Here are some pics showing the proper orientation of a plano-convex lens (curved side towards the laser) and the effect of different focal lengths.
K40 machines typically use Plano-convex lenses with a 50.8 mm (2") focal length as it offers a good balance with a small spot size for engraving and a reasonable depth of field (DOF) for cutting.
Trying to wrap my brain around the lens in terms of aligning our lasers (how much, if any, and what type of misalignment is tolerable).
If the lens is at an angle to the incoming beam the focused spot will still be in line with the unfocused beam, not with where the lens is “pointing”?
If the beam is hitting the lens perpendicular but off center the focused spot will still be in line with the unfocused beam, or will the lens “pull” the spot towards its center, or off to the side?
I see it working this way, if the beam is perpendicular and the full width(covers all 7 vectors) into the lens then the point of most energy is where the most vector lines intersect.
If your beam is a smaller diameter but in the center of the lens and perpendicular then follow the vectors where are incident with the beam and where the most of them intersect is your focal point.
So a properly oriented lens will have the same focal point if the beam is of a smaller diameter than the lens, is straight on or over to one side. BUT, the burn-through hole will not be straight through the material if there’s much thickness to it. Beam vectors at one edge of the lens will cause the hole burned to enter at the top of the material at the focal point but it will exit the material laterally shifted to the opposite side of the lens from where it enters the lens. This is usually not desired.