Pulsed source in Finite Difference Time Domain simulations

I want to simulate a resonator cavity with a medium inside with FDTD method. The medium has a resonant frequency $\omega_R$. I want to find ratio of outgoing $P_{out}$ power to the incoming power $P_{in}$ at the frequency $\omega$:

$A(\omega)= \frac{P_{out}}{P_{in}}$

Currently, I use a continuous wave (CW) source at a fixed frequency. I slowly ramp the amplitude of the source and wait until the the steady state. I added a small damping to the medium. But even with the damping the resonance causes numerical problems, since the amplitude becomes very large at the resonance.

The question:

Is possible to use a pulsed source and still be able to obtain $A(w)$? This way the amplitude, will not become too large and numerical problems will be avoided. I am worried that the steady state is not achieved when the pulse is used as a source.

Appendix:

The simulations are in 2D. The medium is a magnetized plasma. The geometry is shown in the figure below.

• "media" is the plural of "medium" – flippiefanus Jun 18 '18 at 10:49

It sounds to me like you are interested in the resonator quality factor $$Q$$ (please correct me if I'm wrong), which can be roughly defined as $$Q = \frac{\mbox{stored energy}}{\mbox{energy dissipated per period}}.$$
Since you are using a pulsed source, I guess you want to use the ring-down method to estimate $$Q$$: switching-off the driving signal leads to a decay of the stored energy (due to the dissipation) and the decay time $$\tau$$, the $$1/e$$ time, is proportional to $$Q$$, $$Q = \tau\cdot\omega_R,$$ with the resonance frequency of the system $$\omega_R$$.