I am trying to align the Michelson interferometer using 780 nm LED. But I am not getting any interference pattern in the CCD. Initially I align the interferometer using Laser to get the equal path length. Then I have used LED. Please let me know what I am missing. To be noted, coherence length for my LED is 9.5 um. However, my reference I am using a manual stage so there 1 revolution is 250 um.
Getting the two arms of your interferometer equal to within the coherence length is crucial. If you can obtain fringes using a laser, that is not enough to ensure that the path lengths are the same, because a laser typically has at least centimeters of coherence length, whereas the coherence length of your LED source is only 9.5 microns. If the resolution of your stage is really 250 microns per revolution, you will need to search for fringes using increments of about 3 degrees of revolution.
(I would only want to comment but my reputation is not high enough so I will write as an answer.)
1) First thing is the change in collimation and alignment when you swap between your laser and your LED. Usually you see clearly fringe contrast across the entire beamspot when the beam is collimated (i.e. there is only single k-vector along the arm). More generally, you see clear contrast across the beamspot when both the reference beam and the test beam have the same beam divergence (i.e. they have the same k-vector spread).
2) LED is hard to collimate, because it is an extended light source. In fact, it is physically impossible to perfectly collimate LED, because of the conservation of optical extent (known as etendue), analogous to conservation of phase space in Hamiltonian mechanics. I'm worried that when you switch to LED, maybe your reference beam and probe beam have different divergences.
3) If both of your arms are long and they are standing on a relatively thin optical breadboard (let's say 0.5 inch), I will be worried about vibration noise. Usually it's very low frequency ( O(10 Hz)), but the amplitude will be a couple of fringes.