  We study the optical pulse response of a superconducting half-wavelength
coplanar waveguide (CPW) resonator. We apply a short optical pulse to the
center strip of the CPW resonator, where the current distribution shows
antinodes or nodes for different resonance modes, and measure the frequency
response. We develop a time-dependent variable inductance circuit model with
which we can simulate the optical pulse response of the resonator. By fitting
this model to experimental data, we extract the temporal kinetic inductance
variations, which directly reflect the quasi-particle recombination with time
and diffusion in space. We also retrieve the spatial size of the quasi-particle
distribution and the quasi-particle diffusion constant. Our study is very
useful for the design of photon-counting kinetic inductance detectors, and the
method developed in this work provides a useful way to study the quasi-particle
dynamics in the superconductor.
