Theoretical model

In the corresponding theoretical investigation, which we recently carried out28, our starting point was the nonlinear Maxwell-Bloch system of equations that contains the core relations of semi-classical laser theory. Using the newly developed steady-state ab-initio laser theory29 allowed us to rewrite these equations in their steady-state form, where the lasing modes and frequencies as well as the nonlinear terms, which describe their interactions, appear explicitly. The key insight that we derive from these expressions is that, in a parameter window around the first laser threshold, where the nonlinear terms are still weak, an EP can be induced in the corresponding solutions of these equations by changing the pump applied to the laser. A system that is particularly convenient for realizing such a pump-induced EP turns out to be a device consisting of two coupled lasers. In this specific case, it is sufficient to control the overall pump strength applied to each of the two individual lasers, rather than the spatial pump profile that would have to be carefully modified for setups using just a single laser.

For the purpose of illustrating this idea conceptually, we drastically reduce here the nonlinear lasing equations for this coupled laser (provided in ref. 28) to a simple linear form, where each mode in the laser equations is replaced by a resonant level. Restricting ourselves to only two such levels A and B, that is, one in each of the two laser resonators, we end up with a 2 × 2 matrix description. The diagonal elements in this matrix describe the levels A and B with frequencies a, b and effective gain or loss α, β. The off-diagonal elements γ stand for their coupling,

Consider now the situation of two levels with identical frequencies (a=b) and real coupling constant γ. Initially, one of the levels (A) experiences strong net-gain (α>0) and the second one (B) features strong net-loss (β<0), such that the condition α−β>>|γ| is fulfilled. For this case, the non-Hermitian matrix H has one eigenvalue in the upper half of the complex plane and one in the lower half, corresponding to one mode above and the other mode below the lasing threshold (as given here by the real axis). If, in a next step, we continuously add more gain to level B until both levels have the same gain (that is, we sweep β→α), we find the following curious behaviour (see Fig. 1a): first, the two eigenvalues approach each other such that the lasing level is pulled towards the lasing threshold (corresponding to a decrease of light emission). After passing the EP, where the two eigenvalues coalesce, this behaviour changes abruptly and the eigenvalues are repelled in their real parts and then both shifted beyond the lasing threshold (corresponding to an increase of light emission). We may thus conclude that such a device displays exactly the desired reversal of system properties associated with an EP. Note that this behaviour is robust in the sense that it also survives a slight detuning between the two level frequencies (see Fig. 1b) as well as a shift of γ into the complex plane (see Fig. 1c). In both cases, the eigenvalues then just pass the vicinity of an EP.

Figure 1: Movement of complex eigenvalues in the vicinity of an EP. The behaviour of a laser with two coupled modes can qualitatively be understood by the eigenvalues of the 2 × 2 matrix H given in the text. In a, the parametric dependence of these eigenvalues is shown for zero energy splitting between the two modes (a=b=1) and coupling strength γ=0.15. When one level is strongly amplified (α=0.1) and the other level is swept from being strongly attenuated (β=−0.4) to being strongly amplified (β→α, see arrows), the two eigenvalues first approach each other until they coalesce in an EP. After passing the EP, they are repelled in their real parts and shifted upwards in the complex plane. The real axis (dashed line) represents the lasing threshold and whenever it is crossed by an eigenvalue from above (below) a laser mode turns off (on). Note, that the coalescence of eigenmodes at the EP prohibits an unambiguous labelling of the modes. (b) The characteristic reversal of the eigenvalue movement observed in a is also seen for a finite frequency splitting (a=1.0, b=1.002), leading to an avoided eigenvalue crossing in the complex plane. (c) Numerical results for the complex resonance frequencies of the experimental device obtained with three-dimensional finite-element calculations. The movement of resonance frequencies under variation of the imaginary part of the index of refraction in disk B shows a similar avoided level crossing as in b. (Note that the avoidance of the level crossing as observed here for the coupling of the same modes in two identical disks is due to a complex value of the effective coupling constant γ, see refs 48, 49 for details.) An almost degenerate mode pair with a similar eigenvalue movement has been omitted for clarity. Full size image

We emphasize at this point that the 2 × 2 matrix description, which we employed above to illustrate our strategy, constitutes a crude approximation of the full problem as realized in the experiment. In particular, all nonlinear effects related to mode competition, gain pulling, as well as any temporal instabilities and the influence of other than the two considered modes are entirely neglected in this toy model. Also the frequency- and pump-dependence of the coupling between the two modes as well as all effects specific to the quantum cascade lasers (QCLs), which we work with in the experiment, are completely ignored, such that this model only has qualitative rather than quantitative predictive power. Still, as we will show below, our simple 2 × 2 matrix model can capture the essential physics of an EP quite well. To model the experiment quantitatively correctly, however, an appropriate treatment of the three-dimensional nature of the resonators and of their coupling is required (as provided below). An accurate description of the laser’s above-threshold behaviour necessitates, in addition, a fully nonlinear calculation (as provided in ref. 28).

Coupled microcavity QCLs

A natural experimental arena in which the above lasing effect could be realized is given by coupled ridge or microcavity lasers30,31 in each of which one level can be brought to lase. A direct implementation of such a photonic molecule laser, however, meets a number of challenges: first of all, the coupled laser needs to operate in the single-mode regime, as the onset of additional modes would give rise to nonlinear mode-competition effects, which may overshadow the effect of an EP. Owing to this requirement, the gain coefficient α>0 from above is restricted to rather small values for which only one mode is lasing in the first disk. Second, to realize the initial gain/loss-configuration α−β>>|γ|, from which we start the pump sweep (β→α), the loss coefficient β<0 must be very negative (for α being small), corresponding to strong absorption in the second disk. Third, the laser geometries as well as the coupling gap between them need to be engineered on the scale of the lasing wavelength such as to obtain sufficiently similar resonator frequencies as well as the right coupling strength.

Systems that we found to fulfil all of the above stringent criteria are photonic molecule QCLs operating in the THz regime30,32,33. Here, the gain is produced by transitions between quantized energy levels of semiconductor quantum wells, allowing us to adjust the emission wavelength by the quantum well widths. In our experiment, the emission wavelength (~100 μm) is comparable to the size of the actual coupled microdisk lasers, such that these devices feature both a stable regime of single mode operation as well as a much higher fault tolerance with respect to geometric imperfections as compared with corresponding lasers emitting in the visible spectrum of light. Furthermore, QCLs are electrically pumped devices; they provide a symmetric Lorentzian gain profile34 (in contrast to bandgap semiconductor lasers) and do not suffer from surface recombination. Specifically, we fabricated pairs of disk-shaped lasers, which we placed in close vicinity to each other in order to achieve sufficiently strong mode coupling. The active region of the laser is sandwiched, on top and at the bottom, by two metal layers, which act both as a waveguide and as a contact for electrically pumping the device. Owing to their finite conductivity, these metal layers provide much of the required loss already quite naturally. As, however, even higher loss values are necessary to observe an EP, we reduced the thickness of the active region and added an additional absorption layer (see Methods section). Figure 2a shows an image of a fabricated device.

Figure 2: Photonic molecule laser and its pump dependence. (a) Image of the studied photonic molecule quantum cascade laser taken with a scanning electron microscope (disk radius r=47 μm, height h=3.5 μm, inter-cavity distance d=2 μm, Ti absorption layer thickness d abs =90 nm). (b) Configuration for spatially integrated measurements. A high collection efficiency is obtained by guiding the emitted light to the detector using a hollow metallic waveguide. (c) Measured intensity output from the photonic molecule laser in a (integrated over all frequencies and emission directions) as a function of the electric field strength applied to the two individual disks (in the dark blue region the laser is off). The upper right corner contains the non-monotonic pump dependence expected for an EP. When the field strength in one of the disks is fixed and the other disk is steered through the EP’s vicinity (see white dashed lines), this results in a characteristic reversal of the laser’s pump dependence (see inset for the corresponding intensity curves). (d) Numerical results from the three-dimensional simulations: The maximum of the positive imaginary parts of the complex resonance frequencies is shown as a function of the amplification in each disk, as given by the corresponding imaginary parts of the refractive indices (only the n=3 modes are considered which are lasing in the experiment). Note the excellent agreement that we find between these calculations and the experimental data in c. Full size image

Measurements

In a first step, we performed measurements on the total emitted light intensity of this coupled laser device. Using the setup depicted in Fig. 2b, we collected the emitted laser light through an oversized hollow metallic waveguide, which guides the light to the detector resulting in almost 50% collection efficiency. Owing to the symmetry of the setup, the emitted radiation, which leaves the waveguide in the other direction, has the same intensity and is thus not detected. An advantage of this spatially and spectrally integrated measurement is that the results are insensitive to specific details of the far-field emission pattern and of the emission frequency of the laser. The pump dependence of this photonic molecule laser that we measure in this way is shown in Fig. 2c, where the emitted laser light intensity (see false colour plot) is displayed as a function of the bias fields F A and F B applied to disks A and B, respectively. In this plot, the specific pump trajectory, which induces an EP along the discussion above, is realized both as a vertical and as a horizontal line (see insets). In both of these configurations, the starting point is such that the applied bias is above threshold in one disk and below in the other disk. When gradually adjusting the lower field value to the higher value, the laser, indeed, shows the characteristic behaviour, which we predicted: after a regime where an increase of the bias field does not influence the output intensity at all, we first observe a strong reduction in the emitted light intensity, followed by an increase beyond the initial value. The observation of a reversal in the laser’s pump dependence constitutes a clear signature of the presence of an EP near those parameter values where the reversal occurs. To rule out that the observed behaviour is caused by some other mechanism, we carried out a number of additional checks.

As a first test, we performed extensive numerical simulations of the studied setup to verify the presence of the EP in the complex eigenvalue surfaces explicitly (see Methods section). As the coupling between two disks is spuriously overestimated in two-dimensional scalar calculations, we performed three-dimensional vectorial simulations of the photonic molecule device. To emulate the effect of the varying pump strength in the experiment, we solved the Helmholtz equation for the resonances of the photonic molecule under variation of the imaginary part of the index of refraction n A and n B of the two respective microdisks. The first insight derived from these calculations is that the modes which start lasing first in the experiment are whispering gallery modes with radial quantization number n=3 (see insets in Fig. 3 and Supplementary Figs 1 and 2). For these modes, a variation of the imaginary part of the refractive index in one disk, for example, Im(n B ), yields the expected avoided level crossing (see Fig. 1c) as previously obtained with the 2 × 2 matrix model (see Fig. 1b). When varying the imaginary parts of both refractive indices, Im(n A ) and Im(n B ), we obtain the same characteristic dependence on these parameters (see Fig. 2d) as observed in the experiment when varying the applied field strengths F A and F B (see Fig. 2c).

Figure 3: Emission spectrum and lasing modes. (a) Experimental emission spectrum recorded with the device in Fig. 2a, as a function of the pump strength in disk A with disk B at a constant pump value slightly above the lasing threshold. The colour gradient is proportional to the measured intensity and shows that along this pump sweep a single lasing line first turns off and then re-emerges with a shifted frequency (see frequency splitting indicated by the double-arrow). Note that the measured line width is limited here by the resolution of the spectrometer. (b) When varying instead the pump strength in disk B with disk A slightly above threshold, the equivalent behaviour is observed, however, with a frequency shift of opposite sign. The insets depict the electric field profiles (|E(x)|) of the involved modes as calculated at the bottom and top end of the pump sweeps shown in a,b. Full size image

As a final test, we also successfully verified that our experiment reproduces the real frequency shift which modes experience when passing the vicinity of the EP. Following the graphical illustration in Fig. 1, this shift should be observable as a frequency splitting between the lasing modes before and after they pass the EP. To check this behaviour explicitly, we recorded spatially integrated but spectrally resolved emission data, where the pump bias applied on one disk was fixed at a value slightly above threshold and the bias applied on the other disk was swept through. The recorded data shown in Fig. 3a clearly demonstrate that the laser line has a well-resolved splitting between the lasing frequency before and after the EP-induced laser shutdown. The insets of this figure illustrate the spatial mode patterns that are realized at the respective pump conditions. To verify the left–right symmetry of our setup, we also performed the same measurement with the role of the two disks interchanged (see Fig. 3b). Also in this case, we find the desired frequency splitting as well as the laser shutdown at the EP. As already suggested by the plots in Fig. 1, we observe in Fig. 3a,b that the mode frequencies can be shifted to higher as well as to lower values when passing the EP. The data contained in Fig. 3 also clearly demonstrate that the entire recorded pump sweep across the EP involves just a single lasing line, which is switched off and then on again. Nonlinear mode competition effects are therefore ruled out as an explanation for the shutdown of the laser. Such nonlinear effects do, however, become important when going farther above threshold, as at the end of the pump sweep, where both disks are pumped equally and strongly. In this case, the linear theory in Fig. 1c predicts two modes above the laser threshold, whereas in the experiment only a single mode is observed. Our numerical results suggest that the suppression of the second mode is due to modal cross-saturation (spatial hole burning).