Since this is a single mode diode, you should get a short focal length aspheric lens for it. Ones like the G2 can be had on eBay for $8.00 or less and you should get an increase in power of ~20%. Since they are short focal length lenses, the beam diameter remains small when compared to a longer focal length lens like Sanwu's G7. Search for 405nm lenses and look for single element ones on eBay. These lenses are always housed in a brass barrel which is discernible from the black housings of multi-element lenses.
Feebay has them and they will say G2 aspherical lens , it should say 10mm barrel length single lens . Thats what you want.. Try techhood or existotem on line . Or hunt for them they aren't a rare lens. If I knew that you were going to buy the 3E I would have sold you a G2 I have in stock and shipped it with the copper sink.
Yes, and it is black in color. The short focal length aspheric lenses all come in a brass housing. Many are already AR coated for 405nm. But, it never hurts to check what kind of AR coating a lens has that you are ordering from a seller on eBay.