Let's recall the easy way to make a diffraction pattern from a double slit. You just shine a laser on it. Why does this create a diffraction pattern? Well we know that the intensity of the light is determined by the electric field, so to figure out why the intensity of the light is the way it is, we just need to figure out why the electric field is the way it is.
To determine what the electric field is on the far side of the double slit (away from the laser) we only need to know what the electric field is at the two slits. This is because the only way information about the electric field can get to the other side is through the slits. Since the laser produces an electric field which oscillates sinusoidally at a specific frequency, we know that the field at each slit must be oscillating sinusoidally with the same frequency as the laser and hence each other. Also since the distance from the laser is the same, the electric field oscillations at the two slits are in phase (even if the distances to the laser were different, they would still be out of phase by a constant phase, and you would still see a diffraction pattern). So two things are necessary: the light needs to be oscillating sinusoidally at a single frequency (I will call this the monochromicity condition), and the oscillation at both slits needs to be offset by a constant phase (I will call this the phase coherence condition).
I will not explain why these two conditions are sufficient for diffraction (I assume you know that already), but let's see if they are necessary.
Suppose first that the monochromicity condition is met, but the phase coherence condition is not met, so that the phase difference between the two silts varies as a function of time. This could be accomplished for example by holding your double slit up to the sun but using a band-pass filter to filter the sunlight. In this case, I claim you would see no diffraction pattern because as the phase difference between the two slits varies, the location of the diffraction maxima and minima will quickly vary, and the diffraction pattern gets washed out. Thus the phase coherence condition is met.
Is the monochromicity condition necessary. Let's suppose you shine a green laser and a red laser on a diffraction grating at the same time. The electric field at the slits will be a sum of two sine functions. What pattern do you see? You can decompose the electric field on the screen into a piece from the red laser and a piece from the green laser. Each piece individually would give you a nice diffraction pattern, but they interfere with each other creating a complicated pattern of maxima and minima. Moreover, they oscillate at different frequencies, so these maxima and minima will move around very quickly and the pattern will wash out to something that doesn't look much like a diffraction pattern. As the number of lasers increases, this effect would get worse and worse. So the monochromicity condition is necessary.
Now let's take our favorite incoherent light source, an incandescent light bulb, and shine it on a double slit. What happens? Well the light bulb is emitting essentially as a black body so it is emitting radiation at all frequencies and so does not satisfy the monochromicity condition. Also since the light bulb is a large object composed of many independent sources with different path lengths to the slit, the phases at the slits will be uncorrelated and we will not satisfy the coherence condition.
After all this reading we are ready to see the magic. By first passing the light from the light bulb through a single slit, it is possible to arrange things so that the electric field in the double slits satisfies both conditions.
Let's start with the monochromicity condition. We know that a single slit produces a single slit diffraction pattern. Actually the electric field is a superposition of electric fields oscillating at all frequencies, and so the electric field will be a superposition of many interference patterns. But we can put the slits of the double slit at the maxima of the interference pattern for the wavelength we are interested in. Then the electric field from most the other frequencies will be filtered out because they will be closer to their diffraction minima, and we will be left with a sufficiently monochrome electric field to produce a diffraction pattern.
What about phase coherence? Well that is easy. The light takes the same amount of time to propogate from the single slit to either double slit, so the electric field at either of the double slits will always be the same, so they must be in phase.
Since the electric field at the double slit satisfies both the monochromicity and phase coherence conditions, you will see the double slit diffraction pattern.
Suppose the amplitude of the wave from slit $1$ arriving at a point is $A_1$ and the amplitude of the wave from slit $2$ arriving at the same point is $A_2$ and let $A_2>A_1$.
The relative phase between the waves from the two slits arriving at that point depends on the path difference between the slits and the point and not on the amplitude of the waves.
So if the path difference is an integer number of wavelengths then the two waves will arrive exactly in phase with one another and the resultant amplitude will be $A_2+A_1$ which means that the intensity will be proportional to $(A_2+A_1)^2$ at a maximum.
If the path difference is such that the waves arriving at a point are exactly out of phase with one another the resulting amplitude is $A_2-A_!$ which means that the intensity is proportional to $(A_2-A_1)^2$ at a minimum.
The minimum intensity is only zero if the amplitudes of the two waves is the same $(A_2=A_1)$.
So starting from equal intensity (amplitude) from each slit by decreasing the intensity from one slit you decrease the maximum intensities of the interference pattern but increase the minimum intensities of the fringe pattern - the contrast between the bright and dark fringes decreases.
Best Answer
The purpose of the double slits is to split the incoming wave front of a given frequency wave and so in effect you have two coherent sources, which are the slits, emitting light of that wavelength.
As the sources are coherent when the light from those sources overlap an interference pattern is formed.
Light of each wavelength coming through the slits produces its own interference pattern whose size depends on the wavelength of the light.
The zero order fringes (no path difference between the slits) are independent of the wavelength and so will occur at the same position.
If white light is incident on the double skits then that zero order fringe will be white.
The blue light fringes will have a spacing which is less than that for red light and so the first order fringes (one wavelength path difference) will occur in slightly different positions.
So the first order fringe will be relatively broad and bluish nearest the zero order fringe and reddish on the other side.
As one goes further away from the zero order fringe the displacement of the various colours becomes greater so that after a few fringes there is no clear fringe pattern.