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Abstract
Non-Hermitian optics has emerged as a feasible and versatile platform to explore many extraordinary wave phenomena and novel applications. However, owing to ineluctable systematic errors, the constructed non-Hermitian phenomena could be easily broken, thus leading to a compromising performance in practice. Here we theoretically proposed a dynamically tunable mechanism through GST-based phase-change material (PCM) to achieve a reconfigurable non-Hermitian system, which is robust to access the chiral exceptional point (EP). Assisted by PCM that provides tunable coupling efficiency, the effective Hamiltonian of the studied doubly-coupled-ring-based non-Hermitian system can be effectively modulated to resist the external perturbations, thus enabling the reconfigurable chiral EP and a tunable non-reciprocal transmission. Moreover, such tunable properties are nonvolatile and require no static power consumption. With these superior performances, our findings pave a promising way for reconfigurable non-Hermitian photonic devices, which may find applications in tunable on-chip sensors, isolators and so on.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, non-Hermitian optics, mostly known as parity-time (PT) symmetric optics, has emerged as one of the leading fields in photonics. Non-Hermiticity in optics could be effectively mimicked and implemented by spatially modulating loss and gain profiles through integrated optical systems [1–6], which provides a feasible and versatile platform to explore various non-Hermitian physics that originate from quantum mechanics [7]. One of the typical features in non-Hermitian systems is the phase transition, which results from the spontaneous phase breaking around the so-called exceptional point (EP). EP is the degeneracy point, at which the eigenvalues and related eigenvectors simultaneously coalesce. Numerous intriguing phenomena around EP have been discovered, such as particle sensing [8], optical gyroscope [9–11], bistable state switching [12], asymmetric mode conversion [13–16], single-mode lasing [17–19] and so on. Owing to the Kerr nonlinearity or gain saturation in the active medium, another interest of research in non-Hermitian optics is to realize non-reciprocal transmission [20–23], which has potential applications in light isolation and circulation for chip-scale optical communications and computing. Owing to their remarkable features, EPs and non-reciprocal transmission have been widely investigated to realize a variety of photonic devices with novel functionalities, such as directional lasing [8], EP-based sensors [9–12], non-reciprocity-based isolators [21–25]. However, owing to the perturbation of key parameters caused by ineluctable systematic errors and other factors, the practical performances of these integrated photonic devices with a fixed structure will be compromised. It is highly desired to introduce a dynamically tunable mechanism for integrated photonic devices, which is significant to improve device performance or access a robust wave phenomenon of interest. Meanwhile, the dynamically tunable property can offer more opportunities for a fixed photonic device with various functionalities.
Traditionally, the dynamically tunable method is implemented by utilizing electro-optic or thermo-optic modulation, whereas it is not applicative for on-chip photonic circuits due to the high static power consumption and volatility. Recently, great attention has been devoted to phase-change material (PCM) [26–30], especially Ge2Sb2Te5 (GST), as it can provide steady and reversible material states to dynamically tune the optical response of fixed photonic devices without the need of continuous power supply. Recent studies have demonstrated that this tunable scheme has a fast switching speed and could be precisely controlled by thermal [31–34], electrical [35,36] or optical excitation [37–39]. Moreover, PCM could be easily integrated with micro-nano structures to realize numerous reconfigurable devices, such as Fresnel zone plates [40], meta-lens [41], programmable mode convertor [42]. With these superior properties, PCM may offer a promising route to achieve the desired reconfigurability in a non-Hermitian system.
In this work, we proposed and investigated a non-Hermitian optical system integrated with GST-based PCM, which can realize reconfigurable chiral EP and tunable non-reciprocal transmission. The non-Hermitian optical system, composed of doubly-coupled rings coupled with one bus waveguide, is designed based on the erbium-doped lithium niobate (LN) platform [43–46]. Two pairs of resonant modes in doubly-coupled rings enable an active four-mode system with two unbalanced gains, and the non-Hermitian system could be regarded as the combination of a PT-symmetric system and a purely gain system. In spite of a four-mode system, degenerate second-order EP with chiral property can be discovered. Furthermore, by applying laser pulses (either writing or erasing) to engineer phase-change regions of GST films [39,40], the coupling ratio between the bus waveguide and two ring resonators could be dynamically changed to modulate the effective Hamiltonian of the system, which is responsible for reconfigurable chiral EP and tunable non-reciprocal transmission. Our results show that such a dynamically tunable non-Hermitian system can provide a feasible and versatile platform for studying non-Hermitian physics and the associated wave phenomena, enabling the design for photonic device with various functionalities.
2. Model of the reconfigurable non-Hermitian system made of doubly-coupled ring resonators
Figure 1 shows the schematic diagram of our proposed non-Hermitian structure, in which the LN-based waveguide system etched from a z-cut (erbium-doped) LN film is laid on SiO2 substrate with a thickness of d2 = 10 µm, and the etching depth and the non-etching depth of the LN film are h = 400 nm and d1= 100 nm. The LN-based waveguide system consists of doubly-coupled-ring resonators with a coupling strength of $\kappa $ and a bus waveguide for coupling the input probe light. Each ring cavity has a gain rate of ${g_0}/[{1 + I_{1(2 )}^{f,b}(t )/{I_0}} ]$, where ${g_0}$ is the small signal gain; $I_{1(2 )}^{f,b}(t )$ is the time-dependent mode intensity in ring resonators for the forward or backward stimulation and ${I_0}$ represents the gain saturation intensity of a given gain medium. In addition, two cross-coupling (CC) waveguides are patterned with thin layer of GST films (the golden color represents the original low-loss amorphous GST, i.e., aGST, while the gray color represents the phase-change state related to the dissipative crystalline GST, i.e., cGST) to achieve tunable coupling between the bus waveguide and the ring resonators. All these parameters of LN-based waveguide are illustrated by the sectional view (see the inset of Fig. 1). The top widths ${w_s}$ of the bus waveguide and the ring waveguides are set as 1100 nm. To meet the phase-matching condition at 1550 nm for TE polarization, the top width ${w_c}$ of the CC waveguide and ${w_G}$ of the GST film are numerically calculated to be 1007 nm and 907 nm respectively (see Supplement 1).
Fig. 1. Schematic diagram of a reconfigurable non-Hermitian optical structure, consisting of a bus waveguide, two cross-coupling waveguides patterned with GST films and two ring resonators with a coupling strength of $\kappa $. By emitting laser pulses on the GST film to engineer the phase-change region of GST film (the ratio of cGST to aGST), the tunable coupling rate between the bus waveguide and ring resonators can be achieved, which leads to the redistribution of mode intensities at the clockwise (CW) and counter-clockwise (CCW) directions in the two ring resonators for both the forward and the backward stimulated cases. The sectional view in the inset shows the detailed parameters of the coupling region from the bus waveguide to ring cavity-1(2), where ws = 1100 nm, wc = 1007 nm, wG = 907 nm, $s$=400 nm, h = 400 nm, and $\theta = {75^ \circ }$, d1= 100 nm, d2 = 10 µm.
For the forward case (see the red arrow in Fig. 1), the incident probe light from the left side enters into the bus waveguide and stimulates the counter-clockwise (CCW) mode ($a_1^{ccw}$) in the ring-1 through the CC waveguide-1. Subsequently, the mode $a_1^{ccw}$ is coupled into the ring 2 with the coupling strength κ, leading to the clockwise (CW) mode in the ring 2, denoted as $a_2^{cw}$. Simultaneously, part of input light can directly travel along the bus waveguide and then is coupled into the CCW mode ($a_2^{ccw}$) in the ring-2 through the CC waveguide-2, which can further induce the CW mode ($a_1^{cw}$) in the ring-1 with the coupling strength κ. Obviously, when the system is at steady state, the intensities of these modes in ring resonators and the output at the right port strongly depend on the coupling rate ${\mu _1}$ (${\mu _2}$) between the bus waveguide and ring resonator-1 (2), in which GST film play a significant role. If GST film is aGST, the super-modes in such coupling region can fulfill the phase-matching condition, so that the power can be transferred between the bus waveguide and the ring resonators through CC waveguide. By contrast, once the GST film is cGST, the great mode variations in CC waveguide can change the original phase-matching condition, thus leading to an isolation between the bus waveguide and the ring resonator. By emitting laser pulses (either writing or erasing) on the GST films covering on CC waveguides, a dynamically-tunable coupling rate ${\mu _1}$ or ${\mu _2}$ is feasible by controlling the ratio of cGST to aGST in the GST films, which is depicted in Fig. 1. The distinct change in the coupling rates will bring the redistribution of mode intensities, thus leading to the gain discrepancy in the ring resonators. Consequently, the effective Hamiltonian of the system can be effectively modulated to implement the reconfigurable chiral EP and tunable non-reciprocal transmission.
3. Realization of tunable coupling rate
As illustrated in Fig. 2(a), we will reveal how the GST film is engineered to achieve tunable characteristics of the coupling rates (${\mu _1}$ and ${\mu _2}$) between the bus waveguide and ring resonators. Initially, the entire GST films are at aGST state, i.e. ${x_1} = {x_2} = 0$, where ${x_{1(2 )}}$ represents the length of phase-changed region with cGST state on the CC waveguide-1 (2). The length of CC waveguide is designed to fulfill the phase-matching condition of the super-modes in each coupling region, i.e. the coupling length $L_c^{aGST}$, so that the power can be maximally transferred from the bus waveguide to the ring regions through CC waveguides. In this way, the coupling rate ${\mu _{1(2 )}}$ in spatial domain has the maximum value of ${({{\mu_{1(2 )}}} )_{max}} \approx 1$, which could be equivalently transformed into the coupling rate in time domain, i.e., $\rho _{1(2 )}^t \approx {\mu _{1(2 )}}{v_g}/L$ [47], where ${v_g}$ is the group velocity of light and L is the perimeter of each ring. Note that for obtaining the ${({{\mu_{1(2 )}}} )_{max}}$, the change of gap s will result in the variation of $L_c^{aGST}$, so as to affect the maximum value of $\rho _{1(2 )}^t$. Based on numerical simulations using COMSOL Multiphysics (Supplement 1), we show the relationship between the coupling length $L_c^{aGST}$ and the gap s in Fig. 2(b), in which the coupling length $L_c^{aGST}$ grows with the increase of the gap. However, if the GST film is totally changed into to the cGST state, we show the simulated result of the ratio $L_c^{cGST}/L_c^{aGST}$ changed with the gap $s$ in Fig. 2(b). It is clearly found that the coupling length $L_c^{cGST}$ is much larger than $L_c^{aGST}$ for a fixed gap, enabling great opportunities for tuning the coupling rate $\rho _{1(2 )}^t$. For example, when s is set as 400 nm, the corresponding $L_c^{aGST}$ is about 263 $\mu{m}$, while $L_c^{cGST}$ is about 80 times the value of $L_c^{aGST}$. A tiny change of ${x_{1(2 )}}$ can lead to the light isolation in this area (${x_{1(2 )}}$), which shortens the effective coupling distance of CC waveguide. As a result, only part of the power from the bus waveguide can be transmitted into the ring region through CC waveguide at the original $L_c^{aGST}$, thus giving rise to the diminution of $\rho _{1(2 )}^t$. Therefore, the coupling rate $\rho _{1(2 )}^t$ can be dynamically regulated by controlling the position ${x_{1(2 )}}$ of phase-changed area of GST. By operating super-mode analysis [48,49], the coupling rate can be derived as,
Fig. 2. (a) Schematic diagram of the tunable coupling rate between the bus waveguide and ring 1 (ring 2), by controlling the length ${x_1}$ (${x_2}$) of cGST on CC waveguide 1(2). (b) The dependences of the coupling length $L_c^{aGST}$ and the ratio $L_c^{cGST}/L_c^{aGST}$ on the gap s. (c) The coupling rate in time domain over the length of phase changed region. More details are presented in the inset.
4. Reconfigurable chiral exceptional point
To disclose the physical behaviors of light propagation in this tunable non-Hermitian structure, the time-dependent dynamics in the frame rotating with the resonance frequency of cavities can be expressed as the coupled equations below:
Based on Eqs. (2)–(5), the Hamiltonian $\mathrm{{\cal H}}$ of such an active four-mode system can be derived as the sum of two types of Hamiltonians, which is given by:
Beyond that, the spectral bifurcation existing around this chiral EP, obviously seen from Eqs. (7–9), is the square root dependence, rather than linear dependence. Such dependency property is exactly analogous to that of a common EP, which can as well enhance the detection sensitivity of EP-based devices [50]. Prior theoretical studies have presented excellent sensitivity enhancement of more than six orders [10,51,52], but recent experimental demonstrations only show modest sensitivity enhancements of about 4 times [11] and 20 times [53]. The primary cause lies in that EP is susceptible to any undesired, yet inevitable systematic errors in practice, such as fabrication imperfections or thermal fluctuations and the like. These errors can lead to slight shifts of the system parameters from the original values, in particular for the coupling rate or coupling strength, accordingly resulting in a considerable EP deviation that may make the EP-based devices deviate from the EP operation region and ultimately compromise the performances [10,54,55]. Nevertheless, in our scheme, the intrinsic chirality of this degenerated EP can strengthen the robustness against some slight noise around EP. Furthermore, the effective Hamiltonian ${\mathrm{{\cal H}}_{PT}}$ depicted in Eq. (6) can be modulated by regulating the phase-changed regions of GST films, thereby providing a feasible way to dynamically control the chiral EP of the system. If there are disturbances to the system parameters, this active tuning property can reconfigure new chiral EP states and bring the devices back to EP operation region.
For instance, the system is at EP state initially, where the corresponding positions of phase-changed regions are ${x_2} = 190\; \mu{m}$ on CC waveguide-2 and ${x_1} = 171.6\; \mu{m}$ on CC waveguide-1. Under this condition, as shown in Fig. 3(a1), the mode intensities in the two ring resonators are time-depended and can reach to stable values due to the gain saturation. Once the coupling rate $\rho _2^t$ suffers some fluctuations, it will have a slight change. Since the coupling rate is associated with the length of cGST in our scheme, such a change of $\rho _2^t$ can be regarded as a variation of ${x_2}$. Consequently, the length ${x_2}$ is equivalent to be altered to a new position, e.g., ${x_2} = 180\; \mu{m}$. In this case, the related mode intensity distribution will greatly change, as seen in Fig. 3(a2), which modifies the gains in the two ring resonators. The corresponding gain difference $\mathrm{\Delta }g$ can be described by an implicit function $\mathrm{{\cal F}}[{\rho_1^t({{x_1}} )\; ,\rho_2^t({{x_2}} )} ]$ related to the coupling rates, and can be numerically calculated. As a result, owing to the change of coupling rate in this non-Hermitian system, the previous EP state disappears, as demonstrated in Fig. 3(b1), where the normalized gain difference $\mathrm{\Delta }g/2\kappa $ shifts from 1 (EP state A) to 0.54 (non-EP state B). However, by means of the nonvolatile and reversible properties of GST, a new EP state C corresponding to current ${x_2}$ can be reconfigured in this system, when the phase-changed length ${x_1}$ is engineered to be 165.2 $\mu{m}$. In this way, the device will return back to EP region and take advantage of the EP-assisted enhancement. The spectral singularities of the above two EP states (A and C) are shown in Figs. 3(b2) and (b3), where both the real and the imaginary parts of the eigenvalues coalesce to be identical. Simultaneously, by calculating the normalized gain difference $\mathrm{\Delta }g/2\kappa $ with the change of ${x_1}$ and ${x_2}$, the EP solution in such reconfigurable system can be found from the contour line of $\mathrm{\Delta }g/2\kappa = 1$ in Fig. 3(c), where the effective parameter range of ${x_1}$ $({{x_2}} )$ for EP is from 158 $\mu{m}$ (165 $\mu{m}$) to 175 $\mu{m}$ (190 $\mu{m}$). In addition to the coupling rate above, systematic errors may as well bring about the shift of the coupling strength $\kappa $ between the two rings. Supposing $\kappa $ with a 10% deviation, the reconfigurable property herein can also help the system to reach new EP state just by regulating the length of phase-changed region ${x_1}$ with additional adjustment of −1.6 $\mu{m}$ (1.7 $\mu{m}$) for the positive (negative) 10% deviation (Supplement 1). Hence, such a reconfigurable feature is beneficial for the EP-based on chip device in a fixed structure, which can enhance its reusability and adaptability to resist undesired external perturbations.
Fig. 3. The distribution of mode intensities in the two ring resonators under the phase-change situations of (a1) ${x_2} = 190\; \mu{m}$ and (a2) ${x_2} = 180\; \mu{m}$. (b1)-(b3) The achievement of reconfigurable EP. Here, the previous EP state A will be transformed to a non-EP state B due to the variation of ${x_2}$, and a new EP state C can be reconfigured by dynamically modulating ${x_1}$. These figures show the dependences of (b1) normalized gain, (b2) the real part and (b3) the imaginary part of the eigenvalue on the variation of ${x_1}$. (c) Two-dimensional map of the corresponding relationship between ${x_1}$ and ${x_2}$ for the realization of reconfigurable chiral EP. Here, the relative parameters are set as ${\gamma _0} = 200\; MHz$, ${g_0} = \kappa = 10{\gamma _0}$, ${I_0} = {10^6}$.
5. Tunable non-reciprocal transmission
Besides aforementioned reconfigurable chiral EP, tunable non-reciprocal transmission can be implemented in this non-Hermitian system. In general, the breaking of Lorentz reciprocity is requisite for the generation of non-reciprocity. Conventionally, this prerequisite is achieved by use of magneto-optic materials, which are challenging to incorporate with integrated optics devices. By comparison, our scheme may be feasible and convenient for the implementation of compact and tunable optical isolators. In this section, we will reveal non-reciprocal mechanism in the same system.
As shown in Fig. 4(a), the non-reciprocal configuration in the non-Hermitian system is schematically depicted. Firstly, if no gain saturation is considered, the gain in ring resonators is constant either for the forward or the backward stimulation, which can be defined as $g_{1(2 )}^f(t )= g_{1(2 )}^b(t )= {g_{1(2 )}}$. Consequently, the evolution process of these mode fields for a group of inputs (the components of the vector $\overrightarrow {{S_{in}}} $) acting on this structure are described by the following dynamical equations:
Fig. 4. (a) Schematic diagram of non-reciprocal transmission with two reversely stimulated directions, where the forward and backward channels of light flow are presented. (b) An equivalent system with one input port and one output port
However, such reciprocity can be easily broken in our scheme. Owing to the gain-saturation nonlinearity in the two active ring cavities, the identical $S_{in}^f$ and $S_{in}^b$ will bring about unequal mode intensity term of ${|{B_{1(2 )}^{cw}} |^2} + {|{B_{1(2 )}^{ccw}} |^2}$, which is related to the gains in the ring resnantor-1(2). This feature will make the gain in the same ring to be inconsistent under two opposite excitation directions, i.e. $g_1^f \ne g_1^b$ and $g_2^f \ne g_2^b$, which can break the Lorentz reciprocity to achieve a non-reciprocal transmission i.e. ${t_f} \ne {t_b}$. Our simulated results are depicted in Fig. 5(a) and Fig. 5(b), where the intensity of each resonant mode for the forward and backward cases is numerically calculated under a certain phase-change condition of ${x_1} = 184.6\; \mu{m}$, ${x_2} = 165\; \mu{m}$. The mode intensity distributions in these two cases have a large difference, and the gain related terms are also unequal. The corresponding time evolution process of the forward and backward outputs are exhibited in Fig. 5(c), where the transmission non-reciprocity is clearly presented. To describe quantitatively such a non-reciprocity, the isolation rate is presented by the log ratio of $10{\log _{10}}({{{|{S_{out}^f} |}^2}/{{|{S_{out}^b} |}^2}} )$ and is about −65.3 dB.
Fig. 5. Distinct mode intensity distributions and gain related terms in the two ring resonators respectively for (a) the forward case and (b) the backward case. (c) Non-reciprocal transmission with an isolation rate of −65.3 dB under the condition of ${x_1} = 184.6\; \mu{m}$, ${x_2} = 165\; \mu{m}$.
In this way, the tunable non-reciprocity can be implemented by controlling the phase-change regions of GST films. As demonstrated above, the gains in the two ring cavities strongly depend on the coupling rates. Consequently, the elements of $\hat{S}$-matrix in Eq. (10) and Eq. (11) can be dynamically engineered by controlling the phase-changed regions of GST films. We numerically show the isolation rate with the change of ${x_1}$ and ${x_2}$ in Fig. 6(a), where the tuning range for isolation rate is operated from about −90 dB to 90 dB. For a fixed ${x_2}$, as ${x_1}$ continues to rise, a gradually reversal isolation rate from the forward to the backward can be achieved, implying that we can manipulate the interconversion of the system between reciprocity and non-reciprocity by dynamically engineering the GST films. In practical cases, due to fabrication errors or thermal fluctuations, the intrinsic resonance frequencies of the two ring cavities will slightly shift, which may have an impact on such non-reciprocal transmission. By considering a distinct detuning frequency from 0 to 100 MHz, we show the corresponding isolation rate with the change of ${x_1}$ in Fig. 6(b), where ${x_2}$ is set to be 165 $\mu{m}$ and the frequency shifts $\mathrm{\Delta }{\omega _1}$ and $\mathrm{\Delta }{\omega _2}$ in both rings are identical. We can find the increasing frequency shift can decline gradually the isolation rate and make its maximum value gentler with the change of ${x_1}$. However, the forward and backward transmissions remain non-reciprocal and tunable (Supplement 1). Consequently, by means of the phase-change feature of GST films, one can, in principle, control freely the gain factors of the ring resonators within a certain range, thereby effectively adjusting the isolation rate on the same one chip.
Fig. 6. (a) Tunable non-reciprocal transmissions with the modulations of phase-changed lengths ${x_1}$ and ${x_2}$, in which the isolation rate can range from about −90 dB to 90 dB. (b) The impact of resonance frequency shifts in the two ring resonators on the non-reciprocal transmission. The shift frequencies are set to be 0, 33 MHz, 66 MHz, and 100 MHz, respectively.
6. Conclusion
In summary, a GST-based reconfigurable non-Hermitian system has been theoretically investigated via the doubly-coupled ring resonators. In such a system, four resonant modes provide two degenerated EPs around which the responses to a same directional perturbation will perform an obvious chirality. More importantly, by controlling the phase-changed regions of GST films, the effective Hamiltonian of such system can be modulated, and thus the chiral EP is capable of being dynamically and reversibly engineered. Such reconfigurable property together with the intrinsic chirality are of great benefit for EP-based devices with fixed structures to maintain good performances when suffering system disturbances. Besides, relying on the same system, GST-assisted tunable non-reciprocal transmission can also be realized by breaking the Lorentz reciprocity. In comparison to general methods, the isolation rate can be actively tuned over a broad region, which is more propitious for the on-chip integration. Despite the studies herein computed with TE polarization, analogous characteristics can be obtained as well if carrying out the design for TM operation. Furthermore, this reconfigurable scheme needs no static power consumption and the tuning properties are nonvolatile, which is superior to the conventional modulation means. We believe these reported results here may open up a new space of design freedom on reconfigurable (quasi-) PT symmetric photonics with the incorporation of GST, which will also shed light on the potential applications in reconfigurable non-Hermitian optics, such as sensors, isolators and so on.
Funding
National Natural Science Foundation of China (11904169, 91950106); China Postdoctoral Science Foundation (2020M681576); Postdoctoral Science Foundation of Jiangsu Province (2020Z224); Natural Science Foundation of Jiangsu Province (BK20190383).
Acknowledgment
This work is supported by Top-notch Academic Programs Project of Jiangsu Higher Education Institutions.
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.
Supplemental document
See Supplement 1 for supporting content.
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