## Abstract

Controlling and reversing the effects of loss are major challenges in optical systems. For lasers, losses need to be overcome by a sufficient amount of gain to reach the lasing threshold. In this work, we show how to turn losses into gain by steering the parameters of a system to the vicinity of an exceptional point (EP), which occurs when the eigenvalues and the corresponding eigenstates of a system coalesce. In our system of coupled microresonators, EPs are manifested as the loss-induced suppression and revival of lasing. Below a critical value, adding loss annihilates an existing Raman laser. Beyond this critical threshold, lasing recovers despite the increasing loss, in stark contrast to what would be expected from conventional laser theory. Our results exemplify the counterintuitive features of EPs and present an innovative method for reversing the effect of loss.

## Achieving gain despite increasing loss

When energy is pumped into an optically active material, the buildup (or gain) of excitations within the material can reach a critical point where the emission of coherent light, or lasing, can occur. In many systems, however, the buildup of the excitations is suppressed by losses within the material. Overturning conventional wisdom that loss is bad and should be minimized, Peng *et al.* show that carefully tweaking the coupling strength between the various components of a coupled optical system can actually result in an enhancement of the optical properties by adding more loss into the system (see the Perspective by Schwefel). The results may provide a clever design approach to counteract loss in optical devices.

Dissipation is ubiquitous in nature; the states of essentially all physical systems thus have a finite decay time. A proper description of this situation requires a departure from conventional Hermitian models with real eigenvalues and orthogonal eigenstates to non-Hermitian models featuring complex eigenvalues and nonorthogonal eigenstates (*1*–*3*). When tuning the parameters of such a dissipative system, its complex eigenvalues and the corresponding eigenstates may coalesce, giving rise to a non-Hermitian degeneracy, also called an exceptional point (EP) (*4*). The presence of such an EP has a dramatic effect on the system, leading to nontrivial physics with interesting counterintuitive features such as resonance trapping (*5*), a mode exchange when encircling an EP (*6*), and a singular topology in the parameter landscape (*7*). These characteristics can control the flow of light in optical devices with both loss and gain. In particular, waveguides with parity-time symmetry (*8*), where loss and gain are balanced, have attracted enormous attention (*9*, *10*), with effects such as loss-induced transparency (*11*), unidirectional invisibility (*12*), and reflectionless scattering (*13*, *14*) having already been observed.

Theoretical work indicates that EPs give rise to many more intriguing effects when they occur near the lasing regime; for example, enhancement of the laser linewidth (*15*, *16*), fast self-pulsations (*15*), and a pump-induced lasing death (*17*). Realizing such anomalous phenomena, however, requires moving from waveguides to resonators, which can trap and amplify light resonantly beyond the lasing threshold. With the availability of such devices (*18*, *19*), we discuss here the most counterintuitive aspect, namely, that close to an exceptional point, lasing should be inducible solely by adding loss to a resonator.

Our experimental system (*20*) consists of two directly coupled silica whispering-gallery-mode resonators (WGMRs), μR_{1} and μR_{2}, each coupled to a different fiber-taper, WG1 and WG2 (Fig. 1A and supplementary text S1). The resonance frequencies of the WGMRs were tuned to be the same via the thermo-optic effect, and a controllable coupling strength κ was achieved between the WGMRs by adjusting the interresonator distance. To observe its behavior in the vicinity of an EP, the system was steered parametrically via κ and an additional loss induced on μR_{2} by a chromium (Cr)–coated silica-nanofiber tip (Fig. 1, B and C), which features strong absorption in the 1550-nm band. The strength of was increased by enlarging the volume of the nanotip within the μR_{2} mode field, resulting in a broadened resonance linewidth with no observable change in resonance frequency (Fig. 1D). A small fraction of the scattered light from the nanotip coupled back into μR_{2} in the counterpropagating (backward) direction and led to a resonance peak whose linewidth broadened as the loss increased (Fig. 1E). The resonance peak in the backward direction was ~1/10^{4} of the input field, confirming that the linewidth broadening and the decrease of the resonance depth in the forward direction were due to via absorption and scattering losses, but not due to backscattering into the resonator.

In the first set of experiments, WG2 was moved away from μR_{2} to eliminate the coupling between them. We investigated the evolution of the eigenfrequencies and the transmission spectra * from input port 1 to output port 2 by continuously increasing while keeping κ fixed. In this configuration, losses experienced by μR*_{1} and μR_{2} were and , where is the WG1-μR_{1} coupling loss, and and include material absorption, scattering, and radiation losses of μR_{1} and μR_{2}. The coupling between the WGMRs led to the formation of two supermodes with complex eigenfrequencies whose real and imaginary parts are respectively denoted by and (*20*). Here, is the resonance frequency of the solitary WGMRs; and , respectively, quantify the total loss and the loss contrast of the WGMRs; and reflects the transition between the strong and the weak intermode coupling regimes due to an interplay of the interresonator coupling strength κ and the loss contrast (supplementary text S1). In the strong-coupling regime, quantified by and real β, the supermodes had different resonance frequencies (mode splitting of 2β) but the same linewidths quantified by χ. This was reflected as two spectrally separated resonance modes in [Fig. 2A (i)] and in the corresponding eigenfrequencies [Fig. 2B (i)]. Because our system satisfied , introducing to μR_{2} increased the amount of splitting until (that is, ) was satisfied [Fig. 2, A (ii) and B (ii)]. Increasing beyond this point gradually led to an overlap of the supermode resonances [Fig. 2A (iii)], such as to necessitate a fit to a theoretical model to extract the complex resonance parameters (supplementary text S1 and S2) (*20*). At where , the supermodes coalesced at the EP. With a further increase of , the system entered the weak-coupling regime, quantified by and imaginary β, which led to two supermodes with the same resonance frequency but different linewidths [Fig. 2, A (iv) and B (iv)]. The resulting resonance trajectories in the complex plane clearly displayed a reversal of eigenvalue evolution (Fig. 2B): The real parts of the eigenfrequencies of the system approached each other while their imaginary parts remained equal until the EP. After passing the EP, their imaginary parts were repelled, which resulted in an increasing imaginary part for one of the eigenfrequencies and a decreasing imaginary part for the other. As a result, one of the modes became less lossy, whereas the other became more lossy (supplementary text S4).

By repeating the experiments for different κ and , we obtained the eigenfrequency surfaces , whose real and imaginary parts are shown in Fig. 2, C and D. The resulting surfaces exhibit a complex square root–function topology with the special feature that a coalescence of the eigenfrequencies can be realized by varying either κ or alone, which leads to a continuous thread of EPs along what may be called an exceptional line. As expected, the slope of this line is such that stronger κ requires higher to reach the EP (supplementary text S4).

Our second set of experiments was designed to elucidate the effect of the EP on the intracavity field intensities. For this we used both WG1 and WG2, introducing an additional coupling loss to μR_{2} (that is, ). We tested two different cases by choosing different mode pairs in the resonators (*20*). In case 1, the mode in μR_{1} had a higher loss than the mode in μR_{2} ; in case 2, the mode in μR_{2} had a higher loss . The system was adjusted so that two spectrally separated supermodes were observed in the transmission spectra and as resonance dipped and peaked (supplementary text S3). No resonance dip or peak was observed at port 3. Using experimentally obtained and , we estimated the intracavity fields and and the total intensity as a function of (Fig. 3, A to C, and supplementary text S1, S2, S5, and S6). As was increased, first decreased and then started to increase despite increasing loss. This loss-induced recovery of the intensity is in contrast to the expectation that the intensity would decrease with increasing loss and is a direct manifestation of the EP.

The effect of increasing on and at is depicted in Fig. 3, A and B. When and the system was set in the strong-coupling regime, the light input at μR_{1} was freely exchanged between the resonators, establishing evenly distributed supermodes. As a result, the intracavity field intensities were almost equal. As was increased, and decreased continuously at different rates until reached a minimum at . The rate of decrease was higher for because of increasingly higher loss of μR_{2}. Beyond , until the EP was reached at , the system remained in the strong-coupling regime, but the supermode distributions were strongly affected by , which led to an increase of and, hence, of , whereas no appreciable change was observed for (*20*)*.* Increasing further pushed the system beyond the EP, thereby completing the transition from the strong-coupling to the weak-coupling regime during which increased and kept increasing, whereas continued decreasing. This behavior is a manifestation of the progressive localization of one of the supermodes in the less lossy μR_{1} and of the other supermode in the more lossy μR_{2}. We conclude that the nonmonotonic evolution of for increasing values of is the result of a transition from a symmetric to an asymmetric distribution of the supermodes in the two resonators (supplementary text S5 and S7).

The data shown in Fig. 3, A and B, also demonstrate that the initial loss contrast of the resonators affects both the amount of required to bring the system to the EP and the intensity values themselves (*20*): Increasing in case 2 increased to a higher value than that at ; in case 1, on the other hand, stayed below its initial value at . Finally, Fig. 3C shows that the intracavity field intensities at and coincide when ; that is, after the EP transition to the weak coupling regime (*20*). This is a direct consequence of the coalescence of eigenfrequencies at (supplementary text S5).

Whispering-gallery-mode microresonators (*21*) combine high-quality factor *Q* (long photon storage time, narrow linewidth) and high-finesse *F* (strong resonant power build-up) with microscale mode volume *V* (tight spatial confinement, enhanced resonant field intensity) and are thus ideal for studying quantum electrodynamics (*22*), optomechanics (*23*), lasing (*24*), and sensing (*25*–*27*). The ability of WGMRs to provide high intracavity field intensity and long interaction time reduces thresholds for nonlinear processes. Therefore, loss-induced reduction and the recovery of intracavity field intensities should have a direct effect on the thermal nonlinearity (*28*) and the Raman lasing (*24*, *29*) in WGMRs.

Thermal nonlinearity in WGMRs is due to the temperature-dependent resonance-frequency shifts caused by material absorption of the intracavity field and the resultant heating (*20*, *28*). In silica WGMRs, this is manifested as thermal broadening (or narrowing) of the resonance line when the wavelength of a probe laser is scanned from shorter to longer (or longer to shorter) wavelengths. In our system, thermal nonlinearity was observed in as a shark-fin feature (Fig. 3D). With an input power of 600 μW, thermal broadening kicked in and made it impossible to resolve the individual supermodes [Fig. 3D (i and ii)]. When was introduced and gradually increased, thermal nonlinearity and the associated linewidth broadening decreased at first but then gradually recovered [Fig. 3D (iii and iv)]. This aligns well with the evolution of the total intracavity field as a function of loss (supplementary text S8).

Finally, we tested the effect of the loss-induced recovery of the intracavity field intensity on Raman lasing in silica microtoroids (*29*, *30*). The threshold for Raman lasing scales as (where *g*_{R} is the Raman gain coefficient), implying the importance of the pump intracavity field intensity in the process. With a pump in the 1550-nm band, Raman lasing in the silica WGMR takes place in the 1650-nm band. Figure 4 depicts the spectrum and the efficiency of Raman lasing in our system. The lasing threshold for μR_{1} was ~150 μW (Fig. 4B, blue curve). Keeping the pump power fixed, we introduced μR_{2}, which had a much larger loss than μR_{1}. This effectively increased the total loss of the system and annihilated the laser (Fig. 4A, gray curve). Introducing to μR_{2} helped to recover the Raman laser, whose intensity increased with increasing (Fig. 4A). We also checked the lasing threshold of each of the cases depicted in Fig. 4A and observed that as was increased, the increased at first but then decreased (Fig. 4B).

These observations are in stark contrast with what one would expect in conventional systems, where the higher the loss, the higher the lasing threshold. Surprisingly, in the vicinity of an EP, less loss is detrimental and annihilates the process of interest; more loss is good because it helps to recover the process. This counterintuitive effect happens because the supermodes of the coupled system readjust themselves as loss is gradually increased. When the loss exceeds a critical value, one supermode is mostly located in the subsystem with less loss and, thus, the total field can build up more strongly (*20*). As our results demonstrate, this behavior also affects nonlinear processes, such as thermal broadening and Raman lasing, that rely on intracavity field intensity.

Our system provides a comprehensive platform for further studies of EPs and opens up new avenues of research on non-Hermitian systems and their behavior. Our findings may also lead to new schemes and techniques for controlling and reversing the effects of loss in other physical systems, such as in photonic crystal cavities, plasmonic structures, and metamaterials.

## Supplementary Materials

## References and Notes

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**Acknowledgments:**Ş.K.O. and L.Y. conceived the idea and designed the experiments. B.P. and Ş.K.O. performed the experiments with help from H.Y. and F.M. Theoretical background and simulations were provided by B.P., Ş.K.O., S.R., M.L., C.M.B., and F.N. All authors discussed the results, and Ş.K.O., S.R., and L.Y. wrote the manuscript with input from all authors. This work was supported by Army Research Office (ARO) grant no. W911NF-12-1-0026. C.M.B. was supported by U.S. Department of Energy grant no. DE-FG02-91ER40628. F.N. is partially supported by the RIKEN iTHES Project, Multidisciplinary University Research Initiatives (MURI) Center for Dynamic Magneto-Optics, Grant-in-Aid for Scientific Research (S). S.R. and M.L. are supported by the Vienna Science and Technology Fund (WWTF) project no. MA09-030 and by the Austrian Science Fund (FWF) project no. SFB-IR-ON F25-P14, SFB-NextLite F49-P10.