1. Introduction

In the past few years, parity-time (PT) symmetry has been recognized as a means to control the modal content of microcavity lasers [1–6]. In this regard, active coupled microring arrangements subject to a non-uniform pump distribution have been demonstrated to support only one longitudinal mode, while effectively suppressing emission from other cavity resonances [3]. This behavior is attributed to the unequal PT-symmetry breaking thresholds for various modes of the cavity, due to the intrinsic lineshape of the gain system. Nevertheless, such lasers, even when operating in the single longitudinal mode regime, are in principle capable of generating light, traveling in two opposite directions. In most actual settings, these two counter-propagating degenerate modes couple to each other through scatterings or imperfections, leading to further frequency splitting and spectral complexity. Of interest in many applications is to design lasers that only support a single longitudinal mode that circulates in the cavity in a unidirectional fashion [7,8].

Unidirectional lasing in ring-type cavities has been a subject of intense research for several decades. Of course, in free space settings, one can incorporate an isolator to force the laser to operate in a desired direction. However, later on, it was shown that active configurations can lase in a unidirectional fashion even in the absence of an isolator, as long as one can break the rotational symmetry of the structure. This concept was first demonstrated in a pioneering work by Hohimer et al, where unidirectional single-mode lasing was achieved in an integrated platform without resorting to an isolator [7]. In their work, they used an S-shaped waveguide inside the ring to provide an asymmetric coupling of light between the clockwise (CW) and the counterclockwise (CCW) modes. Depending on the chirality of the S-bend, this resulted in the suppression of one of the counter-propagating modes. Since then, different variations of this technique have been pursued by several groups, where the geometry of the cavity is deformed to facilitate asymmetric wave propagation [9–16]. More recently, however, ring cavities are being designed so that hidden symmetries of the structure are instead broken in order to achieve unidirectional lasing [17–19]. In particular, it has been shown that a single PT-symmetric ring resonator subject to a periodic gain-loss and index modulation in the azimuthal direction [17], as well as a ring resonator with two Rayleigh scatterers in the periphery [18,19] can support unidirectional propagation. Albeit, in both cases the direction of power flow in the ring cannot be determined a priori to device fabrication.

In this work, our goal is to investigate the interplay between hidden and apparent symmetry violations as a means for enforcing single-mode unidirectional lasing operation in microring configurations in a deterministic fashion. We will study PT-symmetric coupled cavity systems that are subject to gain and loss, where the unidirectional lasing is achieved by incorporating S-bends with opposite chirality in the two adjacent resonators. The S-bend breaks the chiral symmetry of the ring system by allowing coupling from CCW (CW) to CW (CCW), while prohibiting the energy exchange in the opposite direction. We revisit this concept from a fresh prespective, where we show that this unidirectional oscillation is associated to the emergence of a new type of exceptional point. We provide experimental results that corroborate this unidirectional lasing behavior below and above the PT-symmetry breaking point. Furthermore, we show how this approach can be applied to extended coupled systems with non-local couplings. Such unit cells are used as a building block of photonic topological insulator lasers [20]. The paper is structured as follows. In section 2, we explain the mechanisms that lead to unidirectional lasing in a single ring resonator with an S-bend. In section 3, unidirectional single-mode lasing will be studied in PT-symmetric arrangements. Section 4 extends this approach to ring resonators coupled through a link. Finally, we conclude the paper in Section 5.

2. Unidirectional propagation in a single ring resonator

To enforce unidirectional lasing in ring resonators, we adopt a variation of the S-bend technique introduced in [7,21]. The S-bend is tapered at both ends in a way to prevent back-reflection. These tapered sections operate effectively as photon dumpers, which is essential for attaining unidirectional lasing. A schematic of such an S-bend ring with a bus waveguide is displayed in Fig. 1. The bus waveguide is equipped with a pair of second-order grating couplers in order to probe the emission directionality of the ring laser. To analyze the modal behavior of a ring with an S-bend, one can use temporal coupled mode formulations to describe the evolution dynamics of the clockwise and counterclockwise fields ($E_{CW}$ and $E_{CCW}$, respectively) inside the ring resonator as follows:

 

Fig. 1 Schematic of a ring laser with an S-bend inside. CW and CCW field amplitudes are labeled with their respective directions of propagation. A bus waveguide with a second-order grating is placed beside the ring to monitor the output power from the two counter-propagating directions.

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On the other hand, one may resort to a more detailed spatial coupled mode theory to analyze the behavior of this configuration [16]. As it is shown in Fig. 1, starting at the coupling region between the ring with the incorporated S-bend and the bus waveguide, the clockwise and counterclockwise field amplitudes of the n-th roundtrip are denoted as $E_{\text{n},cw}$ and $E_{\text{n},ccw}$. The outcoupled field amplitudes in the bus waveguide are $E_{out,cw}$ and $E_{out,ccw}$. We assume all the shown coupling regions have identical field through- and cross- coupling coefficients, σ and κ, respectively [16]. Then the n + 1th roundtrip amplitudes are given by:

To quantify the performance of the S-bend, we can define a unidirectionality factor $U$ which is given by the ratio of the outcoupled CCW power to the CW power in the bus waveguide:

Combining Eq. (3) and Eq. (4), we can find that unidirectional CCW emission occurs as ($U$ → ∞). Under this condition, the lasing mode is expected to be unidirectional if the threshold condition of the ring laser is met.

To further clarify the effect of the S-bend waveguide in promoting unidirectional lasing, we used a set of computational tools to perform finite-difference-time-domain (FDTD) and finite element method (FEM) simulations. Using FDTD, we examined temporal evolution of a pulse launched in the S-cavity, both in the CW and CCW directions. Figure 2(a) shows the time dependence for the CCW excitation, where an exponential decay in the pulse peaks is observed. Using this method, the differential photon lifetime can be deduced from the corresponding lifetimes of the two counter-propagating modes of the chiral cavity. The field distribution of the prevalent spinning mode in this active S-resonator is shown in Fig. 2(b), featuring a high degree of power-recirculation through the S-bend that is responsible for the spin-like mode discrimination. In the CCW excitation, the S-bend remains dark in its middle part, indicating no power redirection in this case. The modal response of the device is further simulated using finite element method. Figure 2(c) depicts a top view of the normalized intensity of the electric field in the ring with an S-bend. Here, the relatively uniform intensity of the electric field, rather than the usual standing wave pattern, is a direct indication of the unidirectional power flow in the ring resonator. While a standing wave pattern indicates interference between the two counter propagating modes, it should be noted that the uniform field distribution by itself does not provide adequate information to determine the chirality of the prevailing traveling wave mode [22].

 

Fig. 2 (a)Time evolution of the electrical field and mode analysis in an S-bend microring resonator. (b) Field distribution in an individual S-element as obtained from FDTD-simulations. (c) Normalized intensity of the electrical field in the ring with an S-bend obtained using FEM.

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Next, we experimentally verify the unidirectional light generation in an active microring resonator (radius: 20 μm, width: 500 nm, height: 210 nm) with an S-bend. This structure is fabricated on an InGaAsP quantum well wafer. The cross section of the microring cavity used in our experiment is shown in Fig. 3(a). The widths and heights of the rings are chosen so as to ensure single transverse mode operation (TE₀-like) as shown in Fig. 3(b). To avoid scattering losses, the coupling between the S-bend and the ring is provided through proximity (the nominal separation between the S-bend and the microring is 100 nm). In our experiments, the ring is optically pumped by a pulsed fiber laser with an average pump intensity of $5kW/c{m}^{{}_{{}_{}^2}}$_{at 1064 nm (pulse duration: 15 ns, repetition rate: 290 kHz). The direction resolved emission spectrum is collected through a bus waveguide (Fig. 3(c, d)), and is sent to a spectrometer through second-order gratings incorporated on each side of the waveguide. In our measurements, we intentionally left the bus waveguide and its associated gratings unpumped. The total power directed from each grating is calculated by integrating the power spectrum over a 60 nm wavelength range. The intensity profiles of the rings are also collected using a camera. Figures 3(e, g) and 3(f, h) compare the intensity profiles and emission spectra of the ring cavities without and with an S-bend, respectively. Our experimental studies show a 28 dB extinction ratio between the two counter-propagating modes in a cavity with an incorporated S-bend. This is in sharp contrast to a standard resonator, where a less than 0.1 dB power extinction ratio is measured.}

 

Fig. 3 Lasing emission without/with S-bend. (a) Cross section of the microring cavity. (b) Normalized electric field magnitude of the TE polarized mode of the waveguide. (c, d) Schematic of microring resonators without and with an S-bend. (e, f) Intensity profiles of the resonators and the outcoupling gratings. (g, h) Emission Spectra, demonstrating a 28 dB extinction ratio of the clockwise mode with respect to the counter-clockwise mode when incorporating the S-bend.

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3. Unidirectional single-mode emission in PT-symmetric microring lasers

Microring resonators exhibit a number of desirable features like high quality factor, small footprints and simplicity of fabrication [23]. These properties make them promising candidates for on-chip microscale lasers. However, these cavities also tend to support multiple longitudinal modes with comparable quality factors over the gain bandwidth of most III-V semiconductor gain systems – an aspect that leads to low spectral purity and output power fluctuations [24]. To resolve this issue, in the past few years, our group has shown how in a non-Hermitian system, a selective breaking of PT-symmetry can lead to robust, lossless and self-adapting single frequency lasing [3]. In these studies, PT-symmetry was established by applying a non-uniform pump distribution to a system of coupled active microrings. In the unbroken PT-symmetric regime, the entire spectrum of the system is real-valued. On the other hand, above a certain gain/loss contrast, which depends on the coupling strength between two microring resonators, PT-symmetry is spontaneously broken. In this case, the spectrum becomes partially complex. In other words, some of the modes experience either gain or loss, while the rest of them remain neutral. When used strategically, this property can result in single longitudinal mode operation in microring lasers [2–5,25,26].

In terms of mode directionality however, the standard coupled PT-symmetric laser, similar to a single ring, supports CW and CCW modes in both rings. This bi-directionality may have dire consequences in some applications. For example, the roughness of the sidewall of microring resonators results in coupling between the modes in opposite directions, thus leading to the formation of a set of supermodes with both CW and CCW components. This phenomenon is highly detrimental in ring laser gyroscopes as it limits the minimum detectable rotation rate – an effect that is known as lock-in. This can also affect the detection limit of non-Hermitian microcavity sensing systems [27,28]. Here, we show how one can avoid the complications arising due to the coupling between two counter-propagating modes by incorporating an active S-bend element in each of the microrings in the PT-symmetric arrangement. Clearly, in such coupled systems, the CW mode in one resonator couples to the CCW mode in the other ring. Therefore, the chirality of the S-bends must be flipped from one resonator to the other (Fig. 4). Figure 4(a) shows an SEM image displaying how the S-bends are implemented in a PT-symmetric photonic molecule. Figure 4(b) depicts how this system is expected to behave when both rings are pumped.

 

Fig. 4 Design of unidirectional PT-symmetric lasers. (a) SEM image of coupled microresonators incorporated with an S-bend inside the cavities. (b) Schematic of unidirectional light emission in a PT-symmetric microring system.

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In order to experimentally verify these aspects and particularly to study single-mode unidirectional emission from the PT-symmetric lasers, we fabricated two coupled microring resonators with S-bend elements. The rings are accompanied by a pair of symmetrically positioned bus waveguides. We characterized this arrangement under evenly-pumped as well as PT-symmetrically pumped conditions. In the case of the PT-symmetric configuration, one of the rings is left unpumped by partially blocking the pump laser using a knife edge. The rings in this study have a radius of 10 μm, a width of 500 nm, and a depth of 210 nm. In addition, the minimum separation between the rings as well as the minimum separation between the S-waveguides and the rings are both set at a 100 nm. Figure 5 displays the experimental results concerning the behavior of such a system below and above the PT-symmetry breaking point. Our experiments indicate that the S-bends indeed globally enforce unidirectional light emission over the entire parameter space (different pumping contrasts). Above the PT-symmetry breaking point, lasing occurs exclusively in the active ring, where single-mode operation is achieved with a side mode suppression ratio of nearly 20 dB.

 

Fig. 5 Unidirectional PT-symmetric microring lasers. (a, b) Camera images from the coupled microcavities and outcoupling gratings. (c, d) Corresponding spectra (log scale) below and above the PT-symmetric breaking point.

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4. Coupled rings through a link

To further extend the use of an S-bend in enforcing chiral emission from ring lasers, we also studied systems of resonators comprised of two rings (radii: 10 μm) with embedded identical chiral S-bend elements. These two rings are coupled to each other through an intermediate link cavity as shown in Fig. 6(a). Such structures have been recently used as a building block for non-magnetic topological insulator lasers [20]. The intensity profile (Fig. 6(b)) recorded on an NIR camera shows that the incorporation of S-bends in this system (only in the rings) also leads to unidirectional light emission.

 

Fig. 6 Chiral emission behavior in more complex systems. (a) Schematic of resonators comprised of two rings coupled through an intermediate link cavity (S-bends are imbedded only in the ring resonators). (b) Intensity profile of the resonators and outcoupling gratings (radii: 10 μm).

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5. Conclusion

In conclusion, we have demonstrated that the introduction of an S-bend in an active microroing resonator enables unidirectional light generation. This unidirectional behavior is an outcome of the non-Hermiticity. It indicates the emergence of a new type of exceptional point in such arrangements. This technique can enforce PT-symmetric lasers to operate in a unidirectional fashion and as a result may have practical ramifications in demonstrating PT-symmetric micro-scale gyroscopes and lasers. The S-bend elements may additionally be used in active topological photonic systems as a strategy to endow photons with an “effective charge” as shown in recent studies.