emissions from turboa fan engine

I suggest you do some web searching regarding formation of NOx, my memory was that it's primarily temperature driven. Being continuous cycle doesn't directly impact it that I'm aware of.

So a gas turbine running hotter will generate more NOx.

At my house, the trivial PM experiment is looking up when a MDXX/DC9 derivative is taking off and about to go over head. You bet they smoke, but less than they used to.

I put my boiler's pilot together without the gas orifice. The rich flame literally choked it with soot, and made a CO hazard until I fixed it.

Another easy experiment is lighting a welder's torch, acetylene first. The floating black soot from a continuous flame is I think one source of the carbon used in carbon fibre. Turning the O2 on makes the soot go away. My Mercedes 240D can lays a little smoke and soot because it's probably running on original 34 year old injectors, that only worked so well to begin with. Newer heads, injectors, arrangements and Common rail diesels with multi-event piezoelectric injectors smoke a lot less or a whole lot less. Stoich or lean 02 is required, and good atomomization/evaporation is required for a clean burn, because a burning droplet has a boundary layer of combustion product and less O2 around it. In a jet engine, combustion chamber design, fuel injector design, fuel control and maintenance all probably matter to limit PM/soot.

As for NOx, this looks like an interesting paper. It says that the flight profile can be optimized wrt NOx production:

The thermo-chemistry in a turbine engine combustor is quite complicated. But in general, NOx formation does not occur below flame temperatures of approx. 2800degF.

Higher combustion temperature and NOx emissions is why they no longer sell air-cooled car engines like the Volkswagen engine.

Piston aircraft engines, being air cooled, also have higher NOx emissions, and the avgas they burn still contains some lead.