  We present a method for the measurement of small optical absorption
coefficients. The method exploits the deformation of cavity Airy peaks that
occur if the cavity contains an absorbing material with a non-zero
thermo-refractive coefficient dn/dT or a non-zero expansion coefficient ath .
Light absorption leads to a local temperature change and to an
intensity-dependent phase shift, i.e. to a photo-thermal self-phase modulation.
The absorption coefficient is derived from a comparison of time-resolved
measurements with a numerical time-domain simulation applying a Markov-chain
Monte-Carlo (MCMC) algorithm. We apply our method to the absorption coefficient
of lithium niobate (LN) doped with 7mol% magnesium oxide (MgO) and derive a
value of alphaLN = (5.9 +/- 0.9) *10^-4/cm . Our method should also apply to
materials with much lower absorption coefficients. Based on our modelling we
estimate that with cavity finesse values of the order 10^4, absorption
coefficients of as low as 10^-8 /cm can be measured.
