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Abstract
In recent years, notions drawn from non-Hermitian physics and parity-time (PT) symmetry have raised considerable attention in photonics, enabling various novel structures with entirely new and unexpected features. Here we propose, design, and optimize a compact passive PT-symmetric grating to achieve asymmetric reflection and diffraction based on a silicon-on-insulator (SOI) platform. The structure is composed of two sets of interleaved tailored gratings, which are all well-defined on the top of a silicon waveguide. Without additional loss or gain materials, the effective index and the scattering loss of the waveguide mode are modulated by the structure design. To our knowledge, it is the first time that the scattering loss arising from grating elements is regarded as an efficient way to realize PT-symmetric structure. The complicated multi-parameter optimization process of the proposed PT-symmetric grating is completed by using the particle swarm optimization (PSO) algorithm. In the simulation, asymmetric reflection with high contrast ratio is realized. We also find that the waveguide-to-free-space diffraction from one side of the structure is significantly suppressed, leading to asymmetric diffraction. Moreover, we investigate the fabrication tolerance of the proposed PT-symmetric grating. Our work provides a new perspective for exploring and creating complicated on-chip PT-symmetric devices.
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1. Introduction
Optical losses are considered one of the major challenges in integrated optics, as they affect the performance of many optical devices. Therefore, considerable research efforts have been made to eliminate or mitigate underired optical losses [1]. In recent years, non-Hermitian notions in optics have opened a new avenue for exploiting optical losses, which have led to a number of fascinating optical devices and phenomena. In particular, the concept of PT symmetry stems from the discussion of operators in quantum mechanics, which offers a new approach to exploiting optical losses through tailored structural design [2–5]. In 2007, D. N. Christodoulides et al. introduced PT symmetry into optical systems and formulated a coupled-mode theory (CMT) for PT-symmetric optical waveguide couplers. Thanks to the equivalence between the Schrödinger equation in quantum mechanics and the wave equation in optics, complex PT-symmetric structures can be realized by spatial modulation of the refractive index with balanced gain/loss [6].
The progress in the field of non-Hermitian physics creates opportunities to study novel photonic integrated devices by using gain and loss as new degrees of freedom [7–20]. On the experimental front, PT symmetry has been investigated in various photonic integrated platforms including both silicon, organic and InP materials. Numerous optical structures associated with PT symmetry have been demonstrated, such as compact dimer structures. The first passive and the first active PT-symmetric optical systems based on coupled waveguides were experimentally demonstrated, which were used to realize optical analogies of spontaneous PT symmetry breaking in complex index potentials [7,8]. PT symmetry has also been realized with active-passive-coupled microresonators, where tunable mode splitting and optical isolation were observed [11,17]. Further theoretical and experimental studies of PT symmetry reveal many other interesting optical phenomena [21–31]. An innovative phenomenon named unidirectional invisibility was experimentally demonstrated at the exceptional point in PT-symmetric grating, where reflection from one direction is almost suppressed totally [32–35]. Such a PT-symmetric grating was further employed to achieve unidirectional single-mode orbital angular momentum (OAM) vortex lasing based on an InP-based microring resonator [36]. Despite impressive operation performance, the realization of on-chip PT symmetry remains challenging. It is worth stressing that it is still particularly difficult to integrate and control nanoscale optical gain and loss with high resolution in the fabrication process. This is because the optical gain and loss are limited to certain materials and are not generally compatible with all platforms. For example, the optical gain can be implemented through III–V semiconductors via optical or electrical pumping [37], and optical loss usually arises from material absorption, such as chromium (Cr). Therefore, the heterogeneous integration of different semiconductor materials will significantly increase the complexity of the fabrication process. Therefore, alternative designs of passive PT-symmetric structures with reduced complexity are highly desirable.
In this paper, we propose, design, and optimize a compact passive PT-symmetric grating structure based on a silicon photonic platform. The proposed PT-symmetric structure consists of two sets of interleaved tailored gratings, which are all well-defined on the top of a silicon waveguide. The scattering loss arising from the grating elements is engineered for the design of PT-symmetric structure, which provides a new degree of freedom for the design of non-Hermitian structures. By adjusting the period, the etching depth, and the widths of the gratings, the complex effective index of the waveguide mode could be modulated. In the optimization process, the particle swarm optimization (PSO) method is used for such a multi-parameter PT-symmetric grating structure. In the simulation, both unidirectional reflection and asymmetric waveguide-to-free-space diffraction with high contrast ratio are verified, which indicates the successful theoretical design of PT-symmetric structure. In our design, without the integration of additional gain or loss materials, the fabrication complexity could be significantly simplified. Our work may provide a new perspective for exploring the advantages and applications of chip-scale non-Hermitian photonic devices.
2. Principle, design and optimization
Before the notion of PT symmetry was introduced into optics, the complex grating structure has been studied. In 2005, Mykola Kulishov et al. analyzed such kind of grating systematically, and revealed its striking characteristic named unidirectional reflectionless. A complex grating with the modulated dielectric permittivity might be used to break the space-time reversibility between the mode interaction [35]. This leads to an asymmetrical behavior in this mode coupling process. Specifically, the complex grating will induce and amplify coupling from the forward-propagating mode into the backward-propagating mode when the light is launched from one end of the grating but the same is not true when the light is launched from the opposite end. Figure 1(a) shows the detail of an ideal PT-symmetric grating, which has complex modulation of the refractive index in the propagation. In Fig. 1(a), the process of unidirectional reflectionless is illustrated, where the forward reflection is significantly larger than the backward reflection. In the bottom of Fig. 1(a), the red line shows the real part nR of complex refractive index modulation, and the blue line represents the imaginary part nI (gain/loss). The ideal dielectric permittivity modulation can be expressed as follows:
Fig. 1. (a) Diagram illustrating the process of unidirectional reflectionless and schematic showing ideal PT-symmetric grating with complex refractive index modulation. (b) A quasi-PT-symmetric grating structure with asymmetric reflection.
To realize an equivalent complex index modulation, Feng et al. designed a passive grating structure by using additional Si and loss materials germanium (Ge)/chrome (Cr) bilayer combo structures on top of the waveguide to mimic the microscopic parity-time modulation on a macroscopic scale. As shown in Fig. 1(b), two sets of interleaved passive gratings are used to form a quasi-PT-symmetric grating. The dielectric permittivity modulation can be written as:
Substituting Eq. (2) into Eq. (3), one can obtain the values as follows:
Here, we consider that the constant C in Eq. (2) is set to be exp(-iπ/4), thus, the dielectric permittivity modulation is expressed by:
Based on the above analysis, we propose a new guided-mode modulation method by meticulous structural design and algorithm optimization. Without additional loss materials, the scattering loss could be considered as an efficient way to form a PT-symmetric structure. As shown in Fig. 2, a passive PT-symmetric grating structure based on silicon platform is designed, which is composed of two sets of interleaved tailored gratings. One grating is formed with additional silicon on the top of the waveguide,which can lead to the mode effective index increasing and the scattering loss. Another grating with periodic trenching etching can decrease the mode index and also enhance the scattering loss. Two complex gratings are consistent with the modulations in Eq. (5). To simplify the design, rectangular-function modulated optical potentials using additional silicon and trench etching are employed, which could achieve the same unidirectional effect with sinusoidal-function modulation [33].
Fig. 2. (a) Structure of the proposed passive PT-symmetric grating. (b) Schematic cross section of the grating, where h is the height of the silicon waveguide, h1 and h2 are the heights of two sets of grating respectively. T, T1, T2 and T3 represent the period of PT-symmetric grating, the width of the additional silicon, the width of the trenches and the distance of two sets of grating, respectively.
As shown in Fig. 2, the studied passive PT-symmetric grating is a 500-nm-wide and 220-nm-thick silicon waveguide with periodic etching. The waveguide supports a fundamental mode (quasi-TE mode) with an effective index of neff =2.37 around the wavelength of 1550 nm. As depicted in Fig. 2(a), two sets of grating are interleaved on the top of the silicon waveguide. The height h of the silicon waveguide is 220 nm. The period T is chosen to be 654 nm for the central wavelength of 1550 nm, corresponding to the second Bragg order. The number of grating units is set to 30. Considering that the mode effective index and the scattering loss could be greatly affected by the grating size, which should be meticulously designed. However, the complex effective index of the waveguide mode is actually difficult to calculate accurately due to microscopic structure perturbation. In order to obtain optimal structure parameters, the proposed PT-symmetric grating is optimized by PSO method. The distance between the additional silicon and the trench etching is T/8 (i.e. T3 = T/8) in Fig. 2(b). The widths (T1, T2) and heights (h1, h2) of the grating elements needs further optimization.
Recently, many iterative optimization algorithms have been used to optimize structural parameters of functional photonic integrated devices, such as PSO, genetic algorithm (GA), and direct binary search (DBS) algorithm [38–41]. It is known that PSO has the characteristics of fast convergence and simple operation. Here, we use PSO algorithm to optimize the proposed multi-parameter PT-symmetric structure, which has simpler rules than genetic algorithm (GA) and is more suitable for multi-parameter continuous optimization than direct binary search (DBS) algorithm. The entire optimization procedure is shown in Fig. 3. A flow chart of PSO process is provided in Fig. 3(a). Initializing generation means randomly generating 20 grating structures with T1, T2, h1, h2 to serve as 20 populations of the first generation. The figure-of-merit (FOM) of the device designed by PSO is defined as:
where tF and tB are the reflectivities when light is injected into the designed structure from forward and backward directions. In the simulation, the maximum values of tF and tB over an operating bandwidth of 60 nm are chosen for the FOM calculation. For an ideal PT-symmetric grating, the FOM is 1. In the initial step of the optimization process, particle’s velocity and position in the 4-dimensional parameter space are randomly selected. In PSO algorithm, x means particle’s position and v means particle’s velocity. We choose h2 as x1, h1 as x2, T2 as x3, and T1 as x4. Then v1 and v2 are randomly selected from -10 to 10 when x1 is randomly selected from 20 nm to 200 nm, and x2 is randomly selected from 240 nm to 420 nm, while v3 and v4 are randomly selected from -4 to 4 when x3 and x4 are randomly selected from 80 nm to 220 nm.Fig. 3. (a) Flow chart for the optimization process. (b) Figure of merit at each iteration for the device with 20 populations. The dots are the individual values, and the blue line of the inset shows the maximum FOM values for each iteration. The optimized device has a maximum FOM value of 0.8845.
We simulate the proposed device by using three-dimensional finite difference time domain (3D-FDTD) method. The simulation mesh size is 44 nm×44 nm×10 nm, and the power cutoff condition is 0.00001% of initial power. In the simulation, the structural parameters are adjusted and optimized by PSO algorithm. When the FOM increases, new populations would be generated and used for FOM calculation. Otherwise, the parameters go back to last state. When the FOM exhibits no great improvement after 10 iterations, the whole optimization will end. Figure 3(b) shows the convergence to a FOM of 0.8845, and the inset shows the details. It can be seen that the FOM growth trend slows down after 5 iterations. The final optimization parameters are obtained when 30 iterations are finished. The width T1 and depth h1 of additional silicon are 130 nm and 330 nm, the width T2 and depth h2 of trench etching are 190 nm and 190 nm.
3. Simulation results
3.1 Asymmetric reflection
In this part, we first simulate two gratings separately. Due to the left-right symmetry of the conventional grating structures, the transmission and reflection spectra are the same on both sides. The transmission and reflection spectra of the grating with trench etching are plotted in Fig. 4(a). One can see that the maximum reflectivity is about 0.4 around the wavelength of 1543 nm. In Fig. 4(b), the simulation results for the grating with addition silicon are presented. The center wavelength moves to 1567 nm, and the maximum reflectivity is about 0.35. It is worth noting that trench etching and additional silicon would decrease and increase the waveguide effective index, respectively, and thus shift the resonance peak.
Fig. 4. (a)(b) Numerically calculated transmission and reflection spectra for conventional gratings with trench etching and additional silicon, respectively. (c) Simulated reflection spectra of the designed PT-symmetric grating in forward and backward directions. (d) Calculated contrast ratio of reflectivities, showing high contrast ratios over the studied wavelength range from 1520 nm to 1580 nm.
Different from the uniform gratings with the same reflection spectra on both sides, when two complex gratings are combined to form a passive PT-symmetric grating, the structural symmetry is broken and asymmetric reflection could be realized. Figure 4(c) shows the simulated results of the optimized PT-symmetric grating. One can see that the reflectivities are significantly distinguished in the forward and backward directions with high extinction ratio in the studied wavelength range from 1520 nm to 1580 nm. The resonance peak of the forward reflection spectrum is around 1550 nm owing to the balanced index modulation introduced by both trench etching and additional silicon. In such a passive system, optical loss is no longer compensated without optical gain. The corresponding S-matrix is not unitary and a similar exceptional point still exists, which causes unidirectional invisibility. It is evident that the designed on-chip PT-symmetric grating successfully mimics the unidirectional effect inherently associated with the exceptional point. The complex refractive index modulation essentially creates the asymmetric coupling coefficients, thus leading to nontrivial asymmetric reflection. The contrast ratio between the forward and backward reflection is defined as C = (RF-RB)/(RF+RB), where RF and RB represent the forward and backward reflectivities, respectively. The calculated contrast ratio is plotted in Fig. 4(d), showing that the reflection in the backward direction is significantly suppressed. At the wavelength of 1551 nm, the forward reflectivity is 0.779, while the backward reflectivity is 0.009, and the contrast ratio is 0.977. Compared with previously reported device [33], the reflectivity and contrast ratio of the optimized PT-symmetric grating have been significantly improved.
The asymmetric reflection from the designed PT-symmetric grating can also be visualized from the light fields propagating inside the waveguide. We simulate the transmitted light fields inside the waveguide grating at 1550 nm when light beams are launched from both sides. As shown in Fig. 5, the forward propagating light beam and its reflection form a strong interference, whereas the reflection is barely seen with backward incidence. These results show that the designed on-chip waveguide grating successfully mimics the unidirectional effect inherently associated with the exceptional point.
Fig. 5. (a)(b) Simulated transmitted light fields at 1550 nm when light beams are launched from both sides. The incidences are set at boundaries of the designed PT-symmetric grating along + x/-x in the forward/backward directions.
3.2 Asymmetric diffraction
As shown in Eq. (4), when δ is close to 1, the second-order Fourier expansion coefficient A-2 is much larger than A2, which indicates asymmetric in-plane reflection. Meanwhile, the first-order Fourier expansion coefficient A-1 is smaller than A1, thus leading to asymmetric out-of-plane vertical diffraction. The simulated diffraction efficiencies with forward and backward incidences are plotted in Fig. 6(a), where the vertical diffraction efficiency with forward incidence is much smaller than that of the backward direction around the wavelength of 1550 nm. The contrast ratio between the forward and backward diffractions is calculated and plotted in Fig. 6(b), and the maximum value is 0.75 around 1550 nm. The simulated electric field distributions of out-of-plane emissions at 1550 nm are shown in Figs. 6(c) and 6(d). Since the transmitted guided mode is the fundamental transverse electric (TE) mode, the diffracted light fields are polarized in y-direction. As a comparison, it can be seen that the electric field amplitude of the forward diffraction pattern in Fig. 6(c) is smaller than that of the backward diffraction in Fig. 6(d), which agrees well with the results given in Fig. 6(a). Therefore, both asymmetric in-plane reflection and chip-to-free-space diffraction can be realized based on the proposed PT-symmetric grating.
Fig. 6. (a) Simulated diffraction spectra of the designed PT-symmetric grating with forward and backward incidences. (b) Calculated contrast ratio of the diffraction between forward and backward propagation. (c)(d) Electric field distributions of out-of-plane emissions with forward and backward incidences at 1550 nm.
4. Fabrication tolerance
Since it is difficult to control the etching of the microscopic structures during fabrication, we further analyze the effect of the structure parameter deviations on device performance, including the widths (T1, T2) and heights (h1, h2) of the grating elements. In Figs. 7(a) and 7(b), the forward reflection spectrum shifts slowly when the height and width of the additional silicon vary in the range of ±30 nm. The corresponding contrast ratios are presented in Figs. 7(c) and 7(d). One can see that the average values of the contrast ratios are larger than 0.85, which indicates that the contrast ratio is not very sensitive to the parameter deviations of the additional silicon. In Fig. 8(a), when the trench depth varies in the range of ±20 nm, the forward reflection spectrum shifts drastically and the maximum reflectivity also varies greatly. In Fig. 8(b), the forward reflection spectrum shifts slowly when the trench width varies in the range of ±30 nm. The corresponding contrast ratios of reflectivities are presented in Figs. 8(c) and 8(d). When the deviation Δh2 of etching depth is 20 nm, the average value of the contrast ratio shown in Fig. 7(c) is reduced to 0.6. When the deviation ΔT2 of etching width is ±30 nm, the average value of the contrast ratio is still more than 0.9 in Fig. 7(d). Therefore, by comparison, it can be found that the device performance is very sensitive to the trench depth, and the corresponding fabrication tolerance is about ±20 nm.
Fig. 7. (a)(b) Simulated forward reflection spectra of the device by changing the height and width of the additional silicon, respectively. (c)(d) Calculated contrast ratios of reflectivities of the device by changing the height and width of the additional silicon, respectively.
Fig. 8. (a)(b) Simulated forward reflection spectra of the device by changing the height and width of the trenches, respectively. (c)(d) Calculated contrast ratios of reflectivities of the device by changing the height and width of the trenches, respectively.
5. Discussion
Compared with previous reports, we have made some improvements in structure design and performance. First, we introduce the scattering loss to PT-symmetric optical system by tailoring microscopic structure design. Thus, the absorption materials could be avoided, and the fabrication process could be simplified. Second, we use the PSO algorithm to accelerate the complicated multi-parameter optimization process and improve device performance, which may open up a door to develop more superior on-chip non-Hermitian photonic integrated devices for wide optical applications. Last, we further reveal that asymmetric diffraction can be also achieved. Thanks to its intrinsic asymmetric properties, the proposed passive PT-symmetric grating could be further developed and employed for functional integrated devices such as grating couplers and unidirectional lasers. Specifically, the proposed PT-symmetric grating could help reduce reflection from grating couplers because the backward reflection is greatly suppressed. Moreover, the proposed device could be elegantly exploited to establish unidirectional single-mode lasing in microring resonators, which indicates a likely perspective for fostering a new generation of active photonic devices [36]. Furthermore, the scattering loss induced by gratings could be also used to explore more complicated non-Hermitian photonic devices, such as anti-PT symmetry and high-order exceptional point [42–44].
6. Conclusion
Optical loss is usually undesirable for optical systems and micro-scale optical devices. Fortunately, the concepts from non-Hermitian and PT symmetry provide a new perspective for the design of optical devices by employing various optical losses. We propose, design and optimize a compact passive silicon PT-symmetric grating to achieve asymmetric reflection and diffraction by using PSO algorithm. By combining two sets of interleaved tailored gratings, equivalent PT-symmetric grating structure is designed and optimized. In the simulation, asymmetric reflection and diffraction with high contrast ratio have been realized. It is believed that the proposed passive PT-symmetric structure would be further developed and employed for more complicated functional integrated devices.
Funding
National Key Research and Development Program of China (2018YFE0201800); China Postdoctoral Science Foundation (2021M690390); National Natural Science Foundation of China (62071042, 62105028).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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