My design is not EXACTLY like the multi-$1,000 models, but it will let you do a lot of the same work. Figure on 'manual' rather than 'computer-operated'. This means you don't have to learn CAD, or even have a computer. It can also be used as a portable (bonus!). Did I mention it only costs $60?
When I saw the cutting laser that Stephanie Maksylewich made, I knew all that was missing was some way to control where it points and burns, and a variable AC power supply to control the intensity. See the video here.
I decided to mount a cutting laser on a pantograph, (and see below) used for centuries to copy, enlarge, and shrink an existing drawing. But instead of a pencil lead making the duplicate, we'll use the laser to cut or engrave.
I built a variable power supply out of 10 simple electrical pieces, and a pantograph out of four yardsticks and some nuts and bolts.
Remember to always treat this device with respect -- it's an industrial-strength handheld power tool, even if it looks homebrewed.