Is coherence always necessary for interference? I thought so, but I came across a problem: If you try to measure an x-ray spectrum, you usually do so by using bragg reflection on a crystal, i.e. interference. But x-rays from a usual x-ray-tube aren't coherent. How does that work? Or is a mere collimator sufficient to make it coherent? But than: It is possible to isolate wavelength from a diode (for example) with a prism-monochromator, although that as well is a non-coherent light source. What's the error in reasoning here?
In experiments with single photons or single electrons behind an obstacle after a while - sending a lot of particles one by one - an intensity distribution in the form of fringes is observable. This observation is valid for single shot particles even behind a single sharp edge. This intensity distribution is called interference.
Interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude.
So if one explains the case of fringes from single particles behind a single edge by interference this process should happens in interaction of the particle with itself. Then the phenomena of fringes behind a slit or a lot of slits are reducible to the case of the self-interaction for every particle.
Without an obstacle you observe the particles of a beam for example with a Gaussian distribution. No wave behavior of the particles is observable [left image). Behind an obstacle the particles remember their wave behavior and appear on the observation screen - still as undivided particles - with a wavelike distribution (right image).
If you try to measure an x-ray spectrum, you usually do so by using bragg reflection on a crystal, i.e. interference. But x-rays from a usual x-ray-tube aren't coherent.
If you want to get “nice” fringes, the only thing you have to care about is to get a beam with small bandwidth of wavelengths. But “nice” depends from what you want. For example from unfiltered sunlight you still get fringes, this time with different and shifted intensity distributions from different colors:
Is coherence always necessary for interference?
Reading the things above perhaps you could agree, that not?
Each light is coherent within a certain length that for lasers is much longer than for light bulbs. If you increase the length difference, you'd lose interference from a bulb, but not from a laser. Each point in the emitter is in phase with itself for a short while and then the phase randomly changes. For short distance differences, two beams are still within the time period of the same emission phase, but for a longer difference the beams are from different emission phases and no longer interfere. It is different in lasers, because the emission there is stimulated thus keeping the phase for a much longer time depending on the type of the laser. See Coherence Length
Just to add a couple points to the previous answer, interference in general refers to nothing more than the addition of waves. While this general definition makes no mention of coherence, time-averaged effects do require some degree of coherence.
Luckily, as mentioned before, all waves are mutually coherent with themselves. Another example of this is Fourier transform infrared spectroscopy, which usually uses an incoherent lamp as a light source. When the light is interfered with itself, with no relative shifts between the two interfering beams, there is always a time-averaged interference peak no matter how incoherent the light is. But if the light is quite incoherent, even small relative shifts will kill the time-averaged effect.
This same type of thing happens with your x-ray diffraction example, for which spatial (rather than temporal) coherence is relevant. Sure, the x-ray source is generally incoherent. But any beam is coherent with itself, and spatial translations on the order of the atomic spacing are so small that they're certainly within the "coherence length" (i.e. the distance over which the beam continues to look like itself).
Regarding what I gather to be your final question, you can indeed improve the temporal coherence of an incoherent source by subjecting it to narrow-band filtering. However, it would retain its spatial incoherence unless you also spatially filtered it (with a pinhole and some collection optics for example).