Laser Photonics Calculating Electron Concentration, Gain, and Intensity

Calculating Electron Concentration, Gain, and Intensity

Our goal in this section is to develop a set of equations, including a rate equation, similar to the equations we had for gas lasers.   This will allow us to calculate relevant quantities such as the electron concentration in the gain region, optical gain, and optical intensity; and laser parameters like the threshold pump current.

Let’s first look at what has not changed:  we still have a Fabry-Perot cavity (we’ll discuss the details of how that’s made in the next section), so we still have the same equation for the threshold gain:

\displaystyle \gamma_{th} = -\frac{1}{L} \ln {(R_1 R_2)}

Previously, the gain \gamma depended on the inversion (N_2 - N_1).  For a semiconductor laser, it depends on the concentration of (conduction band) electrons n and (valence band) holes p in the gain region.  Calculating the gain turns out to be somewhat complex, as we need to sum over all possible combinations of conduction and valence band states that could contribute, including broadening.  For now, let’s use a simple phenomenological expression to relate the gain \gamma  to the electron concentration n:

\displaystyle \gamma = \gamma_0 \ln {(n / n_{tr})}

where \gamma_0 and n_{tr} are parameters of the semiconductor material used for the gain region.  n_{tr} is the transparency concentration: the concentration of electrons in the conduction band necessary for the semiconductor gain region to achieve transparency (stimulated emission and absorption equally likely).  Of course, once the laser achieves threshold, the gain and the electron concentration will clamp equal to their threshold values, as before.  The threshold electron concentration is related to the threshold gain as

\displaystyle \gamma_{th} = \gamma_0 \ln {(n_{th} / n_{tr})}

or

\displaystyle n_{th} = n_{tr} e^{(\gamma_{th} / \gamma_0)}

Finally, we need a rate equation for the electron concentration, similar to the rate equations for the excited state populations we used previously.  In semiconductor physics, this equation is called the continuity equation, and can be written as:

\displaystyle \frac{dn}{dt} = \frac{\eta_i J}{q L} - \frac{n}{\tau} - \frac{\gamma I_{\nu}}{h \nu}

where J is the electrical current density being injected into the gain region by the diode, and \eta_i is the injection efficiency (the fraction of the injected carriers which are ‘trapped’ in the gain region).  (1 / \tau) is the rate of spontaneous and dark recombination of electrons in the gain region; i.e. the total rate of any recombination process except stimulated emission.  \tau is therefore the electron lifetime.

It’s worthwhile to compare this to our old rate equations.  The first term on the right hand side is essentially the pump rate, which we previously called R.  The second term is the spontaneous emission rate plus the rate of any non-radiative recombination that may occur.  The third term represents the stimulated emission rate minus the absorption rate, or the net stimulated emission rate.

As before, we can solve this rate equation for certain situations.  Firstly, we can try to find the threshold pump rate, or in this case the threshold current density.  We consider steady state, and we neglect the optical intensity at threshold, to obtain

\displaystyle J_{th} = \frac{q L n_{th}}{\eta_i \tau}

Once we are above threshold, we know that n = n_{th} and \gamma = \gamma_{th}.  In steady state, then, we have

\displaystyle 0 = \frac{\eta_i J}{q L} - \frac{n_{th}}{\tau} - \frac{\gamma_{th} I_{\nu}}{h \nu}

\displaystyle 0 = \frac{\eta_i J}{q L} - \frac{\eta_i J_{th}}{q L} - \frac{\gamma_{th} I_{\nu}}{h \nu}

Solving for the intensity,

\displaystyle I_{\nu} = \frac{h \nu \eta_i}{q L \gamma_{th}}(J - J_{th})