• Photonics Research
  • Vol. 10, Issue 4, 980 (2022)
Hongbin Ma1、2、3, Dongdong Li1、2、3, Nanxuan Wu1、2、3, Yiyun Zhang1、2、3, Hongsheng Chen1、2、3、4、*, and Haoliang Qian1、2、3、5、*
Author Affiliations
  • 1Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China
  • 2ZJU-Hangzhou Global Science and Technology Innovation Center, Key Laboratory of Advanced Micro/Nano Electronic Devices & Smart Systems of Zhejiang, Zhejiang University, Hangzhou 310027, China
  • 3International Joint Innovation Center, ZJU-UIUC Institute, Zhejiang University, Haining 314400, China
  • 4e-mail: hansomchen@zju.edu.cn
  • 5e-mail: haoliangqian@zju.edu.cn
  • show less
    DOI: 10.1364/PRJ.450747 Cite this Article Set citation alerts
    Hongbin Ma, Dongdong Li, Nanxuan Wu, Yiyun Zhang, Hongsheng Chen, Haoliang Qian. Nonlinear all-optical modulator based on non-Hermitian PT symmetry[J]. Photonics Research, 2022, 10(4): 980 Copy Citation Text show less

    Abstract

    All-optical modulators with ultrahigh speed are in high demand due to the rapid development of optical interconnection and computation. However, due to weak photon–photon interaction, the advancement of all-optical modulators is consequently hampered by the large footprint and high power consumption. In this work, the enhanced sensitivity around an exceptional point (EP) from parity-time (PT) symmetry theory is initiatively introduced into a nonlinear all-optical modulator design. Further, a non-Hermitian all-optical modulator based on PT symmetry is proposed, which utilizes the large Kerr nonlinearity from indium tin oxide (ITO) in its epsilon-near-zero (ENZ) region. The whole system is expected to operate around EP, giving rise to the advantages of nanoscale integration and large modulation depth. This presented modulator with high efficiency and high-speed all-optical control can be commendably extended to the design methodology of various nanostructures and further prompt the development of all-optical signal processing.

    1. INTRODUCTION

    Optical modulators play a crucial role in photonic integrated circuits (PICs) for optical interconnection and computation under the circumstance of ever-growing data traffic [13]. In recent years, a variety of electro-optic modulators have been reported, which are implemented on several platforms such as an InGaAsP/Si hybrid metal-oxide–semiconductor [4,5] and classical nonlinear crystals that include lithium niobate [6,7]. However, most of these modulators have issues of large footprint (tens or hundreds of micrometer level) or limited modulation-speed at the GHz level. Over the past few years, all-optical modulators have emerged with higher operating speeds than their electronic counterparts [8,9]. Yet, the weak photon–photon interaction remains an impediment for all-optical device design. Thus, for higher compactness and efficiency, optimized photonic structures with strong optical nonlinear interaction are necessary. Currently, epsilon-near-zero (ENZ) materials that exhibit a vanishing real part of the permittivity in a certain wavelength range have emerged with ultrahigh nonlinear response [10]. As a key member of ENZ materials, ITO has been extensively applied in optical modulators with its active electro-variable refractive index [1113]. However, the large optical third-order nonlinearity from ITO at the ENZ region, which may bring large modulation depth and high modulation speed, has hardly been utilized in modulators hitherto. Moreover, in addition to the nonlinear coefficient of the material itself, efficient nonlinearity is also affected by the photonic structural design. Under this circumstance, recent exploration of the enhanced sensitivity from an exceptional point (EP) in parity-time (PT) symmetry theory may offer a novel approach.

    The concept of PT symmetry effectively describes the existence of completely real spectra for non-Hermitian Hamiltonians with complex conjugation potential as V(x)=V*(x) [14]. PT-symmetric Hamiltonians can undergo a phase transition to a symmetry-broken regime, where complex eigenvalues appear formally through a parametric variation [15]. This kind of transition point could be manifested by the properties of EP, which are initiated in the studying of non-Hermitian operators. At the EP, both the eigenvalues and eigenstates coalesce [16]. It has been proved that an elaborate parametric design would lead systems to operate around EP and exhibit enhanced sensitivity [1720]. For the past decade, non-Hermitian physics and PT symmetry have prompted novel micro-nano devices with high performance, for instance, nonreciprocal resonators [21,22], PT-symmetric lasers [23,24], and non-Hermitian electromagnetic metasurfaces [25]. Tunable effects of PT symmetry have exhibited signs of applications, including 1D and 2D photonic lattice geometries through tuning the complex refractive index [2628], single-mode laser through manipulating the gain/loss of a microring resonator [29], and light-light switching through interferometrical manipulation [30]. However, most of these cases contain the gain media that are difficult to be applied in high-speed all-optical control [3133]. Further, PT symmetry theory has yet to be practically introduced into the design of nonlinear all-optical modulators. It is known that, during the optimization of a modulator, plenty of attempts and simulations are always repeated with different geometric parameters in order to attain the optimum performance. By contrast, the EP may reasonably explicate the optimizing process and make it straightforward to improve the modulation depth and reduce the device footprint.

    Here, we propose a non-Hermitian all-optical modulator based on PT symmetry utilizing the large optical nonlinearity from ITO in its ENZ region to tune the coupling coefficient between two waveguides. This optical modulator is designed to operate around EP and achieve THz-level switching through an external femtosecond pump laser, leading to the advantages of nanoscale integration, large modulation-depth, and high-speed all-optical control. To the best of our knowledge, it is the first time that the practical application based on PT symmetry is demonstrated for nonlinear all-optical modulators. The presented structure is composed of conventional materials such as silicon and cobalt, which have good compatibility toward CMOS fabrication technologies. Also, our work identifies a method of optimization for nanostructures using PT symmetry theory, which could be directly extended to the PIC platform and provides a new impetus for the development of all-optical applications.

    2. THEORY DEMONSTRATION AND THE PROPOSED MODULATOR

    Coupled-waveguides system with its representation of the complex eigenvalues as Riemannian surfaces. (a) Two coupled waveguides, as lossless and lossy ones, respectively, form an integrated system that possesses a Hamiltonian H0. (b), (c) Riemannian surfaces of the complex eigenvalues [real (b) and imaginary (c) parts] of the matrix H0 versus the (κ0,γ0) parameters, where the blue regions represent PT-broken and red ones are PT-symmetric regions. In addition, the boundary is considered as lines of EPs. Notably, these Riemannian surfaces contain the PT-symmetric coupled subsystem and decaying subsystem for a realistic manifestation. The existence of a decaying subsystem causes a tilt angle for the surface in (c).

    Figure 1.Coupled-waveguides system with its representation of the complex eigenvalues as Riemannian surfaces. (a) Two coupled waveguides, as lossless and lossy ones, respectively, form an integrated system that possesses a Hamiltonian H0. (b), (c) Riemannian surfaces of the complex eigenvalues [real (b) and imaginary (c) parts] of the matrix H0 versus the (κ0,γ0) parameters, where the blue regions represent PT-broken and red ones are PT-symmetric regions. In addition, the boundary is considered as lines of EPs. Notably, these Riemannian surfaces contain the PT-symmetric coupled subsystem and decaying subsystem for a realistic manifestation. The existence of a decaying subsystem causes a tilt angle for the surface in (c).

    The proposed modulator integrated on SiO2 substrate is presented in Fig. 2(a), which comprises an ITO layer sandwiched by a lossless propagating waveguide and a hybrid lossy waveguide. The propagating waveguide is totally composed of silicon (Siwide), while the hybrid lossy waveguide consists of silicon (Sinarrow) and cobalt. Here, cobalt is exploited to break through the limitation of Kramers–Kronig relations owing to its similar real part of refractive index to silicon and a large extinction coefficient, which reduces the detuning between two waveguides while leading to heavy loss [38,39]. In order to utilize the large Kerr coefficient from ITO and distinguish the signal and pump, the signal wavelength is set as 1240 nm (ENZ wavelength), while the pump is chosen at 1250 nm according to one of the previous reports on ITO’s Kerr nonlinearity, where the change of the refractive index is from n0=0.42+0.42i to nI=1.14+0.27i with 30° incident pump [40]. Note that the operating wavelength at 1240 nm is deduced from current data, and this wavelength could be moved to 1550 nm telecom wavelength through changing ITO’s annealing temperature and doping concentration [41]. In our work, ITO is expected to realize the alteration of a coupling coefficient through its different refractive index with and without the pump laser. With regard to the modulation mechanism, on the one hand, ITO would involve in the light–matter interaction when a pump laser is applied, leading to an increased real part of refractive index and a decreased imaginary part. With the reduced difference between the refractive indexes of ITO and silicon, propagating light will be coupled into the hybrid lossy waveguide and decay consequently, which results in an off-state. On the other hand, the system is working on-state without a pump. These two states are plotted as power-flow at xy and yz planes, as shown in Figs. 2(b) and 2(c). Apparently, a portion of power-flow is distributed in ITO and the hybrid lossy waveguide at the off-state, indicating the light-field coupling and sharp attenuation. Here, the on-state transmitted power efficiency (the ratio of output and input power) is about 50.69%, while the off-state one is 10.68%, which gives the extinction ratio of 6.46 dB.

    Schematics of the proposed modulator, internal electric-field, and power-flow distribution. (a) Structure of modulator integrated on SiO2 (blue) substrate. ITO (yellow) is sandwiched between the propagating waveguide composed of Siwide and hybrid lossy waveguide composed of Sinarrow and cobalt (gray). Note that both red regions represent silicon. Inset shows the cross-section at the yz plane. Widths of propagating and hybrid lossy waveguides are set to be the same, namely, Wpropagrating=Wlossy=280 nm. Besides, WITO=40 nm and WCo=224 nm are the width of ITO and cobalt, respectively. Furthermore, ITO and two waveguides are designed with the same height h=200 nm. Last, the modulation length is considered to be within one wavelength, Lm=1240 nm. (b), (c) Power-flow in the modulator at on-state and off-state from xy plane and yz plane views. The black dashed lines in (b) mark the location of the yz-plane view in (c), where the coupled energy in hybrid lossy waveguide reaches the maximum. (d) Field enhancement at the central cross-section at xy plane in ITO. Pump laser (1250 nm, 30° incident angle) marked as red spiral curves is imposed on ITO. EITO means the electric field at ITO and E0 is the incident electric field.

    Figure 2.Schematics of the proposed modulator, internal electric-field, and power-flow distribution. (a) Structure of modulator integrated on SiO2 (blue) substrate. ITO (yellow) is sandwiched between the propagating waveguide composed of Siwide and hybrid lossy waveguide composed of Sinarrow and cobalt (gray). Note that both red regions represent silicon. Inset shows the cross-section at the yz plane. Widths of propagating and hybrid lossy waveguides are set to be the same, namely, Wpropagrating=Wlossy=280  nm. Besides, WITO=40  nm and WCo=224  nm are the width of ITO and cobalt, respectively. Furthermore, ITO and two waveguides are designed with the same height h=200  nm. Last, the modulation length is considered to be within one wavelength, Lm=1240  nm. (b), (c) Power-flow in the modulator at on-state and off-state from xy plane and yz plane views. The black dashed lines in (b) mark the location of the yz-plane view in (c), where the coupled energy in hybrid lossy waveguide reaches the maximum. (d) Field enhancement at the central cross-section at xy plane in ITO. Pump laser (1250 nm, 30° incident angle) marked as red spiral curves is imposed on ITO. EITO means the electric field at ITO and E0 is the incident electric field.

    It has been proved that, for an ITO–air interface case, the electric-field components (normal to the interface) in ITO are inversely proportional to the permittivity of ITO because of the continuous displacement field; thus, the electric field would be enhanced at the ENZ region with an obliquely incident light [40]. The calculation result of electric-field enhancement in ITO at the xz plane is shown in Fig. 2(d). The maximum electric-field enhancement inside ITO reaches up to 31 times, and the average enhancement is about 14 times. This enhancement boosts the third-order optical nonlinear interactions in ITO, contributing to enlarging the variation of the linear refractive index at lower pumping power [42]. Note that the nonlinear optical coefficients of silicon [43] and cobalt [44] are several orders of magnitude smaller than that of ITO at the ENZ wavelength; thus, the nonlinear responses of silicon and cobalt are not under consideration.

    3. RESULTS AND SYSTEM ANALYSIS

    For the whole system of the non-Hermitian modulator, here the conventional coupled-mode-theory is used (see Appendix C for its reasonability in this system), and the optical-field dynamics can be represented by coupled mode equations as idA1(x)dx=(β1iγ1)A1(x)+κ12A2(x),idA2(x)dx=(β2iγ2)A2(x)+κ21A1(x),where A1(x) is the optical field amplitude, β1 and γ1 are the propagation constant and propagation loss in propagating waveguide, respectively, while A2(x), β2, and γ2 correspond to the ones in a hybrid lossy waveguide. In addition, κ12 means the energy coupled from a hybrid lossy waveguide to propagating waveguide and κ21 vice versa. Note that κ12 would equal κ21 only when the two waveguides are set to the same [45]. Here, the detuning parameter of the waveguides is set as δ=β1β2, which is close to zero when the refractive indexes of two waveguides are equivalent. In the calculation, due to the similarity of cobalt and silicon for their real parts of refractive index, δ is supposed to be near-zero, leading to an approximative condition of κ=κ12=κ21 (κ will be referred to as the coupling coefficient in the following discussion). Subsequently, system’s Hamiltonian comes to be a new one as H1=[β1iγ1κκβ2iγ2].Similarly, the eigenvalue of the whole system can be calculated as λ1,2=12[(β1iγ1)+(β2iγ2)]±124κ2(γ1γ2)2.Hence, a quantitative judgment method could be deduced to determine which state the modulator operates at. When κ>12|γ1γ2|, the system is PT-symmetric. On the contrary, when κ<12|γ1γ2|, it is at the PT-broken state. Certainly, the point where both of them are equal is expected as EP.

    To search for the EP, the simulation of WCo-dependent transmitted power efficiency of the propagating waveguide with and without a pump has been carried out and plotted, as shown in Fig. 3(a). In order to keep the approximation (δ0) valid, the sum of WCo and the width of Sinarrow are set to be invariant, which means the condition of Wpropagrating=Wlossy=280  nm is always the same. Clearly, the dashed and solid blue curves decrease sharply when WCo reaches up to 160 nm. Further, the ascending transmission efficiency above 224 nm induced by the increasing loss corresponds to the “loss-induced transparency” [36]. It can be inferred that EP should be around 224 nm where the extinction ratio reaches the maximum. Note that the abnormal increasing trend of the transmitted power efficiency before 160 nm is mainly caused by the influence of the gap mode as produced in Sinarrow (see in Appendix B). To study the coupling mechanism around EP, the quantitative calculation for coupling coefficient κ with and without pump is plotted in Fig. 3(b). For a better comparison, the coupling-coefficient ratio (division of coefficients with pump and without pump) is calculated and shown as the red curves. The coupling-coefficient ratio around EP presents a sharp peak, corresponding to an increase of extinction ratio.

    Analysis of PT-symmetric and PT-broken regions and eigenvalue spectra of the whole modulator system. (a) Transmitted power efficiency varies with WCo from 14 to 280 nm. The blue dashed and solid curves mean systems with and without pump, respectively. The red curve represents the calculated extinction ratio. (b) The calculated coupling coefficients κ with pump (blue dashed curve) and without pump (blue solid curve). The range of WCo is captured from 112 to 266 nm to study the change of κ around EP. The red curve presents the ratio of coefficients with and without pump. (c), (d) The WCo dependent κ (black curve) and the propagation loss of hybrid waveguide γ2 (blue curve) (c) without pump and (d) with pump. The red curve that represents |γ1−γ2|/2 is designed to compare with κ to judge whether the system is in the status of PT-symmetric (blue regions) or PT-broken (gray regions). (e), (f) The relative-loss γ=|γ1−γ2| dependent eigenvalue spectra of the system. Four legends with different colors mark the eigenvalues with and without pump. Due to the fluctuation of coupling coefficient, the spectra are not smooth as expected.

    Figure 3.Analysis of PT-symmetric and PT-broken regions and eigenvalue spectra of the whole modulator system. (a) Transmitted power efficiency varies with WCo from 14 to 280 nm. The blue dashed and solid curves mean systems with and without pump, respectively. The red curve represents the calculated extinction ratio. (b) The calculated coupling coefficients κ with pump (blue dashed curve) and without pump (blue solid curve). The range of WCo is captured from 112 to 266 nm to study the change of κ around EP. The red curve presents the ratio of coefficients with and without pump. (c), (d) The WCo dependent κ (black curve) and the propagation loss of hybrid waveguide γ2 (blue curve) (c) without pump and (d) with pump. The red curve that represents |γ1γ2|/2 is designed to compare with κ to judge whether the system is in the status of PT-symmetric (blue regions) or PT-broken (gray regions). (e), (f) The relative-loss γ=|γ1γ2| dependent eigenvalue spectra of the system. Four legends with different colors mark the eigenvalues with and without pump. Due to the fluctuation of coupling coefficient, the spectra are not smooth as expected.

    To further verify the accurate location of EP, calculation for propagation loss is absolutely necessary. Considering the existence of attenuation at on-state (mainly caused by loss of ITO), an approximative approach is adopted to divide ITO in half and distribute them into two waveguides (see in Appendix C for the calculation method). Then, the propagation loss of a hybrid lossy waveguide is presented as γ2, and the γ1 is for the propagation loss of the propagating waveguide caused by the half of ITO. The PT-symmetric and PT-broken regions attained from the comparison of 12|γ1γ2| and κ are plotted in Fig. 3(c) (without pump) and Fig. 3(d) (with pump). Deduced from the intersection point of the red and black curves, the location of EP has a change from 184 nm (without pump) to 201 nm (with pump). The system has been actually PT-broken, even when extinction ratio has not risen to the maximum. In fact, the sensitivity is significantly enhanced near the transition point from the PT-symmetric to PT-broken regime, not actually the EP [17]; further, the near-EP region has been commonly utilized in device design [4648]. Thus, performing the mode switching or modulation operation does not require crossing the EP, and it would be reasonable to set modulator’s operating point at the PT-broken state (near EP) in consideration of high extinction ratio. Given all this, the real and imaginary spectra of systems’ eigenvalues with and without a pump have been calculated according to Eq. (6) in Figs. 3(e) and 3(f). For a better demonstration, here γ=|γ1γ2| is utilized as the unit of horizontal coordinate-axis. The calculated results commendably match the analysis of Riemannian surfaces in Fig. 1.

    The influence of WITO on the modulator’s extinction ratio and on-state transmitted efficiency is studied as shown in Figs. 4(a) and 4(b). The trends of these two performance parameters are nearly opposite under the variation of WITO (the larger extinction ratio, the lower on-state transmitted efficiency), which suggests that it should be necessary to balance these two parameters. Two ultimate situations when WITO is 30 and 70 nm are plotted in Figs. 4(c) and 4(d). For the case of WITO=30  nm, the extinction ratio reaches 10.16 dB, and the on-state transmitted efficiency is 42.5% [Fig. 4(c)], while the extinction ratio is 3.19 dB with on-state transmitted efficiency up to 63.9% under 70 nm WITO [Fig. 4(d)]. Another decisive factor of the overall performance of modulators is modulation speed. For our structure, the modulation speed mainly depends on the group velocity in the modulation region and lifetime of the nonlinear response. As the calculation process shows in Appendix D, the propagating time is about 21.2 fs, while the recovery time of the nonlinear response in ITO is 360 fs. Thus, the recovery time is the predominant factor, which allows an all-optical modulation speed up to 2.78 THz, promising for advancement of the terahertz modulators [49,50].

    Analysis of the balance between extinction ratio and on-state transmitted efficiency. (a), (b) The WCo-dependent extinction ratio and on-state transmitted efficiency under different WITO from 30 to 70 nm. Note that the legends are the same for both (a) and (b). (c), (d) Two ultimate states of power-flow distribution at the xy plane where WITO is 30 nm in (c) and 70 nm in (d).

    Figure 4.Analysis of the balance between extinction ratio and on-state transmitted efficiency. (a), (b) The WCo-dependent extinction ratio and on-state transmitted efficiency under different WITO from 30 to 70 nm. Note that the legends are the same for both (a) and (b). (c), (d) Two ultimate states of power-flow distribution at the xy plane where WITO is 30 nm in (c) and 70 nm in (d).

    4. DISCUSSION AND CONCLUSION

    As described, the offset between EP and the locations of maximum extinction ratio is an inevitable challenge due to the loss of tunable material. To overcome this problem, a new method may be possibly employed to replace ITO by multiple metallic-quantum-wells (MQWs) [51,52], which could largely enhance the optical nonlinearity at the lower-loss wavelength region. On the other hand, for our tentative structure that demonstrates the application of PT symmetry on nonlinear all-optical modulator, there is still room for improvement. For example, the hybrid lossy waveguide could be placed at both sides of the propagating waveguide to achieve shorter modulation length and a more compact device, while it will introduce the difficulty to analyze the coupling coefficient at the same time. In addition, it is also reasonable to consider the resonant ring structure [53] to further improve the performance. Furthermore, the plasmonic structures with extreme field-confinement may prospectively contribute to the advancement of extinction ratio [54,55]. In regard to fabrication of this non-Hermitian modulator, the silicon waveguide can be fabricated by the electron-beam lithography and etching process. The cobalt waveguide and ITO slot can be processed by the film deposition after the electron-beam lithography. These several rounds of processes are compatible with the standard CMOS technology, thus enabling the prospect of integration with other PIC platforms [5658]. At last, due to the 14-times electric-field enhancement provided by the sandwich structure (waveguide, ITO, waveguide) at the ENZ wavelength, and comparing with four-times enhancement in Ref. [40], the actual requirement of incident intensity would be reduced to about 42.8  GW/cm2. As an example of a femtosecond laser with 100 fs pulse width, 80 MHz repetition rate, and the focused spot radius of Lm, the averaged power of 16.56 mW is supposed to be applied. Moreover, the transmitted efficiency versus pump intensity in Appendix A shows that the required power could be reduced further with slight sacrifice of extinction ratio.

    In summary, we propose an all-optical modulator based on non-Hermitian PT symmetry utilizing the nonlinearity from ITO at the ENZ region. Further, it has been demonstrated that applying the supreme sensitivity around EP to the all-optical modulator would lead to performance improvement and superiority. For the presented structure, the modulation depth comes up to 5.21  dB/μm, while the on-state transmitted efficiency is 50.69%. Additionally, the modulator has a compact device footprint of 1.24  μm×0.156  μm and an ultrahigh modulation speed that is up to the THz level. Moreover, the proposed device commendably proves the practical application of EPs in on-chip integrated modulators, which would be referred and extended to the design methodology of various nanostructures and further prompt the development of all-optical communication and high-speed signal processing.

    Acknowledgment

    Acknowledgment. H. Q. and H. M. conceived the idea. H. M. conducted the numerical simulations. H. M., D. L., and H. Q. contributed extensively to the writing of the manuscript. H. M., D. L., N. W., Y. Z., H. C., and H. Q. analyzed data and interpreted the details of the results. H. C. and H. Q. supervised the research.

    APPENDIX A: SIMULATION FOR THE WHOLE STRUCTURE

    Numerical simulations using the finite element method, including the power flow in the waveguides, calculation of coupling coefficient, and loss coefficient, are performed with COMSOL Multiphysics. The 2D mode analysis presented as electric-field distribution for the input port of propagating waveguide is plotted in Fig. 5, which gives the effective mode index as 2.042.98×105i. This calculated effective mode index is utilized for the input that is applied to the port of a propagating waveguide in 3D simulation.

    Electric-field distribution of the input port for propagating waveguide.

    Figure 5.Electric-field distribution of the input port for propagating waveguide.

    Transmitted efficiency versus pump intensity.

    Figure 6.Transmitted efficiency versus pump intensity.

    APPENDIX B: GAP MODE ANALYSIS

    Figure 7(a) is the schematic of the whole system at the xy plane, where the hybrid lossy waveguide is marked by the gray dashed rectangle. The electric-field distribution with different widths of cobalt WCo is plotted in Figs. 7(b)–7(i). The electric field is mainly distributed in the gap between ITO and cobalt (i.e., Sinarrow). This gap mode contains two parts of an electric field, E1 and E2, which are illustrated as black and white dashed ellipses in Fig. 7(b). Deduced from the change of the E1, when the WCo increases (before 160 nm) and the gap becomes narrower, E1 becomes weaker while E2 is stronger. Although E2 is becoming more obvious, it is not in a predominant role until the WCo increases to about 160 nm. Under this circumstance, when the coupled electric field in the gap only has E2, the loss of cobalt would be efficiently utilized. Then, enlarging the WCo can better reflect the increase of total propagation loss for the hybrid waveguide.

    Normalized electric-field distributions in the hybrid lossy waveguide. (a) Schematic of the whole system at the xy plane. Hybrid lossy waveguide composed of Sinarrow and cobalt is marked as the gray dashed rectangle; width of cobalt is marked as WCo. (b)–(i) Electric-field distributions in hybrid lossy waveguide under different WCo, which is from 40 to 250 nm (as the example when ITO is under pump). Note that the electric field is mainly distributed in Sinarrow and sorted into two parts, E1 and E2, which are marked as the black and white circles, respectively, in (b).

    Figure 7.Normalized electric-field distributions in the hybrid lossy waveguide. (a) Schematic of the whole system at the xy plane. Hybrid lossy waveguide composed of Sinarrow and cobalt is marked as the gray dashed rectangle; width of cobalt is marked as WCo. (b)–(i) Electric-field distributions in hybrid lossy waveguide under different WCo, which is from 40 to 250 nm (as the example when ITO is under pump). Note that the electric field is mainly distributed in Sinarrow and sorted into two parts, E1 and E2, which are marked as the black and white circles, respectively, in (b).

    APPENDIX C: CALCULATION METHOD FOR COUPLING COEFFICIENT AND PROPAGATION LOSS

    The modulator’s Hamiltonian is deduced from the coupled mode theory (CMT). Actually, using the conventional CMT for the coupled-waveguides here may cause 1% to 2% error according to Ref. [35]. However, this difference has little influence on the location of the PT-transition point (about few-nanometers offset). In addition, the sensitivity is enhanced around EP, and the whole system working conditions are demonstrated through the full-wave simulation. Moreover, we chose “conventional CMT” to better understand and demonstrate the idea of non-Hermitian modulator straightforwardly.

    For the non-Hermitian PT-symmetry system, loss and the real part of refractive index are equally important; further, the coupling coefficient should be calculated at the existence of cobalt. Here, the coupling coefficient is normally characterized by the coupling length, which is defined as the distance where the maximum guided power is transferred from one waveguide to the other. Accordingly, the coupling coefficient can be represented as κ=π2L0,where L0 is the coupling length. The simulation structure is shown in Fig. 8(a), and the modulation length is intentionally extended to 1600 nm for the calculation of coupling length. The transmitted power at each location of hybrid lossy waveguide is calculated through the surface integral of power flow. The results are plotted in Figs. 8(b) and 8(c).

    Normalized transmitted power in hybrid lossy waveguide that is used for calculation of coupling coefficients. (a) Schematic of the simulation structure. The red cross section in hybrid lossy waveguide is used to calculate the surface integral of power flow at each location. (b), (c) Situations without and with pump, respectively. The x-label is the transmitted distance, and each curve means the situation for the specified width of cobalt (from 28 to 252 nm). The coupling length is exactly the distance where the transmitted power reaches the maximum.

    Figure 8.Normalized transmitted power in hybrid lossy waveguide that is used for calculation of coupling coefficients. (a) Schematic of the simulation structure. The red cross section in hybrid lossy waveguide is used to calculate the surface integral of power flow at each location. (b), (c) Situations without and with pump, respectively. The x-label is the transmitted distance, and each curve means the situation for the specified width of cobalt (from 28 to 252 nm). The coupling length is exactly the distance where the transmitted power reaches the maximum.

    Simulation structures for the calculation of propagation loss and their normalized transmitted power (as the example when ITO is under pump). (a) The propagating waveguide with a half of ITO. (b) The hybrid lossy waveguide with a half of ITO. The imaginary part of the refractive index in the left part (marked as lossless part by gray dashed rectangle) is deliberately set to be zero for the more realistic simulation of the propagating and coupling mechanism, which leads to a sharp decay when light propagates to hybrid lossy waveguide. (c), (d) The decay curves of the transmitted power in (a) and (b). The initial positions (x=0) are where the propagating waveguide has ITO for (c) or at the interface of the lossless part and hybrid lossy waveguide for (d). Note that curves in (d) are presented under different widths of cobalt (from 28 to 252 nm).

    Figure 9.Simulation structures for the calculation of propagation loss and their normalized transmitted power (as the example when ITO is under pump). (a) The propagating waveguide with a half of ITO. (b) The hybrid lossy waveguide with a half of ITO. The imaginary part of the refractive index in the left part (marked as lossless part by gray dashed rectangle) is deliberately set to be zero for the more realistic simulation of the propagating and coupling mechanism, which leads to a sharp decay when light propagates to hybrid lossy waveguide. (c), (d) The decay curves of the transmitted power in (a) and (b). The initial positions (x=0) are where the propagating waveguide has ITO for (c) or at the interface of the lossless part and hybrid lossy waveguide for (d). Note that curves in (d) are presented under different widths of cobalt (from 28 to 252 nm).

    APPENDIX D: MODULATION SPEED

    For the presented modulator, the modulation speed generally depends on the propagation time in the modulation region and the lifetime of nonlinear response. The speed of propagation could use the concept group velocity Vg for the description. Group velocity can be interpreted as the speed of information in a wave, which is defined as [59] Vgωβ,where ω is frequency and β is the propagation constant of the propagating waveguide. The frequency-dependent β is calculated as shown in Fig. 10, where the range of frequency is from 1.497×1015  rad/s to 1.545×1015  rad/s (i.e., around the signal wavelength 1240 nm). Thus, Vg is calculated as a result of 5.846×107  m/s, and the propagation time in the modulation region is 21.21 fs. By contrast, the recovery time of the nonlinear response in ITO is 360 fs [40]. Thus, the recovery time decides the modulation speed mostly, giving rise to a result of 2.78 THz.

    Calculation for propagation speed, which is the ratio of the apparent change in ω to the associated change in β.

    Figure 10.Calculation for propagation speed, which is the ratio of the apparent change in ω to the associated change in β.

    References

    [1] G. T. Reed, G. Mashanovich, F. Y. Gardes, D. J. Thomson. Silicon optical modulators. Nat. Photonics, 4, 518-526(2010).

    [2] P. Minzioni, C. Lacava, T. Tanabe, J. Dong, X. Hu, G. Csaba, W. Porod, G. Singh, A. E. Willner, A. Almaiman. Roadmap on all-optical processing. J. Opt., 21, 063001(2019).

    [3] D. A. Miller. Attojoule optoelectronics for low-energy information processing and communications. J. Lightwave Technol., 35, 346-396(2017).

    [4] T. Hiraki, T. Aihara, K. Hasebe, K. Takeda, T. Fujii, T. Kakitsuka, T. Tsuchizawa, H. Fukuda, S. Matsuo. Heterogeneously integrated III–V/Si MOS capacitor Mach–Zehnder modulator. Nat. Photonics, 11, 482-485(2017).

    [5] J.-H. Han, F. Boeuf, J. Fujikata, S. Takahashi, S. Takagi, M. Takenaka. Efficient low-loss InGaAsP/Si hybrid MOS optical modulator. Nat. Photonics, 11, 486-490(2017).

    [6] M. He, M. Xu, Y. Ren, J. Jian, Z. Ruan, Y. Xu, S. Gao, S. Sun, X. Wen, L. Zhou. High-performance hybrid silicon and lithium niobate Mach–Zehnder modulators for 100 Gbit s−1 and beyond. Nat. Photonics, 13, 359-364(2019).

    [7] M. Xu, M. He, H. Zhang, J. Jian, Y. Pan, X. Liu, L. Chen, X. Meng, H. Chen, Z. Li, X. Xiao, S. Yu, S. Yu, X. Cai. High-performance coherent optical modulators based on thin-film lithium niobate platform. Nat. Commun., 11, 3911(2020).

    [8] X. Guo, R. Liu, D. Hu, H. Hu, Z. Wei, R. Wang, Y. Dai, Y. Cheng, K. Chen, K. Liu. Efficient all-optical plasmonic modulators with atomically thin van der Waals heterostructures. Adv. Mater., 32, 1907105(2020).

    [9] M. Klein, B. H. Badada, R. Binder, A. Alfrey, M. McKie, M. R. Koehler, D. G. Mandrus, T. Taniguchi, K. Watanabe, B. J. LeRoy. 2D semiconductor nonlinear plasmonic modulators. Nat. Commun., 10, 3264(2019).

    [10] O. Reshef, I. De Leon, M. Z. Alam, R. W. Boyd. Nonlinear optical effects in epsilon-near-zero media. Nat. Rev. Mater., 4, 535-551(2019).

    [11] R. Amin, R. Maiti, Y. Gui, C. Suer, M. Miscuglio, E. Heidari, R. T. Chen, H. Dalir, V. J. Sorger. Sub-wavelength GHz-fast broadband ITO Mach–Zehnder modulator on silicon photonics. Optica, 7, 333-335(2020).

    [12] M. G. Wood, S. Campione, S. Parameswaran, T. S. Luk, J. R. Wendt, D. K. Serkland, G. A. Keeler. Gigahertz speed operation of epsilon-near-zero silicon photonic modulators. Optica, 5, 233-236(2018).

    [13] X. Liu, K. Zang, J.-H. Kang, J. Park, J. S. Harris, P. G. Kik, M. L. Brongersma. Epsilon-near-zero Si slot-waveguide modulator. ACS Photon., 5, 4484-4490(2018).

    [14] C. M. Bender, S. Boettcher. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett., 80, 5243-5246(1998).

    [15] M.-A. Miri, A. Alu. Exceptional points in optics and photonics. Science, 363, eaar7709(2019).

    [16] T. Kato. Perturbation Theory for Linear Operators, 132(2013).

    [17] Z.-P. Liu, J. Zhang, Ş. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, Y.-X. Liu. Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition. Phys. Rev. Lett., 117, 110802(2016).

    [18] W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, L. Yang. Exceptional points enhance sensing in an optical microcavity. Nature, 548, 192-196(2017).

    [19] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, M. Khajavikhan. Enhanced sensitivity at higher-order exceptional points. Nature, 548, 187-191(2017).

    [20] Y.-H. Lai, Y.-K. Lu, M.-G. Suh, Z. Yuan, K. Vahala. Observation of the exceptional-point-enhanced Sagnac effect. Nature, 576, 65-69(2019).

    [21] B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, L. Yang. Parity–time-symmetric whispering-gallery microcavities. Nat. Phys., 10, 394-398(2014).

    [22] L. Shao, W. Mao, S. Maity, N. Sinclair, Y. Hu, L. Yang, M. Lončar. Non-reciprocal transmission of microwave acoustic waves in nonlinear parity–time symmetric resonators. Nat. Electron., 3, 267-272(2020).

    [23] H. Jing, S. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, F. Nori. PT-symmetric phonon laser. Phys. Rev. Lett., 113, 053604(2014).

    [24] H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, M. Khajavikhan. Parity-time–symmetric microring lasers. Science, 346, 975-978(2014).

    [25] Z. Li, G. Cao, C. Li, S. Dong, Y. Deng, X. Liu, J. S. Ho, C.-W. Qiu. Non-Hermitian electromagnetic metasurfaces at exceptional points. Prog. Electromagn. Res., 171, 1-20(2021).

    [26] K. G. Makris, R. El-Ganainy, D. Christodoulides, Z. H. Musslimani. Beam dynamics in PT symmetric optical lattices. Phys. Rev. Lett., 100, 103904(2008).

    [27] S. Longhi. Bloch oscillations in complex crystals with PT symmetry. Phys. Rev. Lett., 103, 123601(2009).

    [28] Y. Yan, N. C. Giebink. Passive PT symmetry in organic composite films via complex refractive index modulation. Adv. Opt. Mater., 2, 423-427(2014).

    [29] L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, X. Zhang. Single-mode laser by parity-time symmetry breaking. Science, 346, 972-975(2014).

    [30] H. Zhao, W. S. Fegadolli, J. Yu, Z. Zhang, L. Ge, A. Scherer, L. Feng. Metawaveguide for asymmetric interferometric light-light switching. Phys. Rev. Lett., 117, 193901(2016).

    [31] C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, D. Kip. Observation of parity–time symmetry in optics. Nat. Phys., 6, 192-195(2010).

    [32] S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, Y. S. Kivshar. Nonlinear switching and solitons in PT-symmetric photonic systems. Laser Photon. Rev., 10, 177-213(2016).

    [33] M. Kulishov, J. M. Laniel, N. Bélanger, D. V. Plant. Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission. Opt. Express, 13, 3567-3578(2005).

    [34] Ş. K. Özdemir, S. Rotter, F. Nori, L. Yang. Parity–time symmetry and exceptional points in photonics. Nat. Mater., 18, 783-798(2019).

    [35] W.-P. Huang. Coupled-mode theory for optical waveguides: an overview. J. Opt. Soc. Am. A, 11, 963-983(1994).

    [36] A. Guo, G. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. Siviloglou, D. Christodoulides. Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett., 103, 093902(2009).

    [37] J. Wiersig. Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection. Phys. Rev. Lett., 112, 203901(2014).

    [38] P. Johnson, R. Christy. Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd. Phys. Rev. B, 9, 5056-5070(1974).

    [39] A. D. Rakić, A. B. Djurišić, J. M. Elazar, M. L. Majewski. Optical properties of metallic films for vertical-cavity optoelectronic devices. Appl. Opt., 37, 5271-5283(1998).

    [40] M. Z. Alam, I. De Leon, R. W. Boyd. Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region. Science, 352, 795-797(2016).

    [41] J. H. Ni, W. L. Sarney, A. C. Leff, J. P. Cahill, W. Zhou. Property variation in wavelength-thick epsilon-near-zero ITO metafilm for near IR photonic devices. Sci. Rep., 10, 713(2020).

    [42] R. W. Boyd. Nonlinear Optics(2020).

    [43] L. Zhang, A. M. Agarwal, L. C. Kimerling, J. Michel. Nonlinear group IV photonics based on silicon and germanium: from near-infrared to mid-infrared. Nanophotonics, 3, 247-268(2014).

    [44] K. S. Rao, R. A. Ganeev, K. Zhang, Y. Fu, G. S. Boltaev, P. Krishnendu, P. V. Redkin, C. Guo. Laser ablation–induced synthesis and nonlinear optical characterization of titanium and cobalt nanoparticles. J. Nanopart. Res., 20, 285(2018).

    [45] A. Yariv. Optical Electronics(1991).

    [46] J. Zhang, B. Peng, Ş. K. Özdemir, K. Pichler, D. O. Krimer, G. Zhao, F. Nori, Y.-X. Liu, S. Rotter, L. Yang. A phonon laser operating at an exceptional point. Nat. Photonics, 12, 479-484(2018).

    [47] H. Park, S.-G. Lee, S. Baek, T. Ha, S. Lee, B. Min, S. Zhang, M. Lawrence, T.-T. Kim. Observation of an exceptional point in a non-Hermitian metasurface. Nanophotonics, 9, 1031-1039(2020).

    [48] H. Hodaei, M. A. Miri, A. U. Hassan, W. Hayenga, M. Heinrich, D. Christodoulides, M. Khajavikhan. Parity-time-symmetric coupled microring lasers operating around an exceptional point. Opt. Lett., 40, 4955-4958(2015).

    [49] L. Wang, Y. Zhang, X. Guo, T. Chen, H. Liang, X. Hao, X. Hou, W. Kou, Y. Zhao, T. Zhou. A review of THz modulators with dynamic tunable metasurfaces. Nanomaterials, 9, 965(2019).

    [50] R. Degl’Innocenti, S. J. Kindness, H. E. Beere, D. A. Ritchie. All-integrated terahertz modulators. Nanophotonics, 7, 127-144(2018).

    [51] H. Qian, S. Li, C.-F. Chen, S.-W. Hsu, S. E. Bopp, Q. Ma, A. R. Tao, Z. Liu. Large optical nonlinearity enabled by coupled metallic quantum wells. Light Sci. Appl., 8, 13(2019).

    [52] H. Qian, S. Li, Y. Li, C.-F. Chen, W. Chen, S. E. Bopp, Y.-U. Lee, W. Xiong, Z. Liu. Nanoscale optical pulse limiter enabled by refractory metallic quantum wells. Sci. Adv., 6, eaay3456(2020).

    [53] L. Chen, Q. Xu, M. G. Wood, R. M. Reano. Hybrid silicon and lithium niobate electro-optical ring modulator. Optica, 1, 112-118(2014).

    [54] M. Taghinejad, H. Taghinejad, Z. Xu, Y. Liu, S. P. Rodrigues, K. T. Lee, T. Lian, A. Adibi, W. Cai. Hot-electron-assisted femtosecond all-optical modulation in plasmonics. Adv. Mater., 30, 1704915(2018).

    [55] F. Cheng, C. Wang, Z. Su, X. Wang, Z. Cai, N. X. Sun, Y. Liu. All-optical manipulation of magnetization in ferromagnetic thin films enhanced by plasmonic resonances. Nano Lett., 20, 6437-6443(2020).

    [56] K. Wang, H. Qian, Z. Liu, P. K. Yu. Second-order nonlinear susceptibility enhancement in gallium nitride nanowires. Prog. Electromagn. Res., 168, 25-30(2020).

    [57] M. E. Ramon, A. Gupta, C. Corbet, D. A. Ferrer, H. C. Movva, G. Carpenter, L. Colombo, G. Bourianoff, M. Doczy, D. Akinwande. CMOS-compatible synthesis of large-area, high-mobility graphene by chemical vapor deposition of acetylene on cobalt thin films. ACS Nano, 5, 7198-7204(2011).

    [58] M. Si, J. Andler, X. Lyu, C. Niu, S. Datta, R. Agrawal, P. D. Ye. Indium–tin-oxide transistors with one nanometer thick channel and ferroelectric gating. ACS Nano, 14, 11542-11547(2020).

    [59] S. Ellingson. Electromagnetics, 1(2018).

    Hongbin Ma, Dongdong Li, Nanxuan Wu, Yiyun Zhang, Hongsheng Chen, Haoliang Qian. Nonlinear all-optical modulator based on non-Hermitian PT symmetry[J]. Photonics Research, 2022, 10(4): 980
    Download Citation