The designer of the dual line tube, Dr. Knollenberg, says 2P4 -> 1S4 for this transition and mentions it in his paper "Prospects for the Helium Neon Laser Through the End of the Century*" only he says 610 nm, and does not list the specific wavelength...
Knollenberg is a founder of PMS, which Spun Off REO... He ought to know, and he mentioned this in 1987....
It does not explain a 0.5 nm discrepancy between what Sam and others have measured, and the assignment of the line, but if the tube designer says 2P4->1S4 in Paschen Notation, I'm inclined to stick with it. He was actively trying to make as many different HENE lines lase as possible, and he was good at it, for commercial reasons. Kaiser's spectrum places it right over 610, so given instrument error, 609.61 is about right. Bloom's new measurement is similar.
It took me more than two years to run this down... And I'm about to try to get in touch with a retired REO employee who would know...
*in SPIE Vol 741, Design of Optical Systems Incorporating Low Power Lasers, (1987)
Damn hard to get book...
I really had to work thru a lot of reference material to get this, and when you search Neon 2P4 1S4 on Google, if you have a university level library access, you find another paper measuring the sidelight of this transition by another scientist..... That scientist shows no population inversion at 609.61 but he has 6328 lasing. Dr. Knollenberg says that he has to partially suppress 612 nm using careful optics design to get dual line lasing with 604. So it is possible that you need a very carefully tailored optic to see the 609.61 line, because of gain competition between lines. As well his having to use a partially multimode or multimode tube, Which explains why no one else shows this as a working laser transition.
In the paper, Dr. Knollenberg also explains a lot about the external mirror particle sensing tubes, both the "Active" one Brewster Tube, and the "Passive" three mirror laser, as well as mentioing seeing seven lines in the three mirror laser... He then expanded the three mirror laser to IR as well, but he does not go into detail on that.
He also explains a lot of the mechanics of dual line tubes, as well as what you have to do to get 543 and 594...
There is the possibility of an error in Moore's wavelength tables, which is what everybody uses to find transition wavelengths. But at this rate, it might take me another six months to run this down to more digits, if I can at all... Right now it is 609.616 or 609.618 nm.
But the line is documented in a published work... And yes, Sam knows, I called him.
Steve