Plane Wave Expansion Research Articles

Ultrafast Optical Encoder Using Photonic Crystal Ring Resonator

ABSTRACT In this paper, we design and simulate a two-dimensional photonic crystal-based high speed logical encoder for optical computing application. The encoder consists of a ring resonator, four waveguides and two reflectors arranged in a hexagonal lattice platform. The optical encoder works on the interference effect which guides the propagation of light in the desired path. The light behavior is simulated with four different logic states. The Plane Wave Expansion (PWE) method is used to analyze the band gap and Finite Difference Time Domain (FDTD) analyzes the performance characteristics of the encoder, such as bit rate, delay time, and contrast ratio. The proposed structure has a footprint area of 143.84 µm2. The small compact size structure with simple defect paths reduces the encoder delay time, which in turn increase the data bit rate. The response time of the proposed encoder is 0.119 ps and bit rate is 8.403 Tbps. The encoder is designed with very low output power in “0” logic state and high output power in “1” logic state leading to high contrast ratio of 22.57 dB. Therefore, highly efficient 4 × 2 optical encoder is suitable to build in optical integrated circuits.

Dirac points and flat bands in two-dimensional magnonic crystals with honeycomb–kagome structure

Based on the model of magnonic crystals (MCs) with honeycomb structure, we propose another model of two-dimensional MCs with honeycomb–kagome structure that is a periodic magnetic composite system composed of Fe, Co, or Py ferromagnetic cylindrical scatterers arranged in the EuO matrix as the honeycomb–kagome structure. The band structures of magnons in these systems are studied numerically by using the plane-wave expansion method. The results show that the Dirac points of magnons will be generated at the Brillouin region points if the scatterers are close-packed, that is to say, the edges of cylindrical scatterers are in contact with each other. The frequency of Dirac points can be indirectly adjusted by changing the radius ratio of close-packed cylinders. In addition, in the case of a large difference in the radius between the close-packed cylindrical scatterers, there will be a magnonic flat band in the band structure, which is a phenomenon of so-called compact localized states different from the impurity state in the crystal, and it is formed by the highly interference superposition of spin waves in the honeycomb–kagome structure. The research on the generation and modulation of magnonic Dirac points and flat bands not only expands the research content of condensed matter topological physics but also provides a promising platform for the application of artificial MCs in the fabrication of spin-wave topological devices.

Efficient method for modeling large-scale arrays of optical nanoresonators based on the coupling theory of quasinormal mode.

We propose an efficient method for calculating the electromagnetic field of a large-scale array of optical nanoresonators based on the coupling theory of quasinormal mode (QNM). In this method, two approaches of the scattered-field reconstruction and stationary-phase-principle calculated plane-wave expansion are developed to obtain the regularized QNM (RQNM) in different regions. This accurate and efficient calculation of RQNM resolves the far-field divergence issue of QNMs in the QNM-coupling theory, thus enabling a rapid computation of the electromagnetic field of a large-scale array of optical nanoresonators, which is a challenging task for full-wave numerical methods. Using this method, we consider the numerical example of the radiation problem of a single point source in a large-scale periodic array of optical nanoantennas. In comparison to full-wave numerical methods, this method significantly reduces the computation time by 1∼2 orders of magnitude while maintaining accuracy. The high computational efficiency and physical intuitiveness of the method enables to clarify the impact of array size (exceeding 50 × 50 wavelengths), period and field-coupling range (far beyond the tight-binding approximation) on the optical response. The proposed method and results can provide an efficient tool and guidance for the design of large-scale arrays of optical nanoresonators.

Tunable Ultralow-Frequency Bandgaps Based on Locally Resonant Plate with Quasi-Zero-Stiffness Resonators

In order to suppress the transverse vibration of a plate, a quasi-zero-stiffness (QZS) resonator with tunable ultralow frequency bandgaps was introduced and analyzed. The resonator was designed by introducing the quasi-zero-stiffness systems into mass-in-mass resonators. The plane wave expansion method was employed to derive the bandgap characteristics of the locally resonant (LR) plate with QZS resonators, and corresponding simulations were carried out by finite element method (FEM). The results show that an LR plate with a QZS resonator can provide two bandgaps, and the ranges of the bandgaps agree well with the vibration attenuation bands calculated by FEM. Owing to the introduction of the QZS system, the bandgaps can be easily transferred to a lower frequency or even an ultralow frequency. The damping of the QZS resonators can effectively broaden the vibration attenuation bands. In addition, the differentiated design of the bandgap frequencies can be realized to obtain broadband low-frequency transverse wave suppression performance. Finally, a mechanical structure design scheme was proposed in order to achieve flexible adjustment of the bandgap frequency, which significantly increases the engineering applicability of QZS resonators.

2D FDTD Electromagnetic Simulation of an Ultracompact All Optical Logic Gate Based on 2D Photonic Crystal for Ultrafast Applications

In this paper, the concept of photonic crystals (PhCs) is fundamental to designing and simulating an all-optical logic gate device. We proposed an all-optical switch composed of two-dimensional (2D) photonic crystal waveguides with a central photonic crystal ring resonator (PCRR). The new all-optical NAND logic gate device comprises two linear waveguides coupled to each other through a single compact PCRR. The plane wave expansion (PWE) and finite-difference time-domain (FDTD) methods are applied to simulate the properties of the system. The structure is implemented on the operational wavelength of 1700 nm on an air wafer of only 12 μm × 12 μm. Indeed, the simulation results show that the proposed all-optical NAND gate is a strong candidate for ultrafast photonic integrated circuits (PICs) for applications in optical communications, being advantageous with high transmitting power, with simple design, and without the use of optical amplifiers and nonlinear materials.

Highly sensitive gamma ray dosimeter based on dual-core photonic crystal fiber

This paper introduces the application of a photonic crystal fiber (PCF) sensor to be used as a gamma ray dosimeter. This paper proposes an efficient analysis of a highly sensitive surface Plasmon resonance for gamma ray dose detection in nuclear research reactors. The proposed structure makes use of a Poly-Vinyl Alcohol (PVA) polymer, which has high sensitivity to an external electric field and gamma rays making it suitable for nuclear reactor applications. At resonance frequency, the peak of spectral loss for dosimeter sensors refers to gamma irradiation dose values. Therefore, study the different geometrical parameters for the proposed PCF sensor is of significant importance to achieve high sensitivity with low spectral losses. The full-vectorial finite-element method has been used for analysis with the plane wave expansion method. The sensitivity of the suggested sensor achieves up to 0.006 µm per gamma-ray dose unit (μm/Gy) with low losses.

One-dimensional photonic crystal with tilted termination and its angular filtering properties for radiative cooling.

Photonic crystals are known for their band-gap structures. Due to their band-gaps, they can act as filters in both temporal and spatial domains. However, in most cases, due to their physical symmetry, their angular responses are symmetrical. Here, a structure based on a 1D photonic crystal is introduced and analyzed, which has an asymmetric angular selectivity. The structure is analyzed using the plane wave expansion method. The properties of the structure are expressed and verified by a commercial full-wave simulator software. Based on the analysis and its results, some simple design rules are derived. By using the extracted rules and some approximations, the potential of the structure to be used in radiative coolers, which are not completely toward the sky, is introduced. It is shown that if the structure is used as windows in buildings, it can save up to tens of watts per square meter in energy consumption for air conditioning. Finally, the whole structure including the radiative cooler is simulated, and the results support the calculations and approximations.

Topological localized modes in moiré lattices of bilayer elastic plates with resonators

We investigate the existence of higher order topological localized modes in moiré lattices of bilayer elastic plates. Each plate has a hexagonal array of discrete resonators and one of the plates is rotated at an angle (21.78∘) which results in a periodic moiré lattice with the smallest area. The two plates are then coupled by inter-layer springs at discrete locations where the top and bottom plate resonators coincide. Dispersion analysis using the plane wave expansion method reveals that a bandgap opens on adding the inter-layer springs. The corresponding topological index, namely fractional corner mode, for bands below the bandgap predicts the presence of corner localized modes in a finite structure. Numerical simulations of frequency response show localization at all corners, consistent with the theoretical predictions. The considered continuous elastic bilayered moiré structures opens opportunities for novel wave phenomena, with potential applications in tunable energy localization and vibration isolation.

First-principles study of the electronic structure and optical properties of C-doped SnS2.

Density functional theory (DFT) was used to investigate the effects of varying carbon doping concentrations on the electronic and optical properties of SnS2-doped systems. The findings show that a doping concentration of 3.7% in SnS2 results in the highest structural stability and the lowest formation energy. A pure SnS2 monolayer is an indirect bandgap semiconductor, and the result reveals that increasing carbon doping correlates with a gradual reduction in the system's bandgap. The density of states analysis reveals that the valence band comprises C-2p, S-3p, and Sn-5p orbitals, whereas the conduction band consists of S-3p, Sn-5s, and C-2p orbitals. Furthermore, doping concentration appears to cause a redshift in both the absorption coefficient and reflection peaks, which both decrease as doping concentration increases. The calculations for this study were performed using DFT within the CASTEP module of Materials Studio Segall et al.J Phys: Condens Matter 14(11):2717, 2002. The system parameters and structures were optimized to determine the electronic structure and optical properties. Geometric optimization and calculations were carried out with the generalized gradient approximation plane-wave pseudopotential method and the Perdew-Burke-Ernzerhof functional Perdew et al.Phys Rev Lett 80(4):891-891, 1998. The parameters for structural optimization included a plane-wave expansion cutoff energy set at 500eV and a k-point mesh of 6 × 6 × 1 for Brillouin zone integration. The electronic convergence criteria were established at 1.0 × 10-5eV/atom for the unit cell energy and 1.0 × 10-6 eV/atom for self-consistency. The internal stress deviation was maintained below 0.05 GPa, the atomic force interactions were kept under 0.03eV/Å, and atomic displacements during geometric optimization were confined to less than 0.001Å. To calculate the properties of the SnS2 monolayer, a vacuum spacing of 15Å along the z-axis was introduced to prevent interactions between adjacent layers.

Frequency encoded tristate Pauli X-gate using SOA assisted photonic band gap crystal

The quantum optical tristate Pauli-X gate can broadly be used due to its very fast speed of operation up to THz order, very fast information processing rate, and a wide domain of application. This Pauli X-gate is also important as it can undergo inversion operation. In this proposed scheme, the quantum tristate Pauli-X logic gate has been constructed by a closely packed square lattice of 2D air photonic band gap crystal (PhCs) composed of GaAsInP-dopped rods. The photonic band gap crystal-based tristate Pauli-X gate is performed also as an all-optical reversible quantum logic gate which has the character of low-powered output, very fast speed and a densely packed design structure. This photonic crystal-based structure is also verified by simulation experiment that this proposed scheme performs as the path of two input signals are cross-exchanged to one another at the output, retaining the path of the other input signal same. In this paper, the frequency encoding technique is used in all the designs to establish the logic of all-optical tristate Pauli X-gate towards the need of obtaining a three-input-three-output inversion system. The frequency encoding technique and the photonic crystal-based three Semiconductor Optical Amplifiers (pc-SOAs) are used here in all the schemes to establish the logic of the tristate Pauli-X gate scheme. Tristate Pauli-X gates have been designed and analyzed by Finite-Difference Time-Domain (FDTD) and Plane Wave Expansion (PWE) techniques. The Plane Wave Expansion (PWE) technique is used here in each system for determining the frequency range of band gap structure of transverse electric (TE) mode in the units of μm-1. All of these devices perform within the frequency range 0.629818≤ω2πc≤0.888037 and 1.17551≤ω2πc ≤ 1.34921 μm-1. The wafer dimension of each photonic band gap crystal-based device is 19.20 μm (length) × 16.20 μm (width) providing a very fast response time and very fast information handling (bit rate) capacity, which is found good for these THz Photonic crystal devices. Simulation results show that the proposed gate provides a response period of 0.291 ps and a bit rate of 3.44 Tbps, operating with a low input power of 3.45 mW. The reversible gate exhibits a response period of 0.289 ps and a bit rate of 3.46 Tbps, operating also with a low input power of 2.70 mW. Additionally, the alternative Pauli-X gate demonstrates a response time of 0.291 ps and a bit rate of 3.44 Tbps, operating with a low input power of 2.83 mW.

Design and analysis of corner scatter inclusion in photonic crystal-based ring resonator

One of the significant challenges in integrated optical systems is achieving miniaturization. Traditional components can limit the effective utilization of chip area. Devices with distinctive thickness and length characteristics, including the Mach Zehnder interferometer (MZI), the directional coupler, and waveguides, contribute to this issue, limiting the efficient use of chip area. Photonic crystals offer a solution to this problem. The proposed design results in two optical bandgap ranges, from 0.456586 to 0.675819 μm–1 and from 1.12855 to 1.16338 μm–1, both in TE mode with gap widths of 0.219233 and 0.34826 μm–1, respectively. The corresponding wavelength ranges are from 1479 to 2190 nm and 859 to 886 nm. The analysis of field propagation in the photonic crystal ring resonator (PCRR) is carried out using the finite-difference time domain (FDTD) method, while the bandgap analysis is performed by the plane wave expansion (PWE) method. This work primarily focuses on the incorporation of corner scatters in the PCRR. Corner scatters play a crucial role in guiding the field smoothly inside the ring, thus preventing the localization of light within the structure. After the simulation, various attributes were compared for both structures. At a resonant wavelength of 1550 nm, the input intensity is measured as 0.0591 a.u., and the output intensities for with and without corner scatter are 0.0458 and 0.0539 a.u., respectively.

A novel design of an all‐optical D flip‐flop based on 2D photonic crystals

AbstractIn this study, we endeavored to create and simulate a D‐type flip‐flop utilizing two‐dimensional photonic crystals featuring a square lattice. This flip‐flop, composed of just two intersecting waveguides, is both compact and straightforward. Constructed from 10 × 10 dielectric rods suspended in air, it boasts low output power in logic mode 0, creating a significant distinction between optical powers in modes 0 and 1, lending it a logical aspect. Optimized for operation at a communication wavelength of 1.55 µm, this flip‐flop proves suitable for optical communication systems. The third window optical fiber communication system is suitable for 1.55 μm wavelength because the fiber has the lowest loss. Our analysis involved calculating the band structure via the plane wave expansion method and determining the optical power distribution using the finite difference time domain method.

Transmission and control of band gap vortex solitons in fractional-order diffraction honeycomb lattices

In this paper, the existence and transmission characteristics of gap vortex optical solitons in a honeycomb lattice are investigated based on the fractional nonlinear Schrödinger equation. Firstly, the band-gap structure of honeycomb lattice is obtained by the plane wave expansion method. Then the gap vortex soliton modes and their transmission properties in the fractional nonlinear Schrödinger equation with the honeycomb lattice potential are investigated by the modified squared-operator method, the split-step Fourier method and the Fourier collocation method, respectively. The results show that the transmission of gap vortex solitons is influenced by the <inline-formula><tex-math id="M3">\begin{document}$ {\mathrm{L}}\acute{{\mathrm{e}}}{\mathrm{v}}{\mathrm{y}} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20232005_M3.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20232005_M3.png"/></alternatives></inline-formula> index and the propagation constant. The stable transmission region of gap vortex soliton can be obtained through power graphs. In the stable region, the gap vortex soliton can transmit stably without being disturbed. However, in the unstable region, the gap vortex soliton will gradually lose ring structure and evolves into a fundamental soliton with the transmission distance increasing. And the larger the <inline-formula><tex-math id="M4">\begin{document}$ {\mathrm{L}}\acute{{\mathrm{e}}}{\mathrm{v}}{\mathrm{y}} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20232005_M4.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20232005_M4.png"/></alternatives></inline-formula> index, the longer the stable transmission distance and the lower the power of the bandgap vortex soliton. When multiple vortex solitons transmit in the lattice, the interaction between them is influenced by the lattice position and phase. Two vortex solitons that are in phase and located at adjacent lattices, are superimposed with sidelobe energy, while two vortex solitonsthat are out of phase are cancelled with sidelobe energy. These vortex solitons will gradually lose ring structure and evolve into dipole modes in the transmission process. And they are periodic rotation under the azimuth angle modulating. When two vortex solitons located at non-adjacent lattice, vortex solitons can maintain a ring-shaped structure due to the small influence of sidelobes. When three gap vortex solitons are located at non-adjacent lattices, the solitons can also maintain their ring-like structures. However, when there are more than three gap vortex solitons, the intensity distribution of vortex solitons are uneven due to the sidelobe energy superimposed. These vortex solitons will form dipole modes and rotate under the azimuthal angle modulating in the transmission process. These results can offer theoretical guidance for transmitting and controlling the gap vortex solitons in the lattice.

All-optical simultaneous OR and NAND gates using photonic crystal ring resonator and Kerr effect

All-optical simultaneous OR and NAND gates design is proposed using a simple nonlinear photonic crystal ring resonator based on the chalcogenide glass (Ag20As32Se48) material which is one of the promising materials for all-optical devices. The structure consists of an octagonal ring resonator between two waveguides, which uses the switching threshold mechanism based on the Kerr effect to perform two simultaneous logic gates functions. The plane wave expansion (PWE) method is used to obtain the band diagram in the proposed structure, and the two-dimensional finite difference time domain (2D-FDTD) method is used in the simulation to evaluate the performance of the proposed design. The resonance wavelength is 1551.3 nm, with a high transmission and coupling efficiency of about 100%. The proposed optical OR and NAND logic gates have high contrast ratios of 21.15 dB and 28.19 dB, respectively, with a quality factor of 596.65. The operating power intensity of the proposed structure is 1 kW μm−2, and the threshold power intensity is obtained at 2.8 kW μm−2. The proposed gates provide transmitted power of not less than 0.5. The size of the structure is 20.5 μm × 18.17 μm. The proposed structure is compact, works on low operational power intensity, and has the ability for dense integration. The simplicity and small size of the structure make it easy to fabricate for future integration in all-optical circuits.

Design of a High Q-Factor Label-Free Optical Biosensor Based on a Photonic Crystal Coupled Cavity Waveguide.

In recent years, there has been a significant increase in research into silicon-based on-chip sensing. In this paper, a coupled cavity waveguide (CCW) based on a slab photonic crystal structure was designed for use as a label-free biosensor. The photonic crystal consisted of holes arranged in a triangular lattice. The incorporation of defects can be used to design sensor devices, which are highly sensitive to even slight alterations in the refractive index with a small quantity of analyte. The plane wave expansion method (PWE) was used to study the dispersion and profile of the CCW modes, and the finite difference time domain (FDTD) technique was used to study the transmission spectrum, quality factor, and sensitivity. We present an analysis of adiabatically coupling light into a coupled cavity waveguide. The results of the simulation indicated that a sensitivity of 203 nm/RIU and a quality factor of 13,360 could be achieved when the refractive indices were in the range of 1.33 to 1.55.

Ultra compact, high contrast ratio all optical NOR gate using two dimensional photonic crystals

An all-optical NOR gate is implemented using a two-dimensional photonic crystal waveguide. The paper provides a structure of an all-optical NOR gate constructed by introducing a line and point defect. The proposed NOR gate uses a hexagonal lattice structure with E shaped waveguide. The contrast ratio, response time and bit rate of three input NOR gate are 29.59 dB, 0.8 ps and 1.19 Tbps, respectively. The proposed gate is constructed with the operating wavelength of 1550 nm. The structure has been simulated and analyzed using Finite Difference Time Domain (FDTD) and Plane Wave Expansion (PWE) methods. It offers high contrast ratio, better response time with ultra-compact size. Hence it is suitable for high speed optical integrated circuits and switching devices.

Study on bandgap extension and defect states of hybrid primitive structure phononic crystals

Based on the two-dimensional square lattice phononic crystal of the steel-gas system, a hybrid primitive phononic crystal is constructed by replacing the single-cylinder primitive with a tangent double-cylinders primitive structure and inserting the tangent double-cylinders primitive in the center of the lattice. The plane wave expansion method is used to calculate the band structure according to the variation of the extremal positions of the band. It is found that compared with the simple single-cylinder primitive lattice structure, the new structure can significantly separate the degenerate states of the band, and obtain a huge bandgap with a width of 5 times the simple single-cylinder lattice at low filling ratios. Its mechanism can be revealed by the inconsistency of the sound pressure field in the boundary direction. Meanwhile, adding the double-cylinders primitive to the center of the simple single-cylinder lattice produces an effect similar to the interstitial impurity defect, which can excite high-quality localized states of defects in the bandgap. The filling ratio of the defect body mainly affects the position of the localized states and the locality of sound pressure, whereas the orientation of the defect body has almost no effect on the distribution of sound field and the position of defect states.

Analysis of Reduction Effectiveness and Design of Locally Resonant Metamaterial Barriers for Train Vibration

Train vibrations are the primary concern in environmental engineering and civil engineering. It is significantly imperative to find new methods for reducing and isolating vibrations. The locally resonant metamaterials (LRMs) propose a novel method and concept for reducing train vibration. However, the accurate and quick design structures of LRMs based on vibration characteristics are still an issue. Thus, this study presents a novel inverse design model of three-component locally resonant metamaterial barriers (LRMBs) for vibration reduction based on deep learning. The bandgap characteristics and vibration modes of the LRMB are investigated by using the improved plane wave expansion (IPWE) and finite element method (FEM). Besides, the gradient-combined LRMBs are proposed based on time–frequency features of measured vibration caused by trains and the novel inverse design model, and a two-dimensional finite element model coupling with infinite element boundaries is established to study the reduction efficiency of the gradient-combined LRMBs. And the performances of different LRMBs are fully analyzed in time and frequency domains. The results show that the novel inverse design model can be successfully used to design the LRMB based on vibration features. Moreover, the gradient-combined LRMBs have better isolation performance.

Theoretical investigation and optimization of rotation sensing in the new photonic crystal gyroscope based on the Sagnac effect using nonlinear photonic resonators

In this research, the angular rotation speed in a passive photonic gyroscope based on the combination of side nanoring resonators and compensating waveguides has been analyzed by creating nonlinear effects in the control factors of the rings using the Sagnac effect. This structure consists of a central waveguide, two identical square resonators, and an almost U-shaped waveguide. The U-shaped waveguide causes coupling between the two resonators in a counterclockwise (CCW) mode. In this structure, a phase shift has been created in the output from the interference of two clockwise (CW) and CCW waves inside the resonators, and according to this phase shift and the central wavelength, the angular rotation speed has been estimated. In the proposed design of the gyroscope, by managing the nonlinear effects in the radius and refractive index (RI) of the coupling and inner rods, we have been able to control the changes in power, phase, and wavelength of the output from the device. With the increase in the intensity of power, the output power has an increasing slope at first, and at the point of creating a nonlinear effect in the sensor, the output power slope decreases. Also, this nonlinear effect directly affects the output phase of the structure. The maximum angular rotation speed in this gyroscope was [Formula: see text]/s. By changing the RI of the inner rods from 3.2 to 3.7, the maximum output-to-input power ratio changes from 0.38 W/[Formula: see text]m2 to 0.75 W/[Formula: see text]m2. By changing the radius of the coupling rods from 93 nm to 97 nm, the maximum power ratio decreases from 0.78 W/[Formula: see text]m2 to 0.55 W/[Formula: see text]m2. The field distribution profile and photonic bandgap in this gyroscope have been analyzed using the finite-difference time-domain (FDTD) and plane-wave expansion (PWE) methods, respectively. Also, the gyroscope has a footprint of 163.5 [Formula: see text]m2.

Graphene-based phononic crystal lenses: Machine learning-assisted analysis and design

The advance of artificial intelligence and graphene-based composites brings new vitality into the conventional design of acoustic lenses which suffers from high computation cost and difficulties in achieving precise desired refractive indices. This paper presents an efficient and accurate design methodology for graphene-based gradient-index phononic crystal (GGPC) lenses by combing theoretical formulations and machine learning methods. The GGPC lenses consist of two-dimensional phononic crystals possessing square unit cells with graphene-based composite inclusions. The plane wave expansion method is exploited to obtain the dispersion relations of elastic waves in the structures and then establish the data sets of the effective refractive indices in structures with different volume fractions of graphene fillers in composite materials and filling fractions of inclusions. Based on the database established by the theoretical formulation, genetic programming, a superior machine learning algorithm, is introduced to generate explicit mathematical expressions to predict the effective refractive indices under different structural information. The design of GGPC lenses is conducted with the assistance of the machine learning prediction model, and it will be illustrated by several typical design examples. The proposed design method offers high efficiency, accuracy as well as the ability to achieve inverse design of GGPC lenses, thus significantly facilitating the development of novel phononic crystal lenses and acoustic energy focusing.