Introduction

In the past few years, the use of non-Hermitian systems and the creation of exceptional points (EPs) have various applications in frontier researches, ranging from classical applications to quantum systems1,2,3,4. Extensive researches have been done to investigate the behavior of these types of systems around exceptional points theoretically and experimentally5,6,7,8. Due to the attractiveness of demonstrating an exceptional point in non-Hermitian parity-time and anti-parity-time symmetric systems, especially in optical media, by creating designed gain and loss profiles, where the way of lasing in the system is going to be changed, interests in EPs have been reignited9,10. In most of the recent researches, the main focus has been on the systems with second-order exceptional points11,12. Very few investigations have been done on the representation of higher order exceptional points practically and theoretically, which leads to a greater sensitivity7,13. Compared to second-order EP systems that have one exceptional point, this triple system has more sensitivity due to having two exceptional points and can be used in ultrahigh precision applications. To show the third-order exceptional points, ternary systems are needed to investigate theoretically and experimentally. On the other hand, creating a single-mode spectrum for lasers with multiple microcavities can be a challenging issue due to optical mode competition in the cavities. This causes reduction in laser stability, so it is necessary to achieve a stable single-mode lasing by using different methods for many ultra-highly sensitive applications14,15,16,17. For addressing this problem and achieving high laser mode selectivity, broken parity-time and anti-PT symmetry systems allow us to simultaneously create higher-order exceptional points and obtain a single-mode spectrum18,19,20. Microresonators, especially microrings arrangements provide appropriate capabilities for discovering physics of exceptional points. They are a suitable platform for breaking parity-time and perturbing anti-PT symmetry for obtaining a single-frequency lasing spectrum by applying the spatial modulation of pump21. Unique properties of microrings, including high quality factor (Q), easier fabrication, smaller size, and their wide applications in ultrahigh precision sensing, spectroscopy, optical on-chip filters and lasers, and gyroscopes have made them an exciting candidate22.

Here, we present the observation of higher-order exceptional points based on triple size-matched coupled microrings configuration and perturbation in an anti-parity-time system for achieving a single-mode lasing23. Due to the coupling mode theory and active gain spectrum of the material, these ternary systems have side modes and even high-order modes. To further increase the mode selectivity and achieve a pure single-mode lasing, we used the broken PT and anti-PT symmetry method by introducing perturbation and controlling the gain and loss in ternary systems. Introducing perturbation or controlling the gain and loss in the PT or anti-PT symmetry systems means that a lot of optical modes can be damped or disappeared by selectively pumping of the microresonators. There is a simple way to study PT-symmetric optical resonators using the coupled mode theory. In particular, a useful formalism to study energy exchanges between the optical resonators was proposed in previous works. Now, we can present these formulas for three coupled resonators here as:

$$\frac{{{\text{ida}}_{1} }}{{{\text{dt}}}} = -\upomega _{1} {\text{a}}_{1} - {\text{i}}\uplambda _{1} {\text{a}}_{1} +\upkappa {\text{a}}_{2}$$
(1)
$$\frac{{{\text{ida}}_{{2}} }}{{{\text{dt}}}}{ = } -\upomega _{{2}} {\text{a}}_{{2}} - {\text{i}}\uplambda _{{2}} {\text{a}}_{{2}} { + }\upkappa {\text{a}}_{{1}}$$
(2)
$$\frac{{{\text{ida}}_{{3}} }}{{{\text{dt}}}} = -\upomega _{{3}} {\text{a}}_{{3}} - {\text{i}}\uplambda _{{3}} {\text{a}}_{{3}} { + }\upkappa {\text{a}}_{{1}}$$
(3)

where \({a}_{n}\) represent the energy amplitude in cavities, \({\omega }_{n}, {\lambda }_{n}\) show resonance frequency, net gain/loss in micro-cavities, respectively and \(,\upkappa\) is coupling factor between micro resonators. Now, as can be seen theoretically, the controlling the gain and loss in the PT or anti-PT symmetry systems can damp optical modes except a desired mode. These two equations can be particularized in two special cases:

  • ω1 = ω2 = ω3 = ω0, with k a real value (k = κ);

  • ω3 \(\ne\) ω2 \(\ne\) ω3, with k an imaginary value (k = iκ).

The first case corresponds to an effective PT-symmetric Hamiltonian, whereas the second case becomes anti-PT-symmetric.

The PT symmetry laser with higher-order exceptional point can be easily realized by directly coupling three or more optical resonators. The anti-PT-configurations can also be realized by indirectly dissipative coupling, which needs three coupled resonators with leaky coupling shown in Fig. 1 b. In this configuration the microring resonator number 1, plays the role of dissipative coupling, leads to an imaginary coupling24,25,26.

Fig. 1
figure 1

Different configurations of ternary systems, (a) PT-symmetric system and (b) Anti-PT-symmetric system with dissipative coupling.

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For better understanding this issue, in ternary PT symmetry systems, when all of three coupled microrings experience the same amount of gain, the three eigenvalues of the system are real. If the radii of all the three microrings are equal to 30 μm, then splitting between the excited modes appears. Due to the coupling coefficient and the path difference between the three microcavities shown in Fig. 1 a, each eigenmode has three peaks in the output spectrum. If the radii of the microrings are not equal, demonstrated in Fig. 1 b, where two of the radii are slightly detuned which has the difference of 0.1 µm to provide the condition of anti-PT symmetry, and the radius of the third microring is 10 μm smaller for leaky coupling (R1 = 30, R2 = 40, R3 = 40.1 μm).

Figure 1 illustrates the different ternary systems, which are investigated in this paper and the various coupling coefficient in these systems, so lasers operate in multimode situation. In this paper, we show in practical, theoretical and simulated way how in these types of ternary systems, pure single-mode spectrum can be achieved by breaking parity-time symmetry and perturbing the anti-PT symmetry system and examining higher order exceptional points27,28. After solving Eq. (1)-(3), which has three eigenfrequencies, it is observed that the real or imaginary parts of eigenfrequencies are divided to three areas after passing two points. These two points are called exceptional points (EP2 and EP3).

In order to break the PT-symmetry, it is necessary to examine different schemes of inducing gains and losses to the ternary systems. According to solving Eq. (1)-(3), by inducing losses to the system, when the difference between loss and gain exceeds a specific limit (EP), which is proportional to κ2/3, the eigenvalues reach the completely broken PT symmetry region after passing the third-order EP (EP3). In this state, two of the microcavities experiences attenuation, while the other experiences amplification. The imaginary parts of the eigenfrequencies indicate the amplifying or decaying of the optical field, and as a result, a single-mode operation is obtained19,29. In this case, mode management can be performed by applying gain and loss changes to the microcavities.

In this paper, we achieved a single-mode lasing by using broken parity-time symmetry in structures with triple microrings around third-order EP. Another approach for achieving single-mode lasing is applying perturbation to an anti-PT symmetry laser. This perturbation can be applied by three different mechanisms: 1- changing the amount of gain or loss in microrings, 2- coupling, and 3- resonances. In case 1, this gian-loss difference can be modeled by a new term of εγ in the Hamiltonian of the system and the new Hamiltonian becomes: H = HAPT + Hεγ. In fact, εγ is defined as a loss in the system by covering one or two of microresonators. In the configuration of symmetric triple microrings, each eigenmode has three peaks in the output spectrum due to localized and non-localized modes in three microrings7, and it is essential to use the broken PT-symmetry method for achieving the single-mode lasing. Due to the increase in the different types of losses in the system by increasing in the number of microrings with the same radius, only one main mode is excited. Finally, these proposed structures have been evaluated experimentally. Based on the results, this work can be a suitable road map for working with various microrings arrangements with different physical higher-order EPs and can be effectively used in spectral modulation.

Results

Spectral properties of PT-symmetric and APT-symmetric systems

Considering the non-Hermitian Hamiltonian 3 \(\times\) 3 matrixes in triple coupled microrings as follows:

$${\text{H}}_{{{\text{PT}}}} {\text{ + H}}_{{\upvarepsilon _{\uplambda } }} { = }\left( {\begin{array}{*{20}c} {\upomega _{{1}} {\text{ + i}}\uplambda _{{1}} + {\text{i}}\upvarepsilon _{\uplambda } } & {\upkappa _{{{12}}} } & {\upkappa _{{{13}}} } \\ {\upkappa _{{{21}}} } & {\upomega _{{2}} {\text{ + i}}\uplambda _{{2}} + {\text{i}}\upvarepsilon _{\uplambda } } & {\upkappa _{{{23}}} = 0} \\ {\upkappa _{{{31}}} } & {\upkappa _{{{32}}} = 0} & {\upomega _{{3}} {\text{ + i}}\uplambda _{{3}} } \\ \end{array} } \right)$$
(4)
$${\text{H}}_{{{\text{APT}}}} + {\text{H}}_{{\upvarepsilon _{\uplambda } }} = \left( {\begin{array}{*{20}c} {\upomega _{1} + {\text{i}}\uplambda _{1} + {\text{i}}\upvarepsilon _{\uplambda } } & {{\text{i}}\upkappa _{12} } & {{\text{i}}\upkappa _{13} } \\ {{\text{i}}\upkappa _{21} } & {\upomega _{2} + {\text{i}}\uplambda _{2} + {\text{i}}\upvarepsilon _{\uplambda } } & {\upkappa _{23} = 0} \\ {{\text{i}}\upkappa _{31} } & {\upkappa _{32} = 0} & { -\upomega _{3} + {\text{i}}\uplambda _{3} } \\ \end{array} } \right)$$
(5)

where ω1-3 is the main resonance frequency in three microrings, λ1-3 represents the net loss or gain in each microring, and κ12 = κ21 = κ, κ13 = κ31 is constant for simplifying the equation. κ12, κ13, κ23 are the coupling coefficients between microring 1 and 2, 1 and 3, 2 and 3, respectively and εγ is the model of gain–loss difference caused by perturbation. For a better understanding of this issue, Fig. 2 is provided; the eigenfrequencies of the supermodes are illustrated after solving Eq. (4), which has three complex answers drawn in Fig. 2 a, b. According to the Hamiltonian matrix, the condition for high-order EP broken parity-time symmetry is having real value coupling, and \(\upomega _{2} =\upomega _{3} ,\uplambda _{2} = -\uplambda _{3}\), which is observed in three coupled size-matched structure. It is immediately seen that, in order to have an anti-PT-symmetric system, the Hamiltonian requires \(\upomega _{2} = -\upomega _{3} ,\uplambda _{2} =\uplambda _{3}\) and leaky coupling for κ with imaginary value30. The leaky coupling is equivalent to having an imaginary coupling strength. For this, the anti-PT configuration can be realized by indirectly dissipative coupling with the third microring which has smaller radius. In the microrings, when the radius is smaller, various optical losses are greater. After considering the anti-PT condition ({PT,H} = 0), and for being \(\upomega _{{2}} \ne\upomega _{{3}}\), there should be slightly detuning between microrings number 2 and 3 and as a result, different propagation constant (β) is observed for different excited modes. According to the Hamiltonian (5), by implementing loss perturbation to this system, more modes are suppressed, leading to a single-mode lasing. All these conditions can be executed by the configuration of Fig. 1 b with coupled size-mismatched microrings.

Fig. 2
figure 2

a, b shows the imaginary and real parts of the eigenfrequencies in the vicinity of third-order exceptional points (EPs) according to κ and the introduced loss, respectively. With a further increase in losses, the system enters the broken PT symmetry regime; the imaginary part starts to split, but the real part is still the same. In this regime, two of the three supermodes experiences annihilation, while the other experiences amplification and leads to a single-mode operation.

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Figure 2 (a) Imaginary and (b) real parts of the eigenfrequencies as function of coupling coefficient and introduced loss in triple size-matched mutual microrings. Above the EP3, the real parts of the eigenfrequencies are the same, while the imaginary parts are bifurcated. (c) Imaginary and (d) real parts of the eigenfrequencies as functions of coupling coefficient and introduced loss in triple size-mismatched mutual microrings. The inset shows the enlarged view of splitting around EPs.

Because of the anti-PT effect and the amount of loss in the three coupled asymmetric microrings, the number of excited modes is reduced and by applying gain–loss perturbation to this system, one of the eigenmodes is completely disappeared and leads to a single-mode lasing as shown in Fig. 2c, d theoretically.

The investigation of the broken parity-time symmetry is illustrated in ternary systems in Fig. 3 a-c schematically. As can be seen, many transverse modes can be excited in a single microring (Fig. 3 a). By applying a uniform pump on the triple coupled microrings, the multi-splitting of the eigenmodes occur (Fig. 3 b). Only one dominant mode can be excited and a single-mode lasing is achieved using broken parity-time symmetry method (Fig. 3 c). In this moment, the gain difference between the main mode and the side modes increases. In a PT symmetric setting, the coupling coefficient (κ) plays the role of a virtual loss, and all undesirable modes must be below its corresponding threshold; therefore, a selective breaking of PT symmetry can systematically increase the available amplification for single-mode operation31. After attaining eigenfrequencies of the system, it can be seen that the gain–loss variations are related to the coupling coefficient and these variations can be modeled by the coupling strength shown by dashed line in Fig. 3 b-e.

Fig. 3
figure 3

The eigenmodes of the system in (a) one microring (b) triple size-matched coupled microrings (c) broken PT symmetry triple coupled microring configuration (d) triple size-mismatched coupled microrings, and (e) perturbed triple size-mismatched coupled microrings.

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Figure 3 d, e shows the schematic diagram of an anti-PT symmetry system. In such these systems, because of indirect coupling and slightly detuning in the radius of microrings (R2, R3), there are splitting in each eigenmodes (Fig. 3 d). After passing the EP by applying perturbation to the system, a single- mode lasing is achieved shown in Fig. 2 c theoretically and Fig. 3 e schematically.

Methods

Simulations

To indicate the effect of the broken parity-time symmetry and the gain–loss perturbation of APT in ternary systems, we simulated the asymmetric and symmetric triple microrings structures with finite difference time domain (FDTD) method in Fig. 4 a, c, e, and g with radii of R1 = R2 = R2 = 30 µm, and R1 = 30, R2 = 40, and R3 = 40.1 µm, respectively. The electric field distributions are illustrated in Fig. 4 b, d, f, and h, respectively. It can be observed when the gain is considered for three microrings, there are a lot of eigenmodes in the systems. Because of coupling factor, whereas the gain–loss is introduced to the system; most modes are suppressed and finally a single-mode lasing is observed. The gain–loss could be realized by setting the imaginary part of the refractive index of the cavity. The model of the perfectly match layer (PML) is used to absorb the outgoing waves and Maxwell’s equations are solved in FDTD.

Fig. 4
figure 4

Lasing spectra of (a), (b) symmetric gain-gain-gain triple microrings with radii of 30 µm and its electric field distribution at 611.4 nm (c), (d) symmetric gain–loss-loss triple microrings with radii of 30 µm and its electric field distribution at 611.7 nm (e), (f) asymmetric gain-gain-gain triple microrings with radii of 40, 40.1, and 30 µm and its electric field distribution at 605.3 nm (g), (h) asymmetric gain–loss-loss triple microrings with radii of 40, 40.1, and 30 µm and its electric field distribution at 605.4 nm simulated with FDTD method; the insets show an enlarged view of the spectrums.

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Due to the 2-dimensionality of the simulation and not taking into account all types of losses such as propagation, bending and scattering losses, the simulation of the single-mode spectrum is different from the experimental results.

It should be noted that the amount of gain difference (∆g) between the primary mode and the next largest competing mode is significant. If this amount is small, it is challenging to obtain the single-mode operation. The mode selectivity is improved by the broken parity-time symmetry effect and applying perturbation32. Regarding the shift of the spectrum, it should be noted that in the first case, there are three identical microrings with the same radius, but in the second case, the radii are different. Changing in the radius causes changes in the amount of loss in the system. Depending on the coupling level, the damping of side modes is occurred and a fundamental mode is excited, and because we have gain difference for eigenmodes in two systems, this shift occurs.

Experiments

To investigate the effect of broken parity-time symmetry, a symmetric triple microrings structure with a radius of 30 µm and the anti-parity-time effect, an asymmetric triple microrings configuration with radii of 30 and 40, 40.1 µm were fabricated on the SiO2 substrate. The fabrication process of both structures is done in the same conditions as follows:

We used Rhodamine-B doped SU-8 as a gain material. A 2 µm layer is deposited on a layer of silicon dioxide as a substrate by spin coating. The thickness was measured by Filmetrics (F10-RTA) device. Then we utilized a direct-writing lithography method. The beam of 400 nm laser is focused on the Rhodamine-B doped SU-8 using a 50 × lens. Laser power should be adjusted according to the width of the microrings. To create a width of 1.5 µm, the writing power of the laser was set to 0.5 mW. When the sample has been exposed to the laser beam, the desired pattern remains, and in the rest of the places, the solver removes the material. We utilized the setup in33,34 to characterize these structures. We also used a knife-edge placed on a 3D stage with high precision to cover the desired microring.

We first examine the effect of broken parity-time symmetry using symmetric mutual triple microrings with different air gaps to demonstrate single-mode spectrum. To better understand the issue, we first uniformly pumped three microrings with the same radius (R1, 2, 3 = 30 µm) with a 532 nm pulsed pump. Depending on the coupling strength and the air gap between three microrings, the splitting of the eigenmodes is changed7. Figure 5 a-f shows the lasing spectra and the procedure of achieving a single-mode lasing spectrum for triple-coupled microrings experimentally with different air gaps: 0.4, 0.5, respectively based on broken PT symmetry.

Fig. 5
figure 5

The lasing spectra of evenly and partially pumped symmetric triple microrings with different air gaps (d): (a-c) 0.4, and (d-f) 0.5 nm, respectively. It shows the procedure of single-mode operation with introducing loss gradually; the insets show images of pumping and lasing for different schemes experimentally. (g) Scanning electron microscope (SEM) images of the coupled microrings. Air gap of 0.4 μm; Insets show the enlarged view of air gap.

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As mentioned, the coupling coefficient is dependent on the air gap between two microrings, and the amount of splitting in the excited modes is also dependent on the coupling coefficient. As explained before, to realize the status of gain and loss in the structure of three symmetric microrings, first the three microrings are pumped uniformly in gain-gain-gain mode, which we call it, ‘evenly pumping’. In the next step, only the gain is introduced to two of the microrings which mean that only two microrings is illuminated. After this step, only one of the microrings is pumped and as a result, the parity-time symmetry is going to be completely broken and many modes are suppressed, so the single-mode lasing operation is achieved.

The linewidth and SMSR (side-mode suppression ratio) are 0.6 nm and 8.48 dB for the evenly pumping scheme, 0.4 nm and 22 dB for the one microring pumping scheme, respectively.

The laser spectra of unbroken and broken parity-time symmetry triple microrings laser are shown in Fig. 6 under different pump energies density. To realize the output spectrum of this structure, two different schemes of pumping are introduced: uniform pumping of three microrings (evenly pumping), non-uniform pumping of only one microring. The output spectra of lasing are presented in Fig. 6 a, b. Because the observed frequencies depend on the pump energy, a nonlinear behavior of the system is seen. In Fig. 6 a, the system has three supermodes due to two coupling regions. By covering two of microrings in Fig. 6 b, only one of the supermodes is excited and two of them are disappeared because of experiencing different gains in the system. Figure 6 c, and d show the normalized output emission intensity of the structure under evenly and partially pumping schemes as a function of pumped energy, respectively. In these figures, the slopes changes because of loss in the system. The increased slope in the evenly pumped case is due to the tripled active area. The normalization is done based of the maximum intensity of output spectrum. In the case where only one microring is pumped, the threshold of lasing has not changed so much, which is possible to lead to a better side-mode suppression ratio. The doted blue line in Fig. 6 a shows the amplified spontaneous emission (ASE) spectrum of the material, which is the gain area of the material, in fact, the material can have eigenmodes at this area.

Fig. 6
figure 6

The lasing spectra of (a) evenly pumped, amplified spontaneous emission (ASE) spectrum (doted blue line) (b) partially pumped symmetric triple microrings. The normalized output emission intensity of triple (c) unbroken, and (d) broken parity-time symmetric coupled microrings laser as a function of pumped energy.

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To evaluate the anti-parity-time symmetry effects, an asymmetric triple microrings structure with radii of 30 and 40, 40.1 µm and 0.5 µm air gap is presented. Figure 7 shows how the system works and the changes to emission spectra of triple asymmetric microrings when the perturbation is applied to the APT symmetry system by non-uniform pump distribution. This non-uniform pump distribution causes a perturbation by inducing more loss to the system. For our experiments, we fabricated three mutual microrings (radii: 30, 40, 40.1 μm, widths: 1.5 μm, heights: 2 μm). Figure 7 a illustrates the partially distributed pumping of the structure and the loss introduced to two of the microrings by covering them which leads to a single-mode lasing. Figure 7 c is the scanning electron microscope (SEM) image of the coupled microrings.

Fig. 7
figure 7

The lasing spectra of (a) partially pumped (b) evenly pumped asymmetric triple microrings; the insets show images of lasing for different schemes experimentally. (c) Scanning electron microscope (SEM) images of the coupled microrings.

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With the increase in the number of microrings, many side modes are eliminated due to high losses in the ternary systems, and depending on the type of structure, only one or two multi-peak eigenmodes appear in the output spectrum due to the presence of two coupling regions.

As can be seen in Fig. 7 a, when two of the microrings are blocked by knife-edge and the loss is increased in the system, after the system passes the EP, only a dominant mode with a higher quality factor appears and leading to a single-mode lasing operation under different pump energies density (Fig. 8 b). We made a comparison between the laser spectra of unperturbed and perturbed anti-parity-time triple microrings laser in Fig. 8 under different pump energies density. To realize the output spectrum of this structure, two different schemes of pumping are introduced: uniform pumping of three microrings (evenly pumping), non-uniform pumping of only one microring. The output spectra of lasing are presented in Fig. 8 a, b. After passing the EP in the anti-parity-time symmetry system, the single-mode lasing is attained (Fig. 8 b). Because of mode competition, when only one microring is pumped and the gain–loss contrast is enhanced, as a result, different modes experience different gains, and a dominant mode is selected to lase as shown schematically in Fig. 3 e. Figure 8 c, and d show the normalized output emission intensity of the structure under evenly and partially pumping schemes as a function of pumped energy, respectively. In the case where only one microring is pumped, the threshold of lasing has not changed so much, which is possible to lead to a better SMSR. In a general view, we explain the system behavior around exceptional points, but in the system that the emitted spectrum depends on the pump, denoting nonlinearity that is not considered in the model.

Fig. 8
figure 8

Lasing spectra of asymmetric triple microrings with radii of 30, 40, 40.1 µm when three microrings pump (a) evenly, (b) partially. The normalized output emission intensity of (c) unperturbed, and (d) perturbed anti-parity-time triple asymmetric coupled microrings laser as a function of pumped energy.

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The linewidth and SMSR are 0.6 nm and 1.43 dB for the evenly pumping scheme, 0.4 nm and 13.44 dB for the one microring pumping scheme, respectively.

Discussion

In conclusion, we demonstrated the most recently two important effects in optical cavities well known as broken parity-time symmetry in the vicinity of third-order exceptional point and the perturbed anti-PT symmetry effects which lead to a single-mode operation in the monolithically integrated coupled Rhodamine-B doped SU-8 microresonators. We found that different pumping schemes cause a different lasing in triple symmetric and asymmetric microrings because of the variation in the gain in the excited modes. The SMSR of the single-mode laser reaches more than 22 dB. These compact devices are an appropriate platform for applications like on-chip optical sources, ultrahigh precision sensing, spectroscopy, and optical on-chip filters and lasers.