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Although upstream manufacturers with small- and medium-sized are gradually willing to invest green efforts for stimulating market demand, they have been encountering the challenge of securing sufficient working capital to develop the green supply chain. Thus, this paper systematically incorporates two types of prepayment policies including risk-free (RF) and risk-taking (RT) into a retailer's dominated channel. Via deriving operational and financing equilibrium of the green supply chain, a series of interesting findings can be offered. Specifically, (1) this paper identifies a threshold value regarding the manufacturer's own capital, and proposes two scenarios for assisting the retailer to decide whether offers the manufacturer prepayment policy. (2) The effectiveness of RF for the capital-constrained manufacturer depends on how well its green effort can be implemented. That is, provided that the quality effect is large enough, the manufacturer can get more upfront capital from the retailer, which may entirely cover its total production and green effort costs. (3) Under RT, if the manufacturer's capital is relatively lower, RT enables the manufacturer to obtain sufficient capital and the retailer is willing to share partial of the manufacturer's default risk. Otherwise, RT may not be the best prepayment policy for the retailer. [ABSTRACT FROM AUTHOR]

Under increasingly stringent environmental regulations, inadequate green input levels from manufacturers may lead to substandard emissions and production shutdown, which further results in the disruption risk of the supply chain. This work investigates a green supply chain (GSC) consisting of one environmentally regulated manufacturer and one capital-constrained retailer who faces stochastic market demand. The manufacturer needs to make decisions on the green input level, which is related to the investment cost as well as supply disruption risk. The retailer has to determine product order quantities and financing decisions. We derive the operational equilibriums for the GSC system under three scenarios: no financing, trade credit financing (TCF), and bank credit financing (BCF), and recommend the optimal financial selection for the retailer via the comparison of three financial modes. The analytical and numerical results reveal that the manufacturer should improve the green input level within the financial capability to enhance the sustainable operation level of the supply chain. In addition, we find that the capital-constrained retailer will choose financing, since either BCF or TCF will result in a higher profit than no financing. Moreover, we obtain the threshold of green input level, with which we can decide whether to choose TCF or BCF under the given corresponding parameters. [ABSTRACT FROM AUTHOR]

Phase‐matching of high harmonics with abruptly auto‐focusing ring‐Pearcey–Gaussian beams is investigated. The slowly diffracted propagation after the auto‐focusing of the beams contributes to a slow decrease in both the intensity and the phase with similar trends along the propagation axis. For a specific beam intensity and a harmonic order, the contributions from the atomic and geometric phases can cancel each other within a sufficiently long propagation distance, resulting in the nearly perfect phase‐matching of the harmonic of which the order depends linearly with the beam intensity. The phase‐matching property indicates a dramatic phase‐matching control scenario: one harmonic can be specifically enhanced by adjusting the beam intensity. A numerical simulation is performed to show that the yield of the selected harmonic continuously increases at a distance up to the centimeter range. These findings are helpful for easy tuning and effective generation of the harmonics. [ABSTRACT FROM AUTHOR]

We propose and validate an approach to obtain an ultra-broadband terahertz absorber by taking the advantages of the absorption of the surface plasmon resonance in combination with the absorption of the intrinsic mode. The absorber is composed of a gold (Au) bottom layer and a patterned graphene embedded in TOPAS polymer, exhibiting ultra-broadband absorption by coupling the surface plasmon resonance and the intrinsic modes. The single-layer graphene is patterned to C4 symmetric regular polygons which are arranged in circle circumscribed polygons (CUPs) or circle inscribed polygons (CIPs) at the same periodicity for comparison. With the increase of the side number (N), the strongest and weakest coupling of the surface plasmon resonance and the intrinsic modes was achieved at N = 4 (square) and ∞ (circle) for CUP and CIP arrangements, respectively. The optimum absorption bandwidth of 2.54 THz with relative absorption bandwidth of 109.5% above 90% absorption efficiency has been achieved for the circles with a diameter of 12.4 μm arranged at a gap distance of 1.0 μm. This design shows broader relative absorption bandwidth than previously reported single-layer graphene THz absorbers. Therefore, the proposed method would be highly valuable to design and manufacture ultra-broadband THz absorbers. [ABSTRACT FROM AUTHOR]

We propose a photonic crystal fiber with a high refractive index ring, which supports up to 180 orbital angular momentum (OAM) modes without any high order radial modes from 1.5 um to 1.7 um. The finite element method is employed to numerically study the properties of the designed fiber. Further research shows that the refractive index differences of all the eigenmodes are above 2 × 10−3, which indicates that the corresponding eigenmodes can be well-separated and ensures the stable transmission of OAM modes. In addition, it is quite interesting that the high order radial modes are significantly suppressed via increasing the radius of the cladding in the design, while most of the OAM modes are without phase distortion, which is beneficial for the (de)multiplexing of the OAM modes. Particularly, the mode quality of all the eigenmodes is higher than 94.9% and the chromatic dispersion is flat with minimum value of the dispersion variation is 0.36 ps/(km·nm). Moreover, the designed fiber possesses various advantages such as low nonlinear coefficient (< 0.7 W/km) and confinement loss (on the order between 10−12 and 10−7 dB/m). The proposed fiber has shown great potential for high capacity OAM mode division multiplexing communications. [ABSTRACT FROM AUTHOR]

The dynamics of the inward-focusing ring Pearcey–Gaussian (PeG) beam are investigated in nonlocally defocusing nonlinear media. The propagation of inward-focusing ring PeG beams exhibits an anomalous abrupt focusing behavior, which is also controllable by carefully adjusting the degree of the nonlocality. Interestingly, the position of the focus spot changes exponentially with the amplitude of the beam. Such findings have benefits for optical tweezers in nonlinear regimes. [ABSTRACT FROM AUTHOR]

The dynamics of the inward-focusing ring Airy beam in nonlocally defocusing nonlinear media is investigated in detail, and a bottle beam shape can be observed by adjusting the amplitude of the inward-focusing ring Airy beam and the nonlocality of the media. The gradient force of the bottle beam is numerically studied. It is worth mentioning that the bottle beams are formed in the nonlinear regime, which indicates that much higher intensity is allowable for these beams than those in the linear regime. Our approach and results may pave the way to the optical tweezers in the nonlinear regime. [ABSTRACT FROM AUTHOR]

We proposed and numerically demonstrated an analog of electromagnetically induced transparency (EIT) in a Chinese character 王-shaped all-dielectric metasurface in the near-infrared region. Through asymmetric adjustment, the high transmittance EIT-like optical response of both single and double transparency windows can be realized. The EIT-like optical responses are clearly interpreted by the destructive interference between the bright dipolar moment and trapped magnetic dipolar mode through the asymmetric metasurface, and the dark dipolar moment causes the transparency window to split into two peaks. This all-dielectric structure shows a good application prospect in the field of refractive index sensors and slow-light transmission. [ABSTRACT FROM AUTHOR]

We theoretically investigate the UV-assisted high-order harmonic generation (HHG) process in a mid-infrared polarization gating (PG) field. The ionization and recombination steps of the HHG process can be controlled simultaneously by the UV and PG pulses. The influences of the delay between the UV and PG pulses on the phase-locking degree and the intensity of the harmonics are presented quantitatively. The results show that the harmonics are phase-locked as well as the intensity is significantly enhanced by 3 or 4 orders at a proper delay. Then a phase-locked ultra-broadband supercontinuum with a bandwidth of 240 eV is produced efficiently, supporting the generation of a Fourier-transform-limited pulse duration below one atomic unit of time. In addition, efficient isolated 110 as pulses with tunable central wavelengths can be obtained directly by selecting a different range of the supercontinuum. [ABSTRACT FROM AUTHOR]

In this paper, we numerically and theoretically study the tunable plasmonically induced transparency (PIT) effect based on the graphene metasurface structure consisting of a graphene cut wire (CW) resonator and double split-ring resonators (SRRs) in the middle infrared region (MIR). Both the theoretical calculations according to the coupled harmonic oscillator model and simulation results indicate that the realization of the PIT effect significantly depends on the coupling distance and the coupling strength between the CW resonator and SRRs. In addition, the geometrical parameters of the CW resonator and the number of the graphene layers can alter the optical response of the graphene structure. Particularly, compared with the metal-based metamaterial, the PIT effect realized in the proposed metasurface can be flexibly modulated without adding other actively controlled materials and reconstructing the structure by taking advantage of the tunable complex surface conductivity of the graphene. These results could find significant applications in ultrafast variable optical attenuators, sensors and slow light devices. [ABSTRACT FROM AUTHOR]

In this paper, we propose a metamaterial structure for realizing the electromagnetically induced transparency effect in the MIR region, which consists of a gold split-ring and a graphene split-ring. The simulated results indicate that a single tunable transparency window can be realized in the structure due to the hybridization between the two rings. The transparency window can be tuned individually by the coupling distance and/or the Fermi level of the graphene split-ring via electrostatic gating. These results could find significant applications in nanoscale light control and functional devices operating such as sensors and modulators. [ABSTRACT FROM AUTHOR]

It is shown that the unstable evolutions of the Hermite-Gauss-type stationary solutions for the nonlocal nonlinear Schrödinger equation with the exponential-decay response function can evolve into chaotic states. This new kind of entities are referred to as chaoticons because they exhibit not only chaotic properties (with positive Lyapunov exponents and spatial decoherence) but also soliton-like properties (with invariant statistic width and interaction of quasi-elastic collisions). [ABSTRACT FROM AUTHOR]

By extending the (1+1)-dimensional [(1+1)-D] perturbation-iteration method suggested by Hong et al. [J. Opt. Soc. Am. B 35 (2018) 317–322] to the (1+2)-D case, we develop an arithmetic to the numerical integration for solition solutions of the (1+2)-D nonlocal nonlinear Schrödinger equation, and obtain the Gaussian-like solitons in Cartesian coordinate system and the Laguerre-Gaussian-type-like solitons in cylindrical coordinates system. Moreover, some more accurate soliton solutions were obtained through optimizing the calculation method of the nonlinear refractive index. [ABSTRACT FROM AUTHOR]

• It is the first demonstration of triple high Q-factor and highly tunable quasi bound states in the continuum in all-dielectric metasurface. • The modulation depth, resonant frequency and linewidth of the sharp triple Fano resonances can be engineered by adjusting the asymmetry parameter. In nanophotonics, exciting multiple high quality factor (Q-factor) and high tunable Fano resonances is important for many applications. In this paper, an all-dielectric metasurface consists of Si layer and three Si rectangular bars with different widths and lengths is proposed. The triple high Q-factor and high tunable the quasi bound states in the continuum (quasi-BIC) modes with distinct Fano features can be achieved by breaking the C 2 rotational symmetry of the unit cell. With adjusting the asymmetry parameter, the modulation depth, resonant frequency and linewidth of the sharp triple Fano resonances can be engineered, and the Q-factor can reach 10 7 . In addition, the triple quasi-BIC are both mainly generated by the toroidal dipole oscillation through analyzing the result of electromagnetic distribution and the far-field scattering of Cartesian multipoles with asymmetric nanostructure. The proposed nanostructure has numerous potential applications on nonlinear optics, filters, and multi-channel biosensors. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of the inward-focusing ring Pearcey–Gaussian beam (IRPB) propagated in a type of defocusing nonlinear media with periodic-oscillatory nonlocality. An interesting phenomenon is found that the IRPB exhibits a multi-focusing property during propagation, which can be attributed to the periodic evolution of the defocusing nonlocality. Further, it is shown that the multi-focusing behavior can be controlled by adjusting the initial field parameters (the radius and amplitude of the beam). Such finding will provide a theoretical basis for the generation and regulation of multi-focusing beams in nonlinear regime, which may benefit for optical tweezers and particle manipulation. [ABSTRACT FROM AUTHOR]

We investigate the dynamics of non-uniform breathing solitons in multimode fibers modeled by the oscillating nonlinearities. By using the split-step Fourier-transform method to solve the nonlinear Schrödinger equation with the oscillating nonlinear coefficient, the evolution of non-uniform breathing solitons with irregular breathing periods in the multimode fiber is revealed under the action of second-order dispersion. Further, the number and depth of non-uniform breathers can be manipulated by adjusting the power of the pulse and the oscillation frequency of the nonlinear coefficient. Moreover, we also discuss the evolution of breathers under different dispersion conditions. Our study benefits for the better understanding of the complex dynamics of breathing solitons in multimode fibers. [ABSTRACT FROM AUTHOR]

The generation of tunable extreme-ultraviolet (EUV) vortex beams is highly sought after for optoelectronic applications in the EUV region. In this study, we investigate the generation of vortex high-order harmonics using a ring Pearcey-Gaussian vortex beam as the driving source. We analyze the beam's spatial structure through phase-matching conditions and simulate high-order harmonic generation by solving the Maxwell wave equations. The beam's self-focusing characteristics and low-diffraction properties after focusing significantly enhance harmonics near the 53rd order, indicating the generation of a narrowband EUV vortex. Our findings underscore the advantages of using a ring Pearcey-Gaussian vortex beam for narrowband EUV vortex generation, paving the way for creating tunable vortex high-order harmonics or attosecond pulses with innovative vortex beams.

A novel approach, to the best of our knowledge, for generating short microwave pulse trains based on a hybrid mode-locked optoelectronic oscillator (HML-OEO) is proposed and demonstrated. In the proposed scheme, a saturable absorber (SA) device is inserted into the active mode-locked OEO (AML-OEO) to compress the pulse width of the microwave pulse trains. Numerical simulations and experimental results show that the HML-OEO generates a short microwave pulse train with a repetition rate of 98.994 kHz through fundamental frequency mode locking, and its pulse width is compressed by about 50 % compared to the AML-OEO. Additionally, in the experiment, microwave pulse trains with different repetition rates are generated by second-, third-, fourth-, and fifth-order harmonic mode locking, respectively. Compared to the AML-OEO, the HML-OEO achieves pulse compression effects of 49.3 % , 49.8 % , 49.4 % , and 49.9 % , respectively. Notably, compared to the AML-OEO, the proposed scheme also exhibits outstanding performance in frequency stability.

Circular Airy pulsed beams are introduced to significantly optimize the acceleration of neutral Rydberg atoms. Compared with the conventional pulsed Gaussian beams used in the previous report, the circular Airy structure abruptly self-focuses and subsequently propagates with weak diffraction, resulting in a much higher accelerating efficiency for both radial and longitudinal velocities, as well as a longer accelerating range along the propagation axis. The parameter dependencies of the beams on the acceleration are also analyzed.

Soliton self-mode conversion is a versatile technique that allows for both wavelength changes and mode transformations. This process can be controlled by adjusting the input power, with higher power resulting in a stronger nonlinear effect that facilitates soliton self-mode conversion. Our research has demonstrated that soliton self-mode conversion is a viable method for achieving spatiotemporal coupling. This technique can be applied in optical fibers to link two pulses, resulting in distinct spatial distributions that can be controlled by adjusting the initial time intervals.

In this Letter, we introduce a new, to the best of our knowledge, class of accelerating surface plasmonic wave: the Olver plasmon. Our research reveals that such a surface wave propagates along self-bending trajectories at the silver-air interface with various orders, among which Airy plasmon is regarded as the zeroth-order one. We demonstrate a plasmonic autofocusing hot-spot by the interference of Olver plasmons and the focusing properties can be controlled. Also, a scheme for the generation of this new surface plasmon is proposed with the verification of finite difference time-domain numerical simulations.

We investigate the modal energy flow of the femtosecond-pulsed beam self-cleaning on LP 11 mode with the influence of different factors such as the initial fraction of LP 11 mode, initial peak power, distribution of high-order modes and the numerical aperture of the fiber. It is interesting that there is a critical value of the initial peak power, Pcr, which is the watershed, not only in the quantitatively dominant transverse mode converting from LP 11 mode to LP 01 mode, but also in the behavior of HOMs of the transition from Attractor to chaos. Our simulation results may provide a novel perspective to understanding the beam self-cleaning on LP 11 mode.

A rational number harmonic mode-locked dual-loop optoelectronic oscillator (RHML-DL-OEO) is proposed and experimentally demonstrated. In the proposed system, an external radio frequency (RF) signal and a feedback oscillating microwave signal drive two arms of a dual-drive Mach-Zehnder modulator (DMZM). Mode locking is realized by frequency detuning. The larger effective free spectrum range (FSR) and higher side-mode suppression result from the Vernier effect effectively suppress supermode noise and intermodulation distortions (IMDs). Experimental results demonstrate that the microwave frequency comb (MFC) signals with repetition frequencies of 901.8 kHz, 2.3046 MHz and 5.3106 MHz are generated by 9th-, 23rd- and 53rd-order rational number harmonic mode-locking, respectively. Compared with the rational number harmonic mode-locked optoelectronic oscillator based on single-loop structure, the supermode noise suppression ratios of the scheme we propose are improved by 30.5 dB, 27.6 dB and 20.3 dB, respectively. Furthermore, the performance of single sideband (SSB) phase noise is also investigated.

The electromagnetically induced transparency (EIT) effect realized in a metasurface is potential for slow light applications for its extreme dispersion variation in the transparency window. Herein, we propose an all-dielectric metasurface to generate a double resonance-trapped quasi bound states in the continuum (BICs) in the form of EIT or Fano resonance through selectively exciting the guiding modes with the grating. The group delay of the EIT is effectively improved up to 2113 ps attributing to the ultrahigh Q-factor resonance carried by the resonance-trapped quasi-BIC. The coupled harmonic oscillator model and a full multipole decomposition are utilized to analyze the physical mechanism of EIT-based quasi-BIC. In addition, the BIC based on Fano and EIT resonance can simultaneously exist at different wavelengths. These findings provide a new feasible platform for slow light devices in the near-infrared region.

This erratum corrects the range of the degree of nonlocality σ, in which the out-of-phase bound-state solitons can stably exist in our paper [Opt. Express24, 28784 (2016)10.1364/OE.24.028784].

In this work, we demonstrate an analogue of electromagnetically induced reflection (EIR) effect with hybrid structure consisting of a silica (SiO 2 ) square array layer embedded in graphene-dielectric-Au film constructed F-P cavity. It is shown that the SiO 2 square array and F-P cavity create transverse waveguide with high quality factor (Q-factor) and longitudinal F-P modes, and their destructive interference effectively forms the EIR-like effect, which benefits for obtaining high group delay. In addition, the C4 symmetric structure ensures the polarization-independent for this EIR-like effect. With high Q-factor at the reflection window, the ultra-high group delay as high as 245 ps can be obtained. This structure will be useful to develop the EIT-like devices with excellent performance such as high group delay, polarization and incident insensitivity, and environmental stability.

Multimode fibers (MMFs) support abundant spatial modes and involve rich spatiotemporal dynamics, yielding many promising applications. Here, we investigate the influences of the number and initial energy of high-order modes (HOMs) on the energy flow from the intermediate modes (IMs) to the fundamental mode (FM) and HOMs. It is quite surprising that random distribution of high-order modes evolves to a stationary one, indicating the asymptotic behavior of orbits in the same attraction domain. By employing the Lyapunov exponent, we prove that the threshold of the HOMs-attractor is consistent with the transition point of the energy flow which indicates the HOMs-attracotr acts as a "valve" in the modal energy flow. Our results provide a new perspective to explore the nonlinear phenomena in MMFs, such as Kerr self-cleaning, and may pave the way to some potential applications, such as secure communications in MMFs.

We investigate the phase-matching of the high harmonics (HHG) driven by the circular Airy-Gaussian beams (CAiGB), which abruptly auto-focus and subsequently propagate without diffraction. The results show that the harmonics corresponding to both short and long quantum paths can be well phase-matched after the focusing point of the CAiGB. Therefore, the effective interaction length of HHG for CAiGB is much longer than that for the conventional Gaussian beams with the same size of the waist. Our numerical simulations reveal that the harmonics continuously gain up to 1 cm of the propagation distance. This work provides a route to enhance the conversion efficiency of HHG by the coherent control of abrupt auto-focusing beams.

In this paper, we investigated the geometric parametric instability (GPI) in graded-index multimode fibers through the multimode generalized nonlinear Schrödinger equation. Our results clearly and intuitively indicate that the generations of GPI sidebands are nearly synchronous in the spectrums of all modes, and the shapes of these spectrums are nearly the same. The numerical results show that the energies of the GPI sidebands come from the pump sideband, and these sidebands are carried by similar spatial beam profiles due to the similar modal components. We also found that the large modal dispersion has an influence for the symmetry of these GPI sidebands.

Based on the extended Huygens-Fresnel integral, we have analytically and numerically investigated the propagation properties of a radial phased-locked rotating elliptical Gaussian (RPLREG) beam array in turbulent atmosphere. The average intensity and effective beam sizes in x and y directions of a RPLREG beam array are derived, and their evolution behaviors are analyzed in this paper. Our numerical results indicate that the propagation of a RPLREG beam array in turbulent atmosphere depends on the beam parameters including w , r , N and the structure constant C n 2 of atmospheric turbulence. The results show that the beam arrays of intensity distribution with smaller radius r or larger initial beam size w are very alike, and stronger atmospheric turbulence makes the RPLREG beam evolve into a Gaussian-like beam more rapidly as the propagation distance increases. This research may be useful for optical communications and remote sensing in turbulent atmosphere.

A new class of partially coherent radially and azimuthally polarized rotating elliptical Gaussian (PCRPREG and PCAPREG) beams is introduced. The analytical expressions of the PCRPREG and PCAPREG beams propagating through anisotropy oceanic turbulence are derived based on the extended Huygens-Fresnel principle and the spatial power spectrum of oceanic turbulence. The effects of beam waist size w 0 , coherence width σ 0 , propagation distance z and oceanic turbulence parameters on the evolution statistics properties of PCRPREG and PCAPREG beams are studied in detail by numerical simulation. Our results indicate that with the increase of the propagation distance in the far field region, the normalized initial profile with a doughnut-like distribution of PCRPREG and PCAPREG beams gradually converts into a flat-topped one, and finally evolves into a Gaussian-like beam profile. We also find that the salinity-induced turbulence fluctuation makes a greater contribution to the decrease of beam quality compared with the temperature-induced turbulence fluctuation. Furthermore, the full width at half maximum becomes wider for the larger propagation distance z and wavelength λ or the smaller dissipation rate ε. Our work will pave the way for the development of underwater optical communication and underwater laser radar in oceanic environment.

In this paper, we propose a metamaterial structure for realizing the electromagnetically induced transparency effect in the MIR region, which consists of a gold split-ring and a graphene split-ring. The simulated results indicate that a single tunable transparency window can be realized in the structure due to the hybridization between the two rings. The transparency window can be tuned individually by the coupling distance and/or the Fermi level of the graphene split-ring via electrostatic gating. These results could find significant applications in nanoscale light control and functional devices operating such as sensors and modulators.

In nonlocal nonlinear media with a sine-oscillation response function, two kinds of spatial solitons with complicated structure, in-phase and out-of-phase bound-state solitons, are obtained numerically in the case of the weak nonlocality. The in-phase bound-state soliton exhibits the symmetrical profile and the nonzero central value, and the out-of-phase bound-state soliton has the antisymmetrical profile and the zero central value. The two kinds of bound-state solitons form the degenerate modes subject to the same dependance of soliton power on the propagation constant. For those solitons there exist two abnormal properties: both the soliton propagation constants and the slope of the power versus propagation constants are negative. Both the in-phase and the out-of-phase bound-state solitons are stable by the linear stability analysis, and the stability of the two kinds of solitons obey an inverted Vakhitov-Kolokolov stability criterion. We also discuss the way how to obtain a set of the soliton solutions from one numerical soliton solution via the transform invariance of the nonlocal nonlinear Schrödinger equation and the nonlinear Schrödinger equation.

We investigate the dynamics of a decelerating Airy pulse in the highly noninstantaneous Kerr media. It is found that the deceleration of the Airy pulse can be counteracted by the highly noninstantaneous nonlinearity. When the power of the pulse is specifically chosen, the deceleration of the Airy pulse can be totally restrained, and an Airy-type solitary wave is observed within several dispersion lengths.