Horizons in World Physics. Volume 301 opens with a discussion on how symplectic difference schemes are of great interest because they possess a number of fundamental properties of Hamiltonian differential equations. In particular, they inherit integral invariants such as momentum, angular momentum, and integrals, which are determined by the symplectic structure of the phase space, including the phase-space volume. Following this, the authors describe the dependence of amplified spontaneous emission on the length through the geometrical gain coefficient. In addition, its accuracy in various media with excitation lengths of 200 and 84 cm for N2 and KrF lasers, respectively, is described, and even in small 2-mm-long polymer or few-centimeter-long Ar x-ray lasers. Later, a simple two-dimensional nonstationary problem is formulated for describing the dispersion of a pollutant in a limited area. Its solutions, as well as solutions of an adjoint problem, are used to obtain dual estimates of the pollutant concentration at a point. One study focuses on the dynamics of charged particles subjected to the Lorentz force inside particle accelerators and the correct derivation of their equations of motion and, ultimately, of their trajectories. As we will see, the reason for the increased challenge is the presence of accelerated motion. Many geometrical optics models have been proposed to describe the propagation of paraxial Gaussian beam. However, those paraxial ray-optics models are inapplicable in the paraxial condition. As such, the penultimate chapter introduces a skew line ray-based model to represent the propagation properties of nonparaxial Gaussian beam under the oblate spheroidal coordinates. The free-space evolution of light beam’s complex wavefront, including amplitude and phase, is derived via this model.

Horizons in World Physics. Volume 301 opens with a discussion on how symplectic difference schemes are of great interest because they possess a number of fundamental properties of Hamiltonian differential equations. In particular, they inherit integral invariants such as momentum, angular momentum, and integrals, which are determined by the symplectic structure of the phase space, including the phase-space volume. Following this, the authors describe the dependence of amplified spontaneous emission on the length through the geometrical gain coefficient. In addition, its accuracy in various media with excitation lengths of 200 and 84 cm for N2 and KrF lasers, respectively, is described, and even in small 2-mm-long polymer or few-centimeter-long Ar x-ray lasers. Later, a simple two-dimensional nonstationary problem is formulated for describing the dispersion of a pollutant in a limited area. Its solutions, as well as solutions of an adjoint problem, are used to obtain dual estimates of the pollutant concentration at a point. One study focuses on the dynamics of charged particles subjected to the Lorentz force inside particle accelerators and the correct derivation of their equations of motion and, ultimately, of their trajectories. As we will see, the reason for the increased challenge is the presence of accelerated motion. Many geometrical optics models have been proposed to describe the propagation of paraxial Gaussian beam. However, those paraxial ray-optics models are inapplicable in the paraxial condition. As such, the penultimate chapter introduces a skew line ray-based model to represent the propagation properties of nonparaxial Gaussian beam under the oblate spheroidal coordinates. The free-space evolution of light beam’s complex wavefront, including amplitude and phase, is derived via this model.

Horizons in World Physics. Volume 301 opens with a discussion on how symplectic difference schemes are of great interest because they possess a number of fundamental properties of Hamiltonian differential equations. In particular, they inherit integral invariants such as momentum, angular momentum, and integrals, which are determined by the symplectic structure of the phase space, including the phase-space volume. Following this, the authors describe the dependence of amplified spontaneous emission on the length through the geometrical gain coefficient. In addition, its accuracy in various media with excitation lengths of 200 and 84 cm for N2 and KrF lasers, respectively, is described, and even in small 2-mm-long polymer or few-centimeter-long Ar x-ray lasers. Later, a simple two-dimensional nonstationary problem is formulated for describing the dispersion of a pollutant in a limited area. Its solutions, as well as solutions of an adjoint problem, are used to obtain dual estimates of the pollutant concentration at a point. One study focuses on the dynamics of charged particles subjected to the Lorentz force inside particle accelerators and the correct derivation of their equations of motion and, ultimately, of their trajectories. As we will see, the reason for the increased challenge is the presence of accelerated motion. Many geometrical optics models have been proposed to describe the propagation of paraxial Gaussian beam. However, those paraxial ray-optics models are inapplicable in the paraxial condition. As such, the penultimate chapter introduces a skew line ray-based model to represent the propagation properties of nonparaxial Gaussian beam under the oblate spheroidal coordinates. The free-space evolution of light beam’s complex wavefront, including amplitude and phase, is derived via this model.