Anomalous Single-Mode Lasing Induced by Nonlinearity and the Non-Hermitian Skin Effect

Bofeng Zhu, Qiang Wang, Daniel Leykam, Haoran Xue, Qi Jie Wang, and Y. D. Chong

Phys. Rev. Lett. 129, 013903 – Published 29 June 2022

Abstract

Single-mode operation is a desirable but elusive property for lasers operating at high pump powers. Typically, single-mode lasing is attainable close to threshold, but increasing the pump power gives rise to multiple lasing peaks due to inter-modal gain competition. We propose a laser with the opposite behavior: multimode lasing occurs at low output powers, but pumping beyond a certain value produces a single lasing mode, with all other candidate modes experiencing negative effective gain. This phenomenon arises in a lattice of coupled optical resonators with non-fine-tuned asymmetric couplings, and is caused by an interaction between nonlinear gain saturation and the non-Hermitian skin effect. The single-mode lasing is observed in both frequency domain and time domain simulations. It is robust against on-site disorder, and scales up to large lattice sizes. This finding might be useful for implementing high-power laser arrays.

Received 3 March 2022
Accepted 7 June 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.013903

Physics Subject Headings (PhySH)

Laser dynamics Nonlinear optics Skin effect

Condensed Matter, Materials & Applied PhysicsGeneral PhysicsNonlinear Dynamics

Authors & Affiliations

Bofeng Zhu^1,2, Qiang Wang ¹, Daniel Leykam³, Haoran Xue ¹, Qi Jie Wang ^1,2,4,*, and Y. D. Chong^1,4,†

¹Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
²School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 637371, Singapore
³Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore
⁴Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore 637371, Singapore

^*qjwang@ntu.edu.sg
^†yidong@ntu.edu.sg

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Vol. 129, Iss. 1 — 1 July 2022

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Images

Figure 1
(a) Schematic of a 2D square lattice exhibiting SSEM lasing. The sites (red circles) have uniformly pumped saturable gain and linear loss, with asymmetric nearest neighbor couplings $t_{1} \pm δ_{t}$ (orange and blue arrows). (b) Schematic of a possible realization based on a lattice of coupled ring resonators, with gain (pink) and loss (gray) on the coupling rings. One circulation direction is assumed, indicated by the arrows. (c) Output intensity $I_{out}$ versus pump strength $Γ$ for a $6 \times 6$ lattice with $t_{1} = 0.1$ , $δ_{t} = 0.04$ , and background loss $γ_{0} = 0.2$ . Blue circles show time domain results, with initial conditions $ψ_{m n} (t = 0) = (α_{m n} + i β_{m n}) f_{0}$ , where $(m, n)$ is the site index, $α_{m n}, β_{m n}$ are drawn independently from the standard normal distribution, and $f_{0} = 0.01$ is a scale factor; $I_{out}$ is obtained by averaging over an interval $t \in [20000, 50000]$ . The orange line shows the results of a self-consistent single-mode frequency domain calculation in the large- $Γ$ regime. The threshold $Γ_{th} = 0.2$ is indicated, while blue and red arrows mark the $Γ$ values used in the next subplot. (d) Output spectrum for $Γ = 0.24$ (blue dashes) and $Γ = 0.7$ (red line), using the same lattice parameters as in (c).
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Figure 2
(a) Effective gain $Im [E]$ versus pump strength $Γ$ for the various lattice eigenmodes. Hollow circles are from the eigenenergies of the nonlinear Hamiltonians obtained via time domain simulations, using time-averaged intensities to calculate gain saturation. Orange lines are frequency domain results obtained in the single-mode and below-threshold regimes. (b) IPR, a measure of localization, for the eigenmodes at the lasing threshold $Γ_{th}$ . The arrows indicate two modes at opposite band edges, which are related by $C T$ . (c) $Re (E)$ versus $Γ$ for the two modes indicated in (b). With increasing $Γ$ , they undergo a $C T$ transition, and one of the resulting modes is the SSEM. (d) Intensity distributions of the modes shown in (c) at different $Γ$ . The modes in each $C T$ pair have the same intensities, so only one is plotted. Like the original band edge modes, the SSEM is delocalized, even for large $Γ$ . (e) Phase diagram for the laser, indicating the below-threshold, multimode, and single-mode regimes, determined by time domain simulations for different values of the coupling asymmetry $δ_{t}$ and pump strength $Γ$ . The single-mode regime is divided into SSEM lasing and $C T$ -broken lasing, as described in the text. All other lattice parameters are the same as in Fig. 1.
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Figure 3
(a) Output spectrum for $δ_{t} = 0$ , obtained from a time domain simulation with lattice size $6 \times 6$ , $Γ = 1$ , and all other parameters the same as in Fig. 1. Since the couplings are symmetric, the NHSE is absent. Multimode lasing is observed for all values of pump strength $Γ > Γ_{th}$ we tested. (b) Gain difference $Δ γ$ between the single lasing mode and the next-highest-gain mode, plotted against pump strength $Γ$ in the SSEM lasing regime for lattices of varying size $L \times L$ . Larger $Δ γ$ implies that the laser is further from the multimode regime. Evidently, $Δ γ$ increases with both $Γ$ and $L$ , and its increase with $Γ$ , for fixed $L$ , is also consistent with Fig. 2. (c) Effects of on-site disorder. Left panel: output spectra, plotted with different vertical offsets, in lattices with real on-site detunings $f δ_{m n}$ , where $f$ is a scale factor and $δ_{m n}$ is drawn independently from the standard normal distribution for each site. The lattice is pumped at $Γ = 1$ , with all other parameters the same as in Fig. 1. The results are shown for several values of $f$ , using the same $δ_{m n}$ distribution for each case. For sufficiently weak disorder, single-mode lasing is observed, though the lasing mode is not pinned to zero. For strong disorder, multimode lasing occurs. Right panel: the random disorder profile $δ_{m n}$ used to generate these results.
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