The two-stroke diesel cycle goes like this:
1. When the piston is at the top of its travel, the cylinder contains a charge of highly compressed air. Diesel fuel is sprayed into the cylinder by the injector and immediately ignites because of the heat and pressure inside the cylinder. This is the same process described in How Diesel Engines Work.
2. The pressure created by the combustion of the fuel drives the piston downward. This is the power stroke.
3. As the piston nears the bottom of its stroke, all of the exhaust valves open. Exhaust gases rush out of the cylinder, relieving the pressure.
4. As the piston bottoms out, it uncovers the air intake ports. Pressurized air fills the cylinder, forcing out the remainder of the exhaust gases.
5. The exhaust valves close and the piston starts traveling back upward, re-covering the intake ports and compressing the fresh charge of air. This is the compression stroke.
6. As the piston nears the top of the cylinder, the cycle repeats with step 1.
From this description, you can see the big difference between a diesel two-stroke engine and a gasoline two-stroke engine: In the diesel version, only air fills the cylinder, rather than gas and air mixed together. This means that a diesel two-stroke engine suffers from none of the environmental problems that plague a gasoline two-stroke engine. On the other hand, a diesel two-stroke engine must have a turbocharger or a supercharger, and this means that you will never find a diesel two-stroke on a chain saw -- it would simply be too expensive.