The aim of this work is to analyze the influence of various parameters
on the macroscopic laser equations. This requires a numerical description
and a technical programming implementation, which will also be performed.
To ensure an understanding of both the physical and the mathematical and
numerical bases both are presented in detail. Here, the semi-classical
theory of the laser light is to be reproduced, which connects Maxwell's
electromagnetic theory to a two-level quantum system. This ultimately leads
to three coupled differential equations that link the physical quantities
electric field intensity, polarization, and population inversion. Directly
after that the numerical theory of ordinary and partial differential
equations is developed. Here, various one-step process and subsequently
explicit and implicit multistep methods are presented in the context of
ordinary differential equations first. The numerical theory of partial
differential equations is limited to the method of finite differences;
explicit, implicit, and mixed methods are presented in the examples of the
wave and the diffusion equation. In addition, the theoretical concepts of
the stability study as well as various stability criteria are given. The
numerical implementation of the laser equations is described building on
this. The working methods of the author are emphasized, thus allowing a
simpler understanding. For this reason, the physical preparation - the
rescaling of the equations - is discussed. This allows a unitless and thus
mathematical and numerical greatly simplified handling of the equation.
Then follows the numerical stability study of various methods, applied to
the partial laser initially decoupled equation. This study is also intended
to bring the operation of the author in the foreground and is therefore not
reflected in the strict mathematical proof structure. Rather, the stability
conditions are derived in opposite proof direction, increasing the clarity
and the reading flow. According to the physical interpretation of the
values of the decoupled equation, the three coupled equations are
implemented computationally exhaustive. Of the manifold of values phenomena
of dissymetric occurrence of a population inversion and the dependency of
the laser process of damping terms are examined.