Multiple off axis transverse modes, a horrible mode structure, both transverse mode and longitudinal modes. As well as a very broad wavelength spread, are hurting you. Trying to get good fringes with a modehopping laser that may have a 2-3 nm wide spectrum is very tough.
Can you get your hands on a HENE? Very narrow.... Maybe ~10 Ghz total line-width on a good day with a short cavity.
Or a Tem00 Long cavity DPSS green? ~1.1 nm FHWM.
Anything other then a direct diode laser at that point.
If I was STUCK with 406 or 445 blue for experimental reasons with a interferometer... Well, I'd try to find another way..
If there is no escape, try adding some Littmann-Metcalf style grating feedback to your blue diode. Do not feed 100% of the diode's energy back into itself, it will blow the faucet off the diode. But you can narrow the spectral width by grazing incidence grating feedback and adjusting current. That can be further improved by controlling the diode temperature. That will not help you with the beam profile problems. It will however sharpen the fringes greatly. Any technique you use to clean up a multimode will cost you dearly in output power.
In fact, I'd tune the laser with the grating. Then I'd up collimate, kill the nasty edges of the beam with a circular mask, then try to spatial filter. Or use anamorphic prism correction to circularize the beam, mask it and spatial filter.
By doing the above steps, I'm basically building a clone of Coherent's "Cube" series of lasers. That is a lot of work to perfect. 405 Cubes without the extra wavelength stabilization show up on Ebay for reasonable costs.
Your basically trying to make a silk purse out of a sow's ear, when using a multimode blue/uv diode.
Books you might want to find to read up on theory:
The Late Tony Siegmans' book, a bit tough to read:
Lasers: Anthony E. Siegman: 9780935702118: Amazon.com: Books
William Silfvast's book, wonderfully easy:
Laser Fundamentals: William T. Silfvast: 9780521541053: Amazon.com: Books
This gentleman is the master of the Homemade ECDL, a good read:
Laser-Projects Page
Laser-Projects Page
Before I went to all that trouble, I'd do some math:
There is always the chance that you need to change pinhole size or get a much better focusing lens. A few rings are pretty normal with even the best spatial filter. If I had followed the pinhole sizing requirements on the CVI Melles Griot or Newport App notes, and I still had lots of rings, I'd start narrowing the wavelength and "pre-cleaning" the beam.
I wish you patience and good luck.
Steve