Who discovered that electromagnetic wave doesn't need a medium? I have read that physicists in 19 century searching for the aether. They thought that light must have some medium to carry. 
When did they know that light and other electromagnetic wave doesn't need a medium?
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When did they know that light and other electromagnetic wave doesn't need a medium?

Physicists had an inkling early in the 20th century with the development of Planck's law in 1900, Einstein's development of special relativity in 1905, and Einstein's explanation of the photoelectric effect, also in 1905. This strongly suggested that electromagnetism was carried by a particle, the photon.
Stronger evidence came in the 1920s with a number of experiments and a better form of quantum mechanics. (The old quantum theory such as the Bohr model was fundamentally flawed.) Even stronger evidence came in the 1940s and 1950s with the development of quantum electrodynamics.
What about the Michelson-Morley experiment? Contrary to popular current portrayal, that experiment was not perceived at the time as evidence that the luminiferous aether did not exist. It instead was perceived as showing that the luminiferous aether, if it did exist, did not work the way physicists of the latter half of the 19th century initially thought it would work.
A good number of physicists during the final decades of the 19th century worked on reconciling the conflict between Newtonian mechanics and Maxwell's electrodynamics. Most of them postulated a rather weird aether. It wasn't until 1905 that Einstein showed that the aether wasn't needed, and even then, that did not quite explain what electromagnetism was.
A: The Michelson-Morley experiment was the first good evidence they had that there is no luminiferous aether, and was conducted in 1887. (Michelson had an experiment in 1881 trying to do the same, but it was flawed)
