Mode-locking in anti-PT symmetric frequency lattices

Song, Yiling; Ke, Shaolin; Chen, Yuelan; Wang, Mingfeng

doi:10.1063/5.0146246

Active mode-locking (ML) is an important technique in laser science, which greatly shortens the laser pulse. Here, we construct an anti-parity-time (anti-PT) symmetric Su–Schrieffer–Heeger frequency lattice by two ring resonators with antisymmetric amplitude (AM) modulations. We find that the temporal width of the generated pulse can be greatly shortened by the phase-mismatching of the AM modulations. In addition, the pulse shortening shows extremely high sensitivity to the phase transition point, at which the anti-PT symmetry of the system is completely broken. This work exploits the concept of anti-PT symmetry in a laser field to realize ML, and will have broad application prospects in ultrafast spectroscopy and ultra-high sensitive sensors.

In real life, physical systems always exchange energy with surrounding environments, which can be described by a non-Hermitian Hamiltonian.^1–7 Recently, the exotic phenomena in the non-Hermitian physics with parity-time (PT) symmetry,^4,8–10 anti-parity-time (anti-PT) symmetry,^11–15 and non-Hermitian skin effect (NHSE)^5,16,17 have attracted intense interest. The most special feature for the PT and anti-PT symmetric systems is that they can possess real eigenvalues, in the cases of the PT symmetry maintained and the anti-PT symmetry broken, despite they are non-Hermitian systems. The eigenvalues and the eigenmodes of the systems will be coalesce at their exceptional points (EPs).^10,11,18 After the phase transition with an EP, the band structures of the PT and anti-PT symmetric systems will show time-windows with gain, which indicates their abilities to generate a short pulse.^19–22 In addition, thanks for the sensitivity of the EPs, the systems with PT and anti-PT symmetry have application prospects for sensors with ultrahigh sensitivity.^23–25

In the last few decades, the concept of the synthetic dimensions has been a versatile platform for the studies of the physical system with a complex structure such as non-Hermitian topological photonics.^26–28 The ring resonator with dynamic modulation will make the resonant frequency modes coupled and form a virtual frequency dimension.^26,29,30 The previous works have applied the dynamic modulated ring resonator to the actively mode-locked system, which have shown that the PT phase transition can be used to shorten the pulse width of a laser,³¹ and a turbulent behavior of the laser pulse emission will take place when the PT symmetry is broken.³² For an anti-PT symmetric system, the existence of a time-window with gain also suggests its potential applications in mode-locking (ML), which will generate ultrashort pulses and will have major applications in ultrafast science such as ultrafast spectroscopy and high-speed optical communications.

Here, we construct an anti-PT symmetric Su–Schrieffer–Heeger (SSH) lattice in the frequency dimension based on two ring resonators with antisymmetric amplitude (AM) dynamic modulations. The energy band of the anti-PT symmetric system can be flexibly controlled by the forms of the applied AM modulations. That is to say, the time-window with gain of the anti-PT symmetric double-ring system can be adjusted conveniently to generate ultra-short pulses and realize active ML. Moreover, the shortening of the pulse shows extremely high sensitivity in the vicinity of the phase transition point at which the anti-PT symmetry of the system is completely broken.

In order to construct an anti-PT symmetric system, two ring resonators with antisymmetric AM modulations are mutually coupled, as shown in Fig. 1(a). The two rings are designed to have the same length; thus, they support a series of discrete frequency modes (⁠

ω_{n} = ω_{0} + n Ω

⁠) with the same free spectral range (FSR)

Ω = 2 π v_{g} / L

⁠, where v_g and L are the group velocity of light and the length of the ring, respectively. In an experiment, the AM modulation can be achieved by an AM modulator with period-modulated loss along with an amplifier with constant gain over time.²⁹ Here, the dynamic modulations applied to each ring have opposite phase-mismatching and are in the following forms:

\begin{matrix} J^{A} (t) = - J^{B} (t) = - i 2 J_{1} cos [(2 κ + 2 δ) t] \\ - i 2 J_{2} cos [(Ω - 2 κ - 2 δ) t], \end{matrix}

(1)

where J₁ and J₂ are the modulated amplitudes. κ and δ are the coupling strength between the two rings and the phase-mismatching of the modulations, respectively. As the two rings are mutually coupled, the resonant frequency ω_n of the two rings hybridizes into symmetric (⁠

ω_{n +} = ω_{n} + κ \equiv ω_{a_{m}}

⁠) and antisymmetric (⁠

ω_{n -} = ω_{n} - κ \equiv ω_{b_{m}}

⁠) supermodes, respectively. Then the applied antisymmetric AM dynamic modulations will make the frequency sites to be coupled, and a frequency lattice will be formed. Here, we ignore the group velocity dispersion and apply the representation transformation and rotation wave approximation to the coupled mode equations of the double-ring system. The corresponding dynamics of the system in the frequency domain is described as^30,33

i T_{R} \frac{d}{d T} a_{m} (T) = δ a_{m} (T) + i J_{1} b_{m} (T) + i J_{2} b_{m + 1} (T),

(2)

i T_{R} \frac{d}{d T} b_{m} (T) = - δ b_{m} (T) + i J_{1} a_{m} (T) + i J_{2} a_{m - 1} (T),

(3)

where a_m and b_m correspond to the amplitudes of the m-th symmetric and antisymmetric supermodes, respectively.

T_{R} = 2 π / Ω

is the round trip time. T is the slow time variable, which labels the number of round trips that the pulse goes through. Equations and indicate that the symmetric (a_m) and antisymmetric (b_m) supermodes in the frequency dimension form a strip with nonuniform alternating spacing

2 κ

and

Ω - 2 κ

⁠. In addition, the phase-mismatching of the AM modulations makes the two sites in a unit cell to have the opposite on-site potentials δ, as described in Fig. 1(b). The detailed derivations for the anti-PT symmetric SSH frequency lattice and the validity of the tight-binding model with the rotating wave approximation (RWA) are shown in the supplementary material. Therefore, a non-Hermitian SSH lattice with alternating imaginary coupling strengths is obtained in the frequency dimension, and the couplings and the on-site potentials can be controlled by the modulated amplitudes flexibly.

FIG. 1.

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(a) Two ring resonators with antisymmetric AM dynamic modulations are mutually coupled with a strength of κ. (b) When the two ring are coupled, the resonant frequency modes of the system form an anti-PT symmetric SSH lattice with nonuniform alternating spacing $2 κ$ and $Ω - 2 κ$ and opposite on-site potentials δ.

To explore the active ML of the anti-PT symmetric system, we transform the frequency domain to the time domain by expanding the electric field amplitude

Ψ_{A} (T, t)

and

Ψ_{B} (T, t)

by

Ψ_{A} (T, t) = \sum_{m} a_{m} (T) e^{- imΩt}

and

Ψ_{B} (T, t) = \sum_{m} b_{m} (T) e^{- imΩt}

⁠, where t is the fast time variable that varies over the cavity round trip interval (⁠

- T_{R} / 2 \leq t < T_{R} / 2

⁠). Then, the dynamics of the double-ring system in the time domain can be transformed from Eqs. and as

\begin{matrix} i T_{R} \frac{d}{d T} (\begin{matrix} Ψ_{A} (T, t) \\ Ψ_{B} (T, t) \end{matrix}) = H (t) (\begin{matrix} Ψ_{A} (T, t) \\ Ψ_{B} (T, t) \end{matrix}), \end{matrix}

(4)

H (t) = (\begin{array}{l} δ & i J_{1} + i J_{2} e^{i Ω t} \\ i J_{1} + i J_{2} e^{- i Ω t} & - δ \end{array}) .

(5)

The Hamiltonian H(t) is anti-PT symmetric, i.e.,

(P T) H = - H (P T)

⁠, where the parity operator P is defined as

P H (t) P^{- 1} = σ_{x} H (- t) σ_{x}

(σ_x is the Pauli matrix), and the time-reversal operator T is defined as

T H (t) T^{- 1} = H^{*} (- t)

⁠.^14,34 The Bloch momentum of the anti-PT symmetric system is represented by t.

Moreover, from Eq. , we can know that the anti-PT symmetric system has frequency mode translation symmetry and time translation symmetry along the frequency axis [

H (t) = H (t + T_{R})

].³⁵ Thus, we are able to define the energy band of the double-ring system by the eigenvalues

λ_{\pm}

⁠. By analogy with the crystal lattice of a solid, the band energy of the above anti-PT symmetric double-ring system is expressed as

λ_{\pm} = \pm \sqrt{δ^{2} - J_{1}^{2} - J_{2}^{2} - 2 J_{1} J_{2} cos (Ω t)} .

(6)

Here,

Ω t

is equal to the wavevector along the frequency axis. Equation suggests that the band structure of the system has relationship with the phase-mismatching δ, and the point of

δ = J_{2} + J_{1}

is the phase transition point for the completely broken of the anti-PT symmetric system. In addition, the

δ = J_{2} + J_{1}

is also a transition point for vanishing of the time-window of gain. Therefore, we pay close attention to the evolution of the band structure for various values of δ around

J_{2} + J_{1}

⁠. For intuition, the real and imaginary parts of

λ_{\pm}

for various δ values are plotted in Figs. 2(a) and 2(b), respectively. We can see that when

δ \leq J_{2} - J_{1}

(marked by red dot), the eigenvalues of the system are pure imaginary, meaning that the anti-PT symmetry is kept for all t. In this case, the anti-PT symmetric double-ring system exhibits a time-window of gain for the whole of the band structure. As δ increases and is above

J_{2} - J_{1}

⁠, the anti-PT symmetry will be partly broken at

Ω t = \pm π

⁠. As shown in Fig. 2(c), when

δ = 0.9 (J_{2} + J_{1}) > J_{2} - J_{1}

⁠, the band structure of the anti-PT symmetric system is complex and displays a time-window with gain with the center of

Ω t = 0

⁠. When δ is in close proximity to

J_{2} + J_{1}

[Fig. 2(d)], the time-window with gain gets very narrow, indicating the arising of pulse shortening. The narrowing of the time-window with gain will continue until

δ = J_{2} + J_{1}

[marked by red dot in Figs. 2(a) and 2(b)]. Once the value of δ is above

J_{2} + J_{1}

⁠, the anti-PT-symmetry is completely broken for all t, and the eigenvalues become real. In this situation, the time-windows for gain and loss are disappeared, and the further increasing of δ will widen the bandgap of the anti-PT symmetric system, as shown in Figs. 2(e) and 2(f). In conclusion, by adjusting the phase-mismatching δ, the band structure of the anti-PT symmetric system can be controlled flexibly, which indicates the regulation of the ML in the double-ring system.

FIG. 2.

Real (a) and imaginary (b) parts of eigenvalues λ± for various values of δ. The red dot and black dot mark the parameters for δ=J2−J1 and δ=J2+J1, respectively. Real (blue) and imaginary (red) parts of eigenvalues λ± for δ=0.9 (J2+J1) (c), δ=0.999 (J2+J1) (d), δ=1.005 (J2+J1) (e), and δ=1.1 (J2+J1) (f). The parameter values used are J1=0.1 and J2=0.05.

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Real (a) and imaginary (b) parts of eigenvalues $λ_{\pm}$ for various values of δ. The red dot and black dot mark the parameters for $δ = J_{2} - J_{1}$ and $δ = J_{2} + J_{1}$ ⁠, respectively. Real (blue) and imaginary (red) parts of eigenvalues $λ_{\pm}$ for $δ = 0.9 (J_{2} + J_{1})$ (c), $δ = 0.999 (J_{2} + J_{1})$ (d), $δ = 1.005 (J_{2} + J_{1})$ (e), and $δ = 1.1 (J_{2} + J_{1})$ (f). The parameter values used are $J_{1} = 0.1$ and $J_{2} = 0.05$ ⁠.

As mentioned above, the existence of a time-window with gain suggests the ability of the anti-PT symmetric system to realize ML. To confirm that, we provide a direct calculation for the pulse width and saturated gain of the anti-PT symmetric double-ring system. The dynamics of the pulse envelope circulating in the double-ring system at successive round trips is given by the master equation^19,32,36

\begin{matrix} T_{R} \frac{\partial}{\partial T} Φ (T, t) = (g - l + \frac{g}{Ω_{g}^{2}} \frac{\partial^{2}}{\partial t^{2}}) Φ (T, t) - i H (t) Φ (T, t), \end{matrix}

(7)

where

Φ (T, t) = (\begin{matrix} Ψ_{A} (T, t) \\ Ψ_{B} (T, t) \end{matrix})

and g and l are the saturated gain and loss per transit in the rings, respectively. H(t) is the anti-PT symmetric Hamiltonian defined in Eq. . Ω_g is the gain bandwidth of the rings. For a slow-gain medium, saturated gain g obeys the rate equation

\frac{d g}{d T} = - γ_{∥} (g - g_{0} + g P)

⁠, where g₀ is the small-signal gain from pumping,

γ_{∥} = T_{R} / τ

is the ratio between cavity transit time T_R and upper laser level lifetime τ

(τ ≪ T_{R})

⁠, and

P = \frac{1}{T_{R}} \int_{- T_{R}}^{T_{R}} d t | Φ {(T, t)}^{2} |

is the average laser power normalized to the saturation power.

In the case of

δ < J_{2} + J_{1}

⁠, the time window with gain for the anti-PT symmetric double-ring system is centered at

Ω t = 0

⁠, as shown in Figs. 2(c) and 2(d). Therefore, we focus on the analysis in the vicinity of

Ω t = 0

⁠. In addition, the eigenvalues

λ_{\pm}

near

Ω t = 0

have two components, which are corresponding to the time-windows for gain and loss, respectively. As a result, the steady state of the anti-PT symmetric system will be dominated by the component, which experiences gain, as the component with loss will decay as the evolution over time. Thus, we perform a Taylor expansion near

Ω t = 0

and obtain the pulse evolution equation as

\begin{matrix} T_{R} \frac{\partial}{\partial T} Ψ (T, t) = [g - l + \frac{g}{Ω_{g}^{2}} \frac{\partial^{2}}{\partial t^{2}} + \sqrt{- δ^{2} + {(J_{1} + J_{1})}^{2}}] Ψ (T, t) \\ - [\frac{J_{1} J_{1}}{2 \sqrt{- δ^{2} + {(J_{1} + J_{1})}^{2}}} Ω^{2} t^{2}] Ψ (T, t) . \end{matrix}

(8)

The steady solution of Eq. is a Gaussian of the form of

Ψ (T, t) = e^{- t^{2} / 2 Δ t_{APT}^{2}}

⁠, and the calculated saturated gain is

g_{s} = l - \sqrt{- δ^{2} + {(J_{1} + J_{2})}^{2}} + \frac{g_{s}}{Ω_{g}^{2}} {[\frac{J_{1} J_{2} Ω_{g}^{2} Ω^{2}}{2 g_{s} \sqrt{- δ^{2} + {(J_{1} + J_{2})}^{2}}}]}^{1 / 2},

(9)

which indicates that the saturated gain is directly proportional to the value of phase-mismatching δ for the regime of

δ < J_{2} + J_{1}

⁠. The corresponding pulse width is

Δ t_{APT} = {[\frac{2 g_{s} \sqrt{- δ^{2} + {(J_{1} + J_{2})}^{2}}}{J_{1} J_{2} Ω_{g}^{2} Ω^{2}}]}^{1 / 4} .

(10)

Equation suggests that the width of the pulse will be greatly shortened as the value of δ is close to

J_{1} + J_{2}

⁠, which agrees with the narrowed time-window for gain with the center of

Ω t = 0

in Fig. 2. To demonstrate that, we checked the theoretical analysis by numerical simulations with the initial condition of small random amplitude of the intracavity field

Ψ (0, t)

at the initial round trip. As shown in Fig. 3(a), for

δ = 0.9 (J_{2} + J_{1})

⁠, the numerically computed evolution of the normalized pulse intensity in Eq. is centered at

Ω t = 0

⁠, just meeting with our expectation. As the value of δ increases and is close to

(J_{1} + J_{2})

⁠, the pulse width

Δ t_{A - P T}

becomes quite narrow [Fig. 3(b)], corresponding to the narrowed time-window for gain in Fig. 2(d). Therefore, the output pulses generated by the double-ring system are greatly shortened with increasing phase-mismatching δ in the case of

δ < J_{1} + J_{2}

⁠. To make sure that the numerically computed results under the RWA are valid, we show the corresponding numerically computed evolutions for the full time-dependent dynamical equations in the supplementary material, which are consistent with the evolutions in Fig. 3.

FIG. 3.

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Numerically computed evolution of the normalized pulse intensity for the tight-binding model with the RWA. (a) $δ = 0.9 (J_{2} + J_{1})$ ⁠, (b) $δ = 0.999 (J_{2} + J_{1})$ ⁠, (c) $δ = 1.005 (J_{2} + J_{1})$ ⁠, and (d) $δ = 1.1 (J_{2} + J_{1})$ ⁠. The parameter values used for the simulations are $γ_{∥} = 10^{- 3}$ ⁠, l = 0.5, $g_{0} = 0.5, J_{1} = 0.1, J_{2} / J_{1} = 0.5, J_{1} / κ = 0.05, and Ω_{g} / Ω = 200$ ⁠.

When

δ \geq J_{1} + J_{2}

⁠, the anti-PT symmetry is completely broken for all t in the anti-PT symmetric system. In this regime, active ML also occurs due to the effective phase modulation introduced by the modulators.³⁷ In this case, the ML will take place at

Ω t = 0

and

\pm π

⁠. Here, we focus on the analysis in the vicinity of

Ω t = 0

and π, as the solution near

Ω t = - π

is similar to that for

Ω t = π

⁠. Thus, the steady-state solutions are still a Gaussian and in the forms of

Ψ (T, t) = e^{- (1 + i) t^{2} / 2 Δ t_{APT}^{2}}

and

Ψ (T, t) = e^{- (1 + i) {(t - T_{R} / 2)}^{2} / 2 Δ t_{APT}^{2}}

for

Ω t = 0

and π, respectively. By performing a Taylor expansion near

Ω t = 0

and π, we obtain the saturated gains for the two cases as

g_{s}^{0} = l + \frac{g_{s}}{Ω_{g}^{2}} {[\frac{J_{1} J_{2} Ω_{g}^{2} Ω^{2}}{4 g_{s} \sqrt{δ^{2} - {(J_{2} + J_{1})}^{2}}}]}^{1 / 2},

(11)

g_{s}^{π} = l + \frac{g_{s}}{Ω_{g}^{2}} {[\frac{J_{1} J_{2} Ω_{g}^{2} Ω^{2}}{4 g_{s} \sqrt{δ^{2} - {(J_{2} - J_{1})}^{2}}}]}^{1 / 2},

(12)

respectively. Since

Ω_{g} ≫ Ω

⁠, the second terms are negligible. Thus, the saturated gains

g_{s}^{0}

and

g_{s}^{π}

are approximately equal to the loss l. The corresponding pulse widths for

Ω t = 0

and π are

Δ t_{APT}^{0} = {[\frac{4 g_{s} \sqrt{δ^{2} - {(J_{2} + J_{1})}^{2}}}{J_{1} J_{2} Ω_{g}^{2} Ω^{2}}]}^{1 / 4},

(13)

Δ t_{APT}^{π} = {[\frac{4 g_{s} \sqrt{δ^{2} - {(J_{2} - J_{1})}^{2}}}{J_{1} J_{2} Ω_{g}^{2} Ω^{2}}]}^{1 / 4},

(14)

respectively. Equations and indicate that the further increasing of δ will increase the pulse widths for both two cases, and the pulse width for

Ω t = 0

is narrower than that for

Ω t = π

⁠. In addition, Eq. suggests that the pulse can be greatly shortened at the phase transition point of

δ = (J_{2} - J_{1})

for

Ω t = π

⁠. However, from the band energy in Fig. 2, we can know that the broad time-window with gain will make the energy of the light for

Ω t = π

to be obscured by the larger gain near

Ω t = 0

⁠. Figure 3(c) shows the numerically computed evolution of the normalized pulse intensity for

δ = 1.005 (J_{1} + J_{2})

⁠, and the pulse centered at

Ω t = 0

is much narrower than that for

Ω t = π

⁠. When

δ = 1.1 (J_{1} + J_{2})

⁠, the corresponding pulse width near

Ω t = 0

gets wider, as shown in Fig. 3(d), in agreement with our predictions. The corresponding numerically computed evolutions for the full time-dependent dynamical equations are also shown in the supplementary material.

Moreover, we plot the pulse width parameter $Δ t_{APT}$ and the saturated gain g_s for the anti-PT symmetric coupled-ring system, as the value of δ changes from 0 to $2 (J_{1} + J_{2})$ ⁠. In Fig. 4, g_s and $Δ t_{APT}$ are normalized to the corresponding values for δ = 0. In the regime of $δ \leq J_{1} + J_{2}$ ⁠, the analytical solution for Eq. (10) [blue solid curve in Fig. 4(a)] exhibits a narrowing of the temporal width of the pulse with the increasing of δ, which agrees well with the numerical simulation results for $Δ t_{A - P T}$ (marked by red circle). We can notice that the analytical solutions of the $Δ t_{APT}$ for $Ω t = π$ are disappeared as $δ \leq J_{1} + J_{2}$ ⁠, which is because that the time-window with gain makes the energy of the light for $Ω t = π$ to be obscured by the larger gain. Meanwhile, the numerical simulation results show that the saturated gain increases gradually with δ and are consistent with the calculated g_s in Eq. (9), as shown for the regime of $δ \leq J_{1} + J_{2}$ in Fig. 4(b). The narrowing of the pulse and the increasing of the saturated gain simultaneously end when the value of δ is larger than $J_{1} + J_{2}$ ⁠. The calculated pulse width parameters $Δ t_{APT}$ for $δ \geq J_{1} + J_{2}$ in Eqs. (13) and (14) are plotted in Fig. 4(a) and show that the further increasing of δ broadens the width of the pulse. In addition, the pulse width in the vicinity of $Ω t = 0$ [green solid curve in Fig. 4(a)] displays a great shortening as δ is approaching to $J_{1} + J_{2}$ ⁠, and is much narrower than that for $Ω t = π$ [black solid curve in Fig. 4(a)]. Therefore, the temporal width of the pulse generated by the double-ring system is sensitive to the phase transition point of $δ = J_{1} + J_{2}$ ⁠, at which the anti-PT symmetry of the system is completely broken, while the saturated gain for the regime of $δ \geq J_{1} + J_{2}$ is almost constant, in accordance with the analytic theories for Eqs. (11) and (12). Moreover, the numerical simulation results for $δ > J_{1} + J_{2}$ (red circle) are still in agreement with our theoretical analysis. Thus, we can conclude that our analytic theory can well describe the active ML of the anti-PT symmetric double-ring system. In addition, the output pulses experience a great shortening in the vicinity of the phase transition point for the completely anti-PT symmetry broken, which shows high sensitivity and suggests the application in the sensitive sensor.

FIG. 4.

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Compare analytical solutions for (a) $Δ t_{APT}$ and (b) g_s to the numerical simulation results, as δ changes from 0 to $2 (J_{1} + J_{2})$ ⁠. Blue solid curve labels the solutions for Eqs. (9) and (10), green solid curve labels the solutions for Eqs. (11) and (13), black solid curve labels the solutions for Eqs. (12) and (14), and the numerical simulation results are marked by red circle.

In summary, we constructed an anti-PT symmetric SSH lattice in the frequency dimension based on a double-ring system with the antisymmetric AM dynamic modulations. In the time domain, the time-window with gain of the anti-PT symmetric system can be controlled by the phase-mismatching of the AM modulations, which is flexibly and conveniently able to be used to shorten the temporal width of the pulse. In addition, the shortening of the pulse is highly sensitive to the phase transition point, at which the anti-PT symmetry is completely broken. This work exploits the concept of anti-PT symmetry in lasing systems to realize the active ML and shows broad application prospects in ultrafast spectroscopy and ultra-high sensitive sensors. Additionally, this work provides a convenient way to construct various non-Hermitian topological insolators in the synthetic frequency dimension, which is difficult to be realized in realistic systems. Then, we can expect to explore the interesting physical phenomenon of the synthetic non-Hermitian topological lattices in further studies.

SUPPLEMENTARY MATERIAL

See the supplementary material for the detailed derivations of the anti-PT symmetric SSH frequency lattice, the validity of the tight-binding model with RWA, and the numerically computed evolutions for the full time-dependent dynamical equations.

This work was supported by the Natural Science Foundation of Zhejiang Province, China (Grant No. LQ23A040001), the general project of Zhejiang Provincial Education Department (No. Y202146469), and the Open Foundation Project of Hubei Provincial Key Laboratory of Optical Information and Pattern Recognition, Wuhan Institute of Technology (No. 202203).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Yiling Song: Conceptualization (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Software (equal); Writing – original draft (lead); Writing – review & editing (lead). Shaolin Ke: Funding acquisition (supporting); Investigation (supporting). Yuelan Chen: Investigation (supporting). Mingfeng Wang: Investigation (supporting); Project administration (supporting).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Mode-locking in anti-PT symmetric frequency lattices

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