Properties of CQD microdisks

To fabricate CQD microdisk lasers, we have adopted a facile one-pot method to synthesize high quality compositional gradient core/alloyed-shell CQDs with an average size of 7.5 ± 0.8 nm (see Methods for detailed synthesis procedure). The gradient shell helps to significantly suppress the Auger recombination, which has been the main obstacle to achieve lasing in CQD films in the past. We then utilize a hybrid top-down and bottom-up approach that we previously developed to fabricate the microdisk laser arrays29 (details described in Methods). In brief, this approach combines standard photolithography that provides up-scalability and precise control of spatial distribution to these CQD assemblies, as well as layer-by-layer deposition to integrate tiny CQDs into photoresist-templated trenches along with the crosslinking of ligands to enhance the mechanical integrity (Supplementary Fig. 1). Figure 1a demonstrates a transmission electron microscopy (TEM) image of closely packed CQDs and Fig. 1b shows a scanning electron microscopy (SEM) image of the microdisk structure.

Fig. 1 Emission characteristics of isolated microdisk resonators. a Transmission electron microscopy (TEM) micrograph of oleic acid-capped CdSe/Cd 1−x Zn x Se 1−y S y quantum dots. b Scanning electron microscopy (SEM) image of a microdisk (scale bar is 20 μm). c Laser emission spectra from a microdisk pumped at various intensities: 16 μJ/cm2 (black), 29 μJ/cm2 (red), 66 μJ/cm2 (blue), 116 μJ/cm2 (green). The inset shows the FFT of the emission spectrum vs. the optical path length n eff πD, where n eff = 1.85 is the effective mode index and D = 25 μm. d Collection-angle dependence of the emission in the plane of the microdisk in b with fluorescent image at the center. e Laser emission spectrum from a different microdisk that contains a defect, pumped at 116 μJ/cm2. The different shaded regions are Lorentzian fits to the two laser modes series. f Same as in d but for the spectrum shown in e Full size image

Figure 1c shows the spectral evolution of the laser emission from an exemplary microdisk with increasing pump intensity. At low intensity, only a broad photoluminescence band (FWHM~25 nm) is observed, displaying a peak near 630 nm. Above the threshold intensity of 29 μJ/cm2, narrow laser modes emerge from the PL spectrum (Supplementary Fig. 2). The free-spectral range is Δλ≈2.6 nm and well described by the formula for WGM’s, namely Δλ = λ2/n eff (πD) where n eff is the effective mode index and D is the disk diameter (D = 25 ± 1 μm). By converting the spectral units to μm−1 and taking the Fourier transform, we extract a series of harmonics of the optical path length n eff (πD) (inset), giving the effective index n eff = 1.85 ± 0.05 which is in agreement with the CdSe/Cd 1−x Zn x Se 1−y S y CQD film refractive index (n = 1.86 ± 0.05) determined by ellipsometry30,31. In Fig. 1d, we show the angular dependence of the laser emission in the plane of the microdisk, which is isotropic, signaling a homogenous intensity distribution along the microdisk circumference. Additionally, the fluorescent image of the disk under lasing condition shows that the field amplitude is predominantly located along the perimeter.

In contrast, in Fig. 1e we present the emission spectra from another microdisk of the same size. In this case we are able to resolve two sets of modes, each having Δλ = 2.6 nm. The additional set of modes is split-off from the other WGM modes by ~0.8 nm. The linewidths of all modes are comparable (~0.5 nm) and the splitting does not vary much with increasing pump fluence. In the fluorescent image of Fig. 1f, it is seen that a few particularly bright spots have now appeared on the microdisk circumference.

Microscopic characterization reveals that these bright spots are due to fabrication defects (Supplementary Fig. 3). These defects then act as asymmetric scattering centers that increase out-coupling in the vertical direction giving them their bright appearance. In addition, we see that the defects strongly disrupt the emission isotropy, leading to strong directionality of the emission diametrically across from the defects. Atomic force microscopy has been used to investigate the formation of possible defects during fabrication (Supplementary Fig. 4). There are generally two types of defects in our microdisks that can induce the obtained mode-splitting. The first type is the small CQD aggregates that forms near the disk circumference. These aggregates may originate from the imperfections that exist in CQD solutions, which act as seeds to affect the drying process. The second type of defect is the difference in circumference height due to inhomogeneous drying. Both type of defects may act as localized perturbations that can introduce asymmetric back-scattering between clockwise (CW) and counter-clockwise (CCW) WGM propagation, lifting the degeneracy between them. This degeneracy lifting has been widely discussed in the context of nanoparticle sensing, where the mode-splitting is employed to register the detection event32.

Spectral properties of coupled microdisks

Considering the various phenomena that have been displayed in the non-Hermitian coupled WGM resonators, it is interesting to consider to what degree these interactions may be preserved in the presence of disorder. This information is imperative if such effects are to be eventually employed in a broad range of photonic applications. We therefore took to the investigation of coupled-pairs of our CQD micro-resonators fabricated from a facile, scalable process. The spacing between the two microdisks was 396 ± 20 nm apart (Supplementary Fig. 5), which is less than the emission wavelength and should provide adequate evanescent coupling between the WGM of the two microdisks. The coupling between microdisks was further verified using finite difference time domain (FDTD) simulations (Supplementary Figs. 6, 7). In Fig. 2a–c, we show the emission spectra obtained when the pair is placed at the center of the beam spot, so that both individual microdisks are pumped evenly (Fig. 2b), and when the left or right disks are pumped exclusively (Fig. 2a, c). This configuration resembles the PT-symmetric laser system that was theoretically proposed21 and experimentally demonstrated to achieve selective mode-filtering22. In all three cases we see a distinct laser emission spectrum, and we particularly note the appearance of mode-splitting in Fig. 2c when only the right disk is pumped. It is therefore surprising that the emission spectrum of the evenly-pumped pair shows no trace of the mode-splitting in the same spectral region (Fig. 2b). This is in stark contrast to the majority of single, isolated microdisks whose spectra contain parasitic mode-splitting. These results suggest that the coupling between the microdisks in the pair is helpful in reestablishing spectral purity in the laser emission spectrum.

Fig. 2 Emission characteristics of coupled pairs of microdisk resonators. a–c Laser emission spectra from a coupled microdisk pair, where the pair is placed at different locations in the pump beam spot, such that only the left or right microdisk is pumped (a and c), or the pair is pumped evenly, (b), as illustrated schematically above each plot. The insets show the fluorescent image from the microdisks in each case. d–f Laser emission spectra from three different microdisk pairs pumped at 116 μJ/cm2. In each figure the spectrum from the evenly-pumped pair (blue) is compared to a spectrum from the asymmetric pumping scheme (red). The evenly-pumped pair consistently shows a reduction in mode-splitting compared to the asymmetric case. g Frequency distribution of the spectrally averaged modal splitting parameter, \(\left\langle {\phi ^\lambda } \right\rangle\), (defined in Eq. 1) for the symmetric (blue) and asymmetric pumping schemes (red) Full size image

In Fig. 2d–f we demonstrate that this phenomenon is robust. It is observed in numerous samples in which at least one of the microdisks exhibits mode-splitting during the asymmetric pumping condition. To facilitate comparison over a large number of coupled-microdisks, we define an empirical parameter, ϕλ, to quantify the splitting between individual broken-degeneracy WGM mode-pair,

$$\phi ^\lambda = \frac{{2\left| {\lambda _1 - \lambda _2} \right|}}{{\Delta \lambda }}$$ (1)

where λ 1 and λ 2 are the peak wavelengths, and the wavelength splitting is normalized by half the free-spectral range Δλ. We then take the average \(\left\langle {\phi ^\lambda } \right\rangle\) of all mode-pairs in the spectrum (see Supplementary Fig. 8). Figure 2g shows the frequency distribution of \(\left\langle {\phi ^\lambda } \right\rangle\) obtained from both the symmetric and asymmetric pumping configurations. Indeed, the wide variability of the emission from asymmetrically pumped pairs, which echoes the variation in the emission spectrum of isolated microdisks (Supplementary Fig. 9), collapses to a narrower distribution with significantly reduced parasitic mode-splitting when the pair is pumped evenly. The results show that the coupling between microdisks in the presence of symmetric gain consistently corrects the sample variability that naturally results from a scalable, high-throughput fabrication procedure for micro-photonic elements such as these. As we show below, this is possible due to the presence of an EP in the Hamiltonian of the system that provides a channel through which intra-cavity mode splitting can be modulated through inter-cavity coupling.

Mode coalescence revealed through spatial gain variation

A detailed look at this behavior is provided by an examination of the laser emission modes under the application of spatial gain variation (Fig. 3). In these measurements, the microdisk pair is swept through the pump beam spot incrementally. The offset between the pump spot and the pair center, which we call ΔD p , then becomes a proxy for the gain/loss variation, Δg AB = g B –g A between the two microdisks, where we refer to the microdisk that shows parasitic splitting as microdisk A with gain, g A and its neighbor microdisk B with gain, g B . As seen in Fig. 3a, when ΔD p is large and negative, only microdisk A is pumped, and clear mode-splitting can be observed in the laser emission spectrum. Then as the pair of coupled microdisks is moved towards the center of the excitation beam spot and gain is added to microdisk B, we find that the localized intra-cavity modes gradually move closer in frequency and eventually coalesce near but not quite at the center. This is very surprising, as these modes are localized in microdisk A and the gain/loss differential does not change between them. Here, the application of gain to microdisk B modulates the splitting between the parasitic defect modes of microdisk A through the evanescent coupling between the two microdisks.

Fig. 3 Laser emission behavior of a microdisk pair under spatial gain variation a False-color contour plot of the emission intensity vs. wavelength and relative distance, ΔD p , between the center of the pair to the center of the beam spot, as the pair is shifted through the pump beam spot, which is schematically illustrated on the left. As the microdisk pair nears the center, the split modes merge as clearly seen for several mode pairs in the range of 635–645nm. b Mode splitting in the defected microdisk vs. ΔD p as obtained from the dashed-boxed region in a. The solid lines are eigenvalue dynamics as a function of the gain differential ∆g AB between the coupled microdisks in a three-mode Hamiltonian that shows coalescence of the intra-cavity modes in the defected microdisk due to the gain variation between the coupled microdisks. The model parameters are κ = 3.8 × 1011 s−1, γ 23 = 1.2 × 1012 s−1 and γ 13 = 0 s−1 Full size image

Eigenvalues of the three-mode non-Hermitian Hamiltonian

To understand this behavior in more detail, we consider a solution to the eigenvalue problem described by the following Hamiltonian:

$$\begin{array}{*{20}{c}} {\hat H = \left( {\begin{array}{*{20}{c}} {\omega {\prime}_1 + ig_A} & \kappa & {\gamma _{13}} \\ \kappa & {\omega {\prime}_2 + ig_A} & {\gamma _{23}} \\ {\gamma _{13}} & {\gamma _{23}} & {\omega {\prime}_3 + ig_B} \end{array}} \right)} \end{array}$$ (2)

where \(\omega {\prime}_{i = 1,2,3}\) are the real parts of the eigenfrequencies in the absence of coupling and g i = 1,2,3 are the imaginary parts, where we set g 1 = g 2 = g A , and similarly we set g 3 = g B for the mode 3 in microdisk B. κ is used to represent the intra-disk coupling between CW and CCW modes in microdisk A (modes 1 and 2), whereas γ i3 describes inter-disk coupling of ith mode of microdisk A to the mode 3 of microdisk B (see Fig. 4a). This is an extension of the 2 × 2 Hamiltonians often considered in non-Hermitian systems, where an interaction exists between one mode from each resonator22. In our case the defect-induced mode-splitting produces a scenario in which three modes interact, and the resulting Hamiltonian is 3 × 3. In order to simulate the portion of the spatial gain variation experiment in which ΔD p increases from −25 μm to zero we keep g A constant, g A = g (microdisk A is entirely covered by the pump beam) while sweeping g B from –g to g (microdisk B goes from unpumped to completely pumped). Thus, Δg AB ranges from –2 g to zero.

Fig. 4 Calculated eigenvalue dynamics of the three-mode system. a Schematic diagram of the modes considered and parameters governing their interaction. For microdisk A, a defect on the circumference (shown as the grey area) induces splitting between modes due to their mutual asymmetric backscattering determined by the parameter κ. These two modes can then couple to the degenerate modes in microdisk B through the inter-disk coupling parameters, γ 13 and γ 23 . We consider the case for which \(\gamma _{13} \ll \kappa \ll \gamma _{23}\). b, c Real and Imaginary parts of the eigenvalues of the three-mode Hamiltonian vs. the normalized gain differential, Δg AB /g between microdisks A and B. The dashed lines indicate the locations of the exceptional points EP 1 and EP 2 . The limiting values of the eigenvalue splitting are 2κ at Δg AB = −2 g and 2γ 23 at Δg AB = 0, while the minimum splitting between EP1 and EP2 is γ 13 /2 Full size image

Considering the boxed region in Fig. 3a, the experimentally observed phenomenon of intra-cavity mode coalescence can be satisfactorily replicated as demonstrated in Fig. 3b. The simulations show a typical branch root behavior of an EP that is the origin of the observed coalescence. By setting \(\omega {\prime}_1 = \omega {\prime}_2 = \omega {\prime}_3 = \omega _0\), where ω 0 is the average frequency of all three modes, the splitting at large Δg AB is determined by the intra-cavity mode coupling κ. The values of ω 0 , g, and κ are then taken from experiment30, leaving only γ 23 and γ 13 as free parameters. As discussed below, the key to the observation of the coalescence of intracavity modes is the existence of a large coupling anisotropy, \(\gamma _{23} \gg \gamma _{13}\), with an exact EP obtained in the limit γ 13 →0. Similar agreement is obtained between the experimental results and the model for each of the modes for which the coalescence is observed by adjusting the coupling term γ 23 (Supplementary Fig. 10).