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Relaxation Oscillation of Semiconductor Laser

Figure 6: Left: Changes in the amplitude A vs. time. Right: Changes in the number of the carriers n vs. time.

A desirable feature of a laser is the constant amplitude, shown in Figure [6]. Right after the laser turns on, the amplitude varies for a while and then gets stabilized to a constant. We call the frequency before the laser gets stabilized the relaxation oscillation frequency, which we are going to calculate in this section.

The reason we want to obtain the value of the relaxation oscillation frequency is that physically, we easily get resonance when we add a frequency similar to that of the system; in resonance we can obtain large responses.
When we add the term for the TM injection, we will get various different behaviors from the SRE system, depending on the injection strength and the detuning between the injection and the free running laser. When their detuning has a value similar to the relaxation oscillation frequency of the laser, we often observe interesting non-linear behavior.

To calculate the relaxation oscillation frequency( tex2html_wrap_inline1512 , hereafter), it is convenient to use the laser equation in terms of amplitude.


where tex2html_wrap1518 . tex2html_wrap1519 and tex2html_wrap1520 have damped oscillations as seen in Figure [6]. The frequency changes along with time. The relaxation oscillation frequency is the frequency seen when the system relaxes close to its stable state. So we will find a steady state solution and add a small perturbation in order to calculate the relaxation oscillation frequency. First, when tex2html_wrap1521 , we have tex2html_wrap1522 . We can also find the steady state solutions other than this trivial one by letting tex2html_wrap1523 and tex2html_wrap1524 . Substituting tex2html_wrap1522 , the equations can be written as


From Eqns. (31) and (30) , we obtain the following solution.


For ease of computation, that is, in order to transform this complicated system into a simple 2nd order ODE, we changed the variables into non-dimensional quantities.


We can also write the steady state solutions in terms of tex2html_wrap1526 and tex2html_wrap1527 .


where tex2html_wrap1528 and tex2html_wrap1529 . We differentiate tex2html_wrap1530 and tex2html_wrap1531 with respect to tex2html_wrap1527 and obtain the following differential equation.


Let tex2html_wrap1533 and tex2html_wrap1534 be small perturbation terms and add them to tex2html_wrap1535 and tex2html_wrap1536 respectively.


Substituting Eqns. (34),(35) into Eqns. (32),(33), the equations are rewritten as


In the equations, since we assume p,q are small, tex2html_wrap1537 is so small that it can be neglected. Substituting tex2html_wrap1536 in, we obtain


Then we differentiate Eqn. (37) with respect to tex2html_wrap1527 and substitute p from Eqn. (37) again to obtain the 2nd order differential equation of tex2html_wrap1534 with respect to tex2html_wrap1527 .


The equation above is dimensionless. But the value of tex2html_wrap_inline1512 has tex2html_wrap_inline1582 dimension. So We write the equation in terms of t again to obtain an ODE in terms of t.


The solution for the above ODE is


where tex2html_wrap_inline1512 is


To calculate a numerical value, we put tex2html_wrap1542 and tex2html_wrap1543 . The tex2html_wrap_inline1512 for our model is


next up previous
Next: Periodic Behavior in the Up: Periodic (Transverse Magnetic) Injection Previous: Periodic (Transverse Magnetic) Injection