Warning! Science rant:
A projected, mid-air, self-perpetuating laser system is absurd. A laser requires a power source, a lasing substance, and a lens system. Air is only a fuel in nuclear fusion. The system would have to ignite fusion [1, 1970], (can that level of symmetry even be achieved using single-origin airy beams [2, 2007]?), and then contain it solely using inter-plasma magnetic fields, which are normally turbulent [3, 2012]. An optical trap, (in this case, a 3d grid of laser beams whose cells are only large enough to contain about 1 oxygen or nitrogen molecule [4, 1997]), could form an artificial crystal mid-air, but could that ever be shaped into a lensing meta-material [5, 2012]?
If so, assuming the projector could ignite and contain fusion, then lens the resulting radiation to create a contained, lasing region mid-air [6, 2014], it would need to generate additional air-lenses whose configuration could reproduce said system in its entirety, including the system-reproducing array. It sounds recursive, but this may not be theoretically impossible: After all, every living cell contains sufficient information to reproduce itself, but it at least implies staggering complexity. Then, at the first, smallest deviation, (perhaps a mutation, to maintain the biological analogy), the system would collapse. Although, on the bright side, it probably would not ignite Earth's atmosphere.
But, that's just my two cents.
Sources
[1] Holcomb, Robert. "Controlled Fusion: Plasma Heating With Lasers." Science 167.3921 (1970): 1112. Publisher Provided Full Text Searching File. Web. 20 Aug. 2015.
[2] Siviloglou, G A, et al. "Observation Of Accelerating Airy Beams." Physical Review Letters 99.21 (2007): 213901. MEDLINE. Web. 20 Aug. 2015.
[3] Mondal, Sudipta, et al. "Direct observation of turbulent magnetic fields in hot, dense laser produced plasmas." Proceedings of the National Academy of Sciences of the United States of America 2012: 8011. JSTOR Journals. Web. 20 Aug. 2015.
[4] Ashkin, Arthur. "Optical Trapping and Manipulation of Neutral Particles Using Lasers." Proceedings of the National Academy of Sciences of the United States of America 1997: 4853. JSTOR Journals. Web. 20 Aug. 2015.
[5] Kundtz, Nathan, and David R. Smith. "Extreme-Angle Broadband Metamaterial Lens." Nature Materials 9.2 (2010): 129-132. Academic Search Complete. Web. 20 Aug. 2015.
[6] Laurain, Alexandre, Maik Scheller, and Pavel Polynkin. "Low-Threshold Bidirectional Air Lasing." Physical Review Letters 113.25 (2014): 253901-1-253901-5. Academic Search Complete. Web. 20 Aug. 2015.
(End of science rant.)