|MadSci Network: Engineering|
An intense-enough laser can not only break any chemical bond, but can even go so far as to strip the electrons off atoms and create a plasma.
Although, I suppose there is a difference between "cut" and "melt"... A long-pulse laser will merely heat up the diamond and melt it, or perhaps send a strong shock through it and break the thing. But a short-pulse laser (sub-picosecond) will make a nice little hole -- right now the best way to make high-precision, clean holes is by using a short-pulse laser. You could use a short pulse laser to cut a diamond, although I'm not positive it's actually been done.
As for what intensities you need... You can turn things into a plasma around 10^12 Watts per square centimeter. This is tough with a long-pulse laser, but with sub-picosecond pulses you can get these intensities with a laser energy of less than a milliJoule per pulse.
Here's an article on laser cutting at Livermore Labs.
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