Efficiency of using adaptive artificial boundary conditions at computer simulation of contrast spatio‐temporal laser‐induced structures in a semiconductor

Abstract

Many problems of modern laser physics are governed by equations or sets of equations in an unbounded domain. For solving these problems using the computer simulation, it is necessary to introduce the bounded domain, which size should be extended significantly to avoid the spurious wave reflection from the domain boundaries. Alternatively, the artificial (non‐reflective or transparent) boundary conditions should be stated. This approach is also effective for enhancing computation performance at the numerical solution of the nonlinear partial differential equations (PDEs). In the current paper, we investigate the laser pulse propagation in a semiconductor, governed by the Schrödinger equation, under the appearance of spatio‐temporal contrast structures of semiconductor characteristics. Their evolution is described by a set of PDEs. The optical pulse is partly reflected from the boundaries of these structures. Consequently, even a little reflection of the optical pulse from the artificial boundaries can essentially distort the numerical solution. Thus, these artificial boundary conditions must possess a high quality to minimize their reflection coefficients. With this aim, we propose the method for constructing adaptive artificial boundary conditions and discuss their advantages.

Cross‐impact of beam local wavenumbers on the efficiency of self‐adaptive artificial boundary conditions for two‐dimensional nonlinear Schrödinger equation

“…then the ABCs ( 14) and ( 15) are adaptive. In Trofimov et al [72], we demonstrated that the efficiency of nonadaptive ABCs stated for the 1D linear problem directly depends on the diffraction coefficient D A z : At its rather large value, the nonadaptive ABCs become inefficient. In the current work, we consider 2D problem and study numerically the influence of LWNs k x l , k x r , computed on transverse coordinate, on the accuracy of the adaptive ABCs stated on the longitudinal coordinate.…”

“…In Trofimov et al [72, 74], we demonstrated that the laser‐induced contrast dynamic structures of the charged particles' concentration appear in a semiconductor under certain conditions. Some parts of the optical beam are reflected from the boundaries of these domains.…”

“…As can be seen in Figure 1, the optical beam center is located outside (on the left) of the semiconductor, and it propagates in the positive direction along the z ‐coordinate. When the optical beam falls on a semiconductor and then passes through it, a part of the optical beam is reflected from the contrast structure, induced by optical radiation (Figure 1) [72] and propagates in the opposite direction to the direction of the falling beam. At a certain time moment, this beam reaches the left AB ( ), at which the adaptive or nonadaptive ABC is stated.…”

“…Let us notice that for an adequate description of the optical pulse interaction with a semiconductor, it is necessary to consider both longitudinal and transverse diffraction. Therefore, the Schrödinger equation should contain the second order derivative on the z-coordinate [70,72]. In this case, the complex amplitude slowly varies only both in time and along a spatial coordinate that is transverse to the pulse propagation direction.…”

“…At constructing adaptive ABCs, the main question is how does adequately compute changing LWNs involved in them at changing time and spatial coordinate? In Trofimov et al [72], we proposed a method for the LWN computation based on using parts of the problem's invariants considering 1D Schrödinger equation. Below we extend this approach both for the 2D Schrödinger equation and write time‐dependent formulas for LWNs for both coordinates, and the algorithm of their computation is described as well as demonstration of the efficiency of the approach based on computer simulation is presented.…”