Group-velocity Dispersion Regime Research Articles

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>

Experimental observation of stimulated Raman scattering enabled localized structure in a normal dispersion FP resonator

Dissipative Kerr nonlinear cavities attract intense interest due to their rich dynamics and capability to generate broadband, low-noise, mode-locked optical frequency combs for applications in optical communications, dual-comb spectroscopy, photonic lidar, etc. Different from the Kerr effect, which is an electronic response, stimulated Raman scattering (SRS) is a delayed response to molecular vibrations in materials. In microcavities, when driven in an anomalous group velocity dispersion (GVD) regime, SRS typically leads to a self-frequency shift of solitons and generation of breather solitons, which have been verified both theoretically and experimentally. However, when driven in a normal GVD regime, recent theoretical work predicts that SRS can cause the locking of switching waves (SWs) and thus support bright moving localized structure (LS), which we term as SRS enabled localized structure (SRS-LS). Limited by the design of suitable experimental parameters, experimental observation of this SRS-LS is not achieved yet. Here, we provide numerical investigation and, to our knowledge, the first experimental observation of the SRS-LS in a fiber Fabry–Perot (FP) resonator with ultra-low normal GVD. Such SRS-LS exhibits localized temporal features with strong oscillations at ∼13THz local frequency on the top of a flat-top pulse. The corresponding spectrum is a low-noise and broadband Kerr comb with a typical platicon-like spectrum in the center and two Raman Stokes and anti-Stokes peaks located near ±13THz away from the center. With such an SRS enabled broadband Kerr comb, we have achieved an SRS-LS spectrum with a repetition rate of ∼3.68GHz and a −30dB spectral width of 255 nm. The corresponding comb tooth count is >8500, covering the S + C + L telecommunication bands. Moreover, the formation process of such SRS-LS is also revealed, and it is found that the GVD plays a key role in its generation. Our work will help to advance the study of the dynamics of optical frequency combs under the influence of SRS, as well as provide a broadband coherent mode-locked optical source for wide applications.

Coherent Combining for High‐Power Kerr Combs

AbstractKerr frequency combs enable the miniaturization of comb sources for applications such as data communications and spectroscopy. However, due to the high field confinement and effective nonlinearity, Kerr combs typically operate with low output comb powers. While nonsolitonic Kerr combs operating in the normal group velocity dispersion (GVD) regime can access high pump‐to‐comb conversion efficiencies and relatively flat spectral profiles, many frequency comb applications require even higher comb‐line powers that are otherwise fundamentally unattainable without the use of optical amplifiers. A high efficient coherent beam combining of Kerr combs is demonstrated via on‐chip synchronization to greatly enhanced comb‐line powers. Two normal GVD Kerr combs are coherently combined for a 2.9 dB increase in comb power compared to a single comb while maintaining a high total pump‐to‐comb conversion efficiency of 26%. Lastly, it is shown that via spectral engineering of two normal GVD combs, it should be possible to significantly increase the spectral flatness of a combined comb. This platform enables fully integrated systems with powers that can readily exceed that of an individual comb system by more than an order of magnitude.

Modified nonlinear saturable effects on instability characteristics in three-core oppositely directed coupler with Kerr nonlinearity

This work shows that it is possible to observe the modulational instability (MI) zones caused by the modified form of nonlinear saturation effect in triangular design of a three-core oppositely directed coupler. In this design, the first and third channels are made of a material with a low (negative) refractive index, while the second one is made of a high (positive) index. The governing equations are modified coupled nonlinear Schrödinger equations with saturable nonlinearity terms and coupling terms. In the presence of modified saturable nonlinearity, we can determine the dispersion relation for the optical coupler. We use a linear stability approach to investigate the modulation instability characteristic gain in the normal and anomalous group-velocity dispersion regimes. We greatly widened the spectrum of physical system parameters to consider various contingencies. Using nonlinear and power factors, MI can be controlled. Nonlinear saturation effect normally can suppress the MI gain and bandwidth in optical fiber system. But, as compared to conventional saturable nonlinearity, the modified saturation nonlinearity more significantly changes the MI dynamical behaviour. The analytical results show that new instability bands are created in a system parameter's normal and anomalous dispersion regimes. As a result of the effects of modified nonlinear saturation, we describe new methods for creating and managing the soliton through MI in three-core oppositely directed couplers with two negative index material channels.

Dissipative Kerr single soliton generation with extremely high probability via spectral mode depletion

Optical Kerr solitons generation based on microresonators is essential in nonlinear optics. Among various soliton generation processes, the single soliton generation plays a pivotal role since it ensures rigorous mode-locking on each comb line whose interval equals the free spectral range (FSR) of the microresonator. Current studies show that single soliton generation is challenging due to cavity instability. Here, we propose a new method to greatly improve single soliton generation probalility in the anomalous group velocity dispersion (GVD) regime in a micro-ring resonator based on silicon nitride. The improvement is realized by introducing mode depletion through an integrated coupled filter. It is convenient to introduce controllable single mode depletion in a micro-ring resonator by adjusting the response function of a coupled filter. We show that spectral mode depletion (SMD) can significantly boost the single soliton generation probability. The effect of SMD on the dynamics of optical Kerr solitons generation are also discussed. The proposed method offers a straightforward and simple way to facilitate robust single soliton generation, and will have an impact on the research development in optical Kerr soliton generation and on-chip optical frequency mode manipulation.Graphical abstract

Impact of conventional saturable function on modulation instability in triangular three-core oppositely directed coupler with one positive and two negative index material channels

Given the triangular design of a three-core oppositely directed coupler, this work shows that it is possible to observe the modulational instability (MI) zones caused by the nonlinear saturation effect. Two channels in this design are built from a material with a low refractive index (Channel 1, 3), whereas one is built from a material with a high index (Channel 2). The governing coupled nonlinear Schrödinger equations are modified with the presence of saturable nonlinearity and coupling terms. In the presence of typical saturable nonlinearity, we are able to determine the dispersion relation for the optical coupler under investigation. Using a linear stability analysis, we look at the modulation instability characteristic gain in the normal and anomalous group-velocity dispersion regimes. We significantly increased the physical system parameter range to account for different eventualities. The analytical results demonstrate that new instability bands are created in system parameters anomalous and normal dispersion regimes, in contradiction to Shafeeque et al. (2016). MI can be managed using nonlinear and power factors. We present novel techniques for produce and control the MI in oppositely directed three-core couplers in the impacts of nonlinear saturation.

Stokes Localized Structure in Kerr Resonators

We theoretically demonstrate generation of the Stokes temporal localized structure (TLS) by exploiting its Raman interaction in space and time within an optical potential well shared with another dark pulse. Excitation of the Stokes TLSs is feasible in both the normal and anomalous group-velocity dispersion regime, with a single modulated pump source. Stimulated Raman scattering constitutes influences on the stability and physical feature of the generated Stokes TLS (STLS). Particularly, breathing STLSs are also observed due to the periodic energy transfer between the primary and the Stokes fields. These findings could deepen the understanding of complex nonlinear dynamics in resonators and facilitate the excitation of different types of TLSs in potential platforms.

Modulation instability effects of the conventional Kerr type nonlinearity on the three-core oppositely directed coupler

The present design deals with a three-core system in which two channels are made of negative index material and one made of positive index material. The nonlinear Schrodinger equation is modified by adding the Kerr-type nonlinearity and coupling coefficient terms. We derive the dispersion relation for the optical coupler under investigation in the presence of added terms. Using a linear stability analysis, we examine the modulation instability characteristic gain in both the normal and anomalous group-velocity dispersion regimes. We study the system in diversified physical parameter space to unveil many possibilities. The analytical results demonstrate that, in contrast to Shafeeque et al. (2016), some novel instability bands are produced in the normal and anomalous dispersion regimes. We observe the modulational instability (MI) occurring due to Kerr-type nonlinearity. It is also evident that MI can be controlled by varying pump power and nonlinear factors. As a result, we observe new possibilities for producing and managing the MI in three-core oppositely directed couplers with the impact of opposite directionality.

Dispersion-Suppressed Mode Depletion by Exceptional Points for On-Chip Nonlinear Optics

Dispersion-suppressed mode depletion in a resonant cavity is usually challenging due to the entanglement of amplitude and phase response dictated by the Kramers-Kronig relationship. In this study, we examine the dispersion-suppressed mode depletion induced by parity--time- (PT-) symmetric dual resonators by taking advantage of the exotic property at the exceptional point (EP). We show that this technique can modify the linear spectrum of an optical resonator for relaxed phase-matching conditions in nonlinear optics, thus profoundly altering the nonlinear behavior of integrated Kerr combs. Indeed, dissipation-induced modulation instability (MI) is predicted and numerically verified in normal group-velocity dispersion (GVD) microresonators, and the single-soliton-generation probability is greatly enhanced in the anomalous GVD regime. The merging of two vibrant disciplines, i.e., non-Hermitian optics and nonlinear optics, especially Kerr microcombs, can deepen our understanding of non-Hermitian optical frequency combs and opens broad applications ranging from fiber-optic communications, microwave photonics, and ultrafast optics to quantum state manipulations.

The role of septic saturation on the cross-phase instability in two-core birefringent oppositely directed PIM-NIM coupler

We examine the cross-phase modulational instability in a two-core birefringent coupler made of a positive and negative index material. The modified form of generalized nonlinear Schrödinger equation contains a nonlinear septic term, cross-phase modulation, intrapulse Raman scattering, self-steepening, and conventional type of saturation nonlinearity. We investigate normal and anomalous group-velocity dispersion regimes and cases of birefringent effects such as zero-birefringence, linear birefringence, and circular birefringence. We observe that the MI gain spectrum shows significant changes in the presence of cross-phase modulation with a nonlinear saturation effect and non-Kerr nonlinearity. In addition, the self-steepening brings out new sidebands and shifts the existing sidebands to bring them together, providing a new pathway to generate solitons or ultrashort pulses in the presence of nonlinear septic saturation. The intrapulse Raman scattering effect causes conventional and Raman bands, extending the modulational instability (MI) domain range. The saturable nonlinearity suppresses the MI gain and reduces the bandwidth of the spectrum for both normal and anomalous dispersion regimes. But, in the case of nonlinear saturation with non-Kerr nonlinearity, the MI gain spectra exhibit efficient modification in a two-core birefringent oppositely directed coupler (ODCs) with adjustable effects of SS, IRS, and cross-phase modulation. Therefore, manipulating MI and solitons in a two-core ODCs is better performed when the birefringence is linear rather than circular. Here we present how to control and generate the MI and solitons in birefringent ODCs, emphasizing the negative index material (NIM) channel.

Modulational instability of truncated Airy pulses in cubic–quintic nonlinear optical waveguides with multiphoton absorptions

In this work we study the modulational instability (MI) phenomenon of truncated Airy pulses in a cubic–quintic nonlinear optical waveguide including nonlinear multiphoton absorptions as the two-photon absorption (TPA) and the three-photon absorption (3PA) in the anomalous group-velocity dispersion (AGVD) regime of chromatic dispersion. The investigation is done through a lossless and dimensionless cubic–quintic nonlinear Schrödinger equation (CQNLSE) where we launch a perturbed truncated Airy pulse. The perturbation is defined through the linear analysis of the steady-state of a continuous-wave (CW) in which we have chosen to operate with the optimal frequencies of the MI gain. The data obtained through this analysis led to generate the destabilization of the pulse through its lobes starting with the dominant peak. This MI is improved by the cooperating nonlinearities while its distance of achievement is reduced in this case growing from the main lobe to secondary sides lobes of tail. The MI is avoided in the competing case of nonlinearities for small peak power because the negative quintic Kerr nonlinearity (QKN) reduces or almost cancels the cubic Kerr nonlinearity (CKN) effect. If the peak power of the pulse increases, the deleterious effect of the competing QKN decreases while the AGVD and the CKN take over to bring the MI into existence breaking the Airy pulse into sub-pulses. This feature becomes more and more noised as the peak power increases. The TPA comes as a limiting factor of the MI for both the cases of cooperating and competing nonlinearities. The introduction of 3PA performs this TPA effect in the case of cooperating nonlinearities while in the competing case, the 3PA surprisingly cancels the TPA, increasing the AGVD effect. Since the same feature has already been observed recently in Ref. Heuteu et al, J. Opt. 23, 095503 (2021) for the supercontinuum generation (SCG) of Airy pulses, it is therefore shown that the 3PA could be used for better control the TPA effect even in the MI of this kind of pulses. Moreover, it is also shown that the increase of the Airy pulse duration drastically enhances its MI occurrence with noised outputs for large values of pulse width. However, reducing the pulse width also reduces its input energy, consequently the Airy pulse does not undergo the MI phenomenon but simply manifests the property of acceleration.

Quartic Kerr cavity combs: bright and dark solitons.

We theoretically investigate the dynamics, bifurcation structure, and stability of localized states in Kerr cavities driven at the pure fourth-order dispersion point. Both the normal and anomalous group velocity dispersion regimes are analyzed, highlighting the main differences from the standard second-order dispersion case. In the anomalous regime, single and multi-peak localized states exist and are stable over a much wider region of the parameter space. In the normal dispersion regime, stable narrow bright solitons exist. Some of our findings can be understood using a new, to the best of our knowledge, scenario reported here for the spatial eigenvalues, which imposes oscillatory tails to all localized states.

Active tuning of dispersive waves in Kerr soliton combs.

Kerr soliton combs operate in the anomalous group-velocity dispersion regime through the excitation of dissipative solitons. The generated bandwidth is largely dependent on the cavity dispersion, with higher-order dispersion contributing to dispersive-wave (DW) generation that allows for power enhancement of the comb lines at the wings of the spectrum. However, the spectral position of the DW is highly sensitive to the overall cavity dispersion, and the inevitable dimension variations that occur during the fabrication process result in deviations in the DW emission wavelength. Here, we demonstrate active tuning of the DW wavelength, enabling post-fabrication spectral shaping of the soliton spectrum. We control the DW position by introducing a wavelength-controllable avoided mode crossing through actively tuning the resonances of a silicon nitride coupled microresonator via integrated heaters. We demonstrate DW tuning over 113 nm with a spectral power that can exceed the peak soliton spectral power. In addition, our modeling reveals buildup and enhancement of the DW in the auxiliary resonator, indicating that the mode hybridization arising from the strong coupling between the two resonators is critical for DW formation.

Supercontinuum generation in single-crystal YAG fibers pumped around the zero-dispersion wavelength.

Yttrium aluminum garnet (YAG) is a common host material for both bulk and single-crystal fiber lasers. With increasing interest in developing optical technologies in the short-wave infrared and mid-infrared wavelength range, YAG may be a potential supercontinuum medium for these applications. Here, we characterize femtosecond laser pumped supercontinuum generation with 1200-2000 nm pump wavelengths (λp) for undoped, single-crystal YAG fibers, which are representative of the normal, zero, and anomalous-dispersion regimes. Supercontinuum was observed over the spectral region of about 0.2 to 1.6λp. Z-scan measurements were also performed of bulk YAG, which revealed little dispersion of the nonlinear index of refraction (n2) in the region of interest. The measured values of n2 (∼1×10-6cm2/GW) indicate a regime in which the nonlinear length, LNL, is less than the dispersion length, LD, (LNL≪LD). We report intensity clamping of the generated filament in the normal group velocity dispersion (GVD) regime and an isolated anti-Stokes peak in the anomalous GVD regime, suggesting further consideration is needed to optimize supercontinuum generation in this fiber medium.

Saturable higher nonlinearity effects on the modulational instabilities in three-core triangular configuration couplers

This paper demonstrates the possibility to observe the modulational instability (MI) regions due to modified nonlinear saturability over and above other high-order nonlinearities for the triangular configuration of a three-core oppositely directed coupler. In this configuration, two channels are positive-refractive-index material, and one is a negative-index material. The governing equations are the modified coupled nonlinear Schrödinger equations, to which we add the higher-order nonlinearity terms and coupling terms. This equation is further modified with the saturable nonlinearity term. In the presence of several nonlinearities, we derive the dispersion relation for the optical coupler under consideration. We use a linear stability analysis to study the modulation stability characteristic gain in both the normal and anomalous group-velocity dispersion regimes. We have widely varied the parameter range of the physical system parameters to accommodate different possibilities. In the normal region, the saturable nonlinearity aids in increasing the bandwidth of the MI region, while in the anomalous regime, the bandwidth reduces. In the normal dispersion case, the simultaneous presence of all nonlinearities in the optical coupler does not provide a suitable scope for pulse propagation. On the other hand, in the anomalous regime, these nonlinearities favor for pulse propagation.

Modulation instability in nonlinear metamaterials modeled by a cubic-quintic complex Ginzburg-Landau equation beyond the slowly varying envelope approximation.

Considering the theory of electromagnetic waves from the Maxwell's equations, we introduce a (3+1)-dimensionsal cubic-quintic complex Ginzburg-Landau equation describing the dynamics of dissipative light bullets in nonlinear metamaterials. The model equation, which is derived beyond the slowly varying envelope approximation, includes the effects of diffraction, dispersion, loss, gain, cubic, and quintic nonlinearities, as well as cubic and quintic self-steepening effects. The modulational instability of the plane waves is studied both theoretically, using the linear stability analysis, and numerically, using direct simulations of the Fourier space of the proposed nonlinear wave equation, based on the Drude model. The linear theory predicts instability for any amplitude of the primary wave. Also, in the linear stability analysis, self-steepening effects of different orders are confronted and one discusses their effects on the behavior of the gain spectrum under both normal and anomalous group-velocity dispersion regimes. Analytical results are equally confronted to direct numerical simulations and fully agree with the predictions from the gain spectra. Modulational instability is manifested by clusters of solitons and multihump and dromion-like structures, whose emergence and features depend not only on system parameters, such as the cubic and quintic self-steepening coefficients, but also on the propagation distance under a suitable balance between nonlinear and dispersive effects.

Few-cycle optical pulses in negative index materials with dispersive permittivity and permeability

Propagation of few-cycle optical pulses in nonlinear optical Kerr (cubic) and non-Kerr (quintic) type metamaterials, exhibiting frequency-dependent dielectric susceptibility and magnetic permeability, is considered. Considering the theory of electromagnetic waves from Maxwell’s equations, a new nonlinear evolution equation describing the combined influences of higher-order nonlinearities and higher-order linear and nonlinear dispersions, appropriate for electromagnetic ultrashort pulse propagation in negative index materials, is derived beyond the slowly varying envelope approximation. A fully numerical simulation of the newly derived model equation, based on the lossy Drude model, shows the propagation of soliton-like stable few-cycle optical pulses under some parameter values. The change in types of self-steepening parameters induces structural changes of the initial input pulse, characterized by a soliton molecule made of either asymmetric or symmetric optical pulses. Also, the mutual balancing between Kerr and non-Kerr nonlinearities and higher-order dispersions is found to support the formation of soliton-molecules in both the normal and anomalous group velocity dispersion regimes.

Coherence of bulk-generated supercontinuum

We have developed a numerical framework that allows estimation of coherence in spatiotemporal and spatiospectral domains. Correlation properties of supercontinuum (SC) pulses generated in a bulk medium are investigated by means of second-order coherence theory of non-stationary fields. The analysis is based on simulations of individual space–time and space–frequency realizations of pulses emerging from a 5 mm thick sapphire plate, in the regimes of normal, zero, and anomalous group velocity dispersion. The temporal and spectral coherence properties are analyzed in the near field (as a function of spatial position at the exit plane of the nonlinear medium) and as a function of propagation direction (spatial frequency) in the far field. Unlike in fiber-generated SC, the bulk case features spectacularly high degrees of temporal and spectral coherence in both the spatial and spatial-frequency domains, with increasing degrees of coherence at higher pump energies. When operating near the SC generation threshold, the overall degrees of temporal and spectral coherence exhibit an axial dip in the spatial domain, whereas in the far field, the degree of coherence is highest around the optical axis.

Turn-key, high-efficiency Kerr comb source.

We demonstrate an approach for automated Kerr comb generation in the normal group-velocity dispersion (GVD) regime. Using a coupled-ring geometry in silicon nitride, we precisely control the wavelength location and splitting strength of avoided mode crossings to generate low-noise frequency combs with pump-to-comb conversion efficiencies of up to 41%, which is the highest reported to date for normal-GVD Kerr combs. Our technique enables on-demand generation of a high-power comb source for applications such as wavelength-division multiplexing in optical communications.

Cross-phase modulation instability under delayed response and walk-off effects in the anomalous regime of dispersion

The dynamics of the modulation instability induced by cross phase modulation is studied by considering the influence of the walk-off and noninstantaneous response effects for two copropagating optical fields travelling in the anomalous regime of dispersion. To do so, we make use of extensions of the nonlinear Schrdinger equation jointly with the Debye model for polarization, which is shown to be effective and simplified when compared to the implementation of other methods for the noninstantaneous nonlinear response. In analyzing the sideband formation, two bands are observed with different behaviors with respect to the way the maximum gain and the respective frequency vary with increasing phase mismatch. Further, we also show the manner in which the maximum gain as well as its corresponding frequency scale with the delay parameter τ is substantially different from the cases of both fields experiencing the normal group-velocity dispersion regime and the case of mixed regimes. These facts may give rise to many new possibilities for the evolution of the modulated wave when compared to the nonanomalous cases studied in the literature.