Low-Threshold Lasing with Frozen Mode Regime and Stationary Inflection Point in Three Coupled Waveguide Structure

TL;DR

We address the challenge of achieving low-threshold lasing in slow-light photonic structures by designing a three-waveguide unit cell that supports stationary inflection points (SIPs) in the dispersion. Using transfer-matrix methods and Bloch analysis, the authors show that the SIP condition corresponds to a triply degenerate coalescence of eigenvalues/eigenvectors and can be tuned to occur at multiple frequencies in the Brillouin zone, including two almost-overlapping SIPs. Finite-length devices with $N$ unit cells exhibit progressively sharper, higher-Q resonances near the SIP frequency, and lasing threshold scales as $N^{-3}$, beating regular band-edge lasers whose threshold decays as $N^{-1}$. These SIP-based lasers offer a scalable, integrated route to low-threshold, slow-light lasers with controllable dispersion features, suitable for compact photonic systems.

Abstract

The frozen mode regime is a unique slow-light scenario in periodic structures, where the flat-bands (zero group velocity) are associated with the formation of high-order stationary points (aka exceptional points). The formation of exceptional points is accompanied by enhancement of various optical properties such as gain, Q-factor and absorption, which are key properties for the realization of wide variety of devices such as switches, modulators and lasers. Here we present and study a new integrated optical periodic structure consisting of three waveguides coupled via micro-cavities and directional coupler. We study this design theoretically, demonstrating that a proper choice of parameters yields a third order stationary inflection point (SIP). We also show that the structure can be designed to exhibit two almost-overlapping SIPs at the center of the Brillouin Zone. We study the transmission and reflection of light propagating through realistic devices comprising a finite number of unit-cells and investigate their spectral properties in the vicinity of the stationary points. Finally, we analyze the lasing frequencies and threshold level of finite structures (as a function of the number of unit-cells) and show that it outperforms conventional lasers utilizing regular band edge lasing (such as DFB lasers).

Figures (10)

• Figure 1: Three periodic waveguides coupled with ring resonator and directional coupler structure. The boundaries of the unit cell are marked by a dashed red line.
• Figure 2: The two sections of a unit cell: (a) the directional coupler, and (b) the Add- drop multiplexer of length $2d$.
• Figure 3: 'set 1' parameters dispersion relation (a) Logarithmic scale (using base of ten) of the determinant $|\textbf{M}-\textbf{I}\, e^{-ik\,2d}|$, for varying angular frequency and Bloch wavenumber. The SIP point is denoted with a red circle. (b) normalized $k$'s related to the eigenvalues of the transfer matrix $\textbf{M}$ at a range of frequencies near the SIP. The magenta lines indicates the propagating modes, marked as $\{k_p\}$. The light blue lines indicates the decaying modes, marked as $\{k_d\}$. Black-dotted lines and solid lines represent the imaginary and real parts of the normalized wave numbers, respectively.
• Figure 4: 'set 2' parameters dispersion relation (a) Logarithmic scale (using base of ten) of the determinant $|\textbf{M}-\textbf{I}\cdot e^{-ik\,2d}|$, , for varying angular frequency and Bloch wavenumber. The SIP is marked with a red circle (b) normalized $k$'s related to the eigenvalues of the transfer matrix $M$ at a range of frequencies near the SIP. The magenta lines indicate propagating modes, marked as $\{k_p\}$. The light blue lines indicate decaying modes, marked as $\{k_d\}$. Black-dotted lines and solid lines correspond to the imaginary and real parts of the normalized wave numbers, respectively.
• Figure 5: (a) Transmission and reflection from six ports of a finite-length waveguide structure with 60 unit cells. (b) Transmission from input port 3 to output port 3, near the SIP frequency. The parameters in 'set 1' are used.