steady state semiconductor rate equations – Laser diode equation – equations

Laser diode equation

double heterostructure laser diode

\(N_{ph}\) is the coherent photon concentration

Under steady state operation

Rate of electron injection by current I= Rate of spontaneous emissions+ Rate of stimulated emissions

\( \frac{I}{eLWd} = \frac{n}{\tau_r} + CnN_{ph}\)

Under steady state conditions

Rate of coherent photon loss in the cavity = Rate of stimulated emissions

threshold condition

stimulated emission just overcomes the spontaneous emission and the loss mechanisms inherent in \(\tau_{ph}\). This occurs when \(n\) reaches \(n_{th}\)

\( \frac{N_{ph}}{\tau_{ph}} = CnN_{ph} \rightarrow n_{th} = \frac{1}{C\tau_{ph}} \)

threshold condition

coherent radiation gain in the active layer by stimulated emission balances cavity losses plus losses by spontaneous emission

\( N_{ph}\approx 0 \)

\( \frac{1}{eLWd} = \frac{n}{\tau_r} + CnN_{ph} \rightarrow I_{th} = \frac{n_{th}eLWd}{\tau_r}\)

find \(N_{ph}\)

\(I_{th} = \frac{n_{th}eLWd}{\tau_r}\)      \(\frac{I}{eLWd} = \frac{n_{th}}{\tau_r} + Cn_{th}N_{ph}\)

\(N_{ph} = \frac{\tau_{ph}}{eLWd}(I - I_{th})\)

reach laser diode equation

• We want to find optical output power \(P_o\)
• It takes \(\Delta t = nL/c\) seconds for photons to cross the laser cavity lenght \(L\)
• Only half of the photons(\(1/2N_{ph}\)) in the cavity would be moving toward the output face of the crystal
• Only a fraction \((1-R)\) of the radiation power will escape
• \(P_o = \frac{(\frac{1}{2}N_{ph})(Cavity\ Volume)(Photon\ energy)}{\Delta t}(1 - R)\)
• \(Cavity\ Volume = WLd\)
• \(Photon\ energy = hc/\lambda\)

optical gain curve

find optical gain curve

To find the optical gain curve \(g( \upsilon)\) we need to consider the density of states in the CB and the VB, their occupation statistics through the Fermi-Dirac function, and poositions of \(E_{Fn}\) and \(E_{FP}\)(depend on the injected carrier concentrations n).

optical gain curve