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Loss-induced suppression and revival of lasing

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Science  17 Oct 2014:
Vol. 346, Issue 6207, pp. 328-332
DOI: 10.1126/science.1258004

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.

Science, this issue p. 328; see also p. 304

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 (13). 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), μR1 and μR2, 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 Embedded Image induced on μR2 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 Embedded Image was increased by enlarging the volume of the nanotip within the μR2 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 μR2 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 ~1/104 of the input field, confirming that the linewidth broadening and the decrease of the resonance depth in the forward direction were due to Embedded Image via absorption and scattering losses, but not due to backscattering into the resonator.

Fig. 1 Coupled WGM microresonators and the effect of loss.

(A) Illustration of the coupled resonators μR1 and μR2 with the fiber-taper couplers WG1 and WG2. (B) Optical microscope image of the resonators with WG1 and the Cr nanotip. ain, input field at WG1; a1 and a2, rotating intracavity fields of μR1 and μR2. (C) Scanning electron microscope image of the Cr nanotip. (D and E) Transmission spectra in the forward (D) and backward (E) direction.

In the first set of experiments, WG2 was moved away from μR2 to eliminate the coupling between them. We investigated the evolution of the eigenfrequencies and the transmission spectra Embedded Image from input port 1 to output port 2 by continuously increasing Embedded Image while keeping κ fixed. In this configuration, losses experienced by μR1 and μR2 were Embedded Image and Embedded Image, where Embedded Image is the WG1-μR1 coupling loss, and Embedded Image and Embedded Image include material absorption, scattering, and radiation losses of μR1 and μR2. The coupling between the WGMRs led to the formation of two supermodes with complex eigenfrequencies Embedded Image whose real and imaginary parts are respectively denoted by Embedded Image and Embedded Image (20). Here, Embedded Image is the resonance frequency of the solitary WGMRs; Embedded Image and Embedded Image, respectively, quantify the total loss and the loss contrast of the WGMRs; and Embedded Image 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 Embedded Image (supplementary text S1). In the strong-coupling regime, quantified by Embedded Image 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 Embedded Image [Fig. 2A (i)] and in the corresponding eigenfrequencies [Fig. 2B (i)]. Because our system satisfied Embedded Image, introducing Embedded Image to μR2 increased the amount of splitting until Embedded Image (that is, Embedded Image) was satisfied [Fig. 2, A (ii) and B (ii)]. Increasing Embedded Image 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 Embedded Image where Embedded Image, the supermodes coalesced at the EP. With a further increase of Embedded Image, the system entered the weak-coupling regime, quantified by Embedded Image 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).

Fig. 2 Evolution of the transmission spectra and the eigenfrequencies as a function of loss γtip and interresonator coupling strength κ.

(A) Transmission spectra Embedded Image showing the effect of loss on the supermodes. Blue and red curves denote the experimental data and the best fit using a theoretical model (supplementary text S1 and S2), respectively. (B) Evolution of the eigenfrequencies of the supermodes in the complex plane as Embedded Image was increased. Open circles and squares are the eigenfrequencies estimated from the measured Embedded Image. Dashed red and blue lines denote the best theoretical fit to the experimental data. (C and D) Eigenfrequency surfaces in the (Embedded Image,Embedded Image) parameter space (supplementary text S4).

By repeating the experiments for different κ and Embedded Image, we obtained the eigenfrequency surfaces Embedded Image, 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 Embedded Image 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 Embedded Image 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 Embedded Image to μR2 (that is, Embedded Image). We tested two different cases by choosing different mode pairs in the resonators (20). In case 1, the mode in μR1 had a higher loss than the mode in μR2 Embedded Image; in case 2, the mode in μR2 had a higher loss Embedded Image. The system was adjusted so that two spectrally separated supermodes were observed in the transmission spectra Embedded Image and Embedded Image as resonance dipped and peaked (supplementary text S3). No resonance dip or peak was observed at port 3. Using experimentally obtained Embedded Image and Embedded Image, we estimated the intracavity fields Embedded Image and Embedded Image and the total intensity Embedded Image as a function of Embedded Image (Fig. 3, A to C, and supplementary text S1, S2, S5, and S6). As Embedded Image was increased, Embedded Image 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.

Fig. 3 Loss-induced enhancement of intracavity field intensities and thermal nonlinearity in the vicinity of an exceptional point.

(A and B) Intracavity field intensities of the resonators at Embedded Image (blue, Embedded Image of μR1; green, Embedded Image of μR2; red, total Embedded Image) for (A) case 1 and (B) case 2. Normalization was done with respect to the total intensity at Embedded Image. a.u., arbitrary units. (C) Total intracavity field intensities Embedded Image at eigenfrequencies Embedded Image (black) and Embedded Image (red) for case 1 (see supplementary text S5 for case 2). Normalization was done with respect to the intensity at the EP. (D) Effect of loss on nonlinear thermal response of coupled resonators (supplementary text S8): (i) solitary resonator, (ii) coupled resonators with Embedded Image, and (iii and iv) coupled resonators with increasing Embedded Image(20). Circles in (A) to (C) and squares in (C) were calculated from experimentally obtained transmissions Embedded Image and Embedded Image, whereas solid and dashed curves are from the theoretical model (supplementary text S1 to S3).

The effect of increasing Embedded Image on Embedded Image and Embedded Image at Embedded Image is depicted in Fig. 3, A and B. When Embedded Image and the system was set in the strong-coupling regime, the light input at μR1 was freely exchanged between the resonators, establishing evenly distributed supermodes. As a result, the intracavity field intensities were almost equal. As Embedded Image was increased, Embedded Image and Embedded Image decreased continuously at different rates until Embedded Image reached a minimum at Embedded Image. The rate of decrease was higher for Embedded Image because of increasingly higher loss of μR2. Beyond Embedded Image, until the EP was reached at Embedded Image, the system remained in the strong-coupling regime, but the supermode distributions were strongly affected by Embedded Image, which led to an increase of Embedded Image and, hence, of Embedded Image, whereas no appreciable change was observed for Embedded Image (20). Increasing Embedded Image further pushed the system beyond the EP, thereby completing the transition from the strong-coupling to the weak-coupling regime during which Embedded Image increased and kept increasing, whereas Embedded Image continued decreasing. This behavior is a manifestation of the progressive localization of one of the supermodes in the less lossy μR1 and of the other supermode in the more lossy μR2. We conclude that the nonmonotonic evolution of Embedded Image for increasing values of Embedded Image 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 Embedded Image required to bring the system to the EP and the intensity values themselves (20): Increasing Embedded Image in case 2 increased Embedded Image to a higher value than that at Embedded Image; in case 1, on the other hand, Embedded Image stayed below its initial value at Embedded Image. Finally, Fig. 3C shows that the intracavity field intensities at Embedded Image and Embedded Image coincide when Embedded Image; that is, after the EP transition to the weak coupling regime (20). This is a direct consequence of the coalescence of eigenfrequencies Embedded Image at Embedded Image (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 (2527). 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 Embedded Image 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 Embedded Image 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 Embedded Image (where gR 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 μR1 was ~150 μW (Fig. 4B, blue curve). Keeping the pump power fixed, we introduced μR2, which had a much larger loss than μR1. This effectively increased the total loss of the system and annihilated the laser (Fig. 4A, gray curve). Introducing Embedded Image to μR2 helped to recover the Raman laser, whose intensity increased with increasing Embedded Image (Fig. 4A). We also checked the lasing threshold of each of the cases depicted in Fig. 4A and observed that as Embedded Image was increased, the Embedded Image increased at first but then decreased (Fig. 4B).

Fig. 4 Loss-induced suppression and revival of Raman lasing in silica microcavities.

(A) Raman lasing spectra of coupled silica microtoroid resonators as a function of increasing loss. dBm, decibel-milliwatts. (B) Effect of loss on the threshold of Raman laser and its output power. The inset shows the normalized transmission spectra Embedded Image in the pump band obtained at very weak powers for different amounts of additional loss. Loss increases from top to bottom. The curves with the same color code in (A), (B), and the inset of (B) are obtained at the same value of additionally introduced loss.

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

www.sciencemag.org/content/346/6207/328/suppl/DC1

Supplementary Text

Figs. S1 to S16

Table S1

References (3134)

References and Notes

  1. Supplementary materials are available on Science Online.
  2. 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.
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