The nonlinear optics (NLO) often offers a deeper penetration depth then the linear optics techniques. There are still some analysis problems to study with.  Typical used NLO consists of 2nd-harmonic generation (hyper-Rayleigh scattering), 3rd-harmonic generation, sum/difference generation, etc.   There are also some other non-frequency mixing processes: Kerr effect, multi-photon absorption, stimulated Brillouin scattering, etc. 

1.  The modeling of fluorescent ultrasound modulates optical tomography could be readily combined with the frequency-mixing nonlinear optics. The idea is to study the effects of ultrasound modulation interacting with the frequencies of light. [Westervelt model]

2. The photoacoustic tomography and its variants could be ready to combine with the multi-photon absorption non-linearity. The uniqueness of a positive solution seems true but the fixed point approach does not fully solve that. [partly solved]

3. In multi-photon imaging, it seems an angularly singular source permits a unique solution for small scattering since the scattering-free case can be solved explicitly. For small scattering, it remains to show the uniqueness through a fixed point argument. [two photon case solved]

4. As a follow-up for 3. Does a general multi-photon absorption (MPA) case also work?  [Intuitively, this should work in general if a nonlinear term is a monotone not introducing ambiguity. The growing condition still applies.]

5. It seems the proof for convergence of nonlinear MPA radiative transfer to nonlinear MPA diffusion is not proved.  The Hilbert expansion seems working. [typical cases solved if boundary source is isotropic, Milne case seems OK as well.]

6. Derive the transport model (if it is possible) for the second harmonic generation (SHG) propagation in random media for weakly coupling regime. 

7. How about the same question for Kerr nonlinear effect (with random fluctuations). The Born approximation (linearization regime) seems to work, but it violates the purpose of the nonlinear wave if the solution is too close to the linear version.