Special relativity and diffracting beams

[calacone]

Say there is a beam of light at 1 um (f=300 THz), and there is a particle traveling at constant speed along the axis of the beam, which is, say, a few mm or cm in diameter. Say the particle is going fast enough so that the shift brings the light down to the MHz level (v/c ~ 1- O(10^-16)). Then the wavelength as measured by the particle will be something like a few hundred meters. But how can this happen, if the beam is localized in the transverse direction at the mm or cm level?

If it is a gaussian beam, then I think the beam divergence would make the beam appear consistent with the wavelength observed by the particle, even if the beam divergence is small in our rest frame (I haven't checked this). What about diffraction free beams? Or other odd arrangements that allow the particle to see a highly localized spot at MHz frequencies (say, focusing mirrors spaced periodically along the axis of the beam path but offset to either side, so that the beam periodically intercepts the path of the particle, or something like this).

(sorry in advance for not checking the math, nor checking previous forum posts for similar questions)

 

[anuttarasammyak]

Then the wavelength as measured by the particle will be something like a few hundred meters. But how can this happen, if the beam is localized in the transverse direction at the mm or cm level?

Long wavelength and small bundle beam are compatible.

 

[Sagittarius A-Star]

But how can this happen, if the beam is localized in the transverse direction at the mm or cm level?

The Abbe diffraction limit assumes an isotropic wavelength. The relativistic movement of a reference frame in longitudinal direction does not change the wavelength in transverse direction.

If the beam is not perfectly collinear, the beam has a wider angle in the rest-frame of the moving particle, due to the (inverse) relativistic headlight effect (aberration).