# Intermode coupling in a fiber loop laser at low temperatures

Eyal Buks<EMAIL_ADDRESS>Andrew and Erna Viterbi Department of
Electrical Engineering, Technion, Haifa 32000, Israel

###### Abstract

We experimentally study an unequally-spaced optical comb (USOC), which is
generated by a unidirectional fiber loop laser operated at low temperatures.
The underlying mechanism responsible for USOC formation is explored using both
close and open loop measurements. The role played by dispersion is
investigated using radio frequency spectrum measurements. By integrating a
saturable absorber into the loop, a lasing state is revealed, in which mode
locking coexists with the USOC.

Figure 1: Temperature dependency. (a) The experimental setup. The X box
represents a tunable attenuator, which is used for the measurements presented
in Fig. 2, a wavelength tunable OF, which is used for the measurements
presented in Fig. 7, and a SA, which is used for the measurements presented in
Fig. 8. Spectrum measurements presented in (b) and (c) are performed during
cooling down. OSA data (here and in all other figures) are presented in dBm
units. The EDF length is $l_{\mathrm{EDF}}=5\operatorname{m}$, and the diode
current is (b) $I_{\mathrm{D}}=200\operatorname{mA}$ and (c)
$I_{\mathrm{D}}=120\operatorname{mA}$.

Introduction \- Fiber loop lasers are widely employed for a variety of
applications. Commonly the loop is made of an undoped single mode fiber (SMF)
section, and a doped section, which is used for optical amplification. An
unequally-spaced optical comb (USOC) has been recently found in the optical
spectrum of a fiber loop laser having an amplifying Erbium doped fiber (EDF),
which is cooled down to cryogenic temperatures Buks_128591 . The underlying
mechanism responsible for USOC formation has remained mainly unknown. Here we
report on measurements, performed with both open and close loop
configurations, which provide some insight into the process of USOC formation,
and which allow the extraction of parameters of the fiber loop under study.

A SMF (both doped and undoped) can be characterized by a loss coefficient
$\alpha$, a group velocity $v_{\mathrm{g}}$, a group velocity dispersion
(GVD), and by a complex nonlinear coefficient
$\gamma=\gamma^{\prime}+i\gamma^{\prime\prime}$, which is proportional to the
SMF third-order susceptibility Agrawal_Nonlinear_fiber_optics . The real part
$\gamma^{\prime}$ of $\gamma$ is the nonlinear dispersion coefficient
Montagna_871 , and the imaginary part $\gamma^{\prime\prime}$ of $\gamma$ is
the nonlinear absorption coefficient Jachpure_024007 ; Maeda_1187 . For
saturable absorption (SA) $\gamma^{\prime\prime}<0$, whereas
$\gamma^{\prime\prime}>0$ for reverse saturable absorption (RSA). In a fiber
loop laser, SA commonly promotes mode locking (ML) and the formation of
optical pulses Kapitula_740 ; Dennis_1469 ; Huang_034009 , whereas processes
giving rise to RSA (e.g. two-photon absorption) suppress ML Wang_106805 .
Changeover from SA to RSA has been observed in Wei_7704 .

Fiber parameters are extracted in the current study using a variety of
measurement techniques. The effect of EDF emission and absorption spectra is
studied by varying the temperature and fiber loop loss. The effect of
nonlinear dispersion is explored by measuring both the loop frequency as a
function of optical wavelength and the lasing linewidth. Both EDF gain and
nonlinear response are measured using an open loop configuration. We show that
some of the experimental results can be accounted for using a simple
theoretical model, which assumes that dispersion can be disregarded. However,
on the other hand, this assumption is invalidated by some of our experimental
results, which indicate that dispersion plays an important role.

Experimental setup \- The experimental setup is described by the sketch shown
in Fig. 1(a). A cryogen free cryostat is used to cool down an EDF having
length denoted by $l_{\mathrm{EDF}}$, absorption of $30$ dB
$\operatorname{m}^{-1}$ at $1530\operatorname{nm}$, and mode field diameter of
$6.5\operatorname{\mu m}$ at $1550\operatorname{nm}$. The EDF, which is
thermally coupled to a calibrated silicon diode thermometer, is pumped using a
$980\operatorname{nm}$ laser diode (LD) biased with current denoted by
$I_{\mathrm{D}}$. Results are presented below for two fiber loops, one has
$l_{\mathrm{EDF}}=10\operatorname{m}$, and the other one has
$l_{\mathrm{EDF}}=5\operatorname{m}$.

The cold EDF is integrated with a room temperature fiber loop using a
wavelength-division multiplexing (WDM) device. A 10:90 output coupler (OC),
and two isolators [labeled by arrows in the sketch shown in Fig. 1(a)], are
integrated in the fiber loop. The output port of the 10:90 OC is splitted
using a $50:50$ OC to allow probing the optical signal using both an optical
spectrum analyzer (OSA) and a photodetector (PD), which is connected to either
a radio frequency spectrum analyzer (RFSA), or to an oscilloscope. An optional
tunable optical filter (OF) having a central wavelength denoted by
$\lambda_{\mathrm{F}}$, and a linewidth of
$\delta_{\mathrm{F}}=1.2\operatorname{nm}$ (full width at half maximum) is
connected between the $50:50$ OC and the PD.

Temperature dependency \- Key properties of EDF can be controlled by varying
the temperature Kobyakov_1 ; Le_3611 ; Thevenaz_22 ; Aubry_2100002 ;
Pizzaia_2352 ; Liu_102988 ; Lopez_085401 ; Antuzevics_1149 ; Chu_966 . EDF
operating at low temperatures can be used for some applications, including
multimode lasing perez2013multi ; haken1985laser  and quantum information
storage Saglamyurek_83 ; Staudt_720 ; Wei_2209_00802 ; Ortu_035024 ;
Liu_2201_03692 ; Veissier_195138 ; Saglamyurek_241111 ; Shafiei_F2A ;
Buks_L051001 .

The measured optical spectrum of our device as a function of the temperature
$T$ with diode current of $I_{\mathrm{D}}=200\operatorname{mA}$ is shown in
Fig. 1(b). The wavelength at which lasing peaks is denoted by
$\lambda_{\mathrm{g}}$. The spectrum shown in Fig. 1 indicates that
$\lambda_{\mathrm{g}}$ decreases as the temperature is lowered Kagi_261 . As
is discussed below, this process is attributed to the temperature dependency
of the EDF emission and absorption spectra.

Following Ref. Franco_1090 , consider an infinitesimal EDF section of length
$\mathrm{d}z$. The signal gain along this section is expresses as
$1+\kappa\mathrm{d}z$, where
$\kappa=l_{\mathrm{E}}^{-1}\left(1-p_{\mathrm{g}}\right)-l_{\mathrm{A}}^{-1}p_{\mathrm{g}}$,
$l_{\mathrm{E}}^{-1}$ ($l_{\mathrm{A}}^{-1}$) is the emission (absorption)
inverse length (which may depend on both wavelength $\lambda$ and temperature
$T$), $p_{\mathrm{g}}$ is the ground state population fraction, and
$1-p_{\mathrm{g}}$ is the excited state population fraction (note that $0\leq
p_{\mathrm{g}}\leq 1$, and that due to the isotropic nature of spontaneous
emission, which yields weak coupling to the fiber propagating mode, its
contribution to the signal is disregarded). The total EDF gain
$g_{\mathrm{F}}$ is given by
$g_{\mathrm{F}}=\exp\int_{0}^{l_{\mathrm{EDF}}}\kappa\mathrm{d}z\;$. The
evolution of pump intensity $I_{\mathrm{P}}$ along the EDF is governed by
$\mathrm{d}I_{\mathrm{P}}/\mathrm{d}z=-l_{\mathrm{P}}^{-1}p_{\mathrm{g}}I_{\mathrm{P}}$,
where $l_{\mathrm{P}}^{-1}$ is the inverse pump decay length, thus
$g_{\mathrm{F}}$ can be expressed as
$g_{\mathrm{F}}=\exp\left(l_{\mathrm{EDF}}/l_{\mathrm{E}}+l_{\mathrm{f}}^{-1}\zeta_{\mathrm{P}}\right)$,
where $l_{\mathrm{f}}^{-1}=l_{\mathrm{E}}^{-1}+l_{\mathrm{A}}^{-1}$, and where
$\zeta_{\mathrm{P}}=l_{\mathrm{P}}\mathrm{\log}\left(I_{\mathrm{P}}\left(l_{\mathrm{EDF}}\right)/I_{\mathrm{P}}\left(0\right)\right)$.
The total loop gain $g$ is expressed as
$g=g_{\mathrm{F}}/\alpha_{\mathrm{FL}}$, where $\alpha_{\mathrm{FL}}\geq 1$ is
the fiber loop loss coefficient. The lasing wavelength $\lambda_{\mathrm{g}}$
is determined from the conditions that $\log g=0$ (i.e. $g=1$) and
$\mathrm{d}\log g/\mathrm{d}\lambda=0$. By eliminating the term
$\zeta_{\mathrm{P}}$ from these two conditions one finds that
$\mathrm{d}\varrho/\mathrm{d}\lambda=0$ at the lasing wavelength
$\lambda=\lambda_{\mathrm{g}}$, where $\varrho$ is given by

$\varrho=\left(\eta_{\mathrm{EDF}}-1\right)l_{\mathrm{f}}\log\alpha_{\mathrm{FL}}\;,$
(1)

and where
$\eta_{\mathrm{EDF}}=l_{\mathrm{EDF}}/\left(l_{\mathrm{E}}\log\alpha_{\mathrm{FL}}\right)$
is the effective and normalized EDF length.

Manipulating the effective and normalized EDF length $\eta_{\mathrm{EDF}}$ by
changing the loop loss coefficient $\alpha_{\mathrm{FL}}$ is explored below in
the next section (see Fig. 2), whereas in the current section
$\eta_{\mathrm{EDF}}$ is controlled by exploiting the temperature dependence
of the emission length $l_{\mathrm{E}}$ (see Fig. 1). As the temperature is
lowered the emission length $l_{\mathrm{E}}$ increases, and consequently the
effective and normalized EDF length $\eta_{\mathrm{EDF}}$ decreases. In the
short EDF limit, for which $\eta_{\mathrm{EDF}}\ll 1$, it is expected from the
condition $\mathrm{d}\varrho/\mathrm{d}\lambda=0$ [see Eq. (1)] that lasing
will occur in the band where emission cross section peaks near
$\lambda_{\mathrm{g}}\simeq 1540\operatorname{nm}$ Desurvire_547 ;
Desurvire_246 ; Zyskind_869 ; Hickernell_19 . This behavior is demonstrated by
the plot shown in Fig. 1(b) for temperatures below $10\operatorname{K}$. The
overlaid red rectangle in Fig. 1(b) indicates the region shown in Fig. 1(c)
with higher resolution, inside which USOC occurs.

Attenuator \- To explore the dependency on the loop loss
$\alpha_{\mathrm{FL}}$ coefficient, a tunable attenuator is integrated into
the loop [in the location of the X-box shown in Fig. 1(a)]. The plot in Fig 2,
which displays the measured optical spectrum as a function of attenuation,
demonstrates the strong USOC dependence on loop loss $\alpha_{\mathrm{FL}}$.
This dependency diminishes as the wavelength increases.

The condition $\mathrm{d}\varrho/\mathrm{d}\lambda=0$ implies in the short EDF
limit that
$\left(\mathrm{d}\lambda_{\mathrm{g}}/\mathrm{d}\alpha_{\mathrm{FL}}\right)^{-1}=-\alpha_{\mathrm{FL}}\log\alpha_{\mathrm{FL}}\left(\log
l_{\mathrm{f}}^{\prime}\right)^{\prime}$, where prime denotes a derivative
with respect to wavelength $\lambda$ [see Eq. (1)]. This relation together
with the experimental results shown in Fig. 2 yield the value of $1/\left(\log
l_{\mathrm{f}}^{\prime}\right)^{\prime}=0.3\operatorname{nm}$. Note that this
value is about $10$ times smaller than the value of $1/\left(\log
l_{\mathrm{f}}^{\prime}\right)^{\prime}$ that is extracted from room
temperature absorption and emission measurements that are reported in Ref.
Franco_1090  for a similar EDF.

Figure 2: In loop attenuator. The EDF length is
$l_{\mathrm{EDF}}=5\operatorname{m}$, the temperature is
$T=2.8\operatorname{T}$, and diode current is
$I_{\mathrm{D}}=150\operatorname{mA}$. Attenuation is changed from high ($-20$
dB) to low ($-1$ dB).

USOC \- The USOC $n$’th peak wavelength is denoted by $\lambda_{n}$, where
$n=0,1,2,\cdots$ [see Fig. 1(c) and Fig. 3(a)]. The frequency $f_{n}$
associated with the $n$’th peak is given by $f_{n}=c/\lambda_{n}$, where $c$
is the speed of light in vacuum, and the corresponding normalized frequency
detuning $i_{n}$ is defined by
$i_{n}\equiv\left(f_{0}-f_{n}\right)/f_{\mathrm{L}}$, where
$f_{\mathrm{L}}=c/\left(n_{\mathrm{F}}l_{\mathrm{L}}\right)$ is the loop
frequency, $n_{\mathrm{F}}=1.45$ is the fiber refractive index, and
$l_{\mathrm{L}}$ is the fiber loop total length.

The detuning sequence $\left\\{i_{n}\right\\}$ is found to be well describe by
the following empirical law Buks_128591

$i_{n}=\nu\log p_{n}\ ,$ (2)

where $\nu$ is a positive constant, and $p_{n}$ is the $n$’th prime number.
The comparison between the measured values of $i_{n}$ and the calculated
values of $\nu\log p_{n}$ [see Eq. (2)] yields a good agreement [see Fig. 3
(b)]. The level of disagreement is quantified by the parameter
$\varepsilon=n_{\mathrm{m}}^{-1}\sum_{n=1}^{n_{\mathrm{m}}}\left|\left(i_{n}-\nu\log
p_{n}\right)/i_{n_{\mathrm{m}}}\right|$, where $n_{\mathrm{m}}$ is the number
of peaks that can be reliably resolved. The level of disagreement
$\varepsilon$ is shown in Fig. 3 (c) as a function of the elapsed time $t$
after switching on the diode current. The plot reveals that during the first
10 minutes after switching on the disagreement parameter $\varepsilon$ rapidly
drops down towards a value of about $0.005$.

As can be seen from the empirical law given by Eq. (2), all frequency spacings
$f_{n^{\prime}}-f_{n^{\prime\prime}}$ between pairs of USOC peaks are unique
[recall the fundamental theorem of arithmetic, and that $\log x+\log
y=\log\left(xy\right)$]. As is discussed below, this observation suggests a
connection between USOC formation and intermode coupling. The loop optical
complex amplitude at time $t$ is expressed as $g\left(f_{\mathrm{L}}t\right)$,
where the function $g\left(x\right)$ is Fourier expanded as
$g\left(x\right)=\sum_{n=-\infty}^{\infty}c_{n}e^{2\pi inx}$, where $c_{n}$ is
the complex amplitude of the $n$’th loop mode. When dispersion can be
disregarded, the time evolution of the (assumed slowly varying) amplitude
$c_{n}$ is governed by $\dot{c}_{n}=-\partial_{n}^{\ast}\mathcal{H}+\xi_{n}$,
where $\partial_{n}$ denotes the Wirtinger derivative for the $n$’th mode, and
where the terms $\xi_{n}$ represent noise Haus_1173 . The contribution of
intermode coupling to the total Hamiltonian function $\mathcal{H}$ is given by
$\mathcal{H}_{\mathrm{c}}=\left(\gamma^{\prime\prime}v_{\mathrm{g}}/2\right)\left\|g\right\|_{4}^{4}$,
where $\left\|g\right\|_{p}$ denotes the $L_{p}$ norm of $g$. Note that
$\left\|g\right\|_{2}^{2}$ is the total optical intensity, and that
$\left\|g\right\|_{4}^{4}=\sum_{n^{\prime}-n^{\prime\prime}+n^{\prime\prime\prime}-n^{\prime\prime\prime\prime}=0}c_{n^{\prime}}c_{n^{\prime\prime}}^{\ast}c_{n^{\prime\prime\prime}}c_{n^{\prime\prime\prime\prime}}^{\ast}$.
It was shown in Ref. Buks_128591  that for a fixed $\left\|g\right\|_{2}^{2}$
(total optical intensity), $\left\|g\right\|_{4}^{4}$ is nearly minimized when
USOC is formed, provided that $\gamma^{\prime\prime}>0$ (i.e. RSA occurs in
the USOC band). This observation suggests a connection between USOC formation
and RSA, however, it has remained unclear how the assumption that dispersion
can be disregarded can be justified.

Typically, USOC is highly stable and nearly temperature independent below
about $5\operatorname{K}$ [see Fig. 1(b)]. In contrast, strong temperature
dependency and significant drift in time of USOC peaks are observed at higher
temperatures.

Further insight can be gained from the degree of polarization (DOP) of light
emitted from the output port. DOP is measured by integrating a rotating
quarter wave plate with the coherent optical spectrum analyzer. Using this
method we find a DOP value of about $0.1$ in both USOC and in the continuous
(above $1542\operatorname{nm}$) lasing bands. Note that commonly DOP is below
about $0.15$ in fiber lasers, unless a polarization-maintaining fiber is used
Liem_CMS4 .

Figure 3: USOC. (a) OSA signal $I_{\mathrm{OSA}}$ as a function of wavelength
$\lambda$. EDF length is $l_{\mathrm{EDF}}=10\operatorname{m}$, diode current
is $I_{\mathrm{D}}=200\operatorname{mA}$, and temperature is
$T=3.2\operatorname{K}$. (b) The measured values of normalized frequency
detuning $i_{n}=\left(f_{0}-f_{n}\right)/f_{\mathrm{L}}$ (red cross symbols),
and the calculated values of $\nu\log p_{n}$ with $\nu=7888$ (blue solid line)
[see Eq. (2)]. (c) The level of disagreement $\varepsilon$ [of fitting using
Eq. (2)] as a function of the elapsed time $t$ after switching on the diode
current.

RF spectrum \- Further insight can be gained from probing the PD signal using
a RFSA [see Fig. 1(a)]. The RF spectrum shown in Fig. 4(a) and (b), which is
taken without the tunable OF [see Fig. 1(a)], contains an equally spaced comb
of peaks centered at frequencies $n_{\mathrm{L}}f_{\mathrm{L}}$, where
$f_{\mathrm{L}}=4.705\operatorname{MHz}$ is the loop frequency, and where
$n_{\mathrm{L}}$ is a positive integer (peaks with $n_{\mathrm{L}}\lesssim
2500$ can be resolved). The measured first harmonic peak (i.e.
$n_{\mathrm{L}}=1$) is shown in Fig. 4(c) as a function of the tunable OF
wavelength $\lambda_{\mathrm{F}}$. This plot reveals that the measured value
of the loop frequency $f_{\mathrm{L}}$ inside the USOC band
($\lambda_{\mathrm{F}}\lesssim 1542\operatorname{nm}$) drops well below its
measured value in the continuous band ($\lambda_{\mathrm{F}}\gtrsim
1542\operatorname{nm}$).

The observed drop of $f_{\mathrm{L}}$ in the USOC band can be generally
attributed to both chromatic and nonlinear dispersion. The contribution of
chromatic dispersion can be estimated from the wavelength dependency of the
EDF absorption coefficient $\alpha_{\mathrm{EDF}}$ Le_3611 . The Kramers-
Krönig formula, which relates $\alpha_{\mathrm{EDF}}$ and the EDF refractive
index $n_{\mathrm{EDF}}$ through a Hilbert transform Hutchings_1 ; Sai_136 ;
Hickernell_19 ; Montagna_871 ; Arkwright_798 , can be used to calculate
$n_{\mathrm{EDF}}$, which, in turn, yields the corresponding group index
Gehring_3752 . Comparing the calculated group index to the data shown in Fig.
4(c) indicates that the observed drop in $f_{\mathrm{L}}$ is far larger and
far sharper compared to what is theoretically expected (when only chromatic
dispersion is taken into account). This observation leads to the conclusion
that the contribution of chromatic dispersion to the observed drop is
relatively small. In other words, the dominant underlying mechanism
responsible for the observed drop in $f_{\mathrm{L}}$ inside the USOC band is
nonlinear dispersion, which is characterized by the real part
$\gamma^{\prime}$ of the nonlinear coefficient $\gamma$.

Figure 4: RFSA. The measurements presented in (a) and (b) are performed
without the tunable OF. EDF length is $l_{\mathrm{EDF}}=10\operatorname{m}$,
diode current is $I_{\mathrm{D}}=200\operatorname{mA}$, and temperature is
$T=4.1\operatorname{K}$. (c) The measured first harmonic peak (in dBm units)
as a function of the tunable OF wavelength $\lambda_{\mathrm{F}}$.

EDF gain \- Open loop measurements are performed in order to characterize the
EDF gain $g_{\mathrm{EDF}}$, and intermodulation response (see next section).
In these measurements the loop is opened in the location of the X-box shown in
Fig. 1(a).

To measure the EDF gain, a narrow band laser having a tunable wavelength
$\lambda_{\mathrm{L}}$ is employed. Open loop EDF gain $g_{\mathrm{EDF}}$ is
plotted in Fig. 5 as a function of wavelength $\lambda$ for different values
of the laser input power $P_{\mathrm{in}}$. For the band $\lambda\gtrsim
1535\operatorname{nm}$, the measured EDF gain $g_{\mathrm{EDF}}$ decreases as
the optical input power $P_{\mathrm{in}}$ is increased (see Fig. 5). This
observation suggests that RSA and gain saturation occur in this wavelength
band (inside which USOC is observed when the loop is closed).

Figure 5: EDF gain. EDF length is $l_{\mathrm{EDF}}=10\operatorname{m}$, diode
current is $I_{\mathrm{D}}=150\operatorname{mA}$, and temperature is
$T=2.9\operatorname{K}$. Open loop EDF gain $g_{\mathrm{EDF}}$ as a function
of wavelength $\lambda$ for different values of the laser input power
$P_{\mathrm{in}}$ (values of $P_{\mathrm{in}}$ are indicated by the figure
legend).

Intermodulation \- The technique of intermodulation is employ to characterize
intermode coupling. In this technique two (phase locked) monochromatic tones
are injected into the system under study (i.e. the EDF). The first one, which
has a relatively large amplitude, and a wavelength denoted by
$\lambda_{\mathrm{p}}$, is commonly refereed to as the pump. The second one is
a relatively low-amplitude signal tone having wavelength
$\lambda_{\mathrm{s}}=\lambda_{\mathrm{p}}-\lambda_{\mathrm{d}}$, where
$\lambda_{\mathrm{d}}$ is the detuning wavelength, which is assumed to be
small $\left|\lambda_{\mathrm{d}}\right|\ll\lambda_{\mathrm{p}}$. A driven
ferrimagnetic sphere resonator is employed for the generation of this two-tone
input signal Nayak_193905  [see Fig. 6(a)].

Frequency mixing between the pump and signal input tones occurring in the EDF
gives rise to an idler tone at the output [see Fig. 6(b)]. The intensity of
the idler peak at wavelength
$\lambda_{\mathrm{i}}=\lambda_{\mathrm{p}}+\lambda_{\mathrm{d}}$ is shown in
Fig. 6(c) as a function of pump wavelength $\lambda_{\mathrm{p}}$ and diode
current $I_{\mathrm{D}}$. The results presented in Fig. 6(c) suggest that
intermode coupling is relatively strong in the region where USOC occurs.

The intermodulation gain $g_{\mathrm{I}}\equiv P_{\mathrm{i}}/P_{\mathrm{s}}$,
is the ratio between the generated idler power $P_{\mathrm{i}}$ and the input
signal power $P_{\mathrm{s}}$ Yurke_5054 . The intermodulation gain
$g_{\mathrm{I}}$ is expected to depend on the nonlinear dispersion coefficient
$\gamma^{\prime}$. Measurements of EDFs similar to the ones used in the
current study yield values for $\gamma^{\prime}$ in the range of
$2-30\operatorname{W}^{-1}\operatorname{km}^{-1}$ (at room temperature, and in
the telecom band) Agrawal_Nonlinear_fiber_optics ; shaulov1998modeling ;
Roy_6791 . On the other hand, when nonlinear absorption
$\gamma^{\prime\prime}$ is disregarded, the extraction of $\gamma^{\prime}$
from the measured intermodulation gain $g_{\mathrm{I}}$ [see Fig. 6(c)] and
the EDF gain $g_{\mathrm{EDF}}$ (see Fig. 5) yields values for
$\gamma^{\prime}$ at least one order of magnitude larger (in the band where
USOC occurs). This observation suggests that other nonlinear processes (e.g.
nonlinear absorption) play an important role in the optical band where USOC is
formed (when the loop is closed).

Figure 6: Intermodulation gain - EDF length is
$l_{\mathrm{EDF}}=10\operatorname{m}$, and temperature is
$T=2.9\operatorname{K}$. (a) The two-tone input spectrum generated using a
driven ferrimagnetic sphere resonator Nayak_193905 . (b) The output spectrum.
(c) The idler (wavelength $\lambda_{\mathrm{i}}$) peak intensity (in dBm
units) as a function of pump wavelength $\lambda_{\mathrm{p}}$ and diode
current $I_{\mathrm{D}}$.

In loop OF \- Nonlinear dispersion tends to broaden lasing linewidth Roy_6791
; Lapointe_232 ; Babin_1729 ; Feng_155 . To explore this, an OF having a
tunable wavelength $\lambda_{\mathrm{F}}$, and linewidth of
$\delta_{\mathrm{F}}=1.2\operatorname{nm}$, is integrated into the fiber loop
[in the location of the X-box shown in Fig. 1(a)].

The lasing linewidth $\delta_{\mathrm{L}}$ is related to the nonlinear
dispersion coefficient $\gamma^{\prime}$ by

$\delta_{\mathrm{L}}=\sqrt{\gamma^{\prime}l_{\mathrm{L}}P_{\mathrm{C}}}\delta_{\mathrm{F}}\;,$
(3)

where $P_{\mathrm{C}}$ is the intra-cavity optical power Lapointe_232 . This
relation allows the extraction of $\gamma^{\prime}$ from the measured lasing
linewidth $\delta_{\mathrm{L}}$. The measured optical spectrum is shown in
Fig. 7 as a function of $\lambda_{\mathrm{F}}$. The region where
$\lambda_{\mathrm{F}}$ is tuned to the band where USOC is formed (in the
absence of the OF) is shown in higher resolution in the inset of Fig. 7. The
measured lasing linewidth $\delta_{\mathrm{L}}$ is about $2\operatorname{pm}$
in the range
$\lambda_{\mathrm{F}}\in\left[1548\operatorname{nm},1565\operatorname{nm}\right]$,
whereas $\delta_{\mathrm{L}}\simeq 200-500\operatorname{pm}$ for
$\lambda_{\mathrm{F}}$ in the band where USOC occurs (in the absence of the
OF) [see Fig. 1(b)]. The relation (3) yields the value of
$\gamma^{\prime}=1.4\operatorname{W}^{-1}\operatorname{km}^{-1}$ for the range
$\lambda_{\mathrm{F}}>1548\operatorname{nm}$, whereas significantly larger
values for $\gamma^{\prime}$ are obtained for the band where USOC occurs. This
observation further supports the hypothesis that other nonlinear processes
play an important role in the band where USOC is formed [note that nonlinear
absorption is disregarded in the derivation of Eq. (3)].

Figure 7: In loop OF. EDF length is $l_{\mathrm{EDF}}=10\operatorname{m}$,
diode current is $I_{\mathrm{D}}=120\operatorname{mA}$, and temperature is
$T=2.9\operatorname{K}$.

SA \- To further explore the role played by nonlinear absorption
(characterized by the imaginary part $\gamma^{\prime\prime}$ of the nonlinear
coefficient $\gamma$), a SA made of graphite Lin_880 ; Lin_045109  is
integrated into the fiber loop [in the location of the X-box shown in Fig.
1(a)]. The measured optical spectrum is shown as a function of diode current
$I_{\mathrm{D}}$ in Fig 8(a). The USOC region is shown in higher resolution in
Fig 8(b).

The effect of the integrated SA is explored by measuring PD time traces for
different values of the OF wavelength $\lambda_{\mathrm{F}}$ [see Fig. 8(c)].
Note that the OF is installed outside the loop [see Fig. 1(a)]. The periodic
short pulses that are shown in Fig. 8(c3) and (c4) indicate that ML occurs in
the continuous band ($\lambda_{\mathrm{F}}\gtrsim 1542\operatorname{nm}$). On
the other hand, inside the USOC band ($\lambda_{\mathrm{F}}\lesssim
1542\operatorname{nm}$) no pulses are observed [see Fig. 8(c1) and (c2)], thus
the USOC band does not participate in the ML. The observed coexistence of USOC
and ML suggests that the coupling between the USOC and continuous bands is
relatively weak. This weakness is partially attributed to the above-discussed
drop in the loop frequency $f_{\mathrm{L}}$ that is observed in the USOC band
[see Fig. 4(c)], which, in turn, gives rise to phase mismatch Shibata_1205 ;
schneider2004nonlinear , and suppression of coherent coupling between the USOC
and continuous bands Taheri_013828 ; Godey_063814 ; Mcmillan_411 .

Figure 8: Graphite SA. EDF length is $l_{\mathrm{EDF}}=5\operatorname{m}$, and
temperature is $T=3.2\operatorname{T}$. OF wavelength [labeled by vertical
dotted lines in (a)] is (c1) $\lambda_{\mathrm{F}}=1539.8\operatorname{nm}$,
(c2) $\lambda_{\mathrm{F}}=1541.3\operatorname{nm}$, (c3)
$\lambda_{\mathrm{F}}=1544\operatorname{nm}$ and (c4)
$\lambda_{\mathrm{F}}=1548\operatorname{nm}$, and diode current [labeled by
horizontal dotted lines in (a) and (b)] for the time traces shown in (c1),
(c2), (c3) and (c4) is $I_{\mathrm{D}}=1000\operatorname{mA}$.

Discussion \- Further study is needed to reveal the underlying mechanisms
responsible for USOC formation. On the one hand, a simple theoretical model
that disregards dispersion, suggests a connection between USOC formation and
RSA. On the other hand, our experimental findings invalidate the assumption
that dispersion can be disregarded. Theoretical modeling of the system under
study is highly challenging in the presence of dispersion, because explicit
time dependency cannot be eliminated by a transformation into a rotating
frame. Future work will be devoted for the development of a theory capable of
quantitatively reproducing the highly rich and convoluted behaviors that are
experimentally observed.

Acknowledgments \- Useful discussions with Luca Leuzzi, Vassilios Kovanis and
Kerry Vahala are acknowledged. We thank Michael Shlafman for his help in
graphite preparation.

Disclosures \- The authors declare no conflicts of interest.

## References

  * (1) Eyal Buks,  “Low temperature spectrum of a fiber loop laser”,  Physics Letters A, vol. 458, pp. 128591, 2023.
  * (2) Govind P Agrawal,  “Nonlinear fiber optics”,  in Nonlinear Science at the Dawn of the 21st Century, pp. 195–211. Springer, 2000.
  * (3) M Montagna, S Selleri, and M Zoboli,  “Nonlinear refractive index in erbium-doped optical amplifiers”,  Optical and quantum electronics, vol. 27, pp. 871–880, 1995.
  * (4) Deeksha Jachpure and R Vijaya,  “Saturable absorption and its consequent effects in bistable erbium-doped fiber ring laser”,  Journal of Optics, vol. 24, no. 2, pp. 024007, 2022.
  * (5) Yoshinobu Maeda,  “Mechanism of the negative nonlinear absorption effect in a five-level system of the er 3+ ion”,  Journal of applied physics, vol. 83, no. 3, pp. 1187–1194, 1998\.
  * (6) Todd Kapitula, J Nathan Kutz, and Björn Sandstede,  “Stability of pulses in the master mode-locking equation”,  JOSA B, vol. 19, no. 4, pp. 740–746, 2002.
  * (7) Michael L Dennis and Irl N Duling,  “Experimental study of sideband generation in femtosecond fiber lasers”,  IEEE Journal of Quantum electronics, vol. 30, no. 6, pp. 1469–1477, 1994.
  * (8) Shen Huang, Guodong Shao, Yufeng Song, Luming Zhao, Deyuan Shen, and Dingyuan Tang,  “Dark solitons embedded in a stable periodic pulse train emitted by a fiber ring laser”,  Journal of Physics: Photonics, vol. 2, no. 3, pp. 034009, 2020.
  * (9) Gang Wang, Yuxuan Ma, Ce Shang, Haojing Huang, Zherui Lu, Shuaixin Wang, Jingxuan Sun, Chenghong Zhang, and Bo Fu,  “Influence of reverse saturable absorption effect on conventional and dissipative solitons fiber lasers”,  Optics & Laser Technology, vol. 137, pp. 106805, 2021.
  * (10) Rongfei Wei, Hang Zhang, Xiangling Tian, Tian Qiao, Zhongliang Hu, Zhi Chen, Xin He, Yongze Yu, and Jianrong Qiu,  “Mos 2 nanoflowers as high performance saturable absorbers for an all-fiber passively q-switched erbium-doped fiber laser”,  Nanoscale, vol. 8, no. 14, pp. 7704–7710, 2016.
  * (11) Andrey Kobyakov, Michael Sauer, and Dipak Chowdhury,  “Stimulated brillouin scattering in optical fibers”,  Advances in optics and photonics, vol. 2, no. 1, pp. 1–59, 2010\.
  * (12) Julien Le Gouët, Jérémy Oudin, Philippe Perrault, Alaeddine Abbes, Alice Odier, and Alizée Dubois,  “On the effect of low temperatures on the maximum output power of a coherent erbium-doped fiber amplifier”,  Journal of Lightwave Technology, vol. 37, no. 14, pp. 3611–3619, 2019.
  * (13) Luc Thevenaz, Alexandre Fellay, Massimo Facchini, Walter Scandale, Marc Nikles, and Philippe A Robert,  “Brillouin optical fiber sensor for cryogenic thermometry”,  in Smart Structures and Materials 2002: Smart Sensor Technology and Measurement Systems. International Society for Optics and Photonics, 2002, vol. 4694, pp. 22–27.
  * (14) Marine Aubry, Luciano Mescia, Adriana Morana, Thierry Robin, Arnaud Laurent, Julien Mekki, Emmanuel Marin, Youcef Ouerdane, Sylvain Girard, and Aziz Boukenter,  “Temperature influence on the radiation responses of erbium-doped fiber amplifiers”,  physica status solidi (a), vol. 218, no. 15, pp. 2100002, 2021.
  * (15) João Paulo Lebarck Pizzaia, Rodolpho Ladislau Silva, Arnaldo Gomes Leal-Junior, and Carlos Eduardo Schmidt Castellani,  “Temperature sensor based on an erbium-doped fiber sagnac interferometer”,  Applied Optics, vol. 61, no. 9, pp. 2352–2356, 2022.
  * (16) Haochong Liu, Wei He, Yantao Liu, Yunhui Dong, and Lianqing Zhu,  “Erbium-doped fiber laser based on femtosecond laser inscribed fbg through fiber coating for strain sensing in liquid nitrogen environment”,  Optical Fiber Technology, vol. 72, pp. 102988, 2022.
  * (17) J Lopez, H Kerbertt, M Plata, E Hernandez, and S Stepanov,  “Two-wave mixing in erbium-doped-fibers with spectral-hole burning at 77k”,  Journal of Optics, vol. 22, no. 8, pp. 085401, 2020.
  * (18) Andris Antuzevics,  “Epr characterization of erbium in glasses and glass ceramics”,  Low Temperature Physics, vol. 46, no. 12, pp. 1149–1153, 2020.
  * (19) Tongwei Chu, Pengpeng Wang, and Cunguang Zhu,  “Modeling of active fiber loop ring-down spectroscopy considering gain saturation behavior of edfa”,  Journal of Lightwave Technology, vol. 38, no. 4, pp. 966–973, 2020\.
  * (20) Rosa Ana Perez-Herrera, Manuel Lopez-Amo, SW Harun, and H Arof,  “Multi-wavelength fiber lasers”,  in Current Developments in Optical Fiber Technology. In-Tech, 2013\.
  * (21) Hermann Haken,  Laser light dynamics, vol. 2,  North-Holland Amsterdam, 1985.
  * (22) Erhan Saglamyurek, Jeongwan Jin, Varun B Verma, Matthew D Shaw, Francesco Marsili, Sae Woo Nam, Daniel Oblak, and Wolfgang Tittel,  “Quantum storage of entangled telecom-wavelength photons in an erbium-doped optical fibre”,  Nature Photonics, vol. 9, no. 2, pp. 83–87, 2015.
  * (23) Matthias U Staudt, Sara R Hastings-Simon, Mikael Afzelius, Didier Jaccard, Wolfgang Tittel, and Nicolas Gisin,  “Investigations of optical coherence properties in an erbium-doped silicate fiber for quantum state storage”,  Optics Communications, vol. 266, no. 2, pp. 720–726, 2006.
  * (24) Shi-Hai Wei, Bo Jing, Xue-Ying Zhang, Jin-Yu Liao, Hao Li, Li-Xing You, Zhen Wang, You Wang, Guang-Wei Deng, Hai-Zhi Song, et al.,  “Storage of 1650 modes of single photons at telecom wavelength”,  arXiv:2209.00802, 2022.
  * (25) Antonio Ortu, Jelena V Rakonjac, Adrian Holzäpfel, Alessandro Seri, Samuele Grandi, Margherita Mazzera, Hugues de Riedmatten, and Mikael Afzelius,  “Multimode capacity of atomic-frequency comb quantum memories”,  Quantum Science and Technology, vol. 7, no. 3, pp. 035024, 2022\.
  * (26) Duan-Cheng Liu, Pei-Yun Li, Tian-Xiang Zhu, Liang Zheng, Jian-Yin Huang, Zong-Quan Zhou, Chuan-Feng Li, and Guang-Can Guo,  “On-demand storage of photonic qubits at telecom wavelengths”,  arXiv:2201.03692, 2022.
  * (27) Lucile Veissier, Mohsen Falamarzi, Thomas Lutz, Erhan Saglamyurek, Charles W Thiel, Rufus L Cone, and Wolfgang Tittel,  “Optical decoherence and spectral diffusion in an erbium-doped silica glass fiber featuring long-lived spin sublevels”,  Physical Review B, vol. 94, no. 19, pp. 195138, 2016.
  * (28) Erhan Saglamyurek, Thomas Lutz, Lucile Veissier, Morgan P Hedges, Charles W Thiel, Rufus L Cone, and Wolfgang Tittel,  “Efficient and long-lived zeeman-sublevel atomic population storage in an erbium-doped glass fiber”,  Physical Review B, vol. 92, no. 24, pp. 241111, 2015.
  * (29) Sara Shafiei, Erhan Saglamyurek, and Daniel Oblak,  “Hour-long decay-time of erbium spins in an optical fiber at milli-kelvin temperatures”,  in Quantum Information and Measurement. Optical Society of America, 2021, pp. F2A–4.
  * (30) Eyal Buks,  “Tunable multimode lasing in a fiber ring”,  Physical Review Applied, vol. 19, no. 5, pp. L051001, 2023.
  * (31) Nobuyuki Kagi, Akira Oyobe, and Kazunori Nakamura,  “Temperature dependence of the gain in erbium-doped fibers”,  Journal of lightwave technology, vol. 9, no. 2, pp. 261–265, 1991\.
  * (32) Pierluigi Franco, Michele Midrio, A Tozzato, Marco Romagnoli, and F Fontana,  “Characterization and optimization criteria for filterless erbium-doped fiber lasers”,  JOSA B, vol. 11, no. 6, pp. 1090–1097, 1994.
  * (33) Emmanuel Desurvire and Jay R Simpson,  “Evaluation of 4 i 15/2 and 4 i 13/2 stark-level energies in erbium-doped aluminosilicate glass fibers”,  Optics letters, vol. 15, no. 10, pp. 547–549, 1990.
  * (34) E Desurvire, JL Zyskind, and JR Simpson,  “Spectral gain holeburning at 1.53 m in erbium-doped fiber amplifiers”,  IEEE Photon. Technol. Lett, vol. 2, no. 4, pp. 246–248, 1990.
  * (35) JL Zyskind, E Desurvire, JW Sulhoff, and DJ Di Giovanni,  “Determination of homogeneous linewidth by spectral gain hole-burning in an erbium-doped fiber amplifier with geo/sub 2: Sio/sub 2/core”,  IEEE Photonics technology letters, vol. 2, no. 12, pp. 869–871, 1990.
  * (36) Robert K Hickernell, Kazumasa Takada, Makoto Yamada, Makoto Shimizu, and Masaharu Horiguchi,  “Pump-induced dispersion of erbium-doped fiber measured by fourier-transform spectroscopy”,  Optics letters, vol. 18, no. 1, pp. 19–21, 1993.
  * (37) Herman A Haus,  “Mode-locking of lasers”,  IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 6, pp. 1173–1185, 2000.
  * (38) A Liem, J Limpert, T Schreiber, M Reich, H Zellmer, A Tunnermann, A Carter, and K Tankala,  “High power linearly polarized fiber laser”,  in Conference on Lasers and Electro-Optics. Optica Publishing Group, 2004, p. CMS4.
  * (39) D Crichton Hutchings, M Sheik-Bahae, David J Hagan, and Eric W Van Stryland,  “Kramers-krönig relations in nonlinear optics”,  Optical and Quantum Electronics, vol. 24, pp. 1–30, 1992.
  * (40) Tianqi Sai, Matthias Saba, Eric R Dufresne, Ullrich Steiner, and Bodo D Wilts,  “Designing refractive index fluids using the kramers–kronig relations”,  Faraday Discussions, vol. 223, pp. 136–144, 2020.
  * (41) John W Arkwright, P Elango, Graham R Atkins, T Whitbread, and JF Digonnet,  “Experimental and theoretical analysis of the resonant nonlinearity in ytterbium-doped fiber”,  Journal of lightwave technology, vol. 16, no. 5, pp. 798–806, 1998\.
  * (42) George M Gehring, Robert W Boyd, Alexander L Gaeta, Daniel J Gauthier, and Alan E Willner,  “Fiber-based slow-light technologies”,  Journal of Lightwave Technology, vol. 26, no. 23, pp. 3752–3762, 2008.
  * (43) Banoj Kumar Nayak and Eyal Buks,  “Polarization-selective magneto-optical modulation”,  Journal of Applied Physics, vol. 132, no. 19, pp. 193905, 2022.
  * (44) Bernard Yurke and Eyal Buks,  “Performance of cavity-parametric amplifiers, employing kerr nonlinearites, in the presence of two-photon loss”,  J. Lightwave Tech., vol. 24, pp. 5054–5066, 2006.
  * (45) Gary Shaulov,  Modeling of mode-locked fiber lasers,  Rensselaer Polytechnic Institute, 1998.
  * (46) V Roy, M Piche, F Babin, and GW Schinn,  “Nonlinear wave mixing in a multilongitudinal-mode erbium-doped fiber laser”,  Optics express, vol. 13, no. 18, pp. 6791–6797, 2005.
  * (47) Marc-André Lapointe and Michel Piché,  “Linewidth of high-power fiber lasers”,  in Photonics North 2009. SPIE, 2009, vol. 7386, pp. 232–239.
  * (48) Sergey A Babin, Dmitriy V Churkin, Arsen E Ismagulov, Sergey I Kablukov, and Evgeny V Podivilov,  “Four-wave-mixing-induced turbulent spectral broadening in a long raman fiber laser”,  JOSA B, vol. 24, no. 8, pp. 1729–1738, 2007.
  * (49) Yujun Feng, Xiaojun Wang, Weiwei Ke, Yinhong Sun, Kai Zhang, Yi Ma, Tenglong Li, Yanshan Wang, and Juan Wu,  “Spectral broadening in narrow linewidth, continuous-wave high power fiber amplifiers”,  Optics Communications, vol. 403, pp. 155–161, 2017.
  * (50) GR Lin and YC Lin,  “Directly exfoliated and imprinted graphite nano-particle saturable absorber for passive mode-locking erbium-doped fiber laser”,  Laser Physics Letters, vol. 8, no. 12, pp. 880, 2011.
  * (51) Yung-Hsiang Lin and Gong-Ru Lin,  “Kelly sideband variation and self four-wave-mixing in femtosecond fiber soliton laser mode-locked by multiple exfoliated graphite nano-particles”,  Laser Physics Letters, vol. 10, no. 4, pp. 045109, 2013.
  * (52) Nori Shibata, RALFP Braun, and ROBERTG Waarts,  “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber”,  IEEE Journal of Quantum Electronics, vol. 23, no. 7, pp. 1205–1210, 1987.
  * (53) Thomas Schneider,  Nonlinear optics in telecommunications,  Springer Science & Business Media, 2004.
  * (54) Hossein Taheri, Pascal Del’Haye, Ali A Eftekhar, Kurt Wiesenfeld, and Ali Adibi,  “Self-synchronization phenomena in the lugiato-lefever equation”,  Physical Review A, vol. 96, no. 1, pp. 013828, 2017.
  * (55) Cyril Godey, Irina V Balakireva, Aurélien Coillet, and Yanne K Chembo,  “Stability analysis of the spatiotemporal lugiato-lefever model for kerr optical frequency combs in the anomalous and normal dispersion regimes”,  Physical Review A, vol. 89, no. 6, pp. 063814, 2014.
  * (56) Alex McMillan, Yu-Ping Huang, Bryn Bell, Alex Clark, Prem Kumar, and John Rarity,  “Four-wave mixing in single-mode optical fibers”,  in Experimental Methods in the Physical Sciences, vol. 45, pp. 411–465. Elsevier, 2013.