ALL light diverges. Laser light is no different.
You can think of the light coming from your diode as an "X." It is actually a solidly filled cone, but it is easier to visualize the X.
When the light leaves the diode it begins to diverge (spread out). The lens changes the direction of the divergence from spreading to coming together.
Now the X is starting from the edge of your lens. The "center" (crossing point) of the X is your focus point. When you focus (the center of the X) very close to your lens, the light from the edge of the lens is at a very steep angle to get them together quickly. This is great for burning, but the light leaving the focus point is still traveling at that same steep angle. This causes the spot to get very large very quickly.
The farther away from the lens that you focus the light, the more shallow the angle of divergence. Now visualize the center of the X being very far away. It is still moving together, but it is at a very shallow angle. The beam will appear to stay at the same width as the front of the diode for a long time. This is referred to as "focused to infinity."
Most green lasers you have seen are focused to infinity. They "look like" they are not diverging only because the X is very long and narrow. Is is also why it is more difficult to burn stuff with a green laser. The light is not focused (center of the X) close enough to burn stuff.
If you focus your violet laser very far away, the beam will look thin all the way away. If you focus it up close, it will be big far away.
Peace,
dave
**EDIT** Here's a simple diagram