Generalized Eigenfunction Method Research Articles

Extraordinary optical characteristics in a one-dimensional parity-time symmetric structure of quasiperiodic two-material Octonacci optical waveguide network

We explore Exceptional Points (EPs) within a one-dimensional quasiperiodic Octonacci optical waveguide network. This network consists of segments, each composed of two-material subsegments featuring gain and loss materials, creating a parity-time-symmetric structure. We investigate the trend of EPs across generations 3 to 7 of the Octonacci sequence using the generalized eigenfunction method and transfer matrix method. Our findings reveal that as generations progress, band gaps become more pronounced, and the number of EPs significantly increases within these gaps. In particular, the third generation only supports unidirectional transparency. Bidirectional transparency appears in the fourth generation and becomes more pronounced in later generations. Coherent perfect absorption-lasing points are first seen in the seventh generation. These findings have implications for the design of all-optical devices and wideband and/or narrowband optical filters.

Extraordinary optical characteristics of one-dimensional double anti-PT-symmetric ring optical waveguide networks

In this paper, a unique one-dimensional double anti-PT-symmetric ring optical waveguide network (1D DAPTSROWN) is designed whose refractive index satisfies n(x)=−n∗(−x). The extraordinary optical characteristics of the network are studied by means of the generalized Floquet–Bloch theorem, three-material network equation and generalized eigenfunction method. Due to the physical difference between the anti-PT-symmetric and PT-symmetric optical waveguide networks, the previous method of the PT-symmetric optical waveguide network is not applicable to the anti-PT-symmetric optical waveguide network designed in this paper. Thus, based on the previous method and the network equation, we propose a connection function to identify the extremum mode points in the anti-PT-symmetric optical waveguide network. At the extremum mode point, the optical waveguide network can produce extraordinary ultra-strong transmission, ultra-strong reflection, and ultra-strong photonic localizations. The corresponding calculated values in this paper arrives at 1028, which is much larger than 1012 (the best result reported previously). This shows that the anti-PT-symmetric network may have potential in applications in designing optical filters, optical amplifiers and so on.

Characteristics and mechanism of all-optical switching based on a one-dimensional two-connected periodic triangle optical waveguide network.

In this study, an all-optical switch is designed using a one-dimensional two-segment-connected periodic triangular optical waveguide network, and its switching characteristics and mechanism are investigated. The performance of the switch is numerically calculated by using the network equation and the generalized eigenfunction method and we find it relatively excellent. Its switching efficiency ratio reached 3.7202×1016, which is 5 orders of magnitude larger than the best reported result. The switching threshold control energy is approximately 1.8×10-20J, which is 1 order of magnitude larger than the best reported result. The switch size is approximately 0.0672µm2 and the integration degree is up to 14per/µm2, and it can be used for micrometer chip integration. The switching time is close to 209 fs, which is the same order of magnitude as the previously reported results. In addition, the all-optical switching designed in this study not only exhibits excellent switching performance and a novel working mechanism, but also provides a new technology for the design of pump-free all-optical switching devices.

Characteristics and mechanism of all-optical switching based on one-dimensional periodic two-segment-connected tetrahedral optical waveguide network

In this work, we design all-optical switching based on a one-dimensional (1D) periodic two-segment-connected tetrahedron optical waveguide network. The network equation and generalized eigenfunction method are used for studying the switching characteristics and mechanism. We found that the all-optical switch designed using a tetrahedral optical waveguide network constructed by polystyrene had an ultrahigh switching efficiency of about 7 × 1017, an ultrafast switching time of about 34 fs, an ultralow threshold control energy of about 11 zJ, and the switching volume was about 0.05μm3. The tetrahedral primitive cells contain a high density of equilateral triangle primitives that are capable of generating a strong anti-resonance state, which leads to a deep and wide photonic band gap. Moreover, the integer ratio of the waveguide length of the tetrahedral optical waveguide network is broken, which generates an extremely narrow gap in the passband, and strong photonic localization is generated near the nonlinear material. This research presents a new method for the design of all-optical switching with higher performance indicators, provides the possibility for further practical use of all-optical switching, and also deepens people’s understanding of optical waveguide networks.

Extraordinary propagation characteristics of electromagnetic waves in one-dimensional anti-PT-symmetric ring optical waveguide network**Project supported by the National Natural Science Foundation of China (Grant Nos. 11674107, 61475049, 11775083, 61875057, 61774062, and 61771205), the Natural Science Foundation of Guangdong Province, China (Grant No. 2015A030313374), and the Special Funds for the Cultivation of Guangdong College Students’ Scientifific and Techonlogical Innovation, China (Grant No.

In this paper, we design a one-dimensional anti-PT-symmetric ring optical waveguide network (1D APTSPROWN). Using the three-material network equation and the generalized Floquet–Bloch theorem, we investigate its photonic mode distribution, and observe weak extremum spontaneous anti-PT-symmetric breaking points (WBPs) and strong extremum spontaneous anti-PT-symmetric breaking points (SBPs). Then the transmission spectrum is obtained by using the three-material network equation and the generalized eigenfunction method. The 1D APTSPROWN is found to generate ultra-strong transmission near SBPs and ultra-weak transmission near WBPs and SBPs, with the maximal and minimal transmissions being 4.08× 1012 and 7.08× 10−52, respectively. The maximal transmission has the same order of magnitude as the best-reported result. It is not only because the distribution of photonic modes generated by the 1D APTSROWN results in the coupling resonance and anti-resonance, but also because the 1D APTSROWN composed of materials whose real parts of refractive indices are positive and negative has two kinds of phase effects, which results in the resonance and anti-resonance effects in the same kind of photonic modes. This demonstrates that the anti-PT-symmetric and PT-symmetric optical waveguide networks are quite different, which leads to a more in-depth understanding of anti-PT-symmetric and PT-symmetric structures. This work has the potential for paving a new approach to designing single photon emitters, optical amplifiers, and high-efficiency optical energy saver devices.

The singularity expansion method and near-trapping of linear water waves

AbstractThe problem of near-trapping of linear water waves in the time domain for rigid bodies or variations in bathymetry is considered. The singularity expansion method (SEM) is used to give an approximation of the solution as a projection onto a basis of modes. This requires a modification of the method so that the modes, which grow towards infinity, can be correctly normalized. A time-dependent solution, which allows for possible trapped modes, is introduced through the generalized eigenfunction method. The expression for the trapped mode and the expression for the near-trapped mode given by the SEM are shown to be closely connected. A numerical method that allows the SEM to be implemented is also presented. This method combines the boundary element method with an eigenfunction expansion, which allows the solution to be extended analytically to complex frequencies. The technique is illustrated by numerical simulations for geometries that support near-trapping.

The time-dependent motion of a floating elastic or rigid body in two dimensions

The linear time-dependent motion of a floating elastic or rigid body, subject to some initial displacement, which subsequently evolves freely is considered. The solution is derived by a Fourier transform and by the generalized eigenfunction method. Compared to other solutions, such as the Cummins method, the present solution requires neither time-stepping nor high-frequency calculations. A series of new identities for the frequency-domain problem are also presented. The Fourier transform solution allows an approximate solution to be calculated by an expansion over the complex resonances known as the singularity expansion method. Simple expressions for the singularity expansion method approximation are given. The method is illustrated with a series of numerical calculations.

Large photonic band gap and strong attenuation of multiconnected Peano network

In this paper, by means of the network equation and generalized eigenfunction method we investigate the optical transmission spectra and attenuation behavior of electromagnetic (EM) waves in multiconnected Peano networks composed of one-dimensional (1D) waveguides. It is found that for some two-segment-connected networks a very large photonic band gap (PBG) can be created in the middle of a frequency period and the width of the large PBG can be controlled by adjusting the matching ratio of waveguide length, d 2 : d 1. When d 2 : d 1 = 2 : 1, the width of the large PBG is bigger than half of frequency period. The influence of generation on the width and attenuation of the large PBG are also studied and the numerical results demonstrate that the first-generation Peano network with d 2 : d 1 = 2 : 1 is a good selectable structure for the designing of optical devices with large PBG and strong attenuation.

Generalized eigenfunction method for floating bodies

We consider the time domain problem of a floating body in two dimensions, constrained to move in heave and pitch only, subject to the linear equations of water waves. We show that using the acceleration potential, we can write the equations of motion as an abstract wave equation. From this we derive a generalized eigenfunction solution in which the time domain problem is solved using the frequency-domain solutions. We present numerical results for two simple cases and compare our results with an alternative time domain method.

Quantum Waveguide Properties of Bethe Lattices with a Ring

Based on waveguide theory we investigate electronic transport properties of Bethe lattices with a mesoscopic ring threaded by a magnetic flux. The generalized eigen-function method (GEM) is used to calculate the transmission and reflection coefficients up to the fifth generation of Bethe lattices. The relationships among the transmission coefficient T, magnetic flux ϕ and wave vector kl are investigated in detail. The numerical results are shown by the three-dimensional plots and contour maps. Some resonant-transmission features and the symmetry of the transmission coefficient T to flux ϕ are observed and discussed.

Quantum waveguide theory of a fractal structure

Abstract The electronic transport properties of fractal quantum waveguide networks in the presence of a magnetic field are studied. A Generalized Eigen-function Method (GEM) is used to calculate the transmission and reflection coefficients of the studied systems unto the fourth generation Sierpinski fractal network with node number N = 123 . The relationship among the transmission coefficient T , magnetic flux Φ and wave vector k is investigated in detail. The numerical results are shown by the three-dimensional plots and contour maps. Some resonant-transmission features and the symmetry of the transmission coefficient T to flux Φ are observed and discussed, and compared with the results of the tight-binding model.

Electronic transport properties of the Bethe lattices

In the framework of single-electron tight-binding model, the electronic transport properties of the Bethe lattices are studied by the use of Generalized Eigenfunction Method. The influence of the impurity on electronic transmission coefficients of the Bethe lattices with different size are examined. Because of its special topological structure and quantum coherent effect, the transmission coefficient shows an energy dependence and an interesting peak–valley structure. Its values in different exit channels are calculated and compared.

Quantum waveguide properties of a comb structure with a ring

Based on waveguide theory we study the electronic transport properties of the comb structure with a mesoscopic ring threaded by a magnetic flux. In the framework of the quantum waveguide model the network equation is deduced and the generalized eigenfunction method is used to calculate the transmission and reflection coefficients of the studied systems. Some resonant-transmission features and the symmetry of the transmission coefficient T to magnetic flux $\ensuremath{\Phi}$ and wave vector k are observed and discussed.

Electronic transport properties of Sierpinski lattices in a magnetic field

We have studied the electronic transport properties of open Sierpinski gasket systems in the presence of a magnetic field. The real-space renormalization-group scheme combined with the generalized eigenfunction method is used to calculate the transmission and reflection coefficients. Three kinds of electronic exits are investigated upto the thirtieth generation Sierpinski lattice. Some resonant-transmission features and the symmetry of the transmission coefficient T to the magnetic flux Φ are observed and discussed.

Photonic band structure of Sierpinski waveguide networks

The photonic band structure and transmission properties of Sierpinski fractal networks (SNs) made of one-dimensional waveguides are studied with the generalized eigenfunction method. In the absence or presence of dissipation and in different exit situations, we have numerically calculated the transmission coefficient as a function of frequency in the range 0\char21{}500 MHz for the first four generations of SNs. As the number of generations is increased, the structures in the transmission spectra show explicitly the evolution of discrete eigenmodes and the corresponding photonic band gap structures in such fractal networks. The gap structures are not altered by the presence of dissipation and are independent of the exit situation. An interesting anomaly due to dissipation is found at certain frequencies where a resonant peak is split into two peaks with zero transmission in the valley.

Electronic transport properties of Sierpinski lattices

We have studied the electronic transport properties of open Sierpinski gasket systems connected to two electron reservoirs in the presence of a magnetic field. In the framework of a tight-binding model, the systems are composed of one-dimensional ordered chains. A generalized eigenfunction method, which allows one to deal with finite systems consisting of a large number of lattice sites (nodes), is used to calculate the transmission and reflection coefficients of the studied systems. The numerical results show that there are two kinds of symmetries of the transmission coefficient T to magnetic flux Phi, and there are antiresonant state regions (T = 0) and resonant states (T = 1). It is different from the open ring systems now the electronic energies of resonant states do not coincide with the eigenenergies of the isolated Sierpinski gasket systems. It is also found that the transmission behavior of the single exit systems is much more complicated than that of two exit systems. [S0163-1829(99)03640-1].