Numeric solution to the laser rate equations

Numeric solution to the laser rate equations - An example

Steady State Solution

A numeric solution to the rate equations can easily be obtained under steady state conditions as all parameters can be calculated from the carrier density in a straight forward way.

Starting from a set of values for the carrier density one first calculates the gain at each of those densities. If the gain is linear with the carrier density the gain is obtained from:

We will use this expression in this example. Other expressions for the gain can be used as well.

The rate equation for the photon density can be solved yielding:

from which the photon density corresponding to each value of the carrier density can be calculated. The output power of the laser at the mirror with reflectivity R₁ is then obtained using:

Finally one finds the current through the laser from:

The P-I curve of the laser can now be plotted, simply by plotting the values of the output power as a function of the values of the laser current. Show below is the P-I curve on a linear and semi-logarithmic scale for a typical GaAs/AlGaAs edge-emitting single-quantum-well laser. The values for all the laser parameters and the material constants are summarized in the table.

Laser and material parameters
Cavity length	L	300 mm
Laser width	W	3 mm
Laser area	A = L x W	9 x 10^-6 cm²
Non-radiative recombination time	t_nr	100 ns
Bimolecular recombination constant	b	5 x 10^-5 cm²/s
Auger recombination constant	c	6.25 x 10^-18 cm⁴/s
Group velocity	v_gr	9 x 10⁹ cm/s
Confinement factor	G	0.02
Differential gain		1.23 x 10^-9 cm
Transparency carrier density	N_tr	1.23 x 10¹² cm^-2
Spontaneous emission factor	b	10^-5
Photon lifetime	t_ph	2.58 ps
Mirror #1 reflectivity	R₁	0.3
Mirror #2 reflectivity	R₁	0.3
Photon energy	hn	1.49 eV
Lasing wavelength	l	830 nm
Waveguide losses	a	3 1/cm

Calculated values
Threshold carrier density	N₀	2.99 x 10¹² cm^-2
Threshold current	I_th	0.68 mA
Modal gain at threshold	g(N₀)	43 1/cm
Differential efficiency		0.47 mW/mA

Table x

Laser parameters and material constants used to generate the figure below

rateeq.xls - lasrate.gif

Fig x

Carrier and photon density as a function of the laser current

rateeq.xls - lasrate1.gif

Fig x

Optical output power (on a semi-logarithmic and on a linear scale) as a function of the laser current