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I know that the speed of light remains the same irrespective of whether the light source is moving or not. But I want to know whether the laser intensity remains the same if the laser source is moving near to the speed of light. Please let me know any experiment related to calculating laser intensity if a laser source is moving.

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Yes. Beside the more well-known frequency Doppler effect that makes the energy per photon different, the intensity is boosted by $$I_1=(\nu_1/\nu_0)^3 I_0$$ where $I_0$ is the rest frame intensity and $\nu_0,\nu_1$ the rest-frame frequency and the received frequency. The reason is that emitted photons will tend to move (as seen by an outside observer) in directions along the direction of motion, making the emitted energy more focused.

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