You should measure the intensity dependent transmission. A square dependence of the reduction of transmission is indicative of a two photon absorption process.

Hi Cavery! Nonlinear absorption is followed typically by TPA. Simple checking  of  the similar "participation"  to measure the level of nonlinear "transparency" of your samples at Z-scan. The transparency  should reach near 100% if TPA dosn't limit this trend. I mean here that TPA in similar condition is resonance process in the frame of 3-level energy scheme. Resonance TPA is called typically as three-step -absorption. The  nonlinear absorption with nanosecond pulse interaction is followed also the thermal defocusing  and  stimulated  thermal scattering -  third order nonlinear effects. In my list of publications in  past 1980-90 You can find  much similar info with dye solutions -see references in RG. Wish any success!

Conducting measurements at multiple pulse energies should reveal higher order effects. If you conduct the measurement at 3 energies, say, and they all yield the same 2PA coefficient you should be good. But if you notice a trend of increasing 2PA coefficient it could indicate excited state absorption, for example. However, there's also the possibility that linear (one-photon) absorption plus excited state absorption is occurring, which will have the same energy dependence as 2PA. (of course there must be come linear absorption, which you can easily measure with a spectrophotometer). The best way to eliminate this possibility is time resolved (pump-probe) measurements, since the electrons excited via one-photon absorption with remain in the excited state for a while (ps-ns). Alternatively, z-scans with different pulse widths can help determine this as well (see section 2.2.2 linked book chapter)

http://link.springer.com/chapter/10.1007%2F978-3-642-04702-2_4

I agree with Mathew Reichert . In order to distinguish higher order effects in Z scan, you need to perform the intensity dependent study. If nonlinear absorption or nonlinear refraction shows an increasing or deceasing trend with intensity, then higher order nonlinear effects are present. If NLO coefficient remain constant at different input intensity then higher order effects are not present.

honestly, there is almost now way to distinguish between "real high order" processes and a cascade of three photon ones ( For instance two photons are absorbed, one stimulated, then re-absorbed as part of another pair with say the pump one, and a fifth photon is stimulated out.. In some very particular cases some polarization studies can help, but those would heavily depend on all kind of internal symmetries of your sample.   

It was a very nice work by Tokmakoff and Fleming (back in mid 90-s I think) particularly regarding this problem.