We derive an equation for the differential cross-section dσ/dΩ of reflected electrons by using Electron Mirror method. The mathematical derivation of differential cross-section equation is based on the Rutherford scattering model as well as the equation of electric potential difference. We describe the interaction of focused electron beams with a polyethylene terephthalate sample using focused ion beam-scanning electron microscope electron mirror image. The electron differential cross-section dσ/dΩ for different working distances h=33, 20, and 10 mm, with a scattering angle θ ranged from 10 deg to 55 deg, incident electrons energies η=500, 750, and 1000 eV, and scanning potential Δφ ranged from 1 to 2 kV has been calculated. According to our findings, the differential cross-section dσ/dΩ of reflected electrons decreases with increasing scattering angle θ and the working distance h and is directly proportional to the difference potential Δφ and the energy of incident electrons η. Distort the electron mirror images with increased Δφ.
We investigate the generation of a chirped pulse in a single-mode, ring-cavity, erbium-doped fiber laser employing carbon nanotubes (CNTs) as a saturable absorber (SA). The pulse propagation is simulated using analytical methods to understand and quantify the role of multiple SA properties, particularly in the propagation dynamics of the laser pulse. The soliton solution is obtained on the basis of nonlinear effects, such as gain dispersion, second anomalous group-velocity dispersion, self-phase modulation, and two-photon absorption for a generalized nonlinear Schrodinger equation. The influences of the SA parameter in the range from 0.1 to 0.4 on the chirp, power, and width of the soliton are calculated. A stable, passively mode-locked fiber laser using CNTs as an SA is modeled. In addition, the power, width, chirp, and phase of the soliton pulses can be tuned by choosing suitable SA parameters.
An analytical solution of the generalized nonlinear Schrodinger equation which is implemented with fiber laser applications has been presented. The solution based on the exp-function method which is depending on time, space and small perturbations has been found. This solution was used to test the behavior and study the propagation characteristics of laser pulses and compared with some of the researches in the same field and the nonlinear effects as gain dispersion, second anomalous group velocity dispersion, self phase modulation, and frequency are investigated. The net results are that the parabolic pulse growth after z=4 m , and generate a periodic pulse train, the power of pulse is increased with increasing the length of fiber laser with reduce its width, the nonlinear effects have a small role on the pulse power, but they effect on the modulation stability of the laser and lead to generate sideband, the behavior of the pulse converted to chaotic when increasing the frequency.