{{10}^{10}}$s-1. We calculate the linewidth including
nonlinearities also using the potential approach (see Eqs. (18) and (19) in
Appendix A), with parameters for the three material systems considered (see
TABLE I in Appendix C), which agrees well with the conventional Langevin
approach (see Appendix C).

Figure 3: (a) Fano laser linewidth
$\Delta{{v}_{FL}}\left({{\omega}_{0}}\right)$ (color scale) as a function of
$r_{B}$ and ${{R}_{12}}$ when nanocavity nonlinearities are accounted for.
Here, the nanocavity is made of InP, and a relatively high pump power of
$R$=1035 m-3s-1 is chosen to better illustrate the nonlinear effects. (b) The
minimum linewidth extracted from Fig. 3(a) (red curve) and Fig. 2(c) (black
curve) as a function of $r_{B}$. The Fano mirror reflectivity
${{\left|{{r}_{R}}\left({{\omega}_{0}}\right)\right|}^{2}}$ (blue curve)
corresponding to the red curve is also plotted. (c) Variation of the Fano
mirror reflectivity with pump power. The pump power has been normalized, with
0 dB corresponding to the threshold. Here $r_{B}$=0 and the nanocavity is made
of InP. (d) Fano laser linewidth versus pump power. The red, green, and orange
curves correspond to the case where nanocavity nonlinearities are accounted
for, and the nanocavity is made of InP, Si, and SiN, respectively. The black
and blue curves correspond to the linear case (nanocavity nonlinearities are
absent), and ${{\left|{{r}_{R}}\left({{\omega}_{0}}\right)\right|}^{2}}$ of
the blue curve is set identical to that of the red curve. Here $r_{B}$=0.

To simplify the situation, we assume that the nanocavity resonance is tuned so
that the nonlinear resonance shift is always compensated at steady-state,
i.e., $\operatorname{Re}\left\\{{{\delta}_{NL}}\right\\}=0$. In practice, this
can be implemented by placing electrodes close to the nanocavity for
temperature or electric-field tuning. Here, considering the lack of analytical
solutions and thus more time-consuming computations when including optical
nonlinearities, we focus on the close-to-optimum point,
$\omega_{r}=\omega_{0}$. With nanocavity nonlinearities, Fig. 3(a), the Fano
laser linewidth exhibits similar dependence on the Fano line shape as in the
linear case; i.e., the narrowest linewidth is still located close to
${{R}_{\max}}$. The smallest linewidth in the linear case (black curve in Fig.
3(b)) agrees perfectly with the approximate solution (Eq. (9)).

The linewidth obtained when considering optical nonlinearities is larger than
the linear one and does not depend monotonically on $r_{B}$. This can be
attributed to two factors: 1) the reduction of the Fano mirror reflectivity
caused by nonlinear absorption in the nanocavity, which lowers the $Q$-factor
of the laser; 2) the additional Langevin noise introduced by nanocavity
nonlinearities causes additional phase fluctuations. To determine which factor
is dominant, we examine
${{\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|}^{2}}$ (see the blue
curve in Fig. 3(b)). As seen,
${{\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|}^{2}}$ varies almost
oppositely to the laser linewidth with $r_{B}$, indicating that the reduced
Fano mirror reflectivity plays an important role. A reduced mirror
reflectivity at large $r_{B}$ is ascribed to the fact that a higher $r_{B}$
can enable a larger ${{R}_{\max}}$ and thus stronger field storage in the
nanocavity, resulting in higher nanocavity absorption. Indeed, the Fano mirror
reflectivity decreases dramatically for high pump powers (Fig. 3(c)).
Therefore, unlike the linear case, the laser linewidth in the nonlinear case
suffers from a trade-off between enhanced field localization and enhanced
nanocavity absorption. This trade-off means that the linewidth does not follow
the inverse-power dependence predicted by the Schawlow-Townes formula [53]
(Fig. 3(d)). The linewidth saturates and eventually increases with increasing
pump power. Simulations of other nanocavities, e.g., the H0 type [36] having a
smaller mode volume and a faster carrier decay rate, give qualitatively the
same result. It should be noted that the nonlinear result, e.g., the red curve
in Fig. 3(d)), agrees well with the linear result when accounting for the
power-dependence of
${{\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|}^{2}}$ (blue curve in
Fig. 3(d)), further confirming that the reduction of the reflectivity due to
nonlinear absorption is the dominant effect causing linewidth rebroadening.
Fig. 3(d) predicts that for an InP nanocavity, it may be difficult to reach
sub-MHz linewidth. However, the problem is significantly reduced by using a
material with smaller nonlinear loss, such as Si or SiN (see the green and
orange curves in Fig. 3(d)). Such structures can be realized using
heterogeneous integration technology [29, 54]. We also find that nonlinear
effects become orders of magnitude weaker if the nanocavity of the Fano mirror
is replaced by a much larger cavity.

##  VI Summary and outlook

In summary, we have presented a general theory of the quantum-limited
linewidth of a Fano laser based on a bound state in the continuum. In
particular, we have developed a potential picture valid for lasers with
strongly dispersive laser mirrors. Based on the theory, we show that the Fano
laser allows orders-of-magnitude linewidth reduction without compromising the
monostability and without significantly increasing the size of the laser. This
enables microscopic lasers featuring similar linewidth-narrowing as
conventional macroscopic external cavity lasers, but with different modal
properties. By breaking the symmetry of the Fano mirror, we find that the Fano
laser linewidth can be reduced by an additional factor of four. This
improvement, however, may be compromised by optical nonlinearities that limit
the minimum linewidth obtained for a given material system.

Our theory provides new insights into the stability and coherence of
microscopic lasers. The potential model accounts for global dynamics of the
system that cannot be inferred from small signal analysis. The model thus
facilitates the incorporation of other degrees of design freedom and physics,
exemplified by the mirror symmetry breaking and optical nonlinearities
considered here, which have been largely neglected in previous linewidth
investigations. Therefore, the developed theory can be used to investigate
other configurations, e.g., considering other dispersive laser mirrors, such
as those enabled by multiple Fano resonances, Autler-Townes splitting, or
electromagnetically induced transparency [14, 55, 16, 56].

###### Acknowledgements.

The authors acknowledge the support of the Danish National Research Foundation
via NanoPhoton – Center for Nanophotonics (Grant No. DNRF147), and the
European Research Council (ERC) under the European Union Horizon 2020 Research
and Innovation Program (Grant No. 834410 Fano). Y. Y. acknowledges the support
from Villum Fonden via the Young Investigator Program (Grant No. 42026).

##  Appendix A The effective potential of lasers with dispersive mirror

By separating the amplitude and phase of ${{A}^{+}}\left(t\right)$ in Eq. (1),
i.e.,
${{A}^{+}}\left(t\right)=\left|{{A}^{+}}\left(t\right)\right|{{e}^{j{{\phi}^{+}}\left(t\right)}}$,
we arrive at two differential equations

$\displaystyle\frac{d}{dt}\left|{{A}^{+}}\left(t\right)\right|$
$\displaystyle={{G}_{N}}\Delta
N\left(t\right)\left|{{A}^{+}}\left(t\right)\right|-{{\gamma}_{in}}\left|{{A}^{+}}\left(t\right)\right|$
$\displaystyle+\operatorname{Re}\left\\{K{{A}^{-}}\left(t\right)/{{A}^{+}}\left(t\right)\right\\}\left|{{A}^{+}}\left(t\right)\right|+{{F}_{\left|{{A}^{+}}\right|}}\left(t\right),$
(13) $\displaystyle\frac{d}{dt}{{\phi}^{+}}\left(t\right)$
$\displaystyle=-\alpha{{G}_{N}}\Delta N\left(t\right)$
$\displaystyle+\operatorname{Im}\left\\{K{{A}^{-}}\left(t\right)/{{A}^{+}}\left(t\right)\right\\}+{{F}_{{{\phi}^{+}}}}\left(t\right),$
(14)

where ${{F}_{\left|{{A}^{+}}\right|}}\left(t\right)$,
${{F}_{{{\phi}^{+}}}}\left(t\right)$ are the Langevin noise terms of the field
amplitude and phase. The steady-state, in the absence of noise, is found by
solving $d\left|{{A}^{+}}\left(t\right)\right|/dt=0$ and
$d{{\phi}^{+}}\left(t\right)/dt={{\omega}_{r}}-\omega$, leading to
$\alpha{{G}_{N}}\Delta
N=\alpha{{\gamma}_{in}}-\alpha\operatorname{Re}\left\\{K{{r}_{R}}\left(\omega\right)\right\\}$
and

$\displaystyle{{\omega}_{r}}=\omega-\alpha{{\gamma}_{in}}+\alpha\operatorname{Re}\left\\{K{{r}_{R}}\left(\omega\right)\right\\}+\operatorname{Im}\left\\{K{{r}_{R}}\left(\omega\right)\right\\}.$
(15)

Here, ${{r}_{R}}\left({{\omega}_{r}}\right)={{A}^{-}}/{{A}^{+}}$, and $\omega$
is the oscillation frequency of the entire laser system. Based on the small
perturbation approach, by using
${{r}_{R}}\left(\omega\right)=\left|{{r}_{R}}\left(\omega\right)\right|{{e}^{j{{\phi}_{R}}\left(\omega\right)}}$,
from Eq. (15) one finds the following relation between changes in
${\omega}_{r}$ and ${\omega}$

$\displaystyle\Delta{{\omega}_{r}}=$
$\displaystyle\Delta\omega+{\gamma}_{in}{{\left.\left(\alpha\frac{1}{\left|{{r}_{R}}\left(\omega\right)\right|}\frac{\partial}{\partial\omega}\left|{{r}_{R}}\left(\omega\right)\right|+\frac{\partial}{\partial\omega}{{\phi}_{R}}\left(\omega\right)\right)\right|}_{{{\omega}_{r}}}}\Delta\omega.$
(16)

If the laser has a dispersionless mirror, i.e,
$\partial\left|{{r}_{R}}\left(\omega\right)\right|/\partial\omega,\
\partial{{\phi}_{R}}\left(\omega\right)/\partial\omega=0$, this reduces to the
case of the equivalent FP laser, with mirrors that have the same reflectivity
as the Fano laser evaluated at its operation point but with no frequency
dependence, and a cavity length given by that of the Fano cavity (see Fig.
1(a)). In this case, one has $\Delta{{\omega}_{r}}=\Delta\omega$. The factor
by which the linewidth of the composite cavity laser is reduced compared to
the FP laser counterpart is given by the ratio
${{\left(\Delta\omega/\Delta{{\omega}_{r}}\right)}^{2}}$ [39], where
$\Delta\omega$ is the change of the oscillation frequency of the composite
laser system upon a change $\Delta{{\omega}_{r}}$ of the oscillation frequency
of the corresponding FP laser. Using Eq. (16), one gets

$\displaystyle\frac{\Delta{{v}_{FL}}}{\Delta{{v}_{FP}}}$
$\displaystyle={{\left(\frac{\Delta\omega}{\Delta{{\omega}_{r}}}\right)}^{2}}=\frac{1}{{\eta}^{2}}$
$\displaystyle=1/{{\left(1+\frac{{{\gamma}_{in}}}{2\pi}\left({{L}_{1}}\left(\delta\right)+{{L}_{2}}\left(\delta\right)\right)\right)}^{2}},$
(17)

where
${{L}_{1}}\left(\delta\right)=2\pi\alpha{{\left|{{r}_{R}}\left(\omega\right)\right|}^{-1}}{{\left.\frac{\partial}{\partial\omega}\left|{{r}_{R}}\left(\omega\right)\right|\right|}_{\omega={{\omega}_{r}}}}$
is the normalized amplitude derivative and
${{L}_{2}}\left(\delta\right)=2\pi{{\left.\frac{\partial}{\partial\omega}{{\phi}_{R}}\left(\omega\right)\right|}_{\omega={{\omega}_{r}}}}$
is the normalized phase derivative. For the Fano laser with the Fano mirror
reflectivity of Eq. (6), we have

$\displaystyle\frac{\partial}{\partial\omega}{{\phi}_{R}}\left(\omega\right)$
$\displaystyle=\left(-\frac{{{\delta}_{r}}\left(\omega\right)}{{{\delta}_{r}}{{\left(\omega\right)}^{2}}+{{\left({{\delta}_{i}}\left(\omega\right)-{{\gamma}_{t}}\right)}^{2}}}-\frac{r_{B}^{2}\left(P{S}_{e}-r_{B}^{2}{{\delta}_{r}}\left(\omega\right)\right)}{{{\left(P{S}_{e}-r_{B}^{2}{{\delta}_{r}}\left(\omega\right)\right)}^{2}}+{{\left(r_{B}^{2}\left({{\delta}_{i}}\left(\omega\right)-{{\gamma}_{v}}\right)-{{\gamma}_{2}}+{{\gamma}_{1}}\right)}^{2}}}\right)\frac{\partial}{\partial\omega}{{\delta}_{i}}\left(\omega\right)$
$\displaystyle+\left(\frac{{{\delta}_{i}}\left(\omega\right)-{{\gamma}_{t}}}{{{\delta}_{r}}{{\left(\omega\right)}^{2}}+{{\left({{\delta}_{i}}\left(\omega\right)-{{\gamma}_{t}}\right)}^{2}}}-\frac{r_{B}^{2}\left(r_{B}^{2}\left({{\delta}_{i}}\left(\omega\right)-{{\gamma}_{v}}\right)-{{\gamma}_{2}}+{{\gamma}_{1}}\right)}{{{\left(P{S}_{e}-r_{B}^{2}{{\delta}_{r}}\left(\omega\right)\right)}^{2}}+{{\left(r_{B}^{2}\left({{\delta}_{i}}\left(\omega\right)-{{\gamma}_{v}}\right)-{{\gamma}_{2}}+{{\gamma}_{1}}\right)}^{2}}}\right)\frac{\partial}{\partial\omega}{{\delta}_{r}}\left(\omega\right),$
(18)

and

$\displaystyle{{\left|{{r}_{R}}\left(\omega\right)\right|}^{-1}}\frac{\partial}{\partial\omega}\left|{{r}_{R}}\left(\omega\right)\right|$
$\displaystyle=\left(-\frac{{{\delta}_{r}}\left(\omega\right)}{{{\delta}_{r}}{{\left(\omega\right)}^{2}}+{{\left({{\delta}_{i}}\left(\omega\right)-{{\gamma}_{t}}\right)}^{2}}}-\frac{r_{B}^{2}\left(P{S}_{e}-r_{B}^{2}{{\delta}_{r}}\left(\omega\right)\right)}{{{\left(P{S}_{e}-r_{B}^{2}{{\delta}_{r}}\left(\omega\right)\right)}^{2}}+{{\left(r_{B}^{2}\left({{\delta}_{i}}\left(\omega\right)-{{\gamma}_{v}}\right)-{{\gamma}_{2}}+{{\gamma}_{1}}\right)}^{2}}}\right)\frac{\partial}{\partial\omega}{{\delta}_{r}}\left(\omega\right)$
$\displaystyle-\left(\frac{{{\delta}_{i}}\left(\omega\right)-{{\gamma}_{t}}}{{{\delta}_{r}}{{\left(\omega\right)}^{2}}+{{\left({{\delta}_{i}}\left(\omega\right)-{{\gamma}_{t}}\right)}^{2}}}-\frac{r_{B}^{2}\left(r_{B}^{2}\left({{\delta}_{i}}\left(\omega\right)-{{\gamma}_{v}}\right)-{{\gamma}_{2}}+{{\gamma}_{1}}\right)}{{{\left(P{S}_{e}-r_{B}^{2}{{\delta}_{r}}\left(\omega\right)\right)}^{2}}+{{\left(r_{B}^{2}\left({{\delta}_{i}}\left(\omega\right)-{{\gamma}_{v}}\right)-{{\gamma}_{2}}+{{\gamma}_{1}}\right)}^{2}}}\right)\frac{\partial}{\partial\omega}{{\delta}_{i}}\left(\omega\right),$
(19)

where ${{\delta}_{r}}\left(\omega\right)$
(${{\delta}_{i}}\left(\omega\right)$) is the real (imaginary) part of
$\delta+{{\delta}_{NL}}\left(\omega\right)$, and its specific form depends on
the type of optical nonlinearities in the nanocavity,
${\delta}_{NL}\left(\omega\right)$.

In the linear case where ${{\delta}_{NL}}=0$
(${{\delta}_{r}}\left(\omega\right)={{\omega}_{0}}-\omega={{\omega}_{0}}-{{\omega}_{r}}$,
${{\delta}_{i}}\left(\omega\right)=0$), the above equations reduce to

$\frac{\partial}{\partial\omega}{{\phi}_{R}}\left(\omega\right)=\frac{{{\gamma}_{t}}}{{{\delta}^{2}}+\gamma_{t}^{2}}-\frac{r_{B}^{2}\left(r_{B}^{2}{{\gamma}_{v}}+{{\gamma}_{2}}-{{\gamma}_{1}}\right)}{{{\left(P{S}_{e}-r_{B}^{2}\delta\right)}^{2}}+{{\left(r_{B}^{2}{{\gamma}_{v}}+{{\gamma}_{2}}-{{\gamma}_{1}}\right)}^{2}}},$
(20)

and

$\displaystyle{{\left|{{r}_{R}}\left(\omega\right)\right|}^{-1}}\frac{\partial}{\partial\omega}\left|{{r}_{R}}\left(\omega\right)\right|$
$\displaystyle=\frac{\delta}{{{\delta}^{2}}+\gamma_{t}^{2}}+\frac{r_{B}^{2}\left(P{S}_{e}-r_{B}^{2}\delta\right)}{{{\left(P{S}_{e}-r_{B}^{2}\delta\right)}^{2}}+{{\left(r_{B}^{2}{{\gamma}_{v}}+{{\gamma}_{2}}-{{\gamma}_{1}}\right)}^{2}}},$
(21)

in which
${S}_{e}={{t}_{B}}\sqrt{4{{\gamma}_{1}}{{\gamma}_{2}}-t_{B}^{2}{{\left({{\gamma}_{1}}+{{\gamma}_{2}}\right)}^{2}}}$.
The laser stability can be investigated through a conventional small-signal
analysis, or investigated in the time-domain, where we solve Eqs. (1)-(4)
numerically by treating the Langevin noise terms as random sources with normal
distribution [57]. From Eqs. (13) and (14), and neglecting amplitude
fluctuations, one gets

$\displaystyle\frac{d}{dt}{{\phi}^{+}}\left(t\right)$
$\displaystyle=-\alpha\left({{\gamma}_{in}}-\operatorname{Re}\left\\{K{{r}_{R}}\left(\omega\right)\right\\}\right)+\operatorname{Im}\left\\{K{{r}_{R}}\left(\omega\right)\right\\}+{{F}_{{{\phi}^{+}}}}\left(t\right)$
$\displaystyle=-\alpha{{\gamma}_{in}}+{{\gamma}_{in}}\left|\frac{{{r}_{R}}\left(\omega\right)}{{{r}_{R}}\left({{\omega}_{r}}\right)}\right|(\alpha\cos\left({{\phi}_{R}}\left(\omega\right)-{{\phi}_{R}}\left({{\omega}_{r}}\right)\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\sin\left({{\phi}_{R}}\left(\omega\right)-{{\phi}_{R}}\left({{\omega}_{r}}\right)\right))+{{F}_{{{\phi}^{+}}}}\left(t\right).$
(22)

Figure 4:  (a) The Fano mirror reflectivity
(${{\left|{{r}_{R}}\left(\delta\right)\right|}^{2}}$) for different
reflectivities of the PTE ($r_{B}^{2}$) and different decay ratios
(${{R}_{12}}={{\gamma}_{1}}/{{\gamma}_{2}}$). Here,
$\delta={{\omega}_{0}}-\omega$, with ${{\omega}_{0}}$ being the nanocavity
resonant frequency. (b) The phase change of the Fano mirror reflection
($\arg\left\\{{{r}_{R}}\left(\delta\right)\right\\}$) for different
reflectivities of the PTE and the decay ratios (the phase change was
normalized so that $\arg\left\\{{{r}_{R}}\left(0\right)\right\\}=0$). (c) The
mirror reflectivity ${{\left|{{r}_{R}}\left(\delta\right)\right|}^{2}}$, (d)
normalized amplitude derivative ${{L}_{1}}\left(\delta\right)$, and (e)
normalized phase derivative ${{L}_{2}}\left(\delta\right)$ of the Fano mirror,
as a function of $\delta/{{\gamma}_{t}}$ and ${{R}_{12}}$. Here
$r_{B}^{2}$=0.5 and $\alpha$=3. In (d) and (e), the ranges of color scales are
restricted to the interval from 0 to 5$\times$10-11 s for a clearer
illustration.

Next, we extend the approach taken in Ref. [19] for external cavity lasers to
the general scenario. We introduce ${{\phi}^{+}}\left(t-{{\tau}_{D}}\right)$
which is the phase delayed by a time, $\tau_{D}$. Such a delay is caused by
field dwelling in the external cavity in conventional external cavity lasers
or field storage in the nanocavity in the Fano laser. For example,
${{\tau}_{D}}=1/{{\gamma}_{D}}=\partial{{\phi}_{R}}\left(\omega\right)/\partial\omega=1/{{\gamma}_{t}}$
is the time delay at ${{\omega}_{r}}={{\omega}_{0}}$ when ${{\gamma}_{v}}=0$
and ${{\gamma}_{2}}={{\gamma}_{1}}$. By defining
$\Delta={{\phi}^{+}}\left(t\right)-{{\phi}^{+}}\left(t-{{\tau}_{D}}\right)$,
using $\Delta\approx\left({{\omega}_{r}}-\omega\right){{\tau}_{D}}$ and
$\left\langle\partial{{\phi}^{+}}\left(t-{{\tau}_{D}}\right)/\partial
t\right\rangle\approx{{\gamma}_{D}}\Delta$, we derive from Eq. (22) the
following equation for $\Delta$

$\displaystyle\frac{d}{dt}\Delta$
$\displaystyle=\frac{d}{dt}{{\phi}^{+}}\left(t\right)-\frac{d}{dt}{{\phi}^{+}}\left(t-{{\tau}_{D}}\right)$
$\displaystyle=\frac{d}{dt}{{\phi}^{+}}\left(t\right)-\left(\left\langle\frac{d}{dt}{{\phi}^{+}}\left(t-{{\tau}_{D}}\right)\right\rangle+{{F}_{{{\phi}^{+}}}}\left(t-{{\tau}_{D}}\right)\right)$
$\displaystyle=-\frac{dV}{d\Delta}+{{F}_{\Delta}}\left(t\right),$ (23)

where

$\displaystyle\frac{dV}{d\Delta}$
$\displaystyle=\Delta{\gamma_{D}}+\alpha{\gamma_{in}}$
$\displaystyle-{\gamma_{in}}\left|{r_{R}}\left(\omega\right)/{r_{R}}\left({\omega_{r}}\right)\right|(\alpha\cos\left({\phi_{R}}\left(\omega\right)-{\phi_{R}}\left({\omega_{r}}\right)\right)$
$\displaystyle+\sin\left({\phi_{R}}\left(\omega\right)-{\phi_{R}}\left({\omega_{r}}\right)\right),$
(24)

and
${F_{\Delta}}\left(t\right)={F_{\phi^{+}}}\left(t\right)-{F_{\phi}^{+}}\left(t-{{\tau}_{D}}\right)$
is a Langevin term with a correlation strength
$\left\langle{F_{\Delta}}\left(t\right){F_{\Delta}}\left(t^{\prime}\right)\right\rangle=4\pi\Delta{v_{FP}}\left({{\omega}_{r}}\right)\delta\left(t-t^{\prime}\right)$.
Eq. (23) is analogous to the equation of motion for a particle with coordinate
$\Delta$ moving with strong friction in a potential $V$ and exposed to a
fluctuating force ${F_{\Delta}}\left(t\right)$ [20]. The coordinate $\Delta$
plays the role of the instantaneous laser frequency as well as the phase
change over the time interval ${{\tau}_{D}}$. Taking the Fano mirror
reflectivity (Eq. (6)) and considering the simple case of
${{\gamma}_{1}}={{\gamma}_{2}}={{\gamma}_{c}}/2\approx{{\gamma}_{t}}/2$, we
obtain by integrating Eq. (24)

$\displaystyle V$
$\displaystyle=\frac{1}{2}{{\gamma}_{D}}{{\Delta}^{2}}+\alpha{{\gamma}_{in}}\Delta$
$\displaystyle-\frac{{{\gamma}_{in}}}{2{{\gamma}_{D}}}\frac{2{{r}_{B}}\left({{\gamma}_{D}}\Delta+{{\delta}}\right)\left({{\gamma}_{t}}-\alpha{{\delta}}\right)+2{{\gamma}_{t}}\left({{t}_{B}}\left(\alpha{{\gamma}_{t}}+{{\delta}}\right)-{{r}_{B}}\left({{\gamma}_{t}}-\alpha{{\delta}}\right)\right)\tan^{-1}\left(\frac{{{\gamma}_{D}}\Delta+{{\delta}}}{{{\gamma}_{t}}}\right)}{{{t}_{B}}{{\gamma}_{t}}-{{r}_{B}}{{\delta}}}$
$\displaystyle+\frac{{{\gamma}_{in}}}{2{{\gamma}_{D}}}\frac{{{\gamma}_{t}}\left({{r}_{B}}\left(\alpha{{\gamma}_{t}}+{{\delta}}\right)+{{t}_{B}}\left({{\gamma}_{t}}-\alpha{{\delta}}\right)\right)\ln\left\\{\gamma_{t}^{2}+{{\left({{\gamma}_{D}}\Delta+{{\delta}}\right)}^{2}}\right\\}}{{{t}_{B}}{{\gamma}_{t}}-{{r}_{B}}{{\delta}}},$
(25)

where
${{\gamma}_{D}}=1/\left(\partial{{\phi}_{R}}\left(\omega\right)/\partial\omega\right)$.
For $\alpha=0$, ${{r}_{B}}=0$ (${{t}_{B}}=1$), and
${{\omega}_{r}}={{\omega}_{0}}$, we have ${{\gamma}_{D}}={{\gamma}_{t}}$, so
Eq. (25) reduces to

$\ \ \ \ \ \
V=\frac{1}{2}{{\gamma}_{D}}{{\Delta}^{2}}+{{\gamma}_{in}}\frac{1}{2}\ln\left\\{1+{{\Delta}^{2}}\right\\}.$
(26)

The potential Eq. (26) for the Fano laser is different from the potential for
an external cavity laser [20], which (for $\alpha=0$) assumes the form

$V=\frac{1}{2}{{\gamma}_{D}}{{\Delta}^{2}}-\kappa{{\gamma}_{in}}\cos\left({{\theta}_{0}}+\Delta\right).$
(27)

From Eq. (24), by considering $\partial\omega/\partial\Delta=-{{\gamma}_{D}}$
when $\omega\to{{\omega}_{r}}$, we find the curvature of the potential $V$ in
the minimum point

$\displaystyle{{\left.\frac{1}{{{\gamma}_{D}}}\frac{{{d}^{2}}V}{d{{\Delta}^{2}}}\right|}_{\omega={{\omega}_{r}}}}$
$\displaystyle\approx
1+{{\gamma}_{in}}{{\left.\left(\alpha{{\left|{{r}_{R}}\left(\omega\right)\right|}^{-1}}\frac{\partial}{\partial\omega}\left|{{r}_{R}}\left(\omega\right)\right|+\frac{\partial}{\partial\omega}{{\phi}_{R}}\left(\omega\right)\right)\right|}_{\omega={{\omega}_{r}}}}$
$\displaystyle=\eta={{\left(\frac{\Delta{{v}_{FP}}}{\Delta{{v}_{FL}}}\right)}^{\frac{1}{2}}}.$
(28)

Eq. (28) shows that the curvature of $V$ at $\omega={{\omega}_{r}}$,
normalized by ${{\gamma}_{D}}$, is identical to the linewidth reduction factor
for the Fano laser compared to the FP laser counterpart.

##  Appendix B Coupled-cavity system versus the Fano system with partially
transmitting element

For simplicity, we consider a Fano laser system with PTE, where ${{r}_{B}}$=1
(${{t}_{B}}$=0) and $R_{12}=R_{max}$. We neglect the linewidth enhancement
factor and optical nonlinearities in the nanocavity. By replacing
${{A}_{FP}}\left(t\right)=j{{A}^{+}}\left(t\right)/\sqrt{{{\gamma}_{in}}}$,
and assuming that the laser oscillates at a frequency where the Fano mirror
reflectivity is close to unity
($\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|\to 1$ with
${{\gamma}_{v}}\to 0$) so that the inverse of the photon lifetime of the Fano
cavity can be simplified as
${{\gamma}_{p}}=2{{\gamma}_{i}}+2{{\gamma}_{in}}\ln\left\\{1/\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|\right\\}\approx
2{{\gamma}_{i}}+2{{\gamma}_{in}}\left(1-\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|\right)$,
Eqs. (1) and (4) can be reduced, in a matrix form, to

$\frac{d}{dt}\left(\begin{matrix}{{A}_{c}}\left(t\right)\\\
{{A}_{FP}}\left(t\right)\\\
\end{matrix}\right)=\mathbf{M}\left(\begin{matrix}{{A}_{c}}\left(t\right)\\\
{{A}_{FP}}\left(t\right)\\\ \end{matrix}\right),$ (29)

where

$\mathbf{M}=\left(\begin{matrix}-j{{\delta}_{0}}-{{\gamma}_{t}}&{{\kappa}_{c}}\\\
{{\kappa}_{c}}&g-{{\gamma}_{in}}\\\ \end{matrix}\right).$ (30)

Here $g={{G}_{N}}\left(N-{{N}_{0}}\right)-{{\gamma}_{i}}$, and
${{\kappa}_{c}}$ is, in general, complex but approaches
$\sqrt{2{{\gamma}_{in}}{{\gamma}_{c}}}$ for ${{\gamma}_{v}}\ll{{\gamma}_{c}}$.
By using $\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|\to 1$ and
${{G}_{N}}\left(N-{{N}_{0}}\right)\to{{\gamma}_{i}}$, Eq. (29) leads to two
eigenfrequencies

$\displaystyle{{\omega}_{e,1}}={{\omega}_{0}}-\frac{1}{2}({{\delta}_{0}}-j\left(2{{\gamma}_{in}}+{{\gamma}_{t}}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \
+j\sqrt{8{{\gamma}_{in}}{{\gamma}_{c}}-{{\left({{\delta}_{0}}+j\left(2{{\gamma}_{in}}-{{\gamma}_{t}}\right)\right)}^{2}}}),$
$\displaystyle{{\omega}_{e,2}}={{\omega}_{0}}-\frac{1}{2}({{\delta}_{0}}-j\left(2{{\gamma}_{in}}+{{\gamma}_{t}}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \
-j\sqrt{8{{\gamma}_{in}}{{\gamma}_{c}}-{{\left({{\delta}_{0}}+j\left(2{{\gamma}_{in}}-{{\gamma}_{t}}\right)\right)}^{2}}}).$

We can get the $Q$-factors of the eigenmodes of the Fano laser system as
${{Q}_{e,1/2}}=-\operatorname{Re}\left\\{{{\omega}_{e,1/2}}\right\\}/\left(2\operatorname{Im}\left\\{{{\omega}_{e,1/2}}\right\\}\right)$,
and for ${{\delta}_{0}}=0$, we get ${{Q}_{e,1}}=\infty$,
${{Q}_{e,2}}={{\omega}_{0}}/\left(2\left(2{{\gamma}_{in}}+{{\gamma}_{c}}\right)\right)$.
The corresponding eigenvectors are

${\mathbf{v}_{e,1}}=\left(\begin{matrix}\sqrt{2{{\gamma}_{in}}/{{\gamma}_{t}}}\\\
1\\\
\end{matrix}\right),{\mathbf{v}_{e,2}}=\left(\begin{matrix}-\sqrt{{{\gamma}_{t}}/\left(2{{\gamma}_{in}}\right)}\\\
1\\\ \end{matrix}\right).$ (31)

As seen, the eigenmode 1, with ${{Q}_{e,1}}=\infty$, corresponds to the Fano
mode where the field is concentrated in the nanocavity (since
${{\gamma}_{in}}\gg{{\gamma}_{t}}$). This agrees with our previous analysis
[13]. The eigenmode 2 corresponds to the case where the field is concentrated
in the WG part. Interestingly, the eigenmode 2 has a low $Q$-factor even
though the reflectivity of the PTE is unity, i.e., the WG is closed at the
right end. Such a low $Q$-factor can be ascribed to the fact that when the
field is concentrated in the WG part while the nanocavity is almost empty, the
phase change of the WG field induced by the reflection off the right Fano
mirror has a phase difference of $\pi$ compared to the case where the
nanocavity field is well established. This means that eigenmode 2 dose not
meet the resonant condition of the FP cavity defined by the left end of the WG
and the right PTE. So the field dissipates quickly. Therefore, the Fano mode
can be still classified as a BIC even when the PTE completely blockes the WG.

From Eq. (29), since ${{\kappa}_{c}}$ is almost purely real for the Fano
laser, it contrasts with ordinary coupled-cavity systems where
${{\kappa}_{c}}$ is imaginary [33]. This distinguishes the Fano system from
the case of Autler-Townes splitting [58], which corresponds to a mode doublet
with similar $Q$-factors. Our system rather bears resemblance to the Parity-
Time system [59, 60] working in the Parity-Time broken regime where the two
eigenmodes split in loss. However, a fundamental difference is that the lasing
mode of the Fano laser is the eigenmode 1, with the field concentrated in the
passive (low loss) nanocavity region, while it is distributed evenly between
the passive and active regions in the Parity-Time symmetric regime, or
concentrated in the active (high loss) region in the Parity-Time broken
regime. This is because a real $\kappa_{c}$ in the Fano laser enables the loss
of one cavity to be compensated by the feedback from the other one, i.e., the
lasing is promoted by decreasing the “mirror” loss of the mode through field
destructive interference between ${{A}_{c}}$ (discrete mode) and ${{A}_{FP}}$
(quasi-continuum mode). For the Parity-Time laser (in the Parity-Time broken
regime), instead, the loss is compensated by an enhanced modal gain as in that
regime, $\left|{{\kappa}_{c}}\right|$ is usually small. A smaller
$\left|{{\kappa}_{c}}\right|$ will localize a larger portion of the field in
the active region, i.e., the lasing is promoted by increasing the lateral
optical confinement factor. It should be noted that the BIC of our Fano system
can transit to a conventional coupled-cavity system when ${{t}_{B}}$=0 and
${{\gamma}_{v}}\gg{{\gamma}_{c}}$. In this case, ${{\kappa}_{c}}\to
j{{\kappa}_{c}}$, i.e., ${{\kappa}_{c}}$ becomes imaginary, which means in
practice that the nanocavity and WG have a large spatial separation.

##  Appendix C Laser linewidth based on the Langevin approach

Based on Eqs. (1)-(4) and Eqs. (11) and (12), neglecting the Langevin noise
forces, , we obtain the steady-state solutions of the Fano laser system (for
${{\omega}_{r}}={{\omega}_{0}}$) by solving the following set of algebraic
equations

$\displaystyle{{N}_{c}}={{G}_{T}}{{\left|{{A}_{c}}\right|}^{4}}/{{\gamma}_{nc}},$
$\displaystyle{{r}_{R}}\left({{\omega}_{0}}\right)={{r}_{B}}+\frac{2{{\gamma}_{1}}{{e}^{2j{{\theta}_{1}}}}}{j{{\delta}_{0}}+{{\gamma}_{t}}+{{K}_{T}}{{\left|{{A}_{c}}\right|}^{2}}+U{{\left|{{A}_{c}}\right|}^{4}}},$
$\displaystyle
N={{N}_{0}}+\left({{\gamma}_{i}}+{{\gamma}_{in}}\ln\left\\{1/\left|{{r}_{R}}\left({{\omega}_{0}}\right)\right|\right\\}\right)/{{G}_{N}},$
$\displaystyle\left|{{A}^{+}}\right|$
$\displaystyle=\sqrt{\frac{\hbar{{\omega}_{0}}{{V}_{a}}\left(R-{{\gamma}_{N}}N\right)}{\left(1-{{\left|{{r}_{R}}\left({{\omega}_{0}}\right)\right|}^{2}}\right)\left({{\gamma}_{i}}/\left({{\gamma}_{in}}\ln\left\\{1/\left|{{r}_{R}}\left({{\omega}_{0}}\right)\right|\right\\}\right)+1\right)}},$
$\displaystyle{{A}^{-}}={{r}_{R}}\left({{\omega}_{0}}\right)\left|{{A}^{+}}\right|,\
{{A}_{c}}=\left({{A}^{-}}-{{r}_{B}}\left|{{A}^{+}}\right|\right)/\sqrt{2{{\gamma}_{1}}},$
$\displaystyle{{\phi}^{+}}=0,\
{{\phi}^{-}}=-j\ln\left({{A}_{-}}/\left|{{A}_{-}}\right|\right),$
$\displaystyle{{\phi}_{c}}=-j\ln\left({{A}_{c}}/\left|{{A}_{c}}\right|\right).$

Here, we define $U={{G}_{T}}{{K}_{A}}/{{\gamma}_{nc}}$, and
${{\left|{{A}_{c}}\right|}^{2}}$ is the real and positive solution ($X$) of
the following equation

$\displaystyle{{U}^{2}}{{X}^{5}}+2U{{K}_{T}}{{X}^{4}}+\left(K_{T}^{2}+2U{{\gamma}_{t}}\right){{X}^{3}}+2{{K}_{T}}{{\gamma}_{t}}{{X}^{2}}$
$\displaystyle+\left(\delta_{0}^{2}+\gamma_{t}^{2}\right)X-2{{\gamma}_{1}}{{\left|{{A}^{+}}\right|}^{2}}=0.$
(33)

We assume that the Fano resonance can be tuned so that the nonlinear resonance
shift of the nanocavity is always compensated at steady-state, i.e.,
$\operatorname{Re}\left\\{{{\delta}_{NL}}\right\\}=0$. Next, we separate the
amplitude and phase of the fields, i.e.,
${{A}^{\pm}}\left(t\right)=\left|{{A}^{\pm}}\left(t\right)\right|{{e}^{j{{\phi}^{\pm}}\left(t\right)}}$,
${{A}_{c}}\left(t\right)=\left|{{A}_{c}}\left(t\right)\right|{{e}^{j{{\phi}_{c}}\left(t\right)}}$,
expand the perturbation of the dynamical variables to first-order around their
steady states, i.e.,
$\mathbf{H}\left(t\right)=\mathbf{H}+\Delta\mathbf{H}\left(t\right)$ with
$\mathbf{H}={{\left[\left|{{A}^{+}}\right|,\ \left|{{A}^{-}}\right|,\
\left|{{A}_{c}}\right|,\ {{\phi}^{+}},\ {{\phi}^{-}},\ \ {{\phi}_{c}},\ N,\
{{N}_{c}}\right]}^{T}}$ being the steady-state values. After that, by Fourier
transforming the perturbations to the frequency domain, we arrive at the
relation

$\mathbf{O}\Delta\mathbf{H}\left(\omega\right)=\mathbf{F}\left(\omega\right).$
(34)

Here, $\mathbf{O}$ is a coefficient matrix depending on the laser parameters,
the nonlinear coefficients, and steady-state values [61].
$\mathbf{F}\left(\omega\right)$ is the Langevin noise terms
[$F_{\left|A^{+}\right|}\left(\omega\right)$, 0,
${F_{\left|{A_{c}}\right|}}\left(\omega\right)$,
$F_{\phi^{+}}\left(\omega\right)$, 0, $F_{\phi_{c}}\left(\omega\right)$,
$F_{N}\left(\omega\right)$, 0]T whose correlation strengths can be evaluated
by inspecting the average particle exchange rates into and out of various
reservoirs [61]. For simplicity, if neglecting the shot noise associated with
nanocavity resonance shift, after some algebra, we get

$\displaystyle\left\langle{{F}_{\left|{{A}^{+}}\right|}}{{F}_{\left|{{A}^{+}}\right|}}\right\rangle={{G}_{N}}\left(N-{{N}_{0}}\right){{n}_{sp}}/{{\varsigma}_{s}}\left({{\omega}_{0}}\right),$
$\displaystyle\left\langle{{F}_{{{\phi}^{+}}}}{{F}_{{{\phi}^{+}}}}\right\rangle=\left\langle{{F}_{\left|{{A}^{+}}\right|}}{{F}_{\left|{{A}^{+}}\right|}}\right\rangle/{{\left|{{A}^{+}}\right|}^{2}},$
$\displaystyle\left\langle{{F}_{N}}{{F}_{N}}\right\rangle=4{{G}_{N}}\left(N-{{N}_{0}}\right){{\varsigma}_{s}}\left({{\omega}_{0}}\right){{\left|{{A}^{+}}\right|}^{2}}\left({{n}_{sp}}-1\right)/V_{c}^{2}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ +2R/{{V}_{c}},$
$\displaystyle\left\langle{{F}_{\left|{{A}^{+}}\right|}}{{F}_{N}}\right\rangle$
$\displaystyle=\frac{{{G}_{N}}\left(N-{{N}_{0}}\right)\left({{\varsigma}_{s}}\left({{\omega}_{0}}\right)\left(1-2{{n}_{sp}}\right){{\left|{{A}^{+}}\right|}^{2}}-{{n}_{sp}}\right)}{{{\varsigma}_{s}}\left({{\omega}_{0}}\right){{V}_{c}}\left|{{A}^{+}}\right|},$
$\displaystyle\left\langle{{F}_{\left|{{A}_{c}}\right|}}{{F}_{\left|{{A}_{c}}\right|}}\right\rangle=\hbar{{\omega}_{0}}\left({{K}_{A}}{{N}_{c}}+{{K}_{T}}{{\left|{{A}_{c}}\right|}^{2}}+{{\gamma}_{v}}\right)/2,$
$\displaystyle\left\langle{{F}_{{{\phi}_{c}}}}{{F}_{{{\phi}_{c}}}}\right\rangle=\left\langle{{F}_{\left|{{A}_{c}}\right|}}{{F}_{\left|{{A}_{c}}\right|}}\right\rangle/{{\left|{{A}_{c}}\right|}^{2}}.$

Here $\left\langle{}\right\rangle$ indicates a statistical ensemble average,
which is identical to a time average in the present case of an ergodic system.
We have neglected the Langevin noise terms due to the free carriers generated
in the nanocavity. The phase $\Delta{{\phi}_{c}}\left(\omega\right)$ can be
obtained by solving Eq. (34), and the laser frequency fluctuation is
${{v}_{c}}\left(\omega\right)=-j\omega\Delta{{\phi}_{c}}\left(\omega\right)/\left(2\pi\right)$,
which can be expressed analytically in terms of the Langevin noise sources

$\displaystyle{{v}_{c}}\left(\omega\right)={{\zeta}_{1}}{{F}_{\left|{{A}^{+}}\right|}}\left(\omega\right)+{{\zeta}_{2}}{{F}_{\left|{{A}_{c}}\right|}}\left(\omega\right)+{{\zeta}_{3}}{{F}_{{{\phi}^{+}}}}\left(\omega\right)$
$\displaystyle+{{\zeta}_{4}}{{F}_{{{\phi}_{c}}}}\left(\omega\right)+{{\zeta}_{5}}{{F}_{N}}\left(\omega\right).$

The noise frequency spectrum can thus be obtained as

$\displaystyle{{S}_{v}}\left(\omega\right)=\frac{1}{2\pi}\int{\left\langle{{v}_{c}}\left(\omega\right){{v}_{c}}{{\left({{\omega}^{\prime}}\right)}^{*}}\right\rangle
d{\omega}^{\prime}}$
$\displaystyle={{\left|{{\zeta}_{1}}\right|}^{2}}\left\langle{{F}_{\left|{{A}^{+}}\right|}}{{F}_{\left|{{A}^{+}}\right|}}\right\rangle+{{\left|{{\zeta}_{2}}\right|}^{2}}\left\langle{{F}_{\left|{{A}_{c}}\right|}}{{F}_{\left|{{A}_{c}}\right|}}\right\rangle$
$\displaystyle+{{\left|{{\zeta}_{3}}\right|}^{2}}\left\langle{{F}_{{{\phi}^{+}}}}{{F}_{{{\phi}^{+}}}}\right\rangle+{{\left|{{\zeta}_{4}}\right|}^{2}}\left\langle{{F}_{{{\phi}_{c}}}}{{F}_{{{\phi}_{c}}}}\right\rangle+{{\left|{{\zeta}_{5}}\right|}^{2}}\left\langle{{F}_{N}}{{F}_{N}}\right\rangle$
$\displaystyle+\left({{\zeta}_{1}}\zeta_{5}^{*}+\zeta_{1}^{*}{{\zeta}_{5}}\right)\left\langle{{F}_{\left|{{A}^{+}}\right|}}{{F}_{N}}\right\rangle,$
(36)

which depends on the Langevin noise correlation strengths. The laser linewidth
is finally found as

$\Delta{{v}_{FL}}=2\pi{{S}_{v}}\left(0\right).$ (37)

Noting that by neglecting the Langevin noise correlation terms in Eq. (36),
except the dominating terms,
$\left\langle{{F}_{\left|{{A}^{+}}\right|}}{{F}_{\left|{{A}^{+}}\right|}}\right\rangle$
and $\left\langle{{F}_{{{\phi}^{+}}}}{{F}_{{{\phi}^{+}}}}\right\rangle$, the
expression for the Fano laser linewidth reduces to Eq. (7) in the main text.
Here, the nonlinear coefficients ${{K}_{K}}$, ${{K}_{T}}$, ${{K}_{D}}$,
${{K}_{A}}$, ${{G}_{T}}$, obtained based on the calculated nanocavity
nonlinear mode volumes combined with material parameters [36, 62, 63], are
listed in TABLE I.

Table 1: Nonlinear coefficients for the simulations Coefficients | InP | Si | SiN
---|---|---|---
${{K}_{K}}$ (W-1s-2) | -8.8$\times$1023 | -1.2$\times$1023 | -1.2$\times$1023
${{K}_{T}}$ (W-1s-2) | 1.85$\times$1024 | 2.9$\times$1022 | 6.6$\times$1021
${{K}_{D}}$ (m3/s) | 1.95$\times$10-12 | 4.8$\times$10-13 | 0
${{K}_{A}}$ (m3/s) | 2.13$\times$10-13 | 6.9$\times$10-14 | 0
${{G}_{T}}$ (W-2m-3s-3) | 3.35$\times$1061 | 5.3$\times$1059 | 1.2$\times$1059

##  Appendix D Output power of the Fano laser

When the Fano laser operates around the nanocavity resonance, the output power
transmitted through the Fano mirror (termed as through-port) is limited due to
the very high reflectivity of the Fano mirror.

Figure 5:  (a) The ratio of output powers in the left port and the through-
port (${{R}_{LT}}$, black curve) and the ratio of output powers in the cross-
and through-port (${{R}_{CT}}$, red curve), as a function of $r_{B}$ when the
Fano laser operates at the frequency point of the optimum linewidth
${{\omega}_{m}}$ (as $R_{12}$=$R_{max}$). (b) The output power ratio (log
scale) between the left- and cross-port as a function of $r_{B}$ and $R_{12}$.
The Fano laser operates at the reflectivity peak of the Fano mirror.

Therefore, it is preferable to use another channel for the output, for
example, e.g., either the left mirror (the left end of the WG) by reducing
(increasing) the left mirror reflectivity (transitivity), $r_{L}$ ($t_{L}$),
or through a cross-port by placing an additional WG adjacent to the nanocavity
with a coupling efficiency ${{\gamma}_{3}}$ [24]. These channels are expected
to have a higher external quantum efficiency than the through-port [24]. The
ratio of the output power of the left mirror, $P_{L}$, with respect to the
through-port, $P_{T}$, can be obtained as

${{R}_{LT}}\left({{\omega}_{r}}\right)=\frac{{{P}_{L}}}{{{P}_{T}}}=\frac{\left(1-r_{L}^{2}\right)\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|}{{{r}_{L}}\left(1-{{\left|{{r}_{R}}\left({{\omega}_{r}}\right)\right|}^{2}}\right)},$
(38)

and the ratio of the output power of the cross-port, $P_{C}$, with respect to
the through-port, is

$\displaystyle{{R}_{CT}}\left({{\omega}_{r}}\right)=\frac{{{P}_{C}}}{{{P}_{T}}}$
$\displaystyle=\frac{4{{\gamma}_{1}}{{\gamma}_{3}}}{{{\left|{{t}_{B}}\left({{\omega}_{0}}-{{\omega}_{r}}\right)-j{{t}_{B}}\left({{\gamma}_{v}}+{{\gamma}_{3}}\right)+S_{e}/t_{B}\right|}^{2}}}.$

Fig. 5 compares ${{R}_{LT}}$ and ${{R}_{CT}}$. Here we choose ${{r}_{L}}$=0.99
and ${{\omega}_{0}}/\left(2{{\gamma}_{3}}\right)$=1$\times$105 so that the
maximum of the Fano mirror reflectivity
$\left|{{r}_{R}}\left(\omega\right)\right|={{r}_{L}}$ (when $r_{B}$=0). As
seen, both ${{R}_{LT}}$ and ${{R}_{CT}}$ are much larger than unity, and the
cross-port, in general, exhibits the highest external quantum efficiency when
working around the frequency point of the optimum linewidth ${{\omega}_{m}}$
(Fig. 5(a)). When working around the peak of the Fano mirror reflectivity, the
left mirror can give a higher external quantum efficiency (Fig. 5(b)).

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