Laser speckles relative to laser? I know there are subjective laser speckles forming in the eye as well as objective laser speckles forming on the illuminated surface. What I wonder is, are there any speckles that are constrained to the location of the laser?
Meaning that if I would move the laser by 1mm perpendicular to the beam’s direction, the speckle pattern would move 1mm in that direction too?
 A: The "traditional" subjective and objective speckles do not work as requested. The subjective speckles remain anchored to the rough surface which diffuses the light. Instead the objective speckles change shape when the beam is moved, but do not translate.
I do not know if this can help, but, in the case of subjective speckles, by moving the surface, the speckles move in the same way (same direction and speed).
I think that there is no way to do exactly this:

if I would move the laser 1mm perpendicular to the beam directions, the speckle pattern would move 1mm in that direction too?

However, a similar effect can be created by using a converging or diverging beam. As the scattering elements are hit be different parts of the beam, they "feel" different direction of the light, hence generating a translation in the generated pattern.
Alternatively, it is possible to connect the change of the position of the beam with a change of the speckle pattern. For example, the objective speckles generated by a couple of spots (two beams impinging on the scattering surface) have a correlation function which tells us the distance between the two spots. Moving one of the beams, also moves the corresponding spot and changes the correlation function of the speckles.
By means of heterodine detection of the speckle pattern, it is possible to completely reconstruct the spot that generates the objective speckles. This includes the position of the of the spot, connected to the position of the beam. But, in this case, the properties of the speckle pattern are not exploited.
