Saturation Amplitude Research Articles

Silicon rich nitride: a platform for controllable structural colors

Abstract High refractive index dielectric materials like silicon rich nitride (SRN) are critical for constructing advanced dielectric metasurfaces but are limited by transparency and complementary metal oxide semiconductor (CMOS) process compatibility. SRN’s refractive index can be adjusted by varying the silicon to nitride ratio, although this increases absorption, particularly in the blue spectrum. Dielectric metasurfaces, which utilize the material’s high dielectric constant and nano-resonator geometry, experience loss amplification due to resonance, affecting light reflection, light transmission, and quality factor. This study explores the impact of varying the silicon ratio on structural color applications in metasurfaces, using metrics such as gamut coverage, saturation, and reflection amplitude. We found that a higher SRN ratio enhances these metrics, making it ideal for producing vivid structural colors. Our results show that SRN can produce a color spectrum covering up to 166 % of the sRGB space and a resolution of 38,000 dots per inch. Fabricated samples vividly displayed a parrot, a flower, and a rainbow, illustrating SRN’s potential for high-resolution applications. We also show that SRN can provide a better CIE diagram coverage than other popular metasurfaces materials. These findings highlight the advantages of SRN for photonic devices, suggesting pathways for further material and application development.

Excited ion-scale turbulence by a magnetic island in fusion plasmas

The characteristics of ion-scale turbulence in the presence of a magnetic island are numerically investigated using a gyrokinetic model in fusion plasma. We observe that in the absence of the usual ion temperature gradient (ITG) drive gradient, a magnetic island and its flatten effect could drive ITG instability. The magnetic island (MI) not only drives high-n modes of ITG instability but also induces low-n modes of vortex flow. Moreover, as the magnetic island width increases, the width of the vortex flow also increases. This implies that wider islands may more easily induce vortex flows. The study further indicates that the saturated amplitude and transport level of MI-induced ITG turbulence vary with different magnetic island widths. In general, larger magnetic islands enhance both particle and heat transport. When the magnetic island width reaches to 21ρi, the turbulence-driven transport becomes the same level with the cases that ITG is driven by pressure gradients. Our findings indicate the presence of intricate nonlinear effects in the modulation of plasma turbulence by MIs. These effects are of significant importance for comprehending the phenomenon of nonlinear coupling in forthcoming tokamaks such as ITER.

The effects of low vs. high rating of perceived exertion clamp exercise on performance, neuromuscular, and muscle oxygen saturation responses in females.

This study examined the time course of changes in force, relative to critical force (CF), electromyographic amplitude (EMG AMP), neuromuscular efficiency (NE), and muscle oxygen saturation (SmO2), as well as time to task failure (TTF) and performance fatigability (PF) during isometric handgrip holds to failure (HTF) anchored to the rating of perceived exertion (RPE) at 3 and 7. Ten females completed pre-test maximal voluntary isometric contractions (MVICs), submaximal HTF at four percentages of MVIC, an HTF at RPE = 3 and 7, and post-test MVICs. Analyses included paired samples t-tests, repeated measures ANOVAs and planned comparisons. TTF was not different between RPE 3 (540.4 ± 262.1s) and 7 (592.2 ± 299.6s), but PF for RPE 7 (42.1 ± 19.1%) was greater than RPE 3 (33.5 ± 15.4%) (p < 0.05). There were RPE-dependent decreases in force, EMG AMP, and NE across three discernable phases during the HTF (p < 0.01), but there were no significant changes in SmO2 across time. Although there were overall similar patterns across time for force, neuromuscular, and muscle metabolic responses between the RPE holds, the greater PF at RPE 7 than RPE 3 may be explained by the longer sustained time above CF at RPE 7, resulting in greater accumulation of intramuscular metabolites and afferent feedback. Throughout each trial, it is possible that force was adjusted to avoid the sensory tolerance limit, and the task was ended when force could no longer be reduced to maintain the assigned RPE, resulting in a similar TTF for RPE 7 and RPE 3.

Nonlinear simulations of GAEs in NSTX-U

A set of nonlinear simulations has been performed in order to study the nonlinear evolution of unstable global Alfvén eigenmodes in the National Spherical Torus Experiment-Upgrade (NSTX-U). Results of the single toroidal mode number, n, simulations are compared with a full nonlinear simulation (all toroidal harmonics included). In single-n simulations, the conservation of two integrals of motion of a particle in a cyclotron resonance with a monochromatic wave is demonstrated, resulting in a one-dimensional evolution of the particle distribution in (E,μ,pϕ) phase-space. Nonlinear simulations (both single-n and full nonlinear) show a significant redistribution of the resonant fast ions, especially in the pitch parameter. Thus, the changes in the resonant particle's parallel and perpendicular energies can be several times larger than the total particle energy change, with only a small fraction transferring into the excitation of the mode itself. This implies that even a relatively small amplitude mode can significantly modify the beam distribution in the resonant region. For the NSTX-U case considered, the single-n simulation results are close to full nonlinear simulation only for the most unstable mode, in which case the saturation amplitudes and changes in the fast ion distribution are comparable. In contrast, peak amplitudes of subdominant modes in all-n simulations are smaller by a factor of 3–10 compared to single-n runs due to the flattening of the beam ion distribution by the fastest growing mode.

A piezoelectric nonlinear energy sink shunt for vibration damping

The theoretical study and experimental validation of a nonlinear shunt circuit for piezoelectric vibration damping is investigated here. The circuit consists of a resistor, an inductor and a nonlinear cubic voltage source. The shunt acts as an electrical analog to the mechanical nonlinear energy sinks (NESs). These mechanical NESs are passive vibration absorbers that typically have a cubic nonlinear stiffness. They have attractive properties such as saturation of the host system’s vibration amplitude and strongly modulated response. This increases its operational frequency bandwidth and robustness against variations in the properties of host systems compared to linear vibration absorbers. However, the nonlinear nature may induce isolated responses in the host system that induce high vibration amplitudes. This paper investigates if these attractive properties also occur in the electric nonlinear energy sink shunt. An analytical expression for the frequency response is derived through the complexification-averaging method. Bifurcations in the frequency response reveal the occurrence of a quasi-periodic vibration energy exchange between the host system and the voltage over the electrodes of piezoelectric material. This is the main mechanism behind the amplitude saturation of the host system. Other bifurcations also reveal the existence of isolated responses. The nonlinear shunt is then realized with analog multipliers and a synthetic inductor and its performance in vibration damping is experimentally verified for a cantilever beam.

Beam‐Driven Electron Cyclotron Harmonic and Electron Acoustic Waves as Seen in Particle‐In‐Cell Simulations

AbstractRecent study has demonstrated that electron cyclotron harmonic (ECH) waves can be excited by a low energy electron beam. Such waves propagate at moderately oblique wave normal angles (∼70°). The potential effects of beam‐driven ECH waves on electron dynamics in Earth's plasma sheet is not known. Using two‐dimensional Darwin particle‐in‐cell simulations with initial electron distributions that represent typical plasma conditions in the plasma sheet, we explore the excitation and saturation of such beam‐driven ECH waves. Both ECH and electron acoustic waves are excited in the simulation and propagate at oblique wave normal angles. Compared with the electron acoustic waves, ECH waves grow much faster and have more intense saturation amplitudes. Cold, stationary electrons are first accelerated by ECH waves through cyclotron resonance and then accelerated in the parallel direction by both the ECH and electron acoustic waves through Landau resonance. Beam electrons, on the other hand, are decelerated in the parallel direction and scattered to larger pitch angles. The relaxation of the electron beam and the continuous heating of the cold electrons contribute to ECH wave saturation and suppress the excitation of electron acoustic waves. When the ratio of plasma to electron cyclotron frequency ωpe/ωce increases, the ECH wave amplitude increases while the electron acoustic wave amplitude decreases. Our work reveals the importance of ECH and electron acoustic waves in reshaping sub‐thermal electron distributions and improves our understanding on the potential effects of wave‐particle interactions in trapping ionospheric electron outflows and forming anisotropic (field‐aligned) electron distributions in the plasma sheet.

Cross-scale interaction between microturbulence and meso-scale reversed shear Alfvén eigenmodes in DIII-D plasmas

This paper reports global nonlinear gyrokinetic simulations that couple meso-scale reversed shear Alfvén eigenmodes (RSAEs) driven by energetic particles (EPs) and ion temperature gradient (ITG) microturbulence driven by thermal plasma, using equilibrium and profiles from DIII-D discharge #159243. In simulations focusing only on the ITG, electrostatic ITG drives a huge thermal ion heat transport, which is reduced by a factor of 10 to a level close to the experimental value in electromagnetic simulation due to finite β effect. In the simulations coupling the RSAE and ITG, ITG can scatter the resonant EP nonlinearly trapped by the RSAE and damp the zonal flows generated by the RSAE. The regulation of the RSAE by the ITG greatly reduces the initial saturation amplitude of the RSAE but increases the RSAE amplitude and associated EP transport to experimental levels in the quasi-steady state. The RSAE effects on the ITG, specifically the stronger zonal flows generated by the RSAE and the RSAE frequency modulation of the ITG-induced thermal ion heat transport, in turn, leads to a reduction of the thermal ion heat transport by more than a factor of 2 . For a stronger background ITG, the regulation of the RSAE by the ITG is stronger, while the RSAE effects on the ITG are weaker. This work highlights the importance of cross-scale coupling in the dynamics of the AE turbulence and EP transport.

Magnetic analysis of permeability configurable nanocrystalline flake ribbons for medium frequency energy conversion applications

The novel permeability configurable nanocrystalline flake ribbons (NFR) utilizing Fe-based nanocrystalline material has been increasingly adopted in diverse energy conversion applications. However, the application-oriented magnetic properties are rarely examined. In this paper, the flexible NFR materials with different permeabilities are measured while the influence of permeability configuration is analyzed regarding the initial and amplitude permeabilities, B–H curves, and core losses. A measurement method using a full bridge inverter and LCL compensation is designed to test the magnetic material under various flux densities and in the frequency range between 50 kHz–150 kHz. In addition, ferrite materials with similar permeabilities are also tested for comparison. Results reveal that the NFR materials exhibit superior stability in terms of amplitude permeability changes and saturation capability compared with ferrites. Core loss grows with increased crushing, but permeability becomes more stable at higher frequencies. Hence, NFR materials can be matched to various applications based on their characteristics with different permeabilities. Additionally, the NFR materials have shown lower loss densities than their ferrite counterparts at high flux density conditions. Hence, the NFR material can be beneficial to achieve higher power density. Through analysis and experiments, this paper provides measurement optimization bases for practical energy conversion applications.

Overdamped Ising machine with stochastic resonance phenomena in large noise condition

Gain-dissipative Ising machines (GIMs) are dedicated devices that can rapidly solve combinatorial optimization problems. The noise intensity in traditional GIMs should be significantly smaller than its saturated fixed-point amplitude, indicating a lower noise margin. To overcome the existing limit, this work proposes an overdamped bistability-based GIM (OBGIM). Numerical test on uncoupled spin network show that the OBGIM has a different bifurcation dynamics from that of the traditional GIM. Moreover, the domain clustering dynamics on non-frustrated network proves that the overdamped bistability enables the GIM to suppress noise-induced random spin-state switching effectively; thus, it can function normally in an environment with a relatively large noise level. Besides, some prevalent frustrated graphs from the SuiteSparse Matrix Collection were adopted as MAXCUT benchmarks. The results show that the OBGIM can induce stochastic resonance phenomenon when solving difficult benchmarks. Compared with the traditional GIM, this characteristic makes the OBGIM achieve comparable solution accuracy in larger noise environment, thus achieving strong noise robustness.

Early Alterations in Inner-Retina Neural and Glial Saturated Responses in Lens-Induced Myopia.

Myopic marmosets are known to exhibit significant inner retinal thinning compared to age-matched controls. The purpose of this study was to assess inner retinal activity in marmosets with lens-induced myopia compared to age-matched controls and evaluate its relationship with induced changes in refractive state and eye growth. Cycloplegic refractive error (Rx), vitreous chamber depth (VCD), and photopic full-field electroretinogram were measured in 14 marmosets treated binocularly with negative contact lenses compared to 9 untreated controls at different stages throughout the experimental period (from 74 to 369days of age). The implicit times of the a-, b-, d-, and photopic negative response (PhNR) waves, as well as the saturated amplitude (Vmax), semi-saturation constant (K), and slope (n) estimated from intensity-response functions fitted with Naka-Rushton equations were analyzed. Compared to controls, treated marmosets exhibited attenuated b-, d-, and PhNR waves Vmax amplitudes 7 to 14days into treatment before compensatory changes in refraction and eye growth occurred. At later time points, when treated marmosets had developed axial myopia, the amplitudes and implicit times of the b-, d-, and PhNR waves were similar between groups. In controls, the PhNR wave saturated amplitude increased as the b + d-wave Vmax increased. This trend was absent in treated marmosets. Marmosets induced with negative defocus exhibit early alterations in inner retinal saturated amplitudes compared to controls, prior to the development of compensatory myopia. These early ERG changes are independent of refraction and eye size and may reflect early changes in bipolar, ganglion, amacrine, or glial cell physiology prior to myopia development. The early changes in retinal function identified in the negative lens-treated marmosets may serve as clinical biomarkers to help identify children at risk of developing myopia.

Stresses in Saturated and Unsaturated Subgrade Layer Induced by Railway Track Vibration

The theoretical and practical studies of the cyclic loads resulting from the movement and passage of trains on the unsaturated subgrade to determine the effect of the degree of saturation and moisture content on the foundations and infrastructure of the railway lines, especially the settlement in the railway lines as a result of the development of the train loads. Thirty-six laboratory experiments were carried out using models that simulate a railway with nearly half the scale of the real one, using an iron box of (1.5 × 1.0 × 1.0) meters and a layer of clay soil with a thickness of 0.5 m representing the base layer, were constructed inside it. Above it, there is a layer of crushed stone representing a 0.2 m thick ballast, topped by a rail line of 0.8 m long installed on three sleeper beams with dimensions of 0.9 m (0.1 × 0.1 m). The subgrade layer has been constructed at different saturation degrees as follows: 100, 80, 70, and 60%. The tests were carried out using different load amplitudes and frequencies. These experiments investigated the effect of the subgrade degree of saturation on the value of the stresses generated on the surface and the middle (vertical and lateral stresses) and the settlement of the subgrade. In the case of unsaturated subgrade soil, an increase in load frequency has a clear effect on increasing the generated stresses in the subgrade layer, especially with lower saturation levels. However, the results and measurements of these experiments found that the load frequency almost had no effect on the values of the stresses generated on the surface and inside the subgrade layer with a 100% degree of saturation. The results of the investigation demonstrated that, while load frequency had a minimal effect on track-panel settlement, it increased with the load amplitude and subgrade soil saturation degree. The change of settlement of the track panel with the number of cycles has a high rate at the beginning; after a while from that, it decreases gradually until, after some value of the number of cycles, the settlement changes at a very low rate and gradually.

Estimating the synaptic density deficit in Alzheimer's disease using multi-contrast CEST imaging.

In vivo noninvasive imaging of neurometabolites is crucial to improve our understanding of the underlying pathophysiological mechanism in neurodegenerative diseases. Abnormal changes in synaptic organization leading to synaptic degradation and neuronal loss is considered as one of the primary factors driving Alzheimer's disease pathology. Magnetic resonance based molecular imaging techniques such as chemical exchange saturation transfer (CEST) and magnetic resonance spectroscopy (MRS) can provide neurometabolite specific information which may relate to underlying pathological and compensatory mechanisms. In this study, CEST and short echo time single voxel MRS was performed to evaluate the sensitivity of cerebral metabolites to beta-amyloid (Aβ) induced synaptic deficit in the hippocampus of a mouse model of Alzheimer's disease. The CEST based spectra (Z-spectra) were acquired on a 9.4 Tesla small animal MR imaging system with two radiofrequency (RF) saturation amplitudes (1.47 μT and 5.9 μT) to obtain creatine-weighted and glutamate-weighted CEST contrasts, respectively. Multi-pool Lorentzian fitting and quantitative T1 longitudinal relaxation maps were used to obtain metabolic specific apparent exchange-dependent relaxation (AREX) maps. Short echo time (TE = 12 ms) single voxel MRS was acquired to quantify multiple neurometabolites from the right hippocampus region. AREX contrasts and MRS based metabolite concentration levels were examined in the ARTE10 animal model for Alzheimer's disease and their wild type (WT) littermate counterparts (age = 10 months). Using MRS voxel as a region of interest, group-wise analysis showed significant reduction in Glu-AREX and Cr-AREX in ARTE10, compared to WT animals. The MRS based results in the ARTE10 mice showed significant decrease in glutamate (Glu) and glutamate-total creatine (Glu/tCr) ratio, compared to WT animals. The MRS results also showed significant increase in total creatine (tCr), phosphocreatine (PCr) and glutathione (GSH) concentration levels in ARTE10, compared to WT animals. In the same ROI, Glu-AREX and Cr-AREX demonstrated positive associations with Glu/tCr ratio. These results indicate the involvement of neurotransmitter metabolites and energy metabolism in Aβ-mediated synaptic degradation in the hippocampus region. The study also highlights the feasibility of CEST and MRS to identify and track multiple competing and compensatory mechanisms involved in heterogeneous pathophysiology of Alzheimer's disease in vivo.

Sawtooth suppression by flux pumping on HBT-EP

This study examines the mechanisms underlying sawtooth suppression in the High Beta Tokamak-Extended Pulse (HBT-EP) device. It is observed that strong-intensity sawtooth activities correlate with reduced-amplitude magnetohydrodynamics (MHD) edge modes which are identified as m/n=3/1 external kink modes, while sawtooth suppression correlates with larger and saturated edge mode amplitudes. To further investigate these correlations, the plasma–wall coupling was manipulated by adjusting the positions of the conducting walls in HBT-EP. It was found that strong sawtooth events occur when the normalized wall radius b/a is within a critical value. This implies that the plasma–wall distance must be sufficiently small to ensure effective stabilization of the edge mode. Even slight differences in major radius result in significantly different discharge styles, categorized as ‘sawtoothing discharges’ and ‘sawtooth-suppressed discharges’ respectively. Through a series of mode structure analyses, we confirm the coexistence and coupling of the m/n=1/1 helical core, m/n=2/1 tearing mode, and m/n=3/1 external kink mode during sawtooth-suppression, and that this coupling induces anomalous current broadening. Based on these findings, we conclude that sawtooth suppression in the HBT-EP tokamak is consistent with the process of magnetic flux pumping.

Quasi-steady-state (QUASS) reconstruction enhances T1 normalization in apparent exchange-dependent relaxation (AREX) analysis: A reevaluation of T1 correction in quantitative CEST MRI of rodent brain tumor models.

The apparent exchange-dependent relaxation (AREX) analysis has been proposed as an effective means to correct T1 contribution in CEST quantification. However, it has been recognized that AREX T1 correction is not straightforward if CEST scans are not performed under the equilibrium condition. Our study aimed to test if quasi-steady-state (QUASS) reconstruction could boost the accuracy of the AREX metric under common non-equilibrium scan conditions. Numerical simulation and in vivo scans were performed to assess the AREX metric accuracy. The CEST signal was simulated under different relaxation delays, RF saturation amplitudes, and durations. The AREX was evaluated as a function of the bulk water T1 and labile proton concentration using the multiple linear regression model. AREX MRI was also assessed in brain tumor rodent models, with both apparent CEST scans and QUASS reconstruction. Simulation showed that the AREX calculation from apparent CEST scans, under non-equilibrium conditions, had significant dependence on labile proton fraction ratio, RF saturation time, and T1. In comparison, QUASS-boosted AREX depended on the labile proton fraction ratio without significant dependence on T1 and RF saturation time. Whereas the apparent (2.7 ± 0.8%) and QUASS MTR asymmetry (2.8 ± 0.8%) contrast between normal and tumor regions of interest (ROIs) were significant, the difference was small. In comparison, AREX contrast between normal and tumor ROIs calculated from the apparent CEST scan and QUASS reconstruction was 3.8 ± 1.1%/s and 4.4 ± 1.2%/s, respectively, statistically different from each other. AREX analysis benefits from the QUASS-reconstructed equilibrium CEST effect for improved T1 correction and quantitative CEST analysis.

Saturation of Fishbone Instability by Self-Generated Zonal Flows in Tokamak Plasmas.

Gyrokinetic simulations of the fishbone instability in DIII-D tokamak plasmas find that self-generated zonal flows can dominate the nonlinear saturation by preventing coherent structures from persisting or drifting in the energetic particle phase space when the mode frequency down-chirps. Results from the simulation with zonal flows agree quantitatively, for the first time, with experimental measurements of the fishbone saturation amplitude and energetic particle transport. Moreover, the fishbone-induced zonal flows are likely responsible for the formation of an internal transport barrier that was observed after fishbone bursts in this DIII-D experiment. Finally, gyrokinetic simulations of a related ITER baseline scenario show that the fishbone induces insignificant energetic particle redistribution and may enable high performance scenarios in ITER burning plasma experiments.

Electric-field induced droplet stable vertical vibration: Experiments and numerical simulations

In this study, stable vertical vibration of a single droplet was stimulated through Electrowetting on Dielectric (EWOD) on a Polydimethylsiloxane-coated Indium Tin Oxide glass (PDMS-ITO) surface in an oil atmosphere. The mechanism of the vibration is clarified as a self-adjusted energy earning and losing equilibrium, and a small friction at the contact line is crucial for this stable vibration. We investigated the impact of needle tip electrode height, different voltage forms (AC and DC voltages), and voltage magnitude on the droplet vibration state. As decreasing the needle tip electrode height, there is a transition of the droplet from a vibrating state to a static equilibrium state. During the vibration, the vibration period remains constant regardless of tip height, but the contact angle changes synchronously with the voltage. The electric signal in the circuit is also measured to provide solid evidence for this synchronicity. Finally, the saturation voltage of the contact angle and amplitude is about 120 V. The intrinsic connection between contact angle saturation and amplitude saturation is clarified by studying the surface energy of the droplet. A theoretical model is constructed to numerically simulate the shape and amplitude of the droplet’s vibrations.

Feedback enhanced Dyakonov–Shur instability in graphene field-effect transistors

Graphene devices are known to have the potential to operate THz signals. In particular, graphene field-effect transistors (gFETs) have been proposed as devices to host plasmonic instabilities in the THz realm; for instance, Dyakonov–Shur (DS) instability which relies upon dc excitation. In this work, starting from a hydrodynamical description of the charge carriers, we extend the transmission line description of gFETs to a scheme with a positive feedback loop, also considering the effects of delay, which leads to the transcendental Laplace-transform transfer function, with complex frequency s, with terms of the form e−assechk(s)/s , for a given a∈R0+ arising from the delay time and with k∈N . Applying the conditions for the excitation of DS instability, we report an enhanced voltage gain in the linear regime that is corroborated by our simulations of the nonlinear hydrodynamic model for the charge carriers. This translates to both greater saturation amplitude—often up to 50% increase—and faster growth rate of the self-oscillations. Thus, we bring forth a prospective concept for the realization of a THz oscillator suitable for future plasmonic circuitry.

Saturation of spiral instabilities in disc galaxies

ABSTRACT Spiral density waves can arise in galactic discs as linear instabilities of the underlying stellar distribution function. Such an instability grows exponentially in amplitude at some fixed growth rate β before saturating non-linearly. However, the mechanisms behind saturation, and the resulting saturated spiral amplitude, have received little attention. Here, we argue that one important saturation mechanism is the non-linear trapping of stars near the spiral’s corotation resonance. Under this mechanism, we show analytically that an m-armed spiral instability will saturate when the libration frequency of resonantly trapped orbits reaches $\omega _\mathrm{lib} \sim \mathrm{a\, \, few}\times m^{1/2} \beta$. For a galaxy with a flat rotation curve, this implies a maximum relative spiral surface density $\vert \delta \Sigma /\Sigma _0\vert \sim \mathrm{a\, \, few} \times (\beta /\Omega _\mathrm{p})^2 \cot \alpha$, where Ωp is the spiral pattern speed and α is its pitch angle. This result is in reasonable agreement with recent N-body simulations, and suggests that spirals driven by internally-generated instabilities reach relative amplitudes of at most a few tens of per cent; higher amplitude spirals, like in M51 and NGC 1300, are likely caused by very strong bars and/or tidal perturbations.

Model-Free Prescribed-Time Control Under Input Amplitude and Rate Saturations for Uncertain Mechatronic Systems With Mismatched Disturbances

Model-Free Prescribed-Time Control Under Input Amplitude and Rate Saturations for Uncertain Mechatronic Systems With Mismatched Disturbances

Effect of ion channel guiding on the saturation mechanism of single and two-stream free-electron lasers based upon a rectangular hybrid wiggler

The paper studies a simulation of a one-dimensional nonlinear free electron laser that includes a square hybrid wiggler, prebunched electron beam, and ion-channel guiding mechanism. The Lorentz equations of motion and the Maxwell equations were combined to derive a set of non-linear differential equations solved numerically within the framework of slowly varying amplitude and wavenumber. The relativistic electron beam is assumed to be cold, and self-fields of the electron beam and the slippage of the radiation wave are ignored. The appropriate parameters are chosen in such a way that the Raman regime is being considered, where there is high density and low energy. The effect of ion-channel densities on the saturation of the device is studied for both groups Ι and ΙΙ orbits. It is found that as the ion channel density increases, the range of saturation radiation for group Ι circuits increases, too. In contrast, the saturated radiation amplitude for group ΙΙ orbits is decreased with increasing the ion-channel density. Moreover, the results of the numerical calculations show that the saturation length for an FEL with a prebunched beam is considerably shorter than that of a uniform electron beam. A free electron laser (FEL) model called the two-stream FEL, which uses two relativistic electron beams of different energy, is also studied. Comparing the radiation amplitude between FEL and TSFEL shows that TSFEL is more efficient than FEL.