Notes on PDH waveform

3867.txt

A PDH error signal can be calibrated (into Hz per volt) simply by
observing the peak-to-peak amplitude of the PDH signal as the
cavity is swept through resonance.  Here's the math:

An impedance-matched, lossless cavity has amplitude reflectivity of

    r_c(f) = i f / (i f + pcav)

where pcav is the cavity pole, and f is the detuning frequency (i.e.
the difference between the frequency of the light and the resonant
frequency of the cavity).

The PDH error signal near resonance is well approximated by the
imaginary part of r_c(f).

    Im{ r_c(f) } = f pcav / (f^2 + pcav^2)

This function has a maximum value of 0.5.  We'll normalize our PDH
error signal so that it has maximum/minimum values of +/- 1.

    PDH(f) = 2 f pcav / (f^2 + pcav^2)

The calibration of the PDH error signal in lock is the first derivative
of this, taken around f=0.  It turns out that this value is just the inverse
of the cavity pole:

    (d/df) PDH(f) = 2/pcav    at f=0

So we just measure the peak-to-peak voltage of the PDH error signal,
divide by two to convert from peak-to-peak to amplitude, and then
multiply by the slope (2/pcav) for a normalized PDH signal:

    Error Pt Calibration = (PkPk/2) * (2/pcav)  in Volts per Hz

(Note that the slope of the error signal through f=0 is twice the "average"
slope taken directly through the turning points.)

Finally, it also turns out that the maximum and minimum values of
the PDH error signal -- the 'turning points' -- occur at f=+/-pcav (for
an impedance matched cavity):

    (d/df) PDH(f) = 0             at f = pcav

This can be cross-checked with the FSR or the PZT calibration as
a sanity check.

-- 
Tobin Fricke 2013-02-13