I don't think laser propulsion is going to work as far as creating linear increases in velocity with time for any sustained period of time. I think it's impossible - energy conservation won't allow for it. I believe that the rate of velocity increases will progressively slow.
It's not that a constant thrust laser couldn't potentially make this happen, it's just that it would require quadratically increasing power consumption to do it. This is analogous to automobile acceleration and how fuel consumption is mostly straightlined with kinetic energy, as force gets applied to increasingly greater distances. Velocity is the result.
Kinetic energy is inherently incompatible with classical mechanics in that it exists hand-in-hand with relative time, and without relative time, kinetic energy doesn't happen. It is a failing of 19th century physics that it went ahead and constructed a model in which kinetic energy existed alongside absolute time rather than fully consider that time may be relative.
But we need not repeat the failures of previous times today. Any discusion of kinetic energy, classical kinetic energy, that doesn't mention relative time is, at best, incomplete.
Time dilates mostly at the quadratic at low velocities. Time literally passes slower for an object travelling at classical velocities by an amount that progresses quadratically with velocity increases.
That quadratic is this quadratic.
See https://www.quora.com/Since-time-is-relative-how-much-time-has-passed-for-Voyager-1-in-comparison-to-Earth for a great example of this. The symmetries of special relativity are such that this very real dilation effect can often be lost in the minutiae of the details of reference frame specific math. But it is what connects various reference frames that most interests me, and the model in my head is that it is the relative rate of the passing of time that does the connecting.
I believe that time is real and that it is real in a way that means it actually passes. And the motion of electromagnetic radiation is itself the very essence of time passing. This is an unorthodox viewpoint - probably one worthy of elevating me to crank status - but I am convinced of it, and it is very obvious to me that the quadratic of kinetic energy is in fact the very small amounts of time dilation occurring to this passing of time.
It has to be this way since the infinitesimal of acceleration is light speed invariant. Only by a changing of the relative rate that time is passing can an accelerating object's invariant perspectives of the speed of light both before and after acceleration be made sense of.
Kinetic energy's quadratic is not explained by Doppler redshift. That effect is a linear one with respect to velocity, and it vanishes when the relative velocity of an object returns to zero, with no quadratic energy transfer to account for. This effect is a natural partner of the second term in the Lorentz transformation of time.
Conversely, the quadratic energy increases that correspond to those same linear velocity increases associated with Doppler are in fact "real" in a way where the energy needs to transfer, or be accounted for, when the attained relative velocity is again negated. The quadratic is "squarely" rooted in the Lorentz transformation for time's first term - gamma.
See specifically this comment to another answer by the asker of this very question for context on what we are really pursuing here:
To focus the question a little more: suppose the rocket starts some distance from Station A, at rest relative to each other. After T seconds of constant acceleration, the rocket impacts station A and it's kinetic energy is released, so it's not just theoretical. The amount of KE at the point of impact seems to increase quadratically with acceleration time, tho it's produced by a constant-power photon engine whose energy input/output increases linearly with acceleration time. All this at tiny velocities with negligible relativistic effects. – Zeph
The quadratic isn't explained by Doppler, it's explained by time dilation. Time dilation that is empirically proven to exist in a real way, to the best of my knowledge.
This will work only if quadratically increasing amounts of fuel are consumed to apply force over the ever-increasing distances that the acceleration is occurring over, corresponding to an ever-increasing number of points in space over which infinitesimal amounts of time dilation accrue.