The beam will never become "less Gaussian", in the sense that the beam will always obey I(z) = I(0)exp(-ar(z)^2). However what will change is the value of a, depending on the nature of the optical path length and the phase relationships of the beam vis a vis the power variations along the path. If the various effects Bob mentions (temperature gradients etc) imbue optical power to the beam path, then the Gaussian nature alters, but is still essentially Gaussian. This can be simulated in a systems analasys methodology by having a "box" in the system to simply model all the phase changes caused by power along the beam path. That is:BobH wrote:The illumination will become "less gaussian" at the recording plane if the beam is being steered around by "rarefactions in the air" (like temperature gradients or air flow, not refraction through dust particles), or the other things mentioned above that can redirect the beam slightly over time. The beam itself remains gaussian, like Don says.
phase of beam at some (x,y) on it's surface -> beam surface acted on by a phase transform -> alteration of phase proifile at (x,y)
If you then integrate over the beam surface (or use Gausses Theorerm in phase space), you get the alteration of the 'a' in the Gauss exponent.
For example, if the (Gaussian) beam is expanded onto a collimated mirror from a distance f away, the beam will copntinue to propagate as a planar wavefront, but the intensity variation will still follow the exp(-ar^2) law. The only way to diminish the Gaussian nature of the beam is to attenuate it as a function of the beam radius. That is, the attenuation A =(alpha)* f(r). If (alpha)f(r) = exp(-ar^2), for a specific value of a, then the beam converts from a Gaussian profile to a constant profile. By appropriately "tuning" both the value of alpha and the value of f(r), you can get pretty much any profile you desire, so long as the profile is "well behaved". For any odd profiles that don't follow a given function, you'd have to replace the f(r) with a Fourier series of the desired functional crross section.
Whether or not the beam will (approximately) seem to lose it's Gaussian nature at the recording plane, depends on the effects mentioned, ie the optical power created by temperature gradients etc, and the position of the recording plane. For example, if the recording plane is about 10 mm from a 100X objective, the beam will probably will retain it's Gaussian characteristic (depending, of course, on the value of lambda) across a recording plane dimension of 4 x 5 in. Of course, such stringent requirements are rarely met in a display hologram.