A photon rocket is a hypothetical rocket that uses thrust from emitted photons (radiation pressure by emission) for its propulsion.^{[1]}
The standard textbook case of such a rocket is the ideal case where all of the fuel is converted to photons which are radiated in the same direction. In more realistic treatments, one takes into account that the beam of photons is not perfectly collimated, that not all of the fuel is converted to photons, and so on; see e.g. nuclear photonic rocket.
In the Photonic Laser Thruster, collimated photons are reused by mirrors, multiplying the force by the number of bounces.
Speed
The speed an ideal photon rocket will reach, in the absence of external forces, depends on the ratio of its initial and final mass:

v = c \frac{\left(\frac{m_{i}}{m_{f}}\right)^{2}1}{\left(\frac{m_{i}}{m_{f}}\right)^{2}+1}
where m_{i} is the initial mass and m_{f} is the final mass.
The gamma factor corresponding to this speed has the simple expression:

\gamma = \frac{1}{2}\left(\frac{m_{i}}{m_{f}} + \frac{m_{f}}{m_{i}}\right)
Derivation
We denote the fourmomentum of the rocket at rest as P_{i}, the rocket after it has burned its fuel as P_{f}, and the fourmomentum of the emitted photons as P_{\text{ph}}. Conservation of fourmomentum implies:

P_{\text{ph}} = P_{i}  P_{f}
squaring both sides (i.e. taking the Lorentz inner product of both sides with themselves) gives:

P_{\text{ph}}^{2} = P_{i}^{2} + P_{f}^{2}  2P_{i}\cdot P_{f}
According to the energymomentum relation, the square of the fourmomentum equals the square of the mass, and P_{\text{ph}}^{2}=0 because all the photons are moving in the same direction. Therefore the above equation can be written as:

0 = m_{i}^{2} + m_{f}^{2}  2 m_{i}m_{f}\gamma
Solving for the gamma factor gives:

\gamma = \frac{1}{2}\left(\frac{m_{i}}{m_{f}} + \frac{m_{f}}{m_{i}}\right)
References

^ McCormack, John W. "5. PROPULSION SYSTEMS". SPACE HANDBOOK: ASTRONAUTICS AND ITS APPLICATIONS. Select Committee on Astronautics and Space Exploration. Retrieved 29 October 2012.
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