Semiconductor Lasers

Abstract

In the last chapter, we considered a model for a two-level system in a resonator and derived the following set of dynamical equations for the system’s time evolution:

$$\frac{{d{{b}_{\lambda }}}}{{dt}} = ( - i{{\omega }_{\lambda }} - {{k}_{\lambda }} - i\sum\limits_{\mu } {g_{{\mu \lambda }}^{*}{{\alpha }_{\mu }} + {{F}_{\lambda }}(t)}$$

(a)

$$\frac{{d{{\alpha }_{\mu }}}}{{2dt}} = ( - i\omega - \gamma ){{\alpha }_{\mu }} + i\sum\limits_{\lambda } {{{g}_{{\mu \lambda }}}{{b}_{\lambda }}{{d}_{\mu }} + {{\Gamma }_{{\mu - }}}} \left( t \right)$$

(b)

$$\frac{{d{{d}_{\mu }}}}{{dt}} = \frac{{{{d}_{0}} - {{d}_{\mu }}}}{{{{\tau }_{d}}}} + 2i\sum\limits_{\lambda } {\left( {g_{{\mu \lambda }}^{*}b_{\lambda }^{*}\alpha _{\mu }^{*}} \right) + {{\Gamma }_{d}}\left( t \right)}$$

(c)

To apply such a set of equations to any problem, however, it is necessary to know the parameters ω _λ ,κ _λ ,g _μλ ,ω,γ,and τ _d with some accuracy as well as to understand the pump mechanism well enough to be able to calculate in some detail the dynamic behavior of d ₀ with the external pump. An additional problem arises with semiconductors in that one cannot consider individual atoms in the material as the electrons are not localized in the lattice. Therefore, the index μ is no longer a valid index on which to sum. These issues will be taken up sequentially in the sections of this chapter. In the first section of the effective index method, calculation of ω _λ ,κ _λ ,and the mode patterns u _λ (r) will be considered in some detail. The second section will deal with a very simple model of band structure for the purpose of describing how one can replace the atomic coordinate _µ with the electron wave vector k. The third section will discuss the concepts of Fermi levels and quasi-Fermi levels with attention directed toward elucidating the meaning of the time constants τ _d and τ _α = 1/γ, as well as finding a dynamical model for the d ₀ in terms of measurable parameters. The fourth section of the chapter will cast the Maxwell-Bloch equations in their normal semiconductor rate equation form while giving some discussion of the polarizability χ in the equations. The fifth and concluding section of the chapter will discuss something about laser structure and doping profile.