These past two weeks, I've been working on simulating a basic FabryPerot cavity. I finished up a simulation involving static, nonsuspension mirrors last week. It was supposed to output the electric field in the cavities given a certain shaking (of the mirrors), and the interesting thing was that it outputted the real and imaginary components seperately, so I ended up with six different bode plots. Since we're only interested in the real part, bodes 2, 4, and 6 can be discarded (see attachment 1). There was a LOT of splitpeak behavior, and I think it has to do either with matlab overloading or with the modes of the cavity being very close together (I actually think the first is more likely since a smaller value of T_1 resulted in actual peaks instead of split ones).
At any rate, there really wasn't much I could improve on that simulation (neither was there any point), but I attach the subsystem governing the electric field in the cavity as a matter of academic interest (see attachment 2). So I moved onto simulations where the mirrors are actually suspended pendulums as they are in reality.
A basic simulation of the suspended mirrors gave me fairly good results (see attachment 3). A negative Q resulted in a phase flip, detuning the resonance from the wrong side resulted in a complete loss of the resonance peak, and the peak looked fairly consistent with what it should be. The simulation itself is pretty bare bones, and relies on the two transfer functions P(s) and K(s); P(s) is the transfer function for translating the force of the shaking of the two test masses (lumped together into one transfer function) into actual displacement. Note that s = i*w, where w is the frequency of the force being applied. K(s), on the other hand, is the transfer function that feeds displacement back into the original applied forcebased shaking. Like I said, pretty bare bones, but working (see attachment 4 for a bode plot of a standard detuning value and positive Q). Tweaking the restoring (or anti restoring, depending on the sign of the detuning) force constant (K_0 for short) results in some interesting behavior. The most realistic results are produced for K_0 = 1e4, when the gain is much lower overall but the peak in resonance gets you a gain of 100 in dB. For those curious as to where I got P(s) and K(s), see "Measurement of radiationpressureinduced optomechanical dynamics in a suspended FabryPerot cavity" by Thomas Corbitt, et. al.
I'm currently working on a more realistic simulation, with frequency and force noise as well as electronic feedback (via transfer functions, see attachment 5). The biggest thing so far has been trying to get the electronic transfer functions right. Corbitt's group gave some really interesting transfer functions (H_f(s) and H_l(s) for short; H_f(s) gives the frequencybased electronic transfer function, while H_l(s) gives the lengthbased electronic transfer function), which I've been trying to copy so that I can reproduce their results (see attachment 6). It looks like H_l(s) is a lowpass Butterworth filter, while H_f(s) is a Bessel filter (order TBD). Once that is successful, I'll figure out what H_f(s) and H_l(s) are for us (they might be the same!), add in degrees of freedom, and my first shot at the OSEM system of figuring out where the mirror's position is.
