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.
Moreover, we plot the pulse width parameter and the saturated gain gs for the anti-PT symmetric coupled-ring system, as the value of δ changes from 0 to . In Fig. 4, gs and are normalized to the corresponding values for δ = 0. In the regime of , 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 (marked by red circle). We can notice that the analytical solutions of the for are disappeared as , which is because that the time-window with gain makes the energy of the light for 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 gs in Eq. (9), as shown for the regime of 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 . The calculated pulse width parameters for 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 [green solid curve in Fig. 4(a)] displays a great shortening as δ is approaching to , and is much narrower than that for [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 , at which the anti-PT symmetry of the system is completely broken, while the saturated gain for the regime of is almost constant, in accordance with the analytic theories for Eqs. (11) and (12). Moreover, the numerical simulation results for (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.
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.
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).
Conflict of Interest
The authors have no conflicts to disclose.
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).
The data that support the findings of this study are available from the corresponding author upon reasonable request.