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Abstract
Recently, optical systems with parity-time (PT) symmetry have attracted considerable attention due to its remarkable properties and promising applications. However, these systems usually require separate photonic devices or active semiconductor materials. Here, we investigate PT symmetry and exceptional points (EPs) in monolithically integrated graphene-assisted coupled microresonators. Raman effect and graphene cladding are utilized to introduce the balanced gain and loss. We show that PT-symmetry breaking and EPs can be achieved by changing the pump power and the chemical potential. In addition, the intracavity field intensities experience suppression and revival as the graphene-induced loss increases. Due to the unique distribution of optical field, tunable nonreciprocal light transmission is theoretically demonstrated when introducing the gain saturation nonlinearity. The maximum isolation ratio can reach 26 dB through optimizing the relevant parameters. Our proposed scheme is monolithically integrated, CMOS compatible, and exhibits remarkable properties for microscale light field manipulation. These superior features make our scheme has promising applications in optical communication, computing and sensing.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since Bender et al. showed that the eigenvalues of non-Hermitian can be entirely real if it commutes with the parity-time (PT) operator [1], PT symmetry has been explored in different physical systems, including electronics [2], acoustics [3–5] and photonics [6–10]. Owing to the flexibility in modulating the dielectric permittivity, optical systems provide a suitable platform for achieving PT symmetry by introducing balanced gain and loss. One of the most striking features of the non-Hermitian system is the presence of an exceptional point (EP), at which the eigenvalues and corresponding eigenstates become degenerate [11]. In addition, PT symmetry is spontaneously breaking above this point, thus totally new phenomena will arise [12–18]. Optical systems with PT symmetry and EPs have led to observe many intriguing effects, ranging from power oscillations [19–21], non-reciprocal light transmission [22–26], single-mode lasers [27–29] to loss-induced revival of lasing [30,31].
The necessary condition to achieve a PT-symmetric system is the design of a symmetric refractive index while with an antisymmetric gain/loss. Whispering-gallery-mode resonators, benefit from high quality factors and small mode volumes, provide a promising solution. Dissipation arises from material absorption and radiation loss, while gain can be implemented by using active materials, such as erbium-doped silica, semiconductors, or through nonlinear processes. A typical PT-symmetric system consists of two coupled optical microresonators. One resonator generates gain by optical or electrical pumping, and the other resonator provides an equal amount of loss. In order to observe PT-symmetry breaking and EPs, it requires to adjust the coupling strengths between the two microresonators [23,24]. Thus, such a system requires two separate photonic devices, which leads to increased fabrication complexity and sensitivity to external disturbances. Observations on phase transition by tuning the gain and loss simultaneously based on indium phosphide quantum wells [27], or using a nanofiber tip to control the loss have been reported [30]. However, these methods are incompatible with mature complementary metal-oxide-semiconductor (CMOS) technology. It is then an intriguing question whether there is a suitable scheme to demonstrate PT symmetry and EPs in a monolithically integrated photonic platform.
In this work, we investigate PT-symmetry breaking in monolithically integrated graphene-assisted coupled microresonators. Raman effect is exploited to generate the optical gain, while graphene layers are covered on the microring resonator to provide the symmetric loss. One of the unique properties of graphene is that the Fermi level can be controlled by electric bias [32–34]. The results show that, the gain and loss of the coupled microresonators are tuned by changing the pumping power and the conductivity of graphene to observe PT-symmetry breaking. Furthermore, we find that with a fixed gain, observation of EPs and enhancement of intracavity field intensities are enabled through tuning the absorption of the passive resonator. Our study also shows that by considering the gain saturation nonlinearity, nonreciprocal light transmission is achieved based on this system. We believe that our work provides a new approach towards on-chip optical filed manipulation, amplification and routing, which could develop compact devices for photonic communication, computing and sensing.
2. Theoretical model and PT symmetry
Aluminum nitride (AlN) with large bandgap and six Raman-active phonons, has emerged as a novel platform for nonlinear photonic devices with negligible multiphoton absorption [35–38]. As shown in Fig. 1, the proposed theoretical model consists of two coupled AlN microrings. The active microring (µR1) is optically pumped to produce effective Raman gain in the 1550 nm band. The passive microring (µR2) with graphene layers on part of the microring waveguide has adjustable loss. The two microrings support similar resonance frequencies at 1550 nm, but pump field only exists in µR1 due to there is no coupling at the pump wavelength. Moreover, each resonator is evanescently coupled to a bus waveguide.
Fig. 1. Schematic of graphene-assisted coupled microresonators. Solid and dashed arrows represent the forward and backward signals, respectively.
The coupled-mode equations describing the optical field dynamics can be written as
We first study the modulation characteristic of a graphene-assisted microring. Figure 2(a) shows the cross-section structure of a graphene-AlN hybrid waveguide on a sapphire (Al2O3) substrate, and two layers of graphene are separated by a thin layer of insulating material silica (SiO2). Graphene’s conductivity can be tuned by applying voltage on aurum (Au) electrodes, thus propagation loss of the hybrid waveguide is tuned accordingly. The conductivity of graphene consists of interband and intraband parts [39]
Fig. 2. (a) Schematic cross-section of the hybrid waveguide. (b) Real (black) and Imaginary (red) parts of the effective refractive index as a function of the chemical potential. (c) Calculated propagation loss. The insets show electric field profiles of TE mode at 0.2 and 0.5 eV, respectively. (d) Output Stokes power (black) at 1550 nm and Raman gain (red) versus the pump power at 1406 nm, indicating a lasing threshold of 11.3 mW.
The interband conductivity follows Kubo’s model:
The intraband part can be described as:
As shown in Fig. 2(b), the imaginary part of neff has a step-like falling at the chemical potential threshold (µc = 0.4 eV). When the bias voltage which determines the chemical potential is lower than the threshold, interband transition occurs as the incident photons are absorbed. As a result, graphene exhibits a high loss state, corresponding to a large imaginary part. On the contrary, when the bias voltage is larger than the threshold, there is no interband transition due to Pauli blocking. In this condition, graphene turns to be transparent, leading to a low loss state, corresponding to a small imaginary part. The propagation loss is then obtained through
where k0 is the vacuum wavenumber, Lgra is the coverage-length of graphene. As shown in Fig. 2(c), the loss can be changed from 0.5 GHz to 3.5 GHz as the chemical potential decreases.Then we study the Raman gain of an AlN microring. Stimulated Raman scattering is a nonlinear process that provides an attractive way to generate optical gain at desired wavelengths. An incident high-energy photon (ωp) is scattered into a low-energy photon (ωs) through the generation of an optical phonon Ω = ωp – ωs. For microresonator-based Raman scattering, it is essential to match the free spectrum range with the Raman shift. It has been showed that the $E_2^{high}$ phonon contributes to the Raman scattering for TE pump, corresponding to the Raman shift of 19.8 THz [41]. We design the active microring with a 60-µm radius, the pump at 1406 nm and the Stokes light at 1550 nm can be resonant simultaneously. First-order Raman scattering in a microcavity can be described by coupled-mode equations [42]
After considering the Raman gain and graphene-induced loss, the eigenvalues of this system can be obtained. Figures 3(a) and 3(b) profile the real and imaginary parts of the eigenfrequencies when tuning the gain and loss. We observe a complex square root function of the surfaces, it is obvious that there is an exceptional line, more than one EP can be achieved by changing either g0 or γgra alone. The slope of this line indicates that the smaller gain requires the higher loss to reach the EP. Figures 3(c) and 3(d) give typical eigenfrequency evolution curves when the system is exactly PT symmetric ${\left({{\gamma _{\textrm{gra }}} + \gamma _1^\prime = {g_0} - \gamma _2^\prime } \right)}$, corresponding to the red lines in Figs. 3(a) and 3(b). There are two distinct regions in the frequency space. For a small gain/loss (g/γ < µ), the system is in the unbroken symmetry regime, the real parts of the eigenvalues split but their imaginary parts are equal. It implies that the two supermodes have different resonance frequencies but the same linewidths. At the EP (g/γ = µ), the two supermodes coalesce with the same eigenfrequencies. With a further increase in g/γ, the system enters the broken symmetry regime, the imaginary part starts to split but the real part is still the same. In this regime, one supermode becomes lossy while the other experiences amplification.
Fig. 3. (a, b) Real and imaginary parts of eigenvalues as functions of the Raman gain and the graphene-induced loss. (c, d) Evolution of the eigenfrequencies when the system is PT symmetric. The parameters used here are γ1 = γ2 = 0.61 GHz, κ1 = 0.3 GHz, κ2 = 0.61 GHz, µ = 0.3 GHz and λ0 = 1550 nm.
The symmetry transition can be explained as the influence of the gain/loss on the inter-cavity coupling strength. If there is the smaller gain/loss, we can consider the system is under strong coupling. The pump power in the active resonator can transfer rapidly to compensate the dissipation in the passive resonator, thus the system attains real eigenfrequencies. On the contrary, if there is the larger gain/loss, the coupling between the two resonators is week, and the pump power cannot transfer fast enough to the passive resonator, leading to an unbalanced state as well as complex eigenfrequencies.
3. Loss-induced enhancement of the field intensity
As illustrated in Fig. 3, when considering the intrinsic loss and coupling loss, PT symmetry can only be achieved in a small parameter space, which limits the scope of applications. However, the tunable loss provides an alternative way to manipulate optical field.
For the system with a fixed gain, there still exists an EP for the optcal modes. Figure 4(a) shows the evolution of the eigenfrequencies as a function of the graphene-induced loss γgra. With the increasing of the loss, the real parts approach each other and finally become the same. However, the imaginary parts change differently from the PT symmetry. One of them keeps decreasing, but the other first decreases and then starts increasing. It indicates that both of the supermodes are lossy, after crossing the EP, the loss of one supermode become decreasing. But it cannot go below the initial loss since the fixed Raman gain is not able to compensate the system loss.
Fig. 4. (a) Evolution of the eigenfrequencies as a function of the graphene-induced loss when the gain is fixed. (b) Normalized intracavity field intensities in the coupled microresonators. Green: Field intensity in the passive resonator (I2). Blue: Field intensity in the active resonator (I1). Red: Total field intensity. Here g0 = 0.7 GHz, the other parameters are the same as in Fig. 3.
Figure 4(b) shows the evolution of normalized intracavity field intensities. When γgra = 0.5 GHz, the two resonators possess almost the same intracavity field intensity, indicating that the supermodes distribute equally. As γgra is increased, both I1 and I2 start decreasing with different rates. The difference is due to the continuously increasing total loss of the passive resonator. However, passing a critical point, the intensities are found to increase, which indicates that the additional loss is large enough to affect the supermode distribution. The trend continues until reaching the EP, beyond which I1 still increases whereas I2 starts decreasing. This is the result of the varying eigenfrequencies, the mode located in the active resonator possesses a lower loss. We can also find that the total intracavity field intensity has the same trend as I1 and it can exceed the initial value. As γgra continues to increase, the total intracavity field intensity approaches the field intensity in the active resonator. The loss-induced enhancement is different from the conventional system where the intracavity filed intensity will decrease with the increased loss. It is a direct manifestation of PT non-Hermiticity.
4. Nonreciprocal light transmission
On-chip optical nonreciprocity is highly demanding for high-speed signal processing. Serval methods have been explored to achieve non-reciprocal devices, including magneto-optic effect [43,44], spatiotemporal modulation [45–47], optomechanical interactions [48–50], and optical nonlinearity [51,52]. However, those approaches require large sizes or high energy consumption. The monolithically integrated PT-symmetric system we proposed here can be utilized to attain nonreciprocal light transmission.
It is well known that optical nonlinearity can be strongly enhanced in a PT-symmetric system. We utilize the gain saturation effect to break the Lorentz reciprocity in our system. Thus, the Raman gain in Eq. (1) is replaced by $g = {g_0}/({1 + {{|{{a_1}/{a_{\textrm{sat}}}} |}^2}} )$, where g0 is the gain as the intracavity signal field amplitude becomes zero, asat is the gain saturation threshold. As shown in Fig. (1), the transmission from port 1 (4) to port 4 (1) is defined as the forward TF (backward TB). Due to the PT-symmetry breaking, which leads to the strong localization in the active resonator, making the nonreciprocal light transmission achieved. The isolation ratio is defined as
Figure 5 (a) shows the transmission spectra for forward and backward directions, the normalized transmittances at zero detuning are 0.28 and 0.01, respectively, correspond to an isolation ratio of 14.5 dB. Owing to the gain saturation effects, the gain provided by the Raman effect strongly depends on the optical field intensity. The signal light injected from port 1 directly enters to the active resonator with a high power, it experiences a modest gain and then is absorbed by the passive resonator. On the contrary, when the signal light is injected from port 4, it first experiences a strong absorption before enters the active resonator. Such a low power cannot lead to the gain saturation, thus the backward signal is amplified. This is the basic principle to break Lorentz reciprocity based on this PT-symmetric system. Moreover, the unique field distribution in the symmetry breaking regime would enhance the nonreciprocity.Fig. 5. (a) Transmission spectra for forward (red) and backward (blue) directions in the symmetry breaking regime. The signal power is 25 µW. (b) Isolation ratio as a function of the coupling strength and the graphene-induced loss. Here κ1 = 0.7 GHz, κ2 = 5 MHz, g0 = 2 GHz, |asat/sin|2 = 100s, other parameters remain unchanged.
The isolation ratio could be improved through optimizing the coupling strength or tuning the additional loss. As shown in Fig. 5 (b), when the loss induced by graphene is increased, the system possesses large asymmetric transmissions. Similarly, it allows much larger backward transmission than forward under weaker coupling. The maximum isolation ratio can reach up to 26 dB. Our proposed nonreciprocal device has the advantages of small footprint, tunable isolation ratio and low energy consumption. Despite the nonlinearity would undergo dynamic reciprocity if signals are injected from both directions [53], the system could still be useful for pulse signal processing and LiDAR [54].
5. Conclusion
In summary, the PT symmetry and EPs have been studied in the monolithically integrated graphene-assisted coupled microresonators. Unlike previous demonstrations, tunable Raman gain and graphene-induced loss are utilized to achieve symmetry breaking. We found that there exists an exceptional line in the parameter space, which increases the degree of freedom to achieve EPs. With a fixed loss, we obtained the restraint and enhancement of the intracavity field intensities. Such a counterintuitive phenomenon is attributed to the unique distribution of the supermodes around the EP. Moreover, nonreciprocal light transmission is observed when considering the gain saturation mechanism in the active resonator. The isolation ratio could be tuned through optimizing the coupling strength and changing the graphene-induced loss. Our proposed scheme is compatible with the CMOS process and does not require two separate substrates, which is crucial for monolithic integration. This compact device has potential applications in on-chip optical signal processing, computing and sensing.
Funding
National Natural Science Foundation of China (11774110, 91850115); Fundamental Research Funds for the Central Universities (HUST: 2019kfyRCPY092, 2019kfyXKJC036); State Key Laboratory of Advanced Optical Communication Systems and Networks (2021GZKF003); State Key Laboratory of Applied Optics (SKLAO2021001A10); State Key Laboratory of Information Photonics and Optical Communications (IPOC2019A012); Key Research and Development Program of Hubei Province (2020BAA011).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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