There is no single method.

It all depends on what would you like to study in the nanomaterials which I presume is in a powdered form.

Its physical structure, chemical microstructure, crystallographic structure, or any specific reaction-response from the nanomaterial upon irradiation with specific photons.

Photo response is a general word, you have to be a little more specific on your interest.

each method provides information on a specific aspect, e.g., IR spectroscopy, Far-IR spectroscopy, Raman spectroscopy, X-ray diffraction, XPS, and so on .... there are many different methods.

Subhashree Das

One can check the photo response of nanomaterial by checking its catalytic activity in the presence and absence of light. Here is the articles that you can referee to study the photocatalytic activity of nanomaterials.

Article Photo Enhanced Detoxification of Chromium (VI) by Formic Aci...

Also, you can enhance the recyclability of nanomaterials by more than 100 cycles by immobilizing them in beads if you are going to use them as a catalyst. Here is a link to an article that suggests and explains how to enhance the recyclability of nanomaterials.

"Immobilisation of palladium nanostructures in polyethersulfone beads: recyclable catalyst for chromium(VI) remediation"

https://www.publish.csiro.au/en/en19035

In case you are looking for a response from your nano materials:

Nanomaterials in powders can be studied by photoluminescence spectroscopy, and

1) the nano materials can be suitably doped with rare earth ions, and can study down-conversion and light upconversion effects. Lots of applications under UV and IR light irradiation on nano materials.

2) For Bio imaging applications you need very fine nano materials (< nm).

3) You can form bulk specimens (ceramics) using your nano materials, and study photo conducting effects, and gas sensing effects with UV light irradiation.

4) Dynamic light scattering studies to study the nanomaterials