Anomalous Dispersion Regime Research Articles

Wave patterns of the coupled nonlinear Schrödinger equations in photonic crystal fibers with four-wave mixing

Abstract In this paper, we examine the behavior of modulation instability within photonic crystals. The model employed is the coherent coupled nonlinear Schrödinger equation, incorporating weak birefringence and four-wave mixing, which arises at the edge of the optical mode. The linear analysis is used to derive the modulation instability spectrum. Throughout the modulation instability spectrum, we identify both stable and unstable modes, thereby confirming the breakdown of the plane wave. For certain four-wave mixing parameters, the amplitude of the modulation instability spectrum and its bandwidths expand, creating an opening for localized structures to emerge. Another aspect of this study has been demonstrated in normal and anomalous dispersion regimes where an increasing initial amplitude of the plane wave is fulfilled. Specifically, numerical simulations highlight the occurrence of Benjamin-Feir instability, where wave patterns emerge under the influence of four-wave mixing. Additionally, solitonic waves are generated, demonstrating the presence of Akhmediev breathers and other modulated structures, confirming that photonic crystals with four-wave mixing are conducive to these formations. The findings from this study could inform future research in the development of nonlinear photonic waveguides.

Observation of widely separated soliton molecules in a mode-locked Yb-doped fiber laser operating with net anomalous dispersion

This paper investigates the generation and characterization of soliton molecules (SM) in a semiconductor saturable absorber mirror (SESAM) mode-locked Yb-doped fiber laser with a linear cavity. By adjusting the pump power, both single-pulsing (SP) and double-pulsing (DP) states can be achieved. The observed soliton molecules are highly stable, with minimal temporal jitter despite being separated by 1.56ns. While SM have been observed in many different cavities this is one of the few reports in a Fabry–Perot cavity rather than a ring cavity and again most previous observations of soliton molecules having been in the anomalous dispersion regime. This configuration thus provides detailed and valuable insights into the behaviour of SM, shedding light on their formation, stability, and dynamics. This study contributes to understanding long range soliton interactions in fiber lasers, with potential implications for applications such as multi-bit transmission, bit storage, and pump-probe experiments to study the dynamics of chemical reactions.

In-fiber waveguide-based mode-locker for generating diverse ultrafast pulses

The pursuit of an optimal mode-locker in nonlinear photonics is crucial for advancing high-performance laser technologies. Addressing the market demands and technical complexities in creating highly integrated and robust saturable absorbers, we present a femtosecond laser-inscribed in-fiber straight waveguide, seamlessly integrating a few-mode fiber segment into a single-mode fiber. This design reduces costs, simplifies fabrication, and enhances system integration and stability. As a saturable absorber, it enables nonlinear polarization rotation and multimode interference, which are essential for ultrafast pulse generation. It exhibits slight polarization-dependent loss and significant modulation depth, effectively inducing mode-locking. Based on the waveguide, 1.36 ps soliton pulses with a 2.9 nm bandwidth at a central wavelength of 1573 nm are initially achieved in an anomalous dispersion regime. Transitioning to the normal dispersion regime, at 190 mW pump power, the system produces Q-switched mode-locked pulses at 1574 nm. Increasing the pump power to 295 mW results in 712 fs noise-like pulses at 1572 nm, featuring an 8 nm bandwidth and 4.57 MHz repetition rate. This advancement in fiber lasers, facilitated by hybrid nonlinear effects in the waveguide, represents a significant milestone, promising for nonlinear optics and photonics due to its simplicity, cost-effectiveness, compactness, robustness, and high damage threshold.

Low-dispersive silicon nitride waveguide resonators by nanoimprint lithography

In this study, we demonstrate the fabrication of waveguide resonators using nanoimprint technology. Without relying on traditionally costly lithography methods, such as electron-beam lithography or stepper lithography, silicon nitride (Si3N4) resonators with high-quality factors up to the order of 105 can be realized at C-band by nanoimprint lithography. In addition, by properly designing the waveguide geometry, a low-dispersive waveguide can be achieved with waveguide dispersion at around −35 ps/nm/km in the normal dispersion regime, and the waveguide dispersion can be further tuned to be 29 ps/nm/km in the anomalous dispersion regime with the polymer cladding. The tunability of nanoimprinted devices is demonstrated by the aid of microheaters, realizing on-chip optical functionalities. This work offers the potential to fabricate low-dispersive waveguide resonators for integrated modulators and filters in a significantly cost-effective and process-friendly scheme.

Design and analysis of the C6H6-filled circular photonic crystal fiber with ultra-flat dispersion for broadband supercontinuum generation with peak power of 130 W and 250 W

This article introduces a new model of a circular silica-based photonic crystal fiber with a hollow core filled with C6H6. The difference in the air hole size and the distance between them in the first ring around the core has a profound effect on the dispersion, leading to ultra-flat dispersion with values as low as ±0.996 ps nm−1· km in wavelength range 0.74 µm. The high nonlinear coefficient of several 1000 W−1 · km−1 and the low confinement loss of a few tens of dB m−1 suggest proposing three fibers with dispersion and nonlinear properties suitable for broadband supercontinuum generation at low peak power. The influence of peak power on the broadening of the supercontinuum spectrum is also investigated. Fibers with a flat all-normal dispersion profile provide a smooth spectrum with bandwidths of 1.215 and 1.626 µm at 30 dB with a peak power of 250 W. A fiber with an anomalous dispersion regime generates a supercontinuous spectrum, broadening to 3.868 µm in the mid-infrared region (2.467 µm bandwidth at 30 dB) under laser pulse excitation with 130 W peak power. Our results provide further insights into the generation of broadband mid-infrared supercontinuum using liquid-core silica-fibers, which have great potential for applications in the fields of optical communications and optical sensing.

Cerium oxide/polydimethylsiloxane polymer deposited onto tapered fiber for 1.97 µm mode-locked thulium-doped fiber laser

This work presents the use of cerium oxide (CeO2) as saturable absorber (SA) in the mode-locked pulse fiber laser generation. The CeO2 materials were prepared through sonication which then incorporated into a polydimethylsiloxane (PDMS) polymer to be coated on a tapered fiber. The CeO2/PDMS-SA exhibited saturation intensity of 44 MW/cm2 and modulation depth of 3.4%, confirming its nonlinear intensity-dependent transmission characteristics. Integrated into a thulium-doped fiber laser cavity, the CeO2/PDMS-SA enabled mode-locking at a pump power of 420 mW, leading to a fundamental mode-locking operation up to 1484 mW within the net anomalous dispersion regime. The laser emitted pulses at wavelength of 1973 nm with signal-to-noise ratio of 54 dB, repetition rate of 9.04 MHz and pulse width of 1.49 ps. The CeO2/PDMS-SA mode-locked fiber laser demonstrated stability of an average 3 dB bandwidth of 2.89 ± 0.08 nm, average central wavelength of 1973.05 ± 0.14 nm and average pulse width of 1.5 ± 0.01 ps for the duration of 120 min. Notably, the CeO2/PDMS-SA demonstrated a high pulse energy of 10.62 nJ and average output power of 96 mW. This investigation suggests their viability for use in future studies. The results also demonstrated the potential of CeO2 for use in various photonics applications in the “eye-safe” region.

Energy Dissipation Engineering for Widely Tunable (1.2–2.1 µm) Optical Parametric Oscillation in Integrated Chalcogenide Microresonators

AbstractWidely separated optical parametric oscillation (OPO) in an integrated Kerr microresonator offers a compact, highly coherent, and low‐power laser source, accessing previously inaccessible frequencies. Yet, extending the sideband of OPO from telecom to beyond 2 µm is still challenged by competing nonlinear effects and reduced Q factors at these extended wavelengths. In this work, efficient signal‐idler separations reaching 99 THz (1250–2130 nm) are demonstrated with a modest 8.1 mW threshold power, leveraging the heightened nonlinearity of GeSbS chalcogenide glass in the integrated microresonators. By pioneering energy dissipation control via pulley couplers, competing nonlinear phenomena are effectively suppressed, enabling the first demonstration of a widely separated OPO (wOPO) in the anomalous dispersion regime. Moreover, the wOPO features a robust existence region across a variety of device geometries, resilient to microresonator width alterations of up to 300 nm. These findings underscore the potential of integrated microresonators to adeptly bridge the telecom band to the MIR spectrum, achieving this transition with minimal power and controlled nonlinearities, a promising avenue for contemporary integrated photonics platforms utilizing heterogeneously integrated pump lasers.

Modulation instability in nonidentical metamaterial waveguide arrays by graph Laplacian approach

We theoretically investigate modulation instability (MI) in a nonidentical waveguide array, which is made up of positive and negative index metamaterial waveguides. The unit cell of the optical waveguide array consists of three waveguides arranged in a triangular manner. Waveguides 1 and 3 are made up of positive index material (PIM) channels and waveguide 2 is by negative index material (NIM) channels, as a result, they show different light propagation characteristics. We model this array of waveguides using a generalized nonlinear Schro¨dinger equation, replacing the Laplacian operator with the graph Laplacian. Following linear stability analysis, we will discuss MI for different values of transverse wave number, as it determines the order of the Brillouin zone. We also discuss the effect of input power on periodic MI in normal and anomalous dispersion regimes. Thus we report a comprehensive study on the MI and hence the better ways to generate and manipulate the solitons or ultra-short pulses in NIM PIM waveguide arrays.

Optical Dispersion Data Analysis of Single‐Crystal CH3NH3PbBr3 for Optimized Perovskite Solar Cell Active Layer Absorptance

Proper derivation of CH3NH3PbX3 (MAPbX3; where X = Cl−, Br−, I−) optical constants is a critical step toward the development of high‐performance perovskite devices. To date, the optical dispersions at all wavelengths have been inconsistently characterized by under‐approximating or omitting anomalous spectral features. Herein, a rigorous optical dispersion data analysis of single‐crystal MAPbBr3 involving variable‐angle spectroscopic ellipsometry data appended with transmission intensity data is presented. This approach yields a more robust derivation of the refractive index and extinction coefficient for both anomalous (absorptance) and normal (no absorptance) optical dispersion regimes. Using the derived optical constants, illustrative modeled perovskite solar cell device designs are presented in relation to nonrealistic designs prepared using representative optical constants reported in the literature. In comparison, the derived optical constants enables the modeling of layer thicknesses to maximize absorption by the active layer (MAPbBr3) and minimize parasitic optical absorptance by the nonactive layers at broad angles of incidence (≈0°–70°). This robust derivation of MAPbBr3 optical constants is expected to impact the optical dispersion data analysis of all perovskite analogs and expedite targeted development of, for example, solar cell, light‐emitting diode, photo‐ and X‐ray/γ‐ray detector, and laser system technologies.

Influence of Raman gain on dynamics of spatiotemporal chaos in optical ring microresonators

We theoretically investigate on the evolution of spatiotemporal chaos under the influence of Raman gain, in an optical ring microresonator. The dynamics of the system is shown to be governed by the Lugiator–Lefever equation, with an additional real and imaginary Raman gain term. In the absence of pump field intensity and Raman gain, we observe an increase in the amplitude and frequency of the traveling optical field as the pump detuning frequency increases. However this field gradually vanishes as time increases, owing to the intra-cavity loss. By exploring an appropriate ansatz which incorporate a chirping parameter, the modified Lugiator–Lefever equation is transformed to a set of four coupled first-order nonlinear ordinary differential equations. Fixed point stability analysis strongly suggest that variation in Raman gain has the propensity of inducing the co-existence of bright and dark solitons in the anomalous dispersion regime. Results of numerical simulations underscore that the structural profiles of the optical field amplitude are greatly modified by the Raman effects. Variation in Raman gain equally leads to a bifurcation analysis of the system, in which the Lyapunov exponent diagrams reveal regions of period doubling and chaos. We finally observe that for a fixed value of the real part of Raman gain coefficient, an increase in the effects of soliton self-frequency shift generally drives the system to chaotic regime.

Impact of weak nonlocal nonlinearity on modulational instability and chirp-free soliton trains in parabolic law optical fiber

This study considers the generation and evolution of chirp-free soliton trains in a focusing cubic-quintic nonlinear optical fiber media (by utilizing experimental parameters), under the influence of weak nonlocal nonlinearity. Analysis of modulational instability (MI) reveals that the MI gain exist only in the anomalous dispersion regime, with the quintic nonlinear term increasing the maximum gain and sideband frequencies. The nonlocal nonlinearity parameter generally suppresses the impact of MI, because the maximum gain is greatly reduced with increase in nonlocality. By using the subsidiary ordinary differential equation method, stationary periodic soliton trains are obtained; with the intensity of the optical beam greatly reduced and frequency increased as nonlocality is gradually stepped up. However, an increase in nonlocality instead reduces the frequency of the traveling optical signals; derived via an efficient transformation method. Our results strongly suggest that nonlocal nonlinearity has potential applications in the control of laser beams; as nonlinear periodic optical signals can be easily transformed to very weak quasi-plane waves.

Optimization of dispersions in benzene-core square photonic crystal fibers for mid-infrared supercontinuum generation with very low peak power

An ultra-broadband, very low peak power supercontinuum source is proposed based on a novel benzene-filled structured photonic crystal fiber. The modification of the structural parameters has been highly effective in improving dispersion properties, an ultra-flat all-normal dispersion is achieved with fluctuations of ±1.174 ps/(nm.km) in the 0.458 µm wavelength region. Three fibers with suitable optical properties best meet the requirements of supercontinuum generations with low peak power based on numerical simulation results. With anomalous dispersion regime, the first fiber offers the supercontinuum spectra in the mid-infrared region spanning from 0.784 µm to 3.973 µm (2.539 µm bandwidth at 30 dB) using laser pulses excited at 1.55 µm has a peak power of 150 W. A pulse with a peak power of 280 W propagates in a 5 cm length of the second fiber results in a supercontinuum spectrum covering from 0.744 µm to 1.859 (bandwidth 1.063 µm at 30 dB) at a pump wavelength of 1.3 µm with all-normal dispersion regime. A smooth, highly coherent supercontinuum spectrum spanning from 0.843 µm to 2.517 µm (bandwidth at 30 dB of 1.608 µm) is achieved when the third fiber has all-normal ultra-flat dispersion profile pumped at 1.75 µm wavelength with laser pulse peak power of 350 W. With very low peak power, these proposed three-fiber based broadband supercontinuum sources could be a low-cost alternative to traditional glass-core fiber, which has great potential for mid-infrared spectroscopy.

Impact of the Higher-Order Reactive Nonlinearity on High-Amplitude Dissipative Solitons

In this work, the impact of the higher-order reactive nonlinearity on very high-amplitude solitons of the cubic–quintic complex Ginzburg–Landau equation is investigated. These high amplitude pulses were found in a previous work in the normal and anomalous dispersion regimes, starting from a singularity found by Akhmediev et al. We focus mainly in the normal dispersion regime, where the energy of such pulses is particularly high. In the presence of the higher-order reactive nonlinearity effect, pulse formation are observed for much higher absolute values of dispersion. Under such effect, the amplitude and the energy of the VHA pulses decrease, while their spectral range shrinks. Numerical computations are in good agreement with the predictions based on the method of moments, in the absence of the higher-order reactive nonlinearity effect. However, in the presence of this effect such agreement becomes mainly qualitative. A region of existence of the very high-amplitude pulses was found in the semi-plane defined by the normal dispersion and nonlinear gain.

Cnoidal waves and their soliton limits in single mode fiber lasers

Cnoidal waves are a type of nonlinear periodic wave solutions of the nonlinear dynamic equations. They are well known in fluid dynamics, but it is not the case in optics. In this paper we show both experimentally and numerically that cnoidal waves could be formed in a fiber laser either in the net normal or net anomalous cavity dispersion regime, especially because, as the pump power is increased, the formed cnoidal waves could eventually evolve into a train of bright (in the net anomalous cavity dispersion regime) or dark (in the net normal cavity dispersion regime) solitons. Numerical simulations of the laser operation based on the extended nonlinear Schrödinger equation (NLSE) have well reproduced the experimental observations. The result not only explains why solitons can still be formed in a fiber laser even without mode locking but also suggests a new effective way of automatic stable periodic pulse train generation in lasers with a nonlinear cavity.

Raman-assisted Kerr platicon and perfect soliton crystal formation in the cavity

In this paper, we investigate the soliton evolution dynamics influenced by the stimulated Raman scattering effect. The strong Raman gain ensures the breathing platicon and stable platicon formation at the cavity with normal quartic dispersion. The switching waves benefitted from the power transformation between pump mode and Raman gain spectrum could modulate the temporal waveform at the breathing platicon existence region. The oscillated characteristics of the temporal waveform mainly rely on the central frequency of the Raman gain spectrum. Meanwhile, at the anomalous quadratic dispersion regime, the Raman effects could capture the perfect soliton crystal formation which provides one novel method for the implementation of the harmonic mode-locking process. Our findings provide a viable way to demonstrate the cavity soliton dynamics and soliton-based applications.

28 THz soliton frequency comb in a continuous-wave pumped fiber Fabry–Pérot resonator

We report the generation of an optical frequency comb featuring a 28 THz bandwidth, sustained by a single 80 fs cavity soliton recirculating in a fiber Fabry–Pérot resonator. This large spectrum is comparable to frequency combs obtained with microresonators operating in the anomalous dispersion regime. Thanks to the compact design and the easy coupling of the resonator, cavity solitons can be generated in an all-fiber experimental setup with a continuous wave pumping scheme. We also observe the generation of a dispersive wave at higher frequencies, which is supported by higher-order dispersion. These observations align remarkably well with both numerical simulations and the established theory of cavity solitons.

Design of optical frequency comb based on dual frequency pumped normal dispersion silicon carbide microresonator

<sec>The scheme of generating optical frequency comb (OFC) mainly includes mode-locked laser, electro-optic modulation comb, and nonlinear Kerr micro-resonator comb. The OFC with frequency spacing on the order of 10–200 GHz can be employed in optical communication, microwave photonics, and other fields. Silicon carbide (SiC) has aroused the considerable research interest in integrated nonlinear photonics owing to its high second nonlinear coefficient and third order nonlinear coefficient, low optical loss, without multiphoton absorption loss owing to the wide bandgap. Single soliton microcomb in anomalous group velocity dispersion regime based on a 4H-SiC-on-insulator thin film has been demonstrated with the relative lower pump to comb efficiency, while the OFC in normal dispersion regime based on the SiC microresonator has not been reported. The pump conversion efficiency of OFC in the normal dispersion regime is high, and the pump frequency detuning range for the OFC generation is large, which is conducive to the OFC generation and long-term stable operation. Since there is no modulation instability effect in normal dispersion regime, the key to generating the OFC in normal dispersion regime is that the initial state needs the assistance of a multi-frequency laser (or four-wave mixing sideband). The phase-locked dual-frequency laser can be regarded as a pulse pump laser source with wide pulse duration, which can be realized by integrated distributed feedback laser.</sec><sec>In this paper, a scheme of generating OFC by pumping the normal dispersion SiC microresonator with phase locked dual-frequency laser is proposed. The flat normal dispersion in 1550 nm band is realized through dispersion engineering of the SiC microresonator. The effective mode field area of the TE<sub>0</sub> fundamental mode at 1550 nm in the optimized SiC ridge waveguide is about 0.94 μm<sup>2</sup>, and the nonlinear coefficient is about 3.69 <inline-formula><tex-math id="M2">\begin{document}$ {{\mathrm{W}}}^{-1}{\cdot} {{\mathrm{m}}}^{-1} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20231442_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20231442_M2.png"/></alternatives></inline-formula>. Meanwhile, dispersion parameters of the microresonator with 100 GHz FSR are also obtained. The OFC generation pumped by a phase-locked dual-frequency laser based on normal dispersion SiC microresonator is simulated through using the Lugiato-Lefever equation. The evolution process of the OFC in time and frequency domain related to the pump detuning is studied. The effects of several parameters such as the pump power, microresonator waveguide loss, microresonator dispersion, proportion of the dual-frequency laser, and the frequency interval of dual-frequency laser on the performance of the OFC are also investigated. The conclusions can be obtained through the OFC generation simulation as follows, 1) When the microresonator waveguide loss is larger, the pump detuning range for the OFC generation becomes smaller, and the pulse peak power under the same pulse intensity filling rate decreases. 2) When the input pump power is larger, the pump detuning range for the OFC generation becomes larger, the pulse peak power under the same pulse intensity filling rate increases, and the corresponding spectrum becomes wider. 3) With the increase of absolute dispersion value, the spectrum bandwidth of the generated OFC decreases obviously. 4) The power proportion of dual-frequency laser has little influence on the OFC generation. 5) The frequency spacing of the generated OFC can be tuned through changing the frequency spacing of the two phase-locked lasers with integral multiple of free spectral range.</sec><sec>The OFC with spectrum bandwidth of about 70 nm can be generated in a range of 1500—1600 nm through the simulation. The simulation results are beneficial to promoting the research and practical application of high repetition rate broadband optical frequency comb in a 1550 nm band based on the normal dispersion silicon carbide microresonator.</sec>

Effect of the Dispersion Orders on the Widths of the Coexistence Domain and Combs Spectra of Bright and Dark Solitons in Microresonators

Using the Lugiato–Lefever model, we base on the hysteresis approach to analyze the coexistence domain of bright and dark solitons in the zero, normal, and anomalous dispersion regimes. Our results also highlight that the fourth-order dispersion term affects the width of the frequency combs of both dark and bright solitons. It also allows the appearance of dispersive waves on the soliton spectra that disappear for high values of the fourth-order dispersion followed by the soliton destabilization into harmonic oscillations and oscillation packages.

Temporal evolution and spectral characteristics of dual field in a double-pumped microcavity

To generate dual combs for various precision measurements, the temporal evolution and spectral characteristics of dual fields in a double-pumped microcavity are investigated. Results show that by using dual orthogonally polarized pumps, the dual fields can be controlled by regulating the two detunings. The coexistence of solitons and Turing patterns, which is equivalent to dual-frequency combs with a large repetition frequency difference, can be excited. Two positive detuning parameters are similar, and two soliton pulses with identical free spectral ranges and different intensities are formed. Moreover, two weaker positive detunings are beneficial for the dual Turing patterns. Furthermore, breathers and multiple pulses can be excited under special conditions. The effects of dispersion and pump intensity on the dual fields are also studied. Dual combs exist in an anomalous dispersion regime and the strong negative dispersion only leads to DC fields. For the pump intensity, with the increase of pumping power, one field experiences breathers and variable multi-pulse in sequence while the other field maintains Turing patterns. The results of this study provide a new approach to excite dual-frequency combs by using a single microcavity.

Effects of coupling coefficient dispersion on the Fermi-Pasta-Ulam-Tsingou recurrence in two-core optical fibers

In a two-core optical fiber (TCF), the linear coupling coefficient between the two cores in general varies with the optical wavelength, a phenomenon known as coupling coefficient dispersion (CCD). Here we study the effects of CCD on the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) phenomena in a TCF. Modulation instability tends to be a precursor in the formation of breathers. The subsequent decay of breathers and re-emergence of modulation instability lead to FPUT type periodic patterns. For the symmetric or antisymmetric continuous-wave (CW) state of a TCF, CCD does not affect the modulation instability gain spectra. However, if asymmetric perturbations are applied to a symmetric CW state, CCD can drastically modify FPUT. In the anomalous dispersion regime, without CCD, only FPUT dynamics without energy transfer between the two cores occurs. With CCD effects, FPUT dynamics with strong energy transfer between the two cores can arise. As CCD increases, FPUT dynamics without energy transfer never vanishes, but the corresponding number of FPUT cycles can be drastically reduced to only one. In the normal dispersion regime, no FPUT dynamics with energy transfer between the two cores occurs. The major CCD effect is to significantly decrease the FPUT cycles, and only one FPUT cycle can be observed at relatively large magnitude of CCD values. As the magnitude of CCD increases further, more FPUT cycles arise again. Moreover, larger CCD will lead to higher (lower) propagation speed of the wave patterns in the anomalous (normal) dispersion regime respectively. To enhance theoretical insight, we perform a Floquet analysis for the linearized stability equations with periodic coefficients based on the FPUT dynamics in the normal dispersion regime. The monodromy matrix and eigenvalues can be computed explicitly. Larger eigenvalues correlate with stronger instability and more rapid disintegration of the periodic patterns, agreeing well with the trends of the numerical simulations.